Universal Absence of Walker Breakdown and Linear Current-Velocity Relation via Spin-Orbit Torques in Coupled and Single Domain Wall Motion

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by  Vetle Risinggård

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]
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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