Physical electrostatics of small field emitter arrays/clusters

TL;DR

This paper analyzes the electrostatic effects in small field emitter arrays using the FSEPP model, deriving formulas for charge interactions, and highlighting the model's accuracy and potential improvements for understanding apex field enhancement factors.

Contribution

It provides new analytical formulas for charge-blunting and neighbor-field effects in small emitter arrays using the FSEPP model, and discusses limitations of existing models.

Findings

Mutual charge-blunting dominates in two-emitter systems.

AFEF reduction varies as 1/c for small spacing and as 1/c^3 for large spacing.

FSEPP model accuracy is around 30%, and it guides better fitting formulas.

Abstract

This paper improves understanding of electrostatic influences on apex field enhancement factors (AFEFs) for small field emitter arrays. Using the "floating sphere at emitter-plate potential" (FSEPP) model, it re-examines the electrostatics and mathematics of three simple systems of identical post-like emitters. For the isolated emitter, various approaches are noted. On need consider only the effects of sphere charges and (for separated emitters) image charges. For the 2-emitter system, formulas are found for "charge-blunting" and "neighbour-field" effects, for widely spaced and "sufficiently closely spaced" emitters. Mutual charge-blunting is always dominant, with a related (negative) fractional AFEF-change {\delta}_two. For sufficiently small emitter spacing c, |{\delta}_two| varies as 1/c; for large spacing, |{\delta}_two| decreases as 1/c^3. In a 3-emitter linear array, differential charge-blunting and differential neighbor-field effects occur, but the former are dominant, and cause the "exposed" outer emitters to have higher AFEF ({\gamma}_0) than the central emitter ({\gamma}_1). Formulas are found for the exposure ratio {\Xi}={\gamma}_0/{\gamma}_1, for large and for sufficiently small separations. The FSEPP model for an isolated emitter has accuracy around 30%. Line-charge models (LCMs) are an alternative, but an apparent difficulty with recent LCM models is identified. Better descriptions of array electrostatics may involve developing good fitting equations for AFEFs derived from accurate numerical solution of Laplace's equation, perhaps with equation form(s) guided qualitatively by FSEPP-model results. In existing fitting formulas, the AFEF-reduction decreases exponentially as c increases, which differs from FSEPP-model formulas. FSEPP models might provide a useful guide to the qualitative behaviour of small field emitter clusters larger than those investigated.