# Optical interband absorption in a quantum well including excitonic effects¶

This tutorial presents calculation of interband absorption spectrum in a quantum well including excitonic effects.

There is a separate tutorial that discusses the calculation of the exciton binding energy and exciton Bohr radius of an infinite quantum well: Exciton energy in quantum wells - Tutorial

In this tutorial we calculate the absorption spectrum of a 10 nm GaAs quantum well. The purpose is to calculate the absorption spectrum for a simple model and model that includes excitonic effects on the absorption spectrum.

The absorption spectrum has been calculated using a simple model assuming a parabolic energy dispersion. In order to keep things simple, i.e. to be able to compare our results with analytical formula, we used the same effective mass for electrons and holes ($$m_{\rm e} = m_{\rm h} = 0.065~m_{\rm 0}$$).

The excitonic binding energy $$E_{\rm b}$$ has been calculated to be -9.5 meV. Therefore, the absorption spectrum that includes excitonic contributions starts at an energy roughly 10 meV below than band gap. The exciton Bohr radius $$\lambda$$ was found to be 13.1 nm.

For Lorentzian broadening we use a linewidth of FWHM = 6 meV, and for Gaussian broadening we use FWHM = 10 meV. The FWHM(Voigt) depends in a complicated way on FWHM(Lorentzian) and FWHM(Gaussian).

Property

Symbol

unit

analytical calculation

nextnano GmbH

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

effective hole mass

mh

m0

0.0665

0.0665

refractive index

nr

3.3

3.3

linewidth (FWHM) Lorentzian

$$\Gamma_{\rm L}$$

meV

n/a

6

linewidth (FWHM) Gaussian

$$\Gamma_{\rm G}$$

meV

n/a

10

temperature

T

K

300

300

The Coulomb enhancement factor is given by $$S_{\rm 2D}=\frac{\exp \left(\pi/\sqrt{\Delta}\right)}{\cosh \left(\pi/\sqrt{\Delta}\right)}$$, where $$\Delta$$ is the total excess kinetic energy of the electron–hole pair normalized to $$E_{\rm b}/4$$ [LeverJLT2010].

We observe two major contributions to the absorption spectrum:

• A distinct peak a few meV (corresponding to the exciton binding energy $$E_{\rm b}$$) lower than the absorption edge (band gap). This is the signature of the bound exciton.

• Sommerfeld enhancement: In the continuum part of the absorption spectrum, it is scaled via the Coulomb enhancement factor $$S_{\rm 2D}$$.

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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: 1D_InterbandAbsorption_InfiniteWell_Exciton_nextnano3.py

The following figures have been generated using nextnano³.

The absorption spectrum has been calculated using a simple model assuming a parabolic energy dispersion.

Infinite QW (single-band)

Optical absorption spectrum of bulk crystal and of a quantum well

Optical absorption of a 10 nm quantum well

Optical absorption of a 10 nm quantum well using different broadening functions

Optical absorption of a 10 nm quantum well showing the different contributions to the excitonic absorption

We acknowledge funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 101017194 (SiPho-G).

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

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