Analytic Solutions for Space-Charge-Limited Current Density From a Sharp Tip

While analytic equations for space-charge limited current density (SCLCD) have been derived for planar and nonplanar geometries, the SCLCD for emission from a sharp tip has not been derived. In this article, we use variational calculus (VC) to derive an exact analytic equation for SCLCD for a 1-D tip-to-tip geometry, represented by hyperboloids in the prolate spheroidal coordinates, and recover the SCLCD for a tip-to-plate geometry as a special case. We then consider circles in the extended Poincaré disk, which is the stereographic projection of hyperboloids onto a plane, as conformal transformations to derive the SCLCD for a misaligned tip-to-tip geometry, where the axes of rotation of the hyperboloids are displaced by a small distance. This mapping technique is also applied to study the effect of a small angle tilt in a tilted tip-to-tip geometry, where the axes of rotation of the hyperboloids meet at an angle.