论文标题

截短的晶体结构的缺陷共振

Defect resonances of truncated crystal structures

论文作者

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

论文摘要

晶体材料原子结构中的缺陷可能会产生电子结合状态,称为\ emph {缺陷状态},该状态迅速衰减从缺陷中衰减。缺陷状态的简化模型通常假设缺陷被无限的完美结晶材料包围。实际上,周围的结构必须是有限的,在某些情况下,结构可能足够小,即边缘效应很重要。在这项工作中,我们研究了这些边缘效应,并证明了以下结果。假设拥有正能缺陷状态的一维无限晶体材料与缺陷截止了距离$ m $。然后,对于足够大的$ m $,存在共鸣\ emph {指数关闭}(以$ m $为单位)到绑定的状态特征值。因此,截断的结构具有呈指数长的寿命的亚稳态状态。我们的方法允许将共振频率和关联的共振状态计算为$ e^{ - m} $中的所有订单。我们期望在光子晶体的背景下,这种结果特别感兴趣,在光子晶体的背景下,缺陷状态用于引导和结构相对较小。最后,在一个温和的额外假设下,我们证明,如果缺陷状态具有负能量,则截短的结构将具有指数关闭能量的结合状态。

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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