An analytical solution for microtip field emission current and effective emission area

Abstract

Classical field emission theory relates the material work function and applied electric field at the emitter surface to tunneling current density, based upon the one-dimensional, planar Fowler–Nordheim equation. The extension of this theory to nonplanar, tip-anode geometries is complicated by the spatial variation of the electric field and resultant current density along the microtip surface. To relate this spatially dependent electric field to the applied voltage in this tip-to-plane geometry, experimenters have adopted empirically determined field enhancement factors (i.e., F=βV) and emission area terms to explain quantitative measurements of microtip current. In this work, the nature of the field enhancement and area terms are defined analytically by employing an exact three-dimensional electric field solution in prolate spheroidal coordinates and examining the relationship between total tip current and current density. The analytical and numerical results based on this model provide a physical context for understanding the validity of the generally used field enhancement factor β.