lower than unity phonon occupation number. Thus, we cannot compare the two cases quantitatively. In the case where we consider both acoustic and optical phonon scattering, we simply divide the scattering strengths $D_{\text{AP}}$ and $D_{\text{OP}}$ by a factor of two, which (as we show later) gives similar conductances for their pristine channels as in the acoustic phonon scattering case.

In order to calculate the power factor $G S^2$, where $G$ is the channel conductance and $S$ the Seebeck coefficient, we use the fact that the Seebeck coefficient is the average energy of the current flow [@Kim11] (see Appendix B for derivation details) $$S = \frac{1}{q T L} \int_{0}^{L} \langle E(x) - E_F \rangle dx ,
\label{eq13}$$ where $q$ is the carrier charge ($q=-\vert e \vert$ for electrons and $q=\vert e \vert$ for holes) and $\langle E ( x ) \rangle$ is the energy of the current flow along the transport direction, defined as: $$\langle E ( x ) \rangle = \frac{\int I_{ch} (E, x) E dE}{\int I_{ch} ( E, x) dE}  ,
\label{eq14}$$ where $I_{ch}(E,x)$ is the energy and position resolved current. This is the quantity we integrate in order to get in Eq. (\[eq7\]) the total current as $J 