the third power in channel width, and thus we only consider narrow channels of widths as given above. However, using wider structures should not affect our final conclusions. Throughout the paper we assume room temperature $T = 300$ K.

In the case that we consider electron-optical phonon scattering only, we use an optical phonon energy of $\hbar\omega = 0.06$ eV (a value similar to that of silicon longitudinal optical phonons, see for example [@Jacoboni83]), and simply use the same value of the scattering strength in the simulations, i.e., $D_0 = D_{\text{OP}}=D_{\text{AP}}=0.0026$ eV$^2$. This gives an energy relaxation length of $\lambda_{\text{E}} = 13$ nm, which, in the well region of length $60$ nm, dictates semi-relaxation of the electron energy, after they pass over the SL barriers. We consider this semi-relaxation of energy because it has been shown that it provides optimal conditions for the $PF$ of SL materials [@Neo13ntech; @Thes15JAP; @Kim09]. Note, however, that in this way the conductance of the channel with only optical phonon scattering will be in general larger compared to that with only acoustic phonon scattering. This is due to the fact that electron scattering with optical phonons has weaker phonon absorption rates, as a consequence of 