A quantum dipole–modified work function for a simplified electron emission barrier

Abstract

Changes in electron density near the surface of a metal gives rise to the largest part of its work function, but a small contribution arises from the penetration of electrons into the classically forbidden region and the dipole component which results. An analytic model of the dipole term is difficult for the image charge potential, but it is shown, first, that a slab plus triangular barrier (STB) potential has a straightforward solution and, second, that it fits well with a general “shape factor” method to find the transmission probability (and, from it, the current density). The shape factors for rectangular, triangular, quadratic, STB, and image charge potentials are compared: it is shown that, just as the STB potential is intermediate between the rectangular and triangular potentials, the image charge potential is intermediate between the triangular and quadratic potentials. The evaluation of parameters needed for a general thermal-field-photoemission equation is demonstrated. Finally, the STB model enables a shifted electron density representation that is useful to estimate the increasing dipole component to the work function and the corresponding decrease in current density for high fields. Modifications to treat semiconductors are briefly described.