Modified Fowler–Nordheim field emission formulae from a nonplanar emitter model

Field emission formulae, current–voltage characteristics and energy distribution of emitted electrons, are derived analytically for a nonplanar (hyperboloidal) metallic emitter model. The traditional Fowler–Nordheim (F–N) formulae, which are derived from a planar emitter model, are modified, and the assumption of the planar emitter in the F–N model is reconsidered. Our analytical calculation also reveals the backgrounds of the previous numerical discussion by He et al. on the effect of the geometry of emitter on field emission. The new formulae contain a parameter which characterizes the sharpness of the hyperboloidal emitter, and experimental data of field emissions from clean tungsten emitters and nanotip emitters are analyzed by making use of this feature.