Image charge at a metal surface

Abstract

In this work we provide some new insights into what constitutes the image charge at a metal surface, and therefore into the corresponding image potential. These conclusions are based on a study of the average exchange (hole) charge density as an electron is removed from within the metal to infinity outside. We show, and then explain, that the average exchange charge density is not localized to the surface region in the asymptotic limit when the electron is far outside, and that it has a width dependent upon the position of the electron from the surface. The farther out the electron, the wider the hole. As the electron is removed to infinity, the exchange density spreads out over the entire crystal. Consequently, the image charge and potential in this limit are independent of exchange and due entirely to correlation effects. However, the width of the exchange hole, though large, is still fairly localized for positions of the electron even up to two Fermi wavelengths from the surface. Thus the average exchange charge density contributes to the potential only at and near the metal surface. We have performed our calculations for the electronic wave functions generated by the linear-potential model of a surface. As a consequence, we have in addition been able to study the behavior of the exchange hole for nonuniform systems with electronic densities that vary both more rapidly and more slowly than metallic surface densities. In fact, we show how the commonly accepted concept of an exchange hole localized at the surface is easily arrived at by demonstrating that the results obtained in the unphysical infinite-barrier limit of the present model are contrary to those for the more realistic wave functions. Finally, on the basis of the present work and that of others, we draw additional conclusions with regard to the image correlation charge and surface exchange potential.