The formal derivation of an exact series expansion for the principal Schottky–Nordheim barrier function v, using the Gauss hypergeometric differential equation

Abstract

The standard theory of Fowler–Nordheim tunnelling and cold field electron emission (CFE) employs a mathematical function v, sometimes called the principal field emission elliptic function, but better called the principal Schottky–Nordheim barrier function. This function arises when the simple-JWKB (Jeffreys–Wentzel–Kramers–Brillouin) method is applied to solve the Schrödinger equation approximately, for the image-rounded tunnelling barrier introduced by Schottky and then used by Nordheim in late 1928. An exact series expansion was recently found for v, as a function of a complementary elliptic variable l′ equal to y2, where y is the Nordheim parameter. The expansion was originally found by using an algebraic manipulation package. It was subsequently discovered that v(l′) is a particular solution of the ordinary differential equation (ODE) l′(1 − l′) d2v/dl′2 = (3/16)v. This ODE is now recognized to be a special case of the Gauss hypergeometric differential equation. This paper uses known special-case solutions of the hypergeometric equation to formally derive the series expansion for v(l′). It notes how to derive the defining ODE, and then uses an 1876 result from Cayley to derive the boundary condition that dv/dl′ satisfies as l′ tends to zero. It then establishes the series expansion for v(l′), by applying this and the boundary condition v(0) = 1. This mathematical derivation underpins earlier results, including good approximate expressions for v(l′). Its outcome proves that terms involving ln l′ are part of a mathematically correct solution, but fractional powers of l′ are not. It also implies that simple Taylor-expansion methods cannot easily generate good approximation formulae valid over the whole range 0 ⩽ l′ ⩽ 1; this may also apply to barriers of other shapes. This derivation should bring closure to the particular line of mathematical analysis of CFE theory initiated by Nordheim in 1928. It is hoped that the derivation might also serve as a model for analysing other tunnelling-barrier problems.