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 functionvis expressed as a functionv(l′) of the mathematical variablel′≡y2, whereyis 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 factorν=v(f). This paper separates mathematical and physical descriptions of standard CFE theory, reformulates derivations to be in terms ofl′ andf, rather thany, and gives a fuller account of SN barrier mathematics.v(l′) is found to satisfy the ordinary differential equationl′(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 forvand dv/dl′. The previously reported formulav≈1−l′+(1/6)l′ ln l′ is a good low-order approximation, with |ϵ|<0.0025. Withl′=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 functionw. 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.