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

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 ′≡ y 2 , 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 ′)d 2 v /d l ′ 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 d v /d l ′. The previously reported formula v ≈1− l ′+(1/6) l ′ ln l ′ 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.