Analysis of nonuniform field emission from a sharp tip emitter of Lorentzian or hyperboloid shape

For a sharp tip emitter, due to the non-uniform emission feature and the electron beam expansion in the vacuum, it is difficult to precisely determine the average field enhancement factor $βc$ as well as the effective emission area S_eff for a single field emitter. In this paper, we conduct a numerical experiment to simulate the electron field emission from a sharp tip emitter (Lorentzian or hyperboloid shape). By collecting the emission current I_tot at the finite anode area S_tot, we establish the criteria in using Fowler-Nordheim plot to estimate both $βc$ and S_eff, which agree well with our initial emission condition. It is found that the values of $βc$ and S_eff depend on the emitter's properties as well as the size of the anode area S_tot. In order to determine the precise value of $βc$⁠, S_tot must be large enough to collect all the emitted electrons from the sharp tip (e.g., I_tot reaches maximum). As an example, a Lorentzian type emitter with an aspect ratio of 10 (height over width), the effective enhancement factor is about $βc=33$ as compared to the maximal enhancement of 35 at the apex. At similar maximal enhancement factor at the apex (=360), both types of emitters will give different average field enhancement dependent on the collecting area. The extension of this simple model to a statistical more complicated model to simulate field emission from a cathode consisting of many field emitters is also briefly discussed. This paper should be useful to analyze and characterize field emission data together with experimental measurement.