Fast and accurate determination of the curvature-corrected field emission current

TL;DR

This paper introduces a fast analytical method to accurately calculate curvature-corrected field emission currents, reducing computational effort while maintaining errors below 6% for typical emitter geometries and fields.

Contribution

It provides an analytical expression for curvature-corrected emission current, validated against numerical benchmarks, enabling rapid calculations for large-area emitters.

Findings

Errors below 3% for R_a ≥ 5nm and E_a in 3-10 V/nm range

Analytical method speeds up computation significantly

Validated accuracy with numerical integration benchmarks

Abstract

The curvature-corrected field emission current density, obtained by linearizing at or below the Fermi energy, is investigated. Two special cases, corresponding to the peak of the normal energy distribution and the mean normal energy, are considered. It is found that the current density evaluated using the mean normal energy results in errors in the net emission current below 3% for apex radius of curvature, $R_a \geq 5$nm and for apex fields $E_a$ in the range $3-10$ V/nm for an emitter having work-function $\phi = 4.5$eV. An analytical expression for the net field emission current is also obtained for locally parabolic tips using the generalized cosine law. The errors are found to be below 6% for $R_a \geq 5$nm over an identical range of apex field strengths. The benchmark current is obtained by numerically integrating the current density over the emitter surface and the current density itself computed by integrating over the energy states using the exact Gamow factor and the Kemble form for the WKB transmission coefficient. The analytical expression results in a remarkable speed-up in the computation of the net emission current and is especially useful for large area field emitters having tens of thousands of emission sites.