Simple good approximations for the special elliptic functions in standard Fowler-Nordheim tunneling theory for a Schottky-Nordheim barrier

Abstract

The discovery is reported of simple, good approximate formulas for special elliptic functions that appear in the standard theory of Fowler-Nordheim (FN) [Proc. R. Soc. London, Ser. A 119, 173 (1914)] tunneling through an image-rounded Schottky-Nordheim [W. Schottky, Z. Phys. 15, 872 (1923); L. W. Nordheim, Proc. R. Soc. London, Ser. A 121, 626 (1928)] barrier and in the standard FN equation. The FN-exponent correction factor v can be written as v(y)≈1−y2+(1∕3)y2lny, where y is the Nordheim parameter. This formula has a respectable mathematical basis, predicts exact values of v(y) to within 0.33% in 0⩽y⩽1, and can be rewritten to give (after nearly 80years) a simple, reliable algebraic formula for the explicit dependence of v on barrier field. Significant consequences are expected.