Spin-orbit-coupling induced torque in ballistic domain walls: Equivalence of charge-pumping and nonequilibrium magnetization formalisms

Abstract

To study the effect of spin-orbit coupling (SOC) on spin-transfer torque in magnetic materials, we have implemented two theoretical formalisms that can accommodate SOC. Using the "charge-pumping" formalism, we find two contributions to the out-of-plane spin-transfer torque parameter β in ballistic Ni domain walls (DWs). For short DWs, the nonadiabatic reflection of conduction electrons caused by the rapid spatial variation of the exchange potential results in an out-of-plane torque that increases rapidly with decreasing DW length. For long DWs, the Fermi level conduction channel anisotropy that gives rise to an intrinsic DW resistance in the presence of SOC leads to a linear dependence of β on the DW length. To understand this counterintuitive divergence of β in the long DW limit, we use the "nonequilibrium magnetization" formalism to examine the spatially resolved spin-transfer torque. The SOC-induced out-of-plane torque in ballistic DWs is found to be quantitatively consistent with the values obtained using the charge-pumping calculations, indicating the equivalence of the two theoretical methods.

Spin transfer and spin-orbit torques in in-plane magnetized (Ga,Mn)As tracks

“…4(a)], β/α < +0.7-5 for ≈ 15 nm (C ⊥ tracks), and finally β/α ≈ −1 for ≈ 5 nm [out-of-plane magnetized (Ga,Mn)As]. Although this agrees with the tendency of the SOI-induced torques to increase with domain-wall width calculated in ballistic nickel domain walls [58], it disagrees with the prediction of Nguyen et al [20].…”

“…In metals it also covers the appearance of a DW resistance at abrupt interfaces [55][56][57][58], leading to a momentum transfer force [7,[59][60][61][62] never clearly identified experimentally [63]. Two contributions have been identified in the spin-relaxation nonadiabatic torque.…”

Topological Hall and spin Hall effects in disordered skyrmionic textures

“…It has been recently shown that momentum scattering against defects and impurities has a dramatic impact on spin transport in any realistic magnetic textures [49,50]. As a matter of fact, since spin transport in magnetic textures presents striking similarities with spin transport in spin-orbit coupled band structures, momentum scattering breaking the coherent spin precession around the local magnetic field results, for instance, in an enhanced nonadiabaticity parameter [51].…”

Calculating the transport properties of magnetic materials from first principles including thermal and alloy disorder, noncollinearity, and spin-orbit coupling

“…The development of a new theoretical framework 32 for calculating magnetization damping and its implementation in the framework of density functional theory [33][34][35][36][37] has motivated systematic reinvestigation of the damping in alloys 38,39 and of the temperature dependence of damping in permalloy 40 allowing quantitative confrontation of theory and experiment without invoking adjustable parameters such as the relaxation time in the torque correlation method (TCM). [29][30][31]41 In this paper we describe in detail a method we recently used to calculate the resistivity ρ, spin flip diffusion length (SDL), and Gilbert damping parameter for Ni 1−x Fe x substitutional alloys, 33 the resistivity and damping for the itinerant ferromagnets Fe, Co and Ni with thermal disorder, 34 the resistance 42 and anisotropic damping 43 of magnetic domain walls, the nonadiabatic STTs in ballistic systems, 44 interface-enhanced damping, 45 thermal disorder effects in transport 46 and a novel interface spin Hall effect. 47 It extends earlier work [48][49][50] by including SOC and non-collinearity.…”