Guillaume Weick

Quantum transport
From quantum dots to molecules
Sfb 658 Kolloquium, Berlin, 14.05.2009
The mesoscopic regime
system size L
microscopic
(quantum/relativistic)
macroscopic
(classical/statistical)
mesoscopic

A
nm
m
mm
(
F
L)
The mesoscopic regime
system size L
microscopic
(quantum/relativistic)
macroscopic
(classical/statistical)
mesoscopic

A
nm
m
mm
(
F
L)
L

!
length below which interference (quantum) effects appear
!
destroyed by interactions with environment (via e-e, e-ph, mostly inelastic),
but not by scattering with impurities (elastic)
Phase coherence length L

(or phase breaking time )

= L

/v
F
MESOSCOPICS:
rather a regime than only a
lengthscale (small size and low T)
Choi et al., PRB 87
Classical transport
Drude conductivity:
V
, L

A
|j| = I/A
V = |E|L
G =
A
L
j = (e)nv
= /v
F
v
avg
=
(e)E
m
v
0
v
0
+
(e)Et
m

Drude
=
ne
2

m
j = E
I = GV
Weak localization

A
L L

V
Classical
!
Drude P
cl
= |A
1
|
2
+|A
2
|
2
Quantum
P
q
= |A
1
+A
2
|
2
= |A
1
|
2
+ |A
2
|
2
+ 2|A
1
A
2
| cos
disorder averaging: cos = 0 in general: P
q
= P
cl
A
1
A
2
Weak localization

A
L L

V
Classical
!
Drude P
cl
= |A
1
|
2
+|A
2
|
2
Quantum
P
q
= |A
1
+A
2
|
2
= |A
1
|
2
+ |A
2
|
2
+ 2|A
1
A
2
| cos
disorder averaging: cos = 0 in general: P
q
= P
cl
A
1
= A
2
P
q
= 2P
cl
!
(slighty) decreases the conductance: weak localization correction
exept if
:
A
2
A
1

A
L L

V
A
2
A
1
Weak localization
How do we measure that?
B
A
1
A
1
e
i
= 2

0
A
2
A
2
e
i
P
q
= |A
1
|
2
+ |A
2
|
2
+ 2|A
1
A
2
| cos 2
averaging: constructive interferences disappear!

A
L L

V
A
2
A
1
Weak localization
How do we measure that?
________________________________________________________ 17
Mg L
00
0~ 0
~O.O5
0 O.5~ 0
0
0
0
0.11
~ 0 o 6.4K 0.11 ~
9.4K
0.21
A0.5T 19.9 K 1.01
H
Fig. 2.11. The magneto-resistance of a thin Mg-film for different temperatures as a function of the applied field. The units of the field are given on
the right of the curves. The points represent the experimental results. The full curves are calculated with the theory. The small spin-orbit scattering
is taken into account.
2 ~ (T/K) 2.5 3
-24
In(t1is)
20
t1[10
12s] -25
10 Mg
5 -26
R 84.60
2 -27
1
5 10 T[K] 20
Fig. 2.12. The inelastic lifetime ~ of a Mg-film as a function of temperature. It obeys a r2-law.
Bergmann, Phys. Rep. 84
magnetic eld
!
kills weak localization
Quantum point contacts
B.J. van Wees et al., PRL 88
W
W
L , L L

Quantum point contacts

Ohms law: G =
W
L
???
Quantum point contacts
B.J. van Wees et al., PRL 88
W
W
L , L L

!
Quantization of transport in integer steps of the quantum of conductance
G
0
=
2e
2
h
(h/e
2
26 k)
Intermezzo: Landauer formula
1D:
T(E)
I

L

R

R
= eV
I
LR
=

0
dE

2(E)f
L
(E)

ev(E)T(E)

(E) =

m/2E/h v(E) =

2E/m
=
2e
h

0
dE f
L
(E)T(E)
I
RL
=
2e
h

0
dE f
R
(E)T(E)
I = I
LR
I
RL
=
2e
h

0
dE

f
L
(E) f
R
(E)

T(E)
low T, low bias:
G =
2e
2
h
T(E
F
)
one-channel
Landauer formula
Quantum point contacts
B.J. van Wees et al., PRL
W
U(x, y) = U(y)

k,n
e
ikx

n
(y)
E
k,n
=

2
k
2
2m
+
n

n

n
2
W
2
Figure 14: Two-dimensional lead of width W. The transverse modes are
quantized by the lateral connement.
2.3.2 Multi-Channel Two-Point Conductance
A more realistic situation is the one of quasi one-dimensional leads of
nite cross-section connecting the quantum scatterer to the reservoirs.
We discuss now the simplest example of a two-dimensional strip along
the x-axis having the width W in y-direction, as sketched in gure 14.
Assuming a perfect lead, the connement potential
U(x, y) = U(y) (56)
only depends on the transverse coordinate y. Then, the Schr odinger
equation

2
2m
(x, y) +U(y)(x, y) = E(x, y) (57)
can be separated and one gets as solutions the product wave functions
(x, y) exp(ikx)
n
(y) (58)
and the corresponding energies
E
k,n
=

2
k
2
2m
+E
n
, (59)
where the
n
(y) and E
n
are the one-dimensional eigenfunctions and
the eigenenergies, respectively, of conned particles in the transverse
potential U(y). The longitudinal part, in contrast, reects free prop-
agation with wave-number k along the x-direction. The quantized
transverse parts of the wave-functions are analogous to the electro-
magnetic modes in wave-guides and are called channels or transverse
modes in this context.
The dispersion relation (59) exhibits several branches, one per
transverse quantum number n. Each branch is equivalent to an ideal
one-dimensional system except for the energy oset E
n
. Neglecting
28
x
y
k
E
k,n
n = 1
n = 2
n = 3
0
E
F
G =
2e
2
h
N
c
Quantum point contacts
B.J. van Wees et al., PRL
W
U(x, y) = U(y)

k,n
e
ikx

n
(y)
E
k,n
=

2
k
2
2m
+
n

n

n
2
W
2
Figure 14: Two-dimensional lead of width W. The transverse modes are
quantized by the lateral connement.
2.3.2 Multi-Channel Two-Point Conductance
A more realistic situation is the one of quasi one-dimensional leads of
nite cross-section connecting the quantum scatterer to the reservoirs.
We discuss now the simplest example of a two-dimensional strip along
the x-axis having the width W in y-direction, as sketched in gure 14.
Assuming a perfect lead, the connement potential
U(x, y) = U(y) (56)
only depends on the transverse coordinate y. Then, the Schr odinger
equation

2
2m
(x, y) +U(y)(x, y) = E(x, y) (57)
can be separated and one gets as solutions the product wave functions
(x, y) exp(ikx)
n
(y) (58)
and the corresponding energies
E
k,n
=

2
k
2
2m
+E
n
, (59)
where the
n
(y) and E
n
are the one-dimensional eigenfunctions and
the eigenenergies, respectively, of conned particles in the transverse
potential U(y). The longitudinal part, in contrast, reects free prop-
agation with wave-number k along the x-direction. The quantized
transverse parts of the wave-functions are analogous to the electro-
magnetic modes in wave-guides and are called channels or transverse
modes in this context.
The dispersion relation (59) exhibits several branches, one per
transverse quantum number n. Each branch is equivalent to an ideal
one-dimensional system except for the energy oset E
n
. Neglecting
28
x
y
k
E
k,n
n = 1
n = 2
n = 3
0
E
F
G =
2e
2
h
N
c
Quantum point contacts
B.J. van Wees et al., PRL
W
U(x, y) = U(y)

k,n
e
ikx

n
(y)
E
k,n
=

2
k
2
2m
+
n

n

n
2
W
2
Figure 14: Two-dimensional lead of width W. The transverse modes are
quantized by the lateral connement.
2.3.2 Multi-Channel Two-Point Conductance
A more realistic situation is the one of quasi one-dimensional leads of
nite cross-section connecting the quantum scatterer to the reservoirs.
We discuss now the simplest example of a two-dimensional strip along
the x-axis having the width W in y-direction, as sketched in gure 14.
Assuming a perfect lead, the connement potential
U(x, y) = U(y) (56)
only depends on the transverse coordinate y. Then, the Schr odinger
equation

2
2m
(x, y) +U(y)(x, y) = E(x, y) (57)
can be separated and one gets as solutions the product wave functions
(x, y) exp(ikx)
n
(y) (58)
and the corresponding energies
E
k,n
=

2
k
2
2m
+E
n
, (59)
where the
n
(y) and E
n
are the one-dimensional eigenfunctions and
the eigenenergies, respectively, of conned particles in the transverse
potential U(y). The longitudinal part, in contrast, reects free prop-
agation with wave-number k along the x-direction. The quantized
transverse parts of the wave-functions are analogous to the electro-
magnetic modes in wave-guides and are called channels or transverse
modes in this context.
The dispersion relation (59) exhibits several branches, one per
transverse quantum number n. Each branch is equivalent to an ideal
one-dimensional system except for the energy oset E
n
. Neglecting
28
x
y
k
E
k,n
n = 1
n = 2
n = 3
0
G =
2e
2
h
N
c
E
F
Quantum point contacts
B.J. van Wees et al., PRL
W
U(x, y) = U(y)

k,n
e
ikx

n
(y)
E
k,n
=

2
k
2
2m
+
n

n

n
2
W
2
Figure 14: Two-dimensional lead of width W. The transverse modes are
quantized by the lateral connement.
2.3.2 Multi-Channel Two-Point Conductance
A more realistic situation is the one of quasi one-dimensional leads of
nite cross-section connecting the quantum scatterer to the reservoirs.
We discuss now the simplest example of a two-dimensional strip along
the x-axis having the width W in y-direction, as sketched in gure 14.
Assuming a perfect lead, the connement potential
U(x, y) = U(y) (56)
only depends on the transverse coordinate y. Then, the Schr odinger
equation

2
2m
(x, y) +U(y)(x, y) = E(x, y) (57)
can be separated and one gets as solutions the product wave functions
(x, y) exp(ikx)
n
(y) (58)
and the corresponding energies
E
k,n
=

2
k
2
2m
+E
n
, (59)
where the
n
(y) and E
n
are the one-dimensional eigenfunctions and
the eigenenergies, respectively, of conned particles in the transverse
potential U(y). The longitudinal part, in contrast, reects free prop-
agation with wave-number k along the x-direction. The quantized
transverse parts of the wave-functions are analogous to the electro-
magnetic modes in wave-guides and are called channels or transverse
modes in this context.
The dispersion relation (59) exhibits several branches, one per
transverse quantum number n. Each branch is equivalent to an ideal
one-dimensional system except for the energy oset E
n
. Neglecting
28
x
y
k
E
k,n
n = 1
n = 2
n = 3
0
G =
2e
2
h
N
c
E
F
Quantum point contacts
B.J. van Wees et al., PRL
W
U(x, y) = U(y)

k,n
e
ikx

n
(y)
E
k,n
=

2
k
2
2m
+
n

n

n
2
W
2
Figure 14: Two-dimensional lead of width W. The transverse modes are
quantized by the lateral connement.
2.3.2 Multi-Channel Two-Point Conductance
A more realistic situation is the one of quasi one-dimensional leads of
nite cross-section connecting the quantum scatterer to the reservoirs.
We discuss now the simplest example of a two-dimensional strip along
the x-axis having the width W in y-direction, as sketched in gure 14.
Assuming a perfect lead, the connement potential
U(x, y) = U(y) (56)
only depends on the transverse coordinate y. Then, the Schr odinger
equation

2
2m
(x, y) +U(y)(x, y) = E(x, y) (57)
can be separated and one gets as solutions the product wave functions
(x, y) exp(ikx)
n
(y) (58)
and the corresponding energies
E
k,n
=

2
k
2
2m
+E
n
, (59)
where the
n
(y) and E
n
are the one-dimensional eigenfunctions and
the eigenenergies, respectively, of conned particles in the transverse
potential U(y). The longitudinal part, in contrast, reects free prop-
agation with wave-number k along the x-direction. The quantized
transverse parts of the wave-functions are analogous to the electro-
magnetic modes in wave-guides and are called channels or transverse
modes in this context.
The dispersion relation (59) exhibits several branches, one per
transverse quantum number n. Each branch is equivalent to an ideal
one-dimensional system except for the energy oset E
n
. Neglecting
28
x
y
k
E
k,n
n = 1
n = 2
n = 3
0
G =
2e
2
h
N
c
E
F
Quantum dots
E

R
T
L R
eV
V
g
T

0
V
g
V
V = 2V
g
V = 2V
g
I
Rate equations:

P
0
=
01
P
0
+
10
P
1

P
1
=
10
P
1
+
01
P
0
Quantum dots
E

R
T
L R
eV
V
g
T

0
Rate equations:

P
0
=
01
P
0
+
10
P
1

P
1
=
10
P
1
+
01
P
0
V
g
V
V = 2V
g
V = 2V
g
dI/dV
Single-molecule junctions
E

R
T
L R
eV
V
g
0
, T
V
g
V
V = 2V
g
V = 2V
g
dI/dV

Rate equations:

P
q
0
=

qq

01
P
q
0
+

q
10
P
q

P
q
1
=

qq

10
P
q
1
+

q
01
P
q

0
Conclusion
What is Quantum Transport?

Can only appear in the mesoscopic regime and below

Interference effects
(e.g., weak localization correction in disordered conductors)

Quantization of states in conned system results in quantization of transport

Quantum dots: single-electron transistor

Single-molecule junctions: additional effects due to vibrations of the molecule

Y. Imry, Introduction to Mesoscopic Physics (Oxford University Press, 2002)

S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge University

E. Akkermans, G. Montambaux, Mesoscopic Physics of Electrons and Photons

T. Dittrich, P. Hnggi, G.-L. Ingold, B. Kramer, G. Schn, W. Zwerger, Quantum

D. Weinmann, The Physics of Mesoscopic Systems, Proceedings of the VIIth

N.W. Ashcroft, N.D. Mermin, Solid State Physics (Saunders College, 1976)
Molecular electronics:

M. Galperin, M.A. Ratner, A. Nitzan, Molecular transport junctions: vibrational

G. Weick, M.G. Schultz, F. von Oppen, Quantum Transport through Single-