and so in Eqs. (\[eq8\]) and (\[eq9\]) we let $D_0 n_{\text{B}} \rightarrow D_{\text{AP}}$ (making use of the commonly employed equipartition approximation – see details in Appendix A) while in the case of optical phonons $D_0 \rightarrow D_{\text{OP}}$, which are taken to be constant throughout the channel.

The strength of the phonon scattering is adjusted such that the mean-free-path of electrons is $\lambda = 15$ nm. The way we calibrate this, is that we initially simulate a channel with length $L = 15$ nm in the ballistic regime, and then we increase the electron - acoustic phonon scattering strength $D_{\text{AP}}$ in the NEGF formalism, until the channel conductance drops to 50% of its ballistic value ($D_{\text{AP}}=0.0026$ eV$^2$). This effectively fixes a mean-free-path of $15$ nm for the channel (under acoustic phonon scattering conditions alone), a value that is comparable to that of common semiconductors such as silicon [@Neo11PRB]. The nanostructured geometries that we consider are shown in Fig. 1. The channels have length $L=100$ nm and width $W=15$ nm. Thus, with a mean-free-path of $\lambda = 15$ nm, the channel we consider is long enough for the transport to be diffusive. Note that the computational cost of NEGF simulations scales with 