Cycle-free approximations to amorphous semiconductors

The Hall-Weaire tight-binding semiconductor Hamiltonian is solved when the geometric structure has no closed cycles and is homogeneous, using a method developed by Onsager for ionic energies in ice. The solution yields two bands and two δ functions in the density of states in agreement with the general theorem of Weaire. This solution is proposed to be a reasonable first approximation for the band structures of amorphous semiconductors. The Hamiltonian is also solved when the underlying structure is the inhomogeneous Cayley tree for which surface states predominate. In this case the bands have the property of being nowhere continuous. Instead of just two δ functions outside the bands, there are sequences of bound-state δ functions which bridge the energy gap between bands when the model parameters fall in a finite interval.