Electron states near surfaces of solids

The mathematical theory of electron eigenstates near the surfaces of solids is developed in stages. The first stage is the study of the eigenstates of solids which terminate at a surface but are otherwise unperturbed; it is proved that near the surface the three-dimensional band edges are “softened” and that van Hove's singularities in the density of states are eliminated. A set of “vacuum states” lying primarily outside the solid is constructed out of plane waves orthogonalized to the eigenstates of the solid. This set of vacuum states is not orthonormal, and it must be orthonormalized by a procedure different from that of Schmidt (which is unsuitable). Effects of surface perturbations are studied. An exact method is elaborated for obtaining eigenfunctions in closed form, for a variety of perturbing potentials that extend arbitrary distances from the surface and include interband matrix elements. It consists of calculating the effects of one surface layer at a time and cumulating the results. The intrinsic instability of certain surfaces against the formation of bands of surface states is shown to be the consequence of the vanishing of a bulk quantity α_zz, one of the components of the inverse-effective mass tensor, at certain points in the B.Z.