2.16. Optical spectra

2.16.1. Fermi’s golden rule

The nextnano++ tool has another important calculation scheme of optical properties, which is specified in the section optics{ quantum_spectra{ } }. Here nextnano++ calculates them using the Fermi’s golden rule (time-dependent perturbation theory) with 8-band k.p model.

• Optical absorption coefficient

• Real/imaginary part of the dielectric constant

• Refractive index

• Optical gain as a negative part of optical absorption coefficient

• Spontaneous emission rate

• Transition intensity (optical matrix element)

For further detail about this section, please see Optical absorption for interband and intersubband transitions.

This page will summarize theory, that is currently distributed on the following pages:

• Intersubband transitions in InGaAs/AlInAs multiple quantum well systems

• Optical intraband transitions in a quantum well - Momentum matrix elements and selection rules

• Optical absorption for interband and intersubband transitions

2.16.2. Broadening

Broadening of the optical spectra is a convolution of the calculated spectra with a lineshape function.

\[S(E) = \int dE' L(E-E') S_0(E')\]

where \(S_0(E)\) is the calculated spectrum of some optical property without broadening, \(L(E-E')\) is the lineshape function and \(S(E)\) is the broadened spectrum.

In nextnano++ three different types of broadening are implemented: Gaussian, Lorentzian and Voigt.

Gaussian line function is:

\[\mathcal{L}_{G}(E-E_0)=\frac{1}{\sqrt{2\pi}\sigma}\exp{\big(-\frac{(E-E_0)^2}{2\sigma^2}\big)}\]

Lorentzian line function is:

\[\mathcal{L}_{L}(E-E_0)=\frac{1}{\pi}\frac{\Gamma/2}{(E-E_0)+(\Gamma/2)^2}\]

The Voigt profile is a convolution of a Gaussian and a Lorentzian:

\[\mathcal{L}_{V}(E-E_0)=\int dE' \mathcal{L}_{G}(E-E_0-E')\mathcal{L}_{L}(E')\]

Gaussian and Lorentzian broadening are specified by their respective full width at half maximum (FWHM) parameters \(\text{FWHM}_G\) and \(\text{FWHM}_L\). The standard deviation \(\sigma\) of the Gaussian is related to its FWHM by \(\text{FWHM}_G/ = 2\sqrt{2\ln{2}}\cdot \sigma\), while the Lorentzian FWHM is simply \(\text{FWHM}_L = \Gamma\). In the input file, the parameters are specified as:

optics{
    quantum_spectra{
        ...
        energy_broadening_gaussian = ... # FWHM in eV
        energy_broadening_lorentzian = ... # FWHM in eV
    }
}

In case both Gaussian and Lorentzian broadening are specified in the input file, it results in a Voigt broadening.