Determining the field enhancement factors of various field electron emitters with high numerical accuracy

Theoretical analysis of field electron emission must consider many parameters, one of the most critical being the field enhancement factor (FEF). In a single tip form, the FEF can vary several orders of magnitude and depends only on the system geometry, when the gap length between the emitter and counter-electrode is much greater than the height of the emitter. In this work, we determine very accurate analytical expressions for the FEF of five emitters with various shapes, which are often considered in the literature: Ellipsoidal, Hemisphere-on-Cylindrical post, Hemisphere-on-Orthogonal cone, Paraboloidal, and Hyperboloidal. We map the FEF as a function of the aspect ratio with an error smaller than 2% to serve as a quick reference database. Additionally, we calculate the electric field distribution over the emitters, which can give an insight into the effective notional emission area and the influence of the emitter’s base.