Electronic states on a fractal: The consequences of self-energy variation

Abstract

A tight-binding model on the Koch curve with different self-energies on certain sites is investigated. The energy spectrum and the wave functions are analyzed by means of transfer matrices. An exact analysis of the band splitting process leads to a precise classification of the band-edges states, which fall into three infinite subsets formed by extended, localized, and fast growing self-affine states. Numerical investigation indicates that the model also admits an infinite number of exotic states. The present results are compared with those of a previously analyzed tight-binding model with next-nearest-neighbor interactions on the same fractal. A comparative discussion of these results distinguishes spectrum and state properties dependent on the details of the model from those which are common to both models and depend only on the geometry of the fractal.