Field emission from an elliptical boss: Exact versus approximate treatments

Abstract
Field emission from tips, in comparison to planar surfaces, is complicated by current enhancement effects and changes in the image charge. We show that the approximations typically used for the calculation of the transmission coefficient are inappropriate for small spherical surfaces but adequate for elliptical surfaces. A simple analytical formula for the average emitting area may be derived based on the Fowler–Nordheim equation under conditions of field emission. By use of the elliptical approximation, a simple scale factor is sufficient to deal with spherical and pointed tips.