the specific 2D simulator, which we developed in order to capture phonon scattering, as well as its convergence details, are described in several works of ours and of others [@Koswatta07; @Datta05; @Anantram08; @Foster17; @Var17pssa]. However, in order to be able to better discuss certain characteristics of e-ph scattering in the context of this work, we include here a brief description of the model with notation adopted to our system.

In the NEGF method a system/channel, described by a Hamiltonian $H$, is connected to two contacts (left and right), which are represented by self-energy functions $\Sigma_{\text{L}}$ and $\Sigma_{\text{R}}$. The Hamiltonian is constructed using a 2D effective mass (single-orbital) tight-binding grid uniformly spaced in the $x$- and $y$-directions, resulting in a banded matrix. The 2D channels we consider within the effective mass approximation, have a uniform $m^\ast = m_0$ in the entire channel, where $m_0$ is the rest mass of the electron. Thus, we do not consider a specific material, rather our study aims in providing first-order qualitative guidance into the design of high power factor nanostructured materials. These self-energies represent the influence of the semi-infinite Left and Right leads on the channel, respectively. Note that $\Sigma_{\text{L}}$ and $\Sigma_{\text{R}}$ are energy dependent, 