Predicting space-charge affected field emission current from curved tips

TL;DR

This paper introduces a simple, physically motivated model to predict the impact of space charge on field emission currents from curved tips, aligning well with detailed simulations and applicable to various emitter geometries.

Contribution

A new model based on Gauss's law is developed to estimate space charge effects on field emission from curved tips, validated against advanced simulation results.

Findings

Model predictions agree with molecular dynamics results for planar emitters.

Model predictions align with particle-in-cell simulation results for curved emitters.

Applicable to large area emitters with known enhancement factors.

Abstract

Field emission studies incorporating the effect of space charge reveal that for planar emitters, the steady-state field $E_P$, after initial transients, settles down to a value lower than the vacuum field $E_L$. The ratio $\vartheta = E_P/E_L$ is a measure of the severity of space charge effect with $\vartheta = 0$ being most severe and $\vartheta \simeq 1$ denoting the lack of significant effect. While, $E_L$ can be determined from a single numerical evaluation of the Laplace equation, $E_P$ is largely an unknown quantity whose value can be approximately found using physical models or can be determined `exactly' by particle-in-cell or molecular dynamics codes. We propose here a simple model that applies to planar as well as curved emitters based on an application of Gauss's law. The model is then refined using simple approximations for the magnitude of the anode field and the spread of the beam when it reaches the anode. The predictions are compared with existing molecular dynamics results for the planar case and particle-in-cell simulation results using PASUPAT for curved emitters. In both cases, the agreement is good. The method may also be applied to large area field emitters if the individual enhancement factors are known, for instance, using the hybrid model (D.Biswas, J. Vac. Sci. Technol. B 38, 063201 (2020)).