Intersubband transitions in InGaAs/AlInAs multiple quantum well systems
This tutorial calculates the eigenstates of a single, double and triple quantum wells.
It compares the energy levels and wave functions of the single-band effective mass approximation with the 8-band
Input files for both the nextnano++ and nextnano³ tools are available.
The following input files were used:
Single Quantum Well
1DSirtoriPRB1994_OneWell_sg_self-consistent_nn*.in
(single-band effective mass approximation)1DSirtoriPRB1994_OneWell_kp_self-consistent_nn*.in
(8-band )1DSirtoriPRB1994_OneWell_sg_quantum-only_nn*.in
(single-band effective mass approximation)1DSirtoriPRB1994_OneWell_kp_quantum-only_nn*.in
(8-band )
Two coupled Quantum Wells
1DSirtoriPRB1994_TwoCoupledWells_sg_self-consistent_nn*.in
(single-band effective mass approximation)1DSirtoriPRB1994_TwoCoupledWells_kp_self-consistent_nn*.in
(8-band )1DSirtoriPRB1994_TwoCoupledWells_sg_quantum-only_nn*.in
(single-band effective mass approximation)1DSirtoriPRB1994_TwoCoupledWells_kp_quantum-only_nn*.in
(8-band )
Three coupled Quantum Wells
1DSirtoriPRB1994_ThreeCoupledWells_sg_self-consistent_nn*.in
(single-band effective mass approximation)1DSirtoriPRB1994_ThreeCoupledWells_kp_self-consistent_nn*.in
(8-band )1DSirtoriPRB1994_ThreeCoupledWells_sg_quantum-only_nn*.in
(single-band effective mass approximation)1DSirtoriPRB1994_ThreeCoupledWells_kp_quantum-only_nn*.in
(8-band )
This tutorial aims to reproduce Figs. 4 and 5 of [SirtoriPRB1994].
This tutorial nicely demonstrates that for the ground state energy the single-band effective mass approximation is sufficient whereas for the higher lying states a nonparabolic model, like the 8-band
Layer sequence
We investigate three structures:
a single quantum well
two coupled quantum wells
three coupled quantum wells
Material parameters
We use In0.53 Ga0.47 As as the quantum well material and Al0.48 In0.52 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 publication [SirtoriPRB1994] lists the following material parameters:
conduction band offset |
Al0.48 In0.52 As / In0.53 Ga0.47 As |
0.510 eV |
conduction band effective mass |
Al0.48 In0.52 As |
0.072 m0 |
conduction band effective mass |
In0.53 Ga0.47 As |
0.043 m0 |
The temperature is set to 10 Kelvin.
Method
Single-band effective mass approximation
Because our structure is doped, we have to solve the single-band Schrödinger-Poisson equation self-consistently. 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.
For nextnano++ we use:
# '0' solve Schrödinger equation only # '1' solve Schrödinger and Poisson equations self-consistently $SELF_CONSISTENT = 1 run{ !IF($SELF_CONSISTENT) poisson{ } quantum_poisson{ iterations = 50 } # Schrödinger-Poisson !ELSE quantum{ } # Schrödinger only !ENDIF } quantum { region{ ... Gamma{ # single-band num_ev = 3 # 3 eigenstates }
For nextnano³ we use:
# 'QUANTUM_ONLY': solve Schrödinger and Poisson equations self-consistently # 'SELF_CONSISTENT': solve Schrödinger equation only %QUANTUM_ONLY = .FALSE. %SELF_CONSISTENT = .TRUE. $simulation-flow-control !IF %QUANTUM_ONLY flow-scheme = 3 # Schrödinger only !IF %SELF_CONSISTENT flow-scheme = 2 # Schrödinger-Poisson !IF %QUANTUM_ONLY raw-potential-in = yes !IF %SELF_CONSISTENT raw-potential-in = no $quantum-model-electrons ... model-name = effective-mass # single-band number-of-eigenvalues-per-band = 3 # 3 eigenstates
Note
Single-band eigenstates are two-fold 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.
8-band k.p approximation
Old version of this tutorial:
Becauce both, the single-band and the 8-band
For nextnano³ we use:
$simulation-flow-control !IF %QUANTUM_ONLY flow-scheme = 3 # Schrödinger only !IF %QUANTUM_ONLY raw-directory-in = raw_data/ !IF %QUANTUM_ONLY raw-potential-in = yes $quantum-model-electrons ... model-name = 8x8kp # 8-band k.p number-of-eigenvalues-per-band = 6 # 6 eigenstates
Note
One
New version of this tutorial:
We provide input files for:
self-consistent single-band Schrödinger equation (because the structure is doped)
single-band Schrödinger equation (without self-consistency)
8-band
single-band Schrödinger equation (without self-consistency)
For a), although the structure is doped, the band bending is very small.
Thus we omit for the single-band /
Results
Single quantum well
Figure 2.4.343 shows the lowest two electron eigenstates for an In0.53 Ga0.47 As / Al0.48 In0.52 As quantum well structure calculated with single-band effective mass approximation and with a nonparabolic 8-band
The energies (and square of the wave functions
Our calculated value for the intersubband transition energy

Figure 2.4.343 Conduction band edge, Fermi level and confined electron states of a quantum well
The calculated intersubband dipole moments are:
= 1.55 nm (single-band)
For comparison:
The influence of doping on the eigenenergies is negligible (smaller than 1 meV).
Two coupled quantum wells
Figure 2.4.344 shows the lowest three electron eigenstates for an In0.53 Ga0.47 As / Al0.48 In0.52 As double quantum well structure calculated with single-band effective mass approximation and with a nonparabolic 8-band
The energies (and square of the wave functions
Our calculated values for the intersubband transition energies

Figure 2.4.344 Conduction band edge, Fermi level and confined electron states of two coupled quantum wells
The calculated intersubband dipole moments are:
= 1.61 nm (single-band)
= 1.11 nm (single-band)
For comparison:
The influence of doping on the eigenenergies is almost negligible (between 0 and 2 meV).
Three coupled quantum wells
Figure 2.4.345 shows the lowest four electron eigenstates for an In0.53 Ga0.47 As / Al0.48 In0.52 As triple quantum well structure calculated with single-band effective mass approximation and with a nonparabolic 8-band
The energies (and square of the wave functions
Our calculated values for the intersubband transition energies

Figure 2.4.345 Conduction band edge, Fermi level and confined electron states of three coupled quantum wells
The calculated intersubband dipole moments are:
= 1.81 nm (single-band)
= 0.77 nm (single-band)
= 0.30 nm (single-band)
For comparison:
The influence of doping on the eigenenergies is almost negligible (between 0 and 4 meV).
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Automatic documentation: Running simulations, generating figures and reStructured Text (*.rst) using nextnanopy
The following documentation and figures were generated automatically using nextnanopy.
The following Python script was used: intersubband_MQW_nextnano3.py
The following figures have been generated using nextnano³. Self-consistent Schrödinger-Poisson calculations have been performed for three different structures.
Single Quantum Well
Two coupled Quantum Wells
Three coupled Quantum Wells
The single-band effective mass and the 8-band
The absorption spectrum has been calculated using a simple model assuming a parabolic energy dispersion.
The dipole moment
Quantum Well (single-band)

Figure 2.4.346 Conduction band edge, Fermi level and confined electron states of a quantum well

Figure 2.4.347 Calculated spatially resolved absorption spectrum
Quantum Well (k.p)

Figure 2.4.348 Conduction band edge, Fermi level and confined electron states of a quantum well

Figure 2.4.349 Calculated spatially resolved absorption spectrum

Figure 2.4.350 Calculated absorption spectra
Two Coupled Quantum Wells (single-band)

Figure 2.4.351 Conduction band edge, Fermi level and confined electron states of two coupled quantum wells

Figure 2.4.352 Calculated spatially resolved absorption spectrum:math:alpha(x,E) of two coupled quantum wells
Two Coupled Quantum Wells (k.p)

Figure 2.4.353 Conduction band edge, Fermi level and confined electron states of two coupled quantum wells

Figure 2.4.354 Calculated spatially resolved absorption spectrum:math:alpha(x,E) of two coupled quantum wells

Figure 2.4.355 Calculated absorption spectra
Three Coupled Quantum Wells (single-band)

Figure 2.4.356 Conduction band edge, Fermi level and confined electron states of three coupled quantum wells

Figure 2.4.357 Calculated spatially resolved absorption spectrum
Three Coupled Quantum Wells (k.p)

Figure 2.4.358 Conduction band edge, Fermi level and confined electron states of three coupled quantum wells

Figure 2.4.359 Calculated spatially resolved absorption spectrum

Figure 2.4.360 Calculated absorption spectra
Automatic documentation: Running simulations, generating figures and reStructured Text (*.rst) using nextnanopy
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Last update: nn/nn/nnnn