ELECTRONIC-SPECTRA OF STRONGLY MODULATED APERIODIC STRUCTURES

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 (V(n) = +/- V), the hopping integral t being a constant. We investigate the strong-modulation regime (V much greater than 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 pp-labeling theory; the quantitative predictions for the density of states and the binding energy are extensively compared with numerical data.