the channel length, and is almost identical to the SL case (solid-red line). In the case of optical (inelastic) scattering conditions (dotted-black line), $\langle E ( x ) \rangle$ is relaxing in the region between the SLs, as expected. However, the presence of NIs reduces the number of available states that carriers can fall into after emitting phonons of energy $\hbar \omega$. In addition, since NIs disturb the low energy electrons, the average energy of the current $\langle E ( x ) \rangle$ slightly increases compared to the SL alone case [@Foster17]. Consequently, the degree of energy relaxation in the presence of NIs (dotted-black line) is smaller than that in the absence of NIs (dashed-dotted magenta line-repeated here from Fig. 2(a)), signaling a higher Seebeck coefficient. In Fig. 2(c), we show the corresponding case where the SL regions and the NIs form potential wells, and thus, less obstruction of transport is expected. Indeed, in this case $\langle E ( x ) \rangle$ is uniform throughout the channel, not only in the case of elastic scattering, but also in the case of inelastic scattering, as the wells are too narrow for the electrons to relax into.

![\[fig3\] (Colour online) Row-wise, summary 