Electronic Properties of an Amorphous Solid. II. Further Aspects of the Theory

A further study is made of the properties of the simple tight-binding Hamiltonian for which Weaire has recently shown that a band gap exists in a tetrahedrally bonded solid regardless of its structure. An exact transformation of the density of states is found which relates it to that generated by a much simpler Hamiltonian, providing, at once, an alternative proof of Weaire's result and a powerful tool for future study of this Hamiltonian. Various generalizations and extensions of the model are discussed. These include the definition of a Hamiltonian appropriate to a compound semiconductor and the generalization of the proof of the existence of a gap to cover this case. The resulting structure-independent formula for the gap, in terms of its homopolar and heteropolar parts, bears a close resemblance to that used in Phillips's semiempirical theory of tetrahedrally bonded semiconductors.