Phenomenology of Current-Induced Spin-Orbit Torques

Kjetil M.D. Hals and Arne BrataasDepartment of Physics, Norwegian University of Science and Technology, NO-7491, Trondheim, Norway

Currents induce magnetization torques via spin-transfer when the spin angular momentum isconserved or via relativistic spin-orbit coupling. Beyond simple models, the relationship betweenmaterial properties and spin-orbit torques is not known. Here, we present a novel phenomenology ofcurrent-induced torques that is valid for any strength of intrinsic spin-orbit coupling. In Pt|Co|AlOx,we demonstrate that the domain walls move in response to a novel relativistic dissipative torque thatis dependent on the domain wall structure and that can be controlled via the Dzyaloshinskii-Moriyainteraction. Unlike the non-relativistic spin-transfer torque, the new torque can, together with thespin-Hall effect in the Pt-layer, move domain walls by means of electric currents parallel to the walls.

I. INTRODUCTION

Electric currents can be used to manipulate the mag-netization in ferromagnets. This phenomenon arises fromthe intricate coupling between the quasi-particle flow andthe collective magnetic degrees of freedom, and it hasenabled the development of spin-torque oscillators, spin-logic devices, and spin-transfer torque (STT)-RAM;1 thelatter entered the marketplace at the end of 2012. In itssimplest manifestation, current-driven excitations of theferromagnets are caused by the STT mechanism, in whichspin angular momentum is transferred from the spin-polarized conduction electrons to the magnetization. Re-cent theory2–7 and experiments8–13 have shown that cur-rents can also cause magnetization torques via relativis-tic, intrinsic spin-orbit coupling (SOC), often referred toas spin-orbit torques (SOTs). Relativistic SOTs providealternative and efficient routes for the manipulation ofthe magnetization that, in contrast to non-relativisticSTTs, require neither spin-polarizers nor textured fer-romagnets.

SOTs exist in systems with structural asymmetry,which induces a strong intrinsic SOC. Theory predictsthat SOC in combination with an applied electric fieldproduces an out-of-equilibrium spin density that in turnyields a torque on the magnetization.2–8 The effect ofan SOT was first detected in strained (Ga,Mn)As.8 Theobserved switching of samples with uniform magnetiza-tion markedly differs from the typical behavior in metal-lic ferromagnets, wherein a heterogeneous magnetic tex-ture is required to induce an STT.1,14 A similar switch-ing of a single-domain ferromagnet has also been ob-served in an ultrathin Co layer sandwiched between Ptand AlOx.9,10,15 Both SOTs2,9,10 and the spin-Hall ef-fect (SHE) in combination with conventional STTs15–17

have been proposed to be responsible for the switchingCo magnetization. In the latter case, the SHE in thePt layer produces a spin-current that diffuses into theCo layer, causing an STT on the magnetization.15 In-terestingly, in this system, experiments have observedfast current-driven motion of ferromagnetic domain walls(DWs).11,17,18 An SOC-induced antisymmetric exchangeinteraction, known as the Dzyaloshinskii-Moriya interac-tion (DMI),19,20 stabilizes chiral DWs21,22 with a high

DW mobility whose spin texture is very robust with re-spect to high current densities.11,17,18,22–25 Highly effi-cient, current-driven DW motion has also been observedin trilayers of Co and Ni interfaced with Pt.26,27 In bothPt|Co|AlOx and Co|Ni|Co systems, the DWs move athigh speeds and occasionally in a direction opposite ofthat expected for the non-relativistic bulk STT.

The aforementioned experiments demonstrate that ul-trathin metallic ferromagnets in asymmetric heterostruc-tures are important in the design of future spintronic de-vices wherein the magnetization is efficiently controlledby electric currents. The interplay between SOC ef-fects, such as the DMI, SOTs, and SHE, is believed tobe responsible for the remarkable behavior of these sys-tems. However, a detailed understanding of their prop-erties requires improved theoretical models that exceedthe present phenomenological framework used to modelcurrent-induced magnetization dynamics.

In this Letter, we formulate a general phenomenol-ogy of current-induced torques in systems with arbitrar-ily strong spin-orbit interaction. We apply the formal-ism to compute the current-driven DW drift velocity inPt|Co|AlOx trilayers and to identify a novel SOT. Theeffect of the new torque can be manipulated via the DMIand can, together with the SHE, induce DW motion evenwhen the electric currents are applied parallel to theDWs. This opens a completely new path for controllingthe dissipative current-induced torque by engineering theinterfaces in ferromagnetic heterostructures.

The magnetization dynamics of metallic ferromag-nets is well-described by the Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation:14

m = −γm×Heff + m×αm + τ . (1)

Here, γ is (minus) the gyromagnetic ratio, m(r, t) =M/Ms (Ms = |M|) is the unit direction vector of themagnetization M(r, t), and Heff = −δF [M]/δM is theeffective field determined by the magnetic free-energyfunctional F [M]. The friction processes in the magneticsystem are modeled by the Gilbert damping that is pa-rameterized by the symmetrical, second-rank tensor α.The last term on the right-hand side of Eq. (1) is 14

τ (r, t) = − (1− βm×) (vs ·∇) m, (2)

arX

iv:1

307.

0395

v3 [

cond

-mat

.mes

-hal

l] 2

6 A

ug 2

013

2

and describes the current-induced torques. The torquesof Eq. (2) contain reactive and dissipative contributions.The term proportional to the parameter β representsthe dissipative torque because it breaks the time-reversalsymmetry of Eq. (1). The vector vs is directed along theout-of-equilibrium current density J and is proportionalto the current density, the equilibrium spin density, andthe spin polarization of the induced current. Eq. (2) isfully rotationally symmetric in both the spin space andthe coordinate space. This can be seen by applying aglobal rotation operator R to the magnetization and thespatial vectors: m = Rm, vs = Rvs, and ∇ = R∇.Separate rotations of the spin space and the coordinatespace leave the magnitude of the torque invariant:

| (1− βm×) (vs ·∇) m| = | (1− βm×)(vs · ∇

)m|.

Thus, the STT given by Eq. (2) is completely decoupledfrom the symmetry of the underlying crystal lattice.

However, in systems with strong intrinsic SOC, Eq. (2)is not sufficient to describe the magnetization dynamics.The only possible effect of the intrinsic SOC included inEq. (2) is the renormalization of the β parameter. Theintrinsic SOC mediates a coupling between the spin spaceand the crystal lattice. As a consequence, the system isonly invariant under proper and improper rotations thatleave the crystal structure unchanged and that act si-multaneously on the spin space and on the coordinatespace. The intrinsic SOC therefore introduces severalnew torques that are dependent on the orientation of themagnetization with respect to the crystal lattice. Ex-amples of such SOTs have been derived theoretically inRefs. 2–4,6,12. However, to date, a general theoreticalframework has not been developed that systematicallyand correctly extends the LLG phenomenology in Eq. (1)to include the intrinsic SOC effects.

II. THEORY

To derive our theory, we consider the most general formof the current-induced torque in the local approximation:

τ (r, t) = m×Hc[m,∇m,J ], (3)

where Hc is an effective field induced by the out-of-equilibrium current density that is dependent on the lo-cal magnetization direction and the local gradients of themagnetization. Here, ∇m is a shorthand notation for allthe different combinations of ∂jmi (∂jmi ≡ ∂mi/∂rj).In the absence of an out-of-equilibrium current density,Hc vanishes. Because our interest is the linear responseregime, we expand Eq. (3) to the first order in J

τ (r, t) = m× ηJ . (4)

Here, the second-rank tensor (η[m,∇m])ij =

(∂Hc,i/∂Jj)J=0 operates on the vector J . η(r, t)completely determines the symmetry of the local torque

τ (r, t) and includes all of the current-induced torque ef-fects. Our main aim is to derive a general and simplifiedexpression for η(r, t).

x

z

y

x

y

J

m

∂m/∂x

τ║

τ┴

x

z

yJ

b

a

c

J

~

x

y

m

∂m/∂y

τ║τ┴

J ~

~

~

~~d

FIG. 1: (color online). The magnetic textures and currentdensities in (a) and (b) are related by a rotation of 90 degreesabout the z axis. Due to the four-fold symmetry of the under-lying crystal lattice, the two cases are equivalent. Neumann’sprinciple implies that the local torques on the two texturesare related by a four-fold rotation. This is illustrated in (c)and (d), which show the torque, the gradient, and the direc-tion of the magnetization at two equivalent points indicatedby the black arrows.

We will deduce the second-rank tensor η from Neu-mann’s principle, which states that ”any type of symme-try which is exhibited by the point group of the crystalis possessed by every physical property of the crystal”.28

An illustration of how this principle imposes symmetryrelations on the torque in Eq. (4) is presented Fig. 1.As an example, let us assume that the material has afour-fold symmetry axis along z. Figs. 1a - b show two

3

systems containing a DW along the y and x axis, re-spectively, with a current density applied perpendicularto the DW. The two systems in Figs. 1a - b are re-lated by a rotation R4z of 90 degrees about the z axis,which leads to the following transformations: r = R4zr,∇ = R4z∇, J = R4zJ , and m = |R4z|R4zm. Be-cause the magnetization is a pseudovector, its trans-formation rule includes the determinant |R4z|. Due tothe four-fold symmetry of the system, the two casesin Figs. 1a - b are equivalent, and Neumann’s princi-ple implies that the torques induced on the two mag-netic textures are related by R4z, i.e., τ = |R4z|R4zτ(Figs. 1c - d). Thus, by applying Eq. (4), we find that

m×η[m, ∇m]J = |R4z|R4z (m× η[m,∇m]J ), whichyields the following symmetry relations for η:

η[m, ∇m] = |R|Rη[m,∇m]RT . (5)

This is the first central result of our work. We have re-moved the subscript of R denoting the symmetry oper-ation because Eq. (5) holds for any symmetry operationthat is an element of the crystal’s point group. AlthoughEq. (5) determines many symmetry aspects of the torque,the dependence of torque on the magnetization directionand the gradients of the magnetization means that a largenumber of functions can satisfy Eq. (5). This situationclosely resembles the case of the large number of termsthat can describe the magnetocrystalline anisotropy en-ergy of a system. Although the anisotropy energy shouldobey the symmetries of the material, it is common to ap-proximate the dependence by the first harmonics of thisdependence.

Thus, to simplify the tensor form of η, we assume thatits components can be expressed as a series expansion inincreasing powers of mi and ∂jmi

ηij = Λ(r)ij +Λ

(d)ijkmk+βijkl∂kml+Pijklnmk∂lmn+... (6)

In Eq. (6), and in the following discussion, summationover repeated indices is implied. The power expan-sion in Eq. (6) represents a systematic scheme for de-riving current-induced torques of increasing degrees ofanisotropy. The number of terms in the expansion re-quired to obtain a sufficiently accurate description of themagnetization dynamics can be determined from experi-mental data. As a demonstration, we limit our discussionto the terms explicitly written in Eq. (6) because theylead to a simple extension of Eq. (1), and similar to thecase of magnetic anisotropy, we expect the first harmon-ics to provide a sufficient description of many materials.

In Eq. (6), Λ(r) and Λ(d) describe the reactive and dis-sipative SOTs, respectively, that are present even in sys-tems with uniform magnetization. The tensors P and βare generalizations of the reactive and dissipative torquesgiven in Eq. (2). Our theory therefore demonstates thatin systems with strong intrinsic SOC, the reactive torqueis generalized to become a tensor of rank five, while thedissipative torque parameter β becomes a tensor of rankfour. From our derived symmetry relations in Eq. (5)

and the expansion of η in Eq. (6), we find that the ten-sors Λ(r), Λ(d), β, and P must satisfy the symmetryrelations

Λ(r)ij = |R|Rii′Rjj′ Λ

(r)

i′ j′, (7)

Λ(d)ijk = Rii′Rjj′Rkk′ Λ

(d)

i′ j′k′, (8)

βijkl = Rii′Rjj′Rkk′Rll′βi′ j′k′ l′ , (9)

Pijkln = |R|Rii′Rjj′Rkk′Rll′Rnn′Pi′ j′k′ l′n′ . (10)

Eqs. (7) - (10) represent our second central result, andthey reduce the number of independent tensor coefficientsand greatly simplify the forms of the tensors. For ex-ample, in systems that are invariant under the inver-sion operator Rij = −δij , Eqs. (7) - (8) imply that

Λ(r) = Λ(d) = 0. This means that SOTs are absentin inversion-symmetric systems with uniform magnetiza-tion.2–4,8

The tensors determined from Eqs. (5) - (6), togetherwith Eq. (4), represent a general phenomenology ofcurrent-induced magnetization dynamics that correctlyaccounts for intrinsic SOC effects. The torques originat-ing from the low-order tensors given by Eqs. (7) - (10)lead to a generalization of Eq. (2). The torques in Eq. (2)are obtained from our formalism by assuming a full ro-tational symmetry under separate rotations of the spinspace and the coordinate space. Under these symmetryrequirements, Eqs. (7) - (10) imply that Λ(r) = Λ(d) = 0,Pijkln = Pδjlεikn, and βijkl = Pβδjkδil, which lead tothe non-relativistic STT in Eq. (2) when PJ = vs. Here,δjn and εilk are the Kronecker delta and the Levi-Civitatensor, respectively.

III. APPLICATION: FERROMAGNETICHETEROSTRUCTURE

Next, we consider the ferromagnetic heterostructurePt|Co|AlOx and apply the phenomenology to the studyof current-induced DW motion. The growth of the ultra-thin Co layer, i.e., the stacking direction, occurs alongthe crystallographic direction [111].29 We assume a fullyepitaxial system. The single Co layer has a trigonal crys-tal structure that is described by the centrosymmetricpoint group D3d, in which the three-fold axis of rotationis along the [111] direction.29 The Pt and AlOx layersbreak the inversion symmetry of the system and reducethe symmetry group to C3v. The SOC in the Co layer is,therefore, assumed to be invariant under the action of C3v

when the symmetry operations act simultaneously on thespin and coordinate spaces. Thus, the allowed torques onthe magnetization in the Co layer are determined fromEq. (5) with R ∈ C3v. In the following description, wedenote the [111] direction as the z axis, and the x axis isdefined such that the xz plane corresponds to one of thethree equivalent reflection planes of C3v.

The magnetic Co layer is modeled by the free energy

4

density22

F = (J/2)∂im · ∂im +D (mz∂imi −mi∂imz) +

(Kν/2) (m · ν)2 − (Kz/2)m2

z. (11)

Here, i ∈ {x, y}, J is the spin stiffness, Kz > 0 andKν > 0 are anisotropy constants, and the term pro-portional to D is the DMI. ν = [cos(φ0), sin(φ0), 0]describes the direction of the DW. To model the DWdynamics, we apply a collective coordinate descriptionin which the DW is determined by the ansatz m =[sech(µ) cos(φm), sech(µ) sin(φm), tanh(µ)], where µ =(ν − rw)/λw. Here, λw is the DW length, which is as-sumed to be static, and rw = rw(t) and φm = φm(t) arethe collective coordinates that describe the DW positionand the tilting angle of the texture, respectively. In theabsence of any external fields, the equilibrium value ofφm is given by

cos(φeqvm − φ0) = πD/2Kνλw. (12)

If πD/2Kνλw ≥ 1 or πD/2Kνλw ≤ −1, the equilibriumangle is φeqv

m = φ0 or φeqvm = φ0 + π, respectively.

We believe that the DW dynamics are dominated bythe lowest-order terms given by Eqs. (7) - (10) in com-bination with the SHE torque; higher-order terms canbe included to refine the results. In contrast, a recentexperiment on a homogenous Pt|Co|AlOx system indi-cates that higher-order terms in the series expansionof η in Eq. (6) may influence the dynamics of a sin-gle magnetic domain.12 The SHE torque is modeled by

the Slonczewski torque τ she = τ(0)shem × (s ×m), where

s = J × z is proportional to the polarization of the spincurrent induced within the Pt layer by the electric cur-rent density J = J0[cos(φe), sin(φe), 0].15 For the sym-metry group C3v, the tensors Λ(r), Λ(d), and β are de-scribed by one, five, and 14 independent tensor coeffi-cients, respectively, and their explicit forms can be foundin Ref. 28. In Eq. (10), we keep only the fully isotropicpart Pijkln ∼ δjlεikn. Its value is largely determinedby the spin polarization of the induced current and theequilibrium spin density. The remaining part of Pijkln isgoverned by the SOC and is therefore assumed to providea negligible correction. A similar simplification cannot beapplied to the βijkl tensor because even the isotropic partβijkl ∼ δjkδil is proportional to the spin-flip rate, whichis primarily controlled by the SOC. The Gilbert dampingtensor is diagonal and determined by the two parametersαxx = αyy = α⊥ and αzz.

The equations of motion for rw and φm are30

(Γij −Gij) aj = γFi +

5∑a=1

L(a)i , (13)

where ai ∈ {rw, φm}. The matrices Γij and Gij aregiven by Gij =

∫m · [(∂m/∂ai) × (∂m/∂aj)]dV and

Γij =∫

(∂m/∂ai) · α(∂m/∂aj)dV, and Fi is the forcedue to the effective field, Fi =

∫Heff · (∂m/∂ai)dV. The

current-induced dynamics is described by the five terms

L(a)i =

∫(∂m/∂ai) · [m× τ (a)]dV, where the superscript

(a) denotes the four torques determined from Eqs. (7) -(10) and the SHE torque. Solving Eq. (13) in the linearresponse regime, we compute our final central result, thesteady-state DW velocity

rw =1

2

J0

α⊥ + 2αzz[τsoc cos(φe − φeqv

m ) (14)

+βiso cos(φe − φ0) + βm cos(φe + φ0 − 2φeqvm )],

where τsoc = (3πλw/2)(Λ(d)zyy − Λ

(d)xxz + 2τ

(0)she), βiso =

βxxyy+βxyxy+2βxyyx+4βzxxz, and βm = βxxyy+βxyxy.The first term in Eq. (14) arises from the SHE torqueand the dissipative homogeneous SOT. The term propor-tional to βiso represents a conventional dissipative STTterm in which the current-driven DW motion dependsonly on the relative angle between the applied currentand the DW direction.

The expression for rw of Eq. (14) also contains a newdissipative current-induced torque that has not been pre-viously reported. The term proportional to βm describesa dissipative torque that is dependent on the DW struc-ture through the angle φeqv

m . This is a purely relativistictorque because its magnitude depends on the tensor co-efficients βxxyy and βxyxy, which vanish in the absence ofintrinsic SOC. The angle φeqv

m is dependent on the DMIand can be manipulated by engineering the Pt|Co andCo|AlOx interfaces, as demonstrated experimentally inRef. 27. The βm term therefore opens an exciting newpath for controlling the effective dissipative torque on theDW via the DMI. In principle, even the sign of the to-tal dissipative torque arising from the βijkl tensor canbe controlled by the DMI if βm is of the same order ofmagnitude as βiso.

Another interesting phenomenon observed fromEq. (14) is the current-driven DW motion induced byan electric current applied perpendicular to the texturedirection, i.e., J ⊥ν. Depending on the value of φeqv

m ,both the τsoc term and the βm term can induce a DWmotion in this case. However, the effect requires that|πD/2Kνλw| < 1. When J ⊥ν, the torque efficiencyof the τsoc term is maximized for φeqv

m = φe (mod π),i.e., a Bloch wall structure, whereas the βm term at-tains its maximum efficiency for the angles φeqv

m = φ0 +π/4 (mod π/2). For J ||ν, Eq. (14) implies that a Blochwall motion is only induced by the two β terms, whereasboth the τsoc term and the β terms contribute to the mo-tion of Neel walls, i.e., when φeqv

m = φ0 (mod π). This isin agreement with previously reported results.17,27

In polycrystalline or disordered systems, the discretethree-fold rotation symmetry is averaged out, and a fullrotational symmetry about the z axis can be assumed.However, this symmetry still allows for all tensor coeffi-cients in Eq. (14) to be present, and the equation remainsunchanged.

5

IV. SUMMARY

In summary, we developed a novel phenomenology ofcurrent-induced magnetization dynamics that is valid forany strength of the intrinsic SOC. The formalism is ap-

plied to the study of current-induced DW motion in theferromagnetic heterostructure Pt|Co|AlOx. Our resultsshow that the current-driven DW motion is dependenton a novel dissipative STT that can be controlled via theDMI.

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