and non-Hermitian. They are formed using the first/last channel layers from which electrons are injected into the channel (and thus, they have the same size as these layers), and are calculated using the Sancho-Rubio iterative scheme [@Sancho_Rubio_85]. They are added to the first/last layer elements of the channel Hamiltonian. The e-ph scattering process in the device enters the NEGF formalism through the self-energy function $\Sigma_{\text{S}}$. One can view the scattering process as just another contact described by $\Sigma_{\text{S}}$, similar to the actual contacts described by $\Sigma_{\text{L}}$ and $\Sigma_{\text{R}}$, however, $\Sigma_{\text{S}}$ is added to all diagonal elements of the Hamiltonian.

The retarded Green’s function for the device is given by [@Datta05] $$G(E) = \left[ \left( E + i \eta^+ \right) I - H - \Sigma(E) \right]^{-1} ,
\label{eq1}$$ where $\eta^+$ is an infinitesimal positive number which pushes the poles of $G$ to the lower half plane in complex energy, $I$ is the identity matrix, and $\Sigma(E)$ is the sum of the self-energies $$\Sigma(E) = \Sigma_{\text{L}}(E) + \Sigma_{\text{R}}(E) + \Sigma_{\text{S}}(E) .
\label{eq2}$$ It proves useful and convenient to define the in-scattering self-energies due to contacts as: $$\Sigma_{\text{L, R}}^{\text{in}} (E) = - 2 \text{Im}  \left[ \Sigma_{\text{L, R}} (E) \right] f_{\text{L, R}} (E) ,
