# 1.14. 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 (single-band 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

[ChuangOpto1995]

nextnano

quantum well width

L

nm

10.0

10.0

barrier height

E b

eV

infinite quantum well model

1000

effective electron mass

me

m0

0.0665

0.0665

refractive index

nr

3.3

3.3

doping concentation (n-type)

ND

cm-3

1 $$\cdot$$ 1018

1 $$\cdot$$ 1018

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 Schrödinger-Poisson equation self-consistently. 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 N2 decreases the absorption. Nevertheless, the agreement is reasonable.

Property

Symbol

unit

[ChuangOpto1995]

nextnano

energy level

E1

meV

56.5 (exact)

energy level

E2

meV

226 (exact)

transition energy

E21

meV

169.5 (exact)

166.5

dipole moment

x21

nm

-1.8 (exact)

-1.82

EF - E1

eV

78

28.2

subband density

N1

cm-2

7.19 $$\cdot$$ 1011

9.92 $$\cdot$$ 1011

subband density

N2

cm-2

3 $$\cdot$$ 109

peak in absorption

$$\alpha$$peak

cm-1

1.015 $$\cdot$$ 104

0.986 $$\cdot$$ 104

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 E21.

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. Self-consistent Schroedinger-Poisson calculations have been performed for an infinite quantum well.

A single-band 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) Figure 1.14.1 Conduction band edge, Fermi level and confined electron states of an infinite quantum well (QuantumWellWidth = 10 nm) Figure 1.14.2 Calculated absorption $$\alpha(E)$$ of an infinite quantum well (QuantumWellWidth = 10 nm) Figure 1.14.3 Calculated position resolved absorption $$\alpha(x,E)$$ of an infinite quantum well (QuantumWellWidth = 10 nm)

Infinite Quantum Well (QuantumWellWidth = 13 nm) Figure 1.14.4 Conduction band edge, Fermi level and confined electron states of an infinite quantum well (QuantumWellWidth = 13 nm) Figure 1.14.5 Calculated position resolved absorption $$\alpha(x,E)$$ of an infinite quantum well (QuantumWellWidth = 13 nm)

Infinite Quantum Well (QuantumWellWidth = 16 nm) Figure 1.14.6 Conduction band edge, Fermi level and confined electron states of an infinite quantum well (QuantumWellWidth = 16 nm) Figure 1.14.7 Calculated position resolved absorption $$\alpha(x,E)$$ of an infinite quantum well (QuantumWellWidth = 16 nm)

Infinite Quantum Well (QuantumWellWidth = 19 nm) Figure 1.14.8 Conduction band edge, Fermi level and confined electron states of an infinite quantum well (QuantumWellWidth = 19 nm) Figure 1.14.9 Calculated position resolved absorption $$\alpha(x,E)$$ of an infinite quantum well (QuantumWellWidth = 19 nm)

Parameter sweep: Well width

Figure 1.14.10 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 E21. Figure 1.14.10 Calculated absorption $$\alpha(E)$$ of an infinite quantum well for different well widths Parameter sweep: Doping concentration Figure 1.14.11 shows the absorption for different doping concentrations (Variable: $DopingConcentration). The peak absorption coefficient increases with the doping concentration ND. Figure 1.14.11 Calculated absorption $$\alpha(E)$$ of an infinite quantum well for different doping concentrations

Automatic documentation: Running simulations, generating figures and reStructured Text (*.rst) using nextnanopy

— End —