Generation and manipulation of current-induced spin-orbit torques

An emerging field of spintronics, spin-orbitronics, aims to discover novel phenomena and functionalities originating from spin-orbit coupling in solid-state devices. The development of spin-orbitronics promises a fundamental understanding of spin physics in condensed matter, as well as smaller, faster, and far-more energy-efficient spin-based devices. Of particular importance in this field is current-induced spin-orbit torques, which trigger magnetic dynamics by the transfer of angular momentum from an atomic lattice to local magnetization through the spin-orbit coupling. The spin-orbit torque has attracted extensive attention for its fascinating relativistic and quantum mechanical nature, as well as prospective nanoelectronic applications. In this article, we review our studies on the generation and manipulation of current-induced spin-orbit torques.     

Spin force and torque in non-relativistic Dirac oscillator on a sphere

The spin force operator on a non-relativistic Dirac oscillator (in the non-relativistic limit the Dirac oscillator is a spin one-half 3D harmonic oscillator with strong spin–orbit interaction) is derived using the Heisenberg equations of motion and is seen to be formally similar to the force by the electromagnetic field on a moving charged particle. When confined to a sphere of radius R, it is shown that the Hamiltonian of this non-relativistic oscillator can be expressed as a mere kinetic energy operator with an anomalous part. As a result, the power by the spin force and torque operators in this case are seen to vanish. The spin force operator on the sphere is calculated explicitly and its torque is shown to be equal to the rate of change of the kinetic orbital angular momentum operator, again with an anomalous part. This, along with the conservation of the total angular momentum, suggests that the spin force exerts a spin-dependent torque on the kinetic orbital angular momentum operator in order to conserve total angular momentum. The presence of an anomalous spin part in the kinetic orbital angular momentum operator gives rise to an oscillatory behavior similar to the Zitterbewegung. It is suggested that the underlying physics that gives rise to the spin force and the Zitterbewegung is one and the same in NRDO and in systems that manifest spin Hall effect.     

自旋不是轨道角动量的相对论效应——与许方官和高春媛商榷

对《大学物理》2000年第11期上发表的《氢原子的磁矩－对自旋的讨论之一》一提出了不同的看法，认为该错把相对论的流密度当成了非相对论的流密度，前不能在非相对论近似下过滤到后，在相对论情况下，自旋和轨道耦合在一起，不能互相独立，在非相对论情况下自旋独立于轨道运动而存在；因而，自旋不是轨道角动量的相对论效应。     

Consistency in formulation of spin current and torque associated with a variance of angular momentum

Based on the Noether's theorem, we develop systematically and rigorously the spin-dependent formulation of the conservation laws. The effect of the electronic polarization due to the spin-orbit coupling is included in the Maxwell equations. The polarization is related to the antisymmetric components of spin current, and it provides a possibility to measure the spin current directly. The variances of spin and orbit angular momentum currents imply a torque on the "electric dipole" associated with the moving electron. The dependencies of the torque on the polarization and the force on the motions of spin-polarized electrons in a two-dimensional electron gas with the Rashba spin-orbit coupling are discussed.     

The effect of inertia on the Dirac electron,the spin Hall current and the momentum space Berry curvature

We have studied the spin dependent force and the associated momentum space Berry curvature in an accelerating system. The results are derived by taking into consideration the non-relativistic limit of a generally covariant Dirac equation with an electromagnetic field present, where the methodology of the Foldy–Wouthuysen transformation is applied to achieve the non-relativistic limit. Spin currents appear due to the combined action of the external electric field, the crystal field and the induced inertial electric field via the total effective spin–orbit interaction. In an accelerating frame, the crucial role of momentum space Berry curvature in the spin dynamics has also been addressed from the perspective of spin Hall conductivity. For time dependent acceleration, the expression for the spin polarization has been derived.     

对角动量算符的讨论——对自旋的讨论之三

从算符^／J的构成上证明,所谓的总角动量J实质上是相对论性的轨道动量,自旋是相对论性轨道角动量的一个组成部分。     

自旋光电流与电流诱导的电子自旋极化——半导体自旋电子学的新进展

文章通过自旋光电流与Kerr效应的实验，介绍在自旋一轨道耦合的体系中，如何通过自旋注入来驱动电子的运动产生电流，又如何反过来通过电子的运动或电流在系统中产生自旋激化．实验结果表明，自旋轨道耦合不但给我们提供了对自旋的操控手段，而且还给我们提供了用非磁性材料且无外加磁场条件下作为自旋源的新途径．     

Orbital-angular-momentum based origin of Rashba-type surface band splitting

We propose that the existence of local orbital angular momentum (OAM) on the surfaces of high-Z materials plays a crucial role in the formation of Rashba-type surface band splitting. Local OAM state in a Bloch wave function produces an asymmetric charge distribution (electric dipole). The surface-normal electric field then aligns the electric dipole and results in chiral OAM states and the relevant Rashba-type splitting. Therefore, the band splitting originates from electric dipole interaction, not from the relativistic Zeeman splitting as proposed in the original Rashba picture. The characteristic spin chiral structure of Rashba states is formed through the spin-orbit coupling and thus is a secondary effect to the chiral OAM. Results from first-principles calculations on a single Bi layer under an external electric field verify the key predictions of the new model.     

Relativistic electron vortex beams: angular momentum and spin-orbit interaction

Motivated by the recent discovery of electron vortex beams carrying orbital angular momentum (AM), we construct exact Bessel-beam solutions of the Dirac equation. They describe relativistic and nonparaxial corrections to the scalar electron beams. We describe the spin and orbital AM of the electron with Berry-phase corrections and predict the intrinsic spin-orbit coupling in free space. This can be observed as a spin-dependent probability distribution of the focused electron vortex beams. Moreover, the magnetic moment is calculated, which shows different g factors for spin and orbital AM and also contains the Berry-phase correction.     

Probing Dual Asymmetric Transverse Spin Angular Momentum in Tightly Focused Vector Beams in Optical Tweezers

The spin-orbit interaction (SOI) of light generated by tight focusing in optical tweezers is regularly employed in generating angular momentum - both spin and orbital - the effects being extensively observed in trapped mesoscopic particles. Specifically, the transverse spin angular momentum (TSAM), which arises due to the longitudinal component of the electromagnetic field generated by tight focusing is of special interest, both in terms of fundamental studies and associated applications. This study provides an effective and optimal strategy for generating TSAM in optical tweezers by tightly focusing first-order radially and azimuthally polarized vector beams with no intrinsic angular momentum (AM) into a refractive index stratified medium. The choice of such input fields ensures that the longitudinal spin angular momentum (LSAM) arising from the electric (magnetic) field for the radial (azimuthal) polarization is zero. As a result, the effects of the electric and magnetic TSAM are exclusively observed separately in the case of input first-order radially and azimuthally polarized vector beams on single optically trapped birefringent particles. This research opens up new and simple avenues for exotic and complex particle manipulation in optical tweezers.     

On the Linear Forms of the Schrödinger Equation

Generalising the linearisation procedure used by Dirac and later by Lévy-Leblond, we derive the first-order non-relativistic wave equations for particles of spin 1 and spin 3/2 starting from the Schrödinger equation. By the introduction in the momentum of a correction linear in coordinates, we establish the wave equation of the radial harmonic oscillator with spin-orbit coupling.     

Wigner's pseudo-particle relativistic dynamics in external potential field

The Wigner's pseudo-particle formalism has been generalized to describe quantum dynamics of relativistic particle in external potential field. As a simplest application of the developed formalism the time evolution of the 1D relativistic quantum harmonic oscillator been considered. Due to the complex structure of the evolution equation for Wigner function, the only numerical treatment is possible by combining Monte Carlo and molecular dynamics methods. Relativistic dynamics results in appearance of the new physical effects as opposed to non-relativistic case. Interesting is the complete changing of the shape of the momentum and coordinate distribution functions as well as formation of ‘unexpected’ protuberances. To analyze the influence of relativistic effects on average values of quantum operators, the dependencies on time of average momentum, position, their dispersions and energy have been compared for the non-relativistic and relativistic dynamics.     

The ideal relativistic spinning gas: Polarization and spectra

We study the physics of the ideal relativistic rotating gas at thermodynamical equilibrium and provide analytical expressions of the momentum spectra and polarization vector for the case of massive particles with spin 1/2 and 1. We show that the finite angular momentum J entails an anisotropy in momentum spectra, with particles emitted orthogonally to J having, on average, a larger momentum than along its direction. Unlike in the non-relativistic case, the proper polarization vector turns out not to be aligned with the total angular momentum with a non-trivial momentum dependence.     

铁磁/重金属双层薄膜结构中磁性状态的稳定性分析

本文在理论上研究了铁磁/重金属双层薄膜结构中自旋霍尔效应自旋矩驱动的磁动力学. 通过线性稳定性分析, 获得了以电流和磁场为控制参数的磁性状态相图. 发现通过调节电流密度和外磁场, 可以获得不同的磁性状态, 例如: 平面内的进动态、平面内的稳定态以及双稳态. 当外磁场的方向在一定的范围时, 通过调节电流密度可以实现磁矩的翻转和进动. 同时, 通过数值求解微分方程, 给出了这些磁性状态磁矩的演化轨迹.     

1-碘丙烷分子的电子动量谱学研究：自旋-轨道耦合效应与分子内轨道相互作用

利用不对称不共面电子动量谱仪,在2.5 keV碰撞能量下,采用高精度的SAC-CI方法计算了1-碘丙烷分子束缚能谱,同时采用Hartree-Fock、B3LYP/aug-cc-pVTZ(C,H)6-311G^**(I)方法计算其电子动量分布. 并对电离能峰进行了标示. 结合非相对论与相对论计算方法以及自然键轨道分析,对最外层两个轨道(碘的5p孤对)的自旋-轨道耦合效应与分子内轨道相互作用进行了比较. 两种相互作用对电子动量分布的不同影响是可观的. 实验结果与相对论计算的结果一致,表明1-碘丙烷分子内自旋-轨道耦合效应占主导.     

WKB analysis of relativistic Stern–Gerlach measurements

Spin is an important quantum degree of freedom in relativistic quantum information theory. This paper provides a first-principles derivation of the observable corresponding to a Stern–Gerlach measurement with relativistic particle velocity. The specific mathematical form of the Stern–Gerlach operator is established using the transformation properties of the electromagnetic field. To confirm that this is indeed the correct operator we provide a detailed analysis of the Stern–Gerlach measurement process. We do this by applying a WKB approximation to the minimally coupled Dirac equation describing an interaction between a massive fermion and an electromagnetic field. Making use of the superposition principle we show that the +1 and −1 spin eigenstates of the proposed spin operator are split into separate packets due to the inhomogeneity of the Stern–Gerlach magnetic field. The operator we obtain is dependent on the momentum between particle and Stern–Gerlach apparatus, and is mathematically distinct from two other commonly used operators. The consequences for quantum tomography are considered.     

电子的相对论平均场理论与一阶、二阶Rashba效应

用多体平均场意义下电子的Dirac方程讨论了电子自旋动力学及其相关问题. 在大分量Dirac方程的非相对论展开中讨论了电子自旋动力学的高阶效应,并且在二维情形下得到了包括一阶和二阶Rashba效应的电子自旋动力学哈密顿量,求出了相应的包括二阶Rashba效应的哈密顿量的能量和波函数的本征值解,由此讨论了二阶Rashba效应修正的物理含义和大小. 关键词： 二阶Rashba效应 自旋电子学 Dirac方程 相对论平均场理论     

二维自旋轨道耦合电子气中持续自旋螺旋态的稳定性的研究

本文利用半经典的自旋密度矩阵方法对二维自旋轨道耦合电子气中持续自旋螺旋态的稳定性进行了一些研究, 重点研究了自旋螺旋态的寿命与其波矢、载流子迁移率、温度、自旋轨道耦合强度、外电场强度等因素之间的关系, 并将部分理论计算结果与最近的一些相关实验结果进行了比较,发现两者之间大致是符合的.     

Thomas-Fermi description of nuclear systems including spin-orbit interactions

The Thomas-Fermi statistical method is generalized to include spin-orbit interactions. The momentum distributions are given by toroids, different for two particle spin orientations. A system of two coupled differential equations is derived by a variational procedure for the densities of the two populations. From these equations the polarization at the surface of nuclear matter is calculated, as well as the change of the nuclear surface tension due to spin-orbit coupling. Within the statistical framework the coupling strength of the spin-orbit potential is found to be in reasonable agreement with experiment by using only the experimental single-particle level order of the shell model which implies an excess of states with spin parallel to the orbital angular momentum.     

Is There Really a Spin Crisis?

The matrix element of quark axial vector current is shown to be different from the nonrelativistic quark spin sum for a nucleon at rest. The nucleon spin content discovered in polarized deep inelastic scattering is shown to be accommodated in a constituent quark model with 15% sea quark component mixing. The relativistic correction and sea quark pair excitation inherently related to quark axial vector current reduce the nucleon axial charge and this reduction is compensated by the relativistic quark orbital angular momentum exactly and in turn keeps the nucleon spin 1/2 untouched. Nucleon tensor charge has similar but smaller relativistic and sea quark pair excitation reduction.