nextnano^{3}  Tutorial
next generation 3D nano device simulator
1D Tutorial
Intersubband transitions in InGaAs/AlInAs multiple quantum well systems
Authors:
Stefan Birner
If you want to obtain the input files that are used within this tutorial, please contact stefan.birner@nextnano.de.
> 1DSirtoriPRB1994_OneWell_sg_selfconsistent_nn3.in
/ *_nnp.in  (singleband effective mass approximation)
> 1DSirtoriPRB1994_OneWell_sg_quantumonly_nn3.in
/ *_nnp.in  (singleband effective mass approximation)
> 1DSirtoriPRB1994_OneWell_kp_quantumonly_nn3.in
/ *_nnp.in  (8band k.p)
> 1DSirtoriPRB1994_TwoCoupledWells_sg_selfconsistent_nn3.in /
*_nnp.in  (singleband
effective mass approximation)
> 1DSirtoriPRB1994_TwoCoupledWells_sg_quantumonly_nn3.in
/ *_nnp.in  (singleband
effective mass approximation)
> 1DSirtoriPRB1994_TwoCoupledWells_kp_quantumonly_nn3.in
/ *_nnp.in  (8band k.p)
> 1DSirtoriPRB1994_ThreeCoupledWells_sg_selfconsistent_nn3.in / *_nnp.in
 (singleband effective
mass approximation)
> 1DSirtoriPRB1994_ThreeCoupledWells_sg_quantumonly_nn3.in /
*_nnp.in  (singleband effective
mass approximation)
> 1DSirtoriPRB1994_ThreeCoupledWells_kp_quantumonly_nn3.in
/ *_nnp.in  (8band k.p)
Input files for both the nextnano^{3}
and nextnano++ software are available.
Intersubband transitions in InGaAs/AlInAs multiple quantum well systems
This tutorial aims to reproduce Figs. 4 and 5 of
C. Sirtori, F. Capasso, J. Faist
Nonparabolicity and a sum rule associated with boundtobound
and boundtocontinuum intersubband transitions in quantum wells
Physical Review B 50 (12), 8663
(1994)
This tutorial nicely demonstrates that for the ground state energy the
singleband effective mass approximation is sufficient whereas for the higher
lying states a nonparabolic model, like the 8band k.p approximation, is
necessary.
This is important for e.g. quantum cascade lasers where higher lying
states have a dominant role.
Step1: Layer sequence
We investigate three structures:
a) a single quantum well
b) two coupled quantum wells
c) three coupled quantum wells
Step 2: Material parameters
We use In_{0.53}Ga_{0.47}As as the quantum well material and
Al_{0.48}In_{0.52}As as the barrier material. Both materials are
lattice matched to the substrate material InP. Thus we assume that the InGaAs
and AlInAs layers are unstrained with respect to the InP substrate.
The paper
C. Sirtori, F. Capasso, J. Faist
Nonparabolicity and a sum rule associated with boundtobound
and boundtocontinuum intersubband transitions in quantum wells
Physical Review B 50 (12), 8663
(1994)
lists the following material parameters:
conduction band offset 
Al_{0.48}In_{0.52}As/In_{0.53}Ga_{0.47}As 
0.510 eV 
conduction band effective mass 
(In_{0.53}Ga_{0.47}As) 
0.043 m _{0} 
conduction band effective mass 
(Al_{0.48}In_{0.52}As) 
0.072 m _{0} 
The temperature is set to 10 Kelvin.
Step 3: Method
Singleband effective mass approximation
Because our structure is doped, we have to solve the singleband
SchrödingerPoisson equation selfconsistently.
The doping is such that the electron ground state is below the Fermi level and
all other states are far away from the Fermi level, i.e. only the ground state
is occupied and contributes to the charge density.
$simulationflowcontrol
flowscheme = 2
rawpotentialin = no
$quantummodelelectrons
...
modelname
= effectivemass
numberofeigenvaluesperband = 3
! Note: Singleband eigenstates are twofold spin
degenerate.
The Fermi level is always equal to 0 eV in our simulations and the band profile
is shifted accordingly to meet this requirement.
8band k.p approximation
Old version of this tutorial:
Becauce both, the singleband and the 8band k.p ground state energy
and the corresponding wave functions are almost identical, we can read in the
selfconsistently calculated electrostatic potential of the singleband
approximation and calculate for this potential the 8band k.p eigenstates
and wave functions for k_{} = 0.
$simulationflowcontrol flowscheme =
3 rawdirectoryin =
raw_data/ rawpotentialin =
yes
$quantummodelelectrons ... modelname
= 8x8kp numberofeigenvaluesperband = 6
! Note: One k.p eigenstate for each spin component.
New version of this tutorial:
We provide input files for:
a) selfconsistent singleband Schrödinger equation (because the structure
is doped)
b) singleband Schrödinger equation (without selfconsistency)
c) 8band k.p singleband Schrödinger equation (without
selfconsistency)
For a), although the structure is doped, the band bending is very small.
Thus we omit for the singleband / k.p comparison in b) and c)
the selfconsistent cycle.
Step 4: Results
 Single quantum well
The following figure shows the lowest two electron eigenstates for an In_{0.53}Ga_{0.47}As/Al_{0.48}In_{0.52}As
quantum well structure calculated with singleband effective mass
approximation and with a nonparabolic 8band k.p model. The energies (and
wave functions, i.e. psi²) for the ground state are identical in both models
but the second eigenstate differs substantially. Clearly the singleband model
leads to an energy which is far too high for the upper state. Our calculated
value for the intersubband transition energy E_{12} of 255 meV
compares well with both, the calculated value of Sirtori et al. (258 meV) and
their measured value (compare with absorption curve in Fig. 4 of their paper).
The calculated intersubband dipole moments are:
z_{12} = 1.55 nm (singleband)
z_{12} = ... nm (8band k.p)
(Sirtori's paper: 1.53 nm (exp.), 1.48 nm (th.)).
The influence of doping on the eigenenergies is negligible (smaller than 1
meV).
 Two coupled quantum wells
The following figure shows the lowest three electron eigenstates for an In_{0.53}Ga_{0.47}As/Al_{0.48}In_{0.52}As
double quantum well structure calculated with singleband effective mass
approximation and with a nonparabolic 8band k.p model. The energies (and
wave functions, i.e. psi²) for the ground state are very similar in both models
but the second and especially the third eigenstate differ substantially.
Clearly the singleband model leads to energies which are far too high for the
higher lying states. Our calculated values for the intersubband transition
energies E_{12} = 150 meV and E_{13} = 267 meV compare well
with both, the calculated values of Sirtori et al. (150 meV and 271 meV) and
their measured values (compare with absorption curve in Fig. 5 (a) of their
paper).
The calculated intersubband dipole moments are:
z_{12} = 1.61 nm (singleband)
z_{13} = 1.11 nm (singleband)
z_{12} = ... nm (8band k.p)
z_{13} = ... nm (8band k.p)
(Sirtori's paper: z_{12} = 1.64 nm (exp.), z_{12} = 1.65
nm (th.))
The influence of doping on the eigenenergies is almost negligible (between 0
and 2 meV).
 Three coupled quantum wells
The following figure shows the lowest four electron eigenstates for an In_{0.53}Ga_{0.47}As/Al_{0.48}In_{0.52}As
triple quantum well structure calculated with singleband effective mass
approximation and with a nonparabolic 8band k.p model. The energies (and
wave functions, i.e. psi²) for the ground state are similar in both models but
the second and especially the third and forth eigenstates differ
substantially. Clearly the singleband model leads to energies which are far
too high for the higher lying states. Our calculated values for the
intersubband transition energies E_{12} = 118 meV, E_{13} =
261 and E_{14} = 370 meV compare well with both, the calculated values
of Sirtori et al. (116 meV, 257 meV and 368 meV) and their measured values
(compare with absorption curve in Fig. 5 (b) of their paper).
The calculated intersubband dipole moments are:
z_{12} = 1.81 nm (singleband)
z_{13} = 0.77 nm(singleband)
z_{14} = 0.30 nm(singleband)
z_{12} = ... nm (8band k.p)
z_{13} = ... nm (8band k.p)
z_{14} = ... nm (8band k.p)
(Sirtori's paper: z_{12} = 1.86 nm (exp.), z_{12} = 1.84
nm
(th.))
The influence of doping on the eigenenergies is almost negligible (between 0
and 4 meV).
 Please help us to improve our tutorial! Send comments to
support
[at] nextnano.com .
