Screening near a metal surface: Plasmon effects

We present a formulation of the problem of the screening of an electron by a metal surface in which electron-gas dispersion is fully taken into account. This is done by introducing a model Hamiltonian that treats on an equal footing the interaction of an electron with all the normal modes of a metal with a surface. The coupling functions describing this interaction are obtained in terms of general response properties of the bounded electron gas. As an example, we obtain explicit results for the coupling functions for a simplified model of the metal surface response, in which only collective modes (bulk and surface plasmons) contribute to the imaginary part of the density response function. We establish the plasmon-pole approximation for the surface problem, by showing that our model response function exactly satisfies the f-sum rule for an inhomogeneous electron gas (in the particular case of a sharp electron density profile at the surface). We present explicit results for the "image potential" acting on a charge as a function of its distance from the surface (the charge can be either outside or inside the metal). Our results stress the importance of including electron-gas dispersion (spatial dispersion) in the problem. In particular, model Hamiltonians for the electron-metal surface system that do not include electron-gas dispersion in the coupling functions, give a poor description of the screening at distances from the surface of the order of the electron screening length.