  The physics of energy filtering in electronic transport through nanoscale
barriers is a fundamental aspect in the context of electronic engineering of
nanostructured thermoelectrics. In the context of thermoelectric generators, it
aims to engineer the Seebeck coefficient to favorably increase the power factor
and ultimately the power generated. In this work, we employ the incoherent
non-equilibrium Green's function formalism to investigate in detail the physics
of energy filtering and how it leads to a direct enhancement in power
generation across nanostructured thermoelectrics featuring a single planar
energy barrier. In particular, we reinforce that the enhancement in the
generated power via energy filtering at a particular operating efficiency is a
characteristic of incoherent scattering and is absent in ballistic devices. In
such cases, by assuming an energy dependent relaxation time, $\tau(E)=kE^r$, we
show that there exists a minimum value $r_{min}$ for which the thermoelectric
power generation is enhanced and thereby leading to a degradation in power
generation for $r<r_{min}$. For bulk generators, we delve into the details of
intermode scattering and show that such scattering processes between electrons
in higher energy modes and lower energy modes have a finite contribution to the
enhancement in the generated power. We also discuss realistic aspects such as
finite width of energy barriers and imperfect energy filtering due to partial
reflections. In particular, we show that such imperfect filtering and partial
transmission of electrons near the top of the barrier affects the enhancement
in the generated power drastically in the high efficiency regime of operation.
Analysis of the results obtained in this work should provide general design
guidelines for nanostructured enhancement in power generation via energy
filtering.
