Simple derivation of the formula for Sommerfeld supply density used in electron-emission physics and limitations on its use

Abstract

In a free-electron model, an electron crossing a mathematical plane inside a conductor can be characterized by the energy components associated with its motion normal and parallel to the plane. These components define a two-dimensional “energy-space.” The “supply density” is defined as the electron current crossing the plane, per unit area of the plane, per unit area in energy-space, when the relevant electron states are fully occupied. For a bulk free-electron conductor, the supply density is the same at all points in energy-space and has been called the “Sommerfeld supply density” (zS). This is given by zS=4πeme/hP3, where e is the elementary positive charge, me is the electron mass, and hP is Planck’s constant. This result is often a convenient starting point for developing basic theories of electron emission. A simple proof of it is recorded here. For small electron emitters, it can be a poor approximation to assume that the supply density is constant in energy-space. Consequently, if an emitter is sufficiently small, then the emission will not be well described by the usual basic emission equations. Criteria for assessing what counts as “sufficiently small” are discussed.