The quantum theory of one-electron states at surfaces and interfaces

Abstract

The matching of the wavefunction at an arbitrary surface is formulated in terms of Green functions. If $G_{1}$ and $G_{2}$ are the resolvents for media 1 and 2, respectively, and $G_{\text{s}}$ is the resolvent for a system consisting of 1 and 2 in contact through an interface (not necessarily abrupt), then $G_{\text{s}}$ is fully obtained in terms of $G_{1}$ and $G_{2}$. By taking suitable matrix elements one obtains: (i) a secular determinant for the matching eigenvalues (surface states); (ii) a formula for the wavefunction everywhere, and (iii) a formula for the density of states. The case of a film is studied in the same way. The formalism is illustrated with a few examples. This paper deals with the physical information contained in the matrix elements of $G_{\text{s}}$ between points which are either on the surface or else on the same side, but the method yields also the matrix elements across the surface, which are important for the theory of dielectric responses in a system with an interface.