Generalizing Fowler-Nordheim Tunneling Theory for an Arbitrary Power Law Barrier

TL;DR

This paper extends Fowler-Nordheim tunneling theory to arbitrary power law barriers, deriving an exact analytical formula using hypergeometric functions that accounts for screened interactions and compares favorably with numerical results.

Contribution

The authors derive a novel exact formula for tunneling through power law barriers, generalizing Fowler-Nordheim theory with hypergeometric functions and analyzing its dependence on the power law exponent.

Findings

Exact analytical formula for generalized Fowler-Nordheim tunneling

Benchmarking shows improved approximation over previous models

Analysis of transmission dependence on power law exponent

Abstract

Herein, the canonical Fowler-Nordheim theory is extended by computing the zero-temperature transmission probability for the more general case of a barrier described by a fractional power law. An exact analytical formula is derived, written in terms of Gauss hypergeometric functions, that fully capture the transmission probability for this generalized problem, including screened interaction with the image potential. First, the quality of approximation against the so far most advanced formulation of Fowler-Nordheim, where the transmission is given in terms of elliptic integrals, is benchmarked. In the following, as the barrier is given by a power law, in detail, the dependence of the transmission probability on the exponent of the power law is analyzed. The formalism is compared with results of numerical calculations and its possible experimental relevance is discussed. Finally, it is discussed how the presented solution can be linked in some specific cases with an exact quantum-mechanical solution of the quantum well problem.