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Electronic structure and transport in approximants of the Penrose tiling

We present numerical calculations of electronic structure and transport in Penrose approximants. The electronic structure of perfect approximants shows a spiky density of states and a tendency to localization that is more pronounced in the middle of the band. Near the band edges the behavior is more similar to that of free electrons. These calculations of band structure and in particular the band scaling suggest an anomalous quantum diffusion when compared to normal ballistic crystals. This is confirmed by a numerical calculation of quantum diffusion which shows a crossover from normal ballistic propagation at long times to anomalous, possibly insulator-like, behavior at short times. The time scale t*(E) for this crossover is computed for several approximants and is detailed. The consequences for electronic conductivity are discussed in the context of the relaxation time approximation. The metallic like or non metallic like behavior of the conductivity is dictated by the comparison between the scattering time due to defects and the time scale t*(E).