Improved Fowler–Nordheim equation for field emission from semiconductors

Typical qualitative experimental characterizations of current versus voltage [I(V)] for field emitters fundamentally rely on the Fowler–Nordheim (FN) equation relating the current density J to the field at the surface F. The particular form of the J(F) relationship is due to taking the zero temperature limit of a linearized Wentzel–Kramers–Brillouin (LWKB) analysis, which works well for metals, but for semiconductors it is unsatisfactory, as LWKB does not correctly calculate the energy distribution of the emitted electrons, because the electron energy is too close to an extremum of the potential barrier, especially at high fields. Consequently, estimations of the current density from semiconductors can be substantially in error. We show how to modify the WKB approach so as to minimize its deviation from the exact solution of Schrodinger’s equation, resulting in a modified WKB equation, and from it, a modified FN relation. J(F) will be compared for different semiconductors, and a comparison with metals is given.