Basic Principles of QCLs PDF
Basic Principles of QCLs PDF
Basic Principles of QCLs PDF
Chapter
Carlo Sirtori
Matériaux et Phénomènes Quantique
Université Paris 7, 75251 Paris, France;
THALES Research & Technology
91767 Palaiseau cede, France
Roland Teissier
CEM2, Université de Montpellier 2
34095 Montpellier, France
1.1 Introduction
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
2 Chapter One
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
4 Chapter One
dots; the improvement of the wall plug efficiency and the comprehen-
sion of the gain saturation that limits the peak output power.
That said, we are also convinced that the main challenge for this field
lies in the development of real-world applications and the
establishment of a commercial market.
Quantum Engineering
1800 1900 2000
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
ples of operation are not based on the physical properties of the con-
stituent materials, but arise from the layer sequence forming the
heterostructure.
Quantum engineering associated with the recent progress of the
epitaxial growth techniques allows one to ascribe within a
semiconductor crystal artificial potentials with the desired electronic
energy levels and wave functions. This approach is the basis for
modifying, in a unique way, the optical and transport properties of
semiconductors, opening avenues to artificial materials and the
creation of useful devices. A remarkable illustration of this concept is
the QC terahertz laser, in which by judiciously introducing less than
2% of Al atoms into a GaAs crystal we transform a piece of bulk
semiconductor into a performing far-infrared laser!
Another crucial aspect of QC lasers, related to the quantum
engineering, is that the fundamental principles of this device are
essentially independent of the specific semiconductor system used. As
of today, QC lasers have been demonstrated using basically three
material systems: GaInAs/AlInAs//InP, which is the original system
and the one that still gives the best performance for lasers in the mid-
infrared range; GaAs/AlGaAs//GaAs, which is the material system for
the terahertz laser; and AlSb/InAs//InAs (or //GaSb), which is the most
recently exploited and whose very high conduction band discontinuity
gives hope for short-wavelength QC lasers. Interestingly, in these
three material systems, QC lasers have been fabricated, for instance,
at 10 m, confirming that the emission wavelength is totally
independent of a particular transition intrinsic to the compounds
material. This is unique, and no other laser, semiconductor or not, has
this property.
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
6 Chapter One
2
1
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
In the ideal case Și = 1. Deviations from this value can be due either to
thermal activation from the injector to continuum or highly excited
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
8 Chapter One
J
3 τ3
Injector 2
τ2
1
Active
QW
If one assumes that level e2 is populated only through the direct scat-
tering of electrons from level e3, the electron density in e2 is simply
given by
IJ2
n2 = n3 (1.3)
IJ32
where IJ32 is the mean scattering time from e3 to e2 and IJ2 the lifetime of
electrons in e2. In this picture, the population inversion reads
Și J IJ2
n3 í n2 =
e 3 (
IJ 1í
IJ32 ) (1.4)
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
In the most unfavorable case in which all the electrons flow sequentially
from e3 to e2 (equivalent to saying that IJ3 = IJ32), n2 has its maximum
value
IJ 2Ș i J
n2 = (1.5)
e
and
Și J
n3 í n2 =
e
(IJ3 í IJ2) (1.6)
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
10 Chapter One
600
400
Energy (meV)
200
Δ
0
-200
0 200 400 600 800
Z (Å)
Figure 1.4 Schematic conduction band diagram of a portion of the laser heterostructure
at threshold bias. The wavy lines indicate the moduli squared of the calculated wave
functions. Indicated also is ǻ, the energy that separates the ground state of the laser
transition from the lower state of the injector.
between all quantum levels of the active zone have shown fair agree-
ment with experimental characteristics, such as current-voltage curves
(Donovan et al. 2001). However, not all scattering mechanisms are
included in these calculations, and typically electron-electron inter-
action and interface roughness scattering are not taken into account.
Another approach is a nonequilibrium Green’s function theory (Lee
and Wacker 2002). It allows all the important scattering mechanisms
to be included and accesses the current-voltage characteristics and gain
spectra of QCLs. Its advantage over semiclassical rate equation
approaches is clear for long-wavelength QCLs (terahertz), where the
extension of the wave functions is easily beyond the coherence length
of a quantum state in a semiconductor.
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
E0 ( ) ( )
A = í İ (e i Ȧ t í q썉r í e í i Ȧ t í q썉r ) (1.7)
2Ȧ
The action of this incident wave on the electron eigenstates of the het-
erostructure is given by the time-dependent hamiltonian H, written by
using the dipolar approximation and the effective mass description
(Bastard 1988) as
e
H= í A 썉 p = V (e iȦt í e í iȦt ) (1.8)
m*
2ʌ
Wif ( ല Ȧ ) = | f | V | i |2 į ( E f í Ei ± ല Ȧ ) (1.10)
ല
Using the momentum matrix element, we have
2 2
2 ʌ e E0
Wif ( ല Ȧ ) = | f | İ 썉 p | i |2 į ( Ef í Ei ± ല Ȧ) (1.11)
ല 4m* 2Ȧ2
ȥi (r) = uȣ i f i (r)
1 ik 쌩 i 썉r쌩 (1.12)
f i ( r) = e Ȥi ( z )
S
where uȣ i is the periodic part of the Bloch function, k쌩i and r쌩 are the
two-dimensional wave and position vectors in the plane of the layers
of area S, and Ȥi(z) is the envelope function, which describes the
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
12 Chapter One
f | İ 썉p |i = f f | İ 썉 p | fi
1
= Ȥ f | Ȥi
S d x d y e í i k 쌩 f 썉r쌩
(İ x p x + İ y p y ) e i k 쌩 i 썉r쌩
+ į ( k 쌩 f í k 쌩 i )İ z Ȥ f | p z | Ȥ i
(1.13)
Since the envelope functions are orthogonal, the term Ȥ f | Ȥi is null if
the final subband is different from the initial one. Consequently, the
transition rate from state |i to state | f is given by
2 2
2ʌ e E0 2
Wif (ലȦ) = İ | Ȥf | p z | Ȥ i | |2
ല 4m 2Ȧ2 z (1.14)
× į(k 쌩 f í k 쌩i )į( Ef í Ei ± ലȦ)
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
■
The optical transitions are vertical in k-space (k쌩f = k쌩i) and, in the
parabolic band approximation, the transition energy Eif = Ei í Ef
and the transition rate Wif do not depend on the in-plane wave vector.
■
The matrix element depends exclusively on the envelope functions
and can be tailored by designing the shape of the wave functions in
coupled well structures.
e 2 E02| zif |2
Wif max = (1.17)
2Ȗല
1
P= İ ncE02 (1.18)
2 0
and the number of photons of energy ƫȦ crossing the structure per unit
time is
2
1 İ0ncE0 (1.19)
ĭ= wL p
2 ലȦ
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
14 Chapter One
z
y
x
Propagation direction
Lp
w
Figure 1.5 Schematic geometry of the device used to derive the material gain. The growth
direction is z, and y is the direction of light propagation.
where n3w dy (and n2w dy) is the total number of electrons of level
e3 (and e2) in the slice of length dy. The first term corresponds to the
stimulated emission of photons due to the presence of electrons in e3,
while the second corresponds to the absorption of photons due to the
presence of electrons in e2. The propagation gain (often also referred to
as material gain) is defined as the variation of the photon flux divided
by the number of photons (definition equivalent to the absorption
coefficient)
dĭ / dy
G= (1.21)
ĭ
By using Eqs. 1.19 and 1.20 and the intersubband transition rate
(Eq. 1.17), one gets
2e 2 z32 2Ȧ
G=
İ0nc2ȖL p 3
(n í n2) (1.22)
With the expression of the population inversion in Eq. 1.4 and by using
the wavelength Ȝ = 2ʌc / Ȧ (in vacuum) of the propagating light, one
finally gets the usual expression of the gain of a QC laser, proportional
to the current density,
G = gJ (1.23)
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
4ʌe z32 2 IJ2
g=
İ0nȜ2ȖL p (
Ș i IJ3 1 í
IJ32 ) (1.24)
eല 1 IJ2
g = Și f 32IJ3 1 í (1.26)
İ0cm0 2ȖnL p IJ32
G = īg J (1.27)
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
16 Chapter One
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
293 K
CW operation 10
400 λ = 6 μm 298 K
Optical power (mW)
298 K 8
303 K
300
308 K
Voltage (V)
6
313 K
200
318 K 4
323 K
100 328 K 2
333 K
0 0
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Current (A)
Figure 1.6 Continuous-wave light versus current (L – I) curve of an HR-coated, 9-ȝm-
wide and 3-mm-long buried heterostructure laser at various heat sink temperatures.
The V–I curve at 298 K is also shown. (Reprinted with permission from Evans et al. 2004a.
Copyright © 2004, American Institute of Physics.)
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
18 Chapter One
-10
20 K
Modulation response
-20
300 mA
-30
250 mA
-40
200 mA
-50
150 mA
-60
0 1 10
Modulation frequency (GHz)
Figure 1.7 High-frequency modulation response traces of an 8-m QC laser at 20 K for
different values of the drive current ranging from very near threshold (150 mA), up to
one order of magnitude higher photon density (300 mA). These traces were normalized
to the experimental frequency response of the receiver and reflect only the modulation
response of the QC laser. (After Paiella et al. 2001.)
Active region. The active region of the first QCL was based on a simple
three-quantum-well (QW) scheme, where the laser transition was
diagonal between two adjacent QWs. It emitted at a wavelength of
4.3 m at cryogenic temperatures (Faist et al. 1994). Then more vertical
transitions were explored, in order to improve the intersubband gain
(Sirtori et al. 1998). The most efficient design was commonly considered
to be a vertical intersubband transition in two coupled QWs, with a
third very thin QW in the injection barrier. The role of the latter QW is
to selectively enhance the amplitude of the excited state of the laser in
the injection barrier, to increase resonant tunneling injection while
preventing direct injection into the lower states.
For high-temperature and high-power operation of QC lasers, the
efficiency of electron extraction from the lower state of the active
quantum wells is an important issue. To overcome the bottleneck in
electron extraction from the active region, new designs have been
proposed. Superlattice-based active regions (Scamarcio 1997) have
produced high-efficiency QCLs, thanks to a very rapid carrier
extraction in the superlattice miniband. This was, however, to the
detriment of electron injection into the upper state e3. The most efficient
scheme, called bound-to-continuum (Faist et al. 2001), combines
efficient electron injection into a bound state e3 and rapid extraction
from a delocalized lower state.
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
optical losses of the laser cavity. The undisputable champion for high
performance in the mid-infrared range is the InP material used as op-
tical cladding layers. It is a binary compound that provides a large
dielectric refractive index contrast with the active region. In addition,
InP has high electrical and thermal conductivity, without the need for
a high doping level and therefore allows very low free-carrier optical
losses. For this reason, QCLs made of materials grown on InP have
considerable advantages. Alternative materials can, however, have
their interest in specific cases. This material issue for QCL is discussed
in greater detail in the following sections.
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
20 Chapter One
+35°c
+20°c
Device size:
0.1 0°c
Intensity (arb. units)
1.5 mm x 30 μm
-20°c
-40°c
0.01
0.001
0.0001
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
operation, these devices have shown lines with intrinsic width well
bellow 1 kHz (Myers et al. 2002).
1.3.2 Frontiers
Beyond the conventional mid-infrared QC lasers of increasing matu-
rity, new frontiers are being explored. Thanks to the flexibility of the
active region and the cascading scheme, new functionalities or phe-
nomena can be explored inside the laser cavity, such as nonlinear optics
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
22 Chapter One
or Raman lasing (Troccoli et al. 2005b). However, one of the most at-
tractive research areas is at the frontiers of the emission spectrum, to
extend the spectral width of QC laser operation. The long-wavelength
side is the emerging field of terahertz lasers, which opens new perspec-
tives in terms of imaging or molecular spectroscopy. On the other end
of the spectrum, short-wavelength QCLs (Ȝ < 4 m) could be the candi-
date to fulfill the lack of semiconductor laser sources operating at room
temperature in the 3 to 4 m range.
New physics. The high intensity of optical field in the cavity of QCLs
allows generation of intracavity nonlinear effects. Nonlinearity can
originate either from the semiconductor crystal itself or from specifi-
cally designed structures inserted in series with the active region.
Frequency doubling has been demonstrated (Owschimikow et al. 2003)
as well as sideband generation (Dihlon 2005).
The coherent light generated in the QCL cavity can also be used to
pump intersubband transitions and generate emissions at different
wavelengths. This principle gave birth to the intracavity Raman laser
(Troccoli 2005b).
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
4 20
0.5
2 10
1
T = 10 K
0 0
0 100 200 300
2
Current density (A/cm )
Figure 1.9 Electrical and optical characteristics (continuous-wave) of a 100-m-wide
ridge terahertz laser at 10 K. The length of the device is 2 mm. In the insert is the high-
resolution emission spectrum.
An additional relevant issue for the terahertz range is the fact that
conventional laser waveguides are not suitable, owing to large free-
carrier absorption losses and practical limitations on the thickness of
epilayer growth. For these long-wavelength lasers, the optical
confinement is never achieved by using dielectric claddings, but by
metallic layers very much as in the case of microwave strips. Even if
operating at low temperature, terahertz QC lasers are very promising
candidates to become compact sources for imaging systems, or local
oscillators for heterodyne detection (Gao et al. 2005).
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
24 Chapter One
2.5
GaP
AlAs
2.0 AlAsSb
AlSb
Energy gap (eV)
AlInAs
1.5
GaAs
InP
1.0
GaSb
GaInAs
0.5
InAs
0.0 InSb
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
26 Chapter One
conduction band edge in InAs. Therefore, the emission range for this
novel material system is probably restricted to wavelengths greater
than 3 m. Other specific issues for the realization of short-wavelength
QCLs are presented in Sec. 1.4.
A second specificity of antimonides is the very small effective mass
in InAs. This leads to significantly stronger intrinsic intersubband
optical gain, compared to other material systems (see Fig 1.13 below).
Hence, from a material point of view, it is clear that antimonides are a
very attractive system for mid-infrared QCLs.
In spite of the very high conduction band discontinuity, the first QC
laser demonstrated in this material system was at 10 m (Ohtani and
Ohno 2003). More recently the potential of this heterostructure has
been better exploited and lasers emitting at ~4 ȝm have been
demonstrated up to room temperature (Teissier et al. 2004). Even
though the performances of these devices are very encouraging, they
have not yet reached those of the QC laser based on InP. We cannot
address any fundamental reasons for this, but just remind that these
very recently born devices need more work and optimization.
Si/SiGe. For Si/SiGe the steps toward a unipolar laser are less ad-
vanced. Nevertheless, intersubband electroluminescence has been re-
cently observed from QC active regions realized on metamorphic
substrate Si0.5Ge0.5 (Diehl et al. 2002). The realization of QC lasers
in Si/SiGe would represent a major breakthrough simply because it
would represent the first laser based on Si. In this material system,
intersubband transitions occur in the valence band, which is a serious
complication from the point of view of the theoretical description.
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
4ʌe 2e 1
g = | z |2 IJ = E IJ | z |2
İ0nȜ 2ȖL p 21 2 İ0ncല 2ȖL p 21 2 21
(1.29)
eല 1
= f IJ
İ0cm0 2ȖnL p 21 2
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
28 Chapter One
which means that the gain coefficient is proportional to (m* )í3/2 and that
the ratio of the gain coefficients of the two materials InAs and GaAs can
be evaluated to
3/2
g InAs m*GaAs 0.067 3/2
= = 싉 5 (1.31)
g GaAs m*GaInAs 0.023
GaAs InAs
m* = 0.067 m* = 0.024
E3 3 h 2π2
E21 =
E2 2 m* L2
E1
L = 115 Å L = 189 Å
Figure 1.11 Quantum wells made of different materials. Notice that if we fix the energy
separation E21 the width of the quantum well is inversely proportional to m* .
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
0.12 0.12
Effective mass (m0)
0.08 0.08
0.04 GaAs
0.04 AlGaAs
AlInAs
GaInAs
AlAsSb
InAs
AlSb
0 0
0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8
Energy (eV) Energy (eV)
Figure 1.12 Energy dependence of the effective mass in the wells (left) and in the barriers
(right) for the different material systems. For the barrier materials the zero of the energy
is set at the bottom of the conduction band of the associated well materials, so that one
can read the value of the masses on the same energy scale on both panels.
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
30 Chapter One
120 120
GaAs/AlGaAs
3
InAs/AlSb 2 2
GalnAs/AllnAs 1
90 90
GalnAs/AlAsSb Studied transition Studied transition
τ • f (ps)
60 60
GalnAs/AlAsSb
30 30
GaAs/AlGaAs
GalnAs/AllnAs
InAs/AlSb
0 0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
E21 (eV) E32 (eV)
Figure 1.13 The product f·IJ between levels 2 and 1 is plotted on the left, and between
levels 3 and 2 on the right. The quantity f·IJ directly proportional to the gain.
Escape to the continuum. When a confined level gets closer to the top of
the barriers, the probability of thermal activation of electrons to a de-
localized state increases dramatically (Fig. 1.14). The scattering rate of
an electron out of the confined state e3 is controlled by the occupation
probability in the subband at an energy greater than that of the top of
the barrier, and by the typical scattering time IJscatt from this subband.
The model to describe this effect is identical to the one used for calcu-
lating dark current in quantum well infrared photodetectors (QWIPs)
(Liu et al. 1993).
The two-dimensional density of electrons localized in the quantum
well and with energy greater than ǻEc is given by
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
?Ec
e3
e2
e1
Figure 1.14 Schematic representation of a QC laser active region. The arrow indi-
cates the path of electrons that, due to thermal effects, are activated into the continuum
states.
ǻEc í e3
n2D = n3exp í ( kTe ) (1.32)
where n3 is the total sheet density in level e3, and Te is the electron
temperature. It has been measured (Spagnolo et al. 2004) to be
sensibly greater than the lattice temperature by a value on the order
of 100 K.
The leakage current density to the continuum is then simply given
by
qn2D
Jesc = (1.33)
IJscatt
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
32 Chapter One
IJ 3 ( 0)
IJ3 = (1.34)
1 + 2 / exp(ലȦLO / kT ) í 1
2mB* ( E )
kB = (ǻEc í E ) (1.35)
ല2
3
λ ~ 4.3 μm
(e3 - ΔEc = 30 meV)
2 λ ~ 4.5 μm
(e3 - ΔEc = 50 meV)
J esc/Jc
1
λ ~ 5 μm
(e3 - ΔEc = 100 meV)
0
0 50 100 150 200 250 300 350 400
Temperature (K)
Figure 1.15 Ratio between the current in the active region and the leakage current due
to the thermal leakage in the continuum. It is clear that for activation energy smaller
than 50 meV, room temperature operations are completely hindered by the thermal
activated current.
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
100
Level splitting (meV)
10
1 InAs/AlSb GaAs/AlGaAs
InGaAs/AlAsSb
InGaAs/InAlAs
0 20 40 60 80 100 120
Barrier thickness (Å)
Figure 1.16 Calculated splitting of the levels of two coupled quantum wells, giving a
picture of barrier transparency for different material systems. The confinement energy
is kept constant at a value of 250 meV from the bottom of the QW.
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
34 Chapter One
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
Lattice matched
InGaAs / InAlAs GaAs / Al0.35Ga0.65As
L
InAs / AlSb
X
InGaAs / AlAsSb
ΔEc = 2000 meV
Strain compensated L
Inx Ga1-x As / In1-y Aly As
Figure 1.17 Band offsets of the different material systems used for QC lasers. Both ī and
relevant lateral valleys (X or L) are shown for the wells and the barriers.
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
36 Chapter One
1.4.5 Waveguide
The waveguide design and its realization are crucial steps to obtain
high-performance QC lasers. The vertical confinement of the optical
mode propagating in the cavity of the laser is obtained, as in most
edge-emitting semiconductor lasers, by a multilayer structure: a core
waveguide made of the active region and spacer layers, surrounded
by optical cladding layers made of low-refractive-index material.
This is another aspect for which the material system impacts QCL
performances.
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
Figure 1.18 Schematic diagram of the ī and L conduction band profile in a short-
wavelength InAs/AlSb QC structure, showing the relevant states issued from both
valleys. The arrows show the possible intervalley leakage current mechanism.
AR
|E ( z )|2 d z
ī = +
(1.38)
|E ( z )|2 d z
í
Refractive index
Mode intensity
Active region
Cladding
Cladding
Spacer
Spacer
Figure 1.19 Refractive index profile of a typical QCL waveguide, in the growth
direction z. The electric field intensity of the fundamental TM (electric field in the z
direction) mode is also shown.
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
38 Chapter One
Į w + Įm
īg Jth = Į w + Įm Jth = (1.39)
īg
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
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Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
40 Chapter One
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
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Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
Quantum Cascade Lasers: Overview of Basic Principles of Operation and State of the Art
44 Chapter One