Defect resonances of truncated crystal structures

TL;DR

This paper analyzes how finite size effects influence defect states in crystalline materials, showing that truncation creates long-lived resonances or bound states, with implications for photonic crystal applications.

Contribution

It provides a rigorous analysis of edge effects on defect states in truncated crystals, including exponential approximation of resonances and bound states.

Findings

Resonances exponentially close to defect eigenvalues for large truncation distance M.

Metastable states with exponentially long lifetime in truncated structures.

Existence of bound states with exponentially-close energy if defect state has negative energy.

Abstract

Defects in the atomic structure of crystalline materials may spawn electronic bound states, known as \emph{defect states}, which decay rapidly away from the defect. Simplified models of defect states typically assume the defect is surrounded on all sides by an infinite perfectly crystalline material. In reality the surrounding structure must be finite, and in certain contexts the structure can be small enough that edge effects are significant. In this work we investigate these edge effects and prove the following result. Suppose that a one-dimensional infinite crystalline material hosting a positive energy defect state is truncated a distance $M$ from the defect. Then, for sufficiently large $M$, there exists a resonance \emph{exponentially close} (in $M$) to the bound state eigenvalue. It follows that the truncated structure hosts a metastable state with an exponentially long lifetime. Our methods allow both the resonance frequency and associated resonant state to be computed to all orders in $e^{-M}$. We expect this result to be of particular interest in the context of photonic crystals, where defect states are used for wave-guiding and structures are relatively small. Finally, under a mild additional assumption we prove that if the defect state has negative energy then the truncated structure hosts a bound state with exponentially-close energy.