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

A spectral classification of general one‐dimensional binary aperiodic crystals (BACs) based on both their diffraction patterns and energy spectrum measures is introduced along with a systematic comparison of the zeroth‐order energy spectrum main features for BACs belonging to different 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 off‐diagonal terms aperiodically distributed. An algebraic approach highlighting chemical correlation effects 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 defines 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 specific particular choices of the corresponding model parameters only. The transmission coefficient 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.