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Role of incoherent scattering on energy filtering in nanostructured thermoelectric generators

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, $τ(E)=kE^r$, 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.