Effects of electronic energy-band structure on the energy distribution of field-emitted electrons

Current three-dimensional formulations of field emission of (noninteracting) electrons are critically examined and compared. It is shown that the kinetic-theory description for the energy distribution of field-emitted electrons (FEED) can be brought into a complete formal agreement with the theory of Penn and Plummer (PP). The reinterpretations of the parameters in the PP theory enable us to demonstrate its relationship to a nearly-free-electron theory. It is shown that the claim that FEED measures the "one-dimensional local density of states" as defined by PP is an approximation which may neglect some possibly significant band-structure effects in transition metals. In FEED these manifest themselves in the contribution of states with reduced two-dimensional wave vectors differing from zero. The relationship between Penn and Plummer's theory and a recent, more exact application of Appelbaum and Brinkman's transfer-Hamiltonian formalism, by Nicolaou and Modinos is discussed. A new expression for FEED is derived from Feuchtwang's many-body theory of tunneling. The limitations of the kinetic formulation are contrasted with a complete transfer-Hamiltonian formulation such as Nicolaou and Modinos's and a more exact tunneling theory by Feuchtwang. It is shown that the kinetic theory does not allow for either inelastic tunneling or surface-band-structure effects. It is also shown that the kinetic formulation of FEED involves a "coarse average" over an appropriate "surface" plane of the local density of states in contrast to the more complete theories in which the FEED depends on the full spectral density.