Transverse dielectric response of a semi-infinite metal: Surface effect

The transverse dielectric response of a semi-infinite solid is studied in terms of the current-current correlation function. An expression is given for the conductivity tensor as a generalization of the corresponding bulk formula, when translational invariance normal to the surface is absent, but translational invariance parallel to the surface is retained. The $f$-sum rule is generalized and then used to establish the gauge invariance of the result for conductivity. The transverse conductivity is nonlocal and anisotropic even in the long-wavelength limit owing to the presence of the surface. Specific surface features that influence the result are isolated and studied in a simple, convenient manner. The importance of including finite electronic lifetime effects is emphasized.