Universal scaling of electrostatic effects of a curved counter-electrode on the emitter field enhancement

Experiments on field electron emission from single-tip nanoemitters have typically been carried out using a counter-electrode with a finite curvature radius R, positioned at a distance dgap from the emitter's apex. The effects of the counter-electrode's curvature on the apex field enhancement factor (γCa) of the emitter are still not understood. In this Letter, we theoretically explore how the apex field enhancement factor of an emitter, represented by a hemisphere on a cylindrical post (HCP) with apex radius ra=50 nm, is influenced by the curvature of a sphere-shaped counter-electrode. Importantly, our results show that for HCPs with sharpness aspect ratios typically between 102 and 103, there is a universal scaling such that γCa=γPaΨ(R/dgap), where γPa represents the apex field enhancement factor for the emitter assuming a planar counter-electrode, and Ψ(R/dgap) is a universal scaling function such that Ψ∼1 for R/dgap≫1 and Ψ∼(R/dgap)α, with α close to unity, for R/dgap≪1. These findings help partially explain discrepancies observed in orthodox field electron emission experiments, where it was reported that the effective γCa values extracted from the current–voltage characteristics of single-tip carbon nanotubes typically underestimate the theoretical γPa values when R∼dgap≫ra, a trend that is predicted by our results.