Optical interband absorption in a quantum well including excitonic effects¶
This tutorial calculates the optical interband absorption 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 optical absorption 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 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 optical 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 


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 

effective hole mass 
m_{h} 
m_{0} 
0.0665 
0.0665 

refractive index 
n_{r} 
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:
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, the absorption 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 the nextnano³ software.
The absorption has been calculated using a simple model assuming a parabolic energy dispersion.
Infinite QW (singleband)
Optical absorption of bulk 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 (SiPhoG).
Automatic documentation: Running simulations, generating figures and reStructured Text (*.rst) using nextnanopy
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