Electronic and vibrational spectra of two-dimensional quasicrystals

Abstract

The tight-binding electronic structure of two-dimensional quasicrystals is studied numerically for three patterns of Penrose tiling with up to 426 vertices. According to the range of interactions, three different models are considered. For the simplest model, two different interactions are assigned to long and short edges of the Penrose tile. Energy spectra show several significant gaps whose width and position depend on the relative strength of the interactions. The cumulative density of states is linear in energy at the band edge, indicating the existence of the Van Hove singularities. The energy spectra for other models show similar band gaps and singularities, though the density of states is asymmetric. Participation ratios are examined. When the relative strength of interactions becomes small, significant numbers of states become localized. Lattice vibration perpendicular to the plane is studied in the harmonic approximation for the simplest model. The vibrational spectra show gaps and singularities similar to the electronic spectra.