Statistical Analysis of Field-Emission Currents

The current due to cold tunneling of electrons from a metallic surface exposed to high electric fields, regularly named “dark current,” is commonly described in modern literature using an analytical approximate solution provided by Murphy and Good [Phys. Rev. 102, 1464 (1956)]. This expression, which corrects earlier work by Fowler and Nordheim, is a Fowler-Nordheim-type equation: $I$∼$E$$^{2}$ exp⁡(−$a$/$E$), where I is the dark current, $E$ is the local electric field, and a is a system-specific constant. In this paper, a numerical approximation, rather than the analytical one given by Murphy and Good, is presented. This approximation is accurate over a wide range of fields, and is used to derive the effective field enhancement factor β. On the basis of this approximation, and considering local field and current fluctuations, two alternative methods for β estimation are presented. These methods allow instantaneous field-specific estimation of β, rather than the average estimate derived with current methods. The applicability of fluctuation-based methods is demonstrated by numerical simulation in a variety of conditions. The methods are applied to estimate β using fluctuation analysis in experimental measurements that were not dedicated for this purpose. An open-source code for the implementation of fluctuation-derived β estimation is provided, with an analysis of possible future experimental opportunities using dedicated experiments.