Most models of solid-state thermionic devices assume that all electrons with
energy in the direction of transport greater than the barrier height are
transmitted and utilize the Richardson equation. Here we consider a number
of thermionic systems where the electron energy spectrum differs from the
Richardson model. The electron energy spectra for maximum refrigeration
coefficient of performance and maximum power are presented. We then
consider multilayer solid-state nanostructures with currents not given by the
Richardson equation and discuss the optimization of their energy spectrum.
Nanometre gap vacuum thermionic refrigerators are also treated, where
significant current is provided by below the barrier tunnelling. Finally,
equations are developed for devices that select electrons for emission
according to their total momentum, rather than simply the value in the
direction of transport as is the case with conventional devices.