\label{eq3}$$ where $\text{Im}[...]$ is the imaginary part and $f_{\text{L, R}}$ is the Fermi distribution for the left and right leads. Similarly, the out-scattering self-energies are defined as: $$\Sigma_{\text{L, R}}^{\text{out}} (E) = - 2 \text{Im}  \left[ \Sigma_{\text{L, R}} (E) \right]  \left[ 1 - f_{\text{L, R}} (E) \right] .
\label{eq4}$$ With $\Sigma_{\text{L, R}}^{\text{in/out}} (E)$ one can express the electron and hole correlation functions as: $$G^{\text{n}} (E) = G(E) \Sigma_{\text{L, R}}^{\text{in}} (E) G^\dagger (E) ,
\label{eq5}$$ $$G^{\text{p}} (E) = G(E) \Sigma_{\text{L, R}}^{\text{out}} (E) G^\dagger (E) .
\label{eq6}$$

Assuming that the system consists of 2D grid/lattice points with uniform spacing $a$, and making the nearest neighbour tight-binding approximation, the current density between grid points $j$ and $j+1$ is given by $$\begin{aligned}
\nonumber J_{j, j+1} = \frac{i e}{\hbar}
\\* &&\hspace*{-1in} \times (2)  \int_{-\infty}^{\infty} \frac{d E}{2 \pi} \left[ H_{j+1,j} G_{j, j+1}^{\text{n}} (E) - H_{j,j+1} G_{j+1, j}^{\text{n}} (E) \right]   ,
\label{eq7}$$ where $H_{j+1, j} = H_{j, j+1}^\dagger$ are the hopping matrix elements of the Hamiltonian, and $(2)$ is for the two spin directions.

A second source for in-scattering and out-scattering of electrons from an occupied state is the e-ph interaction. The self-energy at point $j$ and energy $E$ has two terms corresponding to scattering from $(j, E+\hbar 