Interaction of a point charge with a metal surface: theory and calculations for (111), (100) and (110) aluminium surfaces

The asymptotic form of the image interaction is derived for a classical external point charge at a distance z1 outside a periodic metallic surface, generalizing to real metals the analytical result of Lang and Kohn for jellium. The centre of gravity zc of the induced charge coincides with the position of the image plane z0 in the limit of linear response. However, whereas zc is shown to depend through non-linear response on the magnitude of the external charge q, z0 is independent of q. We show that surface periodicity does not modulate z0, but adds to the image interaction a periodic component decaying exponentially with z. In addition, the long-ranged effect of non-linearity on the interaction energy is a term attractive to a positive external charge and proportional to q3/(z1 - z0)4. We report first-principles pseudopotential calculations of the interaction energy with three aluminium surfaces: (111), (100) and (110). A supercell geometry is used. A discrete classical model without adjustable parameters reproduces the effect of the surface periodicity on the image interaction at each of the three surfaces.