Exact solutions for the field electron emission achieved from a flat metal using the standard Fowler–Nordheim equation with a correction factor that accounts for the electric field, the work function, and the Fermi energy of the emitter

Abstract

The author uses a transfer-matrix technique to simulate field electron emission from a flat metal. The author compares in particular the results provided by this numerical scheme with those predicted by the standard Fowler–Nordheim equation. This comparison aims at establishing the influence of different approximations introduced in the standard Fowler–Nordheim theory (in particular the use of the Jeffreys–Wentzel–Kramers–Brillouin approximation for evaluating the transmission coefficient of the surface barrier and the series expansion of this coefficient when integrating over the normal-energy distribution of the incident electrons). In addition to the field and work function considered in previous work, the author explores the dependence of the emission current on the Fermi energy of the emitter. This physical parameter, which is related to the density of free carriers in the emitter, does not appear in the final form of the standard Fowler–Nordheim equation. It is therefore discarded from most analysis of field-emission data. The author shows, however, by a series of arguments that the emission currents are affected by the Fermi energy of the emitter. The author finally establishes a correction factor to be used with the Murphy–Good expression that accounts for the field, for the work function, and for the Fermi energy of the emitter and provides the exact solution for the emission achieved from a flat metal.