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.

Theory of Field Emission

“…For large values of βµ 0 , asymptotic expansions which allow iterative solution may be employed, such as [22,25,29] …”

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

“…Equation (2.19) is equivalent to eqn (34) in Jensen & Ganguly (1993) and eqn (242) in Jensen (2007). For use below, note that defining u 0 = c k F −1/3 c 1/2 yields the exact result…”

“…Over the last 20 years, Jensen has developed effective numerical methods for solving transmission problems, based on modified Airy functions 'Zi' and 'Di' (Jensen & Ganguly 1993;Jensen 2001Jensen , 2003Jensen , 2007. Although some small discrepancies have been detected (see electronic supplementary material, ESM3), these publications contain between them formulae equivalent to several of those derived below.…”

“…Gadzuk & Plummer (1973) outlined a simpler treatment based on the Airy functions Ai and Bi as introduced by Jeffreys (1928Jeffreys ( , 1942) (see Olver 2010 for their properties). A third generation of treatments was initiated by Jensen & Ganguly (1993), who published-without a fully detailed derivation-a revised formula for D ET in terms of Ai and Bi (their eqn (34)). We have found mathematical discrepancies in the pre-1993 treatments (see electronic supplementary material, ESM3).…”

A new method to detect geometrical information by the tunneling microscope

“…For a comparison of different approaches to the calculation of the tunneling through onedimensional potential barriers, see Ref. 16.…”