Spectral Classification of One‐Dimensional Binary Aperiodic Crystals: An Algebraic Approach

Abstract

A spectral classi…cation of general one-dimensional binary aperiodic crystals (BACs) based on both their di¤raction patterns and energy spectrum measures is introduced along with a systematic comparison of the zeroth-order energy spectrum main features for BACs belonging to di¤erent spectral classes, including Fibonacci-class, precious means, metallic means, mixed means and period doubling based representatives. These systems are described by means of mixed-type Hamiltonians which include both diagonal and o¤-diagonal terms aperiodically distributed. An algebraic approach highlighting chemical correlation e¤ects present in the underlying lattice is introduced.Close analytical expressions are obtained by exploiting some algebraic properties of suitable blocking schemes preserving the atomic order of the original lattice. The existence of a resonance energy which de…nes the basic anatomy of the zeroth-order energy spectra structure for the standard Fibonacci, the precious means and the Fibonacci-class quasicrystals is disclosed. This eigenstate is also found in the energy spectra of BACs belonging to other spectral classes, but for speci…c particular choices of the corresponding model parameters only. The transmission coe¢ cient of these resonant states is always bounded below, although their related Landauer conductance values may range from highly conductive to highly resistive ones, depending on the relative strength of the chemical bonds.

Tight-Binding Modeling of Nucleic Acid Sequences: Interplay between Various Types of Order or Disorder and Charge Transport

“…Table 2, where several substitutional sequences studied in the literature are listed, together with their substitution rules and matrices). Limit-quasiperiodic structure representatives are the mixed means sequences with n ≥ m [95].…”

Real Space Theory for Electron and Phonon Transport in Aperiodic Lattices via Renormalization

“…The initial condition is TM 0 = A, and thus TM 3 = ABBABAAB has 2 3 atoms, being the eight most left atoms in Figure 2b. The TM sequence accomplishes the Pisot conjecture, but it has a null substitution matrix determinant, det(M) = 0, as periodic lattices [80]. In consequence, it is not a quasiperiodic system, but exhibits an essentially discrete diffraction pattern, and then TM heterostructures can be regarded as an aperiodic crystal according to the definition of crystals given by the International Union of Crystallography [81].…”

Localization Properties of Non-Periodic Electrical Transmission Lines

“…2 , the Copper mean σ 1,2 = 2, the Bronze mean σ 3,1 = 1 2 3 + √ 13 , the Nickel mean [73] for a spectral classification of one-dimensional binary aperiodic crystals, studying the eigenvalues λ ± and the determinant |M| of the substitution matrix M. The Fibonacci tight-binding quantum disordered systems have been studied exhaustively by [37,38,[40][41][42][43][44][45][46]59,60,63,71]. For the diagonal disordered case, the global number of sub-bands is exactly four.…”

“…A spectral classification of one-dimensional binary aperiodic crystals as a function of the substitution matrix M is shown in Ref. [73].…”

“…In addition, a combination of these measures is possible. Using spectral measures, Maciá [71,73] introduced a classification chart which includes periodic, amorphous and aperiodic systems. In this chart the abscissa is represented by the lattice Fourier transform and the ordinate is represented by the energy spectrum.…”