Three-Dimensional Scattering Analysis of Field Emission from a Tip. I. The Unperturbed Model

Abstract

A formulation of field emission from a spherical tip is developed in terms of scattering theory. It is shown that in three dimensions the usual "transmission coefficient" is replaced by a differential scattering probability. The statistical average over the Fermi distribution is obtained without recourse to the usual kinetic arguments, which are shown to lack general validity. The zero-order problem is taken to be emission from a spherical well through a Coulomb barrier. This problem is solved exactly. The energy distribution of the field-emitted electrons is calculated for this model and found to be always larger than that calculated for the one-dimensional triangular barrier. In a second publication, more realistic models are analyzed in terms of a distorted-wave Born approximation which utilizes the solutions of the present problem.