Localization of electrons and influence of surface states on the density of states in a tight-binding model on the Penrose tiling

The distribution of participation numbers of electronic states in the vertex model of the Penrose tiling (PT) is calculated for a series of approximants and eight fixed values of the energy by means of contour integrations in the first Brillouin zone. Localization of electrons is seen to depend on energy. The local density of states at the surface of a small approximant of the PT is compared to the density of states (DOS) of the same approximant with periodic boundary conditions. The large difference between the two curves indicates that surface states might be responsible for the fact that experimentally surface sensitive techniques do not resolve spikes in the DOS of quasicrystals. The DOS of the (8,5) approximant of the three-dimensional PT is shown to be less spiky than expected.