Intersubband absorption of an infinite quantum well¶
This tutorial calculates the intersubband absorption of a GaAs quantum well with infinite barriers.
Input files for both the nextnano++ and nextnano³ software are available.
The following input file was used:
1D_IntersubbandAbsorption_InfiniteWell_GaAs_Chuang_sg_nn3.in
(singleband effective mass approximation)
This tutorial aims to reproduce the example discussed on p. 376f of Section 9.6.2 Intersubband Absorption Spectrum of [ChuangOpto1995].
Structure¶
Property 
Symbol 
unit 
nextnano 


quantum well width 
L 
nm 
10.0 
10.0 

barrier height 
E _{b} 
eV 
infinite quantum well model 
1000 

effective electron mass 
m_{e} 
m_{0} 
0.0665 
0.0665 

refractive index 
n_{r} 
3.3 
3.3 

doping concentation (ntype) 
N_{D} 
cm^{3} 
1 \(\cdot\) 10^{18} 
1 \(\cdot\) 10^{18} 

linewidth (FWHM) 
\(\Gamma\) 
meV 
30 
30 

temperature 
T 
K 
300 
300 
[ChuangOpto1995] models the infinite quantum well using the analytical solution while we are using a numerical model with a barrier height of 1000 eV.
Results¶
[ChuangOpto1995] uses the analytical infinite quantum well model and calculates the energy levels, and the intersubband dipole moment exactly. Our calculated transition energies differ by 3 meV which is acceptable as we use a finite grid spacing of 0.05 nm. Our calculated dipole moment is also reasonable. More difficult are the densities. In our calculation we solve the SchrödingerPoisson equation selfconsistently. For that reason, the quantum well bottom is not entirely flat but slightly bended. At T = 300 K, the second subband shows a small density which is larger than in the model of [ChuangOpto1995]. The difference in subband densities leads to a slight deviation for the peak of the absorption curve because the occupation of the second level N_{2} decreases the absorption. Nevertheless, the agreement is reasonable.
Property 
Symbol 
unit 
nextnano 


energy level 
E_{1} 
meV 
56.5 (exact) 

energy level 
E_{2} 
meV 
226 (exact) 

transition energy 
E_{21} 
meV 
169.5 (exact) 
166.5 
dipole moment 
x_{21} 
nm 
1.8 (exact) 
1.82 
E_{F}  E_{1} 
eV 
78 
28.2 

subband density 
N_{1} 
cm^{2} 
7.19 \(\cdot\) 10^{11} 
9.92 \(\cdot\) 10^{11} 
subband density 
N_{2} 
cm^{2} 
3 \(\cdot\) 10^{9} 

peak in absorption 
\(\alpha\)_{peak} 
cm^{1} 
1.015 \(\cdot\) 10^{4} 
0.986 \(\cdot\) 10^{4} 
The following figures show the
lowest eigenstates (probability densities) of the infinite quantum well
absorption spectra \(\alpha(\omega)\) in units of cm^{1}
position dependent absorption spectra \(\alpha(\omega ,x)\) in units of cm^{1}
The peak in the absorption spectra occurs at the transition energy E_{21}.
Then we perform two parameter sweeps:
We vary the quantum well width (Variable:
$QuantumWellWidth
).We vary the doping concentration (Variable:
$DopingConcentration
).
Results and explanations for the sweeps can be found further below.
— Begin —
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_InfiniteQW_nextnano3.py
The following figures have been generated using the nextnano³ software. Selfconsistent SchroedingerPoisson calculations have been performed for an infinite quantum well.
A singleband effective mass approach has been used, i.e. not \(\mathbf{k} \cdot \mathbf{p}\).
The absorption has been calculated assuming a parabolic energy dispersion \(E(k)\).
Infinite Quantum Well (QuantumWellWidth = 10 nm)
Infinite Quantum Well (QuantumWellWidth = 13 nm)
Infinite Quantum Well (QuantumWellWidth = 16 nm)
Infinite Quantum Well (QuantumWellWidth = 19 nm)
Parameter sweep: Well width
The following figure shows the absorption for different quantum well widths (Variable: $QuantumWellWidth
). The larger the well, the closer the energy level spacings. Therefore the peak occurs at smaller energies. The larger wells show absorption also for transitions other than E_{21}.
Parameter sweep: Doping concentration
The following figure shows the absorption for different doping concentrations (Variable: $DopingConcentration
). The peak absorption coefficient increases with the doping concentration N_{D}.
Automatic documentation: Running simulations, generating figures and reStructured Text (*.rst) using nextnanopy
— End —