Numerical simulation of field emission and tunneling: A comparison of the Wigner function and transmission coefficient approaches

Abstract

Quantum transport through one-dimensional potential barriers is usually analyzed using either the transmission coefficient (TC) or the Wigner distribution function (WDF) approach. Fast, accurate, and efficient numerical algorithms are developed for each and are compared for (a) calculating current-field relationships for field-emission potentials with silicon parameters (and current-voltage relationships for resonant tunneling diodes), (b) their ability to accommodate scattering, self-consistency, and time dependence, and for (c) the behavior of their ‘‘particle trajectory’’ interpretations. In making the comparisons, the concern will be on the ability of each method to be incorporated into a larger ensemble-particle Monte Carlo simulation; it is argued that, in this regard, the WDF approach has significant advantages. Since the TC calculations rely on the Airy function approach, a detailed comparison of this method is made with the widely used Wentzel–Kramers–Brillouin and Fowler–Nordheim approaches for the general problem of field emission from a material into the vacuum.