Current-induced spin-orbit torques in ferromagnetic and antiferromagnetic systems
A. Manchon, J. Zelezný, I.M. Miron, T. Jungwirth, J. Sinova, A. Thiaville, K. Garello, P. Gambardella
CCurrent-induced spin-orbit torques in ferromagnetic and antiferromagneticsystems
A. Manchon ∗ King Abdullah University of Science and Technology (KAUST),Physical Science and Engineering Division (PSE),and Computer, Electrical,and Mathematical Science and Engineering (CEMSE),Thuwal, 23955-6900,Saudi Arabia
J. ˇZelezn´y
Institute of Physics,Academy of Sciences of the Czech Republic, 162 00 Praha,Czech Republic
I. M. Miron
University of Grenoble Alpes,CNRS, CEA, INAC-SPINTEC,F-38000 Grenoble,France
T. Jungwirth † Institute of Physics,Academy of Sciences of the Czech Republic, 162 00 Praha,Czech RepublicSchool of Physics and Astronomy,University of Nottingham,Nottingham NG7 2RD,United Kingdom
J. Sinova
Institut f¨ur Physik,Johannes Gutenberg Universit¨at Mainz,55128 Mainz GermanyInstitute of Physics,Academy of Sciences of the Czech Republic, 162 00 Praha,Czech Republic
A. Thiaville ‡ Laboratoire de Physique des Solides,Univ. Paris-Sud,CNRS UMR 8502 - 91405 Orsay Cedex,France
K. Garello
IMEC, Kapeeldreef 75, 3001 Leuven,Belgium
P. Gambardella § Department of Materials,ETH Z¨urich, H¨onggerbergring 64,CH-8093 Z¨urich,Switzerland (Dated: April 25, 2019)
Spin-orbit coupling in inversion-asymmetric magnetic crystals and structures hasemerged as a powerful tool to generate complex magnetic textures, interconvert chargeand spin under applied current, and control magnetization dynamics. Current-induced a r X i v : . [ c ond - m a t . m e s - h a ll ] A p r spin-orbit torques mediate the transfer of angular momentum from the lattice to thespin system, leading to sustained magnetic oscillations or switching of ferromagnetic aswell as antiferromagnetic structures. The manipulation of magnetic order, domain wallsand skyrmions by spin-orbit torques provides evidence of the microscopic interactionsbetween charge and spin in a variety of materials and opens novel strategies to designspintronic devices with potentially high impact in data storage, nonvolatile logic, andmagnonic applications. This paper reviews recent progress in the field of spin-orbitronics,focusing on theoretical models, material properties, and experimental results obtainedon bulk noncentrosymmetric conductors and multilayer heterostructures, including met-als, semiconductors, and topological insulator systems. Relevant aspects for improvingthe understanding and optimizing the efficiency of nonequilibrium spin-orbit phenomenain future nanoscale devices are also discussed. CONTENTS
I. Introduction 2II. Overview 5A. Magnetization dynamics induced by spin-orbittorques 5B. Non-uniform magnetic textures 7C. Microscopic origin of spin-orbit torques 8D. Spin-orbit torques in antiferromagnets 9E. Spin-orbit torques in topological materials 10F. Inverse effect of the spin-orbit torque 11III. Theory of spin-orbit torques 11A. Kubo linear response: intraband versus interbandtransitions 13B. Symmetry of spin-orbit torques 14C. Spin-orbit torques due to the spin Hall effect 17D. Spin-orbit torques due to the inverse spin galvaniceffect 191. Inverse spin galvanic torque in a magnetictwo-dimensional electron gas 212. Non-centrosymmetric bulk magnets 213. Spin-orbit torques in magnetic textures 22E.
Ab initio modeling of spin-orbit torques in bilayersystems 22F. Spin-orbit torques in antiferromagnets 23G. Spin-orbit torques in topological insulators 24H. Other spin-orbit torques 251. Anisotropic magnetic tunnel junctions 252. Spin-transfer torque assisted by spin-orbitcoupling 25I. Open theoretical questions 26IV. Spin-orbit torques in magnetic multilayers 27A. Phenomenological description 27B. Measurement techniques 291. Harmonic Hall voltage analysis 292. Spin-torque ferromagnetic resonance 303. Magneto-optic Kerr effect 31C. Materials survey 321. Ferromagnet/nonmagnetic metal layers 322. Ferrimagnet and antiferromagnet/nonmagneticmetal layers 383. Ferromagnet/semiconductor layers 394. Ferromagnet/topological insulator layers 405. Two-dimensional alloys and oxide interfaces 416. Metallic spin-valves 42 ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] 7. Established features and open questions 42D. Magnetization dynamics 43E. Magnetization switching 451. Switching mechanism 472. Switching speed 473. Zero field switching 48F. Memory and logic devices 50V. Spin-orbit torques in noncentrosymmetric magnets 51A. nonmagnetic GaAs structures 51B. Bulk ferromagnetic (Ga,Mn)As and NiMnSb 52C. Collinear antiferromagnets 54D. Antiferromagnetic topological Dirac fermions 56E. Magnonic charge pumping in (Ga,Mn)As 57F. Established features and open questions 57VI. Spin-orbit torques and non-uniform magnetic textures 58A. Domain wall dynamics under current 581. Steady domain wall dynamics 582. Precessional domain wall dynamics 59B. In-plane magnetized samples 601. Soft samples (X domains) 602. Anisotropic samples with Y domains 61C. Perpendicularly magnetized samples 611. Demonstrations of spin-orbit torques incurrent-induced domain wall motion 612. Domain wall motion under spin-orbit torque 633. Two-dimensional effects in current-induced domainwall motion 634. Domain wall motion under combined spin transferand spin-orbit torques 645. Motion of magnetic skyrmions under spin-orbittorques 656. Impact of disorder 66D. Antiferromagnetic and ferrimagnetic systems 66VII. Perspectives 67List of abbreviations 68Acknowledgements 68References 68 I. INTRODUCTION
The interplay between spin and orbital degrees of free-dom in condensed matter physics has been intensivelystudied for more than a century, starting from the semi-nal experiments of Barnett (Barnett, 1915) and Einsteinand de Haas (Einstein and de Haas, 1915). At the timeof these pioneering experiments on the transfer betweenmagnetic and lattice angular momenta, the electron’sspin was unknown and the phenomena could only beexplained on the level of macroscopic angular momen-tum conservation principles. A microscopic insight intospin-orbit coupling emerged later from the relativisticquantum-mechanical Dirac equation. In magnetic ma-terials, the relativistic spin-orbit coupling is now under-stood to play a fundamental role in a number of phenom-ena, including magnetocrystalline anisotropy, magneti-zation precession damping (St¨ohr and Siegmann, 2006),anomalous Hall effect (Nagaosa et al. , 2010), anisotropicmagnetoresistance (McGuire and Potter, 1975), and spinrelaxation (Dyakonov, 2008). In nonmagnetic semicon-ductors, the correlation between nonequilibrium chargeand spin currents has been extensively studied sincethe 1970s (Aronov and Lyanda-Geller, 1989; D’yakonovand Perel’, 1971; Ivchenko and Pikus, 1978), allowingfor the manipulation of spin states using both electri-cal and optical techniques (Ganichev et al. , 2002; Katoand Awschalom, 2008; Rashba and Sheka, 1991). In the past fifteen years, the interest in materials with strongspin-orbit coupling has substantially intensified. Het-erostructures, surfaces and interfaces displaying unprece-dentedly large spin-momentum locking have been recog-nized as powerful platforms for investigating the rela-tivistic motion of electrons in condensed matter systems(Hasan and Kane, 2010; Manchon et al. , 2015) as well asthe formation of chiral magnetic textures (Nagaosa andTokura, 2013; Soumyanarayanan et al. , 2016). In thiscontext, recent predictions (Bernevig and Vafek, 2005;Garate and MacDonald, 2009; Manchon and Zhang,2008, 2009; Obata and Tatara, 2008; Tan et al. , 2007;ˇZelezn´y et al. , 2014) and observations of current-inducedmagnetization dynamics mediated by spin-orbit couplingin ferromagnets and antiferromagnets (Ando et al. , 2008;Chernyshov et al. , 2009; Liu et al. , 2011; Miron et al. ,2010, 2011b; Wadley et al. , 2016) have revolutionized thefield of spintronics, leading to new opportunities to inte-grate electronic and magnetic functionalities in a widevariety of materials and devices.
FIG. 1 (Color Online) Materials in which spin-orbit torques have been observed range from metallic heterostructures involvingtransition metals, topological insulators and other heavy element substrates, to bulk non-centrosymmetric ferro- and antifer-romagnets. Spin-orbit torque is a promising candidate mechanism to drive disruptive spintronics devices such as magneticmemories, nano-oscillators, race-track storage devices, as well as interconnects and spin logic gates.
Research in spintronics explores the possibilities to addthe spin degree of freedom to conventional charge-basedmicroelectronic devices or to completely replace chargewith spin functionalities (Wolf et al. , 2001; Zutic et al. ,2004). Over the past three decades of research and de-velopment, spintronics has offered means to replace mag-netic fields for reading and writing information in nano-magnets by more scalable current-induced spin-torques (Brataas et al. , 2012a; Chappert et al. , 2007). Spin trans-fer torques (STT), which mediate the transfer of spinangular momentum between two magnetic layers hav-ing noncollinear magnetizations (Berger, 1996; Ralph andStiles, 2008; Slonczewski, 1996), are currently the methodof choice for controlling the bit states in magnetic ran-dom access memories (MRAMs) (Apalkov et al. , 2016;Kent and Worledge, 2015). In STT, spin-orbit couplingalready plays an important, but passive role: by induc-ing spin relaxation and magnetic damping, it enables thespin-polarization of the charge current passing throughthe reference layer and permits the magnetization switch-ing. This review focuses on a new family of spin torques,whose physical origin is the transfer of orbital angularmomentum from the lattice to the spin system. Thesetorques rely on the conversion of electrical current to spin(Gambardella and Miron, 2011; Sinova et al. , 2015) andare called spin-orbit torques (SOTs) in order to underlinetheir direct link to the spin-orbit interaction.Because of the ubiquity of spin-orbit coupling, SOTsprovide efficient and versatile ways to control the mag-netic state and dynamics in different classes of materi-als, as schematically shown in Fig. 1. Several micro-scopic mechanisms can give rise to SOT. In one picture,a charge current flowing parallel to an interface with bro-ken inversion-symmetry generates a spin density due tospin-orbit coupling, which in turn exerts a torque on themagnetization of an adjacent magnetic layer via the ex-change coupling (Manchon and Zhang, 2008). Severalnames have appeared in the literature for this modelmechanism, such as the Rashba-Edelstein effect (Edel-stein, 1990) or the inverse spin galvanic effect (iSGE)(Belkov and Ganichev, 2008). In this review we will usethe term iSGE-SOT.In the other model scenario, spin-orbit coupling gen- erates a spin current in the nonmagnetic metal layer dueto the spin Hall effect (SHE) (Dyakonov and Perel, 1971;Hirsch, 1999; Sinova et al. , 2015). The spin current prop-agates towards the interface, where it is absorbed in theform of a magnetization torque in the adjacent ferromag-net (Ando et al. , 2008; Liu et al. , 2011). The SHE-SOTand iSGE-SOT can act in parallel. This is reminiscentof the early observations in semiconductors of the SHEand iSGE as companion phenomena, both allowing forelectrical alignment of spins in the same structure (Kato et al. , 2004a,b; Wunderlich et al. , 2004, 2005).Considering the SOT as originating from either theiSGE or SHE model scenarios can provide a useful phys-ical and materials guidance. The necessary conditionfor the iSGE-induced non-equilibrium spin polarizationis the broken inversion symmetry, which is automaticallyfulfilled in the above mentioned interfaces. However, alsouniform crystals can have unit cells that lack a center ofsymmetry. The initial discovery of the iSGE-SOT wasmade in such a crystal, namely in the zinc-blende dilutedmagnetic semiconductor (Ga,Mn)As (Chernyshov et al. ,2009) and later also reported in asymmetric metal multi-layers (Miron et al. , 2010). This line of research was sub-sequently extended to crystals whose individual atomicpositions in the unit cell are locally non-centrosymmetric,leading to the discovery of a staggered iSGE polariza-tion and current-induced switching in an antiferromagnet(Wadley et al. , 2016; ˇZelezn´y et al. , 2014).
JULY 2017 | PHYSICS TODAY rent that flows upward into the recording ferro-magnet; the SHE e ff ectively turns the heavy-metallayer into a spin injector. The switching of the record-ing ferromagnetic bit is then due to a transfer ofspin angular momentum from carriers to magneti-zation as in STT. In the ISGE variant (figure 3c),a charge current generates a nonequilibrium spinpolarization at the interface between the heavy-metal layer and the ferromagnet, rather than a spin current. The current-induced polarization canswitch the ferromagnetic bit.We emphasized above that spin–orbit couplingmust be present for either the SHE or ISGE mech-anism to work. In addition, it turns out that inver-sion symmetry must be broken.
In typical ap-plications, the breaking is achieved, as in figures 3b and 3c, bya bilayer structure involving the heavy metal and the recordingferromagnet. In such architectures, the SHE and the ISGE areo " en inseparable companions. Their relative contributions toSOT depend on the details of the materials and the interface between the heavy metal and recording ferromagnet.The application of the SHE and the ISGE to SOT is an amaz-ing turn of events in the world of spin–orbit coupling. Manyphysicists had thought of spin–orbit coupling as an e ff ect thatdestroys spin polarization by facilitating spin-flip sca ering.However, with the SHE and the ISGE, the whole thing is turnedaround: Via spin–orbit coupling, the la ice generates spin po-larization instead of destroying it. Remarkably, SOT can be evenmore e ffi cient than STT in the sense that SOT switching can befaster and can require less current. Those features make SOT particularly a ractive for fast processor memories and suggestthat SOT will be a technology at the top level of the computermemory hierarchy.What is the source of SOT’s superior switching? In STT, eachelectron can transfer only one quantum unit of spin angularmomentum as it travels from the reference ferromagnet to therecording ferromagnet. In the SHE and ISGE writing mecha-nisms, each sca ering of a carrier electron generates a small FIGURE 2. TWENTY - FIRST CENTURY MRAM.
The modern magnetic random access memory comprises myriad bits, each ofwhich includes a reference magnetic layer separated from a recordingferromagnet by a nonmagnetic spacer. The reference layer is static,but the recording ferromagnet is switchable, as indicated by the two directions of spin arrows.
FIGURE 3. FLIPPING THE BIT.
In the spin-transfer torque mechanism, (a) a current (gray arrow) of polarized electrons from a reference ferromagnet passes down through a spacer into a recording ferromagnet. Within a few atomic monolayers of entering the recording magnet, the flowing electrons align with the instantaneous recording magnetization (large purple arrow in the recording medium). Thisalignment results in a torque (curved white arrow) on the recording ferromagnet that ultimately causes the recording magnetization to flip from its original orientation (large red arrow). In the snapshot shown here, the recording magnetization is about ⁄ of the way to beingflipped. Note that the time scale for the full reversal is much greater than the time needed for the current to flow from the reference ferromagnet through the recording ferromagnet. A second mechanism, spin–orbit torque, can be driven by the spin Hall effect (SHE) or bythe inverse spin galvanic effect (ISGE). (b) In the SHE variant, as current flows along the contact and the heavy-metal layer, a spin current is generated that flows upward into the recording ferromagnet and flips its magnetization. (c)
In the ISGE mechanism, electrons become polarized at the interface of a heavy metal and a ferromagnet; the polarized electrons then switch the magnetization of the recording ferromagnet. In structures such as those shown in panels b and c, with heavy-metal and ferromagnetic recording layers, both the SHE and the ISGE contribute to spin–orbit torque.
FIG. 2 (Color Online) STT versus SOT switching of a magnetic tunnel junction. (a) In the STT case, a current of spin-polarizedelectrons (gray arrow) flows from a reference ferromagnet through a spacer layer into the recording ferromagnetic layer. Withina few atomic monolayers of entering the recording magnet, the flowing electrons align with the instantaneous magnetizationdue to the exchange interaction (large purple arrow in the recording medium). This alignment results in a torque (curvedwhite arrow) on the recording ferromagnet that ultimately causes the magnetization to switch from its original orientation(large red arrow). In the snapshot shown here, the magnetization is about 2 / The notion of the SHE-induced SOT, on the other hand, led to systematic studies correlating trends inthe magnitude and sign of the SOT in ferromag-netic/nonmagnetic metal bilayers (Ando et al. , 2008; Liu et al. , 2012b; Pai et al. , 2012) with the magnitude andsign of the SHE in the nonmagnetic material calculatedby ab-initio methods (Freimuth et al. , 2010; Tanaka et al. ,2008) or measured by spin absorption in nonlocal spinvalves (Idzuchi et al. , 2015; Morota et al. , 2011). How-ever, in the commonly used bilayers with a nm-scalespin-diffusion length and nm-thick magnetic film, thedistinction between SOTs generated by ”bulk”-SHE or”interface”-iSGE remains principally blurred. Moreover,the experimentally observed complex SOT phenomenol-ogy in the bilayer structures is often not captured by ei-ther of the two idealized model scenarios (Garello et al. ,2013; Kim et al. , 2013a). Other studies have pointed outfurther contributions to the SOT due to interface oxida-tion (An et al. , 2018a; Demasius et al. , 2016; Miron et al. ,2011a; Qiu et al. , 2015) and spin-dependent scatteringof the spin-polarized current flowing in the ferromagnet(Amin et al. , 2018; Saidaoui and Manchon, 2016), whichcan add to the SHE-induced SOT.Independently on their origin, SOT allow for new de-vice architectures and efficient control of the magnetiza-tion. Figure 2 compares the out-of-plane current geom-etry employed in STT-MRAMs for both the write andread operations of a magnetic tunnel junction (MTJ),shown in (a), with the in-plane writing geometry enabledby SOT based on either iSGE (b) or SHE (c). SOT-induced magnetization switching, first demonstrated byMiron et al. (2011a) and Liu et al. (2012b), allows fordecoupling the write and read current paths, with greatadvantages in terms of endurance of the junction andswitching speed relative to STT (Cubukcu et al. , 2015;Fukami and Ohno, 2017; Prenat et al. , 2016). Differentlyfrom STT, SOT allows also for the switching of mag-netic insulators (Avci et al. , 2017a) and antiferromagnets(Wadley et al. , 2016), as well as for the generation of co-herent spin waves (Collet et al. , 2016; Demidov et al. ,2012) and interconversion of electric and magnon cur-rents (Cornelissen et al. , 2015; Goennenwein et al. , 2015;Kajiwara et al. , 2010) in single-layer ferromagnets andferrimagnets.In this article we review the present theoretical under-standing of the SOT in various types of material systemsand summarize the experimentally established SOT phe-nomenology. We also discuss links of SOT to other cur-rently highly active research fields, such as the topologi-cal phenomena in condensed matter, and outline foreseentechnological applications of the SOT. A brief overviewis given in Section II. Readers interested in a more de-tailed discussion of theoretical and experimental aspectsof SOT are referred to the subsequent sections.
II. OVERVIEWA. Magnetization dynamics induced by spin-orbit torques
The dynamics of the recording layer subject to STT orSOT is governed by the Landau-Lifshitz-Gilbert (LLG)equation, d m dt = − γ m × B M + α m × d m dt + γM s T , (1)where γ > × s − T − for free electrons), α is theGilbert damping parameter (dimensionless), M s is thesaturation magnetization and m = M /M s is the magne-tization unit vector. The first term accounts for the pre-cession of the magnetization m about the effective field, B M , defined as the functional derivative of the magneticenergy density E , B M = − δ E /δ M . The second term ac-counts for the relaxation of the magnetization towardsits equilibrium position, and the third term representsthe other torques T that may not derive from an energydensity, notably the torques induced by an electrical cur-rent. Such torques are by definition orthogonal to themagnetization m and adopt the most general form T = τ FL m × ζ + τ DL m × ( m × ζ ) . (2)Here ζ is a unit vector that depends on the microscopicmechanism at the origin of the torques, and the coeffi-cients τ FL , DL may depend on the magnetization angle.In the STT configuration depicted on Fig. 2(a), ζ is thepolarization vector and is oriented along the magnetiza-tion direction of the reference layer. In the case of SOT, ζ is determined by the charge-spin conversion process in-duced by spin-orbit coupling. In the literature, τ DL isusually referred to as the longitudinal (Slonczewski-like)component, which lies in the ( m , ζ ) plane. In contrast, τ FL is normally referred to as the perpendicular (or trans-verse) component, which lies normal to the ( m , ζ ) plane.The directions of these two torque components are rep-resented in Fig. 3. To understand the impact of thesetorques on the magnetization dynamics, it is instructiveto remark that the perpendicular torque, τ FL , acts onthe magnetization like an effective magnetic field [firstterm in Eq. (1)], while the longitudinal torque, τ DL , actslike an effective magnetic damping [second term in Eq.(1), to the lowest order in α ]. Because of these simi-larities, these two torque components, τ FL and τ DL , arealso commonly called field-like and damping-like terms,respectively, a denomination we adopt in this review.Initially, the damping-like component of the SOT wasidentified by measuring the damping of the ferromagneticresonance (FMR) in nonmagnetic metal/ferromagneticmetal bilayers (NM/FM) (Ando et al. , 2008). A changeof the damping factor was induced in the experiment byan in-plane dc current and interpreted as a consequence FIG. 3 (Color Online) Directions of the field-like anddamping-like SOT components. In the present configurationand for clarity, we take ζ = y . of the SHE-SOT. This was a new concept in which a dcelectrical current driven through a conductor adjacent tothe ferromagnet controls the FMR damping, in contrastto traditional means of controlling the FMR frequencyby the dc current-induced Oersted field. Since the SHEin nonmagnetic metals was an emerging topic at the timeof these pioneering SOT experiments, one of the key per-ceived merits of the SOT then was in providing an ex-perimental measure of the spin Hall angle in the non-magnetic material (Ando et al. , 2008). From our presentperspective, however, we emphasize that such measure-ments have to be taken with great caution as the ”spinHall angles” inferred from these experiments are only ef-fective parameters capturing, besides the bulk SHE, alsothe iSGE and other potential spin-orbit coupling and spincurrent contributions originating from the spin-orbit cou-pling from the interface (Amin and Stiles, 2016a,b; Kim et al. , 2017b; Lifshits and Dyakonov, 2009; Miron et al. ,2011a; Saidaoui and Manchon, 2016; Sinova et al. , 2015).While in experiments pioneered by Ando et al. (2008)the FMR is generated externally and the SOT only mod-ifies the dynamics, Liu et al. (2011) demonstrated thatthe SOT itself can drive the FMR when an alternatingin-plane current is applied to the NM/FM bilayer. Themethod was again conceived to provide additional meansto utilize ferromagnet dynamics for measuring the SHE inthe adjacent nonmagnetic metal layer (Liu et al. , 2011).A remarkable turn of events appeared, however. Miron et al. (2011a) and, subsequently, Liu et al. (2012b) ob-served that SOTs can not only trigger small angle FMRprecession but, for large enough electrical currents, it canfully and reversibly switch the ferromagnetic moments.The roles of the ferromagnetic and nonmagnetic metalliclayers got reversed: In the original experiment by Ando et al. (2008), the ferromagnet provided the tool and therelativistic effects in the nonmagnetic metal layer werethe object of interest. From now on, the new means tomanipulate the magnetization took central stage.Phenomenologically, SHE-SOT may appear as a merecounterpart of the STT (Ando et al. , 2008; Liu et al. ,2012b). At first sight, the spin current injected from thenonmagnetic metal layer due to the SHE just replaces the spin-injection from the reference to the recording fer-romagnet in the STT stack (see Fig. 2). However, thechange in the writing electrical current geometry fromout-of-plane in the STT to in-plane in the SOT has ma-jor consequences for the operation of memory devices aswell as for the transport properties of layered structures.In the STT, each electron injected perpendicular to theplane of the heterostructure transfers one quantum unitof spin angular momentum as it travels from the refer-ence to the recording ferromagnet. This transfer can beenhanced by using resonant tunneling (Theodonis et al. ,2007; Vedyayev et al. , 2006), which is difficult to real-ize experimentally. In the relativistic SOT utilizing noreference spin polarizer and where electrons are injectedin the plane of the heterostructure, the spin angular mo-mentum generated from the linear momentum in betweencollisions gives a little kick in every collision or acceler-ation that the electron feels, all along the plane. Thisconfiguration inherently enables the effective transfer ofmore than one spin unit per electron and allows for ex-erting spin torque on large sample areas. This fact hasopened an entirely new space for material and device op-timization of the switching process in SOT MRAMs.Present MRAM bit cells utilize the tunneling magne-toresistance (TMR) for readout (Chappert et al. , 2007).The TMR effect is maximized when the recording andreference magnetizations switch between parallel and an-tiparallel configurations. For STT writing in such a de-vice, however, the injected spin from the reference fer-romagnet with a precisely aligned or anti-aligned orien-tation to the recording magnetization exerts no torque.This implies that the STT mechanism relies on thermalfluctuations of magnetization and the associated incu-bation time for initializing the magnetization dynamicsslows down the switching process. A means to limit theincubation time is to engineer the polarizing and record-ing layers with orthogonal magnetizations. In this config-uration, however, the polarizing layer cannot be used as aTMR sensor to probe the magnetic state of the recordinglayer and a third reference layer needs to be inserted inthe device for this purpose (Kent et al. , 2004; Ye et al. ,2014).In the SOT approach, the orientation of the current-induced spin polarization that exerts the torque in therecording ferromagnet is independent of the magnetiza-tion in the reference ferromagnet of the TMR stack, andcan be engineered to be misaligned with the recordingmagnetization. Therefore, the in-plane writing currentgeometry can make the SOT more efficient and fasterthan STT (Aradhya et al. , 2016; Baumgartner et al. ,2017; Fukami and Ohno, 2017; Garello et al. , 2014; Pre-nat et al. , 2016).Sharing the read and write current paths in STT-MRAMs is also problematic (Apalkov et al. , 2016; Kentand Worledge, 2015). The distributions of read and writecurrent values need to be well separated to avoid unde-sired writing while reading the memory. However, highwriting currents go against energy efficiency. They alsorequire thin tunnel barrier separating the recording andreference ferromagnetic layers, resulting in reliability is-sues due to barrier damage at high writing currents.Moreover, optimizing the tunnel barrier (and other com-ponents of the STT-MRAM stack) for writing can havedetrimental effect on the magnitude of the readout TMR.In contrast, the three-terminal SOT-MRAM bit cell withseparate write and read paths allows for optimizing sep-arately these two basic memory functionalities and toremove the endurance issue by not exposing the tunnelbarrier to the writing current. These advantages come atthe expense of a larger area of the three-terminal SOT-MRAM cell (that is, a lower memory density) comparedto the two-terminal STT-MRAM. Overall, SOT-MRAMscan find a broad utility and appear to be particularly wellsuited for the top of the memory hierarchy, namely forthe embedded processor caches (Fukami and Ohno, 2017;Hanyu et al. , 2016; Prenat et al. , 2016). B. Non-uniform magnetic textures
FIG. 4 (Color online) (a) Domain wall racetrack memorywith red and blue regions representing areas that are oppo-sitely magnetized. Adapted from Parkin and Yang (2015).Illustration of left-handed chiral N´eel domain walls in aNM/FM bilayer. The effective field B of the damping-likeSOT moves adjacent up-down and down-up domains (withvelocity v DW ) in the same direction. Adapted from Emori et al. (2013). (c) Skyrmions in a 2D ferromagnet with uni-axial magnetic anisotropy along the vertical axis. Magneti-zation is pointing down on the edges and pointing up in thecenter. Moving along a diameter, the magnetization rotatesby 2 π around an axis perpendicular to the diameter due tothe DMI. Adapted from Fert et al. (2013). Non-uniform magnetic textures are the basis of theracetrack memory concept illustrated in Fig. 4 (Parkinand Yang, 2015; Parkin et al. , 2008). A domain-wall race-track memory consists of a series of alternating up anddown magnetization domains that can be synchronouslyshifted along the corresponding multi-bit track and bythis sequentially read by a single magnetoresistive sensor[see Fig. 4(a)].An applied uniform (easy-axis) magnetic field cannotbe used to operate the racetrack since it favors one of the two types of domains and thus pushes neighboringdomain walls in opposite directions. Initially, this prob-lem was resolved by replacing magnetic field with theSTT that is induced by an in-plane current driven alongthe racetrack (Parkin et al. , 2008). The physics is analo-gous to STT switching by a vertical current in an MRAMstack where the preferred magnetization direction is con-trolled by the direction of the applied spin current. In theracetrack, one direction of the applied electrical currentmoves electrons from, say, the up-domain to the down-domain at one domain wall, and from down-domain tothe up-domain at the neighboring domain wall. As a re-sult, the sense of the spin current is opposite at the twodomain walls. It implies that, say, the up-domain is pre-ferred at the first domain wall while the down domain ispreferred at the second domain wall and the two domainwalls then move in the same direction.At first sight, SOT in a racetrack fabricated from aNM/FM bilayer cannot be used to synchronously movemultiple domain walls along the track. For example, afield-like SOT due to the iSGE would act as a uniformmagnetic field. Also the damping-like SOT due to theSHE seems unfavorable as it is driven by a uniform ver-tical spin current. This makes the SHE-SOT fundamen-tally distinct in the domain wall racetrack geometry fromthe STT mechanism that exploits the repolarization ofthe in-plane spin current when carriers enter successivedomains.Remarkably, theory and experiment have shown thatthe damping-like SOT can also move neighboring domainwalls in the same direction, provided that the walls are ofN´eel type and have the same spin chirality (Emori et al. ,2013; Ryu et al. , 2013; Thiaville et al. , 2012). Chiral N´eeldomain walls are stabilized by the Dzyaloshinskii-Moriyainteraction (DMI) which relies on the interfacial spin-orbit coupling and broken inversion symmetry, similarlyto the SOT. In this chiral case, the effective field drivingthe damping-like SHE-SOT in the domain wall is orientedalong the easy axis in a direction that alternates from onedomain wall to the next so that current drives them inthe same direction [see Fig. 4(b)]. Moreover, in analogyto switching in MRAMs, the racetrack SOT can be moreefficient than STT, resulting in higher current-induceddomain wall velocities (Miron et al. , 2011b; Yang et al. ,2015).In an alternative racetrack memory concept, the one-dimensional (1D) chiral domain walls are replaced withthe skyrmion topological 2D chiral textures [see Fig. 4(c)](Fert et al. , 2013). While current-driven depinningcan be achieved at substantially lower current densityin skyrmion lattices (Jonietz et al. , 2010), individualmetastable skyrmions are expected to behave as point-like particles and are in principle less sensitive to theboundaries and pinning to boundary defects as comparedto domain walls (Sampaio et al. , 2013). Research is cur-rently focusing on current-driven motion of individualskyrmions (Jiang et al. , 2017b; Legrand et al. , 2017; Litz-ius et al. , 2017; Woo et al. , 2016).
C. Microscopic origin of spin-orbit torques
We mentioned in the introduction that two main modelmechanisms have been proposed to generate SOT. SHEoriginates from asymmetric spin deflection in the bulkof, e.g., a heavy metal induced by spin-orbit coupling.Such a deflection induces a pure spin current, transverseto the direction of the applied electrical current, that issubsequently absorbed in the adjacent magnetic layer, asdepicted in Fig. 2(b). The SHE-SOT model mechanismshares with the STT the basic concept of the angularmomentum transfer from a carrier spin current to mag-netization torque. As a consequence, the dominant com-ponent of the SHE-SOT in this picture is damping-likeand takes the form (Ando et al. , 2008), T = ( j SHEs /t F ) m × ( m × ζ ) , (3)in units of eV/m . Here j SHEs is the SHE spin currentdensity absorbed by the recording magnet of thickness t F ,and ζ is a unit vector of the in-plane spin-polarizationof the out-of-plane SHE spin current. The magnitudeof the injected SHE spin current density into the mag-net is modelled as j SHEs = ( (cid:126) / e ) ηθ sh σ N E , where η isthe spin-injection efficiency across the NM/FM interface,also called transparency, θ sh = σ sh /σ N is the spin-Hallangle in the nonmagnetic material of spin-Hall conduc-tivity σ sh [expressed in units of Ω − m − ] and electricalconductivity σ N , and E ⊥ ζ is the applied in-plane elec-tric field. The SHE-SOT, being damping-like, directlycompetes with the damping term in the LLG equationof magnetization dynamics. This situation favors thecurrent-induced switching of in-plane magnetized layers,as, for a damping-like torque, the critical current has toovercome the magnetic anisotropy barrier multiplied bythe Gilbert damping factor α , the latter being typically (cid:28) odd in momen-tum k . An example of such a spin texture is given in Fig.5 for the prototypical case of the Pt/Co interface. Thisinterfacial spin texture exhibits several features similar FIG. 5 (Color online) Spin texture in momentum space cal-culated for the interfacial Co layer of Pt(8ML)/Co(2ML) slabusing density functional theory. ML stands for monolayer.Both in-plane spin components, (a) S x and (b) S y , are odd inmomentum space, enabling iSGE. Adapted from Haney et al. (2013a). to the ideal case of Rashba spin-orbit coupling (Man-chon et al. , 2015) and promotes iSGE. The iSGE-SOTresembles at first glance a mechanism in which the ap-plied current generates a field rather than a damping-liketorque.In the iSGE mechanism in a NM/FM bilayer, the car-rier spin density and the corresponding non-equilibriumeffective magnetic field acting on the magnetization formdirectly at the inversion-asymmetric interface. Thedamping-like SHE-SOT, on the other hand, has beenprimarily viewed as a consequence of the spin cur-rent pumped from the bulk of the nonmagnetic mate-rial (which can be centrosymmetric) to the ferromagnetwhere it transfers its angular momentum to the mag-netization. In the SHE, however, the spin current alsoyields a non-equilibrium spin density at the edges of thenonmagnetic material where the inversion-symmetry isbroken. This implies an alternative picture of the SHE-SOT caused by the non-equilibrium spin density at theNM/FM interface. Correspondingly, the SHE can be alsoexpected to contribute to the field-like SOT. Vice versa ,as further discussed in Section III, the iSGE mechanismcan yield not only field-like but also damping-like SOTterms (Kurebayashi et al. , 2014; Miron et al. , 2011a).While the original iSGE models consider the effect ofa uniform spin density on the magnetization dynamics,additional torques arise in models where the spin densitygenerated at the interface is allowed to diffuse away fromthe interface (Amin and Stiles, 2016a,b; Haney et al. ,2013b; Manchon, 2012). An example of numerical re-sults is shown on Fig. 6, where the torque magnitudeis plotted against the nonmagnetic metal thickness inthe case of pure SHE and pure iSGE (Amin and Stiles,2016b). For the reasons mentioned above, the decompo-sition into T FL and T DL does not allow to disentanglethe microscopic iSGE and SHE mechanisms of the SOT.Moreover, the factors τ FL and τ DL can depend on the an-gle of m (Garello et al. , 2013). This makes not only themicroscopic analysis but also the phenomenological LLGdescription of the SOT more complex. FIG. 6 (Color online) Torque components as a function ofthe nonmagnetic metal thickness, in the case of SHE (left)and iSGE (right). Both mechanisms produce field-like anddamping-like components. From Amin and Stiles (2016b).
In general, SOT can be directly linked to the appliedelectric field E by a linear-response expression, T = χ T E (Freimuth et al. , 2014b). Alternatively, SOT can be writ-ten as T = M × B T , where B T ≈ − ∆ S /M s is an effectivecurrent-induced spin-orbit field, ∆ is the exchange cou-pling between carrier spins and magnetic moments, and S = χ S E is the current-induced carrier spin density ex-pressed again in the linear response. The different torqueterms in the LLG equation are obtained from the expan-sion, χ S,ij = χ (0) S,ij + χ (1) S,ij,k m k + χ (2) S,ij,kl m k m l + · · · , where m i are the components of the magnetization unit vector.Here the response function coefficients for each order in m are independent of m and their matrix form reflectsthe underlying crystal symmetry of the considered ma-terial or structure (Hals and Brataas, 2013a,b; Wimmer et al. , 2016; ˇZelezn´y et al. , 2017). For example, the field-like SOT corresponds to the zeroth order term while thedamping-like SOT term appears in the first order of theexpansion of χ S .Note that an analogous expansion can be written for χ T and that the approaches using χ T or χ S expansion arein principle equivalent. Using χ S appeals to the two-stepphysical picture of the SOT in which, first, the appliedcurrent polarizes the carriers (which can also appear innonmagnetic metals) and, second, the non-equilibriumcarrier spins generate the torque on magnetic momentsvia exchange coupling ∆. When considering χ T , thephysical intuition based on SHE, iSGE or other non-equilibrium spin density phenomena may be less appar-ent but the experimentally measured quantity, which isthe SOT, is accessed directly. In microscopic theories,SOT has been calculated either from χ S or χ T . In theformer case, one obtains a spin-density averaged over theunit cell which is then multiplied by an averaged ex-change field to obtain the net torque. In the latter case,the cross product of the spin density and exchange fieldis calculated locally and then averaged over the unit cellto get the torque. Hence, using χ T always represents themore rigorous approach.In the Kubo linear response formalism, the microscopic expression for χ S (or χ T ) can be split into the intra-band contribution (Boltzmann theory) and the inter-band term (Garate and MacDonald, 2009). The formerone scales with conductivity, i.e., diverges in the absenceof disorder, and contributes to the field-like SOT (Man-chon and Zhang, 2008). The latter one is finite in thedisorder-free intrinsic limit (Freimuth et al. , 2014a; Kure-bayashi et al. , 2014) and contributes to the damping-like SOT. As a result, the field-like SOT tends to dom-inate the damping-like SOT in clean systems while thetrend reverses in more disordered structures (Freimuth et al. , 2014b; Li et al. , 2015a). This is an example ofbasic guidelines that theory can provide when analyzingSOT experiments. We emphasize, however, that otherterms beyond the lowest order field-like and damping-like torques can also significantly contribute to the totalSOT, as seen in experiments (Fan et al. , 2014b; Garello et al. , 2013).Finally, we note that unlike the rigorous and system-atic methods based on the response functions χ S or χ T ,considering the SHE spin current as an intermediate stepbetween the applied electrical current and the resultingSOT is an intuitive but not rigorous approach. This isbecause other mechanisms beyond the bulk-like SHE cancontribute, and because in spin-orbit coupled systemsthe spin current is not uniquely defined, in contrast tothe well-defined and directly measurable spin density ortorque. As a result, the ”Hall angle” θ sh inferred fromEq. (3), relating the measured torque to a hypotheti-cal SHE spin current, should not be understood in theoriginal sense of the term ”Hall angle” but rather as aneffective experimental parameter providing a simple, andtherefore rather vague, characterization of the charge-to-spin conversion efficiency in a given structure. For similarreasons, the spin current approach has not been appliedfor the systematic crystal and magnetization symmetryanalysis of the series of SOT terms identified in experi-ment. From now on, to avoid unnecessary confusion weuse ξ to designate the charge-to-spin conversion efficiency(see Section IV.A) and θ sh in the specific context of SHE. D. Spin-orbit torques in antiferromagnets
For antiferromagnets, the STT or SOT phenomenologyis modified by considering a current-induced spin den-sity at a particular atomic site that tends to produce atorque which acts locally on the magnetic moment cen-tered on that site (Gomonay and Loktev, 2014; Jung-wirth et al. , 2016; MacDonald and Tsoi, 2011; ˇZelezn´y et al. , 2014). In analogy to ferromagnets, the localtorques acting on the a -th antiferromagnetic sublatticemagnetization, M a , have a field-like component of theform T a = M a × B a , with B a ∼ ζ a , and a damping-likecomponent T a = M a × B (cid:48) a , with B (cid:48) a ∼ M a × ζ a , respec-tively. Note that in a rigorous systematic theory, these0and all other torque terms acting in an antiferromag-net can be again obtained from the linear response ex-pressions in which the coefficients of the magnetization-expansion of χ T,a (or χ S,a ) reflect local crystal symme-tries of the a -th antiferromagnetic sublattice (ˇZelezn´y et al. , 2017). Assuming a collinear antiferromagnet, twomodel scenarios can be considered for the field-like anddamping-like SOTs: One with ζ = ζ = ζ and the otherone with ζ = − ζ .The former case corresponds, e.g., to injection of uni-formly polarized carriers from an external reference fer-romagnet, from a nonmagnetic SHE material, or to thegeneration of a uniform spin density at a nonmagneticmetal/antiferromagnetic metal (NM/AF) interface byiSGE [see e.g. (Manchon, 2017)]. The field-like torque inthe antiferromagnet would then be driven by a uniformnon-staggered effective field B = B ∼ ζ , i.e., wouldbe equally inefficient in switching an antiferromagnet asa uniform external magnetic field acting on an antifer-romagnet. On the other hand, the local non-equilibriumeffective field, B (cid:48) a ∼ M a × ζ , driving the damping-liketorque has an opposite sign on the two spin sublatticessince M = − M . This staggered effective field cantsthe magnetizations of the two sublattices and triggersthe dynamics of the antiferromagnetic order resultingin current-driven switching and excitations, somewhatsimilar to what is obtained in ferromagnets subject todamping-like torque (Cheng et al. , 2016; Gomonay andLoktev, 2010; Khymyn et al. , 2017).The microscopic realization of the second scenario inwhich ζ = − ζ is illustrated in Fig. 7 (Ciccarelli et al. ,2016; Jungwirth et al. , 2016; ˇZelezn´y et al. , 2014). It isthe staggered counterpart of the uniform iSGE spin den-sity discussed above. As mentioned in the previous sec-tion, iSGE only exists in non-centrosymmetric systems.For instance, the unit cell of zinc-blende GaAs [Fig. 7(a)]lacks a center of inversion, enabling an electrical currentto induce a non-equilibrium uniform spin density in thebulk crystal. In contrast, the related diamond latticeof, e.g., Si [Fig. 7(b)] has global inversion symmetry andtherefore cannot promote a net iSGE spin-density whenintegrated over the unit cell. However, the two identicalatoms in the unit cell sitting on the inversion partner siteshave locally non-centrosymmetric environments. As a re-sult, the diamond lattice is an example where the iSGEcan generate local non-equilibrium spin density with op-posite sign and equal magnitude on the two inversion-partner atoms while the global spin density integratedover the whole unit cell vanishes. Here a uniform elec-trical current induces a non-equilibrium staggered spindensity in the bulk crystal.In Si there is no equilibrium antiferromagnetic or-der that could be manipulated by these local staggerednon-equilibrium spin densities. However, antiferromag-nets like CuMnAs shown in Fig. 7(c), share the crys-tal symmetry allowing for the current-induced staggered FIG. 7 (Color online) (a) Global uniform non-equilibriumspin density generated by electrical current in a nonmagneticlattice with global inversion-asymmetry (e.g. GaAs) due tothe iSGE. (b) Local staggered, antiferromagnetic-like non-equilibrium spin density in a nonmagnetic lattice with localinversion-asymmetry (e.g. Si) due to the iSGE. Red dot showsthe inversion-symmetry center of the Si lattice. The two Siatoms on either side of the center occupy inversion-partnerlattice sites with locally asymmetric environments. In GaAslattice, the inversion-symmetry center is absent since the twoinversion-partner sites in the unit cell are occupied by differentatoms. (c) Local staggered non-equilibrium spin density in-ducing a local staggered effective field in an antiferromagneticlattice with local inversion-asymmetry (e.g. CuMnAs). Thinarrows represent the current-induced staggered effective fieldand thick arrows the antiferromagnetic moments. Adaptedfrom Jungwirth et al. (2016). spin density whose sign alternates between the inversion-partner atoms. Moreover, one inversion-partner latticesite is occupied by the magnetic atom belonging to thefirst antiferromagnetic spin sublattice and the other in-version partner is occupied by the magnetic atom be-longing to the second spin sublattice. As a result, thecorresponding field-like N´eel SOT can reorient antiferro-magnetic moments with an efficiency similar to the re-orientation of ferromagnetic moments by an applied uni-form field. This scenario has been confirmed experimen-tally in CuMnAs and Mn Au memory devices (Bodnar et al. , 2018; Meinert et al. , 2018; Wadley et al. , 2016;Zhou et al. , 2018).
E. Spin-orbit torques in topological materials
The distribution of spin texture in momentum spaceis a crucial ingredient to understand SOT. In semicon-ductor materials where the iSGE and SHE were initiallydiscovered, and even more in metal structures, multiplebands cross the Fermi level and their respective contribu-tions to the current-induced spin density tend to compen-1sate each other. Also the spin-textures are more complex,which can further reduce the net effect.
FIG. 8 (Color online) Charge current-induced surface spindensity in topological insulator. (a) Schematic illustration ofthe spin-momentum locked helical spin texture of the surfacestates in topological insulator: clockwise spin texture abovethe Dirac point while anticlockwise spin texture below theDirac point. (b) Schematic of surface spin density on twoopposite surfaces for a charge current flowing along − x direc-tion (i.e., I x <
0, yellow thick arrow) and for a charge currentflowing along + x direction (i.e., I x >
0, green thick arrow).
From this perspective, topological insulators (Hasanand Kane, 2010; Pesin and MacDonald, 2012b) are re-garded as optimal materials for the SOT. The surfacestates of a three-dimensional (3D) topological insulatorform a Dirac cone with a single Fermi surface and a he-lical locking of the relative orientations of the spin andthe momentum [see Fig. 8(a)]. Indeed, SOT-FMR mea-surements in a metallic ferromagnet interfaced with atopological insulator showed an exceptionally large spinconversion efficiency ξ (Mellnik et al. , 2014). However,compared to common nonmagnetic metals, the increaseof ξ in the studied topological insulators turned out tobe primarily due to its decreased electrical conductiv-ity while the inferred effective spin-Hall conductivity wassimilar to the nonmagnetic metals (see Table II).Interfacing a highly resistive topological insulator witha low resistive metal FM has also a practical disadvan-tage that most of the applied electrical current is shuntedthrough the metallic magnet and does not contribute tothe generation of the spin density at the topological in-sulator surface. A possible remedy is in using an insu-lating magnet. An example is a study of highly efficientmagnetization switching at cryogenic temperatures in atopological insulator/magnetic topological insulator bi-layer, in which the inferred spin conversion efficiency ξ was three orders of magnitude larger than in nonmag-netic metals (Fan et al. , 2014b; Fan and Wang, 2016).In the above studies, Dirac quasiparticles exhibitingstrong spin-momentum locking are considered to enhancethe efficiency of the SOT control of magnetic moments. Vice versa , a scheme has been recently proposed for the electric control of Dirac band crossings by reorientingmagnetic moments via SOT (ˇSmejkal et al. , 2017). In-stead of 2D surface states of a topological insulator, thesepredictions consider Dirac bands in the bulk of a topo-logical 3D semimetal. Since Dirac bands can only existin systems with a combined space-inversion and time-inversion ( PT ) symmetry, ferromagnets are excluded.On the other hand, the combined PT -symmetry in anantiferromagnet is equivalent to a magnetic crystal sym-metry in which antiferromagnetic spin sublattices occupyinversion-partner lattice sites. This in turn allows for anefficient SOT, as discussed in the previous section. F. Inverse effect of the spin-orbit torque
The Onsager reciprocity relations imply that there isan inverse phenomenon to the SOT, which we call thespin-orbit charge pumping (Hals et al. , 2010; Kim et al. ,2012a; T¨olle et al. , 2017). The underlying physics of thespin-orbit charge pumping generated from magnetizationdynamics is the direct conversion of magnons into chargecurrents via spin-orbit coupling, as illustrated on Fig. 9.This effect evolves from the spin pumping predicted byBrataas et al. (2002); Tserkovnyak et al. (2002b) whenSOC is included, either in the bulk of the nonmagneticmetal or at the interface. Thus, any external force thatdrives magnetization precession can generate spin-orbitcharge pumping. Similarly to the SOT, two model micro-scopic mechanisms can be considered for the spin-orbitcharge pumping: one due to the inverse effect of the iSGE(Ciccarelli et al. , 2014; Rojas-S´anchez et al. , 2013), calledthe spin galvanic effect (SGE), and the other one due tothe inverse SHE (Saitoh et al. , 2006). Together with thenon-local detection in a lateral structure (Valenzuela andTinkham, 2006), the spin-orbit charge pumping acrossthe NM/FM interface provided the first experimentaldemonstration of the inverse SHE (Saitoh et al. , 2006).Since then it has evolved into one of the most commontools for electrical detection of magnetization dynamics.
III. THEORY OF SPIN-ORBIT TORQUES
In this section we review the progress that has beenmade towards the theoretical understanding of SOTs inboth layered heterostructures and bulk materials. Themost general treatment of the SOT that has been con-sidered so far is based on the (spin-)density functionaltheory, in which the system is described by a Hamilto-nian for non-interacting particlesˆ H = ˆ K + ˆ V eff ( r ) + ˆ σ · Ω xc ( r ) + ˆ H so , (4)where ˆ K is the kinetic energy, ˆ V eff is the effective crystalpotential, Ω xc is the exchange-correlation field, and ˆ H so FIG. 9 (Color online) A spin current is generated by spin-pumping at the NM/FM interface (grey arrows). The timedependent spin density σ ( t ) of this current (indicated as adark grey arrow) rotates almost entirely in the y − z plane.The small time averaged dc component (yellow arrow) ap-pears along the x axis. Both components lead to charge cur-rents in the nonmagnetic metal and can be converted into acand dc voltages, U ( t ) and U DC , by placing probes along the x and y direction, respectively. From Wei et al. (2014). is the spin-orbit coupling. Assuming this form of theHamiltonian, the torque on magnetization at point r isgiven by (Haney et al. , 2008; Manchon and Zhang, 2011) T ( r ) = − S ( r ) × Ω xc ( r ) , (5)where S = (1 /V ) (cid:104) ˆ σ (cid:105) is the current-induced spin density,and V is the volume of the unit cell. This equation isvalid for the STT as well as for the SOT. When spin-orbit coupling is neglected, the torque can be equivalentlyexpressed as a divergence of a spin current (Ralph andStiles, 2008) T i ( r ) = − ∇ · J i s , (6)where J i s = ( (cid:126) / (cid:104){ ˆ σ i , ˆ v }(cid:105) is a vector representing the i -th spin component of the spin current tensor. The j -thcomponent of the vector J i s is noted j j s ,i and denotes aspin current polarized along the i -th direction and prop-agating along the j -th direction. In the absence of spin-orbit coupling, the torque is thus directly given by theabsorption of the spin current. However, when spin-orbitcoupling is not neglected, the spin angular momentum isnot a conserved quantity and the spin current in Eq. (6)is then not uniquely defined, while Eq. (5) remains valid.The total torque is obtained by integrating Eq. (5)over the whole unit cell, and a local torque is obtainedby integrating over a particular magnetic atom. Thistorque can then be inserted into LLG equation to evalu-ate the magnetic dynamics induced by the SOT. Whenusing this approach it is necessary to ensure that thedynamics of the non-equilibrium carrier spins is muchfaster than the dynamics of magnetic moments arisingfrom equilibrium electrons; otherwise the dynamics ofthe two could not be separated. This is well justified in ferromagnets whose magnetization dynamics lies in theGHz range but it could become an issue when discussingantiferromagnets whose dynamics can reach several THz.We also note that Eq. (5) assumes that Eq. (4) accu-rately describes the electronic system. This is reasonablefor most materials of interest, namely metals, but failsin strongly correlated systems. In these systems, moresophisticated many-body approaches are necessary. Sofar SOTs have been studied only using non-interactingmodel (free electron or k.p ) Hamiltonians or Kohn-ShamHamiltonians originating from density functional theory.At weak applied electric fields, the SOT is well de-scribed by linear response theory, T = χ T E , where theresponse tensor χ T can be calculated using Eq. (5).Equivalently, the torque can be rewritten as, T = M × B T , with the effective field obtained from the linear re-sponse expression, B T = χ B E . In many calculations ofthe SOT, especially those based on model Hamiltonians,an approximation is used in which the effective magneticfield is made directly proportional to the current-inducedspin density, B T ≈ − ∆ S /M s . Here ∆ is an exchangecoupling energy corresponding to exchange between thecarrier spins and magnetic moments, and S is again eval-uated using linear response, S = χ S E .As discussed in Section II, the origin of the SOT in thebilayer systems is often attributed to two different effects,the SHE and the iSGE, where the SHE-SOT is assumedto originate from the absorption of a spin current gener-ated in the nonmagnetic metal [see Fig. 10(a)] and theiSGE-SOT is due to spin density generated locally in theferromagnet or at the interface [see Fig. 10(b)]. Equa-tion (5) shows however that the torque always originatesfrom a current-induced spin density. Thus the SHE-SOTcan be more fundamentally understood not in terms ofthe absorption of a spin current but in terms of a spindensity induced by the spin Hall current. Consequently,both contributions can be treated on the same footingand there is no clear way how to theoretically separatethem.Still, it is intuitively appealing to separate the totaltorque into a contribution associated with the absorptionof a spin current, as given by Eq. (6), and a contributiondue to a locally generated spin density, described by Eq.(5). One could then attribute the former contributionto SHE and the latter one to iSGE. However, such anapproach has several drawbacks. First, spin currents arenot necessarily due to the bulk SHE alone and substantialcontributions can also come from the interface with theferromagnet (Amin and Stiles, 2016a,b; Ghosh and Man-chon, 2018; Kim et al. , 2017b; Wang et al. , 2016b). Sec-ond, even in bulk non-centrosymmetric materials whereSOT is considered of purely iSGE origin, local spin cur-rents within the unit cell can contribute to the torque.Third, in a slab geometry, interface and bulk are not welldefined notions, and the terminology of what should bereferred to as iSGE or SHE becomes unclear (Freimuth3 et al. , 2014b). Conventionally, iSGE refers to spin den-sity generated internally in the material. However, eventhe spin density induced by SHE in the bilayers couldbe referred to as iSGE, since it is also a spin densityinduced by a charge current. In conclusion, althoughmodels based on bulk SHE or iSGE due to interfacialRashba spin-orbit coupling can be useful to explain someaspects of the experiments, in real systems there is notmuch point in trying to rigorously parse the torque intothese two contributions.In many experimental studies, the origin of the SOTis analyzed in terms of its symmetries. The damping-liketorque T DL is often referred to as spin Hall or Slon-czewski torque , and the field-like torque T FL as spin-orbit or Rashba torque . This is based primarily on theassumption that any torques associated with transferand absorption of spin-angular momentum would havedissipative-like character and the one arising from theiSGE would be primarily of field-like character. Thisis however not the case, as interband transitions, spin-dependent scattering, spin relaxation, spin precessionand size effects significantly complicate the SOT scenario.Hence, symmetry considerations alone cannot disentan-gle directly the two contributions. However, symmetryanalysis remains a powerful tool. SOTs obey Neumann’sprinciple and must be invariant under the symmetry op-erations of the material system. This can restrict signifi-cantly the forms of the response coefficients, and aids theformulation of the proper phenomenological descriptionof the SOTs, reflecting the underlying crystal symmetryof the considered material or structure (Hals and Brataas,2013a; Wimmer et al. , 2016; ˇZelezn´y et al. , 2017).This section is organized as follows. We review the lin-ear response formalism commonly used for microscopiccalculations of the SOT in Subsection III.A, and the gen-eral symmetry properties of SOT are then discussed inSubsection III.B. Because of the great challenge of in-corporating the full complexity of the bilayer systems atonce, almost all theoretical studies have been focused ei-ther on iSGE in model systems or on the SHE mechanismonly, with a handful of them attempting a comprehensivemodeling. In Subsection III.C we review calculations ofthe SOT in bilayer systems based on the SHE mechanism.In Subsection III.D we review calculations of the SOT inbulk systems which includes the 2D Rashba model and3D non-centrosymmetric materials. Microscopic calcula-tions carried out using density functional theory calcu-lations for bilayer structures are presented in SubsectionIII.E. In Subsection III.F we review calculations of theSOT in bulk antiferromagnets, and a discussion of SOTsin topological insulators and other systems is presentedin Subsections III.G and III.H, respectively.
FIG. 10 (Color online) Two main model spin-charge conver-sion mechanisms at NM/FM interface: (a) iSGE and (b) SHE.Both mechanisms produce damping-like and field-like torques.The small red and blue arrows denote the non-equilibriumspin density accumulating at the interfaces, and their corre-sponding spatial distribution is sketched as a shaded area onthe structure’s side. The large red and blue arrows representthe field-like and damping-like torques, respectively.
A. Kubo linear response: intraband versus interbandtransitions
From a microscopic linear response perspective, basicquantum mechanics states that the statistical average ofan operator ˆ O reads O = (cid:80) n, k (cid:104) n, k | ˆ O| n, k (cid:105) f n, k , where f n, k is the carrier distribution function and | n, k (cid:105) is thequantum eigenstate of the system. Under a small pertur-bation, such as an external electric field, both the distri-bution function f n, k and the eigenstates | n, k (cid:105) are modi-fied, giving rise to different nonequilibrium contributionsto the observable O as consistently modeled by quantumfield theory (Mahan, 2000; Rammer and Smith, 1986).Within the constant relaxation time approximation, thedistribution function and eigenstates become f n, k → f n, k − τ (cid:104) n, k | e E · ˆ v | n, k (cid:105) ∂∂(cid:15) f n, k , (7) | n, k (cid:105) → | n, k (cid:105) − (cid:88) n (cid:48) (cid:104) n (cid:48) , k | e E · ˆ r | n, k (cid:105) (cid:15) n, k − (cid:15) n (cid:48) , k | n (cid:48) , k (cid:105) , (8)where f n, k is the Fermi-Dirac distribution, E is the elec-tric field, ˆ v and ˆ r are the velocity and position operators, (cid:15) n, k is the eigenenergy associated with the unperturbedeigenstate | n, k (cid:105) , and e > O = O Intra + O Inter , where O Intra = − τ (cid:88) n, k Re (cid:104) n, k | e E · ˆ v | n, k (cid:105)(cid:104) n, k | ˆ O| n, k (cid:105) ∂∂(cid:15) f n, k , (9) O Inter = − (cid:126) (cid:88) n,n (cid:48) , k Im (cid:104) n, k | e E · ˆ v | n (cid:48) , k (cid:105)(cid:104) n (cid:48) , k | ˆ O| n, k (cid:105)× ( f n, k − f n (cid:48) , k )( (cid:15) n, k − (cid:15) n (cid:48) , k ) . (10)The first contribution, Eq. (9), is proportional to therelaxation time ∼ τ and only involves intraband tran-sitions, | n, k (cid:105) → | n, k (cid:105) . The second one, Eq. (10), isweakly dependent on disorder and sometimes called in-trinsic . It only involves interband transitions | n, k (cid:105) →| n (cid:48) , k (cid:105) . The intrinsic contribution can be related to theBerry curvature of the material that connects intrinsictransport properties to the topology of the phase space(Sinova et al. , 2015; Xiao et al. , 2010). Equations (9),(10) are valid only under the assumption of a constantand large relaxation time. More generally, the linear re-sponse can be expressed in terms of the Kubo-Bastinformula (Freimuth et al. , 2014b; Wimmer et al. , 2016) O = O I ( a ) + O I ( b ) + O II , (11) O I ( a ) = eh (cid:90) ∞−∞ dε ∂∂ε f ε Tr (cid:104) ˆ O ˆ G Rε ( E · ˆ v ) ˆ G Aε (cid:105) c , (12) O I ( b ) = − eh (cid:90) ∞−∞ dε ∂∂ε f ε ReTr (cid:104) ˆ O ˆ G Rε ( E · ˆ v ) ˆ G Rε (cid:105) c , (13) O II = eh (cid:90) ∞−∞ dεf ε ReTr (cid:104) ˆ O ˆ G Rε ( E · ˆ v ) ∂∂ε G Rε − ˆ O ∂∂ε ˆ G Rε ( E · ˆ v ) ˆ G Rε (cid:105) c , (14)where ˆ G R ( A ) ε denotes the retarded (advanced) Green’sfunction respectively and (cid:104) ... (cid:105) c denotes an average overdisorder configurations. Tr is the trace over spin, mo-mentum and orbital spaces. For concreteness, the oper-ator ˆ O is simply the spin operator ˆ σ or the spin currentoperator J i s defined above. This formula is often sim-plified by assuming that the only effect of disorder is toinduce a constant energy broadening Γ = (cid:126) / τ , such thatˆ G R ( A ) ε = (cid:126) ( ε − ˆ H ± i Γ) − . In the limit of large relaxationtime, Γ → et al. , 2015). Thus fora complete treatment, more sophisticated approaches arenecessary. B. Symmetry of spin-orbit torques
As mentioned above, the torque can always be rewrit-ten in terms of an effective magnetic field B T , T = M × B T . The symmetry of the SOT can be studied eitherin terms of the linear response tensor χ T or, equivalently,in terms of χ B . Here we focus on the effective field sinceits symmetry relations are simpler. In terms of symme-try, the effective magnetic field is equivalent to the iSGE,i.e., the tensors χ B and χ S have the same form (althoughthey are not necessarily proportional, as often assumed inmodel calculations). To understand the symmetry prop-erties of the SOT, it is convenient to parse the effectivefield into two parts, even and odd under time-reversal(or, equivalently, under the reversal of all magnetic mo-ments). This is similar to the case of conductivity inmagnetic systems (Grimmer, 1993). However, unlike forconductivity, the even and odd parts do not correspondto the symmetric and anti-symmetric parts of the effec-tive field tensor. Thus a separate linear response tensorhas to be assigned to each part, B eveneff = χ even B E , (15) B oddeff = χ odd B E . (16)The same parsing can also be done for the torque. Wenote that the odd part of the torque corresponds to theeven part of the effective field and vice versa . Notice-ably, the odd and even parts have very different proper-ties and correspond to different contributions of the Kuboformula: the intraband formula, Eq. (9), corresponds tothe even field, whereas the interband formula, Eq. (10),corresponds to the odd field. Similar separation can bedone for the full Kubo-Bastin formula (Freimuth et al. ,2014b). Furthermore, such a separation is also commonlydone for experimental measurements of SOT (see Sec-tions IV and VI). Since the following applies equally to χ B and χ S we denote the tensor simply by χ . Follow-ing the Neumann’s principle, the tensors χ have to beinvariant under all symmetry operations of the crystal.The two parts transform differently for symmetry opera-tions that contain time-reversal symmetry. For a symme-try operation represented by a matrix R (ˇZelezn´y et al. ,2017), χ even = det( R ) R χ even R − , (17) χ odd = ± det( R ) R χ odd R − , (18)where ± refers to a symmetry operation with and withouttime-reversal, respectively, and det( R ) is the determinantof R . By considering all the symmetry operations in themagnetic point group of the given crystal, the generalform of the response tensors is found from these equa-tions. It is also possible to treat the whole tensor to-gether without separating it into the even and odd parts,although then some information about the structure ofthe torque is lost. See (Wimmer et al. , 2016) for a tableof total χ T tensors for all the magnetic point groups.In systems with more than one magnetic atom in theunit cell, such as antiferromagnets, it is furthermore use-ful to study the symmetry of SOT on each magnetic site.5Then Eqs. (17), (18) are modified as follows (ˇZelezn´y et al. , 2017), χ even a (cid:48) = det( R ) R χ even a R − , (19) χ odd a (cid:48) = ± det( R ) R χ odd a R − , (20)where a denotes a given site and a (cid:48) is the site to which site a transforms under symmetry operation R . In this case itis necessary to consider the full magnetic space group andatomic positions of magnetic moments. The symmetry of χ a is determined by symmetry operations that leave site a invariant (such symmetry operations form the so-calledsite symmetry group), whereas the symmetry operationsthat transform a to a different site a (cid:48) relate tensor χ a totensor χ a (cid:48) .A key conclusion that can be made from Eqs. (17), (18)is that there can be no net SOT (or iSGE) if the systemhas inversion symmetry. However, from Eqs. (19), (20)we see that even in a system with inversion symmetrythere can still be a local SOT if the inversion symmetry isbroken locally, i.e., there can be SOT on site a , if there isno inversion symmetry which would leave site a invariant.To understand the dependence of SOT on the directionof magnetic moments, it is helpful to expand the SOTin the direction of magnetic moments. For a collinearmagnetic material, χ ij ( n ) = χ (0) ij + χ (1) ij,k n k + χ (2) ij,kl n k n l + . . . , (21)where n is the magnetic order parameter (the magnetiza-tion direction in ferromagnets, or the N´eel order param-eter in antiferromagnets). The even terms in the expan-sion correspond to the even effective field and converselythe odd terms correspond to the odd field. The symme-try of the n -independent expansion tensors in Eq. (21) isdetermined by the symmetry group of the nonmagnetic system. For a global SOT in a ferromagnet or a localSOT in a bipartite antiferromagnet the following trans-formation rule is found for the expansion tensors, χ ( ν ) ij,mn... = det( R ) ν − R ik R − Tjl R − Tmo R − Tnp . . . χ ( ν ) kl,op... , (22)For the global case, the nonmagnetic point group has tobe used, whereas for the local case, the nonmagnetic sitesymmetry group has to be used instead. Since there areonly 21 nonmagnetic point groups with broken inversionsymmetry, it is feasible to calculate all allowed leadingterms of the expansion (21). This was done for the ze-roth, first, and some second order terms in Refs. (Cic-carelli et al. , 2016; ˇZelezn´y et al. , 2017). The results forthe zeroth and first order terms are given in Table I.The lowest order even field is typically given by χ (0) ,which corresponds to a field-like torque. In some casessuch a term is, however, prohibited by symmetry andthe lowest order even field is second order in magnetiza-tion. This is the case of the cubic zinc-blende or half-heusler crystal ferromagnets with space group F ¯43m, for instance. Under strain, however, these materials ex-hibit an even field at the zeroth order in magnetization(in other words, a field-like torque) (Chernyshov et al. ,2009). In contrast, MnSi and its parent compounds adoptthe P FIG. 11 (Color online) Various types of the field-like torquesas a function of the electric field direction. The red arrows de-note the corresponding effective field direction for (a) Rashba,(b) Dresselhaus, (c) generalized Rashba, (d) generalized Dres-selhaus, and (e) Weyl coupling schemes.
More generally, the connection between the SOT fieldand the applied electric field can be categorized in threedifferent types illustrated in Fig. 11: Rashba and Dres-selhaus coupling schemes, and a coupling such that theSOT field is collinear to the electric field that we refer toas Weyl coupling [Fig. 11(d)]. These denominations aretaken in analogy with the spin-textures in the momen-tum space of the Rashba, Dresselhaus and Weyl spin-orbit coupling further discussed in Subsection III.D. TheRashba and Dresselhaus fields are confined to a plane andonly appear for electric field lying in the plane. They dif-fer in how the effective field is changed when the electricfield is rotated. In the case of standard Rashba coupling,the effective field rotates in the same direction as theelectric field [Fig. 11(a)], whereas in the case of standardDresselhaus coupling the effective field rotates in the op-posite direction [Fig. 11(b)]. The generalized Rashbacoupling differs from a conventional Rashba coupling inthat the angle between the electric and effective field isnot necessarily 90 ◦ [Fig. 11(c)]. The generalized Dres-selhaus coupling differs from the conventional Dressel-haus coupling in that the effective field is not necessarilyparallel or perpendicular to the electric field along thecrystalline axes [Fig. 11(d)].As seen in Table I, χ (1) has always some non-zero com-ponents. These generate the lowest order odd field. Itoften has a damping-like character, i.e., can be written as B T ∼ m × ζ , where ζ is a vector independent of magne-tization. However in some cases the first-order field doesnot have the damping-like form. An example of a systemwhere no damping-like torque is allowed by symmetryis again cubic zinc-blende or half-heusler crystals. Evenif the damping-like torque is allowed by symmetry therecan be other first-order contributions. Magnetic dynam-ics induced by such torques can differ from the effect of6a damping-like torque and has not been studied so far. Crystal system Point group χ (0) χ (1) triclinic 1 x x x x x x x x x ˆ n x x + ˆ n y x + ˆ n z x ˆ n x x + ˆ n y x + ˆ n z x ˆ n x x + ˆ n y x + ˆ n z x ˆ n x x + ˆ n y x + ˆ n z x ˆ n x x + ˆ n y x + ˆ n z x ˆ n x x + ˆ n y x + ˆ n z x ˆ n x x + ˆ n y x + ˆ n z x ˆ n x x + ˆ n y x + ˆ n z x ˆ n x x + ˆ n y x + ˆ n z x monoclinic 2 x x x x x ˆ n y x ˆ n x x + ˆ n z x ˆ n y x ˆ n x x + ˆ n z x ˆ n y x ˆ n x x + ˆ n z x ˆ n y x ˆ n x x + ˆ n z x ˆ n y x m x x x x ˆ n x x + ˆ n z x ˆ n y x ˆ n x x + ˆ n z x ˆ n y x ˆ n x x + ˆ n z x ˆ n y x ˆ n x x + ˆ n z x ˆ n y x ˆ n x x + ˆ n z x orthorhombic 222 x x
00 0 x n z x ˆ n y x ˆ n z x n x x ˆ n y x ˆ n x x mm2 x x ˆ n z x n x x n z x ˆ n y x ˆ n x x ˆ n y x ˆ n z x tetragonal 4 x − x x x
00 0 x ˆ n z x − ˆ n z x ˆ n x x − ˆ n y x ˆ n z x ˆ n z x ˆ n x x + ˆ n y x ˆ n x x − ˆ n y x ˆ n x x + ˆ n y x ˆ n z x -4 x x x − x
00 0 0 ˆ n z x ˆ n z x ˆ n x x + ˆ n y x ˆ n z x − ˆ n z x ˆ n x x − ˆ n y x ˆ n x x + ˆ n y x ˆ n x x − ˆ n y x x x
00 0 x − ˆ n z x − ˆ n y x ˆ n z x n x x − ˆ n y x ˆ n x x − x x ˆ n z x n x x n z x ˆ n y x ˆ n x x ˆ n y x ˆ n z x -42m x − x
00 0 0 n z x ˆ n y x ˆ n z x n x x ˆ n y x ˆ n x x trigonal 3 x − x x x
00 0 x ˆ n x x + ˆ n y x + ˆ n z x ˆ n x x − ˆ n y x − ˆ n z x ˆ n x x − ˆ n y x ˆ n x x − ˆ n y x + ˆ n z x − ˆ n x x − ˆ n y x + ˆ n z x ˆ n x x + ˆ n y x ˆ n x x − ˆ n y x ˆ n x x + ˆ n y x ˆ n z x x x
00 0 x ˆ n y x ˆ n x x − ˆ n z x − ˆ n y x ˆ n x x + ˆ n z x − ˆ n y x ˆ n x x − ˆ n y x ˆ n x x − x x ˆ n y x + ˆ n z x ˆ n x x ˆ n x x ˆ n x x − ˆ n y x + ˆ n z x ˆ n y x ˆ n x x ˆ n y x ˆ n z x hexagonal 6 x − x x x
00 0 x ˆ n z x − ˆ n z x ˆ n x x − ˆ n y x ˆ n z x ˆ n z x ˆ n x x + ˆ n y x ˆ n x x − ˆ n y x ˆ n x x + ˆ n y x ˆ n z x -6 ˆ n x x + ˆ n y x ˆ n x x − ˆ n y x n x x − ˆ n y x − ˆ n x x − ˆ n y x
00 0 0 x x
00 0 x − ˆ n z x − ˆ n y x ˆ n z x n x x − ˆ n y x ˆ n x x − x x ˆ n z x n x x n z x ˆ n y x ˆ n x x ˆ n y x ˆ n z x -6m2 ˆ n y x ˆ n x x n x x − ˆ n y x
00 0 0 cubic 23 x x
00 0 x n z x ˆ n y x ˆ n z x n x x ˆ n y x ˆ n x x x x
00 0 x − ˆ n z x ˆ n y x ˆ n z x − ˆ n x x − ˆ n y x ˆ n x x -43m n z x ˆ n y x ˆ n z x n x x ˆ n y x ˆ n x x TABLE I: Zeroth and first order terms in the expansion (21) for the point groupswith broken inversion symmetry. The tensors χ (1) have the spin-axis directionincluded: χ (1) ij = χ (1) ij,k ˆ n k . The x parameters can be chosen arbitrarily for eachtensor. The tensors are given in cartesian coordinate systems defined in (ˇZelezn´y et al. , 2017). The lowest order terms frequently describe qualitative aspects of the torque both in experiments and in the7theoretical calculations. The usefulness of the lowest or-der term is illustrated by the fact that materials withvery different electronic structures but same symmetryhave very similar SOTs. For instance this is the case offerromagnetic (Ga,Mn)As and NiMnSb, or systems mod-eled by the 2D Rashba Hamiltonian, or antiferromagnetsMn Au and CuMnAs, discussed in Section III.F. For anaccurate quantitative description of the SOT, higher or-der terms can be important. These are not tabulated butcan be produced by the publicly available code that wasused for generating Table I (Zelezn´y, 2017). This codecan be also used to determine the full tensors χ even a and χ odd a for a given crystal. For the case of an interface withinversion symmetry breaking only (e.g., in the case of aRashba 2D gas), one obtains χ (0) = x − , χ (1) = x − ˆ n z n z n x ˆ n y (23) C. Spin-orbit torques due to the spin Hall effect
The SHE-SOT contribution in bilayer systems arisesfrom the absorption of angular momentum coming froma SHE spin current generated outside the ferromagnet, e.g., in the proximate nonmagnetic metal layer (Dyakonovand Perel, 1971). This is effectively the mechanism ofSTT where the polarizing ferromagnet in a trilayer de-vice is replaced in this instance by the nonmagnetic metal(Brataas et al. , 2012b; Stiles and Zangwill, 2002). Inanalogy to STT, the SHE-SOT mechanism in commonmetal structures is primarily damping-like in character,assuming a full absorption of the carrier spin angularmomentum in the ferromagnet. Therefore in many ex-periments, the damping-like SOT is associated with SHE,and the extracted spin Hall angle is calculated on the ba-sis that this is the only contribution to the damping-likeSOT component. Since this is generally not the case, thespin Hall angle values extracted from these experimentsshould be considered only as effective phenomenologicaldescriptions of the SOT efficiency. On the other hand, inmany experiments a clear correlation between the mag-nitude and sign of SHE, e.g., obtained by non-local mea-surements (Morota et al. , 2011), and damping-like SOTis observed. We do not review here the calculations ofthe SHE, which has been done elsewhere (Sinova et al. ,2015), and focus instead on effective theoretical treat-ments using the spin Hall angle as a phenomenologicalparameter.The SHE-SOT is present in structures where the ferro-magnet is adjacent to a nonmagnetic (Pt, W, Ta, WTe ,conductive Bi Se , etc.) or magnetic metal (IrMn, PtMnetc.). To model this torque, one needs to compute thespin density originating from this metal and diffusing intothe ferromagnet. The simplest method is to solve the drift-diffusion equation in the presence of spin-orbit cou-pling and match the spin currents and accumulations atthe boundary between the ferromagnet and the nonmag-netic metal using, for instance, the spin mixing conduc-tance (Amin and Stiles, 2016b; Chen et al. , 2015b; Haney et al. , 2013b). The charge and spin currents in a non-magnetic metal with spin-orbit coupling read (Dyakonovand Perel, 1971; Pauyac et al. , 2018; Shchelushkin andBrataas, 2005; Shen et al. , 2014) j c /σ N = − ∇ µ c + θ sh ∇ × µ , (24)(2 e/ (cid:126) ) J i s /σ N = − ∇ µ i − θ sh e i × ∇ µ c − θ sw ∇ × ( e i × µ ) , (25)where σ N is the bulk conductivity and θ sw is thespin swapping coefficient (Lifshits and Dyakonov, 2009;Pauyac et al. , 2018) (we comment on the spin swap-ping term in more detail at the end of this subsection). µ c = n/e N and µ = S /e N are the charge and spinchemical potentials, respectively, with N the density ofstates at the Fermi level. Formally, the drift-diffusionapproach for the current-in-plane geometry is only ap-plicable as long as the mean free path is much shorterthan the layer thickness and assuming uniform spin Hallangle and conductivity in the nonmagnetic metal. Thismodel also neglects interfacial spin-flips, or spin-memoryloss (Bass and Pratt, 2007; Belashchenko et al. , 2016;Dolui and Nikolic, 2017). As discussed further below,this assumption is not always accurate. Using the spinmixing conductance, g ↑↓ , as a boundary condition, thespin transfer arises from the absorption of the incomingtransverse spin current at the interface, T = J s /t F , t F being the thickness of the magnet. It is composed of twocomponents, as described in Eq. (2), which read (Haney et al. , 2013b) τ DL = (cid:126) θ sh e ˜ g ↑↓ r + | ˜ g ↑↓ | (1 + ˜ g ↑↓ r ) + ˜ g ↑↓ i (cid:18) − cosh − t N λ sf (cid:19) σ N E, (26) τ FL = − (cid:126) θ sh e ˜ g ↑↓ i (1 + ˜ g ↑↓ r ) + ˜ g ↑↓ i (cid:18) − cosh − t N λ sf (cid:19) σ N E, (27)Here, we omitted the spin swapping term in Eq. (25), and ζ (cid:107) z × j c , z being normal to the interface. We also definethe applied electric field E = − ∂µ c /∂x , and the reducedmixing conductance ˜ g ↑↓ = g ↑↓ λ sf / [ σ N tanh( t N /λ sf )],while t N and λ sf are the thickness and spin relax-ation length of the nonmagnetic metal, respectively. Fi-nally, ˜ g ↑↓ r and ˜ g ↑↓ i refer to the real part and imagi-nary part of ˜ g ↑↓ , respectively. In the limit of smallimaginary part of the mixing conductance, τ DL ∝ ηθ sh and τ FL ∝ ηθ sh g ↑↓ i /g ↑↓ r . The transparency coefficient η = g ↑↓ r / [ g ↑↓ r + σ N tanh( t N /λ sf ) /λ sf ] accounts for the spincurrent transmission through the interface, with η → g ↑↓ r (cid:29) σ N /λ sf . Equations (26), (27) assume that all8the spin current impinging on the interface is either re-flected back to the nonmagnetic metal or absorbed by theferromagnet. Allowing for spin-memory loss at the inter-face opens an additional spin dissipation channel (Bassand Pratt, 2007; Belashchenko et al. , 2016; Dolui andNikolic, 2017) and reduces the effective spin mixing con-ductance, leading to an underestimation of the spin Hallangle (Berger et al. , 2018b; Rojas-S´anchez et al. , 2014).These expressions, although quite extensively usedto interpret experimental data, must be handled withcare as they disregard any corrections emerging fromthickness-dependent conductivity, spin Hall angle, andspin-dependent scattering at interfaces, and assume thesimplest form of the interfacial spin mixing conductance.To overcome these limitations, several methods have beenemployed, such as the Boltzmann transport equationwhere SHE is explicitly contained in the collision inte-gral (Amin and Stiles, 2016b; Engel et al. , 2005; Haney et al. , 2013b), the Kubo formula with real-space Green’sfunction in a slab geometry (Chen and Zhang, 2017),the tight-binding model with random impurity poten-tial (Saidaoui and Manchon, 2016) or transport calcu-lations based on first principles (Dolui and Nikolic, 2017;Freimuth et al. , 2014b, 2015; Wang et al. , 2016b).Several additional features beyond the ”conventional”SHE model have been identified. First, because the thick-ness of the nonmagnetic metal is of the same order as themean free path (5-10 nm), the conductivity depends onthe thickness of the slab (Sondheimer, 1952). In fact, it iswell-known that spin transport in current-in-plane con-figuration is governed by mean free path effects, ratherthan by the spin diffusion length (Camley and Barnas,1989; Zhang and Levy, 1993). A direct consequenceis that the spin Hall current estimated using the drift-diffusion model, Eq. (26), (27), is generally overestimated (Chen and Zhang, 2017).A second important aspect was revealed by ab-initio calculations (Freimuth et al. , 2015; Wang et al. , 2016b).These studies suggest that the SHE itself can be signif-icantly enhanced close to the interface. The computedinterfacial Hall angle can be an order of magnitude largerthan the bulk spin Hall angle, and possibly dominate thetotal SHE signal. On the other hand, increasing the dis-order leads to a progressive reduction of the interfacialspin Hall angle (Freimuth et al. , 2015). Realistic model-ing based on ab-initio simulations are further discussedin Subsection III.E.Finally, it has been recently realized that the preces-sion of spin currents around the spin-orbit field can sub-stantially impact the SOT. This mechanism was origi-nally proposed by Lifshits and Dyakonov (2009) in a dif-ferent context and called spin swapping . In this mech-anism, a primary spin current with spin polarization J s ∼ σ precesses around the spin-orbit field B so , re-sulting in a secondary spin current J s ∼ σ × B so . Thissecondary spin current can be absorbed by an adjacent FIG. 12 (Color online) Spin precession around the spin-orbitfield. (a) Schematics of extrinsic spin swapping effect in abilayer composed of a nonmagnetic metal (blue) and a fer-romagnet (yellow) with magnetization m . The spin currentflowing from the ferromagnet into the nonmagnetic metal ispolarized along m and precesses about the impurity-drivenspin-orbit field normal to the scattering plane ζ . It producesa secondary spin current polarized along m × ζ (Saidaoui andManchon, 2016). (b) Ratio between the magnitude of thefield-like and damping-like torques τ DL /τ FL as a function ofdisorder strength and spin-orbit coupling in the extrinsic spinswapping scenario. The ratio is given in logarithmic scale andthe dashed line indicates τ DL = τ FL . Adapted from Saidaouiand Manchon (2016). ferromagnet, resulting in additional SOT components.Several flavors of this scenario have been proposed, de-pending on the source of the primary spin current and onthe nature of the spin-orbit field. Saidaoui and Manchon(2016) suggested that spin-polarized electrons scatteringoff the ferromagnetic layer inject a primary spin current J s ∼ m in the nonmagnetic metal. This primary spincurrent precesses around the spin-orbit field oriented nor-mal to the scattering plane, which produces a secondaryspin current J s ∼ m × ζ [Fig. 12(a)]. Once absorbedinto the ferromagnet, this spin swapping spin current in-duces a field-like SOT. This mechanism only survives aslong as the nonmagnetic metal thickness is comparableto the mean free path, as shown in Fig. 12(b).While the SHE-SOTs discussed above occur in bi-layers composed of a spin current source (e.g, a heavymetal) and a spin current absorber, another configura-tion involving a single ferromagnet was recently inves-tigated. By computing the spin diffusion equation in acentrosymmetric ferromagnet with spin-orbit coupling,Pauyac et al. (2018) showed how the interplay betweenSHE, spin swapping, and spin precession about the mag-netic exchange can lead to local torques acting at theedges of the magnet. These local torques can nucleatereversed magnetic domains and initiate current-drivenmagnetization dynamics (Wang et al. , 2019).9 D. Spin-orbit torques due to the inverse spin galvanic effect
In this section we review calculations of the SOT in2D and 3D bulk magnetic systems. Such a torque isconsidered to be due to the iSGE, which refers to theelectrical generation of spin density when a current flowsin a system lacking (bulk or interfacial) inversion sym-metry. Its reciprocal effect, the SGE, is the generation ofa charge current in the presence of non-equilibrium spindensity (generated, e.g. by photoexcitation). Microscop-ically, iSGE and SGE are associated with the presenceof a spin-orbit coupling that is odd in momentum k , dueto inversion symmetry breaking (Manchon et al. , 2015;Winkler, 2003). As a consequence, the spin texture inmomentum space becomes antisymmetric in k as shownin Fig. 5 for the Pt/Co interface. To keep the resultstractable, theories usually consider simpler forms of odd-in- k spin-orbit coupling, valid close to high symmetrypoints. For instance, strained zinc-blende crystals dis-play a k -linear Dresselhaus spin-orbit coupling close tothe Γ-point (Dyakonov et al. , 1986),ˆ H D = β (ˆ σ x p x − ˆ σ y p y ) , (28)whereas interfaces display a so-called Rashba spin-orbitcoupling (Bychkov and Rashba, 1984; F.T. Vasko, 1979),ˆ H R = α R ˆ σ · ( p × z ) . (29)The coefficients β and α R are the Dresselhaus andRashba parameters, respectively. In Weyl semimetals,the low energy bulk Hamiltonian directly connects thespin with the linear momentum, ˆ H W = v ˆ σ · p (Wan et al. , 2011; Weyl, 1929). Although the spin-momentumlocking scheme of real materials is in general much morecomplex (see Fig. 5), these various forms of spin-orbitcoupling have been widely used theoretically to studythe SGE and iSGE.SGE was first predicted by Ivchenko et al. (1989) andobserved by Ganichev et al. (2001, 2002). The iSGE hasbeen predicted originally by Ivchenko and Pikus (1978),followed by Aronov and Lyanda-Geller (1989) and Edel-stein (1990), and observed in non-centrosymmetric sys-tems such as tellurium (Vorobev et al. , 1979), strainedsemiconductors (Kato et al. , 2004a) and quantum wells(Ganichev et al. , 2004a; Silov et al. , 2004; Wunderlich et al. , 2004, 2005). More recently, current-driven spindensity has also been observed at the surface of transi-tion metals (Stamm et al. , 2017; Zhang et al. , 2014). Inmagnets lacking inversion symmetry, such as zinc-blendesemiconductors (Bernevig and Vafek, 2005; Garate andMacDonald, 2009; Hals et al. , 2010), or magnetic 2D elec-tron gas with Rashba spin-orbit coupling (Manchon andZhang, 2008; Obata and Tatara, 2008; Tan et al. , 2007),the current-driven spin density can be used to control themagnetic order parameter.The iSGE-induced SOT can be derived from the dy-namics of the carrier spin density S , brought out of equi- librium by the applied electric field, in the presence ofboth magnetic exchange and spin-orbit interaction. Forthe sake of the discussion, let us consider the followingmodel Hamiltonianˆ H = ˆ H + ˆ H ex + ˆ H so , (30)where ˆ H is the spin-independent part, ˆ H ex = (∆ /
2) ˆ σ · m is the s - d exchange. The Heisenberg equation for the spinmotion reads d S dt = ∆ (cid:126) S × m + 1 i (cid:126) (cid:104) [ ˆ σ , ˆ H so ] (cid:105) . (31)Here (cid:104)· · ·(cid:105) represents quantum-mechanical averaging overthe non-equilibrium carrier states and (cid:104) ˆ σ (cid:105) = S . The SOTis obtained by taking the steady-state solution of Eq. (31)( d S /dt = 0) into Eq. (5), T = ∆ (cid:126) m × S = 1 i (cid:126) (cid:104) [ ˆ σ , ˆ H so ] (cid:105) . (32)The right side of Eq. (32) shows explicitly the spin-orbitcoupling origin of the SOT. For discerning qualitativelydistinct SOT contributions (i.e. the extrinsic versus in-trinsic terms introduced in Subsection III.A), we will nowuse the middle expression.Let us first discuss the extrinsic (intraband) contri-bution to the spin density which, in the limit of spin-independent disorder, corresponds to the usual Boltz-mann contribution. In the limit ˆ H ex (cid:28) ˆ H so , this term isindependent of the s - d exchange (Edelstein, 1990; Man-chon and Zhang, 2008). For illustration, we considerRashba spin-orbit coupling, Eq. (29), such that the spinsalign perpendicular to the wavevector, σ k ∼ z × k , asillustrated in Fig. 13(a). In the absence of the elec-tric field, (cid:104) k (cid:105) = 0, and the equilibrium distribution ofthese eigenstate spin vectors adds up into a zero netspin density [Fig. 13(a), top panel]. Under the appliedelectric field, however, the states are repopulated with adeficit/excess of left/right moving carriers with respectto the applied electric field [Fig. 13(a), bottom panel].The steady state non-equilibrium distribution is reachedwhen balancing the carrier acceleration in the electricfield with scattering against disorder, see Eq. (9). Dueto the non-centrosymmetric spin texture of the eigen-states, the non-equilibrium distribution leads to a non-zero net spin density aligned perpendicular to the elec-tric field, S ∼ τ α R z × E . In analogy to the Boltzmanntheory of conductivity, the spin density is proportionalto the momentum lifetime τ and, hence, associated withan extrinsic iSGE. Since we neglected ˆ H ex in the car-rier Hamiltonian, the iSGE generated spin density S inthis approximation is independent of m and, when in-troduced into the middle expression of Eq. (32), yields afield-like SOT, T FL ∼ m × ( z × E ). Incorporating theexchange field only creates small angular dependance ofan otherwise constant spin density.0 FIG. 13 (Color online) (a) Top panel: Rashba spin-texture for one of the chiral states in equilibrium with zero net spin-density.Bottom panel: Non-equilibrium redistribution of eigenstates in an applied electric field resulting in a non-zero spin-density dueto broken inversion symmetry of the spin-texture. When combined with the exchange coupling of the carrier spin-density tomagnetization, this mechanism corresponds to the extrinsic (Boltzmann transport), field-like iSGE-SOT. (b) Top panel: Amodel equilibrium spin texture in a 2D Rashba spin-orbit coupled system with an additional time-reversal symmetry breakingexchange field of a strength much larger than the spin-orbit field. In equilibrium, all spins in this case align approximatelywith the x -direction of the exchange field (magnetization). Bottom panel: In the presence of an electrical current along the x -direction the Fermi surface (circle) is displaced along the same direction. When moving in momentum space, electronsexperience an additional spin-orbit field (purple arrows). In reaction to this non-equilibrium current induced field, spins tiltand generate a uniform, non-equilibrium out-of-plane spin density. (c) Top panel: Same as in (b) for y -direction of the exchangefield. Bottom panel: Same as in (b) but now with the current induced spin-orbit field align with the exchange field, resultingin zero tilt of the carrier spins. (b) and (c) illustrate the intrinsic (Berry curvature) damping-like iSGE-SOT. Adapted fromKurebayashi et al. (2014). Let us now consider the intrinsic (interband) contri-bution, assuming the same Rashba spin-orbit coupling.Such a term is labeled intrinsic because it has a weakdependence on scattering in metallic systems. This con-tribution can be also derived from an intuitive pictureof the Bloch dynamics of carrier spins. To do so we con-sider for simplicity the limit ˆ H ex (cid:29) ˆ H so (i.e., the oppositelimit than considered above). In equilibrium, the carrierspins are then approximately aligned with the exchangefield, σ k ≈ s m , independent of their momentum. Thisis depicted in Figs. 13(b,c), bottom panels, for m (cid:107) E and m ⊥ E , respectively. The Bloch equations describethe carrier spin dynamics during their acceleration in theapplied electric field, i.e., between the scattering events.Without loss of generality, we take E = E x x . For m (cid:107) E ,the equilibrium effective magnetic field acting on the car-rier spins due to the exchange term is, B eqeff ≈ (∆ , , dp x dt = eE x , and the effective mag-netic field acquires a time-dependent y -component dueto ˆ H so for which dB eff , y dt = ( α R / (cid:126) ) dp x dt , as illustrated inFig. 13(b). For small tilts of the spins from equilibrium,the Bloch equation d σ k dt = (cid:126) ( σ k × B eff ) yields σ k ,x ≈ s , σ k ,y ≈ sB eff , y /B eqeff , and σ k ,z ≈ − (cid:126) s ( B eqeff ) dB eff , y dt = − s ∆ α R eE x . (33)The non-equilibrium spin orientation of the carriers ac- quires a time and momentum independent σ k ,z = σ z component. For a general angle θ m − E between m and E we obtain, σ k ,z ( m ) = σ z ( m ) ≈ s ∆ α R eE x cos θ m − E . (34)The total non-equilibrium spin density, S z =2 g D ∆ σ z ( m ), is obtained by integrating σ k ,z overall occupied states ( g D is the density of states). Thenon-equilibrium spin density produces an out-of-planefield which exerts a torque on the in-plane magnetiza-tion. From Eqs. (32) and (34) we obtain an intrinsicdamping-like SOT (Kurebayashi et al. , 2014), T DL = ∆ (cid:126) ( m × S z z ) ∼ m × [ m × ( z × E )] . (35)It is worth pointing out the analogy and differencesbetween the intrinsic iSGE and the intrinsic SHE (Mu-rakami et al. , 2003; Sinova et al. , 2004). In the SHEcase where ˆ H ex = 0 in the paramagnet, B eqeff depends onthe angle θ k of the carrier momentum with respect to E which implies a momentum-dependent z -component ofthe non-equilibrium spin, σ k ,z ≈ sα R k α R eE x sin θ k . (36)The same spin rotation mechanism that generates theuniform bulk spin density in the case of the intrinsic1iSGE in a ferromagnet [Fig. 13(b)] is responsible for thescattering-independent spin current of the SHE in a para-magnet (Sinova et al. , 2004). Note that the SHE spin cur-rent yields zero spin density in the bulk while a net spindensity accumulates only at the edges of the conductionchannel.
1. Inverse spin galvanic torque in a magnetic two-dimensionalelectron gas
Because of the symmetry present in most bilayer sys-tems considered in experiments, the Rashba spin-orbitcoupling given by Eq. (29) is the natural model to study,and therefore the iSGE-SOT has been alternatively calledRashba-SOT in this context. As discussed above, theRashba torque can possess two components correspond-ing to the field-like and damping-like torques, see Eq. (2).While the origin of the field-like torque is well understoodand consistently attributed to the extrinsic intrabandiSGE (Rashba-Edelstein) effect (Edelstein, 1990), sev-eral mechanisms contribute to the damping-like torque.The different contributions have been investigated usingsemiclassical Boltzmann transport equation (van der Bijland Duine, 2012; Kim et al. , 2013b, 2012b; Lee et al. ,2015; Manchon and Zhang, 2008; Matos-Abiague andRodriguez-Su´arez, 2009; Tan et al. , 2007), or quantum-mechanical Kubo formula approaches (Freimuth et al. ,2017a; Li et al. , 2015a; Pesin and MacDonald, 2012a;Qaiumzadeh et al. , 2015; Wang et al. , 2012b, 2014).As discussed above, interband transitions produce anintrinsic damping-like torque in the limit of weak scat-tering (Freimuth et al. , 2014b; Kurebayashi et al. , 2014),and can be related to the Berry curvature of the elec-tronic band structure in the mixed spin-momentum phasespace (Freimuth et al. , 2014b; Kurebayashi et al. , 2014;Lee et al. , 2015; Li et al. , 2015a). As a result, one can ex-pect ”hot spots” in the band structure, i.e., points whereneighboring bands get very close to each other, to givevery large contribution, similarly to the case of intrin-sic SHE (Tanaka et al. , 2008). Notice that in the spe-cific case of the pure 2D Rashba gas, at the first orderin the Rashba parameter, vertex corrections cancel theintrinsic damping-like torque unless the momentum re-laxation time is spin-dependent (Freimuth et al. , 2017a;Qaiumzadeh et al. , 2015), similar to the cancelation oc-curring for the intrinsic SHE in pure 2D Rashba gas (In-oue et al. , 2004). Nevertheless, such cancellations arehighly sensitive to this specific model band structure andcan be considered as accidental, as discussed by Sinova et al. (2015) in the context of intrinsic SHE.Extrinsic iSGE mechanisms related to spin scatteringwere also theoretically shown to generate a damping-like component of the SOT. In (Kim et al. , 2012b), thedamping-like term arises from the momentum-scatteringinduced spin relaxation, an effect initially proposed in metallic spin-valves and domain walls (Zhang et al. , 2002;Zhang and Li, 2004). In fact, when a non-equilibriumspin density S is injected into a magnet, spin relaxationgenerates a corrective term of the form ∼ β sf m × S , where β sf is a parameter that depends on the ratio between spinprecession and spin-flip scattering. In other works, thiscomponent is obtained within a quantum kinetic formal-ism and ascribed to spin-dependent carrier lifetimes orto a term arising from the weak-diffusion limit which inthe leading order is proportional to a constant carrier life-time (Pesin and MacDonald, 2012a; Wang and Manchon,2012; Wang et al. , 2014).Finally, we stress out that the coefficients τ FL , DL arein principle angular dependent and display terms pro-portional to sin n θ m − E , n ∈ N ∗ . This angular depen-dence reflects the distortion of the band structure whenchanging the magnetization direction (Lee et al. , 2015),as well as the anisotropic spin relaxation in the sys-tem due to D’yakonov-Perel’s mechanism (Ortiz Pauyac et al. , 2013).
2. Non-centrosymmetric bulk magnets
Dilute magnetic semiconductor (Ga,Mn)As has been atest-bed material for observing and exploring the bulkSOT. Hence, unlike the case of bilayer systems, alltorques observed in these bulk materials arise internallywith no contribution from externally injected spin cur-rents. Current-driven torques in dilute magnetic semi-conductors were first studied by Bernevig and Vafek(2005). The authors considered the Kohn-L¨uttingerHamiltonian in the spherical approximation, augmentedby a k-linear spin-orbit coupling term arising from strainof the form λ · ˆ J , where ˆ J is the total angular momentumoperator, λ x = C ( (cid:15) xy k y − (cid:15) xz k z ) and λ y,z are obtainedfrom cyclic permutation of indices. The current-drivenspin density reads S = − eτ (cid:126) (cid:32) (cid:88) s = ± (cid:112) n/π ( γ + 2 sγ ) / (cid:33) / ( e E · ∇ k ) λ , (37)where n is the charge density. Because in their calcu-lation they did not consider an exchange coupling di-rectly, the torque induced by this iSGE is therefore afield-like torque. The intrinsic damping-like torque orig-inating from interband transitions was later proposed byKurebayashi et al. (2014) to interpret the experimentalobservation of such a damping-like torque in (Ga,Mn)As,see Fig. 14. The theoretical investigation of SOT indilute magnetic semiconductors was also pursued by Li et al. (2015a, 2013). Besides some subtleties related tothe complex band structure, the numerical results ob-tained by these authors qualitatively confirm the general2picture obtained in the context of the magnetic Dressel-haus and Rashba gas. FIG. 14 (Color online) Microscopic modeling of the intrinsicSOT in bulk (Ga,Mn)As. The solid lines correspond to thenumerical results and the symbols correspond to the experi-mental data. The dashed lines correspond to the free electronapproximation. From Kurebayashi et al. (2014).
Apart from (Ga,Mn)As, the SOT has been studied inseveral other bulk systems. One of them is the ferromag-net NiMnSb (Ciccarelli et al. , 2016). This half-heuslermaterial has the same crystalline symmetry as zinc-blende (Ga,Mn)As; however, it is not a dilute-momentrandom alloy like (Ga,Mn)As, but a dense-moment or-dered compound. Despite these differences, the SOTsfound in NiMnSb are quite similar to those in (Ga,Mn)Asbecause the torque is mostly determined by the lowestorder terms in the magnetization expansion, Eq. (21),which are the same for the two systems.
3. Spin-orbit torques in magnetic textures
When itinerant electrons flow in a magnetic domainwall, their spin slightly misaligns with the local texture,resulting in STT (Tatara and Kohno, 2004; Zhang andLi, 2004) (see Section VI for details). When spin-orbitcoupling is present, the flowing spins experience an addi-tional precession about the spin-orbit field, resulting inenhanced spin torque (Nguyen et al. , 2007; Stier et al. ,2013; Stier and Thorwart, 2015; Yuan and Kelly, 2016).An interesting consequence is the emergence of additionaltorque components that are proportional to the magne-tization gradient (Hals and Brataas, 2013b). Some ofthese contributions can be directly assigned to the pres-ence of anomalous Hall effect and anisotropic magnetore-sistance (van der Bijl and Duine, 2012), while other gen-uinely come from the interplay between magnetic textureand precession around the spin-orbit field. For instance,Kim et al. (2013b, 2012b) showed that in the presence ofRashba spin-orbit coupling, a 2D magnetic texture (i.e. amagnetic skyrmion or vortex) experiences a torque of theform T ∝ ( ∇ · m )[( z × u ) · ∇ ] m , where u is the directionof injection. Such SOTs open interesting perspectivesfor the electrical manipulation of magnetic textures, dis-cussed in Section VI, but have received little attention todate. E. Ab initio modeling of spin-orbit torques in bilayersystems
Although following after the studies in bulk magnets,the SOTs have been most extensively studied experi-mentally in NM/FM bilayer (or multilayer) structures(Section IV). Theories of SOT in bilayer systems basedon iSGE and SHE as exposed in the previous sectionspresent two major limitations. First, both mechanismsformally apply in very distinct situations: in the widelyused diffusive model, SHE in the nonmagnetic metal isconsidered uniform, neglecting semiclassical size effectsand possible variation of the spin Hall angle close to theinterface (Freimuth et al. , 2015; Wang et al. , 2016b), asdiscussed in Subsection III.C. In contrast, iSGE in mag-netic multilayers is often modeled using the Rashba in-teraction, Eq. (29), which applies to 2D gases and ide-ally sharp interfaces only. Both approaches overlook thedetails of the interfacial orbital overlap, which can bequite subtle in transition metal interfaces and lead to en-hanced orbital moment and related spin-orbit phenom-ena (Bl¨ugel and Bihlmayer, 2007; Grytsyuk et al. , 2016;Marmolejo-Tejada et al. , 2017).
FIG. 15 (Color online) Layer-resolved field-like torque in nor-mal Pt/Co(111) (black symbols), when turning off the in-duced magnetization on Pt (red symbols) and when turningoff the spin-orbit coupling on Pt (blue symbols). From Haney et al. (2013a)
To overcome these issues, SOTs in Co/Pt bilayer sys-tems have been computed within the relaxation time ap-proximation using an ab initio density functional the-ory description of the whole bilayer structure (Freimuth et al. , 2014b, 2015; Haney et al. , 2013a). Haney et al. (2013a) focus on the extrinsic iSGE, disregarding theSHE and intrinsic contributions to iSGE. In spite ofthe high complexity of the band structure (see Fig. 5),these calculations confirm the intuitive picture elabo-rated based on the Rashba model. In particular, theyshow that SOT is mostly driven by spin-orbit couplingin Pt, while the influence of induced magnetization isnegligible, see Fig. 15. Moreover, the torque acquires3a non-trivial angular dependence, and depends dramat-ically on the quality of the interface. Using a similarmethod, neglecting intrinsic contributions to both SHEand iSGE, Amin et al. (2018) computed the interfacialspin current in Co/Cu bilayers, obtaining both field-likeand damping-like components, as well as an additionaltorque, T ∝ ζ × ( m × ζ ). These results are confirmed byFreimuth et al. (2018).Alternatively, Freimuth et al. (2014b) and G´eranton et al. (2015) computed the full Kubo-Bastin formula,thereby accounting for both intrinsic SHE and intrinsiciSGE. These calculations confirmed that SOTs are com-posed of both field-like and damping-like torques, thelatter being produced by interband transitions only, seeFig. 16. An interesting aspect revealed through thesecalculations is the high sensitivity of SOTs to interfacialengineering. In fact, the authors found that by cappingthe Co layer by either Al or O atoms, the damping-liketorque is only slightly affected (its magnitude changes upto 50% - see Fig. 16) while the field-like torque is dramat-ically altered and can even change its sign. In a follow-upwork, Freimuth et al. (2015) reported an enhancement ofSHE close to the interface, also predicted by Wang et al. (2016b) for Pt/NiFe. These studies suggest that the as-sumption of uniform spin Hall angle made in the diffusiveapproach may be valid in the strong disorder limit only.Similar results have been recently obtained by Mahfouziand Kioussis (2018) in Co/Pt and Co/Pd bilayers.Whereas all these studies are based on the relaxationtime approximation (i.e., the effect of impurities is cap-tured by a homogeneous broadening), Wimmer et al. (2016) calculated the torque in Pt/Fe x Co − x /Cu su-perlattices using the Kubo-Bastin formula and coherentpotential approximation to account for the alloy disor-der. This allows for treating extrinsic scattering mech-anisms (i.e., side jump and skew scattering) within theframework of density functional theory. The influenceof impurities and phonon scattering on the SOT hasbeen investigated by G´eranton et al. (2016) within theKorringa-Kohn-Rostoker Green’s function method. Fi-nally, Belashchenko et al. (2019) investigated the angu-lar dependence of the SOTs in Co/Pt bilayers using atwo-terminal non-equilibrium Green’s function approachwith real-space Anderson disorder, uncovering a planarHall-like contribution. F. Spin-orbit torques in antiferromagnets
Since the first proposal of STT in antiferromagnets(MacDonald and Tsoi, 2011; N´u˜nez et al. , 2006), sev-eral configurations have been theoretically investigatedto enable large spin torque efficiency. In the course of thesearch for such torques it was realized that in order toefficiently manipulate the order parameter of a collinear,bipartite antiferromagnet one needs a torque that corre-
FIG. 16 (Color online) Layer-resolved damping-like torque t even xy (blue symbols) and spin current q even xy (red symbols) in(a) Pt/Co, (b) Pt/Co/O and (c) Pt/Co/Al. From Freimuth et al. (2014b) sponds to a staggered effective magnetic field, i.e., a fieldwith an opposite sign on the two magnetic sublattices.Such a field, unlike a homogeneous field, couples directlyto the N´eel order. The torque resulting from a staggeredfield has been referred to as N´eel SOT (ˇZelezn´y et al. ,2017, 2014).The damping-like torque due to a spin current (froma SHE or a ferromagnetic polarizer) is a N´eel torque,assuming that it has the same form on each magneticsublattice as in ferromagnets, i.e., T a = τ DL m × ( m × ζ ).This form of the torque has been predicted theoretically(Cheng et al. , 2014; Gomonay et al. , 2012; ˇZelezn´y et al. ,2014) and it was shown that it can indeed efficiently ma-nipulate the antiferromagnetic order (Gomonay et al. ,2012; Gomonay and Loktev, 2010). A recent drift-diffusion theory confirmed that spin current injected froman adjacent ferromagnetic polarizer or induced by SHEindeed generates such a N´eel damping-like torque (Man-chon, 2017). An experimental indication of the presenceof the SHE generated SOT in an NM/AF bilayer wasreported in Reichlova et al. (2015).The bulk SOT can also have the N´eel order form ifthe current-induced spin density has an opposite sign onthe two sublattices (ˇZelezn´y et al. , 2014). In a collinearantiferromagnet, the two sublattices with opposite mag-netizations are connected by some symmetry operation.Typically, this is either a translation or an inversion. Thissymmetry operation combined with time-reversal is thena symmetry of the magnetic system which, using Eqs.(19) and (20), relates the current-induced spin densitieson the two sublattices (ˇZelezn´y et al. , 2017). If the sub-lattices are connected by translation then χ even B,a = χ even B,b , (38) χ odd B,a = − χ odd B,b , (39)4where a, b denotes the two sublattices. If they are con-nected by inversion χ even B,a = − χ even B,b , (40) χ odd B,a = χ odd B,b . (41)Thus in both cases there exists both a staggered com-ponent and a uniform component of the current-inducedspin density and the corresponding effective field. For themagnetic dynamics only the staggered component willhas an appreciable effect for the achievable magnitudesof the effective fields (a few mT). Since the componentthat is staggered is different in the two cases, the dynam-ics will differ. As discussed in Subsection III.B, the evenfield typically has a field-like character, whereas the oddfield is commonly damping-like. Thus in systems wheremagnetic sublattices are connected by translation we canexpect an efficient damping-like torque, whereas in sys-tems where the sublattices are connected by inversion afield-like torque is expected.SOTs in antiferromagnets have been first studied intwo tight-binding models (ˇZelezn´y et al. , 2017, 2014): (i)the antiferromagnetic 2D Rashba gas and (ii) the bulkMn Au. Both systems possess collinear antiferromag-netism. They together illustrate the two main types ofsymmetry discussed above. In the Rashba model the twosublattices are connected by translation and thus the low-est order N´eel order SOT has a damping-like character.In the Mn Au crystal, on the other hand, the two sub-lattices are connected by inversion and the lowest orderN´eel torque has consequently a field-like character. Mi-croscopic calculations based on the Kubo formula withconstant relaxation time indeed show that the N´eel SOTin the Rashba model is primarily damping-like, whereasin Mn Au it is predominantly of field-like character.The origin of the field-like torque in the Mn Au crys-tal can be understood in terms of the symmetry of thenonmagnetic crystal. Without magnetism, the crystalhas inversion symmetry and thus there is no net current-induced spin density. Yet, the Mn sublattices each havelocally broken inversion symmetry and thus can havecurrent-induced spin-densities that have to be preciselyopposite. An intuitive explanation of this behavior is thatthe local inversion breaking is opposite for the two sub-lattices and thus the induced spin-densities are also oppo-site. When magnetism is added these opposite spin den-sities generate a staggered effective field. In the Rashbamodel on the other hand, the inversion breaking is thesame for both sublattices and thus the field generatingthe field-like torque is not staggered. On the other hand,the field generating the damping-like torque is staggered,since it is proportional to the magnetic moment which isstaggered in the antiferromagnet. The field-like torquein Mn Au has a Rashba-like symmetry, i.e., the effec-tive field is on each sublattice proportional to ζ . Thisis because the local symmetry of the Mn sublattices is the same as that of the 2D Rashba model. Additionalsymmetry analysis for various types of crystalline antifer-romagnets has been provided by Watanabe and Yanase(2018) and ˇZelezn´y et al. (2017).Following the calculations based on tight-binding mod-els, the SOT was also calculated in antiferromagnets us-ing density functional theory. Such calculations weredone for Mn Au (ˇZelezn´y et al. , 2017) and CuMnAs(Wadley et al. , 2016), which has a symmetry analogous toMn Au. These results agree well with the tight-bindingcalculations in term of the magnetization and current de-pendence and in addition show a relatively large torque.The magnitude of the effective field is around 2 mT per10 Acm − current density for Mn Au and 3 mT per 10 Acm − for CuMnAs. The switching attributed to thisfield-like N´eel order torque has been observed in CuM-nAs (Wadley et al. , 2016) and subsequently in Mn Au(Bodnar et al. , 2018; Meinert et al. , 2018; Zhou et al. ,2018).
G. Spin-orbit torques in topological insulators
Topological insulators are a class of materials display-ing intriguing properties such as insulating bulk and con-ductive chiral and helical surfaces (Hasan and Moore,2011; Qi and Zhang, 2011; Wehling et al. , 2014). Consid-ering the large spin-charge conversion efficiency recentlyreported in these systems (see Subsection IV.C.4), theydeserve special attention. The category of topologicalmaterials we are interested in are characterized by time-reversal symmetry and helical surface states: their lowenergy surface states are represented by a Dirac Hamil-tonian of the form ∼ p i σ j (see Fig. 8). When elec-trons flow on the surface of these systems, they acquirea non-equilibrium spin density, similar to the case of the2D Rashba gas, as demonstrated in a Bi Se slab us-ing ab initio calculations (Chang et al. , 2015). Since thestrength of the spin-momentum coupling is quite large( ∼ × − eV m at Bi Se surfaces, comparable toBi/Ag surfaces, and two orders of magnitude larger thanin InAlAs/InGaAs 2D gases), iSGE is expected to bevery large. In addition, the absence of bulk conductionin ideal topological insulators further strongly enhancesthe spin-charge conversion efficiency.Spin-charge conversion processes in topological insu-lator/insulating ferromagnet bilayers have been studiedby several authors (Fujimoto and Kohno, 2014; Lin-der, 2014; Sakai and Kohno, 2014; Taguchi et al. , 2015;Tserkovnyak et al. , 2015). The low energy Hamiltonianreads ˆ H = v ˆ σ · (ˆ p × z )+ ∆2 ˆ σ · m , where the first term mod-els the Dirac cone and the second term is the exchange.This model applies when the Dirac states are preserved,so typically when the topological insulator surfaces areinterfaced with magnetic insulators (Katmis et al. , 2016;Lang et al. , 2014; Li et al. , 2015b). The eigenenergies5read (cid:15) sk = s (cid:114) ( vk x + ∆2 m y ) + ( vk y − ∆2 m x ) + ∆ m z . (42)When m z (cid:54) = 0 the surface states are gapped, whereaswhen m z = 0, the origin of the band dispersion is onlyshifted in the k -plane. If the Fermi energy lies in thegap, quantum anomalous Hall effect emerges, accompa-nied by a quantized magnetoelectric effect, S = − e (cid:126) πv E (Nomura and Nagaosa, 2011; Qi et al. , 2008). On theother hand, when the Fermi level lies above the gap, thesystem is metallic and the SOT possesses both field-likeand damping-like components of the form (Garate andFranz, 2010; Ndiaye et al. , 2017) T = τ FL m × ζ + τ DL m z m × ( z × ζ ) , (43)where ζ || z × E . While the field-like torque arises fromthe conventional extrinsic iSGE, the damping-like torquearising from the intrinsic interband contribution is pro-portional to m z and therefore vanishes when the magne-tization lies in the plane of the surface, in sharp contrastwith the usual damping-like torque given in Eq. (2) (Ndi-aye et al. , 2017).The calculations discussed above are based on the 2DDirac gas model, i.e, assuming that the transport is con-fined to the interface and that surface states remain in-tact in the presence of the proximate ferromagnetic layer.Such a model presents two major drawbacks though.First, orbital hybridization between the transition metaland the topological insulator substantially alters the sur-face states at Fermi energy. The presence of magneticadatoms shifts the Dirac cone downward in energy (Hon-olka et al. , 2012; Scholz et al. , 2012; Ye et al. , 2012),and favor the presence of additional metallic bands withRashba-like character across the Fermi level (Marmolejo-Tejada et al. , 2017; Zhang et al. , 2016a).A second limitation comes for the 3D nature of thetransport. Indeed, most experiments involve topologicalinsulators with sizable bulk conductivity, suggesting thatbulk states might participate to the spin-charge conver-sion mechanism. Spin transport in such systems has beenrecently investigated using drift-diffusion model (Fischer et al. , 2016), non-equilibrium Green’s function technique(Mahfouzi et al. , 2016) and Kubo formula on a slab geom-etry (Ghosh and Manchon, 2017b). The first two studiesshow that spin diffusion in the ferromagnet and spin-flipscattering at the interface can enhance the damping-liketorque. The latter work accounts for interfacial and bulktransport on equal footing and demonstrates that a largedamping-like torque is driven by the Berry-curvature ofthe interfacial states, whereas the SHE of the bulk statesis inefficient.Finally, spin-orbit charge pumping, the reciprocal ef-fect of SOT, has also been investigated theoreticallyin topological insulators (Mahfouzi et al. , 2014; Ueda et al. , 2012), providing a charge current of the form j c = χ DL m z ∂∂t m + χ FL z × ∂∂t m (Ndiaye et al. , 2017),where τ DL , FL = χ DL , FL E . A direct consequence of thiscurrent is the induction of an anisotropic magnetic damp-ing on the ferromagnetic layer (Yokoyama et al. , 2010).SOT and spin-orbit charge pumping have also beenstudied in various configurations involving 2D topolog-ical insulators (Soleimani et al. , 2017). These studiesreveal that SOT experiences a significant enhancementdepending on the topological phase (Li and Manchon,2016; Mahfouzi et al. , 2010): the emergence of edge cur-rents promotes a quantized charge pumping when themagnetization is perpendicular to the plane. Such inves-tigations have been recently extended to antiferromag-netic 2D topological insulators, where time-reversal com-bined with a half unit cell translation is a symmetry ofthe system which preserves topological protection, de-spite the broken time-reversal symmetry of the magneticstate (Ghosh and Manchon, 2017a).SOTs have also been theoretically studied in mag-netic 2D hexagonal lattices such as, but not limited to,graphene, silicene, germanene, stanene, transition metaldichalcogenides etc. (Dyrda(cid:32)l and Barnas, 2015; Li et al. ,2016a). The parametric dependencies of the torque inthese materials do not significantly differ from the oneobtained with the Rashba model. Nonetheless, in thesesystems the low-energy transport occurs mostly throughtwo independent valleys, which opens the possibility toobtain valley-dependent SOTs. H. Other spin-orbit torques
1. Anisotropic magnetic tunnel junctions
MTJs composed of a single ferromagnet with inter-facial spin-orbit coupling display tunneling anisotropicmagnetoresistance, i.e., a change of resistance when vary-ing the magnetization direction (Gould et al. , 2004; Park et al. , 2008), see Fig. 17(a). One naturally expects thatspin-polarized electrons impinging on the spin-orbit cou-pled interface precess about the spin-orbit field, result-ing in a torque on the local magnetization (Manchon,2011a). This SOT is of the form given by Eq. (2) with ζ = z . The field-like torque possesses an equilibrium con-tribution [which is nothing but the perpendicular mag-netic anisotropy (Manchon, 2011b)] and the damping-liketorque is purely non-equilibrium. Both torques are linearas a function of the bias voltage, but their magnitude isquadratic in the Rashba parameter, see Fig. 17. A simi-lar idea has been proposed by Mahfouzi et al. (2012) byconsidering a topological insulator as a tunnel barrier.
2. Spin-transfer torque assisted by spin-orbit coupling
When a spin-polarized current penetrates into a fer-romagnet with spin-orbit coupling, the spin momentum6
Normal Metal Ferromagnet I n s u l a t o r Spin-orbit
FIG. 17 (Color online) Left: Schematics of a MTJ composedof a ferromagnet and a nonmagnetic metal separated by a tun-nel barrier. Spin-orbit coupling is present at the interface be-tween the ferromagnet and the tunnel barrier. Right: Rashbadependence (a) and bias dependence (b) of the damping-liketorque. Adapted from Manchon (2011a) precesses around an effective field that is the sum of theexchange and spin-orbit fields. This precession results inadditional angular dependences of the SOT in Rashba(Lee et al. , 2015) and Kohn-L¨uttinger systems as dis-cussed above (Kurebayashi et al. , 2014; Li et al. , 2013).Interestingly, Haney and Stiles (2010) showed that in ametallic spin-valve where spin-orbit coupling is present,such a precession results in an overall STT enhancement.Considering the general Hamiltonian, Eq. (30), withˆ H so = ξ ˆ L · ˆ σ , the total angular momentum ˆ J = ˆ L + ˆ σ obeys the continuity equation d ˆ J dt − ∇ · J J = − ˆ τ STT − ˆ τ lat , (44)where J J is the current density tensor for the total angu-lar momentum, ˆ τ STT is the STT and ˆ τ lat = i (cid:104) [ ˆ H, ˆ L ] (cid:105) / (cid:126) isthe mechanical torque. The latter is nothing but the pre-cession of itinerant spins about the spin-orbit field, suchthat the total spin torque in a spin-valve survives awayfrom the interface, see Fig. 18(a). Due to this additionalprecession, the total torque extends over the whole thick-ness of the free layer, as displayed in Fig. 18(b). A similareffect has been identified in magnetic domain walls wherespin-orbit coupling enhances spin reflection and therebySTT (Nguyen et al. , 2007; Yuan and Kelly, 2016). (a) (b) FIG. 18 (Color online) (a) Spatial profile of the transversespin density injected in the free layer of a (Ga,Mn)As-basedMTJ in the presence and absence of spin-orbit coupling in theband structure; (b) Spin torque efficiency as a function of thefree layer thickness. Adapted from Haney and Stiles (2010).
I. Open theoretical questions
After a decade of theoretical progress, the key mech-anisms giving rise to SOTs are now well understood atthe qualitative level. Although the involved mechanismsare very different (SHE, iSGE etc.), they can all be uni-fied using Eq. (5), i.e. any SOT can be associated witha current-induced spin density. The various mechanismsdiffer in the way this spin density is generated. In theiSGE, the spin density is generated locally, whereas in theother mechanisms (SHE, spin swapping etc.), it is due tospin currents that transfer spin angular momentum fromone part of the system to another. In addition, when amagnetic texture is present, the SOTs acquire new com-ponents that depend on the spatial gradient of the mag-netization. The key ingredients in all these mechanismsare inversion symmetry breaking and the spin-orbit cou-pling. The general form of the torques can be determinedusing symmetry analysis.Nonetheless, a number of challenges remain to be ad-dressed on the theory side. First of all, quantitative agreement between theory and experiments is still miss-ing. While important progress has been made using den-sity functional theory (Freimuth et al. , 2014b), most cal-culations adopt a simplified, unrealistic model of scat-tering. Including realistic disorder (structural imper-fections, interfacial roughness), but also scattering fromphonons and magnons will certainly improve such cal-culations. Moreover, understanding the interplay be-tween bulk transport and interfacial effects, as well asthe impact of interfacial orbital hybridization in mag-netic multilayers will lead to the design of better SOTdevices. These tasks require the development of accu-rate first-principles quantum transport methods (Nikoli´c et al. , 2018).Comprehensive first-principles models would be par-ticularly useful in the context of novel materials. Amongthese, topological materials such as topological insulatorsand Weyl semimetals display exotic surface states withstrong spin-orbit coupling and are regarded as promis-ing for SOT generation. However, we lack an accurateunderstanding of how proximate transition metals mod-ify these surface states (Marmolejo-Tejada et al. , 2017;Zhang et al. , 2016a) and how bulk and surface transportcooperate to produce large SOTs (Ghosh and Manchon,2018). Another important class of materials exhibitingremarkable properties is the antiferromagnets. While thebasic principles of SOTs and current-driven dynamics areunderstood, a proper description of the magnetic textureand dynamics of realistic, disordered antiferromagnets isstill lacking. In addition, non-collinear antiferromagnetssuch as Mn X compounds, host a variety of novel phe-nomena, such as anomalous Hall effect (AHE) (Nakatsuji et al. , 2015) or ”magnetic SHE” (Kimata et al. , 2019;Zelezn´y et al. , 2017), that could be exploited in the con-text of SOT. These various aspects call for further theo-7retical endeavor.Although the single-particle density functional theoryprovides a good description of the electronic structureof most transition metals and semiconductors, it fails indescribing strongly correlated systems, such as Mott orKondo insulators (Cohen et al. , 2008; Jones, 2015). Uti-lizing more advanced many-body approaches, such as thedynamical mean field theory, might thus be necessaryfor accurate description of the SOT in such materials.These calculations are, however, numerically very expen-sive. We note also that Eq. (5) relies on a density func-tional theory description, and a more general expressionthat would be valid even in a many-body system has notbeen established yet.The SOT in bilayer systems is often explained in termsof a spin current, which provides useful (but sometimesmisleading) insight into the physics at stake. This con-cept is, however, controversial on a theoretical level.In the presence of spin-orbit coupling, the spin angu-lar momentum is not a conserved quantity and adopt-ing the conventional definition of the spin current tensor, j j s ,i ∼ { ˆ s i , ˆ v j } , can give rise to dissipationless equilibriumspin currents (Rashba, 2003). For instance, in centrosym-metric crystals Shi et al. (2006) circumvent this hurdleby defining the spin current tensor j j s ,i ∼ ( d/dt ) { ˆ s i , ˆ r j } ,accounting for a ”torque dipole” term that ensures over-all spin conservation. Until now, this new definition hasnot been widely adopted and the question of its applica-bility to heterostructures remains open. Another way toconsider this problem would be to compute the currentof total angular momentum (An et al. , 2012). However,doing so complicates the problem because the total an-gular momentum of the electronic system alone is notconserved as it interacts with phonons and magnons.Along this line of thought, the orbital analogs of spinphenomena, such as the orbital Hall effect (Go et al. ,2018; Tanaka et al. , 2008) and the orbital iSGE (Yoda et al. , 2018), have attracted attention, but their possi-ble connection to the SOT is yet to be explored. Unlikethe SHE and the iSGE, the orbital effects exist even inthe absence of spin-orbit coupling. When spin-orbit cou-pling is present, these effects can in principle couple tothe magnetic moments and thus contribute to the SOT(Go and Lee, 2019). This indicates a route for the op-timization of SOT via orbital engineering. The recentclaim of “maximal” Rashba spin-splitting suggests a di-rection towards this end (Sunko et al. , 2017). IV. SPIN-ORBIT TORQUES IN MAGNETICMULTILAYERS
This section reviews recent experimental progress inthe measurement and characterization of SOT in multi-layer systems. We first introduce the phenomenologicaldescription of SOT commonly used in experiments (Sub-
FIG. 19 (Color online) (a) Spin-orbit torques and correspond-ing effective fields measured in Pt/Co/AlO x layers when themagnetization is tilted parallel to the current direction. (b)Schematic of the coordinate system. section IV.A) and the main techniques employed to mea-sure SOT (Subsection IV.B). Next, we present a surveyof different layered materials, namely nonmagnetic met-als, semiconductors, and topological insulators coupledto either ferromagnets, ferrimagnets, or antiferromagnets(Subsection IV.C), summarizing the most salient featuresof the SOT observed in these systems. Finally, we de-scribe the SOT-induced magnetization dynamics (Sub-section IV.D) and switching (Subsection IV.E), and con-clude by highlighting examples and perspectives for theimplementation of SOT in magnetic devices (SubsectionIV.F). A. Phenomenological description
Current-injection in heterostructures composed of amagnetic layer adjacent to a nonmagnetic conductor witheither bulk or interfacial spin-orbit coupling gives riseto a transverse spin density ζ || z × j c at the interfaceof the magnetic layer. This spin density induces bothdamping-like and field-like SOT components, as shownin Fig. 19(a), and described by Eq. (2). For exper-imental purposes, it is useful to introduce two effec-tive magnetic fields, B DL , FL , which correspond to thedamping-like and field-like torques and are defined by T DL , FL = M × B DL , FL . The advantage of the effectivefield formulation in the SOT characterization is that theiraction on the magnetization can be directly compared tothat of a reference external field of known magnitude anddirection. To the lowest order in the magnetization, fora current j c || x and assuming C v symmetry (see below),Eq. (2) gives B FL = B FL y , (45) B DL = B DL m × y , (46)where the field amplitudes are simply B FL = τ FL and B DL = τ DL if the torques are calculated for the unitary8magnetization m , as assumed in Eq. (2). Thus, for pos-itive values of the SOT coefficients τ FL and τ DL , B FL || y whereas B DL rotates clockwise in the xz plane, corre-sponding to T DL || − y . Figure 19(a) shows the orienta-tion of the torques and effective fields for the model sys-tem Pt/Co/AlO x , in which τ FL > τ DL < et al. , 2013). Typical values of B FL , DL in NM/FM sys-tems are in the range 0.1-10 mT for a current density j c = 10 A/cm . Note also that the Oersted field dueto the current flowing in the nonmagnetic layer producesan additional field B Oe ≈ µ j c t N / y for bottom (top) stacking relative to the magneticlayer.In the typical NM/FM bilayer geometry shown inFig. 19(a), the SOTs are interfacial torques whose mag-nitude, to a first approximation, does not depend on thethickness t F of the ferromagnet. However, the measur-able effects of the SOTs on the magnetization, namely B FL and B DL , scale inversely with t F because the mag-netic inertia is proportional to the volume of the ferro-magnet. Keeping into account the proportional relation-ship between SOTs and injected current, it is thus usefulto define the spin torque efficiencies ξ j DL , FL = 2 e (cid:126) M s t F B DL , FL j c , (47)where M s is the saturation magnetization. The parame-ters ξ j DL , FL represent the ratio of the effective spin cur-rent absorbed by the ferromagnet relative to the chargecurrent injected in the nonmagnetic metal layer, and canthus be considered as effective spin Hall angles for a par-ticular combination of nonmagnetic metal and ferromag-net. In the pure SHE-SOT picture, ξ j DL is equal to thebulk spin Hall angle of the nonmagnetic metal in thelimit of a transparent interface and negligible spin mem-ory loss. Although the SOT efficiencies are useful pa-rameters to compare the strength of the SOT in differentsystems, ambiguities remain on how to estimate j c inlayered heterostructures. While some authors consider j c to be the average current density, others apply a par-allel resistor model to separate the currents flowing inthe nonmagnetic metal and ferromagnetic layers. How-ever, thickness inhomogeneities and interface scatteringcan significantly alter the current distribution in bilayersystems relative to such a model (Chen and Zhang, 2017).Even in homogeneous films, the conductivity is a strongfunction of the thickness (Fuchs, 1938; Sambles, 1983)so that j c changes in the bulk and interface regions ofa conductor. For these reasons, the current normaliza-tion should be performed with care. Alternatively, it is To emphasize the direction of the effective fields in perpendic-ularly magnetized layers, B FL and B DL are sometimes called”transverse field” ( H T ) and ”longitudinal field” ( H L ), respec-tively (Kim et al. , 2013a). possible to measure the torque efficiency per unit electricfield (Nguyen et al. , 2016) ξ E DL , FL = 2 e (cid:126) M s t F B DL , FL E = ξ j DL , FL /ρ, (48)where E = ρj c is the electric field driving the current,which is independent of the sample thickness and can beeasily adjusted in voltage-controlled experiments. Notethat, in the framework of the SHE-SOT model, ξ E DL canbe considered as an effective spin Hall conductivity.Equations (45) and (46) correspond to the lowest or-der terms of the SOT, which are sufficient to describemany experimental results, at least on a qualitative level.On a more general level, however, higher order terms inthe magnetization are allowed by symmetry. The typicalpolycrystalline metal bilayers have C v symmetry, corre-sponding to broken inversion symmetry along the z -axisand in-plane rotational symmetry. For such systems,the torques can be decomposed into the following terms(Garello et al. , 2013), T FL = [ τ { } FL + (cid:88) n ≥ τ { n } FL (sin θ ) n ] m × y (49)+ m × ( z × m ) m x (cid:88) n ≥ τ { n } FL (sin θ ) n − , T DL = τ { } DL m × ( m × y ) (50)+ m x z × m (cid:88) n ≥ τ { n } DL (sin θ ) n − , where θ is the polar angle of the magnetization defined inFig. 19(b). This formula is general and does not dependon the particular mechanism, SHE or iSGE, responsiblefor the spin density. In a material displaying additionalsymmetries, such as epitaxial films or single crystals, ad-ditional angular dependencies arise (Hals and Brataas,2013a; Wimmer et al. , 2016; ˇZelezn´y et al. , 2017). Thiscomplex dependence of the SOT on the magnetizationdirection is best captured by writing the effective fieldsin spherical coordinates, B DL = B θ DL cos ϕ e θ − B ϕ DL cos θ sin ϕ e ϕ , (51) B FL = − B θ FL cos θ sin ϕ e θ − B ϕ FL cos ϕ e ϕ , (52)where e θ and e ϕ are the polar and azimuthal unit vectors,respectively, and B θDL,F L and B ϕ DL , FL are functions of themagnetization orientation, defined by the angles θ and ϕ [see Fig. 19(b)]. In polycrystalline bilayers with C v symmetry, the angular dependence of the polar compo-nents simplifies to a Fourier series expansion of the type B θ DL , FL = B { } DL , FL + B { } DL , FL sin θ + B { } DL , FL sin θ + ... . Theazimuthal components, on the other hand, are found byexperiments to be only weakly angle-dependent and areapproximated by B ϕ DL , FL ≈ B { } DL , FL (Garello et al. , 2013).9 B. Measurement techniques
Experimental measurements of SOT rely on probingthe effect of the electric current on the orientation ofthe magnetization, e.g., by inducing resonant and non-resonant oscillations, switching, or domain wall motion.Schematically, one must first determine the magnetiza-tion angle as a function of the amplitude and phase of theapplied current and, second, extract the effective mag-netic fields that are responsible for the observed dynam-ics. In electrical and optical measurements, the magneti-zation dynamics is detected through changes of the trans-verse or longitudinal conductivity, which are mainly dueto the AHE and anisotropic magnetoresistance (AMR),but include also the linear spin Hall magnetoresistance(SMR) (Kim et al. , 2016; Nakayama et al. , 2013), linearRashba magnetoresistance (Kobs et al. , 2011; Nakayama et al. , 2016), as well as nonlinear magnetoresistance termsproportional to the current-induced spin density (Avci et al. , 2015a,b, 2018; Olejn´ık et al. , 2015; Yasuda et al. ,2017, 2016). Further, current injection always resultsin magnetothermal effects due to the thermal gradientsinduced by Joule heating and asymmetric heat dissipa-tion (Avci et al. , 2014a), which affect the conductiv-ity proportionally to j . The thermal gradients thatdevelop along ( ∇ x T ) or perpendicular to the magneticlayer ( ∇ z T ) contribute to the conductivity through theanomalous Nernst effect (ANE) and, to a smaller ex-tent, through the spin Seebeck effect and the inverse spinNernst effect. The direction of the induced voltage is ∼ ∇ T × m , which modifies both the longitudinal ( ∼ m y )and transverse conductivities ( ∼ m x ). The relative influ-ence of the above effects on SOT measurements dependson the system under investigation and experimental tech-nique. The AMR, AHE, and ANE usually dominate themagnetization dependence of the conductivity and canbe properly separated owing to their different symme-try and magnetic field dependence (Avci et al. , 2014a;Garello et al. , 2013) or frequency-dependent optical re-sponse (Fan et al. , 2016; Montazeri et al. , 2015). In thefollowing, we describe the three main techniques used tocharacterize the SOT measurements: harmonic Hall volt-age analysis, spin-torque ferromagnetic resonance (ST-FMR), and magneto-optical Kerr effect (MOKE). Lessprecise SOT estimates can be obtained from magnetiza-tion switching and domain wall displacements, which arediscussed separately in Subsection IV.E and Section VI.
1. Harmonic Hall voltage analysis
This method detects the harmonic response of themagnetization to a low frequency ac current, typicallyup to a few kHz. Originally, this approach was devel-oped by assuming the simplest form of field-like torque(Pi et al. , 2010) and neglecting the damping-like torque
FIG. 20 (Color online) (a) Schematic of the effect of an accurrent on the magnetization and (b) experimental setup forharmonic Hall voltage measurements. (c) R ωxy and (d) R ωxy ofa Pt(5 nm)/Co(1 nm)/AlO x layer measured with a sinusoidalcurrent of amplitude j c = 10 A/cm and external magneticfield applied at ϕ B = 0 ◦ , ◦ , and 90 ◦ . (e,f) Close up of thecurves in (c,d) showing the field range where the small angleapproximation can be applied (Baumgartner, 2018). and the transverse AMR (the planar Hall effect, PHE).It was soon extended to both components of the torquesaccounting for both the AHE and PHE (Garello et al. ,2013; Hayashi et al. , 2014; Kim et al. , 2013a), as well asfor the torque angular dependence (Garello et al. , 2013;Qiu et al. , 2014) and magnetothermal effects (Avci et al. ,2014a; Ghosh et al. , 2017). SOT measurements are per-formed by analyzing the second harmonic Hall voltagethat arises due to the homodyne mixing of the ac currentwith the Hall resistance modulated by the oscillationsof the magnetization induced by the SOTs [Fig. 20(a)].Since the magnetization dynamics is much faster thanthe current frequency ω , the magnetization is assumed tobe in quasi-static equilibrium at all times, at a positiondetermined by the sum of the anisotropy field, externalmagnetic field, and current-induced fields. To first orderin the current, the time-dependent Hall resistance R xy ( t )is given by R xy ( B ext + B I ( t )) ≈ R xy ( B ext )+ dR xy d B I · B I sin( ωt ) , (53)where B ext is the external magnetic field and B I = B DL + B FL + B Oe is the effective current-induced field dueto the sum of the damping-like and field-like SOT and theOersted field. The Hall voltage V xy ( t ) = R xy ( t ) I sin( ωt )0then reads V xy ( t ) ≈ I [ R xy + R ωxy sin( ωt ) + R ωxy cos(2 ωt )] , (54)where I is the current amplitude, R xy = dR xy d B I · B I , R ωxy = R xy ( B ext ), and R ωxy = − dR ω d B I · B I + R ω ∇ T are thezero, first, and second harmonic components of R xy , re-spectively. The first harmonic term, shown in Fig. 20(c)as a function of external field, is analogous to the dc Hallresistance and given by R ωxy = R AHE cos θ + R PHE sin θ sin(2 ϕ ) , (55)where R AHE and R PHE are the anomalous and planarHall coefficients. This term serves two purposes, namelyto determine the polar angle of the magnetization usingEq. (55) when ϕ = 0 ◦ , ◦ and to measure the susceptibil-ity of the magnetization to the magnetic field, providingself-calibration to the SOT measurement. The secondharmonic term includes the SOT modulation of the Hallresistance as well as an extra contribution due to Jouleheating, R ω ∇ T . In general, the two contributions mayhave a comparable magnitude and must be separated byeither symmetry or magnetic field dependent measure-ments (Avci et al. , 2014a; Ghosh et al. , 2017). Assumingthat R ω ∇ T is negligible or has been subtracted from R ωxy ,it is straightforward to show that R ωxy = A θ B I · e θ + A ϕ B I · e ϕ , (56)where A θ = dR ωxy dB ext [ I sin( θ B − θ )] − and A ϕ = R PHE sin θ d sin(2 ϕ ) dϕ [ I sin θ B cos( ϕ B − ϕ ) B ext ] − . Here θ B and ϕ B are the polar and azimuthal angles of the ap-plied magnetic field. Equation (56) allows one to findthe polar and azimuthal components of B DL and B FL asa function of the magnetization angle by measuring thedependence of R ωxy on B ext . Figure 20(d) shows an exam-ple of R ωxy measured at ϕ B = 0 ◦ and ϕ B = 90 ◦ . Thesecurves are, respectively, odd and even with respect tomagnetization reversal, reflecting the different symme-try of B DL and B FL (Garello et al. , 2013). Because thedamping-like torque is larger when m lies in the xz plane,whereas the field-like torque tends to align m towards y ,measurements taken at ϕ B = 0 ◦ ( ϕ B = 90 ◦ ) reflect thecharacter of the damping-like (field-like) effective fields.In general, four independent measurements at differentazimuthal angles are required to determine the four ef-fective field components in Eqs. (51,52).In uniaxial and easy plane systems the number of in-dependent measurements can be reduced to two, typi-cally at ϕ B = 0 , π or ϕ B = π , π (Avci et al. , 2014a;Garello et al. , 2013). A further simplification is achievedusing the small angle approximation, which is valid forperpendicularly magnetized samples when the magneti-zation deviates by at most a few degrees from the z -axis(Hayashi et al. , 2014; Kim et al. , 2013a). In this case, R ωxy varies linearly with the external field, as shown in FIG. 21 (Color online) (a) Schematic of the circuit used forthe ST-FMR measurement and the sample contact geometry.(b) Measured ST-FMR at room temperature with microwavefrequency ω/ π =58 GHz for Bi Se (8 nm)/Ni Fe (16 nm).A fixed microwave power of 5 dBm is absorbed by the device( I RF =7.7 ± B is oriented at an angle ϕ = π/ ϕ for the symmetric and antisymmetric resonance componentsfor a different sample. Adapted from Mellnik et al. (2014). Fig. 20(f) and the SOTs are extracted by performing twosets of measurement at ϕ B = 0 and π , B DL = − − r ( b x + 2 rb y ) , (57) B FL = − − r ( b y − rb x ) , (58)where r = R PHE /R AHE is the ratio between planar andanomalous Hall coefficients, b i = ∂R ωxy ∂B ext / ∂ R ωxy ∂B is mea-sured for B ext (cid:107) i = x, y , and the partial derivatives arecalculated by linear fits of the curves shown in Fig. 20(f).This approximation provides only the lowest order con-tribution to the SOTs. However, because of its simpleimplementation, it is widely used for characterizing theSOTs in systems with perpendicular anisotropy. Har-monic Hall voltage measurements can also be general-ized to angular scans of the magnetization at constantexternal field, which is particularly suited for in-planemagnetized samples (Avci et al. , 2014a), thus providinga versatile and sensitive method to characterize the SOTsin a variety of systems.
2. Spin-torque ferromagnetic resonance
This method consists in exciting the magnetization ofthe ferromagnet using a radio-frequency (RF) charge cur-rent. The magnetization of the sample is excited throughthe spin torque and exhibits FMR when varying either1the applied magnetic field or the current magnitude. Thisconcept was initially developed in the context of MTJs(Kubota et al. , 2007; Sankey et al. , 2007; Tulapurkar et al. , 2005) and spin-valves (Sankey et al. , 2006) andmore recently extended to the case of ultrathin magneticbilayers (Berger et al. , 2018a; Liu et al. , 2011, 2012b)and bulk non-centrosymmetric magnetic semiconductors(Fang et al. , 2011; Kurebayashi et al. , 2014).The dc voltage that develops across the sample[Fig. 21(a)] arises from the mixing of the RF current andthe RF AMR due to the oscillating magnetization. Itcorresponds to the zero harmonic component in Eq. (54)and here is strongly amplified due to the resonant magne-tization dynamics. This rectified voltage gives informa-tion on the physical parameters of the magnetic materialas well as on the nature of the torques that drive the ex-citation. In the context of an in-plane system with AMRdriven by SOTs, the mixing voltage reads (Liu et al. ,2011; Reynolds et al. , 2017) V mix = − γ I RF ∂∂ϕ R cos ϕ B [ τ DL F S ( B ) + τ FL F A ( B )] , (59) F S ( B ) = αω (2 B + µ M s )( ω − ω ) + α γ ω (2 B + µ M s ) , (60) F A ( B ) = γ B (2 B + µ M s ) − αω (2 B + µ M s )( ω − ω ) + α γ ω (2 B + µ M s ) , (61)where ω is the frequency of the RF current I RF and ω = γ (cid:112) B ( B + µ M s ) is the resonance frequency. The firstcontribution has a symmetric Lorentzian shape ( ∼ F S )that is directly proportional to the damping-like torque,while the second has an antisymmetric shape ( ∼ F A ),providing information about the field-like torque (includ-ing the Oersted field torque). A picture of the experi-mental apparatus is given in Fig. 21(a), together withthe field-dependent and angular-dependent mixing volt-ages in Figs. 21(b,c), respectively. This method is usedextensively to probe torques in magnetic bilayers with in-plane magnetization, as well as in non-centrosymmetricbulk magnets, as explained in Section V. This effect is thereciprocal to spin pumping, where the field-excited pre-cessing magnetization pumps a spin current in the adja-cent nonmagnetic metal (Saitoh et al. , 2006; Tserkovnyak et al. , 2002a).Similar to other techniques, applying this method toultrathin bilayer systems requires extreme care. First,the amplitude of the RF current generating the torquesneeds to be calibrated precisely using a network analyzer.Such a calibration might require thickness-dependentmeasurements to characterize possible size-dependent ef-fects (Nguyen et al. , 2016). Second, Eqs. (59), (61) onlyaccount for the rectification arising from AMR, but othersources such as SMR can also contribute to the mix-ing voltage (Nakayama et al. , 2013), which should beproperly accounted for (Wang et al. , 2016c; Zhang et al. ,2016d). Third, the phase difference between the RF cur- FIG. 22 (Color online) (a) Schematic of a MOKE setupfor SOT detection. (b,c) Differential Kerr angle ∆ θ K mea-sured on a Ta(5 nm)/CoFeB(1.1 nm)/MgO(2.0 nm) trilayerwith perpendicular magnetic anisotropy for B ext || j c (b) and B ext ⊥ j c (c) with current amplitude j c = 4 . × Acm − .Adapted from Montazeri et al. (2015). rent and the RF field can also have significant impacton the output signal (Harder et al. , 2011). We refer theinterested reader to the specialized literature for more in-formation (Harder et al. , 2016). A fourth issue is that thismethod assumes the simplest form of the torques, Eq.(2), neglecting the angular dependence of SOTs (Garello et al. , 2013).
3. Magneto-optic Kerr effect
The magneto-optic Kerr effect (MOKE) allows for de-tecting the in-plane and out-of-plane components of themagnetization through the rotation of the light polariza-tion upon reflection from a magnetic surface (Qiu andBader, 2000). MOKE microscopy, with a wavelength-limited resolution of about 1 µ m, has been used exten-sively to characterize SOT-induced domain nucleationand displacement (Emori et al. , 2013; Miron et al. , 2010;Ryu et al. , 2013; Safeer et al. , 2016) as well as the current-induced spin density in bare Pt and W layers (Stamm et al. , 2017). MOKE-based detection schemes have beenused also to estimate the SOT amplitude by measuringthe oscillations of the magnetization induced by an accurrent in thin metal bilayers (Fan et al. , 2014a).Vector measurements of the SOTs are based on the sep-arate calibration of the first- and second-order magneto-optic coefficients, f ⊥ and f (cid:107) , which parameterize the cou-pling of the light to the out-of-plane and in-plane magne-2tization, respectively (Fan et al. , 2016; Montazeri et al. ,2015). Such a technique measures the damping-like andfield-like components of the SOT as a function of themagnetization angle via the polar and quadratic MOKEresponse, respectively, using only normally incident light[see Fig. 22(a)]. Similar to the Hall resistance, Eq. (53),the Kerr rotation measured during ac current injectioncan be Taylor expanded as θ K ( B ext + B I ( t )) ≈ θ K ( B ext ) + dθ K d B I · B I sin( ωt ) . (62)Here, the first term is the equilibrium Kerr angle given by θ K = f ⊥ m z + f (cid:107) [1 / m y − m x ) sin 2 φ p + m x m y cos 2 φ p ],with φ p the angle between the light polarization and B ext , and the second term results in the differential Kerrsignal ∆ θ K = ( dθ K /dI ) I due to the current-inducedfields. In analogy with the harmonic Hall voltage analysistechnique, measurements of ∆ θ K are mostly sensitive tochanges of m z . Thus, measurements taken with B ext || x reflect the strength of the damping-like effective field,∆ θ K = f ⊥ B DL B ext − B K + f (cid:107) cos 2 φ p ( B FL + B Oe ) B ext , (63)where B K is the magnetic anisotropy field and f (cid:107) (cid:28) f ⊥ .Conversely, measurements taken with B ext || y reflect thestrength of the field-like effective field, B FL = − ∂ (∆ θ K ) /∂B ext ∂ θ K /∂B . (64)Figure 22(b) shows that ∆ θ K exhibits an antisymmetric(symmetric) line shape consistent with the symmetry of B DL ( B FL ) under magnetization reversal, in close anal-ogy with R ωxy [Fig. 20(d)]. SOT vector measurementsperformed by MOKE agree well with harmonic Hall volt-age (Montazeri et al. , 2015) and ST-FMR measurements(Fan et al. , 2016) and can be used to characterize theSOT in metallic as well as insulating ferromagnets. Anadvantage of this technique is that it is less sensitiveto thermoelectric and inductive effects compared to all-electrical SOT probes, and that it offers spatial resolutioncomparable to the wavelength of the probing laser beam. C. Materials survey
1. Ferromagnet/nonmagnetic metal layers
The most studied SOT systems are composed of ametallic ferromagnet deposited on a nonmagnetic metallayer, often capped by an amorphous or crystalline oxidelayer. These systems present strong damping-like andfield-like SOTs, of the order of a few mT per 10 A/cm ( ξ j ≈ .
1, see Table II), are easy to fabricate, and com-patible with established processing of magnetic materials
FIG. 23 (Color online) (a) Damping-like SOT efficiency inX(8 nm)/Co/AlO x (2) trilayers, where X = Ti, Cu, Pd, Ta, W,Pt. The data are measured using the harmonic Hall voltageanalysis method. The Co thickness is 2.5 nm except for the Pdsample where it is 0.6 nm. (b) Room temperature resistivityof the nonmagnetic metal. Adapted from Avci et al. (2015b);Ghosh et al. (2017). for memory applications. An early experimental obser-vation of the damping-like SOT in ferromagnetic met-als was reported by Ando et al. (2008) in a Pt/NiFe bi-layer resonantly excited by an external microwave field,by measuring the change of magnetic damping upon in-jection of a dc current. This effect was attributed tothe SHE of the Pt layer and later extended to the ex-citation of FMR upon injection of an RF current (Liu et al. , 2011). Evidence for the field-like SOT was firstreported by Miron et al. (2010) by observing that thecurrent-induced nucleation of magnetic domains in per-pendicularly magnetized Pt/Co/AlO x wires is either en-hanced or quenched by applying an in-plane magneticfield at an angle of ± ◦ relative to the current. Thiseffect was attributed to the action of a Rashba-like effec-tive field and later quantitatively estimated by harmonicHall voltage analysis measurements (Garello et al. , 2013;Pi et al. , 2010).A major breakthrough was achieved in 2011 whenbipolar magnetization switching was demonstrated inperpendicular Pt/Co/AlO x dots (Miron et al. , 2011a),establishing the relevance of SOT for applications. Theauthors observed that the symmetry of the switching fieldcorresponds to a damping-like torque consistent with ei-ther the SHE or the iSGE, and argued that the SHEof Pt alone could not account for the magnitude of thetorque. Other experiments favored a SHE-only expla-nation of the switching mechanism (Liu et al. , 2012a),triggering an ongoing debate on the origin of the torques(see Subsection IV.C.7). These experiments were rapidly3 Structure MA Method B DL /j B FL /j ξ j DL ξ j FL ξ E DL ξ E FL Nonmagnetic metalsPt(3)/Co(0.6)/AlO x (1.6) (Garello et al. , 2013) OP HHV -6.9 4 0.13 -0.073 3.5 -2.0Pt(3)/CoFe(0.6)/MgO(1.8) (Emori et al. , 2013) OP HHV -5 2 0.064 -0.024Ti(1)/CoFe(0.6)/Pt(5) (Fan et al. , 2014a) IP MOKE 3.2 -0.3 0.074 -0.008Pt(5)/Co(1)/MgO(2) (Nguyen et al. , 2016) OP HHV -4.5 1 0.11 -0.024 2.43 -0.53Pt(5)/ Ni Fe (8)/AlO x (2) (Fan et al. , 2016) IP MOKE -0.49 0.71 0.082 -0.12 2.64 -3.88YIG(50)/Pt(4) (Montazeri et al. , 2015) IP MOKE 0.29 0.03TmIG(8)/Pt(5) (Avci et al. , 2017a) OP HHV 0.59 0.014Ta(4)/CoFeB(1.1)/MgO(1.6) (Liu et al. , 2012b) OP HHV 3.5 -0.13 -0.68Ta(3)/CoFeB(0.9)/MgO(2) (Avci et al. , 2014b) OP HHV 3.2 -2.1 -0.06 0.04 -0.34 0.22Ta(3)/CoFeB(0.9)/MgO(2) a (Garello et al. , 2013) OP HHV 2.4 -4.5 -0.07 0.12 -0.36 0.67Ta(1.5)/CoFeB(1)/MgO(2) a (Kim et al. , 2013a) OP HHV 1.35 -4.46 -0.03 0.11 -0.14 0.48Ta(2)/CoFeB(0.8)/MgO(2) a (Qiu et al. , 2014) OP HHV 4.4 -19.4 -0.11 0.47Ta(5)/CoFeB(1.1)/MgO(2) a (Montazeri et al. , 2015) OP MOKE 2.0 -3.3 -0.05 0.08W(5)/CoFeB(0.85)/Ti(1) a (Pai et al. , 2012) IP ST-FMR -0.33Hf(3.5)/CoFeB(1)/MgO(2) a (Torrejon et al. , 2014) OP HHV 0.8 -2.6 -0.02 0.06Hf(3.5)/CoFeB(1.1)/MgO(2) a (Akyol et al. , 2016) OP HHV 5 -0.17Hf(10)/CoFeB(1.1)/MgO(2) a (Akyol et al. , 2016) OP HHV -1 0.03Hf(1)/CoFeB(1)/MgO(2) (Ramaswamy et al. , 2016) OP HHV -0.24 0.9 0.007 -0.03Hf(6)/CoFeB(1)/MgO(2) (Ramaswamy et al. , 2016) OP HHV 9 -27 -0.28 0.82Pd(7)/Co(0.6)/AlO x (1.6) (Ghosh et al. , 2017) OP HHV -1.3 0.7 0.03 -0.015 1.0 -0.55Oxidized metalsWO x (6)/CoFeB(6)/TaN(2) (Demasius et al. , 2016) IP ST-FMR -0.49SiO /Ni Fe (8)/CuO x (10) (An et al. , 2016) IP ST-FMR 0.08 -0.08Ti(1.2)/Ni Fe (1.5)/AlO x (1.5) (Emori et al. , 2016) IP ST-FMR 0.15 -0.01PtO x (32)/Ni Fe (5)/SiO2(4) (An et al. , 2018a) IP ST-FMR 0.9 -0.2 8.7 -1.8Metal alloysCuAu(8)/Ni Fe (1.5) (Wen et al. , 2017) IP HHV -1.9 0.58 0.01 -0.003 0.33 -0.1Au Pt (4)/Co(0.8)/MgO(2) (Zhu et al. , 2018) OP HHV -8.0 3.2 0.28 -0.11 3.3 -1.3Ni Fe (9)/Ag(2)/Bi(4) (Jungfleisch et al. , 2016) IP ST-FMR 0.18 0.14 b,c AntiferromagnetsIrMn(8)/ Ni Fe (4)/Al(2) (Tshitoyan et al. , 2015) IP ST-FMR -2.2 -1.7 0.22 0.17IrMn [001](6)/ Ni Fe (6)/TaN (Zhang et al. , 2016b) IP ST-FMR 0.20IrMn [111](6)/ Ni Fe (6)/TaN (Zhang et al. , 2016b) IP ST-FMR 0.12IrMn (5)/CoFeB(1)/MgO a (Wu et al. , 2016) OP HHV -1.8 0.7 0.06 -0.02PtMn(8)/Co(1)/MgO(1.6) (Ou et al. , 2016) IP ST-FMR 0.16 -0.04MgO(1.6)/Co(1)/PtMn(8) (Ou et al. , 2016) IP ST-FMR 0.19 (cid:39) et al. , 2015) IP ST-FMR -0.34 b c,d b -0.02 c,d MoS (0.8)/CoFeB(3)/TaO x (3) (Shao et al. , 2016) IP HHV (cid:39) (cid:39) . (cid:39) (cid:39) (0.8)/CoFeB(3)/TaO x (3) (Shao et al. , 2016) IP HHV (cid:39) (cid:39) (cid:39) − . (cid:39) /Ni Fe (6)/Al(1) (MacNeill et al. , 2017) IP ST-FMR 0.04 b b c Topological insulatorsBi Se (8)/Ni Fe (16)/Al(2) (Mellnik et al. , 2014) IP ST-FMR 1 1.3 0.5 0.7Bi Se (20)/CoFeB(5)/MgO(2) (Wang et al. , 2015) IP ST-FMR 0.08 b b Bi Se (10)/Ag(8)/CoFeB(7)/MgO(2) (Shi et al. , 2018) IP ST-FMR 5.3 3.2 0.49 0.3(Bi,Sb) Te (8)/CoTb(8)/SiN x (3) (Han et al. , 2017) OP Coercivity -8 0.4Mn . Ga . (3)/Bi . Sb . (10) (Khang et al. , 2018) OP Coercivity -2300 52 130 a Annealed. b Average value. c Sign uncertain.
TABLE II SOTs in magnetic multilayers. The thickness of the layers is given in nm with the topmost layer on the right. Thefollowing units are used for the effective fields and SOT efficiencies: B DL , FL /j [mT/(10 A/m )], ξ j DL , FL [adimensional], and ξ E DL , FL [10 (Ω m ) − ]. The sign of B DL and B FL is defined as in Eqs. (51), (52). ξ DL > ξ FL < B FL is opposite to the Oersted field. The magnetic anisotropy(MA) of the ferromagnetic layers is indicated as out-of-plane (OP) or in-plane (IP). The values for the OP samples are givenfor the magnetization lying close to the easy axis. All measurements have been carried out at room temperature. Here, HHVstands for Harmonic Hall voltage analysis. et al. , 2014b; Emori et al. , 2013; Garello et al. , 2013; Kim et al. , 2013a; Liu et al. , 2012b) and W/CoFeB/MgO layers (Pai et al. ,2012), which showed that the damping-like SOT corre-lates with the sign of the spin-orbit coupling constantand the SHE of the nonmagnetic metal layer, whereasthe field-like torque has a more erratic behavior depend-ing on the type of ferromagnet and interface structure(Pai et al. , 2015).The largest SOT efficiencies are found in the 5 d met-als, in particular for the highly resistive β -phase of Wand Ta as well for fcc Pt (Fig. 23). In metals wherethe spin Hall conductivity σ sh is of intrinsic origin, thespin Hall angle is directly proportional to the longitudi-nal resistivity, given by θ sh = σ sh ρ . Pt and Pd displaya large SOT efficiency despite their moderate resistivity(Ghosh et al. , 2017; Nguyen et al. , 2016), which is at-tributed to their large intrinsic σ sh and density of statesat the Fermi level (Freimuth et al. , 2010, 2015). Largedamping-like and field-like SOT efficiencies have been re-ported also by replacing the nonmagnetic metal by anintermetallic antiferromagnet such as IrMn (Oh et al. ,2016; Tshitoyan et al. , 2015; Wu et al. , 2016; Zhang et al. ,2016b) and PtMn (Ou et al. , 2016), which allows for in-cluding exchange-biased systems in SOT devices (Sub-section IV.E.3).Enhanced efficiencies can be obtained in multilayerswhere the ferromagnet is sandwiched between two non-magnetic metals with opposite spin Hall angle, giving riseto parallel damping-like torques at opposite interfaces(Woo et al. , 2014; Yu et al. , 2016b). In such systems,the spin current associated with the PHE in the bulk ofthe ferromagnet can give rise to an additional damping-like torque if the spin transfer to the nonmagnetic metalsis asymmetric (Safranski et al. , 2019). Results obtainedon symmetric multilayers such as [Co/Pd] n (Jamali et al. ,2013), on the other hand, are more controversial becauseof the expected compensation of the SOT from the topand bottom interfaces and the missing analysis of ther-mal voltages.In general, significant variations of the torque efficien-cies have been observed depending on multilayer compo-sition, thickness, thermal annealing protocols, interfaceoxidation and dusting, as well as temperature, which webriefly describe below. a. Thickness dependence Assuming that the charge-spinconversion in multilayer systems occurs outside the fer-romagnetic volume, one expects the SOTs to be sim-ply inversely proportional to the ferromagnet thickness( ∼ /t F ), as the effects of the current-induced fields areinversely proportional to the magnetic volume on whichthey act upon, and strongly dependent on the nonmag-netic metal thickness ( t N ) as well as on interfacial proper- PtPdTaW
FIG. 24 (Color online) Damping-like torque efficiency asa function of thickness in NM( t N )/Co/AlO x layers, whereNM=Pt, Pd, Ta, and W (Garello et al. , 2017). The solidlines are fit to the function ξ j DL [1 − sech( t N /λ sf )]. Note thatthe efficiency ξ j DL of W drops abruptly between 5 and 6 nmas the crystal structure changes from the β to the α phase. ties. The influence of t F on the SOT has been systemati-cally investigated in Ta/CoFeB/MgO (Kim et al. , 2013a),NiFe/Cu/Pt (Fan et al. , 2013), Ti/CoFeB/Pt (Fan et al. ,2014a), Co/Pt (Skinner et al. , 2014), Pt/Co/MgO andPt/Co Fe /MgO (Pai et al. , 2015), and Pd/FePd (Lee et al. , 2014a), all deposited on thermally oxidized Si. Kim et al. (2013a) found that B FL decreases strongly while B DL remains approximately constant in Ta/CoFeB/MgOwhen increasing t F from 0.8 to 1.4 nm. Fan et al. (2014a)showed that both fields decrease when increasing t F from0.7 to 6 nm, with B FL dropping significantly faster than1 /t F . The spin torque efficiencies, ξ j DL , FL , have beenfound to decrease in annealed Pt/Co Fe /MgO lay-ers between 0.6 and 1 nm, but to increase in as-grownPt/Co/MgO (Pai et al. , 2015), possibly because the Cothickness has to exceed the spin absorption length (i.e.,the length over which the spin current is absorbed inthe ferromagnet) in order to develop the full torque orbecause of strain relaxation in the Pt/Co layer. Interest-ingly, the sign of the field-like torque is opposite in thesetwo systems. Skinner et al. (2014) have found a sign in-version of the field-like torque in Co/Pt for a 2 nm thickCo layer, which suggests that two mechanisms with dif-ferent dependence on t F compete to determine the totaltorque.The dependence of the SOT on t N has been the fo-cus of many studies aimed at distinguishing the bulkand interfacial nature of the torques. In the simplesttheoretical models, effects coming from the interfacialRashba interaction should be independent of t N , whereaseffects emerging from the bulk SHE should scale as[1 − sech( t N /λ sf )] according to the profile of the spin den-sity in the nonmagnetic metal layer (Liu et al. , 2011).In addition, the Oersted field should increase linearlywith t N . Therefore, assuming that the overall structure5(crystallinity, interface and inter-diffusion processes) isunchanged upon modifying t N , analyzing the thickness-dependence of ξ j DL and ξ j FL should provide informationabout the physical origin of the torques. Figure 24 showsthat ξ j DL of as-grown Co/AlO x layers deposited on β -Ta, β -W, and Pt increases monotonically with t N up tosaturation, which agrees well with the SHE model as-suming a spin diffusion length of the order of 1.5 nmfor all metals. Such a trend is common to a vari-ety of systems based on Ta (Torrejon et al. , 2014), W(Hao and Xiao, 2015), Pt (Nguyen et al. , 2016), and Pd(Ghosh et al. , 2017), suggesting that the SHE is the dom-inant source of the spin current causing the damping-liketorque. Recent theoretical work, however, has pointedout that a similar t N dependence is expected for aRashba-like damping-like torque due to interfacial spin-dependent scattering (Amin and Stiles, 2016b; Haney et al. , 2013b), so that separating the bulk and interfacecontributions to ξ j DL is not straightforward. Moreover, achange of sign of both ξ j DL and ξ j FL has been reported forTa/CoFeB/MgO (Allen et al. , 2015; Kim et al. , 2013a)and Hf/CoFeB/MgO (Akyol et al. , 2016; Ramaswamy et al. , 2016) at t Ta ≈ . t Hf ≈ t N and be propor-tional to the real and imaginary part of the spin mixingconductance of the FM/NM interface, respectively, whichnaturally leads to ξ j DL (cid:29) ξ j FL (Haney et al. , 2013b).Several reports, however, show that ξ j FL (cid:38) ξ j DL in out-of-plane as well as in-plane magnetized layers (Table II)and that the dependence of ξ j FL on t N differs from thatof ξ j DL in systems based on Ta (Kim et al. , 2013a), Pt(Fan et al. , 2014a; Nguyen et al. , 2016), and Pd (Ghosh et al. , 2017), particulary at low thickness ( t N < B , E FL FL B , E DL DL B D L , F L / j ( m T / A c m - ) a B , j FL FL B , j DL DL j D L , F L fit t Pd (nm) B Oe b B D L , F L / E ( - T V - m ) t Pd (nm) D L , F L E ( Ω - m - ) fit FIG. 25 (Color online) SOT efficiency in Pd(t)/Co(0.6)/AlO x trilayers as a function of Pd thickness. (a) ξ j DL , FL and(b) ξ E DL , FL differ significantly from each other due to thestrong decrease of the Pd resistivity with increasing thick-ness (Ghosh et al. , 2017). The field-like torque efficiencyis shown after subtraction of the Oersted field contribution B Oe = µ j Pd t Pd / a perpendicularly magnetized Pd/Co/AlO x layer, where ξ j FL clearly departs from the monotonic increase of ξ j DL as a function of t Pd . Remarkably, the thickness depen-dence changes when the SOT efficiency is normalized tothe electric field, as in Fig. 25(b), showing that ξ E DL and ξ E FL do not saturate up to t N = 8 nm and that ξ E FL ex-trapolates to a finite value at t Pd = 0. The differencebetween ξ E DL , FL and ξ j DL , FL also suggests that the thick-ness dependence should be analyzed with care in filmswhen the resistivity is not homogeneous (Ghosh et al. ,2017; Nguyen et al. , 2016). b. Interfacial tuning The transport of charge and spin inmultilayer systems is strongly affected by interface scat-tering and discontinuities in the electronic band struc-ture, as is well known from early studies of the giantmagnetoresistance (Levy, 1994; Parkin, 1993). Thus,significant variations of the SOTs are expected uponmodification of the interfaces, even when the nonequi-librium spin density originates in the bulk of the non-magnetic metal layer. Experimentally, it has been shownthat the damping-like and field-like SOTs change dra-matically upon annealing and consequent intermixing ofPt/Co/AlO x (Garello et al. , 2013) and Ta/CoFeB/MgO(Avci et al. , 2014b), as well as upon the insertion of differ-ent spacer layers between the ferromagnet and the non-magnetic metal that is considered to be the main sourceof spin density (Fan et al. , 2013; Pai et al. , 2014; Zhang et al. , 2015c). The insertion of a light metal such asCu has been pursued with the intention of removing theinterfacial spin-orbit coupling. Fan et al. (2014a, 2013)measured a field-like torque that decreases smoothly withthe thickness of the Cu spacer in Pt/Cu/NiFe, indicat-ing a nonlocal origin, but also that the ξ j FL /ξ j DL ratio ofCoFeB/Cu/Pt has a discontinuity around t Cu = 0 . et al. , 2014a; Nan et al. , 2015; Rojas-S´anchez et al. , 2014), which is two orders of magnitude smallerthan the spin diffusion length in Cu.The insertion of a spacer layer can also modify the abil-ity of the ferromagnet to absorb the incoming spin cur-rent, by modifying both the transparency (Nguyen et al. ,2015; Zhang et al. , 2015c) and the spin memory loss atthe interface (Berger et al. , 2018b; Dolui and Nikolic,2017; Rojas-S´anchez et al. , 2014; Tao et al. , 2018). Theformer accounts for the spin current backflow in the non-magnetic metal (the larger the backflow, the smaller the6 FIG. 26 (Color online) (a) Effect of gate voltage on thefield-like and damping-like SOT in Pt/Co/Al O (Liu et al. ,2014a). (b) Inversion of the polarity of current-inducedswitching for different thickness of the oxide capping layerin Pt/CoFeB/SiO (Qiu et al. , 2015). spin current transmission into the ferromagnet), whilethe latter opens an additional spin dissipation channel atthe interface (see Subsection III.C). Both effects reducethe effective spin injection. A typical case is that of Hf,which has been shown to improve the SOT efficiency inW/Hf/CoFeB/MgO and Pt/Hf/CoFeB/MgO whilst pro-moting perpendicular magnetic anisotropy (Pai et al. ,2014) and reducing the magnetic damping (Nguyen et al. ,2015). Changes in the SOT efficiency in such cases areusually interpreted in terms of an enhanced spin mixingconductance, which may also explain why the damping-like torque efficiency changes for different ferromagnetscoupled to the same nonmagnetic metal, as observed,e.g., in Pt/Co/TaN ( ξ j DL = 0 .
11) and Pt/NiFe/TaN( ξ j DL = 0 .
05) (Zhang et al. , 2015c). Such a phenomeno-logical parameter, however, accounts for the transmissionof the bulk spin current as much as for the generation ofinterfacial spin currents, so that its use to estimate anasymptotic value of the bulk SHE in nonmagnetic met-als can be questioned. Moreover, the spacer layer itselfcan be regarded as a source of spin current, as has beenshown in the case of Hf (Akyol et al. , 2016; Ramaswamy et al. , 2016).Another interesting aspect is the control of magneticproperties through interfacial oxidation (Manchon et al. ,2008; Monso et al. , 2002; Rodmacq et al. , 2009) or gatevoltage (Bauer et al. , 2015; Maruyama et al. , 2009; Sh-iota et al. , 2012; Wang et al. , 2012a; Weisheit et al. ,2007). Using photoemission spectroscopy, Manchon et al. (2008) showed that both the perpendicular magneticanisotropy and AHE reach a maximum in Pt/Co/AlOxtrilayers when the Co/AlOx interface is optimally oxi-dized. This effect is connected to the dependence of theinterfacial magnetic anisotropy on the electron densityand orbital character of the interface atoms (Dieny andChshiev, 2017; Yang et al. , 2011). It is therefore natu-ral to expect that other spin-orbit coupling properties,such as SOT (Freimuth et al. , 2014b) or DMI (Belabbes et al. , 2016; Srivastava et al. , 2018), can be controlledby tuning the interfacial electron density through oxida-tion or by applying a gate voltage (Emori et al. , 2014;Liu et al. , 2014a; Qiu et al. , 2015). Miron et al. (2011a)first showed that moderate oxidation of Pt/Co/AlO x fa-vors current-induced switching, as recently confirmed inPt/Co/CoO x layers oxidized in air, in which up to a two-fold enhancement of the SOT efficiency was measuredrelative to Pt/Co/MgO (Hibino et al. , 2017). On theother hand, Liu et al. (2014a) demonstrated that bothfield-like and damping-like torques can be modified bygating Pt/Co/Al O multilayers [Fig. 26(a)], obtainingan enhancement of 4% (1%) of the field-like (damping-like) torque for a gate voltage of about 7 V. Since thegate voltage essentially modifies the electric dipole of theCo/Al O interface and leaves the SHE from Pt unaf-fected, this observation provides some indication aboutthe origin of the SOTs in this system. Liu et al. (2014a)estimated that the SHE does not contribute to morethan 20% of the field-like torque, while the interfacialspin-orbit coupling produces about 50% of damping-liketorque. Emori et al. (2014) carried out measurements ongated Pt/Co/GdO x , showing that oxidation of the topCo interface leads to a 10-fold increase of the damping-like torque due to oxygen ion migration, which also af-fects the magnetic anisotropy. Qiu et al. (2015) demon-strated the spectacular impact of interfacial oxidationon SOTs in Pt/CoFeB/SiO , where the oxidation of theCoFeB/SiO layer is varied continuously. They reportedthat the sign of both damping-like and field-like torqueschanges from positive to negative when increasing the ox-idation of CoFeB [see Fig. 26(b)]. The authors attributedthis change of sign to the increase of the orbital momentof Fe and Co upon oxidation (Nistor et al. , 2011; Yang et al. , 2011). This results in an enhancement of the inter-facial SOT at the upper CoFeB/SiO interface that caneven dominate over the SOT arising from the bottomPt/CoFeB.Oxidation of the bottom Pt interface in Pt/Ni Fe bilayers also leads to drastic enhancements of bothdamping-like and field-like torque efficiencies, which canbe controlled by the oxygen flow during sputter deposi-tion as well as by a gate voltage (An et al. , 2018a,b). In-terestingly, the maximum SOT efficiency in this system, ξ j DL = 0 .
92 ( ξ E DL = 9 × Ω − m − ), is reached for a fullyoxidized nonconducting Pt layer. Other reports revealan enhancement of ξ j DL from -0.14 to -0.49 upon oxida-tion of W in W/CoFeB/TaN (Demasius et al. , 2016) andthe emergence of strong SOT in as-grown SiO x /Co/Cu(Verhagen et al. , 2015) and oxidized SiO x /NiFe/Cu lay-ers (An et al. , 2016), with contrasting evidence on therole played by the oxidized interfaces. These experimentsshow that interfacial spin-orbit coupling can produce sig-nificant field-like and damping-like torques, but also thata detailed microstructural analysis of the bulk vs inter-face oxidation is required to understand the role of oxy-7gen in inducing or modifying the SOT.Finally, Qiu et al. (2016) demonstrated a 3-fold en-hancement of the SOT magnitude in a Pt/Co/Ni/Comultilayer by capping the system with Ru. This result isinterpreted in terms of enhanced spin absorption inducedby the negative spin polarization arising at the Co/Ru in-terface (Nozaki et al. , 2004) and could partly explain thevery large SOT magnitude measured in synthetic anti-ferromagnetic domain walls (Yang et al. , 2015). Recentwork on IrMn /NiFe epitaxial layers also shows that ξ j DL has a facet-dependent contribution, which arises from thedifferent orientation of the Mn magnetic moments at dif-ferent interfaces (Zhang et al. , 2016b). c. Angular dependence As mentioned in SubsectionIV.A, the SOTs are anisotropic, i.e., their magnitudechanges depending on the magnetization direction in away that is more complex than described by Eq. (2).In polycrystalline systems with C v symmetry, the mag-nitude of this anisotropy is characterized by the coeffi-cients τ { n } DL , FL in Eq. (49). As measured in Pt/Co/AlO x (Garello et al. , 2013), Pt/Co/MgO (Gweon et al. ,2019), Ta/CoFeB/MgO (Avci et al. , 2014b; Qiu et al. ,2014), and Pd/Co/AlO x (Ghosh et al. , 2017), the SOTanisotropy can be quite large. Figure 27 shows thatboth field-like and damping-like torques increase in abso-lute value when the magnetization points in-plane, whichis the typical behavior observed in metal layers. Theanisotropies of the field-like and damping-like compo-nents differ from each other and can reach up to a factorof 4 depending on the material and annealing conditions. FIG. 27 (Color online) Angular dependence of B DL and B FL measured in as-grown Pt(3)/Co(0.6)/AlO x (Garello et al. ,2013) and Ta(3)/CoFeB(0.9)/MgO (Avci et al. , 2014b) atroom temperature. The angle θ between the magnetiza-tion and the z -axis is determined by anomalous Hall resis-tance measurements. The solid lines are fits to the function B DL , FL θ = B DL , FL0 + B DL , FL2 sin θ . FIG. 28 (Color online) Temperature dependence of (a) B FL /j (∆ H T /j ) and (b) B DL /j (∆ H L /j ) in Ta/CoFeB(1)/MgO(2)layers with different Ta thickness (Kim et al. , 2014). Thebottom panels show a magnified view of the fields for thethinner Ta layers. Solid and open symbols correspond to themagnetization pointing along + z and - z , respectively. The angular dependence of the SOT, although quitegeneral, provides additional clues about the physics tak-ing place in these ultrathin layers. Different physicalmechanisms can generate such an angular dependence:(i) the presence of D’yakonov-Perel relaxation (OrtizPauyac et al. , 2013), (ii) the distortion of the Fermi sur-face when changing the magnetization direction due tostrong spin-orbit coupling (Haney et al. , 2013a; Lee et al. ,2015), and (iii) the angular dependence of the interfa-cial mixing conductance, i.e., the change of spin absorp-tion and reflection as a function of the magnetizationdirection (Amin and Stiles, 2016a; Baek et al. , 2018).Additional effects related to spin scattering in the non-magnetic metal may also be relevant, such as, e.g., spinswapping (Saidaoui and Manchon, 2016). Interestingly,Qiu et al. (2014) reported that the angular dependence ofthe two torque components vanishes when decreasing thetemperature, an observation that highlights the impor-tance of scattering events in the emergence of the angulardependence of the SOTs. Finally, systems characterizedby low crystalline symmetry may display additional con-tributions not included in Eqs. (49) and (50) due to thespecific symmetry of the spin and orbital textures(Chen et al. , 2016; MacNeill et al. , 2017). d. Temperature dependence
A way to obtain informa-tion on the physics governing the SOTs is to mea-sure their magnitude as a function of temperature. InTa/CoFeB/MgO, Qiu et al. (2014) reported that thefield-like torque decreases linearly when reducing thesample temperature, while the damping-like torque re-mains mostly unaffected. A qualitatively similar behav-ior was observed by Kim et al. (2014) in similar struc-tures, i.e., the field-like torque decreases dramaticallywith the temperature, while the damping-like torque in-8creases from 400 to 300 K and saturates at lower temper-atures (Fig. 28). Since the Ta resistivity is almost con-stant between 100 and 400 K, the relative independenceof the damping-like torque on temperature is consistentwith the damping-like torque being driven by the intrin-sic SHE of Ta. In contrast, the strong decrease of thefield-like torque suggests that scattering events involv-ing phonons and magnons (usually stronger at disorderedinterfaces) play an important role in the emergence ofthis component. Studies of the temperature dependenceof the SOTs in Pt-based structures, on the other hand,show that the field-like and damping-like SOTs are bothapproximately constant with temperature in as-grownPt/Co/MgO, whereas both increase with temperature inannealed Pt/CoFeB/MgO (Pai et al. , 2015). In the lattercase, the field-like torque shows a much stronger changecompared to the damping-like torque and even changessign, from parallel to antiparallel to the Oersted field, ataround 125 K.In bilayers including disordered alloys of nonmagneticmetals such as Cu x Au − x , where extrinsic effects dom-inate, the damping-like torque decreases upon reducingthe temperature, consistently with an extrinsic bulk-likeSHE, whereas the field-like torque increases Wen et al. (2017). On the other hand, alloys that present a ferro-magnetic to paramagnetic transition, such as Fe x Pt − x incombination with a ferromagnet such as CoFeB, display amaximum of the damping-like torque near the Curie tem-perature, which is attributed to spin fluctuation enhance-ment of the SHE arising from the interaction betweenthe conduction electrons and the localized magnetic mo-ments (Ou et al. , 2018). Ab-initio calculations addition-ally show that the generation and absorption of spin cur-rents in an ordered FePt alloy are extremely sensitive tothe distribution of defects near the interface (G´eranton et al. , 2016). Overall, these studies support the viewthat intrinsic as well as extrinsic mechanisms contributein different proportion to the field-like and damping-liketorques.
2. Ferrimagnet and antiferromagnet/nonmagnetic metal layers
Ferrimagnetic films were once widely used as record-ing media in bubble memories (Bobeck et al. , 1975) andmagneto-optic memories (Jenkins et al. , 2003). Appli-cations included both insulating garnets (Nielsen, 1976)and amorphous rare-earth 3 d transition-metal alloys(Buschow, 1984). Depending on their composition, ferri-magnets can exhibit a magnetization compensation tem-perature ( T M ) where the magnetizations of the two an-tiparallel coupled sublattices cancel each other and, sim-ilarly, an angular momentum compensation temperature( T A ) where the total angular momentum of the two sub-lattices vanishes (Buschow, 1984; Hirata et al. , 2018;Nielsen, 1976). The frequency of the uniform spin pre- cession mode as well as the magnetic damping are ex-pected to diverge at T A (Stanciu et al. , 2006; Wangsness,1954), which makes these materials extremely interest-ing for ultrafast switching (Mangin et al. , 2014; Stanciu et al. , 2007) as well as fast domain wall motion (Caretta et al. , 2018; Kim et al. , 2017a; Kobayashi et al. , 2005;Siddiqui et al. , 2018).SOT-induced switching of ferrimagnets has been re-ported for amorphous ferrimagnetic alloys, such asTa/TbFeCo (Zhao et al. , 2015), Ta/TbCo (Finleyand Liu, 2016), Pt/GdCo (Mishra et al. , 2017), andPt/GdFeCo (Roschewsky et al. , 2017), as well as rareearth garnets such as Tm Fe O /Pt (Avci et al. ,2017a,b; V´elez et al. , 2019) and Tm Fe O /W (Shao et al. , 2018). In contrast to ferromagnets, the re-duced saturation magnetization of these systems al-lows for switching relatively thick layers, up to 30 nm(Roschewsky et al. , 2017), at current densities of the or-der of 10 A/cm . Moreover, because of the alternanceof magnetic moments with opposite orientation on neigh-boring atomic sites, spin dephasing due to spin preces-sion in metallic ferrimagnets partially cancels out, whichallows a spin current to propagate for several nm insidethese materials, as reported for Pt/[Co/Tb] N multilayersand Pt/CoTb amorphous alloys (Yu et al. , 2019). Theseproperties, combined with the bulk perpendicular mag-netic anisotropy of rare-earth 3 d transition-metal com-pounds, make ferrimagnets very interesting for applica-tions requiring relatively thick magnetic layers.Measurements of the SOT as a function of tempera-ture (Ham et al. , 2018; Ueda et al. , 2017) and composi-tion (Finley and Liu, 2016; Je et al. , 2018; Roschewsky et al. , 2017) show that the damping-like effective fieldtends to diverge as B DL ∝ /M s near T M , whereas ξ j DL isroughly constant across T M , as expected. In some cases,however, a disproportionate scaling between B DL and1 /M s has been observed, leading to a considerable en-hancement of ξ j DL of yet unclear origin (Je et al. , 2018;Mishra et al. , 2017; Yu et al. , 2019). As not only M s and T M , but also the magnetic anisotropy, Gilbert damping,spin-orbit scattering, and spin dephasing depend on thecomposition and thickness of these systems, it is not sur-prising that the simple 1 /M s scaling has no general valid-ity. An interesting point is that, even in systems where B DL ∝ /M s , the threshold switching current does notdecrease at T M , but rather changes smoothly as a func-tion of composition (Je et al. , 2018) or thickness (Yu et al. , 2019). This behavior agrees with a macrospinmodel based on the LLG equation for two antiferromag-netically coupled lattices, which shows that the thresholdswitching current scales with the effective perpendicularanisotropy (Je et al. , 2018). The latter depends on thesum of effective anisotropy energy of each lattice, whichdoes not cancel out at the compensation point.In fully compensated bipartite antiferromagnets, sim-ulations predict that the N´eel order can be manipulated9via damping-like SOT (Gomonay and Loktev, 2010) (seeSubsection III.F). It has been recently shown that currentinjection in Pt/NiO (Chen et al. , 2018b) and Pt/NiO/Pt(Moriyama et al. , 2018) leads to switching of the N´eelvector of up to 90 nm thick films of NiO, independentlyof the strain state and crystallographic orientation (Bal-drati et al. , 2018). Contrasting mechanisms have beenproposed to explain this type of switching, based on thecoherent rotation of the N´eel vector (Chen et al. , 2018b),field-like SOT acting on uncompensated interfacial spins(Moriyama et al. , 2018), as well as rotation of the N´eelvector inside individual domains combined with the dis-placement of the domain walls driven by the damping-likeSOT (Baldrati et al. , 2018).
3. Ferromagnet/semiconductor layers
We now turn from purely metallic systems to ferromag-net/semiconductor bilayers, in which the semiconductorhas a specific crystal structure that brings about addi-tional symmetries on top of the one arising from interfa-cial inversion symmetry. For instance, in zinc-blende lat-tices under strain, such as GaAs, a spin accumulation canbe generated via the iSGE driven by Rashba and Dres-selhaus spin-orbit coupling as well as by the bulk SHE(see Sect. V). Differently from the commonly studiedpolycrystalline NM/FM samples, where the iSGE-basedand the SHE-based mechanisms are indistinguishable inthe lowest order terms (Garello et al. , 2013), the depen-dence of the torques on the angle of the current relativeto the high symmetry directions of the semiconductorcrystal provides a direct means to disentangle the SHEand iSGE contributions. Skinner et al. (2015) provedthis point by investigating the SOTs of an epitaxial Fe(2nm)/(Ga,Mn)As(20 nm) bilayer using the ST-FMR tech-nique. The GaAs host was doped with high enoughconcentration of substitutional Mn acceptors to increasethe semiconductor conductivity, but low enough so that(Ga,Mn)As remains paramagnetic at room temperature.It was then shown that the field-like and damping-liketorques have similar magnitude, with the first originatingfrom the iSGE with Dresselhaus symmetry and the sec-ond from the SHE-like spin current generated inside theparamagnetic p -doped GaAs layer. Chen et al. (2016),on the other hand, showed that the SOT of epitaxialFe films grown on non-conducting GaAs(001) originatefrom the interfacial iSGE and have mixed Rashba andDresselhaus symmetry, which also leads to the emergenceof an unusual crystalline magnetoresistance (Hupfauer et al. , 2015). The interfacial spin-orbit interaction andSOT can further be modulated by applying a gate volt-age across the Schottky barrier at the Fe/GaAs interface(Chen et al. , 2018a).Evidence of strong SOTs due to the iSGE has beenobserved also in heterostructures involving transition FIG. 29 (Color online) (a) Electrical excitation and detec-tion of FMR induced by a 16.245 GHz RF current in a Fe(2nm)/(Ga,Mn)As(20 nm) bilayer. A typical ST-FMR curve(points) is shown as a function of external field. The dc volt-age is fitted (solid green line) by a combination of symmet-ric (red dotted line) and antisymmetric (blue dashed line)Lorentzians. (b) Dependence of the fitted Lorentzian ampli-tudes on the in-plane magnetization angle for a device withcurrent in the [010] direction. (c) iSGE dependence on thedirection of the current. The fitted in-plane field coefficients(representing the field-like torque) for a set of devices in dif-ferent crystal directions. (d) The fitted out-of-plane field co-efficient (representing the damping-like torque) for the samedevices. Adapted from Skinner et al. (2015). metal dichalcogenides and metallic ferromagnets. Vander Waals crystals provide a unique platform for generat-ing SOTs because they have strong spin-orbit coupling, arange of broken crystal symmetries, and can be preparedas monolayer crystalline films by exfoliation or chemi-cal vapor deposition methods (Manchon et al. , 2015; Xu et al. , 2014). Shao et al. (2016) showed that the field-liketorque in 1 nm CoFeB deposited on monolayer MoS andWSe is of the order of 0.1-0.14 mT/(10 A/cm ), inde-pendently of temperature, and is consistent with iSGE-induced spin density, whereas the damping-like torqueis negligibly small. Sizable damping-like SOTs, on theother hand, have been reported for NiFe deposited onMoS (Zhang et al. , 2016c) and on the Weyl semimetalWTe (MacNeill et al. , 2017). The latter case is of par-ticular interest as the surface crystal structure of WTe has only one mirror plane and no two-fold rotational in-variance about the c -axis, which allows for a damping-like torque that is directed out-of-plane when the cur-rent is applied along a low-symmetry axis of the sur-face. Such a damping-like torque is forbidden by sym-metry in NM/FM bilayers, where the direction of theincoming spin polarization is in-plane. The possibilityof controlling the allowed symmetry of the damping-likeSOT in multilayer samples is particularly attractive forcounteracting the torque due to magnetic damping dur-ing magnetization reversal in systems with perpendicular0magnetic anisotropy. Further, the current-induced spindensity in two-dimensional materials is expected to beextremely sensitive to gating, thus allowing for tuningthe SOT efficiency.
4. Ferromagnet/topological insulator layers
Three dimensional topological insulators are materialsthat have insulating bulk and conductive surface states(Hasan and Moore, 2011; Roche et al. , 2015). The sur-face states are protected by time-reversal symmetry andhave a Dirac-like linear dispersion characterized by spin-momentum locking (Fig. 8), a property that makes themvery attractive in the context of SOT and spintronics.Moreover, owing to hexagonal-warping of the Dirac cone(Kuroda et al. , 2010), a current carried by the topologicalsurface states can generate a nonequilibrium spin densitywith both in-plane and out-of-plane components, whichcan induce out-of-plane and in-plane torques onto an ad-jacent magnetic layer.The most thoroughly investigated topological insula-tors to date are the bismuth and antimony chalcogenidesM Q , where M = Bi, Sb and Q = Se, Te. In intrinsicsystems, the Fermi level resides in the bulk energy gapand thus only intersects the topological surface states.However, these materials are narrow gap semiconductorsthat are very sensitive to doping by impurities or crys-talline defects, which typically shifts the Fermi level tothe conduction band. Furthermore, unintentional sur-face doping caused by the formation of extrinsic defectsor the adsorption of impurities leads to the emergence ofa two-dimensional electron gas with strong Rashba-splitbands that are wrapped by the topological Dirac states(King et al. , 2011). These effects have a strong influenceon SOT, which has not yet been fully understood.Topological insulator thin films are usually grown bymolecular beam epitaxy (He et al. , 2013). In order tofavor surface transport over bulk conduction, it is neces-sary to minimize defects such as Se/Te vacancies, dislo-cations, and twin domains. This task has proven to bequite challenging, requiring careful optimization of thelattice matching with the substrate and growth kinetics(Bonell et al. , 2017; Richardella et al. , 2015). Bulk insu-lating materials can be obtained also by growing natu-rally compensated ternary alloys such as (Bi − x Sb x ) Te ,which exploit the tendency of Bi Se and Bi Te to be n -type and of Sb Te to be p -type (Zhang et al. , 2011).Since topological insulators involve heavy elements andspin-momentum locked electron states, they are expectedto show large charge-spin conversion and SOT efficiencywhen interfaced with a magnetic layer. However, threeissues arise when considering these systems. First, theproximity between a ferromagnet and a topological in-sulator induces complex electronic hybridization effects,which go beyond the simple notion of a magnetic ex- change field breaking time reversal symmetry and open-ing a gap in the surface states (Wray et al. , 2011). Us-ing first-principles calculations, Zhang et al. (2016a) andMarmolejo-Tejada et al. (2017) showed that charge trans-fer between Bi Se and 3 d metal layers such as Co, Ni,and Cu, shifts the topological surface states below theFermi energy, where hybridization with the metal bandsdestroys or heavily distorts the helical spin structure.Crucially for SOTs, it was found that proximity spin-orbit coupling also modifies the electronic states of theferromagnet adjacent to Bi Se , leading to a Rashba-like spin texture (Marmolejo-Tejada et al. , 2017). Thus,the properties of magnetic/topological insulators bilay-ers, even in the theoretical approximation of ideal mate-rials and interfaces, cannot be extrapolated from thoseof the parent layers. Second, the interface chemistry be-tween a topological insulator such as Bi Se and typicalcontact metals (Pd, Ir, Cr, Co, Fe, Ni) leads to the for-mation of metal selenides, metallic Bi, or intermetallicalloys, which can evidently alter the properties of thepristine materials (Walsh et al. , 2017). Third, becauseof the competition between bulk and surface conduction,which depends on temperature and extrinsic factors, itis hardly possible to determine the current distributionin magnetic/topological insulators bilayers. This uncer-tainty makes it difficult to identify the electronic statesresponsible for charge-spin conversion as well as to pro-vide consistent estimates of the SOT efficiency in differ-ent systems.Regardless of the role played by the topological surfacestates, mounting experimental evidence suggests thatstrong spin-momentum coupling can be achieved in thesematerials. Spin-charge conversion has been reported byspin pumping for bismuth and antimony chalcogenidelayers adjacent to metallic ferromagnets (Deorani et al. ,2014; Jamali et al. , 2015; Kondou et al. , 2015; Mendes et al. , 2017; Shiomi et al. , 2014) and insulating ferri-magnets (Tang et al. , 2018; Wang et al. , 2016a) as wellas by magnetoresistance measurements (Ando, 2014; Li et al. , 2014; Yasuda et al. , 2016). Current-induced SOTshave been demonstrated by ST-FMR in Bi Se /NiFeand Bi Se /CoFeB bilayers (Mellnik et al. , 2014; Wang et al. , 2015), gate control of the torque efficiency (Fan et al. , 2016), and magnetization switching (Fan et al. ,2014b; Han et al. , 2017; Khang et al. , 2018; Wang et al. ,2017b; Yasuda et al. , 2017). In these experiments, thereported damping-like torque efficiency is widely dis-tributed from 0.01 to 2 for Bi Se , reaching ∼
50 inBi Sb /MnGa (Khang et al. , 2018) and even larger valuesin (Bi . Sb . ) Te /(Cr . Bi . Sb . ) Te (Fan et al. ,2014b). In the latter case, however, the SOT analysisis complicated by nonlinear Hall effects (Yasuda et al. ,2017).Demonstrations of room temperature SOT-drivenswitching with threshold currents that are about one or-der of magnitude smaller compared to NM/FM bilay-1ers are particularly interesting in view of possible appli-cations. Wang et al. (2017b) reported switching of in-plane magnetized Bi Se /NiFe with a critical current of ∼ × A/cm , while Han et al. (2017) demonstratedswitching of perpendicularly magnetized Bi Se /CoTb at ∼ × A/cm and Khang et al. (2018) obtained asimilar switching threshold for the high coercivity sys-tem Bi Sb /MnGa. Whereas all these studies were per-formed on topological insulators grown by molecularbeam epitaxy, Mahendra et al. (2018) used sputteringto grow Bi Se /Ta/CoFeB/Gd/CoFeB heterostructureswith perpendicular anisotropy promoted by the 0.5 nmthick Ta layer. Due to its polycrystalline nature, theBi Se layer is highly resistive, one order of magnitudelarger than (Mellnik et al. , 2014), thereby enabling thecurrent to flow mostly through the interface and in theferromagnetic layer, which enhances the SOT efficiency.However, the role, if any, of the topological surface statesin these sputtered layers remains to be proven, togetherwith the stoichiometric profile of the Bi Se films. Otherstrategies to improve the SOT efficiency in these sys-tems rely on the use of spacer layers, such as Ag, whichfavor the formation of Rashba-split bands with strongspin-momentum coupling (Shi et al. , 2018), the creationof Rashba-Dirac coupled systems (Eremeev et al. , 2015),and the search for novel topological materials (Manna et al. , 2018; Rojas-S´anchez et al. , 2016a).
5. Two-dimensional alloys and oxide interfaces
Spin pumping measurements performed on het-erostructures consisting of a ferromagnetic layer and aninterface alloy with strong Rashba-like spin-orbit cou-pling, such as Ag/Bi (Rojas-S´anchez et al. , 2013), Cu/Bi(Isasa et al. , 2016), and Cu/Bi O (Karube et al. , 2016),have revealed large spin-to-charge conversion efficienciesdue to the SGE. This effect converts a nonequilibrium spin density S into an interfacial 2D charge current ˜ j c (Ivchenko and Pikus, 1978). Owing to the interfacial na-ture of ˜ j c , the spin-to-charge conversion is given by theinverse Rashba-Edelstein ”length” λ IREE = (cid:126) e ˜ j c j s , (65)where j s is the spin current density pumped by the fer-romagnet and associated with the spin density S , ex-pressed in ( (cid:126) / e )A/m , and ˜ j c is measured in A/m. Inthe framework of the Rashba model, it can be shownthat λ IREE = α R τ / (cid:126) , where α R is the Rashba cou-pling strength and τ the momentum relaxation time atthe Rashba-split Fermi surface (Gambardella and Miron,2011; Shen et al. , 2014). Typical values of λ IREE rangefrom 0.1-0.3 nm in NiFe/Ag/Bi (Rojas-S´anchez et al. ,2013; Zhang et al. , 2015b) to -0.6 nm in NiFe/Cu/Bi O (Karube et al. , 2016). Comparison with the inverse SHE in NM/FM bilayersis achieved by converting the effective spin Hall angle (orthe SOT efficiency) into λ IREE by taking λ IREE = θ sh λ sf tanh( t I / λ sf ) , (66)where t I is the ”thickness” of the interface layer in whichthe spin-to-charge conversion takes place (Rojas-S´anchez et al. , 2016b, 2013). The maximum attainable length istherefore λ maxIREE = θ sh λ sf for t I (cid:29) λ sf . For values of θ sh between 0.1 and 0.3, and λ sf = 1 . − λ IREE = 0 . − . λ IREE of the Ag/Bi interface.Current injection in such systems results in sizabledamping-like and field-like SOT due to the iSGE, asdemonstrated by ST-FMR for NiFe/Ag/Bi (Jungfleisch et al. , 2016) as well as for the oxidized heavy metalinterfaces described in Sect. IV.C.1.b (An et al. , 2016,2018a,b; Demasius et al. , 2016; Fujiwara et al. , 2013). Inthis situation, a 2D charge current produces a 3D non-equilibrium spin density, and the relation between the ef-fective spin Hall angle and the Rashba-Edelstein lengthreads (Laczkowski et al. , 2017)1 λ REE = 2 e (cid:126) j s ˜ j c = θ sh t I tanh( t I / λ sf ) . (67)Therefore, the maximum charge-to-spin conversion is (cid:16) λ REE (cid:17) max = θ sh λ sf when t I (cid:28) λ sf . In other words, withinthis picture, a figure of merit of spin-to-charge conversionis θ sh λ sf , while for charge-to-spin conversion it is θ sh /λ sf .Prominent spin-charge interconversion effects are ob-served also in 2D electron gases confined at the inter-face between two insulating oxides, such as LaAlO andSrTiO (Ohtomo and Hwang, 2004). These systems hosta variety of unusual electronic phases (Zubko et al. , 2011)as well as tunable carrier density and Rashba spin-orbitinteraction (Caviglia et al. , 2010). Even in the absence ofheavy metals, the large interfacial electric fields and longelectron relaxation time result in extraordinarily large λ IREE , which can be further modulated by electric gat-ing (Lesne et al. , 2016; Song et al. , 2017). Spin pumpingexperiments on SrTiO /LaAlO /NiFe reveal that λ IREE changes from 2 to -6 nm as the Fermi level is raisedthrough the crystal-field split interface states, namelyfrom a single low-lying band with d xy character to thehigher-lying heavier d xz , yz bands, where α R is largest(Lesne et al. , 2016; Seibold et al. , 2017). The observationof strong SOT in SrTiO /LaAlO /CoFeB at room tem-perature (Wang et al. , 2017a) shows that the spin currentgenerated by the iSGE at the oxide interface can be effec-tively absorbed by a magnetic layer deposited on a fewnm thick LaAlO , likely via inelastic electron tunnelingpromoted by defect states in the oxide layer. Interfacesbetween complex oxides thus represent a notable alterna-tive to heavy metal systems for the generation of SOT,offering additional tools to tune λ REE by controlling the2interplay of crystal field and spin-orbit effects in multi-functional heterostructures.
6. Metallic spin-valves
Recent theoretical (Freimuth et al. , 2017b; Taniguchi et al. , 2015) and experimental works (Baek et al. , 2018;Bose et al. , 2018; Humphries et al. , 2017) pointed outthe possibility to induce SOT in all-metallic spin-valvesby in-plane current injection. These structures, simi-lar to those employed for generating STT (Fig. 2), areFM ref /spacer/FM free trilayers where FM ref is a fixed ref-erence ferromagnet with magnetization along p , FM free is the ferromagnet on which the SOT is measured, andthe spacer is a light metal (e.g., Cu or Ti) such thatno or little SHE or iSGE are expected from it. Accord-ing to Taniguchi et al. (2015), a spin current polarizedalong ζ (cid:107) p is generated by either the AHE or PHE inthe bulk of the reference layer and absorbed by the freelayer, giving rise to both damping-like and field-like SOTsaccording to Eq. (2). However, Baek et al. (2018) andHumphries et al. (2017) pointed out that spin filteringcaused by spin-orbit scattering and spin-orbit precessionexperienced by electrons flowing at the interface betweenmagnetic and nonmagnetic layers give rise to spin cur-rents polarized along ζ (cid:107) z × j c and ζ (cid:107) p × ( z × j c ),respectively, which are potentially stronger than the bulkspin currents generated by the AHE and PHE. These pre-dictions have been verified in trilayers with both in-plane(Baek et al. , 2018; Bose et al. , 2018) and out-of-plane p (Humphries et al. , 2017), for which the SOT symmetry isconsistent with the latter mechanisms and allows also forfield-free switching of the free layer (Baek et al. , 2018).
7. Established features and open questions
The complexity and interplay of the different charge-spin conversion mechanisms outlined in Section III un-derpins an ongoing debate on the origin of SOTs and onstrategies to improve their efficiency. Below, we summa-rize the most important findings drawn from experimen-tal investigations of metallic layers: • In most NM/FM systems, the sign of the damping-liketorque is consistent with that of the SHE of the bulknonmagnetic metal. Additionally, nonmagnetic metalelements with strong SHE present large damping-liketorques. The magnitude and the sign of the damping-like torque can be modified by changing the oxidationstate or the capping layer of the ferromagnet interfacethat is not in contact with the nonmagnetic metal. Sig-nificant damping-like torques have been reported alsofor ferromagnetic layers adjacent to metal alloys andoxide layers with a strong iSGE. • The field-like torque is of the same order of magnitudeas the damping-like torque. The sign and magnitudeof the field-like torque are not consistent with the pre-dictions of the drift-diffusion model based on the bulkSHE. • The damping-like and field-like torques typically in-crease with the thickness of the nonmagnetic metallayer and saturate after a few nm. The dependenceof the two torques on the nonmagnetic metal thicknessis not the same. • The temperature dependence of the field-like anddamping-like torques is different, indicating the distinctrole of electron scattering by magnons or phonons. • Extrinsic effects related to both interface and bulk elec-tron scattering are significant and can give rise to bothdamping-like and field-like torques. The SOTs are typi-cally large in high resistivity metals and correlate withthe presence of strong SMR in NM/FM bilayers andcrystalline AMR in ferromagnet/semiconductor layers. • The angular dependence of the torques shows that in-terfacial spin-orbit coupling, either through D’yakonov-Perel relaxation, Fermi surface distortion or anisotropicmixing conductance, plays a relevant role. • The insertion of a nonmagnetic light metal spacer be-tween the ferromagnet and a nonmagnetic metal re-duces magnetic proximity effects in the nonmagneticmetal, but creates additional interfaces that can con-tribute to the generation of spin currents. Both thedamping-like and field-like torques change upon the in-sertion of nonmagnetic and magnetic spacers. •
2D materials, topological insulators, and oxide het-erostructures provide large SOTs when interfaced withmagnetic layers, consistently with the iSGE arisingfrom spin-momentum locked interface states. Addi-tional contributions to the SOT may result from theSHE in systems with residual bulk conductivity. Thesymmetry of the SOTs generated by single crystal lay-ers is determined by the current injection direction rel-ative to the crystal axes. • Both damping-like and field-like torques can be con-trolled through interface engineering, such as gate volt-age, oxidation, or capping layer, which offers an effi-cient way to improve charge-spin conversion in NM/FMas well as 2D systems.SOT measurements in multilayer systems are often in-terpreted assuming either the SHE-SOT model or theRashba-type iSGE. Such approaches are appealing be-cause of their simplicity, but neglect important aspectsof the generation of SOT. The one-dimensional drift-diffusion theory based on the bulk SHE (SubsectionIII.C) is the most commonly employed model to relatethe torque amplitude to the spin Hall conductivity of thenonmagnetic metal. Such a model includes the proba-bility of spin transmission at the interface through thespin mixing conductance parameter, but neglects the3interface-generated spin density by either the iSGE orspin-dependent electron scattering as well as the spinmemory loss. Another major limitation of this modelis that it assumes constant parameters σ N , λ sf , and θ sh throughout the nonmagnetic metal layer, which is un-justified on both theoretical and experimental grounds.When the thickness of the nonmagnetic metal is compa-rable to the electronic mean free path (of the order ofthe grain size or t N , i.e., a few nm in sputtered samples),semiclassical size effects become important and governthe current distribution in NM/FM bilayers (Camley andBarnas, 1989; Zhang and Levy, 1993). Neglecting theseeffects can lead to a wrong estimation of the SOT effi-ciency and λ sf (Chen and Zhang, 2017). Moreover, ab-initio calculations have shown that the intrinsic spin Hallconductivity can strongly vary close to the interface, lead-ing to an enhancement of the SOT (Freimuth et al. , 2015;Wang et al. , 2016b).On the other hand, most SOT models based on in-terfacial Rashba spin-orbit coupling assume a static spinpolarization localized at a sharp interface between thenonmagnetic metal (or the oxide) and the ferromagnet.Considering the complexity of the real ultrathin mag-netic multilayers involving complex orbital hybridiza-tion, disordered interfaces, and spin-dependent semiclas-sical size effects, it is quite unclear how these two mod-els (bulk SHE and interfacial Rashba-like iSGE) applyto real systems. Spin pumping experiments at Bi sur-faces have been interpreted as evidence for either aninterface-enhanced SHE (Hou et al. , 2012) or the iSGE(Rojas-S´anchez et al. , 2013). Angle-resolved photoemis-sion studies, on the other hand, provide evidence that theiSGE is not a pure 2D effect in metallic thin films: thepresence of magnetic exchange (Krupin et al. , 2005), out-of-plane spin polarization (Takayama et al. , 2011), spin-momentum locked quantum well states in the ferromag-net (Moras et al. , 2015), and topologically protected sur-face states (Marmolejo-Tejada et al. , 2017; Thonig et al. ,2016) significantly alters the Rashba effect at metallic in-terfaces compared to model semiconducting heterostruc-tures.Extrinsic effects such as impurity and interface scat-tering induce additional spin currents that propagatethrough or away the magnetic layer and are polarized indirections different from the standard SHE and Rashbamodels, calling for a generalization of the spin cur-rent sources and spin mixing conductance (Amin andStiles, 2016a,b; Baek et al. , 2018; Chen and Zhang, 2015;Humphries et al. , 2017; Saidaoui and Manchon, 2016).For example, electron scattering from an interface withspins parallel and antiparallel to the local spin-orbit fieldhave different reflection and transmission probabilities,leading to a net spin current polarized parallel to the y = z × j c direction, identical to that of the spin cur-rent due to the SHE. Additionally, if the electrons carrya net spin polarization along p , as in a ferromagnet, pre- FIG. 30 (Color online) (a,c) Two configurations for the SOT-induced nano-oscillator and (b,d) their corresponding excita-tion spectrum. (a,b) Nanopillar deposited on top of a non-magnetic metal (Liu et al. , 2012c), and (c,d) Local injectioninto an extended ferromagnet (Demidov et al. , 2012). cession about the spin-orbit field results in a transversespin current with polarization parallel to the p × y direc-tion (Baek et al. , 2018). The interplay between all theseeffects makes it questionable to draw a clear separationbetween the SHE and iSGE in metallic structures, evenwhen considering idealized theoretical models of theseheterostructures.Finally, the typical SOT bilayers are usually only afew nanometer thick. The phenomenological notion ofan interface between bulk regions, as well as the in-terpretation in terms of bulk SHE, appear unjustifiedbased on both theoretical and practical grounds. A fullquantum-mechanical treatment of the SOT in realisticthree-dimensional structures including disorder is there-fore essential to reach consistency between experimentsand theory. D. Magnetization dynamics
It is well known that, due to the STT, a spin-polarized electric current injected into a ferromagneticlayer through a nanocontact leads to the emission ofspin waves (Berger, 1996; Tsoi et al. , 1998). This ef-fect provides a way to realize tunable spin-torque nano-oscillators, which can serve as active microwave compo-nents in integrated circuits (Demidov et al. , 2010; Kiselev et al. , 2003; Madami et al. , 2011; Rippard et al. , 2004;Tsoi et al. , 2000). The discovery of SOT has led to newparadigms to control the high-frequency magnetizationdynamics by means of dc and ac currents (Demidov et al. ,2017). In contrast to STT, SOTs allow for the compen-sation of magnetic damping and the generation of spinwaves in spatially extended regions of a magnetic mate-rial. Moreover, since the spin and charge currents followseparate paths, the electrical current does not need toflow through the active magnetic layer, allowing for the4excitation of both conducting and insulating magneticmaterials. SOTs thus enable efficient and flexible devicegeometries for the generation and amplification of mag-netic oscillations as well as for the propagation and ma-nipulation of coherent spin waves, opening entirely newperspectives in the field of magnonics (Chumak et al. ,2015).To a first approximation, the effects of the SOTs on themagnetization dynamics are described by Eqs. (1) and(2). Consequently, one expects that the field-like torqueshifts the frequency spectrum of the magnetic layer, sim-ilar to an applied magnetic field, and that the damping-like torque changes the magnitude of the magnetic damp-ing. An early demonstration of the SOT-induced modifi-cation of magnetic damping was reported by Ando et al. (2008) in resonantly excited Pt/NiFe, in which the widthof the FMR line decreased or increased depending on thesign of the dc current injected in the bilayer. This workevidenced variations of the damping constant α by anamount ∆ α = γ πf M s t F (cid:126) e sin ψ ξ jDL j c , (68)where f is the resonance frequency of the magnetic layerand ψ is the angle between the current and the preces-sional axis of the magnetization. Later measurementsshowed how the electrical control of magnetic dampingcan be used to enhance the spin wave propagation lengthin microwave guides (An et al. , 2014; Demidov et al. ,2014b). The variations of ∆ α reported in the literaturerange from a few percent to complete compensation of thedamping, which eventually results in the onset of steady-state auto-oscillations of the magnetization (Demidov et al. , 2012; Liu et al. , 2012c). It shall be noted, how-ever, that the simple linear relationship between damp-ing and current exemplified by Eq. (68) is observed onlyat low currents, whereas nonlinear phenomena and mag-netic fluctuations lead to a more complex behavior as thedamping compensation is approached (Demidov et al. ,2017).A limiting factor for achieving self-sustained oscilla-tions is the degeneracy of spin wave modes. If the sampleis large, a significant amount of modes compete with eachother to absorb the excitations induced by the SOT. Insuch a case, the degeneracy is high and only thermal exci-tations can be electrically controlled rather than current-driven coherent oscillations. Achieving self-sustainedmagnetic oscillations requires to lift the degeneracy byreducing the size of the sample and thereby lowering theexcitation threshold and excitation bandwidth. Current-driven magnetic oscillations where thus reported in a 3-terminal CoFeB/MgO/CoFeB MTJ fabricated on top ofa large Ta buffer layer [see Fig. 30(a)] (Liu et al. , 2012c).Due to the reduced size of the nano pillar ( ∼ × ), current-driven oscillations were detected electri-cally through the MTJ [see Fig. 30(b)] and independent control of the excitation via the currents injected intothe Ta layer and through the MTJ was achieved. Morerecently, Duan et al. (2014a,b) achieved SOT-driven spinwave damping control and auto-oscillations in long andnarrow nanowires ( ∼ . µ m × et al. (2012) by locally injecting a spin current inan extended ferromagnetic layer [see Fig. 30(c)]. Thelocal injection creates a spin wave bullet, i.e., a spinwave packet localized in space through non-linear en-ergy losses (Slavin and Tiberkevich, 2005). This self-localization enables the selection of a small number ofspin wave modes that reveal themselves in the coher-ent auto-oscillation. The microwave spectrum of such anano-oscillator presents features similar to ”traditional”STT point-contact oscillators (Bonetti et al. , 2010; Slavinand Tiberkevich, 2005), namely a spin wave ”bullet”and a propagating spin wave mode (Liu et al. , 2013).These devices, similar to STT point-contact oscillators,are characterized by a strong nonlinearity, which enablestheir efficient synchronization to external RF signals overa broad frequency range (Demidov et al. , 2014a) as wellas the synchronization of different oscillators placed nextto each other at distances of up to several microns (Awad et al. , 2016).A unique feature of SOTs is that they provide inter- FIG. 31 (Color online) (a) Sketch of the measurement config-uration and microscopy image of a device with two connectedmicrodiscs (underneath the circles). The bias field µ H is ori-ented transversely to the dc current I dc flowing in Pt. The in-ductive voltage V y produced in the antenna by the precessionof the YIG magnetization M ( t ) is amplified and monitored bya spectrum analyzer. (b) Power spectral density (PSD) mapsmeasured on a 4 mm YIG/Pt disc at fixed | µ H | =47 mT andvariable I dc . The two panels correspond to two different po-larities of µ H . An auto-oscillation signal is detected abovea threshold current of ± µ H · I d <
0, in agreementwith the symmetry of the torque. Adapted from Collet et al. (2016). et al. (2010) proposed that SOTs can convert a dc elec-tric current flowing in a Pt wire deposited on a YIG filminto a spin wave propagating through the YIG film, whichcan then be detected by a Pt electrode at a different lo-cation using spin pumping. In this experiment, whichremains controversial, SOT needs to be large enough tocompensate the damping of the fundamental FMR spinwave mode. This is particularly difficult to achieve inultralow magnetic damping materials, as SOTs excite abroad range of modes. In YIG, Xiao and Bauer (2012)argued that surface spin waves are preferentially excitedcompared to bulk spin waves, which renders the observa-tion of current-driven auto-oscillations very sensitive toboth the size of the YIG layer and to the quality of theinterface with Pt. Hamadeh et al. (2014) showed thatthe magnetic losses of spin wave modes in micron-sizedYIG(20nm)/Pt(8nm) discs can be reduced or enhanceddepending on the polarity and intensity of the dc currentflowing through Pt, reaching complete compensation ofthe damping of the fundamental mode for a current den-sity of 3 × A cm − , and eventually inducing coher-ent SOT-induced auto-oscillations (Collet et al. , 2016),see Fig. 31. By using Bi-substituted YIG films withperpendicular magnetic anisotropy, it is further possibleto prevent the self-localization of the magnetization os-cillations, thus leading to the propagation of coherentmagnons into an extended magnetic insulator film (Evelt et al. , 2018).An alternative strategy to circumvent the hurdlesposed by the generation of coherent spin waves is toutilize thermal magnons, in the same spirit as in spincaloritronics experiments (Bauer et al. , 2012; Uchida et al. , 2010). In this case, SOTs excite a broad rangeof magnons with characteristic frequencies much higherthan the fundamental FMR mode ( ∼ k B T rather thana few GHz) and able to transmit information over longdistances (Bender et al. , 2012; Zhang and Zhang, 2012).Using a non-local setup consisting of two parallel Ptelectrodes deposited on an extended YIG film, Cornelis-sen et al. (2015) and Goennenwein et al. (2015) demon-strated SOT-driven injection, transmission, and detec-tion of thermal magnons over distances up to 40 µ m,with a crossover between a linear transport regime dom-inated by thermal exchange magnons at low current andnon-linear transport regime dominated by subthermalmagnetostatic magnons at high current (Thiery et al. ,2018). These studies have recently been extended to anti-ferromagnets, such as α -Fe O , where antiferromagneticmagnons can carry spin information over a few tens ofmicrons (Lebrun et al. , 2018).Overall, the SOT approach is very attractive for con-trolling the magnetization dynamics of a broad class offerromagnetic and antiferromagnetic materials, includ- ing both conducting and insulating systems. Because noelectric current is required to flow between the magneticlayer and the spin-orbit coupled electrodes, low-dampingmagnetic dielectric materials can be used as the carri-ers of magnetic information over large distances. More-over, SOTs can be applied to arbitrarily large areas ofa magnetic film, unlike STT, which is limited to pillar-shaped nanostructures, allowing for spin wave amplifica-tion through the compensation of damping. SOTs canthus be utilized for the generation of propagating spinwaves, the enhancement of their propagation range, andtheir detection in a single integrated nanomagnonic de-vice, with the perspective of implementing coherent-wavecomputing and information processing. E. Magnetization switching
The realization of current-driven magnetizationswitching has been a major milestone in the progress to-wards SOT devices. Miron et al. (2011a) and Liu et al. (2012a,b) demonstrated that, in the presence of a con-stant in-plane magnetic field, the magnetization direc-tion of a perpendicularly magnetized ultrathin trilayer(Pt/Co/AlO x and Ta/CoFeB/MgO) can be reversiblyswitched by injecting bipolar current pulses at currentdensities of the order of 10 − A/cm (see Fig. 32).This observation was soon confirmed by several groupsusing different magnetic stacks and heavy metal sub-strates (Avci et al. , 2012, 2014b; Emori et al. , 2013; Pai et al. , 2012; Yu et al. , 2014b), as well as antiferromag-nets (Fukami et al. , 2016b; Oh et al. , 2016; Wadley et al. ,2016), magnetic insulators (Avci et al. , 2017a; Li et al. ,2016b), and topological insulators (Han et al. , 2017; Ma-hendra et al. , 2018; Wang et al. , 2017b). Notably, earlierinvestigations of Pt/Co/Pt and Pt/Co also reported aneffect of very small current densities on the low temper-ature coercivity of Co, albeit mainly attributed to Jouleheating (Lin et al. , 2006; Riss et al. , 2010; Xie et al. ,2008).The switching of a perpendicularly magnetized layercan be qualitatively explained by considering the com-bined action of the damping-like torque and in-plane field B x in a simple macrospin picture, as shown in Fig. 32(e).In the Pt/Co/AlO x stack, a positive current pulse in-duces an effective field B DL , such that the magnetizationcan rotate from up to down if B DL is initially parallel to B x , but cannot rotate from down to up if B DL is antipar-allel to B x . When the current polarity is reversed, thesense of rotation changes, such that bipolar switching isachieved by either current or in-plane field reversal, asshown in Fig. 32(f). More generally, the transferred an-gular momentum is transverse to both the current direc-tion and the normal to the plane, which alone cannot en-sure reversible magnetization switching between the + z and - z directions. Hence, the damping-like torque must6 FIG. 32 (Color online) (a) Schematic of a Co(0.6nm)/AlO x (2nm) dot patterned on top of a 3 nm thick Pt Hall cross. Blackand white arrows indicate the equilibrium magnetization states of the Co layer. (b) Detection scheme and scanning electronmicrograph of the sample. (c) m z measured by the anomalous Hall resistance during a downward sweep of the external field B x applied parallel to the current direction. The field has a 2 ◦ out-of-plane tilt to unambiguously define the residual z component.(d) The same measurement recorded after the injection of positive (black squares) and negative (red circles) current pulses ofamplitude I p = 2 .
58 mA, showing bipolar switching of m z . (e) Macrospin model showing the stable (right) and unstable (left)magnetic configurations depending on the sign of B DL relative to B x . (f) Switching diagram: the dots show the minimumin-plane field at which switching becomes deterministic as a function of the injected current. Dashed (solid) arrows indicatethe magnetization direction before (after) switching. Adapted from Miron et al. (2011b). be supplemented by the in-plane field B x that breaks thesymmetry along the current direction and determines theoutcome of the switching process. In the macrospin ap-proximation, the threshold switching current is given by(Lee et al. , 2013) j sw, ⊥ = 2 e (cid:126) M s t F ξ jDL (cid:18) B K, ⊥ − B x √ (cid:19) , (69)where B K, ⊥ is the perpendicular anisotropy field. In-plane magnetized samples, on the other hand, switch atzero external field as long as the magnetization has anonzero component in the y direction, which can be in-duced by shape anisotropy (Fukami et al. , 2016a; Liu et al. , 2012b). In this case, the threshold current has thesame form as that of the conventional STT switching forfree and fixed layers with in-plane magnetization (Sun,2000), and is given by (Lee et al. , 2013) j sw, || = α e (cid:126) M s t F ξ jDL (cid:18) B K, || + B d (cid:19) , (70)where B K, || is the in-plane anisotropy field and B d the de-magnetizing field. Equations (69) and (70) exemplify therelationship between the power required for switching,the thermal stability of a magnet (determined by B K )and ξ jDL . However, the actual mechanism of SOT switch-ing is more complex than coherent magnetization reversalunder the action of the damping-like torque alone.In realistic systems, j sw depends on the factors ap-pearing in Eqs. (69), (70) as well as on the DMI, do-main pinning field, device geometry, size, temperature,and duration of the current pulses. The temperature,which is determined by the current distribution in thebilayer as well as by the thermal conductivity of the dif-ferent materials in the stack, plays a major role, both in activating the switching as well as in changing criticalparameters such as M s , B K , α , and ξ jDL during switch-ing. These factors vary significantly from experiment toexperiment, so that a comparative estimate of the switch-ing efficiency for different material systems can be highlymisleading. Nonetheless, an approximate figure of meritfor the switching efficiency can be calculated by takingthe threshold power density P sw = j ρ ∝ ρ/ ( ξ jDL ) esti-mated using the macrospin approximation, which is inde-pendent of the device size, temperature, and pulse length.Figure 33(a) presents a comparison of ρ/ ( ξ jDL ) for dif-ferent nonmagnetic material systems (solid bars), basedon the values of ( ξ jDL ) and ρ reported in (b) and (c), re-spectively. Within the confines of such a comparison, Ptand W emerge as the best heavy metal elements, whereastopological insulators offer the largest gains in efficiency.It shall be noted, however, that the current distributionin NM/FM bilayers can significantly alter the efficiency,especially if the resistivity of the ferromagnet is muchsmaller than that of the nonmagnetic material. By us-ing a simple parallel resistor model, the threshold powerdensity can be estimated as P sw ∝ (cid:18) ρ N t F ρ F t N + 1 (cid:19) ρ N ( ξ jDL ) . (71)The open bars in Fig. 33(a) show the change of theefficiency calculated using Eq. (71) for a bilayer with t N = 4 nm, t F = 1 nm, and ρ F = 100 µ Ωcm, as typ-ical, e.g., of CoFeB.7
FIG. 33 (Color online) (a) Switching efficiency ρ/ ( ξ jDL ) (solid bars) calculated using the SOT efficiency (b) and re-sistivity (c) of different nonmagnetic layers. The values aretaken from Table II, with error bars representing experimentalspreads, when available. The open bars represent the switch-ing efficiency calculated for a NM/FM stack including a ficti-tious CoFeB layer according to Eq. (71).
1. Switching mechanism
Although the macrospin model reproduces qualita-tively the stability phase diagram of rather extendedfilms (Liu et al. , 2012a), magnetization switching instructures larger than the width of a domain wall ( (cid:38)
10 nm) occurs by nucleation and expansion of magneticdomains. The magnetization reversal process is thusclosely related to the SOT-driven dynamics of N´eel-typedomain wall in the presence of DMI (see Section VI). Dif-ferent switching models have been proposed based on mi-cromagnetic simulations (Finocchio et al. , 2013; Martinez et al. , 2015; Mikuszeit et al. , 2015; Perez et al. , 2014) andspatially-resolved MOKE measurements (Emori et al. ,2013; Ryu et al. , 2013; Safeer et al. , 2016; Yu et al. ,2014b). In such models, the domain nucleation is eitherrandom and thermally-assisted (Finocchio et al. , 2013;Lee et al. , 2014b; Perez et al. , 2014) or determined bythe combined action of DMI, external field, and edge ef- fects (Martinez et al. , 2015; Mikuszeit et al. , 2015; Pizzini et al. , 2014), followed by domain wall propagation acrossthe magnetic layer driven by the damping-like torque.Time-resolved x-ray microscopy measurements of cir-cular shaped Pt/Co/AlO x dots (Baumgartner et al. ,2017) eventually confirmed the edge nucleation mod-els, further showing that the nucleation point is deter-ministic and alternates between the four quadrants ofa dot depending on the sign of the magnetization, B x ,DMI, damping-like and field-like torque, as illustrated inFig. 34. These measurements also showed that switch-ing is achieved within the duration of the current pulsewith an incubation time below the time resolution of theexperiment ( ≈
100 ps) and fast propagation of a tilteddomain wall across the dot (Baumgartner, 2018) withdomain wall velocities of the order of 400 m/s. As theswitching unfolds along a reproducible and deterministicpath, the timing and the extent of magnetization reversalcan be reliably controlled by the amplitude and durationof the current pulses (Baumgartner et al. , 2017). Mea-surements performed by time-resolved MOKE on largerdots with a thinner Co layer, on the other hand, showsignificant after-pulse magnetic relaxation (Decker et al. ,2017), which is ascribed to long-lasting heating effectsand weaker magnetic anisotropy compared to Baumgart-ner et al. (2017). After-pulse relaxation has been ob-served also in Ta/CoFeB/MgO dots for current pulsesexceeding 2 ns, attributed to domain wall reflection atthe sample edges that is favored by the lower DMI andGilbert damping of Ta/CoFeB/MgO (Yoon et al. , 2017).These different results reveal how the reversal path is de-termined by the balance between damping-like and field-like torques, DMI, magnetic anisotropy, and tempera-ture. For samples matching the width of the current line,the Oersted field can also facilitate or hinder the rever-sal (Aradhya et al. , 2016; Baumgartner et al. , 2017). Inall cases, however, SOT switching is bipolar and robustwith respect to multiple cycling events as well as to thepresence of defects.
2. Switching speed
One of the most attractive features of SOT switchingis the timescale of magnetization reversal. Because theswitching speed scales with the lateral dimensions of thesample, and the domain wall velocity can attain up to750 m/s (Miron et al. , 2011a; Yang et al. , 2015), the re-versal time can be reduced to well below 1 ns in dots of100 nm size (Garello et al. , 2014). Figure 35(a) showsthat the switching probability of perpendicularly mag-netized Pt/Co/AlO x dots has a narrow distribution asa function of pulse length τ p , which decreases to below100 ps as the current density increases. In this study,a switching probability of 100% was demonstrated downto τ p = 180 ps, consistently with reversal due to do-8 FIG. 34 (Color online) (a) Schematics of the tilting of the magnetization at the edges of a Pt/Co/AlO x dot due to the DMI(left), DMI and external field B x (middle), DMI, B x , and current (right). The polar components of the damping-like andfield-like effective fields add up at the nucleation point. (b) Snapshots of the reversal process of a circular dot for differentcombinations of current and field measured by time-resolved scanning x-ray transmission microscopy. The red dot and greenarrows indicate the nucleation point and the domain wall propagation direction, respectively. The pulse duration is 2 ns. (c)Time trace of the average out-of-plane magnetization (black squares) during current injection (red line). The amplitude of thefirst (second) pulse is j p = 3 . × (4 . × ) A/cm ; B x = 0 .
11 T. Adapted from Baumgartner et al. (2017). main nucleation and propagation. The critical switchingcurrent j sw is characterized by a long and a short timescale regime, shown in Fig. 35(b), similar to STT-inducedswitching in metallic spin valves (Liu et al. , 2014b). j sw depends weakly on τ p above 10 ns, as expected for athermally-activated reversal process (Bedau et al. , 2010),and scales linearly with τ − p below about 1 ns, as ex-pected in the intrinsic regime where the reversal time isinversely proportional to the transferred angular momen-tum.Studies of how j sw scales as a function of dot sizehave been performed for Ta and W/CoFeB/MgO dots(Zhang et al. , 2015a, 2018). j sw was found to increaseby one order of magnitude going from micrometer-sizedTa/CoFeB/MgO stripes to 80 nm dots, and to remain ap-proximately constant upon further reduction of the dotsize down to 30 nm [Fig. 35(c,d)]. This behavior wasinterpreted as a signature of incipient monodomain be-havior, even though no precessional switching was ob-served, contrary to the prediction of macrospin models(Lee et al. , 2013; Park et al. , 2014). An additional fea-ture that makes SOT switching very attractive for appli-cations is that the incubation time required to start theprocess appears to be negligible (Garello et al. , 2014).The SOT geometry, in which T DL can be made orthogo-nal to the quiescent magnetization, implies that the mag-netization reacts immediately to the current, contrary toSTT-induced switching, in which T DL is initially zerofor collinear magnetic layers until thermal fluctuationsinduce a misalignment of the free layer magnetizationthat is sufficient to trigger the reversal, leading to ns-long random delays (Devolder et al. , 2008; Hahn et al. ,2016).
3. Zero field switching
A critical issue for perpendicularly magnetized layersis the need to apply an external field B x to uniquely de-fine the switching polarity, as shown in Fig. 32. Although B x by itself cannot switch the magnetization because itis orthogonal to the easy axis, fields ranging from 1 to100 mT are typically required to achieve deterministic FIG. 35 (Color online) (a) Switching probability P of a squarePt(3nm)/Co(0.6nm)/AlO x dot with a lateral size of 90 nmas a function of the current pulse duration τ p at fixed in-plane field B x = 91 mT. (b) Critical current density asa function of pulse duration defined at P = 90 %. Thegreen solid line is a fit to the data in the short-time regime( τ p < τ p ≥ µ s). The blue dash-dotted line repre-sents the intrinsic critical current j c . Adapted from Garello et al. (2014). (c) Scanning electron microscope image of aTa(5nm)/CoFeB(1.2)/MgO dot with a nominal diameter D of 30 nm. (d) Device diameter dependence of the critical cur-rent density at various τ p . Adapted from Zhang et al. (2015a). FIG. 36 (Color online) (a) Left: schematic of the in-plane effective field induced by exchange bias. The colored arrows in the lay-ers indicate the direction of the magnetic moments. Field-free switching of Ta(5 nm)/CoFeB(3 nm)/IrMn(3 nm)/CoFeB/MgOsample as a function of current (Oh et al. , 2016). (b) Magnetization loops of a [Co(0.3)/Ni(0.6)] /Co(0.3) multilayer onPtMn(8nm) measured after the application of current pulses of increasing amplitude up to the maximum specified in thelegend. The black arrow indicates the position from which the measurement starts after initialization by a negative pulse.Adapted from Fukami et al. (2016b). (c) Model representing the uncompensated spin direction in each grain of the antiferro-magnet at the interface with the ferromagnet. Top: situation after field-cooling showing an average exchange bias field (orangearrow). A current pulse along the exchange bias direction (middle) or perpendicular to it (bottom) switches the regions ofthe ferromagnetic layer coupled to only one type of antiferromagnetic domain. Switched regions are indicated in orange andblocked regions are indicated in dark blue. Adapted from van den Brink et al. (2016). reversal, depending on the current density as well as onthe magnetic anisotropy of the layers (Avci et al. , 2014b).Several approaches have been demonstrated to solve thisissue by substituting B x with a real or effective field em-bedded into a device. The first working concept by Miron et al. (2011b) was to deposit two 50 nm thick CoFe lay-ers on either side of the magnetic dot, providing a dipolarin-plane field parallel to the current. This solution, how-ever, is not practical for device integration because itlimits the scalability of a matrix of such dots or MTJs.Lau et al. (2016b) have shown that it is possible toembed an in-plane magnetized CoFe layer directly intothe stack, and provide an effective B x on the perpen-dicular CoFe free layer via interlayer exchange couplingmediated by nonmagnetic Ru or Pt spacers. Such an ap-proach allows for varying the sign of B x upon changingthe spacer thickness, but may not be easily integratedinto standard MTJ architectures. A more straightfor-ward approach relies on the stray field projected byan in-plane magnetized layer placed on top of the freelayer/barrier/reference layer stack (Zhao et al. , 2017),provided that such a field does not reduce the TMR.Alternatively, the ferromagnet can be deposited directlyon top of a few nm-thick antiferromagnet like IrMn orPtMn (van den Brink et al. , 2016; Fukami et al. , 2016b;Oh et al. , 2016). The antiferromagnetic layer providesan in-plane exchange bias field but also the source of thespin density, which enables the switching of perpendicu-lar ferromagnetic layers in zero field at current densitiesof the order of 3 × A/cm . The switching process inferromagnetic/antiferromagnetic systems takes place in astep-wise manner, as schematized in Fig. 36(c), depend-ing on the microstructure of the antiferromagnetic layerand the local direction of the exchange bias field (van denBrink et al. , 2016; Fukami et al. , 2016b; Kurenkov et al. ,2017a). This behavior can be also exploited to introduce analogue memristive properties into three-terminal MTJdevices (Fukami et al. , 2016b).Another practical solution consists in employing amagnetic spin-valve composed of a bottom reference fer-romagnet and a top recording free layer in the current-in-plane configuration (see Sect. IV.C.6). If the mag-netization of the bottom ferromagnet points along thecurrent direction, the spin current resulting from spin-orbit interfacial scattering has a component ζ (cid:107) p × ( z × j c ) ≡ z polarized along the easy axis of the freelayer, which has been show to induce field-free switchingin CoFeB/NiFe/Ti/CoFeB/MgO spin-valves (Baek et al. ,2018). The combination of STT and SOT in perpendicu-lar MTJs also leads to field-free magnetization switchingof the free layer, which is particularly promising for ap-plications (Wang et al. , 2018).Finally, an elegant approach to this problem is to in-troduce lateral symmetry breaking in the magnetic struc-ture. Thickness gradients of the oxide and ferromagneticlayers have been shown to induce an out-of-plane field-like torque (Yu et al. , 2014a,b) or a tilted anisotropy(Torrejon et al. , 2015; You et al. , 2015), both conduciveto zero field switching, whereas asymmetric patterningof the magnetic and conductive layers has been used tocontrol the switching polarity via nonreciprocal domainwall propagation (Safeer et al. , 2016). Recently, artificialnanomagnets consisting of adjacent out-of-plane and in-plane magnetized regions coupled by the DMI have alsobeen shown to exhibit field-free switching, with interest-ing implications to cascade linear and planar arrays ofnanomagnets (Luo et al. , 2019).0 F. Memory and logic devices
SOT-operated devices can find application in mem-ory as well as logic architectures where current-inducedswitching is required to control the magnetization ofone or several magnetic elements (Lee and Lee, 2016).MTJs with in-plane (Liu et al. , 2012b; Pai et al. , 2012;Yamanouchi et al. , 2013) and perpendicular (Cubukcu et al. , 2014; Garello et al. , 2018) magnetization providedthe first demonstration of three-terminal devices in whichthe write operation is performed by SOTs (Fig. 37).MTJs constitute the building blocks of MRAMs, wherethe bit state is encoded in the high (low) TMR corre-sponding to antiparallel (parallel) alignment of the mag-netization of the free and reference layers. The ever in-creasing need for faster data storage and retrieval hasplaced MRAMs in a prime position to replace or comple-ment CMOS-based memory technologies, owing to the in-trinsic nonvolatility, low write energy, low standby power,as well as superior endurance and resistance to radiationof MTJ bit cells compared to semiconductor memories(Apalkov et al. , 2016; Hanyu et al. , 2016). State-of-the-art MRAMs incorporate STT as the writing mechanism(Kent and Worledge, 2015). STT brings great advan-tages in terms of scalability and integration with pe-ripheral electronics, since the critical switching currentscales with the area of the free layer and requires onlytwo terminals to perform the read and write operations[Fig. 37(a)]. However, as the write and read currents flowalong the same path through the oxide tunnel barrier,a compromise between conflicting requirements must beachieved, namely a thin barrier for low current switch-ing and a thick barrier for high TMR. Moreover, becausethe STT reversal process is thermally activated, a largeoverdrive current is required for fast switching, which candamage the tunnel barrier, while the finite probability tonot switch at high currents and to switch at low currentleads to write error rates that are larger than desired (Oh et al. , 2009).Three-terminal MTJ devices based on SOT offer crit-ical advantages in this respect, as the free layer can beswitched without passing a current through the oxide andreference layers [Fig. 37(b)]. The separation of the readand write current paths in the MTJ allows for optimaltuning of the barrier independently of the write processand increases the endurance of the MTJ. Moreover, thedeterministic character of SOT switching enables sub-nsreversal of perpendicular MTJs (Cubukcu et al. , 2015)and low error rates in in-plane MTJs down to 2 ns longcurrent pulses (Aradhya et al. , 2016). The three-terminalconfiguration of MTJs operated by SOT, on the otherhand, implies a larger footprint of the bit cell comparedto two-terminal MTJs. As the actual size of the bit celldepends on the size and number of the transistors re-quired to control data flow, the area penalty depends onthe particular cell architecture and may not be so large.
FIG. 37 (Color online) (a) Left: SOT-induced switching foran in-plane magnetized nanomagnet at room temperature:schematic of the three-terminal MTJ device and the circuitused in the measurements. Right: TMR of the device asa function of the applied dc current. An in-plane externalfield of 3.5 mT is applied to set the device at the center ofthe minor loop, although this is not required for switchingthe in-plane magnetized free layer. Adapted from Liu et al. (2012b). (b) Left: Schematic of a three-terminal MTJ withperpendicular magnetization. Right: TMR as a function ofcurrent amplitude I p injected in the Ta electrode using 50ns long pulses under an in-plane magnetic field of 40 mT.Adapted from Cubukcu et al. (2014). Importantly, the three-terminal configuration also allowsfor voltage control of the magnetic anisotropy of the freelayer, which enables write speed acceleration (Yoda et al. ,2017), lower current thresholds, as well as selective SOTswitching of several MTJs sharing a single write line(Kato et al. , 2018). The analysis of SOT-MRAMs at thecircuit- and architecture-level (Oboril et al. , 2015; Pre-nat et al. , 2016) reveals that this technology can be ad-vantageously introduced in the data cache of processors,offering a strong reduction of the power consumption rel-ative to volatile memories, comparable performances toSTT-MRAMs and significant gains in terms of reliabilityand speed.SOTs hold great promise also for driving magnetic cel-lular automata (Cowburn and Welland, 2000), domainwall logic (Allwood et al. , 2005), and MTJ-based logicdevices (Guo et al. , 2014; Yao et al. , 2012). In the firsttwo types of devices, SOTs offer unique features suchas the clocking of nanomagnetic logic arrays by in-planecurrent injection (Bhowmik et al. , 2014) and the efficientmanipulation of domain walls (Safeer et al. , 2016; Yang et al. , 2015). In hybrid CMOS/magnetic devices based onMTJs, SOTs can perform similar functions as STT (Guo et al. , 2014; Yao et al. , 2012), but also enable novel archi-1tectures. Recent proposals include MTJ devices that ex-ploit gate-voltage-modulated SOT switching for the par-allel initialization of programmable logic arrays (Lee andLee, 2016), four terminal devices that allow for direct cas-cading at high operation gain and low switching power(Kang et al. , 2016), and nonvolatile flip-flops for powergating (Hanyu et al. , 2016; Jabeur et al. , 2014; Kwon et al. , 2014). More futuristic ideas concern ”probabilisticspin logic” systems in which SOTs are used to control thestochastic switching of thermally activated nanomagnetsCamsari et al. (2017) as well as neuromorphic computingarchitectures (Borders et al. , 2018; Locatelli et al. , 2014;Sengupta et al. , 2015). Other unconventional memoryand logic architectures can be envisaged based on purelyplanar structures. In such a case, the SOTs would pro-vide the writing mechanism while the reading operationcan be performed by the AHE (Moritz et al. , 2008) orthe unidirectional SMR (Avci et al. , 2015a, 2018; Olejn´ık et al. , 2015).A critical issue in this wide range of applications isthe dynamic power consumption relative to the thermalstability factor of nanomagnets, ∆ = B K M s V F / k B T ,where V F is the volume of the ferromagnet. In perpendic-ularly magnetized structures with ∆ (cid:38) to a few times 10 A/cm depending on the switching speed [Fig. 35(b)]. However,because the critical current scales with the lateral cross-section of a device, the switching of a 50 nm wide dotis predicted to require less than 200 µ A and a write en-ergy smaller than 100 fJ at 1.5 ns (Cubukcu et al. , 2015),which is close to the best results obtained so far for per-pendicular STT-MRAM devices.Very promising figures of merit in this context havebeen obtained for in-plane CoFeB layers with ∆ (cid:38)
35 bydusting the W/CoFeB interface with Hf, which allows forcritical current densities of the order of 5 × A/cm at 2ns (Shi et al. , 2017). The power dissipated in the currentlines is also a matter of concern, as some of the mostefficient NM/FM combinations are based on the high-resistive phase of W and Ta (Liu et al. , 2012b; Pai et al. ,2012) (Fig. 23). The search for novel SOT materials isthus focusing on systems that combine large charge-spinconversion efficiency with low resistivity or whose mag-netic properties can be strongly modulated by a gate volt-age. While there are still margins of improvement, SOTdevices already offer an unprecedented variety of applica-tions and compatibility with different classes of materials,which extends the range of spintronics well beyond theprototypical spin-valve and MTJ structures of the pasttwo decades. V. SPIN-ORBIT TORQUES INNONCENTROSYMMETRIC MAGNETS
SHE and iSGE are known as distinct but companionphenomena from their initial observations in nonmag-netic semiconductor structures (Belkov and Ganichev,2008; Ganichev et al. , 2004b; Ivchenko and Ganichev,2008; Kato et al. , 2004a,b; Silov et al. , 2004; Wun-derlich et al. , 2004, 2005). As discussed in the pre-vious section, both iSGE and SHE have been utilizedfor electrically generating SOTs in metallic magneticmultilayers. The primary focus of the present sec-tion is to discuss the experiments performed on bulknon-centrosymmetric magnets, including dilute magneticsemiconductors (Chernyshov et al. , 2009; Endo et al. ,2010; Fang et al. , 2011; Kurebayashi et al. , 2014), mag-netic half-heusler compounds (Ciccarelli et al. , 2016) andantiferromagnets (Bodnar et al. , 2018; Meinert et al. ,2018; Wadley et al. , 2016; Zhou et al. , 2018). This typeof systems is particularly interesting as SHE is absent(there is no adjacent nonmagnetic metal), so that theobserved SOTs are solely attributed to iSGE.In analogy to the galvanic (voltaic) cell, the term spingalvanic effect (SGE) was coined for a phenomenon inwhich an externally induced non-equilibrium spin den-sity generates an electrical current (voltage) (Ganichev et al. , 2002). Inversely the iSGE, sometimes also calledthe Rashba-Edelstein effect, then refers to an externallyapplied electrical current that generates a spin density(Aronov and Lyanda-Geller, 1989; Edelstein, 1990; In-oue et al. , 2003; Ivchenko et al. , 1989; Ivchenko andPikus, 1978; Mal’shukov and Chao, 2002). The theoryof iSGE was discussed in details in Subsection III.D.We start in Subsection V.A with initial observationsof the iSGE in nonmagnetic GaAs structures and con-tinue in Subsection V.B by discussing the iSGE inducedSOTs in bulk ferromagnets, namely in the low Curietemperature, dilute-moment semiconductor (Ga,Mn)As,and in the high Curie temperature, dense-moment metalNiMnSb. The physics of staggered iSGE spin densitiesin locally non-centrosymmetric lattices and correspond-ing N´eel SOTs is reviewed in Subsection V.C based onstudies in antiferromagnetic CuMnAs and Mn Au. Weconclude in Subsection V.E by discussing the SGE andspin-orbit-driven magnonic charge pumping phenomenathat are reciprocal to the iSGE and SOT, respectively.
A. nonmagnetic GaAs structures
Initial observations of the iSGE were made in paral-lel with the initial SHE experiments, in both cases insemiconductors and employing optical detection meth-ods (Belkov and Ganichev, 2008; Ganichev et al. , 2004b;Ivchenko and Ganichev, 2008; Kato et al. , 2004a,b; Silov et al. , 2004; Wunderlich et al. , 2004, 2005). In citeWun-2derlich2004,Wunderlich2005, iSGE and SHE were de-tected in the same asymmetrically confined hole gas in aAlGaAs/GaAs semiconductor heterostructure. The ex-periments are shown in Fig. 38. The current-inducedspin density was measured by detecting the circularlypolarized electroluminescence from a built-in planar p-n light emitting diode (LED). Since in this semiconductorheterostructure the iSGE has the Rashba symmetry andthe corresponding in-plane polarization (perpendicular tothe applied electric field) is uniform, the LED was placedacross the hole transport channel and an in-plane obser-vation angle was used [see Fig. 38(a)]. The measurednon-zero circular polarization at zero magnetic field [seeFig. 38(b)] is then a signature of the iSGE spin density ofcurrent-carrying holes that radiatively recombined withelectrons at the detection LED. For comparison, the SHEexperiment is displayed in Figs. 38(c,d). Here oppositeout-of-plane spin densities accumulate only at the edgesand, correspondingly, the detecting LEDs are fabricatedalong the edges of the transport channel and the emittedlight observation angle is out-of-plane.The remarkable strength of these relativistic phenom-ena was already recognized in the initial experiments per-formed in the strongly spin-orbit coupled GaAs valenceband. The effective iSGE fields inferred from Fig. 38(b)are in Teslas. In other words, the ∼ −
10 % spin po-larization was achieved in the microchip at a ∼ µ Acurrent, compared to a ∼
100 A superconducting magnetthat would generate the same degree of spin density inthe semiconductor via an external magnetic field. UsingMaxwell’s equations physics one needs 10 × larger equip-ment with 10 × larger current than using Dirac equa-tion physics in the iSGE (SHE) microchips to achieve thesame polarization in the nominally nonmagnetic system.When the current is switched off these large spin densi-ties immediately vanish, which makes the iSGE and SHEphenomena in nonmagnetic crystals impractical for spin-tronic memory applications. However, shortly after theirinitial discovery, it was realized theoretically (Bernevigand Vafek, 2005; Garate and MacDonald, 2009; Manchonand Zhang, 2008; ˇZelezn´y et al. , 2014) and subsequentlyverified in experiments (Chernyshov et al. , 2009; Cic-carelli et al. , 2016; Wadley et al. , 2016), that iSGE repre-sents uniquely efficient means for electrical writing of in-formation when the non-equilibrium, spin-orbit-inducedcharge polarizations are exchange-coupled to ferromag-netic or antiferromagnetic moments. These are discussedin the following subsections. B. Bulk ferromagnetic (Ga,Mn)As and NiMnSb
One can picture iSGE based on simple symmetry rules.Fig. 39 represents the iSGE spin densities in three se-lected systems: (i) Si diamond lattice, (ii) GaAs zinc-blende crystal and (iii) NiMnSb non-centrosymmetric
FIG. 38 (Color online) (a) Electron micrograph of the de-vice and an optical image of the emitted light in the experi-mental detection of the iSGE by circularly-polarized electro-luminescence. The net in-plane spin polarization is detectedby placing the LED across the transport channel and using anin-plane observation angle. (b) Right: spectral dependence ofthe circular polarization of the emitted light. Left: the de-pendence of the circular polarization on the external in-planemagnetic field. (c,d) Experimental detection of the SHE bytwo LEDs placed along the edges of the conduction channeland using an out-of-plane observation angle. Adapted fromWunderlich et al. (2004, 2005). magnet. A priori, since Si diamond-lattice possesses in-version symmetry, iSGE vanishes globally at the levelof the unit cell. But due to the local inversion symme-try breaking, iSGE generates two spin densities, S A = − S B , pointing in the opposite direction on the twonon-centrosymmetric, inversion-partner sites of the Sidiamond-lattice unit cell, as shown in Fig. 39(a) and dis-cussed theoretically in Ciccarelli et al. (2016) and ref-erences therein. This staggered-symmetry spin densityinduced by the iSGE can generate an efficient SOT incollinear antiferromagnets as further discussed in Sub-section V.C.On the other hand, the zinc-blende lattice of GaAs [or(Ga,Mn)As] and of the closely related half-heusler latticeof NiMnSb are examples of crystals that lack an inver-sion center in the unit cell. This can result in a non-zeronet spin density, illustrated in Figs. 39(b,d), that gener-ates an efficient SOT in ferromagnets, provided that theiSGE-induced spin density is exchange coupled to theferromagnetic moments. As discussed earlier in detail inSubsection III.D, depending on the crystal symmetry, theiSGE can be composed of three distinct terms: general-ized Rashba and Dresselhaus terms, shown in Fig. 39(c),3and a term describing a response collinear to the electricfield. FIG. 39 (Color online) (a) Cartoon representation of op-posite iSGE spin densities generated at the locally non-centrosymmetric inversion-partner lattice cites of the Si lat-tice. (b) Cartoon representation of a net uniform iSGE spindensity generated over a non-centrosymmteric unit cell of azinc-blende GaAs lattice. Exchange coupling between theiSGE spin density of carriers and equilibrium dilute ferromag-netic moments on Mn atoms results in the SOT. (c) Differentsymmetries of iSGE spin density as a function of the electricfield direction corresponding to different non-centrosymmetriccrystal point groups. (d) Same as (b) for a room-temperaturedense-moment ferromagnet NiMnSb. Adapted from Ciccarelli et al. (2016).
The experimental discovery of the iSGE-induced SOTwas reported in a (Ga,Mn)As sample whose image isshown Fig. 40(a) (Chernyshov et al. , 2009; Endo et al. ,2010). The experiment demonstrated not only the pres-ence of the iSGE effective field of the expected Dressel-haus symmetry for the strained (Ga,Mn)As epilayer, butalso demonstrated that iSGE was sufficiently strong toreversibly switch the direction of magnetization. Data inFig. 40(b) were taken at external magnetic field magni-tude and angle fixed close to the switching point betweenthe [010] and [100] easy-axes. The measured transverseAMR, used for the electrical readout, forms a hystere-sis loop as the writing iSGE current is swept between ± et al. , 2011; Kurebayashi et al. , 2014)and presented in Subsection IV.B.2. Here an electric cur-rent oscillating at microwave frequencies is used to create FIG. 40 (Color online) (a) An atomic force micrograph of thesample used to detect the SOT in GaMnAs. (b) R xy showshysteresis as a function of the current for a fixed externalmagnetic field H = 6 mT applied at an angle φ H = 72 ◦ .(c) The magnetization switches between the [010] and [¯100]directions when alternating ± et al. (2009). an oscillating effective SOT field in the magnetic materialbeing probed, which makes it possible to characterize in-dividual nanoscale samples with uniform magnetizationprofiles (Fang et al. , 2011). For detection, a frequencymixing effect based on the AMR was used. When mag-netization precession is driven, there is a time-dependentchange ∆ R ( t ) in longitudinal resistance from the equilib-rium value R (owing to the AMR). The resistance oscil-lates with the same frequency as the microwave current,thus causing frequency mixing, and a directly measurabledc voltage V dc is generated. This voltage provides a probeof the amplitude and phase of magnetization precessionwith respect to the microwave current.The FMR vector magnetometry on the driving SOTfields revealed a dominant Dresselhaus and a weakerRashba contribution [Fig. 41(a)] (Fang et al. , 2011). Byseparating the symmetric and antisymmetric parts of themixing V dc signal [Fig. 41(b)], it was possible to iden-tify both the field-like and the damping-like SOT com-ponents (Kurebayashi et al. , 2014). It was shown that thedamping-like SOT plays a comparably important role indriving the magnetization dynamics in (Ga,Mn)As as thefield-like SOT [Figs. 41(c,d)].The FMR technique was also employed in the study ofthe iSGE-induced SOT in the room-temperature, dense-moment metal ferromagnet NiMnSb, as shown in Fig. 42.In agreement with the symmetry expectations for thestrained half-heusler lattice of the NiMnSb epilayer, and4 FIG. 41 (Color online) Schematic of the (Ga,Mn)As sample,measurement set-up and magnetization precession. The in-jected microwave current drives FMR, which is detected viathe dc voltage V dc across the microbar. θ m − E is the angle ofthe static magnetization direction measured from the currentflow direction. Arrows represent in-plane (blue) and out-of-plane (red) components of the instantaneous non-equilibriumiSGE spin density induced by the microwave current thatdrives the magnetization. (b) A typical ST-FMR signal drivenby an alternating current at 11 GHz and measured by V dc as afunction of external magnetic field. Data were fitted by a com-bination of symmetric (S) and antisymmetric (A) Lorentzianfunctions. (c) Direction and magnitude of the in-plane spin-orbit field (blue arrows) within the microbars (light blue rect-angles). The direction of the electric field is represented by E .(d) Coefficients of the cos θ m − E and sin θ m − E fits to the angledependence of the out-of-plane SOT field for the sample set.In this out-of-plane data, two samples are shown in each mi-crobar direction and are distinguished by blue and red squaredata points. The symmetries expected for the damping-likeSOT, on the basis of the theoretical model for the Dressel-haus spin-orbit Hamiltonian [see Eq. (28)], are shown bylight green shading. Adapted from Kurebayashi et al. (2014). in agreement with the results in the directly relatedzinc-blende lattice of (Ga,Mn)As, the observed field-like component has a dominant Dresselhaus symmetry[Fig. 42(d)]. Unlike (Ga,Mn)As, the damping-like SOTwas not identified in NiMnSb [Fig. 42(b,c)]. This islikely due to the higher conductivity of metallic NiMnSb.While the extrinsic field-like SOT scales with the con-ductivity, the intrinsic contribution to the damping-likeSOT is scattering-independent to lowest order (see Sub-section III.D), implying that the higher conductivity ofthe NiMnSb metal might favor the field-like SOT. C. Collinear antiferromagnets
Compensated two-spin-sublattice antiferromagnetshave north poles of half of the microscopic atomic mo-
FIG. 42 (Color online) (a) Schematic of the NiMnSb epilayersample and measuring set-up. A microwave current is passedin the bar and excites ST-FMR. By measuring the longitudi-nal dc voltage, the magnitude of the spin-orbit driving field isdeduced. (b) The rectified voltage showing FMR for differentfrequencies of the microwave current. The Lorentzians arewell fitted by an antisymmetric line-shape (continuous line)at all frequencies. (c) Power dependence of the symmetricand antisymmetric components of the rectified voltage. (d)Polar plot illustrating the direction of the spin-orbit field forcurrent flowing along different crystal directions of NiMnSb.Adapted from Ciccarelli et al. (2016). ments pointing in one direction and the other half inthe opposite direction. This makes the uniform exter-nal magnetic field inefficient for switching magnetic mo-ments in antiferromagnets. The complete absence of elec-tromagnets or reference permanent magnets in the SOTscheme for writing ferromagnetic memory bits, discussedabove, has served as the key for introducing the physicalconcept for the efficient control of magnetic moments inantiferromagnets (ˇZelezn´y et al. , 2014).Two distinct scenarios have been considered forthe SOT on antiferromagnetic spin sublattices
A/B , ∂∂t m A/B ∼ m A/B × B eff A/B (ˇZelezn´y et al. , 2014). One inwhich the crystal is globally non-centrosymmetric. Herean example is the half-heusler antiferromagnet CuMnSb(Forster et al. , 1968) or any thin-film antiferromagnetwith structural inversion asymmetry. The efficient torquein this case is the damping-like SOT which, assuming e.g.Rashba spin-orbit coupling, is driven by an effective field B eff A/B ∼ ( E × z ) × m A/B . Here B eff A/B is staggered due tothe opposite magnetizations on the two spin sublatticesof the antiferromagnet, m A = − m B . The field-like SOTin these globally non-centrosymmetric crystals in not effi-cient for antiferromagnets since the effective field drivingthe field-like torque, B eff A/B ∼ E × z , is not staggered.In Fig. 39(a) we illustrated that in crystals with twoinversion-partner lattice sites in the unit cell, the iSGEcan generate a staggered spin density. This leads to thesecond scenario in which the field-like component of the5SOT is efficient in an antiferromagnet whose magneticspin sublattices A/B coincide with the two inversion-partner crystal-sublattices. In this case the effective fielddriving the field-like SOT has the staggered form (againassuming the Rashba symmetry): B eff A ∼ E × z and B eff B ∼ − E × z . Mn Au and CuMnAs are examplesof high N´eel temperature antiferromagnetic crystals inwhich this scenario applies (Wadley et al. , 2016; ˇZelezn´y et al. , 2014).Figure 43 illustrates the experimental realization ofelectrical switching by the staggered SOT field in amemory bit cell fabricated from a single-crystal epitax-ial film of a CuMnAs antiferromagnet (Olejnik et al. ,2017; Wadley et al. , 2016). Writing current pulses aresent through the four contacts of the bit-cell to gener-ate current lines in the central region of the cross alongone of two orthogonal axes, representing ”0” and ”1”[Fig. 43(b)]. The writing current pulses give preferenceto domains with antiferromagnetic moments aligned per-pendicular to the current lines (Rashba-like symmetry).Electrical readout is performed by running the probecurrent along one of the arms of the cross and by mea-suring the antiferromagnetic transverse AMR across theother arm [Fig. 43(b)]. The write/read functionality ofthe CuMnAs memory cells was verified to be not sig-nificantly perturbed in a superconducting magnet gener-ating a magnetic field as strong as 12 T (Wadley et al. ,2016). This highlights the efficiency of the staggered SOTfields whose inferred magnitude allowing to switch theantiferromagnetic moments is only in the mT range.The bit-cell write/read signals can be sent at ambientconditions by placing the CuMnAs chip on a standardprinted circuit board connected to a personal computervia a 5 V USB interface (Olejnik et al. , 2017). Fig. 43(c)shows an example of data obtained from this proof-of-concept antiferromagnetic memory device. Apart fromdemonstrating the application potential of antiferromag-nets in spintronics thanks to the SOT, it also illustratesa deterministic multi-level switching of the antiferromag-netic bit cell. Here successive ∼ µ s writing pulses alongone of the current path directions produce reproduciblestep-like changes in the memory readout signal. A pho-toemission electron microscopy study of CuMnAs has as-sociated the multi-level electrical switching signal withthe antiferromagnetic moment reorientations within mul-tiple domains (Grzybowski et al. , 2017).The observation of SOT-driven switching has been re-cently extended to Mn Au, where a large AMR ratio upto 6 % is obtained (Bodnar et al. , 2018; Meinert et al. ,2018; Zhou et al. , 2018). The general switching featuresare quite similar to the ones observed in CuMnAs, reveal-ing the multidomain magnetic structure of the system.Upon increasing the applied current, the N´eel order ofthe different magnetic domains is progressively reorientedunder thermal activation (Meinert et al. , 2018), in sharpcontrast with the fast switching obtained in NM/FM bi-
FIG. 43 (Color online) (a) Optical microscopy image of thedevice containing Au contact pads and the antiferromagnetCuMnAs cross-shape cell on the GaP substrate. (b) Top: Thereadout current (blue arrow) and transverse voltage detectiongeometry. Bottom: Write pulse current lines (red arrows) la-beled ”1” (left) and ”0” (right) and the corresponding pre-ferred antiferromagnetic moment orientations (white double-arrows). (c) Readout signals after repeated four write pulseswith current lines along the [100] direction (”0”) followed byfour pulses with current lines along the [010] direction (”1”).(d) Readout signal as a function of the number of pulses inthe train of pulses for the individual pulse length of 250 ps.Adapted from Olejnik et al. (2017). layers driven by domain wall nucleation/propagation (seeSubsection IV.E). This progressive switching seems to bea specific property of antiferromagnetic materials, as itwas also reported in the case of field-free switching inAF/FM metallic bilayers (van den Brink et al. , 2016;Fukami et al. , 2016b; Oh et al. , 2016) (see SubsectionIV.E.3). In agreement with the multi-domain picture, ex-periments in the AF/FM bilayers showed that the num-ber of intermediate levels decreases with the decreasingsize of the device and finally evolves into a binary modebelow a certain threshold (Kurenkov et al. , 2017b).The multi-level nature of antiferromagnetic bit cellsopens the possibility for combining memory, logic andneuromorphic functionalities (e.g., pulse-counter) withinthe cell (Olejnik et al. , 2017). Another unique merit ofantiferromagnets is the THz scale of the internal spin dy-namics which in combination with the SOT physics opensthe door to ultra-fast switching schemes. Fig. 43(d)shows initial results of experiments in this directiondemonstrating a deterministic memory-counter function-ality in a multi-level CuMnAs memory cell for ∼ ∼
100 ps is6at the limit achievable with common current-pulse se-tups.Subsequently, reversible switching with analogouscharacteristics was demonstrated using 1 ps long writ-ing pulses (Olejn´ık et al. , 2018). A non-contact tech-nique was employed for generating the ultra-short cur-rent pulses in the antiferromagnetic memory cell via THzelectromagnetic transients to overcome the above limitof common contact current-pulse setups. Remarkably,the writing energy did not increase when down-scalingthe pulse-length from ns to ps. This is in striking con-trast to ferromagnetic STT (Bedau et al. , 2010) or SOT(Garello et al. , 2014) memories in which the theoreti-cally extrapolated writing energy at ps would increaseby three orders of magnitude compared to the state-of-the-art ns-switching devices. While readily achievable inantiferromagnets, the ps range remains elusive for ferro-magnets because, in frequency terms, it far exceeds theGHz-scale of the FMR in typical ferromagnets.All the above SOT experiments on antiferromagnetsemployed 90 o switching of the N´eel vector. 180 o switch-ing has been also recently demonstrated in CuMnAs byalternating the sign of the writing current. The read-out of the reversed N´eel vector memory states was per-formed electrically using a second-order magnetoresis-tance whose presence relies on the broken time-reversaland space-inversion symmetries in the antiferromagneticcrystal of CuMnAs (Godinho et al. , 2018). Microscopi-cally, the mechanism of this second-order magnetoresis-tance in CuMnAs was ascribed to a transient tilt of theN´eel vector due to the SOT combined with the AMR. D. Antiferromagnetic topological Dirac fermions
Recently, a new concept has been theoretically pro-posed. It follows from the observation that the staggeredSOT fields can co-exist with topological Dirac fermionsin the band structure of antiferromagnets because of theserendipitous overlap of the key symmetry requirements(ˇSmejkal et al. , 2017). Therefore, one can use SOT toreorient the N´eel vector in antiferromagnets in order tocontrol such topological Dirac fermions. This is illus-trated in Fig. 44(a,b) on examples of the CuMnAs wherethe SOT switching was experimentally verified, as men-tioned above, and of the graphene lattice representing theDirac systems (Castro Neto et al. , 2009): (i) The two-Mn-site primitive cell of CuMnAs favors band crossingsin analogy with the two-C-site graphene lattice. (ii) Inthe paramagnetic phase, CuMnAs has time reversal ( T )and space inversion ( P ) symmetries. It guarantees thateach band is doubly-degenerate forming a Kramer’s pair,in analogy to graphene. In the antiferromagnetic phase,this degeneracy is not lifted because the combined PT symmetry is preserved, although the T symmetry and the P symmetry are individually broken (Chen et al. , 2014; Herring, 1966; ˇSmejkal et al. , 2017; Tang et al. , 2016).(iii) The combined PT symmetry is just another wayof expressing that the two antiferromagnetic spin sub-lattices conincide with the two inversion-partner crystal-sublattices. As explained above, this condition leads toan efficient field-like SOT in bipartite antiferromagnets.An additional crystal symmetry is needed to mediatethe dependence of Dirac quasiparticles on the N´eel vectororientation [Fig. 44(c)]. In graphene there is no symme-try that protects the four-fold degeneracy of Dirac cross-ings of two Kramer’s pair bands in the presence of spin-orbit coupling (Kane and Mele, 2005). In CuMnAs, onthe other hand, the Dirac crossings are protected by anon-symmorphic, glide mirror plane symmetry (Youngand Kane, 2015), G x = (cid:8) M x | (cid:9) , as long as the N´eelvector is aligned with the [100] axis. G x combines the mir-ror symmetry M x along the (100)-plane with the half-primitive cell translation along the [100] axis [Fig. 44(d)].Due to the mirror-reflection behavior of the axial vectorsof magnetic moments [Fig. 44(e)], the G x symmetry, andthus also the Dirac crossing protection, is broken whenthe antiferromagnetic moments are reoriented into a gen-eral crystal direction by the SOT. FIG. 44 (Color online) (a) Mn antiferromagnetic spin sub-lattices of CuMnAs denoted by purple and pink balls withthick arrows. The antiferromagnet order breaks time-reversalsymmetry ( T ) and space-inversion symmetry ( P ), however,the combined PT symmetry is preserved. Staggered current-induced spin density on the sublattices A and B is denotedby cyan and blue arrows. (b) Graphene crystal with two C-sites per unit cell in analogy with the Mn-sites in CuMnAs.(c) Band dispersion of the minimal antiferromagnet modelbased on CuMnAs illustrating the control of the Dirac pointsby the direction of the N´eel vector n . Topological indices ofthe Dirac point are shown in the inset (for the sake of clar-ity the degenerate bands are slightly shifted). (d) Top viewof the model quasi-2D-antiferromagnetic lattice of CuMnAshighlighting the non-symmorphic glide mirror plane symme-try, combining mirror plane ( M x ) reflection with a half-unit-cell translation along the x -axis. (e) An axial vector m undermirror ( M ) reflection. Adapted from ˇSmejkal et al. (2017). E. Magnonic charge pumping in (Ga,Mn)As
We conclude this section by briefly discussing the SGE,which is a reciprocal phenomenon to the iSGE, andits counterpart in magnets termed the magnonic chargepumping (Ciccarelli et al. , 2014). The latter, in turn,is a reciprocal phenomenon to the SOT. Following the-oretical predictions (Aronov and Lyanda-Geller, 1989;Edelstein, 1990; Inoue et al. , 2003; Ivchenko et al. , 1989;Ivchenko and Pikus, 1978; Mal’shukov and Chao, 2002),the SGE was initially observed in an asymmetrically con-fined GaAs quantum well (Ganichev et al. , 2002). Thekey signature of the SGE is the electrical current inducedby a non-equilibrium, but uniform, polarization of elec-tron spins. In the non-equilibrium steady-state, the spin-up and spin-down sub-bands have different populations,induced in Ganichev et al. (2002)’s experiment by a circu-larly polarized light excitation. Simultaneously, the twosub-bands for spin-up and spin-down electrons are shiftedin momentum space due to the inversion asymmetry ofthe semiconductor structure which leads to an inherentasymmetry in the spin-flip scattering events between thetwo sub-bands. This results in the flow of the electricalcurrent.The Onsager reciprocity relations imply that there isalso a reciprocal phenomenon of the iSGE induced SOTin which electrical signal due to the SGE is generatedfrom magnetization precession in a uniform, spin-orbitcoupled magnetic system with broken space inversionsymmetry (see Fig. 45) (Hals et al. , 2010; Kim et al. ,2012a; Tatara et al. , 2013). In this reciprocal SOT effectno secondary spin-charge conversion element is requiredand, as for the SOT, (Ga,Mn)As with broken inversionsymmetry in its bulk crystal structure and strongly spin-orbit coupled holes represents a favorable model systemto explore this phenomenon. The effect was observed in(Ga,Mn)As and termed the magnonic charge pumping(Ciccarelli et al. , 2014). This effect is physically simi-lar to the SGE (or alternatively called inverse Rashba-Edelstein effect) observed at Bi/Ag(111) (Rojas-S´anchez et al. , 2013) or topological insulators surfaces (Shiomi et al. , 2014).
F. Established features and open questions
SOTs in bulk non-centrosymmetric crystals are rela-tively well understood since here the iSGE mechanismis not complemented by the SHE, and ST-FMR exper-iments in external magnetic fields can provide a quan-titatively accurate vector analysis of the SOT fields.Among the remaining open questions is what materialparameters control the relative strengths of field-like anddamping-like SOTs. Regarding the crystal symmetries ofthe SOT, Rashba and Dresselhaus spin-orbit fields havebeen already identified whereas an experimental evidence
FIG. 45 (Color online) (a) A charge current through(Ga,Mn)As results in a non-equilibrium spin polarization ofthe carriers, which exchange-couples to the magnetization andexerts the SOT. An alternating current generates a time-varying torque, which drives magnetic precession resonantlywhen a magnetic field is applied. (b) The reciprocal effectof (a) termed the magnonic charge pumping. From Ciccarelli et al. (2014). of the Weyl symmetry SOT [Fig. 11(e)] is yet to bedemonstrated.Compared to ferromagnets, the research of SOTs inantiferromagnetic crystals is still in its infancy. The in-sensitivity to external magnetic fields makes the exper-imental calibration of the staggered SOT field strengthdifficult to perform. In experiments performed to date,the current-induced switching shows clear signatures ofa heat-assisted mechanism (Bodnar et al. , 2018; Meinert et al. , 2018; Olejn´ık et al. , 2018). On the one hand, thisis favorable for lowering the effective magnetic anisotropybarrier but, on the other hand, it may limit the acces-sible writing frequency and current amplitudes, and fur-ther complicates the experimental determination of thestrength of the SOT fields. Therefore, only the experi-mental switching current amplitudes have been reportedin antiferromagnets so far. In CuMnAs structures, theswitching current densities are in the 10 A/cm range for ∼ ns long writing pulses which is comparable to the com-mon ferromagnetic Co/Pt SOT devices (Garello et al. ,2014; Olejnik et al. , 2017).SOT electrical writing speeds, defined as the inverse ofthe writing pulse length, in the CuMnAs antiferromag-net have been experimentally demonstrated to reach theTHz range, which far exceeds the SOT writing speedsin ferromagnets. Antiferromagnets also offer the pos-sibility to combinate the SOT with topological Diracfermions, which are prohibited by symmetry in ferro-magnets. An indication of the SOT-induced opening andclosing of the Dirac crossing has been already reportedin an experimental and theoretical study of the AMR inMn Au (Bodnar et al. , 2018). The ultimate strength ofthis topological AMR, however, has been predicted forpurely semimetallic antiferromagnets in which the Diracpoints are at the Fermi level and no other trivial bandsare crossing the Fermi energy (ˇSmejkal et al. , 2017). Ifexperimentally demonstrated, it would have importantimplications not only for the basic research of topologi-8cal phenomena in magnetic systems but may also providethe desired large magnetoresistance allowing for a bet-ter scalability of the readout signals in antiferromagneticspintronic devices.
VI. SPIN-ORBIT TORQUES AND NON-UNIFORMMAGNETIC TEXTURES
The electrical manipulation of magnetic textures us-ing SOTs opens stimulating perspectives for applications.In Section IV, we already mentioned that domain wallnucleation and propagation play an important role inthe context of SOT-driven switching. In addition, inten-tional and well-controlled domain wall manipulation con-stitutes the basis of alternative, domain wall-based race-track memories (Parkin and Yang, 2015; Parkin et al. ,2008) and logic concepts (Allwood et al. , 2005). In thiscontext, a major breakthrough has been the recent real-ization and control of individual metastable skyrmionsat room temperatures (Fert et al. , 2017; Jiang et al. ,2017a), which show promising potential for such appli-cations (Fert et al. , 2013; Zhang et al. , 2015d). Nonethe-less, evaluating SOTs in magnetic textures poses a spe-cific challenge compared to the magnetically uniform thinfilms discussed in Section IV. While SOTs induce a rota-tion of the magnetization that can be ’simply’ recordedthrough magnetometry (AMR, AHE or MOKE), in mag-netic textures one can only evaluate the global impactof the SOTs through the texture motion and deforma-tion. This feature transforms the magnetometry issueto a magnetic microscopy issue. The present section ad-dresses SOT-driven domain wall and skyrmion motionand dynamics in detail.Starting with a phenomenological description of theinfluence of current-induced torques on domain wall mo-tion in Subsection VI.A, we then discuss its experimentalobservation in in-plane and perpendicularly magnetizeddomain walls in Subsections VI.B and VI.C, respectively.Recent progress achieved on ferrimagnetic and antiferro-magnetic systems is presented in Subsection VI.D. Notethat the role of domain wall nucleation and propagationin SOT driven switching has been discussed in SubsectionIV.E.
A. Domain wall dynamics under current
The dynamics of magnetic textures is governed by theLLG equation, Eq. (1), at the basis of the continuous the-ory of magnetic structures, called micromagnetics (Hu-bert and Sch¨afer, 1998). In this framework, the localmagnetization vector is written M ( r , t ) = M s m ( r , t ),where the spontaneous magnetization modulus M s de-pends on temperature, whereas the unit vector m speci-fies its local orientation as a function of space and time. The torques induced by an electrical current on the mag-netic texture are of two forms. On the one hand, the STTis generally written, in its local version, as the sum of so-called adiabatic and non-adiabatic terms (Beach et al. ,2008)( γ/M s ) T STT = − ( u · ∇ ) m + β m × [( u · ∇ ) m ] , (72)where the velocity u is proportional to the electrical cur-rent density in the magnetic material, its spin polar-ization etc., and where β is the non-adiabaticity factor(no dimensions). This torque is proportional to the gra-dient of magnetization along the current direction andthus vanishes in the domains. On the other hand, theSOT is expressed by Eq. (2). It does not depend onthe gradient of the magnetization at the lowest order,hence acts also on the magnetization within the domains[for higher order expansion, see van der Bijl and Duine(2012)]. Note that in general when a current is appliedto a magnet/metal bilayer, it flows both into the mag-net, leading to STT, and into the metal, leading to SOTin the magnet as well as to an Oersted field. We thusneed to study the effect of these three torques on domainwalls.A qualitative analysis of these current-induced torquesis instructive. For this we consider, for each torqueterm T , the effective field B T obtained by writing T = M × B T , the evaluation being performed at the centerof the domain wall. From the solved form of Eq. (1),i.e., with ∂ m ∂t only on the left-hand side, one sees thatthe magnetization dynamics is driven by the total effec-tive field B M − B T [where the minus sign is consistentwith Eq. (1)], with on the one hand a precession aroundit driven by the gyromagnetic ratio γ , and on the otherhand a relaxation towards it driven by the damping pa-rameter α . To analyze the impact of current-inducedtorques on the domain wall motion, we need to know thetypes of magnetic domain walls in samples where largecurrent pulses can be applied (typically 10 A/m ). Inorder to promote large current densities while avoidingexcessive sample Joule heating, these samples have theshape of nanostrips, with a width w of about a few hun-dreds of nanometers, and a thickness h of the order of afew nanometers (the thickness being generally thinner forinterfacial SOT). As shown in Fig. 46, a limited numberof domain wall structures has to be considered, accordingto the magnetic anisotropy of the sample.
1. Steady domain wall dynamics
In order to get a steady-state current-induced domainwall motion (CIDM) under a torque T , one needs theeffective field B T to be directed along the domains’ mag-netization (a subtlety exists for the vortex wall, as themagnetization is not uniform outside the vortex core, seebelow). This prescription derives from the analogy with9 TW VW(a): X domainsBW NW(b): Z domainsY-NW Y-BW(c): Y domains
FIG. 46 Schematic of domain wall structures in nanostripsrelevant for SOT studies. (a) When the magnetic easy axisis along the nanostrip ( x axis), typically for small magneticanisotropy, magnetostatics leads to two basic structures, thetransverse wall at small width and thickness (left), and thevortex wall at larger lateral dimensions (right) (McMichaeland Donahue, 1997; Nakatani et al. , 2005). Note that thesedomain walls have a non-zero magnetostatic charge. (b)When the easy axis is perpendicular ( z axis), typically forstrong interface anisotropy, magnetostatics favors the Blochwall (left) but the interfacial DMI can favor the N´eel wall(Heide et al. , 2008; Thiaville et al. , 2012), and fix its chirality(right). A ‘bulk’ DMI would favor the Bloch wall, and fix itschirality. (c) The last case of a transverse easy axis ( y axis)is rare. The associated walls, known for a long time (Hubertand Sch¨afer, 1998), are the N´eel wall (Y-N´eel wall) at smallthickness (left), and the Bloch wall (Y-Bloch wall) at largerthickness (right). In the absence of DMI, the domain wallmagnetization is uncorrelated to the magnetization in the do-mains, so that domain or domain wall magnetization arrowscan be reversed with no change of energy. The dashed linesoutline the shape of the domain walls. the case of an applied magnetic field. It expresses a do-main wall motion controlled by damping: if the effectivefield favors one domain, this domain steadily grows andhence the domain wall moves.The STT torques, Eq. (72), depend on the magnetiza-tion gradient along the current direction, i.e., ∼ ( u · ∇ ) m ( ≡ ∂ m /∂x with the axes convention defined in Fig. 46).From Fig. 46 we see that at the domain wall center thisderivative is along the magnetization of the domain onthe right of the domain wall (an exception to this ruleis afforded by the vortex wall, where the magnetiza-tion streamlines are reoriented by 90 ◦ through the vor-tex structure). By construction, the effective field B T associated with the adiabatic STT is orthogonal to thedomains’ magnetization, so that it cannot lead to steadydomain wall motion. On the other hand, the effectivefield associated with the non-adiabatic STT lies along thedomain magnetization. This explains qualitatively therule for steady STT driven domain wall motion, given bythe velocity formula v = ( β/α ) u in which domain wallsmove along the carriers for positive current polarization(majority spin polarization of the current) and positive β factor (Thiaville et al. , 2005; Zhang and Li, 2004). The same conclusions are reached for the vortex wall case, byconsidering the surrounding of the vortex core instead ofthe domains.We now perform the same analysis for the SOTs. Theeffective field associated with the field-like SOT reads B FL = − ( τ FL /M s ) ζ with ζ || y for a current along x ,considering the Rashba symmetry of the spin-orbit cou-pling. This field is oriented like the main part of theOersted field (as w (cid:29) h the y component of the strayfield dominates the z component). The results for thevarious domain wall structures are summarized in Ta-ble III, a generalization of those of Khvalkovskiy et al. (2013): apart from the obvious case of y easy axis (Obataand Tatara, 2008), no steady domain wall motion is ex-pected. On the other hand, for the damping-like SOTwith B DL = − ( τ DL /M s ) m × ζ , only the N´eel wall for the z easy axis is expected to be set in steady motion.
2. Precessional domain wall dynamics
Another characteristic regime of domain wall motion iscalled precessional , meaning that the domain wall mag-netization is rotating in a given direction around the do-mains’ magnetization. Following very general argumentsinitially due to Slonczewski (1972) according to whichthe domain wall position and the angle of the domainwall magnetization are coupled variables in the Hamiltonsense, a continuously precessing domain wall magnetiza-tion induces an overall domain wall motion.The simplest known case of precession occurs under alarge enough field applied along the domains’ magneti-zation, the field being larger than the so-called Walkerfield. In that case, this precession-induced velocity op-poses that due to the applied field, hence the term ofWalker breakdown stressing that domain wall velocitydrops above the Walker field. The Walker thresholdoccurs because the domain wall structure deformationby domain wall magnetization rotation around the ap-plied field can be counter-balanced by internal energies(anisotropy, demagnetizing field, DMI etc.), up to a cer-tain limit. The same breakdown is therefore also ex-pected when the effective field B T is along the domains’magnetization and large enough [with the subtleties thatfor STT, the velocity increases above the threshold when β < α (Thiaville et al. , 2005; Zhang and Li, 2004), whilefor damping-like SOT the threshold is never reached(Thiaville et al. , 2012)].Domain wall magnetization rotation also occurs by re-laxation towards the current-induced effective fields B T .If these fields are below the ‘breakdown’ threshold, a do-main wall position shift will appear as a result of thedomain wall structure transformation when current is ap-plied. When current goes back to zero, and provided thesample is perfect, the opposite domain wall position shiftwill however occur as the domain wall recovers its initial0 DW STT ad. STT na. FL SOT/ DL SOTOerstedTW N Y N Nodd even even nullVW N Y N Nodd even even oddBW N Y N Nodd even even nullNW N Y N Yodd even even oddY-NW N Y Y Nodd even even oddY-BW N Y Y Nodd even even oddTABLE III Characteristics of the effective field B T associ-ated with the current-induced torques, evaluated at the centerof the domain wall types shown in Fig. 46. For each case, thefirst line indicates (Y/N) whether or not this effective fielddrives the domain wall into steady motion. The second lineindicates (null/odd/even) if this field is zero and, when it isnot, if it is even or odd with respect to the domain wall mag-netization, the case ”odd” leading to long-term precessionaldomain motion. structure. Note that several devices based on an antici-pated stick-slip domain wall motion under application ofdissymmetric pulses with short rise-time and long fall-time have been proposed, based on this phenomenon. Apartial list of cases with domain wall shift was presentedin Khvalkovskiy et al. (2013). The full list is given inTable III. When the effective field related to a current-induced torque is large enough, the domain wall structuregoes to its image where the domain wall magnetizationhas been reversed. Whether this process continues or notdepends on the power to which the domain wall magne-tization enters the expression of the effective field B T . Ifthis power is odd, the opposite field will act on the op-posite domain wall magnetization, leading to indefiniteprecession of domain wall magnetization and hence tolong-term precessional domain wall motion. If the poweris even, however, indefinite precession will not occur andonly a domain wall position shift will occur. These casesare also indicated in Table III. The table shows that field-like SOT (and Oersted field) can only drive domain wallsin the y -easy axis situation, see Y-domain walls in Fig.46.With this analysis in mind, we turn in the next sub-sections to each situation, reviewing the experimental re-ports existing on the subject. B. In-plane magnetized samples
1. Soft samples (X domains)
These samples have been the workhorse of the initialstudies of the STT, leading to the definition of the adia-batic and non-adiabatic STT terms. As Table III shows,such samples are generally not adequate to test the SOT.The vortex wall is a special case in this picture, being acomposite object that can easily deform by lateral mo-tion of the vortex core, inducing a displacement of thewhole wall along the nanostrip [see e.g., (Beach et al. ,2008; Clarke et al. , 2008; Tretiakov et al. , 2008)]. As aresult, under adiabatic STT for example, the vortex coredisplaces laterally (along y ), leading to a longitudinal do-main wall displacement (along x ). The effect is howevertransient as the core eventually stops or disappears atthe nanostrip edge, transforming the vortex wall into atransverse wall. The same effect is expected under SOT.Micromagnetic studies of the effect of disorder onCIDM by STT show that disorder induces, on top ofthe expected current threshold for domain wall motion,a modification of the linear regime (change of slope,offset), as well as a suppression of velocity breakdown(Nakatani et al. , 2003; Thiaville and Nakatani, 2009; Thi-aville et al. , 2005). The modification of the linear regimemay be partly understood by introducting a larger effec-tive damping constant for a magnetic texture (such as adomain wall) moving in a disordered medium (Min et al. ,2010).Up till now, only two studies have considered X do-mains with adjacent heavy metal layers. An early studyon Pt/NiFe (Vanhaverbeke et al. , 2008) investigated theinfluence of the current direction on the domain wall po-larity (i.e. the direction of the domain wall’s transversemagnetization). Another more recent study addressedthermal effects in Ta/NiFe/Pt (Torrejon et al. , 2012).Moreover, typical thicknesses of the ferromagnetic filmwere 10 nm, so that the effect of the interfacial torques isstrongly reduced. Note that the Oersted field effect wasdirectly observed in the case of a bilayer sample (Uhl´ır et al. , 2011) by time-resolved photoelectron emission mi-croscopy using x-ray magnetic circular dichroism, a tech-nique that could be used to measure the field-like SOT insitu . Simulations have shown that field-like SOT modifiesthe STT driven dynamics (Seo et al. , 2012).Trilayer samples, typically Co/Cu/NiFe where easierdomain wall motion and higher velocities have been ob-served, are a special case that could not be understoodin the frame of STT plus Oersted fields. It was thus pro-posed that perpendicular spin currents may play somerole (Pizzini et al. , 2009; Uhl´ır et al. , 2010). Khvalkovskiy et al. (2009) performed a numerical exploration of the ef-fect of various forms of SOT on both transverse wall andvortex wall, taking ζ = x and ζ = z , i.e. the two casesthat are not considered in standard SOT configuration1[the latter case was investigated in (Khvalkovskiy et al. ,2013) for transverse wall]. The results show that indeedin some cases domain wall sustained motion is expected(field-like SOT for ζ = x , damping-like SOT for ζ = z for a vortex wall), but their relation to the experimentalsituation is unclear. Another family of bilayer samplesare the synthetic antiferromagnets. In CoFe/Ru/CoFe,a very low threshold for CIDM has been measured (Lep-adatu et al. , 2017), and attributed to the intrinsic dy-namics of antiferromagnetically-coupled transverse walls,driven by non-adiabatic STT (see Subsection VI.D).
2. Anisotropic samples with Y domains
In the case of Y-domain walls (see Fig. 46) the field-likeSOT is directly active (Obata and Tatara, 2008). Suchsamples require an in-plane anisotropy that is strongerthan the magnetostatic energy cost. This has been re-alized by growing epitaxial layers on single-crystal sub-strates. One example is (Ga,Mn)As grown on (001) GaAs(Thevenard et al. , 2017), where structures with X do-mains and Y domains were compared, on 50 nm thicklayers so that bulk SOT would be active. Large current-induced effects were observed, that strongly differed inthe two cases, but no simple and global understanding ofthe observed effects could be found.Another way to obtain such structures is to use largemagnetostriction materials, as growth-induced stress isrelaxed at the edges of a nanostrip, modifying theanisotropy locally. As a result, transverse Y domainswere observed in Ni Pd films (Chauleau et al. , 2011).No study of CIDM could however be realized on suchsamples, as the Curie temperature was rapidly reached. C. Perpendicularly magnetized samples
A numerical micromagnetic study (Fukami et al. , 2008)demonstrated the interest of perpendicularly magnetizedsamples for CIDM: as the sample thickness is reduced,the energy cost of a N´eel wall relative to the Bloch walldecreases linearly. Thus, the Walker breakdown fieldalso decreases linearly, as well as the current thresholdfor domain wall motion under the adiabatic STT. Inaddition, microscopic STT theories predicted that thenon-adiabatic torque might be larger in narrow domainwalls (Akosa et al. , 2015; Bohlens and Pfannkuche, 2010;Tatara and Kohno, 2004; Waintal and Viret, 2004). Ex-perimentally, studies first focused on the influence of theelectric current on the domain wall depinning (Boulle et al. , 2008; Burrowes et al. , 2010; Ravelosona et al. ,2005). The results seemed encouraging, but there wereonly few systems exhibiting CIDM without the assistanceof external field. One of these systems are the Co/Ni mul-tilayers where the predictions of the adiabatic STT model were most clearly evidenced (Koyama et al. , 2011): (i)the existence of an intrinsic critical current that dependson the geometric structure of the domain wall rather thanthe extrinsic pinning; (ii) the independence of the criticalcurrent on a perpendicular magnetic field.
1. Demonstrations of spin-orbit torques in current-induceddomain wall motion (a)(b)
FIG. 47 (a) Differential Kerr microscopy imaging of do-main wall displacements (stripes of black or white contrast)in an array of Pt/Co 0.6 nm/AlOx 500 nm wide nanostrips,after 20 current pulses ( J = 1 . × A/m , 3 ns dura-tion) (Miron et al. , 2011b). (b) Observation of chiral ef-fects: the velocity of up/down and down/up domain walls(blue and red) is the same, but becomes different when an in-plane field is applied [sample Pt/CoNiCo/TaN, current den-sity J = 1 . × A/m , either positive (triangles) or neg-ative (circles)]. Within the DMI-SOT model, the DMI fieldstrength is indicated by the value of the crossing field, wherethe domain wall velocity changes sign (Ryu et al. , 2013). Among the materials with perpendicular anisotropy,the Pt/Co/AlOx trilayers in particular have attracted alot of interest. The domain wall motion was found tobe significantly faster (Baumgartner et al. , 2017; Miron et al. , 2011b; Moore et al. , 2008) compared to the pre-vious observations in NiFe or Co/Ni films [Fig. 47(a)].Besides the practical importance of fast domain wall mo-tion, the physical parameter determining this improve-ment was the structural inversion asymmetry (Miron et al. , 2009). Indeed, while Pt/Co/AlOx supports fastCIDM, magnetically similar Pt/Co/Pt symmetric layersdo not exhibit any CIDM at all (Cormier et al. , 2010;Miron et al. , 2009). These first observations were ini-tially analyzed within the framework of the STT model,including the influence of the field-like SOT, which wasdiscovered at the same time. It was proposed that thebroken symmetry could accelerate the spin flip rate andenhance the non-adiabatic torque, the field-like SOT sta-2bilizing the Bloch wall structure to prevent the Walkerbreakdown (Miron et al. , 2011b). This idea has been pur-sued by several theoretical investigations of the ability offield-like SOT to delay (Ryu et al. , 2012a) or suppressthe Walker breakdown (Linder and Alidoust, 2013; Ris-ingg˚ard and Linder, 2017; Stier et al. , 2014).At that stage, there was still a major discrepancy be-tween the STT model and the experiment: the domainwalls move in the direction of the electric current and notalong that of the electron flow (Moore et al. , 2009). Thisintriguing observation motivated several theoretical stud-ies, which found that the combination of STT and SOTcould in certain cases produce backwards motion (Boulle et al. , 2014; Kim et al. , 2012b). However these scenariiwere not robust: the backward motion was only obtainedfor certain values of the physical parameters and only ina certain range of current density. In parallel, it was ob-served that nearly symmetric Pt/Co/Pt samples exhibitCIDM if an external in-plane field is applied parallel tothe current, sufficiently large to convert Bloch walls toN´eel walls (Haazen et al. , 2013) [the N´eel wall structureunder in-plane field was later confirmed by anisotropicmagnetoresistance measurements (Franken et al. , 2014)].The damping-like SOT mechanism was shown to be com-patible with all observations, especially (i) the reversalof the domain wall motion upon locating the thicker Ptlayer below or above the Co layer; (ii) the reversal of thedomain wall motion upon change of sign of the in-planefield and (iii) the fact that two successive domain wallsalways move in opposite directions. The latter point is ofcrucial importance: all N´eel walls, having the same mag-netization, feel the same damping-like SOT and are hencedisplaced in opposite directions, like under an easy-axis( z here) field.In this context, a breakthrough was the micromag-netic study (Thiaville et al. , 2012) of the dynamics ofN´eel walls under magnetic field and damping-like SOT,in the case where such walls are stabilized by the in-terfacial DMI. The DMI (Dzyaloshinskii, 1957; Moriya,1960) is an antisymmetric exchange interaction that isallowed when the medium does not have inversion sym-metry. The general form of the DMI energy density reads W DMI = D ij e i · (cid:16) m × ∂ m ∂j (cid:17) , where the coefficient D ij pos-sesses the same symmetries as the SOT response func-tion, χ ij , discussed in Subsection III.B (Freimuth et al. ,2014a). Hence, the generalization of SOT symmetriessuggested by Fig. 11 also applies to DMI. In an isotropicbulk material without inversion symmetry (like a heap ofscrews), to the lowest order in gradient expansion, DMIin continuous micromagnetic form is expressed by an en-ergy density (Bogdanov and Yablonskii, 1989) W = D m · ( ∇ × m ) . (73)Such an interaction favors helicoidal magnetization rota-tions of a given handedness. Referring to Fig. 46, this form of DMI stabilizes chiral Bloch walls or Y-Blochwalls.On the other hand, at the interface between two dis-similar materials where inversion symmetry is struc-turally broken (Fert, 1990), assuming the highest sym-metry ( C ∞ v ) and considering the lowest order in spatialgradient, one obtains (Bogdanov and Yablonskii, 1989;Heide et al. , 2008) W = D m · [( z × ∇ ) × m ] . (74)This interaction, called interfacial DMI, favors cycloidalmagnetization rotations of a given handedness. Againreferring to Fig. 46, this form of DMI stabilizes chiralN´eel walls (but none of the Y-N´eel walls). The imme-diate consequence is that chiral N´eel walls move underdamping-like SOT without any in-plane field, with suc-cessive walls moving in the same direction as their do-main wall magnetizations are opposite. Such a motion,already obtained with STT, is required for domain wallracetrack applications (Parkin et al. , 2008). Another no-table feature of the domain wall dynamics under DMIand damping-like SOT is that the relative sign of do-main wall velocity with respect to that of the current isgiven by the product of the sign of the damping-like SOTand the sign of the DMI.Interfacial DMI was already evidenced in magneticatomic monolayers or bilayers by spin-polarized scan-ning tunneling microscopy that revealed magnetizationcycloids of fixed handedness (Bode et al. , 2007; Meckler et al. , 2009). However, these were situations of very largeDMI so that the uniform magnetic state was destabilized.For the Pt/Co/AlOx case, direct proof that domain wallsare chiral N´eel walls was obtained by NV-center magneticmicroscopy (Tetienne et al. , 2015), and by x-ray magneticcircular dichroism (Boulle et al. , 2016). In addition, spin-polarized low energy electron microscopy has shown thechange of domain wall structure from chiral N´eel wallto achiral Bloch wall as a function of the thickness ofthe magnetic layer (Chen et al. , 2013a,b), confirming theinterfacial DMI description.The prediction of Thiaville et al. (2012) was immedi-ately backed by two experimental papers (Emori et al. ,2013; Ryu et al. , 2013). As the sign of the SHE (henceof the damping-like SOT) was known from other mea-surements, the direction of domain wall motion undercurrent could be related to the sign of DMI [Fig. 47(b)].This sign was later obtained by several other techniques,so that presently estimates of interfacial DMI for a fairnumber of NM/FM interfaces exist. In this picture, thePt/Co interface stands out with one of the largest interfa-cial DMI constant D s ≈ − . et al. ,2015). One of the techniques for determining the DMIconsists in applying an additional in-plane field in orderto compensate the DMI effective field on the domain wall.At this compensation, the domain wall velocity crosseszero (Emori et al. , 2013) [for an example see Fig. 47(b)].3
2. Domain wall motion under spin-orbit torque
We now describe in more detail the dynamics of do-main walls under SOTs and DMI. Once the torques areknown and quantified, the study of their impact on do-main wall motion should ultimately be performed by nu-merical micromagnetic simulations, for the sample pa-rameters and geometrical dimensions. For the physicalunderstanding, however, simplified and as analytical aspossible models are helpful. The simplest model was ex-posed in Subsection VI.A. The next level of complexityis addressed by the so-called q − Φ model, that describesa 1D domain wall dynamics for an assumed domain wallprofile described by only two variables, namely the do-main wall position q and the angle Φ of the domain wallmagnetization within the plane orthogonal to the easyaxis (Schryer and Walker, 1974; Slonczewski, 1972). ForSOT-driven domain wall motion assisted by DMI, themodel was established by Thiaville et al. (2012), andfurther developed to incorporate in-plane fields (Emori et al. , 2013) and STT (Torrejon et al. , 2014). At a higherlevel of complexity, a numerical micromagnetic calcula-tion is performed assuming 1D structure and dynamics,i.e., the magnetization depends only on the x coordinate,the magnetostatic effects being computed for the nanos-trip width w and thickness h . Finally, for ultrathin filmsthe full model consists of 2D numerical micromagnetics.Figure 48(a) shows the predicted velocity versus cur-rent curves, v ( J ), in the case of pure damping-like SOTand for various values of the effective DMI energy den-sity, D = D s /h . The domain wall velocity initially riseslinearly with current, following a slope that does not de-pend on DMI and is given, for DMI dominating the mag-netostatic energy associated to a N´eel wall and using thenotation of Eq. (2), by v = − γ π ∆ W αM s τ DL . (75)Here, ∆ W is the micromagnetic domain wall width pa-rameter. Upon further increase of the current density,the velocity saturates towards a plateau determined bythe DMI strength, v D = γπD/ (2 M s ) (derived in the samelimit). The velocity saturation is physically explained bythe progressive rotation of the domain wall magnetizationfrom N´eel to Bloch around the effective field B DL associ-ated with the damping-like SOT. This rotation leads to areduction of the damping-like SOT on the domain wall, asthe torque vanishes for a Bloch wall. This behavior is ingood overall agreement with experiments [see Fig. 48(b)].With intrinsic curvature and no Walker breakdown, thevelocity versus current behavior, v ( J ), is markedly dif-ferent from that expected for STT.When DMI is not much larger than the magnetostaticenergy associated to the N´eel wall, the situation is morecomplex to analyze, as the velocity v D decreases and be-comes comparable to that induced by STT. Moreover, FIG. 48 Velocity of domain walls in ultrathin Pt/Co/oxidefilms with DMI, under current. (a) Micromagnetic 1D calcu-lations (points) of domain wall velocity versus current density,for various values of effective DMI in a 0.6 nm Co film (Thiav-ille et al. , 2012), considering only damping-like SOT. Curvesshow the q − Φ model results for comparison. (b) Measureddomain wall velocity under current for Pt/CoFe/MgO andTa/CoFe/MgO (Emori et al. , 2013). The CoFe film has 80-20atomic composition and is 0.6 nm thick. Note the log scale forvelocities, and the opposite current signs for the two heavymetal layers. for the q − Φ model, the analytical expressions becomemuch more complex. The analysis of the competition ofDMI versus domain wall magnetostatics, together withthat of damping-like SOT versus
STT, was performed by(Torrejon et al. , 2014) in the case of HM/CoFeB/MgOfor HM=Hf, Ta, TaN, W i.e., the beginning of the 5 d series, using the q − Φ model to analyze the experiments.This showed that the determination of the DMI by the‘crossing field’ technique is strongly affected by the STTwhen DMI is not large.
3. Two-dimensional effects in current-induced domain wallmotion
Unlike in-plane magnetized nanowires, where domainwalls behave as quasi-1D objects, in perpendicular sam-ples domain walls act more like 2D membranes. One ofthe first observations on the influence of the 2D charac-ter on the CIDM in materials with broken inversion sym-metry was the occurrence of a domain wall tilt. Whendomain walls are displaced by sufficiently long currentpulses, their end position is no longer perpendicular tothe wire (at the energy minimum), but tilted at a certainangle (Baumgartner and Gambardella, 2019; Ryu et al. ,2012b) [see Fig. 49(a)]. The fact that this tilt is vis-4ible at rest is a proof that domain wall pinning exists.Boulle et al. (2013) proposed that this tilting arises fromthe competition between the damping-like SOT and theDMI. Because the DMI energy prefers that the domainwall magnetization is perpendicular to the domain wall,the damping-like SOT acting on the domain wall magne-tization modifies the domain wall angle [see also Martinez et al. (2014)]. A direct consequence of this current-driventilting is an additional deformation of the v ( J ) curve atlarge current density. For damping-like SOT only, thisgives rise to a velocity increase close to the threshold fordomain stability given by τ DL max = γB eff K / B eff K beingthe effective perpendicular anisotropy that incorporatesthe demagnetizing field). FIG. 49 (a) Kerr imaging of domain wall tilting producedby current injection (Ryu et al. , 2012b). (b) Kerr images ofcurrent induced domain wall motion in a non-collinear geome-try. (c) Magnetization reversal controlled by geometry (Safeer et al. , 2016).
It was recently shown that the 2D character of CIDMcan be exploited for domain wall manipulation. Sincethe domain wall magnetization is either aligned (Blochwall) or perpendicular (N´eel wall) to the domain wall di-rection, the control of the domain wall tilt allows for set-ting the SOT efficiency by modifying the angle betweenthe electric current and the magnetization of the domainwall. Using this approach, Safeer et al. (2016) have shownthat the current induced domain wall motion in the non-collinear geometry exhibits surprising features [see Fig.49(b)]. Namely, depending on their polarity (up/down ordown/up), the domain walls move faster for a certain signof the electric current. This phenomenon links the po-larity of the domain walls with their direction of motion.Therefore, by controlling the shape of a magnetic layer,one can control its magnetization reversal [Fig. 49(c)].
4. Domain wall motion under combined spin transfer andspin-orbit torques
A detailed study of CIDM in Pt/(Co/Ni) N /Co/MgOas a function of magnetic layer thickness (by varying therepetition number N ) was realized by Ueda et al. (2014). The (Co/Ni) N multilayer system is interesting becausesince magnetic anisotropy arises from the internal Co/Niinterfaces, the total thickness of the multilayer can bechanged while keeping the same magnetic anisotropy,which is not possible for a single Co layer. The domainwalls were observed to move along the electron flow forlarge thicknesses ( N >
N < x , and transverse y ), the crossing field effect [see Fig. 47(b)] was observedin the longitudinal case, in accord with the damping-likeSOT in the presence of DMI. The transverse field was ob-served to linearly modify the velocity of both up/downand down/up domain walls, in the same way. This is alsoconsistent with the DMI and damping-like SOT mech-anism, as the magnetizations of two consecutive chiralN´eel walls precess under the respective fields B DL of thedamping-like SOT towards the same y direction. Thus,for not too large y fields, one polarity increases this ro-tation and hence decreases the domain wall velocities,whereas the other polarity decreases this rotation and in-creases the velocities. From the symmetry of the effects,the authors concluded that the field-like SOT effect wasnegligible. Direct measurements of the two componentsof the SOT confirmed the reduced value of the field-likeSOT. This work clearly evidences the transition frombulk to interfacial CIDM and can serve as a guide forfurther studies of this physics. For example, the absenceof domain wall motion for 3 ≤ N ≤ N multilayers (Bang et al. , 2016).In another study in the same (Co/Ni) N system, thestructure was designed such that SOT acted as a per-turbation with respect to STT (Yoshimura and Koyama,2014). The sample was medium-thick ( N = 4) and thestructure was nominally symmetric with Pt and Ta onboth sides, with the same thicknesses. The domain wallmotion, driven by STT, was modified by applying in-plane fields, both along the current ( x ) or transverse( y ). As expected for adiabatic STT, the motion was sup-pressed by large in-plane fields, as these fields block theprecession of the domain wall moment. The surprise wasthat the domain wall motion windows were not centeredat zero field, with the x -field offset reversing sign be-tween up-down and down-up domain walls. This could5be qualitatively interpreted by (i) a precession dissym-metry under in-plane field that leads to different resi-dence times for N´eel walls of opposite chiralities, and (ii)a non-compensated damping-like SOT due to a measuredimbalance in the conduction of the top and bottom Ptlayers. On the other hand, the independence of the y fieldoffset on the domain wall type (up/down or down/up) isconsistent with an effect of Oersted field and/or field-like SOT. This work, more generally, proposes a way toexperimentally test the presence of the SOT and of theOersted field, as any in-plane field affects the precessionof the domain wall moment triggered by STT. Here we re-fer also to the numerical work by Martinez (2012) on theSTT plus field-like SOT case, for various values on non-adiabaticity, and the micromagnetic simulations analysisby Martinez et al. (2013) and Boulle et al. (2014) of ex-perimental results for Pt/Co/AlOx in terms of STT plusSOT.
5. Motion of magnetic skyrmions under spin-orbit torques
Magnetic skyrmions with non-zero spin winding num-ber are compact magnetic textures with a non-trivialtopology, so that they cannot be removed by a contin-uous transformation, in the continuum limit. Althoughthere are still arguments about the precise meaning ofthis terminology, we adopt here the definition agreed onby a large panel of authors (Hellman et al. , 2017). Thereis currently an increasing interest in the electrical manip-ulation of such objects as they could serve as fundamentalbuilding blocks for data storage and logic devices (Fert et al. , 2013; Tomasello et al. , 2014; Zhang et al. , 2015d).Skyrmions have, in addition to topology and comparedto the magnetic bubbles extensively investigated in thepast (Malozemoff and Slonczewski, 1979), a fixed chiral-ity which is an important asset for SOT as can be inferredfrom the preceding considerations.A physically appealing way to understand how the var-ious characteristics of a skyrmion affect its response tocurrent-induced torques is offered by Thiele’s equationderived from the LLG equation to handle the steady-state motion of rigid textures (Thiele, 1973). Thiele’sequation has been generalized to include STT (Thiav-ille et al. , 2005) and, more recently, also SOT (Sampaio et al. , 2013). It reads, G × ( v − u ) − D ( α v − β u ) + F SOT + F = , (76)where v is the in-plane velocity of the skyrmion center, u is the spin-drift velocity, β is the non-adiabaticity param-eter related to STT [see Eq. (72)], G is the so-called gy-rovector, D is the dissipation tensor introduced by Thiele(from which the damping coefficient α was factored outwhen generalizing to STT), F is the other force appliedto the skyrmion (e.g., pinning), and finally F SOT is the
FIG. 50 (a) N´eel- and (b) Bloch-skyrmions (both with nega-tive polarity), adapted from Fert et al. (2017). (c) Schematicof the forces (black arrow) applied to a skyrmion (circle)by a current density j c , in the case of adiabatic and non-adiabatic STT. The forces are independent on the type of theskyrmion and only depend on its core polarity. (d) Schematicof the forces (black arrows) applied to a skyrmion (circle) bya damping-like SOT with a spin polarization along ζ , as indi-cated by the green arrow. The forces depend on whether theskyrmion is Bloch or N´eel. force that SOTs apply to the skyrmion. Topology ap-pears in the gyrovector G = ( M s h/γ ) 4 πN Sk z , that isalong the film normal and proportional to the topolog-ical (or skyrmion) number N Sk . The latter is simply,for a compact texture, N Sk = Sp with p the polarity ofthe magnetization of the skyrmion center (+1 for + z )and S the winding number of the magnetization (+1 forthe simple skyrmions). The dissipation tensor, diagonalfor high-symmetry textures, is related to the size of theskyrmion [see e.g. Hrabec et al. (2017)]. The force F isnon-zero for example when a confining potential exists,or a small z field gradient. The STT forces on a skyrmionare illustrated in Fig. 50(c).The force from the SOT is computed by projectingthe SOT on the skyrmion displacement (the procedureby which Thiele’s equation is constructed), as a volumeintegral for each component F i, SOT = − (cid:90) d V B SOT · ∂∂i M = τ DL ζ · (cid:90) d V m × ∂∂i m . (77)The field-like SOT gives no contribution to the force asit acts like a constant in-plane field. As for the damping-like SOT contribution, remembering that ζ (cid:107) y for cur-rent along x , Eq. (77) amounts to one term of the DMIenergy density. For the x component of the force (alongthe current), it is the part of the interfacial DMI thatinvolves the x gradients [Eq. (74)]. For the y componentof the force (transverse to the current), it is the partof the bulk DMI that involves the y gradients [Eq. (73)].6Thus, the damping-like SOT force on a skyrmion dependson its chirality and of its type (i.e. Bloch or N´eel), seeFig. 50(d). Because of the gyrotropic term, both STTsand damping-like SOT drive the skyrmion, at some anglebetween the x and y axes, so that skyrmion motion undercurrent alone does not allow to infer its internal structure.Nevertheless, conventional magnetic bubbles, whose low-est energy state is an achiral Bloch skyrmion because ofthe absence of significant DMI, would be sorted accord-ing to their Bloch chirality by damping-like SOT. Thisis in contrast to skyrmions that have a definite chirality(fixed by DMI) and should all follow the same trajectory.Metastable magnetic skyrmions have been recently ob-tained at room temperature in transition metal multilay-ers (Boulle et al. , 2016; Chen et al. , 2015a; Hrabec et al. ,2017; Jiang et al. , 2015; Moreau-Luchaire et al. , 2016;Pollard et al. , 2017; Woo et al. , 2016). Many experi-ments have revealed skyrmions motion under current, ei-ther along or against the direction of the electron flow, inmost cases in agreement with the DMI and damping-likeSOT sign (Hrabec et al. , 2017; Jiang et al. , 2015, 2017b;Litzius et al. , 2017; Woo et al. , 2016; Yu et al. , 2016a).The influence of disorder and thermal fluctuations on thedriven motion of single skyrmions have been investigatedeither using particle-simulations (Lin et al. , 2013) or bymicromagnetic modeling (Sampaio et al. , 2013), demon-strating that skyrmions have the tendency to avoid point-like defects (Iwasaki et al. , 2013a,b). Remarkably, theangle of the gyrotropic deflection (sometimes also calledskyrmion Hall angle) observed in experiments is poorlyreproduced by Thiele’s equation. This discrepancy maybe due to disorder effects, e.g. sliding along grain bound-aries, as suggested by recent simulations (Kim and Yoo,2017; Legrand et al. , 2017; Reichhardt and Olson Re-ichhardt, 2016; Salimath et al. , 2018). Moreover, theskyrmion dynamic deformation leads to an influence ofthe field-like SOT on the deflection angle (Litzius et al. ,2017). Finally, the influence of the gradient of the z -component of the Oersted field deserves further investi-gation (Hrabec et al. , 2017). Altogether, skyrmions ap-pear as favorable objects to be controlled by either STTor SOT since their velocity reach that of magnetic do-main walls in the same structures. Further explorationof their robustness and scalability is currently on-going(Bernand-Mantel et al. , 2018; B¨uttner et al. , 2018).
6. Impact of disorder
In sputtered ultrathin magnetic multilayers, disorderis so strong that CIDM only occurs at large (field orcurrent) drive. At low drive, domain wall motion con-sists of thermally assisted hopping between pinning sites.This regime of domain wall motion is called creep, ordepinning, depending on field magnitude and type ofdomain wall pinning(Gorchon et al. , 2014; Kim et al. , 2009; Metaxas et al. , 2007). Whereas field-driven andSTT-driven creep seemed to be well understood (Chauve et al. , 2000; DuttaGupta et al. , 2016; Jeudy et al. , 2016),the situation has changed with the introduction of SOTand DMI. Several experiments of CIDM have shown thatthe creep regime of domain wall motion deserves furtherstudy in order to be fully understood (Lavrijsen et al. ,2012, 2015; Vanatka et al. , 2015). For instance, it wasrecently proposed that structural inversion asymmetrycould be responsible for a chiral dissipation mechanismaffecting the domain wall dynamics, called chiral damp-ing (Akosa et al. , 2016; Ju´e et al. , 2016b). More recentlyit has been shown that, as the domain wall energy be-comes orientation-dependent under in-plane field (Pelle-gren et al. , 2017) - an effect reinforced by DMI and strik-ingly evidenced by specific domain shapes (Lau et al. ,2016a) -, the simple creep model with uniform domainwall tension fails. Thus, the analysis of creep motionunder in-plane field has to be thoroughly re-examined.In contrast, a thorough study of the influence of disor-der on CIDM in the flow or precessional regimes is stillmissing. Similarly to in-plane materials (see Sec. VI.B.1),one may expect a modification of the threshold valuesfor domain wall motion, a change of slope and/or off-set of velocity in the linear regime, and suppression ofvelocity breakdown. Regarding the apparent offset in ve-locity, a simple model considering the statistical averageof the inverse velocities was shown to reproduce the fea-ture (Feldtkeller, 1968; Ju´e et al. , 2016a). The suppres-sion of velocity breakdown has been observed and nu-merically reproduced for field-driven motion in the pres-ence of interfacial DMI (Ajejas et al. , 2017; Ju´e et al. ,2016a; Pham et al. , 2016; Yoshimura et al. , 2016). Infact, simulations reveal that this effect occurs even in theabsence of disorder, as soon as the domain wall has a suf-ficient length to depart from 1D behavior. The noveltyintroduced by STT, SOT, and DMI is that the domainwall drive depends on the local domain wall orientation,which modifies the overall energetics and dynamics. Inthe creep regime under CIDM and field, these angle de-pendences are exemplified by the formation of triangular-shape domains pinned at nucleation sites (Moon et al. ,2018, 2013).
D. Antiferromagnetic and ferrimagnetic systems
The search for extremely high domain wall velocity hasrecently brought perpendicularly magnetized syntheticantiferromagnet strips, such as (Co/Ni)/Ru/(Co/Ni), tothe forefront. In such systems, Yang et al. (2015) re-ported SOT-driven domain wall velocities as fast as 750m/s and explained the results by the enhanced Walkerbreakdown threshold. Indeed, in the presence of Ru-mediated interlayer exchange coupling (RKKY) the az-imuthal angles of the two antiparallel domain walls sta-7bilize each other such that the domain wall propagatesin the flow regime over a larger range of driving currentdensities [see also (Lepadatu et al. , 2017)]. In a recentwork Qiu et al. (2016) observed that the presence of theRu spacer layer may affect the spin current, leading todifferent SOTs for such trilayers.Similar ideas resulted in the proposition of antifer-romagnetic skyrmions, either in the bilayer form or inbulk antiferromagnets, which display no skyrmion Halleffect and could also reach very high velocity in the lattercase (Barker and Tretiakov, 2016; Tomasello et al. , 2017;Zhang et al. , 2015e). Bulk antiferromagnets are particu-larly interesting for their ability to support THz dynam-ics, and it was recently proposed that antiferromagneticdomain walls driven by SOT could reach extremely highvelocities, while displaying a Lorentz contraction whenreaching the spin wave group velocity (Gomonay et al. ,2016; Shiino et al. , 2016). The investigation of antiferro-magnetic spintronics is still at its infancy though (Baltz et al. , 2018; Jungwirth et al. , 2016), and alternative ma-terials are being explored. From this perspective, ferri-magnets such as FeGdCo offer an appealing platform dueto the tunability of their compensation point. For in-stance, recent studies have demonstrated large enhance-ment of field-driven domain wall velocity and SOT ef-ficiencies close to the angular momentum compensationpoint (Kim et al. , 2017a; Mishra et al. , 2017).
VII. PERSPECTIVES
SOTs offer a powerful and versatile tool to manipu-late and excite magnetic order parameters, and efficientlycontrol magnetic domain walls and skyrmions. A partic-ularly attractive feature of these torques is their abilityto excite any type of magnetic materials, ranging frommetals to semiconductors and insulators, in both ferro-magnetic and antiferromagnetic configurations. This ver-satility has led to groundbreaking accomplishments thatcould not be achieved with STT: the switching of sin-gle layer ferromagnets, ferrimagnets, and antiferromag-nets, as well as the excitation of spin waves and auto-oscillations in planar and vertical device geometries.The discovery of topological materials as spin sourceshas opened appealing avenues for the realization of verylarge charge-to-spin conversion and low critical switchingcurrent. Topological insulators, Dirac semimetals, Weylsemimetals, Kondo topological insulators as well as 2Dmaterials (bismuth chalcogenides, graphene and its sib-lings, transition metal dichalogenides, transition metaltrihalides etc.) present a unique opportunity for the ex-ploitation of exotic spin-charge conversion mechanismsand chiral spin textures.A number of questions remain open, which will havean impact on future developments and materials design.(i) Whereas the basic mechanisms behind SOT seem to be understood, a robust and systematic quantitativeagreement between theory and experiment is still lacking.In particular, understanding the interplay between inter-facial, bulk, but also orbital contributions to SOT, DMIand chiral damping in magnetic multilayers will indicatehow to improve their efficiency.(ii) Besides the two ”flagship mechanisms” that con-trol the current-induced dynamics (iSGE, SHE), novelphenomena have been identified recently: spin swapping(Saidaoui and Manchon, 2016), interfacial spin currents(Amin et al. , 2018), chiral damping (Ju´e et al. , 2016b)etc. What is the actual magnitude of these effects andhow do they influence the magnetization dynamics? Howcan they be best harvested to enhance the operability ofSOT devices?(iii) The electrical control of magnetic domain wallsand skyrmions substantially benefits from SOTs. Nev-ertheless, their behavior in the presence of disorder, andparticularly the creep and depinning regimes, need to bebetter understood. How can these regimes provide in-formation about the nature of the chirality (dissipationand energy)? Several novel torques have been predictedin these textures (van der Bijl and Duine, 2012), but notobserved yet. In addition, topological currents have beenproposed to enhance the mobility of both ferromagneticand antiferromagnetic skyrmions (Abbout et al. , 2018;Akosa et al. , 2017), which calls for experimental verifica-tion.(iv) Antiferromagnets bear outstanding promises dueto the zero net magnetization and their inherent THz dy-namics (Baltz et al. , 2018; Jungwirth et al. , 2016, 2018).However, to date, only a few antiferromagnets have beenelectrically manipulated (CuMnAs, Mn Au and NiO)(Bodnar et al. , 2018; Chen et al. , 2018b; Moriyama et al. ,2018; Wadley et al. , 2016). The next frontier is to ex-tend these observations to more materials, including non-collinear antiferromagnets. The latter present the ad-vantage of displaying AHE as well as MOKE response,enabling for the electrical and optical detection of theirorder parameter’s orientation (Nakatsuji et al. , 2015). Anatural development direction will be to extend theseideas to frustrated magnets that support exotic magneticbehaviors (Balents, 2010).(v) Finally, the search for most efficient sources ofSOTs raises the question about the nature of spin-orbiteffects in the presence of very large spin-orbit coupling.How do concepts such as spin currents, SHE, iSGE, DMIand magnetic damping evolve when the spin-orbit inter-action is comparable to or larger than the crystal field?Is there a limit to the amount of angular momentum thatcan be transferred to the magnetic system, and if so, howcan it be determined? What materials combination pro-duces the largest torque? Similar questions can be askedwhen electronic correlations are important, such as in(Ce,Ca)MnO , Yb Ti O , SmB etc.Besides these important challenges, what makes SOTs8truly attractive is their potential for efficient device op-eration. In a nutshell, SOTs can do everything STT can,with the crucial advantage of decoupling the injectionand detection paths. This unique feature allows for theexcitation and switching of large magnetic surface areas( > µ m ), but also the electrical control of magnetic insu-lators and antiferromagnets, which traditional STT can-not achieve. Its implementation does not only enhancethe performance of devices (speed, power consumption)such as SOT-MRAMs, nano-oscillators or magnetic race-track data storage devices, but it also opens thrilling per-spectives beyond conventional spintronics components(Sato et al. , 2018). For instance, SOT-driven memristorshave been developed to be used as synapses for artifi-cial neural networks (Borders et al. , 2018; Lequeux et al. ,2016), while SHE-SOT can be exploited to build stochas-tic parity-bits for invertible logic (Camsari et al. , 2017).Finally, SOTs could be used to manipulate and exploremore exotic magnetic excitations such as the ones emerg-ing in spin liquids (Balents, 2010), i.e., spinons, magneticmonopoles, anyons, or even Majorana fermions. LIST OF ABBREVIATIONS : One-, two- and three-dimensional
AHE : Anomalous Hall effect
AMR : Anisotropic magnetoresistance
ANE : Anomalous Nernst effect
CIDM : Current-induced domain wall motion
DMI : Dzyaloshinskii-Moriya interaction
FMR : Ferromagnetic resonance iSGE : Inverse spin galvanic effect
LED : Light emitting diode
LLG : Landau-Lifshitz-Gilbert (equation)
MOKE : Magneto-optical Kerr effect
MRAM
Magnetic random access memory
NM/FM : Nonmagnetic metal/ferromagnet
NM/AF : Nonmagnetic metal/antiferromagnet RF : Radio frequency RKKY : Ruderman-Kittel-Kasuya-Yosida (interaction)
SGE : Spin galvanic effect
SHE : Spin Hall effect
SMR : Spin Hall magnetoresistance
SOT : Spin-orbit torque
ST-FMR : Spin torque ferromagnetic resonance
STT : Spin transfer torque
TMR : Tunnelling magnetoresistance
ACKNOWLEDGEMENTS
A.M. was supported by the King Abdullah Univer-sity of Science and Technology (KAUST). T. J. acknowl-edges support from the EU FET Open RIA Grant No. 766566, the Ministry of Education of the Czech Repub-lic Grant No. LM2015087 and LNSM-LNSpin, and theGrant Agency of the Czech Republic Grant No. 14-37427G. J. S. acknowledges the Alexander von HumboldtFoundation, EU FET Open Grant No. 766566, EU ERCSynergy Grant No. 610115, and the Transregional Col-laborative Research Center (SFB/TRR) 173 SPIN+X.K.G. and P.G. acknowledge stimulating discussions withC.O. Avci and financial support by the Swiss NationalScience Foundation (Grants No. 200021-153404 and200020 172775) and the European Commission underthe Seventh Framework Program (spOt project, GrantNo. 318144). A.T. acknowledges support by the AgenceNationale de la Recherche, project ANR-17-CE24-0025(TopSky). J. ˇZ acknowledges the Grant Agency of theCzech Republic grant no. 19-18623Y and support fromthe Institute of Physics of the Czech Academy of Sciencesand the Max Planck Society through the Max PlanckPartner Group programme.
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