Short- and long-range-order features in the electronic structure of bulk and surface vacancies in diamond-structure semiconductors

Some questions concerning the nature of localized electron states at vacancies in covalent semiconductors are examined. In order to make the calculations easier, a honeycomb lattice, instead of a diamond lattice, is first considered. Two simple Hamiltonians are used: a one-state ($s$-like) Hamiltonian and an $s{p}^2$ Hamiltonian of the Weaire-Thorpe-Leman-Friedel type. Short- and long-range-order effects are studied by solving the Hamiltonians within different degrees of approximation: the Bethe lattice, cluster-Bethe lattice, and exact solutions are presented. Vacancies at bulk and (111) surface are discussed. The analysis of the results is facilitated by comparing them with the results obtained by exactly solving a full $s{p}^3$ Hamiltonian for a diamond-structure semiconductor (Si). It is shown that whereas the gap states (dangling-bond-like) are mainly determined by short-range order, the states deep in the valence band, of $A_1$ symmetry, are associated with the rings of atoms characteristic of the diamond structure.