Mathematical Analysis of Domain Wall Motion in Nanowires

铁磁性是一种惊人而微妙的现象。在宏观尺度上可以观察到，它的起源超出了经典物理学的范围，是物质的两个典型量子力学性质的结果，即电子自旋和泡利不相容原理。铁磁性的量子力学起源解释了在微观，甚至纳米尺度上存在铁磁区--具有明显且几乎均匀的磁化作用的区域。在过去的半个世纪里，铁磁区的大小以及操纵(即读写)它们所需的适度能量导致了信息技术中的广泛应用，从磁带驱动器发展到当前的前沿，例如赛道存储器：一种三维存储器，在其上可以围绕纳米线循环读取、移动和写入磁区--“位”。过去二十年见证了微磁学的革命，无论是在基础科学方面，还是随之而来的技术突破。人们很早就知道如何通过外加磁场来控制铁磁区。最近，通过磁化和自旋极化电流的相互作用，发现了一种全新的磁化壁动力学机制。对于几十纳米尺度的长度，有一个成熟且非常成功的微磁学连续统理论，即微磁变分原理及其动态对应的Landau-Lifshitz-Gilbert方程。这一理论在数学上很复杂(方程既是非线性的，也是非局部的)，并涵盖了一系列不同的制度。可以使用现代技术从变分和偏微分方程组中分别解析地研究这些区域。单个原子尺度上的模型必然是量子力学的，需要额外的物理概念和数学设备。本项目的第一个目标是分析研究在电流和外加磁场作用下纳米线和纳米管中的磁化-壁运动。我们将在数学上建立物理行为的存在和广泛的范围，并推导出描述它们的性质的公式。第二个目标是通过对铁磁介质中包括自旋在内的电子动力学的半经典分析，来阐明铁磁体中磁畴-壁传播的基本量子力学机制。

Ferromagnetism is a striking and subtle phenomenon. Observable on the macroscopic scale, its origins lie outside the scope of classical physics, and are consequences of two quintessentially quantum mechanical properties of matter, namely electron spin and the Pauli exclusion principle. The quantum mechanical origin of ferromagnetism accounts for the existence of ferromagnetic domains -- regions of distinct and nearly uniform magnetisation -- on microscopic, indeed nanometer scales. The size of ferromagnetic domains, along with the modest energy required to manipulate (i.e., read and write) them has led to far-reaching applications in information technology, proceeding over the last half-century from magnetic tape drives to the current frontier, for example race-track memory: a three-dimensional memory on which domains -- ''bits'' -- may be read, moved, and written around nanowire loops. The last two decades have witnessed a revolution in micromagnetics, both in fundamental science as well as consequent technological breakthroughs. It has long been understood how ferromagnetic domains can be controlled through external magnetic fields. More recent is the discovery of a wholly new mechanism for domain wall dynamics through the interaction of magnetisation and spin-polarised currents. For length scales down to tens of nanometers, there is a well established and extremely successful continuum theory of micromagnetism, namely the micromagnetic variational principle and its dynamic counterpart, the Landau-Lifshitz-Gilbert equation. The theory is mathematically complex (the equations are both nonlinear and nonlocal), and encompasses a range of diverse regimes. These regimes can be separately investigated analytically using modern techniques from the calculus of variations and partial differential equations. Models on the scale of individual atoms are necessarily quantum mechanical, and require additional physical concepts and mathematical apparatus. The first aim of this project is an analytic study of domain-wall motion in nanowires and nanotubes induced by currents and applied magnetic fields. We will establish mathematically the existence and wide range of physical behaviours, and derive formulas which describe their properties. The second aim is to elucidate the underlying quantum mechanical mechanisms of domain-wall propagation through a semiclassical analysis of the electron dynamics, including spin, in a ferromagnetic medium.