Electronic Properties of an Amorphous Solid. I. A Simple Tight-Binding Theory

Using a simple Hamiltonian of the tight-binding type, rigorous bounds are derived for the density of states of a tetrahedrally bonded solid. These include inner bounds which define a band gap between occupied and unoccupied states. The derivation uses only the assumed perfect coordination of nearest neighbors, and so it holds for all tetrahedrally bonded crystal structures and random networks of the kind proposed for amorphous Si and Ge. Various other results are obtained for the fractional $s$- and $p$- like character of wave functions, the attainment of bounds, and other features of the density of states. A band-structure calculation for the diamond cubic structure serves as a test case.