Reformulation of the standard theory of Fowler-Nordheim tunnelling and cold field electron emission

Abstract

This paper presents a major reformulation of the standard theory of Fowler-Nordheim (FN) tunnelling and cold field electron emission (CFE). Mathematical analysis and physical interpretation become easier if the principal field emission elliptic function v is expressed as a function v(l′) of the mathematical variable l′≡y2, where y is the Nordheim parameter. For the Schottky-Nordheim (SN) barrier used in standard CFE theory, l′ is equal to the 'scaled barrier field' f, which is the ratio of the electric field that defines a tunnelling barrier to the critical field needed to reduce barrier height to zero. The tunnelling exponent correction factor ν=v(f). This paper separates mathematical and physical descriptions of standard CFE theory, reformulates derivations to be in terms of l′ and f, rather than y, and gives a fuller account of SN barrier mathematics. v(l′) is found to satisfy the ordinary differential equation l′(1-l′) d2v/dl′2=(3/16)v; an exact series solution, defined by recurrence formulae, is reported. Numerical approximation formulae, with absolute error |ε|<8×10-10, are given for v and dv/dl′. The previously reported formula v≈1-l′+(1/6) l′lnl′ is a good low-order approximation, with |ε|<0.0025. With l′=f, this has been used to create good approximate formulae for the other special CFE elliptic functions, and to investigate a more universal, 'scaled', form of FN plot. This yields additional insights and a clearer answer to the question: 'what does linearity of an experimental FN plot mean?' FN plot curvature is predicted by a new function w. The new formulation is designed so that it can easily be generalized; thus, our treatment of the SN barrier is a paradigm for other barrier shapes. We urge widespread consideration of this approach. © 2007 The Royal Society.