Electronic spectra of strongly modulated aperiodic structures

Abstract

We consider the tight-binding Hamiltonian on strongly modulated aperiodic chains (e.g., quasiperiodic, self-similar, random). The site energies are distributed according to a given binary sequence (${\mathit{V}}_{\mathit{n}}$=\ifmmode\pm\else\textpm\fi{}V), the hopping integral t being a constant. We investigate the strong-modulation regime (V\ensuremath{\gg}t) by means of degenerate perturbation theory. We thus derive systematic expansions in powers of t/V for the spectrum, the density of states, and other quantities of interest, like the total electronic binding energy. This investigation goes beyond the molecular approximation; it amounts to exploring recursively the pattern of resonances between molecular states along the structure, which can be either ``local'' or ``itinerant.'' The approach is worked out in detail on several classical examples of self-similar chains: Fibonacci, Thue-Morse, period doubling. The hierarchy of gaps thus obtained is put in perspective with the rigorous gap-labeling theory; the quantitative predictions for the density of states and the binding energy are extensively compared with numerical data.