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

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