behavior below). The Seebeck coefficient naturally drops (red bar in second row, first column), but overall the $PF$ is *increased* in the SL structure by $23\%$ compared to the pristine structure. This indicates that the energy filtering, provided by potential barriers that cut lower parts of the Fermi distribution, is more effective at degenerate conditions as long as energy does not relax, as also pointed out in earlier studies [@Kim09; @Neo13ntech; @Thes15JAP]. However, the reason the $PF$ improves originates from the significant increase of the conductivity, rather than from the Seebeck coefficient which is actually degraded. The introduction of NIs in the region between the SL barriers (magenta bars), has a small degrading effect in the electrical conductance of the channel, as the NIs introduce additional scattering. However, this reduction is not strong, and the $PF$ is retained at values higher than those of the pristine material by $19\%$. Thus, we demonstrate here that it is indeed possible to achieve significant $PF$ improvements (rather than reductions) in a hierarchically nanostructured material. This $PF$ improvement combined with the expected very low thermal conductivity, can lead to high $ZT$.

It is important, thus, to clarify the reason behind the increase in $G$, 