Numerical testing of the Fowler–Nordheim equation for the electronic field emission from a flat metal and proposition for an improved equation

Abstract

The author uses a transfer-matrix technique to simulate field electronic emission from a flat metal. He compares, in particular, the results provided by this numerical scheme with those predicted by the standard Fowler–Nordheim equation. He considers for this study electric fields between 1 and 10 V/nm as well as work functions between 1.5 and 5 eV. The results demonstrate that the Fowler–Nordheim theory and the transfer-matrix calculations are globally in good agreement. With the Fermi energy of 10 eV considered in this work, the results provided by the standard Fowler–Nordheim equation are, however, systematically larger than the quantum-mechanical result, especially for low values of the work function and for high electric fields. This is essentially due to the fact the standard Fowler–Nordheim theory relies on the simple Jeffreys–Wentzel–Kramers–Brillouin approximation for evaluating the electronic transmission through the surface barrier of the emitter. A correction factor is thus established that enables the temperature-dependent version of the standard Fowler–Nordheim equation to match the exact quantum-mechanical result.