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

Abstract

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 $d_{\mathrm{gap}}$ from the emitter's apex. The effects of the counter-electrode's curvature on the apex field enhancement factor (${\gamma }_{\mathrm{Ca}}$) 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 $r_{\mathrm{a}}=50$nm, is influenced by the curvature of a spherical-shaped counter-electrode. Importantly, our results show that for HCPs with sharpness aspect ratios typically between $10^2$ and $10^3$, there is a universal scaling such that ${\gamma }_{\mathrm{Ca}}={\gamma }_{\mathrm{Pa}}\Psi \left(R/d_{\mathrm{gap}}\right)$, where ${\gamma }_{\mathrm{Pa}}$ represents the apex field enhancement factor for the emitter assuming a planar counter-electrode, and $\Psi \left(R/d_{\mathrm{gap}}\right)$ is a universal scaling function such that $\Psi \sim 1$ for $R/d_{\mathrm{gap}}\gg 1$ and $\Psi \sim {\left(R/d_{\mathrm{gap}}\right)}^{\alpha }$, with $\alpha$ close to unity, for $R/d_{\mathrm{gap}}\ll 1$. These findings help partially explain discrepancies observed in orhtodox field electron emission experiments, who reported that the effective ${\gamma }_{\mathrm{Ca}}$ values extracted from the current-voltage characteristics of single-tip carbon nanotubes typically underestimate the theoretical ${\gamma }_{\mathrm{Pa}}$ values when $R\sim d_{\mathrm{gap}}\gg r_{\mathrm{a}}$, a trend that is predicted by our results.

This is a revised version of our original Letter submitted to Applied Physics Letters on December 9, 2024. This revised version was resubmitted to Applied Physics Letters on January 20, 2025

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