Multi-Scale Magnonic Crystals and Fractional Schr?dinger Equation-Governed Dynamics

摘要：本项目侧重于利用自然几何图形设计先进材料。与高中的几何课不同，大自然不是由光滑的线条、平面和球体组成的，而是由凹凸不平、皱巴巴、参差不齐的曲线组成的，就像包裹在细胞中的DNA，表面像山地景观，体积像地下水位土壤的多孔性。分数阶导数被用来描述自然界的这些几何形状，但没有人从头开始，一个元素一个元素地建立起这样的几何形状。利用有纹理的磁性薄膜，该团队将构建这样的几何形状，作为一种介于有序和无序之间的新型材料的基础。这些材料将以磁性波（称为自旋波）的形式控制信息的传输，与目前使用的简单几何结构相比，它们在我们的纹理薄膜中移动得更慢（次扩散）或更快（超扩散）。这种新的人造材料的构建将是量子分数阶导数的第一个实验实现，因为自旋波遵循与量子力学的波动物理相同的方程，在这种情况下，分数阶Schrödinger方程，到目前为止是一个纯理论的想法。这项工作的一个关键方面是对研究生进行理论和实验之间的交叉训练，培养一支更强大的劳动力队伍，他们可以以多种方式解决其他方式无法解决的新问题。劳动力可以利用这些知识的领域包括基于多孔、分数材料的电池技术；污染物在土壤和地下水位中的扩散；以及生物物质运输的血管结构，包括基于生物学理念的自愈材料。技术摘要：分数阶导数描述了自然界中凹凸不平、皱巴巴和锯齿状的几何形状，其中整数阶导数导致结果发散，使得传统的导数定义不适用，因为随着“尺子尺寸”的减小，切线和曲率迅速增加。在量子背景下，这种自然几何已经用分数Schrӧdinger方程在理论上进行了描述，但从未进行过实验研究。该团队将探索分数Schrӧdinger方程控制的多尺度材料的基本物理，这些材料由基于磁性薄膜的空间调制磁振子晶体组成。他们将在这个人工晶格中设计和测量可调谐的亚扩散和超扩散输运，作为分数阶Schrӧdinger方程动力学的明确证据。该设计建立在一个新的有序无序晶格调制轴上，一个极端是有序晶格，另一个极端是产生安德森局域化的无序晶格。该项目将为生成分数阶偏微分方程提供一般基础，并导致对目前多孔介质和其他分数阶介质（从电池应用到仿生默里材料到土壤和地下水位中的污染物扩散）的高度经验方法的更深入理解。该项目将通过科罗拉多州立大学吴明忠的实验小组和科罗拉多矿业学院林肯·卡尔的理论小组之间的紧密合作进行。作为一个包含实验和理论凝聚态物理的综合团队，他们提出了一个多方面的方法来满足更广泛的影响目标，以融合实验和理论为中心主题，培养我们的研究生和本科生成为全面发展的科学家。这项DMR资助支持对量子材料的基本理解的研究，特别是在数学和物理科学理事会材料研究部凝聚态物理（CMP）项目的资助下，探索与许多其他凝聚态系统交叉的原理。该奖项反映了美国国家科学基金会的法定使命，并通过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，被认为值得支持。

Nontechnical Abstract:This project focuses on the design of advanced materials utilizing the geometries of nature. Nature, unlike geometry class in high school, is not composed of smooth lines, planes, and spheres, but rather bumpy, wrinkled, jagged curves like DNA wrapped up in the cell, surfaces like a mountainous landscape, and volumes like the porous nature of soil in the water table. Fractional derivatives have been used to describe these geometries of nature, but no one has built such geometries from the ground up, element by element. Using textured thin magnetic films, the team will build such geometries as the basis for a new class of materials intermediate between order and disorder. These materials will control transport of information in the form of waves of magnetism, called spin waves, moving through our textured thin film slower (sub-diffusive) or faster (super-diffusive) than possible in simple geometries used up till now. Construction of this new artificial material will be the first experimental realization of a quantum fractional derivative, because spin waves follow the same equations as the wave physics of quantum mechanics, in this case the fractional Schrödinger equation, up till now a purely theoretical idea. A key facet of this work is cross-training of graduate students between theory and experiment, producing a more robust workforce that can work in multiple modalities to solve new problems not tractable otherwise. Fields in which the work force can excel with this knowledge include battery technology, based on porous, fractional materials; the spread of contaminants in soil and the water table; and vascular structures for transport in biological matter, including self-healing materials based on biological ideas. Technical Abstract:Fractional derivatives describe the bumpy, wrinkled, and jagged geometries of nature, where an integer derivative leads to divergent results rendering traditional definition of a derivative inapplicable, due to a rapid increase in the tangent and curvature with decreasing “ruler size”. Such natural geometries in the quantum context have been theoretically described with the fractional Schrӧdinger equation but never experimentally studied. The team will explore the fractional Schrӧdinger equation-governed fundamental physics in multi-scale materials that consist of magnetic thin film–based, spatially modulated magnonic crystals. They will design and measure tunable sub-diffusive and super-diffusive transport in this artificial lattice as clear evidence of fractional Schrӧdinger equation dynamics. The design builds on a new order-disorder lattice modulation axis, with an ordered lattice at one extreme and a disordered lattice giving rise to Anderson localization at the other. The project will provide a general basis for generating fractional partial differential equations, and lead to a deeper understanding of the present highly empirical approach to porous and other fractional media ranging from battery applications to biomimetic Murray materials to spread of contaminants in soil and the water table. The program will be carried out through tight, integral collaborations between Mingzhong Wu's experimental group at Colorado State University and Lincoln Carr's theoretical group at Colorado School of Mines. Working together as an integrated team encompassing experimental and theoretical condensed matter physics, they propose a multi-faceted approach to meet broader impact goals, centrally themed on blending experiment and theory to train our graduate and undergraduate student to become well-rounded scientists. This DMR grant supports research on fundamental understanding of quantum materials and especially exploring principles that cross-cuts many other condensed matter systems with funding from the Condensed Matter Physics (CMP) Program in the Division of Materials Research of the Mathematical and Physical Sciences Directorate.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.