Band-gap creation by icosahedral symmetry in nearly-free-electron materials.

Abstract

A series of numerical electronic density-of-states calculations is performed for rational approximants to a model one-electron potential based on icosahedrally arranged plane-wave components. It is found that high-order approximants can have band gaps even if the low-order approximants do not; furthermore, the magnitude of the gap increases with the order of the approximant. The results are interpreted via a two- and three-wave analysis of the energy eigenvalues at the pseudo-Jones-zone faces and edges. It is also found that the mechanism of band-gap reduction in the rational approximants is the presence of a small density of gap states. An analytic calculation shows that these gap states result from a splitting of threefold and pseudothreefold states at the valence-band edge when the icosahedral symmetry is broken. The splitting is proportional to the error with which the ratio between the approximant indices approximates \ensuremath{\tau}, the golden mean. Finally, an application to the AlCuLi system is presented.