Quadratic superlattices: A type of nonperiodic lattice with extended states

Abstract

Deterministic nonperiodic lattices often exhibit properties intermediate between those of random and periodic crystals. Properties that have attracted interest include complex heirarchical structure factors, the breaking up of the band into a dense set of critical states and minigaps, and the interplay between extended and localized states. In one dimension, considerable attention has focused on nonperiodic lattices showing properties intermediate between periodic and random. We propose a one-dimensional nonperiodic lattice with lattice sites at ${0}^{2}d, {1}^{2}d, {2}^{2}d$, ... with length $d$ playing the r\^ole of the lattice constant. Here we show that the structure factor ${S}_{0}(k)$, which governs how waves are scattered and is reflected in many physical properties, is simply related to the Jacobi theta function ${\ensuremath{\vartheta}}_{3}(q)$. ${S}_{0}(k)$ is periodic in wave vector $k$ and is singular continuous, consisting of a dense set of peaks with $k$ given by rational-fraction multiples of $\frac{\ensuremath{\pi}}{d}$, though the scaling properties of ${S}_{0}(k)$ are the same as for periodic crystals. The electronic structure considered in a tight-binding model shows a bandlike spectrum, but broken by a dense set of minigaps near peak locations in ${S}_{0}(k)$. These lattices show novel properties such as the coexistence of a singular-continuous spectrum with extended states. Quadratic lattices may enable the physical realization of special functions of mathematical and physical importance as well as for the design of nonperiodic diffraction gratings, antenna arrays, ad coupled optical resonators to produce complex quasirandom behavior and are of interest to study quantum interactions in nonperiodic optical lattices.