Scaling theory of wave confinement in classical and quantum periodic systems

TL;DR

This paper introduces a rigorous, low-cost method to classify wave confinement dimensionality in periodic media across classical and quantum systems, enhancing understanding of defect-induced wave localization.

Contribution

The authors develop a finite-size scaling approach to determine confinement dimensionality from band structure data, applicable to any wave type and dimension, with validation on quantum and classical systems.

Findings

Confirmed electronic confinement in 2D hexagonal BN with nitrogen vacancy.

Identified novel acceptor-like behavior in 3D photonic band gap superlattice.

Method is computationally efficient and broadly applicable.

Abstract

Functional defects in periodic media confine waves - acoustic, electromagnetic, electronic, spin, etc. - in various dimensions, depending on the structure of the defect. While defects are usually modelled by a superlattice with a typical band-structure representation of energy levels, determining the confinement associated with a given band is highly non-trivial and no analytical method is known to date. Therefore, we propose a rigorous method to classify the dimensionality of the confinement. Starting from the confinement energy and the mode volume, we use finite-size scaling to find that ratios of these quantities to certain powers yield the confinement dimensionality of each band. This classification has negligible additional computational costs compared to a band structure calculation and is valid for any type of wave in both quantum and classical regimes, and any dimension. In the quantum case, we illustrate our method on electronic confinement in 2D hexagonal BN with a nitrogen vacancy, which confirms the previous results. In the classical case, we study a three-dimensional photonic band gap cavity superlattice, where we identify novel acceptor-like behavior.