截断晶体结构的缺陷共振
Defect resonances of truncated crystal structures
摘要：
晶体材料的原子结构中的缺陷可能产生电子束缚态，称为\emph{defect states}，其从缺陷处迅速衰减。缺陷态的简化模型通常假设缺陷的所有侧面都被无限完美的晶体材料包围。在现实中，周围的结构必须是有限的，在某些情况下，结构可以足够小，边缘效应是显着的。在这项工作中，我们调查这些边缘效应，并证明了以下结果。假设承载正能量缺陷态的一维无限晶体材料从缺陷截断距离$M$。然后，对于足够大的$M$，存在对束缚态本征值的共振\emph{exponentially close}（在$M$中）。由此可见，截断结构具有指数级长寿命的亚稳态。我们的方法允许共振频率和相关的共振状态计算到所有订单在$e^{-M}$。我们希望这一结果在光子晶体的背景下特别感兴趣，其中缺陷态用于波导和结构相对较小。最后，在一个温和的附加假设下，我们证明了如果缺陷态具有负能量，则截断结构具有指数接近能量的束缚态。
英文摘要：
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.
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