Transmission Resonance in an Infinite Strip of Phason-Defects of a Penrose Approximant Network

TL;DR

This paper presents an analytical method to study transmission resonance phenomena in quasicrystalline approximant networks, revealing conditions for perfect transmission due to topological and interference effects.

Contribution

It introduces an exact analytical approach to compute transfer matrices, band structure, and quantum resistance in quasicrystalline approximants of any dimension, highlighting resonance effects.

Findings

Resonance leads to vanishing reflection coefficients at specific energies.

Analytical expressions for band structure and quantum resistance are derived.

Resonance phenomena relate to the gap structure of the undistorted system.

Abstract

An exact method that analytically provides transfer matrices in finite networks of quasicrystalline approximants of any dimensionality is discussed. We use these matrices in two ways: a) to exactly determine the band structure of an infinite approximant network in analytical form; b) to determine, also analytically, the quantum resistance of a finite strip of a network under appropriate boundary conditions. As a result of a subtle interplay between topology and phase interferences, we find that a strip of phason-defects along a special symmetry direction of a low 2-d Penrose approximant, leads to the rigorous vanishing of the reflection coefficient for certain energies. A similar behavior appears in a low 3-d approximant. This type of ``resonance" is discussed in connection with the gap structure of the corresponding ordered (undefected) system.