Intersubband absorption of an infinite quantum well

Input files for nextnano³:

• 1D_IntersubbandAbsorption_InfiniteWell_GaAs_Chuang_sg_nn3.in

• 1D_IntersubbandAbsorption_InfiniteWell_GaAs_Chuang_kp_nn3.in

Input files for nextnano++:

• 1D_IntersubbandAbsorption_InfiniteWell_GaAs_Chuang_sg_nnp.in

• 1D_IntersubbandAbsorption_InfiniteWell_GaAs_Chuang_Gamma_nnp.in

• 1D_IntersubbandAbsorption_InfiniteWell_GaAs_Chuang_kp_nnp.in

This tutorial presents calculation of intersubband absorption spectrum of a GaAs quantum well with infinite barriers.

Input files for both the nextnano++ and nextnano³ tools 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)	$\mathrm{\Gamma }$	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 bent. 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 spectrum because the occupation of the second level N₂ reduces 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 nextnano³. Self-consistent Schrödinger-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 spectra have been calculated assuming a parabolic energy dispersion $E(k)$.

Infinite Quantum Well (QuantumWellWidth = 10 nm)

Figure 2.4.332 Conduction band edge, Fermi level and confined electron states of an infinite quantum well (QuantumWellWidth = 10 nm)

Figure 2.4.333 Calculated absorption spectra $\alpha (E)$ of an infinite quantum well (QuantumWellWidth = 10 nm)

Figure 2.4.334 Calculated spatially resolved absorption spectrum $\alpha (x,E)$ of an infinite quantum well (QuantumWellWidth = 10 nm)

Infinite Quantum Well (QuantumWellWidth = 13 nm)

Figure 2.4.335 Conduction band edge, Fermi level and confined electron states of an infinite quantum well (QuantumWellWidth = 13 nm)

Figure 2.4.336 Calculated spatially resolved absorption spectrum $\alpha (x,E)$ of an infinite quantum well (QuantumWellWidth = 13 nm)

Infinite Quantum Well (QuantumWellWidth = 16 nm)

Figure 2.4.337 Conduction band edge, Fermi level and confined electron states of an infinite quantum well (QuantumWellWidth = 16 nm)

Figure 2.4.338 Calculated spatially resolved absorption spectrum $\alpha (x,E)$ of an infinite quantum well (QuantumWellWidth = 16 nm)

Infinite Quantum Well (QuantumWellWidth = 19 nm)

Figure 2.4.339 Conduction band edge, Fermi level and confined electron states of an infinite quantum well (QuantumWellWidth = 19 nm)

Figure 2.4.340 Calculated spatially resolved absorption spectrum $\alpha (x,E)$ of an infinite quantum well (QuantumWellWidth = 19 nm)

Parameter sweep: Well width

Figure 2.4.341 shows the absorption spectra 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 2.4.341 Calculated absorption spectra $\alpha (E)$ of an infinite quantum well for different well widths

Parameter sweep: Doping concentration

Figure 2.4.342 shows the absorption spectra for different doping concentrations (Variable: $DopingConcentration). The peak absorption coefficient increases with the doping concentration N_D.

Figure 2.4.342 Calculated absorption spectra $\alpha (E)$ of an infinite quantum well for different doping concentrations

---

Last update: nn/nn/nnnn