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 0) 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 0 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 v=v( f). This paper separates mathematical and physical descriptions of standard CFE theory, reformulates derivations to be in terms of l 0 and f, rather than y, and gives a fuller account of SN barrier mathematics. v( l 0) is found to satisfy the ordinary differential equation l'(1-l') d(2) v/dl('2) = (3/16) v; an exact series solution, defined by recurrence formulae, is reported. Numerical approximation formulae, with absolute error vertical bar epsilon vertical bar < 8 X 10(-10), are given for v and dv/dl'. The previously reported formula v approximate to 1-l' +(1/6) l' ln l' is a good low-order approximation, with vertical bar epsilon vertical bar < 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.