Abstract
Electronic properties of the Fibonacci chain, the Penrose tiling, and the three-dimensional Penrose tiling are studied. The models are quasiperiodic and are expected to show behavior that is generic for quasicrystals. In chapter I surface states of the Fibonacci chain are studied. It is shown that a localization length can be defined, which depends only on the energy of a specific state. Chapters II and III are dedicated to properties of the two-dimensional Penrose tiling. In chapter II the density of states is calculated. The influence of unit-cell size and surface states on the density of states is studied. For one gap in the spectrum the so-called gap labeling is done explicitly. Localization is studied calculating participation ratios for eight values of the energy. It is shown that energies close to the band center lead to localization, whereas states close to the band edge seem to be delocalized. In chapter III the local density of states at a surface is studied. The remaining chapters are about properties of the three-dimensional Penrose tiling. In chapter IV it is shown that the density of states of the ideal tiling is smooth at a resolution of 10 meV. In chapter V the space group of cubic approximants is derived and the density of states of three cubic approximants is presented at a resolution of 0.2 meV. Chapters VI and VII give results for the localization of wave functions and conductivity calculated with the Landauer formula respectively
