# 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.

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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_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) Conduction band edge, Fermi level and confined electron states of an infinite quantum well (QuantumWellWidth = 10 nm) Calculated absorption $$\alpha(E)$$ of an infinite quantum well (QuantumWellWidth = 10 nm) Calculated position resolved absorption $$\alpha(x,E)$$ of an infinite quantum well (QuantumWellWidth = 10 nm)

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

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

Infinite Quantum Well (QuantumWellWidth = 19 nm) Conduction band edge, Fermi level and confined electron states of an infinite quantum well (QuantumWellWidth = 19 nm) Calculated position resolved absorption $$\alpha(x,E)$$ of an 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 E21. Calculated absorption $$\alpha(E)$$ of an infinite quantum well for different well widths 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 ND. 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

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