We consider the effect that the barrier shape has on the electron energy spectrum and lattice thermal conductivity, and together the effect of these coefficient of performance of thermionic refrigerators. Whilst it is shown that wide barriers are also desirable to enhance the electron energy spectrum, the primary motivation to increase barrier width to the maximum allowable value with ballistic transport is to reduce thermal conductivity. It is shown that the barriers which produce the highest electronic coefficient of performance do not necessarily give the highest coefficient of performance when thermal conductivity is considered if electronic heat current is reduced. While mean free path length multibarrier geometries may offer reduced thermal conductivity due to the possibility of interface scattering and phonon miniband formation, this effect needs to be significant to achieve coefficient of performance comparable with a single barrier device. Finally, we show that maximum refrigerator coefficient of performance is achieved by transmitting electrons over a tuned energy range only, which may be approximated by the transmission probability associated with a Gaussian modulated superlattice.
Conventional solid-state and vacuum thermionic devices restrict the flow of electrons between the hot and cold reservoirs according to the magnitude of their momentum in the direction of transport only. Recently it has been suggested that devices may be developed where the filtering of transmitted electrons occurs according to their total momentum. We compare the performance of these two different methods of electron momentum filtering in single barrier and resonant tunneling thermionic refrigeration devices. It is shown that total momentum filtered single barrier refrigerators always outperform conventional single barrier refrigerators due to their larger heat current which is particularly important when the thermal conductivity of the system is significant. We show that whilst conventionally filtered resonant tunneling thermionic refrigerators are outperformed by total momentum resonant tunneling thermionic refrigerators in many conditions, their performance is superior at (1) high temperatures or (2) when the transmission energy is very close to the Fermi energy.