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

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 normal energy distribution and 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 Ra≥5 nm and for apex fields Ea in the range of 3–10 V/nm for an emitter having a work function of ϕ=4.5 eV. An analytical expression for the net field emission current is also obtained for local parabolic tips using the generalized cosine law. The errors are found to be below 6% for Ra≥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.