Mathematical Analysis of Domain Wall Motion in Nanowires

铁磁性是一种引人注目的微妙现象。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特性的结果，即电子自旋和泡利不相容原理。铁磁的量子力学起源解释了铁磁域的存在——在微观，甚至是纳米尺度上具有明显且几乎均匀磁化的区域。铁磁畴的大小，以及操作（即读写）它们所需的适中能量，已经在信息技术中带来了深远的应用，在过去的半个世纪里，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，在其上的畴（“比特”）可以围绕纳米线环进行读取、移动和写入。过去二十年见证了微磁学的革命，无论是在基础科学还是随之而来的技术突破。人们早就知道铁磁畴是如何通过外部磁场来控制的。最近，通过磁化和自旋极化电流的相互作用，发现了一种全新的畴壁动力学机制。对于长度尺度缩小到几十纳米，有一个建立良好且非常成功的微磁性连续介质理论，即微磁性变分原理及其动态对应，Landau-Lifshitz-Gilbert方程。该理论在数学上是复杂的（方程既是非线性的，又是非局部的），并且包含了一系列不同的制度。这些状态可以用变分法和偏微分方程的现代技术分别进行分析研究。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。本课题的首要目标是分析研究纳米线和纳米管在电流和外加磁场作用下的畴壁运动。我们将建立数学的存在和广泛的物理行为，并推导公式来描述他们的性质。第二个目的是通过对铁磁介质中电子动力学（包括自旋）的半经典分析，阐明域壁传播的潜在量子力学机制。

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.