Band-gap Materials, Mesoscopic Structures, and Related Topics

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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.