General form of the tunneling barrier for nanometrically sharp electron emitters

Field electron emission from nanometer-scale objects deviates the predictions of classical theory as both electrostatic potential curves within tunneling region and image planar one. This impels inclusion additional correction terms in barrier. At apex a tip-like rotationally symmetric surface, these are proportional to (single) local emitter curvature. The present paper generalizes this relation, showing that for any geometry, coefficient is given by mean curvature, i.e. average two principal curvatures.