Waves, Novel Two-Dimensional Materials, and Applications

波、新型二维材料及其应用

• 批准号: 1908657
• 负责人: Michael Weinstein
• 金额: $ 67.5万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-08-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1908657&HistoricalAwards=false
• 关键词: Waves; Novel; Two; Dimensional; Materials

At the heart of many realized and envisioned advances in science and technology --from communication networks to quantum computation -- is the ability to robustly transport, channel, and manipulate forms of energy, e.g. electronic and optical, on very small spatial and temporal scales. This is achieved by fabricating novel media in which energy, which propagates as waves, interacts with engineered microfeatures. A current example of a novel, naturally occurring, medium of intense interest is graphene a one-atom thick mono-layer of carbon atoms, arranged in-plane in a honeycomb structure, extending to the macro-scale. Graphene is the most conductive substance known at room temperature (electronically and thermally). This material has been at a center of a revolution in Condensed Matter Physics and Materials Science, known as the Two-Dimensional Material. Many of the novel phenomena observed in graphene relate to general mathematical properties of waves in media with novel symmetry, and hence physicists and engineers have engineered and investigated materials with similar general properties, dubbed "artificial graphene," with a view on applications to optics, communications, information storage and computing. PI Weinstein will study a range of problems in fundamental and applied mathematics related to the existence, stability and transport properties of waves in such novel media, with honeycomb structures as a paradigm for 2D materials, as well as questions concerning the dynamics of waves in nonlinear systems. This work is firmly aligned with Quantum Leap, one of the NSF's 10 Big Ideas. This award will support one graduate student to be trained in a multifaceted approach to applied mathematical research using modeling, analysis and computation, with emphasis on applications of wave phenomena in complex media to the scientifically important questions. PI Weinstein will investigate (A) Energy propagation along line-defects in novel 2D periodic media governed by continuum PDE models, topologically protected states and the relation between bulk properties of periodic media and edge effects, and (B) Dynamics of coherent structures in discrete and continuous nonlinear wave systems, in particular (i) transport in nonlinear lattices and (ii) dynamics of a free boundary problem in compressible fluids. (A) Many remarkable properties of 2D materials (e.g. extraordinary conductivity) are related to the subtle properties of its band structure (the collection of dispersion surfaces and eigenmodes of Floquet-Bloch theory) which dictate energy transport. The conical singularities (Dirac points) of the dispersion surfaces of graphene are at the core of its remarkable electron mobility, and the topological properties of its band structure properties give information about the types of energy-transport can occur when graphene is interfaced with free-space or another material. Thus, it is through a convergence notion from Analysis, PDE and Topology that one can make process understanding these novel materials and their artificial analogues. (B) This research is part of the PI's long-standing research program in nonlinear waves, which now emphasizes i) coherent structure propagation in discrete nonlinear dispersive systems and ii) the nonlinear PDE dynamics of coherent structures with infinitely many internal degrees of freedom, as exemplified by the nonlinear oscillations of a bubble in the fluid.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

在许多实现和设想的科学和技术进步的核心-从通信网络到量子计算-是在非常小的空间和时间尺度上稳健地传输，通道和操纵能量形式的能力，例如电子和光学。这是通过制造新型介质来实现的，在这种介质中，作为波传播的能量与工程微特征相互作用。一种新的、天然存在的、引起强烈兴趣的介质的当前示例是石墨烯，一个原子厚的碳原子单层，以蜂窝结构在平面内排列，延伸到宏观尺度。石墨烯是已知在室温下最导电的物质（电子和热）。这种材料一直处于凝聚态物理学和材料科学革命的中心，被称为二维材料。在石墨烯中观察到的许多新现象与具有新对称性的介质中的波的一般数学性质有关，因此物理学家和工程师设计和研究了具有类似一般性质的材料，称为“人造石墨烯”，以期应用于光学，通信，信息存储和计算。PI Weinstein将研究一系列与这种新型介质中波的存在，稳定性和传输特性相关的基础和应用数学问题，以蜂窝结构作为2D材料的范例，以及有关非线性系统中波的动力学问题。这项工作与NSF的10大想法之一的量子飞跃密切相关。该奖项将支持一名研究生接受多方面的应用数学研究方法的培训，包括建模，分析和计算，重点是复杂介质中波动现象在科学上重要问题的应用。PI Weinstein将研究（A）由连续PDE模型控制的新型2D周期性介质中沿着线缺陷的能量传播，拓扑保护状态以及周期性介质的体属性与边缘效应之间的关系，以及（B）离散和连续非线性波系统中相干结构的动力学，特别是（i）非线性晶格中输运和（ii）可压缩流体中自由边界问题的动力学。 (A)二维材料的许多显著性质（例如非凡的导电性）与其能带结构的微妙性质（Floquet-Bloch理论的色散表面和本征模的集合）有关，这些性质决定了能量传输。石墨烯色散表面的圆锥奇点（狄拉克点）是其显着的电子迁移率的核心，其能带结构特性的拓扑特性提供了有关石墨烯与自由空间或其他材料连接时可能发生的能量传输类型的信息。因此，通过分析，PDE和拓扑学的融合概念，人们可以使过程理解这些新材料及其人工类似物。 (B)这项研究是PI在非线性波方面的长期研究计划的一部分，该计划现在强调i）离散非线性色散系统中的相干结构传播和ii）具有无限多内部自由度的相干结构的非线性PDE动力学，该奖项反映了NSF的法定使命，并被认为是值得支持的，使用基金会的知识价值和更广泛的影响审查标准进行评估。