Band-gap Materials, Mesoscopic Structures, and Related Topics

NSF奖摘要- DMS-0072248数学科学：带隙材料、介观结构及相关主题摘要0072248 Kuchment该项目的目的是发展和推进研究波在带隙或介观介质中传播的分析和数值方法。目前研究的问题包括介观带隙材料和量子线电路的图形模型、先前发现的这类材料的特殊光谱性质、周期性介质中局域杂质的影响、线性缺陷的波导性质以及周期性材料的光谱性质。 光子晶体是一种低损耗的介电材料，由具有不同介电常数值的区域的周期性结构组成（例如，周期性放置在光学致密电介质中的气泡）。 在适当的条件下，这种材料可以表现出带隙，即电磁波不能通过材料传播的频率范围。 光子晶体的声学、弹性、金属和磁性类似物也很重要，尽管研究得少得多。 带隙材料在高效光源、天线、低阈值激光器、反射镜、光信息传输线、量子计算机、声衰减、隔热和许多其他领域有着令人印象深刻的应用。 介观系统是由半导体或超导体雕刻而成的极薄（窄至2nm）表面（量子壁）、线（量子线）或点（量子点）。这些物体享有丰富的应用，甚至比带隙材料的应用更引人注目。 该项目是对复杂的数学问题的调查，在许多情况下，这两种类型的媒体是相似的，在建模这些材料的创建和研究它们的属性。

NSF Award Abstract - DMS-0072248Mathematical Sciences: Band-Gap Materials, Mesoscopic Structures, and Related TopicsAbstract0072248 KuchmentThe aim of the project is to develop and advance analytic and numerical methods for studying wave propagation in a band-gap or mesoscopic medium. Issues under study include graph models for mesoscopic band-gap materials and circuits of quantum wires, unusual spectral properties of such materials discovered previously, effects of localized impurities in a periodic medium, waveguiding properties of linear defects, and spectral properties of periodic materials.BAND-GAP MATERIALS have attracted considerable attention since the idea was suggestedin 1987. A photonic crystal is a low-loss dielectric material, consisting of a periodic structure of regions with different values of the dielectric constant (for example, air bubbles placed periodically into an optically dense dielectric). Under appropriate conditions such a material can exhibit band gaps, that is, ranges of frequencies in which electromagnetic waves cannot propagate through the material. Acoustic, elastic, metallic, and magnetic analogs of photonic crystals are also important, although much less studied. Band-gap materials have impressive applications to high efficiency light sources, antennas, low threshold lasers, mirrors, optical information transmission lines, quantum computers, sound attenuation, heat insulation, and in many other areas. MESOSCOPIC SYSTEMS are extremely thin (as narrow as 2nm) surfaces (quantum walls), wires (quantum wires) or dots (quantum dots) carved out of semi-conductors or super-conductors. These objects enjoy a wealth of applications even more striking than the ones of band-gap materials. This project is an investigation of the intricate mathematical problems, in many cases similar for the two types of media, that arise in modeling the creation of these materials and in studying their properties.