Transmission coefficients for the exact triangular barrier: an exact general analytical theory that can replace Fowler & Nordheim's 1928 theory

In field electron emission theory, evaluating the transmission coefficient D-ET for an exact triangular (ET) potential energy barrier is a paradigm problem. This paper derives a compact, exact, general analytical expression for D-ET, by means of an Airy function approach that uses a reflected barrier and puts the origin of coordinates at the electron's outer classical turning point. This approach has simpler mathematics than previous treatments. The expression derived applies to both tunnelling and 'flyover' (wave-mechanical transmission over the barrier), and is easily evaluated by computer algebra. The outcome is a unified theory of transmission across the ET barrier. In different ranges of relevant physical parameters, the expression yields different approximate formulae. For some ranges, no simple physical dependences exist. Ranges of validity for the most relevant formulae (including the Fowler-Nordheim 1928 formula for D-ET) are explored, and a regime diagram constructed. Previous treatments are assessed and some discrepancies noted. Further approximations involved in deriving the Fowler-Nordheim 1928 equation for current density are stated. To assist testing of numerical procedures, benchmark values of D-ET are stated to six significant figures. This work may be helpful background for research into transmission across barriers for which no exact analytical theory yet exists.