nextnano.NEGF - Software for
Quantum Transport

In this tutorial we present how can be calculated the optical gain upon optical irradiation. The irradiation parameters are the Photon energy of the irradiation, Line width.

The transition rate per volume element can be expressed with the following sum: \[ R = R_{ab} - R_{ba} = \frac{2}{V} \sum_{k_a} \sum_{k_b} \frac{2 \pi}{ \hbar} |H_{ba}| ^2 \delta(E_b - E_a -\hbar \omega)(f_a-f_b) \]

In order to make evaluate the sum much faster we calculate the $H_{ba}$ matrix element at $k_a = 0; k_b = 0$ (Remark: $k_a = k_b$), and we neglect the k dependence of it. Then we can simplify the sum in the following form, if the irradiation has the $\gamma(E, w)$ broadening function, where $E$ is the irradiation energy, and $w$ is the line width.

\[ R(E, w) = C_0(E) \int dE_a dE_b \gamma(E_a-E, w) \cdot H(E_a-E) \cdot [n(E_a) - p(E_b)] \]

Here $C_0(E)$ is an energy dependent constant: \[ C_0 = \frac{\pi e^2 \hbar}{n_r c \epsilon_0 m_0^2 E} \]