Hierarchical mobility edges in a class of one-dimensional generalized Fibonacci quasilattices

Abstract

The electronic energy spectrum and localization of the wave functions for a class of one-dimensional generalized Fibonacci quasilattices, whose substitution rules are A -+ ABB and B~A, have been studied. It has been found that the spectrum has a peculiar trifurcating structure and there are three kinds of wave functions in the spectrum: extended, localized, and intermediate states. The middle part of the central subband in every hierarchy of the spectra is always continuous and the corresponding wave functions are all extended while the rest of the wave functions are intermediate or localized, i.e. , there exist mobility edges in the subband. For the whole spectra the mobility edges possess a type of hierarchical structure.As a one-dimensional version of quasicrystals, the Fibonacci chain has been extensively studied since the experimental discovery by Schechtman et al. ' Its main feature is the Cantor set character of the energy spectrum.For recent years the theoretical interest has been shifting towards other one-dimensional quasiperiodic systems, such as the generalized Fibonacci quasilattices, Thue-Morse and generalized Thue-Morse models, ' '" and the three-tile SML model. ' The so-called generalized Fibonacci sequence is best described by successive application of the substitution rules A -+A 8" and 8~A, with m, n being positive integers, which is denoted as GF(m, n ) here. In this paper we concentrate on the study of GF(1,2) quasilattices, which is also referred to as the twins model or the copper mean lattice.Most studied physical properties of the onedimensional generalized Fibonacci quasilattices are concentrated on the spectral structure. On the other hand, the localization of the electronic states has not been much investigated.But, following the discovery of the quasisemiconductor, ' the localization problem is likely to attract more and more interest. It is well known that for one-and two-dimensional disordered systems the electronic states are all localized, but for three-dimensional ones there are mobility edges, which separate the conducting region (extended state region) from the nonconducting one (localized state region). In a one-dimensional system, no matter whether quasiperiodic or aperiodic, whether there exist mobility edges is a very attractive problem. For the one-dimensional quasilattices, the extended states have been found at individual energies. Recently, Sil et al. have analytically shown that there is an infinite number of extended states in the GF(1,2) chain. For the GF(1,2) model studied in this article, we found that the spectrum has a trifurcating structure. For the central subband of every hierarchy of the spectrum, the middle part is always continuous. Because the continuous spectrum corresponds to the extended state band, we can expect that not only are there mobility edges, but that furthermore the mobility edges would have a hierarchical structure. This conjecture has been confirmed by the numerical calculations. To the best of the authors' knowledge this is the first time...

Perfect self-similarity of energy spectra and gap-labeling properties in one-dimensional Fibonacci-class quasilattices

“…Unlike the Fibonacci chain, their regular selfsimilarity does not show up definitely in the energy spectra. [11][12][13][14][15][16][17] An appealing question is thus raised: does the regular or perfect self-similarity of the energy spectrum also exist in other structures as in the Fibonacci lattice? As an answer, Huang, Liu, and Mo 26 have by chance found a quasilattice, called the intergrowth sequence ͑IS͒ model by them, and proved that it has a perfect self-similar spectrum.…”

“…Among them, so-called generalized Fibonacci sequences, which are given by the substitutions A→A m B n , B→A, have been extensively studied. [10][11][12][13][14][15][16][17] Investigations have revealed their many similar properties as those of the Fibonacci lattice. However, whether these models can be called ''quasiperiodic'' or only ''aperiodic'' is a interesting question.…”

The band–gap structures and recovery rules of generalized -component Fibonacci piezoelectric superlattices

“…The Pisot numbers are λ Above discussion suggests that n-CF lattice covers a wide range of lattice types from periodic, quasiperiodic, critical, and non-quasiperiodic lattices, it offers a natural platform to study the spectra evolution with structural ordering. Though extensive investigations have been carried out on the electronic [4,5,6,7,8], vibrational [9,10,11], and dielectric [12,13,14,15] properties of quasiperiodic structures as a vehicle to study the evolution process from ordered periodic structures to disordered solids, the most previous studies concentrated on the standard Fibonacci lattices [4,5,6,7,8,9,10,11,12,13,14,15] and the generalized 2-component Fibonacci lattices [18,19,20,21,22,23,24], the generalized n-CF lattices are mostly discussed in the context of structural properties [25]. Furthermore, the previous studies mostly dealt with the single degree problems, and studies on the mode-coupling problems with energy transfer between different degrees of freedom are only a few [26,27,28].…”