Analysis of nonuniform field emission from a Lorentzian or Hyperboloid shape emitter

The field emission (FE) process is a type of electron emission, which involves the quantum tunneling of electrons from a surface to vacuum subjected to a strong applied electrical field to lower the surface potential barrier. FE based cathode has been developed to be an alternative choice (compared to thermionic cathode) to produce electron beam required for many applications including coherent radiation sources [1] or as point sources of electrons for applications in high resolution electron microscopy [2]. For one-dimensional cathode with infinitely large emitting area, the FE process is commonly described by the well-known Fowler-Nordheim (FN) law [3]. However for a sharp tip emitter, due to the non-uniform emission feature and the electron beam expansion, it is difficult to precisely measure the average field enhancement factor β_c as well as the effective emission area S_eff for a single field emitter. In this paper [4], we conduct a numerical experiment to simulate the electron field emission emitted from a sharp tip emitter (Lorentzian or Hyperboloid shape). By collecting the emitting current at the anode, we establish the criteria in using Fowler-Nordheim (FN) plot to estimate both β_c and S_eff, which agrees well with our initial emission condition. It is found that the values of β_c and S_eff depend on the type and sharpness of the emitter as well as the size of the anode area to collect the emitting current. Our work may also be extended to a statistical more complicated model to simulate field emission from a cathode consisting of many field emitters.