A further comparison of solid-state thermionic and thermoelectric refrigeration

We show that the expressions for current and heat current calculated via (the non-linearized) ballistic and diffusive transport formalisms reduce to the same form for solid-state devices one electron mean free path in length. The materials parameters for thermionic and thermoelectric devices are also shown to be equal, rather than differing by a multiplicative constant. We derive a simple transport equation that includes both ballistic and diffusive contributions to the current, and, as an example, use this to calculate the maximum temperature difference obtainable for a piece of Bi₂Te₃ as a function of its length, from less than an electron mean-free path to much greater than a mean-free path. Finally we briefly discuss similarities and differences between thermionic and thermoelectric devices in the regime where device length is of the order of a mean-free path length.