Band-gap Materials, Mesoscopic Structures, and Related Topics

NSF奖摘要-DMS-0072248数学科学：带隙材料、介观结构和相关主题该项目的目的是发展和推进研究带隙或介观介质中波传播的解析和数值方法。正在研究的问题包括介观带隙材料和量子线电路的图形模型，以前发现的这类材料的异常光谱性质，周期介质中局域杂质的影响，线性缺陷的波导性，以及周期材料的光谱性质。自1987年提出这一概念以来，带隙材料引起了人们极大的关注。光子晶体是一种低损耗的介质材料，由具有不同介电常数值的区域的周期性结构组成(例如，周期性地将气泡放置在光学致密介质中)。在适当的条件下，这种材料可以表现出带隙，也就是电磁波不能传播通过材料的频率范围。光子晶体的声学、弹性、金属和磁性类比也很重要，尽管研究要少得多。带隙材料在高效率光源、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、隔热等许多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2纳米)的表面(量子墙)、线(量子线)或小点(量子点)。这些物体拥有丰富的应用，甚至比带隙材料的应用更引人注目。这个项目是对复杂的数学问题的研究，在许多情况下，对于这两种类型的介质，在对这些材料的创建进行建模和研究它们的性质时出现了类似的数学问题。

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.