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: $I\sim E^2\exp (-a/E)$, 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 $\beta$. On the basis of this approximation, and considering local field and current fluctuations, two alternative methods for $\beta$ estimation are presented. These methods allow instantaneous field-specific estimation of $\beta$, 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 $\beta$ using fluctuation analysis in experimental measurements that were not dedicated for this purpose. An open-source code for the implementation of fluctuation-derived $\beta$ estimation is provided, with an analysis of possible future experimental opportunities using dedicated experiments.

• Received 21 April 2021
• Revised 27 June 2021
• Accepted 19 July 2021

DOI:https://doi.org/10.1103/PhysRevApplied.16.024007

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Published by the American Physical Society