 
nextnano^{3}  Tutorial
next generation 3D nano device simulator
1D Tutorial
QuantumCascade Laser
Author:
Stefan Birner
If you want to obtain the input files that are used within this tutorial, please contact stefan.birner@nextnano.de.
> 1DQuantumCascadeLaser.in
> 1DQCL_AlGaAs_Sirtori_APL73_1998.in
> 1DQCL_Andrea_Friedrich_NoInjector_InGaAs_APL86_2005_77K_kp.in
> 1DQCL_Andrea_Friedrich_NoInjector_InGaAs_APL86_2005_300K_kp.in
> 1DQCL_Rochat_APL81_2002.in
> 1DQCL_THz_MIT_Sandia_SemicScTech20_2005.in
> THzQCL_Andrews_Vienna_MatSciEng2008_nn3.in / *_nnp.in  input file for nextnano³
and nextnano++ software
> 1DQuantumCascadeLaserSiGe_nn3.in
/ *_nnp.in  input file for nextnano³
and nextnano++ software 
QuantumCascade Laser
This tutorial is based on the quantumcascade structure that has been
presented in the following paper.
300 K operation of a GaAsbased quantumcascade laser at lambda=9 µm
H. Page, C. Becker, A. Robertson, G. Glastre, V. Ortiz, C. Sirtori
Applied Physics Letters 78 (22), 3529 (2001)
Here, we are trying to reproduce Fig. 1 of this paper.
The temperature has been set to 300 K.
The quantumcascade structure consists of a sequence of GaAs wells and Al_{0.45}Ga_{0.55}As
barriers. The sequence is as follows (from 0 nm to 45 nm; it is repeated outside
this region):
4.6 / 1.9 / 1.1
/ 5.4 / 1.1 / 4.8 /
2.8 / 3.4 / 1.7 / 3.0 /
1.8 / 2.8 / 2.0
/ 3.0 / 2.6 / 3.0
The units are [nm] . Blue and bold
scripts indicate Al_{0.45}Ga_{0.55}As
barriers, normal scripts indicate GaAs wells.
In the APL paper, a conduction band offset of 390 meV was used.
Consequently, we modify our default band offset by shiftting the AlGaAs ternary
slightly to also get 390 meV.
$ternaryzbdefault
ternarytype = Al(x)Ga(1x)Aszbdefault
...
bandshift = 0.022719d0
! [eV] ==> to get a band offset of 390 meV for T=300 K and x=0.45 (Al_{0.45}Ga_{0.55}As)
We apply an electric field of 48 kV/cm:
$electricfield
!
electricfieldon =
yes ! 'yes'
/ 'no'
electricfieldstrength = 48d5
! in units of [V/m]  Here: 48 kV/cm
electricfielddirection = 0 0 1 !
[001] direction, i.e. along z axis.
$end_electricfield
!
For simplicity, in contrast to the APL paper, we do not include doping here.
In the original APL paper, the following areas were ntype doped with silicon
with a sheet density of n_{Si} = 3.8 * 10^{11} cm^{2}.
=> between 15.2 nm and 5.6 nm (9.8 nm): '1.8 nm' (barrier), '2.8
nm' (well), '2.0 nm' (barrier) and '3.0 nm' (well)
=> between 29.8 nm and 39.4 nm (9.8 nm): '1.8 nm' (barrier), '2.8
nm' (well), '2.0 nm' (barrier) and '3.0 nm' (well)
We use flowscheme = 21 .
$simulationflowcontrol
! flowscheme = 20 ! ==> apply
electric field and solve Poisson
equation
flowscheme = 21 ! ==>
apply electric field and do not solve Poisson equation
...
These flowschemes includes the following:
1. Calculate the strain (if any).
2. Calculate the piezo and pyroelectric charges (if any).
3. Calculate the conduction and valence band edge profiles by solving Poisson's
equation taking into acount doping (if any), piezo and pyroelectric charges (if
any) and deformation potentials (if strain unequal to zero).
4. Apply the electric field.
5. Calculate the eigenstates and wave functions by solving Schrödinger's equation
(either singleband or k.p).
Note that for flowscheme = 21 , this
is not a selfconsistent calculation of the PoissonSchrödinger equation.
flowscheme = 20 is selfconsistent
but a severe limitation is that population inversion is not taken into account.
In our example, we did not have to calculate the strain. Piezo any
pyroelectric fields do not exist. We do not include doping. We used singleband
(effectivemass) rather than 8band k.p.
The following figure shows the conduction band energy of the Gamma conduction
band edge and the wave functions (psi² = psi squared) of the
ground state 1, the
lower state 2, the excited state 3
and the injector state i.

The figure shows the conduction band edge (black
line) of the quantumcascade structure that has a slope because of the
electric field of 48 kV/cm.
Also shown are four wave functions (psi² = psi squared) that are shifted by
their corresponding eigenenergies. 
The above shown structure of the conduction band edge and the wave functions
is in excellent agreement with Fig. 1 of the following paper:
300 K operation of a GaAsbased quantumcascade laser at lambda=9 µm
H. Page, C. Becker, A. Robertson, G. Glastre, V. Ortiz, C. Sirtori
Applied Physics Letters 78 (22), 3529 (2001)
Note that periodic boundary conditions for the Schrödinger and Poisson
equation do not make sense because of the application of an electric field. Thus
we used Dirichlet boundary conditions. However, this will lead to some
artificial, wrong wave functions at the boundaries because the wave function is
forced to be zero at the boundaries. For the states in the middle of the device
where the wave function decays to zero in any case at the boundaries, the
boundary conditions do not have any influence at all and so these states are
fine.
So the suggestion is to calculate 3 or 5 periods, and then take the energy
levels and wave functions of the center period.
In this way, boundary effects should not be very severe.
$quantummodelelectrons
modelnumber
= 1
modelname
= effectivemass ! 'effectivemass'
or '8x8kp'
boundarycondition001 = Dirichlet !
boundary condition for [001] direction, i.e. along the z
direction
...
Intraband matrix elements
The files
 Schroedinger_1band / intraband_pz1D_cb001_qc001_sg001_deg001_dir.txt and
 Schroedinger_1band / intraband_z1D_cb001_qc001_sg001_deg001_dir.txt and
contain the 'p_{z}' and 'z' intraband matrix elements for all transitions.
Our result for the excited state to
lower state 'z' matrix element is in
excellent agreement with the result of [Page]:
Intersubband dipole moment  < psi_f*  z  psi_i >  [nm]

Oscillator strength []

Energy of transition [eV]

m* [m_0] lifetime [ps]

...
<psi010*zpsi006>
1.6655138016 0.747520328
0.147729769 0.069499455
==> z_{10,6}
= 1.6655138016 [nm] ([Page] : z_{32}
= 1.7 nm)
The transition energy of these two states has been calculated to be
147.7 meV. ([Page, Fig. 3, experiment] : E_{32}
= 160 meV)
For more details, check the tutorial on intraband transitions:
Optical intersubband transitions
in a quantum well  Intraband matrix elements and selection rules
QCL examples
Note: We have nextnano³ input files available for
quantumcascade lasers that are based on the following structures.
Please contact stefan.birner@nextnano.de
if you want to obtain these or any other input files.
 9 µm, i.e. 33 THz or 138 meV
300 K operation of a GaAsbased quantumcascade laser at lambda=9 µm H. Page, C. Becker, A. Robertson, G. Glastre, V. Ortiz, C. Sirtori Applied Physics Letters
78 (22), 3529 (2001)
> 1DQuantumCascadeLaser.in
 9.4 µm or 132 meV
GaAs/Al_{x}Ga_{1x}As quantum cascade lasers C. Sirtori, P. Kruck, S. Barbieri, P. Collot, J. Nagle, M. Beck, J. Faist, U.
Oesterle Applied Physics Letters 73 (24), 3486 (1998)
> 1DQCL_AlGaAs_Sirtori_APL73_1998.in
 10 µm, i.e. 124 meV (77 K)
8.4 µm, i.e. 148 meV (300 K)
Quantumcascade lasers without injector regions operating above room
temperature A. Friedrich, G. Böhm, M.C. Amann, G. Scarpa Applied Physics Letters
86, 161114 (2005)
> 1DQCL_Andrea_Friedrich_NoInjector_InGaAs_APL86_2005_77K_kp.in
> 1DQCL_Andrea_Friedrich_NoInjector_InGaAs_APL86_2005_300K_kp.in
 66 µm, i.e. 4.54 THz or 18.8 meV (nextnano³ calculation:
18.5 meV, i.e. 4.47 THz or 67 µm)
Lowthreshold terahertz quantumcascade lasers
M. Rochat, L. Ajili, H. Willenberg, J. Faist, H. Beere, G. Davies, E.
Linfield, D. Ritchie Applied Physics Letters 81 (8), 1381 (2002)
> 1DQCL_Rochat_APL81_2002.in
Note: The caption in FIG. 1 in this paper must read "from right to left",
rather than "from left to right".
 89.2 µm, i.e. 3.4 THz or 13.9 meV (exp. 14.2 meV) (nextnano³ calculation:
14.02 meV)
Resonantphononassisted THz quantumcascade lasers with metalmetal
waveguides Q. Hu, B.S. Williams, S. Kumar, H. Callebaut, S. Kohen, J.L. Reno Semiconductor Science and Technology
20, S228 (2005)
> 1DQCL_THz_MIT_Sandia_SemicScTech20_2005.in
 107 µm, i.e. 2.8 THz or 11 meV
Doping dependence of LOphonon depletion scheme THz quantumcascade
lasers
A. M. Andrews, A. Benz, C. Deutsch, G. Fasching, K. Unterrainer, P.
Klang, W. Schrenk, G. Strasser
Materials Science and Engineering B 147, 152 (2008)
> THzQCL_Andrews_Vienna_MatSciEng2008_nn3.in  input file for
nextnano³ software
> THzQCL_Andrews_Vienna_MatSciEng2008_nnp.in  input file for nextnano++
software
This input file is parameterized. One can thus use nextnanomat's
Template feature to vary e.g. alloy content or barrier widths.
 9.9 µm, i.e. 30.2 THz or 125 meV (nextnano³ calculation:
124.5 meV)
Intersubband Electroluminescence from SiliconBased Quantum Cascade
Structures G. Dehlinger, L. Diehl, U. Gennser, H. Sigg, J. Faist, K. Ensslin, D.
Grützmacher, E. Müller Science 290, 2277 (2000)
> 1DQuantumCascadeLaserSiGe?nn3.in
Note: The Science article reports a calculated transition energy of 130 meV.
The experimentally measured value, however, was reported to be 125 meV which
is in excellent agreement with the nextnano³ calculations based on the
default parameters in the nextnano³ database.
The eigenstates relevant for the optical transitions are the numbers 9 & 10,
and 31 & 32.
 Please help us to improve our tutorial! Send comments to
support
[at] nextnano.com .
