Modeling, Analysis, and Computation of 2D Layered Materials

用于交错2D材料层的实验方法已经被开发出来，为创建具有所需的电、光、磁和热性能的稳定结构提供了无限的可能性。该项目将开发数学模型和计算方法，以指导寻找和设计具有最佳性能的2D材料。这种方法使人们长期追求的目标成为可能，即对纳米结构进行原子级控制，作为制造具有所需性能和性能特征的设备的构建块。该项目开发的从原子到宏观尺度的数学建模、分析和计算将有助于新材料的设计，从而导致在无公度2D材料中的应用。这一努力将影响具有所需特性和性能的材料和器件的开发，用于广泛的相关应用，包括超高速电子、光电子和磁性器件；非传统光学和光子器件；以及通信设备。对层状无公度异质结构建模的挑战也将促进许多其他非周期材料系统的多尺度模型的发展，如复合材料、原子工程结构和生物材料。2D材料研究是推动新的数学培训和课程的理想平台，这些培训和课程涉及量子电子结构和输运以及材料的力学和拓扑性质的分析、建模和计算。该项目的研究生培训和对代表性不足的学生群体的推广将扩大数学研究社区的多样性。在分层2D材料的数学建模中遇到的主要问题是不同层的晶格周期不匹配，从而导致结构不相称。不使用布洛赫-傅立叶方法的新理论和计算方法将被开发出来，以在目前无法达到的区域内准确地预测材料的性质。这个项目将利用局域组态空间和局域性的概念，给出电子态密度和输运性质(如电导率)的新公式和计算方法。利用动量空间哈密顿量的结构，将发展新的动量空间公式和相应的快速计算方法。还将开发和分析新的模型和计算方法，将我们的电子态密度和电导率模型扩展到包括机械松弛。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Experimental methods for interleaving layers of 2D materials have been developed with endless possibilities for creating stable structures with desired electronic, optical, magnetic and thermal properties. This project will develop mathematical models and computational methods to guide the search and design of 2D materials with optimal properties. This approach makes possible the long-sought goal of atomic-level control of nanostructures as building blocks for creating devices of desirable properties and performance characteristics. The mathematical modeling, analysis, and computation from the atomic to macroscopic scale developed by the project will contribute to the design of new materials leading to applications in incommensurate 2D materials. This effort will impact the development of materials and devices with desired characteristics and performance for a wide range of applications of interest including ultra-fast electronic, opto-electronic, and magnetic devices; nonconventional optical and photonics devices; and communication devices. The challenge of modeling layered incommensurate heterostructures will also promote the development of multiscale models for many other aperiodic materials systems such as composites, atomically engineered structures, and bio-materials. 2D materials research is an ideal platform to motivate new mathematics training and curricula in the analysis, modeling, and computation of quantum electronic structure and transport, and mechanical and topological properties of materials. The project's graduate student training and outreach to underrepresented student populations will broaden the diversity of the mathematical research community.The main issue encountered in the mathematical modeling of layered 2D materials is that the lattice periodicities of different layers do not match and thus lead to incommensurate structures. New theory and computational methods that do not use Bloch-Fourier methods will be developed to accurately predict material properties in currently inaccessible regimes. This project will utilize the notions of local configuration space and locality to give new formulations and computational methods for the electronic density of states and for transport properties such as conductivity. New momentum space formulations and corresponding fast computational methods will be developed that exploit the structure of the momentum space Hamiltonian. Novel models and computational methods will also be developed and analyzed that extend our model for electronic density of states and conductivity to include mechanical relaxation.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.