Physics-based derivation of a formula for the mutual depolarization of two post-like field emitters

TL;DR

This paper derives a simple, physics-based formula for the mutual depolarization effect on the field enhancement factor of two post-like emitters, valid across various separations, aiding experimental modeling.

Contribution

It introduces a novel two-parameter formula for fractional reduction of FEF, applicable to any shape of post-like emitters, extending previous models.

Findings

The formula agrees with numerical data within 1%.

Depolarization decay follows an inverse cube law at large separations.

The formula covers both small and large separation regimes.

Abstract

Recent analyses of the field enhancement factor (FEF) from multiple emitters have revealed that the depolarization effect is more persistent with respect to the separation between the emitters than originally assumed. It has been shown that, at sufficiently large separations, the fractional reduction of the FEF decays with the inverse cube power of separation, rather than exponentially. The behavior of the fractional reduction of the FEF encompassing both the range of technological interest $0<c/h\lesssim5$ ($c$ being the separation and $h$ is the height of the emitters) and $c\rightarrow\infty$, has not been predicted by the existing formulas in field emission literature, for post-like emitters of any shape. In this letter, we use first principles to derive a simple two-parameter formula for fractional reduction that can be of interest for experimentalists to modeling and interpret the FEF from small clusters of emitters or arrays in small and large separations. For the structures tested, the agreement between numerical and analytical data is $\sim1\%$.