cases where the nanofeatures impose potential barriers, or potential wells for charge carriers. We explore the influence of the heights of those barriers, as well as their number density. We then identify the design approaches that will allow for $PF$ immunity in hierarchically nanostructured materials, and in some cases, even significant improvements.

The paper is organized as follows. In Sec. II we describe our NEGF approach including our calibration procedure and indicate the geometries we study. In Sec. III we present and discuss our results, in Sec. IV we present a discussion on optimal nanostructuring, and finally, in Sec. V we conclude.

![\[fig1\] (Colour online) Geometries of the hierarchical nanostructured materials that we consider in this paper. (a) The pristine channel. (b) Channel with SL-type barriers. (c) Channel with SL barriers and quantum-dot potential barrier NIs. (d) Channel with SL-type wells. (e) Channel with SL-type wells and quantum-dot potential well NIs. ](Figure_1.png "fig:"){width="7cm" height="15.2cm"}\

Approach
========

We employ a 2D quantum transport simulator based on the NEGF formalism including electron-phonon (e-ph) scattering in the self-consistent Born approximation. We include both scattering of electrons with acoustic phonons (elastic scattering) and with optical phonons (inelastic scattering). The formalism and the details of 