Comparison of approximations for the principal Schottky-Nordheim barrier function v(f)

The standard Fowler-Nordheim-type (FN-type) equation, as developed by Murphy and Good and recently reformulated by Forbes and Deane, assumes that tunnelling takes place through a triangular barrier rounded by an image-potential-energy term. This barrier was introduced by Schottky and first used in tunnelling theory by Nordheim. It has therefore been called the Schottky-Nordheim (SN) barrier. For cold field electron emission (CFE) from a bulk metal, the standard FN-type equation gives the emission current density J in terms of the local work-function phi and the barrier field.The mathematical function nu was first defined and correctly calculated by Burgess, Kroemer and Houston in 1953. It has sometimes been called a "special field emission elliptic function", but a better name is the "principal Schottky-Nordheim barrier function", because it applies specifically to tunnelling through a SN barrier.The aim of this note is to demonstrate that, when assessed over the whole range 0 < f < 1, approximation outperforms all previous approximations of equivalent complexity. However, over limited ranges of equation performs slightly less well than approximations customised to fit the exact result over the limited range.