A model for effective field enhancement for Fowler-Nordheim field emission

The local field enhancement factor β is often introduced in the Fowler-Nordheim equation to represent the geometrical effects at the surface of the cathode, where β(s)=En(s)/E0 for macroscopic applied field E0. Local variation of β determines the local normal surface electric field, En(s), resulting in local dependence of injection current by the Fowler-Nordheim law. In computational models, it is impractical to determine the time-dependent local surface field each time step on a microscopic space scale. Effective β is introduced in this paper which allows us to study the emission properties at a macroscopic scale. Microscopic (subgrid) local effective β is calculated only at the initial time step, and then the effective β can be recomputed for different surface electrical field through this model. The model allows reduction of dimensionality as well as the ability to include subgrid effects. The model is demonstrated on fundamental cases and compared to a calculation with a mesh fine enough to resolve the geometric features