DEFECT RESONANCES OF TRUNCATED CRYSTAL STRUCTURES

Jianfeng Lu, Jeremy L. Marzuola, Alexander B. Watson

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

Defects in the atomic structure of crystalline materials may spawn electronic bound states, known as 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 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.

Original languageEnglish (US)
Pages (from-to)49-74
Number of pages26
JournalSIAM Journal on Applied Mathematics
Volume82
Issue number1
DOIs
StatePublished - 2022

Bibliographical note

Publisher Copyright:
© 2022 Society for Industrial and Applied Mathematics

Keywords

  • defect states
  • metastable states
  • periodic structures
  • resonances

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