Derivation of a Fowler-Nordheim type equation for highly curved field-emitters

The traditional Fowler-Nordheim (FN) equation has been repeatedly shown to fail for highly curved surfaces. In the past there have been modifications to it mostly for spherical surfaces. In this paper we derive a generalized FN-type equation which is valid for all potentials and surfaces. Both the current density at the emitter apex J(θ=0) and the effective emission area are derived rigorously. From our method the radius of curvature of the emitter can be extracted with sufficient accuracy. An application of our theory to the results of the ETH group verifies this.