Analytic and numerical analysis of average energies of field electrons

Exact quasi-equilibrium analysis for the average energy, <ε>, of the field electrons emitted from the semiconductor as well as the metallic surfaces is discussed. The non-equilibrium effects at high temperatures and strong fields are calculated to investigate the average energy of field electrons for both metals and semiconductors. WKB approximation is used in obtaining the analytic expression of <ε> for both metals and semiconductors. The exact scheme is used in numerical calculation for both field electron energy distribution, j(ε), and <ε> without the WKB approximation. The difference between the two values of <ε> obtained analytically and numerically is found to be small for semiconductors such as Si and III-V nitrides. The obtained analytic form of <ε> can be used in finding the values of the average energy of field electrons.