Influence of barrier form on Fowler-Nordheim plot analysis

TL;DR

This paper investigates how the shape of the tunneling barrier affects Fowler-Nordheim plot analysis, using models for different potential energies and exploring the impact of emitter radius on correction factors.

Contribution

It introduces an initial exploration of barrier form effects on FN plot analysis, validating numerical methods and examining the influence of emitter geometry.

Findings

Numerical and analytical approaches yield equivalent results for Schottky-Nordheim barriers.

Small differences in potential energy models have little impact on correction factors.

Correction factors increase as the emitter radius decreases, especially below 20 nm.

Abstract

Recent research has described an improved method of Fowler-Nordheim plot analysis, based on the definition and evaluation of a slope correction factor and a new form of intercept correction factor. In this improved approach there exists a basic approximation that neglects certain terms in the general theory, and focuses on the influence of the form of the tunneling barrier on the values of basic slope ({\sigma}B) and intercept ({\rho}B) correction factors. Simple formulae exist that allow these to be evaluated numerically for a barrier of arbitrary well-behaved form. This paper makes an initial exploration of the effects of barrier form on FN plot analysis. For a planar emitter, two models for the correlation-and-exchange (C and E) potential energy (PE) are used. For the Schottky-Nordheim barrier, it is shown that numerical and analytical approaches generate equivalent results. This agreement supports the validity of the numerical methods used. Comparisons with results for the Cutler-Gibbons barrier show that small differences in the assumed C and E PE make little difference to values of {\sigma}B and {\rho}B. Schottky's planar image PE has then been used, in conjunction with the electrostatic PE variation associated with a spherical emitter model, to explore the influence of apex radius r_a on correction-factor values, for values of r_a greater than 20 nm. Both {\sigma}B and {\rho}B increase significantly as r_a decreases, especially {\rho}B. At low values of barrier field F, {\sigma}B depends approximately linearly on 1/F, with a slope that depends on r_a. Suggestions are made for how the exploratory work described in this paper might be extended.