Incoherent scattering can favorably influence energy filtering in nanostructured thermoelectrics

Investigating in detail the physics of energy filtering through a single planar energy barrier in nanostructured thermoelectric generators, we reinforce the non-trivial result that the anticipated enhancement in generated power at a given efficiency via energy filtering is a characteristic of systems dominated by incoherent scattering and is absent in ballistic devices. In such cases, assuming an energy dependent relaxation time $\tau(E)=kE^r$, we show that there exists a minimum value $r_{min}$ beyond which generation can be enhanced by embedding nanobarriers. For bulk generators with embedded nanobarriers, we delve into the details of inter sub-band scattering and show that it has finite contribution to the enhancement in generation. We subsequently discuss the realistic aspects, such as the effect of smooth transmission cut-off and show that for $r>r_{min}$, the optimized energy barrier is just sufficiently wide enough to scatter off low energy electrons, a very wide barrier being detrimental to the performance. Analysis of the obtained results should provide general design guidelines for enhancement in thermoelectric generation via energy filtering. Our non-equilibrium approach is typically valid in the absence of local quasi-equilibrium and hence sets the stage for future advancements in thermoelectric device analysis, for example, Peltier cooling near a barrier interface.