Gain spectrum by Fermi’s golden rule

Theory

nextnano++, nextnano³, and nextnano.NEGF treat light as a classical electromagnetic field. It assumes the dipole approximation, which is valid for sub-micron active region sizes and for visible to mid-infrared light.

The electric field amplitude is considered as perturbative. From Fermi’s golden rule, the absorption spectrum, namely (number of photons absorbed per unit volume per unit time) / (number of photons injected per unit area per unit time), is calculated for given electric field polarization \(\vec{\epsilon}\) and photon energy \(\hbar\omega\) [ChuangOpto1995]:

(4.2.6)\[\begin{split}\begin{aligned} \alpha(\vec{\epsilon}, \omega) &= \frac{\pi e^2}{c \varepsilon_0 \sqrt{\varepsilon(\omega)} m_0^2 \omega} \frac{1}{V} \sum_{n > m} \sum_{\mathbf{k}_\parallel} |\vec{\epsilon} \cdot \vec{\pi}_{nm}(\mathbf{k}_\parallel)|^2 [f_m(\mathbf{k}_\parallel) - f_n(\mathbf{k}_\parallel)] \notag\\ & \qquad \times \frac{1}{\sqrt{2\pi}\sigma} \exp{\left[ -\frac{[E_n(\mathbf{k}_\parallel) - E_m(\mathbf{k}_\parallel) - \hbar\omega]^2}{2\sigma^2} \right]} \end{aligned}\end{split}\]

where \(e, c, m_0\) are the elementary charge, vacuum speed of light, and bare electron mass, respectively. \(\varepsilon_0\) and \(\varepsilon(\omega)\) are the vacuum permittivity and dielectric function at the photon frequency. \(\vec{\pi}_{nm}\) is the in-plane wavevector-dependent momentum matrix elements. The summation is taken over all possible transitions with positive photon energy. Due to the dipole approximation, the formula involves only the vertical transitions.

nextnano.NEGF implementation

nextnano.NEGF treats carriers by non-equilibrium Green’s functions. The occupation \(f_n(\mathbf{k}_\parallel)\) is the diagonal elements of the density matrix. The normalization volume \(V\) is the one of a cylinder with the length of one period and radius determined internally from Value. The momentum matrix elements are in the eigenbasis of the reduced Hamiltonian (see SimulationParameter{ } for the mode-space approach in nextnano.NEGF). The standard deviation of the Gaussian distribution is calculated from the input parameter FermiGoldenRule{ Linewidth } which we denote here by \(\Gamma\) (meV):

(4.2.7)\[\sigma = \frac{\Gamma}{2\sqrt{2\log{2}}}\]

nextnano.NEGF outputs the minus of the absorption spectrum, i.e., gain spectrum in the folder (Bias)mV\Gain.