Electronic energy spectra of square and cubic Fibonacci quasicrystals

Abstract

Understanding the electronic properties of quasicrystals, in particular the dependence of these properties on dimension, is among the interesting open problems in the field of quasicrystals. We investigate an off-diagonal tight-binding hamiltonian on the separable square and cubic Fibonacci quasicrystals. We use the well-studied Cantor-like energy spectrum of the one-dimensional Fibonacci quasicrystal to obtain exact results regarding the transitions between different spectral behaviours of the square and cubic quasicrystals. We use analytical results for the addition of one-dimensional spectra to obtain bounds on the range in which the higher-dimensional spectra contain an interval as a component. We also perform a direct numerical study of the spectra, obtaining good results for the square Fibonacci quasicrystal, and rough estimates for the cubic Fibonacci quasicrystal.

Keywords:

• Fibonacci quasicrystals
• electronic spectra
• electronic transport
• quasicrystals

Acknowledgments

We wish to thank David Damanik for showing us the proper way to define the spectrum of the 1D Fibonacci quasicrystal. This research is supported by the Israel Science Foundation through Grant No. 684/06.

Notes

Notes

1. The reader is referred to 5,6 for precise definitions of the terms ‘crystal’ and ‘quasicrystal’.

2. Note that the absence of intervals in the spectrum above does not necessarily correspond to zero total bandwidth. It is in fact possible to use the Cantor set generation process to obtain a totally disconnected set with a finite measure. For example, if at the Nth iteration of the generation process the middle 1/3 ^N part is removed from each of the remaining intervals, one ends with a totally disconnected set whose measure is