K.Ferry TransportSemiconductorMesoscopicDevices

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Transport in Semiconductor
Mesoscopic Devices
(Second Edition)
Transport in Semiconductor
Mesoscopic Devices
(Second Edition)
David K Ferry
School of Electrical, Computer, and Energy Engineering, Tempe, Arizona 85287, USA
IOP Publishing, Bristol, UK

ª IOP Publishing Ltd 2020
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DOI 10.1088/978-0-7503-3139-5
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Contents
Preface to the second edition xii

Preface to the first edition xiv
Author biography xv
1 The world of nanoelectronics 1-1

1.1 Moore’s law 1-2
1.2 Nanostructures 1-4
1.3 Some electronic length and time scales 1-10
1.4 Heterostructures for mesoscopic devices 1-11
1.4.1 The MOS structure 1-11
1.4.2 Fabricating the MOSFET 1-16
1.4.3 The GaAs/AlGaAs heterostructure 1-21
1.4.4 Other important materials 1-23
1.5 Superconductors 1-25
1.5.1 The Meissner effect 1-27
1.5.2 The BCS theory 1-28
1.6 Bits and qubits 1-30
1.7 Some notes on fabrication 1-34
1.7.1 Lithography 1-35
1.7.2 Etching 1-37
1.8 Bottom-up fabrication 1-38
Problems 1-40
References 1-41
2 Wires and channels 2-1

2.1 The quantum point contact 2-1
2.2 The density of states 2-9
2.2.1 Three dimensions 2-10
2.2.2 Two dimensions 2-11
2.2.3 One dimension 2-13
2.2.4 Multiple subbands 2-13
2.3 The Landauer formula 2-15
2.3.1 Temperature dependence 2-19
2.3.2 Scattering and energy relaxation 2-22
2.3.3 Contact resistance and scaled CMOS 2-26
v
Transport in Semiconductor Mesoscopic Devices (Second Edition)
2.4 Beyond the simple theory for the QPC 2-29

2.4.1 High bias transport 2-29
2.4.2 Below the first plateau 2-32
2.5 Simulating the channel: the scattering matrix 2-36
2.6 Simulating the channel: recursive Green’s functions 2-41
Problems 2-45
Appendix A: Coupled quantum and Poisson problems 2-46
Appendix B: The harmonic oscillator 2-51
References 2-54
3 The Aharonov–Bohm effect 3-1

3.1 Simple gauge theory of the AB effect 3-2
3.2 Temperature dependence of the AB effect 3-6
3.3 The AB effect in other structures 3-9
3.4 Gated AB rings 3-10
3.5 The electrostatic AB effect 3-13
3.6 The AAS effect 3-14
3.7 Weak localization 3-15
3.7.1 A semiclassical approach to the conductance change 3-16
3.7.2 Role of the magnetic field 3-20
3.8 Graphene rings 3-21
Problems 3-23
Appendix C: The gauge in field theory 3-23
References 3-27
4 Layered compounds 4-1

4.1 Graphene 4-1
4.2 Carbon nanotubes 4-9
4.3 Topological insulators 4-14
4.4 The metal chalcogenides 4-16
Problems 4-19
References 4-20
5 Localization and fluctuations 5-1

5.1 Localization of electronic states 5-2
5.1.1 The Anderson model 5-3
5.1.2 Deep levels 5-6
5.1.3 Transition metal dichalcogenides 5-10
vi
5.2 Conductivity 5-11

5.3 Conductance fluctuations 5-16
5.4 Correlation functions 5-21
5.5 Phase-braking time 5-25
Problems 5-32
References 5-33
6 The quantum Hall effect 6-1

6.1 The Shubnikov–de Haas effect 6-1
6.2 The quantum Hall effect 6-7
6.3 The Büttiker–Landauer approach 6-9
6.3.1 Two-terminal conductance 6-12
6.3.2 Three-terminal conductance 6-12
6.3.3 The quantum Hall device 6-14
6.3.4 Selective population of edge states 6-15
6.3.5 Nature of the edge states 6-17
6.4 The fractional quantum Hall effect 6-20
6.5 Composite fermions 6-22
Problems 6-25
References 6-27
7 Spin 7-1
7.1 The spin Hall effect 7-2
7.1.1 The spin–orbit interaction 7-2
7.1.2 Bulk inversion asymmetry 7-3
7.1.3 Structural inversion asymmetry 7-6
7.1.4 Berry phase 7-8
7.1.5 Studies of the spin Hall effect 7-12
7.2 Spin injection 7-15
7.3 Spin currents in nanowires 7-17
7.4 Spin qubits 7-20
7.5 Spin relaxation 7-23
Problems 7-25
Appendix D: Spin angular momentum 7-25
Appendix E: The Bloch sphere 7-28
References 7-32
vii
8 Tunnel devices 8-1

8.1 Coulomb blockade 8-3
8.2 Single-electron structures 8-6
8.2.1 A simple quantum-dot tunneling device 8-6
8.2.2 The gated single-electron device 8-10
8.2.3 Double dots 8-16
8.3 Quantum dots and qubits 8-21
8.4 The Josephson qubits 8-28
8.4.1 Josephson tunneling 8-31
8.4.2 SQUIDs 8-31
8.4.3 Charge qubits 8-33
8.4.4 Flux qubit 8-35
8.4.5 The hybrid charge-flux qubit 8-36
8.4.6 Novel qubits 8-37
Problems 8-39
Appendix F: Klein tunneling 8-40
Appendix G: The Darwin–Fock spectrum 8-41
References 8-44
9 Open quantum dots 9-1

9.1 Conductance fluctuations in open quantum dots 9-2
9.1.1 Magnetotransport 9-4
9.1.2 Gate-induced fluctuations 9-7
9.1.3 Phase-breaking processes 9-9
9.2 Einselection and the environment 9-13
9.2.1 Classical orbits 9-14
9.2.2 Coupling the dot to the environment 9-20
9.2.3 Relating classical and quantum orbits 9-23
9.2.4 Pointer state statistics 9-30
9.2.5 Hybrid states 9-32
9.2.6 Quantum Darwinism 9-36
9.3 Imaging the pointer state scar 9-37
Problems 9-40
References 9-41
viii
10 Hot carriers in mesoscopic devices 10-1

10.1 Energy-loss rates 10-2
10.2 The energy-relaxation time 10-10
10.3 Nonlinear transport 10-13
10.3.1 Velocity saturation 10-15
10.3.2 Intervalley transfer 10-18
10.3.3 NDC and NDR 10-20
10.3.4 Velocity overshoot 10-22
Problems 10-24
References 10-24
ix
Preface to the second edition
It generally is regarded as being true that nanostructures may be considered as ideal

systems for the study of the physics of electronic transport. Perhaps this is a self-
fulfilling statement, as I have been involved in the field for my entire career. In the
late 1970s, this area of research was called ‘ultra-small electronics research’, and the
description as one of nanoscale was not applied for a few decades after that. But, it
was interesting that we pursued the use of electron-beam lithography to make things
small. Unfortunately, this endeavor was ended by the success of the microelectronics
industry. For instance, we worked hard in the university environment to make small
transistors with gate lengths on the scale of 20–50 nm. For the past few decades or
so, Intel (and others, of course) has made a number something like a thousand times
the population of the Earth of such devices each day, so this area of research is gone
from the universities.
But this study of small structures is more complex than just the use of nano-
fabrication techniques to make small transistors. Transport is the study of the
motion of charge carriers in the materials of interest, mainly semiconductors in my
world. This transport is characterized by a set of length and time scales. For
example, velocity builds up, and decays, with a time scale governed by the scattering
time that describes the interaction of the carriers with impurities and lattice
vibrations. These times are on the order of a small fraction of a picosecond.
Transport distances are scaled by mean free paths; e.g., the distance that a carrier
travels on average between discrete scattering events. This distance can range from a
few nanometers at room temperature in Si to a few microns at low temperatures in
GaAs. Indeed, the distance a particle may travel before scattering has been
determined to be about 180 nm in an InAs nanowire at room temperature. The
time scales for transport have been probed extensively by the use of femtosecond
pulse length lasers. So, the crucial length and time scales have been accessible for
quite some time and this has allowed the study of the dynamic response of transport
at these fundamental levels.
As a consequence of being able to access the basic length and time scales, it is
possible to create structures in which the underlying physics can be probed in a
meaningful manner. By lowering the temperature to easily achievable cryogenic
levels, scattering can be suppressed significantly, and one can explore the physics
itself. This allows us to explore quantum physics in systems which may seem to be
purely classical at room temperature. The impact of these studies goes beyond a
simple interest in the physics. As mentioned above, the world of the microelectronics
industry has been a leader in the development of nano-transistors, with critical
dimensions at the 10 nm scale, and chips at the so-called 7 nm node (in 2019) of
Moore’s Law. Success has been achieved here because the study of nanostructures
has highlighted many important physical effects prior to their being important in the
integrated circuit. Advances in the study of the underlying physics has provided
important guidance for the continued reductions in device size, but the latter also
provides a pull for continued study of the relevant physics. It is not a one-way street.
x
Rather, this field has prospered from the interplay of science and technology, and,
for example, the computing power that has resulted from the latter has been crucial
for continued development of the former. More importantly, however, nanoelec-
tronics provides the driving technology for much of our high technology life today,
and it holds the promise to keep driving the information growth and processing that
has given us this high technology life.
As remarked in the preface of the first edition, my involvement in this area was
greatly expanded when Jon Bird joined our department. But, our interest in the
continued evolution of semiconductor nanoelectronics and the understanding of the
relevant physics has continued even from before that time. As we move into this
second edition, the understanding of what makes nanoelectronics has moved beyond
just semiconductors and the corresponding materials. Now, a new component of this
field has been driven by the quest to create quantum computers, which so far rely
upon superconducting materials and the relevance of the Josephson tunneling
junction. Consequently, in this second edition, coverage of these topics has been
added to provide a more extensive coverage of the modern field on nanotechnology.
Along the way, some videos from the first edition have been deleted, but those that
remain for the second edition are available at https://iopscience.iop.org/book/978-0-
7503-3139-5.
Some of the problems in the various chapters use tools that are available at
NanoHUB.org, which is a computational science center originally founded by the
National Science Foundation. All the tools are available at their website by creating
an account, which is free, and then selecting ‘tools’ from your sign-in page. The
available tools range from introductory level to quite advanced tools such as for
density functional theory and nonequilibrium Green’s functions. Those needed for
the problems are toward the more introductory level. If problems occur, there is a
reporting system online at their web page.
Finally, I should thank once more Jon Bird for continuing to collaborate and to
provide information about some of the updates. I also have to thank my colleagues
Steve Goodnick and Dragica Vasileska for feedback from their teaching the
material. The staff at IOP Publishing have continued to provide an outstanding
relationship that has existed from the commissioning of the first edition right
through until the present time.
David K Ferry
xi
Preface to the first edition
This book is the result of a project begun some years ago when Jon Bird came to
Arizona State University. At that time, we had a major program on the study of
quantum effects in very small semiconductor structures, an area which has become
known as mesoscopic devices. Over the years, we have developed first at Arizona
State University, and then subsequently when Jon moved to SUNY Buffalo,
graduate level courses in this topic. The present book has grown out of several
versions of lecture notes which were prepared in connection with these courses. Note
that this book is intended to be a textbook for first year graduate students. Thus,
many topics are not covered in the great detail that may be found for example in
Transport in Nanostructures, from Cambridge University Press. To do so would
make the book far too large for the purpose of a textbook and it would probably
overwhelm the students as well.
I need to point out that, in spite of my best lamentations, Jon has continued to
decline my offers to partner in the preparation of this book. Nevertheless, the book
could not be possible without the input from Jon, and I owe a great debt of gratitude
to him for his contributions. I also owe a debt of gratitude to Larry Cooper, long
retired from the Office of Naval Research, without whom our work would not have
prospered as it has. And, without the latter, I would not be in a position to write this
book.
I must also thank the people who have provided various images for inclusion in
the book. It is especially nice to acknowledge those who have provided the videos
that are included in the ebook version. These are Jesper Nygård and the Neils Bohr
Institute, JT Janssen and the National Physical Laboratory, Andrea Morelli and
Andrew Dzurak and the University of New South Wales, Stuart Lindsay, Don
Burgess, and Richard Akis. They all have graciously consented to do so, and I am
eternally grateful for that privilege. Without these images and videos, the book
would not exist.
xii
Author biography
David K Ferry
David K Ferry is Regents’ Professor Emeritus in the School of
Electrical, Computer, and Energy Engineering at Arizona State
University. He was also graduate faculty in the Department of
Physics and the Materials Science and Engineering program at
ASU, as well as Visiting Professor at Chiba University in Japan.
He came to ASU in 1983 following shorter stints at Texas Tech
University, the Office of Naval Research, and Colorado State
University. In the distant past, he received his doctorate from the University of
Texas, Austin, and spent a postdoctoral period at the University of Vienna, Austria.
He enjoyed teaching (which he refers to as ‘warping young minds’) and continues
active research. The latter is focused on semiconductors, particularly as they apply
to nanotechnology and integrated circuits, as well as quantum effects in devices.
In 1999, he received the Cledo Brunetti Award from the Institute of Electrical and
Electronics Engineers, and is a Fellow of this group as well as the American Physical
Society and the Institute of Physics (UK). He has been a Tennessee Squire since 1971
and an Admiral in the Texas Navy since 1973. He is the author, co-author, or editor
of some 40 books and about 900 refereed scientific contributions. More about him
can be found on his home pages http://ferry.faculty.asu.edu/ and http://dferry.net/.
xiii
IOP Publishing
Transport in Semiconductor Mesoscopic Devices

(Second Edition)
David K Ferry
Chapter 1
The world of nanoelectronics
What we call electronics today has a confused beginning. Should it begin with the
first electronic device, the metal-semiconductor junction of Braun in 1876 [1]? Or
should it begin with the first suggestions of a mechanical computing machine, which
dates at least to the analytical engine of Charles Babbage at the middle of the
nineteenth century. If the latter, then we have to look at the theoretical work of Alan
Turing in the 1930s that suggested the use of such a machine to determine if a
number was computable [2]. If the former, then the follow up lies with the
understanding of the vacuum diode [3, 4]. These two threads come together with
the first efforts to develop computing machines at Iowa State University and the
University of Pennsylvania [5, 6]. Yet, the future lay in nanoelectronics, and this
became apparent with the invention of the integrated circuit in the 1950s by Jack
Kilby [7] and Robert Noyce [8]. The rapid development of integrated circuits with
ever smaller critical dimensions led to the evolution of what has become known as
Moore’s Law [9]. Here, Moore suggested that the number of transistors on an
integrated circuit would continue its trend of doubling every 18 months, leading to
the exponential growth that allowed first micro-electronics and then nanoelectronics
to revolutionize everyday life throughout the world. The physical approach which
allowed this was based upon the principles of scaling of dimensions in a manner that
kept the internal electric field constant from one generation to the next [10]. Thus all
dimensions and voltages of a device were reduced by the exact same factor, which
led to the electrostatics of the device being unchanged. As a result, the reduced
dimensionality did not, in principle, change device operation as it was downsized.
This little overlooked fact meant that integrated circuits was a scalable technology,
and this scalability led to the impact of nanoelectronics on the world, a transition
that some have termed the information revolution due to the exponential growth in
computing power [11]. For example, the iPhone 11, introduced in late 2019, uses the
A13 chip based upon the ARMv8.3-A 64 bit 6 core 7 nm node technology,
doi:10.1088/978-0-7503-3139-5ch1 1-1 ª IOP Publishing Ltd 2020

fabricated in Taiwan. The chip computing power is approximately 100 GFLOPS1.

The chip itself is about 1 cm2 and contains 8.5 billion transistors. More interesting
technologically is the fact that some layers of the chip utilize extreme ultra-violet
(EUV) lithography which employs 13.7 nm x-rays.
1.1 Moore’s law

Since the invention of the integrated circuit, the growth in the number of transistors
on the chip has been exponential. This growth has proceeded unabated now for
more than half a century. Only a few years had passed after the invention of the
integrated circuit when Gordon Moore recognized the important driving forces for
the exponential growth. His 1965 paper became the controlling manifesto for the
development of the microchip world [9]. Moore modestly suggested that ‘Integrated
circuits will lead to such wonders as home computers…and personal portable
communications equipment’. Moore and Noyce had joined the original Shockley
semiconductor company, but left with a group to found Fairchild semiconductors to
pursue integrated circuits, and then eventually went on to found Intel.
The key factor in Moore’s law lay in the scaling of transistor sizes. Moore noted
that, if dimensions were reduced by a factor of two every so often, then the number
of transistors per unit area would go up by a factor of four. This scaling is the heart
of the exponential growth in transistor density in modern chips. The development of
a formal scaling theory in which dimensions, dopant densities, and electric fields
were all scaled according to a prescribed plan came somewhat later [10]. But, Moore
recognized that there were more factors involved than merely down-sizing the
transistors. He suggested that the actual size of individual chips would be increased
and that gains could be made through circuit cleverness. So, the eventual factor of
four increase every three years came from these three different ideas, rather than
from merely reducing the size. Examples of the idea of circuit cleverness are the
introduction of complementary metal–oxide–semiconductor (CMOS) technology
itself, the introduction of trench capacitors in which the capacitors were buried in the
silicon rather than being on the surface, and, more recently, the transition to the
finFET (or tri-gate) technology.
In spite of the obvious reductions in the size of individual transistors, these are in
fact results of Moore’s law rather than the driving forces. The actual driving force
does not derive from physical laws but from economics. In a modern integrated
circuit, transistors are laid out on the chip in a planar fashion, much like the houses
in a modern southwestern US city. The drive that continues the growth arises from
the cost reductions associated with a given level of computing power. As the number
of transistors on each chip continues to increase, the number of functional units on
the chip dramatically increases which in turn increases the computing power.
Because the basic cost of manufacturing a single chip has not dramatically increased
over the five plus decades since the invention of the chip, the cost of computing
1
Giga-floating point operations per second
1-2
power has gone down exponentially. It is this economic argument, based upon the
cost of silicon real estate, that seems to drive Moore’s law.
There have been glitches along the way. Intrinsic heating of the chip became a
significant problem in the early 1990s. This led to an extensive industry-wide push
for a new design approach based upon low power. Nevertheless, individual chip size
has not increased significantly since that time, remaining at 1–2 cm2. But, Moore’s
law marches onward, as new technologies are developed to solve problems. Mobility
reductions were overcome with the use of strain in the transistors—n-channel devices
used tensile strain while p-channel devices used compressive strain. The natural
reduction of gate oxide thicknesses with scaling led to a problem of tunnel leakage
through the oxide. This was overcome by the introduction of new dielectrics with
high dielectric constants (currently HfO2), so that thicker materials could be used
while maintaining the gate capacitance. As devices became smaller, the physical
number of dopants in the channel became quite small, which could lead to device-to-
device fluctuations in dopant number [12]. This has been addressed by reducing the
number of dopants considerably in the channel region itself. Hot carrier effects are
ameliorated by careful design of the device. In short, as problems in the transistors
are recognized, the industry finds ways to overcome these problems. The most
dramatic is perhaps the move to tri-gate devices that has recently occurred. Here the
problem is controlling the off/leakage current in the transistors, which has been
addressed by making the transistors vertical so that a ‘wrap around’ gate can be used
to enhance the off state resistance.
Nevertheless, it is easy to recognize that a critical dimension of 25 nm represents a
chain of only 100 silicon atoms. One cannot continue forever along the path of
continuing to down-size the transistor. This recognition raises possible problems
with the CMOS technology itself. Another class of challenges lies in the realm of the
state variable to be used for computing. Can we continue to use charge as is now
done with the transistor? Or, will new paradigms arise which engender new forms of
computing2?
The industry may be at a cross-roads for another technological change and this
may involve a move to nanowires fabricated in a gate-all-around technology, where
the gate wraps completely around the nanowire. This has been studied in research
labs for almost two decades. Nanowires placed parallel to one another on the surface
of silicon is not an attractive approach. This is because, if we are to save silicon
surface area, the nanowire has to have a large diameter, and this means that it
cannot compete with the finFET in saving Si area and achieving high circuit density
[13, 14]. But, another approach is possible, and that is to stack the nanowires
vertically [15], an approach which was followed quickly by others [16, 17].
Naturally, there have been studies to see how these devices compared with others
such as SOI MOSFETs [18] and FinFETs [19, 20]. There is also a consideration
whether or not to incorporate doping in the channel region, and this has been
studied as well for the vertical stacked nanowires [21, 22]. The best sign that this
2
But, quantum computing has not given any indication that it will solve this problem. Current qubits remain
significantly larger than current Si transistors.
1-3
technology is ready for prime time applications is the entry of the major semi-
conductor producers into the field [23].
What the future holds is anyone’s guess, but I suspect that we will continue
onward for many decades with one form or another of Moore’s law. We may well
discontinue reducing the size of the transistor, but perhaps we will begin to layer the
transistors on the chip in the manner of lasagna, as many are suggesting already. In
any of the possible scenarios, there remains a great field of research in what we call
nanostructures that will underlie future advances.
1.2 Nanostructures
If we are to proceed to understand nanostructures, then it will be necessary to have a
full understanding of the basic physics as it relates to those device properties. It is the
objective of this book to provide just such an understanding. The first principle of
nano-device physics is that it is strongly influenced by quantum effects. In this sense,
it differs from the level of understanding that one requires in an undergraduate life.
Here, we need to deal with a variety of effects that become quite important, such as
the quantization of the electronic density of states and the implication that this has
for the electronic properties of nanostructures. A direct result of this quantization
effect, when we are dealing with transport that is nearly ballistic in nature, is the
resulting quantization of the conductance throughout the nanostructure. These two
effects go hand in hand in providing one of the most interesting observations in
nanostructures—the presence of specific modes of propagation, much like a micro-
wave waveguide. Indeed, it is the wave-like nature of the electrons that is being
observed in these experiments. Along this same idea, the presence of quantum
interference is another major observable in nanostructures. That is, the wave can
propagate along a pair of trajectories, much like a two-slit experiment, and the
resulting wave interference is clearly observable in experimental studies. The
tunneling, which can arise from the wave properties of the electrons, is also easily
observable and has led to a number of interesting devices.
To create the devices in which our quantum effects are to be studied, one can
proceed by at least two different approaches. These approaches will be discussed
further below. First, however, we want to note some important points. It is possible
to create nanostructures using top-down microfabrication which follows the same
procedures as used in the microelectronics industry. One difference, however, is the
preferred use of electron-beam lithography for the ease with which it can make
relatively small structures [24]. With such lithography, one can routinely approach
dimensions in the 10–30 nm range, and with some extra effort, lines approaching 5–7
nm have been fabricated [25]. Thus, the standard processes of lithography and
etching can be used to prepare a large number of different nanostructures. On the
other hand, bottom-up processing can be used to create even smaller structures.
Typical approaches utilize either molecular self-assembly [26] or, for example, the
deposition of small quantum dots through strain relaxation of a very thin epitaxial
layer [27]. Then there are entirely new approaches which can arise from the layer
1-4
compounds, such as graphene, which redefine some of our ideas about whether to go
top-down or bottom-up.
Hence, there is a vast range of fabrication tools which can be brought to bear to
create a specific nanostructure needed to study an interesting point in physics. We
will discuss a few of these methods in a later section. But, the field of nanostructures
is truly enormous, covering a wide range of disciplines that range from fundamental
physics, to materials growth and development, and to electronic and optoelectronic
investigations. It is impossible to cover this entire range of topics within a single
book, and even concentrating on transport can lead to a large book [28]. A rather
narrower view will be taken here, and we will focus on a discussion of the basic
effects that are associated with the germane quantum transport effects that can be
found in the electronic properties of various nanostructures. The short term
application of some of these effects is not the primary aim, especially as they relate
to today’s CMOS microelectronics. Rather, the field of mesoscopic physics and
devices goes beyond today’s transistors, and is wide enough to keep us busy
throughout this book. It is the purpose here, however, to try to take a coherent
journey through this field to highlight the common physics and understanding that
underpins this field.
An important point about nanostructured devices is that the critical dimensions
of the structure are comparable to the corresponding de Broglie wavelength of the
electrons. This allows their properties to be strongly influenced by quantum
mechanical effects. For example, if we have an electron in GaAs at a Fermi energy
of, for example, 10 meV, then this corresponds to a momentum of approximately
1.4 × 10−26 kg m s−1 or a wave vector of about 1.3 × 108 m−1. This now corresponds
to a de Broglie wavelength of almost 50 nm. We will see later that this corresponds
to a two-dimensional density of just over 1011 cm−2, which is easily obtained in high
quality GaAs heterostructure layers. One of the nicest demonstrations of the
observability of de Broglie waves was that of the quantum corral by Don Eigler
et al at the IBM Almaden Research Center [29], as shown in figure 1.1. Using a
Figure 1.1. Scanning tunneling microscope image of 48 Fe atoms (the sharp peaks) forming a ring on the
surface of copper [29]. Within the ring, standing waves from the confined copper surface electrons are clearly
visible. The image is used with permission from IBM research.
1-5
scanning tunneling microscope (STM), they arranged iron atoms on the surface of
Cu in the shape of a ring approximately 14.6 nm in diameter. Within the ring, they
could then image the square magnitude of the wave function of electrons on the Cu
surface, even though the relatively higher energy of the electrons at the Cu Fermi
energy have a much shorter wavelength than those in the low density GaAs layer.
Nevertheless, it is absolutely clear that the wave nature of the electrons is exceed-
ingly important in these nanostructures.
As the preceding example indicates, when we confine electrons on the scale of
their wavelengths, or even on larger scales when the motion is coherent, these
electrons are subject to quantization by their confinement. This quantization gives
rise to a dramatic difference in their density of states from that expected in classical
bulk material. One reason for this is that the energy spectrum of the electrons
becomes very different from the quasi-continuous one expected in bulk materials.
The presence of this quantization gives opportunities to probe new and different
physics and applications, some of which might be useful for future device
applications. Moreover, the interaction of the electron with defects and impurities
becomes a much more singular process, in that individual scattering events become
important processes in the transport of the electrons. This disorder, arising from the
impurities or the defects, can introduce new observables in the transport conduc-
tance, which may not be small changes. To understand this, we want to consider
some basic ideas of length and time scales.
In large, bulk conductors, the resistance that exists between two contacts is related
to the bulk conductivity and to the dimensions of the conductor, as expressed by
L
R= , (1.1)
σA
where σ is the conductivity and L and A are the length and cross-sectional area of the
conductor, respectively. If the conductor is a two-dimensional conductor, such as a
thin sheet of metal, then the conductivity is the conductance per square, and the
cross-sectional area is just the width W. This changes the basic formula (1.1) only
slightly, but the argument can be extended to any number of dimensions. Thus, for a
d-dimensional conductor, the cross-sectional area has the dimension A = Ld−1,
where L must be interpreted as a ‘characteristic length’. Then, we may rewrite
equation (1.1) as
L2−d
R= . (1.2)
σd
Here, σd is the d-dimensional conductivity. Whereas one normally thinks of the
conductivity, in simple terms, as σ = neμ, the d-dimensional term depends upon the
d-dimensional density that is used in this latter definition. Thus, in three dimensions,
σ3 is defined from the density per unit volume, while in two dimensions σ2 is defined
as the conductivity per unit square and the density is the sheet density of the carriers.
The conductivity (in any dimension) is not expected to vary much with the
1-6
characteristic dimension, so we may lake the logarithm of the last equation. Then,
taking the derivative with respect to ln(L) leads to
∂ln(R )
=2−d (1.3)
∂ln(L )
This result is expected for macroscopic conducting systems, where resistance is
related to the conductivity through equation (1.2). We may think of this limit as the
bulk limit, in which any characteristic length is large compared to any characteristic
transport length.
In mesoscopic conductors, it has been suggested that the above would no longer
be valid, primarily under the assumption that disorder effects would be proportion-
ally larger in small structures. Let us first consider how this might appear. We have
assumed that the conductivity is independent of the length, or that σd is a constant.
However, if there is surface scattering, which can dominate the mean free path, then
one could expect that the mean free path is l ∼ L. Since l = vFτ, where vF is the Fermi
velocity in a degenerate semiconductor and τ is the mean free time between
collisions, this leads to
nd e 2 τ n e 2L
σd = ⁎
= d⁎ , (1.4)
m m vF
Hence, the dependence of the mean free time on the dimensions of the conductor
changes the basic behavior of the macroscopic result (1.3). This is the simplest of the
modifications. For more intense disorder or more intense scattering, the carriers may
well be localized because the size of the conductor creates localized states whose
energy difference is greater than the thermal excitation, and the conductance will be
quite low. In fact, we may actually have the resistance increasing exponentially with
length according to [30]
R = e αL − 1, (1.5)
where α is a small quantity. We think of the form of equation (1.5) as arising from
the localized carriers tunneling from one site to another, and the last term (−1) is
required to recover zero resistance with zero length. Then, the above scaling
relationship (1.3) is modified to
∂ln(R )
≈ αL . (1.6)
∂ln(L )
In this situation, unless the conductance is sufficiently high, the transport is localized
and the carriers move by hopping. The necessary value has been termed the
minimum metallic conductivity [30], but its value is not given by the present
arguments. Here we just want to point out the difference in the scaling relationships
between systems that are highly conducting (and bulk-like) and those that are largely
localized due to the high disorder.
In a strongly disordered system, such as that discussed above, the wave functions
decay exponentially away from the specific site at which the carrier is present. This
1-7
means that there is no long-range wavelike behavior in the carrier’s character. On

the other hand, by bulk-like extended states we mean that the carrier is wavelike in
nature and has a well-defined wave vector k and momentum ℏk . Most semi-
conducting mesoscopic systems have sufficiently weak scattering that the carriers do
not have localized behavior. Thus, when we talk about diffusive transport, we
generally mean almost-wavelike free states with relatively low scattering rates. Here,
we tend to mean that the concept of an electron mobility is quite valid and that the
scattering occurs often enough to make this the case.
We will examine the idea of localization later (in chapter 5), but here we assume
that the entire conduction band is not localized. Rather, it retains a sufficiently large
region in the center of the energy band that has extended states and a nonzero
conductivity as the temperature is reduced to zero. For this material, the density of
electronic states per unit energy, per unit volume, is given simply by the familiar (we
will return to derive the density of states in section 2.2) dn/dE. Since the conductor
has a finite volume, the electronic states are discrete levels determined by the size of
this volume. These individual energy levels are sensitive to the boundary conditions
applied to the ends of the sample (and to the ‘sides’) and can be shifted by small
amounts on the order of ℏ/τ , where τ is determined here by the time required for an
electron to diffuse to the end of the sample. In essence, one is defining here a
broadening of the levels that is due to the finite lifetime of the electrons in the
sample, a lifetime determined not by scattering but by the carriers’ exit from the
sample. This, in turn, defines a maximum coherence length in terms of the sample
length. This coherence length is defined here as the distance over which the electrons
lose their phase memory, which we will take to be the sample length. The time
required to diffuse to the end of the conductor (or from one end to the other) is L2/D.
where D is the diffusion constant for the electron (or hole, as the case may be) [31].
The conductivity of the material is related to the diffusion constant (we assume for
the moment that the temperature T = 0) as
nd e 2 τ dn
σd (E ) = ⁎
= e 2D , (1.7)
m dE
where we have used the fact that n = (2/d)(dn/dE)E, d is the dimensionality, and
D = vF2τ /d . If L is now introduced as the effective length, and τ is the time for
diffusion, both from D, one finds that
ℏ ℏσ dE
= 2 2 . (1.8)
τ e L dn
The quantity on the left-hand side of equation (1.8) can be defined as the average
broadening of the energy levels ΔEa, and the dimensionless ratio of this broadening
to the average spacing of the energy levels may be defined as
ΔEa ℏσ
= 2 2. (1.9)
dE / dn e L
1-8
Finally, we change to the total number of carriers N = nLd, so that

ΔEa ℏσ
= 2 Ld −2. (1.10)
dE / dn e
This last equation is often seen with an additional factor of 2 to account for the
double degeneracy of each level arising from the spin of the electron. Nevertheless,
this last result agrees with the expected scaling of the resistance in equations (1.2)
and (1.3).
It is now possible to define a dimensionless conductance, called the Thouless
number by Anderson et al [32], in terms of the conductance as
2ℏ
g (L ) = G (L ), (1.11)
e2
where G(L) = σLd−2 is the actual conductance. These latter authors have given a
scaling theory based upon renormalization group theory, which gives us the
dependence on the scale length L and the dimensionality of the system. The details
of such a theory are beyond this book. However, we can obtain the limiting form of
their results from the above arguments. The important factor is a critical exponent
for the reduced conductance g(L) which may be defined by
d [lng(L )]
βd ≡ lim → d − 2, (1.12)
g →∞ dln(L )
which is just equation (1.3) rewritten in terms of the conductance rather than the
resistance. By the same token, one can rework equation (1.6) for the low conducting
state to give
d [lng(L )]
βd ≡ lim → − αL . (1.13)
g →0 dln(L )
What the full scaling theory provides is a connection between these two limits when
the conductance is neither large nor small.
For three dimensions, the critical exponent changes from negative to positive as
one moves from low conductivity to high conductivity, so that the concept of a
mobility edge in disordered (and amorphous) conductors is really interpreted as the
point where β3 = 0. This can be expected to occur about where the reduced
conductance is unity, or for a value of the total conductance of e 2 /πℏ. In less than
three dimensions, there is no critical value of the exponent, as it is by and large
always negative, approaching 0 asymptotically. That is, it has been suggested that all
states are localized in one and two dimensions [32], although experiments tell us
otherwise. What the results (1.12) and (1.13) is that the curve can cross over the
asymptote (1.12) and approach 0 from the positive side for two dimensions. In
studies of high mobility silicon MOS field-effect transistors (MOSFETs) at low
temperatures, it has been demonstrated that there is a phase transition between a
localized regime and a diffusive regime [33–35]. Subsequently, this phase transition
has been observed for a variety of semiconductor heterostructures and for both
1-9
electrons and holes. It is clear that the critical density for the onset of diffusive
transport is affected by the impurities in the semiconductor, as measurements on
very high purity, undoped GaAs/AlGaAs structures suggest that the critical electron
density is below 2.6 × 109 cm−2. Thus, it seems to be clear that any one parameter
scaling theory fails to uncover the crucial physics of any metal–insulator transition
behavior in high quality mesoscopic devices, except in the d = 3 case. Yet, the ideas
of the scaling theory can be a useful guide to what may generally be expected in any
dimensionality, although the exact details may be surprising as in the structures
mentioned.
1.3 Some electronic length and time scales

We have already encountered a number of appropriate length scales with the device
characteristic length L and the mean free path vFτ, where τ is the mean free time
between collisions, and vF is the Fermi velocity. These are connected with the ideas
of mobility μ = eτ/m*, where m* is the effective mass of the electron in the
semiconductor, and the diffusivity D (= μkBT/e), which is given in terms of the
Fermi velocity as vF2τ/d, and d is the dimensionality as above. We have also
discussed briefly the idea of a coherence time, or phase-breaking time τφ which
describes the time over which the wave function retains its coherence. While this is a
vague meaning and description, a better understanding can only arise as we describe
the effect of this phase coherence in the scenario of a number of experiments, in
which the quantum behavior remains well observable for a time scale on the order of
this quantity. This phase-breaking time allows us to connect to a phase-breaking
length through the diffusivity via lφ = Dτφ. We can think of this phase-breaking
length as the average distance which the electrons diffuse before their phase is
disrupted through various scattering events. Naturally, we desire that this length be
larger than, or comparable to, the size of the mesoscopic device under investigation,
which usually implies that the measurements are to be performed at cryogenic
temperatures. Some of the earliest measurements of the phase-breaking length were
performed in thin metallic wires [36, 37], which tend to have more disorder than
semiconductors. In most cases, the phase-breaking time and length were inferred by
fitting to the dependence of the weak localization in these wires on the applied
magnetic field. We will turn to a discussion of this effect in chapter 3.
Other important lengths are the Fermi wavelength, which was introduced earlier,
and the thermal length. This latter is a length that is a little more difficult to grasp, as
it connects the diffusivity with a time defined by the thermal broadening of a typical
energy level. This broadening is used to define a time scale via ℏ/kBT , and this
connects the diffusivity to the thermal length as lT = ℏD /kBT . Again, we will
encounter this length in some detail in the discussion of weak localization and
conductance fluctuations in chapter 3.
Another important time for nonequilibrium systems is the energy-relaxation time
τE, which describes the time scale over which the energy per carrier in the system
returns to its equilibrium value. Usually, this time describes the particular electron–
phonon interactions by which energy is transferred from the electrons to the lattice,
1-10
and so is dominated by inelastic interactions. This raises another issue over the
various time scales. Those discussed in the preceding paragraphs are generally
defined only in equilibrium conditions. For example, the relationship given above
between the diffusion constant and the mobility is only valid in equilibrium (and, as
stated, only for non-degenerate conditions). When the system goes out of equili-
brium, there is no such relationship, and, quite generally, there is no fluctuation-
dissipation theorem from which this relationship is derived [38]. In this case,
however, the thermal length and the phase-breaking length should more likely be
defined in terms of the thermal velocity vT = 2kBT /dm⁎ , and the mean free time or
the phase-breaking time, respectively.
1.4 Heterostructures for mesoscopic devices

In this section, I would like to describe some structures and materials which are
commonly used for mesoscopic devices. First, we treat the two most common device
types—the MOSFET, and the high-mobility, heterostructure device. The most
common type of the former is the Si MOSFET while the most common form of
the latter is the AlGaAs/GaAs heterostructure. Both approaches have been
extensively used to study mesoscopic physics, and our goal here is to describe the
two device structures and discuss a few of their key features, as well as some ideas on
how these structures are fabricated. Then, we turn to superconductors, which have
become of great interest in recent years for their use in possible structures for
quantum computing devices.
1.4.1 The MOS structure

The field effect transistor (FET) is actually the oldest known form of transistor, as it
was originally patented by Lilienfeld in 1926 [39]. But, it was not until 1959 that a
working device was demonstrated [40], as control of the surface was a serious
problem which held up development until after the bipolar transistor. Within a few
years, the MOSFET was the preferred device for the integrated circuit due both to its
planar technology and its generally lower power dissipation. The rest, so they say, is
history. However, it is not generally appreciated that it is also a good device in which
to study mesoscopic physics. Yet, the quantum Hall effect was discovered in a Si
MOSFET [41]. In addition, some of the most extensive early work on conductance
fluctuations [42] and on the disorder induced metal–insulator transition [43] were
both performed in Si MOSFETs. In addition, InSb, GaAs, and Ge have been
studied using mylar as the gate insulator [44], and a wide range of semiconductors
with anodic oxides [45] and deposited oxides [46] have been investigated. So, it seems
clear that the MOSFET is one of the major devices that has been studied for
mesoscopic phenomena.
In figure 1.2, we display the conduction band energy profile for a typical planar
MOSFET. A positive voltage is applied to the drain relative to the source, which is
typically grounded. Between the two is the p-type substrate (the source and drain are
n-type). The oxide is usually silicon dioxide, although these days it is mostly a high
dielectric constant oxide such as hafnium oxide (or hafnium silicate). Between the
1-11
Figure 1.2. The conduction band energy profile for a typical planar MOSFET.
Figure 1.3. Sketch of the band bending in the silicon for a MOS structure under gate bias.
oxide and the p-type regions lies the inversion channel, indicated by the arrow in the
figure. This is an active n-channel which connects the source and drain and is the key
part of the transistor. Normally, one describes the transistor action via classical
statistics, but this is not correct, as we will show. Indeed, the channel is a quantum
object, but this is not very important at room temperature as the quantization is
normal to the current flow, so is a second-order effect. At low temperatures, it is the
quantized channel that is of great interest to researchers.
The inversion channel electrons reside between the potential barrier introduced by
the oxide–semiconductor interface and the confining potential represented by the
conduction band in the silicon. A cut of this confinement is shown in figure 1.3. In
the classical model, the density in the conduction band is related to the separation of
the Fermi energy from the conduction band edge as
⎛ E − EF ⎞
n = Nc exp⎜ − c ⎟, (1.14)
⎝ kBT ⎠
where Nc is the effective density of states, a number on the order of 1019 cm−3 [46]. As
shown in the figure, the conduction band edge is a function of position, so this makes
the density a function of position, and a maximum at the oxide–semiconductor
1-12
interface. From equation (1.14), it is clear that we can write the decay of the density
into the semiconductor in the form
⎛ x⎞
n(x ) = n(0)exp⎜ − ⎟ , (1.15)
⎝ L⎠
where L is a characteristic decay length [47]. In fact, the exponential decay is given
by defining the surface potential φs(x) as the variation of the conduction band away
from its bulk equilibrium value. Then, the decay is determined by the thermal
voltage in equation (1.14), and
⎛ φ (x ) ⎞
n(x ) = n(0)exp⎜ − s ⎟ . (1.16)
⎝ kBT ⎠
Hence, the density has an effective thickness that corresponds to the surface
potential falling by kBT. But, the surface electric field is given as
∂φs (x ) en en(0)L
Es = − ∣x→0 = s ∼ . (1.17)
∂x εs εs
We can put all of this together to obtain a good estimate of thickness of the inversion
layer as
φs (0) k T εs
d eff = ∼ B . (1.18)
eEs e ens
If we take an inversion density of 5 × 1011 cm−2 at room temperature, then we find
that the effective thickness of the inversion layer is about 3.3 nm.
But, quantum mechanically, we need to ask what the corresponding de Broglie
wavelength is for an electron at the Fermi energy in the inversion layer. Suppose we
assume that the average energy of the carrier is just the thermal energy. Then, the de
Broglie wavelength is given as
h h
λd = = . (1.19)
p 2m⁎kBT
Using the transverse mass for transport along the channel, we find that this
wavelength is 18 nm. Now, there is just no way to stuff an 18 nm Marshmallow
into a 3.3 nm hole. The classical idea of band bending is not valid in this quantized
inversion layer, where the potential must be solved in a self-consistent manner.
Before addressing this, let us talk about the phrase ‘transverse mass’. Silicon has a
complicated band structure. The minimum of the conduction band lies along the line
from Γ to X in the Brillouin zone, and is located about 85% of the way to X. Because
of the symmetry of the Brillouin zone, there are six equivalent minima, as shown in
figure 1.4. Each of the six ellipsoids has a longitudinal axis and two transverse axes,
and corresponding values for the mass. In Si, it is generally felt that the effective
mass values are mL = 0.91 m0, mT = 0.19 m0.
1-13
Figure 1.4. A constant energy surface near the minima of the conduction band in silicon consists of six
equivalent ellipsoids oriented along the lines from Γ to X.
Silicon MOSFETs are usually fabricated with the surface normal along a (0,0,1)
direction. Then, the quantization has a beneficial result of splitting the six ellipsoids
into two sets. One pair of ellipsoids has the longitudinal mass normal to the
interface, and this gives a lower-energy set of quantum levels. The other four
ellipsoids show the transverse mass in the direction normal to the interface and, as
this is the smaller mass, will have higher lying quantum levels for motion normal to
the surface. The set of levels corresponding to the two-fold valleys is generally
denoted as E0, E1, E2, … while the set of levels for the four-fold set of valleys is
denoted with a prime on each level. The advantage is that the two-fold set of valleys
now shows the smaller transverse mass in the transport direction, which gives a
higher mobility. The introduction of strain about a decade ago in the industry was
done for the same reason—to separate these valleys and gain a higher mobility.
In figure 1.5, the energy levels for the lowest two states in the two-fold set of
valleys and the lowest energy level for the four-fold set of valleys are plotted for a
range of inversion densities for a silicon substrate doped to 1017 cm−3. The doping
has an effect on the potential, as band bending depletes the substrate so that there is
a contribution to the surface field from this charge. In fact, the surface field has been
estimated to be [48]
e ⎛⎜ n⎞
Es = NAxp + s ⎟ , (1.20)
εs ⎝ 2⎠
where NA is the substrate doping density and xp is the depletion depth of this region.
The factor of ½ arises from the fact that the electric field appears on both sides of the
inversion charge, while it only appears on the oxide side of the bulk depletion
charge. So the results in figure 1.5 are doping-/dependent. These energy levels were
computed using the self-consistent Poisson–Schrödinger solver for the silicon system
developed by Vasileska, and they are referenced to the conduction band minimum at
the oxide–semiconductor interface. The simulation package is called SCHRED 2.0,
1-14
Figure 1.5. The lowest three energy levels, or subbands, for the silicon-silicon dioxide interface with an
acceptor doping of 1017 cm−3. The energy level E0 is shown as the solid red curve, while E1 is depicted by the
upper dashed red curve. The lowest energy level for the four-fold valleys is E0′ and is indicated by the blue line
and symbols.
and is available at NanoHUB.org for anyone to use [49]. Surprisingly, the lowest
state has a ‘thickness’ of about 2.8 nm, which is quite close to the classical value
found above. However, only about 49% of the electrons are in this subband at room
temperature. Some 8.3% are in the second subband of the two-fold valleys, while the
remainder are in the upper, four-fold set of valleys. Of course, at low temperatures,
we expect all of the carriers to be in the lowest energy state, corresponding to a single
highly degenerate subband.
With the momentum normal to the interface quantized, the transport is con-
strained to lie in the plane of the interface. Hence, these electrons form what is
known as a quasi-two-dimensional electron gas (2DEG) [48]. The mobility of these
electrons is limited by scattering from a variety of sources, but primarily by
scattering from ionized impurities, such as the acceptors in the bulk Si. This
impurity scattering is largely an elastic process which does not dissipate energy,
especially at low temperatures. However, the scattering is screened by the electrons
in the inversion layer to some extent. Additional scattering in the silicon system
comes from the roughness at the interface. While this interface is quite good, with
smoothness nearly on the atomic scale, it is random enough to provide significant
scattering of the electrons [50, 51]. Additional scattering processes arise from
the phonons, but the optical phonons are not important at low temperatures, and
the acoustic phonons provide only a weak scattering. In figure 1.6(a), we plot the
mobility as a function of the effective electric field for various temperatures [52]. This
effective field is given by the inversion density, as in equation (1.20). Surface
roughness scattering varies as the square of this field, as in normal perturbation
theory, and the behavior at low temperature agrees well with this interpretation.
1-15
Figure 1.6. (a) Variation of the mobility with the effective field, given by (1.20). The decay with the square of
the field at low temperatures signals surface roughness scattering. (b) Variation of the mobility with
temperature. Various scattering processes are indicated in the plot. (Reprinted with permission from [52].
Copyright 1992 IOP Publishing.)
The contributions from several scattering mechanisms are shown in panel (b) for a
fixed density of 1012 cm−2. Again, it is fairly clear that ionized impurity scattering
dominates the mobility at low temperature, but that surface roughness scattering
plays a role in limiting the mobility.
1.4.2 Fabricating the MOSFET

While this book is not about semiconductor devices per se, one does encounter the
problems of nano-fabrication when trying to create mesoscopic structures with
which to study the physics. In order to fully comprehend the broad range of
problems one encounters in this task, it is useful to actually go through the levels
of detail that are encountered in creating a MOSFET, although this is only one type
of device. Most of the necessary processing is quite similar in any type of
semiconductor device, and this should be kept in mind while reviewing the topic.
To do this, we follow a detailed process flow for a MOSFET with an effective 25 nm
gate length. Although this process is about a quarter of a century old, much of it is
still used in detail today, so it remains surprisingly relevant [53]. Now it is also
important to understand that the end result may not be the proper one. That is, the
device that has been built may not be the one that was either desired or designed.
One would hope that it is quite close, especially if we want to say that it we
understand the various processes from beginning to end.
The first step is isolation of the individual device, which begins with oxidizing the
bare silicon wafer, particularly in the region where the device is to be located, as
1-16
Figure 1.7. Isolation and forming the channel. (a) The oxidized silicon waver. (b) A hole is opened by
lithographic techniques (see text). (c) Dopants are implanted into the waver at the desired location.
shown in figure 1.7. Shown in panel (a) is the oxidized wafer. After this, the sample is
coated with a positive photoresist. With a positive photoresist, the molecules are
usually long-chain molecules for which the light breaks up these chains. Then the
developer dissolves these short chains leaving a hole where the light was projected.
Then the oxide can be etched away leaving a hole, as in panel (b). The remaining
oxide isolates this device from others. To create the p-type layer that will be the
channel region, boron is implanted through the hole at an energy of 30 keV and a
dose of 3.6 × 1013 cm−2. The energy determines the depth of the implant while the
dose will determine the final doping density in the p-type region. Now, we have to
correct a problem. When the Si is implanted, the energetic ions severely damage the
crystal structure of the Si. To repair this damage, the structure must be annealed.
But, the annealing proceeds via an oxide regrowth process. In Si, the fast growth
direction is the (001) direction, so the wafer we use has to be one in which the surface
normal is the (001) direction. If it is any other direction, the fast growth direction will
not be normal to the interface, and we will not be able to anneal out all of the
damage. This follows as the growth proceeds in the (001) direction, which if it is not
the surface normal causes many growth fronts to interfere, leading to dislocations
and grain boundaries. In the case under discussion, the annealing is carried out at
1000 °C for 2 h. Of course, the implanted atoms will diffuse during the anneal
procedure, and this has to be accounted for during the ‘design’ of the device. As a
result of this thermal process, the oxide is regrown over the implanted region and the
surface planarized for the next step.
1-17
The next step is to form the central gate oxide. Once more a positive photoresist is
deposited on the oxidized wafer, and exposed and developed. This time, however,
the new hole is smaller than the previous one and must be located in the center of the
p-type layer formed with the previous hole. This new hole is shown in figure 1.8(a),
and is expanded in panel (b) (red dashed lines indicate the expansion). Once the hole
is opened in the resist, the oxide is etched away so that the surface of the p-region is
exposed. Now, the gate oxide is grown at 800 °C for 8 min, producing a 3 nm
thickness. Over the top of this is deposited/grown polysilicon which will serve as the
actual gate material (dark blue in figure 1.8(b)). This polysilicon is heavily
phosphorous doped to make it n-type and to have a low resistivity. At this point,
it should be pointed out that the shift to an alternate gate material, of high dielectric
constant, was made a few years ago. Typically, this new ‘oxide’ is hafnium oxide.
The technology was developed at Intel (and elsewhere) [54]. The process was
introduced in the 45 nm ‘node’, where the effective gate length is about 25 nm, in
2007–8. In the process flow, the deposition of the polysilicon gate is considered to be
a dummy, in that it will be removed later and replaced with the new oxide. After the
source and drains and stressor components have all been introduced (discussed
below), the polysilicon gates and the gate oxides are removed. Then, the HfO2 is
deposited by atomic layer epitaxy, which basically is a set of chemical reactions that
leave the desired material layer in place [54, 55]. The higher dielectric constant
means that the same gate capacitance can be achieved with a thicker oxide, which is
necessary to prevent tunneling through the thin oxide. The downside is that the gate
itself is no longer polysilicon, but is another material which is chosen to have the
proper work function for the device design. Typically, now different gate materials
are used for the n-channel and the p-channel.
Now that the gate material is deposited, it is time to pattern it to the actual short
gate length desired. This time, a negative photoresist is deposited on the gate
material. With a negative photoresist, we desire the exposing light to cross-link the
material which is dominantly short molecules. Hence, when developed, the exposed
pattern is all that remains. This exposure is typically carried out with an excimer
laser, but even this has a wavelength that is too large to make the small gate desired.
Figure 1.8. (a) The gate region opening is produced in the oxide. (b) The thermal oxide gate is then grown,
followed by deposition of the heavily doped polysilicon gate.
1-18
So, the photoresist that remains, shown in figure 1.9(a) is thinned in an oxygen
plasma (figure 1.9(b)). The resist is thinned until it is 40 nm in length in the source-
drain direction. Hence this is called a 40 nm drawn (lithographically defined) gate.
This pattern is then used to protect the desired gate while the rest of the polysilicon is
etched away with a reactive ion etching process leaving the desired gate in
figure 1.9(c).
It should be remarked that the gate is oriented so that the carriers (in this case,
electrons) will move in the (110) direction in the surface layer. From figure 1.4, and
the discussion about it, it may be recalled that the minimum of the conduction band
lies along the Γ to X line, which is the (100) direction, and lies about 85% of the way
to X. Because of the crystal symmetry, there will be 6 such minima, all of which are
equivalent. A constant of energy surface near this energy will be an ellipsoid of
revolution (a cigar-like shape). The long axis of the ellipsoid is parallel to one of the
(100) directions. Hence, in our device, 4 of these valleys lie in the surface plane, and
2 of them are oriented with the long axis normal to the plane. Now, the (110)
direction makes an equal angle with all 4 of the ellipsoids that lie in the plane, so that
the transport in the device does not see differences in the 4 ellipsoids. Hence, the
transistors are almost always oriented in the (110) direction.
To proceed, we have to face another problem. The natural approach is to now to
implant the source and drain dopants, in order to convert the p-type layer to a
heavily doped n-type layer (creating the n-p-n doping profile down the channel). If
we do this directly at this point, using the gate as a mask to keep the implanted ions
from the region under the gate, these dopants will diffuse into this masked region
during the annealing process after the implant. This will do away with the p-layer
entirely and ruin the transistor. We could place a spacer layer on either side of the
Figure 1.9. (a) The negative photoresist after exposure and development. (b) This resist is then thinned in, for
example, an oxygen plasma. (c) The final gate, after the resisit is stripped away.
1-19
gate to account for the diffusion but this makes the size of this spacer hyper-critical
to the entire process. However, this is the approach we shall follow, but the spacer
will be made larger than the anticipated diffusion along the surface, and it will be
made of a special material. In this case, the spacer is a phospho-silicate glass (PSG).
It is deposited, and again patterned by a lithographic process to result in a structure
such as that of figure 1.10(a), where the PSG is shown in green. This will serve two
purposes. First, it protects the channel area from diffusion of the implanted atoms.
Secondly, it will provide a source of phosphorous atoms that will diffuse out of the
PSG during the anneal and these will form a shallow doped layer that extends from
the source/drain regions to the channel. These regions are known as source(drain)-
extensions. To create the proper source and drain regions, As atoms are implanted at
30 keV with a dose of 5 × 1015 cm−2. The PSG has a phosphorous concentration of
about 3 × 1021 cm−3, and provides a relatively constant source of this dopant during
the anneal. The anneal itself is done by rapid thermal annealing (RTA) at 1000 °C
for 5–10 s. The implantation and anneal of the As gives source and drain regions
that are doped at approximately 5 × 1019 cm−3, and extend about 70 nm below the
surface. The extensions created by the phosphorous out-diffusion from the PSG, are
doped much higher and extend some few nm from the surface, as shown in
figure 1.10(b), which displays what is essentially the completed transistor, lacking
only the metallization layers. One should note that even the shallow phosphorous
diffusion lets the atoms penetrate into the region under the gate. This gives the
distance between the two (red) extensions as about 25 nm, which is called the
effective gate length.
Figure 1.10. (a) The addition of sidewall spacers, formed from PSG (in green), helps in the transistor
fabrication and also helps keep the implanted source/drain atoms from diffusing into the channel regions. The
polysilicon gate is the top blue region, while the narrow grey region is the gate oxide. (b) The finished
transistor. The red regions represent the heavily doped n-type areas that form the source and drain regions of
the transistor.
1-20
Now, there is an additional detail that has appeared in recent years, and that is
the use of strain to change the effective mass, as was pointed out in section 1.4.1. To
accomplish the tensile strain in the n-channel device, a material such as Si3N4 is laid
over the gate stack (the gate itself and the sidewall spacers in figure 1.10) at high
temperature. As it cools, it expands and imposes the tensile strain on the channel
material. For the p-channel device, we need compressive strain. To achieve this, we
do not implant the source and drain as described above. Instead, we use another
lithographic step to open holes in the p-Si layer where the source and drain are
desired to sit. Then, acceptor doped SiGe is regrown in these holes. With the large
lattice constant of this latter material, the expansion imposes compressive forces
upon the channel material between the source and drain.
1.4.3 The GaAs/AlGaAs heterostructure

The most popular material system for mesocopic devices is the GaAs/AlGaAs
heterostructure system. In GaAs, the zinc-blende lattice is a face-centered cubic (fcc)
lattice in which each lattice site is occupied by a diatomic molecule of one Ga atom
and one As atom. If the Ga atom sits on the lattice site, the As atom is displaced one
quarter of the distance across the cube in the body diagonal direction; e.g., (a/4)
(111), where a is the edge of the fcc cube. These two atomic sites are known as the A
site and B site. GaAlAs is an example of a ternary alloy system in which some
fraction of the A site Ga atoms are replaced with Al atoms. Thus, if 30% of the Ga
atoms are randomly replaced with Al atoms, the alloy is referred to as Ga0.7Al0.3As,
or GaxAl1−xAs with x = 0.7. It is important to point out that this is a totally random
alloy in principle, so that there is no clustering or precipitation of various
compounds within the crystal. One reason for the ternary is that the band gap
increases as the percentage of Al increases, thus one can create a series of quantum
barriers and wells with multiple layers of the two compounds. However, AlAs is an
indirect material, and the alloy becomes indirect at around x = 0.55. One usually
stays with the more Ga-rich alloys, especially for optical applications, so that the
band gap is direct at the Γ point. Another important issue is that the lattice constants
of AlAs and GaAs are nearly the same. Hence, the alloy can be grown on GaAs with
almost zero strain in the crystal, and this results in almost atomically sharp
interfaces [56].
The band gap of AlGaAs is larger than that of GaAs, and this difference must be
taken up by band bending at the interface. Part of the band discontinuity is taken up
in the conduction band and part in the valence band, as shown in figure 1.11.
Currently, it is felt that the conduction band discontinuity is about 63% of the total
energy band discontinuity [57]. Prior to bringing the two materials together
(conceptually), the Fermi level will be set in each material by the corresponding
doping. Usually, the GaAs is undoped so that the Fermi level is near mid-gap and
thought to be set by a deep trap level in this material. Once the interface is formed,
however, there must be a single Fermi level that is constant throughout the material,
as no current is flowing. This leads to the band bending as shown in the figure, and
the band discontinuities provide certain offsets that are shown in the figure. In the
1-21
Figure 1.11. The band lineup at the hetero-interface between GaAs (on right) and AlGaAs (on left). The
setback of the dopants is also indicated.
early days, it was common to uniformly dope the GaAlAs, but this is no longer
done. Rather, a single layer of dopant atoms is placed in the GaAlAs a distance dsb
from the interface [58]. Regardless of the doping method, electrons near the interface
will move to the lower-energy states in the quantum well on the GaAs side of the
interface. For the δ-doping case (a single layer of dopants), all of the electrons will
move to the GaAs. In the uniform doping situation, only a small fraction will move.
This technique of getting the electrons into the GaAs is known as modulation
doping [59]. In this approach, the actual ionized dopants are set some distance from
the electrons, so that the Coulomb scattering potential is weakened. Moreover, the
electrons themselves work to screen this scattering potential. These effects lead to
very high mobilities for the electrons in the GaAs. In fact, mobilities above 107 cm2
Vs−1 can be obtained for the electrons in the GaAs at low temperatures [60] and this
is thought to be limited by scattering from the dopants themselves [61]. The usual
dopant for the GaAlAs is silicon, which acts as a donor. However, it can also form a
complex which leads to a trap level, known as the DX center [62]. This trap can be
avoided if the composition of the GaAlAs is kept below about x = 0.25. So this sets
additional limits on the heterostructure.
The GaAs/GaAlAs heterostructure is typically grown on a GaAs semi-insulating
substrate. First, a superlattice formed of thin layers of GaAlAs and GaAs is grown.
This has the double effect of smoothing the surface and trapping dislocations within
the superlattice. Then, a thick undoped GaAs layer is grown, followed by the
GaAlAs layer. For the latter layer, the growth is interrupted to deposit the dopant
layer a desired distance from the interface, and then the additional ternary is grown
to the desired thickness. For the highest mobility layers, dsb can be more than 20 nm
thick. The thicker this set-back layer is, the higher the mobility will be and the lower
the inversion layer density will be. Finally, a heavily doped GaAs ‘cap’ layer is
grown on the surface, which serves two purposes. First, it prevents unwanted
oxidation of the GaAlAs surface. Second, it provides a layer to which it is easier to
make ohmic contacts. The preferred growth method for high quality material is with
molecular beam epitaxy, a growth technique achieved in an ultra-high vacuum. The
atomic constituents are provided by heated sources, known as Knudsen cells, in
which the atomic species are individually vaporized (within the Knudsen cell) and
shutters are used to turn on and off the flow of atoms from the cell. Careful control
1-22
Figure 1.12. (a) The electron density as a function of the setback distance of the δ-doping layer. (b) The energy
of the lowest quantum level as a function of the electron density.
of the deposition rates and the substrate temperature can lead to atomic layer
epitaxy and the ultimate control of the growth. However, this process leads to low
growth rates, below a micron per hour, so it is not conducive for thick structures.
This growth process, and slow grow rate, allows for the design of semiconductor
multi-layers with very precise control of the overall structure for quite specific
applications. Quantum well structures are created by sandwiching a GaAs layer
between two GaAlAs layers, and the bound states in the well can be precisely tuned
by careful control of the thickness of the GaAs layer and the effective barriers
formed by the band offsets arising from a precisely controlled alloy composition.
This process has led to a wide variety of emitters and detectors of radiation over a
wide spectral range.
In figure 1.12, we illustrate the density and quantization energy in the inversion
layer for a GaAlAs/GaAs heterostructure. This is an estimate using the ‘bound state
calculation lab’ at NanoHub.org [63]. The tool was used to find the quantization
energy for a given density, from which the Fermi energy could be found. We used a
δ-doped layer of 1012 cm−12 which was set back a distance dsb and assumed a
conduction band offset of 0.25 eV. It can be seen that the larger one makes the set-
back distance, the lower the electron density becomes. At the same time, this will
increase the mobility by moving the ionized dopant atoms further from the electron
layer.
1.4.4 Other important materials

If we consider just the binary III–V materials, there are already a large number of
possible heterostructures that can be grown. In figure 1.13, we plot the energy gaps
as a function of the lattice constant for the group IV and III–V compounds. Also
shown are some rough connector lines for a few ternary alloys. One popular
substrate is InP, not the least because it is lattice matched to In0.53Ga0.47As which
1-23
Figure 1.13. A wide range of possible materials can be chosen for specific properties of energy gap and lattice
constant to fit desired characteristics and a convenient substrate.
has an energy gap well suited to match the minimum dispersion in quartz fibers, a
match very important for long distance fiber communications. The quaternaries
InGaAsP or InGaAsSb provide the ability to both lattice match InP as a substrate
and to vary the band gap over a very wide range. The ability to thus choose a
material to provide desired characteristics is quite important in these materials. InAs
has become important for THz high-electron-mobility transistors (HEMTs) as well
as with AlSb for spin applications.
The group III nitrides provide another set of materials with a wide range of
attributes. For example, GaN has a band gap of 3.28 eV and an a-plane lattice
constant of 0.316 nm. In AlN, these values are 6.03 eV and 0.311 nm, while in InN,
these values are 0.7 eV and 0.354 nm. Thus, alloys of these materials can span the
entire visible range, and GaN-based systems have found a home in blue and blue-
green lasers. They are also being pursued for high power HEMTs. While one
normally tries to lattice match the various layers in a heterostructure, a controlled
mismatch can be used quite effectively. The group III nitrides tend to be ferro-
electric, which means that they have a built-in polarization in the lattice. The
discontinuity in this polarization at an interface can be used effectively to induce the
inversion layer charge to form without the need for dopants [64].
We cannot ignore the newer two-dimensional materials. The world is full of real
two-dimensional semiconductors, which are called layered compounds. The best
1-24
known on is graphite, in which layers of atomically thin carbon are weakly bound
together to produce the bulk. The single layer of carbon atoms is called graphene,
and it has been isolated only recently [65]. In this single layer graphene, the carbon
atoms are arranged in a hexagonal lattice, which has two atoms per unit cell. The
single layer of graphene is exceedingly strong, and has been suggested for a great
many applications. There are other layered compounds, which have similar
structure, and a well-known group is the transition metal di-chalcogenides
(TMDC), such as MoS2 and WSe2 [66]. In the TMDC, the layer is actually an
atomic tri-layer, with the metal atoms forming the central layer, and the chalcoge-
nide atoms forming the top and bottom layers. Each metal atom has a triangle of
three chalcogenide atoms above it and below it. Yet the basic structure still consists
of hexagonal coordination with two so-called atoms per unit cell—these consist of a
metal atom and an up and down pair of chalcogenide atoms, which sit above one
another to form a pseudo-atom. As a result, both graphene and the main TMDCs
have a quite similar band structure. In the graphene case, the conduction and
valence bands are composed of the pz orbitals of carbon [67]. In the TMDC case, the
conduction and valence bands are determined by the metal d orbitals [66], as these
bands lie in the gap between the bonding and anti-bonding sp hybrids. The TMDC
materials have some interesting properties, particularly with their spins, as they
basically lack inversion symmetry in the plane of the layer. As a result, the spin–orbit
interaction can lead to unique effects such as the spin Hall effect. We will deal with
these materials in subsequent chapters—graphene in chapter 4 and the TMDCs in
chapter 5.
1.5 Superconductors
Superconductors have been an interest set of materials for a great many years.
Kammerlingh Onnes discovered superconductivity in 1911, while examining the
properties of metals at low temperatures [68]. He had succeeded in liquefying He
only three years earlier and was anxious to use this new cryogen. As he cooled
mercury below 4.2 K, he observed a strange phenomenon—the resistance dropped
almost to zero abruptly at this temperature. By the time it had been cooled to 3 K,
the resistance was less than 10−6 Ω. This phenomenon has been observed in more
than one quarter of the elements of the periodic table. And, it has been observed in a
number of compounds at higher temperatures (today referred to as high-temperature
superconductivity). The temperature at which the transition begins is known as the
transition temperature Tc, and the transition itself is recognized as a thermodynamic
phase transition. Nothing much was done with this phenomenon, primarily due to
the scarcity of liquid helium, until after the Second World War. Subsequently, the
needed cryogen began to be available in quantity and cheaply, which led to a relative
boom in studies. Then, these materials became of use in producing magnets, both for
experimental systems for science and for their application in motors and generators.
Perhaps the largest applications are for measuring derivatives of the magnetic field
using superconducting quantum interference devices (SQUIDS) by the military, and
1-25
R(T)
Paraconductivity
1 T/Tc
Figure 1.14. Superconductivity appears below the transition temperature Tc. There is a fluctuation induced
lowering of the metallic resistance just above this temperature, which is called paraconductivity.
now for the burgeoning field of quantum computing. We will deal with the latter in
detail both below and in chapter 8.
The most easily observed characteristic of superconductors is the transition
temperature Tc, below which superconductivity occurs in the material. One can
readily observe the transition temperature when one plots the resistance versus the
temperature, as in figure 1.14. The temperature scale in the figure is normalized to
the transition temperature, Tc. There is a slight rounding of the curve just above Tc,
and the enhanced conduction in this region is often called paraconductivity, in
analogy with paramagnetism, or the enhancement of magnetism. This enhanced
conductivity is attributed to thermal fluctuations at the transition temperature, and
we can think of them as very small regions, which are beginning to exhibit
superconducting behavior. Superconductivity of the entire sample does not occur,
since the small regions are superconducting for only short periods of time and are
generally unconnected. In a sense, these are like the domains exhibited in magnetic
field. Only a few of the domains are superconducting, and then only for short periods
of time. The drop in resistance at T = Tc is exceedingly sharp, and occurs over a
small fraction of a degree, so that the onset of superconducting behavior is easily
observed. The transition temperature is a fundamental property of superconductors.
As we mentioned above, only about one-fourth of the elements of the periodic
table exhibit superconductivity. Such elements as silver and copper are not super-
conductors at the lowest temperatures measured. In fact, it seems that the higher the
conductivity of a material in the normal state, the poorer this material is as a
superconductor. This implies that if a material is a good conductor in the normal
state (T > Tc), it probably has a very low transition temperature or does not exhibit
superconductivity at all. We can understand this effect by thinking about the
dominant interaction mechanism leading to superconductivity. This interaction, at
least in the pure compounds, is generally conceded to be the electron-lattice
interaction, that is, the interaction between two electrons of opposite spin mediated
by interactions with the lattice. The stronger this interaction is, the more likely a
material is to be a superconductor. However, this is related to the dominant
1-26
scattering mechanism in normal metals. If the electrons are strongly scattered and
the conductivity is low, this interaction is strong, and the conductivity is low. Hence
the material is a poor conductor.
An exceedingly useful property in superconductors is the presence of persistence
currents. Consider, for example, a superconducting ring. If a current is induced in the
material in the normal state, and the material is then cooled below Tc, the current in
the ring will persist for a very long time. Since the material has no resistance in the
superconducting state, there is no decay mechanism for the current. Such a
mechanism is exceedingly useful in superconducting magnets. If a fixed value of
magnetic field is required, the superconducting solenoid is energized to the proper
current value by an external power source. Then the current leads are shorted
together at the magnet by a second superconductor. This forms a superconducting
loop, and the external supply can be turned off, while the circulating current in the
loop remains unchanged.
1.5.1 The Meissner effect

A magnetic field can be used to destroy superconductivity. If a magnetic field is
applied to a superconductor, then, for H > Hc, a critical value of the magnetic field
which is different for different superconductors, the superconductivity is destroyed
and the material reverts to the normal state. It is generally found that Hc is a
function of temperature as
⎡ ⎛ T ⎞2 ⎤
Hc = Hc0⎢1 − ⎜ ⎟ ⎥ , (1.21)
⎢⎣ ⎝ Tc ⎠ ⎥⎦
where Hc0 is the critical field at absolute zero. The critical field at T = 0 K is a
function of the critical temperature. The additional magnetic energy of the electrons
serves to break the superconducting bond interaction, just as thermal energy does.
The effect of both types of additional energy terms leads to a result like equation
(1.21). The critical magnetic field will also limit the amount of current which a
superconductor may carry. Since the current itself gives rise to a magnetic field, the
current carried by a superconductor must be less than that which would produce Hc.
For a higher current than this, the magnetic field produced by the current destroys
superconductivity and the material reverts to the normal state. Thus, superconduct-
ing magnets utilize materials with a high critical temperature, such as Nb3Sn, where
Tc = 18.05 K.
The onset of superconductivity leads to a phenomenon called the Meissner effect.
If a superconductor is cooled below Tc, the magnetic flux lines are expelled from the
material. Thus, within a superconductor, the magnetic flux density is zero, B = 0.
Hence, superconductors exhibit perfect diamagnetism, μr = 0. The magnetization
which results from the applied field must completely oppose any applied field. This
occurs up to the critical magnetic field in type I superconductors. But, in many
compound superconductors, there is a lower critical field, Hc1, where the Meissner
effect begins to fail. Thus, there is incomplete expulsion of the flux, and this gets
1-27
weaker and weaker as the external field intensity increases, up to an upper critical
field, Hc2 , where the superconductivity vanishes.
A somewhat different behavior is observed if the superconductor is fabricated as a
thin film. If the films are very thin, the material does not exhibit perfect
diamagnetism. As we shall see in the next section, the film must be of a certain
thickness before it exhibits the properties of bulk superconductors, such as perfect
diamagnetism. In addition, in a thin film, the magnetic field has very little effect and
the critical field is much higher than for the bulk material.
1.5.2 The BCS theory

A number of the preceding experimental observations are consistent with the
existence of an energy gap associated with superconductivity. Theoretically, the
electrons which contribute to superconductivity do so as paired electrons. This set of
two electrons is known as a Cooper pair, and is formed from two electrons which
have opposite spin angular momentum [69]. Thus, if one of the electrons is spin up,
the other is spin down. Normally, electrons repel one another because of Coulomb
forces. But at low temperatures, an additional interaction between the electrons, in
which they interact with the lattice, leads to a weak attractive force between the
electrons. This leads to a pairing of the electrons. This pairing of the electrons results
in a lower energy state than would result from the electrons remaining unpaired.
This, in turn, results in an energy gap of the electrons [70]. The observed energy gap,
though, would correspond to EG = 2Δ, since each of the paired electrons must
receive an additional energy of Δ in order to break the pairing bond, as shown in
figure 1.15.
At absolute zero, all the conduction electrons are paired. At higher temperature,
some of the pairings are broken by thermal agitation, so that some normal electrons
are excited across the energy gap in much the same way that electrons are excited
across the band gap in semiconductors. For temperatures slightly above Tc, all the
pairs are broken and the material exhibits normal resistivity. In the superconducting
state, a mixture of normal electrons and superconducting paired electrons is present
in the material. The normal electrons may be observed in tunneling experiments,
such as we discuss in a later section of this chapter. The energy gap also varies with
E
2∆
EF
Figure 1.15. An energy gap 2Δ opens at the Fermi energy and this gap separates the superconducting electrons
from the normal electrons. At absolute zero of temperature, all the electrons form Cooper pairs. This has the
unusual requirement that the electron number is an even quantity.
1-28
temperature, decreasing with increasing T until, at Tc, the energy gap is zero. The
general behavior of the gap near Tc is given by
T
EG = EG 0 1 − , (1.22)
Tc
where EG0 is the value of the gap at zero temperature. As a first approximation, the
energy gap at zero temperature is related to the energy gap at the transition
temperature as EG0= 3.5kBTc. This result is not exact, but is approximately correct
for a wide range of superconductors.
The basic developments of the BCS theory show that an attractive force between
the electrons can lead to a lower energy state at low temperatures, and, hence, can
lead to an energy gap, as discussed above. For the attractive force, their theory
suggests a second-order interaction between electrons and the lattice. One electron
interacts with the lattice; this interaction deforms the lattice slightly in the
neighborhood of the electron. This deformation is much like the modification of
the density of the electrons near an ionized impurity that leads to electron screening
effects over a length of the order of the Debye length. In the BCS theory, a similar
effect occurs between the electron and the deformed lattice. A second electron
encounters the deformed lattice and interacts with it. Thus, in effect, the second
electron interacts with the first, but does so through the lattice interaction.
Effectively, one could say that the deformed lattice shields the electron’s repulsive
force (which is weaker for electrons with opposite spin) so that the second electron is
attracted to the deformed area, and hence to the first electron. An important aspect
is that the two electrons must have opposite spin, otherwise the Coulomb interaction
between them is just too strong for them to form a pair. The attractive force between
these two electrons, with opposite spin, lowers the energy of the pair. Thus, it costs
energy to break the pair and a gap opens between the paired electrons and the un-
paired electrons, as shown in figure 1.15. The details of the BCS theory are beyond
the scope of this book.
Normally, when a gap opens, the electrons cannot carry any current as they
cannot gain any energy due to the fact that the states below the gap are completely
full. This was important in the band theory of semiconductors. In fact, super-
conductors carry current without dissipation, so there is no need for the paired
electrons to gain energy. In fact, the pair will hold together until a large amount of
energy, given by the gap, is injected into the pair by some mechanism. Hence, there
must be a critical minimum velocity, which corresponds to the presence of an energy
gap. If the velocity of the electrons is less than the value of the critical velocity, no
dissipation, or energy loss, occurs, and the material has zero resistance. However,
for sufficient energy, the pair is broken and normal electrons occur.
The existence of the energy gap can be readily demonstrated by measurements at
microwave or infrared wavelengths [71]. At absolute zero, photons with energy of
ℏω < EG = 2Δ are not absorbed, while those with by ℏω > EG = 2Δ cause tran-
sitions across the energy gap and result in absorption of photons. The photon energy
is absorbed by the electron pair, and this additional energy is sufficient to break the
1-29
bond of the pair. For temperatures above zero, some absorption occurs due to the
presence of a few normal electrons. Since these normal electrons are already excited
across the gap, they are free to absorb any photon that comes along.
1.6 Bits and qubits

When the ENIAC was first developed at the University of Pennsylvania at the end of
the Second World War, it computed with base-10 circuits [72]. The simplest
operation, the addition of two numbers took 0.2 ms. But, calculation in base-10 is
somewhat wasteful of computer resources. It was John von Neumann who suggested
at the time that the binary system should be used (and he also suggested the stored
program approach, which along with binary arithmetic, formed what is called today
the von Neumann architecture [73], the EDVAC was the successor of the ENIAC).
Essentially, all computers from this time forward have used the binary system, as it is
eminently suitable to two-state bits. In this space, we define the two unit values for a
bit as the states 0 and 1. Then, using quantum notation, we define the state of a bit in
this space as
∣ψ ⟩ = a∣0⟩ + b∣1⟩, (1.23)
where
a , b ⊂ (0, 1), a + b = 1. (1.24)
Hence, a and b can be either a 0 or a 1, but the two must be different.
In the computer itself, a particular state is defined by the values of the bits used to
describe the state. Then, one can define a state transition diagram, in which each
possible state is a node, and transitions between these nodes indicate the action taken
when a control signal is applied [74]. Various outputs arise from the set of transitions
and the state of the machine after the transition. Hence, an algorithm is a set of
instructions that guide the system through a set of transitions, changing the various
bits under the guidance of the control signals, to a desired result. Good algorithms
lead to the machine halting at the end of the algorithm and yield a desired output
(and with bad algorithms, the machine sometimes never stops). In fact, the stop state
was required by Turing, as he showed that a number was computable only if the
machine stopped [2]. This same operational model will exist also in the quantum
computer. That is, the qubits will define a state, state transitions will depend upon
control signals.
Let us diverge a bit, as ever so often, someone suggests that the quantum
computer can be reversible and therefore not dissipate energy. But, this is a
perpetual motion machine, even if it is superconducting. Above, we designated
the bit (or qubit) by ∣ψ ⟩. Suppose we have n bits in the machine. Then, there will be
2n combinations of the qubits and this many actual states of the machine. Now we
can express the value of the state by a many-body state with n 0’s or 1’s total. With
this designation, the state transition matrices will be n × n matrices, each selected by
a particular input. However, we can also express the state by a many-body matrix
with 2n − 1 zeroes and a single 1 [75]. This single 1 designates in which state the
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machine exists. Now, if the machine is to perform reversible logic, each state must
have a single successor state and a single predecessor state; e.g., the mapping must be
1:1 in each direction—no fan out or fan in. In this case, the state transition matrix is
such that each row has a single 1 and each column has a single 1, and the matrix is
full rank. It turns out this type of matrix is a characteristic matrix of the cyclic
permutation group of order 2n. Thus, each state lies on a ring upon which it cycles
forever. This machine has no stop state and cannot do computation! And, it is a
perpetual motion machine. Hence, a machine such as this does not meet the general
requirements for computation [76].
When we move from the bit to the qubit, the change is rather subtle. We still have
the two values 0 and 1, which can considered to be coordinate axes representing
these values. But, now the wave function is analog and can take any value on the
unit circle. Hence, equation (1.23) is still valid, but the coefficients now must satisfy
∣a∣2 + ∣b∣2 = 1. (1.25)
Thus, the two coefficients are complex numbers, and equation (1.25) tells us that the
net wave function must have a magnitude of 1, as required by any quantum wave
function. Now, the problem that arises is that this formulation is overly restrictive to
the actual wave function for the quantum qubit. The two states 0 and 1 can be
thought of as representing real energy levels of a two level atom (we will return later
to why we use the atom model, and connect the argument with spin in chapter 7).
For convenience we assume these two states are aligned along the z-axis in spherical
coordinates. From the latter concept, we then tend to write the Hamiltonian
representing this two level system with the Hamiltonian
Δ ⎡1 0 ⎤
H= , (1.26)
2 ⎢⎣ 0 − 1⎥⎦
and the wave function (1.23) is expressed as
⎡ a ⎤ ⎡∣a∣e iφ1 ⎤
ψ = ⎢⎣ ⎦⎥ = ⎢ iφ ⎥ . (1.27)
b ⎣∣b∣e 2 ⎦
Hence, we have mapped the (0,1) states into the states (−1,1), in terms of Δ/2, with Δ
the energy difference between the two states. Hence, the upper level is taken to be the
1 state and the lower level the −1 state.
It is convenient to write the wave function in a slightly different form which
characterizes the system, but make the two phases somewhat less arbitrary. Hence,
we write the two-level system in terms of the density matrix
⎡∣a∣2 ab⁎⎤
ρ=⎢ ⁎ ⎥, (1.28)
⎣ a b ∣b∣2 ⎦
from which we infer that equation (1.25) requires the trace of the density matrix to
satisfy
1-31
Tr{ρ} = 1. (1.29)
In this approach, we introduce the polarization P, and write equation (1.28) as
1
ρ= (I + P · σ ), (1.30)
2
where I is the 2 × 2 unit matrix and the vector σ has the components σx, σy, σz. These
so-called spinors are each a 2 × 2 matrix, which will be discussed in chapter 7. The
important point here is the polarization, which is related to the coefficients in
equation (1.25) as
Px = 2Re(a⁎b)
Py = 2Im(a⁎b) . (1.31)
2 2
Pz = ∣a∣ − ∣b∣
The language in the above description of the density matrix gives the polarization in
the real three dimensional space. But, we can think of the polarization itself as a
vector in an abstract three dimensional Euclidean space. In this space, the state 0 is a
unit vector pointing in the +z direction, while the state 1, is a unit vector pointing in
the –z direction. The polarization itself resides on the unit sphere in this space; this
unit sphere is called the Bloch sphere. Such a Bloch sphere is shown in figure 1.16,
and the two angles are defined as the polar angle and the azimuthal angle, where the
former is the polar angle measured away from the +z-axis, and the latter is the
azimuthal angle and lies in the (x,y) plane and is measured from the x-axis, as is
normal for spherical coordinates. Thus, many qubit operations are easily expressible
as rotations around the three axes of the Bloch sphere.
Figure 1.16. The Bloch sphere allows the polarization to rotate in three directions. Then, many qubit
operations are expressible as rotations around one of the three axes.
1-32
To examine the nature of the polarization itself, let us make some angular
definitions of the various components of the wave function (1.23), as
a = e iγ cos(ϑ /2)
, (1.32)
b = e iγ −ϕsin(ϑ /2)
so that
Px = cos(ϕ)sin(ϑ)
Px = sin(ϕ)sin(ϑ) . (1.33)
Px = cos(ϑ)
In the above discussion, we talked about the qubit as a two-level system,

specifically a two-level atom. This could be two specific levels of a real atom, such
as in the ion trap system, or in an artificial atom such as a quantum dot. The
important point is that the phrase ‘two level atom’ is often a ‘code word’ for the
qubit system. This is because the energy levels in an atom are nonlinearly spaced;
that is, they vary roughly as 1/n2. Hence, the two desired levels are separated by an
energy that is not the same as the separation of any other two levels. If the levels
were linearly spaced, as in a normal harmonic oscillator, one has a problem. For
example, if the lower two levels are those of interest, they are separated by exactly
the same energy as any other two adjacent levels. This means that excitation of the
lower two levels can also be absorbed by any other two levels, and this constitutes a
sizable source of decoherence in the qubit. Hence, one desires the energy levels to be
nonlinearly spaced, so that the two levels of the qubit do not couple to other unused
energy levels. Since, the atom has this property, the phrase ‘two level atom’ is often
applied to any qubit system to indicate it has the desired properties.
One of the most important aspects of the quantum processor is entanglement.
Indeed, Erwin Schrödinger called entanglement the most important aspect of
quantum mechanics [77]. This paper was his response to the Einstein–Podolsky–
Rosen paradox published earlier in the year [78]. His point was that once the two
particles (envisioned by the latter authors) interacted, they were no longer inde-
pendent from one another. Rather, they must now be described by a single entangled
wave function incorporating both particles. And, it was this entanglement that
differentiated the quantum system from the classical one. Needless to say, it is
entanglement that has become the crucial ingredient in quantum computing that
gives rise to the possibility of major speedup in the computation. In the quantum
computer, we want qubits to interact with each other under controlled circum-
stances. When two qubits interact, they become entangled, just as the two particles
in EPR. They remain entangled until some mechanism gives rise to decoherence,
which breaks up the entanglement. As a result, one needs long decoherence times in
the qubits.
In general, the quantum computer works on controlled gates, as these gates
provide the interaction between the qubits. As an example, we take the so-called
CNOT gate, or controlled NOT. In the classical system, any transistor (or CMOS
1-33
gate) is naturally a NOT gate. This is because the transistor inverts its input. If we
take a high voltage as a logical 1 and a low voltage as a logical 0, then a 1 input to
the gate causes the transistor to turn on, which lowers the output voltage, and this
gives a 0 output. Similarly, when the input voltage is low, the transistor is in the off
state and the output voltage is high. Thus, the input is inverted, which means that the
logical output is NOT(input). The CNOT differs from this slightly, in that the action
of the transistor gate is controlled in a manner that makes the classical XOR gate.
Now, the XOR gate has two inputs, which are called a and b. If either a or b is high
(a 1), then the output is also high. But, if both inputs are either high or low, the
output of the gate is low. In other words, the output of the XOR gate is 1 if either a
or b is high, but not when both are high. So, if we take input b as the control input,
when b is high, a is inverted, and we thus have the CNOT. We can write the output
as a wave function, in which the state of the control bit is the first signal and the state
of x is the second variable:
1 1
output → ∣ψ 〉 = (∣0〉∣1〉 ± ∣1〉∣0〉) = (∣01〉 ± ∣10〉). (1.34)
2 2
That is, if the control bit (b) is 0, y is 1 only if x (a) is 1, and if the control bit is 1, y is
1 only if x is 0. The wave function (1.34) is entangled. That is, if we take the control
bit as a two state Hilbert space, and the input bit as a two state Hilbert space, then
the entangled output lies in a tensor product Hilbert space, denoted as H2⊗H2. But
note that the wave is not a simple product, and it cannot be separate into parts that
lie in only one of the individual Hilbert spaces. This is what defines the entangle-
ment. Hence, the wave function in (1.34) is an entangled wave function. The plus/
minus sign gives two possibilities, which both give the same value of 1 for the square
magnitude of the wave function (the factor of ½ provides proper normalization).
The introduction of the control concept is really accomplished by the manner in
which the interaction between qubits is accomplished and managed. Hence, any
algorithm tells us how to manipulate the control signals, which change the
interactions so that entanglement is created, manipulated, or destroyed. This is no
different than the classical computer where bit strings are pushed around, manip-
ulated, and then stored or erased. Here, however, we presumably use entanglement
to increase the power of individual qubits.
1.7 Some notes on fabrication

In creating mesoscopic devices, one generally combines the growth of a specific
material or heterostructure with various processing steps such as those described
above in section 1.4.2. As can be observed there, the two dominant processes that are
universally used are lithography and etching to define the specific structure that is
desired. We cannot do a thorough job of covering these processes, as that would
require a book unto itself. Nevertheless, we give only introduction to the various
processes that can be used to shape mesoscopic structures.
1-34
1.7.1 Lithography
Lithography is primarily the same process as photography in which an image is
exposed and then developed. It does not matter whether one is carrying out optical
lithography, electron-beam lithography, or something more exotic, the approach is
much like the early days of photography. Anyone who has done darkroom work
understands that the developed image results from a combination of the exposure
(amount of light) and the developer. The wafer is our equivalent of the glass plate or
the final print of modern photography. The first step is to spread some ‘goop’ on the
surface of the substrate. This goop is a resist which is spun onto the surface. That is,
the wafer is attached to the rotating plate of a machine which will then spin it at a
high rate of revolution. The liquid containing the resist is dropped onto the spinning
surface and the spinning action will spread the liquid to a nearly uniform thickness.
Then, the wafer is baked to remove the liquid solvent leaving a hard resist coating.
The progress that has been made since Matthew Brady’s time is that the tent has
been replaced with a cleanroom. Then, the resist is exposed, either by photons or
electrons, or some other more exotic form of energy deposition. The developer then
removes unwanted parts of the ‘goop’, leaving the desired image.
In the industry, the exposures discussed in section 1.4.2 have, in recent days, been
produced using excimer lasers as the source, and hard masks in a projection tool
which gives something like a 10:1 reduction in the image size. These tools step across
the wafer to make a great many identical exposures on the typical 300 mm diameter
wafer. Only quite recently has the XUV exposure tool, which uses 13.7 nm x-rays,
been introduced with the new chip in the iPhone 11. Nevertheless, both of these
approaches using large stepping machines are far too expensive for use in research
labs where only 1 or a few devices are desired. Here, one typically uses electron-
beam lithography, or ion-beam lithography, which differ from optical lithography in
only a couple of ways. First, electron-beam lithography uses electrons rather than
photons to deposit the energy in the resist [79]. Similarly, ion-beam lithography
would use ions rather than electrons or photons [80]. The second difference is that
one generally exposes a single pixel at a time in electron-beam, or ion-beam,
lithography so that this is a serial writing method where the beam is raster scanned
across the image area. In optical lithography, the light beam is a large area beam
which usually exposes a chip at a time, and then is stepped across the wafer, as
mentioned. In optical lithography, a photomask is used, and this is generally a metal
coating on a glass plate, in which areas of the metal have been removed,
corresponding to those areas of the chip that are to be exposed to the photons. In
electron-beam lithography, or ion-beam lithography, the beam is turned on or off
for each pixel (technically, the beam is ‘blanked’), and the size of the pixel is set by
the size of the beam spot. Higher resolution requires a smaller spot size.
There are basically two kinds of resists, positive and negative. In a positive resist,
the electron beam breaks long polymer bonds, making this exposed material more
able to be dissolved in a developer. In a negative resist, the electron beam is used to
cross-link the material into a polymer that resists the dissolving action of the
developer. The most common electron beam resist in the mesoscopic world is
1-35
poly-methyl methracrylate (PMMA), and it is usually used as a positive resist that

yields very high resolution. Negative resists are used, for example, in isolating a local
region, such as a mesa, on the chip. Only a small part of the resist over the entire chip
has to be exposed, and, after development, protects the mesa as the surrounding
material is etched away. Positive resist is used where one wants to deposit a metal
wire, so only the position of the wire is exposed. After development, the metal can be
deposited and then dissolving the resist will ‘lift off’ the undesired metal.
Every photosensitive material, including our resists, is characterized by what is
known as a density–exposure (D–E) curve. This curve plots the density of the
remaining resist as a function of the exposure dose. As mentioned above for
photographic development, the D–E curve is both resist and developer sensitive.
That is, like many photographic films, the development can be ‘pushed’ to change
various properties. The exposure–developer combination has a range of
acceptable values, but one can often go outside this range for special effects. Two
other factors also are important. One is the sensitivity of the resist; e.g., how much
energy is required to initiate the bond breaking (or cross-linking). The second is the
molecular weight of the particular resist. In the case of PMMA, one normally uses a
molecular weight approaching a million, although good results can be achieved with
lighter material, and lines as narrow as 7 nm have been written over a range of
molecular weights [25].
The transition from unexposed to fully exposed resist is sensitive to both the resist
and the developer. If we talk about the exposure necessary to give 90% of the
thickness and the exposure necessary to give 10% of the thickness, we can define a
parameter characterizing the D–E curve as
1
γ= . (1.35)
ln(E10 / E 90)
Since we want to have as high a contrast as possible, we want this parameter to be as
large in magnitude as possible. For a positive resist, we have E90 < E10, so the above
γ is positive. For a negative resist, the inequality will be reversed and we have to flip
the ratio to give a positive value for this parameter. It is possible to obtain γ > 10
with special combinations of developers [81, 82]. High resolution is possible to
achieve in a variety of common and uncommon resists [83]. In our discussion above
of lifting off undesired metal, it is clear that high resolution is required. Otherwise,
one will not achieve a complete break from the metal in the exposed grove and the
metal on top of the unexposed resist. Lack of this break will prevent a clean liftoff of
the metal. Using multi-level resists, in which the lower levels of resist have less
resolution and the top layer provides the thin opening, will allow better liftoff of the
metal and better resolution [84].
Other forms of electron-beam lithography have been developed over the years
which do not require the use of a photoresist. Erasable electrostatic lithography uses
the electron beam to deposit charge on a non-conducting surface [85]. This charge
repels electrons and can be used to define a confining potential for electrons in the
GaAs 2DEG. An STM can also be used to oxidize the surface of metals [86] and
1-36
semiconductors [87] to provide a depletion of the electrons under the oxide, which
will also form a confining potential.
1.7.2 Etching
Etching involves the removal of material by a chemical process [79]. This can be
achieved either by a liquid etch, known as wet etching, or by gaseous etching, known
as dry etching, or often as reactive-ion etching. Liquid etches tend to be isotropic,
which means that they etch equally as fast in all directions. For our nanostructures,
the process of mesa isolation, mentioned above, is most effectively achieved by a wet
etch as a great deal of material needs to be removed. As wet etching is a chemical
process, the exact chemistry used will depend upon the material that is being etched.
The etch rates can depend upon the details of the material as well as properties such
as the dopant concentration.
If we are etching grooves or shallow trenches, or transferring a thin line opening
in the resist to the underlying material, anisotropic etching is highly preferable. With
proper control of the etching conditions, anisotropic etching is readily achieved with
dry etching. Here, there are a number of good etching gases that have relatively
similar etch rates for a wide variety of materials.
It is important to understand that etching can be used to define nanostructures.
We recall that the surface of many semiconductors is pinned by defects so that the
Fermi level at the surface is in the band gap, often near the center of the gap. For
example, the Fermi level at the GaAs surface is usually pinned about 0.8–0.9 eV
below the conduction band regardless of whether the surface is a free surface or a
metallized surface. This can work to deplete any electrons in the 2DEG, if the
surface is brought sufficiently close to the electron layer. This is the principle used in
mesa isolation described earlier; we do not need to etch completely into the GaAs
layer. We only need to etch away a sufficient amount of the GaAlAs layer so that the
surface depletion reaches the electron layer.
Figure 1.17. (a) View of patterned Hall bar with an embedded finer structure. (b) The fine structure showing three
individual nanostructures, each composed of a triple dot. Reprinted with permission from Chetan Prasad [88].
1-37
An example of wet etching is shown in figure 1.17, where an etched hall bar is
shown in the GaAs/GaAlAs system [88]. Metal interconnects are also shown, and
the mesa is the dark area in the center of the interconnects. Here, an optical positive
resist was spun to a depth of 1.5 μm and then soft-baked to drive out the solvent.
This was exposed using an optical exposure system and mask. After exposure, the
resist was hard-baked to remove any residual moisture that may have been in the
system, and then the material was etched in a 1:1:150 solution of H2O2:H2SO4:H2O,
which was calibrated to remove about 100 nm min−1. After the etch, the resist mask
was removed and the mesa measured to make sure it was the desired height. This
resist is anisotropic and gives a beveled edge to the mesa, which helps the metal
interconnects run over the mesa edge. The interconnects in the figure were created
with a second step of optical lithography and a liftoff process.
1.7.3 Bottom-up fabrication

In the past few decades, there has been a growing interest in processes which build
the interesting structure from the so-called ground upward. We may call this
explosion of interest as one for the study of self-assembled processes that lead to
nanostructures. This area actually has a relatively long history driven for many years
by the study of self-assembled semiconductor quantum dots that form at hetero-
structure interfaces during molecular beam epitaxy (MBE) [27]. The leading
example of this approach is provided by the self-assembly of InAs, or InGaAs,
quantum dots that form on a GaAs substrate via the Stransky–Krastinov growth
process [27]. In this growth mode, a thin layer, less than a monolayer, of InAs is
grown on top of a GaAs substrate. If the layer is thin enough, the strain developed
from a lattice mismatch will cause the InAs to agglomerate into small three
dimensional quantum dots. Growth of a subsequent layer of GaAs or AlGaAs
seals the dots into the interface between the two materials. These dots have been of
interest for optical applications, such as lasers and LEDs [89], but others have
thought about using the dots for tunneling devices [90]. A more recent use of the self-
assembled quantum dot is for the generation of single, and entangled, photons for
application in quantum computing [91, 92].
While this self-assembly of quantum dots during MBE growth is very interesting,
it is not true self-assembly of nanostructures, since the overall device is still rather
large. Perhaps more fundamental is the use of chemical processes to create nano-
structures, such as graphene nanoribbons [93]. In this fabrication, the topology of
the resulting nanoribbon is set by the initial chemical precursors. The precursor
monomer is illustrated in figure 1.18 which will be used for the fabrication of a
nanoribbon with 7 carbon atoms stretching between the arm-chair edges. The
process begins with chemical monomers, shown in the figure, which is 10,10′-
dibromo-9,9′-bianthryl monomers. Here, the 10 and 10′ denote the particular
carbon atoms (see figure 1.18) on the triple connected benzene rings and the 9
and 9′ denote the positions of the bond that connects the two connected rings. The
dibromo says that we have two bromine atoms, one at each of the first two points.
The connected benzene rings are the anthryl molecules, which are derived from
1-38
6 7
5 8
Br Br
10 9
4 1
3 2
Figure 1.18. Two anthracene molecules are connected and Br atoms added to make the precursor molecule for
generating a graphene nanoribbon.
anthracene. The first step is to deposit these molecules on a gold surface, primarily
by sublimation. This process removes the bromine atoms from the monomers
(known as dehalogenation) which are the building blocks for the final nanoribbon.
The structure is then annealed at 200 °C to lead them to diffuse toward one another
(known as intermolecular colligation) and form C–C bonds between each monomer
and to form polymer chains. Scanning tunneling microscope (STM) images show
that these polymers have protrusions that appear alternately on both sides of the
chain axis with a periodicity of 0.86 nm [93], and steric hindrance between the H
atoms of adjacent anthracene units rotates the H around the σ-bonds leading to
opposite tilts of successive anthracene units with respect to the gold surface. Further
heating to 400 °C leads to dehydrogenation of these H atoms and the final graphene
nanoribbon.
The generation of different numbers of transverse atoms in the final graphene
nanoribbon is accomplished with different numbers of benzene rings in the initial
monomer. However, the 7 atom width seems to be the easiest due to the presence of the
anthracene molecule. For other widths, the precursors are not so easily utilized [94].
Nevertheless, a variety of widths can be fabricated. These nanoribbons can also be
easily doped. One method inserts a third anthryl molecule with B atoms at the 9 and
10 sites, by replacement of the C atom at this point, between the two end anthryl
units [95]. This produces a p-type graphene nanoribbon. N dopants have also been
inserted [96].
The band gap of the resulting nanoribbon is, of course, dependent upon its width.
The lateral width governs the effective quantum well in which the electrons of the
nanoribbon sit. Hence, this tunes the quantum levels and the resultant energy bands
in the direction along the nanoribbon. The band gap for 7 and 13 width nanoribbons
(arm-chair edges along the length) have been reported as 2.5 eV and 1.4 eV,
1-39
respectively [97]. The band gap will also vary with the length of the nanoribbon if the
latter is relatively short, or if H endgroups appear on the nanoribbon [98].
This bottom up synthesis of nanoribbons can be used to generated heterojunc-
tions along the length of the nanoribbon [99]. When sections of a graphene
nanoribbon have different width, edge (arm-chair versus zig-zag), dopants or
termination, these sections can have different electronic topological classes [100].
This can arise because the width leads to different properties as an example. If the
nanoribbon has N atoms (in the above definition of width), then for N = 3p, where p
is an integer) and N = 3p + 1, the corresponding nanoribbon has a band gap
throughout the width. However, if N = 3p + 2, a zero-energy edge state exists along
the ribbon and the latter is metallic in behavior. While both 7 and 9 width
nanoribbons are both gapped, but they possess different topological states. The
7 nanoribbon is topologically trivial (the 2 invariant is 0) while the 9 nanoribbon is
topologically nontrivial with 2 = 1 [101, 102]. So, when we make a nanoribbon
heterostructure 7/9 junctions, there is a change in the 2 invariant across the junction
and this leads to a one dimensional array of interface states which, if aligned
periodically in the superlattice, enable a hierarchy of quantum engineered topo-
logical phase discontinuities, which are quite likely to be very useful in the world of
topological insulators. These interface states should be half-filled states near mid-
gap, and provide a new tool in engineering the electronic properties of the
nanoribbon.
Problems
1. Consider a normal MOS structure on Si at room temperature. Using the
classical approach described in the chapter to determine the effective width
of the inversion layer, compute the width for densities of 5 × 1011, 1 × 1012,
3 × 1012, 5 × 1012, 1 × 1013, all in units of cm−2. Assuming an acceptor
density of 5 × 1017, compute the effective electric field in the oxide. Plot both
the effective thickness and the effective electric field as a function of the
inversion density on a single plot, using a logarithmic axis for the density.
2. Go to nanoHUB.org and launch the ‘bound states calculation lab’. Using the
data for the Si MOS structure above, and using the heavy mass for the Si
inversion layer, enter the effective electric fields computed in the previous
problem. Using the triangular geometry to approximate the inversion layer,
determine the effective thickness of each of the first four subbands as a
function of the inversion density corresponding to the effective electric field
entered in the program.
3. Consider two separate GaAs/AlGaAs quantum wells, A and B, of thickness
L and 2L, respectively. (a) By approximating the AlGaAs barriers of the
wells as having infinite height, determine the ratio EoA/EoB, where are EoA,B
are the lowest quantized energy levels in the wells. (b) Now suppose that the
quantum wells contain electrons and that they both have the same Fermi
energy, EF = 3EoA (measured from the bottom of the well) How many
quantized subbands will be occupied in each of the wells?
1-40
4. Consider a uniformly doped GaAlAs layer, with a composition of x = 0.25,

grown on top of a GaAs layer. If the Fermi energy is pinned 0.85 eV below
the conduction band edge at the surface, what is the minimum thickness of
the GaAlAs layer for it to be completely depleted of electrons? Assume the
donor ionization energy is 50 meV and the donor density is 1018 cm−3.
5. Consider two exposure pixels of an electron-beam lithography system. Each
exposure pixel is characterized by a Gaussian beam with a full-width half-
maximum of 50 nm. If we assume that each Gaussian has a peak amplitude
of unity, compute the exposure (amplitude of the signal) at the mid-point
between the centroids of the Gaussians as they are moved from 20 nm center-
to-center to 200 nm center-to-center.
6. Consider a 2DEG at the interface of a GaAlAs/GaAs heterostructure at
1.2 K. Assume that the electron density is 4 × 1011 cm−2 and the mobility is
500 000 cm2(Vs)−1. (a) Determine the momentum relaxation time from the
mobility, and then determine the diffusion coefficient using the Einstein
relation (you may need to make some corrections for the Fermi–Dirac
distribution function). (b) Estimate the diffusion coefficient from the Fermi
energy and Fermi velocity. Is there a difference in the two values? Why?
7. Using the structure and data from problem 7, estimate the elastic mean free
path for the electrons. If we assume that the device has a transport length of
1 cm, we can use this for the inelastic mean free path. What is the value of the
phase-breaking time in this case?
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1-44
IOP Publishing

(Second Edition)
David K Ferry
Chapter 2
Wires and channels
Perhaps the simplest mesoscopic device that can be conceived actually came after
some years of work in mesoscopic studies. Hence, we would like to deviate from the
historical nature of the field to discuss the topic in a more developmental order. In
this chapter we will attempt to lay some foundations for understanding mesoscopic
transport based upon this developmental order. We will begin with transport in
quasi-one-dimensional (Q1D) structures. Long Q1D structures are nanowires, and
these can be either fabricated on a substrate by top-down processing or exist
naturally, such as with carbon nanotubes. But, short Q1D structures can also be
made, and these are often called quantum point contacts (QPCs), as they are usually
fabricated from metallic gates. For, example, a close examination of figure 1.17(b)
shows that each of the triple quantum dot structures is created using four QPCs,
whose Q1D length varies in the three sets shown. Such a composite structure allows
for studying interesting physics, not only of the QPCs, but of the quantum dots
themselves in chapters 8 and 9. So, we will begin this chapter with the QPC.
Following this, we turn to a discussion of the density of states in nanostructures, and
the forms that can arise with different dimensionality. This will be followed by a
discussion of ballistic behavior and scattering, as well as a consideration of the role
of temperature and magnetic field. Finally, we will introduce two methods of
calculating, or simulating, the transport behavior of various mesoscopic devices—
the scattering matrix and the recursive Green’s function methods.
2.1 The quantum point contact

One of the most important discoveries for the understanding of mesoscopic and
nanoelectronic devices has been the observation of one-dimensional conductance
quantization. What we mean by one-dimensional is that the transport is not just
along a single direction, but is actually characterized as being transport in a narrow

one-dimensional channel, not unlike a small pipe. The observed conductance is

quantized when the transverse dimensions of this ‘pipe’ are comparable to the Fermi
wavelength, and the conductance is found to increase in steps of 2e2/h as the size of
the pipe is slowly increased. Generally, this phenomena is observed in short
quantum wires at low temperatures when the electron transport is largely ballistic
in nature. We will discuss the reason for this limitation later, but one can simply
understand that scattering interferes with the general quantization process.
In figure 2.1(a), we illustrate a mesoscopic structure in which this quantization
can be seen, although here we are interested just in the physics of such a structure.
The heterostructure in the figure is a layer of In0.53Ga0.47As sandwiched between two
layers of In0.52Al0.48As, the latter of which has a much wider band gap, so that the
electrons in the central layer are confined in a vertical quantum well [1]. Then, two
vertical trenches (indicated by the various shades of blue) have been etched through
the layers to provide lateral confinement of the electrons as they pass through the
narrow region between the two trenches, leaving an opening of about 0.6 μm. The
two regions marked ‘A’ and ‘B’ are separately contacted and can be biased to
provide electrostatic confinement to reduce the size of the opening. In figure 2.1(b),
we show a scanning gate microscopy (SGM) image of the transport through the
structure. In this image, the effect of moving a biased scanning probe tip on the
conductance through the structure (from left to right) is shown. The red regions are
the conducting regions, while the blue regions are electrically isolated. By applying a
negative voltage to the two gates (the blue regions), we can squeeze the conducting
region to a smaller size, eventually cutting off the conduction through the
constriction.
The first measurements through a QPC were made by van Wees et al [2] and
Wharam et al [3]. In figure 2.2, we illustrate the quantized conductance through a
QPC defined by two electrostatic Schottky barrier gates [4], for a GaAs/AlGaAs
heterostructure. We can see the physical structure in the upper inset, with the two
split gates (black regions) and source and drain contacts indicated. The overall
behavior with gate bias is shown in the lower inset as the raw data, while the main
panel shows an expanded view of the discrete steps in conductance. To within the
accuracy that can be ascertained from the figure, the steps correspond to plateaus
Figure 2.1. (a) An atomic force microscopy image of a QPC defined by etched trenches. (b) A SGM image of
the transport regions (red) through the structure shown in part (a). (Reprinted from [1] with permission of AIP
Publishing LLC.)
2-2
Figure 2.2. Conductance G(VG) through a QPC, after correction for a series resistance of approximately
700 ohm. In the upper inset is a schematic of the sample with the source and drain contacts indicated and the
split-gate Schottky barriers in black. The lower inset shows the raw data. (Reprinted with permission from [4].
Copyright 1998 the American Physical Society.)
with a conductance that is an integer multiplier of 2e2/h. In this structure, the

negative gate voltage controls the electrostatic width W of the opening between
the metal gates. As the gate voltage is made more negative, the width is reduced and
the conductance is also reduced. The fact that the conductance goes down in steps is
a property of the Q1D nature of the channel that passes through the QPC and the
nearly ballistic nature of the transport.
The QPCs described above provide the prototypical system for the study of Q1D
conductance that can be realized by defining narrow constrictions in a normal quasi-
two-dimensional electron gas (Q2DEG). We recall that the Q2DEG exists at the
interface between GaAs and GaAlAs, or in the quantum well of the InGaAs/InAlAs
system. It has this Q2D behavior as the motion is restricted and quantized in the
direction normal to the heterostructure layers. Hence, the electrons can move only in
the plane of the heterostructure. The transverse gates provide an additional level of
quantization which further restricts the motion of the carriers, limiting them to be
nearly in a single direction. Usually these gates define a constriction whose size is less
than a micron, but hopefully of a size sufficiently small that no impurities exist in this
region, and the transport is ballistic. In the case of electrostatic confinement, such as
the split-gate approach, the potential profile has a saddle point structure.
At the center of the QPC, if we move toward the two gates, the potential
increases. However, if we move along the channel direction, the potential will
decrease. The highest potential in the channel (along the center line) provides a
barrier to transport through the QPC. In most cases, it is reasonable to assume that
2-3
the potential is parabolic in both directions, and this will make the Schrödinger
equation solvable exactly. While one might expect that there are reasons for non-
parabolic behavior of the potential, in fact the self-consistency that constrains the
potential tends to drive it toward a quadratic, parabolic behavior [5]. Thus, we can
write the potential approximately as
m⁎ω 0,2xx 2 m⁎ω 0,2yy 2
V (x , y ) = V0 − + . (2.1)
2 2
where the y-direction is the width of the opening and the x-direction is along the
length of the channel indicated in the upper inset to figure 2.2. The quantities ω0,x
and ω0,y are oscillator frequencies, which are parameters that characterize the
strength of the two parabolic potential variations, while V0 is the height of the saddle
potential center point and m* is the effective mass of the channel material. The form
of the potential equation (2.1) is separable and allows the Schrödinger equation to be
separated into its x- and y-components which can be solved independently.
It is also important to note that in the case of etched trench-isolated gates, as
shown in figure 2.1, the confining potential differs from the parabolic form. Instead,
the potential is controlled from the side of the channel as opposed to arising from a
gate on top of the heterostructure. This lateral potential leads to a confinement
closer to a finite quantum well. As the height of the confining potential is finite, the
separation of the energy levels will differ from those of an infinite quantum well, and
will not have the linear variation that will arise from the parabolic potential of
equation (2.1).
The solution to the Schrödinger equation for the x-motion is quite simple. For
energies above V0, the motion is that of a free electron along the channel, while for
energies below V0, the motion is prohibited, although tunneling can occur near the
saddle center. As we show in appendix B, solving the one-dimensional Schrödinger
equation for the y-direction (the width), yields a series of equally spaced harmonic
oscillator levels which correspond to the one-dimensional subbands that arise in the
channel, with the corresponding energies
⎛ 1⎞
En(kx ) = V0 + ⎜n + ⎟ℏω0,y . (2.2)
⎝ 2⎠
In a long channel, which has the characteristics of a quantum wire, these subband
energies will be more or less uniform along the wire length. In a short QPC, these
values exist only at the center of the QPC, and become closer together as one moves
away from the center in either direction along the length. Eventually, they merge
into the continuum of the Q2DEG when we are well away from the QPC. The gate
voltage varies both the height of the saddle point potential V0, as well as the
harmonic parameter ω0,y . From equation (2.2), it is clear that the increase of the
harmonic parameter ω0,y by the gate potential pushes the energy levels upward, and
increases the spacing between these levels. The electrons can then move through
those channels whose subband energies are less than the Fermi energy in the
Q2DEG away from the QPC.
2-4
Energy (arb. units)
EF
Momentum, kx (arb. units)
Figure 2.3. Dispersion relation for a quantum wire with a parabolic confining potential, according to equation
(2.3).
As remarked, the motion along the length of the QPC is that of a free electron in
one dimension. This is the normal quadratic energy dispersion, and modifies
equation (2.2) to give
⎛ 1⎞ ℏ2k x2
En(kx ) = V0 + ⎜n + ⎟ℏω0,y + , n = 0, 1, 2, …. (2.3)
⎝ 2⎠ 2m⁎
Since the energy in the y-direction is quantized, each subband may be considered to
be a separate one-dimensional channel with essentially free momentum along the
direction of the current flow. In figure 2.3, we illustrate the behavior of these
subbands for momentum along the length of the channel. Here, we show four
subbands below the bulk Fermi level, and two of the many subbands which have
quantized energies above the Fermi level. Thus, there will be four channels flowing
through the QPC. As may be inferred from equation (2.3) and figure 2.3, the number
of occupied subbands is given by determining the largest value of n in equation (2.3)
for which En(kx = 0) is less than the Fermi energy in the region far from the QPC.
For a wire in such a parabolic potential, we can develop an expression for the
effective width of the channel. Using equation (2.1) at x = 0, and equation (2.2), we
can say that
m⁎ω 0,2y ⎛W ⎞2
EF = V0 + ⎜ ⎟ . (2.4)
2 ⎝2⎠
2-5
Using the value of the Fermi wave vector, in the channel center, defined from
ℏ2k F′ 2
EF − V0 = , (2.5)
2m⁎
we find that the width is related to this value as
2ℏk F′ 2
W= . (2.6)
m⁎ω0,y
We may use this relation to obtain an expression for the number of occupied
channels in the wire, through (note that N here is the number of occupied channels,
not the energy level index)
⎛ 1⎞
E F′ = EF − V0 ⩾ ⎜N − ⎟ℏω0,y , (2.7)
⎝ 2⎠
and
⎡1 E F′ ⎤ E F′ E ′W πW
N = int⎢ + ⎥∼ = F = . (2.8)
⎣ 2 ℏω 0,y ⎦ ℏω 0,y 4 2λ F′
Thus, it is clear that the number of modes in the parabolic potential that can
propagate through the QPC is related to the number of half-wavelengths of the
reduced Fermi wavelength that can be fit into the width of the potential at the Fermi
energy.
In the case of the square well potential that arises from the trench-isolated gates of
figure 2.1, the relationship is somewhat simpler, and we have
kF W 2W
N= = . (2.9)
π λF
Thus, a new mode is populated each time the wire width is increased by a half a
Fermi wavelength. Note that here the unprimed quantities are used as there is no
saddle potential minimum in this case, and the Fermi energy has the same reference
point as in the bulk away from the QPC.
Direct experiments for the existence of the modes going through the QPC have
been given through experiments using SGM [6, 7], in which a biased scanning
microscope tip is scanned over the mesoscopic device, the QPC and its surrounding
region in this case, and the conduction modulation caused by the tip measured. The
technique works as the negative bias on the tip causes a local reduction in the
density, and this is reflected as a change in the conductance through the device. The
effect is larger where the local density, or the magnitude squared of the wave
function, is larger so that the scanned measurement image gives an indication of the
spatial extent of the wave function. We can see the concept of the SGM in the video
of figure 2.4. In this video, the propagation through the QPC is shown in the right-
hand panel. The conductance is plotted in the left-hand panel. When the probe
(indicated by the white shape in the right-hand panel) is at the left, it sits over the
2-6
Figure 2.4. The video illustrates the method of operation of the SGM (available at https://iopscience.iop.org/
book/978-0-7503-3139-5. (Video by Richard Akis, included with his permission).
potential barrier (red in the right-hand panel) and the conductance measured
through the QPC is unaffected by it. However, as the probe moves through the
actual channel, the large bias of the tip serves to cut off the transmission through the
QPC and the conductance drops to almost zero. Then, as the probe moves onto the
potential at the right, the conductance rises back to its normal value. The fact that
the conductance is nearly cut off in this video tells us that the probe tip is fairly large.
However, it is possible to adjust the distance of the tip from the surface, and the bias
applied, such that a spatial resolution of 5 nm, or better, can be achieved with this
technique.
Hence, the microscope tip (the movable gate) serves as a probe of the local density
and can be used to map this density. Typically, the highest mode through the QPC is
measured in this technique. In figure 2.5, we show the results of an experimental
study of a QPC, in which the highest mode propagating through the QPC is
alternatively set to be the first mode, the second mode and/or the third mode [8]. In
the second mode, the peak magnitude of the corresponding eigenfunction has two
peaks with non-zero y values, and the transverse momentum constraint is relaxed as
the mode leaves the QPC. Hence, we see two diverging beams of density. In the third
mode, one of the peaks is at zero transverse momentum, so we see three diverging
beams, one of which remains along the axis of the length of the QPC. Also shown in
the figure are simulations of the mode propagation away from the QPC. Although
there is a difference between the experimental observations and the theoretical
predictions, the images are quite close. Presumably, these differences are due to
impurities in the heterostructure which are not present in the simulations. We note in
the figure that the scanning gate does not enter the QPC proper, presumably due to
the use of Schottky split gates on top of the heterostructure. In this situation, the
biased gate must be kept away from the Schottky gates to avoid device failure, and
2-7
Figure 2.5. Experimental images (outer sections of each panel) and theoretical simulation (inner sections) of
the wave functions of electrons passing through a QPC in the first, second, and third modes. (Reprinted [8],
with permission from Elsevier, Copyright 2004).
Figure 2.6. The amplitude squared of the third mode wave function within a QPC. The position scales are in
nanometers, while the vertical amplitude is arbitrary. The simulation was performed by the author with the
scattering matrix wave function approach discussed later in this chapter.
2-8
this is why the trench isolated gates, shown in figure 2.1, are used. In figure 2.6, we
plot the simulation of the third mode in a square-shaped QPC. The wave function is
most well defined within the QPC itself. This simulation was performed using
techniques we will discuss near the end of this chapter.
The existence of the modes gives an indication of just why the conductance
through the QPC shows the steps that are observed in figure 2.2, but it does not
explain why the steps appear as they do, or even why steps are observed at all. What
we know so far is that the steps are observed experimentally, but not why they are
observed. To understand this, we have to dig deeper into the basics of condensed
matter physics and explore both the density of states and the Landauer picture of
transport. Let us now turn to these topics.
2.2 The density of states

The density of states is one of the most important quantities for determining the
electronic properties of condensed matter systems. Basically, we are asking for the
number of states that are available at a given energy in the system. Thus, the density
of states is the number of states per unit energy per unit volume, and is a function of
the energy itself. In three dimensions, the density of states for a semiconductor, or a
free-electron metal, is easily calculated from the quantized solutions of the
Schrödinger equation, and leads to a variation as the square root of the energy.
But, this depends upon the dimensionality of the system, so that the variations will
vary as we move from bulk to two-dimensional to one-dimensional systems.
Moreover, it depends upon the nature of the energy bands, with the traditional
bulk square-root result only arising for parabolic energy bands.
To understand the properties of the density of states in nanostructures, we begin
by determining the properties of the wave functions themselves and how this sets an
important quantization on the momentum states that can exist in the material. The
problem is to solve the Schrödinger equation in the desired material. The most
important initial step, and one that really defines the exact density of states that we
find, is the assumption of periodic boundary conditions on the wave function. In
fact, the electrons are constrained to stay within the volume of the crystal, so that
there are potential barriers that surround the crystal (these are the work function
potential required to be overcome to remove an electron from the crystal). The
reader would be correct in noting that other boundary conditions can be used, but
the best approach is to use the periodic boundary conditions with a length L in each
direction, for reasons that will become clearer below:
ψ (x , y , z ) = ψ (x + L , y , z )
ψ (x , y , z ) = ψ (x , y + L , z ) (2.10)
ψ (x , y , z ) = ψ (x , y , z + L ).
Now, the use of periodic boundary conditions leads to the correct counting of states;
that is, there will be two states (of opposite spin) for each bonding electron in the
basic crystal lattice. For example, in Si or GaAs, the crystal lattice is the fcc lattice
which has four lattice sites per unit cell. But, there is a basis of two atoms at each
2-9
lattice site: either two Si atoms or one Ga and one As atom, so that there are eight
atoms per cell. The periodicity required for the atomic lattice thus gives 4/a3, where a
is the edge of the fcc cell, states, each of which can hold two electrons of opposite
spin. Hence, there are 8/a3 total states. Each atom of the basis has four bonding
electrons, so this gives a total of 8 electrons per lattice site, which just exactly fills all
the states in the band—the valence band, or bonding band in this case. So, these
tetrahedrally coordinated semiconductors are insulators at very low temperatures,
which is exactly what is observed. Using different boundary conditions would not
give this experimentally confirmed result.
Because of the atomic periodicity of the crystalline potential, we find that the
momentum wave numbers are not continuous, but are discretized due to the
boundary conditions. They form a series of discrete points in the three-dimensional
momentum space. Each point in this space corresponds to a particular momentum
state which may be occupied by two electrons of opposite spin, according to the
Pauli exclusion principle. The traditional approach is to use a cubic lattice of edge a,
appropriate to most metals, so that the length L = Na, where N is the number of
atoms in the given direction of the crystal. Then, we find that the quantization of the
momentum, leads to
2πnx N
kx = , nx = 0, ±1, ±2, …
L 2
2πn y N
ky = , n y = 0, ±1, ±2, … (2.11)
L 2
2πnz N
kz = , nz = 0, ±1, ±2, … .
L 2
Of course, nx, ny, nz can be larger than the limits indicated above, but these larger
values are not independent. If we examine our electron waves as having the property
ψ ∼ e ik·r , (2.12)
then we see that the phase rolls over (past 2π) when the larger numbers are used. So
the end result is that the independent values of momentum are precisely equal to the
number of atoms in the lattice (not including the basis).
2.2.1 Three dimensions

In three dimensions, the filling of the lowest energy states at low temperature gives
rise to the formation of the Fermi sphere, whose properties may be easily computed.
In an N electron metal, N/2 distinct wave vectors are required to hold these
electrons. The volume in momentum space for each wave vector is given by our
conditions of equation (2.11) to be (2π/L)3.
Hence, our N/2 states give rise to the requirement that
N Volume of sphere 4π 3 L3
= = kF 3 . (2.13)
2 Volume per state 3 8π
2-10
We can now solve for the Fermi wave number as

⎛ N ⎞1/3
kF = ⎜3π 2 3 ⎟ = (3π 2n )1/3 , (2.14)
⎝ L ⎠
where n is the number of electrons per unit volume. In a simple metal, these are all
free electrons, and they will fill the band up to the half-way point. That is, one-half of
the available states are filled for this simple metal.
At zero temperature, all the states are filled up to the Fermi energy. For our free-
electron metal, this value can be found to be
ℏ2k F2 ℏ2
EF = = (3π 2n )2/3 . (2.15)
2m 2m
This can be inverted to find the density as a function of the Fermi energy, as
1 ⎛ 2mEF ⎞3/2
n= ⎜ ⎟ . (2.16)
3π 2 ⎝ ℏ2 ⎠
This last expression tells us exactly how many electrons fill all the states up to the
Fermi energy. There is no requirement that this be the Fermi energy. If we replace
the Fermi energy by the energy itself, then equation (2.16) tells us how many
electrons can be held in the states up to that energy. Thus, we can find the density of
states by a simple derivative, as
dn 1 ⎛ 2m ⎞3/2 1/2
ρ3 (E ) = = ⎜ ⎟ E . (2.17)
dE 2π 2 ⎝ ℏ2 ⎠
Now, we see that the density of states is proportional to the square root of the energy
in three dimensions.
When we move to semiconductors, or materials other than the simple free-
electron metals, there are few changes unless the parabolic energy dependence is
modified. This parabolic dependence was invoked in the move to equation (2.15), so
is crucial to the development we have used. An additional modification is the
replacement of the free-electron mass by the effective mass appropriate to the band
and semiconductor in use. The effective mass is introduced to equate the wave
momentum to the quasi-particle momentum so as to have consistency (at some level)
between the quantum and semiclassical approaches [9].
2.2.2 Two dimensions

Exactly the same approach can be used in two dimensions. In this case, The Fermi
energy describes a circle in the two-dimensional momentum space, and the number
of states needed to accommodate N/2 distinct wave vectors is given as
N Area of the circle L2
= = πk F2 2 . (2.18)
2 Area per state 4π
2-11
We can now solve for the Fermi wave vector as

⎛ N ⎞1/2
kF = ⎜2π 2 ⎟ = (2πns )1/2 , (2.19)
⎝ L ⎠
where ns is the number of electrons per unit area, or sheet density. At zero
temperature, all the states are filled up to the Fermi energy. For our free-electron
metal, this value can be found to be
ℏ2k F2 ℏ2
EF = = (2πns ). (2.20)
2m 2m
This can be inverted to find the density as a function of the Fermi energy, as
mEF
ns = . (2.21)
π ℏ2
This last expression tells us exactly how many electrons fill all the states up to the
Fermi energy. There is no requirement that this be the Fermi energy. If we replace
the Fermi energy by the energy itself, then equation (2.21) tells us how many
electrons can be held in the states up to that energy. Thus, we can find the density of
states by a simple derivative, as
dn m
ρ2 (E ) = = . (2.22)
dE π ℏ2
Here, the density of states is independent of energy.
Now, the above discussion is for normal semiconductors or metals which have a
band gap. In recent years new materials have been discovered that are characterized
by linear bands [10]. One of these is graphene, which is discussed in some detail
in chapter 4. Another case is topological edge states which possess linear bands as
well [11]. The graphene lattice and Brillouin zone is shown in figure 4.1 and the band
structure is shown in figure 4.2. The important point here is that the conduction and
valence band touch at two points in the primitive Brillouin zone unit cell. These two
points are denoted as K and K′. Near these two points, the energy bands are linear
bands given approximately by
E = ±ℏvF k, (2.23)
where vF is the slope of the linear band and is denoted the Fermi velocity. The upper
sign in equation (2.23) is for electrons and the lower sign is for holes. Here,
3γ0a
vF = , (2.24)
2ℏ
where γ0 is the nearest neighbor coupling energy in a tight-binding computation of
the energy bands [12]. In addition to being linear, the bands are chiral, in that the
wave function for the positive slope band has positive helicity and the negative slope
band has negative helicity. The helicity arises from the pseudo-spin imparted to the
wave function by the atomic contributions to the wave functions; e.g., the unit cell of
the lattice has two atoms in it, which are usually denoted the A and B atoms.
2-12
Usually, one recognizes that the linear bands are Dirac-like, and the Fermi velocity
corresponds to an effective ‘speed of light.’
An important peculiarity of graphene is this Dirac band structure, which in the
Dirac picture corresponds to particles with zero rest mass. That is, the effective mass
at the Dirac point (where the two bands cross and E = 0) is exactly zero. Away from
the Dirac point, however, the particles obtain a dynamic mass which may easily
be found by equating the equivalence of crystal momentum to quasi-particle
momentum [9], as
ℏk ℏ
m⁎ = = πns , (2.25)
vF vF
a result that has been confirmed by cyclotron resonance studies [13, 14]. In a manner
similar to above, we find that equation (2.19) is still valid as it is a normal property in
momentum space. Then, we can use equation (2.23), using only the positive sign, to
get
EF = ℏvF (2πns )1/2 , (2.26)
which leads to
1 ⎛ EF ⎞
2
ns = ⎜ ⎟ , (2.27)
2π ⎝ ℏvF ⎠
and
E
ρ2 (E ) = . (2.28)
π (ℏvF )2
This is a quite different density of states (per unit area) than that for normal 2D or
3D materials, and leads to some interesting behavior, that will be discussed in
chapter 4.
2.2.3 One dimension

Similarly, if we consider an artificial system of free electrons with allowed motion in
one dimension and with only a one-dimensional wave function, such as a nanowire,
the same arguments above lead to the result
⎛ m ⎞1/2 1
ρ1(E ) = ⎜ ⎟ , (2.29)
⎝ 2π ℏ2 ⎠ E
and the density of states varies as the inverse square root of the energy.
2.2.4 Multiple subbands

While we have discussed the systems of interest as two-dimensional and one-
dimensional, the GaAs/GaAlAs heterostructure system, and others of interest, are
more typically fully three-dimensional in character. However, the confinement near
2-13
the interface produces the Q2D character. What we notice in figure 2.2 is the fact
that there are many steps in the QPC conductance, which correspond to many
modes propagating through the structure. In our solution of the harmonic oscillator
in appendix B, we also find many subbands, which correctly reflect the experimental
observations. Now, the question is how do these subbands modify the density of
states found above.
The important view is that each subband corresponds to a channel of transport
through the QPC. The electrons are free to move in any direction parallel to the
heterostructure interface, so this motion appears to be two-dimensional systems.
Each of the subbands produces a similar Q2D layer of electrons. For low energies
(relative to the eigen-energy of the lowest subband), only the first subband is
occupied, and the density of states is the constant value given in equation (2.22).
However, when the Fermi energy is increased such that it exceeds the eigen-energy of
the second subband, then the electrons will be associated with both of the two
subbands, with the result that the net density of states is double at this energy. And,
this multiplicative effect continues as the Fermi energy is raised further. The net
density of states is thus a series (in two dimensions) of steps, the height of which is
equal and given by equation (2.22). A similar behavior occurs in one dimension,
where the electron is free to move along the wire. In figure 2.7, we illustrate this
additivity of the density of states in two and one dimension. In panel (a) we show the
additivity of the density of states for three subbands, each of which arises for a
different eigen-energy. The lowest contributor (blue) arises from the ground state,
and provides the first step in the total density of states. The next contribution comes
from the second subband (red), and provides a second step in the total density of
states. Similarly, the third contribution comes from the third subband (green) and
provides a third step in the total density of states. An important point is that for a
given Fermi energy, the kinetic energy of the carriers in the different subbands will
be different. In the lowest subband, the kinetic energy is measured from the
corresponding lowest eigen-energy. In the second subband, the kinetic energy is
measured from the corresponding second eigen-energy. And, this behavior repeats
Figure 2.7. (a) Additive density of states for three subbands (blue, red, and green) in two dimensions, in units
of m*/πℏ2 . (b) Subband density of states for three subbands (blue, red, green) in one dimension in arbitrary
units. The characteristic shape of the fall of each density of states is due to the square root singularity in
equation (2.29).
2-14
as we march up through the ladder of subbands. Of course, this can greatly

complicate the interpretation of transport measurements when many subbands are
occupied. For this reason, most experiments are designed so that only a single
subband is occupied by electrons, and this of course is the lowest subband.
In panel (b) of this latter figure, we plot the equivalent ladder of contributions to
the density of states for a Q1D structure. Here, we do not add them together, but
display them independently to show how they are replicas of one another. Thus, the
shape of the second subband is identical to that for the first subband, but is shifted to
higher energy according to the two eigen-energies. The total density of states will
thus show a saw tooth shape on a rising background as these individual contribu-
tions are added together, as one can see by adding the three curves of figure 2.7(b)
together.
Because these quasi-low-dimensional systems have these multiple subbands, we
need to account for this behavior in the corresponding total density of states. Hence,
for the Q2D system, equation (2.22) becomes
m
ρ2 (E ) = ∑ Θ(E − En). (2.30)
π ℏ2 En<E
Similarly, for a Q1D system equation (2.30) becomes
⎛ m ⎞1/2 1
⎝ 2π ℏ2 ⎠ ∑ En<E E − En
ρ1(E ) = ⎜ ⎟ Θ(E − En). (2.31)
In these last two equations, Θ(x) is the Heaviside step function, equal to unity when
x > 0, and equal to zero otherwise.
In some nanowires, complications in the density of states can arise due to
confinement in two directions. If the two transverse directions have the same
dimension, then the quantization has degenerate eigenvalues. For example, if we
consider hard wall boundary conditions for a square nanowire, then the lowest
eigenstate has nx = ny = 1, and this state is unique (singly degenerate). The second
eigenstate, however, is doubly degenerate, arising for nx = 2 and ny = 1, or for nx = 1
and ny = 2. This subband density of states will be twice as large as that for the first
subband. The third subband is again singly degenerate, but much higher degener-
acies can arise for higher subbands. Thus, it is not unusual to have the chain of
subband contributions to equation (2.30) be non-uniform among the subbands. We
will see these multi-subband effects in a number of experiments later. Now, however,
it is time to bring all these quantized states together to understand how the steps
occur in figure 2.2.
2.3 The Landauer formula

Many years ago, Rolf Landauer presented an approach to transport, and the
calculation of conductance, that was dramatically different from the microscopic
kinetic theory based on the Boltzmann equation that had been utilized previously
(and is still heavily utilized in macroscopic conductors) [15, 16]. He suggested that
one could compute the conductance of low-dimensional systems simply by
2-15
computing the transmission of a mode from an input reservoir to a similar mode in

an output reservoir. The transmission of this probability from one mode to the other
was then very similar to the computation of a tunneling probability, except that
there was no requirement that the process be one of tunneling. The only real
constraint was that of lateral confinement so that the two reservoirs could be
discussed in terms of their transverse modes. The key property of these two
reservoirs was that they were in equilibrium with any applied potentials. That is,
the electrons in the reservoirs were to be described by their intrinsic Fermi–Dirac
distributions with any applied potentials appearing only as a shift of the relative
energies (which would shift one Fermi level relative to the other). While he originally
considered that the transport was ballistic, this is not required. Rather, the require-
ment is that we can assign a definitive mode to the electron when it is in either of the
two reservoirs, which means that if scattering is present, it must be described
specifically as a transfer of the electron from one internal mode to another.
In figure 2.8, we give a schematic view of the mesoscopic device, in which the
central constriction is connected to a pair of reservoirs which are maintained in
equilibrium. The bias is applied to the right reservoir so that the left Fermi energy
can be used as the reference level for the applied potential (and indeed for the
energies throughout the structure). The right contact now emits carriers into the
constriction with energies up to the local Fermi level plus the applied bias, EF + eV
(note that the energy eV will be negative for a positive voltage). The left contact
emits electrons into the constriction with energies only up to EF. In the following
discussion, we will assume that the applied voltage is quite small, although this also
is not a stringent requirement.
By making the initial assumption that no scattering takes place within the
constriction, we can make a definitive association between the energy of the carriers
and their direction of propagation. That is, electrons injected into the constriction
from the left reservoir have a positive momentum and travel from left to right in the
figure. On the other hand, electrons injected into the constriction from the right
reservoir have a negative momentum and thus travel from the right to the left. We
sketch this bit in figure 2.9, where we again remind ourselves that positive voltage
applied to the right reservoir lowers the energy of that reservoir. In figure 2.9, we
schematically plot the dispersion relation for these electrons. The momentum in the
figure essentially corresponds to the motion of the electrons through the constriction.
Figure 2.8. One can treat the region between the two reservoirs, which may be a quantum wire or a QPC as a
ballistic constriction, although the ballistic requirement is not necessary. The bias is assumed to be applied to
the right reservoir, so that the left reservoir provides the reference level for the energy.
2-16
Figure 2.9. Electrons populate the allowed bands differently for an applied bias (here a negative voltage
applied to the cathode). It can be seen that more electrons are moving to the left than are moving to the right,
giving a current flow in the device.
Those with positive momentum come from the left reservoir and travel to the right,
while those with negative momentum come from the right reservoir and travel to the
left. Because of the applied bias, there are more electrons traveling to the left than
are traveling to the right and this gives us a net current through the constriction. As
discussed previously, the dispersion in momentum actually consists of discrete points
according to equation (2.11). The imbalance between left-going and right-going
electrons means that the constriction is in a condition of non-equilibrium, which is
required to support a net current.
In order to compute the current through the constriction, we need to first evaluate
the charge that is occupied in each channel in the energy range between EF and EF +
eV. As we remarked above, if the applied bias is small, then we can estimate the
excess charge as
e 2V m⁎
δQ ≈ ρ1d (EF ) = e 2V . (2.32)
2 2π 2ℏ2EF
It should be noted that we used only one-half of the density of states given by
equation (2.29) since we are only interested in those electrons that are traveling to
the left (those with positive momentum). Also, because of our definition of the
density of states, equation (2.32) is the charge per unit length. In order to compute
the current, we need only multiply equation (2.32) by the group velocity of the
carriers at the Fermi energy, which is
2-17
1 dE ℏkF 2EF
vg = vF = = = , (2.33)
ℏ sk E =EF m⁎ m⁎
and this leads to the current

m⁎ 2EF 2e 2
Ic = δQvg = e 2V = V. (2.34)
2π 2ℏ2EF m⁎ h
A very interesting effect occurred in this development of the current, and that is the
energy has dropped out of the equation. Hence, the current is completely independ-
ent of the energy, and this is true for each Q1D channel that flows through the
constriction. The result (2.34) thus has no dependence on the channel index; the
current in each channel is identical. This result has been called the equipartition of
the current, which is a unique event for one-dimensional channels. Hence the current
flow is divided equally among the available channels. Since the current in each
channel is equal, the total current carried in the constriction is simply the number of
such channels that are occupied and the current per channel. Hence,
2e 2
Itotal = NIc = N V. (2.35)
h
From this last expression, we obtain the conductance through the constriction as
Itotal 2e 2
G= =N , (2.36)
V h
and this last expression is usually called the Landauer formula. Thus, the
conductance of the quantum wire (our constriction) is quantized in units of 2e2/h
(∼77.28 μS) with a resulting magnitude that depends only upon the number of
occupied channels. Finally, the results of figure 2.2 become clear. The steps in the
conductance occur as individual channels are occupied or emptied (depending upon
a rising or decreasing potential), and the steps occur due to this important behavior
in one-dimensional channels and the cancellation between the energy terms in the
density of states and in the group velocity. The factor of 2 in equation (2.36) arises
from the density of states and is the form for zero magnetic field where we can
assume that the electrons are spin degenerate. Thus, in the experiment with split
gates on the surface of the heterostructure, as the gate voltage is made more positive,
the width of the QPC is increased (and the saddle potential is decreased), which leads
to an increase in the number of channels conducting through the constriction. As
each new channel is occupied, the conductance jumps upward by 2e2/h, and this
increase is the same for each and every channel. To clearly see the steps in the
conductance requires high mobility materials so that the transport is almost ballistic.
As we will see below, the presence of significant scattering will wash out the steps by
introducing transitions in which electrons jump from one channel to another and
this will upset the balance in the channels.
2-18
One method in which the contribution of the separate spin subbands can be
observed is to apply a large magnetic field to a material with significant spin
splitting. The magnetic field lifts the spin degeneracy due to the Zeeman shift in the
electron energy, thus splitting the spin-up and the spin-down electrons. Now, each
subband splits into two distinct subbands, each of which has an opposite spin and
now contributes only e2/h to the total conductance. The same behavior will be
observed as the gate voltage is varied, although there will be twice as many steps,
each of which will be only one-half the previous height [17]. We will return to the
role of the magnetic field in a later section where we see that the spin does not always
follow this simple idea, particularly in the single channel case.
In the above, we did not consider the probability that an electron in one mode in
one reservoir might actually wind up in a different mode in the other reservoir. This
is not reflected in the formula (2.35). Nor, does this latter equation offer the
possibility of partial transmission of a mode, which is clear in the rounding of the
steps in figure 2.2, and will become even clearer in the discussion of the next section.
The transitions from one mode to another will become even more important when
we allow scattering within the channel region. Consequently, it is more intuitive, if
we adapt equation (2.35) to the actual experimental situation in which partial
transmission of a mode is possible. To do this, we rewrite equation (2.35) as
2e 2
Itotal = VT (n ), (2.37)
h
where T(n) is the transmission of the n channels in one reservoir, which we define as
n
T (n ) = ∑ j=1Tmn, (2.38)
with the caveat that m ⩽ n. This caveat is based upon the assumption that we are
dealing with electrons which enter the structure through the left reservoir and exit
through the right reservoir. Thus, the entering electrons occupy modes up to the nth,
which may be partially occupied. They will exit through the same number of modes
in the right reservoir unless there is scattering, in which there may be a different
number occupied in the case of partial occupation of modes. Of course, then the
number of entering modes is an integer, such as when the conductance is on one of
the steps, then we may say that T(n) = N, and we recover equation (2.36). In
equation (2.38), the term Tnm is the probability that an electron entering through
mode n exits through mode m. We will use this notation again below, and several
more times through this book.
2.3.1 Temperature dependence

So far, we have not talked about the transition from the reservoirs to the channel
region. In figure 2.8, this transition is shown as a relatively smooth one with no sharp
edges, and the steps in figure 2.2 are relatively distinct and well formed. But, this is
not always the case. The important point is that the transition should be smooth
enough that no reflections are generated at the transitions. While waves can be
2-19
Figure 2.10. Simulations of a QPC at T = 0 and with pure ballistic transport. Curves ‘a’, ‘b’, ‘c’, ‘d’ are for zero
potential in the channel and 2d/W = 0, 1, 5, 10. Curve e is expanded for a saddle potential of V0 = 2.5Δ and 2d/
W = 10. The first plateau of curve ‘d’ is enlarged at the upper left. The horizontal scale is expanded for this
latter curve with the other curve offset to the right by 1.1 units. (Reprinted with permission [18]. Copyright
1989 the American Physical Society.)
turned around, this must be a smooth process so as not to mix modes at these
transitions. Such a transition is often called an adiabatic transition, and when this is
achieved, nice curves are obtained as in figure 2.2. While the transition is shown as
sharp in the upper inset to this latter figure, split gate Schottky barriers usually do
have relatively smooth transitions due to the slow variations of the electrostatic
barriers.
In figure 2.10, we show a simulation of a QPC with sharp corners and pure
ballistic transport at T = 0 [18]. The model is shown in the lower right inset, and
includes an infinite potential for ∣y∣ > W/2, where W is the width of the opening and
has a channel of length 2d. In the channel, the potential is taken as V0 so as to
simulate the saddle potential. The major reason for the steps and the rounding on the
transitions lies with the properties of the Fermi–Dirac distribution function. At zero
temperatures, the electrons fill the states exactly up to the Fermi energy. As the
temperature is raised, however, electrons begin to be thermally excited above the
Fermi energy, leaving empty states just below the Fermi energy. This, of course,
leads to the well-known form of the Fermi–Dirac distribution
1
fFD (E ) = . (2.39)
1 + exp ( E − EF
kBT )
2-20
If one computes the energy derivative of the Fermi–Dirac distribution, then one
finds a negative peaked function, similar in form to a Lorentzian curve (but, of
course, different from this). The full-width of this peak at half-height is 3.5kBT. So,
as the temperature rises, the transition from a value of unity to zero in equation
(2.39) extends over a greater and greater energy range. For reference, at 4.2 K, the
full-width at half-maximum is just over 1.25 meV. The energy levels that give rise to
the steps in the conductance will have to be spaced considerably wider than this if the
steps are to be observed in an experiment, as obviously they are in figure 2.2.
In the figure, the Fermi energy and the saddle potential are normalized to the
quantity Δ = ℏ2 /8m*W 2 . The Fermi energy is parameterized in the figure as the
quantity ξ = (EF − V0)/Δ. The most striking feature in this figure is the oscillations
that appear at the start of each step. In the most dramatic case, they range in
amplitude from the bottom to the top of the step; e.g., a maximum of 2e2/h. But,
they become damped out as one progresses past the threshold energy for the step.
These oscillations are most distinct on the first step, where there is only a single
longitudinal mode propagating through the channel. Then, they become more
complex as more modes propagate and interfere within the channel. The source of
the oscillations is longitudinal resonances in the transmission through the constric-
tion. With the sharp boundaries between the constriction and the two reservoirs,
there is a distinct mismatch in the wave functions for the allowed modes in the two
regions. One can see such resonances in transmission over a tunneling barrier, and
they are often referred to as ‘over the barrier resonances’. This is exactly the source
for these oscillations, since there is a wave mismatch at the transitions, this sets up
interferences much like in thin dielectric multi-layers. The transmission rises to unity
when the length of the constriction is such that the momentum wave vector is a
multiple of π/4d (remember that the length is 2d). Clearly, as the length of
constriction is raised, the required momentum for resonance is reduced and many
more oscillations are observed. We will see below that raising the temperature and
introducing scattering both work to wash out these oscillations. But, they are seen in
some experiments.
In figure 2.11, we show this temperature behavior for the simulations given in
figure 2.10. Here, the quantity μ is the chemical potential, which is effectively our
Fermi energy. One can see that, for a temperature of only a half percent of the Fermi
energy, the oscillations on the third plateau are completely gone, and only a smooth
step remains. Remarkably, the oscillations completely disappear over a temperature
increase of only a factor of five. As the value of the Fermi energy in the figure is 10
meV, the two non-zero temperatures are 0.1 K and 0.5 K. This reinforces the view
that the steps can be washed out at temperatures of just a few kelvin since the
subband spacing in split gate QPCs may be only a few meV. In figure 2.12, we give
data for a split gate Schottky defined QPC in GaAs/AlGaAs [19]. Here, the role of
the temperature can be clearly observed, as the steps become quite rounded as the
temperature is raised.
2-21
Figure 2.11. Effect of temperature on the simulated step for the third plateau of figure 2.10. Here, μ is 10 meV,
so the center curve is for a temperature of 0.1 K. (Reprinted with permission from [18]. Copyright 1989 the
American Physical Society.)
Figure 2.12. Temperature dependence of the conductance through a QPC. The different sets of temperature
traces are for a different bias at the mid-point of the QPC corresponding to central gate potentials of 0, −1.2,
−1.8, and −2.4 V (left to right). The observed structure below the first plateau will be discussed later.
(Reprinted with permission from [19]. Copyright 2000 the American Physical Society.)
2.3.2 Scattering and energy relaxation

Although we have related the density of states for our quantum wires to their ideal
forms in various dimensions, these are really only approximations to the real density
of states. Certainly these forms are valid at zero temperature and pure ballistic
2-22
transport, but we have already talked about the broadening of the step transitions
due to non-zero temperature. In the presence of scattering, there is a finite lifetime in
any quantum state and this causes a broadening of the density of states. For
example, the step function that arises at the edge of each step can be considered to be
the integral over a delta function which resides at the step position. That is, the step
function in equations (2.30) and (2.31) can be written as
E
Θ(E − En) = ∫ δ(E ′ − En)dE ′ . (2.40)

−∞
When the energy is greater than the subband eigen-energy, then the integral encloses
the delta function and has the value of unity. Otherwise, the integral gives zero, and
these two results define the limiting cases of the step function.
In the presence of scattering the broadening changes the shape of the delta
function giving it a finite height and width in energy. At low temperatures, and near
equilibrium, the shape of the broadened function is nearly a Lorentzian line [20]. The
scattering process is treated via perturbation theory, and this leads to a self-energy
whose imaginary part is given by the total scattering rate, and this leads to
E
ℏ dE ′
Θ(E − EF ) → A(E − EF ) =
πτ
∫ (E ′ − EF )2 + ℏ2 / τ 2
. (2.41)
−∞
In the case of very strong scattering, there can actually be a shift in the eigen-energy
at which the step occurs, much like a heavily damped LC circuit in which this
damping can cause a shift of the resonance. But, the shift is seldom observed as such
strong damping would already have washed out any chance to observe the steps.
In our derivation of the Landauer formula and the conductance quantization, use
was made of the equipartition of current so that each occupied channel carried the
same current. This was a reasonable approximation as long as there was no
scattering between the different subbands in the channel. However, if inter-subband
scattering occurs, we can no longer make this approximation, and we cannot
continue to associate a definite direction of momentum with a given energy level.
Hence, this will also contribute to a reduction of the quantization properties of the
conductance. So, scattering not only smooths the onset of the step, it also provides a
smoothing by introducing scattering between the various subbands in the channel.
It might have occurred to some readers that, when we have purely ballistic
transport, the conductance should have been infinite rather than the finite value
given by the Landauer relation (2.36). In fact, the transition between the reservoir
and the channel has a non-zero resistance which is a contact resistance between the
one-dimensional channel and the reservoirs which give access to it. Normally, the
reservoirs are very wide regions which are approximately Q2D, and the contact
resistance arises from a mode mismatch between these latter regions and the
channel. This occurs even when the transition is very smooth and adiabatic. This
contact resistance goes directly to the question of just where the voltage drop across
2-23
Figure 2.13. The constriction of figure 2.8 is redrawn here with a linear potential drop (bottom portion)
corresponding to a constant electric field.
the constriction actually occurs. We have assumed that the voltage exists between
the two reservoirs, but can we be more specific about this. In a normal resistive
conductor we would expect that this voltage should be dropped uniformly along the
length of the constriction, just like a normal resistor, but this still requires unusual
behavior in the contacts.
Since there is no scattering within the constriction, the voltage drop across it must
correspond to a constant electric field, which can be quite low as we will see. In
reality, the fact that the voltage is not dropped smoothly across the device means
that charge must accumulate at the contacts, which leads to the variations in the
voltage. As a result, the field in the center of the constriction is quite small, and the
voltage drops are forced to occur in the transition regions. Suppose we take the case
of a linear voltage drop across the constriction so that there is a constant electric field
in this region. Then the potential drop will look like that in the bottom panel of
figure 2.13. But notice that, at the two points where the dashed line has been
extended from the transition region, the electric field is discontinuous. Such a
discontinuity in the electric field requires that a sheet of excess charge be present at
this discontinuity. In fact, the charge at the left must be positive charge, while the
charge at the right must be negative charge. So, we must have a depletion of
electrons at the right and an accumulation of electrons at the left. If electrons are the
major charge carrier, then the left hand reservoir is the cathode, through which the
electrons enter the structure. The accumulation of charge at the transition point
contributes to the contact resistance.
There is still an additional problem with the potential as drawn in this figure. The
linear potential drop leads to a sizable electric field, which will accelerate the
electrons. We can see this in figure 2.14, in which we plot the electron energy instead
of the voltage. If the ballistic electron is injected from the left contact, it will gain
2-24
Figure 2.14. In this figure, the energy E(x), rather than the voltage, is plotted through the structure. An
electron entering from the cathode at the left, and traveling ballistically, will gain energy as it moves through
the channel, exiting with an additional energy defined by the applied voltage.
energy from the electric field. That is, the ballistic electron travels at constant energy,
and the potential energy at the cathode is converted to kinetic energy as it travels. By
the time it reaches the anode at the right-hand side, it will have gained kinetic energy
given by eV. A more important aspect is introduced by Kirchoff’s current law. As
the electrons are accelerated by the electric field, their increase in velocity requires
that fewer electrons are present as we move along the channel. That is, Kirchoff’s
current law requires the current to be uniform throughout the structure. Since the
cross-sectional area does not change, this requires the product of density and
velocity to be a constant value. As the velocity rises, the density must decrease. But,
in a semiconductor, the density is usually set by the doping (as we discussed in the
previous chapter in our discussion of heterostructures). As the number of carriers
drops, this creates an additional space charge in the channel, which in turn requires
that the potential be nonlinear; e.g., the potential drop cannot be a linear one. The
result of this argument is that we cannot have the linear potential drop shown in
figures 2.13 and 2.14 if the carriers are moving via ballistic transport. The linear
potential drop only can occur if there is sufficient scattering to assure that the
carriers move with a near-equilibrium energy. Our conclusion then can only be that
the electric field must essentially be quite near zero in the channel, if the carriers are
to move via ballistic transport.
As a result, the potential drop must divide between the cathode and the anode
transition regions, as shown in figure 2.15. While we have drawn the two voltage
drops as nearly the same, there is no real requirement that this be the case. In fact, it
is usually assumed in the transport world that most of the discontinuity is related to
the cathode. But now, ballistic transport can occur through the constriction without
the carriers gaining excess energy from the applied bias. At the same time, we note
2-25
Figure 2.15. Potential drop required for ballistic transport through the constriction.
that the potential drops in the transition regions now require that dipole charge
densities exist at each transition. The potential drop can only occur through the
existence of such dipole charge densities. Originally, in figure 2.13, the dipole was
split between the two transition regions. Now, each transition region has its own
dipole charge density, and this leads to the contact resistance.
These results give rise to an important modification of the Landauer formula. In
equation (2.36), the transmission from one contact to another is defined in the
reservoirs, which means to the left of the cathode transition region and to the right of
the anode transition region. What if we actually do measurements in the channel
itself? If these are two-terminal measurements, in which the same contacts are used
to source the current and to measure the voltage, we will obtain an answer, but the
voltage drop has to occur in these contacts, which have merely replaced the normal
ones. On the other hand, if we use four-terminal measurements, then we will find
that the Landauer formula becomes modified. We will address this modification in a
later section when we deal with the multi-terminal Landauer formula. But, the
modification arises from the presence of these contact resistances. We will return to
these contact resistances below.
2.3.3 Contact resistance and scaled CMOS

When a large level of carrier scattering is present in the channel discussed above,
then it becomes quite likely that many, if not most, of the individual mode
transmissions will deviate from unity due to the greatly broadened probability for
mode occupancy. This can change the nature of the Landauer formula. In order to
simplify the argument, let us first assume that only a few modes are occupied so that
each mode has the same partial transmission Tpc < 1. Then, we can invert the
conductance to talk about the resistance of the device as
2-26
1 h 1 h h 1 − Tpc 1 1
= 2 = 2 + 2 ≡ + . (2.42)
G 2e NTpc 2e N 2e NTpc GC GD
We can see now how the resistance can be divided into a contact resistance, which
can never vanish, and a device resistance which can go away in the absence of
scattering. The contact resistance is connected with the manner in which the actual
voltage drop is proportioned across the charge regions in the transitions between the
reservoirs and the channel.
While we initially developed the Landauer formula with a treatment of transport
in which all the energy relaxation occurred in the reservoirs (true ballistic transport),
this was not really required. The Landauer formula provides a rigorous framework
for the analysis of nanostructures so long as current is conserved and the reservoirs
can be analyzed in terms of modes. This allows us to determine transmission
coefficients, as described above, for transport between these modes from, e.g. the
Schrödinger equation, or from a classical approach such as the Duke tunneling
formula [21]. As long as the scattering can be described as taking a particle from one
channel to another, so that particle conservation accompanies current conservation,
then any level of scattering can be incorporated within the Landauer formula. This
does not mean that energy cannot be transferred to the crystal lattice by the
scattering process. We need only particle and current conservation for the validity of
the Landauer formula. Further equations are required if we desire to examine energy
flow within the nanostructure.
In recent years, it has been popular to try to stretch the Landauer formula to more
common every-day devices, such as the scaled CMOS transistor [22]. It turns out
that this is not a particularly useful or straightforward approach, but one can gain
some insight into the behavior of the MOSFET with this approach. In the
transmission approach, one often jumps to the conclusion that the potential will
have the linear drop shown in figure 2.13 and that the transport is ballistic. For
reasons discussed in connection with this figure in the previous paragraphs, such an
approach is incorrect. The linear potential drop can only occur with considerable
scattering to keep the energy of the carriers from rising in the electric field. Rather
than start with the transmission itself, let us look at the common equation for the
MOSFET and see how we can connect with contact effects and the Landauer
formula. This will shed light on just how the behavior of the common MOSFET is
not that different from our channel discussed in the earlier parts of this chapter.
The common formula for the MOSFET within the gradual channel approxima-
tion is given by an expression for the drain current in terms of the various potentials
applied to the device, as
WμCox ⎛ VD ⎞
ID = ⎜VG − VT − ⎟VD
L ⎝ 2 ⎠
(2.43)
WμCox
= [(VG − VT )2 − (VG − VT − VD)2 ],
2L
2-27
where W and L are the gate width and length (along the channel), Cox is the gate
capacitance per unit area, μ is the electron mobility, VT is the threshold voltage
where charge begins to accumulate in the inversion layer, and VG and VD are the
gate and drain bias voltages (the source is taken to be grounded). In writing the
equation the way in which it is shown in the second line of equation (2.43), we can
immediately make contact with the Landauer formula. The first term in the square
brackets represents current entering the device from the source (left) electrode, while
the second term represents current entering the device from the drain (right)
electrode. Thus, the source and drain are our reservoirs, and we connect with
particle flow exactly as in the Landauer formula.
In thermal equilibrium, with no bias applied to the gate and drain electrodes,
there is no current flow through the device, since the two terms cancel each other.
This cancellation remains the case even with gate voltage applied, but with the drain
voltage set to zero. The gate voltage changes the properties of the channel, but does
not upset the detailed balance of particle flow through the structure. However, when
a small drain bias is applied, the second term in the square brackets is reduced
relative to the first term, a small current begins to flow. As the drain biased is raised,
the current increases. Once the second term is reduced to zero (it is not allowed to go
negative, as this violates the conditions under which it was derived), the current
saturates through the device at the normal value
WμCox
ID,sat = (VG − VT )2 . (2.44)
2L
When the drain bias is applied, the drain region is lowered in energy relative to the
source region, just as shown in figures 2.13–2.15. Saturation occurs when this energy
difference is sufficiently large that electrons in the drain can no longer surmount the
barrier to reach the source. No matter whether the transport is ballistic or not (there
is lots of scattering), the energy regions in the source and drain will be the same.
What will be different between these two cases is how the energy drops between the
source and drain.
Let us now take (2.44) and write it in terms of an ‘equilibrium’ current that
depends upon the gate voltage, as
WμCox
Ieq+ = ID,sat = (VG − VT )2 . (2.45)
2L
Then, we can rewrite the second line of (2.43) as
⎡ ⎛ VD ⎞ ⎤
2
ID = Ieq+⎢1 − ⎜1 − ⎟ ⎥. (2.46)
⎢⎣ ⎝ VG − VT ⎠ ⎥⎦
For small values of the drain voltage, we can approximate write this result in terms
of a resistance (likely seen as a contact resistance)
2-28
VD V −V
RC = ∼ G + T. (2.47)
ID 2Ieq
Even for a very short device with perfect transmission, we cannot get around this
minimum resistance. This resistance characterizes the transition region between the
pure source and the channel (just as in the Landauer approach) and appears
regardless of whether the transport is described by the mobility or is ballistic. In some
sense, this resistance describes a barrier between the source and the channel [23]. The
current flux of equation (2.45) is due to carriers from the source with sufficient
kinetic energy to overcome this barrier between the source and the channel. It is
important to note in equation (2.45) the flux is actually metered by the gate
potential, so that this is a gate-controlled barrier. Once the carriers supplant this
barrier, they are going to flow downhill (down the potential drop) to the drain. They
will flow to the drain regardless of the nature of the transport. The only effect that
ballistic transport can play is to change the shape of the potential landscape between
the source and the drain. In this sense, scattering actually affects this potential
landscape and serves to isolate the drain from the source. In a ballistic device, this
isolation does not occur and the source–channel barrier can be affected by the drain
voltage, an effect known as drain-induced barrier lowering.
2.4 Beyond the simple theory for the QPC

In the previous few sections, it has been shown that the understanding of the
conductance steps in the QPC, or in a longer nanowire, fit nicely with our
understanding of the role played by the density of states within the Landauer
formula. This has led to a beautiful example of condensed matter theory being
demonstrated by the experiments on these devices. But, there is always a chance to
go beyond the simple theory to gain new insight into the devices, or to discover new
things which may not be so well understood.
2.4.1 High bias transport

In deriving the above expressions for the conductance quantization, we made the
assumption that the voltage applied across the device was quite small, in which case
the voltage variation across the device is essentially unaffected by the applied bias.
This supported the idea that transport in the small QPCs is essentially ballistic in
nature, so that the actual voltage was dropped mainly across the transition regions
between the reservoirs and the channel. However, if we apply a larger bias, we can
induce significant band bending in the QPC itself, which of course must be
accompanied by various non-equilibrium charge distributions within the device.
But, these effects can lead to new and different experimental observations of the
quantization phenomena in the observed conductance. For example, the usual
plateaus that appear at integers of 2e2/h evolve with voltage to half-integer values.
We can see this in figure 2.16, which gives the results for a very high quality QPC in
the GaAs/AlAs system. The Si dopants are placed in a thin GaAs quantum well
sandwiched between two AlAs layers, and the 2DEG is located a distance of 160 nm
2-29
Figure 2.16. The pinch-off trace, and the schematic view of the potential, is shown for the QPC. (a) The linear
response for a vanishingly small source–drain bias across the QPC. (b) Response for a larger bias (negative
voltage on source side) where the plateaus have evolved to near the half-integer values. (c) A higher bias such
that the plateaus have evolved further and returned to their integer values. (Reprinted with permission from
[24]. Copyright 2011 IOP Publishing and Deutsche Physikalische Gesellschaft).
below the surface. The Si donors lead to an electron density of 3.5 × 1011 cm−2 with a
mobility of 1−2 × 107 cm2 Vs−1 [24]. In the figure, the conductance through the QPC
and a schematic diagram of how the voltage is dropped across it, are shown for
convenience. In panel (a), the normal results for a very small applied bias are shown.
Here, the steps have their normal plateaus at integer values of 2e2/h. In panel (b),
however, a bias of 2.6 mV has been applied to the drain (note that the figure gives the
values at the source relative to the drain). With this bias, the plateaus have moved
basically to the half-integer values, appearing at 0.5, 1.5, 2.5, and so on. There
remains a small feature at 1.0 though. From the schematic diagram on the right, it
can be seen that the Fermi energies in the two reservoirs are such that a subband
minimum resides between them. Hence, the characteristic of this half-integer plateau
arises when one reservoir injects into one fewer subbands than the other. Finally, in
panel (c) the bias has been increased further, and the integer plateaus have returned.
But, now there are two subband minima between the two Fermi levels. At still higher
values of bias, the plateaus sometimes become washed out due presumably to
heating of the electrons in the QPC [25].
When the source–drain bias is applied, this changes the Fermi potential in the
center of the QPC, which shifts the gate voltages at which the transitions occur.
Tracking the plateaus as one varies the source–drain bias allows a method of
spectroscopy in which we may analyze the subband spacing in the device. This is
illustrated in figure 2.17, where the transconductance is plotted as the two voltages
are varied [24]. The plot uses the source–drain voltage for the vertical axis and the
2-30
Figure 2.17. The transconductance GTC = dG/dVG for one QPC (the two split gates are biased equally). The
source–drain voltage is plotted vertically and the gate voltage horizontally, with the transconductance color
coded according to the scale at the upper right corner. Plateaus in conductance (small transconductance)
appear as dark areas, and the various ones are labels accordingly. (Reprinted with permission from [24].
Copyright 2011 IOP Publishing and Deutsche Physikalische Gesellschaft.)
gate voltage in the horizontal axis, with the transconductance color coded. Bright
yellow is the largest value and black is the smallest value. Hence, the plateaus appear
as dark regions between the brighter ‘lines’ which indicate the transitions between
the plateaus. It can be easily seen how the integer plateaus gradually change with
bias into the half-integer ones, and then change again into integer ones with
increasing source–drain bias. Observation of these returning integer plateaus can
be difficult to see, and the very high mobility in these samples makes this possible.
This high mobility manifests itself as a reduction in the back-scattering of various
modes at the transitions and leads to enhanced ability to do spectroscopy in this
nonlinear transport regime.
Also indicated in the figure is how one can obtain analytical results. Three white
circles are indicated at the transition between the fourth and fifth plateaus at
different values of source–drain bias. We recall from the discussion above, that in
the nonlinear regime, the two Fermi levels (left and right) span subband bottoms. On
the dashed line between the circles, the observed change as the source–drain bias is
increased obviously measures the distance between the subbands. That is, as one
moves across the plateau from low bias to the bias corresponding to the next
transition, one is clearly coupling exactly two new subbands between the Fermi
energies. The transition occurs as the subband crosses one Fermi level, so clearly
one-half of this applied bias corresponds to the subband spacing, as indicated in the
figure. This suggests that, at this bias, the subband spacing is about 2.25 meV. But,
one can go further, as each of the transitions allows us to study the subband spacing
as a function of the two bias voltages. Thus, one can see just how the eigen-energies
2-31
of the harmonic oscillator vary with gate voltage and with source–drain bias voltage.
Thus, for the devices discussed here, the authors have found that the subband
spacing varies from almost 5 meV down to about 1 meV when the QPC is
sufficiently open such that ten modes are propagating through it [24]. It is clear
that, with good quality material, one can fully characterize both the device under
study and the models used to explain the results.
2.4.2 Below the first plateau

It may be clearly noted in figure 2.12 that there is a feature that appears below the
first plateau, near the value of 0.7(2e2/h). This additional shoulder in the conduc-
tance has been seen almost since the first measurements of the quantized con-
ductance through a QPC. This feature cannot be explained by the simple theory,
since it is not correlated with any common behavior, such as the number of modes.
Moreover, this quasi-plateau does not seem to have any universal behavior. Some
researchers have suggested that the 0.7 plateau, as it is usually referred to, merges
with the first plateau as the temperature is lowered and an anomaly at zero source–
drain bias emerges, which they identify with a Kondo-like feature (associated with
charging a localized state in most cases) [26]. On the other hand, there appears to be
no movement at all of the plateau with temperature in figure 2.12, nor is such
behavior seen in many other structures which show conductance quantization, such
as cleaved-edge overgrowth wires [27]. Many of the observations have been brought
together by Pepper and Bird [28] in a Special Issue of the Journal of Physics:
Condensed Matter which collects the work of a great many authors. More recently,
Micolich has provided an extensive review of experimental studies of the 0.7 plateau
[29]. The conclusion which one might draw from these works is that the effect may
not be a single effect, but may be the appearance of several different, but similar,
effects which depend upon the details of the actual device upon which the experi-
ments are being conducted. The discussion here will argue with such a conclusion,
but will present a few of the more universally accepted ideas.
The idea that the 0.7 plateau is, in at least some manner, connected with the
electron spin is supported by observations that the plateau evolves to one at 0.5G0,
where G0 = 2e2/h, as an applied magnetic field is raised to values to show full spin
splitting [30]. In general, there is only a weak variation with the electron density in
the reservoirs [4]. As remarked above, there is some consideration that the 0.7
plateau is related to the Kondo effect [31]. The Kondo effect is often seen in
quantum dots (discussed later) [26]. Such a result is somewhat surprising, since the
Kondo effect is well known to involve the interaction of conduction electrons with a
localized spin (or magnetic moment). In quantum dots, in which the defining QPCs
are pinched off so that only single-electron tunneling can contribute to the transport,
the idea of a localized spin within the dot is easy to connect with a single localized
electron in the highest occupied state of the dot; e.g., a total odd number of electrons
in the dot. For this to occur in a QPC suggests that there must be a localized state
that forms in the QPC below the first plateau, and this has been suggested to form
self-consistently by some simulations [32, 33]. Others, however, have suggested that
2-32
the localized state could produce the 0.7 plateau without the need for a Kondo
effect. But, if the localized state contains a single spin state, as has been suggested
[34, 35], it seems reasonable that Kondo physics could certainly play a part.
As mentioned, simulations in the linear spin-density approximation have shown
that a localized state can form in the QPC below the first plateau [33, 35]. These
features can be associated with the formation of a spin-dependent energy barrier that
arises once one spin is trapped in the localized state. This barrier prevents the other
spin from transporting through the QPC. These barriers rise and fall as a function of
the local density, and with the local potential. By varying the density, the 0.7 plateau
can be induced to evolve into the fully formed first step or to drop near a value of 0.5
[32]. In addition, other features can be observed, such as an anomaly near 0.25G0
that has been observed experimentally [26, 36, 37]. But, what is clear from these
simulations is that the localized state is often not a simple minimum in the potential.
Rather it is much more complicated, perhaps due to the existence of Friedel
oscillations which extend from the minimum of the saddle potential far into the
reservoir regions [38]. Moreover, the local potential tends to have a double
minimum, which leads to three peaks. The amplitude of these peaks varies not
only with the gate voltage, but also with the local density in the QPC (along the wire
for example). In our own simulation, a model potential proposed by Timp [39] was
used in place of the pure harmonic oscillator saddle potential discussed earlier. This
confining potential had the form
⎛ 2x − L 2y + W ⎞ ⎛ 2x + L 2y + W ⎞
V (x , y ) = F ⎜ , ⎟ − F⎜ , ⎟
⎝ 2z 2z ⎠ ⎝ 2z 2z ⎠
(2.48)
⎛ 2x − L −2y + W ⎞ ⎛ 2x + L −2y + W ⎞
+ F⎜ , ⎟ − F⎜ , ⎟,
⎝ 2z 2z ⎠ ⎝ 2z 2z ⎠
where
eVG ⎡⎢ π ⎛ uv ⎞⎤
F (u , v ) = − tan−1(u ) − tan−1(u ) + tan−1 ⎜ ⎟⎥ . (2.49)
2π ⎢⎣ 2 ⎝ 1 + u 2 + v2 ⎠⎥⎦
and z is the distance of the Q2DEG from the surface of the heterostructure. W and L
are the lithographic dimensions of the QPC at the surface of the heterostructure, so
are taken from an experimental structure (140 nm and 350 nm, respectively, in the
simulations discussed here). However, the actual local potential is solved self-
consistently with the density in the linear spin-density approximation [34, 35]. In
figure 2.18, we show the conductance plots for two values of the quantum wire
(average) line density, in order to illustrate the role played by this quantity. In panel
(a), the density is 2.01 × 106 cm−1 (EF = 13.5 meV), and the first plateau has almost
disappeared in favor of the 0.7 plateau. In panel (b), the density is 1.96 × 106 cm−1
(EF = 13.4 meV), and the first plateau is well formed, while the 0.7 plateau has
dropped closer to 0.5. In both cases, the feature near 0.2 is distinct. In these curves,
the solid curve is the total conductance, while the dotted curve is the spin-down
conductance and the dashed curve is the spin-up conductance. The unusual feature
2-33
Figure 2.18. Conductance curves at two different line densities in the QPC: (a) 2.01 × 106 cm−1 and (b) 1.96 ×
106 cm−1. The solid lines are total conductance while the dotted and dashed curves are the spin-down and spin-
up conductances, respectively, as discussed in the text. Data taken from [35].
here is that the spin-up modes are almost completely blockaded until the second
plateau begins to appear. The entire first plateau, with the sub-structure, is entirely
due to the spin up modes. Hence, there appear to be two spin-up modes propagating
to form the first plateau.
As remarked above, the local potential seems to have three peaks in the region of
the center of the QPC. We can see this in figure 2.19, where we plot the local
potential profiles for the two cases shown in figure 2.18. The two panels refer to the
same two panels of the conductance plots. Curves (1) are the potential profiles for
the conductance near the feature at 0.2G0, while curves (2) correspond to the feature
near 0.7G0. The set of curves (3) correspond to other plateaus as indicated by the
arrows. Curves (1) and (3) have been offset for clarity. This three peak structure in
the potential was also reported by Hirose et al [38].
One can probe these anomalies with the SGM discussed previously. We have used
an InGaAs quantum well clad with InAlAs to form the heterostructure, and then
formed in-plane, trench isolated gates such as those shown in figure 2.1. In this way,
the scanning probe tip can be brought into the heart of the QPC without fear of
hitting surface metallic gates. In figure 2.20, we show an SGM image obtained with
the QPC in the 0.7 plateau [40]. Amazingly, essentially the same figure is obtained in
the region 0.2−0.3G0 [1], which suggests a common origin for these two features, just
as found from the simulations mentioned earlier. It is noteworthy that the peaks in
the figure are resistance peaks; e.g., the red color denotes high resistance. These
resistance peaks correspond to the potential peaks that are observed in the
2-34
Figure 2.19. Potential profiles for the conductance plots of figure 2.18. Panels (a) and (b) correspond to the
respective panels in this earlier figure. In each panel, curves (1), (2) and (3) correspond to the features identified
by the arrows in figure 2.18.
Figure 2.20. SGM image of a QPC at the 0.7 plateau. Very similar images are also obtained near the 0.25
plateau that is sometimes observed. The position of the trench isolation for the gates is indicated by the dotted
curves.
simulations. These suggest that the conductance through these quasi-plateaus is

quite probably via tunneling as one might guess. But, it seems to be clear that the
peaks below the first plateau are quite probably connected to the spin of the electron,
and a complete understanding of what is really occurring is still in question.
As mentioned earlier, some investigators do find a simple single localized
minimum in the potential [32]. More recent simulations by Meir have also shown
that a single localized state can exist which allows only a single spin state to
propagate through the QPC [41]. Thus, even the nature of the localized state (or
states) seems to vary with the properties of the particular device under study, and
2-35
even with the local carrier density in the device. Thus, it is not surprising that there is
so much variation in the experimental details, and there remains a significant
likelihood that the structure below the first plateau is certainly a many-body effect,
and may well be not a single effect but many possible ones which all have about the
same likelihood of occurring.
Perhaps, it is best summarized by results from a recent paper, in which one or
more localized states are observed to appear, and these are felt to arise from the
Friedel oscillations in the electron charge density within the QPC [42]. These latter
authors also report that there are spins associated with the localized states, as
discussed above, and the Kondo effect, associated with transport through these
localized spin states, contributes to the formation of the resulting many-body state.
Of course, the self-consistent potential is going to be crucial to the formation of the
localized state(s), and so some authors may not see the role of the Kondo effect, as
has been discussed [29].
2.5 Simulating the channel: the scattering matrix

There are a great many approaches to solve the Schrödinger equation in a finite-
sized system. When the system is large, it is often computationally more useful to use
a recursive approach, in which the solution is generated by propagating a known
contact solution through the structure with a slice by slice recursion. That is, the
properties of a single slice are determined by those of the preceding slices starting
from one reservoir or the other. The best known approach is the mode matching in
which the wave function and its derivative are matched at each interfacial boundary
between slices within the system. Unfortunately, this approach is known to be
unstable for more than a few slices. An important variation of this approach is to
switch the interface matching into the scattering matrix [43, 44], an adaptation
borrowed from earlier microwave theory. We have used this approach for many
years to study quantum dots [45]. A second approach is to use Green’s functions,
derived from the Schrödinger equation, in a recursive approach, and this will be
discussed in the next section. Both of these approaches actually use a Green’s
function in the recursion through the incorporation of Dyson’s equation. In each
case, the transmission through the entire structure is computed, and the conductance
evaluated using the Landauer formula (2.37) and (2.38), with
Tnm = ∣tnm∣2 , (2.50)
where tnm is the complex transmission of the wave function through the device, from
mode n in the left reservoir to mode m in the right reservoir. In fact, a correction
needs to be made to account for the fact that we are dealing with semiconductors
and not metals. Hence, the mode velocities are not just the Fermi velocity. The
velocities in each mode may be different, as they are measured from the subband
minima corresponding to the modes. Thus, we correct equation (2.50) as
v
Tnm = n ∣tnm∣2 , (2.51)
vm
2-36
Figure 2.21. A two-dimensional grid around which the recursion formula is built. One slice is indicated by the
darker shade of blue.
where the v are the mode velocities. In appendix A, we discuss the basic principles of
discretizing the Schrödinger equation and the Poisson’s equation and setting up
slices for both approaches that we will discuss.
As discussed above, the normal approach to matching the wave function and its
derivative at each slice of the structure (across the interface) is generally
unstable when a large number of such interfaces are present. Stability can be
restored by modifying the recursion to one based upon the scattering matrix, which
has long been a staple of microwave systems and entered quantum mechanics
through the Lippmann–Schwinger equation [46]. The strength of this approach lies
in the fact that modal solutions to the Lippmann–Schwinger equation maintain their
orthogonality through the scattering process [46]. In mesoscopic structures and
nanostructures, the connection to microwave theory becomes stronger as the
transport becomes dominated by the modes introduced by the lateral confinement.
Even in the non-recursive mode, the use of scattering states provides a viable method
of building up an orthogonal ensemble, even with weighting of the states by e.g., a
Fermi–Dirac distribution [47]. This is particularly useful in providing the initial
conditions for the wave function in a particular semiconductor device [48]. In the
recursive approach, it reaches new levels of viability when combined with the modal
structure in small devices [43]. In the end, this approach allows one to determine the
transmission from one end to the other and evaluate the conductance via the
Landauer formula.
The approach is based upon the discretization scheme for a uniform array of
points spaced by a (see appendix A). We show such an array in figure 2.21, in which
the array has a slice width in the transverse direction of M + 1 points and a length of
N + 1 slices. A useful fact is that if we just limit our size as shown, the solution
automatically assumes that the wave function will satisfy
ψ (0, j ) = ψ (M + 1, j ). (2.52)
As a result of this, we do not concern ourselves with the first and last points of the
array, and can write the slice wave function as
2-37
⎡ ψ (M , s ) ⎤
⎢ . ⎥
⎢ . ⎥
.
ψs = ⎢ ⎥. (2.53)
⎢ ψ (2, s ) ⎥
⎢⎣ ψ (1, s ) ⎥⎦
Note here, that we have defined i in the y-direction and j in the x-direction, which is
the current flow direction. Note also that the single subscript wave function is the
slice wave function, as opposed to the site wave function which has two variables.
Then, the Schrödinger equation can be rewritten as a matrix equation relating the
slice vectors
H0,jψj − tψj +1 − tψj −1 = EIψj , (2.54)
where t is the diagonal matrix whose elements are the hopping energy
ℏ2
Et = (2.55)
2m⁎a 2
(and not to be confused with the transmission matrix elements, which have
subscripts). In equation (2.54), the Hamiltonian for the isolated slice is written as
⎡ (VMj − 4Et ) Et 0 ⎤
⎢ 0 ⎥
⎢ Et (VM −1j − 4Et ) . ⎥
⎢ . . . ⎥
H0,j = ⎢ . . 0 ⎥, (2.56)
⎢ . (V2j − 4Et ) Et ⎥
⎢ 0 ⎥
⎢⎣ . Et (V1j − 4Et ) ⎥⎦
where the upper right block and lower left block are identically zero, and I is an
identity matrix. We note that the individual elements of the slice wave function are
the corresponding site wave functions. Generally, the transverse wave functions for
the width confinement are found from the corresponding slice Hamiltonian, e.g.,
equation (2.56) with j = −1. This process produces the site wave functions and the
slice wave function, which is doubly degenerate with the ability to propagate to the
right and to the left. These wave functions may also be found through solving the
equation
det[T0 − λI ] = 0, (2.57)
where T0 is a matrix whose rank is twice that of the Hamiltonian (2.56) and is given
as
⎡ U+ U− ⎤
T0 = ⎢ ⎥. (2.58)
⎣ λ+U+ λ−U−⎦
Here, U± are matrices whose rank is equal to that of the Hamiltonian and whose
columns give the slice wave functions in site representation. These are the modes of
2-38
the transverse confinement and the positive and negative signs correspond to
positive (right-directed) propagation and negative (left-directed) propagation.
Similarly, the λ± are the corresponding eigenvalues. These matrices give the site-
to-mode transformations of the various wave functions. There is a corresponding
Fermi energy (the simulation is at zero temperature at this point), so that modes
whose eigenvalues are below the Fermi energy will be propagating, while those
whose eigenvalues lie above the Fermi energy are evanescent. Once, we get slice 0 in
the mode representation, we can proceed with the recursion.
To carry out the recursion, we have to put everything into the scattering matrix
form. The site to mode transformation creates slice wave functions (2.53) in which
each element corresponds to a mode in the system, as suggested above. These modes
can be either propagation or evanescent. To continue, we create two new matrices,
which are the ‘scattering matrices’ C1s and C2s at the general site j = s. The recursion
relation for the are now expressed as
⎡C s+1 C s+1⎤ ⎡ 0 I ⎤⎡C1s C 2s ⎤⎡ I 0 ⎤
⎢ 1 2
⎥=⎢ ⎥⎢ ⎥⎢ ⎥, (2.59)
1 ⎦ ⎣− I t (H0,s − EI )⎦⎣ 0 1 ⎦⎣ P1s P2s ⎦
−1
⎣ 0
where the first matrix on the right-hand side is denoted as Ts, and H0s is the slice
Hamiltonian (2.56). For the zero temperature formulation, C1s is set to a unit matrix,
although for a finite temperature simulation, the elements will be given by the
Fermi–Dirac distribution for the energy of the corresponding eigenvalue. Similar
values are given to C2s at the right-hand end of the system. Now, equation (2.59) is
required to be satisfied, and this allows us to thus determine the propagation
matrices P1s and P2s through the recursions
C1s+1 = P1s = −P2s( −I )C1s

(2.60)
C 2s+1 = P2s = ( −IC 2s + t −1(H0,s − EI ))−1.
The unit matrix (the −I term) is actually the 2,1 element of the Ts matrix which will
be different if there is a magnetic field applied, as it arises from t −1t and will be
different when time reversal symmetry is broken by the magnetic field.
At the end of the sample, the waves are transformed back into the mode
representation, and the transmission used to determine the conductance. Also at
the end of the structure, one determines the wave function at the output in terms of
the mode number and its values at the various sites. Thus, we determine at site i and
mode m the contribution to the wave function as ψj (i , m ). This can now be back
propagated to find the equivalent wave functions at each slice via
ψj (i , m) = P1j + P2jψj +1(i , m), (2.61)
so obviously this recursion now proceeds from right to left. The velocities for each
mode are found from the imaginary part of the eigenvalue for that mode. At the
same time, the density can be determined at each site by the square magnitude of the
wave function at that site (initially, the normalization is to unity, but this can be
2-39
understood to be in terms of the density in the initial contact). A fuller description of

the approach can be found in Akis et al [45].
When we use a vanishingly small bias between the two reservoirs (between the
source and the drain), then we need only compute the transmission and then use the
Landauer formula. However, when a real, non-zero level of bias is applied, the
situation becomes more difficult, on several levels. First, we have to compute both a
source-derived flow and a drain-derived backflow, just as in equation (2.44). Both
flows must be computed to determine the total wave function and the density at each
grid point since the non-zero bias will drive the system out of equilibrium. This
implies that we will then need to use this density to solve self-consistently for the
potential within the device, as the value of the potential at each grid point goes into
the Hamiltonian equation (2.56). Now, we have an iterated loop that solves the
quantum transport simulation and then solves for the self-consistent potential (see
appendix A), and then once again for each until convergence is obtained. The
scattering matrix approach is well suited to this process, as the density is a natural
output of the simulation through equation (2.61). Secondly, we have to account for
the temperature at each end of the device, as we can no longer assume that each
mode has unity amplitude in the reservoir. Rather, the amplitude is determined from
the Fermi–Dirac distribution at the temperature of the reservoir. The drain reservoir
can be at a higher temperature, as this is ultimately where the last of the excess
carrier energy is deposited. This approach has been extended to three dimensions
and nanoscale MOSFETs at room temperature with real scattering [49]. Again, the
scattering matrix approach is well suited to this task, as the Fermi–Dirac functions
are applied in the reservoirs and not within the active region of the device.
By its very form, it is quite simple to modify the recursion method to include
scattering by an on-site self-energy that is added to the diagonal elements of
equation (2.54) and represents the interactions with scattering centers. Generally,
the self-energy has both real and imaginary parts and it is the latter that are of
interest for dissipative scattering. In semiconductors, the scattering is generally quite
weak, and is traditionally treated by first-order, time-dependent perturbation theory,
which yields the common Fermi golden rule for scattering rates. With such weak
scattering, the real part of the self-energy can generally be ignored for the phonon
interactions, and the real part that arises from the carrier-carrier interactions is
incorporated into the solutions of Poisson’s equation. In the many-body formula-
tions of the self-energy, the latter is usually expressed as a two site function [20]
Σ(r1, r2 ), (2.62)
where the two positions are usually three-dimensional vectors. Since we are using
transverse modes in the recursion, we rewrite this expression as
Σ(i , j ; i′ , j ′; x1, x2 ). (2.63)
As usual, the i and j (and their primed versions) are the transverse site and mode,
respectively, and correspondingly the x’s represent positions along the nanowire. We
can then introduce a center-of-mass transformation [50]
2-40
x1 + x2
X= , ξ = x1 − x2 , (2.64)
2
and then Fourier transform on the difference variable to give
1
Σ(i , j ; i′ , j ′; X , kx ) =
2π
∫ dξeik ξΣ(i, j; i′, j′; X , ξ).
x (2.65)
The center-of-mass position X remains in the problem as the mode structure may
change as one moves along the channel. At this point, the left-hand side of equation
(2.65) the self-energy computed by the normal scattering rates, such as is done in
quantum wells and quantum wires [51, 52]. As mentioned above, since scattering in
semiconductors is relatively weak, it is sufficient to compute these using Fermi’s
golden rule, which is an evaluation of the bare self-energy in equation (2.65), rather
than incorporating more elaborate many-body effects. Since the recursion is in the
site representation, we have to reverse the Fourier transform in equation (2.65) to get
the x- axis variation, and do a mode-to-site unitary transformation to get the self-
energy in the form necessary for the recursion. We thus proceed by using the Fermi
golden rule expression for each scattering process of interest and generating a real
space self-energy from it. The imaginary part of the self-energy is related to the
scattering rate via
⎛ 1 ⎞i ′,j ′
Im{Σ(i , j ; i′,j ′; X , kx )} ≡ ℏ⎜ ⎟ . (2.66)
⎝ τ ⎠i ,j
It is the latter scattering rate which we calculate, which is a function of the x-directed
momentum (which is related, in turn, to the energy of the carrier) in a cross-section
of the device, which can be thought of locally as a quantum wire. This scattering rate
must be converted to the site representation with a unitary transformation. At site
s,l,η, the correction due to scattering that gets added to the local potential Vs,l,η is
given by
⎛ ℏ ⎞i ′,j ′
Γ(s , l , η) = Im{Σ} = Us†⎜ ⎟ Us , (2.67)
⎝ τ ⎠i ,j
where Us is the mode-to-site transformation matrix discussed above. The actual

scattering rates are computed in the normal manner [49, 53].
This methodology, based upon the scattering matrix recursion technique, has
been applied to a number of both ballistic and scattering approaches to gated silicon
quantum wire structures with excellent results [54]. In these approaches, it was
possible to directly determine the diffusive to ballistic crossover length.
2.6 Simulating the channel: recursive Green’s functions

A second approach is to use Green’s functions, derived from the Schrödinger
equation, in a recursive approach [55–57]. Again, we have used this technique
successfully in the past [58]. With the recursive Green’s function, the contact area is
2-41
assumed to be a semi-infinite metallic wire of a given width. The modes and

eigenvalues are then computed for this strip for a given Fermi energy. To begin, the
region is discretized as in figure 2.21, and there is a width of the contact given as
W = Ma, where a is the grid spacing. Thus, the transverse eigenfunctions are those of
an infinite potential well, given as
2 ⎛ rπy ⎞
φr (y ) = sin⎜ ⎟ , r = 1, 2, 3…. (2.68)
W ⎝W ⎠
Here, the index r corresponds to the transverse mode in the confined wire contact.
This wave function is evaluated at each grid point i in the transverse direction to
evaluate the wave function at the sites in the zero slice. Once we are given the Fermi
energy, then we can compute the longitudinal wave number from a knowledge of
whether or not the eigenenergy is greater or smaller than the Fermi energy. We first
compute the quantity
⎛ rπ ⎞ EF
ξ = 2 − cos⎜ ⎟ − , (2.69)
⎝ M + 1 ⎠ Et
where Et is the hopping energy (2.55). If ξ ⩽ 1, then the wave is propagating, and we
write the longitudinal wave function in the contact as
ψr(x ) ∼ sin[cos−1(ξ )]
ξ (2.70)
k = , x ⩽ 0.
a
This wave function needs to be properly normalized, of course, but represents the
assumption that the contact leads to a vanishing of the wave function at x = 0 in the
absence of the active channel. If the eigenenergy of the wave function is greater than
the Fermi energy, then we are dealing with an evanescent wave, and the momentum
in equation (2.70) is replaced by
1
λ= [ξ − ξ 2 − 1 ]. (2.71)
a
The recursion begins by establishing a self-energy correction which acts upon the
first slice, but is based upon the left contact and its connection to the first slice. That
is, we define the left self-energy by the action
ΣL = H10G 00H01, ΓL = 2Im{ΣL}. (2.72)
Here, ΣL and ΓL are diagonal matrices, G00, H10, and H01 are M × M matrices. The
matrix G00 is the on-site Green’s function given by
G 00 = (EF I − H0 + iIη)−1, (2.73)
H0 is the site Hamiltonian (2.56), here for the zeroth slice where the potential is zero,
and a small damping factor η has been added to assure convergence of the
matrix inversion process. The use of a positive damping factor assures us that the
2-42
Green’s functions are the retarded (causal) Green’s functions. In the above equation,
I is a unit matrix as before. The two matrices H10 and H01 are inter-slice connection
matrices and are adjoints to each other. For example, H10 contains a hopping energy
on the diagonal connecting a site i along one row with its neighbor on the row to the
right. Equation (2.73) is just used to determine the self-energy coupling between the
contact and the channel, and the actual zeroth slice Green’s function is computed
from
G 00 = (EF I − H0 + ΣL + iIη)−1. (2.74)
Now, we propagate along the active channel with a recursion that computes first the
slice Green’s function and then the connecting Green’s functions. For slice j ⩾ 1, we
can compute the slice Green’s function as
Gj ,j = (EF I − Hj − Hj ,j −1Gj −1,j −1Hj −1,j )−1. (2.75)
Note that we have added a comma to avoid confusion in the subscripts of the
Green’s functions. In this expression, the slice Hamiltonian Hj is just equation (2.56)
and includes the local site potential at each grid point, while the two nonlocal
Green’s functions are just the diagonal hopping terms described previously. These
are equal to one another except when we add a magnetic field, which will be
discussed below. In addition to equation (2.75), we also construct the two
propagating Green’s functions
G 0,j = G 0,j −1Hj −1,jGj ,j , (2.76)
and
Gj ,0 = Gj ,jHj ,j −1Gj −1, 0. (2.77)
At the right contact, we have to connect to the second lead, which we assume occurs
for j = L. Then, the right contact Green’s functions become
GL+1,L+1 = (EF I − HR + ΣR − HL,L+1GL,LHL+1,L )−1
G 0,L+1 = G 0,LHL,L+1GL+1,L+1 (2.78)
GL+1,0 = GL+1,L+1HL+1,LGL,0.
Here, the right self-energy has been computed in exactly the same manner as in
equation (2.72), but with the right contact Hamiltonian. This approach allows one
to actually use leads that have different sizes, neither of which actually needs to be
the same as the active strip. The transmission through the total system is now found
from (there are several variants of this formula available in the literature, for
example, [59])
T = Tr{ΓLG 0,L+1ΓRGL+1,0}. (2.79)
A more extensive derivation of the approach is given in [60]. A magnetic field can be
easily added to the simulation.
2-43
As previously, the recursive Green’s function approach discussed here is well

suited to vanishingly small bias applied between the two reservoirs. Temperature can
be included by computing the transmission for a range of energies around the Fermi
energy and then weighting each transmission by the appropriate value of the Fermi–
Dirac distribution. But, this is an approximation that cannot be extended to larger
values of the applied bias. Nor can the recursive Green’s function approach
discussed here be used in this latter case. The approach here is based upon
equilibrium Green’s functions. Even weighting each value of transmission as
mentioned is an approximation, as reasonable temperatures really require use of
the thermal, or Matsubara, Green’s functions [20]. And, if we go to larger values of
bias applied between the two reservoirs, then the density is driven out of equilibrium,
and one has to go to the more complicated non-equilibrium, or real time Green’s
functions, which can also yield the non-equilibrium distribution function within the
device [51, 61, 62]. This is required to determine the local density, as it does not
follow directly from the calculation as in the case of the previous section. Progress in
this area is best described by OMEN, a powerful simulation package for quantum
device simulation, which can use the real-time Green’s functions [63].
The recursive Green’s function has been used extensively to study conductance
fluctuations, weak localization, and strong localization. We discuss weak local-
ization in chapter 3, strong localization in chapter 5, and conductance fluctuations in
chapter 9. But, we can introduce the idea here by looking at the Hamiltonian (2.56),
and considering the potential to be a random potential, or to have a random
component. The transverse confinement introduces a quantization in the transverse
eigenvalues, some lying below the Fermi energy and others lying above the Fermi
energy as non-propagating evanescent modes. Let us assume that the random
amplitudes of the site potential lie between, e.g., −V0/2 and V0/2, with no uniform
potential present. Now, if V0 is smaller than the Fermi energy, a few modes near this
value will see the random potential, and there will be interference between those
modes that see the potential. This interference can change dramatically for small
changes in the Fermi energy, or a magnetic field, and this can lead to conductance
fluctuations. That is, the conductance changes as the interference changes by a small
amount. This is the conductance fluctuations discussed in chapter 9, although they
are strongly related to the weak localization effect. The latter arises when the
interference leads to back-scattering of some modes. Technically, when there is
back-scattering, without a magnetic field, one can consider a particle can traverse
the back-scattering trajectory in either of two time-reversed directions, and this leads
to interference between the time-reversed paths and, in turn, to a small decrease in
the conductance. This will be discussed further in chapter 3. Finally, when the
potential amplitude gets larger, lower energy modes are in fact cut off, as they
cannot propagate through the random potential. Hence, the number of propagating
modes decreases, and we have strong localization. All of these behaviors can be seen
in a typical nano-ribbon and are easily studied with the recursive Green’s function,
as well as the scattering-matrix approach [64]. We will illustrate these effects with
further examples in the following chapters.
2-44
Problems
1. A Poisson solver that may be used to calculate the energy bands of different
heterostructures may be downloaded at: http://www.nd.edu/∼gsnider/.
Consider a heterostructure comprised of (starting from the top layer): a
5 nm thick GaAs cap layer; 40 nm of undoped AlxGa1−xAs (x = 0.33); 10
nm of AlxGa1−xAs (x = 0.33) doped with Si at a concentration of 1.5 × 1018
cm−3; 20 nm of undoped AlxGa1−xAs (x = 0.33), 100 nm of undoped GaAs;
and a GaAs substrate with an unintentional p-type doping of 5 × 1014 cm−3.
(a) The Poisson solver is a self-consistent program that computes the
band structure by solving two important equations simultaneously.
What are these equations?
(b) Plot the calculated energy bands and ground-state electron wave
function for the heterostructure at 1 K. You may assume full
ionization of the donors.
(c) Explain quantitatively the reasons for the different energy variations
in each of the layers of the heterostructure.
(d) What is the number of two-dimensional subbands that are occupied
at this temperature? Explain this result by using the value of the
electron density determined from the program.
(e) Determine the minimum doping density that may be used to realize a
2DEG in the heterostructure. Plot the energy bands for this doping
condition and the ground state electron wave function.
2. A rectangular quantum wire with cross section 15 × 15 nm2 is realized in
GaAs.
(a) Write an expression for the subband threshold energies of the
quantum wire.
(b) Write a general expression for the electron dispersion in the quantum
wire.
(c) In a table, list the quantized energies and their degeneracies (neglect-
ing spin) for the first 15 energetically distinct subbands.
(d) Plot the density of states of the wire over an energy range that
includes the first fifteen subband energies.
3. In problem 2, we solved for the density of states in a rectangular GaAs
quantum wire with cross section 15 × 15 nm2. For this same wire: (a) Write
an expression for the electron density (per unit length) as a function of the
Fermi energy of the wire. (b) Plot the variation of the electron density as a
function of energy for a range that corresponds to filling the first five
distinct energy levels, taking proper account of level degeneracies.
4. When a voltage (Vsd) is applied across a QPC, it is typical to assume that
the quasi-Fermi level on one side of the barrier is raised by αeVsd while that
on the other drops by (1 − α)eVsd, where α is a phenomenological
parameter that, in a truly symmetrical structure, should be equal to ½.
If we consider a device in which only the lowest subband contributes to
transport, then the current flow through the QPC may be written as:
2-45
⎡ ⎤
2e ⎢
Isd = ∫ T (E )dE − ∫ T (E )dE ⎥ ,
h ⎢⎣ ⎥
⎦
L R
where T(E) is the energy-dependent transmission coefficient of the lowest

subband and L and R denote the left and right reservoirs, respectively. If we
assume low temperatures, we can treat the transmission as a step function,
T(E) = θ(E − E1), where E1 is the threshold energy for the lowest subband.
(a) Write this integral with limits appropriate to determine the current.
(b) Use this information to obtain an expression for the current flowing
through the QPC when the source–drain voltage is such that both reservoirs
populate the lowest subband, and when it populates the subband from just
the higher-energy reservoir.
5. At non-zero temperature, the conductance through a nanodevice is deter-
mined by weighting the transmission at each energy by the value of the
(negative of the) derivative of the Fermi–Dirac distribution. This leads to
the fact that the conductance takes place near the Fermi level. Compute the
derivative of the Fermi–Dirac function and evaluate the full-width at half-
maximum for this function.
6. Consider the potential profile shown in figure 2.14. let us assume that the
total potential drop across the device is 10 mV. Assume that the two
transitions can be approximated by an arctangent function with a full-width
between the 10% and 90% values of the transition of 2 nm. Determine the
dipole charge that exists at each transition, and the corresponding density
profile and value.
7. In treating the QPC with the harmonic oscillator approximation, the
different energy levels are all equally spaced at a given value of the
curvature, which is determined by the gate potential. Using the data shown
in figure 2.17, estimate the energy level spacing as a function of the gate
voltage (horizontal axis) in this device. Include at least the first nine
transitions above the first plateau.
8. Plot the potential profile given in equation (2.48) for a QPC of width
W = 100 nm and length L = 200 nm. You may assume that the 2DEG lies
180 nm below the surface and the bias on the metal gates is −5 V.
9. Consider a MOSFET with a W/L ratio of 50, a mobility of 500 cm2 Vs−1, a
gate SiO2 dielectric thickness of 2 nm, and a threshold voltage of 1 V. Plot
the ‘contact resistance’ between the source and channel as a function of the
gate voltage in the range 0 ⩽ VG ⩽ 5 V.
10. Plot the first three harmonic oscillator wave functions as a function of their
normalized spatial variable.
Appendix A Coupled quantum and Poisson problems

In many real world cases, such as the inversion layer of the Si MOSFET, the shape
of the potential depends upon the actual charge density in the inversion layer.
Moreover, the actual shape of the potential must be computed from the charge
2-46
density via Poisson’s equation. Then, this potential must be used in the Schrödinger
equation to find new values of the energy eigenvalues and the corresponding wave
functions. This loop must be iterated until convergence is obtained. Here, we will
discuss the approach first and then talk about some available software that can
handle the problem for us. To begin with, let us write down the Schrödinger
equation as
ℏ2 2
− ∇ ψ + Vψ = Eψ , (A.1)
2m
while Poisson’s equation is
ρ
∇2 V = − . (A.2)
ε
Here, V is the potential that appears at each site, ρ is the charge density, and ε is the
dielectric permittivity. The wave function ψ corresponds to the eigenenergy E. The
charge density is related to the wave function, as the latter provides the probability
that leads to the charge carrier distribution, as
ρ = −∑ ∣ψi∣2 (A.3)
i
for electrons, and the sum runs over all occupied modes at a site.
It is clear from equations (A.1)–(A.3) that one can iterate easily from wave
function to density to potential to wave function, etc and continue to convergence.
To proceed one has to adopt a gridding scheme, such as that of figure 2.21. Here, all
of the various quantities discussed above are evaluated on the grid points
(complications arise if something like the electric field, which is the gradient of the
potential is wanted, as it has to be evaluated midway between the grid points). If
some arbitrary function is known at a series of equally spaced points, one can use a
Taylor series to expand the function about these points and evaluate it in the regions
between the known points. Suppose we wish to know the solution of the second-
order differential equations above (we begin with only one dimension) in a region
between 0 and L. We can divide this into N + 1 segments of length a. Thus, there are
N interior points and the two end points of x = 0 and x = L. These end points
represent boundary values that will be imposed upon the solutions as necessary. But,
we assume that these segments are all of equal length, and each point is separated
from its neighbor by the distance a. This allows us to develop a finite-difference
scheme for the numerical evaluation of the function. If we first expand the function
in a Taylor series about the points on either side of x0, we obtain
∂f a 2 ∂ 2f
f (x 0 + a ) = f (x 0 ) + a + +…
∂x x0 2 ∂x 2 x0
(A.4)
∂f a 2 ∂ 2f
f (x 0 − a ) = f (x 0 ) − a + + ….
∂x x0 2 ∂x 2 x0
2-47
If we add these two equations together, we can write the second derivative as
∂ 2f f (x0 + a ) + f (x0 − a ) − 2f (x0)

≈ . (A.5)
∂x 2 x0
a2
Using the properties of our grid, we can write x0 as ja, as indicated in figure 2.21.
Hence, we can rewrite equation (A.5) in grid indices as
∂ 2f f j + 1 + f j − 1 − 2f j
≈ . (A.6)
∂x 2 j
a2
We can now represent the one-dimensional Poisson’s equation as

1 ⎡ ρ⎤
[S ][V ] = −⎢ ⎥ + [B ], (A.7)
a 2 ⎣ε⎦
where V, ρ/ε, and B are column matrices of length N. The matrix B contains the
boundary conditions, which are the defined voltages at sites 0 and N + 1, which
appear at the first and last elements of B, the other elements being 0. The matrix S is
a square tri-diagonal matrix of rank N, whose elements are all zero except for the
diagonal and first off-diagonal elements. The diagonal elements are all −2, while the
off-diagonal elements are 1. Inversion of the S matrix used to be difficult due to
memory and speed limitations, so approximate iterative schemes were developed for
the purpose. With modern machines, this matrix can be inverted just once, and then
solving equation (A.7) is simple matrix multiplication.
The Schrödinger equation can be similarly cast into a simple matrix form for the
one-dimensional situation. This becomes an eigenvalue equation with the form
[S ][ψ ] = 0. (A.8)
Once more, S is a tri-diagonal matrix, whose elements are given by
ℏ2 ℏ2
Sj ,j = Vj + − E , Sj ,j ± 1 = − . (A.9)
ma 2 2ma 2
The last term on the right equation is the hopping energy (2.55) and gets its name as
the energy connecting two adjacent sites. Solving equation (A.8) involves finding the
determinant of S which gives an equation whose solutions are the eigenvalues.
Generally, modern techniques diagonalize S which then leaves the eigenvalues on
the main diagonal, and also generate a transformation matrix whose columns are the
site representation for the eigen functions.
But, as mentioned above, the time consuming aspect of this is the iteration
between Poisson’s equation and the Schrödinger equation that is required to reach
convergence of the charge density and the potential consistent with the statistics of
the particular problem. At non-zero temperatures, each eigenstate is occupied
according to the Fermi–Dirac probability for its energy value and the position of
2-48
the Fermi energy. Then, each eigenstate contributes to the density according to its
spatial variation. This provides the spatial variation for the total density which
drives the Poisson equation solution. Once the potential is determined, this drives the
solution to the Schrödinger equation. And, of course, the total density determines
the position of the Fermi energy. This can be an unstable process if it is not put
together carefully. And, as remarked, several people have developed simulation
packages which solve this problem successfully. One such package, mentioned
earlier is SCHRED 2.0 [65], available at NanoHUB.org, a site supported by the
National Science Foundation as a resource for nanoscience and nanotechnology.
SCHRED solves the above two equations self-consistently for a typical MOS
structure. A typical set of input parameters is shown in figure A1. The tool assumes
that the surface normal is one of the three major coordinate axes, and the default is
for the z-direction. The three sets of valleys then have their major axes along the kx,
ky, and kz directions, with the latter the default. One difference is that the tool
numbers the energy levels as 1, 2, … instead of the 0, 1, … that is common in the
literature for silicon (and used above). One drawback of this tool is that the
Figure A1. A screen shot of the parameter panel for SCHRED 2.0, illustrating the ease of setting up the
simulation. The simulator is available at NanoHub.org.
2-49
minimum temperature for which it will simulate the system is 50 K, so it will not be
useful at low temperatures. In figure A2, we show a screen shot of the two lowest
energy wave functions for the conditions simulated in figure 1.5.
Now, when we go to two dimensions, such as for the gridding of the nanowire in
figure 2.21, we have to extend the approach to both x and y derivatives. We can
immediately extend equation (A.6) to
∂ 2f fi ,j +1 + fi ,j −1 − 2fi ,j fi +1,j + fi −1,j − 2fi ,j
≈ + . (A.10)
∂x 2 i ,j
a2 a2
This result has been found using the notation shown in figure A3. The problem is
now that we need a fourth rank tensor for the expansion matrix. To solve this, we
note that figure 2.21 can be considered to be a stack of one-dimensional wires along
the j direction, with the wires denoted by the index i. This suggests to create a two-
dimensional matrix of rank MN. This matrix will be block tri-diagonal, with each
block being the matrix S from equation (A.9) corresponding to wire i. The off-
Figure A2. Screen shot of the two lowest energy wave functions of the z-directed valleys for the simulation of
figure 1.5.
2-50
Figure A3. A five-point discretization for the Schrödinger equation uses a grid in two dimensions.
diagonal blocks with be diagonal matrices whose elements are the hopping energies
from equation (A.9). One important change is that
ℏ2 2ℏ2
→ (A.11)
ma 2 ma 2
in the diagonal blocks. This arises from the factor of 4 in equation (A.1) for the on-
site term, rather than the factor of 2 in equation (A.6); each on-site point has four
nearest neighbors now instead of just two nearest neighbors in the one-dimensional
case.
Appendix B The harmonic oscillator

The Hamiltonian that results from the potential of equation (2.4) is doubly
quadratic; that is, it contains one term quadratic in the momentum operator, arising
from the kinetic energy term, and one term quadratic in the position, from the
quadratic position in the potential. We will reduce the Hamiltonian by introducing a
set of operators that combine position and momentum in a way that leads to a
simpler approach to the problem. We suppose that a proper combination of the
operators will provide a set of new operators that correspond to movement through
the ladder of energy levels (which are yet to be found). We begin by writing the total
Hamiltonian for the time-independent Schrödinger equation as
ℏ2 ∂ 2 m⁎ω 0,2yy 2 p2 m⁎ω 0,2yy 2

H=− + = + . (B.1)
2m⁎ ∂y 2 2 2m⁎ 2
To proceed, let us introduce two complex operators
m⁎ω0,y ⎛ p ⎞
a= ⎜y + i ⁎ ⎟
2ℏ ⎝ m ω0,y ⎠
(B.2)
m⁎ω0,y ⎛ p ⎞
†
a = ⎜y − i ⁎ ⎟.
2ℏ ⎝ m ω0,y ⎠
2-51
With these operators, we find that the Hamiltonian (B.1) can be rewritten as
⎛ 1⎞
H = ℏω0,y⎜a†a + ⎟ . (B.3)
⎝ 2⎠
The momentum and position operators satisfy a basic commutator relationship in
quantum mechanics, since they do not commute with each other. The new operators
in equation (B.2) also will satisfy such a commutator relationship because they do
not commute with one another either. Let us consider the two possible products of
these operators:
1 ⎛ p2 m⁎ω 0,2yy 2 ⎞ 1
†
a a= ⎜ + ⎟⎟ −
ℏω0,y ⎜⎝ 2m⁎ 2 ⎠ 2
(B.4)
1 ⎛ p2 m⁎ω 0,2yy 2 ⎞ 1
aa† = ⎜⎜ + ⎟⎟ + ,
ℏω0,y ⎝ 2m⁎ 2 ⎠ 2
so that
[a , a† ] = aa† − a†a = 1. (B.5)
It is also observed here that the operator products themselves are not operators, but
are what are called c-numbers, or simple numbers, which may be complex in some
cases. This will be important in the following operations by which we determine the
properties of these operators.
Since the product a†a is a c-number, multiplying a function by it simply scales the
amplitude of that function. Hence, if we multiply the nth wave function by it, we can
write the corresponding eigenvalue equation
a†aψn = λnψn, (B.6)
where λn is the eigenvalue, and we must still determine the value for this quantity. It
is easy to show that this value must be a positive quantity. Let us multiply by the
adjoint of this wave function and integrate, while using the properties of adjoint
operators, to yield
∫ ψ †a†aψdy = ∫ (aψ )aψdy ⩾ 0. (B.7)
The adjoint operator is moved to the conjugate term due to the properties of adjoint,
or Hermitian conjugate, operators.
Let us take one of the eigen functions and operate on it with the two operators,
and then apply equation (B.5). This leads to
a†a(a†ψn) = a†(aa† + 1)ψn = (λn + 1)a†ψn
(B.8)
a†a(aψn) = (aa† + 1)aψn = (λn + 1)aψn.
2-52
Thus, we see that operation upon an eigenfunction by the adjoint operator produces
an eigenvalue equation in which the eigenvalue has been increased, or decreased, by
unity. The last form gives us some insight, as there must always be a lowest
eigenvalue and its corresponding eigen function. If we apply the operator a to this
lowest eigen function, we find
aψ0 = 0, (B.9)
since there can be no wave function below the lowest one. Thus, comparison of
equations (B.8) and (B.9) tells us that λ0 = 0, and
λn = n. (B.10)
We can now rewrite (B.3) as
⎛ 1⎞
H = ℏω0,y⎜n + ⎟ . (B.11)
⎝ 2⎠
The conclusions we can draw from the above arguments is that the adjoint operator
a† raises the energy of the harmonic oscillator by one unit of energy ℏω0,y and alters
the wave function to that of the next level up. Similarly, the operator a lowers the
energy of the harmonic oscillator by one unit of energy ℏω0,y and alters the wave
function to that of the next level down. For this reason, the two operators are termed
the raising and lowering operators. With their action, the harmonic operator absorbs
or emits a quantum of energy, respectively.
We can now use equation (B.9) to find the wave function for the ground state. If
we expand the operator in equation (B.9), we obtain
m⁎ω0,y ⎛ p ⎞
aψ0 = ⎜y + i ⁎ ⎟ψ = 0, (B.12)
2ℏ ⎝ m ω0,y ⎠ 0
which leads immediately to

⎛ m⁎ω 0,2yy 2 ⎞
ψ0 = C 0exp⎜⎜ − ⎟. (B.13)
⎝ 2ℏ ⎟⎠
This may be recognized as the lowest order Hermite polynomial with its exponential
weighting function. The constant C0 can be determined by standard normalization
to be
⎛ m⁎ω0,y ⎞1/4
C0 = ⎜ ⎟ . (B.14)
⎝ πℏ ⎠
We can now determine any of the wave functions at the various levels simply by
repeated application of the raising operator, as
ψn = (a†)nψ0. (B.15)
2-53
We now turn to the expectation values for the operators themselves, which will
also give us the full normalization of the various eigenstate wave functions. The
chain generation in equation (B.15) does not automatically guarantee proper
normalization of the wave functions, so that we have to find the correct normal-
ization for all higher lying states. From the basic properties expressed in equations
(B.5) and (B.6), we know that for unnormalized wave functions such that
∫ ψn†ψndy = Cn2, (B.16)
we can write (we take the coefficients as real)
∫ ψn†a†ψkdy = ∫ ψn†ψk+1dy = αn2δn,k+1C n2

(B.17)
∫ ψn†aψkdy = ∫ ψn†ψk−1dy = βn2δn,k−1C n2.
We still have to determine the constants αn and βn, which in fact are related to the
normalization of the eigen functions (the two eigen functions in the above expect-
ation values have different normalizations). For this, we use
∫ (aψn)(aψn)dy = ∫ ψna†aψndy = nCn2 = Cn2−1. (B.18)
This now leads us to be able to say

Cn −1 = n C n. (B.19)
We can now use this property to give the following results
∫ ψka†ψndy = ∫ ψkψn+1dy = n + 1 δk,n+1

(B.20)
∫ ψkaψkdy = ∫ ψkψn−1dy = n δk,n−1.
We can now finally write the exact normalization for the general wave function as
1
ψn = (a†)nψ0. (B.21)
(n + 1)!
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2-56
IOP Publishing

(Second Edition)
David K Ferry
Chapter 3
The Aharonov–Bohm effect
One of the earliest observations of quantum effects in nanostructures, or mesoscopic

devices, was the experimental observation of phase interference in the conductance
through metallic [1] or etched semiconductor rings [2, 3]. In these rings, electron
waves pass from an input port, around the two halves of the ring and interfere at the
output port, presumably due to the Aharonov–Bohm (AB) effect [4]. Interference
between differing waves can occur over distances on the order of the coherence
length of the carrier wave, and the latter distance is generally different from the
inelastic mean free path for quasi-ballistic carriers (those with weak scattering). The
inelastic mean free path is related to the energy-relaxation length le = vτe, where τe is
the energy-relaxation time and v is a characteristic velocity (typically the Fermi
velocity in degenerate material and the thermal velocity in non-degenerate material).
The inelastic mean free path can be quite long, on the order of several tens of
microns for electrons at low temperatures in the inversion channel of a high-
electron-mobility transistor in GaAs/A1GaAs. On the other hand, the coherence
length is often defined for weakly disordered systems by the diffusion constant (as we
discuss below). But, this can be misleading as it has meaning in materials with very
little disorder, where it is often determined by the relevant electron–electron
scattering mean free path. At first thought, one is usually aware that two interacting
electrons do not dissipate either energy or momentum, as both are conserved.
However, this does not mean that each of the two electrons cannot exchange energy
or momentum, and it is this change that breaks up the phase coherence. Hence,
electron–electron scattering is a major source of phase decoherence.
We can illustrate the idea of phase interference by considering two waves (or one
single wave which is split into two parts which propagate over different paths) given
by the general form ψi = Ai e iφi . When the two waves are combined, at some point
different from the origin, the probability amplitude varies as

P = ∣ψ1 + ψ2∣2 = ∣A1∣2 + ∣A2 ∣2 + 4A1*A2 cos(φ2 − φ1). (3.1)
The probability can therefore range from the sum of the two amplitudes to the
differences of the two amplitudes, depending on how the phases of the two waves are
related. In most cases, it is not important to retain any information about the phase
in device problems because the coherence length is much smaller than any device
length scale and because ensemble averaging averages over the phase interference
factor so that it smooths completely away in macroscopic effects. This ensemble
averaging requires that a large number of such small phase coherent regions are
combined stochastically. In small structures this does not occur, and many observed
quantum interference effects are direct results of the lack of ensemble averaging [5].
In this chapter, we start with a discussion of the AB effect, and discuss its
observation in some various systems. We will then discuss the role of the phase-
breaking length and its determination. Following this, we discuss a related effect in
which interference effects are seen via another process which produces a different
dependence upon the magnetic field. In appendix C, we discuss the role of gauge in
mesoscopic systems, and those not familiar with the vagaries of this quantity might
start with the appendix before proceeding to the following section.
3.1 Simple gauge theory of the AB effect

A particularly remarkable illustration of the importance of the phase is the magnetic
AB effect [4]. It was almost immediately verified using a biprism (which causes the
electrons to take two different paths) in a transmission electron microscope (TEM) [6].
Yet, in spite of this first experiment and the familiarity of the effect in today’s
semiconductor world, there were strong debates about whether the effect was even
real or not for nearly the first quarter of a century after the original paper. The
debate was largely settled when the first observation of it was made in electron
holography (electrons within a TEM) [7]. A few years later, the effect was observed
in small metal rings [1, 8], and in semiconductor etched rings [2, 9]. From the
beginning, though, it was inferred that the AB effect was a pure quantum
phenomenon, in which it appears through interference caused by varying the phase
of the wave function. However, it is more proper to think of it as a general wave
mechanical effect which has its analog in classical wave mechanics [10]. For
example, the effect has been studied in water surface waves [11, 12], and via light
scattering from a hydrodynamic ‘vortex’ [13]. We consider it a quantum phenom-
enon in semiconductors solely because we are treating our electrons as quantum
mechanical waves. But, our treatment in this section is in terms of the fields and the
gauge, and is not particularly limited to quantum mechanics.
The basic structure of the experiment is illustrated in figure 3.1. The semi-
conductor structure is formed on an AlGaAs/GaAs heterostructure. A Q1D
conducting channel is fabricated on the surface of a semiconductor by using
electron-beam lithography to deposit a NiCr pattern by liftoff, and then using this
pattern as a mask for reactive-ion etching away parts of the heterostructure to leave
electrons in the ring structure [3]. The waveguide is sufficiently small so that only one
3-2
Figure 3.1. (a) Micrograph of a conducting semiconductor ring etched into an AlGaAs/GaAs heterostructure.
(b) The magnetoresistance and Hall resistance measured for the ring of panel (a). The inset shows the Fourier
transform of the resistance. Reprinted with permission from [3]. Copyright 1988 American Vacuum Society.
or a few electron modes are possible. The incident electrons, from the left of the ring
in figure 3.1(a), have their wave split at the entrance to the ring. The waves
propagate around the two halves of the ring to recombine (and interfere) at the exit
port. The overall transmission through the structure, from the left electrodes to the
right electrodes, depends upon the relative size of the ring circumference in
comparison to the electron wavelength. If the size of the ring is small compared
to the inelastic mean free path, the transmission depends on the phase of the two
fractional paths. In the AB effect, a magnetic field is passed through the annulus of
the ring, and this magnetic field will modulate the phase interference at the exit port.
There are two types of measurements, which are labeled Rxx and Rxy in figure 3.1(b).
3-3
upper
lower
Figure 3.2. A symbolic representation of the two paths electrons can take through the ring.
The Rxx measurement is made directly across the ring and is a direct measure of the
resistance of the ring as measured by the voltage drop across it [3]. The other
measurement is a transverse measurement, which in some sense is a nonlocal one as
it does not measure the voltage drop across the ring, but the effect the ring voltage
has on the rest of the circuit.
We understand the measured behavior from the assumption that the magnetic
field passes vertically through the ring. The vector potential for a magnetic field
passing through the annulus of the ring is azimuthal, so that electrons passing
through either side of the ring will travel either parallel or anti-parallel to the vector
potential, and this difference produces the phase modulation, as indicated by the
ring in figure 3.2. The vector potential will be considered to be directed counter-
clockwise around the ring. (We adopt cylindrical coordinates, with the magnetic
field directed in the z-direction and the vector potential in the θ-direction.) The phase
of the electron in the presence of the vector potential is given by the Peierls’
substitution (see appendix C), in which the normal momentum vector k is replaced
by (p − eA)/ℏ and
1
ϕ = ϕ0 + (p − eA) · r , (3.2)
ℏ
so that the exit phases for the upper and lower arms of the ring can be expressed as
0
⎛ e ⎞⎟
ϕ = ϕ0 + ∫ ⎜k +
⎝
A · a ϑrdφ
ℏ ⎠
π
0
, (3.3)
⎛ e ⎞⎟
ϕ = ϕ0 − ∫ ⎜k −
⎝
A · a ϑrdφ
ℏ ⎠
π
and the net phase difference is just

2π
e e
δϕ =
ℏ
∫ A · a ϑrdφ =
ℏ
∫ B · azdS = 2π ΦΦ0 , (3.4)
0
3-4
where Φ0 = h/e is the quantum unit of flux and Φ is the flux enclosed in the ring. The
phase interference term in equation (3.4) goes through a complete oscillation each
time the magnetic field is increased by one flux quantum unit. This produces a
modulation in the conductance (resistance) that is periodic in the magnetic field,
with a period h/eS, where S is the area of the ring. This periodic oscillation is the AB
effect, and in figure 3.1(b) results are shown for such a semiconductor structure. One
can see that exactly this ‘frequency’ appears in the Fourier transform shown as the
inset to this figure. More interestingly, the transverse resistance, which is not
measured across the ring also shows such an oscillation. This appears to be a
nonlocal effect, but is a reflection of the fact that the voltage oscillations across the
ring directly affect the rest of the circuit.
Now, we can use these results to analyze the experimental data shown in
figure 3.1. The peak of the Fourier transform occurs roughly at 730 T−1, which
corresponds to a radius of about 1.0 μm, which is about that shown in figure 3.1(a).
We can go a little further, however, and note that the full-width of the Fourier peak
is about 100 T−1, from which we can estimate the effective width of the nanowire
forming the ring. Unless the nanowire is a single-mode structure, then one can
conceive of orbits going around the inner diameter as well as the outer diameter, and
anywhere in between. Thus, the width of the Fourier peak is a form of measure of
this spread of orbit areas. The range in the Fourier transform thus suggests a width
of the nanowire of about 70 nm, which is a bit larger than one might guess from the
length scale in the figure. But, we need to remember that the electrical width, which
we are calculating, can be different from the physical width due to edge depletion
and other effects. The authors themselves state that the electrical widths vary from
60 nm to about three times higher, depending upon the carrier density [3]. The value
found here is at the lower end of that range.
We note that the AB effect is a single-electron effect. But, it was recognized early
on that this experiment is a variation on the famous two-slit experiment in quantum
mechanics. In this regard, the upper branch and the lower branch form the two slits
through which the electron can pass. How can one electron decide which path to
take? This is the famous problem of quantum mechanics, and is resolved by
requiring the electron to be a wave which can flow simultaneously through both
slits, or branches. Only then do we obtain the interference at the output of the ring.
This understanding was tested by the experiments in TEMs, where a biprism can be
used to create the two paths for the electrons [6, 7]. As remarked, however, there was
a large debate over whether or not this was a real effect. The observation in
condensed matter came later, as most people thought that the scattering that occurs
in this material would break up the phase coherence needed for the effect. Yet, it can
be found in both metals and semiconductors, which suggests that the phase
coherence time can be much longer than the scattering time. One important
conclusion to be drawn is that the electron phase is not destroyed by elastic
scattering events, as these basically leave the phase undamaged. So, impurity
scattering, for example, is not a major source of phase-breaking collisions.
Electrons which traverse the ring can undergo tens to thousands of collisions due
to impurities, but still clearly show the oscillatory behavior.
3-5
3.2 Temperature dependence of the AB effect

An important feature of the AB effect in many structures, particularly metal rings, is
that they wash out fairly rapidly with increasing temperature. Since there is no
temperature dependence in the basic effect, as shown in equation (3.4), this decay
with increasing temperature must arise from the role of scattering and the
decoherence that it introduces. In particular, the scattering must break up the phase
coherence, so we are concerned with the so-called phase breaking, or phase
coherence, time τφ. Naturally, this is going to be related to a phase coherence
length, which is the distance that the particle travels before the phase is randomized
by the inelastic collisions. These events tend to give the amplitude A of the AB
oscillations a dependence upon this length via a common formula
⎛ L⎞
A(L ) = A0 exp⎜ − ⎟ , (3.5)
⎝ lφ ⎠
where L is the path length of the electrons, or essentially one-half the circumference
of the ring, and lφ is the phase coherence length. In figure 3.3, we show a typical ring
fabricated in an InGaAs/InAlAs heterostructure [14]. The In0.64Ga0.36As quantum
well was slightly strained and was 10 nm thick and located 50 nm from the surface.
The darker areas in the image, delimited by the white edges, are etched trenches
which define the ring structure (lighter gray areas). The electrons are thus forced to
move around the ring between the top contacts and the bottom contacts. In
figure 3.4, typical data obtained are shown, both in the raw form and in the
processed form, in which an oscillatory background has been removed. The Fourier
transform is shown in panel (c), where the clear h/e signal is the strongest component
of the data. Finally, in figure 3.5, the amplitude of the AB signal is shown as a
function of the lattice temperature. Note that the amplitude is plotted on a
logarithmic scale, so the linear decay of the amplitude clearly satisfies the
exponential form (3.5), provided that the coherence length varies linearly with the
temperature, a point we return to below. In the last figure, two different sets of data
are shown, which arise from different cool-downs of the sample in the cryogenic
Figure 3.3. An SEM micrograph of a representative AB ring in an InGaAs/InAlAs heterostructure, with a

schematic of the measurement set up. The darker areas, outlined by the white edges, are etched trenches by
which the ring is formed. The current flows from the top to the bottom as indicated by the source on the left.
(Reprinted with permission from [14]. Copyright 2013 IOP Publishing).
3-6
Figure 3.4. AB resistance oscillations around the ring of figure 3.3. (a) The raw data. (b) Data after removing
the oscillatory background. (c) The Fourier transform of the data. Reprinted with permission from [14],
copyright 2013 by IOP Publishing.
Figure 3.5. The temperature dependence of the amplitude of the AB oscillation, for two different cool-downs
of the same sample. Reprinted with permission from [14], copyright 2013 by IOP Publishing.
refrigerator. Note that the amplitude varied by almost a factor of two between these
two runs. This is not unusual, as warming the sample up and then re-cooling it
corresponds to an annealing cycle. During the warm-up, various impurities can
move around, and affect the details of the sample. This is perhaps one significant
point about mesoscopic experiments: each cool-down presents a different sample
with a different set of characteristics.
Since the transport through the ring is normal scattering limited behavior in a
near equilibrium situation, the mobility and the diffusion constant are related. This
3-7
transport is different from the ballistic or quasi-ballistic only in that many scattering
events occur. While the actual transport may well be quasi-ballistic in nature, the
phase-breaking events depend upon the scattering that occurs. Hence, the mobility,
or the diffusion constant, can be used to define a characteristic phase-breaking
length from the inelastic phase-breaking time, via
lφ = Dτφ , (3.6)
where D is the diffusivity of the semiconductor material. One often thinks about this
type of transport as diffusive transport, although the majority carriers are the
important quantity and their motion is mainly determined by the mobility. But, in
equilibrium, the mobility and diffusivity are connected by the Einstein relation,
although one needs to use a slightly modified form for the degenerate materials. The
modification is a factor that is the ratio of two Fermi integrals, but the exact
integrals are determined by the dimensionality of the sample.
In order to see interference effects, it is necessary that the phase coherence length
be comparable to, or larger than, the size of the device under study, as can clearly be
seen from the form of equation (3.5). It is for this reason that nearly all mesoscopic
experiments are carried out at low temperature, where the scattering is much weaker
than at room temperature. At non-zero temperatures, an additional source of
dephasing arises from the fact that a range of energies, near the Fermi energy, are
involved in the transport. As discussed in the previous chapter, the width of energies
involved is of order kBT. We can therefore define a thermal broadening time by using
the broadening that scattering induced and relating it to the spread in energy as
kBTτT ℏ
∼ 1 → τT ≈ . (3.7)
ℏ kBT
Now, we can define a thermal diffusion length in analogy to the phase coherence
length (3.6) with this new time, as
ℏD
LT = DτT = . (3.8)
kBT
This last contribution to decoherence is an unavoidable result of operating at a non-
zero temperature, as this decoherence effect would exist even if it were possible to
eliminate all the inelastic scattering processes.
The temperature decay of equation (3.5) is said to be a dynamic dephasing, as it is
caused by the scattering processes. On the other hand, the thermal broadening is
thought of as a static dephasing, and contributes differently to the temperature
dependence. In this latter case, one considers each of the different energies as being
uncorrelated with one another, so we have to simply consider the number of such
energies. The important aspect is when the thermal length is comparable to the
sample size L. Thus, we expect that the amplitude will show an additional reduction
which varies as (LT/L)1/2. When this is added to equation (3.5), we get the final
temperature dependence to vary as
3-8
ℏD ⎛ L⎞
ΔG ∼ A0 exp ⎜− ⎟. (3.9)
L2kBT ⎝ lφ ⎠
When dynamic dephasing can be ignored (lφ ≫lT), then we expect the prefactor to
dominate and lead to a temperature which decays as 1/T 1/2. When the reverse is true,
then the exponential term is dominant.
3.3 The AB effect in other structures

In the above two sections, we have pretty much concentrated on standard quantum
rings in which the AB effect is the dominant effect in the conductance. But, we
remember that the effect is basically quite similar to the double-slit experiment in
quantum mechanics, as the passage of the quantum wave through the two slits
actually creates a quantum ring topology. Hence, one can conceive of many different
structures in which quantum interference can be created with the result that one can
see the AB effect in the system response. For example, it has been shown that two
particle correlations can also show the AB effect. This has been shown theoretically
[15] and experimentally [16, 17]. In this last experiment, an electronic analog to the
optical correlation experiment of Brown and Twiss [18] was created. Brown and
Twiss created a new type of radio interferometer, in which the signal was based upon
the correlation between two independent receivers spaced out along a common
baseline for reference. In the experiment, they looked for correlations in the intensity
fluctuations that each receiver measured. Oberholzer et al [17] created an electronic
equivalent to this experiment for a beam of electrons in a Q2DEG. In this latter
experiment, a tunable metallic beam splitter was used to partition the electron beam
into transmitted and reflected partial beams, and the current fluctuations in these
two partial beams were measured. They found that these fluctuations were fully anti-
correlated, which demonstrated that fermions tend to exclude one another and lead
to anti-bunching. In subsequent work, it was shown that almost any electronic
interferometer would exhibit the two-particle effect [19].
As we will see in a later chapter, electrons are confined near the edges of a
rectangular sample when a large magnetic field is applied normal to the sample. This
is a result of the quantum Hall effect. The motion of the electrons for certain
magnetic field is confined to edge states which are channels located at the edges of
the sample. We can create an anti-dot within the sample, which is a local region of
high potential which excludes electrons from the particular local region. Then, the
edge states can be localized around the anti-dot as well, and if tunnel coupling
between the various edge states occurs, one will observe the AB effect in this
geometry as well [20, 21]. More interestingly, if one puts a quantum dot in the center
of the sample in such a manner that most of the edge states cannot propagate
through the quantum point contacts (QPCs) which provide the openings to the dot, as
shown in figure 3.6(a), then there will be trapped edge states within the dot [22]. As these
trapped states essentially form AB loops, their occupation will depend upon the specific
magnetic field, and this will lead to observable oscillations in the transmission through
the dot. A typical conductance sweep in a magnetic field is shown in figure 3.6(b).
3-9
Figure 3.6. (a) Schematic diagram of the edge-state configuration at 4 T for a dot located in a quantum Hall
bar. (b) The magnetoresistance trace reveals AB oscillations. (Reprinted with permission from [22]. Copyright
More recently, the AB effect has been studied in rings formed in InGaAs [23],
where the connection with the Berry phase was more extensively studied. The AB
effect in biexcitons propagating around a GaAs/AlGaAs quantum ring have also
been studied optically [24]. The existence of AB oscillations has also been reported in
graphene p–n junctions [25]. Perhaps more interesting is the observation of AB
oscillation in gate-all-around topological insulator nanowires of Bi2Se3 [26] and
Bi2Te3 [27].
3.4 Gated AB rings

In the general AB rings discussed above, the magnetic field was tuned to provide the
interference and oscillations. But, we note from equation (3.3) that the propagation
through each arm of the ring depends upon the wave vector in that arm. This
suggests that one could use a local gate to tune the wave vector, which would also
lead to oscillatory wave interference as the wave vector is varied. Almost as soon as
the effect was observed in semiconductors, this tuning idea was patented [28]. The
use of a metallic gate on one arm of an electronic double-slit experiment was
demonstrated shortly afterward [29].
We can illustrate how this occurs with a simple description. Consider figure 3.7, in
which we have modified the earlier figure to include a gate over part of one arm of
the ring. Let us assume that the ring is created in a GaAs/AlGaAs heterostructure, in
which the reservoirs are a 2DEG with a density of 4 × 1011 cm−2. This corresponds to
a Fermi energy of about 14.3 meV, and a Fermi wave vector of about 1.6 × 106 cm−1.
We assume that the wire that composes the ring is about 50 nm wide electrically, so
this produces only about 2.3 meV quantization energy in the ring, which lowers the
Fermi momentum in the ring to about 1.5 × 106 cm−1. Now, let us assume that the
ring is 1 μm in diameter. Thus, in the absence of a magnetic field, an electron in each
half of the ring (without gate voltage applied) accumulates about 236 radians of
phase (or about 75π). Of course, the electrons emerge at the exit slit in phase with
each other. Now, let us ask just how much do we need to reduce the density in order
3-10
upper
Gate
lower
Figure 3.7. We have gated one arm of the ring in order to use the gate to control the wave vector and cause
interference.
to shift the phase by π in one arm so that there is destructive interference at the
output port. Let us assume that the angular spread of the gate covers about π/3 of the
ring (about 60°, or one-third of the arm). Then, the electrical length of the gate is
πr/3, where r is 0.5 μm. Hence, we need
2π
Δk · = π → Δk = 6 × 106 m−1 (3.10)
3
which is a very small reduction in the wave number (about 4%). So, we should be able to
obtain an entire series of oscillations for a modest variation in the gate voltage. Of
course, we have assumed ideal conditions here, and the width of the ring as well as
scattering will degrade the signal substantially, just as for the magnetic tuning of the
rings. In actual fact, there has been little work on studying the role of gating on such
rings, although several theoretical analyses have been performed [30–32]. Such experi-
ments as there are have not shown strong voltage-dependent oscillations [29, 33].
Typical of the experiments are those of Krafft et al [33], for which the device is
shown in figure 3.8(a). Here the ring is fabricated in an AlGaAs/GaAs hetero-
structure, with the Q2DEG located about 60 nm below the surface. Measurements
show that the electron density is about 3.7 × 1011 cm−2 at the 50 mK temperature of
the dilution refrigerator. What can be seen in the figure is that the ring is defined
through the use of two gates, A and B, which are isolated from the ring itself via in-
plane trenches which are etched into the heterostructure with wet chemical etching.
The transport measurements were performed with four terminal connections, and
the various leads can be seen in the figure. Typically, current was passed between
terminals 1 and 4, while terminals 2 and 3 were used to measure the voltage
(resistance, as a constant current of 10 nA was passed through the system). The ring
structure shows clear, if weak, oscillations as a function of the applied magnetic
field, as shown in figure 3.8(b). From the inset to this latter figure, one can see that
there is a strong single peak in the Fourier transform of the oscillations, with the
peak corresponding to a period of about 3.1 mT, which gives a ring radius of
650 nm. This corresponds fairly well with the length key in figure 3.8(a), although
the wide width of the ring compared to its radius leads to significant broadening of
3-11
Figure 3.8. (a) Electron-beam micrograph of the ring structure and the in-plane gates A and B. (b) Resistance
of the ring structure of (a) as a function of the magnetic field, with the voltage applied to gate B varied as a
parameter. The inset shows the Fourier transform for zero applied gate voltage. (Reprinted with permission
from [33]. Copyright 2001 Elsevier.).
the Fourier peak, for reasons discussed in section 3.1. In figure 3.8, the voltage
applied to gate B is varied as a parameter. As this voltage is raised to a positive
value, there are observable changes in the various curves, which can be interpreted
as a phase shift of the h/e oscillation. At 0.2 V, the peaks at B = 0 are diminished,
while those near ± 1.5 mT increase in amplitude, and the authors suggest this arises
from a phase shift of π [33]. It is possible to see a gradual shift in a number of the
peaks as the voltage is raised, which supports the idea of the tuning of the oscillation
phase by changing the wave momentum in the arm with the biased gate. But, the
gate tuning effects do not produce as striking a resistance oscillation as that due to
the magnetic field, and this seems to be generally true in such measurements.
More recently, the interaction between the AB effect in a gated ring and the
Coulomb blockage (discussed in chapter 8) has been discussed [34]. Of more interest
is the use of a scanning gate microscope (as discussed in chapter 2) to provide the
3-12
gate bias on the AB ring [35]. The use of the scanning gate allows one to taylor the
potential landscape and observe interference fringes in the transport through the ring
as a function of voltage and magnetic field. As a result, one can create a ring with a
specific number of modes in each arm of the AB ring. Studies of the AB effect in
semiconductors still appear regularly [36].
3.5 The electrostatic AB effect

The magnetic AB effect is often called the vector AB effect due to it arising from the
vector potential, as in equation (3.3). There is another AB effect, which is often
called the electrostatic, or scalar, AB effect. In the scalar AB effect, the interference
between two propagating electrons, or electron waves, is modulated by the scalar
potential generated by an electric field that does not itself exist in the regions where
the electrons propagate. While the electric field does not exist in the region where the
waves are propagating, the scalar potential does, which is in complete analogy to the
vector form [4]. It is important to point out that this scalar form is not the same as
the gate modulation we discussed in the last section. In the last section, the electric
field was in the region in which the waves were propagating and merely modulated
the properties of the waves rather than the phase directly. The electrostatic AB effect
has become of interest for semiconductors, since it has been suggested as a source for
novel device concepts [37].
In the scalar AB effect, the presence of the scalar potential adds to the energy of
the particle or wave, and thus can contribute to a phase shift as
−eVt eVL eVm*L
Δφ = ∼ =− 2 , (3.11)
ℏ ℏvF ℏ kF
where V is the voltage and L is an effective length. If the electrons, or waves, are
confined in a one-dimensional wire, the density and the Fermi wave vector are
related by kF = πn/2. We can use this in equation (3.11) to give the phase shift as a
function of the applied voltage, as
2eVam*L
Δφ = − . (3.12)
ℏ2nπ
If we assume GaAs with a wire density of 5 × 107 cm−1, and a length of 2 μm, then
we would find a phase shift of about 220 rad V−1, which is significant.
But, reality must be called into play here. According to a recent article, the scalar
AB effect has never been experimentally confirmed [38]. As we pointed out in the
introduction to this chapter, there was considerable debate about the reality of the
AB effect when it was first put forward, and this debate lasted for almost a quarter of
a century. Walstad [39] has re-examined in particular the electrostatic AB effect. He
concludes that the electrostatic AB effect rests upon a theoretical error on the part of
the original authors, and that this effect just does not exist, and claims of such are
often mistaken observations of the magnetic AB effect [40]. So, apparently, the
debate has not died down, and there is still significant doubt about such an effect.
3-13
Nevertheless, there are still scientists that give interesting predictions for observa-
tions of the electrostatic AB effect [41–43].
3.6 The AAS effect

Most of the treatment above has dealt with semiconducting rings which are nearly
ballistic in nature. But, there is another effect which can occur in such rings when
they are heavily disordered. In the normal AB effect, the waves go around each side
of the ring and interfere at the output port, giving an oscillatory transmission. In a
strongly disordered ring, it is possible for the two trajectories to deflect past the
output port and continue on around the ring, where they eventually interfere at the
input port, giving an oscillatory reflection signal whose periodicity is h/2e, since the
two trajectories enclose twice as much area. The effect is known as the AAS effect
for the three who developed the theory—Altshuler, Aronov, and Spivak [44]. They
developed the theory based upon dirty metals; e.g., Fermi liquids with very high
electron densities, but it has been seen in other systems as well. Shortly after the
theoretical work, the AAS oscillations were seen, apparently for the first time, in a
Mg cylinder, in which the current flowed from one side to the other [45], although it
was quickly measured in gold and copper rings [46].
Interestingly enough, in small gold rings, both the AB oscillations and the AAS
oscillations can be seen simultaneously [47]. In these rings, the h/2e oscillations are
typically seen near zero magnetic field, while the h/e oscillations appear at a higher
magnetic field. This arises because the AAS oscillations are strongly damped at high
magnetic fields. We noted above that the two trajectories both went completely
around the ring, making a full circuit. In this situation, these two trajectories can be
time-reversed paths, as they are mirror images (across the center line of the ring). In
that sense, the AAS effect is really the continued interference between these two
time-reversed paths as they interact everywhere around the ring. The magnetic field,
however, breaks time-reversal symmetry, and so one expects that these oscillations
will be heavily damped as the amplitude of the magnetic field increases. Hence, at
higher magnetic field, only the AB oscillations remain. This allows one to distinguish
between the AAS effect and any second harmonic signal in the Fourier transform of
the normal AB effect.
It has also been shown that the AAS effect can also be seen in diffusive systems
such as semiconductors [48, 49]. It has been seen in an AlGaAs/GaAs hetero-
structure in which InAs quantum dots were embedded near the interface to provide
correlated disorder in the system [50]. More recently, however, both oscillations
have been seen in InAs/AlGaSb and InSb/InAlSb heterostructures [51]. Similarly,
both types of oscillation have been seen in heterostructures of AlGaN/GaN [52].
It is instructive to look a little closer at a typical good quality semiconductor
heterostructure. In the work of Lillianfeld et al [51], a 15 nm InAs quantum well was
located under a 20 nm Al0.2Ga0.8Sb cap layer and is unintentionally doped. At low
temperature, the electron density was found to be about 8.5 × 1011 cm−2. The sample
composed a 7 × 7 array of rings whose average radius was 350 nm and had a ring
width of 130 nm. Measurements were made at temperatures in the range 0.4–10 K.
3-14
Figure 3.9. (a) The magnetoresistance of the InAs/AlGaSb rings at 0.4 K. The higher frequency oscillations at
low B are the AAS oscillations with period h/2e. At higher values of B, the AB oscillations dominate for ∣B∣ >
50 mT. (b) The Fourier transform of the data for the range ∣B∣ < 100 mT, so that the AAS period is the strong
signal at about 0.2 mT. Digital filtering of the signal isolates the (c) h/e and (d) h/2e components of (a).
(Reprinted with permission from [51]. Copyright 2010 Elsevier).
As previously, both types of oscillations were found to occur, with the AAS
oscillations existing at low values of the magnetic field and going away at higher
magnetic field. Typical data are shown in figure 3.9 for the InAs rings. Digital
filtering of the magnetoresistance signal allows one to separate the AAS component
from the normal AB component, with the former lying mainly below ∣B∣ < 50 mT.
These two different components are shown in panels (c) and (d) of the figure. It may
clearly be seen that the AAS signal is heavily damped above a few tens of mT, and
that the AB oscillations dominate over the larger magnetic field range.
3.7 Weak localization

In the previous section, the AAS effect arose from two trajectories moving around a
ring in opposite directions such that they interfered with one another all around the
ring. This resulted in an enhanced back-scattering which was oscillatory in the
magnetic field for small magnetic fields. Now, this effect does not require the ring.
Disorder can lead to a set of localized scatterers which produce exactly the same
back-scattering. In the bulk, this process is known as weak localization. The idea is
illustrated in figure 3.10, where we show a set of scatterers and two time-reversed
trajectories in which the particles scatter from each site and eventually return to their
original direction as a back-scattering. Since we cannot know which direction the
particle scatters around the ‘ring’, both directions are possible and represent
the time-reversed paths. The presence of weak localization results in a reduction
in the actual conductivity of the sample, that is, weak localization is a negative
3-15
Figure 3.10. The red and blue paths are the time-reversed set of trajectories which scatter from the set of
scatterers shown here. They interfere along their entire path, producing weak localization from the net back
scattering.
contribution to the conductivity due to the back scattering effect. This is usually a
small but significant effect if there is sufficient disorder in the system.
The weak localization correction is damped by the application of a magnetic field.
This follows as the magnetic field breaks time-reversal symmetry, and causes the two
reverse paths to diverge, hence reducing the quantum interference. This divergence
arises from the Lorentz force, which produces a force normal to both the magnetic
field and the local velocity of the particle. Since the two paths have opposite
velocities, the Lorentz force on the two will have opposite directions, hence causing
the paths to diverge and breaking up the interference effect. We will see below that a
critical magnetic field can be defined as that at which the weak localization
correction has been reduced to one-half of its peak value, and this is called the
correlation magnetic field. It will be argued that this value of critical magnetic field is
just enough to enclose one flux quanta (h/e) within the area of the phase-coherent
loop.
3.7.1 A semiclassical approach to the conductance change

The conductivity can be generally related to the current–current correlation
function, a result that arises from the Kubo formula, but it also is dependent
upon an approach involving the retarded Langevin equations [53]. In general, the
conductivity can be expressed simply as
∞ ∞
ne 2 ne 2 ne 2τ
σ=
m* v 2(0)
∫ v(t )v(0) dt ∼
m*
∫ e−t /τdt =
m*
, (3.13)
0 0
where it has been assumed that the velocity decays with a simple exponential, and
the conductivity is measured at the Fermi surface (although this is not strictly
required for the definitions used in this equation). The exponential in equation (3.13)
is related to the probability that a particle diffuses, without scattering, for a time τ,
which is taken as the mean time between collisions. In one dimension, we can use the
facts that D = vF2τ (or more generally, D = vF2τ /d , where d is the dimensionality of
the system), le = vFτ, and n1 = kF/π to write equation (3.13) as
3-16
∞
e2 1 −t /τ
σ1 =
πℏ
2D ∫ 2le
e dt . (3.14)
0
In two dimensions, we use the fact that D = vF2τ /2 and n2 = k F2 /2π to give
∞
e2 kF −t /τ
σ2 =
πℏ
2D ∫ 2le
e dt . (3.15)
0
We note that the integrand now has the units L−d in both cases. It is naturally
expected that this scaling will continue to higher dimensions. However, the quantity
inside the integral is no longer just the simple probability that a particle has escaped
scattering. Instead, it now has a prefactor that arises from critical lengths in the
problem. To continue, one could convert each of these to a conductance by
multiplying by Ld−2. This dimensionality couples with the diffusion constant and
time integration to produce the proper units of conductance. To be consistent with
the remaining discussion, however, this will not be done.
In weak localization, we seek the correlation function that is related to the
probability of return to the initial position. We define the integral analogously to the
above as a correlator, specifically known as the particle–particle correlator. Hence,
we define
∞ ∞
1 −t/τ
C (τ ) = ∫ W (t )dt ∼ ∫ 2Ld
e dt , (3.16)
0 0
where W(τ) is the time-dependent correlation function describing this return, and the
overall expression has exact similarities to the above equations (in fact, the
correlator in the first two equations is just the integral). What is of importance in
the case of weak localization is that we are not interested in the drift time of the free
carriers. Rather, we are interested in the diffusive transport of the carriers and in
their probability of return to the original position. Thus, we will calculate the weak
localization by replacing C or W by the appropriate quantity defined by a diffusion
equation for the strongly scattering regime. Hence, we must find the expression for
the correlation function in a different manner than simply the exponential decay due
to scattering.
Following this train of thought, we can define the weak localization correction
factor through the probability that a particle diffuses some distance and returns to
the original position. If we define this latter probability as W(t), where t is the time
required to diffuse around the loop, we can then define the conductivity correction in
analogy with equation (3.15) to be
∞
2e 2
Δσ = −
h
2D ∫ W (t )dt , x(t ) → x(0), (3.17)
0
3-17
and the prefactor of the integral is precisely that occurring in the first two equations
of this section. (The negative sign is chosen because the phase interference reduces
the conductance.) In fact, most particles will not return to the original position. Only
a small fraction will do so, and the correction to the conductance is in general small.
It is just that small fraction of particles that actually does undergo back-scattering
(and reversal of momentum) after several scattering events that is of interest. In the
above equations, the correlation function describes the decay of ‘knowledge’ of the
initial state. Here, however, we use the ‘probability’ that particles can diffuse for a
time t and return to the initial position while retaining some ‘knowledge’ of that
initial condition (and, more precisely, the phase of the particle at that initial
position). Only in this case can there be interference between the initial wave and
the returning wave, where the ‘knowledge’ is by necessity defined as the retention of
phase coherence in the quantum sense. While we have defined W(t) as a probability,
it is not a true probability since it has the units L−d, characteristic of the conductivity
in some dimension (DWt is dimensionless). In going over to the proper conductance,
this dimensionality is correctly treated, and in choosing a properly normalized
probability function, no further problems will arise. This discussion has begun with
the semiclassical case, but now we are seeking a quantum mechanical memory term,
exemplifying the problems in connecting the classical world to the quantum
mechanical world. In fact, Alt’shuler, Aronov, and Spivak derived precisely this
correlation function in a full Green’s function approach in deriving the effect named
for them [44]. We will not go to that extreme, as the essence can be illustrated quite
simply.
It has been assumed so far that the transport of the carriers is diffusive, that is,
that the motion moves between a great many scattering centers so that the net drift is
one characterized well by Brownian motion. By this, we assume that quantum
effects cause the interference that leads to equation (3.17), and that the motion may
be described by classical motion. This means that kle < 1, where k is the carrier’s
wave vector (usually the Fermi wave vector) and le is the mean free path between
collisions, which is normally the elastic mean free path (which is usually shorter than
the inelastic mean free path). This means that the probability function will be
Gaussian (characteristic of diffusion), and this is relatively easily established by the
fact that W(t) should satisfy the diffusion equation for motion away from a point
source (at time t = 0), since the transport is diffusive. This means that [54]
⎛∂ ⎞
⎜ − D∇2 ⎟W (t ) = δ(t )δ(r ), (3.18)
⎝ ∂t ⎠
which has the general solution

1 ⎛ r2 ⎞
W (r , t ) = exp ⎜− ⎟. (3.19)
(4πDt )d /2 ⎝ 4Dt ⎠
In fact, this solution is for unconstrained motion (motion that arises in an infinite
d-dimensional system). If the system is bounded, as in a two-dimensional quantum
well or in a quantum wire, then the modal solution must be found. At this point we
3-18
will not worry about this. Our interest is in the probability of return, so we set r = 0.
There is one more factor that has been omitted so far, and that is the likelihood that
the particle can diffuse through these multiple collisions without losing phase
memory. Thus, we must add this simple probability, which is an exponential. This
leads us to the probability of return after a time t, without loss of phase, being
1 ⎛ t ⎞
W (r , t ) = exp ⎜− ⎟, (3.20)
(4πDt )d /2 ⎝ τφ ⎠
where the phase-breaking time τφ has been introduced to characterize the phase-
breaking process. We note at this point that the dimensionality of W(t) is L−d (Dt
has the dimensions of L2), which is the dimensionality of the integrand for the
conductivity, not the conductance. Thus, this fits in with the discussion above.
One further modification of this simple semiclassical treatment has been suggested
by Beenakker and van Houten [55]. This has to do with the fact that we do not expect
to find these diffusive effects in ballistic transport regimes. Thus, it can be expected
that on the short-time basis, these effects go away. Here, ‘short time’ is appropriate in
that collisions must occur before diffusive transport can take place. If there are no
collisions, there is little chance for the particle to be back-scattered and to return to the
original position. Thus, these authors suggest modifying equation (3.20) to account for
this process. This gives the new form for the probability of return to be
1 ⎛ t ⎞
W (r , t ) = exp ⎜ − ⎟(1 − e−t /τ ). (3.21)
(4πDt )d /2 ⎝ τφ ⎠
At this point, we have slipped in the only quantum mechanics in the current
approach. This quantum mechanics is connected with the phase of the electrons and
is described by the phenomenological phase relaxation time τφ. We have not actually
carried out a quantum mechanical calculation, yet we have introduced all the
necessary phase interference through this phenomenological term. The actual
quantum calculations are buried at this point, but it is important to recognize
where they have entered in the discussion.
The dimensionality correction to the probability of return will go away if we work
with the total conductance, rather than the conductivity. However, as above, we will
not make this change. We now use equation (3.21) in equation (3.17). This gives the
conductivity corrections for weak localization to be
⎧ 1
⎪ (
1 + τφ / τ − 1 , d = 3)
⎪ 2πlφ
⎪ 1 ⎛⎜ τφ ⎞
e2 ⎪ ln 1 + ⎟ , d = 2.
Δσ = − ⎨ 2π ⎝ τ⎠ (3.22)
πℏ ⎪
⎪ ⎛ τ ⎞
⎪ lφ⎜⎜1 − ⎟, d=1
⎪
⎩ ⎝ τφ + τ ⎟⎠
3-19
Figure 3.11. Low-field magnetoresistance for a Si (111) MOSFET in a perpendicular magnetic field for various
temperatures. For the younger readers, 1 kG = 0.1 T. (Reprinted with permission from [44]. Copyright 1982
the American Physical Society).
It is clear that the important length in this diffusive regime is the phase coherence
length lφ = Dτφ . To be sure, this result is the simplest one that can be obtained
within reasonable constraints. Nevertheless, the results are quite useful and show
that the weak localization reduction of conductance is relatively universal in its
amplitude, but also that it has an adjustment depending upon the ratio of the
important time scales in the transport problem. Nevertheless, the quantum mechan-
ics is buried in the ad hoc introduction of the phase coherence time τφ. Without this
introduction, none of the above formulas would be meaningful.
In figure 3.11, we show data taken from studies of the weak localization in a Si
MOSFET structure [56]. In this case, the surface is a (111) plane, and the electron
density was about 4.5 × 1012 cm−2. The measurements were performed in a dilution
refrigerator, and several curves at different substrate temperatures are shown in the
figure. It is clear that there is a distinct peak structure which occurs at B = 0, and
which decays away rather quickly in magnetic field. As the temperature rises, the
phase-breaking time obviously gets shorter and the enhancement peak at B = 0 is
reduced in amplitude. All of this fits well with the arguments presented above.
3.7.2 Role of the magnetic field

In the presence of a magnetic field, the diffusive paths that return to their initial
coordinates will enclose magnetic flux. To examine this behavior is only slightly
more complicated than that of the above treatment, and we will take the magnetic
field in the z-direction and in the Landau gauge A = (0,Bx,0). We will also consider
only a thin two-dimensional slab with no z variation at present, so that equation
(3.18) can be written as
3-20
⎡∂ ⎛ 2e A ⎞2 1⎤
⎢ − D⎜∇ − i ⎟ + ⎥W (t ) = δ(t )δ(r ), (3.23)
⎣ ∂t ⎝ ℏ ⎠ τφ ⎦
where we have incorporated the vector potential via the Peierls’ substitution. In
addition, the phase-breaking time has been incorporated. If we take this in the two-
dimensional system of the normal heterostructure interface, there will be no z
variation, and we can use the normal Landau gauge for the magnetic field. Finally,
we assume that the diffusive propagator is a free plane wave in the y-direction. Then,
the expanded form of the above equation becomes
⎡∂ ∂2 ⎛ 2eBx ⎞2 1⎤
⎢ − D 2 − D⎜ik − i ⎟ + ⎥W (t ) = δ(t )δ(r ). (3.24)
⎣ ∂t ∂x ⎝ ℏ ⎠ τφ ⎦
If we now once again take the x → 0 limit, then the result is a simpler one-
dimensional diffusion equation in which the magnetic field and phase-breaking
terms can be combined into the relationship
1 1 1 eBD
→ + Dk 2 = + (2n + 1) , (3.25)
τφ τφ τφ ℏ
where it has been assumed that the relevant momentum is the Fermi momentum
which is defined by the cyclotron orbits that may result from the magnetic field. Of
course, this treatment is oversimplified, and a more extensive exact treatment is
required. Such a treatment is beyond the level that we are discussing here, but it has
been shown that [57]
e2 ⎡ ⎛ 1 ℏ ⎞ ⎛1 ℏ ⎞ 1 ⎛⎜ τφ ⎞⎟⎤
Δσ ∼ ⎢Ψ⎜ + ⎟ − Ψ⎜ + ⎟ + ln ⎥, (3.26)
π ℏ ⎢⎣ ⎝ 2 4eDBτφ ⎠ ⎝2 4eDBτ ⎠ 2 ⎝ τ ⎠⎥⎦
where Ψ(·) is the digamma function. In figure 3.12, we plot magnetoresistance data, for
a range of temperatures, taken from an AlGaAs/GaAs heterostructure sample [58]. The
AlGaAs layer was 60 nm thick and uniformly Si doped, although a 10 nm undoped
spacer layer was placed between the doped layer and the GaAs layer. A doped 20 nm
GaAs cap layer was placed on the top surface to facilitate making ohmic contacts to the
structure. This particular sample had a sheet density of 2.86 × 1011 cm−2. The curves
through the data are fits to the form of equation (3.26), illustrating the agreement that
can be obtained between the theory and the experiment.
3.8 Graphene rings

Observing the AB effect in graphene rings has appeared in recent years. This is more
complicated because of the degenerate pair of valleys in the conduction band, a zero
energy gap, and the complications of the monolayer material. However, thanks to
modern processing technology, this has now become possible. Rings fabricated by
etching also have multiple problems due to scattering at the layer surface from the
substrate and any overlayers as well as by edge roughness due to the etching. The
3-21
Figure 3.12. Conductance of a AlGaAs/GaAs heterostructure sample at various temperatures as a function of

magnetic field. The dots are measured data and the lines are fits using equation (3.26). (Reprinted with
permission from [58]. Copyright 1984 the American Physical Society).
former can be reduced by encapsulation with hexagonal BN layers on both top and
bottom surfaces, and the rings have been formed by reactive ion etching of the
BN/graphene/BN sandwich after using electron beam lithography to pattern the rings
[59]. The dimensions of the ring were determined by scanning force microscopy to be
an inner diameter of the ring of 405 nm and an outer diameter of 755 nm. A scanning
force microscopy image of the graphene ring is shown in figure 3.13(a). Figure 3.13(b)
shows the two-terminal (indicated as V in panel a) conductance G2W as a function of
the applied magnetic field. In this regime, the AB oscillations are fairly evident but
ride upon a modulated background (the red curve in panel b). This red curve is
obtained by filtering out the high frequency components and is thought to be
conductance fluctuations (discussed in chapter 5). By subtracting this red curve
(a different form is found for each gate voltage), the AB oscillations at different gate
voltages are shown in panels c-d. In particular, the blue arrows in panel c suggest a
periodicity of 5.2 mT for the AB oscillations, which is the expected periodicity for a
ring with a mean diameter of 580 nm, which corresponds to the dimensions given
above.
3-22
Figure 3.13. (a) Scanning force microscopy image of the graphene ring and contacts. The dimensions are given
in the text. (b) The solid black line is the two-terminal conductance G2W as a function of magnetic field. The
red line shows the smoothed back-ground conductance which can be removed to high-light the AB oscillations.
These are shown in panels (c) and (d). The blue arrows in (c) show the periodicity of the AB oscillations
yielding a ring diameter of 580 nm, which agrees well with measurements of the ring. Reprinted with
permission from [59], copyright 2017 by the American Physical Society.
Problems
1. Using the understanding in section 3.1, analyze the data in figure 3.5 to
determine the size of the ring in figure 3.4 as well as the electrical width of the
ring.
2. Consider a free electron in a magnetic field. Using the Peierls’ substitution
and the Landau gauge, show that the electron satisfies a Hamiltonian that
has the form of a harmonic oscillator.
3. Consider a gated AB ring such as shown in figure 3.9. The ring is formed in
an InAlAs/InAs heterostructure, in which the 2DEG is in the InAs. In the
absence of the gate, the 2DEG has a density of 5 × 1011 cm−2. When the gate
voltage is applied, the density under the gate drops to 3 × 1011 cm−2. What
should the phase shift be in this situation?
4. In weak localization, the magnetic field dependence is often characterized by
a correlation magnetic field Bc, which is the value of the magnetic field at
which the weak localization correction has dropped to one-half of its peak
value. Using the lowest temperature curve in figure 3.11, estimate the value
of the peak correction to the resistance by using an estimated extension of the
resistance curves at higher magnetic field down to zero field. (a) Estimate the
correlation magnetic field. (b) Using the peak correction at zero magnetic
field, estimate the phase-breaking time (the mobility of the device is
estimated to be about 103 cm2 V−1s−1 at low temperatures).
Appendix C The gauge in field theory

Most students find their introduction to the concept of gauge in the study of
electromagnetic fields, and so we shall start there as well. We begin with one of
Maxwell’s equations, which is written as
3-23
∂B
∇×E=− . (C.1)
∂t
It is through this equation that we introduce the vector potential A, which is related
to the magnetic field via
B = ∇ × A. (C.2)
Using this formula in (C.1) leads to the result
∂
∇×E=− (∇ × A). (C.3)
∂t
Many people jump to the conclusion that the electric field is just the partial
derivative of the vector potential with respect to time, but this overlooks some
important considerations. Strictly speaking, we have to integrate equation (C.3) to
obtain the result
∂A
E=− + C , ∇ × C = 0, (C.3)
∂t
where the last equation is required in order to satisfy equation (C.3) At this point, we
assure ourselves of this last requirement by defining C as
C ≡ −∇φ , (C.5)
which introduces the scalar potential φ. These all now lead to one common form for
the electric field
∂A
E=− − ∇φ . (C.6)
∂t
We normally see this without the vector potential in our studies of semiconductor
devices, but the vector potential term is important in, e.g., the AB effect. How we
chose to represent the electric field, and the connections between the vector and
scalar potentials, is referred to as a gauge condition.
To understand the conditions on the two potentials, we begin with the second of
Maxwell’s main equations and one of the constituent equations, as
∂E
∇ × B = με + μJ
∂t , (C.7)
ρ
∇·E=
ε
where μ and ε are the permeability and dielectric permittivity of the material in
which the fields are present. Here, J is the current density flowing in the material and
ρ is the charge density, which are the two quantities which can give rise to the fields.
We transform these two equations by introducing the two potentials. Doing this
with the first of equations (C.7) leads to
3-24
∂ ρ
∇2 φ + (∇ · A) = − . (C.8)
∂t ε
This form differs from that of the Poisson equation which is usually used to find the
local potential and the electric field in self-consistent studies. We will recover
the Poisson equation from this equation through a choice of gauge. We now turn to
the second of equations (C.7), which gives
∂ ⎛ ∂A ⎞
∇ × (∇ × A) = −με ⎜ + ∇φ ⎟ + μ J , (C.9)
∂t ⎝ ∂t ⎠
which can be rearranged to give

∂ 2A ⎛ ∂φ ⎞
∇2 A − με = −μJ + ∇⎜∇ · A − με ⎟ . (C.10)
∂t 2 ⎝ ∂t ⎠
If we could set the term in parentheses to zero, we would have the inhomogeneous
wave equation for the vector potential with the current density as the driving term.
At the same time, setting the relation to zero allows equation (C.8) to become the
inhomogeneous wave equation for the scalar potential with the charge density as the
driving term. Thus, this act uncouples the two potentials, and each of these will have
its own driving term. This condition is known as the Lorentz gauge, or sometimes
simply as the gauge equation
∂φ
∇ · A − με = 0. (C.11)
∂t
We have to remember, however, that this is a choice of convenience and is not a
required condition. The fact that it is usually made in the study of electromagnetic
fields does not make it a firm fact or theorem. Nevertheless, when this gauge
equation is imposed, it is a rigid constraint upon the solutions of the two wave
equations.
There are further possibilities that can be imposed. For example, a further
approximation is to invoke the Coulomb gauge, or the electrostatic gauge as it is
sometimes called, in which we set
∇ · A = 0, (C.12)
for which equation (C.11) then leads to
∂φ
= 0. (C.13)
∂t
When this condition is assumed, we note that equation (C.8) now becomes the more
familiar Poisson equation. Once more, we note that this familiar result arises from a
choice of gauge; it is not automatically true and basic. So, when we solve the Poisson
equation for a device, we are assuming that only low frequency effects are of interest,
and that the potential and electric field instantaneously follow variations in charge
(or that the propagation delays are much shorter than any time constant in the
3-25
system). That is, in assuming (C.13), we are assuming that the charge density in the
device is static and time invariant. Then, the scalar potential follows the charge
density change instantaneously, in clear violation of relativity. So, if one wants to
use the Coulomb gauge in device simulation, it must first be ascertained that any
charge changes, due to the imposition of self-consistency, must be slow enough to
validate use of this gauge.
Having discussed the various gauges that are commonly used above, we have
come nowhere near exhausting the number of gauge choices that one can make.
When we study the magnetic field effect on various mesoscopic devices, there are
two more usual gauge choices that are made. These arise from the manner in which
we can force the magnetic field and the vector potential to satisfy equation (C.2). We
have already met one of these in chapter 3, where we introduced the Landau gauge
A = Bxa y , (C.14)
which could also have been written as
A = −Byax . (C.15)
The choice of which of these to use in a particular situation is one of convenience.
But, in many applications, such as quantum dots to be seen in a later chapter, it is
convenient to combine these two forms into the symmetric gauge
1
A=
2
(−Byax + Bxay). (C.16)
In quantum mechanics, it is quite useful that we make the wave function gauge
invariant, especially if we want to talk about the wave function as a field. Normally,
we invoke gauge invariance through the condition that we create a new function Λ,
which we use to change the vector potential through
A → A + ∇Λ . (C.17)
In order to keep the gauge condition (C.11) satisfied, we then have to change the
scalar potential as
∂Λ
φ→φ− . (C.18)
∂t
In classical mechanics, as well as in quantum mechanics, the electromagnetic
interactions are taken into account by a change in the momentum, in which we make
the Peierls’ substitution as
p→p − e A, (C.19)
so that the introduction of a magnetic field to the system is accommodated by
introducing the vector potential to the momentum in the Hamiltonian. Then, to
keep the wave function gauge invariant, we require that a gauge shift such as
equation (C.17) lead to a phase shift of the wave function according to
3-26
ψ (r) → e ie Λ/hψ (r). (C.20)

At this point, we have covered the normal discussion that is given to most graduate
students on the use of gauge. However, it is important to understand that we have
come nowhere close to covering the entire topic. Most discussions proceed with the
use of the two potentials—the vector and scalar potentials. But, it has been known
for a very long time that these two potentials can be determined from a single
vector quantity that is both time and position varying [60], which is often called the
Hertz potential. But, this can be supplemented by quantities such as the polar-
ization P in discussions of the dielectric function. Many other potentials, some
complex, have been introduced for particular applications, such as diffraction.
Needless to say, there are many advanced forms of this, nearly all of which are
beyond the level of discussion here. Another point worth discussing is the
aforementioned dielectric function. It would be a major mistake to assume a
constant for ε. In particular, we know that in condensed matter systems, there are
many factors which contribute to the dielectric function [61], even if we evaluate
this function within linear response. There are also nonlinear and inhomogeneous
effects which cannot be handled easily and directly affect the manner in which
electrons or wave propagate within the solid [62].
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IOP Publishing

(Second Edition)
David K Ferry
Chapter 4
Layered compounds
Since a single layer of graphene was first isolated and studied, it has been known to
have some remarkable properties, primarily because of its Dirac-like linear bands
with zero gap, which has led to new physics [1]. Graphene is a single layer of carbon
atoms arranged in a hexagonal honeycomb structure, and is obtained by extracting
such a single layer from bulk graphite, which is a layered compound. Earlier, there
was interest in carbon nanotubes (CNTs), and this interest still remains, but the
nanotube has been recognized as a rolled up layer of graphene. If we use a single
layer of graphene, we obtain a single-walled nanotube; multi-layers of graphene give
multi- walled nanotubes. Because of the promise of graphene, other layered
materials, such as the transition-metal dichalcogenides (TMDC), have come back
into the spotlight, and it has been recognized that these materials have considerable
promise for electronic applications. In this chapter, we will examine graphene,
CNTs, and some of the newer interest layered compounds.
4.1 Graphene
As we remarked above, graphene is a single layer of carbon atoms arranged in a
hexagonal honeycomb structure. The proper unit cell contains two atoms per unit
cell, which are denoted as atom A and atom B, as shown with two different colors in
figure 4.1(a). This leads to an equivalent Brillouin zone with two minima at K and K′,
as shown in figure 4.1(b). These latter two points are known as the Dirac points for
reasons that become clear below. The C–C nearest-neighbor distance is a = 0.142
nm, but the length of the primitive lattice vector shown in the figure is 0.246 nm (the
two vectors are shown in the figure and have lengths a(3, ± 3 )/2, while the
reciprocal lattice vectors are 2π (1, ± 3 )/3a in the x–y coordinates). Carbon atoms
possess four valence electrons. Three of these form tight in-plane bonds, known as σ
bonds, with three neighboring atoms in the graphene plane. The fourth bond is the

Figure 4.1. The lattice (a) and Brillouin zone (b) of graphene. The δi are nearest-neighbor vectors whose length
is the distance a = 0.142 nm. The lattice vectors ai and reciprocal lattice vectors bi are also indicated.
Figure 4.2. The conduction (reddish) and valence (blue) bands of graphene. They have zero gap at the points K
and K′, identified in figure 4.1(b). There are six of these points around the hexagon.
pz orbital, which is oriented normal to the lattice plane. The σ bonds form a deep
valence band which will not play a role in the conductivity. The pz orbitals form the
π band and this constitutes the band of interest. The nature of graphene is such that
the most common method of computing the band structure is simple tight-binding,
in which the nearest-neighbor interaction is dominant, and is usually denoted by the
general matrix element γ0. Each atom has three nearest neighbors of the opposite
type and six second neighbors of the same type, and the tight-binding formulation
leads to [2]
⎛ 3 ⎞ ⎛ 3k a ⎞
E (k ) = ±γ0 3 + 2cos ( )
3 kya + 4cos⎜
⎝ 2
kya⎟ cos⎜ x ⎟ .
⎠ ⎝ 2 ⎠
(4.1)
This band structure is shown in figure 4.2. At the K and K′ points the bands touch.
Expanding around these points, for small wave vectors away from them, we find that
the energies are approximately [1]
3γ0a
E=± k. (4.2)
2
4-2
The bands are linear bands; that is, they vary linearly with the wave vector k, as
opposed to the quadratic behavior found in normal semiconductors. Moreover,
these bands are chiral, in that the positive slope band has positive helicity and the
negative slope band has negative helicity. The helicity arises from the pseudo-spin
describing the two atomic contributions to the wave function, so that the wave
function is a two component (spinor) entity. One component has a phase shift
relative to the other which leads to the helicity; opposite helicities have opposite
signs of this phase shift.
We can see a little more of the unusual nature of graphene when we note that
these linear bands are Dirac-like in that we can write the energy
E = ±ℏvF k, (4.3)
as discussed in section 2.2.2 for the density of states. Here, the Fermi velocity, or
effective ‘speed of light’, is given as
3γ0a
vF = ∼ 8 × 107 cm s−1. (4.4)
2ℏ
This is obtained by fitting to angle-resolved photo-emission data [3]. Because of the
zero gap, the carriers in graphene are often referred to as ‘massless chiral Dirac
fermions’. This connotation recognizes that there is no rest mass contribution to the
Dirac-like bands, that the two atomic contributions to the wave functions impose
chirality on these wave functions, and that they are indeed fermions. While the rest
mass vanishes, this should not be construed as them being massless particles. Indeed,
from equation (4.3), we can immediately determine the effective mass of, e.g., the
electrons as [4]
1 1 ∂E v ℏk
= 2 = F, m*= . (4.5)
m* ℏ k ∂k ℏk vF
This mass, and this band structure, give a different energy dependence for the normal
density of states in two dimensions. The proper value was found in section 2.6, and for
graphene, the number of carriers per unit area, per unit energy, is given as
E
ρ2 (E ) = . (4.6)
π (ℏvF )2
While the density of states and the effective mass both vanish as the energy moves to
the Dirac point (where the two bands touch at the K and K′ points), the density is not
observed to vanish, although this is what one expects. Rather, it is found that a
random potential exists in graphene which leads to electron–hole ‘puddles’ forming
at energies near the Dirac point. Zhang et al [5] used STM to probe the local
potential in graphene, and demonstrated the existence of these electron and hole
‘puddles’ near the Dirac point. These puddles were shown to be related to the
impurities external to the graphene sheet. Gibertini et al [6] estimate, from their
simulations, that the size of the puddles is a few nanometers. Deshpande et al [7]
4-3
have used scanning tunneling spectroscopy, finding that the fluctuations of the
surface topography show that puddle-like regions are of the order of 5–7 nm in
extent. The simulations of Rossi and Das Sarma [8] suggest a similar size range for
the puddles.
A typical conductivity curve for graphene is shown in figure 4.3, taken at room
temperature. The graphene sheet was extracted from highly oriented pyrolytic
graphite by mechanical exfoliation using the standard sticky tape approach. It was
then deposited on an oxidized Si wafer with 290 nm of thermal oxide. Standard
optical lithography was used to define the source and drain contacts, and the
conductivity was then measured as the bias applied to a contact on the reverse side of
the Si wafer was varied [9]. A constant 10 mV source–drain bias was applied. The
variation of the back gate voltage is coupled to the density in the graphene sheet by
the capacitance between the back gate electrode and the graphene. This variation
can then sweep the Fermi energy throughout the graphene bands. There is a broad
minimum around 45–50 V on the back gate, which is normally assumed to indicate
the region where the Dirac point is located. The shift away from zero gate voltage is
due to acceptor charges located either in the oxide or on the oxide surface under the
graphene. The solid curve in the figure is a theoretical fit to the data which assumes
that there are about 3.4 × 1012 cm−2 ionized acceptors, so that a positive gate bias is
necessary to eliminate the associated holes in the graphene [10]. The fact that the
conductivity does not go to zero at the Dirac point is direct evidence for the puddling
discussed in the previous paragraph. Generally, the conductivity tends to rise
linearly with gate voltage as one moves away from the Dirac point, which is mainly
a result of the capacitive nature of the effect of the gate bias, where the density will
increase linearly with the gate voltage. The linear rise of the conductivity then means
that the mobility is relatively constant, at least within this density range.
Figure 4.3. Conductivity in a sheet of graphene placed upon an oxidized Si wafer, with 10 mV bias applied.
The minimum in the conductivity signals the location of the Dirac point. (Reprinted with permission from [9].
Copyright 2009 the American Vacuum Society.)
4-4
The conductivity and the mobility of the carriers in graphene are dominated by
the various phonons in graphene [10], as well as by the Coulomb scattering from the
impurities and the flexural modes of the rippled graphene sheet [11]. In addition,
there can be scattering from point defects (short-range scatterers) [12, 13] as well as
corrugations [14] and steps in the thin layer [15]. Very high values of mobility have
been reported for free-standing graphene sheets, especially at low temperatures.
However, when the graphene is placed on a substrate such as an oxidized Si wafer,
or SiC or BN, the mobility is not found to be more than a few thousand (cm2(Vs)−1)
at room temperature [16–20]. The variety of different types of non-intrinsic
scattering mechanisms can make it difficult to understand individual mobility
measurements.
Most experimental studies of graphene transport will utilize ribbons of the
material, which are small slices of a nearly uniform width. When graphene is
patterned into a desired width, there are two main terminations, which are referred
to as the arm-chair and zig-zag edges. These two edges are illustrated in figure 4.4.
Energetically, there is a difference in these two edges. Within the tight-binding
formulation of the energy bands discussed above, the zig-zag edges will always retain
their zero energy gap, and are termed metallic edges. The arm-chair edges will be
different, however, and can have either metallic edges with zero gap or semi-
conductor edges in which a small energy gap opens. It has been shown that this gap
will open except when the number of atoms in the ribbon width is given by 3p + 2,
where p is an integer [21]. The tight-binding formulation tends to keep the atoms in
their perfect atomic alignment, but this is not how edges and surfaces of three-
dimensional material behave. It is energetically favorable for a gap to open, as this
lowers the energy of the electrons at the top of the valence band, and this gap usually
is created by a relaxation or reconstruction of the atomic structure at the edge or
bulk surface [22]. Relaxation occurs when the atoms move without changing the
edge (or surface) unit cell, while reconstruction changes the unit cell. These changes
can be calculated by introducing a set of molecular dynamics forces between the
atoms, where these forces are computed from the band structure, usually by a
technique developed by Feynman and Hellman [23, 24]. In general, this approach is
more easily accomplished through the use of pseudopotentials to calculate the band
Arm-chair edge
Zig-zagedge
Figure 4.4. A small section of graphene illustrating the arm-chair (top and bottom) and zig-zag (left and right)
edges for perfectly oriented material.
4-5
structure [4]. In any case, it is found that when this effect is accounted for, a gap
always opens in the bands for a ribbon. It has been found, for example, that for arm-
chair ribbons, the size of the gap varies with EG,3p+1 > EG,3p > EG,3p+2 [25]. Here, the
smallest gap corresponds to the ribbon that would have preferred to remain metallic.
In most materials, however, the formation of the ribbon does not produce perfect
edges, and one does not have either of the two arising for the aligned lattice shown in
the figure. Just as one can have voids and five- and seven-member rings within the
lattice, these can occur at the edge as well [26].
As mentioned above, the zig-zag edges remain metallic, but this is not a simple
conclusion. In fact, the metallic states are located at the edges of the ribbon, and in
this region, the bands are almost flat so that the density of states is large [27]. The
corresponding wave functions are thus localized at the ribbon edges. It is also found
that the width (into the ribbon) increases as the ribbon is made wider. It is thought
that the edge states in these zig-zag ribbons do not derive from either bulk graphite
nor from the dangling bonds. Rather, they seem to be a general property of π
electron networks with a zig-zag edge [27].
Creating constrictions in graphene with which to observe conductance quantiza-
tion is a nebulous process. The reason is the lack of a band gap in the bulk graphene.
Normally, the Schottky gate depletes the electron gas by pushing that region into the
band gap of the material. However, in graphene, the gate merely replaces electrons
by holes, or vice versa. This leads to tunneling right through the voltage-induced
barrier by Klein tunneling [28] (appendix D), a process which has apparently been
experimentally observed in graphene [29, 30]. Moreover, the mobility in graphene
laid on most substrates is quite low, even at low temperatures, which tends to
preclude the existence of the quasi-ballistic transport needed to see the coherent
conductance quantization. Nevertheless, attempts have been made to make gate
defined constrictions [31], but the evidence of conductance quantization is limited.
Another approach is to use bilayer graphene, as an electric field applied vertically
between the two layers can open a gap in the energy spectrum of both layers. This
allows one to try to create actual depletion layers more effectively, and gate
controlled constrictions have been created effectively with this approach [32–34].
Confinement and quantum dot behavior have been observed, but the mobility issue
has not been overcome. The use of suspended graphene is more difficult, but can
overcome the mobility problem. In one such approach [35], a polymer was placed
between the oxidized Si wafer and the graphene sheet. In this case, the oxide was
about 500 nm thick, and the polymer was about 1 μm thick. The polymer was
removed in areas where the graphene is to be suspended and the sheet is patterned by
lithographic methods. This forms a constriction of the order of 250–280 nm wide, as
estimated electrically. Then, the conductance can be measured as the back gate bias
is varied to change the density in the graphene ribbon. Interestingly, the conductance
seemed to be quantized at the values 2Ne2/h (N is an integer), whereas twice this
value would be expected for graphene. Quantization in multiples of 2e2/h were seen
in chapter 2 for GaAs, and other III–V materials, where there is only spin
degeneracy to worry about. However, graphene has the valley degeneracy between
the K and K′ points, and this should produce an extra factor of two, although it is
4-6
known that a magnetic field can lift both degeneracies [36, 37]. It has been
conjectured that the states in a nanoribbon will hybridize the two valley wave
functions [38], this may be the process at work here to lead to the reduced value of
the conductance steps.
The AB effect has been observed in graphene rings [39]. Here, we illustrate more
recent work [40], where the graphene flake was deposited on an oxidized Si wafer in
which the oxide was 295 nm thick. Then, electron-beam lithography and reactive-
ion etching were used to pattern the graphene ring, shown in figure 4.5(a). Raman
scattering was used to provide the evidence that the graphene was a single layer, with
the characteristic G and 2D lines (shown in figure 4.5(b)) providing the evidence. The
resistance of the flake is shown in figure 4.5(c) as a function of the back gate (Si
layer) voltage. The ring had an inner radius of about 200 nm and an outer radius of
about 350 nm, as can be observed in the image of the ring. The leads to the ring are
graphene ribbons of approximately 150 nm width. In figure 4.6(a), the four-terminal
resistance across the graphene ring is shown as a function of the magnetic field. This
resistance consists of several parts, including the two leads and the ring itself. This
resistance was obtained at a back gate voltage of −5.8 V, which corresponds to a
hole density of 1.2 × 1012 cm−2. The AB oscillation signal is obtained by subtracting
the background resistance, and this is shown in figure 4.6(b). This signal has a period
of about 17.9 mT, as indicated by the vertical lines. The oscillatory signal was
Fourier transformed, with the spectrum shown in figure 4.6(c). There is a dominant
single peak around 60 mT−1 which corresponds reasonably well to the periodicity
seen in the oscillatory signal. It is interesting that the width of this peak is narrower
than might be expected from the size of the ring, shown as the gray shaded region in
this last figure. This suggests that the electrical width of the ring is less than the
physical width, which is not unusual.
Weak localization is also seen in graphene [41] (see figure 4.7). Here, we discuss
measurements made on high quality epitaxial graphene grown on the silicon face of
Figure 4.5. (a) A scanning-force micrograph of the patterned graphene layer, showing the ring and its leads
and side gates. (b) The Raman spectrum observed from the same flake prior to patterning, establishing the
single-layer nature of the material. (c) Four-terminal resistance of the ring structure with back gate bias at 0.5
K. (Reprinted with permission from [40]. Copyright 2010 IOP Publishing and Deutsche Physikalische
Gesellschaft.)
4-7
Figure 4.6. (a) Four-terminal resistance measurement across the ring as a function of magnetic field.
(b) Oscillatory signal after the background has been removed. The lines indicate a rough period of 17.9 mT.
(c) The Fourier transform of the oscillatory signal. The main peak near 60 mT−1 is much narrower than that
expected from the size of the ring (gray shaded area). (Reprinted with permission from [40]. Copyright 2010 IOP
Publishing and Deutsche Physikalische Gesellschaft.)
Figure 4.7. (a) Magnetoconductance at three temperatures. (b) A fit (black curve) to the weak localization
signal. (Reprinted with permission from [42]. Copyright 2011 IOP Publishing and Deutsche Physikalische
Gesellschaft.)
6H–SiC substrates [42]. Conventional photolithography was used to pattern the

Hall-bar structures used in the experiments. The sample had an electron density of
6 × 1011 cm−2 and a mobility of about 10 000 cm2 V−1s−1 at low temperature. In
figure 4.7, the relative conductivity Δσ (B ) = σxx(B ) − σxx(0) is shown at three
selected temperatures [42]. The conductivity is deduced from the actual resistivity
measured for a constant current through the sample. Weak localization is observed
at all three temperatures. In order to observe the effect, a very slow sweep of the
magnetic field had to be employed, basically of the order of 100 min per tesla. Any
faster sweep rate in magnetic field was found to reduce the amplitude of the weak
localization peak. In panel (b) of the figure, the peak around zero field has been fit to
an analytic form described by McCann et al [43], which is a modification of that
4-8
na1
A arm-chair line
B
ma2
Figure 4.8. The rolling vector is R = rA − rB, which in turn is defined as multiples of the unit vectors (see
figure 4.1) according to (4.7). The wrapping angle is the angle between this vector and the arm-chair line.
given in the last chapter which is felt to be more appropriate to graphene. This
allowed the authors to extract a phase-breaking time which was temperature-
independent below about 1–1.5 K, and then decreased as T −1/2 at higher temper-
atures. This contributed to a phase-breaking length of just over 1 μm at low
temperature.
4.2 Carbon nanotubes

Another interesting material that has been around longer than graphene is the CNT,
having been first discovered in 1991 [44]. But its properties derive from those of
graphene. If we take the sheet of graphene that is depicted in figure 4.4, and roll it
into a cylinder, then we obtain a CNT. If we use a single layer of graphene, then we
obtain a single-walled CNT. But, if we use multiple layers of graphene, then we
obtain a multi-walled CNT. Like a sheet of paper, however, there are many ways in
which to roll up the sheet of graphene, so we need a method of characterizing just
how the nanotube is rolled. We start to define this method by picking an arbitrary
point on the graphene lattice, and we denote this as point A, as shown in figure 4.8.
In figure 4.1, the lattice vectors for the graphene lattice were defined as extending, for
example, from one A type atom to the nearest other A type atoms on the hexagon
(the nearest neighbors are B type atoms). Hence, we can define multiples of these
lattice vectors using the integers n and m as shown in figure 4.8. Suppose we roll the
graphene sheet so that point A goes to point B (admittedly, this is a very tightly
rolled tube, but it is the example that is important here). We can define the rolling
vector, or chiral vector as
R = rA − rB = na1 + ma2 → 2a1 + 4a2 . (4.7)
where the last form is for the particular case shown in figure 4.8. The axis of the
CNT is perpendicular to the rolling vector, so it would be a vector normal to R.
Since it is usually impossible to count the number of unit cells in a large sheet of
graphene, or a large CNT, another useful quantity is the wrapping angle. We define
this wrapping angle as the angle between the arm-chair line (that defined by either
lattice vector) and the chiral vector R, as [45]
4-9
R
φ = ∠armchair . (4.8)
The arm-chair vector is the case of equation (4.7) when n = m. Hence, we can find
the cosine of the wrapping angle using the properties of the two vectors, as defined
by n and m, to show that
2n + m
cos(φ) = . (4.9)
2 n + m 2 + nm
2
There is, of course, an ambiguity about the sign of the wrapping angle, but a proper
choice follows from the symmetry of the tube. If you assign one sign of the angle,
just interchanging the two ends of the CNT changes the sign. Since the physics of the
CNT does not depend upon the orientation of it, the sign is thus not particularly
important until we apply something like a magnetic field which will break the
symmetry.
Just as the width of a graphene ribbon introduces some changes to the band
structure of the flake, the length of the rolling vector also introduces some changes.
In computing the density of states in chapter 2, the periodicity of the lattice was
important in setting the values of the momentum wave vectors. The rolling vector
has the same effect in a CNT, as this defines the periodicity of the structure (as we
move around the circumference of the tube). The crucial factor here is the difference
between n and m. If n − m = 3p, where p is any integer (or zero), then the CNT will
retain the graphene band structure and have a zero energy gap between the
conduction and valence bands. These tubes will then be metallic. If n − m ≠ 3p,
then the CNT will have a gap between the two bands and will be a semiconducting
tube. Nevertheless, the density of states for the CNT will have the peculiar behavior
of a Q1D conductor given by equation (2.29). However, one has to distinguish here
whether or not we are dealing with a metallic tube or a semiconducting tube, as we
have to be careful with the conversion from wave vector to energy. We note that the
derivation of equation (2.29) involves the fact that the density of states in one
dimension is given as
1 dk
ρ1 = , (4.10)
π dE
which includes a factor of two for spin, but no extra term for valley degeneracy has
been added yet. So, if we have a metallic tube which retains the graphene band
structure, we have the valley degeneracy and the Dirac bands, which lead to
2 2k
ρ1,metallic = = . (4.11)
π ℏvF πE
On the other hand, if the gap is opened, then we have semiconducting behavior with
an easily determined effective mass, and we obtain equation (2.29)
1 2m*
ρ1,semicon. = , (4.12)
πℏ E
4-10
although a factor of two has been added for valley degeneracy. In both cases, there is
a divergence of the density of states as one approaches the band edge. In the
semiconducting tubes, the gap depends upon the diameter of the CNT, with the
latter given by
∣a1∣ 2
d= n + m 2 + nm , (4.13)
π
with the length of a1 given as 0.246 nanometers. If either n or m are zero, then the
CNT is called a zig-zag nanotube. As noted above, if n = m, then the CNT is called
an arm-chair tube.
In a metallic tube, the quantization of the wave number discussed above will lead
to the condition that [45]
ΔkR = 2πs , (4.14)
where R is the magnitude of the chiral vector equation (4.7) and s is an integer,
which defines the particular band, as it derives from the graphene band structure.
Each value of s defines a line (in one dimension) of allowed k vectors that contributes
to one occupied π-band. In semiconducting tubes, we have a similar behavior except
that there is a gap opening. We can redefine the above condition on n and m to be
n − m = 3p + ν . (4.15)
If ν = 0, then we have a metallic tube. But, if ν is ±1, then we have a semiconducting
tube. The integer p is then related to the band index s, and we can generalize
equation (4.14) to handle the semiconducting tubes as well by using
⎛ ν⎞ 2⎛ ν⎞
ksR = 2π ⎜s + ⎟ → ⎜s + ⎟ , (4.16)
⎝ 3 ⎠ d ⎝ 3⎠
where, as before, d is the tube diameter. Now, the wave number k is the momentum
vector for motion around the tube. We recall that the presence of a magnetic field
changes the momentum, described by the Peierl’s substitution in earlier chapters.
Indeed, the Hamiltonian is modified by the vector potential according to this
substitution, and this leads to a gauge variation that applies a phase shift to the wave
function. Because we have applied additional quantization to the wave vector in
creating the tube, this magnetic field will cause a variation in this quantization.
Hence, a magnetic field, applied along the tube axis will lead to a modification of
equation (4.16) to account for this AB phase, as [46]
2⎛ ν Φ⎞
k⊥ → k⊥(φ) = ⎜s + + ⎟, (4.17)
d⎝ 3 Φ0 ⎠
where Φ = B πd 2 /4 is the flux enclosed by the tube and Φ0 = h /e is the quantum flux
unit as before. Thus, we see that the magnetic field modulates the energy bands of
the CNT [47]. The magnetic field can change a metallic tube to a semiconducting
tube, and vice versa. Varying the magnetic field gives this transition as a function of
4-11
the field. To see this effect, however, requires enormous magnetic fields due to the
extremely small cross-section of a CNT. For example, a 10 nm diameter CNT would
require more than 50 T to complete one cycle of the modulation, but this is a doable
field as we will see.
The first experimental search for AB oscillations in a CNT was apparently by
Bachtold et al [48], and these authors did find magnetic modulation of the
conductivity when the magnetic field was aligned with the tube axis. But, these
authors found a magnetic period of h/2e, more related to the disorder-induced AAS
behavior of rings discussed in the last chapter [49]. The outer current-carrying ring
(of a multi-walled CNT) was measured to have a radius of approximately 8 nm by
atomic force microscopy (AFM), and the h/2e behavior gave an inferred radius of
approximately 8.6 nm. Several more recent studies continued to find these AAS
oscillations [50, 51]. The first clear evidence of the AB effect was apparently the work
of Coskun et al [52], who observed it in CNT quantum dots. In this latter case, the
authors used a multi-walled CNT with an outer radius of about 15 nm, but placed
the CNT on top of the metallic contacts; this tends to produce tunneling barriers so
that the interior of the CNT was effectively an isolated quantum dot. A back-contact
gate could be used to modulate the CNT conductance, and the authors observed a
modulation of the quantum dot conductance diamonds (we discuss these in a later
chapter) with the magnetic field, which had the proper h/e periodicity. At about the
same time, another group used a single-walled CNT of about 1 nm radius, and saw
some signatures of the AB effect, but could not see oscillations in this small diameter
tube [53].
As indicated above, it is useful to have a very high magnetic field available for
these studies. Lassagne et al [54, 55] were able to bring a 55 T magnet to the study of
a multi-walled CNT 10 nm in diameter. The tubes were placed on an oxidized Si
wafer so that the voltage applied to the Si would serve as a back gate. Pd metallic
electrodes, spaced by 200 nm, were deposited on top of the CNT, which usually
produces ohmic contacts. Nevertheless, some evidence of Schottky barrier behavior
was observed. Experiments suggested that their tubes were in the ballistic regime so
that good phase coherence could be expected. In figure 4.9, we display the
conductance as a function of the back gate voltage and the magnetic field. The
data suggest an oscillatory behavior with a period of 48 T, which is strong evidence
of the h/e modulation, as the 10 nm diameter tube would be expected to show a
period of 50 T, as discussed above. Subsequent experiments [55] used an 18 nm
diameter CNT with a contact spacing of 150 nm, so that more periods of the
oscillation could be observed in the available 55 T. The measured conductance is
shown in figure 4.10 for several values of the back gate voltage. Here, it is clear that
good h/e oscillations are seen in the data, even at the elevated temperature of 100 K.
These oscillations persist over basically all the values of the gate voltage, even
though the peak positions do shift with the bias, as expected for modulation of the
energy bands by the magnetic field. In all of these experiments, it is evident that the
dominant conduction is in the outer shell of a multi-wall CNT.
4-12
Figure 4.9. Left panel: the magnetoconductance at 100 K for several values of gate voltage as a function of the
magnetic field. At 10 V, the tube is basically biased to a point with the Fermi level in the gap, and the magnetic
field produces the necessary change to the bands to give the h/e modulation. As discussed in the text, 50 T is
just about enough to produce one complete oscillation for this 10 nm tube. The right panel is a 3D
representation of the conductance. (Reprinted with permission from [54]. Copyright 2007 the American
Physical Society.)
Figure 4.10. Magnetoconductance observed in an 18 nm diameter multi-walled CNT at 100 K, for various
back gate voltages. The curves have been offset for clarity. (Reproduced with permission from [55]. Copyright
2009 Elsevier Masson SAS.)
Weak localization is also seen in the CNT, especially when the magnetic field is
oriented normal to the tube axis [50, 56]. Because of the complicated periodic band
structure of the CNT, however, the peak in resistance does not always occur at
B = 0, as it can depend upon gate bias and other properties of the CNT.
4-13
4.3 Topological insulators

Topological insulators are materials in which a surface or interface provides a
localized energy structure that has the Dirac-like bands of graphene [57], but
generated with some additional properties. One prototypical material system is a
heterostructure between HgTe and CdTe. In most semiconductors with a direct
band gap at the Γ point (center of the Brillouin zone), the bottom of the conduction
band is composed of atomic S orbitals from the cations, and often denoted as the Γ6
or Γ1 (the former is the so-called double-group notation used when the spin–orbit
interaction is included) band. On the other hand, the top of the valence band is
usually composed of the anion P orbitals, and denoted as the Γ8 or Γ15 band. In
HgTe, these two roles are reversed, so that the Γ6 band lies below the Γ8. Hence,
HgTe is often referred to as having a negative band gap. In the interface between
these two materials, these bands must cross as they reverse their roles from HgTe to
CdTe. Now, this property has been known for quite some time as the HgTe/CdTe
superlattice band structure was studied at least as early as 1979 [58, 59]. But, for a
topological insulator, one wants more to assure that the zero gap and its properties
are topologically protected from disorder. In figure 4.11, we draw schematically how
the interface bands extend from the bulk bands to provide the Dirac-like bands. One
would like the bands to be such that, for example, one had spin up and the other spin
down. In a topological insulator, the spin of one branch is locked at a right angle to
their momentum (termed spin-momentum locking), so that carriers in the other
branch have different spin and back-scattering is then forbidden. Time-reversal
Figure 4.11. The bulk bands are shown in blue and characterize a normal semiconductor. The surface bands in
red lead to the topological insulator on the surface. The nature of the spin and time-reversal symmetry leads to
the effect.
4-14
symmetry was predicted to lead to these type of edge states in quantum wells of
HgTe placed between layers of CdTe in 1987 [60], and it was experimentally
observed in 2007 [61]. More recently, this type of surface band structure was
predicted to occur in three-dimensional materials such as some Bi compounds [62],
although the general basis for the topological insulator seems to have been put
forward more generally in 2005 [63, 64]. The first experimental evidence for a three-
dimensional topological insulator was in bismuth antimonide, in 2008 [65].
The interest here, of course, is in mesoscopic effects as they may occur in these
materials. And, it is clear that the AB effect can be observed in topological
insulators [66], and has been seen in Bi2Se3 nanoribbons [67]. Bi2Se3 is a layered
compound with a rhombohedral phase, but with covalent bonding in the layer and
weak van der Waals bonding between the layers, much like a nanoribbon, as it is
on these surfaces that the protected state exists (the bulk is an insulator) [67]. The
authors plotted the magnetic field position of each resistance minimum as a
function of its index (a counting of the position away from B = 0). The linearity of
this curve, along with the index as a multiple of h/e supports the interpretation in
terms of the AB effect. The Fourier transform of the resistance trace illustrated a
single dominant AB peak. An interesting observation was the presence of weak
anti-localization at B = 0, as the resistance shows a drop instead of the peak
expected for weak localization [67]. We recall from the previous chapter that weak
localization occurs when the electrons can move around two time-reversed paths
and constructively interfere with back-scattering. However, when the spin–orbit
interaction is important, as it is in the topological insulators, the spin is coupled to
the momentum, as mentioned above. Hence the spins of the carriers in the two
time-reversed paths are opposite to each other. As a result, the two paths interfere
in such a manner that it results in a reduction in the resistance, and weak anti-
localization [68]. The magnetic field still leads to breaking the time-reversal
symmetry and rapid decay of the signal.
More recently, a field-effect transistor has been fabricated with Bi2Se3 for which
the AB effect has been observed at low temperature [69]. In this latter case, the AB
effect was somewhat anomalous in that the AB period corresponded to a minimum
in the magnetoresistance, rather than a maximum, presumably due to the spin–orbit
interaction. We see the effect in figure 4.12. Panels (a) and (b) show the transistor
action at various magnetic fields. There is an obvious turn-on of the transistor at a
particular gate characteristic as well as no real magnetic field modulation of the
output current. Panel (c) shows the presence of the AB oscillations as the magnetic
field is swept from −9 to 9 T, as well as a strong weak localization peak (note the
increase in resistance at zero magnetic field. Finally, panel (d) shows the Fourier
transform illustrating a clear h/e period.
A variation of the topological insulator is the topological crystalline insulator
(TCI) which refers to materials in which the crystalline symmetry leads to
topologically protected surface states with a chiral spin texture (see figure 4.11).
Materials such as SnTe and Pb1−xSnxSe or Pb1−xSnxTe grown along the (001)
direction develop non-trivial surface states [70, 71]. These protected states can even
form at step edges of the grown layer [72]. Indeed, enhanced current has been
4-15
Figure 4.12. (a) Transfer characteristics (IDS − VGS) of the Bi2Se3 nanowire FET at various magnetic field
intensities ranging from −9 T to 9 T. VDS was maintained at 50 mV. Inset: The linear-scale transfer
characteristics. (b) Output characteristics (IDS − VDS) of the Bi2Se3 nanowire FET under different VGS.
(c) Oscillatory features of the magnetoresistance obtained by sweeping the magnetic field between ±9 T.
(d) FFT analysis showing the amplitude as a function of 1/B. The location of h/e was labeled. Reprinted with
permission from [69], copyright 2019 by AIP Publishing.
observed at step edges in epitaxially grown Bi2Te3 [73]. Since this effect is not
observed at step edges of graphite, the authors suggest that there may be a possible
interaction between the spin–orbit coupling and the topological nature of the edge
states.
4.4 The metal chalcogenides

Following the rapid increase of interest in graphene, people began to wonder if there
were layered compounds in which a single layer could be exfoliated and which had a
real band gap. The need for a band gap, of course, arises from the desire to
incorporate these materials into active semiconductor devices, and that meant a need
to be able to turn the device off, something that cannot be easily done with graphene.
It turns out that there are some materials which have these properties, and these are
the transition metal dichalcogenides (TMDC). The chalcogenides have a large
number of stoichiometries and phases, but the best known for transport purposes are
4-16
the dichalcogenides of the form MX2, where M is a transition metal, typically Mo or

Ta, and X is usually S, Se, or Te. The compounds with Mo or W are semiconductors
with a band gap usually near or larger than that of Si. However, the Nb and Ta
based compounds are usually metals. These materials form in a layered compound,
where each layer is typically a layer of the transition metal with a layer of the
chalcogenide both above and below the metal layer [74]. The layer to layer bonding
is weak, as it is in graphene, so that individual layers can be removed by exfoliation,
either mechanically or via a liquid-based procedure, and then placed on a convenient
substrate.
These materials actually have a rather longer history than graphene, as they were
pursued heavily in the mid-1970s and later in a search for charge density waves. The
understanding of this phenomenon begins with the Peierls distortion, in which a one-
dimensional chain of atoms is actually unstable. Peierls showed that a commensu-
rate, periodic distortion of the chain that coincided with the Fermi surface would
lower the overall energy of the electron gas [75]. A metallic one-dimensional material
in which there is one electron per atom fills the band up to π/2a, which leads to a
natural factor of two distortion. Here, two neighboring atoms move slightly closer
together while having a slightly larger distance from their neighboring pairs. It was
later shown that even Q2D materials could undergo this distortion. If a distortion
momentum vector could be found that spanned the Fermi surface, and which was
also some integer multiple of the lattice vectors, then a charge density wave could be
formed. Here, the two opposite wave vectors lead to a standing density wave
composed of two counter propagating electron waves [76]. This couples to the
Peierls distortion of the lattice. Most interestingly, it is the TMDCs that were
suggested as being the proper type of layered compound in which the charge density
wave could be observed [77]. One of the earliest observations of the existence of the
charge density wave was in TaS2 [78]. The electron density of the charge density
wave in TaS2 and TaSe2 has been directly imaged by the use of an STM [79].
As an example, let us look at MoS2, which is a layered compound in where each
layer is composed of Mo atoms at the center of the layer and S atoms displaced
above and below this center. In bulk form, it has an indirect band gap, which
becomes direct only in the monolayer limit [80, 81]. The nature of the three atoms
per unit cell leads to the lack of inversion symmetry in the monolayer. When a
second monolayer is added to the first, it is reversed in direction of the unit cell so
that inversion symmetry is restored. Hence, the bulk is actually stacks of bi-layers.
The band extrema of the monolayers are located at the K and K′ points of the
Brillouin zone, so that there are two minima for the conduction and valence band.
Usually, by the time the second monolayer is placed on the first, the minimum of the
conduction band moves to the T (or Q) valleys, which lie midway between Γ and K,
and the material is indirect. In the monolayer subsidiary valleys of the conduction
arise from the residual valleys of what was the indirect gap. These valleys, referred to
as the T valleys [82]. The conduction band mass is approximately 0.45m0 in the K
valleys and 0.57m0 in the subsidiary valley [81]. These bands are non-parabolic, and
this has to be taken into account for transport. With certain limitations, the material
parameters, phonon energies, and coupling constants are all given in the work of [80].
4-17
One natural question is about the polar LO modes which one would expect in this
material, due to the dissimilar atoms. These have been claimed to exist [80], but there
are some anomalies with them. First, because of the 3 atoms per unit cell, there are 9
phonon modes, so there are two LO-TO pairs. The polar interaction normally would
arise from the LO–TO splitting at the Γ point, but this splitting vanishes at this point
for both the LO1–TO1 and the LO2–TO2 modes [80, 83]. As the electromagnetic
waves couple only at the Γ point, this lack of splitting means that these phonons do
not couple to the electromagnetic wave, and thus give rise to no polarization at this
point. Hence, the dielectric constant is continuous at this energy, as required by the
Lyddane–Sachs–Teller relation, and the polar mode is not generated. On the other
hand, the two modes are split at the K point, but this phonon momentum would be
an intervalley interaction, and this is dominated by the deformation potential
coupled LO mode [80]. In particular, MoS2 and WS2 have been studied for their
electrical properties, but as of this writing, no mesoscopic studies have surfaced.
It is also possible to form trichalcogenides such as NbSe3 or MoS3. Typically,
these materials form what are often called ‘triangle poles’, which are six
chalcogenide atoms surrounding a central metal atom. The pole is a triangular
prism of chalcogenide atoms, one above the metal atom and one below the metal
atom [84]. These structures tend to form chains of such poles, where the poles are
built up parallel to one another, although they typically have several different
phases [85]. These chains tend to make the material look like whiskers.
Importantly, these materials can be superconducting, and can show Q1D charge
density wave behavior [86]. But, it has also been observed that these materials can
form layered compounds which can be effectively exfoliated [87]. Importantly,
these Q1D wires, or ribbons, can be joined to form what are called topological
crystals (see the previous section). That is, the end of the ribbon can be attached to
the beginning of the ribbon to form a loop, which can be relatively wide [88].
Interestingly, the ends can be joined with a twist of ±π to create a real world
Möbius strip, and therefore have only a single surface, hence the name topological
crystal.
As in the dichalcogenides, the trichalcogenides support charge density waves, so
that the mesoscopic effects can be somewhat different. Tsubota et al [89] have grown
TaS3 in a ring geometry without any twist. In fact, the material grows into what may
better be described as tubes, but these were sliced using a focused ion beam to
produce a ring about 27 μm in diameter and with a 1 × 0.1 μm cross-section. Au
electrodes were deposited and current passed through the ring, with a magnetic field
normal to the plane of the ring, so that AB oscillations could be observed. The
results of the measurements are shown in figure 4.13. AB oscillations with a period
of h/2e were observed in the oscillatory current (constant bias voltage is applied).
The rings are large enough that the AAS disorder oscillations can be ruled out [89].
In fact, the AB oscillations are observed up to temperatures as high as 79 K. It is
believed that the current induces a sliding charge density wave of soliton nature
which carries an effective charge of approximately 2e, with the measurements giving
an actual effective charge of ∼1.9e. The model of this behavior is that the charge
density wave is a correlated electron system that can transport the electrons through
4-18
Figure 4.13. (a) Variation of the current as the magnetic field is changed for three different bias voltages, for a
ring of TaSe3. (b) The power spectra of the Fourier transform by two methods: the DTF (left) and maximum
entropy method (right). Adapted with permission from Tsubota et al [89], copyright 2012 IOP Publishing.
a Q1D material in groups, and the time correlated soliton model that follows from
this is based upon coherent, Josephson-like tunneling of microscopic solitons of
charge 2e [90].
Problems
1. Use a small k expansion around the Dirac point in graphene to show that the
bands are linear in this region.
2. The conductivity of graphene is observed to increase linearly as the back gate
voltage is varied (for either electrons or holes). As the back gate voltage has a
capacitance effect on the graphene, the density of electrons or holes must also
vary linearly with the voltage. Using the density of states for graphene,
determine what the energy variation of the mobility must be to give linear
conductivity changes.
3. Graphene can have phonons whose vibration lies either in the plane or
normal to the plane. Explain why the phonons whose vibration is normal to
the plane cannot scatter electrons in the first-order perturbation theory that is
usually used.
4. For the weak localization signals of figure 4.7, use the theory of the last
chapter to estimate the phase coherence time.
5. In multi-walled CNTs, it is usually found that the current is flowing mainly
in the outer layer of the tube. What physical effect could cause this to be the
case?
6. Explain the difference between a charge density wave and a spin density
wave.
4-19
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4-22
IOP Publishing

(Second Edition)
David K Ferry
Chapter 5
Localization and fluctuations
In the previous chapters, we have discussed the role of disorder primarily as creating
scattering centers and affecting phase coherence. But, disorder can also lead to
localization of the carriers and to an effect known as conductance fluctuations.
These are not unrelated effects, although many have thought that they were. In
earlier days, it was assumed that localization, which is often called strong local-
ization in distinction from the weak localization discussed in previous chapters, was
caused by strong disorder. At the same time, it was believed that conductance
fluctuations were caused by phase interference in the potential landscape created by
the weak disorder arising from a large impurity or defect density. In fact, the two are
closely related, which should have been obvious from the fact that the same
theoretical model is used to simulate both effects! This model is the Anderson
model [1], which we will discuss in the next section.
The fundamental problem with disorder is to determine how it affects the allowed
energy levels in the material. Normally, we think of band semiconductors as having
a region of allowed energy states, in which the electron waves are free to propagate,
and a band gap, in which the electron waves become localized and non-propagating.
With disorder in the band semiconductor, the edge between the band and the gap
becomes a dispersed gray area. Usually the width of the band increases allowing
states to broaden into the gap area, which is called band tailing. But many of the
band states in this tail, as well as somewhat into the normal band region, can become
localized. Here, the long-range order of the crystal can be compromised, with the
ultimate limit producing amorphous material. Because of the extending of the band
region and its interaction with localized states, a new parameter arises which plays
the role of the normal band edge, and this is the mobility edge. The mobility edge
now separates the localized states from the conducting, or long-range ordered,
states.

Because of the disorder in the potential, which is assumed to be random, many

small AB-like loops of various size can exist in the normally conducting states. In
addition, many back-scattering loops, like those in weak localization, can also exist.
These various phase coherent loops will randomly appear and disappear as one
varies the Fermi energy or a magnetic field. This then leads to a fluctuation in the
conductance as the interferences within a phase coherent area vary with these
external variations. For some time, it was assumed that these conductance fluctua-
tions were basically different from the localization phenomena, and that the former
were universal, in which they had an amplitude that did not vary with the disorder
strength, or did not vary with choice of perturbation—Fermi energy variation or
magnetic field variation. Today, we know that this is not the case. Rather, the
amplitude of the fluctuations vary with the size of the disorder, just as the
localization behavior does, up to a critical value where a single Landauer channel
is being switched on or off. At that point the amplitude saturates, which would lead
one to believe in a universality if the strength of the disorder is sufficiently large.
Hence, both effects derive from the same source—disorder in the crystal. But, very
high quality material, such as the super high mobility GaAs/AlGaAs heterostructure
generally does not display any conductance fluctuations.
So, in this chapter, we will begin with the Anderson theory of how disorder affects
the electronic states and leads to localization. Then, we see how this leads to
fluctuations, and how these vary with the strength of the disorder. This leads us to a
discussion of the phase coherent area and the phase-breaking time.
5.1 Localization of electronic states

The concept of a rapid transition, at some critical energy, from a set of strongly
localized states with only short-range order to a set of extended states with long-
range coherence of their wave function, is remarkable. Normally, in band semi-
conductors, this is just the band edge and we give it no further thought. Perhaps the
reason lies in the fact that most people are not taught about the complex band
structure with the continuum of localized states that exist throughout the band
gap [2]. But, this idea of such a transition warrants further investigation. Here, we
will follow the approach of Anderson [1], using one version of the several he
discusses. The model adopts a set of atomic sites in a crystal, in which the atomic
energies are randomly, but uniformly, distributed over a fixed energy range, which is
commonly denoted as W. This means that the energy is a random function with a
probability distribution function given by 1/W for energies within a given range,
usually denoted as −W/2 ⩽ E ⩽ W/2. This will have the effect of shifting the zero of
energy to the center of the band, whereas we normally associate it with one of the
band edges. The reason for this will become clear later. We will find that there is a
critical value of the width, such that if W is greater than this critical value, all the
states in the band will be localized. The value of W at this critical value is normally
associated with the Anderson transition.
5-2
5.1.1 The Anderson model

In this approach, each atomic site (which may not lie on a lattice site in the disorder
model) has a ‘site’ energy, which corresponds to the atomic energy level of that
particular atom, and an ‘overlap’ energy describing the interaction of the wave
function of that atom with the wave function of neighboring atoms. For example, in
the previous chapter, our model of graphene assumed the site energy was zero and
the overlap energy was the parameter γ0. There are two general approaches to
disorder that have developed with other approaches to this topic. One is to consider
primarily site disorder, as described by the energy distribution discussed above, or
some other equivalent form. The second is bond disorder which primarily treats a
random variation in the overlap energy. The Anderson model used here is primarily
a site disorder model, but it is mapped into a system in which the lattice is regular
with the atoms located at the lattice sites.
In the presence of the other atoms, one can use normal lowest order perturbation
theory to write the Schrödinger equation in terms of the wave function and the
perturbation of the neighboring sites as
∂ψs
iℏ + Hψs = Eψs = E 0,sψs + ∑s′≠s Vss′ψs′, (5.1)
∂t
where the subscript s refers to the site and the on-site energy E0,s is a random variable
as described above. The quantity Vss ′ is the overlap energy between neighboring sites,
and the sum runs over only those nearest-neighbor sites. This latter energy is usually
relatively constant. The last term on the right-hand side of equation (5.1) is just the
first-order perturbation term, and higher-order terms lead to a series expansion for
either the wave function or the energy. In such a series, the random site energy is just
the zero-order term, while the actual final energy can be expressed as
Vss ′Vs ′ s Vss ′Vs ″ sVs ′ s ″
E = E 0,s + ∑s′≠s E + ∑s′, s″≠s (E + …. (5.2)
0,s − E 0,s ′ 0,s − E 0,s ′)(E 0,s − E 0,s ″)
Unless the energy Es is real, the wave function decays with time since the wave
function is normally connected to the energy via a term exp(iEst /ℏ). Thus, the nature
of the states will be investigated by examining the convergence properties of the
infinite series in equation (5.2). If this series converges, then the energy is real and the
state is localized at that site. On the other hand, if the series does not converge, it
must be assumed that the energy lies within an extended band of energies which
correspond to wave functions that are extended over the entire crystal.
The series in equation (5.2) is a stochastic series since the zero-order on-site
energies are a random variable uniformly distributed between −W/2 and W/2.
Consequently, we can examine the convergence of the series in a statistical sense.
Each term in the series contains VL+1, where L is an integer giving the order of the
term in the series. In a general lattice, each atom has Z nearest neighbors, so that
there are ZL contributions to the Lth term, a point we shall use later. The general
Lth term has the form
5-3
V
Ts ′ = . (5.3)
E 0,s − E 0,s ′
If the values of the site energies on the primed subscripted sites are statistically
uncorrelated, the magnitude of the contribution of a product of such terms can be
estimated by taking the average of its logarithm, as
〈ln∣TsTs ′…Ts ″∣〉 ∼ L〈ln(∣T ∣)〉. (5.4)
This form of the terms tells us that the series is to be interpreted as a geometric series,
which is a special form of a power series that will converge provided that the ratio of
subsequent terms approaches a limit which is less than unity. Thus, we require that if
the coefficients of the series are AL, the series will converge if
AL+1
limL→∞ x < 1, (5.5)
AL
where x is the argument of the series, which here is our energy. We now apply this
convergence criterion to the perturbation series for the energy (5.2). This leads us to
the conclusion that the series will converge if
Z exp(〈ln(∣T ∣)〉 < 1). (5.6)
We now introduce the probability distribution function discussed above in order to
evaluate the expectation value. This leads to
W /2
1 V
〈ln(∣T ∣)〉 =
W
∫−W /2 ln E 0,s − E 0,s ′
dE 0,s ′
⎡ (5.7)
1 4E 0,2 s − W 2 E 0,s 2E 0,s + W ⎤⎥
= 1 − ⎢ln +2 ln .
2 ⎢⎣ 4V 2 W 2E 0,s − W ⎥⎦
It is obvious from the above result that the value of W required for localization
depends upon the energy in which one is interested. The energy in equation (5.7) is
now a smooth variable. If we take the center of the band, where the energy is 0, then
only the first term in the larger parentheses survives, and
⎛ ⎛ W ⎞⎞
Z exp⎜1 − ln⎜ ⎟⎟ < 1 (5.8)
⎝ ⎝ 2V ⎠⎠
or
W
> Z exp(1) ∼ 10.87 (5.9)
2V
is required to completely localize the band (we have taken Z = 4 for the tetrahedrally
coordinated semiconductors). In general, we can say that W > 2.72ΔE, where
ΔE = 2VZ will totally localize the band, so that no extended states exist. At the other
5-4
extreme, we can take W = 0, and we arrive at the equivalent inequality that says
states will be localized if
E 0,s > ΔE /2. (5.10)
But, this is just the normal band requirement, as the bandwidth is basically VZ.
It is of interest to also determine the point at which an energy at the edge of the
distribution, E0,s = W /2 is localized. We can then combine the two logarithm terms,
to find that when W > ΔE exp(1)/4, we start to develop localized states at the edge of
the band. Hence, there is a critical size of the disorder that must be exceeded before
the band states near the edge begin to be localized. These different conditions are
sketched in figure 5.1.
We can illustrate the localizing effect of disorder and the random potential in
another way, and that is to simulate a standard mesoscopic material such as the
AlGaAs/GaAs heterostructure. The conducting layer is a Q2D electron (or hole) gas
located at the interface between the two materials. The simulation is a study of the
conductance of this electron layer, following the approaches of sections 2.7 and 2.8.
In this approach, we discretize the two-dimensional layer with a grid size of 5 nm,
and consider a range of widths, in the range of 0.2–0.6 μm, and lengths, in the range
of 0.3–0.8 μm. We use the Anderson model to impose a random potential at each
grid point according to the uniform distribution discussed above. This gives a
random potential whose peak-to-peak amplitude is W. While this random potential
induces states below the lowest energy of the band (and above the highest energy in
the band), it also localizes a fraction of the normally transmitting states. A critical
energy separating the localized modes and the propagating modes defines what has
been called the mobility edge [2]. Since we are going to vary the amplitude of the
random potential, it is useful to see how this localizes the conductance. We consider
a density of 4 × 1011 cm−2, which gives a Fermi energy about 15 meV in GaAs. Since
we have a finite width of sample, the system is quantized in the transverse direction
Es,c
ΔE
0.5
W
0 1 2 3 ΔE
Figure 5.1. The critical energy at which states become localized (red curve). The localized state are shown as
green shading, while the low disorder range where there is no formation of localized states is indicated by the
dashed line.
5-5
into a set of modes, and the conductance is determined by the number of these
modes whose transverse eigen-energies lie below the Fermi energy. Then, we
determine the fraction of these modes that actually propagate in the presence of
the random potential. This is plotted in figure 5.2, and it may be seen that the
behavior suggested by figure 5.1 is certainly followed. The error bars represent an
average over many sizes of sample, all of which have different implementations of
the random potential. While there may be 30–60 modes propagating normally, once
the disorder becomes sufficiently large, only a very few remain propagating. This is a
significant point, as we will see later in discussions of fluctuations. Nevertheless, the
lowest data point in the figure lies at a peak-to-peak disorder potential of 0.1Et,
where Et is the hopping energy (2.54). Even though this potential reaches peaks more
than twice the Fermi energy, almost all of the modes remain propagating. So, it
takes a significant amount of disorder to really affect the band states.
5.1.2 Deep levels

The levels of interest above were the shallow levels, as they are sometimes called. In
these levels, the impurity ionization energies were only a few, or a few tens, of meV.
These levels are usually treated within the effective mass approximation as a kind of
hydrogenic impurity, and this continues in the localized states. In some defects,
however, the impurity creates a significant lattice distortion about the core of the
1
Fraction of Modes Propagating
0.1
0.01
0.1 1 10
Vpp (eV)
Figure 5.2. Fraction of modes that remain propagating as a function of the peak-to-peak amplitude (W) of the
random potential. The values are averaged over a number of samples of various sizes. The amplitude of the
random potential is normalized to the hopping energy.
5-6
impurity site. The difference between the case above and these so-called deep levels
lies in the central cell potential—the core potential that interacts on a short range
(whereas the Coulombic potential is relatively slowly varying over the unit cell size).
When this core potential is sufficiently strong to affect the electrons, this affect is
much stronger. An electron bound to the deep levels is largely localized by the
strength of the potential to the range of the local unit cell. These levels are often near
the middle of the band gap. Two effects create a sizable contribution to the
understanding of these deep levels and their energy levels. These are the Jahn–
Teller distortion and the Franck–Condon effect.
The Jahn–Teller theorem states that any electronic system with energy levels that
are multiply degenerate may be split when the confining potential is affected by a
different symmetry. Thus, if we put a deep level atom into a crystal such that the
electronic ground state is multiply degenerate, there can be distortion of the lattice
around this atom that raises this degeneracy. For example if an As atom sits on a Ga
site (a so-called anti-site defect), there are two extra electrons bound to the As atom
in the neutral state, and these two atoms have degenerate energy levels. This will lead
to a local distortion around the As atom that raises this degeneracy, an effect that
has been seen experimentally [3].
There is another distortion, however, that can also arise. The strength of the
central cell potential can significantly affect the actual band structure in the
neighborhood of the defect atom. The role that this interaction plays is considerably
different when the defect is neutral than when it is ionized and becomes charged.
Thus, we expect the local band structure and the defect energy levels to change as the
defect is ionized by removing an electron. This means that the optical transition
energy is different for excitation and for relaxation, and this is the Franck–Condon
effect. According to this latter effect, the energy of the localized state will change as
the charge level of the defect is modified, and this arises from a local polarization of
the lattice. In other words, the atoms in the vicinity of the local defect will relax to a
new set of positions, in much the same way in which a surface relaxes. This
relaxation is quite local in the lattice and therefore occupies a large part of the
Brillouin zone; hence the relaxation is often accompanied by the excitation of a large
number of phonons. For this reason, the optical transitions for a deep level are often
accompanied by a number of ‘phonon sidebands’—transitions that differ from one
another by a phonon energy.
The distortion that accompanies the Franck–Condon effect can lead to another
interesting phenomenon. When an electron is excited from a localized deep level, the
lattice relaxation can lead to a reduction in the conduction band energy that gives a
configuration of lower energy. The electron can no longer recombine with the
charged deep level, since it no longer has sufficient energy to make the transition
back; in essence there is an energy barrier preventing the electron from recombining,
as it must now not only have the energy to recombine but must also drive the lattice
relaxation back to its former state. If this kinetic barrier is sufficiently large, we can
have the effect of persistent photoconductivity, which is observed in GaAlAs at low
temperatures [4]. In this latter case, it has been hypothesized that this effect is related
to a donor-vacancy complex, referred to as the D-X center. Optical absorption
5-7
excites electrons from the deep levels, which produces a higher conductivity. When
the light is removed, the semiconductor remains conducting until the temperature is
raised to the order of 200 K or so, where the excited carriers have sufficient thermal
energy to surmount the barrier and recombine.
In both physics and chemistry, the details of the Franck–Condon effect and deep
impurities is based upon what are called reaction coordinates. One of the earliest
papers in this area actually dealt with electron transfer from a donor or acceptor in
which there was a lattice relaxation described by a reaction coordinate [5]. As
discussed above, the local state of the lattice around the impurity changes with the
charge on the impurity, and this is described by the reaction coordinate. This also
changes the energy level, so that optical transitions can change depending upon the
charge state. In this approach, the interactions can be studied by a spin-boson model
that provides a pragmatic, yet realistic formulation for the role of dissipation in the
electron transfer [5]. The role of a driving electric field was also considered in the
study. Others have also studied electron transfer when coupled to collective boson
degree of freedom, including the study of the fluctuations in the system [6]. Again,
these latter studies also employed the reaction coordinate for the impurity system.
Let us now describe the approach with the reaction coordinate model. The idea is
about the same whether we deal with a molecule or with condensed matter. For an
impurity, say a donor, the upper state corresponds in the condensed matter system
with the conduction band, while the lower state is the actual impurity level (in the
molecule, this might actually be an extended state with dispersion). When the
electron is excited from the donor impurity, the local potential changes primarily
due to the short range local potential around the impurity. The potential change
induces a lattice relaxation around the impurity, and this in turn changes the level of
the impurity relative to the conduction band [7], but this is usually treated as a
change in the conduction band. There are two contributions in this interaction. First,
the Jahn–Teller theorem tells us that any electronic system with multiply degenerate
ground states is unstable against a distortion that removes the degeneracy. This leads
to the lattice relaxation, which is of course much more prevalent in deep donors as
opposed to shallow impurity levels. As a result, the atom is actually displaced in
position, and this is the reaction coordinate. But, there is a second distortion, and
that is the change in the band structure between the neutral and the ionized states, as
mention above, and this is the Franck–Condon effect, and this results in a difference
between the optical excitation energy and the thermal excitation energy of the
impurity [3, 8]. The overall situation, as mentioned in the previous paragraph, is best
described by a reaction coordinate diagram, as shown in figure 5.3. The two
potential curves cross at a point x*, which is the point at which electron transfer
occurs. Once the electron is into the second potential V2, it must gain an energy
greater than ε0 in order to recombine with the donor. Actually, if we consider
tunneling, the electron only needs to be excited to positive energy, where it can
tunnel to V1. In semiconductors, this leads to persistent photoconductivity at low
temperatures. Here, the Hamiltonian can be written as [5, 6]
H = HEL + HRC + HB , (5.11)
5-8
Figure 5.3. A diagram of the reaction coordinate x and the various potentials. V1 represents the neutral
impurity, whose energy level is below those shown here. The potential V2 represents the shift of the conduction
band when the neutral impurity level is taken as the reference point for the energy. The energy (ε0 + ε′)
represents the energy the electron must gain in order to recombine with the charged impurity.
where the bare electronic system is written in terms of a two-state pseudo–spin

system as
1 ℏΔ
HEL = − [V1(x ) + V2(x )]σˆz + σˆx. (5.12)
2 2
The two pseudo-spin operators are given by
σˆz = ∣1⟩⟨1∣ − ∣2⟩⟨2∣
(5.13)
σˆx = ∣1⟩⟨2∣ + ∣2⟩⟨1∣ .
The two states are those of the impurity and band combinations. The coupling term
Δ is assumed to be independent of the reaction coordinates. The reaction coordinate
term is given by
⎡ p2 ⎤⎡1 0 ⎤
HRC = ⎢ + V1(x ) + V2(x )⎥⎢ . (5.14)
⎣ 2m ⎦⎣ 0 1 ⎥⎦
The two harmonic potentials, described in figure 5.3, are given as

mω 02 2
V1(x ) = x
2 (5.15)
mω 02
V2(x ) = ( x − x 0 ) 2 − ε0 ,
2
and the bare term HB is the energy of the non-interacting particles. Here, the so-
called reorganization energy is V1(x0), and the excitation energy indicated by the red
arrow in figure 5.3 is the sum of this energy and the offset energy ε0. One notes that
the reaction coordinate x is the primary variable in the Hamiltonian and the
resulting problem being studied. The important point is that when the electron is
transferred from the ‘donor’ molecule (state ∣1⟩) to the ‘acceptor’ molecule (state ∣2⟩),
the relaxation process makes this a ‘one way’ reaction, as the barrier (ε0 + ε′) to the
back reaction is created by the molecular relaxation characterized by the reaction
coordinate. This is what leads to the persistent photoconductivity.
5-9
5.1.3 Transition metal dichalcogenides

We have already shown the role of localization in a GaAs/AlGaAs system in
figure 5.2. A more complicated form can be found in the transition metal
dichalcogenides (TMDCs), discussed in section 4.4. The TMDCs form interesting
two dimensional semiconductor layers that have been shown to be useful for
electronic applications. However, they currently suffer from the presence of defects
that act as both deep levels and random potentials leading to localization. In
particular, it has been shown that charge impurities such as S vacancies in MoS2 and
impurities from the dielectric environment not only have significant impact on the
mobility of the MoS2 devices [9], but also can lead to the formation of an impurity
band tail within the band gap region as well as localized states above the conduction
band edge [10]. The TNDCs tend to have a significant number of chalcogenide
vacancies (e.g., missing S in this case). These vacancies lead to both effects. First, the
vacancy itself tends to produce a deep level that lies near mid-gap in the material.
Secondly, the missing atomic potential produces a modulation of the crystal
potential formed from the pseudo-potentials of the atoms themselves. This modu-
lation creates the random potential that leads to the Anderson model and the
resulting localization.
To gain a quantitative understanding of the effects of these impurity states, we
examine the room temperature device characteristics using a model proposed by
Zhu et al [10]. From studying extensive capacitance voltage measurements between
a back gate on the TMDC and the layer itself (using a SiO2 insulator grown on the Si
back gate), these authors find that there is a mid-gap state that has a population of
approximately 1012 cm−2. This is the deep level, although it is not known if this level
has any lattice relaxation associated with it. They also find an extremely broad
distribution of localized states, as in the Anderson model, and an extremely high
mobility edge for the conduction band. They then adapt a simulation model to fit the
data. In this model, the impurity band tail is incorporated by the following density of
states distribution:
⎧ ⎡E ⎤ EG
⎪ αD0exp⎢ ⎥ , − < E < 0,
⎪ ⎣φ⎦ 2
Dn(E ) = ⎨ (5.16)
⎪ ⎡ E⎤
⎪ D0 − (1 − α )exp⎢⎣ − φ′ ⎥⎦ , E > 0.
⎩
As before, the energies here are referenced to the conduction band edge, in the
absence of the disorder. Here, D0 is the normal two-dimensional density of states
given by (2.21), using an effective mass of 0.4 m0. The (1 − α) term for positive
energy accounts for the states that are removed from the conduction band to form
the band tail states of the first line. The parameter φ is the characteristic width of the
localized states that form the band tail discussed above and is found to be ∼100
meV, and φ′ is chosen so that the two piece-wise functions have a continuous
gradient at E = 0. The parameter α is found to be ∼0.33. From this model and the
5-10
Figure 5.4. Localized states trap electrons into non-mobile states. Here are the total electron density and the
free carrier density for a MoS2 monolayer at 300 K. The parameters are discussed in the text.
measured conductance, the mobility edge is found to lie 10 meV above the
conduction band edge.
To calculate the free carrier density, we consider only the extended states above
the mobility edge energy EM with respect to the conduction band edge:
∞
n free = ∫EM
Dn(E )PFD(E )dE . (5.17)
In different experiments, reported by Xiao et al [11], a slightly different value of the

band tailing, φ ∼ 110 meV, and a positively charged oxide impurity density of
3.5 × 1012 cm−2 were found. As previously, these were found for a layer of MoS2
placed upon an oxidized Si wafer, so that the Si could be used as a back gate to vary
the electron concentration. Using this data and parameters, we plot the total induced
electron concentration as well as the free carrier (those above the mobility edge)
concentration at room temperature in figure 5.4. The difference in these two
densities yields the number of carriers which are trapped in the localized states.
5.2 Conductivity
As indicated in the previous section, the role of disorder is to create band tails as well
as to localize various states, and we saw this in the numerical example for GaAs. We
indicate this in figure 5.5. Here, the normal conduction and valence bands are shown
in blue. Since the total number of states in the band is constant, the presence of
disorder must move some states from the band to the tail in order to have band
tailing. This produces the new band edges shown in red. As is clear from figure 5.1,
there is a threshold in the disorder strength required to produce the band tailing
effects. Normal impurities in the semiconductor do not produce enough disorder to
5-11
Figure 5.5. The normal bands are shown in blue, along with the localized states that arise in the presence of
significant disorder. The mobility edges are shown in green.
cause the band tailing, unless the density is approaching the degeneracy level. In
figure 5.5, we show the localized states that arise from the disorder in brown. They
extend throughout the tails and into the band proper, as was evident in the example
of figure 5.2. The boundary between the localized states and the extended states is
termed the mobility edge, shown as the dashed green vertical lines in figure 5.5. The
localized states now have wave functions that are spatially localized and are not
sufficiently dense to cause overlap of these wave functions. The degree of tailing and
position of the mobility edge both depend upon the degree of disorder in the system.
In many amorphous semiconductors, the tailing from the conduction and valence
bands actually overlaps so that there is no real band gap. However, excitation
studies, such as those for photoconduction, actually measure the mobility edges, as
the localized states do not contribute to normal conduction. Hence, the optical gaps
can be larger than the normal band gaps in crystalline semiconductors.
In crystalline (ordered) semiconductors, the activation energy for the conductivity
arises from the presence of the band gap; that is, the intrinsic electron and hole
population (in non-degenerate material) is given by
⎛ Egap ⎞
ni = NC NV exp⎜ − ⎟, (5.18)
⎝ 2kBT ⎠
where NC and NV are the effective density of states for the conduction and valence
bands, respectively. The band gap variation arises because the density of states in the
ordered material has a sharp cutoff at the band edges. However, in disordered
material, the existence of the tails of localized states below EC and above EV would
appear to eliminate the possibility of a sharp activation energy. This is not really the
case, because of the basic difference in the nature of the extended and localized
states. In the extended states, the wave functions have some coherence (at least in
amplitude) over many ‘unit cells’ of the material, so that one might expect some of
the concepts of effective mass and mobility to be applicable. In this sense, the
mobility exists and may be not too much smaller than in the ordered state. In the
localized states, on the other hand, the wave functions do not have any long-range
coherence and do not overlap the neighboring atoms, or ‘unit cells’, to any great
degree. Therefore, the concept of nearly free motion is just not applicable, and one
must think of the electron moving from one cell to the next by a tunneling process, or
‘hopping’ process, in which the carrier is thermally activated over the potential
5-12
barrier between the atoms, only to be retrapped at the next atom. There is thus a
large barrier preventing motion among the atoms in the localized states, and the
mobility is essentially zero (very small) on the scale of the extended states.
Thus, at the transition energy between the localized and the extended states, it is
to be expected that the mobility will fall by several orders of magnitude. In the
disordered semiconductors, the boundaries between these two types of states are
therefore the boundaries between states for which the carriers are free to move and
those in which the movement is largely forbidden. The disordered semiconductor
therefore has regions of allowed and forbidden mobility, as opposed to allowed and
forbidden states, and the activation energy is that required to take the electron from
an extended state in the valence band to an extended state in the conduction band.
As we mentioned above, the activation energy will usually be larger than the band
gap, as part of the otherwise allowed states in the conduction and valence bands will
in fact have forbidden mobilities due to the disorder-induced localization.
When the energy is in the range of the localized states, conductivity proceeds
mainly by a mechanism of hopping, whereby an electron jumps from one site to a
neighboring site. Repeated hopping leads to the possibility of the carrier transiting
through the entire sample, but the conductivity is reduced by the need to be excited
over a barrier which has to be surmounted at each individual hop, and this barrier
can also be a random function. Since the wave functions are localized on a single
site, the probability of the jump, either by a phonon-assisted transition over the
barrier or tunneling through the barrier, is proportional to the overlap of the wave
function on the two neighboring sites, which falls off as exp(−αR), where R is the
hopping distance and α is a decay constant (which for tunneling can be calculated if
the details of the potential barrier are known). Such hopping is known as nearest-
neighbor hopping [12] and is often found in the case of impurity conduction in highly
doped semiconductors. The conductivity is proportional to the density of states at
the Fermi level and the width of the Fermi–Dirac distribution, the difference in the
probabilities for forward and backward hopping when a field is present, and an
effective velocity that is approximately the distance times a ‘hop frequency’. The
latter is related to the phonon frequency in phonon-assisted hopping. Thus, the
current density is given by [13]
⎛ ΔE ± eFR ⎞
J ∼ ∑ ekBTρ(EF )Rνph exp⎜ −2αR − ⎟
± ⎝ kBT ⎠
(5.19)
⎛ ΔE ⎞ ⎛ eFR ⎞
= 2ekBTρ(EF )Rνph exp⎜ −2αR − ⎟sinh⎜ ⎟,
⎝ kBT ⎠ ⎝ kBT ⎠
where ΔE is the effective barrier height, νph is the phonon-induced attempt

frequency, and F is the electric field. For weak fields, the hyperbolic sine function
can be expanded, and this leads to
⎛ ΔE ⎞
σ = 2e 2ρ(EF )R2νph exp⎜ −2αR − ⎟. (5.20)
⎝ kBT ⎠
5-13
Nearest-neighbor hopping is expected mainly when αR ≫ 1. In this case, the

conductance is greatly reduced by the exponential factor. In this limit, the energy
levels that are allowed are expected to be rather widely spaced, and an estimate of
the barrier may be obtained from the density of states ρ(EF), which is the number of
states per unit volume (in d-dimensional space). Then, the number of states per unit
energy is just Rd ρ(EF), and the average energy spacing per state is ΔE ∼ [Rd ρ(EF )]−1.
But, if this spacing is that observed at a single site, it is comparable to the separation
of the atomic levels at that site and hence is of the order of the bandwidth. This leads
us to the conclusion that such nearest-neighbor hopping is expected only in the case
for which all levels in the conduction band (or the impurity band) are localized, so
that any mobility edge lies in a higher-lying band [12].
On the other hand, it is often the case that αR ⩽ 1, so that the hopping range may
extend well beyond that of the nearest neighbor. This is the regime of variable-range
hopping. Here, the distance over which the hop is expected to occur increases with
increasing temperature, and the exponential argument is a very low power of the
inverse temperature [13, 14]. At a given temperature, the electron will ‘hop’ to a site
that lies somewhere inside a radius of order R, which gives γd(R/R0)d available sites
for the hop (R0 is an average distance for the hops), where γd is a numerical factor
that depends upon the dimensionality (this factor is 4π/3 in three dimensions, π in
two dimensions, and 2 in one dimension). The hop generally occurs to a site for
which the activation energy is the lowest. The latter is given by the same argument of
the previous paragraph, and is ΔE ∼ [γd R 0d ρ(EF )]−1. Thus, the effective barrier
depends upon the distance over which the hop can occur. The probability of a
hop is proportional to
⎛ ΔE ⎞
exp⎜ −2αR − ⎟ (5.21)
⎝ kBT ⎠
as before. Because of the definition of the average energy barrier, we can find a
maximum value in the exponential argument when
ΔE 1
α= = d +1
. (5.22)
2RkBT 2γd R 0 ρ(EF )kBT
The prefactor for this exponential will involve the average hop distance, which
involves the factor R as well, giving
d
〈R〉 ∼ R, (5.23)
d+1
and, as before, we can write the conductivity as
⎛ B ⎞
σ = 2e 2ρ(EF )〈R〉2 νph exp⎜ −2αR − 1/(d +1) ⎟ , (5.24)
⎝ T ⎠
5-14
where
4α d /(d +1)
B= . (5.25)
[2γd ρ(EF )kB ]1/(d +1)
The result of equation (5.24) provides the famous T −1/4 behavior of the conductivity
that is often found in disordered material (in three dimensions). In two dimensions,
the expected behavior is T −1/3, and this has been found in activated transport
conductivity in the inversion layer of MOSFETs at low temperatures [15, 16]. In
these experiments, Na was intentionally introduced into the oxide of the MOSFET.
Then the Na atoms could be induced to drift through the oxide by a large gate
voltage, and this allowed one to control the scattering effect of the Na atoms, and
thereby the degree of disorder introduced into the channel of the transistor. That is,
the charge in the oxide provides the random potential seen by the electrons in the
channel and leads to the disorder. The strength of the disorder is controllable by the
distance the charge resides from the oxide–semiconductor interface. The Na doped
transistor provides a unique opportunity for the evaluation of various models for
hopping conduction because it allows one to vary a number of parameters in the
transport model by the use of substrate bias, gate bias, and the position of the oxide
charge [17]. In figure 5.6, we show the results on the temperature dependence of the
conductivity in such a MOSFET with a gate oxide thickness of 100 nm and a
channel length of 10 μm [17]. Here, the Na concentration at the interface was
7.5 × 1011 cm−2 and the carrier density was 2.8 × 1011 cm−2. The various curves are
Figure 5.6. The temperature dependence of the conductivity in a Si MOSFET with Na in the oxide, as
described in the text. The various curves have different values of the disorder bandwidth ΔE and the value of
α−1 found from fitting to the theory (solid lines). These values are, respectively, a: 7.38 meV and 4.11 nm;
b: 5.49 meV and 4.49 nm; c: 4.36 meV and 4.15 nm; d: 3.74 meV and 4.56 nm; e: 3.21 meV and 4.73 nm. The
substrate biases for these five curves were (in order) 8.0, −3.0, −1.0, 0.0, and 0.5 V, respectively. (Reprinted
with permission from [17]. Copyright 1986 the American Physical Society.)
5-15
for different values of the substrate bias. It is clear that the straight line portions of
the various curves fit to the temperature dependence expected for variable range
hopping in this Q2D system.
5.3 Conductance fluctuations

In our discussion of weak localization in an earlier chapter, it was clear that the
existence of time-reversed paths can lead to quantum interference that causes a
correction to the conductivity. This was also apparent in the AB effect. These paths
are modified by the change in the Fermi energy (changes in the wave phase velocity)
and in the magnetic field (similar changes in the momentum occur through the
vector potential). In fact, the magnetic field destroyed the weak localization while it
made possible the oscillatory terms in the AB effect. So, the effect of the magnetic
field can be more complicated than that of the variation in the Fermi energy. If a
sample is composed of a great many such loops (and not necessarily time-reversed
loops), with each contributing a phase-dependent correction to the conductivity,
then the summation over these loops may or may not ensemble average to zero,
depending upon the number of such loops contained within the sample (and, hence,
upon the size of the sample). On the other hand, in mesoscopic systems, where the
number of such loops is relatively small, ensemble averaging can fail, and these
loops lead to the presence of conductance fluctuations [18, 19]. Since the impurity
distribution is reasonably fixed for a given sample configuration, the particular
oscillatory pattern is often thought of as a fingerprint of an individual sample. That
is, the impurity distribution is different from one sample to another, and therefore
the detailed nature of the interference pattern seen in any one sample is a
characterization of its unique impurity distribution. In figures 5.7 and 5.8, we
show experimental data for fluctuations in AlGaAs/GaAs heterostructures [20, 21].
One view in estimating the amplitude of these fluctuations arises from the
Landauer formula, in which the conductance is quantized for each possible channel
through the sample. Here, the conductivity is given by [22]
2e 2 N
G=
h
∑i,j=1∣tij∣2 , (5.26)
which differs from equation (2.36) in the possibility that the disorder induces
scattering from one channel (mode) to another channel. This is characterized by the
transmission tij which describes the transmission of the wave entering in channel j
and leaving in channel i. In the case of no such inter-channel scattering, the sum
reverts to the value N found in equation (2.36).
It has been suggested that the amplitude of these fluctuations is universal in nature
and arises directly from the universality of the Landauer formula. This suggests that
we could write the amplitude of the fluctuations (determined from the variance of
the conductance) as [23, 24]
g Ce 2
δG = var(G ) ∼ , (5.27)
2 βh
5-16
Figure 5.7. Magnetoconductance for a narrow sample for a variety of temperatures and a gate voltage of
−3.01 V. At zero magnetic field, the base conductance was 10−4 s and varied little over these temperatures.
Figure 5.8. Longitudinal magnetoresistance as a function of the magnetic field at several temperatures. The
sample is depicted in the inset. (Reprinted with permission from [21]. Copyright 1988 the American Physical
Society.)
5-17
where g is a factor accounting for spin and valley degeneracy (we use only spin
degeneracy in equation (2.36)), C is a constant thought to be about 0.73 for a narrow
channel with L ≫ W, and β is unity in zero magnetic field but takes a value of 2 when
the magnetic field lifts the time-reversal symmetry of the system. It would be nice to
adopt this view, but this concept of universality is contrary to our intuition. As we
learned above, the role of disorder comes on gradually as the size of the disorder is
increased. Certainly, in the absence of disorder one would not expect any local-
ization or fluctuations, and there are no reports of conductance fluctuations in the
very high mobility AlGaAs/GaAs heterostructures discussed in chapter 1. Indeed, in
our own numerical studies of conductance fluctuations, it was found that the
amplitude of the fluctuations increased gradually as the size of the random potential
was increased [25]. Certainly, the amplitude tended to saturate once a critical value
of disorder was reached, and this does have a relation to the Landauer formula. If we
consider that the phase interferences have a maximum value when a single channel is
being randomly switched on or off, then the peak-to-peak value of the fluctuation
should be of the order of 2e2/h for a spin degenerate channel. Naturally, the peak
amplitude is one-half of this value and the rms value is reduced further. This leads to
a value for the effective amplitude of
g 2 e2 2e 2
(5.28)
δG = var(G ) ∼ ∼ 0.7 ,
2 2h h
for a spin degenerate channel. And, in a high magnetic field, this would be reduced
further by a factor of two as the spin degeneracy is removed, although this is not seen
in some early experiments [26]. One problem, of course, lies in the fact that most
studies on high mobility heterostructures are not able to perform Fermi energy
sweeps, as they are ungated samples. This was not the case in early studies in Si
MOSFETs where there is a natural gate voltage that can be swept. We will see
equation (5.28) again below in connection with studies of the fluctuations in
graphene.
While the above gives a reasonable approach for the general semiconductor, it is
possible to use advanced simulation techniques to study the fluctuations. Let us
demonstrate this with a study of the conductance fluctuations seen in a graphene
nanoribbon [27, 28]. In the theoretical approach, the disorder arose from impurities
located in the SiO2 sheet upon which the graphene rested. The local potential arising
from these impurities was modeled self-consistently for each different siting of the
impurities, with an impurity density of 3 × 1012 cm−2. This local potential was then
imposed onto an atomic-basis simulation of transport in the graphene nanoribbon [29],
following the approach of section 2.5. The graphene nanoribbon sample used in our
calculation has a width of 199 atoms and a length of 200 columns (100 slices). So the
area of the sample will be around 24.3 nm (width) × 42.4 nm (length) = 1030.32 nm2
(we refer to this as the ‘normal’ size). This means that the number of charged
impurities in the area of the ribbon is 31. Then, these impurity charges are randomly
distributed throughout the area of the graphene nanoribbon. In addition, the
distance between the graphene layer and the impurity charge layer is set to d = a0
5-18
Figure 5.9. Amplitude of the conductance fluctuations obtained with a sweep of the magnetic field for our
calculations (blue circles) and from the experiments of [27] (red diamonds). The latter has been multiplied by a
factor of 14 to account for averaging over phase coherent regions in the experiment. The inset shows actual
fluctuations versus Fermi energy for two different magnetic fields. Reprinted with permission from [28].
Copyright 2016, Institute of Physics Publishing.
and the screening coefficient is ξ = 1/(10 × a0), corresponding to a screening length of

10 × a0. Here, a0 = 0.142 nm is distance between two adjacent carbon atoms in the
nanoribbon. This random potential provides the landscape in which the transport is
computed by the above prescriptions. Because the conductance fluctuations are
random, they will change with each implementation of the impurity potential.
Hence, we use several samples, each of which has the same impurity density.
In the main panel of figure 5.9, we plot the RMS amplitude (δGrms) of the CF
induced by the Fermi energy sweeps, undertaken in the presence of various static
magnetic fields oriented normal to the graphene plane. In the inset to this figure, we
also plot the CF as a function of the Fermi energy for two different values of the
magnetic field. It can clearly be seen that the amplitude of the fluctuation is
significantly smaller with the presence of the magnetic field. Also plotted in the same
figure are the experimental data from [27], which have been multiplied by a factor
of 14. In the experiments, studies of the correlation function of the fluctuations
observed in magnetic field sweeps were used to estimate the phase coherence length
lφ, which was found to be about 200 nm [27]. This distance is much smaller than the
size of the sample over which the measurements were made, and smaller than the
estimated thermal diffusion length. Hence, it is expected that the fluctuations
observed experimentally should be considerably smaller than the theory due to
statistical averaging of many phase coherent regions. The averaging length L in the
measurements is about 2 μm. For the conditions discussed here, the reduction in
amplitude is expected to be about [24] 6(lφ/L)3/2, which suggests that the exper-
imental values should be smaller than the zero-temperature calculation by a factor of
12.9. Here, we focus upon a qualitative comparison with the experiment. Hence, we
5-19
have arbitrarily used a factor of 14 in figure 5.9 to match the experimental and
theoretical data at zero magnetic field, as this gives agreement for the values of
theory and experiment at this point.
Referring to the field-induced reduction in the magnitude of the computed
fluctuations in figure 5.9, we may conclude that this is not due to any symmetry
breaking process (such as that due to the breaking of time reversal symmetry or the
lifting of spin or valley degeneracy), since such mechanisms are not included in our
calculations. Indeed, spin degeneracy should be lifted also when the magnetic field is
applied in the plane of the graphene, but no such evidence of such symmetry
breaking was found in the study of [27]. So, there must be another explanation for
the magnetic-field-induced reduction in the amplitude of the conductance fluctuations.
In this case, it is likely related to the fact that, in a large magnetic field, there can be a
suppression of both elastic and inelastic (phase breaking) back scattering [30]. As
edge states begin to form in the magnetic field, most of the perceived resistance is
found at the current contacts, and the forward and backward edge states are
unequally occupied. Reflection from impurities cannot induce back scattering of the
topologically-protected edge states, and the magnetic field also leads to a reduction
in the number of channels that actually propagate through the sample. The result of
these two effects should be to suppress the amplitude of the fluctuations in the
conductance, and we speculate that it is this behavior that is seen in figure 5.9.
However, we caution that the edge states are nowhere near fully formed in these
samples. The fact that the experimental data falls more rapidly than our calculations
suggest that the reduction of back scattering may well be more effective in the
presence of additional scattering mechanisms, and when there are several phase
coherent regions that lead to the averaging seen in the experimental data.
In figure 5.10, we plot the amplitude of the conductance fluctuations observed in
experimental and simulated magnetic field sweeps as a function of the electron
density. The calculations are actually done as a function of the Fermi energy, while
the experiments are performed by varying the back-gate voltage. We have converted
both variations to one in terms of carrier density, so as to have a common parameter
for comparison. Again, the experimental data has been multiplied by a factor of 14
to account for the multiple phase coherent area averaging [27]. A similar reduction
in the amplitude of the fluctuations with density has been seen for both gate voltage
sweeps and magnetic field sweeps [31]. It may be seen from this figure that the
amplitude of the fluctuations is smaller than that in figure 5.9 for the Fermi energy
sweeps. The difference is of the order of a factor of 3, which has also been seen in
simulations for conductance fluctuations in GaAs [32]. This is a clear indication that
the CF do not possess any ergodic properties, as such a property would lead to equal
amplitude fluctuations for different perturbations, such as sweeps in energy versus
sweeps in magnetic field, or for different samples. In addition, it may be observed
that the amplitude of the computed fluctuations decreases somewhat as the carrier
density is raised, behavior that is captured in experiment also. This might be
explained in the experiments by a decreasing phase breaking time at higher densities
[33, 34], which would lead to a reduction of the phase coherence length. But, this
cannot be the case in the theory, as the phase coherence length is assumed to be set
5-20
Figure 5.10. The average value of the conductance fluctuations that arise in a Fermi energy (or gate voltage in
the experiment) sweep as a function of the electron density. The experimental data (red diamonds) has been
multiplied by a factor of 14 (discussed in the text). Reprinted with permission from [28]. Copyright 2016 by the
Institute of Physics Publishing.
by the simulation size. The observation, in both experiment and theory, of the
density-dependent variations in the fluctuation amplitude points, once again, to the
non-ergodic character of these fluctuations.
When the fluctuations are fully developed, their amplitude is essentially just that
expected by the turning on or off of a single transverse mode in the sample. Thus, we
expect that this will lead to a maximum rms value of the conductance fluctuation as
ηvalleyηspin e 2
δGrms = , (5.29)
2 2 h
where the η are the degeneracy factors for valley and spin, and this equation is just a
generalization of equation (5.28). This value actually agrees well with the earlier
perturbation theory estimates, but is based more on empirical observations than on
analytical methods. It should be remarked that the peak rms amplitude seen in
graphene can be less than this, with this peak value of 0.35(4e2/h) being seen only at
zero magnetic field in figure 5.9.
5.4 Correlation functions

In our discussion of weak localization in chapter 3, the magnetic field led to a decay
in the effect. In the AB effect, the magnetic field led to oscillatory conductance. We
have pointed out that the conductance fluctuations arise from similar considerations,
and that the interactions of many such phase coherent loops lead to the fluctuations.
Thus, we might expect that variation of the magnetic field or the Fermi energy will
affect the amplitude of the interaction and thus lead to the fluctuations. The
randomness of the fluctuations can be examined by the use of a correlation function
of the fluctuations. Variation in the position of the Fermi energy in the density of
5-21
states, whether caused by a gate bias or by a magnetic field, should also cause the
correlation among the phase coherent loops to decay (or at least to change). This can
be examined with the correlation function, which is often defined by
C (ΔE , ΔB ) = 〈G (E + ΔE , B + ΔB )G (E , B )〉 − 〈G (E , B )〉2 , (5.30)
although this must be normalized properly (the value of the correlation function
when ΔE and ΔB are zero must be 1). Normally, the variables are not continuous,
but are a discrete set of data when the data are gathered by a computer. Hence, a
more useful definition is to work with the form
∑i G(Ei + ΔE , Bi + ΔB )G(Ei , Bi ) − ⎡⎣∑i G(Ei , Bi )⎤⎦ .

2
C (ΔE , ΔB ) = (5.31)
Again, this must be properly normalized. It is obvious that the magnetic field
variation causes a change in the correlation function. The energy variation arises
from the fact that the impurities cause an inhomogeneous energy surface and that
different current paths will see different local variations in the Fermi energy. This
means that small changes in the overall Fermi energy, due to an applied gate
voltage, for example, will cause larger variations in the local potential.
Another useful connection is the variance of the conductance. The variance is the
traditional rms value defined as
var(G ) = 〈G 2(E , B )〉 − 〈G (E , B )〉2 . (5.32)
This becomes important as it is thought that the fluctuation is given as [24]
δGrms = var(G ) . (5.33)
In figures 5.9 and 5.10, the fluctuation itself is plotted as a function of magnetic field
and gate voltage (density), respectively. The correlation function is slightly different
from the variance, but the decay of the fluctuation in these figures is related to the
correlation energy ΔBc and the correlation energy (or gate voltage, reduced by the
effective lever arm between a voltage change and the Fermi energy change) ΔEc.
These values are determined by the energy or magnetic field for which the
correlation function has fallen to a value of 0.5 of its initial value (which should
be unity).
The value of the correlation field ΔBc is generally related to the increment of flux
enclosed within a phase coherent area. Some argue that since we are not using two
time-reversed paths, but only a single loop, the magnetic field coupled to the loop is
reduced by a factor of two from that found for weak localization reduction. But,
from the ideas of the AB experiments, this would still lead to the flux being coupled
through a single phase coherent area. Thus, we expect that the correlation magnetic
field would be related to this through
h
ΔBc ∼ , W ≪ lφ ≪ L , (5.34)
eWlφ
5-22
where W and L are the width and length of the sample, respectively, and lφ is the
phase coherence length. More complicated arguments and equations have appeared
in the literature in place of equation (5.34) [23, 24, 35]. Hence, some versions simply
put a factor of β in front of the right-hand side of equation (5.34), with β having a
numerical value of 0.55 commonly being observed, although others have suggested a
value closer to 3 [36]. However, in most cases, determining the value of the phase
coherence length, or the related phase-breaking time, is quite difficult from the
experimental structures, so that a wide range of results can be found in the literature.
For example, a value of ∼0.25 was found for β in a study of a variety of long wires
fabricated in high mobility material [37].
In computational experiments, such as discussed above, the lack of a phase-
breaking process within the ‘sample’ area means that the phase coherent area is that
of the entire sample. That is, in the zero temperature limit where phase breaking
occurs only at the sample contacts, it is expected that the phase coherence length in
equation (5.34) is replaced by the sample length itself. In these experiments, it
becomes possible to estimate the validity of equation (5.34) directly. In figure 5.11,
we plot the values for ΔBc as a function of the amplitude of the random potential
[32]. Here, it may be seen that relation (5.34) holds fairly well except at the lowest
values of the random potential. There is no statistical difference in the energy at
which the magnetoconductance sweep is made. Again, there is a distribution of
values at each choice of the random potential and energy, but when a sufficiently
2.0
1.5
eBA/h
1.0
0.5
0.0
0.10 1.0 10
Peak-to-Peak Amplitude (in units of Ehop)
Figure 5.11. Values of the effective flux enclosed in the sample determined from the correlation magnetic field
in fluctuation correlation functions.
5-23
large ensemble of systems (with varying size and energy) is chosen, the average lies
very close to the value suggested in equation (5.34).
The correlation energy ΔEc is usually discussed in terms of a phase coherence
time, or a diffusion time across the sample. Thus, there should be some correlation
between its value and the sample size. It has been suggested that one can define the
correlation energy as the energy difference for the phase difference achieved after
diffusing a length l1 = Dτ1 is unity. Here, the correlation time has been introduced,
which is the time over which the acquired phase difference between two paths of
energy difference δE, is of order unity. With this complicated set of quantities, the
correlation energy may be expressed as [24]
ℏD
ΔEC = ∼ δE. (5.35)
l12
This all seems to be very complicated, if not confusing. The end result seems to be
that the correlation energy is given approximately by the mean energy separation of
the various energy levels in the device, a quantity that has often been accused of
various sins, but seldom has proved to be of much value in estimating the properties
of mesoscopic devices. In the simulations discussed here [32], it was generally found
that the value for the correlation energy lay in the range of 0.15–0.25 μeV, where the
lower value occurred at smaller values of the random potential. Typically, this value
is a small fraction of the mean energy separation of the modes in the simulation.
Moreover, the observed values seemed to have little correlation with the sample size,
which is quite different from the behavior observed for the correlation magnetic
field.
On the other hand, Thouless [38] suggested that the critical time was that time
required for a particle to diffuse across the entire system in the current direction. If
we take this view, then we can arrive at a correlation energy as [23, 39]
ℏπ 2D
ΔEC = . (5.36)
L z2
This suggests a strong correlation between the observed correlation energy and the
length of the system (Lz = L used earlier). Yet, as remarked above, this does not
seem to be found in our simulations, and the dearth of reports of experimental values
for the correlation energy seem significant. The problem lies with the fact that the
correlation energy increases weakly with the amplitude of the random potential.
Such an increase requires a corresponding increase in the diffusivity according to
either of the above two equations. In turn, this requires an increase in the scattering
time as the amplitude of the random potential is increased, which is counter-
intuitive. In fact, both of these suggested equations rely upon the Thouless
argument, and neither depends upon the amplitude of the random potential.
Hence, it is unlikely that any of these extrinsic arguments apply here. Thus, it
appears that the correlation energy arises from an intrinsic property of the material,
probably related to the scattering and decoherence processes introduced by the
random potential itself, and suggests that the correlation energy is related to the
5-24
Fermi energy change needed to turn on or off a single Landauer mode in the sample,
as suggested by (5.28).
5.5 Phase-braking time

In mesoscopic physics, the size of various effects often depends upon the phase
breaking, or phase coherence, time τφ, or less directly upon the phase coherence
length lφ = Dτ1 . In chapter 3, these quantities were crucial to the description of
weak localization, while just above they appeared in the discussion of the magnetic
correlation ΔBc. But, there were many descriptions of how to fit the weak
localization peak, depending upon the dimensionality and the relative sizes of the
coherence length, the mean free path, the thermal length and the various dimensions
of the device. Similarly, there are various formula for ΔBc depending upon these
same variables in length. Nevertheless, one can estimate the values for the coherence
length and the phase-breaking time from these measurements, especially as the
various forms of the equations do not vary by orders of magnitude.
There are a variety of contributors to the decay of the coherence length itself with
temperature. In many cases, however, these various mechanisms are not separated in
an experimental measurement. Rather, only the change in the coherence length is
measured by measuring, for example, the change in the behavior of the weak
localization or the change in the amplitude of the fluctuations. In this section, we
review a few of the measurements of the coherence length and, more importantly,
the phase-breaking time. It is important to remember, however, that this length is
not itself a key factor, but that it summarizes a variety of temperature variations due
to the diffusion ‘constant’ and to the phase coherence time. In figure 5.12, we show
the inelastic, or phase-breaking, time found in an AlGaAs/GaAs heterostructure
from measurements of the weak localization [40]. The data are taken for three
different carrier densities (three different samples) in these modulation doped
Figure 5.12. The temperature dependence of the inelastic, or phase-breaking, time in an AlGaAs/GaAs
heterostructure for three different densities (three different samples). (Reprinted with permission from [40].
5-25
heterostructures. The values are extracted from a least-square fit to the digamma

function fit to the weak localization lineshape. The solid lines are fits to the data and
have the slope of 1/T, which is the usual behavior for Q2D systems except at very
low temperatures.
In general, one sees a saturation of the phase coherence length and the phase-
breaking time at low temperatures, both in metals [41] and in semiconductors [42].
Usually, the transition from the temperature-independent behavior and the temper-
ature-dependent behavior occurs in the vicinity of 0.1–1.0 K. Where the transition
occurs seems to depend upon the quality of the material and the carrier density.
There is just a hint that this might be ‘occurring for the top curve in figure 5.12. This
can be seen somewhat better in figure 5.13, in which the temperature dependence of
the phase-breaking scattering rate (1/τφ) is plotted for a pair of narrow GaAs wires
formed in an AlGaAs/GaAs heterostructure [43]. As usual, the heterostructure is
grown by molecular-beam epitaxy, and the wires are created by electron-beam
lithography. Data for two wires, with nominal widths of 90 nm and 260 nm, are
shown in the figure. The data are consistent with the phase-breaking rate depending
linearly on the temperature, but the line intercepts the T = 0 edge at a value well
above zero phase-breaking rate. This infers that there is a saturation in this quantity
at the lowest temperatures. Rather than use the standard formulas, these latter
authors Fourier transformed the combination of the weak localization and con-
ductance fluctuation signals, with the magnetic field as the principal variable. By
connecting the amplitude an of each Fourier component fn with the effective area Sn
that arises from the Fourier component through [43]
2e
fn (B ) = B · Sn , (5.37)
h
Figure 5.13. The temperature dependence of the phase-breaking rate for two narrow GaAs ribbon devices.
(Reprinted with permission from [43]. Copyright 1988 Elsevier.)
5-26
they can determine the phase-breaking length from

⎛ 2a Sn ⎞
an(T ) = β exp⎜⎜ − ⎟⎟ , (5.38)
⎝ lφT ⎠
where α ∼ 4.5 is a parameter determined from a statistical analysis for the size of the
loops most likely to contribute to conductance fluctuations.
Recognizing that the correlation function is not always well behaved, it has been
suggested that one use the inflection point of the curve rather than the value at half-
height [36]. These latter authors argue that the half-width is not always the best
choice of metric, and that the inflection point, where the second derivative of the
correlation point is zero, is the field separation at which the correlations change the
fastest [36]. In numerical studies, they have found that the value of the phase-
breaking time found from the inflection point is about a factor of four smaller than
that determined from the half-width. In figure 5.14, we illustrate this with a
correlation function determined from studies of fluctuations in graphene [44]. As
usual, a monolayer of graphene was exfoliated onto an oxidized Si wafer, and the
transport studied at low temperatures. From the conductance fluctuations, the
correlation function could be computed. In figure 5.14, both the correlation function
and the derivative (dotted curve) are shown as a function of the separation magnetic
field. From the derivative, it is easy to find the inflection point of the correlation
function, and this is indicated by a vertical dashed line. The same approach has also
been applied to studies of the weak localization and consistent results are obtained
there as well. In figure 5.15, the phase-breaking rates determined from this inflection
point analysis are shown for the graphene sample. Two curves are shown here, and
the difference is that an in-plane magnetic field of 6 T has been applied for the lower
curve. It is thought that the difference lies in the role of magnetic defects in, or on,
the graphene. Magnetic defects have been studied for some time [45, 46]. By
applying the in-plane magnetic field, it is assumed that these magnetic defects are
polarized and their random potential is thereby removed [44], leading to the lower
value of dephasing rate. It is also clear that there is a saturation in the dephasing rate
at the lowest temperatures (below about 0.1 K in this case), which is supported by
Figure 5.14. A typical autocorrelation function for the magnetic-field-induced conductance fluctuations (solid
curve). The derivative is shown as the dotted curve, and the inflection point of the correlation function is
indicated by the vertical dashed curve. (Reprinted with permission from [44]. Copyright 2013 the American
Physical Society.)
5-27
Figure 5.15. Dependence of the dephasing rate on temperature for a graphene sample. The open circles are for
the absence of an in-plane magnetic field, while the filled circles have an in-plane magnetic field of 6 T. The
dotted line shows the expected results if there were no saturation in the dephasing rate at low temperatures.
some theories of the carrier–carrier interactions [47]. But, as we remarked earlier, the
saturation at low temperatures occurs in nearly all materials that have been studied
to date.
Generally, the conductance fluctuations in a magnetic field are studied at lower
values of the magnetic field. As one increases the magnetic field, a breakdown in the
scaling relationships has been reported to occur [48]. This generally is seen by an
increase in the correlation magnetic field [49], as shown in figure 5.16. In this study,
narrow samples of GaAs wires, formed in an AlGaAs/GaAs/AlGaAs double
heterostructure via electron-beam lithography, were used to study the fluctuations
in a magnetic field. Results for several different temperatures are shown in the figure,
but they seem to be relatively similar. Just above ωcτ = 1, the correlation magnetic
field begins to increase. Here, ωc = eB/m* is the cyclotron frequency and τ is the
scattering time inferred from the mobility (about 8600 cm2 V−1 s−1) of the material.
In this case, the GaAs quantum well was 10 nm thick and the wires were about 700
nm wide. In the standard analysis, this increase in the correlation magnetic field
would lead one to conclude that the phase-breaking time was increasing, but this is
not correct due to the breakdown of the scaling at this magnetic field. However, this
allows a different approach to determine the phase-breaking time.
Generally, in diffusive samples with relatively low mobility, in the higher
magnetic fields, the correlation magnetic field increases with increasing magnetic
field without any increase in the amplitude of the fluctuations. In high mobility
material, however, the increase in the correlation magnetic field is accompanied by
an increase in the amplitude of the fluctuations [50, 51]. Normally, one thinks that
the diffusion constant, part of the phase coherence length, decreases as the magnetic
field increases, but this would lead to a reduction in the coherence length and an
5-28
Figure 5.16. Variation in the correlation magnetic field at high values of the magnetic field, with data shown
for three different temperatures. (Reprinted with permission from [49]. Copyright 1993 IOP Publishing.
Reproduced with permission. All rights reserved.)
Figure 5.17. Schematic for a parabolic potential in the y-direction and a magnetic field normal to this.
increase in the correlation magnetic field, it should also lead to a reduction in the
amplitude of the correlations, which is not observed. There is enough rigor in the
theoretical arguments [39] if one reinterprets the meaning of the phase coherent
areas, which are represented in equation (5.34) by the product Wlφ. The Nottingham
group has argued that there are (at least) two kinds of phase coherent areas to be
considered [52]. One of these is the bulk regions of the entire sample, which is
important at lower magnetic fields. However, once one reaches ωcτ = 1, the current-
carrying trajectories start to be pushed into edge states, and the regions important
for the fluctuations get squeezed into small areas at the edges of the sample. Let us
first examine the edge-state formation.
Consider the diagram shown in figure 5.17. Here, we have a narrow strip of
semiconductor with confinement in the y-direction, indicated by the blue quadratic
potential. The axis of the semiconductor lies along the x-direction, and we will
5-29
consider a magnetic field in the z-direction, as indicated. We have chosen the

quadratic confinement potential for ease of calculation, as we are familiar with the
harmonic oscillator of appendix B. We want to include the magnetic field in this
formulation, so we introduce the field via the Landau gauge, with a vector potential
A = −Byax . (5.39)
As we can see in (C.19) of appendix C, this goes into the momentum via Peierls’
substitution to give the resulting Schrödinger equation
1 1
( −i ℏ∇ − e A)2 ψ (x , y ) + mω 2y 2 ψ (x , y ) = Eψ (x , y ), (5.40)
2m 2
in the two dimensions of an assumed heterostructure interface electron gas. Inserting
equation (5.39) and expanding the first quadratic term leads to
⎡ ℏ2 ⎛ ∂ 2 ∂ 2 ⎞ ieℏB ∂ e 2B 2y 2 ⎤
⎢− ⎜ 2 + 2⎟ − y − ⎥ψ (x , y )
⎣ 2m ⎝ ∂x ∂y ⎠ m ∂x 2m ⎦
(5.41)
1
+ mω 2y 2 ψ (x , y ) = Eψ (x , y ).
2
Let us now introduce the hybrid frequency
Ω2 = ω 2 + ωc2 , (5.42)
so that we can reduce (5.41) to

⎡ ℏ2 ⎛ ∂ 2 ∂ 2 ⎞ ieℏB ∂ ⎤ 1
⎢− ⎜ 2 + 2⎟ − y ⎥ψ (x , y ) + mΩ2y 2 ψ (x , y ) = Eψ (x , y ). (5.43)
⎣ 2m ⎝ ∂x ∂y ⎠ m ∂x ⎦ 2
At this point, we postulate that the electrons are free to move in the x-direction, and
can be represented as plane waves in this direction. Hence, we assert that the wave
function can be written as
ψ (x , y ) = e ikxψ (y ), (5.44)
Now, equation (5.43) can be rewritten as

⎡ ℏ2 ∂ 2 1 ⎤ ⎡ ℏ2k 2 ⎤
⎢− + ℏωcky + mΩ2y 2 ⎥ψ (y ) = ⎢E − ⎥ψ (y ). (5.45)
⎣ 2m ∂x 2
2 ⎦ ⎣ 2m ⎦
By completing the square for the potential terms, we can write this as
ℏ2 ∂ 2ψ (y ) 1 ⎡ ℏ2k 2 ⎛ ωc2 ⎞⎤
− + mΩ 2
( y + y ) 2
ψ ( y ) = ⎢E − ⎜1 − ⎟⎥ψ (y ), (5.46)
⎣⎢ Ω2 ⎠⎥⎦
0
2m ∂x 2 2 2m ⎝
5-30
where
ℏkωc ℏk
y0 = ⎯⎯⎯⎯⎯⎯⎯⎯⎯→ . (5.47)
mΩ2 ωc ≫ ω eB
In this last high magnetic field limit, the correction term to the energy in the square
brackets of equation (5.46) vanishes. This term y0 represents a shift of the electron
trajectory in the magnetic field. Remember that the sign on the electronic charge is
negative so that for electrons moving in the positive x-direction (k > 0), the carriers
are shifted in the negative y-direction, and for electrons moving in the negative
x-direction, the shift is in the positive y-direction. If one thinks about the drift of the
electrons (we have not specified any longitudinal electric field or potential) arising
from the electromagnetic E × B forces, the electric fields are given by the gradients of
the confinement potential, and point toward the center of the potential well. That is,
for y < 0, the electric fields point in the positive y-direction, and conversely for the
other side of the potential well. Hence, the E × B forces produce the velocities in the
directions shown, when one remembers that the charge is negative. Another way of
thinking about this is that the electrons circulate around the magnetic field in a
cyclotron motion. Those shifted in the positive y-direction hit the wall and are
specularly reflected which results in a bouncing orbit moving in the direction
indicated. The net result is that the left-hand side of equation (5.46) gives rise to
the harmonic oscillator energy levels, but the center of the wave functions are shifted
to one side of the potential or the other, depending upon their motion in the
x-direction.
With multiple quantized levels full, the multiple trajectories lead to interference
and conductance fluctuations just as in the case of normal quantum interferences. As
mentioned above, the interferences in the high magnetic field case lead to a different
type of phase coherent area, as shown in figure 5.18, where the multiple trajectories
are pushed to the side of the sample. The Nottingham group has carried out
simulations which suggest that the length of the phase coherent area is changed very
little, but the width of this area is now determined by the cyclotron diameter [52]. As
a result, the correlation magnetic field, which depends upon this phase coherent
area, will increase as the magnetic field increases, and in a linear fashion.
Figure 5.18. Numerical simulation of classical electron trajectories showing their confinement along the edge
of the sample, which leads to enhanced diffusion for a large magnetic field. (Reprinted with permission from
[52]. Copyright 1993 the American Physical Society.)
5-31
Using the ideas from the Nottingham group, we can construct the effective phase
coherent area in a slightly different manner. We note that the phase coherence length
is given simply as before as
lφ = Dτφ , (5.48)
and the inelastic mean free path is

τφ
l inel = vF τ = lφ . (5.49)
τ
In the systems of interest, we have d = 2 for most purposes; that is, we are primarily
considering Q2D systems. For relatively high mobility material, the length of the
average phase retaining trajectory is of the order of Nπrc2 /2, where N is the number
of bounces along the edge of the sample in the high magnetic field case [53]. Here,
the length of this path is Nπrc ∼ linel. This leads to the replacement of the phase
coherent area as
rcl inel ℏ2k F2τφ
Wlφ → = , (5.50)
2 2meB
where we have used rc = kF lB2 = ℏkF /eB . Then, using equation (5.34), the correlation
magnetic field in these high magnetic field regions is approximately
2mheB hB
ΔBc = 2 2
= . (5.51)
eℏ k F τφ EF τφ
A slightly different form is found by the Nottingham group [52], although both
forms retain the linear increase with magnetic field. Hence, by determining the
correlation magnetic field as a function of the magnetic field itself, one can determine
directly the value for the phase-breaking time. The density for the sample in
figure 5.16 was about 3.7 × 1011 cm−2, which corresponds to a Fermi energy of
about 13.3 meV, and this gives a phase-breaking time of about 1.2 × 10−10 s.
Problems
1. An alternative to the uniform distribution of energies assumed in section 5.1
is to take a Lorentzian distribution as
1⎛ Γ ⎞
P (E i ) = ⎜ 2 ⎟.
π ⎝ Ei + Γ2 ⎠
Repeat the arguments of that section and show that the result is a smooth
band with a width of ΔE when Γ = 0, and a band with localized edges leading
to a mobility edge given by
⎛ ΔE ⎞2
E mob = ± ⎜ ⎟ − Γ2
⎝ 2 ⎠
for ΔE > Γ > 0.
5-32
2. From the data in figure 5.6, estimate the hopping range, as a function of
temperature, for each of the curves shown. Work only in the range below
10−8 s, where the curves are linear in the plot.
3. In figure 5.8, the magnetoresistance curves seem to be somewhat asymmetric
around B = 0. Explain why this may be the case.
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5-34
IOP Publishing

(Second Edition)
David K Ferry
Chapter 6
The quantum Hall effect
In mesoscopic systems, a static magnetic field may have a profound effect on the
electronic and transport properties. The application of high magnetic fields to these
mesoscopic devices is an invaluable degree of freedom available to the experimen-
talist in probing the system, and also to the theoretician. Magnetic fields give rise to
new fundamental behaviors not observed in bulk-like systems, such as the quantum
Hall effect [1]. The fundamental quantity characterizing a magnetic field is the
magnetic flux density, B, which in MKS units is measured in Teslas. In the study of
semiconductor mesoscopic devices, low fields usually correspond to fields less than 1
T, which is the regime in which low-field magnetotransport experiments such as Hall
effect measurements are usually performed. Magnetic intensities of 10–20 T are
obtainable in the usual university and commercial research environments, using
superconducting alloy coils with high critical magnetic fields (superconductivity can
be destroyed by a high magnetic field) immersed in a liquid He Dewar. Higher
magnetic fields are obtainable only at a few large-scale facilities scattered around the
world.
6.1 The Shubnikov–de Haas effect

When we consider mesoscopic systems such as quantum wells, wires, and dots, the
effect of a magnetic field may be roughly separated into two cases. In the first, the
magnetic field is parallel to one of the directions of free-electron propagation; in the
other, it is perpendicular to the free-electron motion of the system. Qualitatively,
when free particles are subject to a magnetic field, they experience a Lorentz force
F = e(E + v × B), (6.1)
where v is the velocity of the carrier and E is the electric field. Since the force is
always perpendicular to the direction of travel of the particle, its motion in the

absence of other forces is circular, with angular frequency given by the cyclotron
frequency, which we have seen in earlier chapters, given as
eB
ωc = . (6.2)
m⁎
Quantum mechanically, the circular orbits associated with the Lorentz force must be
quantized in analogy to the orbital quantization occurring about a central potential,
for example, about the nucleus of an atom. Since the particles execute time-
harmonic motion similar to the motion in a harmonic oscillator potential, the
energy associated with the motion in the plane perpendicular to the magnetic field is
expected to be quantized. If we now consider the magnetic field applied perpendic-
ular to the plane of a 2DEG, then the entire free-electron-like motion in the plane
parallel to the interface is quantized, and the energy spectrum becomes completely
discrete. Indeed, we saw this in the last chapter, where we coupled the magnetic
motion to that of a harmonic oscillator. If we take equation (5.46) and remove the
harmonic oscillator by letting ω → 0, Ω → ωc, we then have
ℏ2 ∂ 2ψ (y ) 1
− 2
+ mωc2(y + y0)2 ψ (y ) = Eψ (y ) (6.3)
2m ∂x 2
where
ℏk
y0 = (6.4)
eB
is the position offset. By comparison with equation (B.1) in appendix B, we see that
the magnetic field introduces its own harmonic oscillator. The energy is then given,
in comparison with equation (B.11), to be
⎛ 1⎞
En = ⎜n + ⎟ℏωc , n = 0, 1, 2…. (6.5)
⎝ 2⎠
The various energy levels of this harmonic oscillator are termed the Landau levels.
As the electron is traveling in a closed circular orbit, we can refer to the radius of this
orbit as the cyclotron radius
⎛ 1 ⎞ 2ℏ ℏ
rc = ⎜n + ⎟ = kF ≡ kF l B2. (6.6)
⎝ 2 ⎠ eB eB
This latter form introduces the magnetic length as well.

An important point about the formation of the Landau levels is the effect they
have on the density of states. Let us consider simply a two-dimensional system in
which the magnetic field is oriented normal to the plane of this system. Normally, in
such a two-dimensional system, the density of states is uniform at m*/π ℏ2 , and the
energy is continuous. The formation of the Landau levels in energy means that
the density of states becomes non-uniform, with these states migrating to the
6-2
momentum space orbits of the electrons. This leads to a localized density of states
which is generally of the form [2]
1 1
2 ∑n
ρ(E , En) ≅ ,
2πl B (6.7)
( )
E − En 2
1+ Γ
where En is given by equation (6.5) and the broadening is generally given by

2ℏ2ωc
Γ2 = . (6.8)
πτ
Hence, we see that, in general, the broadening of the levels is proportional to the
square root of the magnetic field. But, as we increase the magnetic field, the levels
move further apart and the density of states peak in each level increases linearly with
the magnetic field.
As the magnetic field is raised, each of the Landau levels rises to a higher energy.
However, the Fermi energy remains fixed, so at a critical magnetic field, the highest
filled Landau level will cross the Fermi energy. In general, at a given magnetic field
and Fermi energy, the Landau levels are filled up to some nmax determined by the
density as
nmax ⎛ 1⎞ ℏ nmax ℏ
nsrc2 = 2∑ ⎜n + ⎟ ⁎ → rc2 = ∑n=0 (2n + 1) . (6.9)
n=0 ⎝ 2 ⎠ m ωc eBns
As the magnetic field increases in value, the number of available states in each
Landau level increases correspondingly. Thus, when the highest occupied Landau
level pushes up through the Fermi energy, the remaining carriers drop into the lower
Landau levels, and this reduces the number of terms in the sum in equation (6.9).
This can only occur due to the increase of the density of states in each Landau level,
so a given Landau level can hold more electrons as the magnetic field is raised. As a
consequence, the average radius (obtained from the squared average of the radius) is
modulated by the magnetic field, going through a maximum each time a Landau
level crosses the Fermi level and is emptied. From equation (6.9), it appears that the
radius is periodic in the inverse of the magnetic field. At least in two dimensions, this
periodicity is proportional to the areal density of the free carriers and can be used to
measure this density. The effect, commonly called the Shubnikov–de Haas effect, is
normally applied by measuring the conductivity in the plane normal to the field. In
figure 6.1, we show schematically how this emptying of the Landau levels occurs. As
shown in the figure, the Fermi energy appears to oscillate as the magnetic field
increases.
The separation of the Landau levels is ℏωc , and all the states in the range
En ± ℏωc /2 are coalesced into the nth Landau level. The density of states in two
dimensions is m* /πℏ2 , so the number of carriers in each Landau level is the product
of this density of states and ℏωc , or m*ωc /π ℏ = 1/πlB2 . Thus, the number of filled
Landau levels is πlB2ns and this leads to a periodicity of
6-3
Figure 6.1. The Landau levels ‘fan out’ as the magnetic field is increased, and the electrons drop from the
topmost filled level into lower levels as the states become available.
⎛1⎞ e
Δ⎜ ⎟ = . (6.10)
⎝ B ⎠ π ℏns
To understand the conductivity oscillations, it is necessary to reintroduce the

scattering process. Without this process, the electrons remain in the closed
Landau orbits. However, the scattering process can cause the electrons to ‘hop’
slowly from one orbit to the next in real space by randomizing the momentum. This
leads to a slow drift of the carriers in the direction of the applied field (we will see
below that the edge states are mainly responsible for this). The drift is slower than in
the absence of the magnetic field because the tendency is to have the carriers remain
in the orbits. Here the scattering induces the motion instead of retarding the motion
as in the field-free case (one can compare figure 5.18, for example). Thus the
conductivity is expected to be less in the presence of the magnetic field. When the
Fermi level lies in a Landau level, away from the transition regions, there are many
states available for the electron to gain small amounts of energy from the applied
field and therefore contribute to the conduction process. On the other hand, when
the Fermi level is in the transition phase, the upper Landau levels are empty and the
lower Landau levels are full. Thus there are no available states for the electron to be
accelerated into, and the conductivity drops to zero in two dimensions. In three
dimensions it can be scattered into the direction parallel to the field (the z-direction),
and this conductivity provides a positive background on which the oscillations ride.
A sample of these oscillations in two dimensions are shown in figure 6.2, which is a
typical measurement of the longitudinal resistance. In this case, the sample is a
quantum well of AlGaAs grown on GaAs. The density of the 2DEG is approx-
imately 3 × 1011 cm−2 and the mobility was 3 × 105 cm2 V−1 s−1, and the
measurement was made at 1.5 K. The Landau index n is shown for several of the
minima. The small dip between the n = 2 and n = 3 minima is the onset of a spin split
level [3], which will be discussed further in the next section.
It may be noted that the resistance goes to zero at certain values of the magnetic
field, but this does not indicate superconductivity. Rather, it is a result of the
conductance going to zero at these magnetic fields. In the Q2D system, the
6-4
Figure 6.2. Typical Shubnikov–de Haas measurements in an AlGaAs/GaAs heterostructure. The various
oscillations arise from the emptying of the Landau levels for two different subbands as the magnetic field is
increased. The integers refer to the particular Landau level.
conductivity is a matrix due to the transverse electric fields that arise from the Hall
effect. This matrix may be written as
⎡ σxx σxy ⎤
[σ ] = ⎢− σ σ ⎥ , (6.11)
⎣ xy xx ⎦
and when this is inverted, the longitudinal resistivity becomes

σ
ρxx = 2 xx 2 (6.12)
σxx + σxy
and the non-zero nature of the transverse (off diagonal) terms leads to zero resistivity
when the longitudinal conductivity vanishes. This just means that the electric field is
perpendicular to the current flow, which leads to vanishing dissipation, although the
material is not a superconductor.
Equation (6.10) has an important point to tell us, and that is that the Shubnikov–
de Haas resistance, the longitudinal resistance of the sample, is periodic in the
inverse magnetic field. To illustrate this, we re-plot the data in figure 6.2 as a
function of the inverse magnetic field in figure 6.3. One can almost pick out the
periodicity of this curve. If there are more than a single subband, it becomes more
complicated. First, the oscillations arising from the second subband have a different
periodicity in an inverse magnetic field, and this oscillation causes interference with
that of the lowest subband. As a result, one must Fourier transform data like
6-5
Figure 6.3. The data from figure 6.2 are re-plotted here as a function of the inverse magnetic field,
corresponding to (6.10). Now, the periodicity is clearly evident.
figure 6.3 and examine the various frequencies that are present in the signal. With
luck, only two frequencies will be present. Hence, the Fourier transform is one of the
most useful tools to use in data analysis.
The second problem that arises is that we have assumed that the density of states
is spin degenerate. At the high magnetic fields that are used in this experiment, this is
not always the case. The Zeeman energy splits the Landau levels into two different
levels, one with spin up and the other with spin down. This splitting is given by
1
δE = ± gμB B, (6.13)
2
where μB is the Bohr magneton (eℏ/2m 0 = 58 μeV T−1) and g is the Landé factor, but
is usually a ‘fudge’ factor that differs for electrons in various semiconductors [4]. In
fact, it is usually found that the g factor dramatically increases with magnetic field
[2, 5]. As the density in each Landau level increases, the electron–electron interaction
gets stronger and this is thought to lead to the enhanced g factor. In practice,
however, as we will see in the next section, the enhanced g factor leads to the spin-
split Landau levels being spaced between the Landau ladder of levels, but the totality
of the levels are not equally spaced. Because of this doubling of the number of levels
at high magnetic fields, a different symbol is usually used, and this is ν . Thus, ν = 1
and 2 refer to the two spin-split levels that emanate from the n = 0 landau level.
ν = 3 and 4 refer to the two spin-split levels that emanate from the n = 1 landau
6-6
level, and so on. We will attach more significance to this parameter in the next
section.’
6.2 The quantum Hall effect

In the previous chapter, we treated a conducting channel which had a harmonic
oscillator potential in the transverse direction and a magnetic field normal to the
channel. In this situation, we discovered that the motion of the electrons was offset
to one side of the channel or the other depending upon the sign of the momentum.
This led to what we called edge states. In that situation, the energy of a particular
level was given by
⎛ 1⎞ ℏ2k 2 ⎛ ω 2 ⎞
En = ⎜n + ⎟ℏωc + ⎜ ⎟, (6.14)
⎝ 2⎠ 2m⁎ ⎝ Ω2 ⎠
where Ω2 = ω 2 + ωc2 . When the magnetic field is large, the levels are primarily
Landau levels, but they continue to rise in energy as they approach the sides of the
confining potential. The confinement energy raises the energy of the Landau levels.
We find essentially the same behavior if the confinement is due to the physical
confinement of a finite width sample of semiconductor material. The potentials at
the edges of the ribbon sample are basically the work function of the material, and
decay rapidly into the sample from the edge. Nevertheless, as we near the edge of the
sample, the Landau levels rise above their values in the interior of the sample. This is
shown schematically in figure 6.4, in which we have identified by the various energy
levels by the index ν which means that these are spin slit levels. When the Fermi
energy lies in one of the bulk levels, then the conductivity is high and one sees the
peaks that appear in figures 6.2 and 6.3. But, when the Fermi level lies as shown in
Figure 6.4. Schematic view of spin-split Landau level behavior near the edge of a finite width sample. While
these levels are shown equally spaced, this is not the real case, but this will not impact the edge state picture.
6-7
figure 6.4, the bulk levels are completely full, and electrons exist up to where the
levels cross the Fermi energy at the edges of the sample. We see that the index ν not
only indicates the upper most filled bulk level, but also indicates the number of edge
states that cross the Fermi energy.
The three edge states that are indicated in figure 6.4 are essentially one-dimen-
sional channels that flow along the edges of the sample. From the discussion in
chapter 2, and the Landauer formula, we would expect each of these spin-split
channels would produce a conductance of e2/h, but as we noted in the previous
section, the net conductivity is zero. We could think about this as arising from the
fact that the currents are oppositely directed on the two sides of the sample and
therefore cancel, but this would be misleading. In fact, the conductance vanishes as
there is no longitudinal electric field when we are in the edge-state regime. The
voltage that may be exist between the ends of the sample must appear entirely as the
Hall voltage.
We can follow the same logic that led to equation (6.10) by noting that the
spacing of the spin-split levels is approximately ℏωc /2 so that the density of carriers
in each level is given by m*ωc /h = eB /h = 1/2 lB2 . The Hall resistance is given by the
product of the Hall constant and the magnetic field (basically Ey/Jx in the normal
configuration in which the current flows in the x-direction, and the Hall voltage is
measured in the y-direction). If we put this density into the Hall constant, then we
find the Hall resistance as
h
RH = Ey / Jx = , (6.15)
νe 2
where ν is our index of the highest filled level and the number of edge states. In
figure 6.5, we plot the Hall resistance for the data used in figures 6.2 and 6.3. There
are clear plateaus in the curve, which are the levels given by equation (6.14). The
highest plateau shown here is for ν = 4 (in the lowest subband). Of course, this result
occurs only when the Fermi energy is not in one of the bulk levels, but the result is
phenomenal in that the Hall resistance is the ratio of two fundamental constants and
a pure number. This is the quantum Hall effect, discovered by Klaus von Klitzing in
1980 [1], and for which he won the Nobel prize. Since 1990, the quantum Hall effect
has been adopted as the standard value for the SI ohm [6] by nearly all countries in
the world. The quantity h/e2 = 25 812.807 Ω (this has now been standardized by the
redefinition of the SI units in 2019) is now referred to as the von Klitzing constant
(RK) and experiments have demonstrated that this value is known to 3.5 parts in
1010. That is, it is known to better than nine significant digits. It is remarkable as well
that this is independent of the material being studied.
One might ask why the Hall resistance is so stable. This is because the set up that
leads to the quantum Hall effect is topologically stable. It is independent of the size
of the sample under test. It is not affected by the surface conditions or the manner of
isolating the device. We have already remarked that it is not dependent upon the
material being studied. Because of the topology of the experiment, we are led to the
result that the Hall resistance is a result of the gauge invariance and depends only
6-8
Figure 6.5. The Hall resistance for the data of figures 6.2 and 6.3. The plateaus that appear are a manifestation
of the quantum Hall effect [1]. The integers ν have been inserted by several of the plateaus. Only one spin-split
level is observable.
upon the ratio of these fundamental constants [7]. The stability of the quantum Hall
effect is remarkable, and because of this it has become the international standard for
resistance. One important aspect of equation (6.17) is that the value depends only
upon fundamental constants, and efforts have been taken to assure that its value is
universal among various materials [8], as mentioned above.
6.3 The Büttiker–Landauer approach

In section 2.3, we developed the Landauer equation for the transport of carriers
through a Q1D system in which the transport was dominated by channels or modes.
The Shubnikov–de Haas effect and the quantum Hall effect both raise the need for a
multi-contact, or multi-probe version of this approach. This extension of the channel
idea was primarily as derived by Büttiker [9, 10]. As a generalization of the two
contact case, current probes are considered phase-randomizing agents that are
connected through ideal leads to reservoirs (characterized by a chemical potential μi,
for the ith probe) which emit and absorb electrons incoherently. In general, the
‘probes’ need not be physical objects. They can be any phase-randomizing entity,
such as inelastic scattering centers distributed throughout the sample [10]. Here, we
will assume that voltage probes are phase randomizing, although there is some
disagreement over the validity of this assumption of phase randomization.
To simplify the initial discussion, we first assume that the leads contain only a
single channel with two states at the Fermi energy corresponding to positive and
6-9
negative velocity. The various leads are labeled i = 1, 2,…, each with corresponding
chemical potential μi. A scattering matrix may be defined which connects the states
in lead i with those in lead j. As discussed in chapter 2, this relationship is Tij = ∣tij∣2
and is the transmission coefficient into lead i of a particle incident on the sample in
lead j, while Rii = 1 − ∣tii∣2 is the probability of a carrier incident on the sample from
lead i to be reflected back into that lead. In the general case, there may be a magnetic
flux Φ present which penetrates the sample. The elements of the scattering matrix
must satisfy the following reciprocity relations due to time-reversal symmetry
tij (Φ) = tji ( −Φ). (6.16)
Because of this symmetry in the transmission itself, both the mode reflection and
mode transmission possess this same symmetry due to the circuit reciprocity
theorem.
In order to simplify the discussion, an additional chemical potential is introduced,
μ0, which is less than or equal to the value of all the other chemical potentials so that
all states below μ0 can be considered filled (at TL = 0 K). With reference then to μ0,
the total current injected from lead i is given as (2e /h )Δμi , where Δμi = μi − μ0
(again assuming that the difference in chemical potentials is sufficiently small that
the energy dependencies of the transmission and reflection coefficients may be
neglected). As we have mentioned, a fraction Rii of the current is reflected back into
lead i. Similarly, lead j injects carriers into lead i as (2e /h )Tji Δμj . The net current
flowing in lead i is therefore the net difference between the current injected from lead
i to that injected back into lead i due to reflection and transmission from all the other
leads, and we can write this current as
2e ⎡ ⎤
Ii =
h ⎢⎣
(1 − Rii )Δμi − ∑j≠i TijΔμj ⎥⎦. (6.17)
Conservation of the particle flux requires that

(1 − Rii ) = Tii = ∑i≠j Tji = ∑j≠i Tij . (6.18)
Therefore the reference potential drops out of the problem and we have
2e ⎡ ⎤
Ii =
h ⎢⎣
(1 − Rii )μi − ∑j≠i Tijμj ⎥⎦. (6.19)
The generalization of equation (6.19) to the multi-channel case is obtained by

simply considering that in each lead there are Ni channels at the Fermi energy μi. A
generalized scattering matrix may be defined which connects the different leads and
the different channels in each lead to one another. The elements of this scattering
matrix are labeled tij,mn, where i and j label the leads and m and n label the channels
in the two leads. The probability for a carrier incident in lead j in channel n to be
scattered into channel m of lead i is given by Tij,mn = ∣tij,mn∣2 . Likewise, the probability
of being reflected within lead i from channel n into channel m is given by
Rii = ∣tii,mn∣2 . The reciprocity of the generalized scattering matrix with respect to
magnetic flux now becomes tij ,mn(Φ) = t ji,nm( −Φ). The reservoirs are assumed to feed
6-10
all the channels equally up to the respective Fermi energy of a given lead. If we
define the reduced transmission and reflection coefficients as
Tij = ∑m,n Tij,mn, Rii = ∑m,n Rii,mn, (6.20)
and the total current in lead i becomes

2e ⎡ ⎤
Ii =
h ⎣⎢
(Ni − Rii )Δμi − ∑j≠i TijΔμj ⎦⎥, (6.21)
which differs from equation (6.19) by the presence of the number of modes entering
the lead and the reflection and transmission terms are now sums over modes
themselves.
Each of the chemical potentials may be associated with a local voltage μi = eVi ,
so that equation (6.21) may be written in matrix form as
I = GV. (6.22)
Here, I and V are column matrices and G is the square conductance matrix. The size
of the conductance matrix is set by the number of probes that exist in the sample,
and the elements in the matrix maintain the same symmetry given by equation
(6.16). Thus the conductance matrix is equal to its transpose under reversal of the
magnetic flux. Of course, such relations are expected in linear response theory for a
system governed by a local conductivity tensor satisfying the Onsager–Casimir
symmetry relation.
The resistance matrix may be defined from the inverse of the conductance matrix,
and this matrix maintains the same symmetry properties as those discussed above.
This result may be used to establish an important property that assures that the
reciprocity relation is maintained. We define the sample specific resistance term Rij,kl
as the resistance obtained by passing current through leads i and j while measuring
the voltage at leads k and l, or
Vk − Vl
Rij ,kl = . (6.23)
Iij
Here, the two voltages are related to specific rows of the resistance matrix, while the
only non-zero current entries are Ii = I and Ij = −I. Thus, we may write equation
(6.23) as
(R ki − R kj )I − (Rli − Rlj )I
Rij ,kl = = R ki + Rlj − R kj − Rli . (6.24)
I
Now, if we reverse the magnetic field (and flux), then we obtain
Rij ,kl (Φ) = Rik + Rjl − Rjk − Ril = R kl ,ij ( −Φ). (6.25)
This well-known experimental result basically states that in the absence of a
magnetic field, the resistance measured by passing current through one pair of
contacts and measuring the voltage between the other two is identical to that
6-11
measured if the voltage and current contacts are swapped. In the presence of a
magnetic field, the more general result (6.25) requires that this be carried out while
simultaneously reversing the flux through the sample. These are the requirements of
the reciprocity theorem in circuits.
6.3.1 Two-terminal conductance

Let us first consider the simplest case, where current is injected through lead 1 and
drained through lead 2. These same two terminals will be used to measure the
voltage. By conservation of particle flux, we must have
N1 − R1 = T21
(6.26)
N2 − R2 = T12.
Using these two relations, we can now write the currents as
2e
I = I1 = T12(μ1 − μ2 )
h
(6.27)
2e
− I = I2 = T21(μ2 − μ1).
h
These two equations imply that T = T12 = T21, and that this transmission must be
symmetric in the magnetic flux because it is diagonal. The measured voltage is the
difference in the chemical potentials, and we achieve the Landauer formula
eI 2e 2
G= = T. (6.28)
μ1 − μ2 h
6.3.2 Three-terminal conductance

The situation with three probes is a little more complicated and instructive. The
situation we consider is shown in figure 6.6, and we assume that no current flows
through terminal 3. Thus, this latter terminal corresponds to an ideal voltage probe.
First consider the resistance R12,13, which signifies the ratio of the voltage measured
between contacts 1 and 3 for a certain current I flowing from contact 1 into contact 2.
Using equation (6.21), the condition of zero current flowing in contact 3 gives
Figure 6.6. A hypothetical three-terminal sample.
6-12
2e
0= [(N3 − R33)μ3 − T31μ1 − T32μ2 ]. (6.29)
h
Since particle conservation implies that N3 − R33 = T13 + T23, we may write this
equation as
T31μ1 + T32μ2
μ3 = . (6.30)
T13 + T23
We now turn to probes 1 and 2, where we may write these equations using equation
(6.21) as
2e
I = I1 = [(N1 − R11)μ1 − T12 μ2 − T13 μ3]
h
(6.31)
2e
− I = I2 = [(N2 − R22 )μ2 − T21 μ1 − T23 μ3].
h
If we rearrange equation (6.30), we can replace μ2 in the second of these equations,
and use the continuity of the flux entering probe 2 to give
2e ⎡ (T13 + T23)(T12 + T32 ) − T23T32 (T + T32 )T31 + T21T32 ⎤
−I = ⎢ μ3 − 12 μ1⎥ . (6.32)
h⎣ T32 T32 ⎦
In the absence of a magnetic field, the various T are symmetrical and the numerators
are the same for both terms. (In the presence of a magnetic field, μ2 is influenced by
the Hall voltage and more care is required in establishing the various resistances.)
Then, we find that
μ1 − μ3 hT
R12,13 = = 232 , (6.33)
eI 2e D
where
D = T13(T12 + T32 ) + T23T12 = T31(T21 + T23) + T23T21. (6.34)
In a similar manner, we could solve for μ1 in equation (6.30) and insert this into
the first of equation (6.31). Then, using the conservation of flux through probe 1, we
could then find that
μ3 − μ2 hT
R12,32 = = 231 . (6.35)
eI 2e D
Finally, using the sum of the two resistances in equations (6.33) and (6.35), we find
the total resistance of the device to be
μ1 − μ2 h(T32 + T31)
R12,12 = = . (6.36)
eI 2e 2D
The net two-terminal conductance of the device with three probes will be given by
the inverse of this last equation.
6-13
6.3.3 The quantum Hall device

By starting from a picture of edge states as ballistic one-dimensional channels, it is
possible to account for the main features of the quantum Hall effect, when we use
the Büttiker–Landauer approach described above. We begin this discussion by
assuming that the value of the magnetic field is such that ν spin-split Landau levels
are occupied and that, within the flat interior of the sample, the Fermi level lies
between the νth and (ν + 1)st Landau level. We consider a conductor with a Hall
bar geometry, and with several Ohmic contacts to its 2DEG, as we illustrate in
figure 6.7. The problem of solving for the Hall resistance of this system is one that
involves calculating the different electrochemical potentials at which the contacts
sit, under conditions of a fixed applied current (I). We are guided in this analysis by
several key assumptions, the first of which is that all edge states leaving any given
contact are assumed to be fully equilibrated with it (that is, it is assumed that the
electrochemical potential of these edge states is the same as that of the contact that
they are leaving).
The other important assumption is that, whenever any contact of the Hall bar
is configured as a voltage probe, it then draws no electrical current. It should be
recognized that this statement applies to the net current flowing through the
voltage probe. This current is comprised of two contributions, the first which is
due to incoming edge states that are equilibrated at the electrochemical potential
of the neighboring contact (downstream in the sense of the edge-state flow), while
the other is carried by edge states leaving the probe. As is common in experi-
ments, we consider that contacts 2, 3, 5, and 6 in figure 6.7 are voltage probes,
while contacts 1 and 4 are used to source and sink the current, respectively. We
can then write the following expressions for the current drawn by the voltage
probes:
Figure 6.7. A descriptive schematic of a typical Hall bar sample. The edge states are indicated by the blue
trajectories. Typical closed cyclotron orbits within the bulk of the sample are indicated in black.
6-14
eν
I2 = 0 = (μ − μ1)
h 2
eν
I3 = 0 = (μ − μ2 )
h 3 (6.37)
eν
I5 = 0 = (μ − μ4)
h 5
eν
I6 = 0 = (μ − μ5).
h 6
As indicated in the first line, we have made the tacit assumption that
T21 = T32 = T43 = T54 = T65 = T16 = ν, and the reverse transmissions are zero.
Hence, these equations lead us to the result that
V3 = V2 = V1
(6.38)
V6 = V5 = V4.
The two current probes of course do draw current through the device, as these two
probes are the actual source and sink of the current. Thus, these two equations can
be written as
2eν
I1 = I = (μ − μ6)
h 1
(6.39)
2eν
I4 = − I = (μ − μ3).
h 4
With these results, we can now calculate the Hall resistance, which can be measured
either between probes 1 and 6 or between 2 and 5. With the first pair, we have
V2 − V6 h V − V6 h
RH = R14,62 = = 2 2 = 2, (6.40)
I νe V1 − V6 νe
where we have used equation (6.38) to set V1 = V2. This last equation is, of course,
just equation (6.15).
The other unique feature of the quantum Hall effect is the vanishing of the
longitudinal resistance for the same ranges of magnetic field where the quantized
plateaus are observed. This feature can also be explained by the picture of non-
dissipative transport via adiabatic edge states. For the geometry of figure 6.7, the
longitudinal resistance (Rxx) may be determined by measuring the voltage drop
between probes 2 and 3, or 5 and 6. Rxx vanishes for either configuration, however,
just as is found in experiments, as the voltage is the same at each of the probes
selected.
6.3.4 Selective population of edge states

The edge-state picture of the quantized resistances in the quantum Hall effect
illustrates clearly that such quantization arises from the perfect transmission of
6-15
carriers from one ideal contact to another due to the suppression of back-
scattering. The resistance is essentially given in terms of a fundamental constant
times the number of transmitted edge states at the Fermi energy. A number of
experiments have probed this picture of high-field magnetotransport through
independent control of the number of edge states that are transmitted through a
potential barrier [11, 12]. Such experiments typically feature a geometry similar to
that shown in figure 6.8, in which the key innovation is that a gate is now
introduced to generate a potential barrier across the primary current path. In
practice, the barrier may be realized by using a split-gate quantum point contact
(QPC), as was discussed in chapter 2. To understand the resulting behavior in this
system, it is important that we appreciate that the edge states corresponding to
different Landau levels propagate along the edges of the sample while following
equipotential paths with distinctly different guiding-center energies. That is, the
edge states arising from the highest occupied Landau level (or its spin-split
counterpart) will lie the furthest from the edge of the sample. Edge states from
lower-lying Landau levels will be closer to the edge. Thus a gate can selectively
reflect some Landau levels while allowing others to propagate.
With these considerations, the problem becomes one of calculating the con-
ductance of the gated device, using similar arguments to those employed in our
analysis of the quantum Hall effect. To further simplify the analysis we are free to
define the voltage of probe 4 to be equal to zero (V4 = 0), so that the voltages of the
remaining probes are then all referenced with respect to this. The most important
difference with our earlier analysis is that now the edge states entering probes 3 and
6 originate from different probes, whereas before the absence of any barrier meant
that they always came from the same probe. Here, we assume that νR edge states
are reflected at the barrier, and replace all of the transmission coefficients by the
number of transmitted and reflected edge states. Then, the various probe currents
become
Figure 6.8. A quantum Hall bar with a gate (orange) placed over a portion of the device to allow controlled
reflection of some edge states. Here, we indicate two edge states in the device, of which only one is allowed
through the gated region.
6-16
e
I1 = I = (νμ − νμ6)
h 1
e
I2 = 0 = (νμ2 − νμ1)
h
e
I3 = 0 = [νμ3 − (ν − νR )μ2 − νR μ5]
h (6.41)
e
I4 = − I = (νμ4 − νμ3)
h
e
I5 = 0 = (νμ5 − νμ4)
h
e
I6 = 0 = [νμ6 − (ν − νR )μ5 − νR μ2 ].
h
By simple inspection of these equations, we note that the previous voltage equality
among the non-current-carrying probes on the two sides of the bar is upset. To
simplify the analysis, we take V4 = 0 to be our reference potential. Hence, we now
have
V5 = V4 = 0
V2 = V1
⎛ ν ⎞
V3 = ⎜1 − R ⎟V2 (6.42)
⎝ ν⎠
νR
V6 = .
ν
With these voltage expressions, we can now compute the longitudinal resistance to
be [12]
V2 − V3 h V − V3 h⎛ 1 1⎞
Rxx = R14,23 = = 2 2 = 2⎜ − ⎟. (6.43)
I νe V1 − V6 e ⎝ ν − νR ν⎠
If we have four edge states in the un-gated region, and we reflect two of them, then
the longitudinal resistance is h/4e2 according to this equation. Similarly, if we have
two edge states in the un-gated region, and reflect one of them, the longitudinal
resistance should be h/2e2 according to this equation. In figure 6.9, we show results
from Haug et al [12] that demonstrate this result and show that the plateaus obtained
are quite flat and precise in their nature. It is another result that shows the clear
nature of the quantization with the edge states.
6.3.5 Nature of the edge states

More information can be obtained about the nature of the edge states by probing
them with a SGM. In this experiment, the AFM tip is metalized and with a bias
voltage applied can be used as a scanning gate to locally probe the conductance and
the carrier density. The sample uses the ideas of figure 6.8 except that in-plane gates
(figure 2.1) are used so that no gate metal can interfere with the AFM tip. The
sample was fabricated in an InAlAs/InGaAs/InAlAs quantum well heterostructure
6-17
Figure 6.9. Resistance across a gated region as a function of the gate bias. The gate width is 10 μm, and the
temperature is 0.55 K. Two different values of magnetic field are used, which lead to bulk filling factors of 2
and 4 as indicated. (Reprinted with permission from [12]. Copyright 1988 the American Physical Society.)
with the InGaAs quantum well lying 45 nm below the surface [13]. Shubnikov–de
Haas measurements showed that there were two subbands occupied with densities of
7.2 × 1011 cm−2 and 2.1 × 1011 cm−2. With the gate regions biased, no edge states of
the second subband were observed to pass through the QPC, so that these were
apparently totally reflected. Studies of the relative effects of tip bias and gate bias
showed that the capacitances differed by about a factor of three, so that quantitative
measurements of the conductance, and its change, could easily be made. With a
magnetic field of 6.6 T applied to the sample, the quantum Hall effect was clearly
established with approximately five edge states established in the lowest subband in
the bulk of the sample. With a gate bias of −7.0 V, these separate so that only one
edge state propagates through the QPC. Using equation (6.43), this leads to a
conductance through the QPC of approximately 1.25e 2 /h. At less negative gate bias,
a second edge state is observed to transit the QPC, and equation (6.43) would
suggest a rise in conductance to approximately 3.3e 2 /h, although a somewhat
smaller value is found in this case. In figure 6.10, we plot a series of line scans
which depict the local conductance as the tip is scanned across the QPC in a
direction normal to the current flow for a tip bias of −1.0 V. It is clear that at the
most negative bias used, the tip bias leads to a vanishing conductance as it is
sufficient to deplete the entire area. The existence of the propagating edge state is
clear in these scans as indicated by the plateaus that exist near 1.25e 2 /h. The
interesting feature is, in fact, the existence of these plateaus that suggest that the
potential drop and density of the edge state are not the simple ideas appearing in
6-18
Figure 6.10. Conductance profiles determined by a scanning gate with a tip bias of −1.0 V at a temperature of
280 mK. The letters a–e depict curves for in-plane gate biases of −3.8, −4.2, −5.0, −5.8, and −7.4 V,
respectively. (Adapted with permission from [13]. Copyright 2005 the American Physical Society.)
Figure 6.11. The top panel is a three-dimensional representation of curve d of figure 6.10. The bottom panel
shows three representations of the edge states with color bars indicating to which portion of the three-
dimensional image they refer.
figure 6.3. Instead, something more complicated is occurring, to which we shall

return shortly.
We can understand what we are seeing in figure 6.10 with a different approach. In
the top panel of figure 6.11, we plot a three-dimensional image of the conductance
for −5.8 V on the in-plane gates (curve d of figure 6.10). Below this panel are three
schematics of the QPC and the edge states with a color bar indicated the portion of
the three-dimensional plot. In the bluish region, the QPC is sufficiently open that
two edge states propagate through the QPC with a conductance as discussed above.
In the yellowish region, however, the QPC has closed down from the tip bias so that
only a single edge state propagates through it. Finally, the reddish portions of the
three-dimensional image are where the tip has closed the QPC so that all edge states
are reflected. The fact that the conductance shows a plateau in the state in which
6-19
only one edge state is propagated is a reflection of the fact that the local potential
should also show such a plateau, which is different than indicated in figure 6.4.
The above results suggest that a proper understanding of the microscopic
structure of the edge states is vital to the quantitative analysis of many experiments.
This understanding has been put forward by Chlovskii et al, who carried out a self-
consistent determination of the electrostatics of the edge states associated with a
potential boundary, such as that formed by a gate [14, 15]. They showed that the
position of the ξth spin-split edge state from the gate edge is approximately
⎡ ν 2 + ξ2 ⎤
x = d ⎢1 + 2 ⎥, (6.44)
⎣ ν − ξ2 ⎦
where ν is the bulk filling factor and 2d is the depletion width around the gate at zero
magnetic field. More importantly, the space occupied by the level as it passes
through the Fermi energy is not the small amount indicated in figure 6.4, but is a
larger space as required by the electrostatics. If the density changed instantaneously
as indicated by this latter figure, this would require a massive change in the potential
to support this discontinuity in density. The screening properties of the 2DEG in the
presence of a magnetic field requires that self-sufficient evaluation of the potential as
the magnetic field is varied. This causes the self-consistent potential to develop a
series of broad terraces that are separated in energy by the same energy as that of the
bulk levels. The electrons sit in the spaces of these terraces and provide compressible
regions of the electron gas in the sense that they correspond to spatial regions where
many electrons are available near the Fermi energy. These regions are characterized
by a metallic conductivity and contribute the screening which, in turn, leads to the
plateaus in potential. The spatial regions between the edge states now correspond to
incompressible regions, as they have a lack of available states close to the Fermi
level. These incompressible stripes are regions of integer filling factor. These plateaus
in potential are precisely the constant conduction stripes that appear in figures 6.10
and 6.10.
6.4 The fractional quantum Hall effect

As remarkable as the quantum Hall effect is, it was discovered in a Si MOSFET
device. As interest moved to heterostructure material systems, where the quality of
the material was much higher, and consequently the mobility was much higher, a
new structure was discovered within the quantum Hall effect. This second quantum
Hall effect is the fractional quantum Hall effect, or FQHE. In general, one expects
the Hall resistance to show the simple plateaus predicted by equation (6.15) and the
longitudinal resistivity (or conductivity) to show a set of zeroes at the plateaus. As ν
is an integer for these plateaus, this has come to be called the integer quantum Hall
effect, or IQHE. Experimentally, in a high quality material such as that of a GaAs/
AlGaAs heterostructure, the behavior is qualitatively different from this simple
picture. The first measurements were made by Störmer and Tsui [16], in which they
saw particularly strong plateaus at values of 1/3 and 2/3 for ν. As the quality of the
2DEG material has improved over time, the richness of the structure in the FQHE
6-20
has also increased [17], with the presence of better defined zeroes at more fractions.
As can be seen from figure 6.12, a quite rich structure exists in the longitudinal
resistance that is accompanied by the appearance of further Hall plateaus as one
enters the regime of fractional filling factor. Crucially, however, these fractions are
found to correspond to very specific combinations of integer numerators and
denominators.
The deep minimum which Störmer and Tsui observed at ν = 1/3, which is much
stronger than any of the other fractional states [16], led to the conclusion by
Laughlin that the electrons had to be condensing into a new collective ground state
[18, 19]. That is, while the IQHE could be understood within a single-particle
picture, the FQHE must be a complicated many-body state. Laughlin predicted that
the new collective state was a quantum fluid for which the elementary excitations,
quasi-electrons and quasi-holes, were fractionally charged [18]. Moreover, Laughlin
proposed a ground state wave function that possessed angular momentum, with the
eigenvalue of 1/ν. Thus, the ν = 1/3 state had an angular momentum of 3ℏ. (For this
discovery, Laughlin shared the 1998 Nobel Prize in physics with Störmer and Tsui.)
This leads one to conclude that the quantity ν has a deeper meaning than simply
Figure 6.12. An overview of the longitudinal resistivity ρxx and the Hall resistivity ρxy for a very high mobility
heterostructure at 150 mK. The use of a hybrid magnet required composition of this figure from four different
field traces (breaks at 12 T are noted). (Reprinted with permission from [17]. Copyright 1987 the American
Physical Society.)
6-21
being an integer that counts filled edge states. If we return to our derivation of the
Hall resistance, we can rewrite the electron density as
B νeB
ns = = . (6.45)
eRH h
Let us now multiply both sides by the area A of the sample, to obtain the total
number of electrons in the sample, as
νeBA Φ
N = nsA = =ν , (6.46)
h Φ0
where Φ is the total flux passing through the sample and Φ0 is the flux quantum h/e.
Hence it is clear that ν is the number of electrons per flux quantum in the system. For
ν = 1, we have one electron for each flux quantum. Again, for ν = 1/3, we have one
electron for every three flux quanta. This then suggests that in the many-body state
for this latter fractional plateau, each electron is likely to be bound to three flux
quanta.
An important consequence of the collective origin of the FQHE is the general
incompressibility of the electronic state, just as in the integer FQE (IQHE). While
such incompressibility is a simple consequence of the Pauli principle for the IQHE,
its existence at the fractional fillings must be a result of the repulsive interactions
between the electrons [20]. Haldane noted that by extending Laughlin’s ideas into a
spherical geometry, one could obtain a translationally invariant version which was
readily extended to an entire hierarchy of fractional states [21]. The entire hierarchy
of fractional (and integer) states can be written as [22, 23], in its most common form,
p
ν= . (6.47)
2ps + 1
States with s = 0 correspond to the IQHE, while states for s > 0 give rise to the set of
FQHE states. It was also suggested that an excitation gap existed between the
ground state of the FQHE and any excited states, and that the kinetic energy needed
to bridge this gap was quenched by the high magnetic field [24]. The cause of such a
gap is presumably due to the many-body correlations arising from the Coulomb
interaction among the electrons, as well as with the flux quanta. An important
further recognition was that the arguments above could be understood on topo-
logical grounds, and that the flux was exceedingly important in this context [25].
6.5 Composite fermions

In the above discussion, it is clear that only fractional factors in which the
denominator was an odd integer appeared in the experimental data. However, in
1987, experimental evidence was first presented in which an even-denominator
plateau was seen to begin to form [17], and there is evidence for this in figure 6.12. In
this work, in fact, evidence appeared for the ν = 1/2, 3/2, and 5/2 plateaus.
Observation of the latter two plateaus was particularly significant, since they
correspond to the observation of fractional Hall quantization under conditions
6-22
where more than one spin-resolved Landau level is occupied. Moreover, it was
particularly surprising that the ν = 5/2 state was the better resolved of the three,
showing the clearest minima in the longitudinal resistivity. Expansion of figure 6.12
around the ν = 5/2 state is shown in figure 6.13. It was also found that tilting the
magnetic field led to a rapid collapse of these even-denominator plateaus, an effect
which is not seen in the odd-denominator fractional states [26]. While there is no
rational reason to exclude the even-denominator fractional states, it does not sit well
with the description introduced above.
Shortly after the experiments, Haldane and Rezayi [27] showed theoretically that
a new incompressible quantum-liquid state of electrons existed which gave half-
integral quantum Hall effect quantization for a non-polarized spin-singlet ground
state. Moore and Read [28] then pointed out that the attachment of two quanta of a
Figure 6.13. Expansion of figure 6.12 around the ν = 5/2 state, showing the detailed number of different
plateaus that are found. This is the box marked ‘a’ in the previous figure. A range of temperatures from 25 to
100 mK are shown. The filling factors are indicated in the longitudinal resistance while quantum numbers p/q
(=ν) are shown in the Hall resistance. (Reprinted with permission from [17]. Copyright 1987 the American
Physical Society.)
6-23
fictitious flux to each electron would lead to an acceptable order parameter for this
spin-singlet state. This so-called fictitious magnetic field comes from a Chern–
Simmons gauge transformation, which introduces a gauge magnetic field [29]. The
important idea here is that the electron, plus the two flux quanta, form what is
known as a composite fermion. In this new quasi-particle, the flux quanta are
actually connected with quantized vortices [30]. Here, the gauge field is just exactly
strong enough to cancel the external field at ν = 1/2. That is, when one rewrites the
total Hamiltonian, with the gauge transformation, in terms of these composite
particles, there is no magnetic field remaining at the value of the applied field for this
filling factor. Thus, the composite fermions represent a system of spinless fermions in
a (net) zero magnetic field. An additional astonishing fact is that, if the density of
electrons ns is held fixed, then the magnetic field corresponding to ν = p /(2p + 1)
(p = 1 for the ν = 1/3 plateau) satisfies
hns
ΔB = B − B1 = , (6.48)
2 ep
where
2hns
B1 = (6.49)
2 e
is the magnetic field at which ν = 1/2. That is, the fractional plateaus correspond to
the integer plateaus for the composite fermions [22]! Experimental studies of the
composite fermion quickly established the reality of the composite fermion by
measurements of the magnetotransport.
When we say that the composite fermion is a quasi-particle, this means that it
behaves like e.g. an electron with a different mass from that at low magnetic field.
Values for this mass have been inferred from Shubnikov–de Haas studies with a
range of values of 0.53–0.92m0 [31], 0.43m0 with a magnetic field dependence [32],
0.91m0 right at the ν = 1/2 plateau, with indications of interactions between the
composite fermions [33]. More recent measurements have focused upon using
cyclotron resonance, with surface acoustic waves, to determine the effective
mass [34] and this has shown both a density dependence and a magnetic field
dependence [35]. In figure 6.14, the cyclotron resonance frequency is shown as a
function of magnetic field around the ν = 1/2 plateau for two different wave numbers
of the rf and for two different densities. At the same cyclotron resonance frequency,
there is a dependence upon the wave number as well as the density. A given
frequency occurs at a different magnetic field for different wave numbers with
surface acoustic waves. In figure 6.15, the dispersion of the cyclotron frequency and
the inferred composite fermion mass are plotted, using a scaling that seems
warranted by the experiments. The value of mass extrapolated to zero wave number
appears to scale as 1.6(nsa B2 )1/2 away from the ν = 1/2 plateau, where aB is the Bohr
radius (about 10 nm in GaAs).
As we have progressed from the IQHE to the FQHE to the composite fermion,
the physics has gotten more complicated. But, this progression may not be complete.
6-24
Figure 6.14. The magnetic field dependence of the composite fermion cyclotron resonance mode for two values
of the wave vector, kSAW = 10.5 × 107 m−1 (open symbols) and 3.9 × 107 m−1 (solid symbols), and two electron
densities: (a) ns = 1.09 × 1011 cm−2 and (b) ns = 0.59 × 1011 cm−2. (Reprinted with permission from [35].
There are indications that they may well be a cascade of more phase transitions near
the ν = 5/2 plateau [36], and that might suggest that there is more exciting physics
still to be discovered.
Problems
1. Consider a 2DEG at zero temperature. Construct plots that show the
variation of the different Landau level energies, as well as the associated
variation of the Fermi level, as a function of magnetic field. Compute these
plots for a 2DEG of density 2 × 1011 cm−2 and 5 × 1011 cm−2. Explain the
difference between the resulting plots.
2. Consider figure 6.13 from [29]. (a) Use the Hall resistance data to determine
the 2DEG density. (b) Use the Shubnikov–de Haas oscillations to construct a
so-called Landau plot, and thus determine the electron density. (c) Confirm
that the plateau in the Hall resistance near 4.2 T is consistent with the
electron density determined for the sample. (d) The Shubnikov–de Haas
oscillations show an additional splitting at high magnetic fields. Explain the
physical origin of this effect.
3. When discussing the Hall effect in a 2DEG, the two-dimensional nature of
current flow leads to the expression J = σE, where σ is the conductivity tensor
(6.11). This matrix, and the associated resistivity matrix given by the inverse
of (6.11). Consider a heterostructure in which two subbands are occupied
6-25
Figure 6.15. Left: k dependence of the cyclotron resonance mode energy of composite fermions (expressed in
terms of inverse cyclotron mass) for ns = 1.09 × 1011 cm−2 and ns = 0.59 × 1011 cm−2. Right: The composite-
fermion cyclotron mass in the limit k → 0 as a function of density. The inset plots the B-field location of the
cyclotron-resonance mode versus the SAW frequency. The dashed line marks the commensurability condition
where the SAW wavelength equals the composite- fermion cyclotron diameter. (Reprinted with permission
from [35]. Copyright 2007 the American Physical Society.)
(with densities n1 and n2 and mobilities μ1 and μ2). In this case, the total
current is obtained by adding the appropriate conductivity matrix elements,
before inverting to get the total resistivity tensor. (a) By defining an effective
number density neff and relating this to the measured Hall resistance show that,
at low magnetic fields (where ωcτ < 1), n eff = (n1μ1 + n2μ2 )2 /(n1μ12 + n2μ2 2 )
(Hint: use the identity ωcτ = μB ). (b) Show that the longitudinal resistivity
develops a parabolic dependence on magnetic field in the presence of the two-
channel transport.
4. let us consider a device such as that in figure 6.9, but where the bottom two
probes are removed (they do not exist). (a) Write expressions for the current
flowing through the four different contacts, in terms of their associated
voltages. (b) Derive an expression for the resistance that would be measured
by passing current between contacts 1 and 4 and measuring the voltage drop
between 2 and 3. (c) What is the measured resistance when all of the edge
states in the 2DEG are transmitted through the constriction? Also, what is
the value of the resistance when a fixed number of edge states pass through
the QPC and the Landau index NL → ∞ (i.e. B → 0)? What do these
observations tell us about the physical origin of the quantized conductance in
one-dimensional conductors?
6-26
References
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[6] Mohr P J and Taylor B N 2000 Rev. Mod. Phys. 72 351
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[30] Heinonen O 1998 Composite Fermions (Singapore: World Scientific)
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6-27
IOP Publishing

(Second Edition)
David K Ferry
Chapter 7
Spin
In the previous chapter, we encountered spin-splitting in the measurements of

transport at high magnetic fields. Such spin-splitting is a result of the Zeeman effect
[1], in which the energy of a free carrier is modified by the magnetic field interacting
with the spin. Normally, the effect is most familiar with optical spectroscopy of
atoms or of impurity levels in a solid. For the free carriers in a semiconductor,
however, the Zeeman effect leads to just two levels, given as the additional energy
1
EZ = gμB S · B = ± gμB Bz , (7.1)
2
where the magnetic field is oriented in the z-direction. Here, μB (= eℏ/2m0) is the
Bohr magneton, 57.94 μVT−1. The factor g is referred to as the Landé g-factor,
which has a value of 2 for a truly free electrons. It differs greatly from this value in
semiconductors, and can even be negative (∼−0.43 in GaAs at low temperatures [2]).
This negative value has the effect of reversing the ordering of the two spin-split
energy levels. As was discussed in the previous chapter, the Zeeman effect leads to
splitting of the Landau levels and can be seen in the Shubnikov–de Haas oscillations
at high magnetic fields. This splitting of the spin degeneracy of the Landau levels led
to the IQHE [3].
While the Zeeman effect is the best known of the spin effects on transport, there
are other effects which have become better known since the intense interest in spin-
based semiconductor devices arose a few decades ago. This interest was spawned by
the idea of a spin-based transistor [4], but has grown over the possibility of spin-
based logic gates which will not be subject to the capacitance limitations of charge-
based switching circuits. Many of these new concepts are dependent upon the
propagation of spin channels, and the use of the spin orientation as a logic variable,
and this has fostered the term spintronics [5].

7.1 The spin Hall effect

The spin of an electron can be manipulated in a large number of ways, but in order
to take advantage of current semiconductor processing technology, it would be
preferable to find a purely electrical means of achieving this. For this reason, a great
deal of attention has centered on the spin Hall effect in semiconductors. The idea
was apparently first suggested by Dyakonov and Perel [6] and later, and independ-
ently by Hirsch [7]. The basic idea was that, in the presence of scatterers, it was
possible for spins of one orientation to be scattered in a different direction than spins
of the opposite orientation. This would lead different spins to accumulate on
opposite sides of the sample, a result of the presence of anisotropic scattering in the
presence of the spin–orbit interaction [8]. Thus, a transverse spin current arises in
response to a longitudinal charge current, without the need for magnetic materials or
externally applied magnetic fields [9, 10]. The spin–orbit interaction is a common
part of the energy bands of a semiconductor, where it splits the otherwise triply
degenerate top of the valence band at the Γ point. But, there are other forms of the
spin–orbit interaction that are of interest in situations in which symmetries are
broken in the semiconductor device. In the spin Hall effect, we achieve edge states,
as in the quantum Hall effect, but here these edge states are spin polarized. The
impurity driven spin separation is known as the extrinsic spin Hall effect, and there
can be an intrinsic spin Hall effect directly from the spin–orbit interaction when
asymmetries exist in the device.
7.1.1 The spin–orbit interaction

The quantum structure of atoms can lead to the angular momentum of the electrons
mixing with the spin angular momentum of these particles. Since the energy bands
are composed of both the s- and p-orbitals of the individual atoms in many of the
semiconductors, it has been found that the spin–orbit interaction affects band
structure calculations. The spin–orbit interaction is a relativistic effect in which the
angular motion of the electron interacts with the gradient of the confining potential
of the atom to produce an effective magnetic field. This field couples to the spin in a
manner similar to the Zeeman effect. Early papers, which used first-principles
calculations of the band structure, clearly demonstrated that the spin–orbit
interaction was important for the detailed properties of the bands [11, 12]. Not
the least of these effects is the splitting of the three-fold bands at the top of the
valence band, producing the so-called split-off band. This latter band lies from a few
meV to a significant fraction of an eV below the top of the valence band in various
semiconductors.
The first inclusion of the spin–orbit interactions in modern pseudopotential calcu-
lations for the band structure of semiconductors is thought to be due to Bloom and
Bergstresser [13], who extended the interaction Hamiltonian of Weisz [14] to
compound semiconductors. We can illustrate how the spin–orbit interaction
modifies the transport by considering a simple Hamiltonian for the electrons in a
periodic potential V(r) as [15]
7-2
p2 ℏ
H (p , r) = + V (r) + ( σ × ∇V ) · p , (7.2)
2m 4m 2 c 2
where p is the normal momentum operator and the last term represents the spin–
orbit interaction. The quantity σ is a vector whose components are the normal Pauli
spin matrices, as
σ = σxax + σya y + σz az , (7.3)
where (see appendix D)
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
σx = ⎢ 0 1 ⎥ , σy = ⎢ 0 − i ⎥ , σz = ⎢1 0 ⎥ . (7.4)
⎣1 0 ⎦ ⎣i 0 ⎦ ⎣ 0 − 1⎦
It is clear from these matrices that the wave function is now more complicated with a
spin component, to which we return below. The direct effect of the spin–orbit
interaction may be qualitatively understood as an extra energy cost for the align-
ment of the intrinsic magnetic moment of the electron with the magnetic field that
arises from its own orbital motion. As a result, this term leads to a modification of
the momentum of the carriers, through a gauge transformation of the wave function
much like the Peierls’ modification from the presence of the magnetic field.
7.1.2 Bulk inversion asymmetry

Bulk inversion asymmetry arises in crystals which lack an inversion symmetry, such
as the zinc-blende materials. In these crystals, the basis pair at each lattice site is
composed of two dissimilar atoms, such as In and P, or Ga and Sb. Because of this,
the crystal has lower symmetry than, e.g., the diamond lattice, where the basis pair is
two Si atoms. Without this inversion symmetry, one still can have symmetry of the
energy bands E(k) = E(−k), but the periodic part of the Bloch functions no longer
satisfies uk(r) = uk(−r). As a result of this, the normal two-fold spin degeneracy is no
longer required throughout the Brillouin zone [16]. The importance of this
interaction is already recognized by people who study the electronic band structure.
The inclusion of this term via the spin–orbit interaction leads to the warped surface
of the valence bands [17]. For the conduction band, the perturbing Hamiltonian can
be written as
HBIA = η⎡⎣{kx, k y2 − k z2}σx + {ky, k z2 − k x2}σy + {kz , k x2 − k y2}σz ⎤⎦ , (7.5)
where kx, ky, and kz are aligned along the [100], [010], and [001] axes, respectively,
and the σi are the Pauli spin matrices introduced in equation (7.4). The terms in curly
brackets are modified anti-commutation relations given by
1
{A , B} = (AB + BA). (7.6)
2
The parameter η comes from the spin–orbit interaction and is given by [18]
7-3
4i ⎡ 1 1 ⎤
η= PP′Q⎢ − ⎥, (7.7)
3 ⎣ (EG + Δ)(Γ0 − Δc ) EG Γ0 ⎦
where EG and Δ are the primary energy gap and the spin–orbit splitting of the
valence band in a material in which the minimum of the conduction band and the
maximum of the valence band both occur at the Γ point in the Brillouin zone. The
quantities P, P′ and Q are matrix elements in the spin–orbit interaction along the
line of
iℏ
P= 〈S∣pz ∣Z 〉, (7.8)
m0
where S and Z are s-symmetry and p-symmetry wave functions and pz is the
momentum operator. Finally, in equation (7.7), Γ0 is the splitting of the two lowest
conduction bands at the zone center and Δc is the spin–orbit splitting of the lowest
conduction band at the zone center. This interaction is stronger in materials with
small band gaps, as may be inferred from equation (7.7). Note that equation (7.5) is
cubic in the magnitude of the wave vector and is often referred to as the k3 term.
While the above expressions apply to bulk semiconductors, much of the interest in
recent years has been directed at Q2D systems in which the carriers are confined in a
quantum well such as exists at the interface between AlGaAs and GaAs. Often, this
structure is then patterned to create a quantum wire. For example, a common
configuration is with growth along the [001] axis, so that there is no net momentum
in the z-direction, and 〈kz〉 = 0, while 〈kz2〉 ≠ 0 is a representation of the quantization
energy in the z-direction. Then, equation (7.5) can be written as
HBIA = η⎡⎣〈k z2〉(kyσy − kxσx ) + kxky(kyσx − kxσy )⎤⎦ . (7.9)
This is an important result. The prefactor of the first term in the square brackets is
constant, and depends upon the material and the details of the quantum well. This
average over the z-momentum corresponds to the different subbands in the quantum
well, so that only a single value will result when the carriers are only in the lowest
subband. But, this structure has now split equation (7.5) into a k-linear term and a k3
term.
To explore equation (7.9) a little closer, let us chose a set of spinors to represent
the spin-up and -down states as follows:
⎡ ⎤ ⎡ ⎤
∣+⟩ = ∣↑⟩ = ⎢1 ⎥ , ∣−⟩ = ∣↓⟩ = ⎢ 0 ⎥ , (7.10)
⎣0⎦ ⎣1 ⎦
as in appendix D. Then, the linear first term in equation (7.9) gives rise to an energy
splitting according to
ΔE1 ∼ − η〈k z2〉(kx ± iky ) (7.11)
in the rotating coordinates discussed in appendix D. Now, the spin ‘up’ state rotates
around the z-axis in a right-hand sense (with the thumb in the z-direction) with the
7-4
spin polarization tangential to the constant energy circle in two dimensions. On the
other hand, the spin ‘down’ state rotates in the opposite direction, but with the spin
polarization still tangential to the energy circle (we will illustrate this rotation
below).
If we ignore the cubic terms in momentum (the last term in equation (7.9)) for the
moment and solve for the eigenvalues of the two spin states, we have
⎡ ℏ2k 2 ⎤
⎢ − η 〈k 2
z 〉( kx + ik y ) ⎥
2m⁎
H=⎢ ⎥, (7.12)
⎢ 2 ℏ 2 2
k ⎥
⎢⎣− η〈k z 〉(kx − iky ) ⁎ ⎥⎦
2m
where we have assumed that the normal energy bands are parabolic for convenience.
Then, we find that the energy levels of the two states are given as
ℏ2k 2
E= ± η〈k z2〉k . (7.13)
2m⁎
Thus, we find that not only is the energy splitting linear in k, but it is also isotropic in
the two-dimensional momentum space. The resulting energy bands for the two states
are composed of two inter-penetrating paraboloids, and a constant energy surface is
composed of two concentric circles (see figure 7.1 below). The inner circle represents
the positive sign in the above equation while the outer circle corresponds to the
negative sign. The eigenfunctions are no longer pure spin states, but are an
admixture given by
1 ⎡ 1 ⎤
φz(±) = ⎢ ⎥, (7.14)
2 ⎣ e ±i ϑ ⎦
where ϑ is the angle that k makes with the [100] axis of the underlying crystal in the
heterostructure quantum well. This angle is the angle defined by the polar
coordinates in the two-dimensional momentum space. Hence, the root with the
upper sign in equation (7.13), which we think of as being mostly spin ‘up’, has the
spin polarization tangential to the inner circle. Correspondingly, the root with the
lower sign in equation (7.13), which we think of as mostly spin ‘down’, has the spin
polarization tangential to the outer circle.
If we now add in the cubic terms in equation (7.9), the energy levels of equation
(7.13) are modified to be
ℏ2k 2 ⎡ ⎛ k4 k2 ⎞ 2 ⎤1/2
E= ± η 〈k 2
z 〉k ⎢ 1 + ⎜ 22 − 4 ⎟sin ϑcos 2 ⎥
ϑ . (7.15)
2m⁎ ⎢⎣ ⎝ 〈k z 〉 〈k z2〉 ⎠ ⎥⎦
This is a much more complicated momentum and angle dependence, but suggests
that the constant energy circles are now warped in the same manner as the valence
band. The transport is no longer isotropic in the transport plane. Similarly, the phase
7-5
Figure 7.1. (a) Energy structure for the Rashba spin–orbit coupled system. The momentum is shown by green
arrows and the spin polarization is shown by the red arrows. (b) When an electric field is applied for some time,
the spin is rotated out of the plane according to the direction of the transverse momentum. (Reprinted with
permission from [10]. Copyright 2004 the American Physical Society.)
on the down spin contribution to the eigenfunction equation (7.14) is no longer

simply the angle, but acquires a ‘wobble’ as it rotates around the circle.
7.1.3 Structural inversion asymmetry

The spin Hall effect most commonly originates from the Rashba form of spin–orbit
coupling [19], which is present in a 2DEG formed in an asymmetric semiconductor
quantum well. Such an asymmetric quantum well is the quasi-triangular well found
at the interface between AlGaAs and GaAs or at the interface of a Si MOSFET
shown in figure 1.2, although the spin–orbit interaction is small in Si (but still very
important). This is known as structural inversion asymmetry. In either of the
quantum wells mentioned above, the structure is asymmetric around the hetero-
junction (or oxide) interface. As a result, there is a relatively strong electric field in
the quantum well, and motion normal to this can induce an effective magnetic field.
This is the structural inversion asymmetry known as the Rashba effect [19]. Just as in
7-6
the bulk inversion asymmetry, the electric field in the quantum well can lead to spin
splitting without any applied magnetic field, due to the spin–orbit interaction. If we
take the z-axis as normal to the heterojunction interface, then the spin–orbit
interaction equation (7.2) can be written as
HSIA = r σ · (k × ∇V ) → [α · (σ × k)] z , (7.16)
where
P2 ⎡ 1 1 ⎤ P′2 ⎡ 1 1 ⎤
r= ⎢ 2 − ⎥+ ⎢ 2 − ⎥ (7.17)
3 ⎣ EG (EG + Δ) ⎦
2
3 ⎣ Γ0 (Γ0 + Δc ) ⎦
2
arises from the spin–orbit interaction in the creation of the band structure, and the
symbols have the same meanings as in equation (7.7). The parameter is composed of
the constants and the electric field, and for the configuration discussed, we have
HSIA = αz(kyσx − kxσy ), (7.18)
where
1 ⎛ ℏ ⎞ ∂V
2
αz = ⎜ ⎟ . (7.19)
4 ⎝ m 0c ⎠ ∂z
If we continue to use the basis set defined in equation (7.10), then the Rashba
contribution to the energy is simply
ER = ∓iαz(kx ± iky ). (7.20)
While the spin states are split in energy, this does not simply add to the bulk
inversion asymmetry. First, the two spin states are orthogonal to each other, and
then they are phase shifted (with opposite phase shift) relative to the previous results.
It is easier to understand the effect of this Rashba term if we diagonalize the
Hamiltonian for the two spin states with the Rashba contribution. We can write the
Hamiltonian, for parabolic bands using equation (7.20) as
⎡ ℏ2k 2 ⎤
⎢ ⁎
αz(kx + iky )⎥
H=⎢ ⎥.
2m
(7.21)
⎢ ℏ2k 2 ⎥
⎢⎣ αz(kx − iky ) ⁎ ⎥⎦
2m
When this Hamiltonian is compared to equation (7.12), the two off-diagonal terms
are shifted not only by the minus sign in front, but also by a phase factor of −π/2
within the term in parentheses in the upper right term and π/2 in the lower left term.
Nevertheless, we can still diagonalize the Hamiltonian to give the new energies
ℏ2k 2
E= ± αzk . (7.22)
2m⁎
7-7
As in the case of the linear term in the bulk inversion asymmetry, the energy splitting
is linear in k, and also isotropic with respect to the direction of k. Thus, the energy
bands also are composed of two inter-penetrating paraboloids, and a constant
energy surface is composed of two concentric circles (see figure 7.1). The inner circle
represents the positive sign in equation (7.22), while the outer circle corresponds to
the negative sign. The two eigenfunctions are given by
1 ⎡ 1 ⎤
φz(±) = ⎢ ⎥, (7.23)
2 ⎣ e i (ϑ∓π /2)⎦
where ϑ is the angle that k makes with the [100] axis of the underlying crystal, within
the heterostructure quantum well described in the previous section. The form of
equation (7.23) clearly shows the phase shift relative to the bulk inversion
asymmetry wave function. The spin direction remains tangential to the two circles,
but pointed in the negative angular direction for the inner circle and in the positive
angular direction for the outer circle. When both spin processes are present, the spin
behavior becomes quite anisotropic in the transport plane [20]. However, the
Dresselhaus bulk inversion asymmetry is generally believed to be much weaker
than the Rashba terms discussed here. The strength of the Rashba effect can be
modified by an electrostatic gate applied to the heterostructure, as it modifies the
potential gradient term in equation (7.19).
7.1.4 Berry phase

In chapter 3, we discussed the Aharonov–Bohm effect in which trajectories passing
through the top or the bottom of a mesoscopic ring would gain a phase difference
when the ring was penetrated by a magnetic field. Here the difference in phase was
given by (3.4) as
2π
e e
δϕ =
ℏ
∫ A · a ϑrdφ =
ℏ
∫ B · azdS = 2π ΦΦ0 . (3.4)
0
Now, this phase will occur whether the two paths cover the two separate paths or a
single trajectory moves completely around the ring. Now, there is a complimentary
quantization that arises from the Schrödinger equation. We can get at this by
assuming a complex wave function and using it in the Schrödinger equation [21–23].
The form of the wave function is given as
ψ (x , t ) = A(x , t )exp(iS (x , t )/ ℏ). (7.24)
The quantity S is known as the action in classical mechanics, and leads to a
continuity equation from which we may conclude that the velocity is related to this
as
1
v= ∇S . (7.25)
m
7-8
Obviously, then ∇S represents the momentum associated with the probability flow.
In essence, this is a treatment of the quantum wave function as a form of
hydrodynamics. But there is more to say about S. Examination of equation (7.24)
tells us that S is defined modulo h (Planck’s constant). Since the momentum is
normally described by ∇S, the periodic nature of S leads to a quantization condition
∮ p · dr = nh, (7.26)
where n is an integer. Thus, S is not determined entirely by the local dynamics, as

would be the classical case, but must also satisfy a topological condition which
follows from its origin as the phase of an independent complex field—the wave
function [24]. Another way of expressing this is that, if there are any vortices that
form in the quantum flow, their angular momentum must be quantized as well.
Equation (7.26) was originally proposed by Einstein [25], and further developed
by Brillouin [26] and Keller [27]. It is usually referred to as EBK quantization due to
these authors. But, it is important to understand how this simple quantization relates
to the topological nature of the wave function. The Aharonov–Bohm effect does not
require quantization of the phase, and experiments have shown a smooth phase
variation as the magnetic field is varied. This means that the form (7.26) is not the
ultimate understanding. This importance was highlighted in discussions of the
geometrical phase associated with the wave function by Berry [28]. We consider
motion around a closed path according to equation (3.4) which has a period of T
around this path such that r(T ) = r(0). The state of the system then evolves
according to the Schrödinger equation. While the energy may be slowly varying in
space and time (despite the constant energy assumption used previously), quite
generally there is no firm relation between the phases at different points in space. If
the system evolves adiabatically, then the states will evolve in a manner in which the
wave function can be written as
t
−i ∫ Edt ′ (7.27)
ℏ
ψ (t ) ∼ e 0 e iγ (t )ψ (r(t )),
which differs from equation (7.24) only by the second exponential. Berry points out
that this phase factor, γ(t), is non-integrable and cannot be written as a function of
the position variable, as it is usually not single-valued as one moves around the
circuit. This phase factor is determined by the requirement that equation (7.27)
satisfies the Schrödinger equation, and direct substitution leads to the relationship
∂γ (t ) ∂r(t )
= i〈ψ (r(t ))∣∇ψ (r(t ))〉 · . (7.28)
∂t ∂t
Integrating this around the closed contour discussed above gives
γ (C ) = i ∮C 〈ψ (r(t ))∣∇ψ (r(t ))〉 · dr. (7.29)
7-9
If we now recognize the momentum operator, this becomes

1
γ (C ) = −
ℏ
∮ 〈p 〉 · dr. (7.30)
This phase is often referred to as the Berry phase, with the previous expectation of
equation (7.28) denoted as the Berry connection. Comparing this to equation (7.26)
tells us that the left-hand side of equation (7.30) may often be an integer.
We can go a little further and relate this to the Aharonov–Bohm effect discussed
above. Let us replace the momentum in equation (7.26) by the adjusted momentum
due to the Peierls’ substitution. Then, equation (7.26) becomes
∮ p · dr − e ∮ A · dr = nh. (7.31)
Let us now take the second term on the left and rewrite it using Stoke’s theorem as
e ∮ A · dr = e ∫ (∇ × A) · nd Ω = eBΩ = hΦΦ0 , (7.32)
where Ω is the area enclosed by the contour discussed above (the ring in the AB
effect), and the other symbols have their normal meanings. Now, we see that the
Aharonov–Bohm effect is an example of the Berry phase and the topology of the
ring. As the flux enclosed by the ring (Φ) changes, this requires the momentum in the
ring to also change in order to maintain the constant right-hand side of equation
(7.31). This change in momentum is sensed outside the ring as a change in
conductance that gives rise to the oscillations characteristic of the Aharonov–
Bohm effect.
A great deal of attention has been given to the Berry phase in quasi-two-
dimensional systems, such as mesoscopic systems. This particularly true in demon-
strating the topological stability of the quantum Hall effect, as illustrated by
Thouless and Kohmoto [29, 30]. These papers demonstrated that the Hall con-
ductivity of a 2DEG in a magnetic field perpendicular to the two-dimensional
transport plane is proportional to the Berry phase through the vector potential
introduced above and seen by the nth subband eigenstate φn . Thus, the Hall
conductivity arises from equation (7.32) as
e2 1
σH = ∑
h n 2πi
∮ d k · A n(k). (7.33)
Different subbands (or Landau levels) may see different values of the vector
potential as they often have different enclosed magnetic fields due to their radii. It
was later shown by Chang and Niu that the velocity operator (7.25) for a semi-
classical wave packet can acquire an anomalous phase from the Berry curvature
q ⎡ ∂u n ∂u n ⎤
Ω n (k ) = i ∑ ⎢ un′ un′ − c. c⎥ , (7.34)
⎢
n ′= 1 ≠ n ⎣
∂k1 ∂k2 ⎥⎦
7-10
where un is the cell-periodic part of the Bloch function and q is the highest occupied
subband [31]. With the spin Hall effect, the spin–orbit coupling can lead to crossing/
anti-crossings of various energy levels, and the anti-crossing effect is enhanced by the
Berry curvature [32], which may be rewritten as
⎡ ∂H ∂H ⎤
q ⎢ 〈u n u n ′〉〈u n ′ u n〉 − c . c ⎥
Ωn(k ) = i ∑ ⎢⎢ ⎥.
∂k1 ∂k2
⎥ (7.35)
n ′= 1 ≠ n ⎢
(E n − E n ′) 2
⎥
⎣ ⎦
At those values where the anti-crossings occur, the denominator becomes small, and
the Berry curvature is greatly enhanced. We have observed this effect in subband
anti-crossings in the spin Hall effect in nanowires [33].
This can be even more pronounced in the transition-metal di-chalcogenides
(TMDCs). These materials were discussed in section 4.4, as layered compounds. The
Brillouin zone is similar to that of graphene and these monolayer materials have a
direct bandgap at the K and K’ points. But, the lack of inversion symmetry leads to
some interesting spin effects. In the TMDCs, in the presence of the spin–orbit
interaction, the Hamiltonian can be written as [34]
Δ λτ
H = at(τkxσx + kyσy ) ± − (σz − 1)sz , (7.36)
2 2
where the σ’s are the Pauli matrices for the pseudo-spin basis functions of the valleys,
τ is the valley pseudo-spin index, a is the lattice constant, t is the nearest neighbor
hopping energy, Δ is the energy gap, 2λ is the spin splitting at the valence band top,
and sz is the Pauli matrix for spin. Hence, each of the two bands is also spin split.
Here, the energy is measured from the mid-gap point. Generally, it is the spin–orbit
interaction that produces the splitting of the doubly degenerate valence and
conduction bands. Each of the two bands (conduction and valence) has a
Kramers degeneracy due to the spin of the carriers. When the spin–orbit interaction
is included, the Kramers doublets are split. Interestingly, the spin orientation is
different at K and K’. The splitting of the two valence bands is about 150 meV in
MoS2 and about 430 meV in WS2 [35]. While the conduction band splitting is small
in MoS2, it is about 30 meV in WS2 and can be as large as 50 meV in many of the
other TMDCs. Thus, it is conceivable that only a single spin state is occupied in each
valley, whether we are considering electrons or holes.
While we think of the spin as being either ‘up’ or ‘down’, the spins actually lie in
the plane of the monolayer material [36]. A constant energy circle is found around
each valley (see figure 7.1) with the momentum normal to this circle. However, the
spin direction is tangent to the circle, and in opposite directions in the two valleys, as
shown in figure 7.1. As a result, if one integrates the spin momentum around the
constant energy circle, a Berry phase is produced and this leads to a Berry curvature
Ω [37]. This potential acts like a pseudo-magnetic field and therefore has to point in
opposite directions for the two valleys in order to maintain time reversal symmetry
7-11
(that is, the pseudo-magnetic field is oriented in the z-direction for the K valleys and
is oriented in the –z-direction for the K’ valleys, where z is normal to the monolayer).
The Berry curvature reacts with a longitudinal electric field to produce a transverse
momentum. This gives rise to what would normally be referred to as the spin Hall
effect when a longitudinal electric field is applied to the crystal [6, 7, 37].
As remarked, the spin is coupled to the valleys due to the difference in the
orientation of the spin splitting [34, 38]. This produces a valley-spin Hall effect,
which again arises from the fact that, in the presence of the longitudinal electric field,
the Berry curvature leads to a transverse velocity E × Ω. Hence, the carriers in the
two valleys will be driven to opposite sides of the ribbon width. The importance of
this lies in the recognition that spin separation may lead to useful information
processing devices [39] and there have been many attempts to make useful devices
using this concept [40]. Xiao et al [34] have evaluated the Berry curvature for the
TMDC conduction band in terms of the material parameters as
2a 2 t 2 Δ
Ω c (k ) = − τ , (7.37)
[Δ + 4a 2t 2k 2 ]3/2
2
where k is the wave number (measured from K or K′) and the other parameters are
as defined above. The values of these parameters are taken from [34], where these
authors give a value for the Berry curvature for WS2 of 9.57 × 10−16 cm2. This
curvature is normal to the monolayer of the TMDC, and as discussed above, is
oppositely directed in the K and K’ valleys.
7.1.5 Studies of the spin Hall effect

One of the more remarkable features of the Rashba spin–orbit term (the structural
inversion asymmetry terms) is that this effect gives rise to the intrinsic spin Hall
effect in a nanowire, in which the longitudinal (charge) current along the nanowire
gives rise to a transverse spin current. In this situation, one spin state will move to
one side of the nanowire, while the other spin state moves to the opposite side. There
can be an intrinsic spin Hall effect that does not rely upon the presence of any
impurities with their spin scattering [10]. We can illustrate this with a consideration
of the energy surfaces. In figure 7.1(a), the nested double parabola discussed in the
previous sections is sketched, and this leads to the constant energy surfaces being
two nested circles in momentum space, as shown in the right panel of figure 7.1(a). In
this latter case, the direction of the momentum in the Rashba spin–orbit coupled
system are indicated by the green arrows, and the two eigen-spinor polarizations are
shown by the red arrows. If we apply an electric field along the x-axis, the current-
carrying direction, this produces a force on the spin, as indicated in figure 7.1(b).
After some time (but small compared to the scattering time), the electrons experience
an effective torque that rotates the spin polarization. This torque rotates the
polarization upward, and giving a positive polarization in the z-direction for positive
y-momentum, and giving a negative polarization in the z-direction for negative y-
momentum. This difference drives spin-up electrons to one side of the sample and
spin-down electrons to the other side, yielding a net spin current in the device. These
7-12
authors suggested that this intrinsic spin Hall effect should have a universal value
of [10]
Js,y e
σsH = − = . (7.38)
Ex 8π
Other studies showed that, in the infinite two-dimensional sample limit, arbitrarily
small disorder introduces a vertex correction that exactly cancels out the transverse
spin current [41]. However, in finite systems such as quantum wires, the spin Hall
effect survives in the presence of disorder and manifests itself as an accumulation of
oppositely polarized spins on opposite sides of the wire [42], and that the observed
value of the spin Hall effect could be larger or smaller than that of equation (7.38),
depending upon the magnitude of the spin–orbit coupling, the Fermi energy, and the
degree of disorder [43]. This has led to the proposal for a variety of devices that
utilize branched, Q1D structures to generate and detect spin-polarized currents
through purely electrical measurements [44], and experiments have been performed
to try to measure these effects [39]. We will return to this in section 7.3. In addition to
Rashba spin–orbit coupling, a term due to the bulk inversion asymmetry of the host
semiconductor crystal, the Dresselhaus spin–orbit coupling [16] discussed above, can
also yield a spin Hall current.
What was probably the first experimental effort was carried out by Awschalom
et al [45]. Here, experiments were based on an n-GaAs layer 2 μm thick, grown on
top of an AlGaAs thin layer itself grown on a semi-insulating GaAs substrate. The
n-layer was doped to 3 × 1016 cm−3 in order to achieve long spin lifetimes.
Measurements were made using Kerr rotation of an optically polarized beam.
The magneto-optical Kerr effect is a rotation of the optical polarization due to an
electric field in the quantum well, which modifies the dielectric function. This electric
field can be produced by the optical beam itself, but the effect is modified by the
presence of a spin polarization. Then, the Kerr rotation can be used to measure the
spin polarization in the sample. When an electric field is applied along the length of
the sample, the spin Hall effect can be developed as discussed above. A magnetic
field is applied transverse to the electric field and in the plane of the sample. The
Kerr signal occurs for small magnetic fields (of order a few mT around zero
magnetic field). It was clear in the measurements that opposite spin directions were
found at the two longitudinal edges of the sample. The effect diminishes as the
magnetic field is increased due to spin precession (see appendix E). Further
information on this spatial imaging of the spin separation was given in a subsequent
paper [46].
A somewhat different approach has been presented by Wunderlich et al [47]. In
this experiment, a p-type layer is produced in an AlGaAs/GaAs heterostructure.
This structure is then patterned into a ribbon so that n-type layers can be placed in
close proximity. These latter layers then produce two light-emitting diodes at the
edges of the ribbon. The spin Hall effect is especially strong in the hole layer, and
this guides the spin polarized carriers to the diodes, but each diode sees a different
spin polarization. This is used to create optical circular polarization from the diodes
7-13
Figure 7.2. The intrinsic spin Hall effect in a GaAs structure. (a) A micrograph of the device structure. The
two LEDs are sensitive to the spin polarization of the holes at the two sides of the p-layer. (b) Reversing the
current in the p-layer reverses the polarization of the LED emission. (c) The two LEDs sense the opposite spins
on either side of the p-layer. (d) The quasi-particle lifetime as a function of the hole density. The color scale
corresponds to the ratio of the measured spin Hall effect to the universal value of (7.24). (Reprinted with
due to the spin polarization of the injected holes. Because of the optical selection
rules, the presence of a particular spin polarization will create a corresponding
circular polarization in the light, and this can be measured. These results are shown
in figure 7.2. The device itself is depicted in panel (a), and the two diodes are labeled
as LED 1 and LED 2. As indicated in panel (b), the optical circular polarization is
reversed when the current through the p-layer is reversed. This indicates that the
opposite spin is pushed toward that LED when the current is reversed, as expected
for the spin Hall effect. Then, the two outputs from the two LEDs are shown in
panel (c), and it is clear that they are affected by opposite spins. Finally, the group
investigated whether the spin Hall effect has a universal value, as given by equation
(7.38). This is shown in panel (d) as a color plot of the quasi-particle lifetime as a
function of the hole density. The color denotes the magnitude of the spin Hall effect
relative to the ‘universal’ value given in equation (7.38). The measured strength is the
white dot at the lower center of the image, and suggests that this measured value is
roughly twice the value expected if it were universal. They term this a ‘strong
intrinsic spin Hall effect.
7-14
7.2 Spin injection

As semiconductor devices have grown continually smaller, it has become apparent
that this trend cannot continue indefinitely. In fact, the nominal ‘size’ in the 2020
generation of integrated circuits from ARM is only 7 nm, which is roughly 30 times
the spacing of the Si atoms. Even though this ‘size’ is not a real physical dimension
any more, it is clear that there is a limit that is rapidly approaching. As a new path to
continued growth in the capabilities of integrated circuits, many researchers have
taken to exploring alternative approaches to computer logic devices, by changing
from the charge basis of our traditional transistors. One possible approach is to use
spin as the variable, and this possibility burst into the integrated circuit world with
the proposal for the ‘spin FET’ [4]. In this proposal, the base material for a HEMT-
like device was InAs, which has a large spin–orbit splitting and so should have a
reasonably long spin lifetime (the time before the spin polarization randomizes). The
source and drain contacts, however, were made of iron, a magnetic material. Thus, a
spin polarized contact would inject a polarized spin into the InAs channel. Of
course, this spin would precess, and the amount of precession would vary with the
electric field provided by the gate bias. Hence, the ability of the polarized spin carrier
to leave the drain contact would depend upon the polarization when this contact was
reached. As a result, the drain current could be switched by the gate bias, as is
normally expected. The key issue, of course, was the ability to successfully inject a
single spin state from the source contact.
Of course, the idea of spin injection did not begin with the above device proposal.
The concept of spin injection into semiconductors seems to have originated with
Aronov [48, 49]. One of the earliest experiments used a permalloy contact on
aluminum to inject from the polarized permalloy into the nonmagnetic aluminum,
where the spin polarization could be analyzed and the spin lifetime determined [50].
In fact, it was later suggested to extend this structure to produce a spin bipolar
transistor [51]. The quest for the spin FET then led to the first experiments to
successfully inject spin polarized carriers into a semiconductor, in this case CdTe [52]
and InAs [53–55], but it was generally found that one needed to make a tunnel
barrier between the magnetic metal and the semiconductor to avoid conductivity
mismatch in the two materials. In fact, an actual spin FET has been realized using
the InAs channel [56], which was further reviewed by Johnson in [40]. In figure 7.3,
we illustrate the device and the measurements. The structure is shown in panels (b)
and (c) and constitutes an InAs quantum well grown on GaAs with GaAlSb
barriers. The InAs channel was patterned to a width of about 0.9 μm. The magnetic
metal layers are placed over this structure and spaced some 10.6 μm. A picture of the
sample is shown in panel (a), and shows that six separate channels are connected in
parallel to create the device, although other devices with a single channel of width
15 μm were also measured, this device is primarily discussed. While the device is
symmetrical, metal F1 was typically used as the injector and metal F2 was used as
the detector, which acts as a spin sensitive potentiometer (or resistance). The
detected signal is shown in panel (d) of the figure for the single-channel device,
and the difference between the up sweep and the down sweep is typical for spin
7-15
Figure 7.3. (a) Micrograph of the structure showing six parallel channels. (b), (c) Structure of the
heterostructure spin system used to propagate spin from one contact to a second contact utilizing an InAs
quantum well. (d) Measurement of the spin propagation detected at F2 for a single channel of 15 μm width (see
the text for the discussion of the hysteretic curves). Other details are given in the text. (Reprinted with
permission from [40]. Copyright (2005) American Chemical Society.)
measurements. It is clear that the spin has been successfully injected and then
detected at a distance away, with the detection being sensitive in this case to the
magnetic field.
To understand the signal in figure 7.3(d), we have to consider the experiment in a
little more detail. When the electrodes F1 and F2 are both ferromagnets, then the
measurement is a voltage that is linearly proportional to the current being fed
through the sample, and this is recorded as a resistance. This voltage (resistance) is
relatively high when the two contact ferromagnets have parallel polarization and
relatively low when their polarizations are opposite. In general, the InAs quantum
well is asymmetrical in the growth direction (normal to the heterostructure
interfaces) and this leads to an electric field in the z-direction (the growth direction),
which produces a spin polarization in the conduction band according to equation
(7.22). Normally, the two concentric circles (figure 7.1(a)) have the same Fermi
energy, but when spins of one polarization are injected into the semiconductor, the
electrochemical potentials of the two spin subbands are altered [57]. When contact
F1 provides spin-down electrons, these propagate ballistically to the other contact,
where the spin-down electrochemical potential is raised, and the spin-up electro-
chemical potential is lowered. When the magnetizations of the two contacts are
parallel, this potential separation is measured and appears as the high resistance
regions of the curves in figure 7.3(b). When the two magnetizations are anti-parallel,
a smaller voltage corresponding closer to the equilibrium state is measured, and this
is reflected in the dips in the curves near B = 0. The two contacts have slightly
different coercive forces, HF1 ≠ HF2, which leads to the fact that one contact will
switch its magnetic state before the other one, and this gives the observed curves.
Spin propagation in Si is more difficult due to the very small spin–orbit
interaction that exists in this material. However, spin injection has been seen in
Si, and detected as either light emission [56] or by the spin-valve effect of the InAs
quantum well discussed in the previous paragraph [58]. In the Si device, an Al–
Al2O3–Al tunnel junction is used to inject unpolarized electrons into a layer of
7-16
CoFe. There is an exponentially different mean free path for the two different spin
orientations in the polarized CoFe film, so that a dominant single-spin polarization
is injected over the barrier into the Si film, where it transports through the layer. The
spin polarization is then measured by a NiFe/Cu layer and the last Si layer. If the
polarization of the NiFe layer is compatible, the spins can propagate through to the
last Si layer. When the polarizations of the two magnetic layers are the same, the
detected current will be higher than when the two polarizations are opposite. This
detected current is changed as the magnetization of the magnetic layers is reversed.
7.3 Spin currents in nanowires

As mentioned in the previous section, there is an interest in spin as a logical variable
for use in future nanoelectronic devices. There is also some interest in using spin as a
quantum bit (qubit) in quantum computing since the natural two state nature of
spin, and its precession on the Bloch sphere are essentially the same as that
conceived for an analog qubit. One suggestion for a semiconductor qubit actually
uses wave propagation in a pair of parallel quantum wires [59, 60]. Such a qubit
would depend upon the interaction of the carrier waves in the two wires, especially at
a designated interaction region. It has been shown that the wave definitely can be
switched from one wire to the other with an applied magnetic field [61], which
supports the idea of using spin as the important variable.
The ability to obtain spin separation in a quantum wire certainly depends upon
the Rashba effect and the spin–orbit interaction. The induced magnetic field can
lead to spin separation through a process which is essentially that of the spin Hall
effect discussed above. In figure 7.4, the spin resolved squared magnitude of the
wave function in a 100 nm wide quantum wire in the lowest subband of an InAs
quantum well heterostructure is shown as a function of the wave momentum down
the wire [62]. Here, the Rashba coefficient (7.19) is 20 meV-nm. The yellow color
represents the spin along the z-direction, while the red corresponds to the opposite
spin direction. It can be seen that there is a change of the spin wave functions as a
Figure 7.4. The spin resolved square magnitude of the wave function in an InAs nanowire as a function of the
momentum along the wire. (Reprinted with permission from [62].)
7-17
function of the momentum, and this arises from the fact that the various subbands
cross and interact in the effective magnetic field, so that a wave can hybridize
between the various subbands. Nevertheless, the ability to spatially separate the two
spins across the nanowire leads to the ability to make a spin filter by splitting the
nanowire in a Y-type branch [44], as shown in figure 7.5(a) (a spin branch cascade is
actually shown in this figure). It is conceived that the individual nanowires would be
InAs with a width of 100 nm, and the branches would have a radius of curvature of
100 nm. In figure 7.5(b), we show the spin resolved square magnitude of the wave
functions in the various channels of the structure at very low temperatures. The color
indications are the same as above, as are the spin–orbit parameters, but the doping is
assumed to be 3 × 1010 cm−2, so that only the lowest subband is occupied, although
there are of course two spin states. In the input wire, the spin Hall effect leads to the
two spin states being spatially separated. When the first branch is reached, the
opposite spins move to opposite sides of the branch and into the different arms.
Because the electrons in the output wires are polarized out of the plane of the
heterostructure and therefore undergo precession as they move down the wires. This
precession leads to the wave function moving from one side of the wire to the other,
a wobbling motion that has often been called zitterbewegung.
The efficiency of the Y-branch switch is best characterized by the degree of spin
polarization of the electrons in the output arms. We find that this polarization can
rise to a value of almost 60%. While small, this is considerably larger than that
usually achieved by spin injection.
Figure 7.5. (a) A two-stage cascade of Y-branch switches which can be used for spin filtering. (b) The spin
resolved squared magnitude of the wave function for the device in (a). (Reprinted with permission from [62].)
7-18
To make it more useful, one can develop a cascade of such Y-branches [63], as
shown above in figure 7.5. As indicated in this figure, the use of a second set of Y-
branch switches really does not add much under the proper conditions, as the results
of the first Y-branch is sufficient to isolate the spin states into only one arm of the
second set of switches. However, under proper conditions, the spin state can be
switched from one arm to the other in the second set of gates, which can be achieved
by changing the propagation length of the intermediate wires. Alternatively, the two
outputs of the second set of Y-branch can be used as a spin polarization detector.
The precession can also be reduced by the use of an in-plane magnetic field so that a
true spin filter can then be created [62].
Attempts to experimentally measure such spin filters have had reasonable success
[64, 65]. One such example is shown in figure 7.6(a) [65]. The device has been
fabricated from an InAs heterostructure by electron-beam lithography and reactive-
ion etching [66]. The common horizontal wire(in the figure) is oriented along the [1,
1, 0] direction of the crystal, as this direction exhibits the highest mobility. The
contacts are the typical AuGe contacts used for many III–V materials. The top gate,
used to adjust the carrier concentration in the InAs quantum well, is separated from
the semiconductors by a polymer layer about 0.24 μm thick. At low temperatures, it
is hoped that the transport through the structure will be ballistic in nature. The
central wire, discussed above, is 1 μm long and has a width of 150 nm. Current is
injected at port A and the current that flows to the other arms is measured. The
corresponding voltages allow a determination of the effective conductance at each of
the other ports. In figure 7.6(b), the conductance from the source to output arms C
(blue) and D (green, as indicated in the left panel) are shown for different
conductances through the central arm. One can see that, as the magnetic field is
varied, the currents in these two arms oscillate, indicating that they are measuring a
Figure 7.6. (a) Micrograph of an InAs heterostructure with patterned Y-branch switches at left and right sides.
The color code and terminals of measured conductances are indicated in the figure. (b) Measured conductances
at terminals C (blue) and D (green) are plotted for various values of the conductance in the central arm. Both
the raw data and the smoothed data (lines) are shown. (Reprinted with permission from [65]. Copyright 2013
AIP Publishing LLC.)
7-19
spin polarization of the output signals. These oscillations in the output of this second
filter are interpreted as being a result of spin polarization in the first stage (left-hand
side) of the Y-branch filter.
7.4 Spin qubits

As we mentioned above, the spin–orbit interaction is relatively small in Si, but it is
large enough to consider making spin-based qubits in this material. The idea for a
qubit based upon Si effectively dates from the suggestion of Kane to use a 31P
dopant in isotopically pure 28Si [67]. In this approach, it was suggested to use the
nuclear spin of the P atom, which is 1/2, and the fact that the particular isotope of Si
does not have a nuclear spin. Then, the electron wave function for the positively
charged donors extends extensively through the conduction band and can mediate
the interaction between the nuclear spins on neighboring P atoms. Such a qubit is
interesting because it has the promise of a long spin-relaxation time (discussed in the
next section), and is scalable [68]. Getting the P atoms into the right position thus
constitutes a serious problem, but the latter authors have demonstrated that it is
possible to create an atomically precise linear array of single P bearing molecules on
a Si surface, and that these P atoms can act as quantum qubits.
The process of getting a P atom to a precise location turns out not to be so
difficult. One method is to hydrogenate the dangling bonds on the Si surface, then to
remove particular hydrogen atoms using a scanning tunneling microscope (in ultra-
high vacuum) [69]. Removing the hydrogen allows the molecule to be chemically
attach to the dangling Si bond at that particular site [70]. In the case mentioned
above, the molecules from phosphine gas are used to deposit the P atom at the
desired position, then the difficult task is to deposit isotopically pure Si while
maintaining the P atom at the desired position [67]. Another approach to locating
the P atom is to pursue single atom implantation [71–73]. This latter technology was
developed some years ago, and qubits based upon P atoms have been fabricated with
both approaches.
The operation of a single donor qubit under control of surface gates and
measured with a single-electron transistor was evaluated by Dehollian et al [74].
In figure 7.7, we give some information about the operation of this donor qubit
formed with the P atom. Here, they use gate set tomography (GST), which is a tool
for characterizing logic operations in the qubit. The qubit itself is formed by the spin
states of an electron bound to a 31P atom, implanted into the isotopically pure silicon
substrate. The spin states are split by an applied magnetic field, and switching is
achieved via coupling of the electron spin to the nuclear spin, resulting in a two spin,
four level system. Qubit preparation is handled by tunneling to/from a single-
electron transistor, which is also used for sensing the state. The aluminum gates are
placed on top of a SiO2 layer. In addition, an electron spin resonance signal is used
for qubit manipulation.
It has also been suggested to use the electron spin rather than the nuclear spin in a
semiconductor nanostructure [75]. One reason to use the electron spin lies in the fact
that the spin resonance transition for the electron can be rather effectively tuned by
7-20
Figure 7.7. Diagram of qubit device and the gate set tomography (GST) model of a qubit. SEM image of the
on-chip gate structure is shown to the left. The aluminum gates have been false colored for clarity. Depicted in
red are the source-drain n+ regions which couple the single-electron transistor to the current measurement
electronics. For initialization and measurement, the donor gates are pulsed such that the various Fermi
energies are μ↑ > μSET > μ↓, inducing spin-dependent tunneling between the donor and the SET. When
applying a gate sequence, the DG are pulsed to higher voltage to prevent the donor electron from tunneling to
the SET. The inset diagram (upper left)—zoomed from the approximate donor location—represents the Bloch
sphere of the qubit, consisting of the spin of an electron confined by an implanted 31P donor, with its nuclear
spin frozen in an eigenstate. The model treats the qubit as a black box with buttons (right) which allow one to
initialize (ρ0), apply each gate in the gate set (Gi,x,y) and measure (M) in the observable basis (∣↑〉or∣↓〉).
Reproduced under the Creative Commons Attribution 3.0 licence from Dehollian et al [74].
electrostatic gates, which are a common part of the single-electron quantum dot
structures (discussed more in section 8.2). The use of the electron spin has been
reviewed recently [76]. It has also been suggested [77] to move to Ge for better
tenability via the bias induced Stark effect, but most work remains in the Si system.
While much of the effort on single impurity qubits has focused on donors, there has
also been some work directed at the use of acceptors as well. There is a feeling that
the dopants for this type of qubit need to be near an interface in order to interact well
with surface gates. Hence, the use of acceptors near the surface of both Ge an Si have
been studied as well [78, 79].
The case for nearly all of the above dopant-based qubits is that the dynamic
variable is the spin itself. It is possible to consider a hybrid qubit in which the
dynamical variable depends upon both spin and charge. The hybrid qubit requires
neither nuclear-state preparation nor micro-magnets for control, and becomes
considerably more amenable to systems [80]. Such a hybrid qubit has been studied
by these latter authors in a traditional double dot system, similar to those in single-
electron transistors (again, discussed more in section 8.2). Three electrons are used in
the system, with the gates tuned so that there are two electrons in the left dot and one
in the right dot to define the (2,1) state. Changing the voltage on gate L to raise the
energy difference between the dots favors the opposite situation and the state (1,2).
The singlet state in the right dot, state (2,1), is taken to be the ∣0⟩ state. The triplet
state in the right dot, the (1,2) state, is taken to be the logical ∣1⟩. The presence of the
extra electron means that the hybrid states of the dots are not pure singlet or triplet
7-21
Figure 7.8. Experimental and theoretical qubit set-up, and resulting energy dispersions. (a) Scanning electron
microscope image of a device nominally identical to the one used in the experiment. The gate voltages applied
to the various leads are tuned to form two quantum dots, located approximately within the dashed circles,
where the red spots represent electrons in a (1,2) charge configuration (one level left dot and two levels right
dot). (b) Schematic of the four pulse sequence employed in the experiments to obtain the qubit parameters.
(c) The experimentally measured dispersion of the quantum-dot hybrid qubit. The red line is a least squares fit
to the data. The inset is the three energy eigenstates determined for the qubit. (d) A schematic cartoon
illustrating the model for the qubit. Reprinted with permission from [82], copyright 2018 by the American
Physical Society.
and fast electric field techniques can be used to manipulate the qubit in either X or Z
rotations on the Bloch sphere (discussed in appendix E). The use of the three electron
system tends to extend the coherence [81]. The singlet-triplet qubit is attractive for
many reasons, particularly for its high fidelity, at least for single qubit operations. In
figure 7.8, we show such a hybrid qubit, with its excitation and control character-
istics [82]. The double-dot system is excited with microwave pulses; four different
pulse sequences are used. These set of pulse sequences allow one to map out the
energy dispersion of the qubit. Also in the figure, Δ1 and Δ2 refer to the tunnel
couplings between the various charge states and ΔR is the energy splitting between
the two basis states for the qubit, which are the singlet and triplet states in the right
quantum dot. Tuning the dispersion was found to be critical to finding the exact
parameters for best qubit operation.
For two qubit operations leading to entanglement, hybrid qubits can be problem-
atic in a sense that there are a range of operations over which the interaction can
7-22
occur and this leads to a preferred set for good performance [83], just as was found
for the double-quantum dot qubit in the previous paragraph [82]. In addition, the
interaction between a hybrid qubit (discussed above) and other types of qubits has
been studied, since such an interaction can usually be tuned between different
operating modes via the gates [84].
Pure spin qubits formed of quantum dots in the Si/SiO2 system have been formed
in a manner to utilize from 1 to 3 electrons in the dots [85]. Then, pulsed electron
spin resonance is used to exercise coherent control over the qubit. Quantum dots
have also been created in the Si/SiGe heterostructure system. High fidelity gating of
a single hybrid qubit has been achieved in this system [86]. The optimal geometry for
the gates for such a spin qubit in this heterostructure (or in GaAs) has been studied
for a single qubit [87] and for a qubit linear array [88].
7.5 Spin relaxation

The decay, or decoherence, of a spin polarization can be instigated by a magnetic
impurity, or by a normal impurity in a material with the spin–orbit interaction. As
our various spin applications discussed above depend upon the presence of this latter
interaction, we can ignore the discussion of magnetic impurities. The key issue then
is the change of polarization that can occur during the scattering of the carrier by the
impurity. In the Elliott–Yafet mechanism [89, 90], a spin flip can occur during
the scattering of the carrier by the impurity. In the presence of spin–orbit scattering,
the electron state is not a pure single polarization, as there is always a small
admixture of the other spin in the wave function. Hence, under the scattering
process, there is a possibility that the overlap of the initial and final state wave
functions will lead to a flip of the spin. Thus, there is a random spin rotation that
results from the scattering and this gradually breaks up the spin coherence. But, the
spin generally is not affected between the scattering processes, so that the spin
lifetime is proportional to the momentum relaxation time. This is not limited to just
‘impurities, but can also arise from boundary scattering, interface scattering, and
phonon scattering [89]. A more complicated interaction can also arise from the
phonon effect on the spin–orbit interaction of the lattice ions [91]. Yafet showed that
the spin relaxation rate (the inverse of the lifetime) basically follows the temperature
dependence of the resistivity of the sample. In a heavily doped semiconductor in
which there is little freeze-out of the carriers, this would lead to an independence of
temperature at low temperatures, as observed experimentally [92]. On the other
hand, the resistivity is strongly affected by the carrier density, which is given by the
ionized impurity density. At higher temperatures, the resistivity decreases as the
carrier density increases (for materials in which the donors or acceptors are not fully
ionized), and hence the spin lifetime will also decrease with temperature. Figure 7.9
shows the spin lifetime in Ge, which was Sb-doped at 5 × 1017 cm−3 [93]. The lifetime
is estimated from a fit to Hanle effect data. Also shown in the figure is a result from a
theoretical work using the Elliott–Yafet mechanism [94]. While the quantitative fit
differs from the theory, the qualitative dependence on temperature is as expected for
this mechanism.
7-23
Figure 7.9. Spin lifetime in Ge as a function of temperature. Here, the dominant interaction is the Elliott–
Yafet mechanism. (Reprinted with permission from [93]. Copyright 2012 AIP Publishing LLC.)
A second important mechanism for spin relaxation is the Dyakonov–Perel

spin relaxation mechanism [95, 96], which occurs in materials which lack an
inversion symmetry (due to either a lack of bulk inversion symmetry or structural
inversion symmetry). This mechanism depends upon the presence of the spin–orbit
interaction in the material to lead to an effective magnetic field which affects the spin
during its transport. In addition, the spin–orbit interaction lifts the degeneracy of the
two spin states, as discussed above in section 7.1. This induced magnetic field leads to
spin precession during the motion of the carrier, and the impurity induces a change in
the polarization which randomizes the precession of the spin. Additionally, the
magnetic field itself can cause spin flip through the spin–orbit interaction. The
randomization of the precession caused by the impurities leads to spin relaxation,
but Dynakonov and Perel showed that the lifting of the spin degeneracy was sufficient
to cause spin relaxation. From section 7.1, we recognize that the interaction term,
given by the appropriate term added to the Hamiltonian for the spin–orbit interaction,
is larger for larger momentum. Since each carrier has a different momentum, the
precession will be different for each electron, leading to the dephasing within the
ensemble. In the Dyakonov–Perel mechanism, the precession angle around any
particular reference axis diffuses so that the square of the precession increases with
time roughly as (t/τ)(ωτ)2, where τ is the momentum relaxation time, and ω is a typical
precession frequency (see appendix E) [97]. The spin relaxation time is then defined to
be the time when this precession angle becomes of order unity, so that it is defined as
τs = 1/ω2τ. Hence, as the mobility and the conductivity rise (for constant carrier
density), τ is getting larger, and the spin relaxation time is getting smaller. One form of
the spin relaxation time for the Dyakonov–Perel mechanism is given as [98]
105ℏ6
τs = , (7.39)
64η 2(EF m⁎)3τ
7-24
where η is the spin–orbit coupling parameter in the bulk inversion asymmetry case.
In heavily doped materials, particularly when they are p-type materials, another
mechanism that can lead to spin relaxation is the Bir–Aronov–Pikus mechanism [99].
This process depends upon the electron and hole interaction, and the spins of the
interacting particles. The spin–orbit interaction is important for both electrons and
holes and the exchange interaction between the electron and hole can create an
effective magnetic field via this process. As before, this magnetic field leads to spin
precession of the electron. However, the spin of the hole varies much faster than the
precession of the electron, and this leads to a fluctuating effective magnetic field.
This, in turn, introduces a fluctuation in the precession of the electron spin. This
process is relevant in those semiconductors where the spin–orbit interaction
introduces a significant overlap between the electron and hole wave functions.
Surprisingly, it is found that the structure of the scattering rate for the electron–hole
exchange interaction is qualitatively the same as found for the carrier–phonon
scattering [100]. This mechanism for spin relaxation has been shown to be important
in Mn-doped GaAs, as the Mn is a acceptor dopant as well as being important in
producing magnetism in this material [101].
Problems
1. Consider two electrons in a state in which the radius (around some orbit
center) is normalized to unity, and the angular wave function is φ(ϑ) = f
(cosϑ). If these two electrons have opposite spins and are located at ϑ = 0 and
π, discuss the anti-symmetry properties of these electrons. Can you infer the
nature of the function which describes these angular variations?
2. In a particular semiconductor with effective mass of 0.04 m0, it is found that
the spin-split Landau levels are such that the spin-splitting (the shift upward
of the upper spin state) is exactly enough that all the levels are equally spaced
at a magnetic field of 10 T. What is the value of g at this magnetic field?
3. It has been reported that, in some semiconductors at high magnetic field, the
lower spin state of one Landau level becomes degenerate with the upper spin
state of the next lower Landau level. What must the value of g be for this
condition to occur?
4. The Kane perturbation model is perhaps the simplest approach to including
both the k · p and spin–orbit interactions. How do the parameters in this
theory compare to those that appear in equations (7.7), (7.8), and (7.17)?
Appendix D Spin angular momentum

As is commonly known, electrons can have two spin orientations. Typically, these
are called spin up and spin down. But, this is for an arbitrary orientation of the
electron’s own magnetic moment. Recognition of the two spin states arises from the
introduction of the Pauli exclusion principle whenever we consider a fully quantized
state. These two spin states must have opposite polarization, hence the designations
as up and down. In fact, the spin can be oriented into any desired direction by the
application of external forces, such as a magnetic field. The most common
7-25
orientation, arising in the Zeeman effect, is along the z-axis which is achieved via an
external magnetic field oriented in this direction. But, other directions are useful in a
variety of applications such as the use of spin orientation in a qubit (quantum bit) for
quantum computation. In this appendix, we wish to spend a little time talking about
the values and orientations of the spin and get to the Pauli spin matrices.
Most people get a smattering of angular momentum in their undergraduate
courses in classical mechanics or atomic physics. For sure, the angular momentum
of an electron orbiting in a centrally symmetric potential, such as the Coulomb
potential around an atom, possesses a quantized value for the spin angular
momentum. In this atomic case, both the total angular momentum L2 and the z-
directed angular momentum Lz can be made to commute. The fact that they could
both be made to commute with the Hamiltonian tells us that they can both be
diagonalized along with the Hamiltonian and therefore could both be measured at
the same time as the total energy. In the present case, the only angular momentum to
be considered is the spin angular momentum, which we take to be oriented in the z-
direction, just as in the atomic case. We expect that, just as in the atomic case, the
total spin S2 will continue to commute with the z-component of spin Sz, just as in the
atomic case. This will be useful in what follows.
Let us begin by recalling that, in any quantum confinement problem such as the
atom or a quantum well, the total wave function is a sum over a set of eigenfunctions
ψi(r). For each of these functions, which are defined by a set of quantized values due
to the confinement, one has not considered the spin. If we now want to also include
the spin, then we need an additional part of each eigenstate wave function.
Typically, this is a multiplicative term describing the spin state. Traditionally, this
is a two component wave function called a spinor. From the Zeeman effect, we know
that one typically denotes the extra energy for the spin-up state by the value 1/2, and
the value of the spin-down state by the value −1/2. We use this to denote the two
possible states and their spinors as
⎛1 ⎞ ⎡ ⎤ ⎛ 1 ⎞ ⎡ ⎤
φ⎜ ⎟ = ⎢ 1 ⎥ , φ⎜ − ⎟ = ⎢ 0 ⎥ , (D.1)
⎝ 2 ⎠ ⎣ 0 ⎦ ⎝ 2 ⎠ ⎣1 ⎦
where the first equation refers to the up state and the second equation refers to the
down state. Thus, the eigenvalues correspond to those adopted in the Zeeman effect,
as mentioned above.
Because the spin angular momentum has been taken to be oriented along the z-
axis, we expect the spin matrix for the z-component of angular momentum must be
diagonal for two reasons. First, it must commute with the total spin and with the
Hamiltonian, and, second, it must produce the eigenvalues found from the Zeeman
effect. Thus, we simply state that
ℏ ⎡1 0 ⎤
Sz = , (D.2)
2 ⎢⎣ 0 − 1⎥⎦
and this gives
7-26
⎛1 ⎞ ℏ ⎡1 0 ⎤⎡1 ⎤ ℏ ⎛1 ⎞
Sz × φ⎜ ⎟ = ⎢ ⎥⎢ ⎥ = φ⎜ ⎟
⎝2⎠ 2 ⎣ 0 − 1⎦⎣ 0 ⎦ 2 ⎝2⎠
(D.3)
⎛ 1⎞ ℏ ⎡1 0 ⎤⎡ 0 ⎤ ℏ ⎛ 1⎞
Sz × φ⎜ − ⎟ = ⎢ ⎥⎢ ⎥ = − φ⎜ − ⎟
⎝ 2⎠ ⎣
2 0 −1 1 ⎦⎣ ⎦ 2 ⎝ 2⎠
as expected. We know from the study of normal angular momentum that the value
of L2 is given as l(l+1)ℏ2. Thus, we expect a similar result for the spin angular
momentum, and using s = 1/2, we get S2 = s(s+1)ℏ2 =3ℏ2/4. More strictly, this value
is for the square of the magnitude of the total spin angular momentum. Again, the
matrix representation of this total spin angular momentum must be diagonal, and
this is given by
3ℏ2 ⎡1 0 ⎤
∣S∣2 = . (D.4)
4 ⎢⎣ 0 1 ⎥⎦
To find the other components of the spin angular momentum, and the other spin
matrices, we introduce the rotating coordinates, as
S+ = Sx + iSy, S− = Sx − iSy . (D.5)
Then, we can write the square magnitude of the total spin angular momentum as
S 2 = Sx2 + S y2 + Sz2 = S+S− + Sz2 − i [Sx, Sy ]. (D.6)
The commutator relations for these spin components are given as

[Sx, Sy ] = ihSz , [Sy, Sz ] = ihSx, [Sz , Sx ] = ihSy . (D.7)
Then, (D.6) becomes

S 2 = Sx2 + S y2 + Sz2 = S+S− + Sz2 + ihSz . (D.8)
Similarly, if we reverse the two rotating terms we get

S 2 = Sx2 + S y2 + Sz2 = S−S+ + Sz2 − ihSz . (D.9)
We can combine these last two equations, and use the results of equations (D.2) and
(D.4) to yield
1 ℏ2
S 2 − Sz2 = (S+S− + S−S+) = . (D.10)
2 2
The operators S+ and S− act as creation and annihilation operators for the spin
angular momentum. That is, operating on a spinor with the first of these operators
will raise the angular momentum, which can only occur if it acts on the spin-down
state and produces the spin-up state, or
⎛ 1⎞ ⎛1 ⎞ ⎡0 1 ⎤
S+φ⎜ − ⎟ = ℏφ⎜ ⎟ → S+ = ℏ⎢ . (D.11)
⎝ 2⎠ ⎝2⎠ ⎣ 0 0 ⎦⎥
7-27
Similarly, acting with the operator S− removes a quantum of angular momentum

and lowers the spin angular. This can occur only if the operator acts upon the spin-
up state and produces the spin-down state, or
⎛1 ⎞ ⎛ 1⎞ ⎡ ⎤
S−φ⎜ ⎟ = ℏφ⎜ − ⎟ → S− = ℏ⎢ 0 0 ⎥ . (D.12)
⎝2⎠ ⎝ 2⎠ ⎣ 1 0⎦
We can now easily invert equation (D.5) to give
1 ℏ⎡ 1⎤
Sx = (S+ + S−) = ⎢ 0
2 2 ⎣1 0 ⎥⎦
. (D.13)
1 ℏ⎡ −i ⎤
Sy = (S+ − S−) = ⎢ 0
2 2 ⎣i 0 ⎥⎦
The Pauli spin matrices are just the matrices that appear in the definitions of the
components of the spin angular momentum, as given by equation (7.4) above.
Appendix E The Bloch sphere

As we discussed in the previous appendix, the spin of the electron is characterized by
its eigenstate, for which the wave function is a two component spinor. In a
computer, the bit of information can also be expressed as a two component spinor,
where the state ∣0⟩ might correspond, for example, to the spin-down state and the
state ∣1⟩ would then correspond to the spin-up state. It makes natural sense then to
spend a little more time with the idea of two level systems [102]. To simplify the
notation slightly, let us use the generalized spinors
⎡ ⎤ ⎡ ⎤
α = ⎢1 ⎥ , β = ⎢0⎥ . (E.1)
⎣0⎦ ⎣1 ⎦
The equation of motion for these spinors is, as usual, given by the Schrödinger
equation in which the Hamiltonian is a 2 × 2 matrix. If there are no external forces
acting upon our simple two level system, then the Hamiltonian is independent of
time and the wave functions evolve as
ψ (t ) = e−iHt/ℏψ (0), (E.2)
where ψ corresponds to either of the spinors α or β. In general, we diagonalize the
Hamiltonian to provide the eigenvalues, which we take to be Eα and Eβ. The total
wave function can be written as a sum of the two spinors as
ψ = c1α + c2 β, ∣c1∣2 + ∣c2∣2 = 1. (E.3)
We can use the above properties of the Hamiltonian and the spinors to determine a
number of intuitive properties. For example, if we want to determine how β evolves
into α, we need only examine the total Hamiltonian, as
7-28
⎡ ⎤
β †e−iHt /ℏα = [ 0 1]e−iHt /ℏ⎢1 ⎥
⎣0⎦
(E.4)
H21
= (e−iEαt /ℏ − e−iEβt /ℏ),
Eβ − Eα
where
⎡ ⎤
H21 = ⟨β∣H ∣α⟩ = [ 0 1]H ⎢1 ⎥ . (E.5)
⎣0⎦
If we examine the magnitude squared value of this term, we discover that it

oscillates, and that the occupation oscillates from one state to the other with a
frequency given by the difference in the eigen-energies of these two states.
The oscillation that occurs above is suggestive, as it points out that we can define
the total state equation (E.3) itself as a single spinor whose components are the
complex coefficients given in this equation. What this means is that the two spinors α
and β define a two-dimensional space for our wave function. Where the quantum
well was an infinite-dimensional space of eigenfunctions, the problem here is limited
to just these two states. Thus, a spinor whose coefficients are the coefficients in
equation (E.3) is a vector in this two component space, as
⎡ c ⎤ ⎡ ∣c1∣e i ϑ1 ⎤
ψ = ⎢⎣c1 ⎥⎦ = ⎢ ⎥. (E.6)
2 ⎣∣c2∣e i ϑ 2 ⎦
But, this description is not unique since the squares of the magnitude must sum to
unity. Thus, the two phases can be rather arbitrary, and we can only specify in detail
the relative phase ϑ1 −ϑ2. So, if we write ϑ2 = ϑ1 + φ, we cannot then tell the
difference between the states ψ and e i ϑ1ψ , which means we have to somehow believe
that these are the same state. In a sense, this all arises from the periodicities of the
angles and that is a property we will use below.
A useful quantity to use in dealing with two level systems is the corresponding
density matrix, defined via
⎡c ⎤ ⁎ ⁎ ⎡ ∣c1∣2 c1c ⁎ ⎤
ρ = ψψ † = ⎢⎣c1 ⎦⎥[c1 c2 ] = ⎢ ⁎ ⎥.
2
(E.7)
2 ⎣c2c1 ∣c2∣2 ⎦
An important property of this matrix is that its trace is unity, as required by equation
(E.3). Now, we come to an important point. A fundamental property of any 2 × 2
matrix, whose trace is unity, is that it can be written in terms of the Pauli spin
matrices and a quantity known as the polarization as
1
ρ= (I + P · σ), (E.8)
2
where the individual Pauli spin matrices are given in equations (D.2) and (D.13).
Comparing this last result with equation (E.7), we can identify the various
component connections through
7-29
1
∣c1∣2 = (1 + Pz )
2
1
∣c2∣2 = (1 − Pz ) . (E.9)
2
1
c1c2⁎ = (c2c1⁎)⁎ = (Px − iPy )
2
We can now invert these equations to identify the polarization components
themselves as
Px = 2Re(c1c2⁎)
Py = − 2Im(c1c2⁎) . (E.10)
2 2
Pz = ∣c1∣ − ∣c2∣
It is easy to see that the trace of the density matrix is unity. Not only is this required
by equation (E.3), but this trace is required to be the sum of the eigenvalues of the
two wave function components. Since these are 0 and 1, we also satisfy this
requirement. This now implies that one of the eigenvectors of this density matrix
is ψ itself. The other eigenvector must be orthogonal to ψ so that their inner product
yields 0, but this other eigenvector is still arbitrary at this point.
An important property arises from the expectation values of the various spinors.
We examine this with the x-spinor as
1
〈σx〉 = Tr{ρσx} = (Tr{σx} + P · Tr{σσx})
2 . (E.11)
1
= PxTr{σx2} = Px
2
This can be repeated for each of the other components, which yields the important
result that
P = 〈σ〉 = Tr{ρσ}. (E.12)
That is, the polarization of our two level system is defined by the expectation value
of the spin vector. The direction of the spin is uniquely connected to the polarization
of the system. Another important point is that since the wave function ψ is an
eigenfunction of the density matrix, it is also an eigenfunction of the last term in the
density matrix
P · σψ = ψ . (E.13)
To examine the nature of the polarization itself, let us make some angular definitions
of the various components of the wave function equation (E.6), as
c1 = e iγ cos(ϑ /2), c2 = e i (γ +φ )sin(ϑ /2), (E.14)
7-30
so that
Px = cos(φ) sin(ϑ)
Py = sin(φ) sin(ϑ) . (E.15)
Pz = cos(ϑ)
We recognize these angles as the angles in a spherical coordinate system which relate
to the normal rectangular coordinates. The angle ϑ is the polar angle and the angle φ
is the azimuthal angle. The spherical system in which we utilize a spherical shell of
unity radius is known as the Bloch sphere, and is shown in figure E1. From the
definition of our spinors and the above coefficients, we recognize that our states may
be defined as
β = ∣0⟩, ϑ = 0, P = az
. (E.16)
α = ∣1⟩, ϑ = π , P = −az
Hence, moving around within the state ψ means that we move around on the surface
of the Bloch sphere by varying the two angles.
One of the properties of the spin that we are familiar with is the fact that it
precesses under a variety of different forces. We can investigate that by introducing a
simple form for the Hamiltonian. This form is suggested by the density matrix itself,
and we will denote it as
1
H= (Q I + Q · σ). (E.17)
2 0
The time rate of change of the polarization is then given by the well-known relation
in quantum mechanics
dP i
= − 〈[σ , H ]〉. (E.18)
dt ℏ
Figure E1. The Bloch sphere is a spherical shell of unit radius, which describes our unique two level system.
7-31
The only component of the Hamiltonian that is important now is the second term in
equation (E.17), as the first term commutes with everything. Then, we need to
evaluate the quantity
σ(Q · σ) − (Q · σ)σ = Q × (σ × σ). (E.19)
If σ were a simple vector, the last term in parentheses would vanish, but this is not a
simple vector. Rather, we find that
ax a y az
( σ × σ) = σx σy σz = 2i σ , (E.20)
σx σy σz
which is surprising but leads to

dP 1 1
= Q × 〈σ〉 = Q × P. (E.21)
dt ℏ ℏ
So, the precession of the spin polarization arises from any vector term in the
Hamiltonian which is not parallel to the polarization itself. It is easy to show that
this motion does not change the amplitude of the polarization, which is of course
required for the spin itself. If the vector Q is a constant amplitude vector, then it
defines a precession energy corresponding to a precession frequency ωQ = Q/ℏ.
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7-35
IOP Publishing

(Second Edition)
David K Ferry
Chapter 8
Tunnel devices
Tunneling is an interesting process. It appears in nearly all books on quantum

mechanics, but it is often overlooked as an important topic. Yet, it is well-known
that the nuclear fusion process that makes our Sun work, and is therefore important
to life on Earth, just would not work without tunneling. So, it seems a little bizarre
that it is not held in higher regard in parts of the physics community. Moreover, it
has become a more important part of the world of devices as these devices have
continued to become ever smaller as a result of the world following Moore’s law. It
is an essential part of every FET, as tunneling is a major source of gate leakage
currents. Then, since Esaki’s discovery of the tunnel diode, tunneling has been
studied on its own for device applications. And, as we continue to investigate
mesoscopic devices, tunneling is recognized as often being an integral part of the
device under study. New devices, such as the resonant tunneling diode (RTD) and
the single-electron transistor (SET) have furthered this interest and the importance
of tunneling.
We want to start by looking at the simple quantum point contact (QPC), because
a great many of the devices in which we will be interested in this chapter use the QPC
as an input or output connection to the environment. If we look at the QPC
conductance in figure 2.2, there are the obvious plateaus corresponding to the full
channels being transmitted through the QPC. Then, there is a little rounding on the
more negative voltage end of each plateau, which arises from the thermal broad-
ening of the Fermi–Dirac distribution. But, the main rise in current between each
plateau is due to tunneling of carriers through the barrier that they see in the QPC.
In the case of Schottky-gate-induced formation of the QPC, the actual potential that
forms between the two gates appears to be a saddle potential such as that shown
schematically in figure 8.1. The transverse potential—the rising potential seen in the
figure as one moves away from the actual saddle maximum—is actually a harmonic
oscillator parabolic potential, as was discussed in section 2.4.1. In the current flow

Figure 8.1. A schematic view of the saddle potential that forms in a quantum QPC due to surface Schottky
barrier gates. The carrier flow direction is indicated by the purplish region coming in from the front.
direction (indicated by the purplish region), the potential decreases in a parabolic

manner, and these opposing parabolic potentials create the saddle shape shown in
the figure. The actual peak of the saddle comes from the bare potential plus the
factor of ℏω/2 that is apparent in the longitudinal energy bands shown in figure 2.3.
As the carriers approach the QPC barrier, they cannot pass the barrier if the Fermi
energy is below the saddle. This is represented by the purplish flow of carriers, which
ends just below the saddle in figure 8.1. Consequently, if they are to pass the QPC,
they must do so by tunneling. As the Fermi energy is increased, relative to the saddle
potential, the tunneling probability increases and the resulting current increases. We
can show this with a simple approach to the tunneling coefficient, which can be
approximated as (see appendix F)
T ∼ exp( − ∫ γ(x)dx), (8.1)
where γ is the decay constant of the quantum wave for energies below the height of
the barrier [1]. The tunneling problem can be represented by the sketch of figure 8.2.
Here, the parabolic potential in the current direction, which decreases away from the
saddle as can be seen in figure 8.1, is indicated by the black line. The Fermi levels on
the two sides of the barrier are shown in blue, with the dashed extension through the
barrier. Hence the barrier has a spatially varying shape as indicated. The points ± x0
are the so-called turning points where the Fermi energy meets the potential. We can
now write the integral in equation (8.1) as
x0
2m⁎
Int ∼ − ∫−x 0 ℏ2
(V0 − αx 2 ) dx , (8.2)
where α characterizes the downward curvature of the potential in the current

direction, and has the units of eV cm−2. If we measure V0 from the Fermi energy,
then the potential is zero at x0. Hence, we can write the integral as
8-2
Figure 8.2. A schematic picture to help in evaluating the tunnel transmission through the QPC potential.
π 2m⁎V02 (8.3)
Int ∼ − .
2 αℏ2
As the gate voltage is made more positive, the curvature of the potential decreases,
but so does the potential amplitude, which leads to an overall increase in the
tunneling current. If we adapt the Landauer formula and use equation (8.1) as the
transmission, then the rising part of the steps in figure 2.2 is given approximately as
⎛ ⎞
2e 2 π 2m⁎V02 ⎟
G∼ exp⎜⎜ − , (8.4)
h ⎝ 2 αℏ2 ⎟⎠
but one must still determine experimentally just how one can determine V0 and α.
The latter can be determined by measuring the harmonic oscillator parameters as in
figure 2.17. From this measurement, one can determine the energy level spacing, and
this can be used with the gate voltage spacing of the steps to estimate the potential
height as a function of gate voltage.
8.1 Coulomb blockade

When the dielectric material in a capacitor becomes too thin, then it is possible for
the charge on the capacitor plate to tunnel through the insulator, and the capacitor
becomes what is known as a leaky capacitor. Under some mesoscopic device
concepts, this tunneling is a desirable effect, and can be used to create some
interesting devices. In other cases, such as the gate oxide in a MOSFET, this is an
undesirable effect. When, the capacitor is made with a small area, then another
effect begins to occur. While the capacitor is small, we continue to describe it as a
macroscopic capacitance associated with the system, although this description may
not be fully valid. The change in electrostatic potential due to a change in the charge
on an ideal conductor is associated with the linear relationship between the charge
and the voltage
8-3
Q = CV , (8.5)
where C is the capacitance, Q is the charge on the conductor, and V the electrostatic
potential that exists between the two ‘plates’ of the capacitor. For an ideal metal,
any charge added to it will rearrange itself such that the electric field inside the metal
vanishes, and the surface of the metal becomes an equipotential surface. Therefore,
the electrostatic potential associated with the metal relative to its reference is
uniquely defined. In our capacitor, we consider two metal conductors connected
by a dc voltage source. This leads to a charge +Q on one conductor and a charge
−Q on the other. The capacitance of the two conductor system is then defined as
C = Q/V12. The electrostatic energy stored in the two conductor system is the work
done in building up the charge Q on the two conductors and is given by
Q2
E= . (8.6)
2C
In the case of very small capacitors, the charging energy given by this latter equation
due to a single electron, e2/2C, becomes comparable to the thermal energy, kBT. The
transfer of a single electron between conductors therefore results in a voltage change
that is significant compared to the thermal voltage fluctuations and creates an energy
barrier to the transfer of electrons. This barrier remains until the charging energy is
overcome by sufficient bias. How small must the capacitor be for such effects to
become important? If the energy stored in the capacitor is about the same as the
thermal energy, then the capacitor has a value of 3 × 10−18 F, or 3 aF, at room
temperature. Of course, at very low temperatures, the capacitance can be signifi-
cantly larger.
Historically, Coulomb blockade effects were first predicted and observed in small
metallic tunnel junction systems. As mentioned already, the conditions in metallic
systems of high electron density, large effective mass, and short phase coherence
length (compared to semiconductor systems) usually allow us to neglect size
quantization effects. The dominant single-electron effect for small metal tunnel
junctions is therefore the charging energy due to the transfer of individual electrons,
e2/2C. The effects of single-electron charging in the conductance properties of very
thin metallic films was recognized in the early 1950s by Gorter [2] and Darmois [3].
It was found that these metal films formed arrays of small islands, and conduction
occurs due to tunneling between these islands. Since the island size is small, the
tunneling electron has to overcome an additional barrier due to the charging energy,
which leads to an increase in resistance at low temperature. Such discontinuous
metal films show an activated conductance, similar to an intrinsic semiconductor.
Neugebauer and Webb [4] developed a theory of activated tunneling in which
the activation energy resembles an energy gap and is therefore referred to as a
Coulomb gap.
There have been many experimental studies of the transport properties of metal
clusters or islands imbedded in an insulator that are then contacted by conducting
electrodes. More interest in the area developed in studies of superconducting tunnel
junctions [5], where the coulomb blockade interacted with the normal Josephson
8-4
Figure 8.3. (a) The I–V characteristics of a tunnel junction between a STM tungsten tip and a stainless steel
surface. The distance of the tip from the surface is reduced for each sweep (moving to the right in the figure).
(b) A single curve from (a) plotted as the dots. The straight lines are guides to the eye, while the quadratic curve
through the origin is a fit to theory. (Reprinted with permission from [8]. Copyright 1988 the American
Physical Society.)
tunneling [6] (we will see this again when we return to superconducting qubits in
section 8.4 below). Very soon, however, the coulomb blockade and the tunneling of
single electrons were observed for normal metal systems [7, 8]. What is normally
seen in these experiments is that no, or a very low, current flows until the applied
voltage reaches e/2C, then the current begins to rise. So, this gives a plateau around
zero bias, and the width of this plateau is typically e/C, corresponding to the energy
in equation (8.6). We illustrate this with measurements made between a tungsten tip
on an STM and a stainless steel metal substrate in figure 8.3 [8]. In panel (a), many
I–V plots are shown corresponding to an increasing distance from the tip to the
surface. In panel (b), one single trace is shown to illustrate the Coulomb blockade,
which leads to the observed plateau. At higher values of the applied bias, the current
approaches a linear dependence on the voltage. The plateau in this curve appears to
be about 120 mV wide, which then corresponds to a capacitance of 1.3 aF. As the
measurements were made at helium temperature, the charging energy (8.6) is
certainly larger than the thermal energy. These measurements appear to be clear
evidence for single-electron tunneling through their structure, by which only a single
8-5
electron tunnels at any one point in time. These results correlate well with the
theory [5].
8.2 Single-electron structures

A single capacitor is seldom used to make a structural device. Certainly, a single
capacitor is used in dynamic random-access memory, but it is coupled there to a
transistor which controls the charging and discharging of the capacitor. In meso-
scopic devices, the idea of a single-electron device, or SET, has appeared, and this
uses (at least) two capacitors connected in series. Between the two capacitors is a
region that can accumulate charge. Typically, this small region is termed a quantum
dot. In metals, it may contain more than a thousand electrons, but in semi-
conductors the charge states may have their own quantization due to the small
size of the dot and the number of electrons can be few, even down to zero.
8.2.1 A simple quantum-dot tunneling device

Let us consider the two-capacitor circuit shown in figure 8.4, in which the ‘island’
consists of a small quantum dot coupled weakly through thin insulators to metal
leads as shown. Typically, the tunnel junction can be considered as a parallel
combination of the tunneling resistance R and the actual capacitance C. In metal
tunnel junctions, the tunneling barrier is typically very high and thin, while the
density of states at the Fermi energy is very high. The tunneling resistance is
therefore almost independent of the voltage drop across the junction, but of course is
much larger than the h/2e2 that corresponds to a single propagating mode. In the
analysis that follows, we ignore this resistance, but will call upon it later. Electrons
that tunnel through one junction or the other are therefore assumed to immediately
relax due to carrier–carrier scattering, so that resonant tunneling through both
barriers is simultaneously neglected. This assumption is made as we are interested in
the charge that can accumulate in/on the quantum dot in the sequential tunneling
approach. That is, charge may tunnel through only one of the two capacitors, and
this will change the amount of charge on the dot. Tunneling represents the injection
Figure 8.4. Circuit for two Coulomb blockaded capacitors and a central quantum dot on which to accumulate
charge.
8-6
of single particles, which involve several characteristic time scales. The tunneling
time (the time to tunnel from one side of the barrier to the other) is the shortest time
(on the order of 10−14 s), whereas the actual time between tunneling events
themselves is on the order of the current divided by e, which for typical currents
in the nA range implies a mean time of several hundred picoseconds between events.
The time for charge to rearrange itself on the electrodes due to the tunneling of a
single electron will be something on the order of the dielectric relaxation time, which
is also very short. Therefore, for purposes of analysis, we can consider that the
junctions in the regime of interest behave as ideal capacitors through which charge is
slowly leaked.
The net charge on the electrodes of the individual capacitors is given by equation
(8.5), which can be rewritten for the present case as
Q1 = C1V1
. (8.7)
Q2 = C2V2
The net charge Qdot on the island is the difference of these two charges. In the
absence of tunneling, the difference in charge would be zero and the island neutral.
Tunneling allows an integer number of excess electrons to accumulate on the island
so that we find
Q dot = −ne = Q2 − Q1. (8.8)
Here, n is the net number of electrons, and can only exist if the charge is different on
the two capacitors. This convention is chosen such that an increase in either n1 or n2,
the number of electrons that define the charge on the two capacitors, corresponds to
increasing either the junction charge Q1 or Q2, respectively, in equation (8.7). The
sum of the two voltages across the two capacitors is just the applied voltage, Va, so
that, using equations (8.7) and (8.8), we may write the voltage drops across the two
tunnel junctions as
1
V1 = (C2Va + ne )
C1 + C2
. (8.9)
1
V2 = (C1Va − ne )
C1 + C2
In the following, we will write CT = C1 + C2 to simplify the equations. The
electrostatic energy stored in the two capacitors is given by
Q12 Q 22 1
E=
2C1
+
2C 2
=
2CT
(C1C2Va2 + Qdot2 ). (8.10)
However, this is not the total energy in the circuit. We must add to this the work
done by the voltage source in transferring the charge to the two capacitors, which
involves the various tunneling currents through them. This additional work is found
by integrating the current over time, as
8-7
Wa = ∫ dtVaI (t ) = VaΔQ. (8.11)
Here, ΔQ is the total charge transferred from the voltage source, including the
integer number of electrons that tunnel into the island and the continuous polar-
ization charge that builds up in response to the change of electrostatic potential on
the island. A change in the charge on the island due to one electron tunneling
through capacitor 2 (so that n′2 = n2 + 1) changes the charge on the island to
Q′ = Q + e, and n′ = n − 1. From equation (8.9), the voltage across junction 1
changes as V ′1 = V1 − e /CT . Therefore, from equation (8.7), a polarization charge
flows in from the voltage source ΔQ = −eC1/CT to compensate. The total work done
to pass in n2 charges through junction 2 can then be written as
C1
Wa(n2 ) = −n2eVa . (8.12)
CT
By a similar approach, we can also find the work done in transferring n1 charges
through junction 1 to be
C2
Wa(n1) = −n1eVa . (8.13)
CT
With this, we can write the total energy in the system as
1
E (n1, n2 ) =
2CT
(C1C2Va2 + Qdot2 ) + eV
CT
a
(C1n2 + C2n1). (8.14)
The condition for Coulomb blockade is based on the change in this electrostatic
energy with the tunneling of a particle through either junction. At zero temperature,
the system has to evolve from a state of higher energy to one of lower energy.
Therefore, tunneling transitions that take the system to a state of higher energy are
not allowed, at least at zero temperature (at higher temperature, thermal fluctuations
in energy on the order of kBT weaken this condition). At high enough temperature,
the thermal fluctuations wash out the Coulomb blockade, and the capacitors become
just leaky capacitors. We find the voltage at which the tunneling of single electrons
can occur from the change in energy of the system when the charge on the dot
changes by ±1. let us assume first that the charge on C2 changes by the addition or
subtraction of a single electron, which leads to the change in energy of
ΔE 2± = E (n1, n2 ) − E (n1, n2 ± 1)
e ⎡ e ⎤ . (8.15)
= ⎢ − ± (ne − VaC1)⎥
CT ⎣ 2 ⎦
Similarly, for a change in the charge on C1 by the addition or subtraction of a single

electron, the change in energy is
8-8
ΔE 2± = E (n1, n2 ) − E (n1 ± 1, n2 )
e ⎡ e ⎤ . (8.16)
= ⎢ − ∓ (ne + VaC1)⎥
CT ⎣ 2 ⎦
For all possible transitions on to or off of the island, the leading term involving the
Coulomb energy of the island causes ΔE to be negative until the magnitude of Va
exceeds a threshold that depends on the lesser of the two capacitances. For
C1 = C2 = C, the requirement becomes simply ∣Va∣ > e/2C. Tunneling is prohibited
and no current flows below this threshold, as evident in the I–V characteristics
shown in figure 8.3 (although the figure is a single capacitor, the effect is the same).
This region of Coulomb blockade is a direct result of the additional Coulomb
energy, e2/2CT, which must be expended by an electron in order to tunnel on to or
off of the island. The effect on the current voltage characteristics is a region of very
low conductance around the origin. For large-area junctions where CT is large, no
regime of Coulomb blockade is observed, and current flows according to the tunnel
resistance.
Figure 8.5(a) shows the equilibrium band diagram for a double-tunnel-junction
system, illustrating the Coulomb blockade effect for equal capacitances. A Coulomb
gap of width e2/2CT has opened at the Fermi energy of the metal island, half of
which appears above and half below the original Fermi energy, so that no states are
available for electrons to tunnel into from the left and right electrodes. In essence,
this energy gap is the charging energy required to put an electron onto the dot. For
large capacitances, this energy is reduced toward zero. Similarly, electrons in the
island cannot tunnel from the island to the metals, because there are no empty states
in the metallic contacts. However, when a bias of Va ⩾ e/2CT is applied, the bands
shift to bring the dot level into alignment with one of the two contacts. Now,
electrons can tunnel through the barriers, and with small capacitors, only a single
electron at a time tunnels. When the first electron tunnels onto the dot, the energy
will be shifted to show a Coulomb blockade to the second electron. This gap
prohibits the tunneling until the voltage is raised to 3e/2CT, as is apparent from
equations (8.15) and (8.16) with n = ±1 in this case.
Figure 8.5. Band diagram of a two-capacitor single-electron circuit (a) in equilibrium, showing the Coulomb
blockade, and (b) with a bias above the tunneling threshold so that the minimum energy can be supplied from
the source.
8-9
An important aspect of the structure we have been discussing is the size of the
quantum dot. As is apparent, the necessary voltage at which current begins to flow is
dependent upon the size of the dot. In general, we can assume that the distance
between the top electrode of C1 and the bottom electrode of C2 is held fixed in any
experimental configuration. Of course, if an STM tip is used, this may not be the
case, but we nevertheless assume that this will be the case for the studies we want to
discuss now. Then, the capacitances will depend upon the size of the quantum dot. If
the dot is made smaller, the capacitances will also become smaller due to the increase
in the thickness of the insulating medium between the top electrode and the dot. This
will lead to an increase in the voltage at which the current starts to rise. And the
reverse will also be the case, as when we make the dot larger, the capacitances will
also become larger, so that the required voltage is decreased. As a result, when a bias
is applied that is larger than the turn-on voltages, the resulting tunnel current can be
a measure of the size of the quantum dot. Thus, the tunnel current can be used to
monitor this size, which may be used for an interesting quantitative sensing circuit.
For example, if single molecules are used as the quantum dot, then these molecules
can be interrogated by the tunnel current [9]. By bonding a pair of recognition
molecules to the two metallic tips that represent the wires from the applied bias
source, recognition can be done on a target molecule which has preferential
hydrogen bonding to particular sites on the recognition molecules [10]. Moreover,
one can identify particular DNA molecules by detecting the direct tunneling current
through them, as the molecules are passed through a nanopore to which the
tunneling electrodes are attached [11, 12]. That is, the different molecules crossing
the DNA double helix are of different sizes, and hence different tunneling currents
are associated with each of these molecules. Measuring the tunneling current as the
DNA strand is pulled through the hole, between the electrodes, the sequence can be
determined.
The above approach has since been extended to amino acids and peptides through
the recognition tunneling method [13]. In this approach, the tunneling leads are
modulated. Since the tunneling current depends exponentially upon both the voltage
applied and the tunneling distance, the modulation leads to a series of current spikes.
As an amino acid is moved into the gap between the electrodes, the amplitude of
these spikes will vary according to the size of the molecule. These current spikes can
then be used as a recognition signal to identify the particular amino acid that is in the
tunneling gap. The process is explained further in the video of figure 8.6, from the
ASU group [13]. The process of making the entire structure is further explained
in [14].
8.2.2 The gated single-electron device

The single-electron circuit that we have been considering in figure 8.4 is interesting,
but it was quickly realized that having an additional bias voltage to control the
actual charge on the quantum dot would be beneficial to creating a SET. Consider
the topologically equivalent SET circuit shown in figure 8.7. In this circuit, a
separate voltage source, Vg, is coupled to the island through an ideal (infinite tunnel
8-10
Figure 8.6. The video describes the recognition of individual amino acids as they are pulled through the gap in
a tunneling structure. Here, the acids play the role of the quantum dot in, e.g., figure 8.4. (The video is from
Lindsay, included with his permission. Available at https://iopscience.iop.org/book/978-0-7503-3139-5.)
Figure 8.7. The simple circuit of figure 8.4 is now modified by the addition of a gate voltage source which is
coupled to the quantum dot by a gate capacitor.
resistance) capacitor, Cg. This additional voltage modifies the charge balance on the
island so that equation (8.7) requires an additional polarization charge that arises
from this new bias source and its coupling capacitance. This new charge is
Qg = Cg(Vg − V2 ). (8.17)
Now, the charge on the quantum dot is also affected, and

Q dot = −ne = Q2 − Q1 − Qg . (8.18)
We can now combine the various equations above to give the new forms of the
voltages across the two capacitors, which now include the effect of the gate and its
capacitance. These equations can be written as
8-11
1
V1 = [(C2 + Cg )Va − CgVg + ne ]
C ′T
1 . (8.19)
V2 = (C1Va + CgVg − ne )
C ′T
C ′T = CT + Cg = C1 + C2 + Cg
As before, we can now write the energy due to the charge on the capacitors,
equivalent to equation (8.10), as
1 ⎡
⎣CgC1(Vg − Va ) +C1C2Va + CgC2V g + Q dot⎤⎦ .
2 2 2 2
E= (8.20)
2C ′T
The work performed by the voltage sources during the tunneling through junctions 1
and 2 now includes both the work done by the gate voltage and the additional charge
flowing onto the gate capacitor electrodes. Equations (8.12) and (8.13) are now
generalized to
n2e
Wa(n2 ) = − (VaC1 + VgCg )
C ′T
ne . (8.21)
Wa(n1) = − 1 [VaC2 + Cg(Va − Vg )]
C ′T
These different contributions to the total energy can now be combined as in equation
(8.14). More important to us, however, are the relevant equations for the change of
the charge on the dot by adding or subtracting a charge from one of the capacitors.
These changes in energy can be written in analogy with equations (8.15) and (8.16)
to be
e ⎡ e ⎤
ΔE 2± = ⎢ − ± (ne − C1Va − VgCg )⎥ , (8.22)
C ′T ⎣ 2 ⎦
and
ΔE1± =
e
C ′T {− 2e ∓ ⎡⎣ne − V C + (C + C )V ⎤⎦}.
g g 2 g a (8.23)
The gate bias now allows us to change the charge on the island, and therefore to shift
the region of Coulomb blockade. Thus, a stable region of Coulomb blockade may be
realized for n ≠ 0. As before, the condition for tunneling at low temperature is that
ΔE1,2 > 0 so that the system goes to a state of lower energy after tunneling. The
conditions for forward and backward tunneling then become
e
− ± (ne − C1Va − VgCg ) > 0
2 . (8.24)
e
− ∓ [ne − VgCg + (C2 + Cg )Va ] > 0
2
8-12
Figure 8.8. Stability diagram for the circuit of figure 8.7. The gate voltage can now be used to create new
regions of Coulomb blockade (in blue). Each diamond is characterized by a different value of the charge on the
quantum dot.
The four equation (8.24), for each value of n may be used to generate a stability plot
in the Va–Vg plane, which shows stable regions corresponding to each n for which no
tunneling may occur. Such a diagram is shown in figure 8.8 for the case of
Cg = C2 = C, C1 = 2C. The lines represent the boundaries for the onset of tunneling
given by given by these equations for different values of n. The trapezoidal shaded
areas correspond to regions where no solution satisfies the equations, and hence
where Coulomb blockade exists. Each of the regions corresponds to a different
integer number of electrons on the island, which is ‘stable’ in the sense that this
charge state cannot change, at least at low temperature when thermal fluctuations
are negligible. The gate voltage then allows us to tune between stable regimes,
essentially adding or subtracting one electron at a time to the island.
It is possible to actually perform spectroscopy on the states of the quantum dot.
In semiconductors, the lower density of states and small size of the quantum dot can
lead to quantization of the wave function within the dot. Hence, the tunneling
characteristics will be modified from the simple charging diagram due to this
quantization. This has been probed using vertical GaAs/AlGaAs heterostructures in
an elegant manner [15, 16]. In these structures, a vertical resonant tunneling
structure is created. This involves two GaAlAs barriers with a GaAs quantum
well in between them. GaAs provides the source and drain cladding layers on either
side of the two GaAlAs layers, as shown in figure 8.9(a) [16]. A schematic of the
potential across the resonant tunneling device, under bias, is shown in figure 8.9(b).
The structure is then patterned into small vertical pillars, as shown in panel (c) of the
figure, and gates can then be deposited around the periphery of the pillars as
8-13
Figure 8.9. (a) Schematic of the vertical resonant single-electron device, showing the surrounding gate
electrode. (b) Plot of the bias across the device. (c) Micrographs of various pillar shapes that have been made.
(Reprinted with permission from [16]. Copyright 2001 IOP Publishing.)
Figure 8.10. Current flowing through the circular vertical single-electron device as the gate voltage is varied.
Each peak corresponds to the tunneling of a single electron. N indicates the number of electrons that reside on
the quantum dot. (Reprinted with permission from [16]. Copyright 2001 IOP Publishing.)
indicated in panel (a). This gate allows one to completely deplete the quantum dot of
electrons, and then allow individual electrons to tunnel into the dot by slowly
increasing (lowering the negative bias of the Schottky barrier gate) the gate voltage.
As each edge of the Coulomb diamond is found, a current peak will occur that
signals the tunneling of a single electron. This is shown in figure 8.10. In this latter
figure, each peak of the current corresponds to adding a single electron to the charge
on the quantum dot. It can be seen that there are larger gaps that appear when
N = 2, 6, 12, … (see the inset), which is taken to be indicative of the Darwin–Fock
8-14
spectrum (see appendix F) for a two-dimensional harmonic oscillator, especially

when a normal magnetic field is applied. In the lowest quantum level of the dot, only
two electrons of opposite spin can be accommodated. In the next level, four electrons
in which each pair has opposite spin can be accommodated, and so on up the ladder
of states. The non-uniform spacing of the tunneling peaks arises from the Coulomb
charging energy plus the quantization energy and the varying many-body energies
that arise as various numbers of electrons occupy the quantum dot.
In the previous example, the current was observed as a series of spikes as one
passed the crossing points for the Coulomb diamonds. This was due to strong
quantization in the central quantum dot. If the quantum dot does not show
quantization, a different behavior can be found in the structure, in which one can
see what is known as a Coulomb staircase. Let us consider the MOS structure shown
in figure 8.11 [17]. Here, the MOS device is fabricated on a p-type substrate with an
initial silicon-on-insulator thickness of 300 nm over a buried oxide of 375 nm
thickness. As indicated in panel (a), three gates are formed in the structure. The two
side gates are biased to deplete the electrons from the inversion layer and thus to
create potential barriers as shown in panel (b). A third, top gate is used to control the
potential in the quantum dot that is created between the two side gates. Here, the
central quantum dot is relatively large and does not show quantized energy levels,
especially as the device will be operating at room temperature, rather than at low
temperatures. The transfer characteristics are plotted in figure 8.12, where both the
drain current and the transconductance are plotted as a function of the bias on the
third gate, which is the central top gate. Initially, this bias is sufficiently negative to
deplete the quantum dot of electrons. As the bias is made more positive, current
begins to flow (red curve), and we can see the characteristic staircase behavior. The
transconductance (black data) is the derivative of the current with gate bias, and
shows peaks at each of the transitions in the staircase. From the peak spacing in the
transconductance, which is about 1 V, it is felt that the gate capacitance is quite
small, and about 0.16 aF. Note that the data are obtained at 300 K, which is the
Figure 8.11. (a) Schematic depiction of the self-aligned double-gate single-electron device. (b) Simulation of
the electrostatic potential along the channel for the device in (a). (Reprinted with permission from [17].
Copyright 2008 IOP Publishing. All rights reserved.)
8-15
Figure 8.12. Transfer characteristics for the device shown in figure 8.11. The current (red) shows a Coulomb
staircase rather than current spikes. The transconductance (black) oscillations indicate a gate capacitance of
about 0.16 aF. (Reprinted with permission from [17]. Copyright 2008 IOP Publishing. All rights reserved.)
reason that the quantization effect is absent, and one therefore does not see current
spikes as observed at low temperature. Hence, it is quite likely that the observed
behavior is that due to single-electron tunneling along the source–drain direction.
One can carry the quantum dot size down to the ultimate of being a single atom,
and this structure has been labeled as a single-atom transistor [18]. One of the
earliest proposals to create a quantum computer in a condensed matter system was
to use single-atom isotopes of 31P implanted into a silicon crystal [19]. This idea
would use the nuclear spin states as the qubit (discussed below in a later section). By
hydrogenating the surface of silicon (100), all the dangling bonds are satisfied. A
single hydrogen can be removed by using an STM to break the individual bond [20].
Then, the surface can be exposed to phosphine, and a single molecule will attach to
the single bare dangling bond exposed by the hydrogen removal. Subsequent
reaction leads to the phosphorus atom moving into the crystal, where it acts as a
single donor atom [18]. The atomic potential of this atom, when placed between the
metallic source and drain electrodes, plays the role of the quantum dot. This
potential can be manipulated by the appropriate placement of one or more side
gates, and thus provide the equivalent behavior of the system in figure 8.7.
8.2.3 Double dots

In the preceding sections, we established a correspondence between Coulomb-
blockaded semiconductor quantum dots and an artificial atom through the spectrum
of charge states that can exist in the dots. We want to extend this analogy to consider
how a pair of dots may be coupled to each other in a simple manner such that they
form what may be viewed as artificial molecules. As with the discussion of real
molecules in nature, where new molecular orbitals are formed as a result of the wave
function overlap between the component atoms, we will see here that such overlap
can give rise to new electronic states in coupled quantum dots. The collective
character of these states has the potential to lead to new classes of electronic devices,
8-16
particularly for application to quantum computing, where one is interested in using

the superposition states of quantum systems as the basis for computing. In
discussions of such collective phenomena, however, there is an important need to
distinguish between essentially classical collective effects that arise from Coulomb
charging between otherwise isolated dots, and true quantum effects that are a
consequence of controlled wave function coupling. In the following discussion, we
first focus on the charging effects in coupled quantum dots. Here, we closely follow
the treatment of van der Wiel et al [21].
Previously, we determined the conditions for which the Coulomb blockade
existed by determining the total energy of the circuit and requiring that this energy
be lowered as a result of single-electron tunneling. The extension of this concept to
the problem of single-electron tunneling through a pair of sequentially coupled
quantum dots is indicated in figure 8.13. In this new circuit, we take the number of
electrons stored on dot 1 (2) to be N1 (N2) (note the difference from the notation n1
and n2 earlier, where the latter numbers denoted the number of electrons to have
tunneled through junctions 1 and 2, respectively). The system now features three
tunnel barriers (as opposed to two in the original problem). The new capacitor
controls the electrostatic coupling between the two dots. As in the case of the SET,
the electrostatic energy of each dot is regulated via an independent gate voltage
(defined here to be Vg1 and Vg 2 ). Each of these gate voltages couples to the respective
charge island via an associated capacitance (Cg1 and Cg 2 ). In order to focus on the
key effects arising from the influence of the electrostatic coupling between the dots,
we assume that the circuit model of figure 8.13 characterizes the system completely.
A classical analysis of the energetics of Coulomb-coupled quantum dots has been
performed by several authors, in which the quantization of energy within the dots is
ignored [22–24]. Following the derivation by van der Wiel et al [21], we begin by
considering the case of linear transport through the double-dot system,
Figure 8.13. Simple circuit in which two quantum dots are coupled with individual gate voltages for each dot.
The coupling between the dots is governed by the central capacitor Cm.
8-17
corresponding to Va = 0. Under such conditions, it can be shown that the

electrostatic energy of the coupled-dot system, containing N1 and N2 electrons on
dots 1 and 2, respectively, may be written as
e 2N12 ⎛ C2 ⎞ e 2N22 ⎛ C1 ⎞
E (N1, N2 ) = ⎜ 2⎟
+ ⎜ 2⎟
2 ⎝ C1C2 − C m ⎠ 2 ⎝ C1C2 − C m ⎠
. (8.25)
e N1N2 ⎛
2
Cm ⎞
+ ⎜ ⎟ + f (Vg1, Vg 2 )
2 ⎝ C1C2 − C m2 ⎠
The notation here differs from that used earlier. Note that, in figure 8.13, the main
capacitors are now labels CL and CR. In equation (8.25), the capacitances C1 and C2
are the total capacitance seen by dots 1 and 2, respectively. These are given by
C1 = CL + Cg1 + Cm
. (8.26)
C2 = CR + Cg 2 + Cm
The last term in equation (8.25) provides the energy changes that the two gate
voltages provide, and may be written as [21]
⎛ C g21V g21 ⎞⎛ C2 ⎞
⎜
f (Vg1, Vg 2 ) = ⎜eN1Cg1Vg1 + ⎟⎟⎜ ⎟
⎝ 2 ⎠⎝ C1C2 − C m2 ⎠
⎛ C g22V g22 ⎞⎛ C1 ⎞
+ ⎜⎜eN2Cg 2Vg 2 + ⎟⎟⎜ ⎟
⎝ 2 ⎠⎝ C1C2 − C m2 ⎠ (8.27)
+ (eN2Cg1Vg1 + eN1Cg 2Vg 2
⎛ Cm ⎞
+ Cg1Vg1Cg 2Vg 2 )⎜ ⎟.
⎝ C1C2 − C m2 ⎠
We can examine the implications of these results by considering two extreme cases.
In the limit of zero coupling between the dots, corresponding to Cm → 0, only the
first two terms on the right-hand side of equation (8.25) survive. This means that the
total electrostatic energy of the system is simply that of the two isolated dots. The
other limit arises in the situation where Cm is the largest capacitance in the system, in
which case C1(2) ≈ Cm and the problem essentially reduces to one involving the
charging of a large single-dot formed by the two smaller ones. From this reasoning,
one can construct the charge-stability diagram in the space of the two gate voltages
for the first case (Cm → 0) as shown in figure 8.14(a). Here, the charging regions
simply are sets of squares in which the allowed number of electrons are shown
schematically in this figure. Hence, a change in one gate voltage only affects the
quantum dot to which it is attached. It does not affect the charge on the other
quantum dot, as there is no connection to that dot. The gate capacitance and the
appropriate other capacitance correspond just to the case of figure 8.4.
For non-zero coupling between the dots (Cm ≠ 0) the contour becomes distorted,
and the original four-fold intersections of the different charge squares in
8-18
Figure 8.14. (a) Charge-stability diagram for the case when there is no coupling between the two quantum
dots. (b) Charge-stability diagram for coupling between the two dots. The simple crossing evolves into a pair of
triple points.
figure 8.14(a) develop instead into a pair of closely sited triple points, as shown in
figure 8.14(b). The existence of these triple points is critical for transport, as they
allow for the flow of current with a small source–drain bias present. This pair of
points allows the electron number on both dots to fluctuate by a single electron. The
origin of these triple points is explained in terms of the mutual capacitance between
the two dots in the system. When this capacitance is non-zero, the charging of one of
the dots by an additional single electron modifies the electrostatic energy of the
other, and this effectively causes a repulsion of the resonance lines in the region
where the resonance lines of both dots intersect. Consider the triple point highlighted
in red in figure 8.14(b) which indicates the values of the gate voltages (Vg1 and Vg 2 ) for
which the four charge populations (N1, N2) = (0,0), (1,0), (0,1), and (1,1) come
together. The flow of current via single-electron tunneling is possible at this
degeneracy point, via the charge sequence (0,0) → (1,0) → (0,1) → (0,0), and so
on. This triple point therefore involves a single-electron transfer through the double-
dot system.
As can be imagined current can flow freely between the pair of triple points. In
figure 8.15, we show the stability diagram which plots the transconductance as a
function of the two gate voltages for a double-dot system in a triple-layer graphene
sheet [25]. The device structure is shown as the inset to the figure, and the different
regimes are separated (and defined) through metal gates used to deplete the carriers
under the gates. The gates were fabricated by standard electron-beam lithography;
the actual gates were between the metallic Schottky barrier regions. Contacts to the
individual regions were then formed as a Cr/Au double layer (10/50 nm, respec-
tively). The triangular-shaped dots are estimated to have areas of 4 × 10−3 μm2 for
dot 1 (left-hand dot in the figure) and 5 × 10−3 μm2 for dot 2 (right-hand dot in the
figure). From the shape of the modified diamonds, the various capacitances have
been estimated, using the approach in [21] (discussed below) to be Cg1= 2.2 aF, Cg 2 =
8-19
Figure 8.15. Charge-stability diagram, plotted as the measured transconductance, for a double-quantum-dot
system on graphene. The gate patterns are shown in the inset to the figure. Current flows mainly between the
paired triple points. (Reprinted with permission from [25]. Copyright 2009 American Chemical Society.)
2.9 aF, and Cm = 10 aF. The total dot capacitances are estimated to be 30 aF and 60
aF for C1 and C2, respectively. This gives a value for CL of 17.8 aF and a value for
CR of 47.9 aF.
The shape of each of the Coulomb ‘diamonds’ that appears in figures 8.14(b) and
8.15 can be used to determine the values of the various capacitances that appear in
figure 8.13. The basic procedure was described in [21]. The key is to measure the
various voltages indicated in figure 8.16, where we show an expanded view of a
single ‘diamond’, outlined in red. These can be simply related to the various
quantities as
e e
ΔVg1 = , ΔVg 2 = ,
Cg1 Cg 2
C eCm
ΔV gm1 = ΔVg1 m = , (8.28)
C2 Cg1C2
C eCm
ΔV gm2 = ΔVg 2 m = .
C1 Cg 2C1
Now, this gives four equations, but there are five capacitances that must be
determined. To fully determine all the capacitances, one must go beyond the linear
theory and apply an applied bias to the ends of the circuit. Of importance is the
parameter α1(2) that provides the scaling between effects induced from the gate and
from the applied bias. These are determined by the capacitances as [21]
8-20
Figure 8.16. Illustration of a single-Coulomb ‘diamond’ to determine the parameters for the double quantum
dot of figure 8.13.
Cg1(2)
α1(2)ΔVg1(2) = eΔVg1(2) = ∣eVa∣ . (8.29)
C1(2)
With these parameters known, the full set of capacitors can be determined. In the
work discussed in figure 8.15, these parameters were determined to be α1 ∼ 0.07 and
α2 ∼ 0.05, both in meV/mV [25].
When the quantum dots are sufficiently small to lead to quantization, the regions
around the triple points can be far more complicated, as excited states can influence
these charging diagrams. Nonlinear effects can occur as well, as already mentioned.
Discussion of the full analysis is beyond the level we consider here, but are discussed
in [21].
8.3 Quantum dots and qubits

In order to understand how quantum dots can be envisioned as representing qubits,
we have to first understand how the quantum bit, or qubit, differs from the normal
computer binary bit. This difference is shown in figure 8.17. In panel (a), we show a
two-dimensional space, in which the axes represent the binary 1 and the binary 0 for
the nth bit of our computer. Now, with normal binary encoding, these two unit
vectors are added to produce the ‘state’ of this bit according to
∣n⟩ = an∣0⟩ + bn∣1⟩, (8.30)
but this equation is subject to the conditions that either an = 1 and bn = 0, or an = 0
and bn = 1. These are the only two possibilities that are allowed. The entire computer
runs on this approach. The standard storage is by bytes, which are eight bits, and
8-21
Figure 8.17. (a) The two-dimensional space for a standard computer binary bit. (b) The modification of this
space for a qubit, where the wave function can have any analog, complex combination of the two states.
this can represent a number between 0 and 255. For example, the number 62 is
represented by the many-bit state ∣0111110⟩. As a further example, the standard
compression of an image is via the jpeg standard, where a pixel, or picture element
has its color encoded as three bytes, one byte each for red, blue, and green. This
combination provides 16.78 million colors. Black is all zeroes and white is all ones.
Any other color is somewhere between these limits.
Now, the desire to use quantum computation lies in the ability to store more
information in each quantum bit, or qubit, and to use entanglement between the bits
to enlarge the information content even more. For this purpose, we use the
properties of the quantum wave function for each bit, as shown in figure 8.17(b).
We still use the same basis set, as indicated in the figure, but now the coefficients
only need to satisfy ∣an∣2 + ∣bn∣2 = 1, which is required by the normalization of the
wave function. Hence, the fact that the wave function is complex introduces the
phase, and the vector orientation of the wave function can lead to a point that is
anywhere on the unit circle (only the first quadrant is shown in the figure). Now,
each qubit is a two level system, corresponding to the two axes, and the occupancy
of each level is governed by the wave function. But, the two level system was
discussed in appendix E in terms of the Bloch sphere. There, in figure E1, we see that
the two axes are represented by the top of the sphere corresponding to the ‘zero’ state
and the bottom of the sphere corresponding to the ‘one’ state. The actual qubit state
can be any point on the surface of the sphere, and quantum computer ‘operations’
generally can be expressed as rotations in this three-dimensional space. These actual
operations, and the concepts of quantum computing, are beyond the material we
want to discuss here, but can be found in several textbooks [26, 27]. When we recall
that the Bloch sphere was introduced to describe a particular two level system, that
of spins, it becomes clear why there is a significant effort to utilize a spin qubit, as
discussed above.
Let us examine how a pair of quantum dots can be formed to represent at least a
binary bit. As we have discussed above, quantum dots can be formed in a very large
8-22
number of ways. Here, we assume that the dots are formed in a GaAs/AlGaAs
heterostructure, and their definition is by a set of Schottky barrier gates, as shown in
figure 8.18(a). The black gates are the basic definition gates for a single large dot,
which is then split into two smaller dots by the blue finger gates. The gates marked as
V2 and V4 are used to fine tune the exact size of the individual dots. The potential
profile is shown in figure 8.18(b), with the deep red color being above the Fermi
energy, and the other colors representing various potential depths. The entire gate
structure is considered to be about 1 μm high by 1.5 μm wide [28]. We will highlight
the 19th eigenstate in the structure, as it shows switching from one dot to the other.
Gate V4 is maintained at a bias of −1 V, and the eigenstates are shown in figure 8.19.
In panel (a), V2 is also set to be −1 V, and the density is clearly in the left dot. On the
other hand, when V2 is reduced slightly, to −1.01 V, then the density switches to the
right-hand dot, as shown in panel (b). It is clear that the small perturbation potential
causes the switching between these two possible states, which can represent the two
possible values for the bit.
Figure 8.18. (a) Assumed gate pattern for the double-dot simulation. (b) Self-consistent potential profile for
the dots. Reprinted from [28].
Figure 8.19. In panel (a) with V2 = −1.0 V, the wave function is localized in the left dot. But, when V2 = −1.01
V, panel (b) shows that the wave function is localized in the right dot. Reprinted from [28].
8-23
The potential above is a double-well potential, with each well appearing as a two-
dimensional harmonic oscillator, although considerably deformed from the text-
book examples, such as discussed in appendix C. If the two wells are exactly the
same, then a given eigenstate in one well will have an exact equivalent in the other
well. These two states will have the exact same energy, and this is not allowed by the
exclusion principle. As a result, there will be a coupling between the wells, which can
be controlled by the blue finger gates in figure 8.18(a). This coupling leads to an
energy splitting corresponding to a lower-energy bonding interaction and a slightly
higher-energy anti-bonding interaction. In each of these new states, the wave
function has an equal fraction in each of the quantum dots. If we were to localize
an electron in the left dot, it will oscillate back and forth between the dots. In order
to localize the wave function in only one of the dots, they have to be not exactly
equal. This inequality is provided by the discretization of the system in the
simulations for figures 8.18 and 8.19. By slightly modifying the left potential, the
localization switches to the other dot. In principle, the various potentials can be
carefully adjusted so that the two states become merged, so that the wave function
oscillates between the two dots, and it is thought that this oscillation can be used to
create a qubit with these two dots.
The idea of using spin to create a qubit with quantum dots was apparently first
proposed by Loss and DiVincenzo [29]. They developed a detailed scheme to achieve
quantum computation with a pair of single-electron quantum dots. The qubit is
realized with the spin of an excess electron in one of the quantum dots. Two-qubit
quantum-gate operation is achieved by merely adjusting the barrier existing between
the two dots (as, for example, adjusting the blue barriers in figure 8.18(a) above). If
the potential barrier is high, the two qubits, one in each dot, do not interact. If the
potential is lowered, then the two qubits are allowed to interact and the spins are
affected by a coupling due to the spin–spin coupling energy. Many approaches burst
upon the scene after this seminal paper.
We can illustrate the double-quantum-dot approach with some relatively recent
work [30]. Here, the structure is fabricated on a GaAs/AlGaAs heterostructure with
Schottky barrier metallic gates used to define the active region, which may be
glimpsed below the magnetic material in figure 8.20(a). As above, this is a gate-
defined double quantum dot, to which has been added a split Co micromagnet (the
yellow regions in this figure). To the outside of the active double dot, a QPC has
been added (indicated in the figure by the current path arrow). When the dot on the
right has an electron (or more) in it, the Coulomb interaction causes the opening for
the QPC to narrow, and the resulting decrease in current can signal the charge state.
This sensor can be used to map out the stability regime for the double-dot system,
shown in figure 8.20(b). Essentially, this is equivalent to the red square region in
figure 8.14(b), except that the applied source–drain region opens the simple pair of
triple points in the triangular structure seen here. Here, NL and NR denote the charge
in the left and right dots. The single-spin rotations and qubit interactions are carried
out in the (1,1) state (lower right of the stability diagram). To rotate each spin,
electrically driven spin resonance is used, although the Co nanomagnets provide a
spatially varying magnetic field due to the shape of these magnets. The spin
8-24
Figure 8.20. (a) A false color image of the device showing the two quantum dots and the electron spin. (b) The
stability diagram around the area of interest. (c) Spin resonance signals for the left and right dots.
(d) Measurement cycle for controlled spin rotations. (Reprinted with permission from [30]. Copyright 2011
the American Physical Society.)
resonance signal is seen in panel (c) of this figure. Panel (d) shows the shape of the
double-well potential, which is controlled by the various gate electrodes (as depicted
in figure 8.18(a)). During operation, a static magnetic field of 2 T in the left dot and
1.985 T in the right dot is applied, and the dot is excited with a microwave signal of
11.1 GHz. As indicated in the figure, various gate voltage pulses are also applied to
sequence the interaction. For stage A in the figure, the two dots are set in the spin
blockade phase, where the spins cannot interact with both spins either up or down.
In stage B, the dots are isolated from one another and one of the spins is rotated by
applying a pulsed microwave signal. Finally, at the second stage A, to the right, the
spin can be read out with the QPC sensor in which the signal is proportional to the
probability that the two spins are oppositely polarized. This demonstrates the ability
to control the spin polarization. Finally, two-qubit gates are used to demonstrate
operations via the interdot spin exchange operation, as suggested by Loss and
DiVincenzo [29].
If all of this seems to be a great deal of work to operate a pair of qubits, it is. Yet,
this technology is still in its infancy, and experiments such as this demonstrate the
feasibility of the scientific basis. In the experiments, the authors demonstrated that
8-25
Figure 8.21. A qubit using a pair of phosphorus atoms, which have been implanted into the silicon substrate.
(a) Schematic of the device showing the control gates and readout SET. (c) Two choices for the qubit state.
(b) The barrier (B) and symmetry (S) gates which control the potential of the two quantum wells. (Reprinted
an all electrical two-qubit gate could be realized and it could perform simple
operations.
It is also possible to marry the previous double-dot qubit with the single-atom
transistor discussed in the last section. A double-dot qubit has been realized using
two single phosphorus atoms recently [31]. The double-well potential is created from
the local potential well created by the individual P atoms. This qubit is a charge-
based qubit in which the charge resides on one of the atoms, while the other is
ionized, thus creating a P–P+ charge system. The qubit is manipulated by a set of
gates and a SET, operated at a high ac frequency, is used for the readout. The
structure is shown in figure 8.21. The device is created in a Si–SiO2 with the pair of P
atoms implanted into the Si substrate. In the figure, the structure, the control gates,
and the SET are shown schematically. The control gates are used to move the charge
from one P atom to the other, and this position can be sensed by the SET.
More recently, the use of a single phosphorus atom has been suggested to be
adequate for a qubit which utilizes the spin of the electron on the atom [32]. The
device is subject to an applied in-plane magnetic field of 1 T which provides well-
defined spin-up and spin-down states of the electron. Transitions between these
states are achieved through the use of a microwave spin resonance signal provided
by an on-chip transmission line, and these transitions can be measured with a SET.
8-26
In the read/initialization stage, a spin-up electron will tunnel into the SET, to later be
replaced by a spin-down electron. This causes a pulse of current through the SET,
while a spin-down electron remains trapped on the donor. During the control stage,
the electron spin states are placed well below the SET state, and cannot interact with
it, while the spins are manipulated by the microwaves through electron spin
resonance.
The case for nearly all of the above dopant-based qubits is that the dynamic
variable is the spin itself, just as for the double dot qubits described earlier. It is
possible to consider a hybrid qubit in which the dynamical variable depends upon
both spin and charge. The hybrid qubit requires neither nuclear-state preparation nor
micromagnets for control, and becomes considerably more amenable to systems [33].
This has been studied by these latter authors for the double quantum dot, which is
defined lithographically by surface gates (as shown in panel (a) of figure 8.22). Here, a
more traditional double dot system, similar to those in SETs, is used for the qubits.
Three electrons are used in the system, with the gates tuned so that there are two
electrons in the left dot and one in the right dot to define the (2,1) state. Changing the
voltage on gate L to raise the energy difference between the dots favors the opposite
situation and the state (1,2). The singlet state in the right dot, state (2,1), is taken to be
the ∣0⟩ state. The triplet state in the right dot, the (1,2) state, is taken to be the logical
∣1⟩. The presence of the extra electron means that the hybrid states of the dots are not
pure singlet or triplet and fast electric field techniques can be used to manipulate the
qubit in either X or Z rotations on the Bloch sphere.
The coherence can be extended when the hybrid qubit is composed of three
electrons in a double quantum dot [34]. Similar surface control gates have been used
in the pure spin qubit case as well [35], where multiple dots are used to suppress
charge noise. This was studied for a pair of singlet-triplet qubits, each of which was
composed of two electrons in a double quantum dot. More recently, the importance
of the spin–orbit interaction in these gate-defined dots has been studied [36, 37], and
the impact of atomic-scale structure in the energy dispersion on decoherence was
investigated [38]. In fact, figure 8.22 is from this last paper, and shows the double
quantum dot qubit and its operation.
The singlet-triplet qubit is attractive for many reasons, as it seems to have a higher
fidelity for single qubit operations. However, for two qubit operations leading to
entanglement, it can be problematic in a sense that there are a range of operations
over which the interaction can occur and this leads to a preferred set for good
performance [39]. Spin qubits formed of quantum dots in the Si/SiO2 system have
been formed in a manner to utilize from 1 to 3 electrons in the dots [40] for this
purpose. Then, pulsed electron spin resonance is used to exercise coherent control
over the qubit. With these structures, these authors then demonstrated the achieve-
ment of valley splitting (normally, in the MOS system, the six ellipsoids of the
conduction band break into a two-fold and a four-fold set due to the inversion
potential, and one seeks to further split the lower lying two-fold set) [41]. The role of
disorder on this valley splitting has also been studied [42].
8-27
Figure 8.22. Experimental and theoretical setup, and resulting energy dispersions. (a) A scanning electron
microscope image of a device nominally identical to the one used in the experiment. The gate voltages are
tuned to form two quantum dots, located approximately within the dashed circles, where red dots represent
electrons in a (1,2) charge configuration. (b) Schematics of the four pulse sequences employed in the
experiments. The three-step sequence is used to obtain the qubit frequency data fQ plotted in (c). The
Ramsey pulse sequence is used to obtain the qubit frequencies and Ramsey decay rates. The Rabi and Larmor
sequences are used to obtain Rabi fringes and fQ. (c) The experimentally measured fQ of a quantum-dot hybrid
qubit as a function of detuning ε (black dots). The solid red line shows the results of a least-squares fit of the
data to theory assuming ε-independent model parameters. Inset: The three energy eigenstates obtained by
diagonalizing the theory Hamiltonian. (d) A schematic cartoon illustrating the theoretical model for both the
quantum-dot hybrid qubit and the single- electron charge qubit, with the low-energy basis states ∣L0 〉, ∣R0 〉,
and ∣R1〉, as appropriate for the hybrid qubit. In our 2D tight-binding simulations, atomic-scale step disorder is
introduced into the top interface as shown here. The lateral confinement potential is taken to be biquadratic,
and the two dots are offset by energy ε. The interdot tunnel couplings are labeled 1 and 2, and we refer to R as
the ‘valley splitting,’ although ∣R1〉 may involve a valley-orbit excitation. (Reprinted with permission from [38].
8.4 The Josephson qubits

Kammerlingh Onnes, after liquefying helium at very low temperatures, spent most
of his time examining the properties of various materials at this low temperature of
4.2 K. In studying Hg, he found a totally unexpected result, in that as he cooled Hg
below the temperature of liquid helium, the resistance dropped dramatically. At 3 K,
the resistance was less than 10−6 ohm. This phenomena is now known as super-
conductivity, and has been observed in a significant fraction of the known elements.
8-28
Table 8.1. Superconductor transition temperatures.
Element Tc (K) Compound Tc (K)
Pb 7.2 Nb3Sn 18.05

V 3.72 V3Ga 16.5
Ti 0.39 V3Si 17.01
Nb 9.1 NbN 16.0
Ta 4.48 InSb 1.9
Hg 4.15 Nb3Al 17.5
Zn 0.85 YBa2Cu3O7 92
Sn 3.72 HgBa2Ca2Cu3O8 134
The onset of superconductivity is considered to be a phase transition, and the critical

temperature below which it occurs is called the transition temperature Tc. If one plots
the resistance as a function of temperature, there is a clear drop in the resistance by
orders of magnitude at this transition temperature. Just above the transition
temperature, there is a gradual drop in resistance, that gives a ‘rounding’ to the
curve, and this is caused by thermal fluctuations at the transition temperature. These
fluctuations are believed to be the fact that not all of the electrons are paired except
at absolute zero temperature. Above Tc, on average none of the electrons are paired,
so that the fraction that are paired is a function of temperature. In this region of
enhanced conductivity above Tc, it is thought that small regions of the material are
beginning to become superconducting, with the entire sample becoming so at further
reductions of the temperature.
In general, metals which have very high conductivity do not become super-
conductors. On the other hand, metals which are poor conductors, such as Pb, Ta,
Nb, Hg, and so on, become quite good superconductors. Some of the transition
temperatures are given in table 8.1. Also shown are a couple of members of a new
class of materials known as high temperature superconductors, which have
transition temperatures of tens of degrees Kelvin, up to ∼200 K at high pressure.
These latter materials tend to be cuprates and ceramics.
A magnetic field can be used to destroy the superconductivity. That is, there is a
critical magnetic field Hc above which the superconductivity vanishes. Moreover, it
is found that this critical field varies with temperature as
⎡ ⎛ T ⎞2 ⎤
Hc = Hc 0⎢1 − ⎜ ⎟ ⎥ , (8.31)
⎢⎣ ⎝ Tc ⎠ ⎥⎦
where Hc0 is the critical field at absolute zero of temperature. Comparing different
material, it is found that in general a higher critical temperature will lead to a higher
critical magnetic field. This critical magnetic field will also limit the amount of
current that the superconductor can carry, since the current gives rise to a magnetic
field by Ampere’s law. However, there are other materials, such as the compounds in
table 8.1, where the superconductivity does not end abruptly. In these materials, a
8-29
lower critical magnetic field signals the beginning of the process and an upper critical
field signals when superconductivity has finally gone away. These materials are
sometimes called hard superconductors.
In superconductivity, it is found that the electrons would pair up as Cooper pairs
[43]. Normally, electrons repel each other due to the Coulomb interaction between
them. However, in metals there are typically something like a few times 1022
electrons cm−3. Such a high electron concentration usually heavily screens the
Coulomb interaction, so that it is a relatively weak interaction, especially if the
electrons are not fairly close to one another. Cooper hypothesized that because of
this weakness, electrons could actually interact with each other if there were another
positive interaction. He suggested that one electron could interact with the atoms of
the crystal in a manner that distorted the local atomic potential, creating a potential
well into which a second electron could be drawn. He thought the process would
work best if the two electrons had opposite spin. The combination of the two
electrons, via the atomic interaction, creates the Cooper pair. Because the two
electrons have opposite spin, the net spin of the Cooper pair is zero. This makes the
Cooper pair into what is called a composite boson. Bosons are not required to satisfy
the Pauli exclusion principle, so there is no limit to the number of bosons that can
exist in any quantum state. Moreover, when we form the Cooper pair, it lies in a
lower energy state—this lowering of the energy is the pair formation energy, which
we called Δ per electron. The result of all of this process is observable by a gap of 2Δ
that opens in the energy spectrum at the Fermi level. A Cooper pair must gain this
energy to transition above the gap and become a pair of free electrons. The
interesting point is that the two electrons in the Cooper pair need not be close to
one another. Instead, it is generally thought that they can be some 100–400 nm
apart, a length that is referred to as the coherence length. This means that there are a
great number of other electrons near to the Cooper pair, or at very low temperature,
the large number of Cooper pairs are all intertwined with one another. The energy
gap that opens is also temperature dependent, varying as
⎛ T⎞
1/2
EG = EG 0⎜1 − ⎟ , (8.32)
⎝ TC ⎠
where this gap is the 2Δ mentioned above.
It has been suggested that the Cooper pairs are the basic unit of superconductivity
[44, 45]. In the BCS theory, named for these three authors, the critical temperature is
found to depend upon the density of states at the Fermi energy and the strength of
the interaction between the lattice and the electron that leads to forming the Cooper
pairs. Because all the states are full below the energy gap, one would not expect
conduction, but the nature of the bosonic Cooper pairs seems to indicate they can
move through the lattice without dissipation. That is, they cannot be scattered, so
move dissipation free. And this seems to be the case. If one creates a coil of
superconducting wire and induces a current through the wire, and then shorts the
leads together, this current will continue to flow without dissipation. This current is
known as a persistent current, and is the heart of superconducting magnets.
8-30
8.4.1 Josephson tunneling

The Josephson junction [46] is a tunnel junction in which the materials on either side
of the insulator are superconductors. In the Josephson junction, these superconduct-
ing leads produce zero loss in dc operation. The formation of the Cooper pairs
lowers the energy of the entire electron gas below the Fermi energy, and this leads to
a gap opening at the Fermi energy, as pointed out above.
Normally, with Josephson junctions, the tunneling is carried out by these Cooper
pairs, but both the Cooper pair tunneling and single-electron tunneling can occur
under the appropriate conditions. The Josephson junction itself operates so that, no
tunneling occurs until eVa > 2Δ at low temperature. As the temperature increases,
both unpaired single electrons and Cooper pairs exist below the gap, and the normal
(unpaired) can tunnel giving a very small current. As with normal tunneling, if the
barrier is relatively thin, say 1–2 nm, then the superconducting wave functions can
extend through the barrier, so that they interact with those wave functions on the
other side. An unusual effect in the Josephson junction is that this coherent mixing
of the wave functions on either side of the junction can produce a current at zero
bias! This is termed the dc Josephson current. This current has a fixed magnitude
that can be modulated by a magnetic field as
⎛ eBA ⎞
IJ (B ) = IJ 0cos⎜ ⎟, (8.33)
⎝ h ⎠
where BA = Φ is the flux flowing through the area of the junction. As before, h/e is
the quantum unit of flux, so that the flux can be quantized in this quantum system.
The interesting aspect is that the current peak IJ0 is proportional to the single-
particle tunneling coefficient through the junction [47]. We note that if a bias is
applied, nothing much happens to the normal tunneling curve, except a few unpaired
electrons may tunnel through. But, this voltage has a significant effect on the
coherent flux of the Josephson current (8.33). This follows from the relationship
dΦ
ℏ = 2eVa , (8.34)
dt
where the factor of 2 arises from the two electrons in the Cooper pair. Now, what
this means is the associated flux is changed by the voltage and this produces an ac
signal, known as the ac Josephson effect. This leads to the important relation
ℏω = 2eVa . (8.35)
Hence, the Josephson junction can be a microwave source. It has also been noticed
that there is a peak in the spectrum at ω = 4Δ/ℏ, which is called a Riedel peak in the
response of the junction [48].
8.4.2 SQUIDs
The Superconducting QUantum Interference Device (SQUID) is basically a
variation of the circuit in figure 8.4, but with current bias instead of voltage bias,
8-31
as shown in figure 8.23 [49]. The two capacitors are Josephson tunnel junctions, and
the quantum dot indicated may not be a microscopic dot at all (but we return to this
below). This is known as a dc SQUID and works with the dc Josephson current
(8.33). Shown is a bias current Ia that flows in from the left and out to the right. This
current splits between the two arms of the interferometer, so that Ia/2 flows through
each capacitor. An important point is that all of the leads are made from
superconducting material. Initially, there is no flux through the superconducting
loop that contains the two tunnel junctions. If we induce a flux in the loop that is less
than Φ0/2, this creates a circulating current, which we designate as Ic, that flows in
the loop. The flux can be created by coupling the loop to an external loop that
provides the flux. Thus, for example, the currents through the two capacitors
become
1
I1 = Ia + Ic
2 (8.36)
1
I2 = Ia − Ic .
2
When the flux is increased to > Φ0/2, the induced circulating current has to change
sign according to equation (8.33). And, when the flux reaches a full flux quantum,
the cycle repeats. As a result, the SQUID is very sensitive to magnetic fields and can
measure fields as small as a few times 10−18 T, so are used in a great many
applications [50]. As the currents oscillate with the flux, the induced voltage, that
arises from the junction resistance and the loop inductance, becomes
R
ΔV = ΔΦ . (8.37)
L
There is also an RF-SQUID which utilizes only a single Josephson tunneling
junction and the superconducting loop [51]. However, most applications of the
SQUID for quantum computing applications use the two junction (or more
junctions) version of the dc SQUID. However, in the drive to make the qubits
small, one actually introduces the quantum dot, as shown in figure 8.23. That is,
Figure 8.23. A DC SQUID showing the bias current and the measured voltage that arises from a variation in
the flux coupled into the loop.
8-32
there is a long part of the loop and a short part of the loop, the latter of which forms
the quantum dot, along which the capacitors also show single-electron(pair)
tunneling. Hence, there is a competition in various energies in the system. There
is the single-electron(pair) energy
Ec = e /2C . (8.38)
(we may assume the two junctions are equal capacitances), where the two arises for
the Cooper pair tunneling. There is also the Josephson energy arising from the
current (8.33) which becomes
EJ = Φ0IJ 0 /2π . (8.39)
The operation of the qubit and the SQUID depends upon the relative size of these
two energies.
To explain this further, one needs to distinguish two types of qubits: microscopic
and macroscopic [52]. The microscopic qubit is based upon an internal charge or
spin, much like the Si qubits of the previous section, or upon two natural levels as in
an atom. Macroscopic qubits are based quantum levels representing collective
degrees of freedom (often called macroscopic quantum states) like the persistent
current in a superconducting ring [53] or excess charge on a superconducting island
[54, 55]. Hence, one might think that the macroscopic form will be physically larger,
but that can be totally misleading. When superconducting components are used in
electrical circuits, then these circuits become quantum objects, which can then be
used in either type of qubit. The pursuit of superconducting qubits can be a Cooper
pair box charge qubit, the persistent current flux qubit, or a hybrid charge-flux qubit
[52, 56].
8.4.3 Charge qubits

The charge qubit is composed primarily of the Cooper pair box, where the box is
formed by a pair of small Josephson junctions so that single electron tunneling
governs the excitation of the box itself, which is isolated by the two junctions. A
single Cooper pair box [57] is a unique artificial solid-state system [54]. Although
there are quite a few electrons on the island (box), they all form Cooper pairs in the
superconducting state and then condense into a single macroscopic quantum state.
This state is separated from the normal electrons by the superconducting gap Δ,
discussed above. The only low-energy excitations arise from Cooper pair tunneling
when the gap is larger than the charging energy EC = e2/2C. On the other hand, if the
charging energy is larger than the gap energy, and the thermal fluctuations are
suppressed, the system can be considered to be a two level system in which the lowest
two energy states differ by one Cooper pair. The separation of the two levels of
interest can be controlled by an additional gate voltage. In figure 8.24, we show such
a charge qubit from the work of Lehnert et al [58].
When the island is small, the charging energy dominates the qubit operation, but
one can write a simple Hamiltonian as [54]
8-33
Figure 8.24. (a) An SET electrometer. The device is made from an evaporated aluminum film (light gray
regions) on an insulating SiO2 substrate (dark gray regions) by the technique of double angle evaporation,
which gives the double image. The aluminum has BCS gap Δ/kB = 2.4 K. (b) A circuit diagram of the box and
RF-SET electrometer. The SET gate voltage Vge, the 500 MHz oscillatory bias, and the dc bias (RFin + dc)
determine the electrometer’s operating point. The charge on the box is inferred from variation in the amount of
applied RF power that is reflected (RFout) from the SET electrometer, which is a sensitive function of SET’s
conductance. The tunnel junctions (crosses in boxes) are characterized by a junction resistance RJ and
capacitance CJ, which enter the box’s Hamiltonian through CT = CC + 2CJ + Cg and RT = RJ/2. (Reprinted
⎛1 ⎞
H = 4EC (N − ng )2 − 2EJ cos(ϑ)cos⎜ φe ⎟ , (8.40)
⎝2 ⎠
where EJ is the Josephson energy, ng is indicative of the gate voltage (through the
capacitor coupling), φe is the external flux and θ is the difference in the Josephson
phases of the two junctions. With the two junctions, the box can be tuned with the
external flux passing by the reservoir, much like a SQUID, which is discussed below.
When the charge qubit is shunted by a capacitor, as a method of reducing current
noise, the device has been called a transmon [59]. The shunt capacitor also reduces
the charging energy and hence increases the size of the Josephson energy relative to
the charging energy. Measurement and control of the box qubit is commonly done
by means of microwave resonators with the techniques usual in quantum electro-
dynamics, as in other superconductor qubits [60]. As a result, it is possible to couple
a pair of transmons to photons and produce nonlinear optical effects such as photon
blockade [61].
8-34
8.4.4 Flux qubit

One of the most interesting aspects of circuits is that a resonant circuit can be a
quantum oscillator, much akin to the harmonic oscillator [1]. In the absence of
dissipation, the Hamiltonian can be written as
q2 Φ2
H= + , (8.41)
2C 2L
where the charge q and flux Φ are the conjugate variables, and satisfy a commutator
relation with one another. Of course, to reach the quantum limit, we must work at
low temperature, which naturally makes a connection with superconducting circuits.
In the presence of dissipation, we need to have a good quality factor Q (= ωτ ≫ 1).
Hence, when we couple a Josephson junction to a superconducting ring, we obtain
the persistent current flux qubit, sometimes called an RF superconducting quantum
interference device (RF-SQUID, discussed above). Here, we need to have EJ ≫ EC
[53, 62], so the charging energy cannot be too large. The eigenstates of the ring
represent two counter-rotating persistent currents, corresponding to a fixed number
of flux quanta (h/e) in the loop. The inductance of the ring gives rise to a parabolic
potential, like the harmonic oscillator, and adding the Josephson oscillating
potential provides the needed nonlinearity to separate off the lowest states from
the linear chain, as discussed earlier. For the two levels of the qubit, the Hamiltonian
can be written as [54]
H = 4EC n 2 + EL(φ − φe + φintσz )2 − EJ cos(φ), (8.42)
where the interaction term arises from the effect of the external bias flux on the two
persistent current states. When φe = Φ0/2, the lower part of the potential is a
symmetric double well creating nearly degenerate ∣L⟩ and ∣R⟩ states which lead to
bonding and anti-bonding hybrids of these two states. These latter two states are
created by the macroscopic tunneling through the Josephson junction which couples
the two persistent current states, and these hybrid states provide the two levels of the
qubit. The Wigner function in number-phase representation shows that the state of
the system evolves into a quantum superposition of two coherent states which clearly
demonstrate interference and negative values for the Wigner function description of
the system. Yet, when the system is represented in the number-phase coordinates, the
Wigner function evolves in a classical manner [62]. This ring-junction system was
studied further to investigate the role of dissipation on the evolution of the Wigner
function description of the system [63]. In this latter case, the dissipation is
incorporated by coupling the system to a reservoir, and this expanded system is
then projected back onto the reduced density matrix for the ring-junction system.
They conclude that the two coherent states survive even in the presence of
dissipation, at least for weak dissipation.
The flux qubit can also be prepared with a three junction ring, much like the
hybrid qubit of the next section. This qubit can then be coupled to a transmission
line resonator to produce cavity QED interactions. The capacitance between the
qubit and the resonator can be controlled by varying the width of the capacitance
8-35
Figure 8.25. Circuit diagram of the quantronium qubit. (Reprinted with permission from [67]. Copyright 2006
the American Physical Society.)
line, so that the coupling depends upon the number of qubits placed in the overall
circuit [64, 65]. We note that this can also be done with single junction qubits [66].
8.4.5 The hybrid charge-flux qubit

If we add a second Josephson junction in the ring of the flux qubit (as in the charge
qubit), we can have both the charge box between the two junctions and flux coupled
through the ring. If the Coulomb energy EC dominates the Josephson energy, we
have the charge qubit limit. If the Josephson energy dominates the Coulomb energy,
we reach the flux qubit limit, but if we have EJ ⩾ EC, we are in the charge-phase, or
hybrid charge-flux qubit regime [52]. In many situations, the system acts as a two-
level atom, which is called a quantronium [55]. A quite similar version of the
quantronium is shown in figure 8.25 [67]. The two lower Josephson junctions create
the charge box part of the circuit, but the ring from these to the larger third junction
creates the supercurrent, flux reservoir. This loop is coupled (not shown) to an
external flux circuit that is used to vary the flux enclosed in this ring. The box charge
is varied through the coupling capacitor and gate voltage. A pair of capacitors is
8-36
used to reduce the ac noise, while the current source provides the current drive to the
qubit itself. The Hamiltonian for the circuit is given as [67]
⎡ ⎛φ + γ ⎞ ⎤
H = EC (N − ng )2 + EJ ⎢1 − cos⎜ ⎟cos(φ )
⎥⎦
⎣ ⎝ 2 ⎠
(8.43)
Q2
+ + EJ 0(1 − cosγ ),
2C
where the different parameters are defined in the figure in terms of the fluxes through
the various junctions with φ = (φ1 + φ2)/2 and ng has its former meaning. The fluxes
are normalized to the flux quantum. The lowest and first excited states of the
quantronium are taken to be the ∣0⟩ and ∣1⟩ states. In this configuration, the circuit is
operated so that the applied flux Φx is set to zero and the normalized gate charge is
set to 1/2 so that the qubit levels are separated by the Josephson junction coupling
energy. Then the Hamiltonian can be written as
⎡ ⎛ γ ⎞⎤
H ′ = EJ ⎢1 − cos⎜ ⎟⎥σz + EC 0N 2 + EJ 0(1 − cosγ ), (8.44)
⎣ ⎝ 2 ⎠⎦
where EC0 = 2e2/C and N is the charge operator for the large Josephson junction.
This Hamiltonian describes a two-level system, with the levels separated by an
energy given by the first term on the right of (8.44). The phase in this term provides
the coupling between the qubit and the readout junction.
As discussed above, the charge-flux qubit is operated normally at this charge
degeneracy point and is control and switched via microwave pulses in order to
demonstrate the presence of Rabi oscillations between the two qubit states. It often
can have a fairly long coherence time, and thus can perform hundreds of well-
controlled single qubit gate operations. In simulations of the circuit, it is found that
during the readout pulse for the circuit of figure 8.25, the coherence time is of the
order of 0.2 ns after the bias current has been applied (the role of the bias current is
to diphase the qubit). The simulation results have been compared with the
experimental data obtained by Vion et al [55].
8.4.6 Novel qubits

It is possible to also make Josephson junctions by using a semiconductor as an
insulator. If the semiconductor can undergo induced superconductivity from a
nearby superconductor, then there is the possibility of using gate control on the
junction. This can become more interesting if there is an interplay between induced
superconductivity, spin–orbit coupling, and topological edge states [68, 69]. In this
quest, quasi-two-dimensional semiconductors have been of intense interest for the
creation of gate controlled junctions, that may have the possibility to create
topological states of interest. The interest in topological quantum computing lies
in the fact that the topological states are protected and such qubits should be free
from random noise. A novel three junction Josephson device, which utilizes a
semiconductor electrode and is gate switchable, has recently been proposed in a
8-37
quest for such topological states [70]. The device is depicted in figure 8.26. The
device uses an InAs quantum well and a 10 nm aluminum superconducting layer to
generate the proximity superconductivity in the InAs. The structure is complex, as
the lattice constants of the InP substrate and the InAs quantum well are dramatically
different. The first layer grown (starting from the substrate) is a graded InxAl1−xAs
layer, where the grading begins at x = 0.52 (lattice matched to the substrate) up to
x = 0.81. Then, a 25 nm In0.81Ga0.19As/In0.81Al0.19As superlattice layer is grown,
followed by 100 nm of In0.81Al0.19As with Si δ-doping at 2 × 1012 cm-2. Then, a 6 nm
bottom barrier of In0.75Ga0.25As is followed by the 7 nm InAs quantum well and a
10 nm In0.75Ga0.25As top barrier. The sample has a measured carrier concentration
of about 1012 cm-2 in the quantum well and a mobility of about 30 000 cm2 V−1 s−1.
Electron-beam lithography and wet etching are used to finish the device fabrication.
The authors performed dc current-biased measurements in a dilution refrigerator
with a base temperature of T ∼ 14 mK. The current applied between terminals 1 and 0
(see figure 8.26) is referred to as I1, the current applied between 2 and 0 as I2, and
the current between 1 and 2 as I12. By making the top-gate voltage more negative, the
electron density in the interstitial 2DEG is gradually depleted, which tunes the
Figure 8.26. (a) False-color scanning electron microscope image of gated three-terminal junction with
measurement schematic. Blue areas are aluminum and grey areas are etched to the insulating buffer layers
to create the device mesas. The Ti/Au top gate (yellow) overlays the 200-nm-wide Y-shaped junction formed by
selectively etching the Al layer only (dotted lines). (b) Cross-sectional view along the purple horizontal line in
(a) (not to scale). (c) Gate dependence of the switching current when biasing between terminals 1 and 2
showing pinch-off occurring at Vg ∼ −4.5 V. (d) Two-terminal I1 versus V1 curve with I2 = 0 showing hysteresis.
8-38
switching current from ∼560 nA to 0 [panel (c)]. As shown in panel (d) the junctions
exhibit hysteresis with respect to the current sweep direction. Hysteretic Josephson I–V
curves can occur either due to the presence of a shunt capacitance (as described by the
resistively and capacitively shunted junction model), or due to Joule self-heating which
sets on as the junction becomes dissipative. Although 2DEG-based lateral Josephson
junctions can in principle have large shunt capacitances due to conducting underlayers
in the heterostructures or capacitive coupling of each superconducting terminal to the
top gate, we estimate that the capacitance in our device is small [70].
The authors then mapped out the device performance by varying the different
excitations at the three terminals. The ability to find novel distinguishing features,
that are due to mesoscopic superconducting transport in such a multiterminal
Josephson junction is hoped to prove useful in future studies aiming at topological
effects [70]. These effects could provide a route forward in the development of qubits
with greater defense against decoherence.
Problems
1. Consider a trapezoidal potential well that is V0 deep on the left-hand side and
V1 deep on the right-hand side. The well is located in the region 0 < x < a.
Using the Wentzel–Kramers–Brillouin (WKB) approximation, determine the
bound states within the well. If V0 = 0.3 V and V1 = 0.4 V, with
m* = 0.067m0 and a = 5 nm, what are the bound state energies?
2. When a voltage (Vsd) is applied across a QPC it is typical to assume that the
quasi-Fermi level on one side of the barrier is raised by αeVsd while that on
the other drops by (1 − α)eVsd, where α is a phenomenological parameter
that, in a truly symmetrical structure, should be equal to 1/2. If we consider a
device in which only the lowest subband contributes to transport, then the
current flow through the QPC may be written as:
2e ⎡ ⎤
Isd =
h⎣
⎢∫L T (E )dE − ∫R T (E )dE ⎥⎦,
where T(E) is the energy-dependent transmission coefficient of the lowest
subband and ‘L’ and ‘R’ denote the left and right reservoirs, respectively. If
we assume low temperatures, we can treat the transmission as a step function,
T(E) = u0(E − E1), where E1 is the threshold energy for the lowest subband.
(a) Write this integral with limits appropriate to determine the current.
(b) Use this information to obtain an expression for the current flowing
through the QPC when the source drain voltage is such that both reservoirs
populate the lowest subband, and when it populates the subband from just
the higher-energy reservoir.
3. Consider an AlGaAs–GaAs–AlGaAs RTD with barrier widths of 5 nm and
a well width of 5 nm. Assume the WKB approach used in the previous
problem. (a) Assuming that the barrier height is 300 meV, estimate ΓL and
ΓR, the effective tunneling rates through the two different barriers, under
8-39
conditions corresponding to the first tunneling resonance. (b) Calculate the

fraction of the current that is coherent if the phase relaxation time is 1 ps.
4. Consider a SET whose individual tunnel barriers have capacitances C1 = C
and C2 = 2C and whose gate capacitance Cg = C/3. You may assume that
any background charge can be taken to be zero in this problem. (a) Write the
set of four equations that can be used to define the charge-stability diagram
for this device. (b) Plot the charge-stability diagram for this device and
indicate the regions where current flow is Coulomb blockaded. (c) An
important parameter for any transistor is its voltage gain, i.e., the change
in source–drain voltage arising from a change in gain voltage. For the SET
considered here, what can you say about its voltage gain?
5. Consider a parabolic quantum dot implemented in a GaAs 2DEG with
ℏω0 = 2 meV, where ω0 is characterized by the harmonic oscillator itself (See
appendix G). Now consider the situation in which 12 electrons are present in
this dot and assume also that spin splitting of any electron states can be
neglected. Plot the energy levels of the dot as a function of magnetic field (for
0 < B < 4 T) and indicate the variation of the highest filled electron state as a
function of magnetic field in the situation where the electrons occupy the
ground state of the dot.
Appendix F Klein tunneling

In classical semiconductors, a potential barrier which is higher than the energy of the
incident particle generally blocks transmission of the particle through the barrier.
However, if the barrier has a finite thickness, then it is possible for the particle to
tunnel through the barrier [1]. This is because, with the wave interpretation, the part
of the wave that penetrates the potential barrier does not fully decay before the back
edge of the barrier is reached. Nevertheless, the transmission probability decays
exponentially both with the height of the potential barrier and with its thickness. In
the relativistic world, however, the Klein paradox leads to a situation in which an
incoming electron can penetrate a potential barrier if the height exceeds the rest
energy mc2 [71]. When this happens, the transparency of the barrier depends only
weakly on the barrier height and actually increases as the barrier height increases.
The physics of the process is that the penetrating electron can couple to positrons
under the barrier to affect the transmission, and matching between the two sets of
wave functions leads to the high transparency [72].
Graphene has the Dirac bands and the zero energy gap leads to zero rest energy.
Consequently, any barrier height would lead to the same behavior as the Klein
paradox, a result that has been worked out for the chiral particles in graphene [73].
Thus, under a wide range of conditions, a potential barrier poses no obstacle to an
electron or hole in graphene, and the concept of a Schottky barrier just does not
work well, as discussed earlier in the chapter. Here, we follow the treatment of [74] to
illustrate the conditions under which Klein tunneling appears. The basic premise is,
of course, that the electrons and holes in graphene accurately follow the properties of
the electrons and positrons that Klein analyzed. Let us consider a barrier of height
8-40
V0, that exists in the region 0 < x < L. An electron at the Fermi energy approaches
the barrier with wave vector kF and with an angle ϕ defined by the longitudinal and
transverse components of the Fermi wave vector through
kx = kF cos(ϕ), ky = kF sin(ϕ). (F.1)
As in the classical tunneling problem, it is assumed that the barrier has infinitely
sharp edges so that no disorder is introduced by these edges. The valley degeneracy
gives the equivalent Dirac spinor of two wave functions corresponding to the
pseudo-spin of the composite wave function. These two wave functions are written
as [70]
⎧(e ikxx + re−ikxx )e ikyy , x < 0,
⎪
ψ1 = ⎨(ae x + be x )e y ,
iq x − iq x ik y
0 < x < L,
⎪ −ikxx ikyy
⎩te e , x > L,
(F.2)
⎧ s(e ikxx+iϕ + re−ikxx−iϕ )e ikyy , x < 0,
⎪
⎨
ψ2 = s′(ae iqx x + i ϑ
+ be − iqx x − i ϑ ik
)e , yy
0 < x < L,
⎪ −ikxx+iϕ ikyy
⎩ ste e , x > L.
Here, we use
(E − V0)2
qx = 2 2
− k y2 ,
ℏ vF
⎛ ky ⎞ (F.3)
ϑ = arctan⎜ ⎟ ,
⎝ qx ⎠
s = sign(E ), s′ = sign(E − V0).
The angle ϑ is the refraction angle of the wave. Matching the various coefficients
lead to the reflection coefficient
sin(ϕ) − ss′ sin(ϑ)
r = 2ie iϕ sin(qxL ) . (F.4)
ss′[e−iqxL cos(ϕ + ϑ) + e iqxL cos(ϕ − ϑ)] − 2i sin(qxL )
The transmission through the barrier is given by T = ∣t∣2 = 1 − ∣r∣2 .
Appendix G The Darwin–Fock spectrum

In appendix B, we introduced the operator solution to the harmonic oscillator. Here,
we treat a cylindrically symmetric quantum dot in two dimensions, which actually
makes use of the harmonic oscillator solutions. In principle, we write the
Schrödinger equation in the (x,y)-plane as [1]
ℏ2 ⎛ ∂ 2 ∂ 2 ⎞ m⁎ω 2 2
H=− ⎜ + ⎟+ (x + y 2 ). (G.1)
2m⁎ ⎝ ∂x 2 ∂y 2 ⎠ 2
8-41
This, of course, just doubles the number of variables in the overall harmonic
oscillator, and one can immediately write the energy as
⎛ 1⎞ ⎛ 1⎞
E = ℏω⎜a x†ax + ⎟ + ℏω⎜a y†ay + ⎟
⎝ 2⎠ ⎝ 2⎠ (G.2)
= ℏω(nx + n y + 1).
But, this is not the most efficient way to proceed, especially when we add a magnetic
field. The cylindrical quantum dot in a magnetic field was solved early on in
quantum mechanics by Darwin [75] and Fock [76]. To proceed, it is better to use the
symmetric gauge (see appendix C), in which the vector potential is written as
1
A= ( −Byax + Bxa y). (G.3)
2
When this is added via the Peierls’ substitution, two additional terms in the
Hamiltonian arise as
⎛ ∂ ∂ ⎞ e 2B 2 2
δH = −ieℏB⎜x −y ⎟+ (x + y 2 ). (G.4)
⎝ ∂y ∂x ⎠ 4
The second term here just adds to the last term in equation (G.1) to give a hybrid
frequency for the combined oscillator as
⎛ ωc ⎞2 eB
Ω= ω2 + ⎜ ⎟ , ωc = . (G.5)
⎝2⎠ m⁎
The first term in (G.4) is a new term that breaks the symmetry of our two-
dimensional harmonic oscillator. If we recognize that L = r × p, then this first term is
just eBLz. It is well known in atomic physics that the magnetic field splits the angular
momentum states, which are usually degenerate. To generate the same thing in this
problem, we need to go away from our x and y oscillators and rewrite everything in
cylindrical coordinates. To achieve this, we introduce a new set of operators as
1 1
a=
2
(ax − iay ) a† =
2
(a x† + ia y† )
(G.6)
1 1
b=
2
(ax + iay ) b† =
2
(ax† − ia y†).
To understand these a little better, we can look at the expansions of these operators
1 †
a†a = na =
2
(ax ax + a y†ay − ia y†ax + iax†ay )
. (G.7)
1
b†b = nb = (a x†ax + a y†ay + ia y†ax − ia x†ay )
2
8-42
Hence, we still achieve the same energy, with the total index
n = nx + n y = na + nb. (G.8)
If we rewrite the coordinate operators in terms of their actual coordinates, then we
can write
Lz = (a†a − b†b)ℏ = (na − nb)ℏ. (G.9)
We now have two sets of operators which work individually on the positive
angular momentum states and the negative angular momentum states. These four
operators work on both the energy and the angular momentum in the following way:
Operator energy Lz
a† ↑ ↑
a ↓ ↓ (G.10)
†
b ↑ ↓
b ↓ ↑
where the arrows indicate a raising or lowering of the state. A given energy level n
has both an energy and a degeneracy of n + 1, where n = 0, 1, 2, …. The angular
momentum here runs from −n to +n in steps of two units, and the magnetic field
splits the degeneracy. If we write mz = na − nb, we can write the total energy as
1
E = ℏΩ(na + nb + 1) + ωcLz
2 (G.11)
1
= ℏΩ(n + 1) + ℏωcmz .
2
The heart of the spectrum of states lies in the hybrid frequency Ω. For small
values of the magnetic field, we can expand this term as
⎡ 1 ⎛ ω ⎞2 ⎤ ω2
Ω ∼ ω⎢1 + ⎜ c ⎟ ⎥ ∼ ω + c , (G.12)
⎣ 2 ⎝ 2ω ⎠ ⎦ 8ω
and the energy is

⎡ ω ⎤
E ∼ ℏω(n + 1) + ℏωc⎢mz + (n + 1) c ⎥ . (G.13)
⎣ 8ω ⎦
So, while the magnetic field does split the angular momentum states, it also gives
them an upward curvature. In the opposite limit, where the magnetic field is large,
we can approximate the hybrid frequency as
ωc ⎡ 1 ⎛ 2ω ⎞ ⎤ ω
2
ω2
Ω∼ ⎢1 + ⎜ ⎟ ⎥ ∼ c + 2 , (G.14)
2 ⎢⎣ 2 ⎝ ωc ⎠ ⎥⎦ 2 ωc
8-43
Figure G1. The various angular momentum levels for the lowest four dot energies. One can see the splitting in
the magnetic field at low values and the tendency to form Landau levels at high values of the magnetic field.
and the energy is given as

⎛ 1⎞ ω2
E ∼ ℏωc⎜na + ⎟ + ℏ (n + 1). (G.15)
⎝ 2⎠ ωc
This last form is interesting, as it indicates that the major feature of the spectrum is
the formation of Landau levels, which all have the similar positive angular
momentum, but the levels are split by the interaction of the confining potential.
We see these two limits in the structure shown in figure G1.
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IOP Publishing

(Second Edition)
David K Ferry
Chapter 9
Open quantum dots
The description of a quantum dot can be applied to a great many types of structure,
some of which are defined by physical characteristics, such as self-assembled dots [1],
and others of which are defined through lithography. This can either be an etched
structure [2], or achieved by the imposition of a self-consistent potential which is
applied through a set of confining gates [3], much like those which form the quantum
point contact (QPC) in chapter 2. Over the last two decades, quantum dots have
proven to be a natural test bed in which to probe the understanding of the transition
between quantum mechanics and classical mechanics. In an open quantum dot, the
interior dot region is coupled to the reservoirs by means of waveguide leads, usually
formed by a pair of QPCs. While it might be expected that the charge fluctuations
which wash out the Coulomb blockade should also obscure the signatures of this
quantum structure, this is not the case. Many states are washed out by interaction
with the environment, but it is the remaining states that provide new insight into the
fundamental quantum physics. To be sure, open quantum systems, and the ability to
make measurements upon them, have been a point of discussion since the early days
of quantum mechanics [4–6]. When open quantum dots were first studied, it was
found that a series of conductance fluctuations could be observed as current passed
through the device [7, 8], much like those of section 5.3. Early on, it was assumed
that these conductance fluctuations were the same as the earlier ones, which arise
from disordered material [9]. Since the material involved in studies of these quantum
dots was not disordered, but of relatively high quality, it was then thought that the
conductance fluctuations arose from chaotic behavior within the quantum dots.
However, there was a significant difference in that the fluctuations observed in these
open quantum dots were nearly periodic in the magnetic field, instead of the
aperiodic behavior expected from chaotic effects, and that this period was very close
to that found for regular classical trajectories in classically confined structures of the
same dimensions [10]. In this regard, it was established that the conductance

fluctuations were really connected with regular orbits within the open dots [11], even
in typically chaotic structures such as stadiums [12]. Later, it was clearly demon-
strated that these regular quantum trajectories were very closely related to
equivalent classical trajectories originating from Kolmogorov–Arnold–Moser
(KAM) islands and rings of attractors (in arrays) that gave rise to classical orbits
[13, 14].
The results of measurements have always been assumed to be classical, as the
results appear in the laboratory itself. Yet, when the quantum nature persisted in
the dots, it was found that the wave function was heavily scarred. Such ‘scars’ are
imprints of a classical orbit which leads to fringes and a larger magnitude of the
wave function along these orbits. The connection between the quantum scarred
wave functions arising from the trajectories and the classical orbits turned out to be
intrinsic to Zurek’s decoherence theory in quantum mechanics [15]. There are two
crucial parts to this theory that make the connection. First, when the quantum
system is opened, a great many of the normal eigenstates become ‘decohered’ by
interacting strongly with the environment (outside the quantum dot in this case).
However, there remained a large number of eigenstates which remained strongly
coherent as they did not mix with the environmental states. These states were termed
the ‘pointer states’ [16], and these would leave an imprint upon the environment in
such a manner that they could be measured. Second, these pointer states would
evolve into the classical regular orbits [15], and should show a classical distribution
of states, as a function of level spacing, rather than the quantum Gaussian
distributions connected with quantum chaos [17].
In an important early study using Wigner functions, Berry [18] showed that the
quantum wave function would be concentrated on the energy surface for which the
classical orbit provides an oscillatory correction. That is, the scar is an enhancement
of the quantum wave function around a periodic orbit, which itself is a property of
the underlying classical system [19]. The existence of this scar is a connection
between the quantum system and the equivalent classical system. It has normally
been felt that such scars are unstable. However, it was found that these scars were
quite stable [20], a result of a process known as quantum Darwinism [16, 21].
9.1 Conductance fluctuations in open quantum dots

As mentioned above, quantum dots have been extensively investigated for a few
decades. Our interest is in the case in which the dots are open and relatively strongly
coupled to the environment. In most of the earlier studies, considerable effort was
expended to try to establish that the basic behavior of these ballistic quantum dots
was governed by universal properties that were generic in nature and independent of
the specific properties of the individual dots. In fact, averaging over, e.g. the gate
voltage, was used to remove the quasi-periodic fluctuations in order to reveal what
was believed to be a chaotic background. In fact, this process removed the significant
signatures of what we now believe to be the pointer states, which are the surviving
quantum states in the dot. In principle, the underlying physics of the ballistic
quantum dots is described by characteristics that are dependent upon the individual
9-2
dot under study, so that there is a universal behavior that is characteristic of such
dots, but it is not the generic behavior described by the pseudo-chaotic mathematics
of these previous studies. Instead, the underlying properties, and the characteristic
transport through the dots, is governed by the basic regular nature of the semi-
classical orbits in the dot [22]. This regular nature is observed as reproducible
fluctuations.
The transport is dominated in the typical quantum dot at low temperature by
reproducible fluctuations which are observed when a magnetic field, or the gate
voltage applied to the dots, is varied. These fluctuations exhibit quasi-periodic
oscillations, whose nearly single-frequency character is easily discernible in the
correlation functions and in the Fourier transforms, and which are endemic to
the semiclassical regular orbits in the dots. Crucial to this result is the excitation of
the ballistic dots by open QPCs, as the latter provide a collimated excitation of the
particles within the dot. This collimation provides a specific excitation of quantum
structure related to the semiclassical orbits.
In figure 9.1, we show a variety of open quantum dots to illustrate that there are
various ways in which to fabricate these structures [23]. In all cases, the dot size is
much smaller than the elastic mean free path, so that the underlying motion is
ballistic in nature. These structures are all fabricated on a GaAs/AlGaAs hetero-
structure, and include: (1) a device with surface Schottky barrier gates to define the
dots [24, 25], (2) a structure in which trenches have been etched into the material so
that other areas of the structure can be used as gates [26], and (3) a structure which
has been etched to define the dot [27]. We have also studied these dots in an InAs/
AlInAs heterostructure and an InGaAs/GaAs heterostructure. The basic observa-
tions are similar in all of these cases, and independent of the material system and
gate technology. For the moment, we will concentrate on dots of the first sort. For
these dots, the gate regions are defined by liftoff of metallic Schottky barrier metals.
The dots are basically square in nature, and sizes ranged from 1.0 to 0.4 μm (the
electrical sizes are somewhat smaller due to edge depletion around the gates). In
nearly all the cases studied, the carrier density was 3–5 × 1011 cm−2 and mobilities
Figure 9.1. Three different methods for forming a quantum dot. Left: we use surface Schottky gates. Middle:
we etched trenches so that other regions of the heterostructure can be used as gates. Right: a dot in which other
regions of the heterostructure have been removed by etching. Left and middle: reprinted with permission from
[30], Copyright 2005 IOP Publishing. Right: reprinted with permission from [27]. Copyright 1997 American
Physical Society.
9-3
are typically 40–200 m2 V−1s−1. The gate design allows electrons to be trapped in the
central square cavity, but with the open QPCs, this density is set by the two-
dimensional electron density outside the dot. This also means that the Fermi energy
in the system is the same and set by these regions outside the dot. Measurements are
typically carried out at 10–30 mK, the base temperature in the dilution refrigerator,
and source–drain voltage is kept well below the thermal voltage with lock-in
techniques utilized.
9.1.1 Magnetotransport
In figure 9.2, the conductance fluctuation that results from a sweep of the magnetic
field for a typical sample is shown. This particular sweep came from a dot with
Schottky gates used to define it. These reproducible fluctuations persist across a wide
range of magnetic fields, and for a variety of gate voltages (applied to the Schottky
gates). At higher magnetic fields, a transition into edge-state behavior and the
quantum Hall effect occur. At still higher magnetic fields, AB oscillations from edge-
state interference within the dot are observed [28], and this allows us to unambig-
uously determine the electrical size of the ballistic cavity. It is found that the basic
nature of the reproducible fluctuations is independent of the details of the sample
material and gate technology. Instead, this behavior seems to be a basic property of
the dot geometry and size, a result of the intrinsic dot properties and quite distinct
from the generic results of chaos theory [22]. Here, the basic oscillatory properties
are universal, although the specific frequency content is very dot-dependent.
Although not shown in the above figure, the low-field magnetoresistance often
exhibits a peak at zero magnetic field, which is unrelated to weak localization but
instead is an intrinsic property of the states within the dot [29]. The actual line shape
of this peak is found to vary with contact opening, and similar behavior has been
observed in simple QPC structures which cannot support chaotic behavior. Thus, it
is improper to assume a priori that such peaks are connected with either chaotic
Figure 9.2. A sweep of the magnetic field produces the conductance fluctuations shown here (a background
conductance has been removed). These are obtained for a 0.4 μm GaAs dot defined by Schottky gates.
9-4
behavior or weak localization. Indeed, phase-space filling can easily be generated by

the integrable motion within these dots due to the magnetic-field-induced precession
of the orbits. Thus, it is not surprising that the quasi-periodic fluctuations observed
in the magnetoconductance are quite different from those observed in chaotic
systems.
We have carried out calculations to simulate the behavior of these dots, using a
stable variant of the transfer matrix approach, which was discussed in section 2.5.
For example, conductance fluctuations as a function of magnetic field are studied for
a 0.3 μm square dot (this size reflects the electrical size, which is less than the actual
structure due to gate depletion) with 0.04 μm port openings. This allows two modes
to enter and exit the dot. Instead of a random aperiodic variation with magnetic
field, a series of nearly periodic oscillations is evident, as may be seen in figure 9.3.
Also apparent are several resonance features. In particular, a set of resonances
occurs at B ~ 0.069, 0.173, 0.283 and 0.397 T, in which the wave function is heavily
‘scarred’, in that the quantum mechanical amplitude appears to follow a single
underlying classical orbit, as shown in figure 9.4. Given that the period for the
reappearance of the diamond is ΔB ∼ 0.11 T, and using the criterion familiar from
the AB effect, that Δφ/φ0 = 2π for the difference in magnetic flux, one obtains A ∼
0.04 μm2 for the enclosed area, which corresponds well with the enclosed area of the
diamond-shaped wave function, depicted in figure 9.4.
Periodic orbits have played a huge theoretical role in the computation of
semiclassical quantization of bound states for many years, dating back to the
Einstein–Brillouin–Keller view of quantization. More recent studies indicate that the
‘imprints’ of these orbits persist up through thousands of states [18], a result that
suggests the closed orbits are quite stable in regular systems and only become
unstable as one passes to the ergodic regime, which seems to be replicated as one
passes from the semiclassical to the quantum regime. The QPC imposes a boundary
condition on the particles, in which the entry angle is determined by the wave
Figure 9.3. The computed conductance fluctuations at T = 0 K for a 0.3 μm dot in GaAs. The circles indicate
the periodic resonances which display the heavily scarred wave function of figure 9.4 below. (Reprinted with
permission from [23]. Copyright 2005 IOP Publishing.)
9-5
Figure 9.4. The squared magnitude of the wave function within the dot whose magnetic field sweep is shown in
figure 9.3. This wave function recurs at several magnetic fields as a resonance. (Reprinted with permission from
[30]. Copyright 2011 IOP Publishing.)
Figure 9.5. This movie illustrates the sharp resonance at which the scar of figure 9.4 recurs in the magnetic
field. The scar appears four times in the sweep of the magnetic field. Available at https://iopscience.iop.org/
book/978-0-7503-3139-5.
mechanical nature of propagation through the QPC. It is this collimation that is

important for exciting a particular set of regular orbits of the particles. From the
study of classical orbits in these dots, we can compute the transmission, and hence
the conductance, as well as the fluctuations. When we say that the transmission, as
denoted in figure 9.3, exhibits resonances, we mean that a particular scarred orbit,
that shown in figure 9.4, recurs on a regular basis [30]. It is this recurrence that gives
rise to the regular and periodic orbits corresponding to a dominant wave function.
The fact that it recurs at a regular set of eigen-energies points strongly to the idea of
quantum Darwinism [21]. To illustrate this recurrence, we show in figure 9.5 a video
which depicts the wave function in the dot as the magnetic field is varied over the
range illustrated in figure 9.3.
9-6
9.1.2 Gate-induced fluctuations

It is also possible to sweep the Fermi energy by varying the gate voltage. In this
situation, making the gate voltage (on the confining gates) more negative. As the
gate voltage is made more negative, the dot is reduced in size and the various energy
levels are pushed up through the Fermi energy. The results are quite similar in
behavior to that seen as the magnetic field is varied. This leads to a series of
conductance oscillations, which ride on top of a monotonic background, and which
disappear at a few degrees Kelvin. The oscillations often are observed over the entire
range of gate voltage and persist to conductance values as high as 15e2/h, which
represents a very open dot. Again, the experimental results and quantum simulations
(discussed below) yield the same dominant frequency in the dots. Here, we will focus
upon a smaller dot, whose gate defined dimension varied from 0.2–0.3 μm, depend-
ing upon the value of the applied gate voltage, a picture of which is shown in the
inset to figure 9.6. The conductance through the quantum dot as the gate voltage is
varied is shown as the main panel in figure 9.6 as the blue curve exhibiting the
fluctuations. It may be seen that the fluctuations ride on a uniformly increasing (for
increasing gate voltage) conductance background. Rather than try to smooth the
curve, the temperature is raised above 2 K, a point at which the fluctuations are
largely damped out, as shown in the red dotted curve in the figure. This latter curve
is then subtracted from the low temperature curve to isolate the fluctuations
themselves, which are plotted as the upper (blue) trace in figure 9.7. These
oscillations are very nearly periodic with a dominant period of about 15 V−1.
Figure 9.6. Conductance as a function of the gate voltage for a dot with staggered QPCs (shown in the inset,
the two gates are kept at the same potential). The solid (blue) curve is the conductance at 10 mK, while the
dashed (red) curve is at 2 K, where the fluctuations have been temperature damped. (Reprinted with
9-7
Figure 9.7. The measured conductance fluctuations are shown in the top (blue) trace, while those calculated
from the recursive scattering matrix are shown in the lower (red) trace. (Reprinted with permission from [30].
To account for the behavior seen in the experiment, we have used our recursive
scattering matrix simulation to study these fluctuations. These begin by first
computing the exact self-consistent potential for the dot at more than 300 gate
voltages. This is accomplished with a three-dimensional Poisson solver, which uses
only experimental values for the various parameters of the heterostructure and gate
layout. Once the potential profile is known, we simulate the quantum transport
through the device. The calculations are performed for a fixed Fermi level which is
defined by its value in the reservoirs to which the dot is attached through the QPC.
The influence of these contacts is included via the exact potential profile of the open
dots. This approach also allows computation of the wave function within the dot.
This latter can be decomposed into the states of the closed dot for later use by an
eigenfunction decomposition. The computed conductance fluctuation is also shown
in figure 9.7 as the lower (red) trace. While there is a variation in amplitude, the
frequency and general periodicity agree quite well between the experiment and
theory. This may be confirmed by comparing the Fourier transforms of these traces,
and this is shown in figure 9.8.
The same self-consistent potential has also been used to study the classical
dynamics of electrons injected into these dots [10]. As with most systems of this
nature, the dynamics is non-hyperbolic in that there are regions of chaotic scattering
which coexist with non-escaping KAM islands surrounding stable orbits in phase
space. Hence, these dots have a mixed phase space [31]. As mentioned above, when
the quasi-periodic fluctuations are removed by averaging over gate voltage, only the
chaotic background remains. It may be presumed that this chaotic sea provides the
background conductance through the dot, such as that given by the 2.0 K curve
9-8
Figure 9.8. Fourier transform of the fluctuations seen in figure 9.7. Those from the experiment are shown as
the solid curve. The simulated fluctuations are given by the dotted curve. (Reprinted with permission from [23].
shown in figure 9.6. It is thought that this chaotic background provides the regions
of chaotic scattering seen in the phase-space portraits of the classical dynamics
within these dots [10]. These phase-space portraits will be discussed below. Hence,
the periodic orbits which are enclosed within the KAM island must correspond to
the quasi-periodic fluctuations that we have been describing here. This was checked
by varying the gate voltage, and observing the change in size of these periodic orbits.
The results indicate that this periodicity agrees exceedingly well with that found for
both the experiment and the quantum simulation. The conclusion is that it is those
scarred quantum wave functions, which give rise to the periodicity, that relax into
the classical periodic orbits on the KAM island. This is important, as it is generally
felt that the pointer states are essentially the classical remains of the quantum states.
9.1.3 Phase-breaking processes

It is already clear from the plots in figure 9.6 that the fluctuations decay relatively
rapidly with temperature. While there may be some semblance left at 2.0 K, the large
amplitude of the fluctuations is basically gone. So the thermal spread of the discrete
energy levels of the selected eigenstates that have not decayed due to interaction with
the environment is still relatively narrow. However, it is relatively hard to estimate
this quantity, as concepts such as the mean level separation (Fermi energy in the dot
divided by the number of electrons) have no meaning when a portion of the
spectrum has been decohered via the interaction with the environment. So, it is not
clear at once whether the thermal effect leads to overlap of the pointer state energy
levels, or leads to a hybridization of these states with the decohered states. However,
we must remember that the states in the closed dot are all a set of orthonormal
9-9
states; that is, there is a well-determined set of energy levels, which are not highly
degenerate. When we open the dot, a large fraction of these states do hybridize with
the environment and are no longer eigenstates of the dot. However, one can study
this effect within the quantum transport simulation that has been mentioned several
times in this chapter. When this is done, it is found that the pointer states are very
stable and persist within the dot [32]. Moreover, these can move right through the
decohered spectrum without any hybridization with these latter states [33], which
just reinforces the fact that these are eigenstates. So, the thermal effects do not arise
from hybridizing the pointer states, and must arise from the broadening of these
states so that they interfere with one another. This implies that there is a fairly
sizable number of these pointer states within the dot. The effective thermal spread at
2.0 K is a fraction of a meV, so one expects there to be sufficiently many pointer
states that their individual level spacing is smaller than this.
Nevertheless, the dots and their pointer states undergo phase-breaking processes
just as those discussed in chapter 5. But, as mentioned there, the dominant phase-
breaking processes may arise from electron–electron interactions. But, these cannot
be interactions where both carriers are within the dot, as these would exist even in
the closed dot, where the eigenstates are stable, time-independent states. Just
opening the dots to the outside world would not change this, but certainly introduces
an interaction between the carriers in the pointer states and those lying outside the
dot. The persistence of the pointer states means that this interaction is weak, and at
most is a small perturbation. Nevertheless, the effective phase-breaking time can be
determined precisely by the same techniques as discussed in chapter 5.
In figure 9.9, the dependence of the phase-breaking time for two dots, whose
lithographic dimensions are 0.6 and 1.0 μm, is shown. The shape of the dot is that of
the left panel of figure 9.1. The phase-breaking time was determined by measuring
the magnetic field dependence of the correlation magnetic field for the fluctuations in
the magnetoconductance, as described in section 5.5. As mentioned, the
Figure 9.9. The variation in the measured phase-breaking time in two different quantum dots. The dotted line
is a guide to the eye indicating a slope of 1/T. (Reprinted with permission from [23]. Copyright 2005 IOP
Publishing.)
9-10
measurements indicate a saturation in the phase-breaking time below a transition

temperature, which is sample-size-dependent, and a decay at higher temperatures.
This saturation at low temperatures is not due to a saturation of the sample
temperature, as the amplitude of the conductance oscillations continue to increase as
the temperature is lowered. This saturation also occurs in other dimensions as well,
although the temperature decay portion has a different exponent for different
dimensionalities. In the figure, the temperature decay is as T −1, which is indicative of
a two-dimensional system. From figure 9.1 it is clear that the dot is connected
through two QPCs to regions which behave as Q2DEGs, and here the temperature
decay appears to correspond to the dimensionality of those regions.
Let us now consider a quantum dot coupled to quantum wires. The gated dot is
defined by electron-beam lithography and then etching away the extra material as
shown in the third panel of figure 9.1. The dot has a lithographic dimension of
0.8 μm2 with ∼150 nm QPC leads positioned at either side. These leads open to
quantum wires which gradually increase in width to 2 μm, over a distance of 10 μm,
before reaching a 2DEG patterned into a Hall bar structure. The wires provide
adiabatic coupling to the dot. The saturated phase-breaking time figure 9.10, at low
temperature, is found to depend on the nature of the coupling between the dot and
the wire, as this is varied by changing the bias applied to a top gate on the structure.
From measurements at high magnetic field of the trapped AB orbits, it is determined
that the electrical dot size depends upon this applied voltage from the top gate. At
low temperatures, the phase-breaking time saturates at a value which depends upon
the actual dot size. At higher temperatures, the phase-breaking time is found to
exhibit an apparent T −2/3 behavior [27], which is consistent with the coupling of the
dot to a one-dimensional quantum wire [34], as shown in the figure. However, some
caution is advised as only a few data points are present. These suggest that the
behavior has this shape, but much more work is necessary to confirm this. One
Figure 9.10. The phase-breaking time in a single etched dot, which is attached to quantum wires as shown in
the third panel of figure 9.1. Here the coupling is varied as the gate voltage is made more negative, which
increases the phase-breaking time in the saturation regime. (Reprinted with permission from [23]. Copyright
2005 IOP Publishing.)
9-11
expects to have the T −2/3 behavior for phase-breaking processes in a one-dimen-

sional system. Once again, we seem to see that the decoherence of the pointer states
is occurring in the environmental system to which the quantum dot is coupled.
Surprisingly, the above behavior persists even when multiple dots are coupled
together as an array of open quantum dots. Such a three dot array is shown in the
inset to figure 9.11. We have carried out measurements of the phase-breaking time in
three or four identical dots in series, separated by evenly spaced QPC. The
lithographic dimensions of the dots were 1.4 × 1.0 μm2 for the three dot array
and 1.0 × 0.6 μm2 for the four dot array [35]. The structures were designed using the
split-gate technique and all gates were tied together to provide uniform formation of
the dots. Measurements were performed in magnetic fields up to 4 T. In general, a
very low sample current was used. The phase-breaking time as a function of the
lattice temperature is shown in the main panel of figure 9.11 for the two dot arrays.
Once again, the temperature decay seems to vary as T−1, in keeping with the fact
that the arrays are embedded in a Q2DEG. The pointer states apparently do not
significantly decohere within one dot or the dot array, but this decoherence occurs
within the environment as represented by this Q2D contact system. This is a very
interesting result, and is in agreement with studies which have shown that the
fluctuations themselves can be composed of orbits which are coherent across the dots
of the array [36]. Indeed, there is some indication in this figure that the phase-
breaking time is longer in the four dot array than in the three dot array. As these two
Figure 9.11. The phase-breaking time for a three dot (open circles) and four dot (closed circles) array. The
inset is a micrograph of the three dot array. (Reprinted with permission from [23]. Copyright 2005 IOP
Publishing.)
9-12
devices were made on the same sample, at the same time, this observation may be
valid, although much more work in this area needs to be undertaken before such a
conclusion can be drawn.
9.2 Einselection and the environment

Decoherence is thought to be an important part of the measurement process,
especially in selecting the classical results, that is, in passing from the quantum states
to the measured classical states of a system [15]. However, the description (and
interpretation) of the decoherence process has varied widely, but key is the
interaction of the system with the environment, as well as the interaction of the
environment with the system. When the quantum dot is closed to the outside world,
a complete set of eigenstates exist within the dot. All of these eigenstates are
orthogonal to one another. Consider the dot shown in figure 9.4, in which the
squared magnitude of one of the pointer states is illustrated. In this dot, the leads are
located at the top of the two sides of the dot. What is clear is that this eigenstate has
no weight near the QPCs. If we divide the entire set of states of the closed dot into
two groups, we can label these groups by their weights near the QPC. Those which
have significant wave function amplitude near the QPC have a chance to hybridize
with the environment states when the QPC is open. The other set of states do not
recognize whether or not the QPC is open, as they have no wave function amplitude
near the QPC. It is this latter group of states which become the pointer states as they
survive the process of the opening of the QPCs.
As the first group of states hybridizes with the environment states, they are no
longer eigenstates of the quantum dot. Moreover, as these states provide continuity
between the regions outside the dot (the environment) and regions within the dot,
these become the current-carrying states. However, the portion of the wave function
of these current-carrying states that lies within the dot remains orthogonal to the
pointer states. Hence, there is no natural connection between the current-carrying
states and the pointer states. Their mutual orthogonality remains. If the position and
orientation of the QPCs is altered, the main effect is to modify the details of the
electron collimation so that a different set of pointer states will occur, giving rise to
different scarring properties [12]. As a result, any coupling between the current-
carrying states and the pointer states must occur via phase-space tunneling. Phase-
space, or dynamical, tunneling through regular phase-space structures, such as
KAM islands [17], fundamentally determines the characteristics of the conductance
fluctuations in these quantum dots. Theoretical analysis, discussed further below,
based on the tunneling mechanism gives quantitative predictions (the average
frequency of the fluctuations) in excellent agreement with experimental measure-
ments [10].
In this section, we will examine the classical set of states that can exist, even in a
closed dot. Then, we will turn to a mathematical approach to showing how the
environment affects the dot when it is opened. This will lead us to examine some new
states that arise in the open dot, but do not appear in the closed dot; these are known
as hybrid states.
9-13
9.2.1 Classical orbits

A special place in the description of the classical motion is given by the specific
features of the phase-space description of such motion. The classical phase space is
coordinate space versus momentum space. Dynamical systems which follow regular
closed orbits trace reproducible paths through this phase space, while chaotic orbits
follow random paths through the phase space. While dynamical systems may be
purely chaotic or regular, the most ubiquitous in nature are those whose phase space
is mixed [31], in which regular and chaotic orbits exist in adjacent regions of phase
space. The nature of each orbit may be categorized within the nonlinear transport
regime through the KAM theory [17], which suggests that for small, smooth
perturbations around a quasi-periodic motion, there will be an invariant torus in
phase space. Such an investigation can lead to an understanding of the nature of the
motion of the classical representation of the quantum wave function, as we will
discuss in this section. But, first we want to examine the classical behavior of the
closed dot [37].
When Bohr first provided a quantum picture of discrete energy levels and atomic
radii, he gave life to the emerging quantum world. Shortly thereafter, Einstein
turned the question around and asked which classical mechanical systems could be
subject to such quantum behavior [38]. The result was that when the Hamiltonian
was rewritten in terms of action-angle variables, then there existed conserved action
integrals. These appear already as in equation (7.26), but he wrote them as
∮C pdx = 2π ℏni ,
i
(9.1)
where ni is an integer, and i refers to the particular conserved action. In equation

(9.1), p is the generalized momentum and x is the generalized coordinate. The paths
Ci refer to extremal orbits on the invariant torus of the system, but they can be
almost any closed path. In two dimensions, we should have two such integrals. Each
of the periodic orbits can have as many integers in its description as there are degrees
of freedom in the Hamiltonian. Further understanding of equation (9.1) has been
provided by Brillouin [39] and Keller [40], so that this equation is often called the
EBK quantization condition. In a more modern scenario, we know in general that
the index on the RHS of equation (9.1) is modified, such as by the addition of a
factor of ½ in WKB approximations. Hence, it has generally been found that
equation (9.1) should be written as
⎛ βi ⎞
∮C pdx = 2π ℏ⎜⎝ni +
i 4⎠
⎟, (9.2)
where βj is the Morse or Maslov index [41]. In the former version, it is related to the
number of conjugate points in the trajectory, and this relates it to WKB theory,
where this index is the number of turning points. What this means is that closed
orbits lead to quantized states in the formal quantum theory. So, already we have a
close connection between the classical closed orbits and the quantum eigenstates. We
also know, from the discussion in chapter 7, that the EBK quantization rule, and
9-14
equation (9.2), are closely related to the Berry phase that is encountered in classical
periodic orbits [18].
The Maslov index is more related to the topology of the system [42], for integrable
systems. The motion takes place on what is called a Lagrangian manifold in phase
space, and the Maslov index relates to the topology of this manifold. For non-
integrable systems, this can be related to the Morse index for a particular trajectory.
The closed trajectories which give rise to equation (9.2) are closely related to the
wave functions in the quantum world. Berry and Balazs [43] showed that the
classical phase space is convenient for semiclassical quantization, and used the
quantum Wigner function [44] for this purpose. Away from turning points, the
Wigner function is found to localize on the classical trajectory. At turning points,
however, the normal WKB approximation breaks down, and a more complicated
approach is required. In these latter regions, the wave function develops complicated
interferences, which these authors refer to as whorls and tendrils. At these regions,
the regular motion is affected by non-integrable regions, and the wave function has
trouble resolving the details of the phase space structure. Nevertheless, the closed
orbits giving rise to equation (9.2) remain related to the regular bound spectrum of
the quantum world [45]. The idea that the regular orbits would be imprinted on the
quantum wave function was given by Heller as the idea of a scar [46]. Berry [18] has
extended this to show that any periodic orbit will yield a scar (wave function) which
is centered on the orbit, but will have fringes whose characteristics will depend upon
whether or not the orbit is on the energy shell.
The above point is important, because it suggests that, as the classical regular
spectrum is dominated by the closed orbits, the frequencies related to these orbits
can be seen in experiment. For any trajectory, we can form the correlation function
in time as
T
1
C (τ ) = limT →∞
2πT
∫0 x(t )x(t + τ )dt . (9.3)
Here, T is the period for a periodic function and in this case the limit can be dropped.
The Fourier transform of C has some very interesting characteristics. If the
trajectories are regular, then the Fourier transform will consist of only a few
frequencies, which are related to the time variation of the angle variables in action-
angle coordinates. That is, these frequencies will provide the dominant variations of
the trajectories in the phase space. It is clear from figure 9.4, that there are very
regular parts of the function. Indeed, this figure may have only a single frequency. It
is clear then that the quantum behavior is dominated by a few (one or two) closed
orbits, and these will be related to the pointer states discussed above.
When we look at the transport through the open dot, the oscillations that are seen
arise from the density of states within the dot. The latter can include regular orbits as
well as a broader background arising from the density of chaotic orbits within and
passing through the dot. Clearly, the oscillations, as discussed in the preceding
sections, are not washed out with the open dot, and we can use this fact to begin to
understand the source of the oscillations and fluctuations. The conductance is easily
9-15
shown to be an integral over the density of states at the Fermi energy, and this
density of states is defined by the orbits within the dot. When the dot is opened, it is
the closed orbits which are confined in the dot, but which connect to the ports
through tunneling to the open dot trajectories. It is important to note that these must
be derived from the orbits of the closed dots and make a large, but finite, number of
orbits around the trajectory before exiting; that is, these trajectories return to a
starting point, as has been conjectured in much semiclassical transport theory [47].
The eigenvalue spectrum of the closed dot is a series of δ-functions located (in
energy space) at each of the resonant eigen-energies of the dot cavity. Carrying this
further, it is then possible to say that the density of states for the closed dot is
ρ(E ) = ∑n δ(E − En). (9.4)
where the En are the various energy levels for the available states. It is assumed here
that the sum runs even over degenerate levels. The connection between the density of
states and semiclassical trajectories is easily obtained through semiclassical quantum
mechanics. The δ-function is replaced by its Fourier representation (in energy space)
via the Poisson summation formula [48]
∞
∞ 1
∑n f (n) = ∑M =−∞∫0 f (n )e i 2πnM dn + f (0).
2
(9.5)
The various integrals can be evaluated via the saddle-point method, and the density
of states can be expanded into the form [48], for a two-dimensional dot of side a
m⁎a 2 ∞ ⎛1 ⎞ a ∞
ρ(E ) = 2
∑M1, M2=−∞J0⎜⎝ SM1M2⎟⎠ − ∑M1=−∞cos(Χ), (9.6)
2π ℏ ℏ 4π ℏ
where
2M1a
Χ= 2m⁎E (9.7)
ℏ
is the effective phase when only a single Mi is present. The general action appearing
in the Bessel function of the first term is given as
1 2a
SM1M2 = 2m⁎E M12 + M22 . (9.8)
ℏ ℏ
Here, M1 and M2 are the multiplicities of the primary periodic orbits defined
through the quantization condition in equation (9.2). As mentioned following
equation (9.7), the second term in equation (9.6) is a boundary correction for those
orbits which do not sample all four walls of the dot.
The major point of these equations is that the density of states has oscillatory
terms, which are needed to reproduce the delta functions in equation (9.4). As the
system is more heavily damped, or the levels broadened, these oscillations are
gradually smoothed and in the ultimate case, only the term for M1 = M2 = 1 survives
9-16
from the first term of equation (9.6). This is the Thomas–Fermi density of states,
given merely by the so-called mean level separation. That is, the mean level
separation is the smoothed value appropriate to the Thomas–Fermi approximation
of a smooth two-dimensional density of states. If we keep the boundary (second)
term in equation (9.6), we obtain the extended Thomas–Fermi approximation
m⁎a 2 m⁎a 2 (9.9)

ρ(E ) = − .
2π ℏ2 2π 2ℏ2E
We note that both the correction term in equation (9.9) and the argument of the
oscillatory terms in equation (9.6) depend upon the length of the orbit, a point to
which we will return later. The important issue here is that the Thomas–Fermi
approximation discards the oscillatory contributions to the density of states, hence
discards the fluctuations that arise from the details of the quantization of the system.
That is, using the mean level separation, or equivalently the Thomas–Fermi
approximation, will miss the important oscillatory terms arising from the trapped
regular orbits of the pointer states.
The fact that the density of states has oscillatory terms is information that can be
probed experimentally. Moreover, the regular and the chaotic states have different
statistics [17]. In particular, the regular states, and their quantum pointer state
counterparts, have classical Poissonian statistics, while the chaotic states are
characterized by one of the Gaussian ensembles (the choice depends upon whether
or not time reversal symmetry is broken by a magnetic field). This difference in
statistics has been quantified in a quantum simulation of the quantum dot [49]. This
further serves as a clear connection between the classical and the quantum behavior.
As mentioned, both equations (9.6) and (9.9) have terms which vary with the area
of the orbit as well as with the length of the orbit. What is often missed in such trace
formulas is that the action integral must be quantized in the dot, and hence only
certain values of the energy (or momentum) are allowed. As a result, only some of
the terms contribute at each energy. It is this quantization of the energy that will
allow a particular term in equation (9.6) to recur as the energy is varied (or as a
magnetic field is varied). It has been suggested that one can introduce the magnetic
field variation by multiplying each term in equation (9.6) by a factor given by [50]
⎛ nΦ ⎞ ea 2B
cos⎜ ⎟, Φ = , (9.10)
⎝ 2 ⎠ M1M2ℏ
but this is not entirely correct. The problem with this formulation may be seen by the
plots in figure 9.12. The two closed trajectories, labeled (a) and (b), both have
M1 = M2 = 1, and therefore have the same length around the orbit. Hence, their
action integral is the same in the absence of a magnetic field, and they are degenerate
at the same energy level. However, these two orbits enclose different areas, so that
their behavior in a magnetic field will be different. In essence, the magnetic field
raises the degeneracy of the levels, and (for a circular dot) the different areas
correspond to different angular momentum in the orbit. In the square, this
degeneracy is still raised, but the results are not thought to be pure angular
9-17
Figure 9.12. Two trajectories embedded in a dot of side a. The trajectories (a) and (b) both have the same
length and action integral in the absence of a magnetic field. However, they have different areas and will
respond differently in a magnetic field. Reprinted with permission from [37], copyright 2012 by IOP
Publishing.
momentum states. Equation (9.10) defines the magnetic dependence only through
M1 and M2, but clearly another quantum number is required to specify the area or
something like the angular momentum.
The multiplicities M1, M2 lead to an interesting interpretation of the orbits which
circulate around the dot multiple times before closing. Quite generally, the wave
function of the closed dot may be expanded in a Fourier series as
⎛ nπx ⎞ ⎛ mπy ⎞
ψ (x , y ) = ∑m, n Amn sin⎜⎝ ⎟sin⎜
a ⎠ ⎝ a ⎠
⎟, (9.11)
and the coefficients are determined by the particular energy eigenstate. The factors in
the parentheses are in fact related to a set of reciprocal lattice vectors, in which the
first Brillouin zone is defined by
π π
− < kx, ky ⩽ . (9.12)
a a
Where there is a reciprocal lattice, there must be a real-space lattice, and this is just
the multiplicity of the simple square unit cell of side a. But, this is just one cell of the
periodic lattice that can be considered to exist. The multiplicities M1, M2 can then be
said to correspond to orbits which traverse more than the single unit cell of this
lattice.
It is easy to understand how this view can arise. Consider, for example, figure 9.13,
in which we embed the dot in a periodic lattice. If we continue the orbit through the
boundary wall, instead of reflecting it back into the dot, we see that the orbit which
starts at point c ends at point c’, which is a translation of the original point c by a set
of vectors corresponding to the sides of the dot, and to the lattice [51]. In a sense then,
all closed trajectories create a lattice in which the basic unit cell has the edge length
of the original dot. A sum over all trajectories represents a sum over all possible
lattice vectors in figure 9.13, and this results in a localized state in momentum space
9-18
Figure 9.13. Periodic lattice corresponding to extension of the basic square unit cell. The red line is the
extension, by reflection, of the enclosed trajectory in the single unit cell. Reprinted with permission from [37].
Copyright 2012 by IOP Publishing.
(due to the principle of closure of a complete set), which is the δ-function on the left-
hand side of equation (9.4). If the longer trajectories are damped by, for example,
phase breaking processes, then the δ-function is broadened. This is just the scattering
broadened density of states, which further contributes to the oscillatory (fluctuating)
density of states. On the other hand, the quantum point contacts can also affect
which trajectories are selected.
The idea that the trajectory is extended by reflection takes on added meaning
when the magnetic field is added. With a magnetic field, the trajectories take on
curvature, and each section of the trajectory is an arc taken from the cyclotron orbit
appropriate for that energy and magnetic field. Now, however, when that segment is
reflected through the boundary of the unit cell, the curvature is reversed. It takes two
reflections at subsequent boundaries before the curvature is returned to the original
value. This means that the unit cell in the presence of the magnetic field is a supercell
of dimension 2a × 2a. To account for the reversed curvature, this means that the
magnetic field is reversed in the second and fourth quadrants of this supercell. As a
result, we have breaking of time-reversal symmetry (which leads to magnetic
ordering) without any net magnetic field in the supercell. Consequently, the electron
states retain their usual Bloch state character. Such a situation was considered by
Haldane for the 2D graphite (what we now know as graphene) structure [52], which
is unique in that the hexagonal lattice supports points in the Brillouin zone where the
conduction and valence bands meet with a zero gap [53]. The square lattice does not
have such peculiarities. Since the net flux in the supercell vanishes, one can chose to
use a periodic vector potential with some flexibility in choice of phase.
A two-dimensional system can be wrapped around a two-torus (a donut), and
with a magnetic field present, trajectories will close only for so-called rational values
of the magnetic field relative to the lattice [54]. The most famous study of these
systems was done by Hofstadter [53], and the fractal-looking band structure has
9-19
been called the ‘Hofstadter butterfly.’ It has been shown that even for open quantum
dots in a real array, the energy structure maintains major elements of the Hofstadter
butterfly in that the energy bands become split when the magnetic field takes rational
values [55], and the current amplitudes (conductance) also obeys a type of Harper’s
equation [56]. This seems to hold even into the edge state regime [57]. The conclusion
is that transport in the single quantum dot has similarities to that expected in a
superlattice of such dots, even in the open dot regime, as we have attempted to show
throughout this chapter.
9.2.2 Coupling the dot to the environment

Decoherence is thought to be an important part of the measurement process,
especially in selecting the classical results; that is, in passing from the quantum states
to the measured classical states of a system [58]. However, the description (and
interpretation) of the decoherence process has varied widely, but key is the
interaction of the system upon the environment, as well as the interaction of the
environment upon the system. Zurek has proposed that the interaction of the system
on the environment leads to a preferred, discrete set of quantum states, known as
pointer states, which remain robust, as their superposition with other states, and
among themselves, is reduced by the decoherence process [59]. This decoherence-
induced selection of the preferred pointer states was termed einselection [58]. While
this describes the physics of einselection, the mathematics can be shown by the use of
projection operators. We give a brief overview here. We consider a system S,
interacting with its environment E, so that the combined system plus environment
(S+E) is either closed, or influenced by external driving fields that are assumed
known and unaffected by the feedback from this combined S+E. The Hilbert spaces
of both the environment and the system are assumed to be finite dimensional,
although this is not critical. These two spaces form a tensor-product Hilbert space of
the system plus environment. The operators in which we are interested are called
superoperators, and exist in an expanded space often called the Liouville space.
When we open the dots, there will be an interaction between the dot and the
environment, so that we can write the total Hamiltonian as [60]
H = HS + HE + Hint, (9.13)
where the three terms represent the system, the environment, and the interaction
between these two, respectively.
There are two crucial steps in defining a reduced density matrix for just the
pointer states. The first is to project out these states via a projection superoperator.
The second is to then trace over the environmental states yielding just the reduced set
of pointer states. This procedure has been known for a considerable time [61–63].
There are many ways to define, or create, the necessary projection operator, which,
as mentioned above, is a commutator-generating superoperator [64]. We want to
choose the particular projection operator such that
ˆ ˆ ˆ = Hˆ red,
PHP (9.14)
9-20
where the last term is the reduced Hamiltonian for the pointer states, and the carets
over the operators indicate that these are superoperators. Now, these projection
2
operators are idempotent (Pˆ = Pˆ ), and consequently have eigenvalues of 0 or 1, and
it is a central tenet of quantum mechanics that we can build the system through
knowledge of the eigenfunctions and their eigen-values. For this subsystem, all the
eigenvectors are stationary states in the Heisenberg representation [65]. We have
used this in an earlier study of open quantum systems [66], and this approach has
been used to create so-called decoherence-free subspaces in quantum information
processing [67–69]. With this approach, we recognize that the pointer states really
are a set of isolated states within the dot, and are not directly coupled.
We begin by writing the Liouville equation in terms of the composite density
matrix ρ, which is defined on a tensor product Hilbert space of the system density
matrix and the environment density matrix, as
ρ = ρE ⊗ ρS , (9.15)
for which the Liouville equation can be written

∂ρ
iℏ = Hˆ ρ . (9.16)
∂t
In particular, the Hamiltonian is a commutator-generating superoperator. Equation
(9.16) is easier to understand when we see that the Hamiltonian is now a 4th rank
tensor, which generates the commutator relation normally seen in this equation via
(Hˆ ρ)kn = ∑rs Hˆ kn,rsρrs = ∑rs (Hkrρrn − ρks Hsn). (9.17)
If the dimension of ρS is ds and the dimension of ρE is de, then the dimension of the
superoperator is d s2d e2 [66]. To simplify the approach, we Laplace transform (9.16),
and then trace over the environment variables to give [64]
(i ℏs − Hˆ S )ρS = TrE {Hˆ intρ} + ρS (0). (9.18)
As discussed above, we now use a projection operator which yields the pointer states
as eigenfunctions. This operator has the basic properties
2
ρS,red = PˆρS , Pˆ = Pˆ , Qˆ = 1ˆ −Pˆ . (9.19)
To proceed, we need only to use an identity that is obtained by projecting the

Liouville equation with both P̂ and Q̂ , solving for Q̂ρs to formally decouple these two
equations and recombining the terms [66, 70]. This identity is
1 ˆ ˆ ˆ ˆ ˆ) 1 ˆ ˆ ˆ ˆ)
= (Pˆ + QRQHP (Pˆ + PHQR
i ℏs − Hˆ i ℏs − Cˆ − PHP
ˆ ˆ ˆ , (9.20)
ˆ ˆ ˆ
+ PHQRQHP ˆ ˆ ˆ ˆ
9-21
where
1
Rˆ = ,
ˆ ˆ ˆ
i ℏs − QHQ (9.21)
Cˆ = PHQRQHP
ˆ ˆ ˆ ˆ ˆ ˆ ˆ.
The last term is a ‘collision’ type term which connects the environment to the device
via the off-diagonal elements of the superoperator. If we now define some further
reduced parameters as
Σ̂ρS,red = TrE {Cˆ ρS,red },

(9.22)
Hˆ ′int = TrE {Hˆ intρS,red },
we can then write the reduced equation as
i ℏρS,red − ρS (0) = Hˆ S,redρS,red + Σ̂ρS,red + Hˆ ′int ρS,red

. (9.23)
ˆ ˆ ˆ ˆ ˆρ (0)}
+ i ℏTrE {PHQRQ S
In general, we are seeking the steady-state, long-time limit, and would ignore the
initial conditions. However, the last term in equation (9.23) has been suggested as
contributing to the random force [71] that appears in e.g. the Langevin equation as
well as to screening [72], so that it may not be proper to totally ignore it. However,
the final long-time limit equation becomes, after inverting the Laplace transform,
∂ρS,red
ih = (Hˆ S,red + Hˆ ′int )ρS,red + Σ̂ρS,red . (9.24)
∂t
The second term on the right hand side is not a scattering, or decoherence term, as
that would appear in the last term on the right. Instead, it represents a weak
interaction between the pointer states and the environment via the decohered states.
That is, it represents primarily the environment interaction on the non-pointer
states, which can then weakly couple to the pointer states. This term must represent
the phase space tunneling by which the pointer states appear in experiment [10]. It is
true, however, that this can bring decoherence into the world of the pointer states via
electron-electron interactions in the environmental states, a point we have reviewed
previously [23]. Normally, the pointer states do not interact with the environment, so
that the last term would vanish, but the interaction of the pointer states, through the
decohered states, to the environment produces the phase breaking discussed in the
previous section. This means that the last term does not vanish, but represents this
phase breaking process through an imaginary term in this self-description. Since this
scattering term is part of the Hamiltonian of the pointer states (the reduced set of
states), it is a diagonal term, but has an imaginary part, which makes the
Hamiltonian non-Hermitian.
It is important to remark here that this form of Hamiltonian is not what one
normally refers to as non-Hermitian. Here, the imaginary term which breaks up the
9-22
Hermitian properties lies on the diagonal of the matrix. In essence, this will also
make the resulting reduced density matrix non-norm conserving as it breaks up the
unity of the trace. But, this is because the germane interactions are from states within
the dot and states of the environment, which can include renormalized decohered
states of the dot. This environment sensitive dephasing interaction is seen in
experiment [23], as discussed above. The steady-state situation has to then account
for lost amplitude in the dot being replaced by electrons injected from the environ-
ment itself. These source terms must be incorporated within the first term on the
right-hand side of equation (9.24) as in any other transport problem.
Now, there are further problems with equation (9.24). While it appears to be quite
simple conceptually, this is not really the case. First, within the partial trace in both
terms of equation (9.22), there is an explicit dependence on the choice of the
projection operator P̂ or, equivalently, on the environment density matrix that
induces the projection operator, so one must make a choice of the latter to actually
be able to use equation (9.24). At the end of the day, the equation of motion for ρS(t)
should not depend on it. And yet it does through its affects on the projection
operator. The rationale for believing that the equation of motion for ρS(t) should not
depend on the details of the environment goes back to our statement above that the
eigen-states of ρS(t) are our pointer states, and these are properties of the specific dot,
and exist in the dot for a variety of different environments. We have previously
shown [66] that quite generally, the eigen-space of the projection operator,
corresponding to those states with eigenvalue 1, must be isomorphic to the states
of ρS(t), which are our pointer states.
Now, in addition, to giving rise to the phase-space tunneling interaction between
the environment states and the pointer states, the interaction term can also give rise
to hybrid states that would not exist either in the environment or in the closed
system. We will see examples of these in the following sections.
9.2.3 Relating classical and quantum orbits

When we study motion in multiple dimensions, the phase space also has more
dimensions. For example, if we study particle flow in two spatial dimensions, then we
have four dimensions in the phase space. How are we to plot these? First, we do not
simply plot the entire phase space, and the Poincaré map becomes essential to
understanding the dynamics of this two-dimensional motion. Typically, a plane is
chosen to represent the plane of the Poincaré section, such as the y = 0 axis. The next
choice is where to choose the critical point. In the following, we will see two-
dimensional Poincaré sections and three-dimensional ones. For the two-dimensional
plots, we chose the time at which the trajectory passes through the y = 0 plane with a
positive velocity in the y-direction. At times, we will also plot the value of vy and this
leads to a three-dimensional Poincaré section. These choices are, of course, arbitrary,
but have become values that are widely used. With this bit of introductory material,
let us now turn to the classical study of the motion within the quantum dot potential.
To study the connection, we first describe a method of carrying out the classical
simulation of orbits. With Schottky barrier confinement walls, such as shown in the
9-23
left panel of figure 9.1, the actual potential within the dot is a relatively soft wall
potential (much like a harmonic oscillator potential). In numerically modeling the
classical motion, a good fit to the experimental magnetoresistance is only obtained
by using such a soft wall potential. Within the dot, the potential is modeled as a two-
dimensional harmonic oscillator via the potential
m⁎ ⎡ 2
⎣ωx (x − x0,i ) + ω y y ⎤⎦ .
2 2 2
Vdot(x , y ) = (9.25)
2
The dot shape is somewhat elliptical, as shown by the different frequencies in the two
directions. Here, the x-axis is taken along the major axis of, for example, a seven dot
array, in which the center of each dot is indicated by x0,i. This array is used to probe
the locality and coupling of various states between the dots and with the environ-
ment. Between the dots, as well as between the end dots and the 2DEG environment,
there is a quantum QPC, which is modeled by the saddle potential
m⁎ ⎡
⎣ −ω x, QPC (x − xQ,j ) + ω y y ⎤⎦ + V0,QPC ,
2 2 2 2
VQPC (x , y ) = (9.26)
2
where V0,QPC is the saddle potential height at the center of the QPC and xQ,j is the
position of the saddle potential peak in QPC j. The various frequencies in the two
potential forms are adjusted so that these potentials match the fully self-consistent
potentials obtained in the quantum calculations [20]. An example of such a potential
is shown in figure 9.14. Each dot has a length of 2ld and each constriction has a
length of 2lc. The classical simulations consider a ballistic motion of an electron in
this potential and a normal (to the plane of the motion) magnetic field Bz, which
produces the cyclotron frequency ωc = eBz/m*. These potentials produce the
equations of motion
d 2x dy
2
= −m⁎ωx2(x − x0,i ) + ωC
dt dt . (9.27)
2
d y dx
= −ω y2y − ωC
dt 2 dt
Figure 9.14. An example of a seven dot confining potential. Coupling between the dots and with the
environment is via the saddle potentials along the major axis.
9-24
within a dot, and
d 2x dy
= m⁎ω x2, QPC (x − xQj ) + ωC
dt 2 dt (9.28)
2
d y dx
= −ω y2y − ωC
dt 2 dt
in the constriction. Just as in a two-dimensional harmonic oscillator, the coupling of
the harmonic motion with the cyclotron motion leads to a pair of hybrid frequencies
[73]
ω±2 = β ± β 2 − ωx2ω y2
1 2 . (9.29)
β=
2
(ωx +ω y2 + ωC2 )
For a constriction region, the motion is somewhat more complicated, with a

modification of these hybrid frequencies as
ω+2 = −β′ + β′2 + ω x2, QPC ω y2 ,
ω−2 = −β′ − β′2 + ω x2, QPC ω y2 , (9.30)

1
β′ = (ω x2, QPC −ω y2 − ωC2 ) .
2
The motion is followed by plotting each passage of the y = 0 plane with a positive
velocity in the y-direction. For each passage of this plane, the x-position and velocity
are plotted. This yields a Poincaré plot that can be used to study the characteristics
of the classical dynamics. As discussed above, a perfectly regular orbit will always
pass through precisely the same spot in this plot. Hence, a regular orbit creates a
single point in the plot. At the other extreme, a chaotic orbit will never pass through
the same point, and so creates a sea of points, which is often referred to as the ‘sea of
chaos’. Other orbits can pass through the plane many times before coming back to
the same point, and these can create closed lines in the Poincaré plot. These latter are
quasi-periodic orbits, which together with the single points usually are surrounded
by regions where no trajectory passage can be found. These are the KAM islands. In
figure 9.15, we show the Poincaré plot for one of the inner dots of the seven dot
array [37]. This is calculated for an off-resonance condition where ω+/ω− = 2.156
(Bz ~ 0.28 T). Here there are two larger KAM islands at the left and right side of the
image, along the vx = 0 axis. These two KAM islands are embedded in a dense sea of
chaos, but each is surrounded by a ‘sticky’ layer [74], that emerges from trajectories
that oscillate between a given dot and one of its QPCs for a long time before finally
entering the sea of chaos. All the trajectories that begin in the sea of chaos are
unstable and escape the dot array, contributing to the current-carrying decohered
states (we illustrate these later).
9-25
Figure 9.15. The Poincaré section for one of the interior dots in the array. This is for an off-resonance
condition, and two major KAM islands can be seen near the left and right edges of the image. (Reprinted with
We can illustrate the nature of the states on the KAM islands a little better by
using the full three-dimensional Poincaré plot where we include also the y-velocity,
instead of just taking the point where it passes through the y = 0 plane. This is shown
in figure 9.16(a), for the first two dots of the array. Three KAM islands now lie on
the top of the three-dimensional surface [37], as well as near the lower front of the
sphere. In the classical phase space, different KAM islands may be degenerate in
energy, as is the case here. Quantum mechanics will generally split this degeneracy
by the presence of eigenstates with significant amplitude (but different phases) on
both KAM islands. This does not occur classically, because there is an energy barrier
between the two islands which cannot be surpassed by the classical motion at
constant energy. In the present case, the pointer states typically correspond to
closed, periodic orbits that are normally classically inaccessible. But, in the open dot,
phase-space tunneling [10] can occur between these KAM islands, and this leads to
the fluctuations that have been discussed above. That is, there is a very good
probability that an incoming electron will pass through the regular regions in phase
space via a tunneling process. In figure 9.16(a), six distinct KAM islands are shown.
Three of these have been labeled as 1–3. The individual orbits for the states lying at
the center of the KAM islands are illustrated in figure 9.16(b). The orbits found in
the center of islands 2 and 3 (in the first and second dots, respectively) are essentially
the same, although they have reversed velocities, so that they can be expected to be
coupled by a quantum wave function. Thus, we expect phase-space tunneling to be
possible between these KAM islands, as shown by the arrow in figure 9.16(a).
It is possible to connect the classical motion to that obtained from a full quantum
simulation of the same seven dot array. In figure 9.17, we do just that. In panel (a),
we plot the magnitude of the quantum mechanical wave function for the condition
of point 2 in figure 9.16(b). Here, the wave is colored according to the x-directed
velocity, red for positive and blue for negative. The arrows indicate the direction of
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Figure 9.16. The Poincaré surface for the first dot and part of the second dot of the seven dot array. The three
KAM islands are identified by the possibility of phase-space tunneling between them. (b) The Poincaré plots
for each of the three islands showing the different shapes and velocity vectors (arrows) in the y-direction.
the total velocity for this eigenstate. In panel (b), we replot the classical motion from
figure 9.16(b) for orbit 2. It is clear that there is a strong one-to-one correlation
between the classical and the quantum motion at this point. In order to better
visualize the motion, we plot the so-called quiver plot for the quantum wave
function. The individual arrows give the direction of the velocity and their size gives
the relative magnitude of this velocity. The fuzziness that appears around the y = 0
axis in panel (a) is clearly associated with the slow velocity as the trajectories cross
this axis, as indicated by the shorter arrows and the broader area over which they are
spread in panel (c).
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Figure 9.17. The magnitude of the quantum wave function (a) and the classical orbit (b) taken from the orbit
labeled 2 in figure 9.15(b). Arrows have been added to these to show the velocity direction. (c) The quiver plot
in which the arrows indicate the velocity direction and magnitude for the quantum wave function.
Figure 9.18. Top: the Poincaré plot for the entire seven dot array. As earlier there is a large sea of chaos, with
two KAM islands located in each dot. Bottom: the magnitude squared of the wave function for the frequency
ratio of 3, showing a skipping orbit that arises from the decohered, current-carrying states. (Reprinted with
As mentioned above, the decohered states are coupled to the environment and
serve as current-carrying states for the background current that flows through the
dot array. These states arise as one varies the ratio of the two hybrid frequencies and
often show up when this ratio is an integer. We can see this for a couple of ratios in
figure 9.18. In panel (a), we plot the Poincaré plot for the entire seven dot array for a
value of ω+/ω− = 1.6269 (Bz ∼ 0.16 T). In each dot, there is a large chaotic sea and
two different types of KAM islands, as discussed above. However, in panel (b), we
show the case for ω+/ω− = 3.0 (Bz ∼ 0.43 T), where a large skipping orbit passes
entirely through the array. In this latter case, the probability obtained from the
quantum wave function is plotted rather than the Poincaré plot. The classical
trajectory for this type of current-carrying state does not contribute to the sea of
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chaos; in fact, at this value of magnetic field, the sea of chaos has largely been
drained (e.g., reduced to a very small part of the phase space). As a result, the phase-
space probability is largely concentrated in the QPCs which connect the dots [13].
For a comparison of the classical phase-space dynamics and the quantum
mechanical analog, the Husumi function is a useful concept [75]. The wave function
obtained from the quantum simulation is smoothed and transformed via a coherent
state, which is a Gaussian wave packet with a finite momentum, such as
⎛ 1 ⎞1/4 ⎛ x2 + y2 ⎞
ξ(x , y ) = ⎜ 2 ⎟ exp⎜ − + ik · r ⎟. (9.31)
⎝ πσ ⎠ ⎝ 2σ 2 ⎠
Here, the wave packet is centered at a convenient point, taken to be the center of
each dot, and k and r are both two-dimensional vectors, and σ is the half-width of
the wave packet. The imaginary term provides a uniform velocity for the packet as
v = ℏk/m*, the customary form in semiconductors. This coherent wave packet is then
used to transform the quantum wave function ψ(x,y) into the Husumi phase-space
function via the integral transform
∫
H (r , k) = ∣ d r′ψ (r′)ξ(r − r′ , k)∣2 . (9.32)
The projection of this onto the Poincaré plot is obtained by taking y = 0 (to
correspond to the equivalent classical plot) and then integrating over ky. In the top
panel of figure 9.18 above, the Husumi function is overlaid on the classical phase
space, and appears as the color parcels over the normal blue–white Poincaré plot.
The wave function for this case is precisely the one obtained for the same set of
hybrid frequencies, potential, and magnetic field as the classical case. Hence, it may
Figure 9.19. A blowup of the first dot from figure 9.17, with the Husumi function overlaid for comparison. The
left side shows the normal Poincaré plot, while the right side shows the plot with the Husumi function overlaid
on it.
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be seen that the maximum of the Husumi function (bright red) lies precisely in the
KAM island where the phase-space point of the classical orbit resides. We show a
blowup of this region in figure 9.19. In the left-hand side of the figure is the blowup
of the classical Poincaré plot. In the right-hand side, the black–white image is
converted to blue–white and the colored Husumi plot is overlaid, so that one can
clearly see the convergence of the classical and quantum pictures. It is clear that the
quantum–classical correspondence is very strong and this strengthens the argument
that the pointer states are the connection to the classical orbits in these open
quantum dots. One further connection arises in the next section.
9.2.4 Pointer state statistics

In the above discussion, we have dealt with an important issue in quantum
measurement theory, namely how the quantum states evolve into classical states,
in this case through the connection between the quantum pointer states and the
classical trajectories in the same potential system. It is important to recall that it has
been suggested that these pointer states are indeed the basis of the transition to
classical behavior, and even possess classical properties [15]. The relationship
between the classical dynamics of a system and the spectral statistics of its quantum
analog has been a primary concern in the study of quantum chaos [17, 41]. Systems
with integrable dynamics are expected to have uncorrelated energy levels that yield
Poisson statistics [41], while completely chaotic dynamics is associated with the
Wigner statistics [76] of one of the random matrix ensembles, the Gaussian
orthogonal ensemble (GOE) when time-reversal symmetry is preserved and the
Gaussian unitary ensemble when it is broken (for example, by a magnetic field).
However, we have shown that when a stadium dot, which is fully chaotic in the
closed state, is opened to the environment and only the pointer states with amplitude
localized to in the interior remain, the pointer state distribution becomes Poissonian,
indicating that the pointer states are intimately associated with the regular orbits
[49]. Here, we will show that the same is true for the nearly square quantum dots
discussed in this chapter.
For the analysis of the spectral statistics of a quantum dot system, a set of states is
chosen which lies in a given energy that covers that expected to be probed in the
experimental situation. Since typically only a limited number of eigenstates occur
over this range, one also varies the perpendicular magnetic field to generate many
different ensembles of eigenstates, thus greatly increasing the number of energy
levels available for the analysis. It is important that the number of QPC modes does
not vary over the range of magnetic field. Moreover, the values used for B should be
sufficiently small that no substantial change in the nature of the electron dynamics is
expected [33]. Once the set of relevant energy levels is determined, the raw energy
levels are mapped onto a dimensionless, unfolded set of eigenvalues which have a
local density of unity [77]. This is achieved through the conversion
E i (B )
ei (B ) = , (9.33)
〈∆E 〉
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where 〈∆E 〉 is the average level spacing for the set of energy levels obtained at a
given magnetic field. This now yields a dimensionless sequence of nearest-neighbor
energy level spacings
si = ei +1(B ) − ei (B ). (9.34)
This is computed for each magnetic field. The statistics of the number of
dimensionless energy levels that have a given spacing si can be determined and
the resulting plot yields the nature of these statistics. In figure 9.20, we plot the
nearest-neighbor distribution function P(s) for the pointer states [78]. It may be seen
that P(s) for pointer states very clearly follows the Poisson distribution (the GOE is
also shown for comparison). If the total set of energy levels is considered, then the
GOE distribution is recovered, as there are many more decohered eigenstates than
pointer states.
It is clear from figure 9.20 that pointer states in an open quantum dot yield
Poisson spectral statistics associated with classically regular behavior. On the other
hand, if we use all the eigenstates of the closed dot, the GOE distribution associated
with quantum chaos is recovered. While there have been previous theoretical
spectral studies that did not observe such a change in behavior, a key factor that
enabled us to see a dramatic shift to a regular distribution is that we studied very
open quantum dots strongly coupled to the external environment. This means that
the width of the QPCs is such that the latter pass several modes, and thus have a very
Figure 9.20. The nearest-neighbor distribution function found for the pointer states is shown by the blue dots.
The red dashed line is a Poissonian distribution, while ‘the green curve represents the GOE distribution.
(Reprinted with permission from [78]. Copyright 2015 the Swedish Academy of Sciences and IOP Publishing.)
9-31
important effect on the nature of its confinement. In this limit, the pointer states that
survive are localized in the interior of the dot, and are typically scarred by relatively
simple periodic orbits.
9.2.5 Hybrid states

In the previous sections, we have talked about states which evolve from the
eigenstates of the closed dot. However, opening the dot to the environment, in
this case the Q2DEG that exists in the remainder of the heterostructure, can lead to
other states, where a particle enters the dot through one QPC and bounces around to
exit at the same QPC, or, in multiple dots, just sit in the connecting QPCs. This
requires a magnetic field in order to make the trajectory curve sufficiently well to
create this unique back-scattered trajectory. Of course, there will be a quantum
equivalent of this trajectory. It is important to note that this is very different from
the weak localization effect, introduced in section 3.7. In weak localization, one
looks at the conductance or resistance at zero magnetic field and the effect arises
from the interference of two time-reversed paths, which are scattered by multiple
impurities until they return to the source. In this case, the magnetic field breaks the
time-reversed symmetry so that the two paths do not trace each other any longer,
and the effect goes away. Here, the magnetic field is critical to the enhanced
resistance that arises from the back-scattering, and there are no time-reversed paths
due to the presence of the magnetic field. This effect was apparently first discovered
by Ochiai et al [79], in an array of quantum dots. Of course, these trajectories can
bounce around and leave through the other end of the dot or array of dots, as
Figure 9.21. (a) The classically calculated trajectories for certain initial conditions in the entrance QPC which
lead to either back-scattered or transmitted trajectories. The conditions of frequencies are ω+/ω− = 2, 3, 4, 5, 6
for the plots from left to right, respectively. Other parameters are given in the text. (b) The quantum
mechanical density probabilities for the same conditions. (Reprinted with permission from [13]. Copyright
9-32
already shown in figure 9.18(b). The choice of whether the entering trajectory leaves
at the entrance of the QPC or at the far end of the array depends upon the potentials
and the exact magnetic field. We illustrate these trajectories for various conditions in
figure 9.21. According to the details of the self-consistent potential, the parameters
used in equations (9.25) and (9.26) are ωx = 1.06 × 1012 s−1, ωy = 0.85 × 1012 s−1, and
ωx,QPC = 2.16 × 1012 s−1. The dots have an extension in the x-direction of 0.32 μm in
the dot and 0.076 μm in the QPC, which corresponds to the Fermi energy. This is less
than the lithographic dimension of 0.4 μm for each dot. The magnetic field for the
first, third, and fifth images are approximately 0.2, 0.5, and 0.7 T. For the
trajectories in the figure, these three images show back-scattered trajectories, which
make successively larger numbers of bounces before returning to the entrance QPC.
On the other hand, the second and fourth images illustrate transmitted trajectories
which exit at the last QPC, and correspond, for example, to the plot shown in
figure 9.18(b). The top row of images are the classical trajectories, while the lower
row of images are the quantum mechanical density probability plots for the same
conditions.
In a dot array, the possibility for back-scattering trajectories can be more
complicated as the reflection may come from a dot other than the one closest to
Figure 9.22. (a) Three different back reflecting trajectories that can arise in a dot array. Of course, each occurs
at a different magnetic field. (b) Expansion of a set of possible trajectories for the ‘b’ described in panel (a).
(Adapted with permission from [79]. Copyright 1997 the American Physical Society.)
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the entrance QPC. We illustrate how multiple back reflections can occur in a dot
array in figure 9.22. In panel (a) of this figure, we illustrate back reflections from the
sequence of dots that can occur. Of course, each back-reflected orbit occurs for a
slightly different magnetic field, as we will show below. In panel (b) of the figure, we
show back-reflected orbits from the second dot, which are basically parallel with
those shown in figure 9.22(a). As may be expected, the details of these various orbits
will be sample-dependent and depend upon the details of both the potentials and the
magnetic field.
By carefully determining the size of the dot array and determining the depletion
inside the metal edges from exact self-consistent potentials, one can model the
various trajectories and compute the conductance and resistance of the dot array
quantum mechanically [20]. This allows a good comparison with the available
experimental data. In figure 9.23, we illustrate this comparison using the data from
Ochiai et al [79]. The theory curve shows more fine structure, presumably due to the
fact that this curve has not been thermal averaged.
In these back-scattered orbits, one might wonder why they have to occur from the
entrance QPC. In fact, if you look at the trajectory in the first panel of figure 9.22(a),
and consider how this might look in an interior QPC, you can reach a striking
conclusion. If one rotates the array around the center point of an interior QPC, you
realize that rotating the trajectory of this latter figure would lead to a trapped
trajectory centered upon the interior QPC. This is not a state that arises from one of
the closed dot states, but is a state that can appear only in the extended dot array.
Nevertheless, it is an eigenstate of the dot array which does not couple to the
environment. The actual trajectory that is found to be stable in these dot arrays is
Figure 9.23. Comparison of a quantum simulation of the back-scattered trajectories with the experimental
data from [79]. The labels correspond to various trajectories shown in figure 9.21. (Adapted with permission
from [79]. Copyright 1997 the American Physical Society.)
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Figure 9.24. (a) Classical trajectories for the bipartite state that exists at a QPC interior to the dot array.
Multiple trajectories correspond to the states in the basin of the attractor. (b) The quantum wave function
corresponding to this bipartite state.
slightly more complex, but not much. The classical and quantum versions of these
internal reflecting states are shown in figure 9.24. These new states are distributed
over two interior dots and have been termed ‘bipartite states’ [14]. These states are
found to be quite robust, and exist over a range of variation in the QPC parameters,
and are also found to correspond to a classically stable trajectory. However, instead
of corresponding to a KAM island state, it is found that the bipartite states
correspond to an island of stable attractors. That is, every attractor has its own
basin, where all the trajectories within this region of phase space become attracted.
In some sense, the basin of attraction corresponds to the size of the bipartite state in
real space. This suggests that the bipartite states also arise from the same einselection
process that is important for the regular pointer states, but are hybrid states that can
only arise in a dot array.
9-35
9.2.6 Quantum Darwinism

The promotion of certain information in a quantum system due to a natural
selection process is known as quantum Darwinism [16, 21]. It is inspired by the
Darwinian concept [80] for the rules of reproduction, heredity, and variation. The
pointer states are characterized not only by their robustness, despite the existing
environment, but also by their ability to create ‘offspring’ of the states, which means
that they advertise information about themselves. This ability makes it possible for
different observers to measure the same information. The resulting objectivity arises
from the classical states. That is, in order to measure a quantum system objectively,
one has to design a system where the transition between the classical and quantum
world is observable. The open quantum dots we have been studying in this chapter
are ideal for this purpose. In particular, the hybrid states are the easiest to study in
this regard.
Blume-Kohout and Zurek [21] studied quantum Darwinism in zero-temperature
quantum Brownian motion, where they partitioned the environment as a whole into
smaller subspaces and then observed the imprint of the pointer states on these
individual subspaces. This observation demonstrates the objective existence of the
pointer states. Similarly, we can study the transmission amplitudes of individual
propagating modes through the QOCs. These modes can be seen as being analogous
to the individual subspaces in [21]. Our calculations basically show that all of the
transmitting modes become resonant at the same time. That is, by looking at the
imprint of the bipartite pointer state, shown in figure 9.24, in the individual
propagating modes, we gain confirmation that offspring of the robust state exist
(other modes yield similar results). This is illustrated in figure 9.25 for a two dot
Figure 9.25. (a) Conductance for a single-dot (red, thick dashed line) and a two-dot array (black, thick line).
Insets: probability density of the quantum wave function (at the solid blue circles) for regular pointer states (i)–
(iii), and for the bipartite states (iv)–(v). (b) probability density of the quantum wave function for the 1st, 2nd,
6th, and 14th modes at (iv). A magnetic field of 0.2 T is present in all cases. (Adapted from [14]. Copyright
2008 by the American Physical Society.)
9-36
array, in which the bipartite state of figure 9.24 can be studied in the region between
the two dots. For every mode, we find a display of the bipartite state. Therefore, the
information is objective [16], and every ‘observer’ is able to see the same result.
Consequently, quantum Darwinism is in action in the open dot array. Now, by
observer, we point out that in transport, most of the conductance occurs at the
Fermi energy. In these open dots, the Fermi energy is set by the quasi-two-
dimensional electron gas that defines the environment. By varying the opening of
the QPC, we are adjusting where the Fermi energy sits relative to the harmonic
oscillator states of the transverse potential of the QPC. Thus, as we sweep from a
negative to a more positive QPC potential, we are moving the Fermi energy from the
lowest harmonic oscillator state to higher lying states. These states are the modes
that we probe with the conductance. Thus, the bipartite state is present in each and
every transverse mode of the QPC. We regard the first mode as the ‘mother’ state,
and the replicas seen in each higher order mode the daughter, or replica, of that
mother mode. Each of these modes represents a different ‘subspace’ of the system,
which is examined as the Fermi energy is swept through the modes. It is thus felt that
this system represents a valid theoretical and experimental test bed for the ideas of
quantum Darwinism.
9.3 Imaging the pointer state scar

In chapter 2, we first discussed the use of SGM, whereby a rastered AFM tip is
modified by metallizing the tip so that a voltage can be applied to the tip [81, 82].
This voltage is used to perturb the local surface potential and carrier density, while
monitoring the conductance of the sample as a function of tip position. This
generates a map of conductance change, which should be proportional to the local
density near the probe tip. The use of this approach with the quantum dot is
illustrated in figure 9.26, where we simulate the conductance change as the biased tip
is rastered over the dot. In this case, the negative bias tends to break up the scar and
push carriers into the current-carrying states, thus raising the conductance. This
development has allowed one to probe the interior of quantum structures with the
intent of mapping the local density around the high regions of probability density
given by the magnitude of the wave function. This is based upon the idea that this
property of the wave function should correlate with peaks in the local density. In
particular, this has allowed the investigation of quantum behaviors within quantum
dots in heterostructures such as are used here [83, 84]. These early attempts did not
yield clear images, perhaps due to the larger effective mass that the electrons have in
GaAs, or the large dot size used. We subsequently achieved a better image of what
was believed to be a real scar of one of the pointer states in a quantum dot in
InAs [85]. InAs has a larger g value than GaAs and will therefore have a more
pronounced response to the magnetic field. To achieve this result, however, the fact
that the scar recurs periodically in the magnetic field was used to make
the measurement. We recall that the pointer states survive opening the dot to the
environment primarily because they do not couple to the environment. With
the SGM, we are bringing the environment into the dot itself. There is certainly
9-37
Figure 9.26. Simulation of scanning conductance microscopy of the scar in an open quantum dot. Here the
biased tip breaks up the scar, pushing carriers into the current-carrying states and raising the conductance.
Animation available at https://iopscience.iop.org/book/978-0-7503-3139-5.
the danger that this will destroy the pointer states during the scan, but this did not
seem to be the case. Another problem is that the classical trajectory correlating with
the pointer state will deform in the presence of the tip voltage. The voltage creates a
local minimum in the potential which deflects the trajectories. Hence, the scar was
imaged by making a large series of SGM scans with a small change in magnetic field
for each scan. This image was then Fourier transformed in magnetic field, and these
Fourier transforms searched for periodic parts. Thus, the transform could be filtered
around discrete frequencies, and the filtered transforms then used to generate a real
space image corresponding to that magnetic frequency. It was these images which
showed the presence of the scars appropriate to the dots.
What we would like to have seen, however, was the scar represented by figure 9.4.
This is clearly one of the dominant pointer states. But, the InAs dot used above had
the QPCs centered along the central line of the dot, and this would couple the scar of
figure 9.4 to the environment. This should destroy that particular pointer state,
although the resultant image obtained had significant similarity to that of figure 9.4.
Nevertheless, a search was continued to find the state for the dot with the QPCs
along the top, as indicated in this previous figure. For this approach, a 12 nm InAs
quantum well was imbedded with an Al0.7Ga0.3Sb strain relaxed bottom cladding
layer and a thin In0.2Al0.8Sb top cladding layer [86]. The sample was then etched to
produce a ribbon of InAs with etched trenches to isolate the dot region within the
ribbon. From Shubnikov–de Haas measurements it was determined that the density
and mobility were 1.5 × 1012 cm−2 and 65 000 cm2 V−1s−1 at 280 mK. The dot size
9-38
Figure 9.27. (a) The measured magnetoconductance through the dot as a function of magnetic field. The inset
is an AFM scan of the dot itself. The dashed rectangle is the SGM scan area. (b) The conductance fluctuations
from a quantum simulation of the dot. (Reprinted with permission from [87]. Copyright 2010 the American
Physical Society.)
after patterning was nominally 1.1 × 1.1 μm2, while the QPCs had 0.45 μm openings
and a length of 0.1 μm. The measured magnetoconductance is shown in figure 9.27,
where an AFM image of the dot is shown as an inset [87]. We also performed a full
quantum simulation of the quantum dot, using the techniques discussed previously,
and the conductance fluctuations that were computed are shown in figure 9.27(b).
The SGM was a metallized AFM tip as mentioned above. In this case, we used a
piezoresistive AFM tip which was coated with 15 nm of Cr. The tip was rastered
across the quantum dot at a lift height of 100 nm and with a bias of −0.7 V. This
provides a small perturbation to the local density, which results in a conductance
change of less than 0.2G0 (where G0 is the ubiquitous 2e2/h associated with quantum
conductance studies). The dashed line in the inset image of figure 9.27 indicates the
scan area over the sample. A series of SGM images was taken every 10 mT from 0 to
350 mT. Each such image consists of 120 raster scans across the dot area. Of course,
features within the scan area are masked by the conductance background. To
remove this, the magnetic Fourier transform approach described previously was
then used. Because the conductance is symmetrical in a magnetic field, a cosine
transform was used. A recurrent feature was found at a magnetic frequency of
13.5 T−1, and the filtered transform at this frequency was inverted to yield a real
space image. This state is shown in figure 9.28(a), and strongly resembles the scarred
state of figure 9.4, although there are significant differences. This leads to the
question of why the SGM images are not better representations of the scarred state
of figure 9.4. The answer lies in the perturbation that is affected to the quantum
wave function by the rastered tip potential. To probe this effect, we have carried out
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Figure 9.28. (a) Resulting SGM image determined from the Fourier analysis. (b) Simulation of the SGM
modulated conductance through the dot. ((a) reprinted with permission from [87]. Copyright 2010 the
American Physical Society. (b) reprinted with permission from [30]. Copyright 2011 IOP Publishing.)
a simulation with the dot and the rastering potential tip. In this case the conductance
is determined at each point of the tip during its raster scan, and the resultant
simulation for the SGM scan is computed. This is shown in figure 9.28(b), where it
can be compared with the experimental image. Note that the amplitude of the wave
function is larger at the four corners, which are points of reflection as there is a
turning point in the classical trajectory. This larger amplitude is also seen in the
experimental image. In areas where the scar is weaker, there is a corresponding
weaker response to the SGM sweep. Another point to note is that there is no signal
around the edges of the dot, where the SGM tip potential cannot be distinguished
from the confining potential. But the strong correlation between the experimental
image and the simulated image is strong support for the type of pointer state shown
in figure 9.4.
Problems
1. In figure 9.20, it is apparent that particles (and the wave function) can
correspond to an entrance into the dot with a direction that is not along the
axis of the QPC. Using the image of figure 2.4, explain how this occurs for
QPC modes other than the lowest.
2. Using the model of equations (9.1) and (9.2), construct a potential for a
single quantum dot with two QPCs aligned along a central axis. Assume that
the Fermi energy within the dot is 15 meV above the bottom of the potential,
while it lies 7 meV above the saddle potential maxima in the QPCs. The size
of the dot at the Fermi energy is to be 0.3 μm in the transverse direction.
Make a three-dimensional plot of this potential. How many modes traverse
the QPCs? What is the width of the QPC at the Fermi energy?
9-40
3. Consider a physical system, which is of course subject to a differential

equation between position and time. By dividing the time into discrete
increments, this equation can be written as
xn+1 = 1 − axn2 + bxn−1,
where n corresponds to the time increment. Starting from say x = 1, make a
plot of xn+1 as a function of xn for a = 1.4 and b = 0.3.
4. Consider a physical system, which is of course subject to a differential
equation between position and time. By dividing the time into discrete
increments, this equation can be written as
xn+1 = rxn(1 − xn),
where n corresponds to the time increment. Plot each value xn+1 as a function
of r, as r is varied from 2.5 to 4 in steps of 0.01. Plot at least 100 values for
each xn+1 at each value of r.
5. Consider the potential surface created in problem 2. At what value of the
magnetic field does a ‘particle’ make a complete cyclotron orbit at the Fermi
surface within the dot? Assume the effective mass is that for GaAs.
6. For the results of problem 5, estimate the mobility that is required for the
‘particle’ to complete the cyclotron orbit at the Fermi surface.
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176801
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IOP Publishing

(Second Edition)
David K Ferry
Chapter 10
Hot carriers in mesoscopic devices
In the preceding chapters, we have often discussed the role of decoherence, which is
a process where the quantum-mechanical interference properties of a mesoscopic
system decay, and the system begins to behave classically. The characterization of
this phenomenon is based upon measurement of the phase-breaking time τφ of the
system, which determines the temperature (and length) scales over which quantum
interference is observed. Generally, the phase-breaking time arises from the inelastic
scattering processes in the device. But, these inelastic processes give rise to another
characteristic time when the system is driven out of equilibrium, for example by
passing a large current through it. In this latter case, the system returns to
equilibrium by a process called energy relaxation. The energy-relaxation time τε is
another parameter of importance as it describes the characteristic time for the
system to return to equilibrium. There have been many experimental and theoretical
studies directed toward understanding the process of dephasing and its variations
with lattice temperature, and these have been discussed in the preceding chapters.
There are few studies, however, in which both the phase breaking and energy
relaxation have been studied in the same system [1–3]. Generally, such measure-
ments provide us with the experimental means to determine the equivalent carrier
temperature that arises from the heating by the current or voltage, and to determine
the effective relaxation times, as well as studying how these two times differ.
The study of carrier heating in mesoscopic devices is relatively old, but not as old
as the study of carrier heating in bulk materials. The latter were first studied to try to
ascertain the breakdown properties of dielectrics [4], as well as silicon [5]. This came
to mesoscopics through studies of the detailed properties of Si MOSFETs at low
temperatures, where electron heating [6] and velocity saturation [7] were observed,
just as in bulk materials, but with different characteristics.
Studies of the phase-breaking time generally involve varying the lattice temper-
ature T and studying how the phase breaking process depends upon this

temperature. The phase breaking process is then obtained from various effects, such
as weak localization or the change in conductance fluctuations as a magnetic field is
varied.
Studies of energy relaxation involve heating experiments where Joule heating of,
for example, a 2DEG is used to extract energy-loss rates and the energy-relaxation
time. In order to analyze the heating, and to determine things like the average power
input per electron and the energy-relaxation time, one needs to have a ‘thermom-
eter’, a mechanism by which the actual electron temperature can be determined from
the measurements. One of the earliest methods was to measure the dependence of the
mobility on both the electric field and the lattice temperature (at low electric field)
[8]. At low temperatures, particularly in bulk material, the mobility is dominated
by scattering from the ionized impurities. In this interaction, only a single temper-
ature—that of the carriers—is involved, so that variation of this temperature by
either of the two excitations leads to a scale that can be used to convert the mobility
at a given electric field to that at a given lattice temperature. Hence, the carrier
temperature is determined. As high mobility material became available, the effect of
the impurities was dramatically reduced by the use of modulation doping and the
resulting increase in effective screening. Thus, the mobility ceased to be an useful
method in mesoscopic studies. Alternatively, it was shown that using the
Shubnikov–de Haas oscillations provided an effective thermometer. Here, varying
the temperature and applied fields could provide a good measure of obtaining the
carrier temperature from the amplitude of the oscillations [8]. This has remained the
method of choice for estimating the carrier temperature. If these are studied at
various temperatures and currents, these two sets of data may be used to extract the
temperature dependence of the energy-loss rate P(T ), or equivalently the power
being dissipated within the mesoscopic system. This appears in the literature in two
forms, the first being the loss per electron, and the second is just the total loss in the
device. Obviously, these differ by the area or volume of the device and the carrier
density at which the measurements are made. This energy-loss rate is normally a
function of a power law of the hot carrier temperature, and the exponent often
considered to be an indicator of the type of electron–phonon scattering that is
dominating the inelastic process in the device. In spite of years of work in both
normal semiconductor systems, as well as mesoscopic devices, there is not yet a clear
understanding of the entire process.
In this chapter, we will present the approach that is most often used in the study of
hot carriers in these systems, along with some of the available experimental data to
illustrate the approach, and that of the general field of hot carriers.
10.1 Energy-loss rates

As mentioned, one traditional approach to the study of energy-loss rates involves
measuring the Shubnikov–de Haas curves of the sample under study, both as a
function of the lattice temperature and the excitation current (or voltage) of the
sample. We illustrate this for a heterostructure grown on a lattice matched InP
substrate. This consists of a 25 nm wide InGaAs quantum-well sandwiched between
10-2
two InAlAs barrier layers. The top barrier layer consists of a 30 nm n-doped layer
and a 10 nm undoped spacer layer (closest to the InGaAs quantum well). The
bottom barrier layer is a 250 nm undoped layer. The doping in the top barrier
provides sufficient carriers in the quantum well to populate two subbands, but this
does not affect the experiment. The total electron concentration was 1.3 × 1012 cm−2
with a mobility of about 140 000 cm2 V−1s−1. Measurements were made at a base
temperature of 4.2 K. A micrograph of the sample is shown in the inset to
figure 10.1. As one can see, it is configured as a Hall bar, with current passed
through the end leads and the voltages measured on the side arms. Data are recorded
for various lattice temperatures in the range 4.2–30 K with an excitation current of
500 nA. Then sample heating experiments are conducted with currents ranging from
0.5–250 μA. The measurements for varying temperature are shown in the main panel
of figure 10.1. These measurements clearly show a dual frequency (in 1/B) behavior,
by which the density in each of the two subbands may be determined. The interest is
in the major peaks that appear in the plot and how these change with temperature,
as indicated by the arrow in the figure. Three of these peaks, for 1.6, 2.0, and 3.2 T,
were carefully measured for their temperature dependence. At each magnetic field,
the neighboring peak and trough are both measured, and this difference is used as
the appropriate measure. Then, the measurements were made at base temperature
with varying current. There is a small magnetic field dependence on the carrier
Figure 10.1. Magnetoresistance measurements in the 2DEG show Shubnikov–de Haas oscillations. As the
temperature rises, the amplitude decays as indicated by the arrow. The curves are for 4.2, 6, 8, 10, 15.5, 20 and
30 K. The inset is a micrograph of the sample. (Adapted with permission from [3]. Copyright 2003 the
American Vacuum Society.)
10-3
heating, so using these three values of magnetic field also in the current-dependent
measurements allows one to both gain more data points and to compensate for the
small field dependence. The resulting change in the amplitude of the oscillations is
used to correlate a particular temperature with a particular current. This comparison
is shown in figure 10.2. Here, one can see that at the lowest current level, there is no
apparent heating of the sample. The straight line drawn in this figure suggests that
the temperature varies approximately as the square root of the current.
When measurements such as these are made at low temperatures, one cannot
generally assert that the phonons are within the equipartition limit, where the full
details of the Bose–Einstein distribution can be ignored. This leads to an important
point, by which we may define the Bloch–Gruneisen temperature. In general, the
largest q vector for a phonon that can be generated in an electron–phonon scattering
event is the one that spans the Fermi surface from one side to the other, basically a
value of 2kF. With this wavelength, the maximum acoustic phonon energy is then
2ℏkFs where s is the sound velocity. The issue is whether or not the exponential in the
denominator of the Bose–Einstein statistic can be expanded to simplify the
expression. The argument of the exponential is the ratio of the phonon energy in
the collision to the thermal energy. Thus, the critical temperature is said to be the
Bloch–Gruneisen temperature
Figure 10.2. Electron temperature as a function of the drive current. The black circles, red squares, and blue
triangles denote data centered at 1.6, 2.0, and 3.2 T. (Adapted with permission from [3]. Copyright 2003 the
American Vacuum Society.)
10-4
2ℏkF s
TBG = , (10.1)
kB
For temperatures below this, the full Bose–Einstein distribution has to be kept, and
this complicates the determination of the loss to phonons. So, for a density of
1012 cm−2, one finds that in a 2DEG for a typical semiconductor, the Fermi wave
number is 2.5 × 106 cm−1. The sound velocity in InGaAs is about 4.7 × 105 cm s−1,
so that the Bloch–Gruneisen temperature is about 10 K. Now, this straddles the
temperature range involved in the experiments, so that it is unsafe to do anything
other than keeping the full Bose–Einstein distribution. When this is done, it has been
shown that the general formula for the energy-loss rate can be written as [9]
P(Te ) = F (Te ) − F (T ) = A(Tep − T p) , (10.2)
where F is a general function, and A is a combination of material parameters and
fundamental constants. The exponent p (and the prefactor A) depend upon the
specific scattering mechanism that governs energy relaxation. For example in
heterostructures for the III–V compound semiconductors, one expects p = 7 for
acoustic phonon interactions, and p = 5 for the screened piezoelectric interaction,
and p = 3 for the unscreened piezoelectric interaction [10].
Using the above approach, the data for the devices discussed above are plotted as
a function of the inferred electron temperature in figure 10.3. The data come from
105
Power Loss Per Electron (eV/s)
104
103
102
5 6 7 8 9 10 20
Temperature (K)
Figure 10.3. The power dissipated per electron for the devices of figures 10.1 and 10.2. The circles are for a
magnetic field of 1.6 T, the squares for 2.0 T, and the triangles for 3.2 T. (Adapted with permission from [3].
10-5
the three values of magnetic field that were used in figure 10.2. It may be seen that
the input power per electron all fits on a single curve, although it is difficult to see
that this curve fits to p = 3 (or any other value). Nevertheless, the slope of the curve
at the higher values of electron temperature do correspond to this value of the
exponent. To see this better, we re-plot the data in figure 10.4 as a function of the
temperature expression in the parentheses of equation (10.2) with p = 3. Here, it may
be seen that the fit to the power law is very good, although there is a little more
scatter in the data. However, it may be seen that slope agrees with each of the three
sets of data for different magnetic fields. Thus, it may be inferred that the major
source of the energy relaxation of the excited carriers is due to phonons via the
unscreened piezoelectric interaction.
In figure 10.5, we show the energy loss per electron for a bilayer of graphene [11].
Here, the samples were prepared by exfoliating Kish graphite onto a doped Si
substrate upon which 300 nm of SiO2 were grown. Monolayer and bilayer devices
were formed in different samples by this process, with the number of layers being
confirmed by optical microscopy and Raman spectroscopy. The samples were
contacted with Cr/Au electrodes in a six probe configuration which allowed the
electric field to be determined in a manner free from the effect of current contact
resistances (as is done in the Hall effect, discussed in chapter 6). The samples
typically exhibited mobilities in the range 103–104 cm2 V−1s−1 at low temperature.
The measurements in the figure are on a bilayer device with a base temperature
106
Energy Loss rate per Electron (eV/s)
105
104
103
102
101 102 103 104
Te3-TL3
Figure 10.4. The power loss per electron from figure 10.3 re-plotted so that it corresponds to the form of
equation (10.2), with the parameter p equal to 3.
10-6
Figure 10.5. The carrier power loss (per electron) as a function of temperature according to the form of
equation (10.2) at different gate voltages (color coded as indicated). The inset compares the data for slopes of
different p values, showing that 3 is the most likely. This is indicated in the main panel as the dotted line.
(Reprinted with permission from [11]. Copyright 2013 American Chemical Society.)
of 1.8 K. In this case, the samples exhibited conductance fluctuations, as discussed in

chapter 5. As these fluctuations varied with the base temperature and the current
passed through the sample, the amplitude of the fluctuations was used as the
thermometer to ascertain the electron temperature as a function of the drive current
through the device. In the main panel of the figure, the input power per electron is
plotted as a function of the temperature relation of equation (10.2) with p = 3, the
data sets corresponding to different gate voltages (bias on the Si substrate), whose
values are shown by the color coding indicated in the figure. The inset shows
different values of p for the same data, with each value shown as a dotted line. The
conclusion is that the best fit is to a value of p = 3. While this is the same exponent
found in the InGaAs heterostructure discussed above, the interpretation of this is
much different. First, graphene is not piezoelectric, so the loss process found above
does not exist in this material. More importantly, the unique Dirac-like band
structure of graphene changes the exponents found for the various scattering
mechanisms. For loss via the acoustic phonons, one expects an exponent of 4 in
graphene [12–14]. In earlier work, the Oxford group had examined the energy-loss
process in chemical-vapor deposition grown graphene on the Si-terminated face of
SiC [15]. In this latter work, the authors claimed a dependence with p = 4, although
some of the data suggest that it was closer to p = 3. Similar results were recently
found in bilayer graphene as well [16]. The data in figure 10.5, however, clearly do
not fit with p = 4, and have a much better fit to p = 3. Very similar behavior was
10-7
found for monolayer graphene in several samples. It has been suggested that a
second-order impurity mediated phonon scattering could lead to an exponent of
3 [17]. However, it was found that the strength of this interaction was much too
weak to explain the data in the figure. More recently, it has been found that
plasmon-mediated energy relaxation can explain the energy-loss rate seen in this
latter figure [18]. Plasmons are excitations of the electron gas itself, so the plasmon
loss process actually only redistributes the energy among the electrons. But, in the
presence of current, the Fermi sea moves to account for the current flow and is no
longer centered about the zero of momentum. At high currents, this becomes highly
accentuated, with many electrons above the equilibrium Fermi energy. Hence, the
plasmon process accounts for scattering of these excited electrons back to states
mostly vacated by the shift in the Fermi sea. This process reduces the current and the
energy input to the electron gas, but ultimate relaxation of the energy must occur by
some other process, such as the acoustic phonons. The derivation of the plasmon
energy loss follows equation (10.2), with the matrix element [18, 19]
2πN ⎧ V (q ) ⎫ ⎛ Ek − Ek±q ⎞
W± = Im⎨ ⎬δ⎜ω − ⎟, (10.3)
ℏ ⎩ ϵ( q , ω ) ⎭ ⎝ ℏ ⎠
where N is either the Bose–Einstein distribution, in the absorption term, or 1 plus

this distribution, in the emission term. The scattering function arises from the
imaginary part of the screened Coulomb potential V (q ) = e /εsq , where e is the
electron charge and εs is the permittivity of graphene [20, 21]. Here, the screening is
done by the full frequency- and momentum-dependent dielectric function. As may
be expected, the dielectric function is dominated by the plasmon pole, but in using
the inverse of the dielectric function in equation (10.3), we incorporate the collision
broadening of this resonance. The broadening arises from the total scattering rates,
which yield a scattering time τ. Carrying out various integrations leads to the
relaxation rate
dE 2.4k B3 3
P (E ) = − = (Te − T 3) . (10.4)
dt 2π ℏ2v F2τ
Since the mobility is easily measured, the relaxation time can be determined for the
sample [18, 19], so that equation (10.4) contains no adjustable constants. The
electron temperature can be determined either from thermometer type measure-
ments discussed above, or from using the definition of the energy relaxation time,
discussed in the next section. If we know the latter quantity from experimental
measurements, then the temperature is given by
kB(Te − T )
P (E ) = . (10.5)
τE
In figure 10.6, we compare the variation of τE(Te), obtained in the experiment of [11],
with that predicted by equation (10.4) for the base temperature of 1.8 K, as used in
the experiments. Theoretical values of τE are determined by taking the
10-8
Figure 10.6. The energy relaxation time for several values of the electron and hole density. Experimental
values from figure 10.5 [11] are plotted as open symbols while theoretical values obtained from equations (10.4)
and (10.5) are the closed symbols. Error bars are shown for the experimental data only. Dotted lines through
the theoretical data are guides to the eye that indicate a linear variation of power input with T 3. In each case,
the red and green curves have been offset by a factor of 10 for clarity as indicated. Reprinted with permission
from [18], copyright 2015 by AIP Publishing.
experimentally-determined energy-loss rate for the given density and Te and

introducing this into equations (10.4) and (10.5) to compute τE. From this figure,
we see that the plasmon-based model reproduces not only the low-temperature
magnitude of τE, for both electrons and holes, but that it also captures its
quantitative dependence on temperature.
The plasmon-loss mechanism rearranges the energy distribution function, but this
energy remains in the electron gas as a whole. Under the excitation current (or
electric field), the distribution function is shifted in momentum space to reflect the
current, and it spreads as the electron temperature rises. The shift can also distort the
distribution to produce an elongated extension in the direction of the field or current.
Between the shift in momentum space, and the elongation, the distribution function
has more carriers in the forward direction than the backward direction. The plasmon
loss takes carriers from the streaming forward direction and moves them to the
10-9
backward direction, in the symmetric terms of the distribution function, on average.

This scattering process relaxes the streaming distribution, but the ultimate result still
remains that another process must pass the energy to the lattice. This can be due to
acoustic-phonon scattering, or even the so-called supercollisions of [17]. However, it
is the plasmon interaction that dominates what is seen experimentally in transport,
as it is this process which moves the current carrying particles out of the forward
extension of the distribution. On the other hand, optical measurements will see only
the symmetric part of the distribution, so may well give other variations with
temperature.
10.2 The energy-relaxation time

The energy input per electron is just one measure of the manner in which the energy
relaxes to the lattice in a timely manner. The strength of this measure is the exponent
that appears when matching to equation (10.2), as this exponent is an indicator of
the dominant relaxation process. However, there is another measure that is
important and that is the energy-relaxation time itself. The idea of an energy-
relaxation time is linked to the study of hot electrons in semiconductors in which the
dynamics of the process is developed by a set of moment equations [22]. That is, the
Boltzmann transport equation is multiplied by powers of the momentum and
integrated to yield a set of equations for the density, the momentum, the energy,
and so on. Each of these is characterized by an appropriate relaxation time, which
can be determined by taking the corresponding momentum of the scattering-induced
changes of the distribution with time. For a simple approach, we can use these
results to write an expression relating the input power per electron to the energy-
relaxation time which is just equation (10.5). This expression now defines the energy-
relaxation time from the measured power input per electron and the inferred electron
temperature achieved at that power input level.
In figure 10.7, we plot the energy-relaxation time that was found for the three
different values of magnetic field in the InGaAs 2DEG discussed above [3]. It may
be seen from this figure that a difference now appears between the lowest magnetic
field and the other two values, which presumably arises from the tendency to form
edge states (associated with the quantum Hall effect) more effectively at the higher
magnetic fields. The edge states are quite similar to one-dimensional quantum wires
in which the motion is only allowed to be in one direction, which hinders back-
scattering processes (discussed in chapter 6). The lines are guides to the eye, with
different slopes, but are indicators of the relaxation process. At higher temperatures,
the fit appears to be close to the T −3 curve, and this behavior has been seen in clean
systems where the energy is thought to be relaxed by three-dimensional electron–
phonon scattering processes [2, 23]. At lower temperature, a transition is seen to an
alternate T −1 behavior at all values of the magnetic field. The cause for this behavior
is not fully understood at present and is not expected from the p = 3 behavior seen in
the input power curves, but it could be the quantum wire behavior of the edge states.
In figure 10.8, the energy-relaxation time is shown for three nanowires fabricated
in the InGaAs heterostructure discussed above [24]. In this case, the wires were
10-10
T-3
T-1
Energy Relaxation Time (s)
T-1
10-7
10-8
2 4 6 8 10 30
Electron Temperature (K)
Figure 10.7. The energy-relaxation time determined for the InGaAs device discussed in the previous section.
The blue diamonds, red squares, and green circles are for a magnetic field of 1.5 T, 2.0 T, and 3.2 T,
respectively. (Adapted with permission from [3]. Copyright 2003 the American Vacuum Society.)
formed by an isotropic etch to remove the InGaAs and upper layers to yield the
wires. Each end of the wire is connected to large two-dimensional reservoirs.
However, only two-terminal measurements can be made on the wire, so that the
results are not guaranteed to be free of contact resistances that form at the reservoir–
wire transition. The measurements in the figure are for three different wires, of 215
nm, 545 nm, and 750 nm widths. The electrical widths were inferred from the
magnetic field at which the onset of Shubnikov–de Haas oscillations were observed
in the narrowest wire. This value was then extrapolated to the other wires. The wider
wires show relaxation behavior that is more two-dimensional than one-dimensional,
but the narrower wire clearly shows a difference, both in the magnitude and the
slope of the relaxation time. The solid curve in the figure shows the T −3 behavior
observed previously, but it is not at all clear whether any of the wires demonstrate
the T −1 behavior (dashed curve). In fact, the narrower wire may be approaching a
constant value at the lowest temperatures. Interestingly enough, the narrow wire
shows a power per electron that appears to also match p = 3, much as the larger
samples discussed above.
The energy-relaxation time found in graphene samples [11] is displayed in
figure 10.9 as a function of the electron density, rather than the temperature.
Here, there are curves for both electrons and holes separately, and the data are
plotted with the input current level as a parameter. The shaded area around the
Dirac point represents a region where the density cannot be accurately ascertained
10-11
Figure 10.8. The energy-relaxation time found in etched InGaAs nanowires for widths of 215 nm (blue
diamonds), 545 nm (black circles), and 750 nm (red squares). The two wider wires are thought to be narrow
two-dimensional devices rather than true one-dimensional wires. (Adapted with permission from [24].
from Hall measurements and is inferred from the capacitance and gate voltages.
Outside this region, the two measurements give comparable values. The data are
shown for a bilayer graphene device, but the variations shown are quite generic and
seen in all the monolayer and bilayer devices studied. The remarkable behavior
indicates that the relaxation time actually decreases rapidly as one approaches the
Dirac point from either the conduction or valence band. While unexpected, this
behavior can be explained by the plasmon-mediated energy-loss mechanism
discussed above, since the total power input is independent of the electron density,
due to the unique nature of the Dirac band structure and the resultant dynamic
effective mass of the carriers. Hence, the power input per electron increases at lower
density, and this leads to a reduced energy-relaxation time from equation (10.5), just
as seen in the figure. It may also be due to the fact that puddles of electrons and holes
form in graphene near the Dirac point, which likely will complicate the relaxation
process [25]. In figure 10.10, the energy-relaxation time is plotted for two electron
densities as a function of the electron temperature, and compared with the
theoretical prediction for the plasmon-mediated process of energy-relaxation. The
open symbols are data taken from that used in figures 10.5 and 10.9, while the closed
symbols are the theoretical values obtained. It is clear that the relaxation time
decreases with increasing electron temperature, but this decay is not a simple integer
value. Rather, it appears the decay in the theory is roughly T −1.75. This is not far
10-12
Figure 10.9. The energy-relaxation time for a bilayer graphene device at a lattice temperature of 1.8 K. The
various curves are for different values of the excitation current, as indicated at the lower left of the figure.
(Reprinted with permission from [11]. Copyright 2013 American Chemical Society.)
from the T −1.5 reported in [11]. It is also clear that the relaxation time decreases
rapidly as a function of the density, just as indicated in figure 10.10.
10.3 Nonlinear transport

When high electric fields are applied to a semiconductor, the resulting current
becomes nonlinear, and is accompanied by a reduction in the differential mobility.
In this situation, the system has moved into the far-from-equilibrium condition. The
distribution function, characterizing the carriers, is quite far from its low-field
Boltzmann form. In this nonlinear transport regime, the ergodic approximation
fails. In particular, it fails because the distribution function itself is not a stationary
process. It varies in time and its form arises from a delicate balance between the
driving forces, the electric field usually, and the dissipative forces, the scattering. As
such, it really has no connection to the equilibrium form, and linear response
approaches generally do not work.
10-13
Figure 10.10. The energy-relaxation in a monolayer of graphene as a function of the electron temperature.
Here, the data are shown as the open symbols for two densities, and are taken from [11]. The closed symbols
are from the plasmon-mediated loss theory [18].
Early studies of high-electric field transport in solids focused mainly on the

breakdown studies of dielectrics [4], although the electrical properties of silicon had
been studied much earlier [26]. Studies of the variation of the velocity with high
electric fields began with Ryder [5], who discovered that the velocity in Si and Ge
saturated at high electric fields. The saturated (or nearly saturated) velocity is an
important parameter for electron device considerations. For example, with a
modern nanoscale MOSFET of say 20 nm gate length, an applied voltage of 1 V
produces an average electric field of 0.5 MV cm−1 in the channel, which is an
incredible electric field, especially when one considers that the saturation velocity
sets in at about 20 kV cm−1 in Si at room temperature. Then, in many so-called
figures of merit for high speed, high frequency, or high power devices, the saturated
velocity is an important parameter as it affects the frequency response and power
handling properties of the device. The saturated velocity is also important as a
mirror into the electron–phonon interactions in the material. Consequently, it
certainly affects the distribution function that is found at each electric field. It is
natural that these effects are important in mesoscopic devices, as the MOSFET
mentioned above can certainly be described as a mesoscopic device, or even a
microscopic device. In this section, we want to describe some of the effects that can
be studied in the nonlinear regime.
10-14
10.3.1 Velocity saturation

(Parts of this section are adapted from Semiconductor Transport (1st Edition), David
Ferry, 2000)
In nonlinear transport, one must deal with a multitude of characteristic times for
the semiconductor system. The importance of the various time scales is obvious due
to the necessity of evaluating the time dependence of the various measurable
properties. The non-equilibrium distribution function evolves(and is therefore
non-stationary) the time scale for variations of the measurable quantity. This
implies that the system is non-ergodic (by which is meant simply that time averages
do not equate to ensemble, or distribution, averages, the latter of which are the
important averages) over this time scale [27]. In fact, it is important to evaluate
carefully the various collection of time scales that are important. In a semi-
conductor, numerous collisions occur and it is these collisions that provide the
mechanism of exchange of energy and momentum and relax these quantities toward
their equilibrium values. There are collisions between the carriers which randomize
the energy and momentum within an ensemble but do not relax either of these
quantities for the ensemble as a whole. There are also elastic collisions between the
carriers and impurities or acoustic phonons which relax the momentum but do little
to relax the energy. Finally, there are inelastic collisions between the carriers and
lattice vibrations which relax both the energy and the momentum. In general, four
generic time scales can be identified [28, 29]:
τc < τ < τR < τH . (10.6)
Here the average duration of a collision is denoted by τc. Generally, this time scale is
quite short and not of importance to most considerations. However, on the scale of
fast femtosecond laser experiments, this may no longer be true. The collision
duration is the time required to establish the energy-conserving delta function (in
the Fermi golden rule for scattering). In distinction, the average time between
collisions, the mean free time, is denoted by the simple τ. For time scales such that t
⩽ τ, the evolution of the system depends strongly upon the details of the initial state.
Generally, τ ≫ τc, but this is not always the case at high electric fields, and the
breakdown of this inequality can lead to new transport effects, which must be
treated in a quantum mechanical manner.
The establishment of equilibrium, or a non-equilibrium steady state, can be
achieved within a few or a few tens of τ, and the characteristic time associated with
this process is the relaxation time τR. Typical quantities characterized by a relaxation
time are the momentum relaxation process, and in high fields the energy relaxation
process. If configuration-space gradients exist, the situation becomes more complex.
Relaxation in momentum space proceeds on the scale of τR and establishes a ‘local’
equilibrium over regions smaller than a macroscopic scale, perhaps only a few mean
free paths in extent. The achievement of a uniform equilibrium or non-equilibrium
steady-state requires a longer time, the hydrodynamic time τH > τR. Only for times
large compared to this hydrodynamic time can the ensemble truly be said to be
10-15
Figure 10.11. Ensemble Monte Carlo simulation of the velocity of electrons in silicon at 300 K. The actual
velocity saturation is slightly higher than that seen experimentally.
stationary, and only for times on this scale are the processes even beginning to
become ergodic. Examples of the hydrodynamic time scale are diffusion times,
arising from recombination processes for excess carriers as well as from local non-
homogeneous carrier distributions and sometimes energy relaxation times.
The main experimental observable of the hot carriers, at least in homogeneous
semiconductors, is the observation of their velocity saturation. When the carriers are
heated by the field, the temperature rises and the distribution spreads with more
carriers at higher energy. Since the scattering rate generally increases with energy,
there is more carrier scattering (to accommodate the relaxation of the energy input
from the field), the mobility is reduced and the velocity does not increase as rapidly.
In fact, in Si, the velocity appears to actually saturate at a value near to 107 cm s−1 at
a lattice temperature of 300 K. We show this in figure 10.11, where the results of a
Monte Carlo simulation are plotted. In this figure, the velocity is plotted for a range
of electric field.
When we study high field effects, many of the important processes set in at 100s
of kV cm−1, and it becomes difficult to make measurements on normal sized
samples. Hence, considerable effort is necessary to make structures in which the
needed high electric fields can be established at reasonable voltages. Such a structure
is shown in figure 10.12, used to study the velocity behavior in GaN [30]. In panel
(a), the dark areas are the GaN, which has been etched to provide the small wire like
structure at the center, and shown expanded in panel (b). In figure 10.13, we show
10-16
Figure 10.12. (a) Fabrication of a sample to prepare a high field region (the narrow ‘wire’). The GaN is the
dark region after etching the shape. (b) A blow up of the constriction over which the high electric field will
exist. Courtesy of J M Barker.
Figure 10.13. The experimental data for a GaN sample is plotted in the red curve as a function of the electric
field. The blue curve is the EMC calculation of the velocity, while the green curve is the fraction of carriers that
remain in the main conduction band valley (the Γ valley).
the velocity measured in these GaN samples as a function of the applied electric
field, for a sample with a constriction of size 15 × 3 square microns. The measured
velocities are compared to those computed via a three valley ensemble Monte Carlo
procedure [31]. Also shown is the fraction of carriers that remain in the main
conduction band valley as a function of the electric field. It may be seen that the
velocity reaches a peak at about 180 kV cm−1, and then begins to decrease. As may
be seen from the green curve, this is explained as arising when the carriers leave the
main Γ valley and move to satellite valleys in the conduction band, where the mass is
heavier and the velocity lower. We return to this point below.
10-17
10.3.2 Intervalley transfer

Transfer between different valleys of the full energy band is one of the oldest and
most interesting aspects of high-electric-field transport in semiconductors. There are
two possible methods by which transfer between nonequivalent sets of valleys can
occur. In one, the symmetry-breaking properties of the high electric field are used to
break the symmetry between valleys that are normally equivalent. This occurs, for
example, in Si at low temperatures when the electric field is not oriented at the same
angle with all six mimima of the conduction band. For this case, the carriers in each
valley are heated differently and a repopulation appears between the various valleys.
The second method by which intervalley transfer can play a major role is when
only a single conduction band minimum, with a small effective mass, is normally
occupied, such as in the case of GaN (the principal minimum is at the Γ point),
discussed in the previous section. With the application of a high electric field, the
carriers are heated to relatively high energies. Then, a fraction of the carriers will
actually be at energies above the minima of a secondary set of valleys [which lie
along the (111) directions at the set of M-L points of the Brillouin zone in wurtzite
GaN, and are some 1.3 eV above the Γ minimum] of the conduction band. Since, the
equivalent M and L minima (the difference in notation lies in the basal plane set or
the z-axis set of wurtzite) have relatively high values of the effective mass, their
density of states is much larger than that of the central minimum, and intervalley
scattering will cause electrons to move to the satellite valleys of the conduction band.
Consider, for example, the conduction band of InSb shown in figure 10.14. There is a
Figure 10.14. The band structure for InSb determined using a nonlocal empirical pseudopotential method that
includes the spin-orbit interactions. The arrows show the possible intervalley transfers for hot electrons in the Γ
valley.
10-18
central valley characterized by a small effective mass of 0.013m0. In addition, there

are subsidiary minima located at both the L point, lying some 0.55 eV higher, and at
the X point, lying some 1.0 eV higher. These valleys have a considerably greater
effective mass and density of states, as the former is Ge-like while the latter is similar
to Si. Under normal circumstances, the central valley is the only one occupied.
However, for an applied field of some 600 V cm−1 at 77 K, electrons begin to
transfer to the L valleys, a process first observed by Gunn in GaAs [32]. It was
identified clearly as intervalley transfer by Ridley and Watkins [33, 34]. In
figure 10.15, we show the average drift velocity and kinetic energy as a function
of the electric field in InSb at a lattice temperature of 77 K.
Intervalley transfer is seen in a great many semiconductors, particularly those
with a direct bandgap, so that the lowest lying conduction band is at the center of the
Brillouin zone. Recent examples include observations of this effect in atomically thin
transition-metal dichalcogenides [35, 36], and in the absorber material on AlInAs
hot-carrier solar cells [37]. As may be seen in figures 10.13 and 10.15, once transfer
begins, the velocity-field curve begins to show negative differential conductance
(NDC). The resulting NDC that occurs when the carriers are transferred from low-
mass, high-velocity states to high-mass, low-velocity states is often referred to as the
Gunn effect and is used in transferred electron devices (TEDs, which are used for
microwave sources). Transfer occurs as the carriers are heated in an external electric
field. Once some of the carriers reach energies near that of the satellite valleys (the L
and X valleys), inter-valley scattering can occur. Since the density of states in the
satellite valleys is much higher than in the central valley, inter-valley scattering from
the Γ valley to the satellite X and L valleys is a dominant process. The transfer in this
Figure 10.15. The average drift velocity and kinetic energy for InSb at a lattice temperature of 77 K are plotted
as a function of the electric field. The drop in velocity and energy above 700 V cm−1 is due to intervalley
transfer. The energy drops as kinetic energy is converted to the potential energy of the satellite valleys.
10-19
particular direction is much more pronounced than the reverse process (recall that
the scattering rate is essentially a direct measure of the density of final states).
Because of the higher mass and density of states in the satellite valleys, the mobility
and velocity are much lower and a negative differential conductivity will occur.
10.3.3 NDC and NDR

Both negative differential conductance (NDC) and negative differential resistance
(NDR) are both extreme nonlinear properties of various materials. The difference
between the two is mainly one of semantics and how one identifies the principle
variable (voltage or current) and the response to this. The non-equilibrium
thermodynamics of these two processes were described quite some time ago by
Ridley [38]. We illustrate this difference in figure 10.16. In NDC, the electric field is
the excitation, while the current density is the response. A given current may appear
at multiple values of the electric field, and this gives a characteristic ‘N’ shape to the
curve. As the curve is the current as a function of the electric field, it naturally
displays the conductance, and gives rise to a region of NDC. On the other hand,
with NDR, the current is the excitation, and the electric field is the response, since a
given electric field may exist at multiple values of the current density. This gives the
curve a characteristic ‘S’ shape. As the electric field is given as a function of the
current, it naturally displays the resistance, and gives rise to a region of NDR.
If we excite a semiconductor with NDC with an increasing voltage, the electric
field is usually quite homogeneous (we neglect contact effects) across the length of
the device until the peak of the current is reached. The NDC region is an
unstable region in which fluctuations in the electric field will lead to an inhomoge-
neous distribution of the electric field. For example, if we try to hold the electric field
in the NDC region (where the current is decreasing with an increasing electric field),
the thermodynamics will try to have one electric field on the low-field linear rising
part of the curve, and a second much higher electric field on the far-right side of
NDR
Current Density
NDC
Electric Field
Figure 10.16. Characteristic strong nonlinearities in the current density versus electric field for the cases of
NDC and NDR.
10-20
figure 10.16. Often there is no point on the curve, and the high electric field ‘domain’
is limited by the applied voltage V through
L
V=− ∫ E (x )dx , (10.7)

0
where the cathode is taken as x = 0 and the anode is taken as x = L. In many cases,
the high field domain is not stationary, but moves with the charge dipoles that
creates the high electric field. Then, it can cycle through the device with a period
L
T= , (10.8)
veff
Where veff is an effective velocity lying somewhere between the peak velocity and the
valley velocity (lowest value in the curve of figure 10.16). As most compound
semiconductors are piezoelectric, the very high electric field can lead to strain in the
crystal, and this can ultimately cause the sample to be fractured, as happens in GaN;
that’s why the experimental data ends where it does in figure 10.13. In other cases,
the high electric field can lead to impact ionization and avalanche breakdown; a
situation usually seen in InSb [39]. The onset of impact ionization can transition the
behavior to the NDR situation.
If we excite a semiconductor with NDR with an increasing current, the current
density is usually quite homogeneous (we neglect contact effects) across the cross-
section of the device until the peak of the electric field is reached. The NDR region is
an unstable region in which fluctuations in the current density will lead to an
inhomogeneous distribution of the current density across the cross-section of the
device. For example, if we try to hold the current density in the NDR region (where
the electric field is decreasing with an increasing current density), the thermody-
namics will try to form a filamentary current in the device. These filaments are
natural formations in these situations and are called micro-plasmas when observed
in the breakdown of p-n junctions [40, 41]. In materials like InSb, the density of the
electron-hole plasma that is generated by avalanche breakdown can even lead to a
plasma ‘pinch’ due to the induced magnetic field from the high current in the
filament [42]. This leads to an even high current density which may lead to intense
local heating and even melting that in turns leads to device destruction from the
trapped gasses, whether in a bulk material such as InSb or in a semiconductor device
during breakdown [43].
An important point about both NDC and NDR is that, if the appropriate voltage
or current is cycled slowly, then the device will exhibit hysteresis. For example, with
NDC, the operating point with rising voltage moves up the linear curve to the peak
of the current density, and then jumps to the curve on the right. In lowering the
voltage, the operating point does not retrace this path. Rather, if follows the right
positive conductance path down to the valley current density and them jumps to the
linear curve. Thus, the operating point exhibits a clockwise hysteresis in the plane of
figure 10.16. On the other hand, for NDR, the current leads the electric field to rise
10-21
to the peak in the linear region, then jump to higher current density as the filament
forms. In lowering the current, the operating point follows the curve down to the
‘valley’ electric field and then jumps to the linear curve, thus exhibiting and anti-
clockwise hysteresis. In particular, this hysteresis in NDR materials has led to
interesting memory devices [44, 45].
10.3.4 Velocity overshoot

In high electric fields, it is quite likely that the momentum relaxation time and the
energy relaxation time will differ, with the latter being the slower process. The
momentum relaxation time τm describes the decay of the velocity (and the velocity
fluctuations about a local near-equilibrium state). In essence it is the shift in
momentum space of the entire distribution that is required to give an average
velocity. However, the nonlinear transport in high electric fields arises primarily
from the change in the distribution function in the presence of the high electric field,
which leads to an increase in the average energy of the carriers. The response of the
distribution function, which results in this increase in average energy, is charac-
terized by its own relaxation time, which is referred to as the energy relaxation time
τE, since the evolution of the distribution function represents the evolution of the
average energy of the carrier ensemble.
If the energy relaxation process is slower than the momentum relaxation process,
the velocity can overshoot its ultimate steady-state value (the saturation velocity) in
high fields. This occurs because the distribution function first shifts (equivalent to the
shift studied in previous chapters) in momentum space as the velocity rises to a value
characterized mainly by its low-field mobility. As the distribution function then
diffuses in energy (or momentum) space to its non-equilibrium form, the mobility
decreases to its ultimate high-field value, with a consequent decrease in the velocity.
It can readily be shown that this ‘overshoot’ behavior requires a more complicated
behavior than the simple behavior characterized by a Langevin equation
dv eE v
= ⁎ − , (10.9)
dt m τm
where we have omitted the random force as it will average to zero. When overshoot
occurs, an equation such as equation (10.9) must have at least two zeros—one at a
time corresponding to the steady state and one at a time corresponding to the peak
velocity. However, the relaxation rate is an increasing function of energy (or
velocity), so that the right-hand side has only a single zero, at the steady-state
product of the field and the mobility. Thus a second time scale must be involved,
which is the characteristic time of the energy relaxation. One approach to this is to
let the motion of the particles be governed by a retarded Langevin equation, written
as [46]
t
dv eE
dt
= ⁎ −
m
∫ γ (t − u )v(u )du , (10.10)
0
10-22
where again the random force term has been omitted. The function γ(t) is a ‘memory
function’ for the non-equilibrium system, and has a time response related to the
energy relaxation time. In fact, this function can be written in the simplest form as
1 −t/τE
γ (t ) = e . (10.11)
τmτE
These results demonstrate a number of important aspects of hot-carrier behavior.
First, the dynamics become retarded with a memory effect because of the extra time
behavior corresponding to the evolution of the distribution function, the energy
relaxation time. This, in turn, opens the door for velocity overshoot to occur.
Moreover, this process, when coupled with the velocity saturation effect, clearly
indicates the far-from-equilibrium nature of this nonlinear transport. In the field
range where velocity saturation and velocity overshoot can occur, the distribution
function is determined by carrier-lattice interactions and by boundary conditions in
the form, for example, of applied fields, and is not simply related to the equilibrium
form. There is one caveat, however, and that is that the transient velocity (which
may be calculated in an ensemble Monte Carlo process) will be a result that depends
upon all the scattering processes.
In figure 10.17, we illustrate the transient response of electrons in graphene to a
high electric field at room temperature. Here, the graphene is deposited on a SiO2
(grown on silicon) substrate and has been patterned in a way to support a high
electric field, as discussed above, although these are theoretical data taken from
an EMC simulation of the transport. Because of the relatively high scattering rates
of the surface optical phonons in SiO2 [47] which strongly affect the graphene
Figure 10.17. Transient velocity for electrons in graphene on SiO2 at various electric fields.
10-23
transport [48], the overshoot lasts for less than a single picosecond. The graphene
shows a peak velocity at about 15 kV cm−1 and the shows the onset of a weak NDC
beyond this field. At the three electric fields in the figure, the 126 meV surface optical
phonon is the major scattering process in graphene, being about 5–10 times more
effective than the low energy 55 meV mode. Although the threshold energy for the
onset of emission of the high-energy phonon is larger than that for the low-energy
phonon, it becomes a much stronger scattering process at the high energy the
carriers reach in high electric field. The saturated velocity of graphene remains at
about 4 × 107 cm s−1 across a wide range of electron densities. In fact, for fields at, or
above 40 kV cm−1, this dominant scattering rate is above 1013 s−1 at room
temperature.
Problems
1. A particular semiconductor has a zero-field mobility composed of ionized
impurity scattering of 3500 cm2 V−1s−1 and of acoustic scattering of 4500
cm2 V−1s−1 at 4.2 K. Impurity scattering varies as T 3/2 while acoustic
scattering varies as T −1/2. We can assume that the average temperature of the
carriers is given by
3
kB(TE − T ) = eμe EτE ,
3
where E is the electric field and τE is the energy-relaxation time and has a
value of 1 ps. Plot the electron temperature and the mobility as a function of
the electric field E.
2. In a particular semiconductor, it is found that the electron temperature varies
linearly with the electric field as
Te = T + αE,
where E is the electric field and α has a value of about 10−3 cm V−1. In
addition, one may represent the mobility as a function of the electric field as
μ0
μe = ,
μE
1+ 0
vsat
where μ0 is 103 cm2 V−1 s−1 and vsat is 107 cm s−1. Using the equation from
the previous problem, plot the electron temperature and the energy-relaxa-
tion time as a function of the electric field. Then, plot the energy-relaxation
time as a function of the electron temperature.
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ISBN 978-0-7503-3139-5 (ebook)

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Preface to the second edition xii

1 The world of nanoelectronics 1-1

2 Wires and channels 2-1

2.4 Beyond the simple theory for the QPC 2-29

3 The Aharonov–Bohm effect 3-1

4 Layered compounds 4-1

5 Localization and fluctuations 5-1

5.2 Conductivity 5-11

6 The quantum Hall effect 6-1

8 Tunnel devices 8-1

9 Open quantum dots 9-1

10 Hot carriers in mesoscopic devices 10-1

It generally is regarded as being true that nanostructures may be considered as ideal

Transport in Semiconductor Mesoscopic Devices

doi:10.1088/978-0-7503-3139-5ch1 1-1 ª IOP Publishing Ltd 2020

fabricated in Taiwan. The chip computing power is approximately 100 GFLOPS1.

1.1 Moore’s law

means that there is no long-range wavelike behavior in the carrier’s character. On

Finally, we change to the total number of carriers N = nLd, so that

1.3 Some electronic length and time scales

1.4 Heterostructures for mesoscopic devices

1.4.1 The MOS structure

1.4.2 Fabricating the MOSFET

1.4.3 The GaAs/AlGaAs heterostructure

1.4.4 Other important materials

1.5.1 The Meissner effect

1.5.2 The BCS theory

1.6 Bits and qubits

In the above discussion, we talked about the qubit as a two-level system,

1.7 Some notes on fabrication

poly-methyl methracrylate (PMMA), and it is usually used as a positive resist that

1.7.3 Bottom-up fabrication

4. Consider a uniformly doped GaAlAs layer, with a composition of x = 0.25,

[18] Tachi K, Barraud S and Kakushima K et al 2011 Microelectron. Reliabil. 51 885

[95] Kawai S, Saito S and Osumi S et al 2015 Nature Commun. 6 8098

Transport in Semiconductor Mesoscopic Devices

2.1 The quantum point contact

doi:10.1088/978-0-7503-3139-5ch2 2-1 ª IOP Publishing Ltd 2020

one-dimensional channel, not unlike a small pipe. The observed conductance is

with a conductance that is an integer multiplier of 2e2/h. In this structure, the

Energy (arb. units)

Momentum, kx (arb. units)

2.2 The density of states

2.2.1 Three dimensions

We can now solve for the Fermi wave number as

2.2.2 Two dimensions

We can now solve for the Fermi wave vector as

2.2.3 One dimension

2.2.4 Multiple subbands

as we march up through the ladder of subbands. Of course, this can greatly

2.3 The Landauer formula

computing the transmission of a mode from an input reservoir to a similar mode in