Energy exchange processes in electron emission at high fields and temperatures

A new more complete theory for energy exchange processes in electron emission is formulated. It is found that the tunneling contribution to the availability of vacant states is necessary to explain the replacement process occurring in the emitter region. The introduction of the tunneling states now makes it possible to obtain both the average energies of the emitted and replacement electrons using the same formalism. At T=0 K, the average energy of replacement electrons, 〈εr〉, is the same as the average energy of the emitted electrons, 〈εe〉. As T increases, 〈εr〉 increases rapidly until it reaches a maximum and then decreases slowly, while 〈εe〉 increases monotonically. When T equals the inversion temperature Ti, 〈εe〉=〈εr〉 and the energy exchange Δε=0. We have also calculated both Δε and Ti as a function of field F. For high temperature and fields, the value of Ti differs considerably from that obtained without the tunneling state contribution and Ti exhibits nonlinear behavior as a function of field. Tunneling state contributions are essential for explaining the steady state condition in the conduction process, especially at very low temperatures. These results are crucial for resolving the controversial problem of the replacement process in electron emission. Contrary to the assertion of Nottingham [Phys. Rev. 59, 907 (1941)] that the replacement energy is the Fermi energy, the current results indicate that the average value can be 10–102 meV less than the Fermi energy in agreement with Fleming and Henderson [Phys. Rev. 58, 887 (1940)]. Nonequilibrium effects evaluated within the relaxation time approximation are significant only for large fields.