# Polytope model and the electronic and structural properties of amorphous semiconductors

## Abstract

A model for describing a new kind of order in amorphous covalently-bonded networks is described. We introduce the concept of propagated short-range order (SRO), which expresses the idea that SRO about a particular atom can constrain the form of SRO around neighboring atoms. Propagated SRO can be achieved by applying a certain network-building rule to every site in the solid. For example, the rule Place eight cubes around each vertex generates a simple-cubic crystal. However, some ways of propagating SRO are incompatible with long-range crystalline order; specific examples of this frustrated propagation are shown in two and three dimensions. Many interesting kinds of propagated short-range order are compatible with translationally ordered structures called polytopes, which are large molecules or crystals in non-Euclidean space. We propose that certain types of polytope order describe intermediate-range order in amorphous semiconductors. Therefore, the quantum numbers labeling the symmetries of these polytopes provide approximate symmetry labels for electronic eigenstates of the amorphous solid. We explain the symmetry groups of the polytopes in detail; these are discrete subgroups of SO(4). We use their irreducible representations to construct the analogy of the energy-band dispersion for the amorphous solid. If patches of polytope order exist in the amorphous solid, then an approximate vertical selection rule should govern optical absorption in these materials, and we suggest it as part of a possible explanation of the depressed optical absorption near the band edge in hydrogenated amorphous silicon. © 1985 The American Physical Society.