UniversalAbsenceofWalkerBreakdownandLinearCurrent–VelocityRelation viaSpin–OrbitTorquesinCoupledandSingleDomainWallMotion Vetle Risinggård and Jacob Linder ∗ DepartmentofPhysics,NTNU,NorwegianUniversityofScienceandTechnology,N-7491Trondheim,Norway (Dated:December21,2016) Weconsidertheoreticallydomainwallmotiondrivenbyspin–orbitandspinHalltorques. Wefindthatit ispossibletoachieveuniversalabsenceofWalkerbreakdownforallspin–orbittorquesusingexperimentally relevantspin–orbitcouplingstrengths. Forspin–orbittorquesotherthanthepureRashbaspin–orbittorque, thisgivesalinearcurrent–velocityrelationinsteadofasaturationofthevelocityathighcurrentdensities.The effectsisveryrobustandisfoundinbothsoftandhardmagneticmaterials,aswellasinthepresenceofthe 7 Dzyaloshinskii–Moriyainteractionandincoupleddomainwallsinsyntheticantiferromagnets,whereitleadsto 1 veryhighdomainwallvelocities. 0 2 n Domain wall motion in ferromagnetic strips is a central ultrathinferromagnetwithaheavymetalunderlayerasshown a themeinmagnetizationdynamics. Thetopichasfundamental inFigure1. Wedescribethedynamicsofthemagnetization J interest,andhasrecentlybeeninstrumentaltothediscoveryof m(r,t)usingtheLandau–Lifshitz–Gilbert(LLG)equation[19, 3 severalnewcurrent-inducedeffectsonthemagnetizationdy- 20], ] namics[1–6].Particularlysincetheproposalofracetrackmem- α ll ories[7]—madepossiblebycurrent-inducedtorques—domain ∂tm =γm×H − mm×∂tm+τ, (1) a wallmotionhasalsohadatechnologicalinterest,motivating h where γ < 0 is the gyromagnetic ratio, m is the saturation thesearchforeverhigherdomainwallvelocities. Theattain- - magnetization,α < 0istheGilbertdamping, H = δF/δm s able velocity of a domain wall driven by conventional spin- − e istheeffectivefieldactingonthemagnetizationandτ isthe m transfertorques(STTs)islimitedbytheWalkerbreakdown[8], current-inducedtorques. ThefreeenergyF oftheferromagnet uponwhichthedomainwallhasanegativedifferentialvelocity . isasum, t withrespecttothecurrent. a m Current-induced torques derived from spin–orbit effects F = dr f + f + f + f , (2) (SOTs)suchasaninterfacialRashbaspin–orbitcoupling[9– Z ex DM a - Z d 11]orthespinHalleffect[4–6,12]haveenabledlargedomain (cid:16) (cid:17) n wall velocities. We consider the dependence of the domain o wallvelocityonthecurrentandfindthatregardlessoftherel- c [ ativeimportanceofthereactiveanddissipativecomponents of the torque it is possible to achieve universal absence of 1 Walkerbreakdownforallcurrentdensitiesforexperimentally v relevantspin–orbitcouplingstrengths. Forspin–orbittorques 6 8 otherthanthepureRashbaSOTs,suchasthespinHalltorques, 7 thevelocitywillnotsaturateasafunctionofcurrent,butin- 0 creaselinearlyaslongasaconventionalspin-transfertorque 0 ispresent. Thisbehaviorisrobustagainstthepresenceofan . 1 interfacialDzyaloshinskii–Moriyainteraction[13,14]andis 0 foundbothinperpendicularanisotropyferromagnetsandin 7 shapeanisotropy-dominatedstrips. 1 Veryrecently,domainwalldynamicsinbilayerracetracks : v antiferromagneticallycoupledbyinterlayerexchangehasbeen i X consideredboththeoreticallyandexperimentally[15–18]. We r findthatthissystemalsoexhibitsuniversalabsenceofWalker a breakdownandasymptoticlineardependenceofthevelocityon Figure1.Domainwalldynamicsinasingleferromagnet.(a)Ultrathin ferromagnetwithaheavymetalunderlayer.Weconsidertransverse thecurrent. Thestrengthoftheinterlayerexchangecoupling (IEC) determines the rate at which the velocity approaches domainwallmotionalongthexaxis.r,σlandσsdenotethethree nontrivialoperationsofthesymmetrygroupC2v.(b)Current–velocity theasymptoticsolution. Thisenablesveryhighdomainwall relationforthreedifferentSOTs. TheRashbaSOTslevelofftoa velocitiesforrelativelysmallcurrentdensities. Thecombina- constant velocity at large currents, whereas the spin Hall torques tionofSOTswiththeinterlayerexchangetorquealsoenables asymptoticallyapproachalinearcurrent–velocityrelation. Dashed otherinterestingeffectssuchasanonmonotonicdependence linesshowtheasymptoticexpansionanddottedcurvesshowtheseries ofthedomainwallvelocityontheratioofthethicknessesof about j =0.Weuseγ= 0.19GHz/T,λ =4nm,Ky =200kJ/m3, − theferromagnets. m = 1MA/m, D = 1.4mJ/m2, α = 0.25, β = 0.5, P = 0.5, − − UniversalAbsenceofWalkerBreakdown. Weconsideran Hx =0,αR=6.3meVnm,θSH=0.1, βSH=0.02andt =1.2nm. 2 oftheZeemanenergyduetoanyappliedmagneticfields,the As can be seen from the LLG equation (1), the magni- isotropicexchange,themagneticanisotropyandtheinterfacial tude m of the magnetization is constant, and the magnetiza- Dzyaloshinskii–Moriyainteraction. tionisconvenientlyparameterizedinsphericalcoordinatesas TheZeemanenergyandtheisotropicexchangecanbewrit- m/m = cosφsinθex +sinφsinθey +cosθez. Usingtheas- tenrespectivelyas[20] f = H m,whereH istheapplied sumptionthatthereisnomagnetictexturealongthe yandthe Z 0 0 − · magnetic field, and [20] fex = (A/m2)[( mx)2 + ( my)2 + zaxes,giving =∂xex,wecanfindthedomainwallprofile ∇ ∇ ∇ ( mz)2], where Ais the exchange stiffness. Inversion sym- byminimizingthefreeenergyorbysolvingtheLLGequation ∇ metrybreakingattheinterfacebetweentheheavymetaland (1)inthestaticlimit,∂tm =0.Ifweassumethattheazimuthal the ferromagnet gives rise to an anisotropic contribution to angleφisindependentofposition,thisgivestheNéelwallso- theexchangeknownastheDzyaloshinskii–Moriyainteraction, lutionθ =2arctanexp[ (x X)/λ]forthepolarangle,where ± − whichfavorsacantingofthespins[13,14,21]. Theresulting X isthedomainwallpositionandλ = A/Kz isthedomain contributiontothefreeenergyis fDM = (D/m2)[mz( m) wallwidth;andφ=nπ,wheren=0,2,4,...ifD <0andwe ∇· − p (m )mz],whereDisthemagnitudeoftheDzyaloshinskii– choose the + sign for θ, and n=1,3,5,... if D >0 and we ·∇ Moriyavector. choosethe signforθ. − Ultrathin magnetic films are prone to exhibit perpendicu- SubstitutionofthisdomainwallprofileintothefullLLG lar magnetization due to interface contributions to the mag- equation (1) using H0 = Hxex and a positive topological netic anisotropy [22]. Consequently, we write the magnetic chargegivesthecollectivecoordinateequations, anisotropyenergyas fa =−Kzmz2+Kym2y,correspondingtoan αX˙ βu easyaxisinthez-directionandahardaxisinthe y-direction. φ˙=+πγ H βH cosφ+ , (6) Thecurrent-inducedtorquesτ areconventionallydivided λ − 2 SH− R λ intospin-transfertorques[23–25], (1+α2)φ˙= α(cid:16)γKy sin2φ+(cid:17)παγ(D−Hxmλ) sinφ (7) − m 2mλ βu u(α+ β) (cid:102) (cid:103) τSTT =u∂xm− m m×∂xm, (3) − λ − π2γ HSH(1−αβSH)−HR(α+ β) cosφ, andspin–orbittorques[4–6,9–12], for the domain wall position X and tilt φ. The equations can be simplified by introducing aj = πγ(H βH ), τR =γm×HRey −γm× m× βHRey/m , (4) bj = βu/λ, c = 2αγKy/m, d = παγ(D 2 HxmSHλ)−/(2mλR), τSH =γm× m×HSHey/(cid:16)m +γm× βSH(cid:17)HSHey. (5) Weja=lke−rπ2bγre[HakSdHo(w1−−nαisβaSbHs)e−nHtwR(hαe+nβth)e]atinmdef−dje=riv−auti(vαe+φ˙βv)a/nλ-. (cid:16) (cid:17) In fact, assuming that the stack can be described using the ishes,resultinginthecondition C2v symmetrygroupasindicatedbythesymmetryoperations 0=csinφcosφ+dsinφ+ j(ecosφ+ f). (8) in Figure 1(a), it can be shown that these torques exhaust the number of possible torque components if the current is If e > f this equation will always have a solution for φ re- onlyappliedinthe xdirectionandthewidth(ydirection)and gardless of the value of j. For realistic material values this the thickness (z direction) of the ferromagnetic layer are so corresponds to a Rashba parameter α > 4µ2/(πeγλ) = R B smallthatthereisnomagnetictexturealongtheseaxes(see 5.5meVnm(pureRashbaSOTs)oraspinHallangle θ > SH SupplementalMaterialfordetails[26]). 4µ Pt/(π~γλ) =0.091(purespinHalltorques). Theabsence B Thereisonereactiveandonedissipativespin-transfertorque, ofWalkerbreakdownforsufficientlystrongRashbaspin–orbit whoserelativemagnitudeisparameterizedby β[23–25]. The couplinghasbeenpointedoutpreviously[27],andcanalsobe strengthoftheSTTsisdeterminedbytheelectriccurrent jand notedinreferences[10,28–30]. itsspinpolarizationPthroughu= µBPj/[em(1+ β2)]where Letuswrite ξ = cosφandη = sinφ,sothat ξ2 +η2 = 1. µB is the Bohr magneton and e is the electron charge. The Solvingequation(8)forηtogetη = j(eξ+ f)/(cξ+d),this Rashba SOTs τ are due to the Rashba spin–orbit coupling − R relationgivesaquarticequation inducedintheferromagnetbythebrokeninversionsymmetry at the heavy metal interface [9–11]. Just as for the STTs, c2ξ4+2cdξ3+[(ej)2+d2 c2]ξ2 − (9) thereisonereactiveandonedissipativetorque(β-term). The +2(efj2 cd)ξ+(fj)2 d2 =0. strength of the Rashba SOTs is determined by the Rashba − − spin–orbit coupling α and the spin current through H = Theexactsolutionsofthequartic(9)arehopelesslycompli- R R α Pj/[2µ m(1+ β2)]. ThespinHalltorquesτ aredueto cated.However,theyallhavethesameseriesexpansionaround R B SH thetransversalspincurrentinjectedintotheferromagnetfrom j =0and j = . Weconsiderfirsttheasymptoticexpansion, ∞ theheavymetalduetothespinHalleffect[4–6,12]. Since whichgives the polarization direction of this current changes sign upon f S time-reversaltheprincipalspinHalltorquetermisdissipative, ξ = + 1 + j−2 , (10) −e j O whilethe β correctionisreactive. Themagnitudeofthespin SH (cid:16) (cid:17) Halltorquesisdeterminedbytheinjectedspincurrentandthe where S represents the solutions of the quadratic equation 1 thicknessoftheferromagnett throughH =~θ j/(2emt). e6ζ2 = d2e4+c2f4+(c2 d2)f2e2+2cdef(f2 e2). The SH SH − − 3 domainwallvelocity,asgivenbyequation(6)thentakesthe form αX˙ af λ = bj− e j+aS1+aO j−1 . (11) (cid:16) (cid:17) Back-substitutionoftheabbreviationsa,b, f andeshowsthat forpureRashbaSOTsthecoefficientofthelineartermreduces tozerobecausetheratioofthereactivetothedissipativetorque isthesamefortheSTTsandtheRashbaSOTs. Thus,forlarge j the domain wall velocity approaches a constant. Instead, for pure spin Hall torques we get the linear term uα(1 + − ββ )/[λ(1 αβ )]. Thismeansthatforlarge jthevelocity SH SH − isactuallyindependentofthesignofthespinHallangleand increaseslinearlywith j. Notetheimportanceofincludingthe STTs—whicharealwayspresent—intheseconsiderations: in theabsenceofSTTs(u 0)bothband f gotozeroandthe → velocitylevelsofftoaconstantforlarge j foranycombination ofSOTs. For completeness, we also consider the series expansion about j = 0,whichgivesξ = 1+[(e f)2/2(c d)2]j2+ − − − (j4) and X˙ = bj aj +[a(e f)2/2(c d)2]j3+a (j5). O − − − O Thekeyobservationhereisthatinthisregimethevelocitydoes dependonthesignofthespinHallangle(a θ forpure SH ∝ spinHalltorques)andincreaseswiththecubeof j. Figure1(b) showsanumericalsolutionofthecoupledequations(6)and (7)asafunctionof j forpureRashbaSOTsandforpurespin Halltorquesbothinthecasesofθ >0andθ <0together SH SH withtheanalyticalsolutionscloseto j =0andforlarge j. We seethatouranalyticalresultssuccessfullyapproximatethefull Figure2.Domainwalldynamicsininterlayerexchangecoupledfer- solutionintheexpectedrangesofvalidity(seeSupplemental romagnets.(a)Twoultrathinferromagnetsseparatedbyaninsulating Materialforadescriptionofthenumerics[26]). spacerwithheavymetalover-andunderlayers.Theferromagnetsare Thein-planehardaxisincludedinthemagneticanisotropy identicalexceptfortheirthicknesses,butthedifferentheavymetals is appropriatefor narrow ferromagneticstrips, which gener- inducedifferentDMIandSOTs.(b)and(c):Domainwalldynamics allyhostNéelwalls. WiderstripsgiveBlochwalls[22],and insingleferromagnetscomparedwithaSAFstructure.Blackcurves correspondtowallsdrivenbySTTsonly,whileorangecurvesinclude bymakingthenecessarymodificationstotheabovecalcula- spinHalltorques. (b)TheIECdelaysWalkerbreakdownforSTT- tions,wefindthatinthiscasethedomainwallvelocityretains driving.WithspinHalltorquesthereisuniversalabsenceofWalker thequalitativefeatureselucidatedabove. Thisisalsotruefor breakdownandtheIECrescalesthecurrentaxis.Boththeresultsfor shapeanisotropy-dominatedstrips,whichhosthead-to-head STTsandspinHalltorquesaretheresultofasuppressionofthetilt walls. This shows that universal absence of Walker break- angleasshownin(c). (d)TheIECgivesthevelocityanonmono- downisarobusteffectthatdoesnotdependonthedetailsof tonicthickness-dependenceresultinginapeakclosetot1/t2 = 1. theferromagneticmaterial,unlikeotherSOTeffectsstudied Results in (d) are shown for j = 3·1013A/m2, corresponding to the dashed vertical line in (b). We use the same parameters as in previously[31]. CoupledDomainWallsinaSAFStructure.Weconsidernext FJig=u(r5em1Jw/imth2t)2/t=1t2t1. =1.2nm(exceptin(d)wheret2isvaried)and anasymmetricstackoftwoultrathinferromagnetsseparated byaninsulatingspacerasshowninFigure2(a). Wedescribe thedynamicsofeachoftheferromagnetsusingseparateLLG twoferromagnets,aswellasontheirstrengthrelativetothe equationssimilarlytowhatwasdoneintheprevioussection, interlayer exchange and on the relative thickness of the fer- butaddtothefreeenergyacouplingterm, romagnets. To limit the scope of the treatment we consider (cid:102) (cid:103) onlythecasewhereD andD havethesamesign. Thus,for dr dr 1 2 FIEC = m(11) m(22) J(r1−r2) m(1)(r1)·m(2)(r2) , (12) oppositetopologicalchargesforthetwowalls,boththeDMI Z Z andtheIECfavorthesamechiralitiesforthewalls. representingtheinterlayerexchange(IEC).Weassumethatthe Followingthesameprocedureasintheprevioussectionwe IECislocalintheplane, J(r r ) = Jδ(x x )δ(y y ). maynowderivefourcoupledcollectivecoordinateequations 1 2 1 2 1 2 − − − Equation (12) then represent the lowest order coupling pro- (seeSupplementalMaterial[26]). Wesolvetheseequations posed by Bruno [32]. The static solution will now depend numericallyfort /t =1inthepresenceofSTTsonlyandin 1 2 onthesignsoftheDzyaloshinskii–Moriyainteractionsinthe thepresenceofspinHalltorquestogetthecurrent–velocity 4 relations shown in Figure 2(b). We see that the presence of theIECdelaysWalkerbreakdownwhenthewallisdrivenby ordinarySTTs,butthesubcriticaldifferentialvelocityremains unaffected. This can also be shown analytically by solving ∗ [email protected] [1] I.M.Miron,P.-J.Zermatten,G.Gaudin,S.Auffret,B.Rodmacq, forthetiltangleofthewallasafunctionofcurrent. Sucha andA.Schuhl,Phys.Rev.Lett.102,137202(2009). calculationshowsthatthetiltangleissuppressedbytheIEC [2] I.M.Miron, G.Gaudin, S.Auffret, B.Rodmacq, A.Schuhl, (butthebreakdownangleisstillπ/4).Back-substitutionofthis S.Pizzini, J.Vogel, andP.Gambardella,Nat.Mater.9,230 angleintothetorqueactingonthewallshowsthatthistorque (2010). isindependentof J,explainingthatthereisnochangeinthe [3] I.M.Miron,T.Moore,H.Szambolics,L.D.Buda-Prejbeanu, S. Auffret, B. Rodmacq, S. Pizzini, J. Vogel, M. Bonfim, differentialvelocity. A.Schuhl, andG.Gaudin,Nat.Mater.10,419(2011). AverysimilarmechanismemergesfromthespinHalltorque [4] S.Emori,U.Bauer,S.-M.Ahn,E.Martinez, andG.S.D.Beach, results,whichshowthatthemaineffectoftheIECistorescale Nat.Mater.12,611(2013). thecurrentaxis. Sucharescalingofthecurrentcanbeunder- [5] K.-S.Ryu,L.Thomas,S.-H.Yang, andS.Parkin,Nat.Nan- stoodintermsoftheexpansion(10)simplyasarescalingof otechnol.8,527(2013). theconstantS andthehigherorderconstantsS ,S ,... Again [6] P.P.J.Haazen,E.Murè,J.H.Franken,R.Lavrijsen,H.J.M. 1 2 3 thisisnothingbutasuppressionofthe tiltangle, as seenin Swagten, andB.Koopmans,Nat.Mater.12,299(2013). [7] S.S.P.Parkin,M.Hayashi, andL.Thomas,Science320,190 Figure2(c).ThisshowsthatuniversalabsenceofWalkerbreak- (2008). downisalsofoundinSAFstructures.Wenoteinparticularthat [8] N.L.SchryerandL.R.Walker,J.Appl.Phys.45,5406(1974). thecombinationofspinHalltorquesandIECproducesmuch [9] A.ManchonandS.Zhang,Phys.Rev.B78,212405(2008). higherdomainwallvelocitiesthaninsingleferromagnetsfor [10] K.-W.Kim,S.-M.Seo,J.Ryu,K.-J.Lee, andH.-W.Lee,Phys. comparativelysmallcurrentdensities. Rev.B85,180404(2012). Inthecaseofasingleferromagneticlayerthevelocityofa [11] A.Chernyshov,M.Overby,X.Liu,J.K.Furdyna,Y.Lyanda- domainwalldrivenbyspinHalltorqueshasamonotonic1/t Geller, andL.P.Rokhinson,Nat.Phys.5,656(2009). [12] K.Ando,S.Takahashi,K.Harii,K.Sasage,J.Ieda,S.Maekawa, thicknessdependence. AsshowninFigure2(d)thevelocityof andE.Saitoh,Phys.Rev.Lett.101,036601(2008). coupledwallsinaSAFstructurepeaksclosetot /t =1when 1 2 [13] I.E.Dzyaloshinskii,J.Exp.Theor.Phys.5,1259(1957). changingthethicknessoflayer2fromt = 2t tot = t /2. 2 1 2 1 [14] T.Moriya,Phys.Rev.Lett.4,228(1960); Phys.Rev.120,91 Thispeakistheresultoftwocompetingmechanisms. Whent2 (1960). islargethewallinlayer2movesslowerthanthewallinlayer1 [15] H.Saarikoski,H.Kohno,C.H.Marrows, andG.Tatara,Phys. andtheeffectivefieldfeltbylayer1duetotheIEC—whichis Rev.B90,094411(2014). proportionaltot —-islarge. Thusthevelocityofthedouble [16] S.Lepadatu,H.Saarikoski,R.Beacham,M.J.Benitez,T.A. 2 Moore,G.Burnell,S.Sugimoto,D.Yesudas,M.C.Wheeler, wallsystemisdeterminedbywall2. Thevelocityincreasesup J.Miguel,S.S.Dhesi,D.McGrouther,S.McVitie,G.Tatara, tot /t 1(thedeviationfrom1isduetotheDMI)following 1 2 ≈ andC.H.Marrows, arXiv:1604.07992. approximatelythe1/t increaseexpectedbyasinglewall. As [17] S.-H.Yang,K.-S.Ryu, andS.Parkin,Nat.Nanotechnol.10, thethicknesschangesfurther,(φ1 φ2)approachesπ. Ascan 221(2015). − beseenfromequation(12)theIECtorquevanisheswhenthe [18] R.Tomasello,V.Puliafito,E.Martinez,A.Manchon,M.Ricci, two magnetizations are colinear, and the velocity decreases M.Carpentieri, andG.Finocchio, arXiv:1610.00894. towardsthesinglewallvalue. [19] L.D.LandauandE.M.Lifshitz,Phys.ZeitschriftderSowjetu- nion8,153(1935);Ukr.J.Phys.53,14(2008). Conclusion. Wehaveshownthatcompletesuppressionof [20] T.Gilbert,IEEETrans.Magn.40,3443(2004). Walkerbreakdownispossibleinawiderangeofdomainwall [21] A.Fert,Mater.Sci.Forum59–60,439(1990). systemsdrivenbyspin–orbittorques,includinghead-to-head [22] R.C.O’Handley,Modernmagneticmaterials.Principlesand wallsinsoftmagnets,BlochandNéelwallsinPMAmagnets, applications(JohnWiley&Sons,2000). in the presence of the DMI and in coupled domain walls in [23] S.ZhangandZ.Li,Phys.Rev.Lett.93,127204(2004). SAFs. ForSOTsotherthanpureRashbaSOTsthisleadstoa [24] G.Beach,M.Tsoi, andJ.Erskine,J.Magn.Magn.Mater.320, linearcurrent–velocityrelationinsteadofasaturationofthe 1272(2008). [25] A. Brataas, A. D. Kent, and H. Ohno, Nat. Mater. 11, 372 velocity for large currents. In combination with IEC, SOT- (2012). driven domain wall motion in SAFs gives rise to very high [26] SeeSupplementalMaterial,whichincludesRefs.[32–38],for domainwallvelocities. detailsonthepossiblespin–orbittorquesinthissystem,ade- Fundingviathe“OutstandingAcademicFellows”program scriptionofthenumericsandforthecollectivecoordinateequa- at NTNU, the COST Action MP-1201, the NT-Faculty, and tionsforthecoupleddomainwalls. theResearchCouncilofNorwayGrantnumbers216700and [27] J.LinderandM.Alidoust,Phys.Rev.B88,064420(2013). 240806,isgratefullyacknowledged.WethankMortenAmund- [28] M.Stier,M.Creutzburg, andM.Thorwart,Phys.Rev.B90, 014433(2014). senforveryusefuldiscussionsofthenumerics. [29] O. Boulle, L. D. Buda-Prejbeanu, E. Jué, I. M. Miron, and G.Gaudin,J.Appl.Phys.115,17D502(2014). [30] P.-B.He,H.Yan,M.-Q.Cai, andZ.-D.Li,Europhys.Lett.114, 67001(2016). [31] A.V.Khvalkovskiy,V.Cros,D.Apalkov,V.Nikitin,M.Krounbi, 5 K.A.Zvezdin,A.Anane,J.Grollier, andA.Fert,Phys.Rev.B T.Suzuki,S.Mitani, andH.Ohno,Nat.Mater.12,240(2012). 87,020402(2013). [36] K.-S.Ryu, S.-H.Yang, L.Thomas, andS.S.P.Parkin,Nat. [32] P.Bruno,Phys.Rev.B52,411(1995). Commun.5,3910(2014). [33] K.M.D.HalsandA.Brataas,Phys.Rev.B88,085423(2013); [37] X.Fan,H.Celik,J.Wu,C.Ni,K.-J.Lee,V.O.Lorenz, and Phys.Rev.B91,214401(2015). J.Q.Xiao,Nat.Commun.5,3042(2014). [34] R.R.Birss,SymmetryandMagnetism,1sted.,editedbyE.P. [38] J.R.DormandandP.J.Prince,J.Comput.Appl.Math.6,19 Wohlfarth,SelectedTopicsinSolidStatePhysics,Vol.3(North- (1980). HollandPublisingCompany,1964). [35] J. Kim, J. Sinha, M. Hayashi, M. Yamanouchi, S. Fukami, SupplementalMaterialfor UniversalAbsenceofWalkerBreakdownandLinearCurrent–VelocityRelation viaSpin–OrbitTorquesinCoupledandSingleDomainWallMotion Vetle Risinggård and Jacob Linder DepartmentofPhysics,NTNU,NorwegianUniversityofScienceandTechnology,N-7491Trondheim,Norway (Dated:December21,2016) SYMMETRYADMISSIBLESPIN–ORBITTORQUES αR,θSHand βSH. AshasbeenshownbyHalsandBrataas[S1],thegeneral- HalsandBrataas[S1]describespin–orbittorquesandgen- izedspin-transfertorquesreducetotheordinarySTTsinthe eralizedspin-transfertorquesintermsofatensorexpansion. nonrelativistic limit. Thus, by using the ordinary STTs we Assumingthelowestordersaresufficienttodescribetheessen- neglectpossiblespin–orbitcouplingcorrectionstothesehigher tialdynamics,thereactiveanddissipativespin–orbittorques orderterms. aredescribedbyrespectivelyanaxialsecondranktensorand a polar third rank tensor while the generalized spin-transfer torquesaredescribedusingapolarfourthranktensorandan EQUATIONSOFMOTIONFORSAFSTRUCTURE axialfifthranktensor. Thetorquesthatariseinagivenstruc- ture are limited by the requirement that the tensors must be invariantunderthesymmetryoperationsfulfilledbythestruc- Weassumethattheinterlayerexchangecoupling(IEC)is ture. Wehaveassumedthatthephysicalsystemsweconsider completely local in the plane [S6], J(r1 r2) = Jδ(x1 − − aredescribedbyC2v symmetry. Combinedwiththefactthat x2)δ(y1 − y2). It follows that the IEC should not affect the thecurrentisappliedinthe xdirectiononlyandthat∂ym =0 domain wall profile, but can only determine the chirality of and∂zm =0,thisimpliesthatthereareonly1relevantnonzero coupled domain walls. Thus, we can use the static solution elementintheaxialsecondranktensor,2elementsinthepolar derivedinthemaintext,θ =2arctanexp[ (x X)/λ],where ± − thirdranktensor,3elementsinthepolarfourthranktensorand λ = A/Kz is the domain wall width. For a single wall the 6elementsintheaxialfifthranktensor[S2]. azimuthal angle φ is given by φ = nπ, where n=0,2,4,... p The3relevantnonzeroelementsofthesecondandthirdrank if D <0 and we choose the + sign for θ, and n=1,3,5,... tensorsgiveriseto3spin–orbittorques. Adetailedanalysis if D >0andwechoosethe signforθ. Twocoupledwalls − showsthatthesetorquecomponentsarecapturedbytheRashba must have opposite topological charges. If D1 and D2 both and spin Hall torques in equations (4) and (5) in the main arenegativetheDMIandtheIECcooperatetogivethestatic text. Asanaside,wenotethatalthoughtheRashbaandspin solution φ1 = 0 (positive topological charge) and φ2 = π Hall effects may not necessarily capture all of the relevant (negativetopologicalcharge). microscopicphysics[S3–S5]thesetorquescanstillbeusedto SubstitutingthisstaticsolutionintotheLLGequationusing modelthedynamicsbecausetheycontain3‘free’parameters, H0 = Hxex givesthecollectivecoordinateequations (cid:102) (cid:103) (1+α2)X˙1 = γKy sin2φ + πγ(D1−Hxmλ) sinφ + γJt2 αU(s)cos(φ φ )+αW(s)+V(s)sin(φ φ ) λ − m 1 2mλ 1 2m 1− 2 1− 2 (cid:102) (cid:103) (S1) u(1 αβ) − + πγ H(1) α+ β(1) +H(1)(1 αβ) cosφ , − λ 2 SH SH R − 1 (cid:16) (cid:17) (cid:102) (cid:103) (1+α2)X˙2 =+ γKy sin2φ + πγ(D2+Hxmλ) sinφ γJt1 αU(s)cos(φ φ )+αW(s) V(s)sin(φ φ ) λ m 2 2mλ 2− 2m 1− 2 − 1− 2 (cid:102) (cid:103) (S2) u(1 αβ) − + πγ H(2) α+ β(2) +H(2)(1 αβ) cosφ , − λ 2 SH SH R − 2 (cid:16) (cid:17) (cid:102) (cid:103) (1+α2)φ˙ = αγKy sin2φ + παγ(D1−Hxmλ) sinφ γJt2 U(s)cos(φ φ )+W(s) αV(s)sin(φ φ ) 1 − m 1 2mλ 1− 2m 1− 2 − 1− 2 (cid:102) (cid:103) (S3) u(α+ β) παγ H(1) 1 αβ(1) H(1)(α+ β) cosφ , − λ − 2 SH − SH − R 1 (cid:16) (cid:17) 2 (cid:102) (cid:103) (1+α2)φ˙ = αγKy sin2φ παγ(D2+Hxmλ) sinφ γJt1 U(s)cos(φ φ )+W(s)+αV(s)sin(φ φ ) 2 − m 2− 2mλ 2− 2m 1− 2 1− 2 (cid:102) (cid:103) (S4) u(α+ β) + + παγ H(2) 1 αβ(2) H(2)(α+ β) cosφ . λ 2 SH − SH − R 2 (cid:16) (cid:17) wherewehaveassumedthatthebulkparametersofthetwofer- NUMERICS romagnets are equal and where s is the separation between the two walls, s = (X1 X2)/λ. The IEC terms are ex- Equations(S1)and(S3)reducetoequations(6)and(7)in − pressedusingthethreefunctionsV(s),U(s)andW(s):V(s) = themaintextwhen J 0. Tosolveequations(S1)through 4ses/(e2s 1),U(s) = 4es(e2s(s 1)+s+1)/(e2s 1)2 (S4) numerically, we r→escale the equations to obtain dimen- − − − − andW(s) = 2(e4s 4se2s 1)/(e2s 1)2. Thesefunctions sionlessvariables. Thedimensionofequations(S1)through − − − areplottedinFigure1. (S4)isHz. Aconvenientscalingfactorwiththesamedimen- sions is µ γm. By dividing equations (S1) through (S4) by 0 µ0γm we get the rescaled variables t˜ = tµ0γm, X˜i = Xi/λ, 2 H˜x = Hx/µ0m, K˜y = Ky/µ0m2, D˜i = Di/µ0m2λ,t˜i = ti/λ, J˜= Jλ/µ m2andu˜=u/µ γmλ. 0 0 Equations (S1) through (S4) are solved using an explicit fourthorderRunge–Kuttaschemewithadaptivestepsizecon- 0 trol, implemented as a Dormand–Prince pair [S7]. Material V(s) parametersarechosentomatchapproximatelyCoorCo/Ni U(s) multilayers,seeTableI. 2 W(s) − 6 4 2 0 2 4 6 − − − separations Figure1.DependenceoftheIECtermsonthedomainwallseparation. [S1] K.M.D.HalsandA.Brataas,PhysicalReviewB88,085423 (2013);PhysicalReviewB91,214401(2015). [S2] R.R.Birss,SymmetryandMagnetism,1sted.,editedbyE.P. TableI.Materialparametersusedforthenumericalsolutionofequa- Wohlfarth,SelectedTopicsinSolidStatePhysics,Vol.3(North- tions(S1)through(S4)andforanalyticalestimatesinthemaintext. HollandPublisingCompany,1964). [S3] J. Kim, J. Sinha, M. Hayashi, M. Yamanouchi, S. Fukami, parameter value unit T.Suzuki,S.Mitani, andH.Ohno,NatureMaterials12,240 (2012). gyromagneticratioγ 0.19 GHz/T [S4] K.-S.Ryu,S.-H.Yang,L.Thomas, andS.S.P.Parkin,Nature domainwallwidthλ 4.0 nm Communications5,3910(2014). hardaxisanisotropyKy 0.20 MJ/m3 [S5] X.Fan,H.Celik,J.Wu,C.Ni,K.-J.Lee,V.O.Lorenz, and saturationmagnetizationm 1.0 MA/m J.Q.Xiao,NatureCommunications5,3042(2014). Dzyaloshinskii–Moriyaconst. D 1.4 mJ/m2 [S6] P.Bruno,PhysicalReviewB52,411(1995). − Gilbertdampingα 0.25 [S7] J.R.DormandandP.J.Prince,JournalofComputationaland spin-polarizationP 0.50 AppliedMathematics6,19(1980). nonadiabacityparameter β 0.50 Rashbaspin–orbitcouplingα 6.3 meVnm R spinHallangleθ 0.1 SH spinHall β-term β 0.02 SH interlayerexchange Jt t 5.0 mJ/m2 1 2 thicknesst 1.2 nm