The circumstances in which I am asking this: I have two materials, copper and cesium, in which the surface of the two are contacting. The Fermi Energy value for copper is 7.0eV and for cesium it is 1.59eV. Copper has an electrical conductivity of copper is 16.78 nΩ·m (at 20 °C and cesium 205 nΩ·m (at 20 °C). The materials are layered in a sandwich with layer of cesium in between two layers of copper. This system is also part of an incomplete circuit, and heat is being conducted into this system. My question is how would the variables of the temperature affect the Fermi Level difference between the two materials, due to thermal excitation, and thus the thermionic emission of electrons from the cesium into the copper? I am replacing the work function variable with the Fermi Level difference in the Richardson-Dushmann equation of thermionic emission due to the fact there is no vacuum and the surface of the cesium is contacting the surface of the copper.