Surface States for a Molecular Orbital Model Hamiltonian: Resolvent Methods and Application to Zinc Blende

Abstract

A careful discussion is given of the application of the resolvent or Koster-Slater method to a surface-state problem described by a model Hamiltonian of the tight-binding type. Because of a special symmetry, peculiar to the surface-state problem, it is possible to reduce the degree of the characteristic determinant by a factor of 2. The usual, "truncation" method of doing this is shown to sometimes lead to spurious solutions. The origin and properties of these are determined and a prescription for recognizing them presented. An alternative, "pre-severed" approach to the reduction of the determinant is described. This latter is less useful for numerical calculation, but has advantages in obtaining analytical results. The general form of the surface-state wave function is derived from the integral representation of the resolvent. It proves possible to characterize this form using only simple properties of the bulk band structure as represented by the model Hamiltonian. This then permits the popular "ansatz" method of treating the surface states to be routinely used in problems having an arbitrary degree of complexity. The results of the general discussion are applied to determine the surface states associated with the (110) cleavage face of a semiconductor having the zinc-blende structure. The "pre-severed" resolvent method is used to derive some preliminary analytical results, and a program is set up for the numerical calculation of the surface-state properties for more realistic models of the surface perturbation.