Physical electrostatics of small field emitter arrays/clusters

Abstract

This paper aims to improve qualitative understanding of electrostatic influences on apex field enhancement factors (AFEFs) for small field emitter arrays/clusters. 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. An adequate approximation is to consider only the effects of sphere charges and (for significantly separated emitters) image charges. For the 2-emitter system, formulas are found for charge-transfer (“charge-blunting”) effects and neighbor-field effects, for widely spaced and for “sufficiently closely spaced” emitters. Mutual charge-blunting is always the dominant effect, with a related (negative) fractional AFEF-change δtwo. For sufficiently small emitter spacing c, |δtwo| varies approximately as 1/c; for large spacing, |δtwo| decreases as 1/c3. In a 3-emitter equispaced linear array, differential charge-blunting and differential neighbor-field effects occur, but differential charge-blunting effects are dominant, and cause the “exposed” outer emitters to have higher AFEF (γ0) than the central emitter (γ1). Formulas are found for the exposure ratio Ξ = γ0/γ1, 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 implementations 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 is different from the FSEPP-model formulas. This discrepancy needs to be investigated, using systematic Laplace-based simulations and appropriate results analysis. FSEPP models might provide a useful provisional guide to the qualitative behaviour of small field emitter clusters larger than those investigated here.