[    {        "title": "1403.5955v1.Existence_Results_for_Some_Damped_Second_Order_Volterra_Integro_Differential_Equations.pdf",        "content": "arXiv:1403.5955v1  [math.AP]  24 Mar 2014EXISTENCE RESULTS FOR SOME DAMPED SECOND-ORDER VOLTERRA\nINTEGRO-DIFFERENTIAL EQUATIONS\nTOKADIAGANA\nInMemoryofProf. YahyaOuldHamidoune\nAbstract. In this paper we make a subtle use of operator theory techniq ues and the well-\nknownSchauderfixed-pointprincipletoestablish theexist enceofpseudo-almostautomor-\nphic solutions to some second-order damped integro-di fferential equations with pseudo-\nalmost automorphic coe fficients. In order to illustrate our main results, we will stud y\nthe existence of pseudo-almost automorphic solutions to a s tructurally damped plate-like\nboundary value problem.\n1. Introduction\nIntegro-differentialequationsplayanimportantrolewhenitcomestomo delingvarious\nnatural phenomena, see, e.g., [ 9,10,15,27,28,29,34,43,49,52,53,57,60,61,62,\n71]. In recent years, noteworthy progress has been made in stud ying the existence of\nperiodic,almostperiodic,almostautomorphic,pseudo-al mostperiodic,andpseudo-almost\nautomorphicsolutionsto first-orderintegro-di fferentialequations,see, e.g., [ 2,17,22,23,\n24,25,35,36,37,44,45,60,61,62]. The most popular method used to deal with the\nexistence of solutions to those first-order integro-di fferentialequations consists of the so-\ncalledmethodofresolvents,see, e.g.,[ 1,14,15,36,37,44,45].\nFixα∈(0,1). LetHbe an infinite dimensional separable Hilbert space over the fi eld\nofcomplexnumbersequippedwith the innerproductandnormg ivenrespectivelyby /an}bracketle{t·,·/an}bracketri}ht\nand/bardbl·/bardbl. The purpose of this paper consists of making use of a new appr oach to study\ntheexistenceofpseudo-almostautomorphicsolutionstoth eclassofdampedsecond-order\nVolterraintegro-differentialequationsgivenby\nd2ϕ\ndt2+Bdϕ\ndt+Aϕ=/integraldisplayt\n−∞C(t−s)ϕ(s)ds+f(t,ϕ), (1.1)\nwhereA:D(A)⊂H/mapsto→His an unbounded self-adjoint linear operator whose spectru m\nconsistsofisolatedeigenvaluesgivenby\n0<λ1<λ2<...<λ n→∞\nasn→∞with eacheigenvaluehavinga finite multiplicity γjequalsto themultiplicityof\nthecorrespondingeigenspace, B:D(B)⊂H/mapsto→His a positiveself-adjointlinearoperator\nsuchthatthereexisttwoconstants γ1,γ2>0andsuchthat γ1Aα≤B≤γ2Aα, thatis,\nγ1/an}bracketle{tAαϕ,ϕ/an}bracketri}ht≤/an}bracketle{tBϕ,ϕ/an}bracketri}ht≤γ2/an}bracketle{tAαϕ,ϕ/an}bracketri}ht\nfor allϕ∈D(B1\n2)=D(Aα\n2), the mappings C(t) :D(A)⊂H/mapsto→Hconsist of (possibly\nunbounded)linear operators for each t∈R, and the function f:R×H/mapsto→His pseudo-\nalmostautomorphicin thefirst variableuniformlyinthesec ondone.\n2000Mathematics Subject Classification. 12H20; 45J05;43A60; 35L71; 35L10;37L05.\nKey words and phrases. second-order integro-di fferential equation; pseudo-almost automorphic; Schauder\nfixed point theorem; hyperbolic semigroup; structurally da mped plate-like boundary value problem.\n12 TOKADIAGANA\nEquationsoftypeEq. ( 1.1)ariseveryofteninthestudyofnaturalphenomenainwhich\nacertainmemorye ffectistakenintoconsideration,see,e.g.,[ 3,6,46,50,51]. In[3,6]for\ninstance,equationsoftypeEq. ( 1.1)appearedinthestudyofa viscoelasticwaveequation\nwithmemory.\nThe existence, uniqueness, and asymptotic behavior of solu tions to Eq. ( 1.1) have\nwidely been studied, see, e.g., [ 3,6,7,8,41,42,46,50,51,54,55,56]. However, to\nthe best of our knowledge, the existence of pseudo-almost au tomorphic solutions to Eq.\n(1.1) is an untreated original problem with important applicati ons, which constitutes the\nmainmotivationofthispaper.\nIn this paper,we are interestedin the special case B=2γAαwhereγ >0 is a constant,\nthatis,\nd2ϕ\ndt2+2γAαdϕ\ndt+Aϕ=/integraldisplayt\n−∞C(t−s)ϕ(s)ds+f(t,ϕ),t∈R. (1.2)\nIt should be mentioned that various versions of Eq. ( 1.2) have been investigated in the\nliterature, see, e.g., Chen and Triggiani [ 11,12], Huang [ 30,31,32,33], and Xiao and\nLiang[65,66,67,68,69,70].\nConsiderthepolynomial Qγ\nnassociatedwiththeleft handsideofEq. ( 1.2),thatis,\nQγ\nn(ρ) :=ρ2+2γλα\nnρ+λn (1.3)\nanddenoteits rootsby ρn\n1:=dn+ienandρn\n2:=rn+isnforalln≥1.\nIntherestofthepaper,wesupposethattheroots ρn\n1andρn\n2satisfy:ρn\n1/nequalρn\n2foralln≥1\nandthat thefollowingcrucialassumptionholds: thereexis tsδ0>0suchthat\nsup\nn≥1/bracketleftig\nmax(dn,rn)/bracketrightig\n≤−δ0<0. (1.4)\nIn order to investigate the existence of pseudo-almost auto morphic solutions to Eq.\n(1.2), our strategy consists of rewriting it as a first-order inte gro-differential equation in\nthe product spaceE1\n2:=D(A1\n2)×Hand then study the existence of pseudo-almost auto-\nmorphic solutions to the obtained first-order integro-di fferentialequation with the help of\nSchauderfixedpointprincipleandthengobacktoEq. ( 1.2).\nRecall thatthe innerproductof E1\n2isdefinedasfollows:\n/parenleftigg/parenleftiggϕ1\nϕ2/parenrightigg\n,ψ1\nψ2/parenrightigg\nE1\n2:=/an}bracketle{tA1\n2ϕ1,A1\n2ψ1/an}bracketri}ht+/an}bracketle{tϕ2,ψ2/an}bracketri}ht\nforallϕ1,ψ1∈D(A1\n2) andϕ2,ψ2∈H. Itscorrespondingnormwill bedenoted /bardbl·/bardblE1\n2.\nLetting\nΦ:=ϕ\nϕ′∈E1\n2,\nthenEq. ( 1.2)canberewrittenin thefollowingform\n(1.5)dΦ\ndt=AΦ+/integraldisplayt\n−∞C(t−s)Φ(s)ds+F(t,Φ(t)),t∈R,EXISTENCE OF PSEUDO-ALMOST AUTOMORPHIC SOLUTIONS 3\nwhereA,Caretheoperatormatricesdefinedby\nA=0I\n−A−Aα,C=C\n0,\nwithdomain D(A)=D(A)×[D(A1\n2)∩D(Aα)]=D(C)(D(A)=D(A)×D(A1\n2)if0<α≤1\n2\nandD(A)=D(A)×D(Aα) if1\n2≤α <1), and the function F:R×E1\n2/mapsto→E:=H×His\ngivenby\nF(t,Φ)=0\nf(t,ϕ).\nIn order to investigate Eq. ( 1.5), we study the first-order di fferential equation in the\nspaceE1\n2givenby,\ndϕ\ndt=(A+B)ϕ+F(t,ϕ),t∈R, (1.6)\nwhereB:C(R,D(A))/mapsto→E1\n2isthelinearoperatordefinedby\nBϕ:=/integraldisplayt\n−∞C(t−s)ϕ(s)ds, ϕ∈C(R,D(A)) (1.7)\nwithC(R,D(A))beingthecollectionofall continuousfunctionsfrom RintoD(A).\nIn order to study the existence of solutionsto Eq. ( 1.6), we will make extensive use of\nhyperbolicsemigrouptoolsandfractionalpowersofoperat ors,andthatthelinearoperator\nBsatisfiessomeadditionalassumptions. Ourexistenceresul twillthenbeobtainedthrough\ntheuseofthewell-knownSchauderfixed-pointtheorem. Obvi ously,onceweestablishthe\nsought existence results for Eq. ( 1.6), then we can easily go back to Eq. ( 1.2) notably\nthroughEq. ( 1.7).\nThe concept of pseudo almost automorphy is a powerful notion introduced in the lit-\nerature by Liang et al.[39,40,63,64]. This concept has recently generated several de-\nvelopmentsandextensions,whichhavebeensummarizedin a n ewbookbyDiagana[ 17].\nThe existence of almost periodic and asymptotically almost periodic solutions to integro-\ndifferential equations of the form Eq. ( 1.5) in a general context has recently been estab-\nlishedin[ 36,37]. Similarly,in[ 45],theexistenceofpseudo-almostautomorphicsolutions\nto Eq. (1.5)was studied. Themain methodused in the above-mentionedpa persare resol-\nvents operators. However, to the best of our knowledge, the e xistence of pseudo-almost\nautomorphic solutions to Eq. ( 1.2) is an important untreated topic with some interesting\napplications. Amongotherthings, we will make extensiveus e of theSchauderfixedpoint\ntoderivesomesufficientconditionsfortheexistenceofpseudo-almostautomo rphic(mild)\nsolutionsto ( 1.6)andthentoEq. ( 1.2).\n2. Preliminaries\nSome of the basic results discussed in this section are mainl ytaken fromthe following\nrecentpapersbyDiagana[ 18,21]. Inthispaper,Hdenoteaninfinitedimensionalseparable\nHilbertspaceoverthefieldofcomplexnumbersequippedwith theinnerproductandnorm\ngivenrespectivelyby /an}bracketle{t·,·/an}bracketri}htand/bardbl·/bardbl. IfAisalinearoperatoruponaBanachspace( X,/bardbl·/bardbl),\nthen the notations D(A),ρ(A),σ(A),N(A), andR(A) stand respectively for the domain,\nresolvent, spectrum, kernel, and the range of A. Similarly, if A:D:=D(A)⊂X/mapsto→Xis4 TOKADIAGANA\na closed linear operator on a Banach space, one denotes its gr aph norm by/bardbl·/bardblDdefined\nby/bardblx/bardblD:=/bardblx/bardbl+/bardblAx/bardblfor allx∈D. From the closedness of A, one can easily see that\n(D,/bardbl·/bardblD) is a Banach space. Moreover, one sets R(λ,L) :=(λI−L)−1for allλ∈ρ(A).\nWe setQ=I−Pfor a projection P. IfY,Zare Banach spaces, then the space B(Y,Z)\ndenotes the collection of all bounded linear operators from YintoZequipped with its\nnatural uniform operator topology /bardbl·/bardblB(Y,Z). We also set B(Y)=B(Y,Y). IfK⊂Xis a\nsubset,we let coKdenotetheclosed convexhullof K. Additionally,Twill denotethe set\ndefinedby,T:={(t,s)∈R×R:t≥s}.If(X,/bardbl·/bardblX)and(Y,/bardbl·/bardblY)areBanachspaces,their\nproductX×Y:={(x,y) :x∈X,y∈Y}is also a Banach when it is equipped with the\nnormgivenby\n/bardbl(x,y)/bardblX×Y=/radicalig\n/bardblx/bardbl2\nX+/bardbly/bardbl2\nYforall (x,y)∈X×Y.\nIn this paper if β≥0, then we setEβ:=D(Aβ)×H, andE:=H×Hand equip them\nwith their corresponding topologies /bardbl·/bardblEβand/bardbl·/bardblE. Recall that D(Aβ) will be equipped\nwiththenormdefinedby, /bardblϕ/bardblβ:=/bardblAβϕ/bardblforallϕ∈D(Aβ).\nIn the sequel, A:D(A)⊂H/mapsto→Hstands for a self-adjoint (possibly unbounded)\nlinear operator on the Hilbert space Hwhose spectrum consists of isolated eigenvalues\n0< λ1< λ2< ... < λ n→∞with each eigenvalue having a finite multiplicity γjequals\nto the multiplicity of the correspondingeigenspace. Let {ek\nj}be a (complete) orthonormal\nsequence of eigenvectors associated with the eigenvalues {λj}j≥1. Clearly, for each u∈\nD(A),whereif\nu∈D(A) :=/braceleftig\nu∈H:∞/summationdisplay\nj=1λ2\nj/bardblEju/bardbl2<∞/bracerightig\n,thenAu=∞/summationdisplay\nj=1λjγj/summationdisplay\nk=1/an}bracketle{tu,ek\nj/an}bracketri}htek\nj=∞/summationdisplay\nj=1λjEju\nwithEju=/summationtextγj\nk=1/an}bracketle{tu,ek\nj/an}bracketri}htek\nj.Note that{Ej}j≥1is a sequence of orthogonalprojectionson H.\nMoreover,each u∈Hcan written as follows: u=/summationtext∞\nj=1Eju.It should also be mentioned\nthattheoperator−Aistheinfinitesimalgeneratorofananalyticsemigroup {S(t)}t≥0,which\nisexplicitlyexpressedintermsofthoseorthogonalprojec tionsEjby,forall u∈H,\nS(t)u=∞/summationdisplay\nj=1e−λjtEju\nwhichin particularisexponentiallystable as\n/bardblS(t)/bardbl≤e−λ1t\nforallt≥0.\n3. SectorialLinearOperators\nThebasicresultsdiscussedinthissectionaremainlytaken fromDiagana[ 17,20].\nDefinition 3.1. A linearoperator B:D(B)⊂X/mapsto→X(notnecessarilydenselydefined)on\na BanachspaceXissaid to be sectorialif the followinghold: thereexist con stantsω∈R,\nθ∈/parenleftbiggπ\n2,π/parenrightbigg\n,andM>0 suchthatρ(B)⊃Sθ,ω,\nSθ,ω:=/braceleftig\nλ∈C:λ/nequalω,|arg(λ−ω)|<θ/bracerightig\n,and (3.1)\n/bardblR(λ,B)/bardbl≤M\n|λ−ω|, λ∈Sθ,ω. (3.2)EXISTENCE OF PSEUDO-ALMOST AUTOMORPHIC SOLUTIONS 5\nExample 3.2. Letp≥1 and letΩ⊂Rdbe open bounded subset with C2boundary∂Ω.\nLetX:=Lp(Ω)betheLebesguespaceequippedwith thenorm, /bardbl·/bardblpdefinedby,\n/bardblϕ/bardblp=/parenleftig/integraldisplay\nΩ|ϕ(x)|pdx/parenrightig1/p.\nDefinethe operator Aasfollows:\nD(B)=W2,p(Ω)∩W1,p\n0(Ω),B(ϕ)=∆ϕ,∀ϕ∈D(B),\nwhere∆=d/summationdisplay\nk=1∂2\n∂x2\nkistheLaplaceoperator. Itcanbecheckedthattheoperator Bissectorial\nonLp(Ω).\nIt is well-known [ 47] that ifB:D(B)⊂X/mapsto→Xis a sectorial linear operator, then it\ngeneratesananalyticsemigroup( T(t))t≥0,whichmaps(0 ,∞)intoB(X)andsuchthatthere\nexistM0,M1>0 with\n/bardblT(t)/bardbl≤M0eωt,t>0, (3.3)\n/bardblt(A−ω)T(t)/bardbl≤M1eωt,t>0. (3.4)\nIn this paper, we suppose that the semigroup ( T(t))t≥0is hyperbolic,that is, there exist\na projection Pand constants M,δ >0 such that T(t) commutes with P,N(P) is invariant\nwithrespectto T(t),T(t) :R(Q)/mapsto→R(Q)isinvertible,andthefollowinghold\n(3.5) /bardblT(t)Px/bardbl≤Me−δt/bardblx/bardblfort≥0,\n(3.6) /bardblT(t)Qx/bardbl≤Meδt/bardblx/bardblfort≤0,\nwhereQ:=I−Pand,fort≤0,T(t) :=(T(−t))−1.\nRecall that the analytic semigroup( T(t))t≥0associated with Bis hyperbolicif and only\nifσ(B)∩iR=∅,see detailsin [ 26, Proposition1.15,pp.305].\nDefinition 3.3. Letα∈(0,1). A Banach space ( Xα,/bardbl·/bardblα) is said to be an intermediate\nspace between D(B) andX, or a space of class Jα, ifD(B)⊂Xα⊂Xand there is a\nconstantc>0suchthat\n(3.7) /bardblx/bardblα≤c/bardblx/bardbl1−α/bardblx/bardblα\nB,x∈D(B).\nConcreteexamplesof XαincludeD((−Bα))forα∈(0,1),thedomainsofthefractional\npowersof B, the real interpolationspaces DB(α,∞),α∈(0,1),defined as the space of all\nx∈Xsuchthat,\n[x]α=sup\n0<t≤1/bardblt1−αBT(t)x/bardbl<∞\nwiththenorm\n/bardblx/bardblα=/bardblx/bardbl+[x]α,\nthe abstract H¨ older spaces DB(α) :=D(B)/bardbl./bardblαas well as the complex interpolation spaces\n[X,D(B)]α.\nFora hyperbolicanalyticsemigroup( T(t))t≥0, onecaneasilycheckthat similarestima-\ntions as both Eq. ( 3.5) and Eq. ( 3.6) still hold with the α-norms/bardbl·/bardblα. In fact, as the part\nofAinR(Q)isbounded,it followsfromEq. ( 3.6)that\n/bardblBT(t)Qx/bardbl≤C′eδt/bardblx/bardblfort≤0.\nHence,fromEq. ( 3.7)thereexistsaconstant c(α)>0suchthat\n(3.8) /bardblT(t)Qx/bardblα≤c(α)eδt/bardblx/bardblfort≤0.6 TOKADIAGANA\nInadditiontothe above,thefollowingholds\n/bardblT(t)Px/bardblα≤/bardblT(1)/bardblB(X,Xα)/bardblT(t−1)Px/bardbl,t≥1,\nandhencefromEq. ( 3.5),oneobtains\n/bardblT(t)Px/bardblα≤M′e−δt/bardblx/bardbl,t≥1,\nwhereM′dependson α. Fort∈(0,1],byEq. ( 3.4)andEq. ( 3.7),\n/bardblT(t)Px/bardblα≤M′′t−α/bardblx/bardbl.\nHence,thereexistconstants M(α)>0andγ>0suchthat\n(3.9) /bardblT(t)Px/bardblα≤M(α)t−αe−γt/bardblx/bardblfort>0.\nRemark3.4.Note that if the analytic semigroup T(t) is exponential stable, that is, there\nexists constants N,δ >0 such that/bardblT(t)/bardbl≤Ne−δtfort≥0, then the projection P=I\n(Q=I−P(t)=0). In that case, Eq. ( 3.9) still holds and can be rewritten as follows: for\nallx∈X,\n(3.10) /bardblT(t)x/bardblα≤M(α)e−γ\n2tt−α/bardblx/bardbl.\nFormoreoninterpolationspacesandrelatedissues,werefe rthereadertothefollowing\nexcellentbooksAmann[ 5] andLunardi[ 47].\n3.1.Pseudo-AlmostAutomorphic Functions. LetBC(R,X)stand for the Banachspace\nofallboundedcontinuousfunctions ϕ:R/mapsto→X,whichweequipwiththesup-normdefined\nby/bardblϕ/bardbl∞:=supt∈R/bardblϕ(t)/bardblforallϕ∈BC(R,X). Ifβ≥0,wewillalsobeusingthefollowing\nnotions,\n/bardblΦ/bardblEβ,∞:=sup\nt∈R/bardblΦ(t)/bardblEβ\nforΦ∈BC(R,Eβ), and\n/bardblϕ/bardblβ,∞:=sup\nt∈R/bardblϕ(t)/bardblβ\nforϕ∈BC(R,D(Aβ)).\nDefinition 3.5. [17] A function f∈C(R,X) is said to be almost automorphicif for every\nsequenceofrealnumbers( s′\nn)n∈N,thereexistsasubsequence( sn)n∈Nsuchthat\ng(t) :=lim\nn→∞f(t+sn)\niswelldefinedforeach t∈R, and\nlim\nn→∞g(t−sn)=f(t)\nforeacht∈R.\nIf the convergenceabove is uniformin t∈R, thenfis almost periodic in the classical\nBochner’s sense. Denote by AA(X) the collection of all almost automorphic functions\nR/mapsto→X. Notethat AA(X)equippedwiththesup-normturnsouttobea Banachspace.\nAmongotherthings,almostautomorphicfunctionssatisfy t hefollowingproperties.\nTheorem3.6. [17]If f,f1,f2∈AA(X),then\n(i)f1+f2∈AA(X),\n(ii)λf∈AA(X)foranyscalar λ,\n(iii)fα∈AA(X)where fα:R→Xisdefinedby f α(·)=f(·+α),\n(iv)the rangeRf:=/braceleftbigf(t) :t∈R/bracerightbigis relatively compact in X, thus f is bounded in\nnorm,EXISTENCE OF PSEUDO-ALMOST AUTOMORPHIC SOLUTIONS 7\n(v)if fn→f uniformlyonRwhereeach f n∈AA(X),then f∈AA(X)too.\nDefinition3.7. LetYbeanotherBanachspace. Ajointlycontinuousfunction F:R×Y/mapsto→\nXis said to be almost automorphic in t∈Rift/mapsto→F(t,x) is almost automorphic for all\nx∈K(K⊂Ybeinganyboundedsubset). Equivalently,foreverysequenc eofrealnumbers\n(s′\nn)n∈N,thereexistsasubsequence( sn)n∈Nsuchthat\nG(t,x) :=lim\nn→∞F(t+sn,x)\niswelldefinedin t∈Randforeach x∈K, and\nlim\nn→∞G(t−sn,x)=F(t,x)\nforallt∈Randx∈K.\nThecollectionofsuchfunctionswill bedenotedby AA(R×X).\nFormoreonalmostautomorphicfunctionsandtheirgenerali zations,wereferthereader\ntotherecentbookbyDiagana[ 17].\nDefine (see Diagana [ 17,19]) the space PAP0(R,X) as the collection of all functions\nϕ∈BC(R,X)satisfying,\nlim\nr→∞1\n2r/integraldisplayr\n−r/bardblϕ(s)/bardblds=0.\nSimilarly, PAP0(R×X) will denotethe collectionof all boundedcontinuousfunct ions\nF:R×Y/mapsto→Xsuchthat\nlim\nT→∞1\n2r/integraldisplayr\n−r/bardblF(s,x)/bardblds=0\nuniformlyin x∈K,whereK⊂Yis anyboundedsubset.\nDefinition 3.8. (Lianget al.[39] andXiao et al. [63]) A function f∈BC(R,X)is called\npseudo almost automorphic if it can be expressed as f=g+φ,whereg∈AA(X) and\nφ∈PAP0(X). Thecollectionofsuchfunctionswill bedenotedby PAA(X).\nThe functions gandφappearing in Definition 3.8are respectively called the almost\nautomorphic andtheergodicperturbation componentsof f.\nDefinition 3.9. LetYbe another Banach space. A boundedcontinuousfunction F:R×\nY/mapsto→Xbelongs to AA(R×X) whenever it can be expressed as F=G+Φ,whereG∈\nAA(R×X) andΦ∈PAP0(R×X). The collection of such functions will be denoted by\nPAA(R×X).\nA substantialresultisthenexttheorem,whichisduetoXiao et al. [63].\nTheorem 3.10. [63]The space PAA (X)equipped with the sup norm /bardbl·/bardbl∞is a Banach\nspace.\nTheorem3.11. [63]IfYisanotherBanachspace, f :R×Y/mapsto→Xbelongsto PAA (R×X)\nandif x/mapsto→f(t,x)isuniformlycontinuousoneachboundedsubset K of Yuniformlyint∈\nR,thenthefunctiondefinedbyh (t)=f(t,ϕ(t))belongsto PAA (X)providedϕ∈PAA(Y).\nFor more on pseudo-almost automorphic functions and their g eneralizations, we refer\nthereadertothe recentbookbyDiagana[ 17].8 TOKADIAGANA\n4. MainResults\nFixβ∈(0,1). Considerthe first-orderdi fferentialequations,\ndϕ\ndt=Aϕ+g(t),t∈R, (4.1)\nand\ndϕ\ndt=(A+B)ϕ+f(t,ϕ),t∈R, (4.2)\nwhereA:D(A)⊂X/mapsto→Xis a sectorial linear operator on a Banach space X,B:\nC(R,D(A))/mapsto→Xis a linear operator, and g:R/mapsto→Xandf:R×X/mapsto→Xare bounded\ncontinuousfunctions.\nTo study the existence of pseudo-almost automorphic mild so lutions to Eq. ( 4.1) (and\nhenceEq. ( 4.2)),wewill needthefollowingassumptions,\n(H.1) Thelinearoperator Aissectorial. Moreover,if T(t)denotestheanalyticsemigroup\nassociatedwithit, we supposethat T(t) ishyperbolic,thatis,\nσ(A)∩iR=∅.\n(H.2) The semigroup T(t) is not onlycompactfor t>0 but also is exponentiallystable,\ni.e.,thereexistsconstants N,δ>0suchthat\n/bardblT(t)/bardbl≤Ne−δt\nfort≥0.\n(H.3) The linear operator B:BC(R,Xβ)/mapsto→X, whereXβ:=D((−A)β), is bounded .\nMoreover,thefollowingholds,\nC0:=/bardblB/bardblB(BC(R,Xβ),X)≤1\n2d(β),\nwhered(β) :=M(β)(2δ−1)1−βΓ(1−β).\n(H.4) The function f:R×Xβ/mapsto→Xis pseudo-almost automorphic in the first vari-\nable uniformly in the second one. For each bounded subset K⊂Xβ,f(R,K)\nis bounded. Moreover, the function u/mapsto→f(t,u) is uniformly continuous on any\nboundedsubset KofXβforeacht∈R. Finally,wesupposethatthereexists L>0\nsuchthat\nsup\nt∈R,/bardblϕ/bardblβ≤L/vextenddouble/vextenddouble/vextenddouble/vextenddoublef(t,ϕ)/vextenddouble/vextenddouble/vextenddouble/vextenddouble≤L\n2d(β).\n(H.5) If( un)n∈N⊂PAA(Xβ)isuniformlyboundedanduniformlyconvergentuponevery\ncompactsubsetofR, thenf(·,un(·))isrelativelycompactin BC(R,X).\nRemark4.1.Notethatif(H.3)holds,thenitcanbeeasilyshownthatthel inearoperator B\nmapsPAA(Xβ) intoPAA(X).\nDefinition 4.2. Under assumption(H.1), a continuousfunction ϕ:R/mapsto→Xis said to be a\nmildsolutiontoEq. ( 4.1)providedthat\nϕ(t)=T(t−s)ϕ(s)+/integraldisplayt\nsT(t−τ)g(τ)dτ,∀(t,s)∈T. (4.3)\nLemma 4.3. [17]Suppose assumptions (H.1)–(H.2) hold. If g :R/mapsto→Xis a bounded\ncontinuousfunction,then ϕgivenby\nϕ(t) :=/integraldisplayt\n−∞T(t−s)g(s)ds (4.4)EXISTENCE OF PSEUDO-ALMOST AUTOMORPHIC SOLUTIONS 9\nforall t∈R, istheuniqueboundedmildsolutionto Eq. ( 4.1).\nDefinition 4.4. Under assumptions (H.1), (H.2), and (H.3) and if f:R×Xβ/mapsto→Xis a\nboundedcontinuousfunction,thena continuousfunction ϕ:R/mapsto→Xβsatisfying\nϕ(t)=T(t−s)ϕ(s)+/integraldisplayt\nsT(t−s)/bracketleftig\nBϕ(s)+f(s,ϕ(s))/bracketrightig\nds,∀(t,s)∈T (4.5)\niscalledamildsolutiontoEq. ( 4.2).\nUnderassumptions(H.1),(H.2),and(H.3)andif f:R×Xβ/mapsto→Xis aboundedcontin-\nuousfunction,it canbeshownthatthefunction ϕ:R/mapsto→Xβdefinedby\nϕ(t)=/integraldisplayt\n−∞T(t−s)/bracketleftig\nBϕ(s)+f(s,ϕ(s))/bracketrightig\nds (4.6)\nforallt∈R, isa mildsolutiontoEq. ( 4.2).\nDefinethe followingintegraloperator,\n(Sϕ)(t)=/integraldisplayt\n−∞T(t−s)/bracketleftig\nBϕ(s)+f(s,ϕ(s))/bracketrightig\nds.\nWe have\nLemma 4.5. Under assumptions (H.1)–(H.2)–(H.3) and if f:R×Xβ/mapsto→Xis a bounded\ncontinuous function, then the mapping S :BC(R,Xβ)/mapsto→BC(R,Xβ)is well-defined and\ncontinuous.\nProof.We first show that Sis well-defined and that S(BC(R,Xβ))⊂BC(R,Xβ). Indeed,\nlettingu∈BC(R,Xβ),g(t) :=f(t,u(t)),andusingEq. ( 3.10),weobtain\n/vextenddouble/vextenddouble/vextenddoubleSu(t)/vextenddouble/vextenddouble/vextenddoubleβ≤/integraldisplayt\n−∞/vextenddouble/vextenddouble/vextenddoubleT(t−s)[Bu(s)+g(s)]/vextenddouble/vextenddouble/vextenddoubleβds\n≤/integraldisplayt\n−∞M(β)e−δ\n2(t−s)(t−s)1−β/bracketleftig\n/bardblBu(s)/bardbl+/bardblg(s)/bardbl/bracketrightig\nds\n≤/integraldisplayt\n−∞M(β)e−δ\n2(t−s)(t−s)1−β/bracketleftig\nC0/bardblu(s)/bardblβ+/bardblg(s)/bardbl/bracketrightig\nds\n≤d(β)/parenleftig\nC0/bardblu/bardblβ,∞+/bardblg/bardbl∞/parenrightig\n,\nforallt∈R, whered:=M(β)(2δ−1)1−βΓ(1−β),andhence Su:R/mapsto→Xβisbounded.\nTo completetheproofit remainstoshowthat Siscontinuous. Forthat,set\nF(s,u(s)) :=Bu(s)+g(s)=Bu(s)+f(s,u(s)),∀s∈R.\nConsider an arbitrary sequence of functions un∈BC(R,Xβ) that convergesuniformly\ntosomeu∈BC(R,Xβ),that is,/vextenddouble/vextenddouble/vextenddoubleun−u/vextenddouble/vextenddouble/vextenddoubleβ,∞→0 asn→∞.10 TOKADIAGANA\nNow\n/vextenddouble/vextenddouble/vextenddoubleSu(t)−Sun(t)/vextenddouble/vextenddouble/vextenddoubleβ=/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\n−∞T(t−s)[F(s,un(s))−F(s,u(s))]ds/vextenddouble/vextenddouble/vextenddoubleβ\n≤M(β)/integraldisplayt\n−∞(t−s)−βe−δ\n2(t−s)/vextenddouble/vextenddouble/vextenddoubleF(s,un(s))−F(s,u(s))/vextenddouble/vextenddouble/vextenddoubleds.\n≤M(β)/integraldisplayt\n−∞(t−s)−βe−δ\n2(t−s)/vextenddouble/vextenddouble/vextenddoublef(s,un(s))−f(s,u(s))/vextenddouble/vextenddouble/vextenddoubleds\n+M(β)/integraldisplayt\n−∞(t−s)−βe−δ\n2(t−s)/vextenddouble/vextenddouble/vextenddoubleB(un(s)−u(s))/vextenddouble/vextenddouble/vextenddoubleds\n≤M(β)/integraldisplayt\n−∞(t−s)−βe−δ\n2(t−s)/vextenddouble/vextenddouble/vextenddoublef(s,un(s))−f(s,u(s))/vextenddouble/vextenddouble/vextenddoubleds\n+d(β)C0/bardblun−u/bardblβ,∞.\nUsing the continuity of the function f:R×Xβ/mapsto→Xand the Lebesgue Dominated\nConvergenceTheoremweconcludethat\n/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\n−∞T(t−s)P(s)[f(s,un(s))−f(s,u(s))]ds/vextenddouble/vextenddouble/vextenddouble→0 asn→∞.\nTherefore,/vextenddouble/vextenddouble/vextenddoubleSun−Su/vextenddouble/vextenddouble/vextenddoubleβ,∞→0asn→∞. Theproofis complete.\n/square\nLemma4.6. Underassumptions (H.1)—(H.4) ,thenS(PAA(Xβ)⊂PAA(Xβ).\nProof.Letu∈PAA(Xβ) and define h(s) :=f(s,u(s))+Bu(s) for alls∈R. Using (H.4)\nandTheorem 3.11itfollowsthatthefunction s/mapsto→f(s,u(s))belongsto PAA(X). Similarly,\nusing Remark 4.1it follows that the function s/mapsto→Bu(s) belongs to PAA(X). In view of\ntheabove,thefunction s/mapsto→h(s)belongsto PAA(X).\nNowwrite h=h1+h2∈PAA(X)whereh1∈AA(X)andh2∈PAP0(X)andset\nRhj(t) :=/integraldisplayt\n−∞T(t−s)hj(s)dsforallt∈R,j=1,2.\nOur first task consists of showing that R/parenleftbigAA(X)/parenrightbig⊂AA(Xβ). Indeed,using the fact that\nh1∈AA(X), for everysequence of real numbers( τ′\nn)n∈Nthere exist a subsequence( τn)n∈N\nanda function f1suchthat\nf1(t) :=lim\nn→∞h1(t+τn)\niswelldefinedforeach t∈R, and\nlim\nn→∞f1(t−τn)=h1(t)\nforeacht∈R.\nNow\n(Rh1)(t+τn)−(Rf1)(t)=/integraldisplayt+τn\n−∞T(t+τn−s)h1(s)ds−/integraldisplayt\n−∞T(t−s)f1(s)ds\n=/integraldisplayt\n−∞T(t−s)h1(s+τn)ds−/integraldisplayt\n−∞T(t−s)f1(s)ds.\n=/integraldisplayt\n−∞T(t−s)/parenleftig\nh1(s+τn)−f1(s)/parenrightig\nds.EXISTENCE OF PSEUDO-ALMOST AUTOMORPHIC SOLUTIONS 11\nFrom Eq. ( 3.10)andthe LebesgueDominatedConvergenceTheorem,it easily follows\nthat\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\n−∞T(t−s)/parenleftig\nh1(s+τn)−f1(s)/parenrightig\nds/vextenddouble/vextenddouble/vextenddouble/vextenddoubleβ≤/integraldisplayt\n−∞/vextenddouble/vextenddouble/vextenddouble/vextenddoubleT(t−s)/parenleftig\nh1(s+τn)−f1(s)/parenrightig/vextenddouble/vextenddouble/vextenddouble/vextenddoubleβds\n≤M(β)/integraldisplayt\n−∞(t−s)−βe−δ\n2(t−s)/bardblh1(s+τn)−f1(s)/bardblds\n→0 asn→∞,\nandhence\n(Rf1)(t)=lim\nn→∞(Rh1)(t+τn)\nforallt∈R.\nUsingsimilar argumentsasaboveoneobtainsthat\n(Rh1)(t)=lim\nn→∞(Rf1)(t−τn)\nforallt∈R, whichyields, t/mapsto→(Sh1)(t) belongsto AA(Xβ).\nThe next step consists of showing that R/parenleftbigPAP0(X)/parenrightbig⊂PAP0(Xβ). Obviously, Rh2∈\nBC(R,Xβ) (see Lemma 4.5). Using the fact that h2∈PAP0(X) and Eq. ( 3.10) it can be\neasilyshownthat( Rh2)∈PAP0(Xβ). Indeed,for r>0,\n1\n2r/integraldisplayr\n−r/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\n−∞T(t−s)h2(s)ds/vextenddouble/vextenddouble/vextenddouble/vextenddoubleβdt≤M(β)\n2r/integraldisplayr\n−r/integraldisplay∞\n0eδ\n2ss−β/vextenddouble/vextenddouble/vextenddouble/vextenddoubleh2(t−s)/vextenddouble/vextenddouble/vextenddouble/vextenddoubledsdt\n≤M(β)/integraldisplay∞\n0eδ\n2ss−β/parenleftigg1\n2r/integraldisplayr\n−r/vextenddouble/vextenddouble/vextenddouble/vextenddoubleh2(t−s)/vextenddouble/vextenddouble/vextenddouble/vextenddoubledt/parenrightigg\nds.\nUsingthe factthat PAP0(X)istranslation-invariantit followsthat\nlim\nr→∞1\n2r/integraldisplayr\n−r/vextenddouble/vextenddouble/vextenddouble/vextenddoubleh2(t−s)/vextenddouble/vextenddouble/vextenddouble/vextenddoubledt=0,\nast/mapsto→h2(t−s)∈PAP0(X)forevery s∈R.\nOne completes the proof by using the Lebesgue Dominated Conv ergenceTheorem. In\nsummary,( Rh2)∈PAP0(Xβ),whichcompletestheproof.\n/square\nTheorem 4.7. Suppose assumptions (H.1)—(H.5) hold, then Eq. ( 4.2) has at least one\npseudo-almostautomorphicmildsolution\nProof.LetBβ={u∈PAA(Xβ) :/bardblu/bardblβ≤L}. Using the proofof Lemma 4.5it followsthat\nBβisa convexandclosedset. NowusingLemma 4.6it followsthat S(Bβ)⊂PAA(Xβ).\nNowforall u∈Bβ,\n/vextenddouble/vextenddouble/vextenddoubleSu(t)/vextenddouble/vextenddouble/vextenddoubleβ≤/integraldisplayt\n−∞/vextenddouble/vextenddouble/vextenddoubleT(t−s)[Bu(s)+g(s)]/vextenddouble/vextenddouble/vextenddoubleβds\n≤/integraldisplayt\n−∞M(β)e−δ\n2(t−s)(t−s)−β/bracketleftig\n/bardblBu(s)/bardbl+/bardblf(s,u(s))/bardbl/bracketrightig\nds\n≤/integraldisplayt\n−∞M(β)e−δ\n2(t−s)(t−s)−β/bracketleftig\nC0/bardblu(s)/bardblβ+/bardblf(s,u(s))/bardbl/bracketrightig\nds\n≤d(β)/parenleftigL\n2d(β)+L\n2d(β)/parenrightig\n=L\nforallt∈R, andhence Su∈Bβ.12 TOKADIAGANA\nTo completetheproof,we haveto provethefollowing:\na) ThatV={Su(t) :u∈Bβ}isa relativelycompactsubsetof Xβforeacht∈R;\nb) ThatW={Su:u∈Bβ}⊂BC(R,Xβ)isequi-continuous.\nTo showa),fix t∈Randconsideranarbitrary ε>0.\nNow\n(Sεu)(t) :=/integraldisplayt−ε\n−∞T(t−s)F(s,u(s))ds,u∈Bβ\n=T(ε)/integraldisplayt−ε\n−∞T(t−ε−s)F(s,u(s))ds,u∈Bβ\n=T(ε)(Su)(t−ε),u∈Bβ\nandhence Vε:={Sεu(t) :u∈Bβ}is relativelycompactin Xβas the evolutionfamily T(ε)\niscompactbyassumption.\nNow\n/vextenddouble/vextenddouble/vextenddoubleSu(t)−T(ε)/integraldisplayt−ε\n−∞T(t−ε−s)F(s,u(s))ds/vextenddouble/vextenddouble/vextenddoubleβ\n≤/integraldisplayt\nt−ε/bardblT(t−s)F(s,u(s))/bardblβds\n≤M(β)/integraldisplayt\nt−εe−δ\n2(t−s)(t−s)−β/bardblF(s,u(s))/bardblds\n≤M(β)/integraldisplayt\nt−ε(t−s)−βe−δ\n2(t−s)/vextenddouble/vextenddouble/vextenddoubleg(s)/vextenddouble/vextenddouble/vextenddoubleds+M(β)/integraldisplayt\nt−ε(t−s)−βe−δ\n2(t−s)/vextenddouble/vextenddouble/vextenddoubleBu(s)/vextenddouble/vextenddouble/vextenddoubleds\n≤M(β)/integraldisplayt\nt−ε(t−s)−βe−δ\n2(t−s)/vextenddouble/vextenddouble/vextenddoublef(s,u(s))/vextenddouble/vextenddouble/vextenddoubleds+M(β)C0/bardblu/bardblβ,∞/integraldisplayt\nt−ε(t−s)−βe−δ\n2(t−s)ds\n≤M(β)L/parenleftig\nd−1(β)+C0/parenrightig/integraldisplayt\nt−ε(t−s)−αds\n=M(β)L/parenleftig\nd−1(β)+C0/parenrightig\nε1−β(1−β)−1,\nandhencetheset V:={Su(t) :u∈Bβ}⊂Xβisrelativelycompact.\nTheproofforb)followsalongthesamelinesasinLi etal.[38,Theorem31]andhence\nisomitted.\nThe rest of the proof slightly follows along the same lines as in Diagana [ 18]. In-\ndeed, since Bβis a closed convex subset of PAA(Xβ) and that S(Bβ)⊂Bβ, it follows that\ncoS(Bβ)⊂Bβ.Consequently,\nS(coS(Bβ))⊂S(Bβ)⊂coS(Bβ).\nFurther, it is not hard to see that {u(t) :u∈coS(Bβ)}is relatively compact in Xβfor\neach fixed t∈Rand that functions in coS(Bβ) are equi-continuouson R. Using Arzel` a-\nAscoli theorem,we deducethat the restrictionof coS(Bβ) to anycompactsubset IofRis\nrelativelycompactin C(I,Xβ).\nIn summary, S:coS(Bβ)/mapsto→coS(Bβ) is continuousand compact. Using the Schauder\nfixed point it follows that Shas a fixed-point, which obviously is a pseudo-almost auto-\nmorphicmildsolutionto Eq. ( 4.2).\n/squareEXISTENCE OF PSEUDO-ALMOST AUTOMORPHIC SOLUTIONS 13\nFixα∈[1\n2,1). In orderto study Eq. ( 1.6),we letβ=1\n2andsuppose that the following\nadditionalassumptionholds:\n(H.6) Thereexistsa function ρ∈L1(R,(0,∞))with/bardblρ/bardblL1(R,(0,∞))≤1\n2d(1\n2)suchthat\n/vextenddouble/vextenddouble/vextenddouble/vextenddoubleC(t)ϕ/vextenddouble/vextenddouble/vextenddouble/vextenddoubleE≤ρ(t)/vextenddouble/vextenddouble/vextenddoubleϕ/vextenddouble/vextenddouble/vextenddoubleE1\n2\nforallϕ∈E1\n2andt∈R.\nCorollary 4.8. Under assumptions (H.1)–(H.2)–(H.4)–(H.5)–(H.6) , then Eq. ( 1.6) (and\nhenceEq. ( 1.5)andEq. ( 1.2))hasatleast onepseudo-almostautomorphicmildsolution .\nProof.It suffices to show thatAandBsatisfy similar assumptions as (H.1)–(H.2)–(H.3)\nandthatFsatisfies similarassumptionsas(H.4)–(H.5).\nStep 1. Assumption (H.6) yields Bsatisfies similar assumption as (H.3), where Bis\ndefinedby\nBϕ(t) :=/integraldisplayt\n−∞C(t−s)ϕ(s)ds.\nIndeed, since the function ρis integrable, it is clear that the operator Bbelong to\nB(BC(R,E1\n2),E) with/bardblB/bardblB(BC(R,E1\n2),E)≤/bardblρ/bardblL1(R,(0,∞)). In fact, we take C0=/bardblρ/bardblL1(R,(0,∞)).\nThe fact the function t/mapsto→Bϕ(t) is pseudo-almost automorphic for any ϕ∈PAA(E1\n2) is\nguaranteedbyRemark 4.1. However,forthesakeofclarity,wewillshowit. Indeed,wr ite\nϕ=ϕ1+ϕ2, whereϕ1∈AA(E1\n2) andϕ2∈PAP0(E1\n2). Using the fact that the function\nt/mapsto→ϕ1(t) belongs to AA(E1\n2), for every sequence of real numbers ( τ′\nn)n∈Nthere exist a\nsubsequence( τn)n∈Nandafunction ψ1suchthat\nψ1(t) :=lim\nn→∞ϕ1(t+τn)\niswelldefinedforeach t∈R, and\nlim\nn→∞ψ1(t−τn)=ϕ1(t)\nforeacht∈R.\nNow\nBϕ1(t+τn)−Bψ1(t)=/integraldisplayt+τn\n−∞C(t+τn−s)ϕ1(s)ds−/integraldisplayt\n−∞C(t−s)ψ1(s)ds\n=/integraldisplayt\n−∞C(t−s)ϕ1(s+τn)ds−/integraldisplayt\n−∞C(t−s)ψ1(s)ds\n=/integraldisplayt\n−∞C(t−s)/parenleftig\nϕ1(s+τn)−ψ1(s)/parenrightig\nds\nandhence\n/vextenddouble/vextenddouble/vextenddouble/vextenddoubleBϕ1(t+τn)−Bψ1(t)/vextenddouble/vextenddouble/vextenddouble/vextenddoubleE≤/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\n−∞C(t−s)/parenleftig\nϕ1(s+τn)−ψ1(s)/parenrightig\nds/vextenddouble/vextenddouble/vextenddouble/vextenddoubleE\n≤/integraldisplayt\n−∞/vextenddouble/vextenddouble/vextenddouble/vextenddoubleC(t−s)/parenleftig\nϕ1(s+τn)−ψ1(s)/parenrightig/vextenddouble/vextenddouble/vextenddouble/vextenddoubleEds\n≤/integraldisplayt\n−∞ρ(t−s)/vextenddouble/vextenddouble/vextenddouble/vextenddoubleϕ1(s+τn)−ψ1(s)/vextenddouble/vextenddouble/vextenddouble/vextenddoubleE1\n2ds14 TOKADIAGANA\nwhichbyLebesgueDominatedConvergenceTheoremyields\nlim\nn→∞/vextenddouble/vextenddouble/vextenddouble/vextenddoubleBϕ1(t+τn)−Bψ1(t)/vextenddouble/vextenddouble/vextenddouble/vextenddoubleE=0.\nUsingsimilar arguments,weobtain\nlim\nn→∞/vextenddouble/vextenddouble/vextenddouble/vextenddoubleBψ1(t−τn)−Bφ1(t)/vextenddouble/vextenddouble/vextenddouble/vextenddoubleE=0.\nForr>0,\n1\n2r/integraldisplayr\n−r/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\n−∞C(t−s)ϕ2(s)ds/vextenddouble/vextenddouble/vextenddouble/vextenddoubleEdt≤1\n2r/integraldisplayr\n−r/integraldisplay∞\n0ρ(s)/vextenddouble/vextenddouble/vextenddouble/vextenddoubleϕ2(t−s)/vextenddouble/vextenddouble/vextenddouble/vextenddoubleE1\n2dsdt\n≤/integraldisplay∞\n0ρ(s)1\n2r/integraldisplayr\n−r/vextenddouble/vextenddouble/vextenddouble/vextenddoubleϕ2(t−s)/vextenddouble/vextenddouble/vextenddouble/vextenddoubleE1\n2dtds.\nNow\nlim\nr→∞1\n2r/integraldisplayr\n−r/vextenddouble/vextenddouble/vextenddouble/vextenddoubleϕ2(t−s)/vextenddouble/vextenddouble/vextenddouble/vextenddoubleE1\n2dt=0,\nast/mapsto→ϕ2(t−s)∈PAP0(E1\n2) forevery s∈R.\nTherefore,\nlim\nr→∞1\n2r/integraldisplayr\n−r/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\n−∞C(t−s)ϕ2(s)ds/vextenddouble/vextenddouble/vextenddouble/vextenddoubleEdt=0\nbyusingtheLebesgueDominatedConvergenceTheorem.\nStep 2. Clearly, the operator Asatisfies similar assumptions as (H.1)–(H.2) in the space\nE1\n2. Indeed,forall ϕ∈D(A),we have\nAϕ=∞/summationdisplay\nn=1AnPnϕ,\nwhere\nPn:=En0\n0EnandAn:=0 1\n−λn−λα\nn,n≥1.\nThecharacteristicequationfor Anisgivenby\nρ2+2γλα\nnρ+λn=0,\nfromwhichwe obtainitseigenvaluesgivenby\nρn\n1=λα\nn/parenleftig\n−γ+/radicalig\nγ2−λ1−2αn/parenrightig\nandλn\n2=λα\nn/parenleftig\n−γ−/radicalig\nγ2−λ1−2αn/parenrightig\n,\nandhenceσ(An)=/braceleftig\nρn\n1,ρn\n2/bracerightig\n.\nUsingEq. ( 1.4)it followsthatthereexists ω>0 suchthatρ(A)containsthehalfplane\nSω:=/braceleftig\nλ∈C:ℜeλ≥ω/bracerightig\n.\nNowsinceρn\n1andρn\n2aredistinctandthateachofthemisofmultiplicityone,the nAnis\ndiagonalizable. Further,it isnot di fficultto see thatAn=K−1\nnJnKn, whereJn,KnandK−1\nn\narerespectivelygivenby\nJn=ρn\n10\n0ρn\n2,Kn=1 1\nρn\n1ρn\n2,\nandEXISTENCE OF PSEUDO-ALMOST AUTOMORPHIC SOLUTIONS 15\nK−1\nn=1\nρn\n1−ρn\n2−ρn\n21\nρn\n1−1.\nForλ∈Sωandϕ∈E1\n2,onehas\nR(λ,A)ϕ=∞/summationdisplay\nn=1(λ−An)−1Pnϕ\n=∞/summationdisplay\nn=1Kn(λ−Jn)−1K−1\nnPnϕ.\nHence,\n/vextenddouble/vextenddouble/vextenddouble/vextenddoubleR(λ,A)ϕ/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nE1\n2≤∞/summationdisplay\nn=1/vextenddouble/vextenddouble/vextenddouble/vextenddoubleKn(λ−Jn)−1K−1\nn/vextenddouble/vextenddouble/vextenddouble/vextenddouble2/vextenddouble/vextenddouble/vextenddouble/vextenddoublePnϕ/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nE1\n2\n≤∞/summationdisplay\nn=1/vextenddouble/vextenddouble/vextenddouble/vextenddoubleKn/vextenddouble/vextenddouble/vextenddouble/vextenddouble2/vextenddouble/vextenddouble/vextenddouble/vextenddouble(λ−Jn)−1/vextenddouble/vextenddouble/vextenddouble/vextenddouble2/vextenddouble/vextenddouble/vextenddouble/vextenddoubleK−1\nn/vextenddouble/vextenddouble/vextenddouble/vextenddouble2/vextenddouble/vextenddouble/vextenddouble/vextenddoublePnϕ/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nE1\n2.\nIt iseasytosee that thereexist twoconstants C1,C2>0suchthat\n/bardblKn/bardbl≤C1|ρn\n1(t)|,/bardblK−1\nn/bardbl≤C2\n|ρn\n1|foralln≥1.\nNow\n/bardbl(λ−Jn)−1/bardbl2=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble1\nλ−ρn\n10\n01\nλ−ρn\n2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n≤1\n|λ−ρn\n1|2+1\n|λ−ρn\n2|2.\nDefinethefunction\nΘ(λ) :=|λ|\n|λ−ρn\n1(t)|.\nIt isclearthatΘiscontinuousandboundedon Sω. Ifwe take\nC3=sup/braceleftigg|λ|\n|λ−λn\nk|:λ∈Sω,n≥1;k=1,2/bracerightigg\nit followsthat\n/bardbl(λ−Jn)−1/bardbl≤C3\n|λ|, λ∈Sω.\nTherefore,onecanfinda constant K≥1such\n/bardblR(λ,A)/bardblB(E1\n2)≤K\n|λ|, λ∈Sω,\nandhencetheoperator AissectorialonE1\n2.\nSinceAissectorialinE1\n2,thenitgeneratesananalyticsemigroup( T(τ))τ≥0:=(eτA)τ≥0\nonE1\n2givenby\neτAϕ=∞/summationdisplay\nn=0K−1\nnPneτJnPnKnPnϕ.16 TOKADIAGANA\nFirst of all, notethat the semigroup( T(τ))τ≥0ishyperbolicas σ(A)∩iR=∅. Inorder\nwords,Asatisfies anassumptionsimilarto (H.1).\nSecondly,usingthe factthat Aisan operatorof compactresolventit followsthat T(τ)\niscompactfor τ>0. Ontheotherhand,we have\n/bardbleτAϕ/bardblE1\n2=∞/summationdisplay\nn=0/bardblK−1\nnPn/bardbl/bardbleτJnPn/bardbl/bardblKnPn/bardbl/bardblPnϕ/bardblE1\n2,\nwithforeach ϕ=/parenleftiggϕ1\nϕ2/parenrightigg\n∈E1\n2,\n/bardbleτJnPnϕ/bardbl2\nE1\n2=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleeρn\n1τEn0\n0eρn\n2τEnϕ1\nϕ2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nE1\n2\n≤ /bardbleρn\n1τEnϕ1/bardbl2\n1\n2+/bardbleρn\n2τEnϕ2/bardbl2\n≤eℜe(ρn\n1)τ/bardblϕ/bardbl2\nE1\n2.\nUsingEq. ( 1.4)it followsthereexists N′≥1suchthat\n/bardblT(τ)/bardblB(E1\n2)≤N′e−δ0τ, τ≥0,\nand henceT(τ) is exponentiallystable, that is, Asatisfies an assumption similar to (H.2)\ninE1\n2.\nStep 3. Thefactthat Fsatisfiessimilar assumptionsas(H.4)and(H.5)isclear.\n/square\n5. Example\nInthissection,wetake α=β=1\n2. LetΩ⊂RNbeanopenboundedsetwithsu fficiently\nsmooth boundary ∂Ωand letH=L2(Ω) be the Hilbert space of all measurable functions\nϕ:Ω/mapsto→Csuchthat\n/bardblϕ/bardblL2(Ω)=/parenleftigg/integraldisplay\nΩ|ϕ(x)|2dx/parenrightigg1/2\n<∞.\nHere, we studytheexistenceof pseudo-almostautomorphics olutionsϕ(t,x) to a struc-\nturally damped plate-like system given by (see also Chen and Triggiani [ 13], Schnaubelt\nandVeraar[ 58],Triggiani[ 59]),\n∂2ϕ\n∂t2(t,x)−2γ∆∂ϕ\n∂t(t,x)+∆2ϕ(t,x)=/integraldisplayt\n−∞b(t−s)ϕ(s,x)ds+f(t,ϕ(t,x)) (5.1)\n+ηϕ(t,x),(t,x)∈R×Ω\n∆ϕ(t,x)=ϕ(t,x)=0,(t,x)∈R×∂Ω, (5.2)\nwhereγ,η >0 are constants, b:R/mapsto→[0,∞) is a measurable function, the function\nf:R×L2(Ω)/mapsto→L2(Ω) is pseudo-almost automorphic in t∈Runiformly in the second\nvariable,and∆standsfortheusualLaplaceoperatorin thespacevariable x.\nSetting\nAϕ=∆2ϕforallϕ∈D(A)=D(∆2)=/braceleftig\nϕ∈H4(Ω) :∆ϕ=ϕ=0 on∂Ω/bracerightig\n,\nBϕ=A1\n2ϕ=−∆ϕ,∀ϕ∈D(B)=H1\n0(Ω)∩H2(Ω),EXISTENCE OF PSEUDO-ALMOST AUTOMORPHIC SOLUTIONS 17\nC(t)ϕ=b(t)ϕforallϕ∈D(C(t))=D(∆),\nandf:R×/parenleftig\nH1\n0(Ω)∩H2(Ω)/parenrightig\n/mapsto→L2(Ω), one can easily see that Eq. ( 1.2) is exactly the\nstructurallydampedplate-likesystemformulatedinEqs. ( 5.1)-(5.2).\nHereE1\n2=D(A1\n2)×L2(Ω)=/parenleftig\nH1\n0(Ω)∩H2(Ω)/parenrightig\n×L2(Ω)anditisequippedwiththeinner\nproductdefinedby\n/parenleftigg/parenleftiggϕ1\nϕ2/parenrightigg\n,ψ1\nψ2/parenrightigg\nE1\n2:=/integraldisplay\nΩ∆ϕ1∆ψ1dx+/integraldisplay\nΩϕ2ψ2dx\nforallϕ1,ψ1∈H1\n0(Ω)∩H2(Ω)andϕ2,ψ2∈L2(Ω).\nSimilarly, D(A1\n2)=H1\n0(Ω)∩H2(Ω)isequippedwith thenormdefinedby\n/bardblϕ/bardbl1\n2=/bardblA1\n2ϕ/bardblL2(Ω):=/parenleftig/integraldisplay\nΩ|∆ϕ|2dx/parenrightig1\n2\nforallϕ∈H1\n0(Ω)∩H2(Ω).\nClearly,−Aη=−(∆2+ηI) is a sectorial operator on L2(Ω) and let ( T(t))t≥0be the\nanalyticsemigroupassociatedwithit. It iswell-knowntha tthesemigroup T(t) isnotonly\ncompactfor t>0butalso isexponentiallystable as\n/bardblT(t)/bardbl≤e−ηt\nforallt≥0.\nUsingthefacttheLaplaceoperator ∆withdomain D(∆)=H2(Ω)∩H1\n0(Ω)isinvertible\ninL2(Ω)it followsthat\n/bardblϕ/bardblL2(Ω)=/bardbl∆−1∆ϕ/bardblL2(Ω)\n≤ /bardbl∆−1/bardblB(L2(Ω))./bardbl∆ϕ/bardblL2(Ω)\n=/bardbl∆−1/bardblB(L2(Ω))./bardblA1\n2ϕ/bardblL2(Ω)\n=/bardbl∆−1/bardblB(L2(Ω))./bardblϕ/bardbl1\n2\nforallϕ∈H2(Ω)∩H1\n0(Ω).\nIfb∈L1(R,(0,∞)),thenusingthe previousinequalityit followsthat\n/bardblC(t)Φ/bardblE=b(t)/bardblϕ/bardblL2(Ω)\n≤b(t)/bardbl∆−1/bardblB(L2(Ω))./bardblϕ/bardbl1\n2\n≤b(t)/bardbl∆−1/bardblB(L2(Ω))./bardblΦ/bardblE1\n2\nforallΦ=/parenleftiggϕ\nψ/parenrightigg\n∈E1\n2andt∈R.\nThissettingrequiresthefollowingassumptions,\n(H.7) Eq. 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Soc. 69(1978),\nno.2, pp. 255–260\nDepartmentof Mathematics ,HowardUniversity,2441 6thStreetN.W.,Washington ,D.C.20059, USA\nE-mail address :tdiagana@howard.edu"    },    {        "title": "2103.00461v1.Stability_for_an_inverse_source_problem_of_the_damped_biharmonic_plate_equation.pdf",        "content": "arXiv:2103.00461v1  [math.AP]  28 Feb 2021STABILITY FOR AN INVERSE SOURCE PROBLEM OF THE DAMPED\nBIHARMONIC PLATE EQUATION\nPEIJUN LI, XIAOHUA YAO, AND YUE ZHAO\nAbstract. This paper is concerned with the stability of the inverse sou rce problem for the damped\nbiharmonic plate equation in three dimensions. The stabili ty estimate consists of the Lipschitz type\ndata discrepancy and the high frequency tail of the source fu nction, where the latter decreases as\nthe upper bound of the frequency increases. The stability al so shows exponential dependence on the\nconstant damping coefficient. The analysis employs Carleman estimates and time decay estimates\nfor the damped plate wave equation to obtain an exact observa bility bound and depends on the\nstudy of the resonance-free region and an upper bound of the r esolvent of the biharmonic operator\nwith respect to the complex wavenumber.\n1.Introduction\nConsider the damped biharmonic plate equation in three dime nsions\n∆2upx,kq ´k2upx,kq ´ikσupx,kq “fpxq, x PR3, (1.1)\nwhereką0 is the wavenumber, σą0 is the damping coefficient, and fPL2pR3qis a assumed to be\na real-valued function with a compact support contained in BR“ txPR3:|x| ăRu, whereRą0 is\na constant. Let BBRbe the boundary of BR. Since the problem is formulated in the open domain,\nthe Sommerfeld radiation condition is imposed usually on uand ∆uto ensure the well-posedness of\nthe problem [17]. This paper is concerned with the inverse so urce problem of determining ffrom\nthe boundary measurements\nupx,kq,∇upx,kq,∆upx,kq,∇∆upx,kq, x P BBR\ncorresponding to the wavenumber kgiven in a finite interval.\nIn general, there is no uniqueness for the inverse source pro blems of the wave equations at a fixed\nfrequency [2,12]. Computationally, a more serious issue is the lack of stability, i.e., a small variation\nof the data might lead to a huge error in the reconstruction. H ence it is crucial to examine the\nstability of the inverse source problems. In [2], the author s initialized the study of the inverse source\nproblem for the Helmholtz equation by using multi-frequenc y data. Since then, it has become an\nactive research topic on the inverse source problems via mul tiple frequency data in order to over-\ncome the non-uniqueness issue and enhance the stability. Th e increasing stability was investigated\nfor the inverse source problems of various wave equations wh ich include the acoustic, elastic, and\nelectromagnetic wave equations [3–6,13,14] and the Helmho ltz equation with attenuation [8]. On\nthe other hand, it has generated sustained interest in the ma thematics community on the boundary\nvalue problems for higher-order elliptic operators [7]. Th e biharmonic operator, which can be en-\ncountered in models originating from elasticity for exampl e, appears as a natural candidate for such\na study [15,16]. Compared with the equations involving the s econd order differential operators, the\nmodel equations with the biharmonic operators are much less studied in the community of inverse\nproblems. We refer to [1,9–11,17] and the references cited t herein on the recovery of the lower-order\ncoefficients by using either the far-field pattern or the Diric hlet-to-Neumann map on the boundary.\n2000Mathematics Subject Classification. 35R30, 31B30.\nKey words and phrases. inverse source problem, the biharmonic operator, the dampe d biharmonic plate equation,\nstability.\n12 P. LI, X. YAO, AND Y. ZHAO\nIn a recent paper [12], the authors demonstrated the increas ing stability for the inverse source prob-\nlem of the biharmonic operator with a zeroth order perturbat ion by using multi-frequency near-field\ndata. The main ingredient of the analysis relies on the study of an eigenvalue problem for the bi-\nharmonic operator with the hinged boundary conditions. But the method is not applicable directly\nto handle the biharmonic operator with a damping coefficient.\nMotivated by [4,8], we use the Fourier transform in time to re duce the inverse source problem\ninto the identification of the initial data for the initial va lue problem of the damped biharmonic\nplate wave equation by lateral Cauchy data. The Carleman est imate is utilized to obtain an exact\nobservability bound for the source function in the framewor k of the initial value problem for the\ncorresponding wave equation, which connects the scatterin g data and the unknown source function\nby taking the inverse Fourier transform. An appropriate rat e of time decay for the damped plate\nwave equation is proved in order to justify the Fourier trans form. Then applying the results in [12]\non the resolvent of the biharmonic operator, we obtain a reso nance-free region of the data with\nrespect to the complex wavenumber and the bound of the analyt ic continuation of the data from the\ngiven data to the higher wavenumber data. By studying the dep endence of analytic continuation\nand of the exact observability bound for the damped plate wav e equation on the damping coefficient,\nwe show the exponential dependence of increasing stability on the damping constant. The stability\nestimate consists of the Lipschitz type of data discrepancy and the high wavenumber tail of the\nsource function. The latter decreases as the wavenumber of t he data increases, which implies that\nthe inverse problem is more stable when the higher wavenumbe r data is used. But the stability\ndeteriorates as the damping constant becomes larger. It sho uld be pointed out that due to the\nexistence of the damping coefficient, we can not obtain a secto rial resonance-free region for the data\nas that in [4,13]. Instead, we choose a rectangular resonanc e-free region as that in [14], which leads\nto a double logarithmic type of the high wavenumber tail for t he estimate.\nThis paper is organized as follows. In section 2, the direct s ource problem is discussed; the\nresolvent is introduced for the elliptic operator, and its r esonance-free region and upper bound are\nobtained. Section 3 is devoted to the stability analysis of t he inverse source problem by using multi-\nfrequency data. In appendix A, we usethe Carleman estimate t o derive an exact observability bound\nwith exponential dependence on the damping coefficient. In ap pendix B, we prove an appropriate\nrate of time decay for the damped plate wave equation to justi fy the Fourier transform.\n2.The direct source problem\nIn this section, we discuss the solution of the direct source problem and study the resolvent of the\nbiharmonic operator with a damping coefficient.\nTheorem 2.1. LetfPL2pR3qwith a compact support. Then there exists a unique solution uof\nSchwartz distribution to (1.1)for every ką0. Moreover, the solution satisfies\n|upx,kq| ďCpk,fqe´cpk,σq|x|\nas|x| Ñ 8,whereCpk,fqandcpk,σqare positive constants depending on k,fandk,σ, respectively.\nProof.Taking the Fourier transform of upx,kqformally with respect to the spatial variable x, we\ndefine\nu˚px,kq “ż\nR3eix¨ξ1\n|ξ|4´k2´ikσˆfpξqdξ, x PR3,\nwhere\nˆfpξq “1\np2πq3ż\nR3fpxqe´ix¨ξdx.\nIt follows from the Plancherel theorem that for each ką0 we have that u˚p¨,kq PH4pR3qand\nsatisfies the equation (1.1) in the sense of Schwartz distrib ution.AN INVERSE SOURCE PROBLEM 3\nDenote\nGpx,kq “ż\nR3eix¨ξ1\n|ξ|4´k2´ikσdξ.\nBy a direct calculation we can write u˚px,kqas\nu˚px,kq “ pG˚fqpxq “1\n2κ2ż\nR3´eiκ|x´y|\n4π|x´y|´e´κ|x´y|\n4π|x´y|¯\nfpyqdy, (2.1)\nwhereκ“ pk2`ikσq1\n4such that ℜκą0 andℑκą0. Since fhas a compact support, we obtain\nfrom (2.1) that the solution u˚px,kqsatisfies the estimate\n|u˚px,kq| ďCpk,fqe´cpk,σq|x|\nas|x| Ñ 8, whereCpk,fqandcpk,σqare positive constants dependingon k,fandk,σ, respectively.\nBydirectcalculations, wemayalsoshowthat ∇u˚and∆u˚havesimilarexponential decayestimates.\nNext is show the uniqueness. Let ˜ u˚px,kqbe another Schwartz distributional solution to (1.1).\nClearly we have\np∆2´k2´ikσqpu˚´˜u˚q “0.\nTaking the Fourier transform on both sides of the above equat ion yields\np|ξ|4´k2´ikσqp{u˚´˜˚uqpξq “0.\nNotice that for ką0 we have |ξ|4´k2´ikσ‰0 for all ξPR3. Taking the generalized inverse\nFourier transform gives u˚´˜u˚“0, which proves the uniqueness. /square\nTo study the resolvent we let\nu˚px,κq:“upx,kq, κ “ pk2`ikσq1\n4,\nwhereℜκą0 andℑκą0. By (1.1), u˚satisfies\n∆2u˚´κ4u˚“f.\nDenote by R“ tzPC:pδ,`8q ˆ p´ d,dquthe infinite rectangular slab, where δis any positive\nconstant and d!1. ForkPR, denote the resolvent\nRpkq:“ p∆2´k2´ikσq´1.\nThen we have Rpκq “ p∆2´κ4q´1. Hereafter, the notation aÀbstands for aďCb,whereCą0\nis a generic constant which may change step by step in the proo fs.\nLemma 2.2. For each kPRandρPC8\n0pBRqthe resolvent operator Rpkqis analytic and has the\nfollowing estimate:\n}ρRpkqρ}L2pBRqÑHjpBRqÀ |k|j\n2e2Rpσ`1q|k|1\n2, j “0,1,2,3,4.\nProof.It is clear to note that for a sufficiently small d, the set tpk2`ikσq1\n4:kPRubelongs to the\nfirst quadrant. Consequently, pk2`ikσq1\n4is analytic with respect to kPR. By [12, Theorem 2.1],\nthe resolvent Rpκqis analytic in Czt0uand the following estimate holds:\n}ρRpκqρ}L2pBRqÑHjpBRqÀ |κ|´2xκyjpe2Rpℑκq´`e2Rpℜκq´q, j “0,1,2,3,4,(2.2)\nwherex´:“maxt´x,0uand xκy “ p1` |κ|2q1{2. On the other hand, letting k“k1`ik2, we have\nfrom a direct calculation that\nk2`ikσ“k2\n1´k2\n2´k2σ` p2k1k2`k1σqi.\nIt is easy to see that if dis sufficiently small, which gives that |k2|is sufficiently small, there is a\npositive lower bound for |k2`ikσ|withkPRand then |κ| ącfor some positive constant c. The\nproof is completed by replacing κwith pk2`ikσq1\n4in (2.2). /square4 P. LI, X. YAO, AND Y. ZHAO\n3.The inverse source problem\nIn this section, we address the inverse source problem of the damped biharmonic plate equation\nand present an increasing stability estimate by using multi -frequency scattering data.\nDenote\n}upx,kq}2\nBBR:“ż\nBBR´\npk4`k2q|upx,kq|2`k2|∇upx,kq|2\n` pk2`1q|∆upx,kq|2` |∇∆upx,kq|2¯\ndspxq.\nThe following lemma provides a relation between the unknown source function and the boundary\nmeasurements. Hereafter, by Remark B.3, we assume that fPHnpBRqwhereně4.\nLemma 3.1. Letube the solution to the direct scattering problem (1.1). Then\n}f}2\nL2pBRqÀ2eCσ2ż`8\n0}upx,kq}2\nBBRdk.\nProof.Consider the initial value problem for the damped biharmoni c plate wave equation\n#\nB2\ntUpx,tq `∆2Upx,tq `σBtUpx,tq “0, px,tq PBRˆ p0,`8q,\nUpx,0q “0,BtUpx,0q “fpxq, x PBR.(3.1)\nWe define Upx,tq “0 whentă0 and denote UTpx,tq “Upx,tqχr0,Tsptqand\nxUTpx,kq “żT\n0Upx,tqeiktdt.\nBy the decay estimate (B.2) we have that Upx,tq PL2\ntp0,`8qand lim TÑ8UTpx,tq “Upx,tqin\nL2\ntpRquniformly for all xPR3. It follows from the Plancherel Theorem that xUTalso converges in\nL2\nkpRqto a function u˚px,kq PL2\nkpRquniformly for all xPR3, which implies that u˚px,kqis the\nFourier transform of Upx,tq.\nDenoteby x¨,¨yandStheusualscalarinnerproductof L2pR3qandthespaceofSchwartzfunctions,\nrespectively. We take u˚px,kqas a Schwartz distribution such that u˚px,kqpϕq “ xu˚,ϕyfor each\nϕPS. In what follows, we show that u˚px,kqsatisfies the equation (1.1) in the sense of Schwartz\ndistribution.\nFirst we multiply both sides of the wave equation (3.1) by a Sc hwartz function ϕand take inte-\ngration over R3. Using the wave equation (3.1) and the integration by parts w ith respect to the t\nvariable over r0,TsforTą0, we obtain\n0“żT\n0xB2\ntU`∆2U`σBtU,ϕyeiktdt\n“eikTxBtUpx,Tq,ϕy ´ikeikTxUpx,Tq,ϕy `σeikTxUpx,Tq,ϕy\n´ xBtUpx,0q,ϕy `AżT\n0p∆2U´k2U´ikσU qeiktdt,ϕE\n. (3.2)\nItfollowsfromthedecay estimate(B.2)that |BtUpx,tq|,|Upx,tq| À p1`tq´3\n4uniformlyforall xPR3,\nwhich give\nlim\nTÑ8eikTxBtUpx,Tq,ϕy “lim\nTÑ8ikeikTxUpx,Tq,ϕy “lim\nTÑ8σeikTxUpx,Tq,ϕy “0.AN INVERSE SOURCE PROBLEM 5\nOn the other hand, we have from the integration by parts that\nAżT\n0p∆2U´k2U´ikσU qeiktdt,ϕE\n“AżT\n0Udt,∆2ϕE\n`AżT\n0p´k2U´ikσU qeiktdt,ϕE\n. (3.3)\nSince lim TÑ`8xUTpx,kq “u˚px,kqinL2\nkpRquniformly for xPR3, we can choose a positive sequence\ntTnu8\nn“1such that lim nÑ8Tn“ `8and lim nÑ8yUTnpx,kq “u˚px,kqpointwisely for a.e. kPRand\nuniformly for all xPR3. Define a sequence of Schwartz distributions tDnu8\nn“1ĂS1as follows\nDnpϕq:“ xyUTn,ϕy, ϕ PS.\nSince lim nÑ8yUTnpx,kq “u˚px,kqfor a.e.kPRand uniformly for all xPR3, we have\nlim\nnÑ8Dnpϕq “ xu˚,ϕy.\nConsequently, replacing TbyTnin (3.3) and letting nÑ 8, we get\nlim\nnÑ8´@żTn\n0Udt,∆2ϕD\n`@żTn\n0p´k2U´ikσU qeiktdt,ϕD¯\n“u˚p∆2ϕq ´k2u˚pϕq ´ikσu ˚pϕq\n“ p∆2´k2´ikσqu˚pϕq,\nwhich further implies by (3.2) that\np∆2´k2´ikσqu˚pϕq “ xf,ϕy\nfor every ϕPS. Then u˚px,kqis a solution to the equation (1.1) as a Schwartz distributio n.\nFurthermore, it follows from the uniqueness of the direct pr oblem that we obtain u˚px,kq “upx,kq,\nwhich gives that upx,kqis the Fourier transform of Upx,tq.\nBy Theorem B.1, we have the estimates\n|B2\ntU|,|BtU|,|Bt∇U|,|Bt∆U|,|∆U|,|∇∆U| À p1`tq´3\n4.\nMoreover, they are continuous and belong to L2\ntpRquniformly for all xPR3. Similarly, we may show\nthat\nyB2\ntU“ ´k2u,yBtU“iku, {Bt∇U“ik∇u,\n{Bt∆U“ik∆u,y∆U“∆u,{∇∆U“∇∆u.\nIt follows from Plancherel’s theorem thatż`8\n0´\n|B2\ntU|2` |BtU|2` |Bt∇U|2` |Bt∆U|2` |∆U|2` |∇∆U|2¯\ndt\n“ż`8\n´8´\n|k2u|2` |ku|2` |k∇u|2` |k∆u|2` |∆u|2` |∇∆u|2¯\ndk. (3.4)\nBy (3.4) and the exact observability bounds (A.1), we obtain\n}f}2\nL2pBRqÀeCσ2ż`8\n´8ż\nBBR´\npk4`k2q|upx,kq|2`k2|∇upx,kq|2\n` pk2`1q|∆upx,kq|2` |∇∆upx,kq|2¯\ndspxqdk\nÀeCσ2ż8\n´8}upx,kq}2\nBBRdk.6 P. LI, X. YAO, AND Y. ZHAO\nSincefpxqis real-valued, we have upx,kq “upx,´kqforkPRand then\nż8\n´8}upx,kq}2\nBBRdk“2ż8\n0}upx,kq}2\nBBRdk,\nwhich completes the proof. /square\nLetδbe a positive constant and define\nIpkq “żk\nδ}upx,ωq}2\nBBRdspxqdω.\nThe following lemma gives a link between the values of an anal ytical function for small and large\narguments (cf. [14, Lemma A.1]).\nLemma 3.2. Letppzqbe analytic in the infinite rectangular slab\nR“ tzPC:pδ,`8q ˆ p´ d,dqu,\nwhereδis a positive constant, and continuous in Rsatisfying#\n|ppzq| ďǫ1, z P pδ,Ks,\n|ppzq| ďM, z PR,\nwhereδ,K,ǫ1andMare positive constants. Then there exists a function µpzqwithzP pK,`8q\nsatisfying\nµpzq ě64ad\n3π2pa2`4d2qeπ\n2dpa\n2´zq,\nwherea“K´δ, such that\n|ppzq| ďMǫµpzq@zP pK,`8q.\nLemma 3.3. Letfbe a real-valued function and }f}L2pBRqďQ. Then there exist positive constants\ndandδ,Ksatisfying 0ăδăK, which do not depend on fandQ, such that\n|Ipkq| ÀQ2e4Rpσ`2qκǫ2µpkq\n1 @kP pK,`8q\nand\nǫ2\n1“żK\nδż\nBBR}upx,kq}2\nBBRdspxqdk, µ pkq ě64ad\n3π2pa2`4d2qeπ\n2dpa\n2´kq,\nwherea“K´δ.\nProof.Let\nI1pkq “żk\nδż\nBBR´\npω4`ω2qupx,ωqupx,´ωq `ω2∇upx,ωq ¨∇upx,´ωq\n` pω2`1q∆upx,ωq∆upx,´ωq `∇∆upx,ωq ¨∇∆upx,´ωq¯\ndspxqdω,\nwherekPR. Following similar arguments as those in the proof of Lemma 2 .2, we may show that\nRp´kqis also analytic for kPR. Sincefis real-valued, we have upx,kq “upx,´kqforkPR, which\ngives\nI1pkq “Ipkq, k ą0.\nIt follows from Lemma 2.2 that\n|I1pkq| ÀQ2eCσ2e4Rpσ`1q|k|, k PR,\nwhich gives\ne´4Rpσ`2q|k||I1pkq| ÀQ2eCσ2, k PR.AN INVERSE SOURCE PROBLEM 7\nAn application of Lemma 3.2 leads to\nˇˇe´4Rpσ`2q|k|IpkqˇˇÀQ2ǫ2µpkq@kP pK,`8q,\nwhere\nµpkq ě64ad\n3π2pa2`4d2qeπ\n2dpa\n2´kq,\nwhich completes the proof. /square\nHere we state a simple uniqueness result for the inverse sour ce problem.\nTheorem 3.4. LetfPL2pBRqandIĂR`be an open interval. Then the source function fcan\nbe uniquely determined by the multi-frequency Cauchy data tupx,kq,∇upx,kq,∆upx,kq,∇∆upx,kq:\nxP BBR,kPIu.\nProof.Letupx,kq “∇upx,kq “∆upx,kq “∇∆upx,kq “0 for all xP BBRandkPI. It suffices to\nprove that fpxq “0. By Lemma 2.2, upx,kqis analytic in the infinite slab Rfor anyδą0, which\nimplies that upx,kq “∆upx,kq “0 for all kPR`. We conclude from Lemma 3.1 that f“0./square\nThe following result concerns the estimate of upx,kqfor high wavenumbers.\nLemma 3.5. LetfPHnpBRqand }f}HnpBRqďQ. Then the following estimate holds:\nż8\ns}upx,kq}2\nBBRdkÀ1\nsn´3}f}2\nHnpBRq.\nProof.Recall the identity\nż8\ns}upx,kq}2\nBBRdk“ż8\nsż\nBBR´\npk4`k2q|upx,kq|2`k2|∇upx,kq|2\n` pk2`1q|∆upx,kq|2` |∇∆upx,kq|2¯\ndspxqdk. (3.5)\nUsing the decomposition\nRpκq “ p∆2´κ4q´1“1\n2κ2“\np´∆´κ2q´1´ p´∆`κ2q´1‰\n,\nwe obtain\nupxq “ż\nBR1\n2κ2´eiκ|x´y|\n4π|x´y|´e´κ|x´y|\n4π|x´y|¯\nfpyqdy, x P BBR.\nFor instance, we consider one of the integrals on the right-h and side of (3.5)\nJ:“ż8\nsk4|upx,kq|dk\n“ż8\nsk4ˇˇˇż\nBR1\n2κ2´eiκ|x´y|\n4π|x´y|´e´κ|x´y|\n4π|x´y|¯\nfpyqdyˇˇˇ2\ndk.\nUsing the spherical coordinates r“ |x´y|originated at y, we have\nJ“1\n8πż8\nsż\nBBRk2ˇˇˇż2π\n0dθżπ\n0sinϕdϕż8\n0peiκr´e´κrqfrdrˇˇˇ2\ndspxqdk.\nBy the integration by parts and noting xP BBRand supp fĂBˆRĂBRfor some ˆRăR, we obtain\nJ“1\n4πż8\nsż\nBBRk2ˇˇˇż2π\n0dθżπ\n0sinϕdϕż2R\nR´ˆR´eiκr\npiκqn´e´κr\np´κqn¯Bnpfrq\nBrndrˇˇˇ2\ndspxqdk.8 P. LI, X. YAO, AND Y. ZHAO\nSincexP BBRand |κ| ěk1{2forką0, we get from direction calculations that\nJÀ }f}2\nHnpBRqż8\nsk2´ndkÀ1\nsn´3}f}2\nHnpBRq.\nTheotherintegrals ontheright-handsideof (3.5)can beest imated similarly. Thedetails areomitted\nfor brevity. /square\nDefine a real-valued function space\nCQ“ tfPHnpBRq:ně4,}f}HnpBRqďQ,suppfĂBˆRĂBR, f:BRÑRu,\nwhereˆRăR. Now we are in the position to present the main result of this p aper.\nTheorem 3.6. Letupx,κqbe the solution of the scattering problem (1.1)corresponding to the source\nfPCQ. Then for ǫsufficiently small, the following estimate holds:\n}f}2\nL2pBRqÀeCσ2´\nǫ2`Q2\nK1\n2pn´3qpln|lnǫ|q1\n2pn´3q¯\n, (3.6)\nwhere\nǫ:“żK\n0}upx,kq}2\nBBRdk“żδ\n0}upx,kq}2\nBBRdk`ǫ2\n1.\nProof.We can assume that ǫďe´1, otherwise the estimate is obvious.\nFirst, we link the data Ipkqfor large wavenumber ksatisfying kďLwith the given data ǫ1of\nsmall wavenumber by using the analytic continuation in Lemm a 3.3, where Lis some large positive\ninteger to be determined later. It follows from Lemma 3.3 tha t\nIpkq ÀQ2ec|κ|ǫµpκq\n1\nÀQ2exptcκ´c2a\na2`c3ec1pa\n2´κq|lnǫ1|u\nÀQ2expt´c2a\na2`c3ec1pa\n2´κq|lnǫ1|p1´c4κpa2`c3q\naec1pκ´a\n2q|lnǫ1|´1qu\nÀQ2expt´c2a\na2`c3ec1pa\n2´Lq|lnǫ1|p1´c4Lpa2`c3q\naec1pL´a\n2q|lnǫ1|´1qu\nÀQ2expt´b0e´c1L|lnǫ1|p1´b1Lec1L|lnǫ1|´1qu,\nwherec,ci,i“1,2 andb0,b1are constants. Let\nL“#”\n1\n2c1ln|lnǫ1|ı\n, k ď1\n2c1ln|lnǫ1|,\nk, k ą1\n2c1ln|lnǫ1|.\nIfKď1\n2c1ln|lnǫ1|, we obtain for sufficiently small ǫ1that\nIpkq ÀQ2expt´b0e´c1L|lnǫ1|p1´b1Lec1L|lnǫ1|´1qu\nÀQ2expt´1\n2b0e´c1L|lnǫ1|u.\nNotinge´xďp2n`3q!\nx2n`3forxą0, we have\nIpLq ÀQ2ep2n`3qc1L|lnǫ1|´p2n`3q.AN INVERSE SOURCE PROBLEM 9\nTakingL“1\n2c1ln|lnǫ1|, combining the above estimates, Lemma 3.1 and Lemma 3.5, we g et\n}f}2\nL2pBRqÀeCσ2´\nǫ2`IpLq `ż8\nLż\nBBR}upx,kq}2\nBBRdk¯\nÀeCσ2´\nǫ2`Q2ep2n`3qc1L|lnǫ1|´p2n`3q`Q2\nLn´3¯\nÀeCσ2´\nǫ2`Q2´\n|lnǫ1|2n`3\n2|lnǫ1|´p2n`3q` pln|lnǫ1|q3´n¯ ¯\nÀeCσ2´\nǫ2`Q2´\n|lnǫ1|´2n`3\n2` pln|lnǫ1|q3´n¯ ¯\nÀeCσ2´\nǫ2`Q2pln|lnǫ1|q3´n¯\nÀeCσ2´\nǫ2`Q2\nK1\n2pn´3qpln|lnǫ1|q1\n2pn´3q¯\nÀeCσ2´\nǫ2`Q2\nK1\n2pn´3qpln|lnǫ|q1\n2pn´3q¯\n,\nwhere we have used |lnǫ1|1{2ěln|lnǫ1|for sufficiently small ǫ1and ln |lnǫ1| ěln|lnǫ|.\nIfKą1\n2c1ln|lnǫ1|, we have from Lemma 3.5 that\n}f}2\nL2pBRqÀeCσ2´\nǫ2`ż8\nKż\nBBR}upx,kq}2\nBBRdk¯\nÀeCσ2´\nǫ2`Q2\nKn´3¯\nÀeCσ2´\nǫ2`Q2\nK1\n2pn´3qpln|lnǫ|q1\n2pn´3q¯\n,\nwhich completes the proof. /square\nIt can be observed that for a fixed damping coefficient σ, the stability (3.6) consists of two parts:\nthe data discrepancy and the high frequency tail. The former is of the Lipschitz type. The latter\ndecreases as Kincreases which makes the problem have an almost Lipschitz s tability. But the\nstability deteriorates exponentially as the damping coeffic ientσincreases.\nAppendix A.An exact observability bound\nConsider the initial value problem for the damped biharmoni c plate wave equation\n#\nB2\ntUpx,tq `∆2Upx,tq `σBtUpx,tq “0, px,tq PBRˆ p0,`8q,\nUpx,0q “0,BtUpx,0q “fpxq, x PBR.(A.1)\nThe following theorem presents an exact observability boun d for the above equation. The proof\nfollows closely from that in [8, Theorem 3.1].\nTheorem A.1. Let the observation time 4p2R`1q ăTă5p2R`1q. Then there exists a constant\nCdepending on the domain BRsuch that\n}f}2\nL2pBRqďCeCσ2`\n}B2\ntU}2\nL2pBBRˆp0,Tqq` }BtU}2\nL2pBBRˆp0,Tqq` }Bt∇U}2\nL2pBBRˆp0,Tqq\n` }Bt∆U}2\nL2pBBRˆp0,Tqq` }∆U}2\nL2pBBRˆp0,Tqq` }∇∆U}2\nL2pBBRˆp0,Tqq˘\n.(A.2)10 P. LI, X. YAO, AND Y. ZHAO\nBefore showing the proof, we introduce the energies\nEptq “1\n2ż\nΩ`\n|BtUpx,tq|2` |∆Upx,tq|2` |Upx,tq|2˘\ndx,\nE0ptq “1\n2ż\nΩ`\n|BtUpx,tq|2` |∆Upx,tq|2˘\ndx,\nand denote\nF2“ż\nBΩˆpt1,t2q`\n|B2\ntUpx,tq|2` |BtUpx,tq|2` |Bt∇Upx,tq|2\n` |Bt∆Upx,tq|2` |∆Upx,tq|2` |∇∆Upx,tq|2˘\ndspxqdt.\nLemma A.2. LetUbe a solution of the damped biharmonic plate wave equation (A.1)with the\ninitial value fPH1pBRq,suppf ĂBR. Let0ďt1ăt2ďTand1ď2σ. Then the following\nestimates holds:\nEpt2q ďe4pt2´t1q2p2Ept1q `F2q, (A.3)\nEpt2q ďep2σ`4pt2´t1qqpt2´t1qpEpt2q `F2q. (A.4)\nProof.Multiplying both sides of (A.1) by pBtUqeθtand integrating over Ω ˆ pt1,t2qgive\nż\nΩˆppt1,t2q´1\n2BtpBtUq2`∆2UBtU`σpBtUq2¯\neθtdxdt“0.\nUsing the integration ∆2UBtUby parts over Ω and noting ∆ UBtp∆Uq “1\n2Bt|∆U|2, we obtain\nżt2\nt1pBtE0ptqqeθtdt`ż\nΩˆpt1,t2qσpBtUq2eθtdxdt\n`ż\nBΩˆpt1,t2qpBνp∆UqBtU´∆UBtpBνUqqeθtdspxqdt“0.\nHence,\nE0pt2qeθt2´E0pt1qeθt1“ż\nΩˆpt1,t2q´θ\n2ppBtUq2` |∆U|2q ´σpBtUq2¯\neθtdxdt\n´ż\nBΩˆpt1,t2qpBνp∆UqBtU´∆UBtpBνUqqeθtdspxqdt“0.\nLettingθ“0, using Schwartz’s inequality, and noting σą0, we get\nE0pt2q ďE0pt1q `ż\nΩˆpt1,t2qp´σqpBtUq2dxdt\n`1\n2ż\nBΩˆpt1,t2q´\npBtUq2` pBtpBνUqq2¯\ndspxqdt\n`1\n2ż\nBΩˆpt1,t2q´\np∆Uq2` pBνp∆Uqq2¯\ndspxqdt\nďE0pt1q `F2.AN INVERSE SOURCE PROBLEM 11\nSimilarly, letting θ“2σ, we derive\nE0pt1qe2σt1ďE0pt2qe2σt2`ż\nΩˆpt2,t1q´σp∆Uq2dxdt\n`1\n2ż\nBΩˆpt1,t2q´\npBtUq2` pBtpBνUqq2¯\ne2σtdspxqdt\n`1\n2ż\nBΩˆpt1,t2q´\np∆Uq2` pBνp∆Uqq2¯\ne2σtdspxqdt\nďE0pt2qe2σt2`1\n2ż\nBΩˆpt1,t2q´\npBtUq2` pBtpBνUqq2¯\ne2σtdspxqdt\n`1\n2ż\nBΩˆpt1,t2q´\np∆Uq2` pBνp∆Uqq2¯\ne2σtdspxqdt.\nwhich gives\nE0pt1q ďe2σpt2´t1qpE0pt2q `F2q.\nThe proof is completed by following similar arguments as tho se in [8, Lemma 3.2]. /square\nNow we return to the proof of Theorem A.2\nProof of Theorem A.2. Letϕpx,tq “ |x´a|2´θ2pt´T\n2q2, where dist pa,Ωq “1,θ“1\n2. Using the\nCarleman-type estimate in [18], we obtain\nτ6ż\nQ|U|2e2τϕdxdt`τ3ż\nQ|BtU|2e2τϕdxdt`τż\nQ|∆U|2e2τϕdxdt\nÀż\nQppB2\nt`∆2qUq2e2τϕdxdt\n`ż\nBQτ6p|Bν∆U|2` |Bt∆U|2` |B2\ntUq|2qe2τϕdspxqdt. (A.5)\nIt is easy to see that 1 ´θ2ε2\n0ďϕon Ω ˆ t|t´T\n2| ăε0ufor some positive εă1. Then we have from\nA.4 that\nτ6ż\nQ|U|2e2τϕdxdt`τ3ż\nQ|BtU|2e2τϕdxdt`τż\nQ|∆U|2e2τϕdxdt\něτ6ż\nΩˆpT\n2´ε0,T\n2`ε0q|U|2e2τp1´θ2ε2\n0qdxdt`τ3ż\nΩˆpT\n2´ε0,T\n2`ε0q|BtU|2e2τp1´θ2ε2\n0qdxdt\n`τż\nΩˆpT\n2´ε0,T\n2`ε0q|∆U|2e2τp1´θ2ε2\n0qdxdt\něτe2τp1´θ2ε2\n0qż\nΩˆpT\n2´ε0,T\n2`ε0qEptqdt\něτe2τp1´θ2ε2\n0qε0p2e´p2σ`4TqTEp0q ´F2q. (A.6)\nMoreover, it follows from (A.4) and ϕď p2R`1q2´θ2T2{4 on Ω ˆ p0,Tqthat\nτ6ż\nQ|U|2e2τϕdxdt`τ3ż\nQ|BtU|2e2τϕdxdt`τż\nQ|∆U|2e2τϕdxdt\nďτ6e2τpp2R`1q2´θ2T2{4qpEp0q `EpTqq\nďτ6e2τpp2R`1q2´θ2T2{4qppe4T2`1qEp0q `e4T2F2q.12 P. LI, X. YAO, AND Y. ZHAO\nBy (A.5) and (A.6), we obtain\nτe2τp1´θ2ε2\n0qε0e´p2σ`1`4TqTEp0q\n`τ6ż\nQ|U|2e2τϕdxdt`τ3ż\nQ|BtU|2e2τϕdxdt`τż\nQ|∆U|2e2τϕdxdt\nď´\nσ2ż\nQ|BtU|2e2τϕdxdt`ż\nBQτ6p|Bν∆U|2` |Bt∆U|2` |B2\ntUq|2qe2τϕdspxqdt\n` pτe2τp1´θ2ε2\n0q`τ6e2τpp2R`1q2´θ2T2{4qe4T2qF2`τ6e2τpp2R`1q2´θ2T2{4qe4T2Ep0q¯\n.(A.7)\nChoosing τsufficiently large, we may remove the first integral on the righ t hand side of (A.7). We\nalso choose T2“4p2R`1q2\nθ2`4ε2\n0andτ“ p2σ`8TqT`lnp2pε0q´1Cq `Cσ2. Noting τ5e´τď5!, we\nhave\nτ5e2τpp2R`1q2´θ2T2{4´1`θ2ε2\n0q`p2σ`8TqT“τ5e´2τ`p2σ`8TqT\nď5!e´τ`p2σ`8TqTďε0\n2C.\nIn addition, since Tď5p2R`1q, it follows that\nτ5e2τpp2R`1q2´1`θ2ε2\n0`p2σ`4TqTqďτ5e2pp2σ`8TqT`Cσ2`Cqp2R`1q2`p2σ`4TqTďCeCσ2.\nUsing the above inequality and the inequality ϕă p2R`1q2onQand dividing both sides in (A.7)\nby the factor of Ep0qon the left hand side, we obtain\nEp0q ďCeCσ2F2.\nSincefis supported in Ω, there holds }U}L2pBBRˆp0,TqqďC}BtU}L2pBBRˆp0,Tqq, which completes the\nproof. /square\nAppendix B.A decay estimate\nWe prove a decay estimate for the solution of the initial valu e problem of the damped plate wave\nequation\n#\nB2\ntUpx,tq `∆2Upx,tq `σBtUpx,tq “0, px,tq PR3ˆ p0,`8q,\nUpx,0q “0,BtUpx,0q “fpxq, x PR3,(B.1)\nwherefpxq PL1pR3q XHspR3q. By the Fourier transform, the solution Upx,tqof (B.1) is given as\nUpx,tq “F´1pmσpt,ξqˆfpξqqpxq,\nwhereF´1denotes the inverse Fourier transform,\nmσpt,ξq “e´σ\n2t\na\nσ2´4|ξ|4´\ne1\n2t?\nσ2´4|ξ|4´e´1\n2t?\nσ2´4|ξ|4¯\n,\nandˆfpξqis the Fourier transform of f, i.e.,\nˆfpξq “1\np2πq3ż\nR3e´ix¨ξfpxqdx.\nLeta\nσ2´4|ξ|4“ia\n4|ξ|4´σ2when |ξ|4ąσ2\n4. Then we have\nmσpt,ξq “$\n’&\n’%e´σ\n2tsinh pt\n2?\nσ2´4|ξ|4q?\nσ2´4|ξ|4, |ξ|4ăσ2\n4,\ne´σ\n2tsinpt\n2?\n4|ξ|4´σ2q?\n4|ξ|4´σ2, |ξ|4ąσ2\n4.AN INVERSE SOURCE PROBLEM 13\nIt is clear to note from the representation of mσpt,ξqthat the solution Upx,tqdepends on both of\nthe low and high frequency of ξ. In fact, the solution Upx,tqbehaves as a “parabolic type” of e´t∆2f\nfor the low frequency part, while for the high frequency part it behaves like a “dispersive type” of\neit∆2f.\nTheorem B.1. LetUpx,tqbe the solution of (B.1). ThenUpx,tqsatisfies the decay estimate\nsupxPR3|Bα\nxBj\ntUpx,tq| À p1`tq´3`|α|\n4}f}L1pR3q`e´ct}f}HspR3q, (B.2)\nwherejPN,αis a multi-index vector in N3such that Bα“ Bα1x1Bα2x2Bα3x3,są2j` |α| ´1\n2andcą0\nis some positive constant. In particular, for |α| “s“0, the following estimate holds:\nsupxPR3|Upx,tq| À p1`tq´3\n4p}f}L1pR3q` }f}L2pR3qq. (B.3)\nRemark B.2. The estimate (B.3)provides a time decay of the order Opp1`tq´3\n4qforUpx,tq\nuniformly for all xPR3, which gives\nsup\nxPR3ż8\n0|Upx,tq|2dtÀż8\n0p1`tq´3{2dtă `8.\nHence, let Upx,tq “0whentă0, thenUpx,tqhas a Fourier transform ˆUpx,kq PL2pRqfor each\nxPR3. Moreover, the following Plancherel equality holds:\nż`8\n0|Upx,tq|2dt“ż`8\n´8|ˆUpx,kq|2dk.\nRemark B.3. To study the inverse source problem, it suffices to assume that fPH4pR3q. In this\ncase, it follows from the above theorem that both B2\ntUpx,tqand∆2Upx,tqare continuous functions.\nMoreover, we have from (B.2)that the following estimate holds:\nsupxPR3|Bj\ntUpx,tq| À p1`tq´3\n4}f}L1pR3q`e´ct}f}HspR3q, j “1,2,\nsupxPR3|Bα\nxUpx,tq| À p1`tq´3`|α|\n4}f}L1pR3q`e´ct}f}HspR3q,|α| ď4.\nProof.Without loss of generality, we may assume that σ“1, and then\nmσpt,ξq “e´1\n2t\na\n1´4|ξ|4´\ne1\n2t?\n1´4|ξ|4´e´1\n2t?\n1´4|ξ|4¯\n.\nFirst we prove (B.2) for j“0. Choose χPC8\n0pR3qsuch that supp χĂBp0,1\n2qandχpξq “1 for\n|ξ| ď1\n4. Let\nUpx,tq “F´1pmpt,ξqχpξqˆfq `F´1pmpt,ξqp1´χpξqqˆfq\n:“U1px,tq `U2px,tq.\nForU1px,tq, sincea\n1´4|ξ|4ď1´2|ξ|4when 0 ď |ξ| ď1\n2, we have for |ξ| ď1\n2that\nmpt,ξq “1a\n1´4|ξ|4e´t\n2p1˘?\n1´4|ξ|4qď2e´t|ξ|4, t ě0.\nFor each xPR3we have\nBαU1px,tq “ż\nR3eix¨ξpiξqαmpt,ξqχpξqˆfpξqdξ,\nwhich gives\nsup\nxPR3|Bα\nxU1px,tq| ďż\n|ξ|ď1\n2|ξ|αe´t|ξ|4|ˆfpξq|dξÀ }ˆf}L8pR3qż\n|ξ|ď1\n2|ξ|αe´t|ξ|4dξ.14 P. LI, X. YAO, AND Y. ZHAO\nSince\nż\n|ξ|ď1\n2|ξ|αe´t|ξ|4dξď#\nC, 0ďtď1,\nt´3`|α|\n4, t ě1,\nand }ˆf}L8pR3qď }f}L1pR3q, we obtain\nsup\nxPR3|Bα\nxU1px,tq| À p1`tq´3`|α|\n4|f}L1pR3q@αPN3. (B.4)\nTo estimate U2px,tq, noting\np1´∆qp\n2U2px,tq “ż\nR3eix¨ξp1` |ξ|2qp\n2mpt,ξqp1´χpξqqˆfpξqdξ,\nwe have from Plancherel’s theorem that\nż\nR3|p1´∆qp\n2U2px,tq|2dx“ż\nR3p1` |ξ|2qp|mpt,ξqp1´χpξqqˆfpξq|2dξ. (B.5)\nIt holds that\n|mpt,ξq| ď$\n’’’’&\n’’’’%te´t\n2p1´?\n1´4|ξ|4qˇˇˇ1´e´t?\n1´4|ξ|4\nt?\n1´4|ξ|4ˇˇˇÀe´t\n8,1\n2ă |ξ| ď?\n2\n2,\n1\n2e´t\n2sint\n2?\n4|ξ|4´1\nt\n2?\n4|ξ|4´1Àe´t\n8,?\n2\n2ă |ξ| ď1,\ne´t\n2?\n4|ξ|4´1|sint\n2a\n4|ξ|4´1| ďe´t\n2?\n4|ξ|4´1, |ξ| ą1.\nHence, when |ξ| ě1\n2we have\n|p1` |ξ|2qmpt,ξq| Àe´t\n8.\nIt follows from (B.5) that\n}U2px,tq}2\nHppR3qďż\n|ξ|ě1\n2|p1` |ξ|2qp\n2mpt,ξqˆfpξq|2dξ\nďe´t\n4ż\nR3|p1` |ξ|2q´1`p\n2ˆfpξq|2dξ“e´t\n4}f}2\nHp´2pR3q.\nOn the other hand, by Sobolev’s theorem, we have for pą3\n2that\nsup\nxPR3|U2px,tq| ď }U2p¨,tq}HppR3qÀe´t\n8}f}Hp´2pR3q.\nMore generally, for any αPN3it holds that\np1´∆qp\n2Bα\nxU2px,tq “F´1pp1` |ξ|2qp\n2mpt,ξqp1´χpξqqyBαfq,\nwhich leads to\nsup\nxPR3|Bα\nxU2px,tq| Àe´t\n8}Bαf}Hp´2pR3qÀe´t\n8}f}HspR3q. (B.6)\nHeres“p´2` |α| ą |α| ´1\n2by choosing pą3\n2. Combining the estimate (B.4) with (B.6) yields\n(B.2) for j“0.\nNext we consider the general case with j‰0. Noting\nBj\ntUpx,tq “ż\nR3eix¨ξBj\ntmpt,ξqˆfpξqdξ,AN INVERSE SOURCE PROBLEM 15\nwe obtain from direct calculations that\nBj\ntmpt,ξq “ Bj´e´1\n2t\na\n1´4|ξ|4´\ne1\n2t?\n1´4|ξ|4´e´1\n2t?\n1´4|ξ|4¯¯\n“jÿ\nl“02´jpa\n1´4|ξ|4ql´1e´t\n2´\ne1\n2t?\n1´4|ξ|4` p´1ql`1e´1\n2t?\n1´4|ξ|4¯\n:“jÿ\nl“0mlpt,ξq.\nHence we can write Bj\ntUpx,tqas\nBj\ntUpx,tq “jÿ\nl“0ż\nR3eix¨ξmlpt,ξqˆfpξqdξ:“jÿ\nl“0Wlpx,tq. (B.7)\nFor each 0 ďlďj, j‰0, using similar arguments for the case j“0 we obtain\nsup\nxPR3|Bα\nxWlpx,tq| ď p1`tq´3`|α|\n4}f}L1pR3q`e´t\n8}f}HspR3q (B.8)\nforsą2l` |α| ´1\n2. Combining (B.7) and (B.8), we obtain the general estimate ( B.2). /square\nRemark B.4. For the damped biharmonic plate wave equation, besides the d ecay estimate (B.2), we\ncan deduce other decay estimates of the Lp-Lqtype and time-space estimates by more sophisticated\nanalysis for the Fourier multiplier mpt,ξq. For example, it can be proved that\n}Upx,tq}LqpR3qÀ p1`tq´3\n4p1\np´1\nqq}f}LppR3q`e´ct}f}Wq,spR3q,\nwhere1ăpďqă `8andsě3p1\nq´1\n2q ´2. We hope to present the proofs of these Lp-Lqestimates\nand their applications elsewhere.\nAcknowledgement\nWe would like to thank Prof. Masahiro Yamamoto for providing the reference [18] on Carleman\nestimates of the Kirchhoff plate equation. The research of PL is supported in part by the NSF grant\nDMS-1912704. The research of XY is supported in part by NSFC ( No. 11771165). The research of\nYZ is supported in part by NSFC (No. 12001222).\nReferences\n[1] T. Aktosun and V. Papanicolaou, Time evolution of the sca ttering data for a fourth-order linear differential\noperator, Inverse Problems, 24 (2008), 055013.\n[2] G. Bao, J. Lin, and F. Triki, A multi-frequency inverse so urce problem, J. Differential Equations, 249 (2010),\n3443–3465.\n[3] G. Bao, P. Li, and Y. Zhao, Stability for the inverse sourc e problems in elastic and electromagnetic waves, J.\nMath. Pures Appl., 134 (2020), 122–178.\n[4] J. Cheng, V. Isakov, and S. Lu, Increasing stability in th e inverse source problem with many frequencies, J.\nDifferential Equations, 260 (2016), 4786–4804.\n[5] M. Entekhabi and V. Isakov, On increasing stability in th e two dimensional inverse source scattering problem\nwith many frequencies, Inverse Problems, 34 (2018), 055005 .\n[6] M. Entekhabi and V. Isakov, Increasing stability in acou stic and elastic inverse source problems, SIAM J. Appl.\nMath., 52 (2020), 5232–5256.\n[7] F. Gazzola, H.-C. Grunau, and G. Sweers, Polyharmonic bo undary value problems, Springer-Verlag Berlin Hei-\ndelberg, 2010.\n[8] V. Isakov and S. Lu, Increasing stability in the inverse s ource problem with attenuation and many frequencies,\nSIAM J. Appl. Math., 78 (2018), 1–18.\n[9] K. Iwasaki, Scattering theory for 4th order differential operators: I-II, Japan. J. Math., 14 (1988), 1–96.16 P. LI, X. YAO, AND Y. ZHAO\n[10] K. Krupchyk, M. Lassas, and G. Uhlmann, Inverse boundar y value problems for the perturbed poly-harmonic\noperator, Trans. Amer. Math. Soc., 366 (2014), 95–112.\n[11] M. Lassas, K. Krupchyk and G. Uhlmann, Determining a firs t order perturbation of the biharmonic operator by\npartial boundary measurements, J. Funct. Anal., 262 (2012) , 1781–1801.\n[12] P. Li, X. Yao, and Y. Zhao, Stability for an inverse sourc e problem of the biharmonic operator, arXiv:2102.04631.\n[13] P. Li and G. Yuan, Increasing stability for the inverse s ource scattering problem with multifrequencies, Inverse\nProblems Imag., 11 (2017), 745–759.\n[14] P. Li, J. Zhai, and Y. Zhao, Stability for the acoustic in verse source problem in inhomogeneous media, SIAM J.\nAppl. Math., to appear.\n[15] N.V. Movchan, R.C. McPhedran, A.B. Movchan, and C.G. Po ulton, Wave scattering by platonic grating stacks,\nProc. R. Soc. A, 465 (2009), 3383–3400.\n[16] J. Rousseau and L. Robbiano, Spectral inequality and re solvent estimate for the bi-harmonic operator, J. Eur.\nMath. Soc., 22 (2020), 1003–1094.\n[17] T. Tyni and V. Serov, Scattering problems for perturbat ions of the multidimensional biharmonic operator, Inverse\nProblems and Imaging, 12 (2018), 205–227.\n[18] G. Yuan and M. Yamamoto, Lipschitz stability in inverse problems for a Kirchhoff plate equation, Asymptotic\nAnalysis, 53 (2007), 29–60.\nDepartment of Mathematics, Purdue University, West Lafaye tte, Indiana 47907, USA\nEmail address :lipeijun@math.purdue.edu\nSchool of Mathematics and Statistics, China Central Normal University, Wuhan, Hubei, China\nEmail address :yaoxiaohua@mail.ccnu.edu.cn\nSchool of Mathematics and Statistics, China Central Normal University, Wuhan, Hubei, China\nEmail address :zhaoyueccnu@163.com"    },    {        "title": "2111.08768v1.Ultrathin_ferrimagnetic_GdFeCo_films_with_very_low_damping.pdf",        "content": "Ultrathin ferrimagnetic GdFeCo \flms with very low damping\nLakhan Bainsla*,1,a)Akash Kumar,1Ahmad A. Awad,1Chunlei Wang,2Mohammad Zahedinejad,1Nilamani\nBehera,1Himanshu Fulara,1Roman Khymyn,1Afshin Houshang,1Jonas Weissenrieder,2and J. \u0017Akerman1,b)\n1)Physics Department, University of Gothenburg, 412 96 Gothenburg, Sweden.\n2)Department of Applied Physics, KTH Royal Institute of Technology, 106 91 Stockholm,\nSweden\nFerromagnetic materials dominate as the magnetically active element in spintronic devices, but come with\ndrawbacks such as large stray \felds, and low operational frequencies. Compensated ferrimagnets provide an\nalternative as they combine the ultrafast magnetization dynamics of antiferromagnets with a ferromagnet-like\nspin-orbit-torque (SOT) behavior. However to use ferrimagnets in spintronic devices their advantageous prop-\nerties must be retained also in ultrathin \flms ( t<10 nm). In this study, ferrimagnetic Gd x(Fe87:5Co12:5)1\u0000x\nthin \flms in the thickness range t= 2{20 nm were grown on high resistance Si(100) substrates and studied\nusing broadband ferromagnetic resonance measurements at room temperature. By tuning their stoichiometry,\na nearly compensated behavior is observed in 2 nm Gd x(Fe87:5Co12:5)1\u0000xultrathin \flms for the \frst time,\nwith an e\u000bective magnetization of Me\u000b= 0.02 T and a low e\u000bective Gilbert damping constant of \u000b= 0.0078,\ncomparable to the lowest values reported so far in 30 nm \flms. These results show great promise for the\ndevelopment of ultrafast and energy e\u000ecient ferrimagnetic spintronic devices.\nI. INTRODUCTION\nSpintronic devices utilize the spin degree of freedom for\ndata storage, information processing, and sensing1,2with\ncommercial applications such as hard drives, magnetic\nrandom access memories, and sensors. Besides conven-\ntional memory applications based on quasi-static opera-\ntion of magnetic tunnel junctions, high frequency spin-\ntronic oscillators3,4have recently been demonstrated for\nanalog computing applications such as bio-inspired neu-\nromorphic computing5,6, logic operations, energy har-\nvesting and Ising Machines.7For the \frst time, such oscil-\nlators are now used in commercial magnetic hard drives\nto facilitate writing to the disc.8The key challenges in\ndeveloping such devices is to \fnd material combinations\nwhich allow for fast operation, low-power consumption,\nnon-volatility, and high endurance. Due to their nat-\nural spin polarization and easy manipulation, ferromag-\nnetic materials (FM) dominate as active elements in these\ndevices.4However, FMs come with drawbacks such as:\n(i) large magnetic stray \felds a\u000becting the operation of\nneighbouring devices; (ii) limited scalability of magnetic\nbits in memory devices; (iii) the operating frequency of\nspin-based oscillators limited by ferromagnetic resonance\nfrequency, and (iv) slow synchronization of such oscilla-\ntors. These shortcomings drive researchers to \fnd more\nsuitable materials for future spintronic devices.\nVery recently, the interest in antiferromagnetic (AFM)\nspintronics9{11increased rapidly, as AFM materials have\nno stray \felds and can o\u000ber ultrafast spin dynamics, in-\ncluding AFM resonance frequencies in the THz region.\nIt was theoretically shown that such high-frequency ex-\ncitations are possible to achieve without any applied\nmagnetic \feld by injecting spin currents into AFM\na)Electronic mail: lakhan.bainsla@physics.gu.se\nb)Electronic mail: johan.akerman@physics.gu.sematerials.12{15Experiments have since demonstrated\npossible THz writing/reading capabilities.16However,\nthe absence of a net magnetic moment in AFMs leads\nto di\u000eculties in the read-out of the spin dynamics, in-\ncluding any microwave output signal from the AFM\noscillators.13{15\nA possible solution is presented by ferrimagnets\n(FiMs), which combine the properties of FMs and AFMs.\nFiMs posses magnetic sub-lattices in the same way as\nAFMs do, but their sub-lattices are inequivalent. The\nmagnetic sub-lattices in FiMs often consist of di\u000berent\nmagnetic ions, such as rare earth (e.g. Gd) and transi-\ntion metal (e.g. Fe, Co) alloys (RE-TM) such as CoGd,\nand as a result, a large residual magnetization remains\ndespite the two opposing sub-magnetizations. The tem-\nperature dependence of RE and TM sub-magnetizations\nin FiM can be quite di\u000berent which result in magneti-\nzations that can increase, and even change sign, with\ntemperature17,18, in stark contrast to the non-monotonic\ndecreasing temperature dependence for FMs and AFMs.\nSimilar e\u000bects could also be seen by varying the com-\nposition of ferrimagnetic alloys instead of changing the\ntemperature.19In addition, the di\u000berent properties of the\ntwo magnetic sub-lattices also results in two compen-\nsation points, namely the magnetization compensation\npointTmand the angular compensation point Ta. AtTm,\nthe two magnetic sub-lattices cancel each other, which re-\nsults in a zero net magnetic moment, while at T a, their\nnet angular momentum vanishes, as in AFMs. Therefore,\natTa, FiMs can have a near-THz resonance as in AFMs,\nwhile still having a net magnetic moment which can lead\nto strong read-out signals, including e\u000ecient microwave\nsignal output from FiM-based oscillators20, as well as ef-\n\fcient control and excitation. FiMs also show high spin\npolarization which also make them suitable candidate for\ne\u000ecient magnetic tunnel junctions.21\nDue to these unique properties, research in FiMs for\nspintronic applications is intensifying22, focusing mainly\non RE-TM based systems such as CoTb23, CoGd24, andarXiv:2111.08768v1  [cond-mat.mtrl-sci]  16 Nov 20212\nFigure 1. (a) Schematic illustration of the coplanar waveguide (CPW), the thin \flm sample and its orientation, the directions\nof the applied magnetic \feld H, the microwave \feld hrf, and the e\u000bective magnetic \feld He\u000bduring FMR measurements.\nInset shows the \flm stack. (b) FMR response (derivative of the FMR absorption) for a 10 nm Gd 12:5Fe76:1Co11:4\flm (S2)\nrecorded at di\u000berent frequencies and \ftted (solid lines) to Eq. 1. While FMR curves were recorded at 1 GHz frequency intervals\nthroughout this study, \fgure (b) only shows curves with \u0001 f= 2 GHz for clarity.\nGdFeCo25and Mn 3\u0000xPtxGa26,27based Heusler alloy.\nAmong these, GdFeCo has been studied the most with\ndemonstrations of fast domain wall motion28and ultra-\nfast spin dynamics17nearTa, large spin-orbit torques\nand their sign reversal,25,29low magnetic damping in\nthick 30 nm \flms,30and sub-picosecond magnetization\nreversal,31to name a few. What is missing, however, is a\ndemonstration that these unique material properties per-\nsist down to much thinner \flms, which will ultimately be\nneeded if FiMs are to be used in spin-Hall nano oscillators\n(SHNOs).4\nIn the present study, we systematically study the\ngrowth and functional properties of ultrathin ferrimag-\nnetic Gd x(Fe87:5Co12:5)1\u0000xthin \flms [referred to as\nGdx(FeCo) 1\u0000xhereafter]. GdFeCo thin \flms in the\nthickness range of 2{20 nm were grown on high resis-\ntance silicon (HR-Si) substrate. The atomic composi-\ntion of Gd x(FeCo) 1\u0000xwas controlled using co-sputtering\nand determined using inductively coupled plasma optical\nemission spectroscopy (ICP-OES). The magnetic prop-\nerties and Gilbert damping were studied using broad-\nband ferromagnetic resonance (FMR) measurements. We\nalso demonstrate ultra low Gilbert damping for 2 nm\nGdFeCo, near the compensation point of Gd x(FeCo) 1\u0000x.\nThese results paves the way for integration of FiMs into\nvarious spintronic devices and applications.\nII. RESULTS AND DISCUSSION\nThe growth conditions for GdFeCo were \frst optimized\nby growing four 10 nm thick Gd 12:5Fe76:1Co11:4\flms on\nHR-Si (100) substrates using di\u000berent MgO seed layer\nthicknesses: 0 nm (S1), 6 nm (S2), 10 nm (S3 & S4);in S4, the seed was annealed at 600C for 1 hour prior\nto GdFeCo deposition to check the e\u000bect of MgO crys-\ntallinity. MgO was chosen as seed since it is insulating\nand therefore will not contribute any spin sinking to the\nmagnetic damping.32\nA. Seed layer dependence on 10nm thick\nGd12:5Fe76:1Co11:4\flms\nFurther details of the growth conditions are given in\nthe experimental section. FMR measurements, on 6 \u00023\nmm2rectangular pieces cut from these \flms, were then\nperformed using a NanOsc PhaseFMR-40 FMR Spec-\ntrometer. The sample orientation on the coplanar waveg-\nuide (CPW), together with the directions of the applied\n\feld, the microwave excitation \feld hrf, and the e\u000bec-\ntive magnetic \feld He\u000b, are shown in Fig. 1(a). Typical\n(derivative) FMR absorption spectra obtained for S2 are\nshown in \fgure 1(b) together with \fts to a sum of sym-\nmetric and anti-symmetric Lorentzian derivatives:33\ndP\ndH(H) =\u00008C1\u0001H(H\u0000HR)\n[\u0001H2+ 4(H\u0000HR)2]2+2C2(\u0001H2\u00004(H\u0000HR)2)\n[\u0001H2+ 4(H\u0000HR)2]2\n(1)\nwhereHR, \u0001H,C1, andC2represent the resonance \feld,\nthe full width at half maximum (FWHM) of the FMR ab-\nsorption, and the symmetric and anti-symmetric \ftting\nparameters of the Lorentzian derivatives, respectively.\nThe extracted values of HRvs.fare shown in \fgure\n2 (b) together with \fts to Kittel's equation:34\nf=\r\u00160\n2\u0019q\n(HR\u0000Hk)(HR\u0000Hk+Meff) (2)3\nFigure 2. (a) Seed layer dependence of frequency vs resonance \feld of the 10 nm thick Gd 12:5Fe76:1Co11:4\flms, here solid\nsymbols and solid lines are the experimental data points and \ftting with equation (2), respectively. (b) Resonance linewidth\n(\u0001H)vs.frequency of the 10 nm thick Gd 12:5Fe76:1Co11:4\flms, here solid symbols and solid lines are the experimental data\npoints and \ftting with equation (3), respectively. The e\u000bective Gilbert damping constant values of all the samples are given in\n\fgure 2 (b). The black and violet dotted lines in \fgure 2(b) shows the \ftting of equation (3) in low and high frequency regions,\nrespectively.\nwhere,\r,HkandMe\u000bare the gyromagnetic ratio, the\nin-plane magnetic anisotropy \feld, and the e\u000bective mag-\nnetization of the sample, respectively, all allowed to be\nfree \ftting parameters. Values for \randHkonly showed\nminor variation between the four samples, with \r=2\u0019=\n29.4-30.0 GHz/T and Hk= 66-104 Oe. Me\u000bvaried more\nstrongly, with values of 0.79, 1.19, 0.71 and 0.76 T ob-\ntained for S1, S2, S3 and S4, respectively.\nThe e\u000bective Gilbert damping constant \u000bcan then be\nobtained from \fts of \u0001 Hvs.fto:35\n\u0001H= \u0001H0+4\u0019\u000bf\n\r\u00160(3)\nwhere the o\u000bset \u0001 H0represents the inhomogeneous\nbroadening. Equation (3) is well \ftted to the experimen-\ntal values, using \u0001 H0and\u000bas adjustable \ftting param-\neters for all the four samples, as shown in the \fgure 2(b).\n\u0001H0= 2{4 mT is essentially sample independent within\nthe measurement accuracy. In contrast, the obtained val-\nues of\u000bvary quite strongly and are given inside \fgure\n2(b). The GdFeCo grown with 6 nm MgO seed layer (S2)\nclearly shows the lowest value of \u000b= 0:0055, although\nthis might be a\u000bected by the slight non-linear behavior\naround 10 to 15 GHz. However, when only the high-\feld\ndata is \ftted, the extracted damping of \u000b= 0.0076 is\nstill the lowest and at all frequencies the linewidth of S2\nlies well below all the other samples. As damping is one\nof the most important parameters for spintronic devices,\nwe hence chose the growth conditions of S2 for all subse-\nquent \flms in this study.B. Thickness dependence on Gd 12:5Fe76:1Co11:4\flms\nAfter optimizing the growth conditions for\nGd12:5Fe76:1Co11:4, the thickness dependence of the\n\flms was studied with the same composition using the\ngrowth conditions of sample S2. The FMR linewidth\n\u0001Hvs. f is shown in \fgure 3(a) and exhibits a relatively\nstrong dependence on thickness. It is noteworthy\nthat the 4 nm \flm shows the narrowest linewidth at\nall frequencies, clearly demonstrating that very low\ndamping can be achieved also in ultra-thin GdFeCo.\nThe extracted Me\u000band\u000bare shown vs.thickness in\n\fgure 3(b), both showing a strong thickness dependence.\nDamping as low as \u000b= 0:0055 is obtained for the 10\nnm thick \flms. If only the high-\feld portion of the data\nis \ftted, the extracted damping increases to 0.0076,\nwhich is still about an order of magnitude lower than\nany literature value on 10 or 30 nm \flms.19,36Both\nthe 10 and 20 nm \flms showed a minor nonlinearity in\n\u0001Hvs.fdata and were therefore analysed by \ftting\nthe data in both the low and the high \feld regions\nseparately, as shown by the dotted lines in \fgure 3(a).\nThe\u000bvalue for the 20 nm \flm increased slightly from\n0.0098 to 0.0109 if only high \feld data is used for\nanalysis. The relatively higher damping for the 20 nm\n\flm might be due to the radiative damping mechanism\nwhich increases proportionally with magnetic layer\nthickness.37We conclude that 2 nm ultrathin \flms can\nindeed be grown with reasonably low damping. Since\nthe damping is strongly thickness dependent in this\nregime, the optimum thickness for devices may likely be\nfound in the 2{4 nm range.4\nFigure 3. (a) FMR linewidth \u0001 Hvs.ffor four Gd 12:5Fe76:1Co11:4\flms with di\u000berent thicknesses, together with linear \fts to\nequation (3). The dotted lines show \fts for the 20 nm \flm in its low and high frequency regions, respectively. (b) E\u000bective\nmagnetization and e\u000bective Gilbert damping constant vs.thickness; lines are guides to the eye.\nC. Composition dependence on 2nm thick \flms\nTo \fnally investigate whether we can achieve a com-\npensated ferrimagnetic behavior also in ultra-thin \flms,\nwe grew 2 nm Gd x(FeCo) 1\u0000x\flms in the composition\nrange 12{27 at.% Gd. The \flms were characterized using\nFMR spectrometry as described above and the extracted\nresults are shown in \fgure 4.\nThe extracted Me\u000band\u000bfollow a similar trend as re-\nported earlier for one order of magnitude thicker GdFeCo\n\flms characterized using an all-optical pump-probe tech-\nnique.17We \frst note that we can indeed reach an es-\nsentially fully compensated antiferromagnetic behavior in\ntwo \flms around a composition of 25 at.% Gd. We have\nmarked this compensation point with xmand a dashed\nline in \fgure 4 (c). Both \flms show very low damping of\n0.0078 and 0.009 respectively. However, just below this\ncomposition, the damping shows a peak, which is con-\nsistent with an angular compensation point, which we\ndenote byxa. It is noteworthy that the extracted damp-\ning value of \u000b= 0.0142 is still more than an order of\nmagnitude lower than \u000b= 0.45 of 30 nm \flms measured\nusing FMR spectrometry19and\u000b= 0.20 of 20 nm \flms\nmeasured using an optical pump-probe technique.17\nIII. CONCLUSION\nIn view of the potential application of compensated\nferrimagnets to spintronic devices, we prepared ferri-\nmagnetic thin \flms of Gd x(FeCo) 1\u0000xon high resistance\nSi(100) substrates and studied them using the FMR mea-\nsurements. Their growth conditions were optimized us-\ning 10 nm thick Gd 12:5Fe76:1Co11:4\flms, after which\nthickness dependent studies were done on the same com-\nposition in the thickness range of 2{20 nm. Composi-\ntion dependence studies were \fnally done on 2 nm thick\nGdx(FeCo) 1\u0000x\flms and an essentially compensated fer-rimagnetic behavior was observed for the \frst time in\nultrathin 2 nm \flms. The angular momentum compensa-\ntion and magnetic compensation points observed in this\nwork are very close to those reported earlier on much\nthicker \flms in the literature. A record low \u000bvalue of\nabout 0.0078 is obtained near the magnetic compensa-\ntion point, which is an order of magnitude lower than\nthe values reported in the literature using similar analysis\nmethods. The observation of compensated ferrimagnetic\nbehavior in ultrathin \flms together with very low value\nof\u000bare promising results for the future development of\nultrafast and energy e\u000ecient ferrimagnetic spintronic de-\nvices.\nEXPERIMENTAL SECTION\nA. Thin \flms growth and composition analysis\nAll the samples were prepared on high resistivity\nSi(100) substrates using a magnetron sputtering sys-\ntem with a base pressure of less than 2 \u000210\u00008torr.\nThin \flms of Gd x(FeCo) 1\u0000xwere deposited using the\nco-sputtering of high purity (more than 99.95%) Gd\nand Fe 87:5Co12:5targets, and composition analysis\nwas done using the inductively coupled plasma mass\nspectroscopy (ICP-MS). Thin \flms stacking structure\nof Si(100)/MgO(t)/Gd 12:5Fe76:1Co11:4(10)/SiO 2(4)\nwere used for seed layer dependence studies, here,\nthe number in the bracket is the thickness of the\nlayer in nm, where t=0, 6 and 10 nm. Four sam-\nples, namely S1 to S4 were prepared to obtain the\nbest conditions to grow Gd 12:5Fe76:1Co11:4(10) \flms.\nFor S1, Gd 12:5Fe76:1Co11:4(10) was grown directly\nover HR-Si (100) substrates, while in both S2 and\nS3 Gd 12:5Fe76:1Co11:4were grown with MgO seed\nlayer of 6 and 10 nm, respectively. All the lay-\ners in S1-S3 were grown at room temperature and5\nFigure 4. (a) Frequency vs.resonance \feld and (b) resonance linewidth vs.frequency, of 2 nm thick Gd x(FeCo) 1\u0000x\flms as\na function of Gd content in atomic %. (c) E\u000bective magnetization and e\u000bective Gilbert damping constant vs.Gd content.\nSolid symbols represent the values obtained by \ftting the experimental FMR data in (a) and (b) using the equation (2) and\n(3), respectively; solid lines in (c) are guides to the eye. xaand xmshow the angular and magnetic compensation points,\nrespectively, obtained from the literature17,19.\nno further heat treatment was given to them. In\nS4, 10 nm MgO seed layer were grown over HR\nSi(100) substrates at RT and followed by a in-situ\npost-annealing at 600C for 1 hour, and after that\nGd12:5Fe76:1Co11:4were deposited. The stacking struc-\nture of Si(100)/MgO(6)/Gd 12:5Fe76:1Co11:4(m)/SiO 2(4)\nwere used for thickness dependence studies, where\nm is the thickness of Gd 12:5Fe76:1Co11:4layer,\nand varied from 2 to 20 nm. For composi-\ntion dependence studies, stacking structure of\nSi(100)/MgO(6)/Gd x(FeCo) 1\u0000x(2)/SiO 2(4) were used,\nwhere xvaried from 12.5 to 26.7. The composition of\nGdx(FeCo) 1\u0000x\flms was varied by changing the sput-\ntering rate of Fe 87:5Co12:5target, while keeping the Gd\nsputtering rate \fxed for most \flms. All the samples for\nthickness dependence and composition dependence were\ngrown at room temperature and no post-annealing was\nused. Layer thicknesses were determined by estimating\nthe growth rate using the Dektak pro\fler on more than\n100 nm thick \flms.B. Inductively coupled plasma mass spectroscopy\n(ICP-MS) measurements\nThe elemental composition (Co, Fe, and Gd) of the\nthin \flm samples was determined by inductively coupled\nplasma optical emission spectroscopy (ICP-OES) using a\nThermo Fisher Scienti\fc iCAP 6000 Series spectrometer.\nEach thin \flm sample was exhaustively extracted in 5 mL\nHNO3 (65%, Supelco, Merck KgaA, Sigma-Aldrich) for\na duration of 30 min. 5 mL ultrapure MilliQ-water (18\nM\ncm) was added to the solution and the extract was al-\nlowed to rest for 30 minutes. The extract was transferred\nto a 100 mL volumetric \rask. The extracted sample was\nthen rinsed for several cycles in ultrapure water. The\nwater used for rinsing was transferred to the same volu-\nmetric \rask. The extract was diluted to 100 mL for ICP\nanalysis. ICP check standards were prepared from stan-\ndard solutions (Co and Fe: Merck, Germany; Ga: Accu-\nstandard, USA). The relative standard deviation (from\nthree individual injections) were within 1%.6\nTable I. The obtained values of e\u000bective Gilbert damping constant \u000bat room temperature (RT) in this work and comparison\nwith the lowest values reported so far in the literature at RT and also at their respective angular momentum compensation\n(Ta) and magnetic compensation (T m) points.\nFilm composition Film thickness \u000b Measurement technique Analysis method Reference\nGd23:5Fe68:9Co7:6 30 \u00180.45 (at RT) FMR Kittel's FMR19\n\u00180.35 (at RT) Pump-probe\nGd22Fe74:6Co3:4 20 \u00180.21 (at T a) Pump-probe -do-17\n\u00180.13 (at T m)\nGd25Fe65:6Co9:4 10 \u00180.07 (at RT) Spin torque FMR -do-36\n\u00190.01 (at RT) Spin torque FMR Ferrimagnetc resonance\nGd23:5Fe66:9Co9:6 30 0.0072 (at RT) Domain wall (DW) Field driven DW30\nmotion mobility\nGd12:5Fe76:1Co11:4 10 0.0055 Broadband FMR Kittel's FMR This work\n0.0076 (HF data) -do- -do- This work\nGd12:5Fe76:1Co11:4 4 0.0064 -do- -do- This work\nGd12:5Fe76:1Co11:4 2 0.0101 -do- -do- This work\nGd23:4Fe67:0Co9:6 2 0.0141 -do- -do- This work\nGd24:4Fe66:1Co9:5 2 0.0078 -do- -do- This work\nC. Ferromagnetic resonance (FMR) measurements\nRectangular pieces of about 6 \u00023 mm2were cut from\nthe blanket \flms and broadband FMR spectroscopy was\nperformed using a NanOsc Phase FMR (40 GHz) system\nwith a co-planar waveguide for microwave \feld excita-\ntion. Microwave excitation \felds hrfwith frequencies up\nto 30 GHz were applied in the \flm plane, and perpendic-\nular to the applied in-plane dc magnetic \feld H. All the\nFMR measurements were performed at the room tem-\nperature. The schematic of FMR measurement setup is\nshown in 1(a), and further details about the measure-\nments are given in Section 2 (results and discussions).\nSUPPORTING INFORMATION\nSupporting Information is available from the Wiley\nOnline Library or from the corresponding author.ACKNOWLEDGEMENTS\nLakhan Bainsla thanks MSCA - European Commission\nfor Marie Curie Individual Fellowship (MSCA-IF Grant\nNo. 896307). This work was also partially supported\nby the Swedish Research Council (VR Grant No. 2016-\n05980) and the Horizon 2020 research and innovation\nprogramme (ERC Advanced Grant No. 835068 \"TOP-\nSPIN\").\nCONFLICT OF INTEREST\nThe authors declare no con\rict of interest.\nAUTHOR CONTRIBUTIONS\nL.B. and J. \u0017A. planned the study. L.B. grew the \flms,\nperformed the FMR measurements and analysed the ob-\ntained FMR data. J.W. helped with ICP-MS measure-\nments and analysis. L.B. wrote the original draft of the\npaper. J. \u0017A. coordinated and supervised the work. All\nauthors contributed to the data analysis and co-wrote\nthe manuscript.7\nDATA AVAILABILITY STATEMENT\nThe data that support the \fndings of this study are\navailable from the corresponding author on reasonable\nrequest.\nREFERENCES\n1S. Wolf, D. Awschalom, R. Buhrman, J. Daughton, v. S. von\nMoln\u0013 ar, M. Roukes, A. Y. Chtchelkanova, and D. 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Shaw, Nature Physics\n12, 839 (2016)."    },    {        "title": "1607.01307v1.Magnetic_moment_of_inertia_within_the_breathing_model.pdf",        "content": "Magnetic moment of inertia within the breathing model\nDanny Thonig,\u0003Manuel Pereiro, and Olle Eriksson\nDepartment of Physics and Astronomy, Material Theory, University Uppsala, S-75120 Uppsala, Sweden\n(Dated: June 20, 2021)\nAn essential property of magnetic devices is the relaxation rate in magnetic switching which\nstrongly depends on the energy dissipation and magnetic inertia of the magnetization dynamics.\nBoth parameters are commonly taken as a phenomenological entities. However very recently, a large\ne\u000bort has been dedicated to obtain Gilbert damping from \frst principles. In contrast, there is no\nab initio study that so far has reproduced measured data of magnetic inertia in magnetic materials.\nIn this letter, we present and elaborate on a theoretical model for calculating the magnetic moment\nof inertia based on the torque-torque correlation model. Particularly, the method has been applied\nto bulk bcc Fe, fcc Co and fcc Ni in the framework of the tight-binding approximation and the\nnumerical values are comparable with recent experimental measurements. The theoretical results\nelucidate the physical origin of the moment of inertia based on the electronic structure. Even though\nthe moment of inertia and damping are produced by the spin-orbit coupling, our analysis shows that\nthey are caused by undergo di\u000berent electronic structure mechanisms.\nPACS numbers: 75.10.-b,75.30.-m,75.40.Mg,75.78.-n,75.40.Gb\nThe research on magnetic materials with particular fo-\ncus on spintronics or magnonic applications became more\nand more intensi\fed, over the last decades [1, 2]. For\nthis purpose, \\good\" candidates are materials exhibiting\nthermally stable magnetic properties [3], energy e\u000ecient\nmagnetization dynamics [4, 5], as well as fast and stable\nmagnetic switching [6, 7]. Especially the latter can be\ninduced by i)an external magnetic \feld, ii)spin polar-\nized currents [8], iii)laser induced all-optical switching\n[9], or iv)electric \felds [10]. The aforementioned mag-\nnetic excitation methods allow switching of the magnetic\nmoment on sub-ps timescales.\nThe classical atomistic Landau-Lifshitz-Gilbert (LLG)\nequation [11, 12] provides a proper description of mag-\nnetic moment switching [13], but is derived within the\nadiabatic limit [14, 15]. This limit characterises the\nblurry boundary where the time scales of electrons and\natomic magnetic moments are separable [16] | usually\nbetween 10\u0000100 fs. In this time-scale, the applicabil-\nity of the atomistic LLG equation must be scrutinized\nin great detail. In particular, in its common formula-\ntion, it does not account for creation of magnetic inertia\n[17], compared to its classical mechanical counterpart of\na gyroscope. At short times, the rotation axis of the\ngyroscope do not coincide with the angular momentum\naxis due to a \\fast\" external force. This results in a\nsuperimposed precession around the angular-momentum\nand the gravity \feld axis; the gyroscope nutates. It is\nexpected for magnetisation dynamics that atomic mag-\nnetic moments behave in an analogous way on ultrafast\ntimescales [17, 18] (Fig. 1).\nConceptional thoughts in terms of \\magnetic mass\"\nof domain walls were already introduced theoretically by\nD oring [19] in the late 50's and evidence was found ex-\nperimentally by De Leeuw and Robertson [20]. More\nrecently, nutation was discovered on a single-atom mag-\nnetic moment trajectory in a Josephson junction [21{23]\nB\nprecession conenutation cone\nmFIG. 1. (Color online) Schematic \fgure of nutation in the\natomistic magnetic moment evolution. The magnetic moment\nm(red arrow) evolves around an e\u000bective magnetic \feld B\n(gray arrow) by a superposition of the precession around the\n\feld (bright blue line) and around the angular momentum\naxis (dark blue line). The resulting trajectory (gray line)\nshows an elongated cycloid.\ndue to angular momentum transfer caused by an elec-\ntron spin \rip. From micromagnetic Boltzman theory,\nCiornei et al. [18, 24] derived a term in the extended\nLLG equation that addresses \\magnetic mass\" scaled by\nthe moment of inertia tensor \u0013. This macroscopic model\nwas transferred to atomistic magnetization dynamics and\napplied to nanostructures by the authors of Ref. 17, and\nanalyzed analytically in Ref. 25 and Ref. 26. Even in the\ndynamics of Skyrmions, magnetic inertia was observed\nexperimentally [27].\nLike the Gilbert damping \u000b, the moment of inertia ten-\nsor\u0013have been considered as a parameter in theoretical\ninvestigations and postulated to be material speci\fc. Re-\ncently, the latter was experimentally examined by Li et\nal. [28] who measured the moment of inertia for Ni 79Fe21\nand Co \flms near room temperature with ferromagnetic\nresonance (FMR) in the high-frequency regime (aroundarXiv:1607.01307v1  [cond-mat.mtrl-sci]  5 Jul 20162\n200 GHz). At these high frequencies, an additional sti\u000b-\nening was observed that was quadratic in the probing fre-\nquency!and, consequently, proportional to the moment\nof inertia\u0013=\u0006\u000b\u0001\u001c. Here, the lifetime of the nutation \u001c\nwas determined to be in the range of \u001c= 0:12\u00000:47 ps,\ndepending not only on the selected material but also on\nits thickness. This result calls for a proper theoretical\ndescription and calculations based on ab-initio electronic\nstructure footings.\nA \frst model was already provided by Bhattacharjee\net al. [29], where the moment of inertia \u0013was derived\nin terms of Green's functions in the framework of the\nlinear response theory. However, neither \frst-principles\nelectronic structure-based numerical values nor a detailed\nphysical picture of the origin of the inertia and a poten-\ntial coupling to the electronic structure was reported in\nthis study. In this Letter, we derive a model for the\nmoment of inertia tensor based on the torque-torque cor-\nrelation formalism [30, 31]. We reveal the basic electron\nmechanisms for observing magnetic inertia by calculat-\ning numerical values for bulk itinerant magnets Fe, Co,\nand Ni with both the torque-torque correlation model\nand the linear response Green's function model [29]. In-\nterestingly, our study elucidate also the misconception\nabout the sign convention of the moment of inertia [32].\nThe moment of inertia \u0013is de\fned in a similar way\nas the Gilbert damping \u000bwithin the e\u000bective dissipation\n\feldBdiss[30, 33]. This ad hoc introduced \feld is ex-\npanded in terms of viscous damping \u000b@m=@tand magnetic\ninertia\u0013@2m=@t2in the relaxation time approach [32, 34]\n(see Supplementary Material). The o\u000b-equilibrium mag-\nnetic state induces excited states in the electronic struc-\nture due to spin-orbit coupling. Within the adiabatic\nlimit, the electrons equilibrate into the ground state at\ncertain time scales due to band transitions [35]. If this\nrelaxation time \u001cis close to the adiabatic limit, it will\nhave two implications for magnetism: i)magnetic mo-\nments respond in a inert fashion, due to formation of\nmagnetism, ii)the kinetic energy is proportional to mu2=2\nwith the velocity u=@m=@tand the \\mass\" m of mag-\nnetic moments, following equations of motion of classical\nNewtonian mechanics. The inertia forces the magnetic\nmoment to remain in their present state, represented in\nthe Kambersky model by \u000b=\u0000\u0013\u0001\u001c(Ref. 32 and 34);\ntheraison d'etre of inertia is to behave opposite to the\nGilbert damping.\nIn experiments, the Gilbert damping and the moment\nof inertia are measurable from the diagonal elements of\nthe magnetic response function \u001fvia ferromagnetic res-\nonance [31] (see Supplementary Material)\n\u000b=!2\n0\n!Mlim\n!!0=\u001f?\n!(1)\n\u0013=1\n2!2\n0\n!Mlim\n!!0@!<\u001f?\n!\u00001\n!0; (2)\nwhere!M=\rBand!0=\rB0are the frequencies re-lated to the internal e\u000bective and the external magnetic\n\feld, respectively. Thus, the moment of inertia \u0013is equal\nto the change of the FMR peak position, say the \frst\nderivative of the real part of \u001fwith respect to the prob-\ning frequency [29, 36]. Alternatively, rapid external \feld\nchanges induced by spin-polarized currents lead also to\nnutation of the macrospin [37].\nSetting\u001fonab-initio footings, we use the torque-\ntorque correlation model, as applied for the Gilbert\ndamping in Ref. 30 and 35. We obtain (see Supplemen-\ntary Material)\n\u000b\u0016\u0017=g\u0019\nmsX\nnmZ\nT\u0016\nnm(k)T\u0017\nnm(k)Wnmdk (3)\n\u0013\u0016\u0017=\u0000g~\nmsX\nnmZ\nT\u0016\nnm(k)T\u0017\nnm(k)Vnmdk; (4)\nwhere\u0016;\u0017 =x;y;z andmsis the size of the mag-\nnetic moment. The spin-orbit-torque matrix elements\nTnm=hn;kj[\u001b;Hsoc]jm;ki| related to the commuta-\ntor of the Pauli matrices \u001band the spin-orbit Hamilto-\nnian | create transitions between electron states jn;ki\nandjm;kiin bandsnandm. This mechanism is equal\nfor both, Gilbert damping and moment of inertia. Note\nthat the wave vector kis conserved, since we neglect non-\nuniform magnon creation with non-zero wave vector. The\ndi\u000berence between moment of inertia and damping comes\nfrom di\u000berent weighting mechanism Wnm;Vnm: for the\ndampingWnm=R\n\u0011(\")Ank(\")Amk(\")d\"where the elec-\ntron spectral functions are represented by Lorentzian's\nAnk(\") centred around the band energies \"nkand broad-\nened by interactions with the lattice, electron-electron\ninteractions or alloying. The width of the spectral func-\ntion \u0000 provides a phenomenological account for angular\nmomentum transfer to other reservoirs. For inertia, how-\never,Vnm=R\nf(\") (Ank(\")Bmk(\") +Bnk(\")Amk(\")) d\"\nwhereBmk(\") = 2(\"\u0000\"mk)((\"\u0000\"mk)2\u00003\u00002)=((\"\u0000\"mk)2+\u00002)3\n(see Supplementary Material). Here, f(\") and\u0011(\") are\nthe Fermi-Dirac distribution and the \frst derivative of it\nwith respect to \". Knowing the explicit form of Bmk, we\ncan reveal particular properties of the moment of inertia:\ni)for \u0000!0 (\u001c!1 ),Vnm=2=(\"nk\u0000\"mk)3. Sincen=m\nis not excluded, \u0013!\u00001 ; the perturbed electron system\nwill not relax back into the equilibrium. ii)In the limit\n\u0000!1 (\u001c!0), the electron system equilibrates imme-\ndiately into the ground state and, consequently, \u0013= 0.\nThese limiting properties are consistent with the expres-\nsion\u0013=\u0000\u000b\u0001\u001c. Eq. (4) also indicates that the time scale\nis dictated by ~and, consequently, on a femto-second\ntime scale.\nTo study these properties, we performed \frst-\nprinciples tight binding (TB) calculations [38] of the\ntorque-correlation model as described by Eq. (4) as well\nas for the Green's function model reported in Ref. 29.\nThe materials investigated in this letter are bcc Fe, fcc\nCo, and fcc Ni. Since our magnetic moment is \fxed3\n-1·10−3-5·10−405·10−41·10−3−ι(fs)\n10−110+0\nΓ (eV)Fe\nCo\nNiTorque\nGreen\n10−21α10−410−21Γ (eV)\nFIG. 2. (Color) Moment of inertia \u0013as a function of the band\nwidth \u0000 for bcc Fe (green dotes and lines), fcc Co (red dotes\nand lines), and fcc Ni (blue dotes and lines) and with two\ndi\u000berent methods: i)the torque-correlation method (\flled\ntriangles) and the ii)Greens function method [29](\flled cir-\ncles). The dotted gray lines indicating the zero level. The\ninsets show the calculated Gilbert damping \u000bas a function of\n\u0000. Lines are added to guide the eye. Notice the negative sign\nof the moment of inertia.\nin thezdirection, variations occur primarily in xory\nand, consequently, the e\u000bective torque matrix element is\nT\u0000=hn;kj[\u001b\u0000;Hsoc]jm;ki, where\u001b\u0000=\u001bx\u0000i\u001by. The\ncubic symmetry of the selected materials allows only di-\nagonal elements in both damping and moment of inertia\ntensor. The numerical calculations, as shown in Fig. 2,\ngive results that are consistent with the torque-torque\ncorrelation model predictions in both limits, \u0000 !0 and\n\u0000!1 . Note that the latter is only true if we assume\nthe validity of the adiabatic limit up to \u001c= 0. It should\nalso be noted that Eq. (4) is only valid in the adiabatic\nlimit (>10 fs). The strong dependency on \u0000 indicates,\nhowever, that the current model is not a parameter-free\napproach. Fortunately, the relevant parameters can be\nextracted from ab-initio methods: e.g., \u0000 is related ei-\nther to the electron-phonon self energy [39] or to electron\ncorrelations [40].\nThe approximation \u0013=\u0000\u000b\u0001\u001cderived by F ahnle et\nal. [32] from the Kambersk\u0013 y model is not valid for all\n\u0000. It holds for \u0000 <10 meV, where intraband transi-\ntions dominate for both damping and moment of inertia;\nbands with di\u000berent energies narrowly overlap. Here, the\nmoment of inertia decreases proportional to 1=\u00004up to a\ncertain minimum. Above the minimum and with an ap-\npropriate large band width \u0000, interband transitions hap-\npen so that the moment of inertia approaches zero for\nhigh values of \u0000. In this range, the relation \u0013=\u000b\u0001\u001c\nused by Ciornei et al [18] holds and softens the FMR res-\nonance frequency. Comparing qualitative the di\u000berence\n10−410−310−210−1−ι(fs)/α\n510+02510+12510+22\nτ(fs)\n5·10−310−22·10−23·10−2\nΓ (eV)−ι\nαFIG. 3. (Color online) Gilbert damping \u000b(red dashed line),\nmoment of inertia \u0013(blue dashed line), and the resulting nu-\ntation lifetime \u001c=\u0013=\u000b(black line) as a function of \u0000 in the\nintraband region for Fe bulk. Arrows indicating the ordinate\nbelonging of the data lines. Notice the negative sign of the\nmoment of inertia.\nbetween the itinerant magnets Fe, Co and Ni, we obtain\nsimilar features in \u0013and\u000bvs. \u0000, but the position of the\nminimum and the slope in the intraband region varies\nwith the elements: \u0013min= 5:9\u000110\u00003fs\u00001at \u0000 = 60 meV\nfor bcc Fe, \u0013min= 6:5\u000110\u00003fs\u00001at \u0000 = 50 meV for fcc\nCo, and\u0013min= 6:1\u000110\u00003fs\u00001at \u0000 = 80 meV for fcc Ni.\nThe crossing point of intra- and interband transitions for\nthe damping was already reported by Gilmore et al. [35]\nand Thonig et al. [41]. The same trends are also repro-\nduced by applying the Green's function formalism from\nBhattacharjee et al. [29] (see Fig. 2). Consequently, both\nmethods | torque-torque correlation and the linear re-\nsponse Green's function method | are equivalent as it\ncan also be demonstrated not only for the moment of\ninertia but also for the Gilbert damping \u000b(see Supple-\nmentary Material)[41]. In the torque-torque correlation\nmodel (4), the coupling \u0000 de\fnes the width of the en-\nergy window in which transitions Tnmtake place. The\nGreen function approach, however, provides a more ac-\ncurate description with respect to the ab initio results\nthan the torque-torque correlation approach. This may\nbe understood from the fact that a \fnite \u0000 broadens and\nslightly shifts maxima in the spectral function. In par-\nticular, shifted electronic states at energies around the\nFermi level causes di\u000berences in the minimum of \u0013in both\nmodels. Furthermore, the moment of inertia can be re-\nsolved by an orbital decomposition and, like the Gilbert\ndamping\u000b, scales quadratically with the spin-orbit cou-\npling\u0010, caused by the torque operator ^Tin Eq. (4). Thus,\none criteria for \fnding large moments of inertia is by hav-\ning materials with strong spin-orbit coupling.\nIn order to show the region of \u0000 where the approxi-\nmation\u0013=\u0000\u000b\u0001\u001cholds, we show in Fig. 3 calculated\nvalues of\u0013,\u000b, and the resulting nutation lifetime \u001cfor a\nselection of \u0000 that are below \u0013min. According to the data\nreported in Ref. 28, this is a suitable regime accessible4\nfor experiments. To achieve the room temperature mea-\nsured experimental values of \u001c= 0:12\u00000:47 ps, we have\nfurthermore to guarantee that \u0013 >> \u000b . An appropriate\nexperimental range is \u0000 \u00195\u000010 meV, which is realistic\nand caused, e.g., by the electron-phonon coupling. A nu-\ntation lifetime of \u001c\u00190:25\u00000:1 ps is revealed for these\nvalues of \u0000 (see Fig. 3), a value similar to that found in ex-\nperiment. The aforementioned electron-phonon coupling,\nhowever, is underestimated compared to the electron-\nphonon coupling from a Debye model (\u0000 \u001950 meV) [42].\nIn addition, e\u000bects on spin disorder and electron corre-\nlation are neglected, that could lead to uncertainties in\n\u0000 and hence discrepancies to the experiment. On the\nother hand, it is not excluded that other second order\nenergy dissipation terms, Bdiss, proportional to ( @e=@t)2\nwill also contribute [32] (see Supplementary material).\nThe derivation of the moment of inertia tensor from the\nKambersk\u0013 y model and our numerics corroborates that\nrecently observed properties of the Gilbert damping will\nbe also valid for the moment of inertia: i)the moment\nof inertia is temperature dependent [41, 43] and decays\nwith increasing phonon temperature, where the later usu-\nally increase the electron-phonon coupling \u0000 in certain\ntemperature intervals [42]; ii)the moment of inertia is\na tensor, however, o\u000b-diagonal elements for bulk mate-\nrials are negligible small; iii)it is non-local [36, 41, 44]\nand depends on the magnetic moment [45{47]. Note that\nthe sign change of the moment of inertia also e\u000bects the\ndynamics of the magnetic moments (see Supplementary\nMaterial).\nThe physical mechanism of magnetic moment of inertia\nbecomes understandable from an inspection of the elec-\ntron band structure (see Fig. 4 for fcc Co, as an example).\nThe model proposed here allows to reveal the inertia k-\nand band-index nresolved contributions (integrand of\nEq. (4)). Note that we analyse for simplicity and clarity\nonly one contribution, AnBm, in the expression for Vnm.\nAs Fig. 4 shows the contribution to Vnmis signi\fcant only\nfor speci\fc energy levels and speci\fc k-points. The \fg-\nure also shows a considerable anisotropy, in the sense that\nmagnetisations aligned along the z- or y-directions give\nsigni\fcantly di\u000berent contributions. Also, a closer in-\nspection shows that degenerate or even close energy levels\nnandm, which overlap due to the broadening of energy\nlevels, e.g. as caused by electron-phonon coupling, \u0000, ac-\ncelerate the relaxation of the electron-hole pairs caused\nby magnetic moment rotation combined with the spin\norbit coupling. This acceleration decrease the moment\nof inertia, since inertia is the tendency of staying in a\nconstant magnetic state. Our analysis also shows that\nthe moment of inertia is linked to the spin-polarization\nof the bands. Since, as mentioned, the inertia preserves\nthe angular momentum, it has largest contributions in\nthe electronic structure, where multiple electron bands\nwith the same spin-polarization are close to each other\n(cf. Fig. 4 c). However, some aspects of the inertia,\n-4-3-2-10E−EF(eV)\n-4-3-2-10E−EF(eV)\nι<0\nι>0\n-4-3-2-10E−EF(eV)\nΓ H N\nk(a−1\n0)(a)\n(b)\n(c)y\nz\nFIG. 4. (Color online) Moment of inertia in the electron band\nstructure for bulk fcc Co with the magnetic moment a) in y\ndirection and b) in zdirection. The color and the intensity\nindicates the sign and value of the inertia contribution (blue\n-\u0013 <0; red -\u0013 >0; yellow - \u0013\u00190). The dotted gray line\nis the Fermi energy and \u0000 is 0 :1 eV. c) Spin polarization of\nthe electronic band structure (blue - spin down; red - spin up;\nyellow - mixed states).\ne.g. being caused by band overlaps, is similar to the\nGilbert damping [48], although the moment of inertia is\na property that spans over the whole band structure and\nnot only over the Fermi-surface. Inertia is relevant in\nthe equation of motion [17, 35] only for \u001c&0:1 ps and\nparticularly for low dimensional systems. Nevertheless,\nin the literature there are measurements, as reported in\nRef. 37, where the inertia e\u000bects are present.\nIn summary, we have derived a theoretical model for\nthe magnetic moment of inertia based on the torque-\ntorque correlation model and provided \frst-principle\nproperties of the moment of inertia that are compared\nto the Gilbert damping. The Gilbert damping and the\nmoment of inertia are both proportional to the spin-\norbit coupling, however, the basic electron band struc-5\nture mechanisms for having inertia are shown to be dif-\nferent than those for the damping. We analyze details\nof the dispersion of electron energy states, and the fea-\ntures of a band structure that are important for having\na sizable magnetic inertia. We also demonstrate that\nthe torque correlation model provides identical results\nto those obtained from a Greens functions formulation.\nFurthermore, we provide numerical values of the moment\nof inertia that are comparable with recent experimen-\ntal measurements[28]. The calculated moment of inertia\nparameter can be included in atomistic spin-dynamics\ncodes, giving a large step forward in describing ultrafast,\nsub-ps processes.\nAcknowledgements The authors thank Jonas Frans-\nson and Yi Li for fruitful discussions. The support of\nthe Swedish Research Council (VR), eSSENCE and the\nKAW foundation (projects 2013.0020 and 2012.0031) are\nacknowledged. The computations were performed on re-\nsources provided by the Swedish National Infrastructure\nfor Computing (SNIC).\n\u0003danny.thonig@physics.uu.se\n[1] S. S. P. Parkin, J. X., C. Kaiser, A. Panchula, K. Roche,\nand M. Samant, Proceedings of the IEEE 91, 661 (2003).\n[2] Y. Xu and S. 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Under the assumption tha t the damping coefficients have some\nsingularities near the transmission point, we prove that th e semigroup corresponding to the system\nis polynomially stable and the decay rates depends on the spe ed of the degeneracy. This result\nimproves the decay rate of the semigroup associated to the sy stem on an earlier result of Z. Liu\nand Q. Zhang in [20] involving the wave equation with local Ke lvin-Voigt damping and non-smooth\ncoefficient at interface.\nContents\n1. Introduction 1\n2. Well-posedness 4\n3. Strong stability 5\n4. Polynomial stability 7\nReferences 14\n1.Introduction\nWe consider one-dimensional wave propagation through N+1 edges (with N≥1) consisting of\nan elastic and a Kelvin-Voigt medium all connected to one tra nsmission point. The later material\nis a viscoelastic material having the properties both of ela sticity and viscosity. More precisely we\nconsider the following initial and boundary-value problem\n(1.1)\n\n¨u0(x,t)−u′′\n0(x,t) = 0 ( x,t)∈(0,ℓ0)×(0,+∞),\n¨uj(x,t)−/bracketleftbig\nu′\nj(x,t)+dj(x)˙u′\nj(x,t)/bracketrightbig′= 0 (x,t)∈(0,ℓ1)×(0,+∞), j= 1,...,N,\nuj(ℓj,t) = 0 t∈(0,+∞), j= 0,...,N,\nu0(0,t) =···=uN(0,t) t∈(0,+∞),\nu′\n0(0,t)+N/summationdisplay\nj=1u′\nj(0,t)+dj(0)˙u′\nj(0,t) = 0t∈(0,+∞),\nuj(x,0) =u0\nj(x),˙uj(x,0) =u1\nj(x) x∈(0,ℓj), j= 0,...,N,\nwhere the point stands for the time derivative and the prime s tands for the space derivative, uj:\n[0,ℓj]×[0,+∞[−→Rforj= 0,...,Nare the displacement of the of the string of length ℓjand the\ncoefficient damping djis assumed to be a non-negative function.\nThe natural energy of system (1.1) is given by\nE(t) =1\n2N/summationdisplay\nj=0/integraldisplayℓj\n0/parenleftbig\n|˙uj(x,t)|2+|u′\nj(x,t)|2/parenrightbig\ndx\nand it is dissipated according to the following law\nd\ndtE(t) =−N/summationdisplay\nj=1/integraldisplayℓj\n0dj(x)|˙u′\nj(x,t)|2dx,∀t >0.\n2010Mathematics Subject Classification. 35B35, 35B40, 93D20.\nKey words and phrases. Network of strings, Kelvin-Voigt damping, non-smooth coeffi cient.\n12 FATHI HASSINE\nThe stability of this model was intensively studied in this l ast two decades:\nOn high dimensional case: Liu and Rao [19] proved the exponen tial decay of the energy providing\nthat the damping region is a neighborhood of the whole bounda ry, and further restrictions are\nimposed on the damping coefficient. Next, this result was gene ralized and improved by Tebou [25]\nwhen damping is localized in a suitable open subset, of the do main under consideration, which\nsatisfies the piecewise multipliers condition of Liu. He sho ws that the energy of this system decays\npolynomially when the damping coefficient is only boundedmea surable, and it decays exponentially\nwhen the damping coefficient as well as its gradient are bounde d measurable, and the damping\ncoefficient further satisfies a structural condition. Recent ly, Using Carleman estimates Ammari et\nal. [2] show a logarithmic decay rate of the semigroup associ ated to the system when the damping\ncoefficient is arbitrary localized. Next, Burq generalized t his result in [7]. This author shows also a\npolynomial decay rate of the semigroup when the damping regi on verifying some geometric control\ncondition of order equal to1\n2and of order equal to1\n4for a cubic domain with eventual degeneracy\non the coefficient damping in case of dimension 2.\nIn case of the interval (or when N= 1): It is well known that the Kelvin-Voigt damping is much\nstronger than the viscous damping in the sens that if the enti re medium is of the Kelvin-Voigt type,\nthedampingforthewave equation not onlyinducesexponenti al energydecay butalsotheassociated\nto semigroup is analytic [14]; while the entire medium of the viscous type, the associate semigroup\nis only exponential stable [8] and does not have any smoothin g property since the spectrum of the\nsemigroup generator has a vertical asymptote on the left han d side of the imaginary axes of the\ncomplex plane. When the damping is localized (i.e., distrib uted only on the proper subset of the\nspatial domain), such a comparison is not valid anymore. Whi le the local viscous damping still\nenjoys the exponential stability as long as the damped regio n contains an interval of any size in the\ndomain, the local KelvinVoigt damping doesn’t follow the sa me analogue. Chen et al. [9] proved\nlack of the exponential stability when the damping coefficien t is a step function. This unexpected\nresult reveals that the KelvinVoigt damping does not follow the ”geometric optics” condition (see\n[5]). And Liu and Rao [18] proved that the solution of this mod el actually decays at a rate of\nt−2. The optimality of this order was proven by Alves et al. in [1] . In 2002, it was shown in [16]\nthat exponential energy decay still holds if the damping coe fficient is smooth enough. Later, the\nsmoothness condition was weakened in [26] and satisfies the f ollowing condition\na′(0) = 0,/integraldisplayx\n0|a′(s)|2\na(s)ds≤C|a′(x)| ∀x∈[0,1].\nThisindicates that theasymptotic behaviorof thesolution dependsontheregularity ofthedamping\ncoefficient function, which is not the case for the viscous dam ping model. Renardy [24] in 2004\nproved that thereal partof theeigenvalues arenot boundedb elow ifthedampingcoefficient behaves\nlikexαwithα >1near theinterface x= 0. Underthis samecondition Liuet al. [17] proved that the\nsolution of the system is eventually differentiable which als o guarantees the exponential stability\nsince there is no spectrum on the imaginary axis and the syste m is dissipative. In 2016, Under\nthe assumption that the damping coefficient has a singularity at the interface of the damped and\nundamped regions and behaves like xαwithα∈(0,1) near the interface, [20] proved that the\nsemigroup corresponding to the system is polynomially stab le and the decay rate depends on the\nparameter α.\nIn case of multi-link structure: In [10] we proved that the se migroup is polynomially stable\nwhen the coefficient damping is piecewise function with an opt imal decay rate equal to 2 (we refer\nalso to [11] for the case of transmission Euler-Bernoulli pl ate and wave equation with a localized\nKelvin-Voigt damping). Recently, Ammari et al. [3] conside r the tree of elastic strings with local\nKelvin-Voigt damping. They proved under some assumptions o n the smoothness of the damping\ncoefficients, say W2,∞, and some other considerations that the semigroup is expone ntially stable\nif the damping coefficient is continuous at every node of the tr ee and otherwise it is polynomially\nstable with a decay rate equal to 2.\nIn light of all the above results it is obvious to say that the a symptotic behavior of the solution\nto system (1.1) depends on the regularity on the damping coeffi cient function. In this paper we\nwant to generalize and improve the polynomial decay rate giv en by Liu and Zhang in [20] when theSTABILITY OF A STAR-SHAPED NETWORK WITH LOCAL KELVIN-VOIGT DAMPING 3\ndamping coefficient has a singularity at the interface and beh aves like xαwithα∈(0,1). Precisely,\nwe make the following assumptions: For every j= 1,...,N\n•There exist aj, bj∈[0,ℓj] withaj< bjsuch that\n(A 1.2) [ aj,bj]⊂supp(dj) and dj∈L∞(0,ℓj).\n•There exist αj∈(0,1) andκj≥0 such that\n(A 1.3) lim\nx→0+dj(x)\nxαj=κj.\n•There exists ηj∈[0,1) such that\n(A 1.4) lim\nx→0+xd′\nj(x)\ndj(x)=ηj.\nRemark 1.1. The typical example of functions djthat satisfies assumptions (A 1.2),(A 1.3)and\n(A 1.4), is when dj(x) =xαjwithαj∈(0,1)for every j= 1,...,N. An interesting example too\nis when we take dj(x) =xα′\nj|ln(x)|βjwithα′\nj∈(0,1)andβj>0in this case assumptions (A 1.2),\n(A 1.3)and(A 1.4)are satisfied with κj= 0,αj=α′\nj−εjfor all0< εj< α′\njandηj=α′\nj.\nLetH=V×N/productdisplay\nj=0L2(0,ℓj) be the Hilbert space endowed with the inner product define fo r (u,v) =\n((uj)j=0,...,N,(vj)j=0,...,N)∈ Hand (˜u,˜v) = ((˜uj)j=0,...,N,(˜vj)j=0,...,N)∈ Hby\n/an}bracketle{t(u,v),(˜u,˜v)/an}bracketri}htH=N/summationdisplay\nj=0/integraldisplayℓj\n0u′\nj(x).˜u′\nj(x)dx+/integraldisplayℓj\n0vj(x).˜vj(x)dx,\nwhereVis the Hilbert space defined by\nV=\n\n(uj)j=0,...,N∈N/productdisplay\nj=0H1(0,ℓj) :uj(ℓj) = 0∀j= 0,...,N;u0(0) =···=uN(0)\n\n.\nBy setting U(t) = ((uj(t))j=0,...,N,(vj(t))j=0,...,N) andU0=/parenleftig\n(u0\nj)j=0,...,N,(u1\nj)j=0,...,N/parenrightig\nwe can\nrewrite system (1.1) as a first order differential equation as f ollows\n(1.11) ˙U(t) =AU(t), U(0) =U0∈ D(A),\nwhere\nA((uj)j=0,...,N,(vj)j=0,...,N) =/parenleftbig\n(vj)j=0,...,N,(u′′\n0,[u′\n1+d1v′\n1]′,...,[u′\nN+dNv′\nN]′)/parenrightbig\n,\nwith\nD(A) =/braceleftbigg\n((uj)j=0,...,N,(vj)j=0,...,N)∈ H: (vj)j=0,...,N∈V;u′′\n0∈L2(0,ℓ0);\n[u′\nj+djv′\nj]′∈L2(0,ℓj)∀j= 1,...,N;u′\n0(0)+N/summationdisplay\nj=1u′\nj(0)+dj(0)v′\nj(0) = 0/bracerightbigg\n.\nTheorem 1.1. Assume that for j= 1,...,Nthe coefficient functions dj∈ C([0,ℓj])∩ C1((0,ℓj))\nare such that conditions (A 1.2),(A 1.3)and(A 1.4)hold. Then, the semigroup etAassociated to\nsystem(1.1)(see Proposition 2.1) is polynomially stable precisely we h ave: There exists C >0such\nthat\n/bardbletAU0/bardblH≤C\n(t+1)2−α\n1−αk/vextenddouble/vextenddoubleU0/vextenddouble/vextenddouble\nD(Ak)∀U0=/parenleftbig\n(u0\nj)j=0,...,N,(u1\nj)j=0,...,N/parenrightbig\n∈ D(Ak)∀t≥0,\nwhereα= min{α1,...,α N}.4 FATHI HASSINE\nRemark 1.2. This theorem reveals that the stability order of the semigrou petAassociated to problem\n(1.1)depends on the behavior of the damping coefficients djdescribed by the parameters αjfor\nj= 1,...,N. This result improves the decay rate of the energy given in [20]from1\n1−αto2−α\n1−α\nand make it more meaningful in fact, as αgoes to1−the order of polynomial stability2−α\n1−αgoes to\n∞which is consistent with the exponential stability when α= 1(see[3, 20, 16, 20] ) and as αgoes\nto0+the order of polynomial stability2−α\n1−αgoes to2which is consistent with the optimal order\nstability when α= 0(see[1, 10, 18] ).\nRemark 1.3. When the coefficient functions djbehave polynomially near 0asxαjwith0< αj<1\nthen from Theorem 1.1 the semigroup etAdecays polynomially with the decay rate given above in\nthe theorem. Moreover, when djdecay faster than xαjand slower than xαj+εfor allε >0this can\nmay be seen for instance with the example of dj(x) =xαj|ln(x)|βjwithβj>0then according to\nRemark 1.1 the semigroup etAdecays polynomially with the decay rate equal to2−α+ε\n1−α+εfor each\nε∈(0,α)whereα= min{α1,...,α N}. Consequently, this decay rate is worse than if the coefficient\nfunctions djbehave near 0likexαjand better than if the coefficient functions djbehave near 0like\nxα′\njfor anyα′\nj>0such that α′\nj< αj.\nThis article is organized as follows. In section 2, we prove t he well posedness of system (1.1). In\nsection 3, we show that the semigroup associated to the gener atorAis strongly stable . In section\n4, we prove the polynomial decay rate given by Theorem 1.1.\n2.Well-posedness\nIn this section we use the semigroup approach to prove the wel l-posedness of system (1.1).\nProposition 2.1. Assume that condition (A 1.2)holds. Then Agenerates a C0-semigroup of\ncontractions etAon the Hilbert space H.\nProof.For any ( u,v) =/parenleftbig\n(uj)j=0,...,N,(vj)j=0,...,N/parenrightbig\n∈ D(A), we have\n/an}bracketle{tA(u,v),(u,v)/an}bracketri}htH=−N/summationdisplay\nj=1/integraldisplayℓj\n0dj(x)|∂xvj(x)|2dx.\nThis shows that the operator Ais dissipative.\nGiven (f,g) =/parenleftbig\n(fj)j=0,...,N,(gj)j=0,...,N/parenrightbig\n∈ Hwe look for ( u,v) =/parenleftbig\n(uj)j=0,...,N,(vj)j=0,...,N/parenrightbig\n∈ D(A)\nsuch that A(u,v) = (f,g), this is also written\n\n\nvj=fj∀j= 0,...,N,\nu′′\n0=g0,\n(u′\nj+djv′\nj)′=gj∀j= 1,...,N,\nuj(ℓj) = 0∀j= 0,...,N,\nu0(0) =···=uN(0),\nu′\n0(0)+N/summationdisplay\nj=1u′\nj(0)+dj(0)v′\nj(0) = 0,\nor equivalently\n(2.2)\n\nvj=fj∀j= 0,...,N,\nu′′\n0=g0,\nu′′\nj=gj−(djf′\nj)′∀j= 1,...,N,\nuj(ℓj) = 0∀j= 0,...,N,\nu0(0) =···=uN(0),\nu′\n0(0)+N/summationdisplay\nj=1u′\nj(0)+dj(0)f′\nj(0) = 0.STABILITY OF A STAR-SHAPED NETWORK WITH LOCAL KELVIN-VOIGT DAMPING 5\nFor this aim we set the continuous coercive and bi-linear for m inV\nL(u,˜u) =N/summationdisplay\nj=0/integraldisplayℓj\n0u′\nj.˜u′\njdx.\nBy Lax-Milligram theorem there exists a unique element ( uj)j=0,...,N∈Vsuch that\n(2.3)N/summationdisplay\nj=0/integraldisplayℓj\n0u′\nj.˜u′\njdx=−N/summationdisplay\nj=1/integraldisplayℓj\n0djf′\nj.˜u′\njdx−N/summationdisplay\nj=0/integraldisplayℓj\n0gj.˜ujdx.\nIt follows that by taking (2.3) in the sens of distribution th at∂2\nxu0=g0inL2(0,ℓ0) andu′′\nj=\ngj−[djf′\nj]′inL2(0,ℓj) for all j= 1,...,N. Back again to (2.3) and integrating by parts we find\nthatu′\n0(0)+N/summationdisplay\nj=1u′\nj(0)+dj(0)f′\nj(0) = 0. This prove that the operator Ais surjective. Moreover, by\nmultiplying the second line by u1of (2.2) and the third line by ujand integrating over (0 ,ℓ0) and\n(0,ℓj) respectively and summing up then by Poincar´ e inequality a nd Cauchy-Schwarz inequality we\nfind that there exists a constant C >0 such that\nN/summationdisplay\nj=0/integraldisplayℓj\n0|u′\nj|2dx≤C\nN/summationdisplay\nj=0/integraldisplayℓj\n0|f′\nj|2dx+N/summationdisplay\nj=0/integraldisplayℓj\n0|gj|2dx\n\nwhich combined with the first line of (2.2) leads to\nN/summationdisplay\nj=0/integraldisplayℓj\n0|u′\nj|2dx+N/summationdisplay\nj=0/integraldisplayℓj\n0|vj|2dx≤C\nN/summationdisplay\nj=0/integraldisplayℓj\n0|f′\nj|2dx+N/summationdisplay\nj=0/integraldisplayℓj\n0|gj|2dx\n.\nThis implies that 0 ∈ρ(A) and by contraction principle, we easily get R(λ−A) =Hfor sufficient\nsmallλ >0. Since D(A) is dense in Hthen thanks to Lumer-Phillips theorem [22, Theorem 1.4.3],\nAgenerates a C0-semi-group of contractions on H. /square\nAs a consequence of Proposition 2.1 we have the following wel l-posedness result of system (1.1).\nCorollary 2.1. For any initial data U0∈ H, there exists a unique solution U(t)∈ C([0,+∞[,H)\nto the problem (1.11). Moreover, if U0∈ D(A), then\nU(t)∈ C([0,+∞[,D(A))∩C1([0,+∞),H).\n3.Strong stability\nThe aim of this section is to prove that the semi-group genera ted by the operator Ais strongly\nstable. In another words this means that the energy of system (1.1) degenerates over the time to\nzero.\nLemma 3.1. Assume that condition (A 1.2)holds. Then for every λ∈Rthe operator (iλ−A)is\ninjective.\nProof.Since 0∈ρ(A) (according to the proof of Theorem 2.3), we only need to chec k that for every\nλ∈R∗we have ker( iλI−A) ={0}. Letλ/ne}ationslash= 0 and ( u,v) =/parenleftbig\n(uj)j=0,...,N,(vj)j=0,...,N/parenrightbig\n∈ D(A) such\nthat\n(3.2) A(u,v) =iλ(u,v)\nTaking the real part of the inner product in Hof (3.2) with ( u,v) and using the dissipation of A,\nwe get\nReiλ/bardbl(u,v)/bardbl2\nH= Re/an}bracketle{tA(u,v),(u,v)/an}bracketri}htH=−N/summationdisplay\nj=1/integraldisplayℓj\n0dj(x)|v′\nj(x)|2dx= 0,\nwhich implies that\n(3.3) djv′\nj= 0 inL2(0,ℓj),∀j= 1,...,N.6 FATHI HASSINE\nInserting (3.3) into (3.2), we obtain\n(3.4)\n\niλuj=vj in (0,ℓj), j= 0,...,N,\nλ2uj+u′′\nj= 0 in (0 ,ℓj), j= 0,...,N,\nuj(ℓj) = 0 j= 0,...,N,\nu1(0) =···=uN(0)\nN/summationdisplay\nj=0u′\nj(0) = 0\nCombining (3.3) with the first line of (3.4), we get\nu′\nj= 0 a.e in [ aj,bj]∀j= 1,...,N.\nSinceuj∈H2(0,ℓj) then following to the embedding H1(0,ℓj)֒→C0(0,ℓj) we have\n(3.5) u′\nj≡0 in [aj,bj]∀j= 1,...,N.\nWhich by the second line of (3.4) leads to\nuj≡0 in [aj,bj]∀j= 1,...,N.\nSo for every j= 1,...,N,ujandvjare solution to the following problem\n\n\niλuj=vj in (0,ℓj),\nλ2uj+u′′\nj= 0 in (0 ,ℓj),\nuj(aj) =u′\nj(aj) = 0,\nand this clearly gives that uj=vj≡0 in (0,ℓj) for every j= 1,...,N. Following to system (3.4)\nwe have then\n(3.6)\n\niλu0=v0 in (0,ℓ0),\nλ2u0+u′′\n0= 0 in (0 ,ℓ0)\nu0(ℓ0) =u′\n0(ℓ0) = 0\nwhich gives also that u0=v0≡0 in (0,ℓ0). This shows that ( u,v) = (0,0) and consequently\n(iλI−A) is injective for all λ∈R. /square\nLemma 3.2. Assume that condition (A 1.2)holds. Then for every λ∈Rthe operator (iλ−A)is\nsurjective.\nProof.Since 0∈ρ(A) (see proof of Theorem 2.3), we only need to check that for eve ryλ∈R∗we\nhaveR(iλI−A) =H. Letλ/ne}ationslash= 0, then given ( f,g) =/parenleftbig\n(fj)j=0,...,N,(gj)j=0,...,N/parenrightbig\n∈ Hwe are looking\nfor (u,v) =/parenleftbig\n(uj)j=0,...,N,(vj)j=0,...,N/parenrightbig\nD(A) such that\n(3.8) ( iλI−A)(u,v) = (f,g),\nor equivalently\n(3.9)\n\nvj=iλuj−fj in (0,ℓj), j= 0,...,N,\n−λ2u0−u′′\n0=iλf0+g0 in (0,ℓ0),\n−λ2uj−(u′\nj+iλdjuj′)′=iλfj+gj−(djf′\nj)′in (0,ℓj), j= 1,...,N.\nWe define for all u= ((uj)j=0,...,N/parenrightbig\n∈Vthe operator\nAu=/parenleftbig\n−u′′\n0,−(u′\n1+iλdju′\n1)′,...,−(u′\nN+iλdNu′\nN)′/parenrightbig\n.\nThanks to Lax-Milgram’s theorem [15, Theorem 2.9.1], it is e asy to show that Ais an isomorphism\nfromVintoV′(whereV′is the dual space of Vwith respect to the pivot space H). Then the\nsecond ant the third line of (3.9) can be written as follows\n(3.10) u−λ2A−1u=A−1/parenleftbig\niλf0+g0,iλf1+g1−(d1f′\n1)′,...,iλf N+gN−(dNf′\nN)′/parenrightbig\n.\nIfu∈ker(I−λA−1), then we obtain\n(3.11)/braceleftbiggλ2u0+u′′\n0= 0 in (0 ,ℓ0),\nλ2uj+(u′\nj+iλdju′\nj)′= 0 in (0 ,ℓj), j= 1,...,N.STABILITY OF A STAR-SHAPED NETWORK WITH LOCAL KELVIN-VOIGT DAMPING 7\nForj= 1,...,Nwe multiply each line of (3.11) by ujand integrating over (0 ,ℓj) and summing up\n(3.12) λ2N/summationdisplay\nj=0/integraldisplayℓj\n0|uj|2dx−N/summationdisplay\nj=0/integraldisplayℓj\n0|u′\nj|2dx−iλN/summationdisplay\nj=1/integraldisplayℓj\n0dj|u′\nj|2dx= 0.\nBy taking the imaginary part of (3.12) we get\nN/summationdisplay\nj=1/integraldisplayℓj\n0dj|u′\nj|2dx= 0.\nThis means that dju′\nj= 0 in (0 ,ℓj) for alli= 1,...,N, which inserted into (3.11) one gets\nλ2uj+u′′\nj= 0 in (0 ,ℓj), j= 0,...,N.\nThen using the same arguments as proof of Lemma 3.1 we find that u= 0. Hence, we proved\nthat ker( I−λ2A−1) ={0}. Besides, thanks to the compact embeddings V ֒→HandH ֒→V′\nthe operator A−1is compact in V. So that, following to Fredholm’s alternative, the operato r\n(I−λ2A−1) is invertible in V. Therefore, equation (3.10) have a unique solution in V. Thus, the\noperator iλI−Ais surjective. This completes the proof. /square\nThanks to Lemmas 3.1 and 3.2 and the closed graph theorem we ha veσ(A)∩iR=∅. This with\nArendt and Batty [4] result following to which a C0-semi-group of contractions in a Banach space\nis strongly stable, if ρ(A)∩iRcontains only a countable number of continuous spectrum of Alead\nto the following\nTheorem 3.1. Assume that condition (A 1.2)holds. Then the semigroup (etA)t≥0is strongly stable\nin the energy space Hi.e.,\nlim\nt→+∞/bardbletAU0/bardblH= 0,∀U0∈ H.\n4.Polynomial stability\nIn this section, we prove Theorem 1.1. The idea is to estimate the energy norm and boundary\nterms at the interface by the local viscoelastic damping. Th e difficulty is to deal with the higher\norderboundarytermattheinterfacesothattheenergyon(0 ,ℓ0)canbecontrolledbytheviscoelastic\ndamping on (0 ,ℓj) for every j= 1,...,N. Our proof is based on the following result\nProposition 4.1. [6, Theorem 2.4] LetetBbe a bounded C0-semi-group on a Hilbert space Xwith\ngenerator Bsuch that iR∈ρ(A). ThenetBis polynomially stable with order1\nγi.e. there exists\nC >0such that\n/bardbletBu/bardblX≤C\nt1\nγ/bardblu/bardblD(B)∀u∈ D(B)∀t≥0,\nif and only if\nlimsup\n|λ|→∞/bardblλ−γ(iλI−B)−1/bardblX<∞.\nAccording to Proposition 4.1 we shall verify that for α= min{α1,...,α N}andγ=1−α\n2−αthere\nexistsC0>0 such that\n(4.1)\ninf\n/bardbl((uj)j=0,...,N,(vj)j=0,...,N)/bardblH=1\nλ∈Rλγ/bardbliλ((uj)j=0,...,N,(vj)j=0,...,N)−A((uj)j=0,...,N,(vj)j=0,...,N)/bardblH≥C0.\nSuppose that (4.1) fails then there exist a sequence of real n umbersλnand a sequence of functions\n(un,vn)n∈N=/parenleftbig\n(u0,n,...,u N,n),(v0,n,...,vN,n)/parenrightbig\nn∈N⊂ D(A) such that\nλn−→ ∞asn−→ ∞, (4.2)/vextenddouble/vextenddouble(un,vn)/vextenddouble/vextenddouble= 1, (4.3)\nλγ\nn/bardbliλn(un,vn)−A(un,vn)/bardblH=o(1). (4.4)8 FATHI HASSINE\nSince, we have\n(4.5) λγ\nnRe/an}bracketle{tiλ(un,vn)−A(un,vn),(un,vn)/an}bracketri}ht=λγ\nnn/summationdisplay\nj=1/integraldisplayℓj\n0dj|v′\nj,n|2dx\nthen using (4.3) and (4.4) we obtain\n(4.6)n/summationdisplay\nj=1/bardbld1\n2\njv′\nj,n/bardblL2(0,ℓj)=o(λ−γ\n2n).\nFollowing to (4.4) we have\nλγ\nn(iλnuj,n−vj,n) =fj,n−→0 inH1(0,ℓj), j= 0,...,N, (4.7)\nλγ\nn(iλnv0,n−u′′\n0,n) =g0,n−→0 inL2(0,ℓ0), (4.8)\nλγ\nn(iλnvj,n−T′\nj,n) =gj,n−→0 inL2(0,ℓj), j= 1,...,N, (4.9)\nwith the transmission conditions\nu0,n(0) =···=uN,n(0), (4.10)\nu′\n0,n(0)+N/summationdisplay\nj=1Tj,n(0) = 0, (4.11)\nwhere for j= 1,...,Nwe have denoted by\n(4.12) Tj,n=u′\nj,n+djv′\nj,n= (1+iλndj)u′\nj,n−λ−γ\nndjf′\nj,n.\nBy (4.6) and (4.7) we find\n(4.13)n/summationdisplay\nj=1/bardbld1\n2\nju′\nj,n/bardblL2(0,ℓj)=o(λ−γ\n2−1\nn).\nOne multiplies (4.8) by ( x−ℓ0)u′\n0,nintegrating by parts over the interval (0 ,ℓ0) and use (4.3) and\n(4.7) we get\n(4.14)/integraldisplayℓ0\n0/parenleftbig\n|u′\n0,n|2+|v0,n|2/parenrightbig\ndx−ℓ0/parenleftbig\n|u′\n0,n(0)|2+|v0,n(0)|2/parenrightbig\n=o(1).\nWe multiply (4.9) by vj,nforj= 1,...,Nand (4.8) by v0,nthen integrating over (0 ,ℓj) forj=\n0,...,Nand summing up to get\n(4.15) iλγ+1\nnN/summationdisplay\nj=0/bardblvj,n/bardbl2\nL2(0,ℓj)+λγ\nnN/summationdisplay\nj=0/an}bracketle{tu′\nj,n,v′\nj,n/an}bracketri}htL2(0,ℓj)+λγ\nnN/summationdisplay\nj=1/bardbld1\n2\njv′\nj,n/bardblL2(0,ℓj)=o(1).\nWe take the inner product of (4.7) with uj,ninH1(0,ℓj) forj= 0,...,Nand summing up,\n(4.16) iλγ+1\nnN/summationdisplay\nj=0/bardblu′\nj/bardbl2\nL2(0,ℓj)−λγ\nnN/summationdisplay\nj=0/an}bracketle{tv′\nj,n,u′\nj,n/an}bracketri}htL2(0,ℓj)=o(1).\nAdding (4.15) and (4.16) and taking the imaginary part of the equality then by (4.6) we arrive at\n(4.17)N/summationdisplay\nj=0/parenleftig\n/bardblu′\nj,n/bardblL2(0,ℓj)−/bardblvj,n/bardblL2(0,ℓj)/parenrightig\n=o(1).\nAt this stage we recall the following Hardy type inequalitie s\nLemma 4.1. [21, Theorem 3.8] LetL >0anda: [0,L]→R+be such that a∈ C([0,L])∩C1((0,L])\nand satisfying\nlim\nx→+∞xa′(x)\na(x)=η∈[0,1).STABILITY OF A STAR-SHAPED NETWORK WITH LOCAL KELVIN-VOIGT DAMPING 9\nThen there exists C(η,L)>0such that for all locally continuous function zon[0,L]satisfying\nz(0) = 0 and/integraldisplayL\n0a(x)|z′(x)|2dx <∞\nthe following inequality holds\n/integraldisplayL\n0a(x)\nx2|z(x)|2dx≤C(η,L)/integraldisplayL\n0a(x)|z′(x)|2dx.\nLemma 4.2. [20, Lemma 2.2] LetL >0andρ1, ρ2>0be two weight functions defied on (0,L).\nThen the following conditions are equivalent:\n(4.18)/integraldisplayL\n0ρ1(x)|Tf(x)|2dx≤C/integraldisplayL\n0ρ2(x)|f(x)|2dx,\nand\nK= sup\nx∈(0,L)/parenleftbigg/integraldisplayL−x\n0ρ1(x)dx/parenrightbigg/parenleftbigg/integraldisplayL\nL−x[ρ2(x)]−1dx/parenrightbigg\n<∞\nwhereTf(x) =/integraldisplayx\n0f(s)dx. Moreover, the best constant Cin(4.18)satisfies K≤C≤2K.\nLetβsuch that1−αj\n2< β <1 for all j= 1,...,Nandδjare positive numbers that will be\nspecified later. Following to Lemmas 4.1 and 4.2 and assumpti ons (A 1.3) and (A 1.4) then for n\nlarge enough\n/bardblvj,n/bardbl\nL2/parenleftBigg/bracketleftBigg\nλ−δj\nn\n2,λ−δj\nn/bracketrightBigg/parenrightBigg≤max\nx∈/bracketleftBigg\nλ−δj\nn\n2,λ−δj\nn/bracketrightBigg/braceleftigg\nx1−β\ndj(x)1\n2/bracerightigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubled1\n2\nj\nx(xβvj,n)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2/parenleftBigg/bracketleftBigg\nλ−δj\nn\n2,λ−δj\nn/bracketrightBigg/parenrightBigg\n≤Cλ−δj(1−β−αj\n2)\nn/parenleftigg/vextenddouble/vextenddouble/vextenddouble/vextenddoubled1\n2\njxβv′\nj,n/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2([0,ℓj])+β/vextenddouble/vextenddouble/vextenddouble/vextenddoubled1\n2\njxβ−1vj,n/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2([0,ℓj])/parenrightigg\n≤Cλ−δj(1−β−αj\n2)\nn/vextenddouble/vextenddouble/vextenddouble/vextenddoubled1\n2\njv′\nj,n/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2([0,ℓj]). (4.21)\nPerforming the following calculation and uses (4.21) one fin ds\n(4.22)\nmin\nx∈/bracketleftBigg\nλ−δj\nn\n2,λ−δj\nn/bracketrightBigg{|vj,n(x)|+|Tj,n(x)|} ≤√\n2λδj\n2n\n/bardblvj,n/bardbl\nL2/parenleftBigg/bracketleftBigg\nλ−δj\nn\n2,λ−δj\nn/bracketrightBigg/parenrightBigg+/bardblTj,n/bardbl\nL2/parenleftBigg/bracketleftBigg\nλ−δj\nn\n2,λ−δj\nn/bracketrightBigg/parenrightBigg\n\n≤√\n2λδj\n2n\n/bardblvj,n/bardbl\nL2/parenleftBigg/bracketleftBigg\nλ−δj\nn\n2,λ−δj\nn/bracketrightBigg/parenrightBigg+/bardblu′\nj,n/bardbl\nL2/parenleftBigg/bracketleftBigg\nλ−δj\nn\n2,λ−δj\nn/bracketrightBigg/parenrightBigg+/bardbldjv′\nj,n/bardbl\nL2/parenleftBigg/bracketleftBigg\nλ−δj\nn\n2,λ−δj\nn/bracketrightBigg/parenrightBigg\n\n≤Cλδj\n2n/parenleftigg\nλ−δj(1−β−αj\n2)\nn/vextenddouble/vextenddouble/vextenddouble/vextenddoubled1\n2\njv′\nj,n/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2([0,ℓj])+ max\nx∈/bracketleftBigg\nλ−δj\nn\n2,λ−δj\nn/bracketrightBigg{dj(x)−1\n2}/bardbld1\n2\nju′\nj,n/bardblL2([0,ℓj])\n+ max\nx∈/bracketleftBigg\nλ−δj\nn\n2,λ−δj\nn/bracketrightBigg{dj(x)1\n2}/bardbld1\n2\njv′\nj,n/bardblL2([0,ℓj])/parenrightigg\n≤Cλδj\n2n/parenleftigg\nλ−δj(1−β−αj\n2)\nn/vextenddouble/vextenddouble/vextenddouble/vextenddoubled1\n2\njv′\nj,n/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2([0,ℓj])+λδjαj\n2n/bardbld1\n2\nju′\nj,n/bardblL2([0,ℓj])+λ−δjαj\n2n/bardbld1\n2\njv′\nj,n/bardblL2([0,ℓj])/parenrightigg\n.10 FATHI HASSINE\nInserting (4.6), (4.13) into (4.22), then we obtain\nmin\nx∈/bracketleftBigg\nλ−δj\nn\n2,λ−δj\nn/bracketrightBigg{|vj,n(x)|+|Tj,n(x)|}=/parenleftbigg\nλ−δj(1−αj\n2−β)−γ\n2n +λδj\n2(αj+1)−γ\n2−1\nn +λδj\n2(1−αj)−γ\n2n/parenrightbigg\no(1),\nfor every βsuch that1−αj\n2< β <1 for every j= 1,...,N. Then we can choose βsuch that\n−δj(1−αj\n2−β)−γ\n2<0 for every j= 1,...,N. Hence, as long as we choose δj>0 andγ >0\nsuch that\n(4.23)δj\n2(αj+1)−γ\n2−1≤0 andδj\n2(1−αj)−γ\n2≤0∀j= 1,...,N.\nthe following estimate holds\n(4.24) min\nx∈/bracketleftBigg\nλ−δj\nn\n2,λ−δj\nn/bracketrightBigg{|vj,n(x)|+|Tj,n(x)|}=o(1).\nAnd consequently, we are able to find ξj,n∈/bracketleftbigg\nλ−δj\nn\n2,λ−δjn/bracketrightbigg\nsuch that\n(4.25) |vj,n(ξj,n)|=o(1) and |Tj,n(ξj,n)|=o(1).\nWe set\nz±\nj,n(x) =iλn/radicalbig\n1+λndj(x)/integraldisplayξj,n\nxvj,n(τ)dτ±vj,n(x),∀x∈[0,ξj,n].\nThen we have\n(4.26)z± ′\nj,n(x) =−iλnd′\nj(x)\n4(1+λndj(x))(z+\nj,n(x)+z−\nj,n(x))∓iλn/radicalbig\n1+λndj(x)z±\nj,n(x)\n∓λ2\nn\n1+λndj(x)/integraldisplayξj,n\nxvj,n(τ)dτ±v′\nj,n(x).\nCombining (4.7) and (4.12) we have\n±v′\nj,n(x) =±iλnu′\nj,n(x)∓λ−γ\nnf′\nj,n(x) =±iλn\n1+iλndj(x)Tj,n(x)±iλ1−γ\nndj(x)\n1+iλndjf′\nj,n(x)∓λ−γ\nnf′\nj,n(x)\n=±iλn\n1+iλndjTj,n(x)∓λ−γ\nn\n1+iλndj(x)f′\nj,n(x). (4.27)\nIntegrating (4.9) over ( x,ξj,n) and multiplying by ±iλn\n1+iλndj(x)then we get\n(4.28)\n∓λ2\nn\n1+iλndj(x)/integraldisplayξj,n\nxvj,n(τ)dτ=±iλn\n1+iλndj(x)(Tj,n(ξj,n)−Tj,n(x))±iλ1−γ\nn\n1+iλndj(x)/integraldisplayξj,n\nxgj,n(τ)dτ.\nInserting (4.27) and (4.28) into (4.26), we find\n(4.29)\nz± ′\nj,n(x) =∓iλn/radicalbig\n1+λndj(x)z±\nj,n(x)−iλnd′\nj(x)\n4(1+λndj(x))(z+\nj,n(x)+z−\nj,n(x))±iλn\n1+iλndj(x)Tj,n(ξj,n)±Fj,n(x)\nwhere\nFj,n(x) =−λ−γ\nn\n1+iλndj(x)f′\nj,n(x)+iλ1−γ\nn\n1+iλndj(x)/integraldisplayξj,n\nxgj,n(τ)dτ.STABILITY OF A STAR-SHAPED NETWORK WITH LOCAL KELVIN-VOIGT DAMPING 11\nSolving (4.29), then for every x∈[0,ξj,n], one gets\n(4.30)\nz±\nj,n(x) =vj,n(ξj,n)e∓(qj,n(x)−qj,n(ξj,n))−/integraldisplayx\nξj,ne∓(qj,n(x)−qj,n(s))iλnd′\nj(s)\n4(1+λndj(s))(z+\nj,n(s)+z−\nj,n(s))ds\n±Tj,n(ξj,n)/integraldisplayx\nξj,ne∓(qj,n(x)−qj,n(s))iλn\n1+iλndj(s)ds±/integraldisplayx\nξj,ne∓(qj,n(x)−qj,n(s))Fj,n(s)ds\nwhere\nqj,n(x) =iλn/integraldisplayx\n0ds/radicalbig\n1+iλndj(s).\nFor allx∈[0,ξj,n] we have\nqj,n(x) =iλn/integraldisplayx\n0eiϕj,n(s)\n(1+(λj,ndj(s))2)1\n4ds\nwhere\nϕj,n(s) =−1\n2arg(1+iλndj,n(s)).\nConsequently,\nRe(qj,n(x)) =±λn/integraldisplayx\n0sin(ϕj,n(s))\n(1+(λj,ndj(s))2)1\n4ds.\nWhens∈[0,ξj,n] and from assumption we have\n(4.31)1\n(1+(λj,ndj(s))2)1\n4=/braceleftiggO(1) if δjαj≥1\nO(λδjαj−1\n2n) ifδjαj<1\nand\n|sin(ϕj,n(s))|=/radicaligg\n1\n2−1\n2/radicalbig\n1+(λndj(s))2\n=/braceleftbigg\nO(λ1−δjαjn) ifδjαj>1\nO(1) if δjαj≤1.(4.32)\nWith 0≤x≤s≤ξj,n, from (4.31) and (4.32) we see that\n|Re(qj,n(x)−qj,n(s))| ≤λn/integraldisplays\nx|sin(ϕj,n(τ))|\n(1+(λndj(τ))2)1\n4dτ\n≤sup\nτ∈(x,s)/braceleftigg\n|sin(ϕj,n(τ))|\n(1+(λndj(τ))2)1\n4/bracerightigg\nλn(s−x)\n=\n\nO(λ−(αj+1)δj+2\nn ) =o(1) if δjαj>1\nO(λ1−δjn) =O(λ1−1\nαjn) =o(1) ifδjαj= 1\nO(λδj(αj−2)+1\n2n ) if δjαj<1.\nThis implies that\n|e±(qj,n(x)−qj,n(s))| ≤1 (4.33)\nproviding that δj>0 satisfying (4.23) and\n(4.34) δj≥1\nαjorδj≤1\n2−αj∀j= 1,...,N.12 FATHI HASSINE\nFrom (4.31) for every x∈[0,ξj,n], we have\n/integraldisplayξj,n\nxds\n|1+iλndj(s)|=/braceleftbiggO(1)(ξj,n−x) if δjαj≥1\nO(λαjδj−1\nn)(ξj,n−x) ifδjαj<1\n=/braceleftigg\nO(λ−δjn) if δjαj≥1\nO(λ(αj−1)δj−1\nn ) ifδjαj<1(4.35)\nand\n/parenleftbigg/integraldisplayξj,n\nxds\n|1+iλndj(s)|2/parenrightbigg1\n2\n=\n\nO(λ−δj\n2n) if δjαj≥1\nO(λ(αj−1\n2)δj−1\nn ) ifδjαj<1.(4.36)\nTherefore, when δjandγsatisfy (4.23) and (4.34), form (4.25), (4.33) and (4.35) we obtain\n|Tj,n(ξj,n)|./vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayx\nξj,ne∓(qj,n(x)−qj,n(s))iλn\n1+iλndj(s)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ |Tj,n(ξj,n)|./integraldisplayξj,n\nxλn\n|1+iλndj(s)|ds\n=o(1)∀x∈[0,ξj,n]. (4.37)\nMoreover, underthesameconditionson δjandγ, dueto(4.7), (4.33)and(4.36)theCauchy-Schwarz\ninequality leads to\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayx\nξj,ne∓(qj,n(x)−qj,n(s))λ−γ\nn\n1+iλndj(s)f′\nj,n(s)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/integraldisplayξj,n\nxλ−γ\nn\n|1+iλndj(s)||f′\nj,n(s)|ds\n≤λ−γ\nn/bardblf′\nj,n/bardblL2(0,ℓj)/parenleftbigg/integraldisplayξj,n\nxds\n|1+iλndj(s)|2/parenrightbigg1\n2\n=\n\no(λ−δj\n2−γ\nn) if δjαj≥1\no(λδj(αj−1\n2)−γ−1\nn ) ifδjαj<1\n=o(1)∀x∈[0,ξj,n], (4.38)\nand due to (4.9), (4.33) and (4.35), we have\n/integraldisplayx\nξj,ne∓(qj,n(x)−qj,n(s))iλ1−γ\nn\n1+iλndj(s)/integraldisplayξj,n\nsgj,n(τ)dτds\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayξj,n\nx/integraldisplayτ\nxe∓(qj,n(x)−qj,n(s))λ1−γ\nn\n1+iλndj(s)gj,n(τ)dτds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤λ1−γ\nn/integraldisplayξj,n\nx/integraldisplayτ\nx|gj,n(τ)|\n|1+iλndj(s)|dτds\n≤λ1−γ\nn/integraldisplayξj,n\nxds\n|1+iλndj(s)|/integraldisplayξj,n\nx|gj,n(τ)|dτ\n=\n\no(λ1−γ−3δj\n2n) ifδjαj≥1\no(λδj(αj−3\n2)−γ\nn ) ifδjαj<1\n=o(1)∀x∈[0,ξj,n]. (4.39)\nCombining (4.38) and (4.39) yields\n(4.40)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayx\nξj,ne∓(qj,n(x)−qj,n(s))Fj,n(s)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=o(1)∀x∈[0,ξj,n].STABILITY OF A STAR-SHAPED NETWORK WITH LOCAL KELVIN-VOIGT DAMPING 13\nFrom assumption (A 1.3) we have d′\nj(s)>0 near 0, then by Cauchy-Schwarz inequality and as-\nsumption (A 1.4) we find\n/integraldisplayξj,n\nxλnd′\nj(s)\n|1+iλndj(s)|ds≤/parenleftigg/integraldisplayξj,n\nxλ2\nnd′\nj(s)\n1+(λndj(s))2ds/parenrightigg1\n2\n./parenleftbigg/integraldisplayξj,n\nxd′\nj(s)ds/parenrightbigg1\n2\n≤/parenleftbig\narctan(λ2\nndj(ξj,n))−arctan(λ2\nndj(x))/parenrightbig1\n2.(dj(ξj,n)−dj(x))1\n2\n=O(λ−αjδj\n2n)∀x∈[0,ξj,n]. (4.43)\nInserting (4.25), (4.33), (4.37), (4.40) and (4.43) into (4 .30), then we get\n|z±\nj,n(x)| ≤o(1)(mj,n+1)∀x∈[0,ξj,n],\nwhere\nmj,n= max\nx∈[0,ξj,n]{|z+\nj,n(x)|+|z−\nj,n(x)|},\nwhich leads to\n(4.44) mj,n=o(1).\nSince we can write\nvj,n(x) =1\n2(z+\nj,n(x)−z−\nj,n(x))\nthen we follow from (4.44) that\n(4.45) /bardblvj,n/bardblL2(0,ξj,n)=1\n2/bardblz+\nj,n−z−\nj,n/bardblL2(0,ξj,n)≤mj,n\n2/radicalbig\nξj,n=o(λ−δj\n2n)\nand\n(4.46) |vj,n(0)|=1\n2|z+\nj,n(0)−z−\nj,n(0)| ≤mj,n\n2=o(1).\nIntegrating (4.9) over (0 ,ξj,n),\n(4.47) iλn/integraldisplayξj,n\n0vj,n(s)ds−Tj,n(ξj,n)+Tj,n(0) =λ−γ\nn/integraldisplayξj,n\n0gj,n(s)ds.\nDue to (4.44) and the fact that a(0) = 0, we have\n(4.48)/vextendsingle/vextendsingle/vextendsingle/vextendsingleiλn/integraldisplayξj,n\n0vj,nds/vextendsingle/vextendsingle/vextendsingle/vextendsingle=1\n2|z+\nj,n(0)−z−\nj,n(0)|=o(1).\nSubstituting (4.9), (4.24) and (4.48) into (4.47) yields\n(4.49) |Tj,n(0)|=o(1).\nSubstituting (4.46), (4.49) into (4.14), by the transmissi on conditions (4.10) and (4.11) we con-\nclude\n(4.50)/integraldisplayℓ0\n0/parenleftbig\n|u′\n0,n|2+|v0,n|2/parenrightbig\ndx=o(1).\nForj= 1,...,Nwe have\n/bardbld1\n2\nju′\nj/bardblL2(0,ℓj)≥ /bardbld1\n2\nju′\nj/bardblL2(ξj,n,ℓj)≥min\nx∈(ξj,n,ℓj)/parenleftbigg/radicalig\ndj(x)/parenrightbigg\n/bardblu′\nj,n/bardblL2(ξj,n,ℓj)\n≥/radicalig\ndj(ξj,n)/bardblu′\nj,n/bardblL2(ξj,n,ℓj)≥Cξαj\n2\nj,n/bardblu′\nj,n/bardblL2(ξj,n,ℓj)≥Cλ−δjαj\n2n/bardblu′\nj,n/bardblL2(ξj,n,ℓj). (4.51)\nFrom (4.23) we haveαjδj\n2≤γ\n2+1 for every j= 1,...,N, then by combining (4.13) and (4.51) one\ngets\n(4.52) /bardblu′\nj,n/bardblL2(ξj,n,ℓj)=o(λαjδj\n2−γ\n2−1\nn ).14 FATHI HASSINE\nTherefore by the trace formula\n(4.53) |uj,n(ξj,n)|=o(λαjδj\n2−γ\n2−1\nn ).\nBy trace formula and (4.12) we have\n(4.54) |uj,n(0)| ≤C/bardblu′\nj,n/bardblL2(0,ℓj)=O(λαjδj\n2−γ\n2−1\nn ).\nFrom (4.12) and (4.7) one has\n(4.55) λn/bardbluj,n/bardblL2(0,ℓj)=O(1).\nMultiplying (4.9) by λ−γ\nnuj,nand integrating over (0 ,ξj,n) then by integrating by parts we arrive\n(4.56)iλn/an}bracketle{tvj,n,uj,n/an}bracketri}htL2(0,ξj,n)+/an}bracketle{tdj,nvj,n,uj,n/an}bracketri}htL2(0,ξj,n)+/bardblu′\nj,n/bardblL2(0,ξj,n)\n+Tj,n(0)uj,n(0)−Tj,n(ξj,n)uj,n(ξj,n) =o(λ−γ\nn).\nInserting (4.25), (4.45), (4.49), (4.53), (4.54) and (4.55 ) into (4.56) one finds\n(4.57) /bardblu′\nj,n/bardblL2(0,ξj,n)=o(1),∀j= 1,...,N.\nSo that, adding (4.52) and (4.57), we obtain\n(4.58) /bardblu′\nj,n/bardblL2(0,ℓj)=o(1),∀j= 1,...,N.\nFrom (4.17), (4.50) and (4.58) leads to\n(4.59) /bardblvj/bardblL2(0,ℓj)=o(1),∀j= 1,...,N.\nThis conclude the prove since now we have proved that /bardbl(un,vn)/bardblH=o(1) from (4.50), (4.58) and\n(4.59) providing that (4.23) and (4.34) hold true.\nFinally, notingthat thebest γandδj, intermofmaximization of γ, that satisfies (4.23)and(4.34)\nare where γ= max/braceleftbigg1−αj\n2−αj, j= 1,...,N/bracerightbigg\nandδj=1\n2−αjforj= 1,...,N. 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Tebou, Stabilization of some elastic systems with localized Kelvi n-Voigt damping, Discrete and continuous\ndynamical systems, 36(2016), 7117–7136.\n[26]Q. Zhang, Exponential stability of an elastic string with local Kelvi nVoigt damping, Z. Angew. Math. Phys.,\n61(2010), 1009–1015.\nUR Analysis and Control of PDEs, UR 13ES64, Department of Mat hematics, Faculty of Sciences\nof Monastir, University of Monastir, Tunisia\nE-mail address :fathi.hassine@fsm.rnu.tn"    },    {        "title": "1704.03326v1.CoFeAlB_alloy_with_low_damping_and_low_magnetization_for_spin_transfer_torque_switching.pdf",        "content": "arXiv:1704.03326v1  [cond-mat.mtrl-sci]  11 Apr 2017CoFeAlB alloy with low damping and low magnetization for spi n transfer torque\nswitching\nA. Conca,1,∗T. Nakano,2T. Meyer,1Y. Ando,2and B. Hillebrands1\n1Fachbereich Physik and Landesforschungszentrum OPTIMAS,\nTechnische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany\n2Department of Applied Physics, Tohoku University, Japan\n(Dated: June 29, 2021)\nWe investigate the effect of Al doping on the magnetic propert ies of the alloy CoFeB. Comparative\nmeasurements of the saturation magnetization, the Gilbert damping parameter αand the exchange\nconstantasafunctionoftheannealingtemperature forCoFe B andCoFeAlBthinfilmsare presented.\nOur results reveal a strong reduction of the magnetization f or CoFeAlB in comparison to CoFeB.\nIf the prepared CoFeAlB films are amorphous, the damping para meterαis unaffected by the Al\ndoping in comparison to the CoFeB alloy. In contrast, in the c ase of a crystalline CoFeAlB film, α\nis found to be reduced. Furthermore, the x-ray characteriza tion and the evolution of the exchange\nconstant with the annealing temperature indicate a similar crystallization process in both alloys.\nThe data proves the suitability of CoFeAlB for spin torque sw itching properties where a reduction\nof the switching current in comparison with CoFeB is expecte d.\nThe alloy CoFeB is widely used in magnetic tunnel-\ning junctions in combination with MgO barriers due to\nthe large magnetoresistance effect originating in the spin\nfiltering effect [1–4]. For the application in magnetic ran-\ndom accessmemories, the switching ofthe magnetization\nof the free layer via spin transfer torque (STT) with spin\npolarised currents is a key technology. However, the re-\nquired currents for the switching process are still large\nand hinder the applicability of this technique. The criti-\ncal switching current density for an in-plane magnetized\nsystem is given by [5]\nJc0=2eαMStf(HK+Hext+2πMS)\n/planckover2pi1η(1)\nwhereeis the electron charge, αis the Gilbert damping\nparameter, MSis the saturation magnetization, tfis the\nthickness of the free layer, Hextis the external field, HK\nis the effective anisotropy field and ηis the spin transfer\nefficiency. Fromtheexpressionitisclearthat, concerning\nmaterial parameters, Jc0is ruled by the product αM2\nS.\nFor out-of-plane oriented layers, the term 2 πMSvanishes\nand then JC0is proportional to αMS[6]. Even in the\ncase of using pure spin currents created by the Spin Hall\neffect, the required currents are proportional to factors\nof the form αnMSwithn= 1,1/2 [7]. A proper strat-\negy to reduce the critical switching currents is then de-\nfined by reducing the saturation magnetization. This can\nbe achieved by the development of new materials or the\nmodification of known materials with promising prop-\nerties. Since the compatibility with a MgO tunneling\nbarrier and the spin filtering effect must be guaranteed\ntogether with industrial applicability, the second option\nis clearly an advantage by reducing MSin the CoFeB al-\nloy. In this case, a critical point is that this reduction\nmust not be associated with an increase of the damping\nparameter α.In the last years, several reports on doped CoFeB al-\nloys have proven the potential of this approach. The\nintroduction of Cr results in a strong reduction of MS\n[8–10], however, it is sometimes also causing an increase\nof the damping parameter [8]. The reduction of MSby\ndoping CoFeB with Ni is smaller compared to a doping\nwith Cr but it additionally leads to a reduction of α[8].\nFIG. 1. (Color online) θ/2θ-scans for 40 nm thick films of\nCo40Fe40B20(top) and Co 36Fe36Al18B10(bottom) showing\nthe evolution of crystallization with the annealing temper a-\nture.2\nFIG. 2. (Color online) Evolution of the saturation magnetiz a-\ntion for CoFeB and CoFeAlB with the annealing temperature\nTann.\nIn constrast, the reduction of magnetization with V is\ncomparable to Cr [9] but to our knowledge no values for\nαhave been published. In the case of doping of CoFeB\nby Cr or by V, a reduction of the switching current has\nbeen shown [8, 9].\nIn this Letter, we report on results on Al doped CoFeB\nalloy thin films characterized by ferromagnetic resonance\nspectroscopy. The dependence of MS, the Gilbert damp-\ning parameter αand the exchange constant on the an-\nnealing temperature is discussed together with the crys-\ntalline structure of the films and the suitability for STT\nswitching devices.\nThe samples are grown on Si/SiO 2substrates us-\ning DC (for metals) and RF (for MgO) sput-\ntering techniques. The layer stack of the sam-\nples is Si/SiO 2/Ta(5)/MgO(2)/FM(40)/MgO(2)/Ta(5)\nwhere FM = Co 40Fe40B20(CoFeB) or Co 36Fe36Al18B10\n(CoFeAlB). Here, the values in brackets denote the layer\nthicknesses in nm. In particular, the FM/MgO interface\nis chosen since it is widely used for STT devices based\non MTJs. This interface is also required to promote the\ncorrect crystallizationof the CoFeB layerupon annealing\nsince the MgO layer acts as a template for a CoFe bcc\n(100)-oriented structure [1–3] with consequent B migra-\ntion.\nThe dynamic properties and material parameters were\nstudied by measuring the ferromagnetic resonance using\na strip-line vector network analyzer (VNA-FMR). For\nthis, the samples were placed face down and the S 12\ntransmission parameter was recorded. A more detailed\ndescription of the FMR measurement and analysis pro-\ncedure is shown in previous work [11, 12]. Brillouin light\nspectroscopy (BLS) was additionally used for the mea-\nsurement of the exchange constant. The crystalline bulk\nproperties of the films were studied by X-ray diffractom-\netry (XRD) using the Cu-K αline.\nFigure 1 shows the θ/2θ-scans for CoFeB (top) andFIG. 3. (Color online) Linewidth at a fixed frequency of 18\nGHz (a) and Gilbert dampingparameter αdependenceon the\nannealing temperature T ann(b). The αvalue for Tann= 500◦\nis only a rough estimation since the large linewidth value do es\nnot allow for a proper estimation. The inset shows the linear\ndependence of the linewidth on the frequency exemplarily fo r\nCoFeAlB annealed at 350◦C and 400◦C. The red lines are a\nlinear fit.\nCoFeAlB (bottom) samples annealed at different tem-\nperatures T ann. The appearance of the CoFe diffractions\npeaks, as shown by the arrowsin Fig. 1 indicate the start\nof crystallization at high annealing temperatures of more\nthan 400◦C. In the case of lower annealing temperatures\nor the as-deposited samples, the FM layer is in an amor-\nphous state. The first appearance of the (200) diffraction\npeak occurs at the same point for both alloys showing a\nverysimilarthermalevolution. Thissimplifiesasubstitu-\ntion ofCoFeB by the Al alloyin tunneling junctions since\nthe same annealing recipes can be applied. This is criti-\ncal since the used values must be also optimized for the\nquality of the tunneling barrier itself or the perpendicu-\nlar anisotropy induced by the FM/MgO interface. The\n(110) CoFe peak is also present for both material compo-\nsitions owing to a partial texturing of the film. However,\nthe larger intensity of the (200) peak is not compatible\nwith a random crystallite orientation but with a domi-\nnant (100) oriented film [13, 14]. This is needed since the\nspin filtering effect responsible for the large magnetore-\nsistance effect in MgO-based junctions requires a (100)3\nFIG.4. (Color online)Dependenceoftheproduct αM2\nSonthe\nannealing temperature T annfor CoFeB and CoFeAlB. This\nquantityisrulingtheswitchingcurrentinin-planemagnet ized\nSTT devices as shown in Eq. 1.\norientation.\nThe dependence of the FMR frequency on the external\nmagnetic field is described by Kittel’s formula [15]. The\nvalue ofMeffextracted from the Kittel fit is related with\nthe saturation magnetization of the sample and the in-\nterfacial properties by Meff=MS−2K⊥\nS/µ0MSdwhere\nK⊥\nSis the interface perpendicular anisotropy constant.\nFor the thickness used in this work (40 nm) and physi-\ncally reasonable K⊥\nSvalues, the influence of the interface\nis negligible and therefore Meff≈MS. For details about\nthe estimation of Meffthe reader is referred to [12].\nFigure 2 shows the obtained values for MSfor all sam-\nples. AstrongreductionforCoFeAlBin comparisonwith\nstandard CoFeB is observed and the relative difference is\nmaintained for all T ann. The evolution with annealing is\nvery similar for both alloys. Significantly, the increase in\nMSstartsforvaluesofT annlowerthan expectedfromthe\nappearance of the characteristic CoFe diffraction peaks\nin the XRD data (see Fig. 1). This shows that the mea-\nsurement of MSis the more sensitive method to probe\nthe change of the crystalline structure.\nFor CoFeB a saturationvalue around MS≈1500kA/m\nis reached at T ann= 450◦C. This is compatible with val-\nues reported for CoFe (1350-1700 kA/m) [16, 17] and\nCoFeB (1350-1500 kA/m) [17, 18]. On the contrary, for\nCoFeAlB the introduction of Al reduces the magnetiza-\ntion of the samples and the annealing does not recover\nto CoFe-like values.\nFigure 3(a) shows the dependence of the magnetic\nfield linewidth on T annmeasured at a fixed frequency\nof 18 GHz. From the linear dependence of this linewidth\non the FMR frequency, the Gilbert damping parameter\nis extracted (as exemplarily shown for the CoFeAlB al-\nloy in the inset in Fig. 3(b)) and the results are shown\nin Fig. 3(b). For T annvalues up to 350◦C, where theFIG. 5. (Color online) Dependence of the exchange con-\nstantAexon the annealing temperature T annfor CoFeB and\nCoFeAlB. The top panels show typical BLS spectra for ma-\nterials (see text).\namorphousphaseisstill dominating, almostnodifference\nbetween both alloys is observed. With increasing tem-\nperature the damping increases for both alloys but the\nevolution is different. For CoFeAlB the increase starts\nalmost abruptly at T ann= 400◦C, reaches a maximum\naroundα= 0.02 and then decreases again to α= 0.012\nfor Tann= 500◦C. In contrast, the increase for CoFeB\nis more smoothly with T annand increases stadily with\nhigher T ann. In fact, due to the large linewidths reached\nfor Tann= 500◦C, the value of αcannot be properly\nestimated and only a lower limit of 0.03-0.04 can be\ngiven. This situation is represented by the dashed line\nin Fig. 3(b). It is important to note here that when the\ncrystallization process is fulfilled (i.e. for T ann= 500◦C)\nαis much lower for the Al doped alloy. This is rele-\nvant for the application in tunneling junctions where a\nfull crystallization is required for the presence of the spin\nfiltering effect originating large magnetoresistance values\nin combination with MgO barriers [4].\nFor further comparison of both alloys, the quantity\nαM2\nShasbeen calculatedand plotted in Fig. 4. As shown\nin Eq. 1, this value is ruling the critical switching current\nin in-plane magnetized systems. We observe for the al-\nloys showing a mostly amorphous phase (T ann<400◦C)\na slight improvement for CoFeAlB in comparison with\nCoFeB due to the lower MS. However, for fully crys-\ntalline films (T ann= 500◦C), the CoFeAlB shows a much\nsmaller value for αM2\nS. Since a full crystalline phase is\nneeded for any application of this alloy in MTJ-based de-\nvices, this denotes a major advantage of this compound\ncompared to standard.\nThe exchange constant Aexis a critical parameter that\nis strongly influenced by the introduction of Al. Its esti-\nmationinrequiredformodelingthespintorqueswitching\nbehaviorofthe alloys. The accessto the constantisgiven4\nby the dependence of the frequency of the perpendicular\nstanding spin-wave (PSSW) modes on the external static\nmagnetic field [19]. As shown in previous works [12, 20],\nitispossibletoobservethePSSWmodesinmetallicfilms\nwith a standard VNA-FMR setup. However, the signal\nis strongly reduced compared to the FMR peak. For the\nsamples presented in this paper, the PSSW peak could\nnot be observed for T ann>400◦C since the increased\ndamping leads to a broadening and lowering of the peak\nwhich prevents the estimation of Aex. For this reason,\nBLS spectroscopy is used for the measurement of the fre-\nquency position of the PSSW modes. This technique has\nalargersensitivityforthePSSWmodesthanVNA-FMR.\nFigure 5(c) shows the evolution of Aexupon annealing\nfor both alloys. For the films dominated by the amor-\nphous phase the value is much lower for CoFeAlB which\nis also compatible with the lower magnetization. How-\never, asthe crystallizationevolves,theexchangeconstant\nincreases stronger than for CoFeB and the same value is\nobtained for the fully crystallized films. This fact points\nto a similar role of Al and B during the crystallization\nprocess: when the CoFe crystallitesform, the light atoms\nare expelled forming a Al-B-rich matrix embedding the\nmagnetic crystallites. This explains also the similar evo-\nlution observed in the XRD data shown in Fig. 1. The\nlower maximal magnetization obtained for the CoFeAlB\ncan be explained by the reduced CoFe content but also\na certain number of residual Al and B atoms in the crys-\ntallites, which may differ for both alloys.\nTheAexvalues for as-deposited CoFeB films are very\nsimilar to previous reports [12, 20, 21]. Concerning the\nvalues for the crystallized samples, since the properties\nare strongly dependent on the B content and of the ra-\ntio between Co and Fe as well as on the exact annealing\nconditions, a comparison with literature has to be made\ncarefully. Nevertheless, the maximal value and the evo-\nlution with T annfor CoFeB is similar to the one reported\nby some of the authors [12]. Also results for alloys with\nthe same B content arecompatiblewith ourdata [22, 23].\nCoFeB films with reduced B content show larger values\n[17], the same is true for CoFe alloys with values between\n3.84-2.61 ×1011J/m depending on the exact stoichiome-\ntry [16, 17]. This may again be a hint that a rest of Al\nor B is present in the CoFe crystallites.\nIn summary, the presented experimental results show\nthat CoFeAlB is a good candidate as alternative to\nCoFeB for spin torque switching devices due to the re-\nduction of the factor αM2\nSwhich dominates the critical\nswitching current. This reduction was found to originate\nfrom a strong reduction of the saturation magnetization\nandadecreaseddampingparameter αforfullycrystalline\nCoFeAlB films. Furthermore, the results reveal a larger\nthermal stability of the damping properties in CoFeAlB\ncompared to CoFeB. The absolute values of MSand the\nexchange constant Aexfor crystalline films point to a for-\nmation of CoFe crystallines with a non-vanishing contentof the lights atoms embedded in a B or Al matrix.\nFinancial support by M-era.Net through the\nHEUMEM project, the DFG in the framework of\nthe research unit TRR 173 Spin+X and by the JSPS\nCore-to-Core Program is gratefully acknowledged.\n∗conca@physik.uni-kl.de\n[1] S.YuasaandD.D.Djayaprawira, J.Phys.D:Appl.Phys.\n40, R337-R354 (2007).\n[2] S. Yuasa,Y. Suzuki, T. Katayama, and K. Ando,\nAppl. Phys. Lett. 87, 242503 (2005).\n[3] Y. S. Choi, K. Tsunekawa, Y. Nagamine, and\nD. Djayaprawira, J. Appl. Phys. 101, 013907 (2007).\n[4] X.-G. Zhang and W. H. Butler, J. Phys.: Condens. Mat-\nter15R1603, (2003).\n[5] Z. Diao, Z. Li, S. Wang, Y. Ding, A. Panchula, E. Chen,\nL.-C. Wang, and Y. Huai, J. Phys. D: Appl. Phys. 19,\n165209 (2007).\n[6] K. L. Wang, J. G. Alzate, and P.K. Amiri, J. Phys. D:\nAppl. Phys. 46, 074003 (2013).\n[7] T. Taniguchi, S. Mitani, and M. Hayashi, Phys. Rev. B\n92, 024428 (2015).\n[8] K. Oguz, M. Ozdemir, O. Dur, and J. M. D. Coey,\nJ. Appl. Phys. 111, 113904 (2012).\n[9] H. Kubota, A. Fukushima, K. Yakushiji, S. Yakata,\nS. Yuasa, K. Ando, M. Ogane, Y.Ando, andT. Miyazaki,\nJ. Appl. Phys. 105, 07D117 (2009).\n[10] Y. Cui, M. Ding, S. J. Poon, T. P. Adl, S. Keshavarz,\nT. Mewes, S. A. Wolf, and J. Lu, J. Appl. Phys. 114,\n153902 (2013).\n[11] A. Conca, S. Keller, L. Mihalceanu, T. Kehagias,\nG. P. Dimitrakopulos, B. Hillebrands, and E. Th. Pa-\npaioannou, Phys. Rev. B 93, 134405 (2016).\n[12] A. Conca, E. Th. Papaioannou, S. Klingler, J. Greser,\nT. Sebastian, B. Leven, J. L¨ osch, and B. Hillebrands,\nAppl. Phys. Lett. 104, 182407 (2014).\n[13] G. Concas, F. Congiu, G. Ennas, G. Piccaluga, and\nG. Spano. J. of Non-Crystalline Solids 330, 234 (2003).\n[14] C. Y. You, T. Ohkubo, Y. K. Takahashi, and K. Hono,\nJ. Appl. Phys. 104, 033517 (2008).\n[15] C. Kittel, Phys. Rev. 73, 155 (1948).\n[16] X. Liu, R. Sooryakumar, C. J. Gutierrez, and\nG. A. Prinz, J. Appl. Phys. 75, 7021 (1994).\n[17] C. Bilzer, T. Devolder, J.-V. Kim, G. Counil, C. Chap-\npert, S. Cardoso, and P. P. Freitas, J. Appl. Phys. 100,\n053903 (2006).\n[18] X. Liu, W. Zhang, M. J. Carter, and G. Xiao, J. Appl.\nPhys.110, 033910 (2011).\n[19] S. O. Demokritov, B. Hillebrands, Spin Dynamics in\nConfined Magnetic Structures I , Springer, Berlin, (2002).\n[20] A. Conca, J. Greser, T. Sebastian, S. Klingler, B. Obry,\nB. Leven, and B. Hillebrands, J. Appl. Phys. 113, 213909\n(2013).\n[21] J. Cho, J.Jung, K.-E.Kim, S.-I.Kim, S.-Y.Park, andM.-\nH. Jung, C.-Y. You, J. of Magn and Magn. Mat. 339, 36\n(2013).\n[22] A. Helmer, S. Cornelissen, T. Devolder, J.-V. Kim,\nW. van Roy, L. Lagae, and C. Chappert, Phys. Rev. B\n81, 094416 (2010).\n[23] H.Sato, M.Yamanouchi, K.Miura, S.Ikeda, R.Koizumi,5\nF. Matsukura, and H. Ohno, IEEE Magn. Lett., 3,\n3000204 (2012)."    },    {        "title": "1807.04977v1.Gilbert_damping_of_high_anisotropy_Co_Pt_multilayers.pdf",        "content": "Gilbert damping of high anisotropy Co/Pt multilayers\nThibaut Devolder\u0003\nCentre de Nanosciences et de Nanotechnologies, CNRS, Univ. Paris-Sud,\nUniversité Paris-Saclay, C2N-Orsay, 91405 Orsay cedex, France\nS. Couet, J. Swerts, and G. S. Kar\nimec, Kapeldreef 75, 3001 Heverlee, Belgium\n(Dated: June 16, 2021)\nUsing broadband ferromagnetic resonance, we measure the damping parameter of [Co(5 Å)/Pt(3 Å)] \u00026mul-\ntilayers whose growth was optimized to maximize the perpendicular anisotropy. Structural characterizations in-\ndicate abrupt interfaces essentially free of intermixing despite the miscible character of Co and Pt. Gilbert damp-\ning parameters as low as 0.021 can be obtained despite a magneto-crystalline anisotropy as large as 106J/m3.\nThe inhomogeneous broadening accounts for part of the ferromagnetic resonance linewidth, indicating some\nstructural disorder leading to a equivalent 20 mT of inhomogenity of the effective field. The unexpectedly rel-\natively low damping factor indicates that the presence of the Pt heavy metal within the multilayer may not be\ndetrimental to the damping provided that intermixing is avoided at the Co/Pt interfaces.\nI. INTRODUCTION\nThanks to their large perpendicular magnetic anisotropy,\ntheir confortable magneto-optical signals and their easy\ngrowth by physical vapor deposition1, the [Co/Pt] multilay-\ners are one of the most popular system in spintronics. Early\nin spintronics history this model system was used to study\nthe physics of domain wall propagation2, for the develop-\nment of advanced patterning techniques3and for the assess-\nment of micromagnetic theories4. More recently they have\nbeen extensively used as high quality fixed layers in per-\npendicularly magnetized tunnel junctions, in particular in the\nmost advanced prototypes of spin-transfer-torque magnetic\nrandom access memories memories5. Despite the widespread\nuse of Co/Pt multilayers, their high frequency properties,\nand in particular their Gilbert damping parameter remains\nlargely debated with experimental values that can differ by\norders of magnitude from60.02 to 20 times larger7and the-\noretical calculations from circa 0.035 in Co 50Pt50alloys8\nto slightly smaller or substantially larger values in multilay-\ners made of chemically pure layers9. Direct measurements\nby conventional ferromagnetic resonance (FMR) are scarce\nas the high anisotropy of the material pushes the FMR fre-\nquencies far above610 GHz and results in a correlatively\nlow permeability that challenges the sensitivity of commer-\ncial FMR instruments10. As a result most of the measure-\nments of the damping of Co/Pt systems were made by all-\noptical techniques11,12in small intervals of applied fields. Un-\nfortunately this technique requires the static magnetization to\nbe tilted away from the out-of-plane axis and this tilt ren-\nders difficult the estimation of the contribution of the ma-\nterial disorder to the observed FMR linewidth using the es-\ntablished protocols13; this is problematic since in Co-Pt sys-\ntems there contributions of inhomogeneity line broadening\nand two-magnon scattering by the structural disorder (rough-\nness, interdiffusion, granularity,...) are often large14,15.\nIt is noticeable that past reports on the damping of Co/Pt\nsystems concluded that it ought to be remeasured in sam-\nples with atomically flat interfaces11. Besides, this measure-\nment should be done in out-of-plane applied field since thiseases the separation of the Gilbert damping contribution to\nthe linewidth from the contribution of structural disorder16.\nIn this paper, we measure the damping parameter of [Co(5\nÅ)/Pt(3 Å)]\u00026multilayers whose growth was optimized to\nmaximize perpendicular anisotropy anisotropy. The sputter-\ndeposition is performed at an extremely low17Argon pressure\nin remote plasma conditions which enables very abrupt inter-\nfaces that are essentially free of intermixing. We show that in\ncontrast to common thinking, the Gilbert damping parameter\nof Co/Pt multilayers can be low; its effective value is 0.021\nbut it still likely16includes contributions from spin-pumping\nthat our protocol can unfortunately not suppress.\nII. EXPERIMENTAL\nOur objective is to report the high frequency properties of\nCo/Pt multilayers that were optimized for high anisotropy.\nThe multilayer is grown by sputter-deposition on a Ru (50 Å)\nbuffer and capped with a Ru(70 Å)/Ta(70 Å)/Ru(100 Å)/Ta(10\nÅ, cap) sequence (bottom to top order). The Ru buffer was\nchosen because it does not mix with Co-based multilayers\neven under tough annealing conditions18. The stacks were\ndeposited by physical vapor deposition in a Canon-Anelva\nEC7800 300 mm system on oxidized silicon substrates at\nroom temperature. The Argon plasma pressure is kept at 0.02\nPa, i.e. substantially lower than the usual conditions of 0.1-0.5\nPa used in typical deposition machines17. As this multilayer\nis meant to be the reference layer of bottom-pinned magnetic\ntunnel junctions, in some samples (fig. 1) the non-magnetic\ncap is replaced the following sequence: Ta cap / Fe 60Co20B20\n/ MgO / Fe 60Co20B20/ Ta / [Co(5 Å)/Pt(3 Å)] \u00024/ Ru sim-\nilar to as in ref. 19 and 20 to form a bottom-pinned mag-\nnetic tunnel junction with properties designed for spin-torque\napplications21. All samples were annealed at 300\u000eC for 30\nminutes in an out-of-plane field of 1 T.arXiv:1807.04977v1  [cond-mat.mtrl-sci]  13 Jul 20182\nRu[Co5Å/Pt3Å]×6 RuMgOFeCoBFeCoBTaRuTa[Co5Å/Pt3Å]×4Co5Å(a)(b)(c)\nFIG. 1. (Color online). Structure and anisotropy of a Co-Pt multi-\nlayer. (a) Transmission Electron Micrograph of a magnetic tunnel\njunction that embodies our Co/Pt as hard multilayer at the bottom of\nthe reference synthetic antiferromagnet, similar to that of ref. 21. (b)\nEasy axis and (c) hard axis hysteresis loops of the hard multilayer\nwhen covered with Ru(70 Å)/Ta(70 Å)/Ru(100 Å)/Ta(10 Å, cap)\nIII. STRUCTURE\nX-ray reflectivity scans (not shown) indicate Bragg reflex-\nions at 2\u0012= 11 , 22.2 and 33.6 deg., consistent with the mul-\ntilayer periodicity of 8 Å. Consistently, the Pt to Co inter-\nmixing is sufficiently low that well formed 3Å Pt spacers can\nbe seen the Transmission Electron Micrograph after anneal-\ning [Fig. 1(a)]. Almost no roughness is observed throughout\nthe Co/Pt multilayer. We emphasize that this quality of inter-\nfaces is almost equivalent to that obtained in Molecular Beam\nEpitaxy conditions22. Indeed Co and Pt are strongly miscible\nsuch that hyperthermal (high energy) deposition techniques\nlike sputter deposition do not easily yield this low degree of\nintermixing, except when the deposition is conducted under\nsufficiently low plasma pressure in remote plasma conditions,\ni.e. when the substrate-to-target distance is large to avoid di-\nrect plasma exposure to the film being deposited.\nIV . ANISOTROPY\nThe magnetic material properties were measured by vibrat-\ning sample magnetometry (VSM) and Vector Network Ana-\nlyzer ferromagnetic resonance23in both easy (z) and hard axis\n(x) configurations. For VNA-FMR the sample is mechanically\npressed on the surface of a 50 microns wide coplanar waveg-\nuide terminated by an open circuit; data analysis is conducted\nfollowing the methods described in ref. 24. The VSM signal\nindicated a magnetization Ms= 8:5\u0002105kA/m if assuming\na magnetic thickness of 48 Å, i.e. assuming that the [Co(5\nÅ)/Pt(3 Å)]\u00026multilayer can be described as a single mate-\nrial. The loops indicate a perpendicular anisotropy with full\nremanence. The reversal starts at 46.8 mT and completes be-\n/s45/s50 /s45/s49 /s48 /s49 /s50/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48/s55/s48\n/s72/s101/s102/s102\n/s107/s49/s43/s72\n/s107/s50\n/s105/s110/s45/s112/s108/s97/s110/s101/s32\n/s102/s105/s101/s108/s100/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41\n/s48/s72/s32/s40/s84/s41/s72/s101/s102/s102\n/s107/s49/s111/s117/s116/s45/s111/s102/s45/s112/s108/s97/s110/s101/s32\n/s102/s105/s101/s108/s100FIG. 2. (Color online). FMR frequencies versus in-plane (cross sym-\nbols) or out-of-plane (square symbols) applied field. The bold lines\nare fits using Eq. 1 and 2, yielding \u00160(Hk1\u0000Ms) = 1:320\u00060:005T\nand\u00160Hk2= 0:120\u00060:015T.\nfore 48 mT with a tail-free square hysteresis loop. Careful\nattempts to demagnetize the sample using an acperpendicular\nfield failed to produce a multidomain state at remanence. This\nindicates that the lowest nucleation field in the whole sample\nis larger that the domain wall propagation field everywhere in\nthe film. This low propagation field indicates qualitatively that\nthe effective anisotropy field is very uniform. The hard axis\nloop indicates an in-plane saturation field of \u00191:3\u00060:1T in\nline with the expectations for such composition3. The round-\ning of the hard axis loop near saturation and its slight hys-\nteretic remanence [Fig. 1(c)] impedes a more precise deduc-\ntion of the anisotropy fields from the sole hard axis loop.\nWe shall instead use the ferromagnetic resonance data\nbecause magnetization eigenfrequencies constitute absolute\nmeasurements of the effective fields acting on the mag-\nnetization. Fig. 2 gathers the measured FMR frequen-\ncies measured for in-plane and out-of-plane applied fields\nfrom -2.5 to 2.5 T. To analyze the microwave susceptibil-\nity data, we assume an energy density that reads E=\n1\n2\u00160Hk1MSsin2\u0012+1\n4\u00160Hk2MSsin4\u0012with\u0012the (suppos-\nedly uniform) angle between the magnetization and the sam-\nple normal. Our convention is that the first and second order\nmagneto-crystalline anisotropy fields Hk1= 2K1=(\u00160MS)\nandHk2= 4K2=(\u00160MS)are positive when they favor per-\npendicular magnetization, i.e. \u0012= 0.\nIn that framework, the ferromagnetic resonance frequencies\nin out-of-plane and in-plane applied fields saturating the mag-\nnetization as:\n!perp=\r0(Hz+Hk1\u0000Ms) (1)\nand\n!in-plane =\r0p\nHx(Hx\u0000Hk1\u0000Hk2+Ms); (2)\nwhere\r0=j\rj\u00160is the gyromagnetic ratio. (For in-plane\nfieldsHxlower thanHx;sat=Hk1\u0000Hk2\u0000Msthe magne-\ntization is tilted. A straightforward energy minimization was3\nused to yield magnetization tilt \u0012that was subsequently in-\njected to a Smit and Beljers equation to yield the FMR fre-\nquency). The best fit to the experimental data is obtained\nfor\u00160(Hk1\u0000Ms) = 1:320\u00060:005 T (corresponding to\nK1= 106J/m3) and\u00160Hk2= 0:120\u00060:015T. Note that\nthe second order anisotropy is small but non negligible such\nthat the effective anisotropy fields deduced from easy and axis\naxis measurements would differ by circa 10% if Hk2was dis-\nregarded.\nV . GILBERT DAMPING\nA. Models\nWe now turn to the analysis of the FMR linewidth (Fig.\n3). As common in FMR, the linewidth comprises an intrin-\nsic Gilbert damping part and an extrinsic additional contribu-\ntion linked to the lateral non uniformity of the local effective\nfieldsHk1\u0000Ms. This can be gathered in a characteristic\nfield \u0001H0measuring the disorder relevant for FMR. In out-\nof-plane field FMR experiments, the proportionality between\neffective fields and resonance frequencies (Eq. 1) allows to\nwrite simply \u0001H0=1\n\r0\u0001!j!!0, and for the perpendicular\nmagnetization we follow the usual convention16and write:\n1\n\r0\u0001!perp= 2\u000b(Hz+Hk1\u0000Ms) + \u0001H0 (3)\nor equivalently \u0001!perp= 2\u000b!perp+\r0\u0001H0.\nFor in-plane magnetization, the intrinsic linewidth above\nthe in-plane saturation field is\n1\n\r0\u0001!Gilbert\nin-plane =\u000b(2Hx\u0000Hk1\u0000Hk2+Ms) (4)\nThe resonance frequency (Eq. 2) is non linear with the ef-\nfective fields such that the non uniformity \u0001H0of the local\neffective fields translates in a linewidth broadening through\nthe term\n1\n\r0\u0001!disorder\nin-plane =d!in-plane\nd(Ms\u0000Hk1)\u0001H0 (5)\nwhere the derivative term ispHx\n2pHx\u0000Hk1\u0000Hk2+Ms. In case of\nfinite disorder, this factor diverges at the spatially-averaged\nin-plane saturation field Hx;sat.\nB. Results\nFor each applied field, the real and imaginary parts of the\ntransverse permeability \u0016(f)were fitted with the one expected\nfor the uniform precession mode25with three free param-\neters: the FMR frequency !FMR=(2\u0019), the FMR linewidth\n\u0001!=(2\u0019))and a scaling (sensitivity) factor common to both\nreal and imaginary parts of \u0016(f)as illustrated in Fig. 3b.When plotting the symmetric lorentzian-shaped imagi-\nnary part of the transverse permeability versus the asymet-\nric lorentzian-shaped real part of the permeability for fre-\nquencies ranging from dcto infinity, a circle of diameter\nMs=[2\u000b(Hz+Hk1\u0000Ms)]should be obtained for a spatially\nuniform sample18. The finite disorder \u0001H0distorts the exper-\nimental imaginary part of the permeability towards a larger\nand more gaussian shape. It can also damp and smoothen the\npositive and negative peaks of the real part of the permeabil-\nity; when the applied field is such that the inhomogeneous\nbroadening is larger than the intrinsic Gilbert linewidth, this\nresults in a visible ellipticity of the polar plot of \u0016(f). In our\nexperimental polar plot of \u0016(f)(Fig. 3a) the deviations from\nperfect circularity are hardly visible which indicates that the\ninhomogeneous broadening is not the dominant contribution\nto the sample FMR linewidth in out-of-plane field conditions.\nTo confirm this point we have plotted in Fig. 3c the de-\npendence of FMR linewidth with FMR frequency for out-of-\nplane applied fields. A linear fit yields \u000b= 0:021\u00060:002and\n\u0001H0\u001940mT. A substantial part of the measured linewidth\nthus still comes from the contribution of the lateral inhomo-\ngeneity of the effective anisotropy field within the film. As a\nresult, low field measurements of the FMR linewidth would\nbe insufficient to disentangle the Gilbert contribution and the\nstructural disorder contributions to the total FMR linewidth.\nThe in-plane applied field FMR linewidth can in principle\nbe used to confirm this estimate of the damping factor. Un-\nfortunately we experience a weak signal to noise ratio in in-\nplane field FMR experiments such that only a crude estimation\nof the linewidth was possible.Within the error bar, it is inde-\npendent from the applied field from 1.7 to 2.5 T (not shown)\nwhich indicates that the disorder still substantially contributes\nto the linewidth even at our maximum achievable field. At\n2.5 T the linewidth was1\n2\u0019\u0001!in-plane\u00193:0\u00060:3GHz. This\nis consistent width the expectations of that would predict 2.2\nGHz of intrinsic contribution (Eq. 4) and 0.4 GHz of intrinsic\ncontribution (Eq. 5).\nVI. DISCUSSION\nWe conclude that the damping of Co/Pt multilayers can be\nof the order of 0.02 even for multilayers with anisotropies\namong the strongest reported (see ref. 26 for a survey of\nthe anisotropy of Co/Pt multilayers). Note that \u000b\u00190:021is\nstill a higher bound, as we are unable to measure and subtract\nthe spin-pumping contribution. Measuring the spin-pumping\ncontribution would require to vary the cap and buffer layer\nthicknesses without affecting the multilayer structure which\nis difficult to achieve. Still, we can conclude that the damp-\ning of Co/Pt multilayers lies in the same range as other high\nanisotropy multilayers like Co/Ni (ref. 18 and 27) and Co/Pd\n(ref. 16) systems.\nThis conclusion is in stark contrast with the common\nthinking7that Co/Pt systems alway s have a large damping.\nThis widespread opinion is based on the standard models\nof magneto-crystalline anisotropy28and damping29that pre-\ndicts that they both scale with the square of the spin-orbit4\n/s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s49/s46/s48/s49/s46/s53/s50/s46/s48/s52/s52 /s52/s54 /s52/s56 /s53/s48 /s53/s50 /s53/s52 /s53/s54/s45/s49/s48/s48/s49/s48/s45/s49/s48 /s48 /s49/s48\n/s45/s49/s48/s48/s72/s97/s108/s102/s32/s70/s77/s82/s32/s108/s105/s110/s101/s119/s105/s100/s116/s104/s32/s40/s71/s72/s122/s41\n/s70/s77/s82/s32/s70/s114/s101/s113/s46/s32/s40/s71/s72/s122/s41/s48/s46/s54/s32/s71/s72/s122/s32/s43/s32/s48/s46/s48/s50/s49 /s102/s80/s101/s114/s109/s101/s97/s98/s105/s108/s105/s116/s121/s32/s40/s114/s101/s97/s108/s32/s112/s97/s114/s116/s41/s80/s101/s114/s109/s101/s97/s98/s105/s108/s105/s116/s121/s32\n/s40/s105/s109/s97/s103/s105/s110/s97/s114/s121/s32/s112/s97/s114/s116/s41\n/s73/s109/s97/s103/s105/s110/s97/s114/s121/s32/s48/s46/s48/s52\n/s32/s50/s102/s32 /s102\n/s32/s70/s114/s101/s113/s46/s32/s40/s71/s72/s122/s41/s32/s80/s101/s114/s109/s101/s97/s98/s105/s108/s105/s116/s121/s82/s101/s97/s108\n/s40/s99/s41/s40/s98/s41/s77/s97/s99/s114/s111/s115/s112/s105/s110/s32/s102/s105/s116/s32/s32\n/s40/s97/s41\nFIG. 3. (Color online). Gilbert damping of the Co/Pt multilayer. (a)\nImaginary part versus real part of the permeability for a field of 0.45\nT applied perpendicularly to the plane. The bold lines are theoretical\nmacrospin permeability curves with linewidth parameters (i.e. effec-\ntive damping) of 0.04. (b) Same data but versus frequency. (c): FMR\nhalf linewidth versus FMR frequency. The bold line is a guide to the\neye with a slope \u000b= 0:021and zero frequency intercept of 0.6 GHz.\ncoupling\u0018, which is particularly large in the Pt atoms. We\nemphasize that this expectation of large damping is not sys-\ntematically verified: in studies that make a thorough anal-\nysis of the effects of structural disorder, no correlation wasfound between anisotropy and damping in comparable mate-\nrial systems11,16. Rather, a large correlation was found be-\ntweenHk1and\u0001H0, indicating that when the anisotropy is\nstrong, any local inhomogeneity thereof has a large impact\non the FMR linewidth. Owing to the difficulty of achiev-\ning well-defined Co/Pt interfaces, we believe that past con-\nclusions on the large damping of Co/Pt systems were based\non systems likely to present some intermixing at the inter-\nface; indeed the presence of impurities with large spin-orbit\ncoupling considerably degrades (increases) the damping of a\nmagnetic material30and synchonously degrades (decreases)\nthe magneto-crystalline anisotropy31.\nVII. CONCLUSION\nIn summary, we have studied high anisotropy [Co(5 Å)/Pt(3\nÅ)]\u00026multilayers grown by low pressure remote plasma\nsputter deposition. The deposition conditions were tuned\nto achieve abrupt interfaces with little intermixing. Broad-\nband ferromagnetic resonance was used to measure the first\nand second order uniaxial anisotropy fields. With the mag-\nnetization measured by vibrating sample magnetometry, this\nyields an anisotropy energy of 1MJ/m3. The inhomogeneous\nbroadening accounts for part of the ferromagnetic resonance\nlinewidth, indicating some structural disorder leading to a\nequivalent 40 mT (or equivalently 600 MHz) of inhomogenity\nof the effective field in out-of-plane applied fields. This FMR-\nrelevant inhomogeneity is comparable to the coercivity of 47\nmT. Despite the large anisotropy a Gilbert damping parameter\nas low as 0.021\u00060.002 is obtained. This unexpectedly rela-\ntively low damping factor indicates that the presence of the Pt\nheavy metal within the multilayer can in some condition not\nbe detrimental to the damping. We interpret our results and\nliterature values by analyzing the consequences of Pt/Co in-\ntermixing: Pt impurities within a Cobalt layer reduce locally\nthe interface anisotropy as they reduce the abruptness of the\ncomposition profile, but they also increase substantially the\nGilbert damping. As a result, a large anisotropy together with\na low damping can be obtained provided that intermixing is\nminimized at the Co/Pt interfaces.\n\u0003thibaut.devolder@u-psud.fr\n1V . Mathet, T. Devolder, C. Chappert, J. Ferré, S. Lemerle, L. Bel-\nliard, and G. Guentherodt, Journal of Magnetism and Magnetic\nMaterials 260, 295 (2003).\n2S. Lemerle, J. Ferré, C. Chappert, V . Mathet, T. Giamarchi, and\nP. Le Doussal, Physical Review Letters 80, 849 (1998).\n3C. Chappert, H. Bernas, J. Ferré, V . Kottler, J.-P. Jamet,\nY . Chen, E. Cambril, T. Devolder, F. Rousseaux, V . Mathet, and\nH. Launois, Science 280, 1919 (1998).\n4L. Belliard, J. Miltat, V . Kottler, V . Mathet, C. Chappert, and\nT. 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Schmidt, Journal of Applied Physics 101, 09D102 (2007).\n13S. Mizukami, Y . Ando, and T. Miyazaki, Physical Review B 66,\n104413 (2002).\n14N. Mo, J. Hohlfeld, M. ul Islam, C. S. Brown, E. Girt, P. Krivosik,\nW. Tong, A. Rebei, and C. E. Patton, Applied Physics Letters 92,\n022506 (2008).\n15A. J. Schellekens, L. Deen, D. Wang, J. T. Kohlhepp, H. J. M.\nSwagten, and B. Koopmans, Applied Physics Letters 102, 082405\n(2013).\n16J. M. Shaw, H. T. Nembach, and T. J. Silva, Physical Review B\n85, 054412 (2012).\n17J. Musil, Vacuum 50, 363 (1998).\n18E. Liu, J. Swerts, T. Devolder, S. Couet, S. Mertens, T. Lin,\nV . Spampinato, A. Franquet, T. Conard, S. Van Elshocht,\nA. Furnemont, J. De Boeck, and G. Kar, Journal of Applied\nPhysics 121, 043905 (2017).\n19J. Swerts, S. Mertens, T. Lin, S. Couet, Y . Tomczak, K. Sankaran,\nG. Pourtois, W. Kim, J. Meersschaut, L. Souriau, D. Radisic, S. V .\nElshocht, G. Kar, and A. Furnemont, Applied Physics Letters\n106, 262407 (2015).\n20T. Devolder, S. Couet, J. Swerts, and A. Furnemont, Applied\nPhysics Letters 108, 172409 (2016).21T. Devolder, J.-V . Kim, F. Garcia-Sanchez, J. Swerts, W. Kim,\nS. Couet, G. Kar, and A. Furnemont, Physical Review B 93,\n024420 (2016).\n22D. Weller, L. Folks, M. Best, E. E. Fullerton, B. D. Terris, G. J.\nKusinski, K. M. Krishnan, and G. Thomas, Journal of Applied\nPhysics 89, 7525 (2001).\n23C. Bilzer, T. Devolder, P. Crozat, C. Chappert, S. Cardoso, and\nP. P. Freitas, Journal of Applied Physics 101, 074505 (2007).\n24C. Bilzer, T. Devolder, P. Crozat, and C. Chappert, IEEE Trans-\nactions on Magnetics 44, 3265 (2008).\n25T. Devolder, Physical Review B 96, 104413 (2017).\n26V . W. Guo, B. Lu, X. Wu, G. Ju, B. Valcu, and D. Weller, Journal\nof Applied Physics 99, 08E918 (2006).\n27J.-M. L. Beaujour, W. Chen, K. Krycka, C.-C. Kao, J. Z. Sun, and\nA. D. Kent, The European Physical Journal B 59, 475 (2007).\n28P. Bruno, Physical Review B 39, 865 (1989).\n29V . Kambersky, Physical Review B 76(2007), 10.1103/Phys-\nRevB.76.134416.\n30J. O. Rantscher, R. D. McMichael, A. Castillo, A. J. Shapiro, W. F.\nEgelhoff, B. B. Maranville, D. Pulugurtha, A. P. Chen, and L. M.\nConnors, Journal of Applied Physics 101, 033911 (2007).\n31T. Devolder, Physical Review B 62, 5794 (2000)."    },    {        "title": "2305.01969v2.Lyapunov_functions_for_linear_damped_wave_equations_in_one_dimensional_space_with_dynamic_boundary_conditions.pdf",        "content": "Lyapunov functions for linear damped wave equations in\none-dimensional space with dynamic boundary conditions.\nYacine Chitoura, Hoai-Minh Nguyenb, Christophe Romanc\naLaboratoire des signaux et syst` emes, Universit´ e Paris Saclay, Centralesupelec CNRS, Gif-sur-Yvette, France.\nbLaboratoire Jacques Louis Lions, Sorbonne Universit´ e, Paris, France.\ncLaboratoire informatique et syst` eme, Aix-Marseille Universit´ e, Marseille, France.\nAbstract\nThis paper considers a one-dimensional wave equation on [0 ,1], with dynamic boundary conditions of second order at x= 0\nandx= 1, also referred to as Wentzell/Ventzel boundary conditions in the literature. In additions the wave is subjected\nto constant disturbance in the domain and at the boundary. This model is inspired by a real experiment. By the means of\na proportional integral control, the regulation with exponential converge rate is obtained when the damping coefficient is a\nnowhere-vanishing function of space. The analysis is based on the determination of appropriate Lyapunov functions and some\nfurther analysis on an associated error system. The latter is proven to be exponentially stable towards an attractor. Numerical\nsimulations on the output regulation problem and additional results on related wave equations are also provided.\nKey words: one-dimensional wave equation, Wentzel boundary conditions, regulation, output feedback control.\nThe wave equation is one of the classical partial differen-\ntial equations. The actual reason is that the wave equa-\ntion is the continuous pendant of Newton’s second law\nof motion, i.e., where momentum is equal to the sum of\nthe forces. As a consequence, it is also linked with the\nEuler-Lagrange framework, and therefore with the prin-\nciple of least action. For stationary systems, the energy\nis conserved and the action (or Lagrangian) is station-\nary. Other physical phenomena are therefore associated\nwith the wave equation such that electromagnetic law,\nand quantum phenomena with the Klein-Gordon equa-\ntion.\nIn the control community, the wave equation has been\nmainly used for the modelization, estimation, and con-\ntrol of mechanical vibration and deformation phenom-\nena. The regulation and control problem applied on the\none-dimensional wave equation with dynamic bound-\nary condition has attracted the attention of many re-\nsearchers in the control community: crane regulation [8],\n[10], [14], and [6], hanging cable immersed in water [5],\nEmail addresses:\nyacine.chitour@l2s.centralesupelec.fr (Yacine\nChitour), hoai-minh.nguyen@sorbonne-universite.fr\n(Hoai-Minh Nguyen), christophe.roman@lis-lab.fr\n(Christophe Roman).drilling torsional vibrations [38], [45] ,[1], [48], piezoelec-\ntric control [24], and flexible structure [18]. There are\nnowadays two main classes of issues : on the one hand,\nlongitudinal variation with for example overhead crane\nand underwater cable, and, on the other hand, torsional\nvariation with drilling string dynamics. The difference\nis on the control objective: one aims at controlling the\nposition in the first case, and instead the velocity in the\nsecond case.\nThe behavior of the wave equation is strongly related to\nits boundary conditions. In the case of classical bound-\nary condition (i.e., Dirichlet, Neumann, Robin) that is-\nsue is well understood in the linear case and without\nhigh-order terms. Particular terms at one boundary can\ncompensate for anti-damping terms at other boundaries\nand even in the domain, for example, see [41], [40] and\n[35]. Moreover, there are cases where even if the energy\nof the one-dimensional linear wave equation decreases\nalong trajectories, it still does not decay exponentially\n[23, Section 4].\nThe wave equation under consideration is subject to two\ndynamic boundary conditions. This model results from\nan identification problem associated with a laboratory\nexperiment [36].\nPreprint submitted to Automatica 6 February 2024arXiv:2305.01969v2  [math.AP]  5 Feb 20241 Problem statement.\nThe considered system is defined for t⩾0 and for x∈\n(0,1), by\n\n\nvtt(t, x) = (a(x)vx)x(t, x)−q(x)vt(t, x) +f(x),(1a)\nvtt(t,1) =−β1vx(t,1)−νvt(t,1) +U(t) +fc,(1b)\nvtt(t,0) = µ1vx(t,0)−γ1vt(t,0) +fac, (1c)\nv(0,·) =v0, v t(0,·) =v1. (1d)\nHere U(t) is the control input and we assume that\n(h1) the function a: [0,1]→R∗\n+is in W1,∞(0,1) and\nthat there exist a,a >0 such that a≤a(·)≤aa.e.\non [0,1]. This function is associated with the mass\nand elasticity of the wave and it is also linked with\nthe velocity.\n(h2) The function q: [0,1]→R∗\n+, describing the in-\ndomain damping is in L∞(0,1) and satisfies q≤\nq(·)≤qa.e. on [0 ,1] for some q,q >0.\n(h3) The constants β1,γ1, µ1are positive real numbers,\nandνis real.\n(h4) The source terms f(·) is in L∞(0,1), and the real\nconstants fc, facare unknown and therefore they\ncannot be used in the computation of the control\nlawU(t).\nThe regularity of a(·) stated in ( h1) follows by classi-\ncal arguments. In detail, for the computation, we need\na(·)vx(t,·) to be in H1(0,1). To be more precise every-\nthing will be the same as in the constant parameters\ncase if a(·)vx(t,·) and vx(t,·) have the same regular-\nity. To get strong solutions for (1), one needs to have\nthat v∈H2(0,1). Next, it can be easily shown that\nifa∈W1,∞and, v∈H2(0,1) then a(·)vx(t,·) is in\nH1(0,1). Note that this is just a sufficient condition for\nthe regularity. We refer the reader to [43, Chapter 21]\nfor more details about the regularity of a. In the sequel,\nwe also need qvt(t,·)2need to be integrable, this means\nq∈L∞(0,1). For fwe actually only need it to be inte-\ngrable, it holds nonetheless L∞(0,1)⊂L1(0,1).\nThe objective of the paper is to regulate vt(t,·) to the\nconstant reference value vref\n1, by means of a proportional\nintegral (PI) control law using the measurement of the\nvelocity collocated with the actuation, vt(1, t), in other\nwords, the control U(t) can take the form\nU(t) :=−k(vt(t,1)−vref\n1)−kiZt\n0(vt(s,1)−vref\n1)ds,(2)\nwhere the constants k, kihave to be chosen. This can\nbe equivalently written as\n\u001aU(t) =−k(vt(t,1)−vref\n1)−kiηv(t), (3a)\n˙ηv(t) =vt(t,1)−vref\n1, η v(0) = 0 . (3b)In the literature boundary conditions of the type (1b)-\n(1c) can be recast as Wentzell’s boundary conditions\n[15]. It involves a modification of the usual state space\nwhich in our case requires the addition of two finite-\ndimensional state variables, in a similar way as in [39],\n[25], [10], [14] and [6]. When the wave equation is more\nthan a one-dimensional, the reader is referred to [15] and\n[4] and references therein.\nThis type of control problem lies in robust output reg-\nulation. There has been an effort to extend the result\nand method from linear finite dimensional systems, to\ninfinite dimensional systems. We refer the reader to [29],\n[27], [28], more recently [47] and reference within all of\nthem. These papers establish general results for exam-\nple [47] deals with non-linear systems. However, they\nare mostly based on either passivity, strong monoticity\nor exponential decay properties. These properties often\nremain to be proven as it is the case of the present pa-\nper. In [44], the authors establish general result on the\nPI control of infinite dimensional systems with the as-\nsumption beforehand on the exponential stability of the\nzero input system.\nThe impact of the in-domain damping q(x) can be an\nissue for the decay rate, as we can have some overdamp-\ning phenomenon. Intuitively, the damping should help\nthe decay rate of the system. But as one can see in [17]\nwhere a semi-linear wave equation is considered, the de-\ncay rate of the non-damped system is finite time, the\naddition of the damping degrade this performance to an\nexponential decay rate. Note that for the present case\nthe in-domain damping is mandatory for the proof.\nThere are specific configurations of (1) that can be solved\nusing more intricate and general control law designs,\nespecially those tailored to address partial differential\nequations coupled with ordinary differential equations\nat the boundary. If disturbances are not considered, [11]\nand its extension [12] can be employed. Additionally,\nassuming that a(·) is constant allows the use of [37], [50],\nor [49].\nThese five papers primarily employ an infinite-\ndimensional backstepping approach, a development\nclosely associated with the influential work of Miroslav\nKrstic [21]. Given that the wave equation can be ex-\npressed as a coupled heterodirectional hyperbolic par-\ntial differential equation (PDE), the main strategy in\nthe aforementioned papers involves using backstepping\ntransformations to decouple or cascade the PDE. This\ntransforms the closed-loop system into a target sys-\ntem, the stability of which is easier to analyze. Notably,\nthe uniqueness of the present paper lies in achieving\nexponential stability without the need for decoupling,\nthereby establishing new potential target systems for\nbackstepping based design.\nThe closest approach associated with the present paper\n2is [45] where the velocity regulation with a PI is consid-\nered. However, the controlled boundary condition con-\nsidered in [45] is not a second order dynamic one, and\nthus is different from the one considered in this paper.\nNevertheless, the boundary condition considered in [45]\nimplies the exponential stability even with small viscous\nanti-damping at the boundary opposite to the actuation.\nIn the case under consideration, only viscous damping\nat the opposite boundary is considered, and exponen-\ntial stability is achieved. In [6] the wave equation is sub-\nject to two dynamic boundary conditions. The authors\nestablish asymptotic stability for the position stabiliza-\ntion and that the decay rate is not exponential, and no\nviscous terms are considered for the zero input system.\nIn [25], for the same model (as [6]) the exponential sta-\nbility toward the origin is obtained but the control law\nneeds the knowledge of vxt(1, t). This can be related to\nthe finite dimensional backstepping done in [14]. Studies\nhave been conducted concerning the potential absence of\nexponential stabilization for wave-like equations, as evi-\ndenced by works such as [26], [31], and references therein.\nIn a broader context, investigations into this issue ex-\ntend to more general setups, as seen in [16], [46], and\nrelated references.\nPI controllers have been successfully and recently used\nin order to regulate linear and non-linear PDE, see [7],\n[22]. An identification procedure has been presented in\n[36] for the system (1) without source terms on exper-\nimental data. This means that the considered problem\ncan be associated with an experimental setup. A first\nstudy has been made on this system in [33] using clas-\nsical form a Lyapunov functional but it failed to prove\nthe exponential stability. Only asymptotic stability was\nestablished, by using the LaSalle invariance principle.\nThis paper provides a new term in the Lyapunov func-\ntional and an associated methodology, for the present\nsetup. The proof of the exponential stability is given in\nSection 3. In Section 4, this proof is compared with exist-\ning results. Next the proof of the robustness of the con-\ntrolled system is given in Section 5. Then in Section 6\nwe study, using the same approach, simpler cases where\none boundary condition is a Dirichlet one and this al-\nlows us to establish the exponential stability of the zero\ninput system in the undisturbed case. The last part of\nthe paper deals with numerical simulations. The numer-\nical scheme is not derived from the usual approxima-\ntion of space and time derivatives. We used the fact that\nthe wave equation can be derived from the Lagrangian\nand the least action principle to approximate the sys-\ntem space energy by a finite dimensional continuous time\nEuler-Lagrange equation. The finite dimensional contin-\nuous time system is then numerically solved by using\nsymplectic integrators. This suggested numerical scheme\nis new up to the authors’ knowledge and provides an\ninteresting alternative compares to more standard dis-\ncretization schemes.Notations: IfIis an interval of real numbers, L2(I;R)\ndenotes (the class of equivalence of) square-integrable\nfunctions from ItoR. Moreover L2([0,1];R) is abusively\ndenoted L2(0,1). Furthermore Hndenotes the Sobolev\nspace Wn,2, i.e.,\nu∈H1⇔u∈L2, u′∈L2, (4)\nin which u′denotes the derivative of u.\n2 Main result\nTo achieve our objective, we perform a change of variable\nin order to obtain an error variable u(·,·) and to prove\nexponential decay of its partial derivatives.\nThe error variable u(·,·) is defined as follows, for every\n(x, t)∈[0,1]×[0,∞)\nu(t, x) :=v(t, x)−tvref\n1\n+Zx\n01\na(s)Zs\n0[−vref\n1q(χ) +f(χ)]dχds\n+a(0)\nµ1[−γ1vref\n1+fac]Zx\n01\na(s)ds, (5)\nη2(t) :=ηv(t)−β1\nkia(1)Z1\n0[−vref\n1q(s) +f(s)]ds\n−β1a(0)\nkiµ1a(1)[−γ1vref\n1+fac] +νvref\n1−fc\nki.(6)\nNote that, for every ( x, t)∈[0,1]×[0,∞),\nu(t, x) =v(t, x)−tvref\n1+F(x), (7)\nut(t, x) =vt(t, x)−vref\n1, (8)\nwhere we have gathered all the uncertainties in the func-\ntionF(·) and it is immediate to deduce from (8) that\nproving exponential decay of ut(in an appropriate sense)\nis equivalent to prove it for vt−vref\n1and hence to achieve\nthe desired control objective.\nFrom now, we will therefore focus on the error variable\nu(·,·). Direct computations yield that it is the solution\nof the following system:\n\n\nutt−(a(x)ux)x=−q(x)ut,\n(t, x)∈R+×(0,1), (9a)\nut(t,1) = η1(t) (9b)\nut(t,0) = ξ1(t) (9c)\n˙η1(t) =−α1η1(t)−α2η2(t)−β1ux(t,1),(9d)\n˙η2(t) =η1(t), (9e)\n˙ξ1(t) =−γ1ξ1(t) +µ1ux(t,0), (9f)\nu(0,·) =u0, u t(0,·) =u1on (0 ,1), (9g)\nη1(0) = η0, η 2(0) = η2,0ξ1(0) = ξ0. (9h)\n3where α1:=k+νandα2:=ki, and kis chosen such\nthatα1is positive.\nConsider the following Hilbert spaces\nXw: =H1((0,1);R)×L2((0,1);R)×R3,(10)\nXs: =H2((0,1);R)×H1((0,1);R)×R3.(11)\nThe wave equation is associated with the following ab-\nstract problem\n\u001a˙X(t) +AX(t) = 0 , (12a)\nX(0) =X0∈Dom(A)⊂Xs⊂Xw, (12b)\nin which\n∀z∈Dom(A),Az:=\n−z2\n−(az′\n1)′+qz2\nα1z3+α2z4+β1z′\n1(1)\n−z3\nγ1z5−µ1z′\n1(0)\n,(13)\nand\nDom(A) :={z∈Xs;z2(1) = z3, z2(0) = z5}.(14)\nOur well-posed result goes as follows.\nTheorem 1 Considering assumption (h1)and(h2), the\nabstract problem (12) is well-posed. In order words for\nany initial data X0∈Dom(A), there exists a unique\nsolution to the abstract problem (12), such that for any\nt≥0,X(t)∈Dom(A)⊂Xsand\nX ∈C0([0,∞);Dom(A))∩C1([0,∞);Xw),(15)\nXwis the state space of weak solutions and the Hilbert\nspace considered and is defined in (10).Xsis the state\nspace of strong solutions and is defined in (11).\nIn addition, for all initial data X0∈Xw, there exists a\nweak solution X(t)∈Xwto the abstract problem (12)\ngiven by\nX(t) =S(t)X0, (16)\nin which Sis the C0-semigroup generated by the un-\nbounded operator A. Moreover, it holds\nX ∈C0([0,∞);Xw). (17)\nThe proof is based on finding a transformation such that\nthe abstract problem is associated with a linear maxi-\nmal monotone operator. Then the conclusion is drawnby using the Hille-Yosida theorem. The part on weak\nsolutions holds true from the fact that Dom( A) is dense\ninXw, and therefore S(t) defined a strongly continu-\nous map from XwtoXw. Details are provided in Ap-\npendix A. The state is\nX(t) := [ u(t,·), ut(t,·),\nη1(t), η2(t), ξ1(t)]∈Dom(A)⊂Xs.(18)\nWe define the energy Euof a solution of (9) as ∀t≥0\nEu(t) :=1\n2Z1\n0(ut(t, x)2+a(x)ux(t, x)2)dx. (19)\nNote that this energy is invariant by translations with\nconstants, i.e., Eu=Evifu−vis a constant func-\ntion. Moreover, the absolutely continuous function\nu(·,1)−η2(·) is constant along a trajectory of (9) and\nequal to u∗where\nu∗:=u0(1)−η2(0). (20)\nOur objective is to establish the exponential stability of\nthe trajectory with respect to the following attractor\nS:={z∈Xw, z1(·) =d, d∈R, z2(·) = 0 ,\nz3= 0, z4= 0, z5= 0}. (21)\nThis attractor is the kernel of the following functional\nΓ(X(t)) :=Z1\n0[u2\nt(t, x) +u2\nx(t, x)]dx\n+η2\n1(t) +η2\n2(t) +ξ2\n1(t), (22)\nindeed it holds\nΓ(z) = 0⇔z∈S. (23)\nWe establish the following result.\nTheorem 2 Consider the 1D wave equation (9)with the\nassumptions (h1),(h2),(h3), and with α2, α1>0. Then,\nthere exist a positive constant ρ, and a positive constant\nMsuch that, for every weak solution X, it holds,\nΓ(X(t))⩽MΓ(X(0))e−ρt. (24)\nand the system is exponentially stable towards the attrac-\ntorS.\nIn addition it holds that max x∈[0,1]|u(t, x)−u∗|tends\nexponentially to zero as ttends to infinity, with a decay\nrate larger than or equal to ρ.\n4Theorem 3 Under assumption (h1)-(h2), and for any\nki=α2>0andβ1, µ1>0, the conclusion on Theorem 2\nstill holds if\nα1>−β1κ2\n2a(1)+κ1κ2, (25)\nγ1>−µ1κ2\n2a(0)+κ1κ2, (26)\nwhere\nκ1:=1\na\u0000\n3a+∥ax∥L∞\n2+q\n2\u0001\n, (27)\nκ2:=2\nq(1 +q\n2+κ1). (28)\nRemark 1 The link between α1andνis defined right\nbelow (9). This theorem means in particular that (24)\nholds in a robust way and the regulation can even admit\nsmall anti-damping at the uncontrolled boundary for cer-\ntain value of µ1,a(·), and q(·).\n3 Proof of Theorem 2\nThis proof follows a standard strategy: the result is first\nestablished for strong solutions by the determination of\nLyapunov functions verifying an appropriate differential\ninequality, and then it is extended to weak solutions by\na classical density argument. Hence, in the sequel, solu-\ntions of (9) are all assumed to be strong.\nWe start with the time derivative of Eualong a strong\nsolution. It holds for t≥0\n˙Eu=−Z1\n0qu2\ntdx+a(1)η1(t)ux(t,1)\n−a(0)ξ1(t)ux(t,0). (29)\nOne also has, for t≥0, after using (9d) and (9e)\na(1)η1(t)ux(t,1) =−a(1)\nβ1η1(t)\u0010\n˙η1(t) +α1η1(t)\n+α2η2(t)\u0011\n=−d\ndt\u0010a(1)\n2β1(η2\n1(t) +α2η2\n2(t))\u0011\n−a(1)α1\nβ1η2\n1(t). (30)\nSimilarly, one also has, for t≥0, after using (9f)\n−a(0)ξ1(t)ux(t,0) =−a(0)\nµ1ξ1(t)\u0010\n˙ξ1(t) +γ1ξ1(t)\u0011\n=−d\ndt\u0010a(0)\n2µ1ξ2\n1(t)\u0011\n−a(0)γ1\nµ1ξ2\n1(t).(31)Define for t≥0\nF(X(t)) := Eu(t) +a(1)\n2β1η2\n1(t) +a(0)\n2µ1ξ2\n1(t).(32)\nThen, by gathering (29), (30) and (31), one deduces that,\nfort≥0,\nd\ndt\u0010\nF+a(1)α2\n2β1η2\n2(t)\u0011\n=−Z1\n0qu2\ntdx\n−a(1)α1\nβ1η2\n1(t)−a(0)γ1\nµ1ξ2\n1(t). (33)\nTo conclude on the exponential stability we also need a\nnegative term in u2\nxandη2\n2. We next consider an extra\nterm which will be added in the candidate Lyapunov\nfunction in the sequel. From (20) it holds that\nη2(t) =u(t,1)−u∗, t≥0. (34)\nSet\nξ2(t) :=u(t,0)−u∗, t≥0. (35)\nOne has, for t≥0, that\nd\ndt\u0010Z1\n0(u−u∗)utdx\u0011\n=Z1\n0u2\nt+Z1\n0(u−u∗)utt\n=Z1\n0u2\nt+Z1\n0(u−u∗)(aux)x−Z1\n0q(u−u∗)ut\n=Z1\n0u2\nt−Z1\n0au2\nx−d\ndt\u0010Z1\n0q\n2(u−u∗)2dx\u0011\n+a(1)η2ux(t,1)−a(0)ξ2ux(t,0). (36)\nUsing (9d) and (9f), one deduces after computations sim-\nilar to those performed to get (30) and (31), that, for\nt≥0,\nη2ux(t,1) =−α2η2\n2(t) +η2\n1(t)\nβ1\n−d\ndt\u0010α1\n2η2\n2(t) +η1(t)η2(t)\nβ1\u0011\n, (37)\n−ξ2ux(t,0) =ξ2\n1(t)\nµ1−d\ndt\u0010γ1\n2ξ2\n2(t) +ξ2(t)ξ1(t)\nµ1\u0011\n.(38)\nWe next define for t≥0\nW(X(t)) =Z1\n0(u−u∗)utdx+Z1\n0q\n2(u−u∗)2dx\n+a(1)\nβ1\u0010α1\n2η2\n2(t) +η2(t)η1(t)\u0011\n+a(0)\nµ1\u0010γ1\n2ξ2\n2(t) +ξ2(t)ξ1(t)\u0011\n. (39)\n5Gathering (36), (37) and (38), it holds for t≥0\n˙W=−Z1\n0au2\nx−a(1)α2\nβ1η2\n2(t) +Z1\n0u2\nt\n+a(1)\nβ1η2\n1(t) +a(0)\nµ1ξ2\n1(t). (40)\nWe finally define the candidate Lyapunov function V\nused for proving Theorem 2, which is positive definite\nfor some constant ℓsuch thatp2q> ℓ > 0 by\nV(X(t)) =F(X(t)) +a(1)α2\n2β1η2\n2(t)\n+ℓW(X(t)),≥0. (41)\nPutting together (32) and (39), it holds for t≥0,\nV(X(t)) =Eu(t)\n+ℓZ1\n0\u0010\n(u−u∗)ut+q\n2(u−u∗)2\u0011\ndx\n+a(1)\n2β1\u0010\nη2\n1+α2η2\n2+ℓ(2η2η1+α1η2\n2)\u0011\n+a(0)\n2µ1\u0010\nξ2\n1+ℓ(2ξ2ξ1+γ1ξ2\n2)\u0011\n. (42)\nand similarly, putting together (33) and (40), it holds\nfort≥0,\n˙V=−Z1\n0(q\n2−ℓ)u2\ntdx−ℓZ1\n0au2\nxdx\n−a(1)\nβ1\u0010\n(α1−ℓ)η2\n1+α2ℓη2\n2\u0011\n−a(0)(γ1−ℓ)\nµ1ξ2\n1. (43)\nThe purpose of Vdefined in (42) compared with Fis\nto make negative terms in u2\nxandη2\n2appear. Next we\ncompare the functional Vto the functional Γ defined in\n(22).\nProposition 1 With the notations above, and Γdefined\nin(22), there exist ℓ > 0and two positive constants\nc, C, ρ > 0such that for every strong solution X(t)of\n(9), one gets, for t≥0,\ncΓ(X(t))≤V(X(t))≤CΓ(X(t)), (44)\n˙V(X(t))≤ −CρΓ(X(t)). (45)\nRemark 2 Using α1andα2as tuning parameters one\ncan show that a necessary condition for\n˙V(X(t))⩽−CρΓ(X(t)). (46)is that\nCρ < min{aq\n4, aq\n2q,aa(0)γ1\naµ1+a(0)}. (47)\nThis upper bound is deduced from the next inequalities\nextracted from (43)and the condition for Vto be definite\npositive.\nq\n2q> ℓ (48)\nq\n2−ℓ > Cρ (49)\nℓa > Cρ (50)\na(0)(γ1−ℓ)\nµ1> Cρ (51)\nMoreover as candCdoes not depend on q. It holds for\nthe decay rate ρ\nρ−→\nq→00. (52)\nThe suggested approach allows us only to conclude for\nstability when q= 0, and in this case we can stop at (33).\nNevertheless following [33] or [6] we could use LaSalle’s\ninvariance principle to establish asymptotic stability. If\nin addition α2= 0, in the case of no integrator the system\nfalls as a one-dimensional particular case of [4, Theorem\n1.2], and therefore the decay rate is at least logarithmic.\nPROOF. Using (20) and (32) one can observe that for\nevery t≥0 and x∈[0,1] it holds\n|u(t, x)−u∗|2≤2|u(t, x)−u(t,1)|2+ 2η2\n2(t)\n≤2Z1\n0u2\nx(t, x)dx+ 2η2\n2(t)\n≤4\naEu(t) + 2η2\n2(t), (53)\nAs an immediate consequence, one gets that, for t≥0,\nZ1\n0(u−u∗)2dx≤4\naEu(t) + 2η2\n2(t), (54)\nξ2\n2(t)≤4\naEu(t) + 2η2\n2(t). (55)\nThe proof of (44) relies now on the combination of (42),\n(54) and (55), several completions of squares and the\nCauchy-Schwartz inequality. As for the argument of\n(45), it is obtained similarly by using (43), (54) and (55),\nRelying on Proposition 1, we complete the proof of The-\norem 2.\n6From (44) and (45), it follows that ˙V≤ −ρVhence\nyielding exponential decrease of Vat the rate ρand the\nsimilar conclusion holds for Γ, thanks to (45). All items\nof Theorem 2 are proven after using (53).\n4 Discussion on the proof of the Theorem 2\nThere exist cases where the linear one-dimensional wave\ndoes not decay exponentially. For example, the solution\nuof the system\n\n\nutt(t, x) =uxx(t, x),∈R+×(0,1), (56a)\nu(t,0) = 0 , (56b)\nutt(t,1) =−ux(t,1)−ut(t,1), (56c)\ndoes not decrease exponentially towards the origin, see\n[23, Section 4]. It follows a t−1sharp decay rate. The\naddition/suppression of one term can make the decay\nrate drastically different, for example\n\n\nutt(t, x) = (aux)x(t, x),∈R+×(0,1),(57a)\nutt(t,0) = ux(t,0), (57b)\nutt(t,1) =−ux(t,1)−ut(t,1)−u(t,1)\nutt(t,1) =−uxt(t,1), (57c)\nis exponentially stable [25], whereas\n\n\nutt(t, x) = (aux)x(t, x),∈R+×(0,1),(58a)\nutt(t,0) = ux(t,0), (58b)\nutt(t,1) =−ux(t,1)−ut(t,1)−u(t,1),(58c)\nis not exponentially stable, see [6]. However the solution\nof (57) need to be more regular, see [25]. The energy of\nthe following two systems\n\n\nutt(t, x) =uxx(t, x),∈R+×(0,1), (59a)\nux(t,0) = ut(t,0), (59b)\nux(t,1) =−ut(t,1), (59c)\nand\n\n\nutt(t, x) =uxx(t, x),∈R+×(0,1), (60a)\nux(t,0) = ut(t,0), (60b)\nutt(t,1) =−ux(t,1)−ut(t,1), (60c)\nare exponentially decreasing [34]. Typically, for both pre-\nvious cases, the exponential decrease and stability can\nbe obtained via Energy/Lyapunov approach using cross\nterms in the following form.\nZ1\n0(1 +x)utuxdx, (61)which can make negative term as u2\nxandu2\ntappear for\nthe Energy/Lyapunov functional derivative. This pervi-\nous term implies boundary terms in the following form\n\u0002\nu2\nx+u2\nt\u00031\nx=0, (62)\nin the case of (59b)-(59c) or (60b)-(60c) we can manage\nto handle this term. However, this is problematic when\nconsidering both boundary conditions as (9d) and (9f).\nIndeed, even when α2= 0, we do not arrive to cope with\nthe term u2\nxboth in 0 and 1. This incapacity to handle\nthe term u2\nxwith both dynamic boundary conditions is\nproperly shown in [33] with more general form of uxut\ncross terms, and considering a large family of reforma-\ntion as hyperbolic PDE for example.\nIn particular the term (62) can be also taken care of if we\nhave damped position terms ( u) on the domain. Indeed\nin this case, this term enable us to use cross terms like\nZ1\n0uut, (63)\nThe exponential stability of the linear wave equation at\nthe origin with both dynamic boundary condition and\ndamped in velocity and position everywhere is estab-\nlished in [32, Chapter 9]. We stress that the paper deals\nwith velocity regulation which has been transformed to\nvelocity exponential stability. The term (63) is close to\nthe one we suggest\nZ1\n0(u−u∗)ut, (64)\nThis mostly corresponds to the beforehand knowledge\nof the limit value of ufor the system. This can be made\nbecause the integrator part of the system captures the\ndistance between the state and the attractor. In our case\nthis term can be added because qis strictly positive, see\n(36).\n5 Proof on Theorem 3\nWe start from the proof of Theorem 2, in (43), then we\ncompute the derivative of the following cross term, using\nintegration by parts\nd\ndt\u0010Z1\n0(1−2x)uxutdx\u0011\n=−a(1)u2\nx(t,1) +a(0)u2\nx(t,0)\n2\n+Z1\n0(2a+(a(1−2x))′\n2)u2\nx−η2\n1(t) +ξ2\n1(t)\n2dx\n+Z1\n0u2\ntdx−Z1\n0(1−2x)quxutdx. (65)\n7The above cross term can be used to make negative terms\ninη2\n1andξ2\n1appear at the cost of positive terms in u2\nt\nandu2\nx.\nConsider that |ℓ2|<√athen\nGu=Vu+ℓ2Z1\n0(1−2x)uxutdx, (66)\nis positive.\nGathering (43) and (65), and using the Young’s inequal-\nity, the derivative of Gualong the trajectory is\n˙Gu≤ −Z1\n0(q\n2−ℓ−ℓ2−ℓ2q\n2)u2\ntdx\n−Z1\n0(ℓa−ℓ2(2a+(a(1−2x))′\n2)−ℓ2q\n2)u2\nxdx\n−\u0010a(1)\nβ1(α1−ℓ) +ℓ2\n2\u0011\nη2\n1−(a(0)(γ1−ℓ)\nµ1+ℓ2\n2)ξ2\n1\n−a(1)\nβ1α2ℓη2\n2 (67)\nThe exponential stability still holds if the following in-\nequalities hold\nq\n2−ℓ−ℓ2(1 +q\n2)>0, (68)\nℓa−ℓ2(3a+a′\n2+q\n2)>0, (69)\n2a(1)(α1−ℓ) +β1ℓ2>0, (70)\n2a(0)(γ1−ℓ) +µ1ℓ2>0. (71)\nA sufficient condition for the four previous inequalities\nto hold is\nℓ2< κ2, (72)\nℓ > κ 1κ2, (73)\n2a(1)(α1−κ1κ2) +β1κ2>0, (74)\n2a(0)(γ1−κ1κ2) +µ1κ2>0. (75)\nwhere κ1andκ2are defined in (27)-(28). This concludes\nthe proof.\n6 Exponential stability for the zero input sys-\ntem with no disturbance.\nIn the following we investigate and establish results on\nassociated problems. We first start with a wave equa-\ntion subject to a Dirichlet’s boundary conditions and a\n2nd order dynamic boundary condition. The second sys-\ntem we add an integral action to the dynamics boundary\ncondition. The third and last system consist of a wave\nequation with both 2nd order dynamics boundary con-\nditions, and correspond to the zero input system with\nno disturbance.Proposition 2 Consider the following 1D wave equa-\ntion\n\n\nutt−(aux)x=−qut,(t, x)∈R+×(0,1),(76a)\nut(t,1) = η1(t), (76b)\n˙η1(t) =−α1η1(t)−β1ux(t,1), (76c)\nu(t,0) = 0 , t≥0, (76d)\nu(0,·) =u0, u t(0,·) =u1,on(0,1), (76e)\nη1(0) = η0. (76f)\nwhere a(·),q(·)are respecting (h1)-(h2), and with α1and\nβ1are strictly positive.\nThe state of this system is\nX2(t) =[u(t,·), ut(t,·), η1(t)]∈Dom(A2), (77)\nwhere A2is the unbounded operator associated with (76).\nThe domain is defined as\nDom(A2) ={z∈X2,s, z1(0) = 0 , z2(1) = z3},(78)\nwhere X2,sis the space of strong solutions, and X2,wis\nthe space of weak solutions defined as\nX2,s=H2×H1×R, (79)\nX2,w=H1×L2×R. (80)\nFinally, consider\nΓ2(X2(t)) =Z1\n0(u2\nt(t, x) +u2\nx(t, x))dx\n+η2\n1(t). (81)\nThen, there exist a positive constant ρand a positive\nconstant Msuch that for every weak solution X2, it holds\nΓ2(X2(t))⩽MΓ2(X2(0))e−ρt. (82)\nAnd the system is exponentially stable towards the origin\nofX2,w.\nIn addition, it holds that max x∈[0,1]|u(t, x)|tends expo-\nnentially to zero as ttends to infinity, with a decay rate\nlarger than or equal to ρ.\nThe following system is when we consider an integral\npart at the dynamic boundary for (76).\nProposition 3 Consider the following 1D wave equa-\n8tion,\n\n\nutt−(aux)x=−qut,(t, x)∈R+×(0,1),(83a)\nut(t,1) = η1(t), (83b)\n˙η1(t) =−α1η1(t)−α2η2(t)−β1ux(t,1), (83c)\n˙η2(t) =η1(t), (83d)\nu(t,0) = 0 , t≥0, (83e)\nu(0,·) =u0, u t(0,·) =u1,on(0,1), (83f)\nη1(0) = η0, η 2(0) = η2,0. (83g)\nwhere a(·),q(·)are respecting (h1)-(h2), and with α1,α2\nandβ1are strictly positive.\nForx∈[0,1], define\nv(x) :=C2Zx\n0ds\na(s), (84)\nC2:=a(1)α2\na(1)α2R1\n0ds\na(s)+β1u∗(1). (85)\nThe state of this system is\nX3(t) = [u(t,·), ut(t,·), η1(t),\nη2(t)]∈Dom(A3), (86)\nwhere A3is the unbounded operator associated with (83).\nThe domain is defined as\nDom(A3) ={z∈X3,s, z1(0) = 0 , z2(1) = z3},(87)\nwhere X3,sis the space of strong solutions, and X3,wis\nthe space of weak solutions defined as\nX3,s=H2×H1×R2, (88)\nX3,w=H1×L2×R2. (89)\nFinally, consider\nΓ3(X2(t)) =Z1\n0((u(t, x)−v(x))2+u2\nt(t, x)\n+u2\nx(t, x))dx\n+η2\n1(t) + (η2(t)−β1C2\na(1)α2)2.(90)\nThen, there exists a positive constant ρand a positive\nconstant Msuch that for every weak solution X2, it holds\nΓ3(X3(t))⩽MΓ3(X3(0))e−ρt. (91)\nAnd the system is exponentially stable towards the at-\ntractor defined as ker (Γ3(·)).\nIn additions, it holds that max x∈[0,1]|u(t, x)−v(x)|tends\nexponentially to zero as ttends to infinity, with a decay\nrate larger than or equal to ρ.Now we consider the case where α2= 0 in (9). This\nsystem has been studied in a more general and multidi-\nmensional setup in [4], the author establishes with lessen\nhypothesis logarithmic decay rates.\nProposition 4 Consider the following 1D wave equa-\ntion,\n\n\nutt−(aux)x=−qut,(t, x)∈R+×(0,1),(92a)\nut(t,1) = η1(t), (92b)\nut(t,0) = ξ1(t), (92c)\n˙η1(t) =−α1η1(t)−β1ux(t,1), (92d)\n˙ξ1(t) =−γ1ξ1(t) +µ1ux(t,0), (92e)\nu(0,·) =u0, u t(0,·) =u1on(0,1), (92f)\nη1(0) = η0, ξ 1(0) = ξ0. (92g)\nwhere a(·),q(·)are respecting (h1)-(h2), and with α1, β1,\nγ1andµ1are positive. The state of this system is\nX4(t) =[u(t,·), ut(t,·), η1(t), ξ1(t)]∈Dom(A4),(93)\nwhere A4is the unbounded operator associated with (92).\nThe domain is defined as\nDom(A4) ={z∈X3,s;\nz2(0) = z4, z2(1) = z3}, (94)\nwhere X3,sis the space of strong solutions, and X3,w\nis the space of weak solutions, both defined in (88)-(89)\nFinally, consider\nΓ4(X4(t)) =Z1\n0(u2\nt(t, x) +u2\nx(t, x))dx\n+η2\n1(t) +ξ2\n1(t). (95)\nThen, there exists a positive constant ρand a positive\nconstant Msuch that, for every weak solution X4, it holds\nΓ4(X4(t))⩽MΓ4(X4(0))e−ρt. (96)\nAnd the system is exponentially stable towards the at-\ntractor S4defined by\nS4={z∈X3,w, z1(·) =d, d∈R, z2(·) = 0 ,\nz3= 0, z4= 0}. (97)\nwhich is the kernel of Γ4(·).\nIn addition, there exists u∗so that max x∈[0,1]|u(t, x)−u∗|\ntends exponentially to zero as ttends to infinity.\nPROOF.\nWe start by proving Proposition 2. As before, the ar-\ngument is based on an appropriate Lyapunov function\n9¯Vu=¯Fu+ℓ¯Wuwhere ℓis a positive constant to be cho-\nsen and\n¯Fu(t) :=1\n2Z1\n0(u2\nt+au2\nx)dx+a(1)\n2β1η2\n1, (98)\n¯Wu(t) :=Z1\n0uutdx+1\n2Z1\n0qu2dx\n+a(1)\nβ1η1u(t,1)+a(1)α1\n2β1u(t,1)2. (99)\nOne gets, using integration by parts, (76c), and (76d)\n˙¯Fu(t) :=−Z1\n0(qu2\nt)dx−a(1)α1\nβ1η2\n1, (100)\n˙¯Wu(t) :=Z1\n0u2\ntdx−Z1\n0au2\nxdx+a(1)\nβ1η2\n1.(101)\nTherefore\n˙¯Vu=−Z1\n0(q−ℓ)u2\ntdx−ℓZ1\n0au2\nxdx\n−a(1)\nβ1(α1−ℓ)η2\n1. (102)\nThe conclusion follows by taking ℓ >0 small enough and\nnoting that, thanks to the Dirichlet boundary condition\n(83e), for every t≥0 and x∈[0,1]\n|u(t, x)|2=|u(t, x)−u(t,0)|2\n≤Z1\n0u2\nx(t, x)dx≤2Eu(t)≤2¯Fu(t).(103)\nOne proceeds by establishing an analog to Proposition 1\nwhere Γ and Vare replaced by Γ 2and¯Vuin order first\nto obtain that˙¯Vu≤ −ρ¯Vufor some positive constant ρ\nindependent of the state and finally to conclude as in the\nfinal part of the argument of Theorem 2.\nWe next turn to the proof of Proposition 3. Using the\nnotations of the proposition, we set\nw(t, x) : = u(t, x)−v(x), t≥0, x∈[0,1],\n¯η2(t) : = η2(t) +β1C2\na(1)α2, t≥0. (104)\nIt is a matter of elementary computations to check that\nwis the solution of (83) with different and corresponding\ninitial conditions with the Dirichlet boundary condition\natx= 0 (since v(0) = 0) and the boundary condition\ngiven by\n˙η1(t) =−α1η1(t)−α2¯η2(t)−β1wx(t,1), (105)\n˙¯η2(t) =η1(t). (106)It holds\nw∗(1) := w(t,1)−¯η2(t)\n=u∗(1)−v(1)−β1C2\na(1)α2\n=u∗(1)−C2(Z1\n0ds\na(s)+β1C2\na(1)α2) = 0 .(107)\nWe have essentially reduced the problem to only deal\nwith solutions of (92) with the Dirichlet boundary con-\ndition at x= 0, with the additional constraint that\nw∗(1) = 0. In that case, we consider the candidate Lya-\npunov function ˜Vw=˜Fw+ℓ˜Wwwhere ℓis a positive\nconstant to be chosen and\n˜Fw(t) : =1\n2Z1\n0(w2\nt+aw2\nx)dx\n+a(1)\n2β1(η2\n1+α2¯η2\n2), (108)\n˜Ww(t) :=Z1\n0wwtdx+1\n2Z1\n0qw2dx\n+a(1)\nβ1\u0010α1\n2¯η2\n2+η1¯η2\u0011\n. (109)\nOne gets\n˙˜Vu=−Z1\n0(q−ℓ)w2\ntdx−ℓZ1\n0aw2\nxdx\n−a(1)\nβ1(α1−ℓ)η2\n1−ℓa(1)α2\nβ1¯η2\n2, (110)\nwhere we have repeatedly used the equality w(t,1) =\n¯η2(t). By following what has been done previously, the\nconclusion follows.\nWe finally prove Proposition 4. As before the argument\nis based on an appropriate Lyapunov function ¯Vudefined\nlater. We first consider Fgiven in (32) and note that for\nt≥0 it holds\n˙F=−Z1\n0qu2\ntdx−a(1)α1\nβ1η2\n1(t)−a(0)γ1\nµ1ξ2\n1(t).(111)\nWe next compute along solutions of (92) the following\ntime derivative\nd\ndt\u0010Z1\n0\u0010\nu(t, x)−u(t,1)\u0011\nut(t, x)dx\u0011\n=\n+Z1\n0u2\ntdx−Z1\n0q\u0000\nu(t, x)−u(t,1)\u0001\nut(t, x)dx\n+\u0010\nu(t,1)−u(t,0)\u0011\na(0)ux(t,0)\n−Z1\n0au2\nxdx−η1Z1\n0ut(t, x)dx. (112)\n10In the above equation, we use (92e) to get rid of ux(t,0)\nand, to obtain for t≥0 that\n\u0010\nu(t,1)−u(t,0)\u0011\nux(t,0) =\n+\u0010\nu(t,1)−u(t,0)\u0011˙ξ1+γ1ξ1\nµ1\n=d\ndt\u0010\u0010\nu(t,1)−u(t,0)\u0011ξ1\nµ1\u0011\n−(η1−ξ1)ξ1\nµ1+γ1ξ1\nµ1\u0010\nu(t,1)−u(t,0)\u0011\n. (113)\nSetting for t≥0\nGu(t) : =Z1\n0\u0010\nu(t, x)−u(t,1)\u0011\nut(t, x)dx\n−a(0)ξ1\nµ1\u0010\nu(t,1)−u(t,0)\u0011\n, (114)\nwe deduce from the above that along with solutions of\n(92) that\n˙Gu=Z1\n0u2\ntdx−Z1\n0au2\nxdx−a(0)ξ1\nµ1(η1−ξ1)\n+a(0)γ1ξ1\nµ1\u0010\nu(t,1)−u(t,0)\u0011\n−η1Z1\n0ut(t, x)dx\n−Z1\n0q\u0010\nu(t, x)−u(t,1)\u0011\nutdx. (115)\nWe finally recall that there exists a positive constant\nCa(independent of the solutions of (92)) such that, for\nt≥0,\nZ1\n0|ut|dx+ max\nx∈[0,1]|u(t, x)−u(t,1)|\n≤Z1\n0(|ut|+|ux|)dx\n≤CaE1/2\nu(t). (116)\nWe now choose ¯Vu=F+ℓGuforℓ >0 small enough. Us-\ning repeatedly the Cauchy-Schwarz inequality, and (116)\nin (111) and (115), one gets for εandℓsmall enough\nthat (44) and (45) hold true, from which one deduces\nItem( i) of Proposition 4.\nFinally, to get Item( ii) of Proposition 4, first notice that\nu(t,1) admits a limit u∗asttends to infinity since, for\nevery t, t′>0 it holds u(t,1)−u(t′,1) =Rt\nt′η1andη1\ndecreases to zero exponentially. The conclusion follows\nnow by using (116).\nRemark 3 In the proofs of all our results, one could use\nthe function Gu(especially the integral term) to obtainthe exponential decrease of Euand some of the compo-\nnents of the Wentzell’s boundary conditions. However,\nthis does not allow one to determine the limit u∗for the\nsolution uin terms of initial conditions. In particular,\nwe are not able to characterize u∗in Proposition 4.\nNote also that\nu(t, x)−u(t,1) =−Z1\nxux(t, s)ds. (117)\nThis can be related with the means of uxand therefore\nwe have extended our Lyapunov function with a space\nmoving evaluation of the mean of the force/torque. Indeed\nuxis associated with the torque or the force in mechanical\nsetup.\n7 Numerical schemes and simulations.\nThere exist several ways to compute numerical approxi-\nmation of the solution of evolution problems associated\nwith partial differential equation, [42]. In the case un-\nder consideration, spectral methods lead to an estima-\ntion of the base function at each time step due to the\ndynamics boundary condition. This requires an impor-\ntant computing power. As we have only one dimension\nin space finite-element methods reduce to finite differ-\nence methods with (possibly unequal) spacial step. Fi-\nnite different methods can be delicate to design in or-\nder to ensure at the same time numerical stability and\ngood approximation. Note that there also exist specific\nschemes based on Riemann invariants [2]. These last\nschemes have good numerical property, but their exten-\nsion to dynamic boundary conditions is not obvious.\nIn this paper, we suggest a new approach, which pro-\nvides numerical scheme stability and therefore achieves\nstructural stability. It is based on the discretization of\nthe Lagrangian associated with the wave equation. This\napproach leads to a special finite difference scheme. As\npreviously said the wave equation in its stationary form\ncan be associated with a Lagrangian. For the case un-\nder consideration (1), (in the stationary case where ν=\nU(t) =fc=γ1=fac= 0), this Lagrangian is given by\nL(v(t,·)) =Z1\n01\n2(v2\nt(t, x)−a(x)v2\nx(t, x))dx\n+1\n2(a(1)\nβ1vt(t,1)2+a(0)\nµ1vt(t,0)2).(118)\nFollowing the strategy in [20] and the least action princi-\nple, the dynamics of the system is associated with a sta-\ntionary action. The action for any time interval is given\nas\nI(v) =Ztf\ntiL(v(t,·))dt. (119)\n11A stationary action means that the first variation is equal\nto zero\nδI(v, δv) = 0 , (120)\nwhere the first variation is defined as\nδI(v, δv) =δI(v+δv)−δI(v) +O(∥δv∥2).(121)\nComputation gives the following stationary system\n\n\nvtt−(avx)x= 0,(x, t) inR+×(0,1),(122a)\nvtt(t,1) =−β1vx(t,1), (122b)\nvtt(t,0) = µ1vx(t,0). (122c)\nThis is the stationary part of (1), as usual the less ac-\ntion principle, the dissipation and the input are added\nafterward to obtain exactly (1). Now consider a discrete\nversion of (118)\nLd(vd(t)[·]) =1\n2N−1X\ni=1[ ˙vd(t)[i]2\n−ai−1\n12(vd(t)[i]−vd(t)[i−1]\ndxi)2\n−ai\n3(vd(t)[i+ 1]−vd(t)[i−1]\ndxi+dxi+1)2\n−ai+1\n12(vd(t)[i+ 1]−vd(t)[i]\ndxi+1)2]dxi\n+1\n2aN\nβ1˙vd(t)[N]2+1\n2a0\nµ1˙vd(t)[0]2.(123)\nThe integral part in v2\nxhas approximated using Simp-\nson’s 1 /3 rule. The derivation of the Euler-Lagrange\nequation can then be done by a symbolic numerical com-\nputation. This gives an autonomous stationary linear fi-\nnite dimensional system:\nE¨vd(t) =Avd(t), (124)\nwith σ(A)∈iR. It holds\nE= diagh\n1\nβ1dx1. . . dx N−11\nµ1i\n. (125)\nThen we add dissipation with a positive symmetric ma-\ntrixR, source term (disturbance and action) and obser-\nvation,\n\u001aE¨vd(t) =Avd(t)−R˙vd(t) +BU(t) +fet,(126a)\ny(t) =C˙vd(t). (126b)\nwith\nfT\net=h\nfcf1f2. . . f aci\n(127)which represents the disturbance, and with\nR= diagh\nν q1. . . q N−1γ1i\n. (128)\nThe control U(t) is computed through\n\u001a˙ηv(t) =y(t)−yref, (129a)\nU(t) =−kiηv(t)−kp(y(t)−yref). (129b)\nAs the main idea of this discretization scheme is to have\na good approximation of the energy, we suggest going\non with this idea using symplectic integrator scheme,\nsee [13] and references within. These methods, like the\nCrank-Nicolson method have the property preserve the\nenergy as time evolves. It is known that for a system\nwhich has an eigenvalue in iRexplicit schemes are un-\nstable, and implicit schemes are exponentially stable see\n[13]. As our system has structurally the zero eigenvalue,\nsymplectic numerical discretization schemes tend to give\nbetter behaviors approximation.\nThe idea of a symplectic scheme is to combine an implicit\nscheme together with an explicit one. This leads to\nvd[k+ 1] = vd[k] + ∆ t˙vd[k+ 1], (130)\nE˙vd[k+ 1] = E˙vd[k] + ∆ t Avd[k]−∆t R˙vd[k+ 1]\n+ ∆t BU[k] + ∆ tf. (131)\nThe second line is implicit, but Rin our case is a diagonal\nmatrix and so the associated inverse matrix is easily\ncomputed\n˙vd[k+ 1] =(1 + ∆ tE−1R)−1( ˙vd[k] + ∆ t E−1Avd[k]\n+ ∆t E−1BU[k] + ∆ tE−1f). (132)\nThere are several key points to note in this last equation.\nFirst, the term (1 + E−1∆tR)−1correspond to a con-\ntraction map in the case where Ris positive, and there-\nfore is associated with dissipation terms. Second, in the\ncase where Rrepresent anti-dissipation term, there exist\ndiscretized steps∆t\ndxwhere the numerical shame is unde-\nfined. Third, where R= 0, these equations are two-step\nexplicit ones. The value selected for the numerical simu-\nlation for the output regulation problem is summarized\nin Table 1.\nThe Figure 1 illustrates the behavior of the output reg-\nulation problem, we observe that boundary velocities of\nthe system goes exponentially towards the constant ref-\nerence. In Figure 2 the time response of the regulation\nproblem objectives are depicted. The in-domain velocity\nconverges in L2norm towards the reference. The con-\ntrol law associated with these time responses are given\nin Figure 3. It is not clear how to select the control gain\nto provide rapidity and robustness. Getting an urge in-\ntegrator gain in order to have the control go faster to-\nwards its steady state may cause some heavy oscillation.\nHowever, as proven the exponential stability still holds.\n12Symbol value Symbol value\nN 199 fc −1\na(x) sin(2 x) + 2 fac 1\nq(x) .01 + .1x2kp 10\nf(x) sin(2πx) ki=α2 20\nβ1 20 vref\n1 5\nµ1 20 vd[·] 0\nν 1 ˙vd[0 :N] 0\nγ1 1 ∆t 0.001\nTable 1\nParameter values for the simulation.\nThe time response of the wave equation velocity is drawn\nas a surface in a 3d perspective in Figure 4. There is\nfirst some important oscillation, with traveling wave go-\ning back and forth from the boundary, then the oscilla-\ntion rapidly goes smaller, and finally the velocity goes\nsmoothly towards the reference. The time response of\nthe position is given in Figure 5. The impact of the con-\nstant disturbance are more visible in this graph. The os-\ncillations observed in Figure 5 are mainly due to the dis-\nturbance which needs a particular distribution of the po-\nsition along the space. Once this particular distribution\nis obtained, the constant disturbance is compensated by\nthe integrator. The last figure, Figure 6 depicts vx(t,·)\nit allows to observe the effect of the disturbance and the\nin-domain damping. The smooth convergence of the ve-\nlocity can be compared with the behavior of vx(t,·).\n0 2 4 6 8 10 12 14\ntime t0.02.55.07.510.0velocityvt(0, t)\nvt(1, t)\nFig. 1. The boundary velocities times responses.\n8 Conclusion\nThis paper presents the first systematic Lypunov anal-\nysis for a 1-dimensional damped wave equation subject\nto various dynamic (or Wentzell) boundary conditions,\nin the case where the damping is everywhere active. As\na particular case, we also provide a regulation law for a\nwave equation (with dynamic boundary conditions) by\nthe means of a PI control. The control law achieved ex-\nponential decay rate towards the constant reference, and\n0.0 2.5 5.0 7.5 10.0 12.5 15.0\ntime t−5.0−2.50.02.55.0objectivevt(0, t)−vref\nt\nvt(1, t)−vref\nt∫x\n0(vt(x, t)−vref\nt)dxFig. 2. The objectives times responses.\n0 2 4 6 8 10 12 14\ntime t20406080100U(t)\nFig. 3. The control law time response.\nFig. 4. The distributed velocity ˙ v(t, x) time response.\nthe rejection of constant disturbance. The possible re-\njection of the disturbance by the integral action can be\nexplained by the interne model principle. The numeri-\ncal simulation shows the behavior of the closed-loop sys-\ntem with unknown disturbance. Future work will be to\nuse some of the exponential decay system study in the\nappendix as the target system for infinite-dimensional\nbackstepping control design. There is also a great inter-\nest towards considering non-linear terms. For example,\nwhat is happening when the damping is non-linear like in\n[17], or even can we generalize towards non-linear waves\n13Fig. 5. The distributed position v(t, x) time response.\nFig. 6. The distributed vx(t, x) time response.\nas\nutt= (a(x)uxp\n1 +u2x)x−q(x)ut. (133)\nMoreover for practical applications there is great inter-\nest studying the wave equation with dynamics bound-\nary condition but with a non-linear friction term at the\nboundary opposite to the control, typically LuGre fric-\ntion term.\nReferences\n[1] Matthieu Barreau, Fr´ ed´ eric Gouaisbaut, and Alexandre\nSeuret. Practical stability analysis of a drilling pipe under\nfriction with a pi-controller. IEEE Transactions on Control\nSystems Technology , 29(2):620–634, 2021.\n[2] Sylvie Boldo, Fran¸ cois Cl´ ement, Jean-Christophe Filliˆ atre,\nMicaela Mayero, Guillaume Melquiond, and Pierre Weis.Wave equation numerical resolution: a comprehensive\nmechanized proof of a c program. 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Springer Science & Business\nMedia, 2007.\n[44] Alexandre Terrand-Jeanne, Vincent Andrieu, Val´ erie\nDos Santos Martins, and Cheng-Zhong Xu. Adding integral\naction for open-loop exponentially stable semigroups and\napplication to boundary control of pde systems. IEEE\nTransactions on Automatic Control , 65(11):4481–4492, 2019.\n[45] Alexandre Terrand Jeanne, Vincent Andrieu, Melaz\nTayakout Fayolle, and Valerie Dos Santos Martins.\nRegulation of inhomogeneous drilling model with a pi\ncontroller. IEEE Transactions on Automatic Control ,\n65(1):58–71, 2019.\n[46] Roberto Triggiani. Lack of uniform stabilization for\nnoncontractive semigroups under compact perturbation.\nProceedings of the American Mathematical Society ,\n105(2):375–383, 1989.\n[47] Nicolas Vanspranghe and Lucas Brivadis. Output regulation\nof infinite-dimensional nonlinear systems: A forwarding\napproach for contraction semigroups. SIAM Journal on\nControl and Optimization , 61(4):2571–2594, 2023.\n[48] Nicolas Vanspranghe, Francesco Ferrante, and Christophe\nPrieur. Velocity stabilization of a wave equation with a\nnonlinear dynamic boundary condition. IEEE Transactions\non Automatic Control , 67(12):6786–6793, 2022.\n[49] Ji Wang and Miroslav Krstic. Delay-compensated control of\nsandwiched ode–pde–ode hyperbolic systems for oil drilling\nand disaster relief. Automatica , 120:109131, 2020.\n[50] Ji Wang and Miroslav Krstic. Output-feedback control of\nan extended class of sandwiched hyperbolic pde-ode systems.\nIEEE Transactions on Automatic Control , 66(6):2588–2603,\n2020.\nA Proof of Theorem 1\nThe proof follows the same lines as the ones exposed in\n[36]. The idea of the proof is to decompose the operator\nAdefined in (13) into a maximal monotone part and a\nremaining part. We should be able to cancel the remain-\ning part with a bijective change of variable. Finally, we\nconclude using the following theorem.\n15Theorem 4 (Hille-Yosida [3, Theorem 7.4 ]) Let\nAbe a maximal operator on the Hilbert space Hthen for\nevery X0∈ D\u0000\nA\u0001\nthere exists a unique solution Xto\nthe following abstract problem.\n\n\ndX\ndt(t) +AX(t) = 0 , (A.1a)\nX(0) = X0. (A.1b)\nwith\nX∈C1([0,∞);H)∩C([0,∞);D\u0000\nA\u0001\n). (A.2)\nNow consider the following operator\n∀z∈ D(G), Gz =\n−z2\n−(az′\n1)′+z2+z1\nβ1z′\n1(1)\n0\n−µ1z′\n1(0)\n, (A.3)\nand the following matrix\nB=\n0 0 0 0 0\n1−q+ 1 0 0 0\n0 0 −α1−α20\n0 0 1 0 0\n0 0 0 0 −γ1\n. (A.4)\nThe domain of Gis equal to the domain of A. One gets\nA=G+B. (A.5)\nGis a monotone part, this is established in the following\nlemma and Bis a bounded operator.\nLemma 5 The unbounded linear operator Gdefined in\n(A.3) is a maximal monotone operator on Xwdefined in\n(11).\nPROOF. Considering the following scalar product onXw\n⟨z, q⟩=Z1\n0(z1ν+z2q2+az′\n1ν′)dx+\na(1)\nβ1z3q3+z4q4+a(0)\nµ1z5q5, (A.6)\n⟨z, Gz⟩=Z1\n0[−z1z2+z2(−(az′\n1)′+z2+z1)\n−a(x)z′\n1z′\n2]dx+a(1)z3z′\n1(1)\n−a(0)z5z′\n1(0), (A.7)\nusing integration by parts and the fact that z∈ D(A),\none obtains\n⟨z, Gz⟩=Z1\n0z2\n2dx⩾0. (A.8)\nThus the operator Gis monotone (see [3, Chapter 7 on\nPage 181]) on the Hilbert Xw. In addition, if we establish\nthat\nR(I+G) =Xw, (A.9)\nthen the operator Gis maximal monotone (see [3, Chap-\nter 7 on Page 181]), Rstands for the range of the oper-\nator. Let y∈Xw, we have to solve\nz∈ D(A), z +Gz=y, (A.10)\nwhich means that\nz1−z2=y1, (A.11)\nz2−(az′\n1)′+z2+z1=y2, (A.12)\nz3+βz′\n1(1) = y3, (A.13)\nz4+ 0 = y4, (A.14)\nz5−µ1z′\n1(0) = y5, (A.15)\nusing the fact that z∈ D(A) one gets\n3z1−(az′\n1)′= 2y1+y2, (A.16)\nβ1z′\n1(1) + z1(1) = ( y3+y1(1)), (A.17)\n−µ1z′\n1(0) + z1(0) = ( y5+y1(0)). (A.18)\nThis is a classical stationary problem (e.g., see [3]) with\nRobin’s boundaries conditions, using standard result (as\ndone in [3, Example 6, On Page 226] ) one gets that as\n2y1+y2∈L2(0,1), (A.16)-(A.18) has a unique solution\nz1∈H2(0,1). Now one can check that the element z=\n(z1, z2, z3, z4, z5) with\n\n\nz1is the solution to (A.16)-(A.18) ,(A.19a)\nz2=z1−y1, (A.19b)\nz3=y3−a(1)z′\n1(1), (A.19c)\nz4=y4z4=y5+a(0)z′\n1(0), (A.19d)\nsatisfies (A.11)-(A.15). Moreover using (A.16)-(A.18) on\n(A.19) one gets that zsatisfying (A.19) is in D(A).\n16Now, we are ready to state the proof of the well posedness\nof (12). Note that the fact that Gis maximal monotone\nimplies that D(A) is dense in Xw(i.e.,D(A) =Xw).\nUsing the bijective change of variable\nze(t) =z(t)eBt, (A.20)\nzis the solution to (12) is equivalent to, ze∈ D\u0000\nA\u0001\nis\nthe solution to\n\n\nd\ndtze(t) +Gze(t) = 0 , (A.21a)\nze(0) = z0, (A.21b)\nwhere Bis defined in (A.4) and Gis defined in (A.3).\nFrom Lemma 5, using Theorem 4 on (A.21), and the\nchange of variable (A.20), one establishes (i). Using ar-\ngument of density of D(A) inXw, and C0-semigroup\ntheory one obtains the regularity of weak solutions.\nNote that we refer the reader to [19], [30] for the notion\nweak solutions. Moreover part of the proof are inspired\nfrom [6] and [9] which in turn originates from [39].\nB Additional materials\nThis section pertains to additional materials that are\nnot included in the accepted version of the paper and\nincludes links to online resources.\nThe first line of Ais\nh\n−a0\n6dx1−2a1dx1\n3(dx1+dx2)2,a0\n6dx1,2a1dx1\n3(dx1+dx2)2, . . .i\n(B.1)\nThe second line is\n\u0014\na0\n6dx1,−a0\n6dx1−a1\n6dx2−a2dx1\n6dx2\n2−2a2dx2\n3(dx2+dx3)2,\na1\n6dx2+a2dx1\n6dx2\n2,2a2dx2\n3(dx2+dx3)2, . . .\u0015\n(B.2)\nThei-line for i∈[3, N−2] for column i−2 ati+ 2 is\n\u0014\n2ai−2dxi−2\n3(dxi−2+dxi−1)2,ai−1dxi−2+ai−2dxi−1\n6dx2\ni−1,\n−2ai−2dxi−2\n3(dxi−2+dxi−1)2−ai−1dxi−2+ai−2dxi−1\n6dx2\ni−1\n−aidxi−1+ai−1dxi\n6dx2\ni−2aidxi\n3(dxi+dxi+1)2,\n+aidxi−1+ai−1dxi\n6dx2\ni,2aidxi\n3(dxi+dxi+1)2\u0015\n(B.3)and zero elsewhere. The N−1 line\n\u0014\n. . . ,2aN−2dxN−2\n3(dxN−2+dxN−1)2,aN−2\n6dxN−1+aN−1dxN−2\n6dx2\nN−1,\n−2aN−2dxN−2\n3(dxN−2+dxN−1)2−aN−2\n6dxN−1−aN−1dxN−2\n6dx2\nN−1\n−aNdxN−1\n6dx2\nN,aNdxN−1\n6dx2\nN\u0015\n(B.4)\nTheNline\n\u0014\n. . . ,2aN−1dxN−1\n3(dxN−1+dxN)2aNdxN−1\n6dx2\nN,\n−2aN−1dxN−1\n3(dxN−1+dxN)2−aNdxN−1\n6dx2\nN\u0015\n(B.5)\nThe reader will find an online environ-\nment for the numerical simulation at\nhttps://colab.research.google.com/drive/\n1m6uhaur3eySqQ6eyjKf6SXxHXxXhSsWd?usp=sharing\nand a git-hub depot of the numerical simulation at\nhttps://github.com/christoautom/wave_1d .\n17"    },    {        "title": "2308.09843v1.Large_thermo_spin_effects_in_Heusler_alloy_based_spin_gapless_semiconductor_thin_films.pdf",        "content": "  \n1 \n Large  thermo -spin effects  in Heusler alloy based spin -gapless \nsemiconductor thin film s \nAmit Chanda1*, Deepika Rani2, Derick DeT ellem1, Noha Alzahrani1, Dario A. Arena1, Sarath \nWitanachchi1, Ratnamala Chatterjee2, Manh -Huong Phan1 and Hari haran  Srikanth1* \n1 Department of Physics, University of South Florida, Tampa FL 33620  \n2 Physics Department, Indian Institute of Technology Delhi, New Delhi - 110016  \n*Corresponding authors: achanda@usf.edu ; sharihar@usf.edu  \n \nKeywords: Longitudinal spin Seebeck effect , Anomalous Nernst effect , Spin gapless \nsemiconductor, Heusler alloy , Magnetic anisotropy, Gilbert damping  \n \nAbstract  \nRecently, H eusler alloys -based spin gapless semiconductors (SGSs) with high Curie \ntemperature ( 𝑇𝐶) and sizeable spin polarization have emerged  as potential candidates for \ntunable spintronic applications. We report  comprehensive investigation of the temperature \ndepe ndent ANE and intrinsic longitudinal spin Seebeck effect (LSSE) in CoFeCrGa  thin films \ngrown on MgO substrates.  Our findings show  the anomalous Nernst coefficient  for the \nMgO/CoFeCrGa  (95 nm) film is ≈ 1.86 μV.K−1 at room temperature which is  nearly two \norders of magnitude  higher than  that of  the bulk polycrystalline sample of CoFeCrGa (≈\n0.018 μV.K−1) but comparable to that of the magnetic Weyl semimetal Co 2MnGa thin film  (≈\n2−3 μV.K−1). Furthermore, the LSSE coefficient for our MgO/CoFeCrGa (95nm)/Pt(5nm) \nheterostructure is ≈20.5 nV.K−1.Ω−1 at room temperature  which is twice larger  than that of \nthe half-metallic ferromagnet ic La0.7Sr0.3MnO 3 thin films (≈9 nV.K−1.Ω−1). We show that \nboth ANE and LSSE coefficients follow identical temperature dependences and exhibit a   \n2 \n maximum  at  ≈225 K which is understood as the combined effects of inelastic magnon \nscatterings and reduced magnon population at low temperatures . Our analys es not only indicate \nthat the extrinsic skew scattering is the dominating mechanism for ANE in these films but also \nprovide critical insight s into the functional  form of the observed temperature dependent LSSE  \nat low temperatures . Furthermore, by employing radio frequency transverse susceptibility and \nbroadband ferromagnetic resonance in combination with the LSSE measurements, we establish \na correlation among the observed LSSE signal, magnetic anisotropy and Gilbert damping of \nthe CoFeCrGa thin films , which will be beneficial for fabricating tunable and highly efficient \nHeusler alloys based spincaloritronic nanodevices.  \n   \n3 \n 1. INTRODUCTION  \nThe p ast few years have witnessed extensive resear ch efforts in the field of spin caloritronics \nfor the development of highly efficient next -generation spin -based electronic devices by \ncombining the versatile advantages of spintronics and thermoelectricity , with the aim of finding \nnovel  avenues for waste heat recovery  and thermoelectric energy conversion1,2. Fundamental \nknowledge  of the interplay between heat, charge, and spin degrees of freedom not only allowed \nus to understand  how thermal gradients can be utilized to manipulate and control the flow of \nspin angular momenta  inside a material a t nanoscale , but also helped the scientific community \nto explore various  intriguing  thermo -spin transport  phenomena, such as the anomalous Nernst \neffect  (ANE)3, spin Nernst effect4, spin Seebeck effect5,6, spin Peltier effect7 and so on . \n \nThe ANE refers to the generation of a transverse  thermoele ctric voltage  in a magnetic \nconductor/semiconductor by the application of  a thermal gradient and an external magnetic \nfield8,9. The ANE has been observed in a large  range of magnetic materials, from  half-metallic  \nferromagnets such as hole -doped manganites10, cobaltite s11–13, spin gapless semiconductors14 \nto ferrimagnets such as iron oxide15, Mn-based nitride16, as well as unconventional magnetic \nsystems with topologically non -trivial phases such as topological full Heusler ferroma gnets3,17–\n19, ferro magnetic Weyl semimetal s20,21, two -dimensional topological van der Waals \nferromagnets22,23, chiral8 and canted24 topological antiferromagnet s etc. In a topological \nmagnetic  material , charge carriers moving  through a periodic potential with strong spin -orbit \ncoupling  (SOC) acquire  an additional anomalous velocity  perpendicular to their original \ntrajectory due to the non -zero Berry curvature at the Fermi level25. This anomalous velocity \ncauses a real space spin selective deflection of the charge carriers and leads to a potentially \nlarge ANE response in these topological magnetic materia ls compared to conventional magnets \n25. In addition to the aforementioned intrinsic origin, ANE can also originate from extrinsic   \n4 \n effects for example, asymmetric skew scattering of charge carriers as observed in Heusler \nferromagnets14,26,27, hole -doped manganites10, cobaltite s11–13, spin gapless semiconductors14, \niron oxide15 etc.  \n \nOn the other hand, the longitudinal spin Seebeck effect (LSSE) refers to the thermal \ngeneration of magnonic spin current in a ferromagnetic (FM) material by the concurrent \napplications of a temperature gradient and an external magnetic field across a FM/heavy met al \n(HM) bilayer structure  and injection of th at spin current to the adjacent HM layer  with strong \nSOC , which is then converted into electrically detectable charge current in the HM layer via \nthe inverse spin Hall effect (ISHE)1,28–30. The bilayer structure consisting of the ferrimagnetic \ninsulator Y 3Fe5O12 (YIG) and Pt is known as the benchmark  system for generating pure spin \ncurrent and hence , LSSE28,30 –34. Apart from YIG, other magnetic insulators for example, the \ncompensated ferrimagnetic insulator Gd 3Fe5O12,35,36 insulating spinel ferrites  CoFe 2O4, \nNiFe 2O437,38, noncollinear antiferromagnet ic insulator LuFeO 339 etc., have also emerged as \npromising spincaloritronic materials . Nevertheless, observation of LSSE is not only restricted \nto magnetic insulators , but it has also been observed in metallic5,40, half-metallic41–43 and \nsemiconducting ferromagnet s44.  \n \nAlthough ANE and LSSE are two distinct types of magnetothermoelectric phenomena, \nthey share common origin for materials exhibiting extrinsic effects dominated  ANE45. In both  \nthe cases, simultaneous application of thermal gradient and external magnetic field generates \nmagnonic excitations . While in the case of ANE, the thermally generated magnons transfer \nspin angular momenta to the itinerant electrons of the FM via the electr on-magnon scattering \nand thereby dynamically spin polarizes them, in the case of LSSE, a spatial gradient of those \nthermally generated magnons leads to magnon accumulation close to the FM/HM interface and   \n5 \n pumps spin current to the HM layer45. Large magnon -induced ANE has been observed in MnBi  \nsingle crystal45. However, observation of large ANE in a FM conductor does not necessarily \nindicate a promise for a large  LSSE, and vice versa. Therefore, it would be technologically \nadvantageous from the perspective of spincaloritronic device applications and thermal energy \nharnessing to search for a FM material that can simultaneously exhibit large LSSE and ANE.  \n \nIn recent years , Heusler alloys -based spin gapless semiconductors (SGSs) have \nemerged  as promising magnetic materials for tunable spintronic applications as they not only \ncombine the characteristics of both half -metallic ferromagnets and gapless semiconductors ,46 \nbut also possess high Curie temperature ( 𝑇𝐶) and substantial spin polarization47–50. We have \nrecently observed large ANE  in the bulk sample of Heusler alloy based SGS : CoFeCrGa with \n𝑇𝐶≈690 K, 14,50,51 which was the first experimental observation of ANE in the SGS family. \nOur fascinating observation motivated us to explore ANE a s well as  LSSE in the CoFeCrGa \nthin films. Although SGS has been theoretically predicted to be a promising candidate for \nspintr onic applications52, there is no previous experimental study  on the  thermo -spin transport \nphenomena, especially LSSE in SGS  thin films . In this paper , we report on the temperature \ndependent ANE and LSSE in the CoFeCrGa  single layer and CoFeCrGa /Pt bilayer films with \ndifferent CoFeCrGa film thicknesses. We found that both ANE and LSSE coefficients  follow \nidentical temperature dependences and exhibit a maximum  at ≈225 K which is understood as \nthe combined effects of inelastic magnon scatterings and reduced magnon population at low \ntemperatures . Our analys es not only indicate that  the extrinsic skew scattering is the dominating \nmechanism for ANE in these films but also provi de critical insight s into the functional  form of \nthe observed temperature dependent LSSE.  Furthermore, we have established a correlation \namong  the observed LSSE signal , magnetic anisotropy and Gilbert damping of the CoFeCrGa \nthin films which will be beneficial for fabricating  tunable and efficient spincaloritronic device s.   \n6 \n 2. EXPERIMENTAL  SECTION  \nThe thin film s of CoFeCrGa w ere grown on single crystal MgO (001) substrates of \nsurface area 5×5 mm2 using an excimer KrF pulsed laser deposition  (PLD) system. The films \nwere deposited at 500  °C and were further annealed in -situ at 500  °C for 30 min to further \nenhance the chemical order and crystallization . The film surface morphology was investigated \nby field emission gun – scanning electron microscop y (FEG -SEM) and atomic force \nmicroscopy (AFM), while the structural properties of the thin films were identified by x -ray \ndiffraction (XRD) using  monochromatic Cu Kα radiation.  \n \nAFM and t emperature dependent magnetic force microscopy ( MFM ) measurements \nwere performed on a Hitachi 5300E system. All measurements were done under high vacuum \n(P ≤ 10-6 Torr). MFM measurements utilized PPP-MFMR  tips, which were magnetized out -of-\nplane with respect to the tip surface via a permanent magnet. Films were first magnetized to \ntheir saturation magnetization by being placed in a 1T static magnetic field, in -plane with the \nfilm surface. After that AC demagnetization of th e film was implement ed before initiating the \nMFM scans. After scans were performed, first a linear background was subtracted which comes \nfrom the film not being completely flat on the sample stage . After that,  a parabolic background \nwas subtracted, which a rises from the nonlinear motion of the piezoelectric crystal that drives \nthe x-y translation.  Phase standard deviation was determined by fitting a Gaussian to the image \nphase distribution and extracting the standard deviation from the fit parameters.  \n \nThe DC magnetic measurements on the samples at temperatures between 100 K and \n300 K were performed using a vibrating sample magnetometer (VSM) attached to a physical \nproperty measurement system (PPMS), Quantum Design. A linear background originating  \nfrom the diamagnetic MgO  substrate was thereby subtracted. Due to a trapped remanent field   \n7 \n inside the superconducting coils, the measured magnetic field was corrected using a \nparamagnetic reference sample.   \n \nThe longitudinal electrical resistivity, longitudinal Seebeck coefficient, and thermal \nconductivity of the bulk samples were simultaneously measured with the thermal transport \noption (TTO) of the PPMS. The electrical resistivity and Hall measurements on th e thin film \nsamples were performed using the DC resistivity option of the PPMS by employing a standard \nfour point measurement technique with sourcing currents of 500 A and 1 mA, respectively.   \n \nThe temperature dependence of the effective magnetic anisotropy fields of the \nMgO/CoFeCrGa films were measured by using a radio frequency (RF) transverse susceptibility \n(TS) measurement technique that exploits a self -resonant tunnel diode oscillator (TDO) circuit \nwith a resonance frequency of ≈12 MHz53,54. The PPMS was used as a platform to sweep the \nexternal DC magnetic field and temperature. During the TS measurement, the MgO/CoFeCrGa \nthin film samples were firmly placed inside an inductor coil (L), which is a component of an \nLC resonator circuit. The coil containing the sample was positioned at the ba se of the PPMS \nsample chamber through a multifunctional PPMS probe in such a way that the axial RF \nmagnetic field generated inside the coil stay ed parallel to the film surface, but perpendicular to \nthe DC magnetic field generated by the superconducting mag net of the PPMS. In presence of \nboth the RF and DC magnetic fields, the dynamic transverse susceptibility of the sample \nchanges which eventually changes the resonance frequency of the LC circuit53. From the \nmagnetic field  dependence of the shift in the resonance frequency recorded by an Agilent \nfrequency counter, we obtained the fie ld dependent transverse susceptibility.  \n   \n8 \n  The ANE and LSSE  measurements were performed  using a custom -designed  setup \nassembled on a universal PPMS sample puck , as shown in our previous reports14,36. For both \nthe measurements , the thin film samples were sandwiched between two copper plates. A single \nlayer of thin Kapton tape was thermally anchored  to the bare surfaces of the top (cold) and \nbottom (hot) copper plates . Cryogenic Apiezon N -grease was used to create good thermal \nconnectivity  between the thin film surface and that of the Kapton tape s. A resistive heater (PT -\n100 RTD sensor) and a calibrated Si -diode thermometer (DT-621-HR silicon diode sensor) \nwere attached to each of th ose copper  plates . The temperature s of both these copper plates  were \nmonitored and controlled individually by employing two distinct  separate temperature \ncontrollers (Scientific Instruments Model no. 9700). The top copper plate was thermally linked  \nto the base of the PPMS universal puck us ing a pair of  molybdenum screws and a 4 mm thick \nTeflon block was thermally sandwiched  between the universal PPMS puck base and the bottom \ncopper plate  to maintain a temperature difference of ~ 10 K between the hot copper plate and \nthe PPMS universal puck base.  The Ohmic contacts for the ANE and LSSE voltage \nmeasurements were made by using  a pair of thin gold wires of 25 µm diameter to the Pt layer \nby high quality conducting silver paint (SPI Supplies). In presence  of an applied temperature \ngradient  along the z-direction , and an i n-plane external DC magnetic field applied along the x-\ndirection, the transverse thermoelectric voltage generated along the y-direction across the Pt \nlayer due to the ISHE ( 𝑉𝐼𝑆𝐻𝐸) and across the CoFeCrGa film itself due  to the ANE was recorded \nwith a Keithley 2182a nanovoltmeter.  \n \nBroadband ferromagnetic resonance (FMR) measurements were performed using a \nbroadband FMR spectrometer (NanOscTM Phase -FMR, Quantum Design Inc., USA) integrated \nto the Dynacool PPMS55. \n   \n9 \n 3. RESULTS AND DISCUSSION  \n3.1. Structural and morphological properties  \nFigure 1 (a) shows the X -ray 2- (out of plane) diffraction pattern for CoFeCrGa (95nm) film \ngrown on MgO (001) substrate. In addition to the peaks  corresponding to the MgO substrate, \nthere are additional (002) and (004) diffraction peaks from the film, indicating the growth in \nthe (001) orientation. The formation of B2 CoFeCrGa structure  is confirmed by t he presence \nof (002) peak . To find the CoFeCrGa (220) peak intensity , a 2- scan was performed with = \n45 as shown in Fig. 1 (b). The lattice parameter as estimated by applying the Bragg equation \nto the (0 22) peak, was found to be 5.76 Å. \nFigure 1. (a) XRD  of MgO/ CoFeCrGa (95nm) film: ω–2θ (ou t-of-plane) scan . (b) The 2θ –θ \nscan of the (022) plane. (c) Phi -scan of the (022) plane. (d) 𝜙 -scan of the (111) plane . (e) FEG -\nSEM , (f) Cross sectional SEM image  and (g) AFM image  for the MgO/ CoFeCrGa (95nm)  film.  \n \n  \n10 \n To further confirm the epitaxial growth of the CoFeCrGa (95nm) film, 𝜙-scan was \nperformed for the (220) and (111) planes by tilting the sample, i.e., = 45 for the (220) plane \nand = 54.7 for the (111) plane ( Fig. 1 (c), and Fig. 1 (d)). The 𝜙 - scans of  both (220) and \n(111) plane show a four -fold symmetry, as four well defined peaks periodically separated from \neach other by 90 were observed. The presence of both (111) and (200) peaks rule out the \npossibility of complete A2 or B2 disorder, however partia l disorder can still be present . The \nchemical composition was interpreted by the scanning electron microscopy energy -dispersive \nspectroscopy (SEM -EDS) measurements an d was found to be Co 1.05Fe1.05Cr0.9Ga0.99, which is \nvery close to the ideal stoichiometric  composition expected for an equiatomic quaternary \nHeusler alloy. The surface morphology of the film obtained from the F EG-SEM image is \nshown in Fig. 1 (e), which indicates that the film is homogenous , which was further confirmed \nby AFM measurements as shown in Fig. 1 (g). A low root -mean -square (RMS) roughness of ≈ \n2.5nm is achieved for the CoFeCrGa film , as noticeable in the AFM image  shown in Fig. 1 (g). \nThe cross -section SEM imag e of the film is shown in Fig. 1 (f), which indicates  that the film \nthickness (~95± 5 nm).  \n \nIn Fig. 2 , we show the temperature dependent magnetic force microscopy ( MFM ) \nimages recorded  on the MgO/ CoFeCrGa (95nm)  film. The MFM image at 300 K ( see Fig. 2(a)) \nshows a bright /dark contrast with highly irregular shape d features indicating cloudlike domain -\nclusters56. Note  that in MFM, the domain -image contrast is determined by the magnetic force -\ngradient (𝑑𝐹\n𝑑𝑧) between the sample and the MFM tip (magnetized ⊥ to the film -surface), which \nis proportional to the perpendicular  component of the stray field  of the film57,58. For our film, \ndue to low bright/dark c ontrast patterns of the MFM images in the T-range: 160K ≤𝑇≤\n300 K (Fig. 2(a)-(e)), the domain boundaries are not as well -defined as observed in  films with \nstrong PMA59.   \n11 \n Figure 2. Magnetic force microscopy (MFM)  images of MgO/ CoFeCrGa (95nm)  film \nmeasured  at (a) T = 300 K, (b) T = 250 K, (c) T = 200 K, (d) T = 180 K, and (e) T = 160 K \nwhile cooling the sample after applying an IP magnetic field (higher than the IP saturation \nfield) and then AC demagnetization of the sample  at 300  K. (f) The RMS value of the phase \nshift, Δ𝜙𝑅𝑀𝑆 as a function of temperature for the MgO/ CoFeCrGa (95nm)  film extracted from \nthe MFM images.  (g) The average domain width as a function of temperature obtained from \nthe MFM images.  \n \nA steep increase in the root mean square ( RMS ) value of the phase shift,57 Δ𝜙𝑅𝑀𝑆≈\n 𝑄\n𝐾[𝑑𝐹\n𝑑𝑧] (Q = quality factor and 𝐾 = spring constant of the tip; hence, Δ𝜙𝑅𝑀𝑆 ∝ average domain \ncontrast58) has also been observed below 300 K  (see Fig. 2(f)). However, Δ𝜙𝑅𝑀𝑆 decreases \nslightly below 180 K. Averag e domain widths were determined by calculating the 2D \nautocorrelation across the MFM image s, then determining the full -width half -max (FWHM) of \narbitrary lines through the 2D autocorrelation spectra.  As shown in Fig. 2(g), the average \ndomain width also increases with decreasing temperature followed by a slight decrease below \n170 K.  \n  \n12 \n 3.2. Magnetic and electrical t ransport properties  \nPrevious studies on bulk CoFeCrGa50,51 as well as MgO/CoFeCrGa  thin films60 reveal that the \nferromagnetic transition temperature of this sample is very high (at least ≥500 K). The main \npanel of  Fig. 3(a) shows the magnetic field  dependence of magnetization, 𝑀(𝐻) of our \nMgO/CoFeCrGa  film measured at selected  temperature s in the  range: 125 K ≤𝑇 ≤300 K in \npresence of an in -plane sweeping magnetic field . The 𝑀(𝐻) loops exhibit very small coercivity \nthroughout the measured temperature range.  \n \nFigure 3. (a) Main panel: magnetic field dependence of magnetization, 𝑀(𝐻) of our \nMgO/CoFeCrGa (95nm) film measured at selected temperatures in the range: 125 K ≤𝑇 ≤\n300 K in presence of an in -plane sweeping magnetic field , inset: temperature dependence of \nthe saturation magnetizati on, MS. (b) T emperature dependence of magnetization, 𝑀(𝑇) \nmeasured in zero -field-cooled warming (ZFCW) and field -cooled -warming (FCW) protocols \nin presence of an external magnetic field : 𝜇0𝐻=0.1 T. (c) N ormalized 𝑀(𝐻) hysteresis loops \nat 𝑇=300 K for the in -plane (IP) and out -of-plane (OOP) configurations . (d) Main panel: \n  \n13 \n temperature  dependence of longitudinal resistivity, 𝜌𝑥𝑥(𝑇) for the MgO/CoFeCrGa  film in the \ntemperature range: 10 K ≤𝑇 ≤300 K, inset shows corresponding temperature dependence \nof electrical conductivity, 𝜎𝑥𝑥(𝑇). (e) The bipolar field scan s (+𝐻𝐷𝐶𝑚𝑎𝑥−𝐻𝐷𝐶𝑚𝑎𝑥 +𝐻𝐷𝐶𝑚𝑎𝑥) \nof ∆𝜒𝑇\n𝜒𝑇(𝐻𝐷𝐶) for MgO/CoFeCrGa (95nm) film measured at T = 20 K for both IP ( HDC is \nparallel to the film surface) and OOP ( HDC is perpendicular to the film surface) configurations.  \n(f) Temperature variations of  the effective anisotropy fields:  𝐻𝐾𝐼𝑃 and 𝐻𝐾𝑂𝑂𝑃for our \nMgO/CoFeCrGa (95nm) film. \n \n As shown in the inset of Fig. 3(a), the saturation magnetization, 𝑀𝑆 increases  almost \nlinearly with decreasing temperature , which is in agreement with the temperature dependent \nΔ𝜙𝑅𝑀𝑆 obtained from the MFM images58. In Fig. 3(b), we show the temperature dependence \nof magnetization, 𝑀(𝑇) measured in zero -field-cooled warming (ZFCW) and field -cooled -\nwarming (FCW) protocols in presence of an external magnetic field : 𝜇0𝐻=0.1 T. It is evident \nthat both ZFCW and FCW 𝑀(𝑇) increases with decreasing temperature down to 10 K below \nwhich it shows a slight up -turn. Furthermore, the ZFCW and FCW 𝑀(𝑇) curves do not exhibit \nany considerable bifurcation at low temperatures which  is indicative of the absence of any \nglassy magnetic ground state. Fig. 3(c) shows the normalized 𝑀(𝐻) hysteresis loops at 𝑇=\n300 K for the in -plane (IP) and out -of-plane (OOP) configurations confirming the soft \nferromagnetic nature of the film along the I P direction, which is consistent with a recent report \non this system60. \n \n The main panel of  Fig. 3(d) demonstrates  the T-dependence of longitudinal resistivity, \n𝜌𝑥𝑥(𝑇) for the MgO/CoFeCrGa  film in the temperature range: 10 K ≤𝑇 ≤300 K. It is \nobvious  that 𝜌𝑥𝑥(𝑇) exhibits semiconducting -like resistivity (𝜕𝜌𝑥𝑥\n𝜕𝑇>0) throughout the \ntemperature range . The  inset of Fig. 3(d) shows  the T-dependence of electrical conductivity , \n𝜎𝑥𝑥(𝑇) for the MgO/CoFeCrGa  film. Note that the values of both  𝜌𝑥𝑥(𝑇) and 𝜎𝑥𝑥(𝑇) for our \nMgO/CoFeCrGa  film are quite close to those reported on the same film with 12 nm thickness60.   \n14 \n Furthermore, the linear temperature coefficient of the resistivity for our MgO/CoFeCrGa  film \nwas found to be ≈−1.37 ×10−10Ω m/K, which is of the same magnitude to  that reported for \ndifferent Heusler alloys -based spin gapless semiconductors ( SGSs), such as Mn2CoAl  \n(−1.4 ×10−9Ω m/K),47  CoFeMnSi  (−7 ×10−10Ω m/K),61 CoFeCrAl  (−5 ×10−9Ω m/\nK),62 and CoFeCrGa (−1.9 ×10−9Ω m/K)60 thin films . \n \n We have also performed radio frequency (RF) transverse susceptibility (TS) \nmeasurements on our MgO/CoFeCrGa  film in the temperature range: 20 K ≤𝑇 ≤300 K to \ndetermine the temperature evolution of effective magnetic anisotropy. This technique can \naccurately determine the dynamical magnetic response of a magnetic material in presence of a \nDC magnetic field ( HDC) and a transverse RF magnetic field ( HRF) with small and fixed \namplitude.63 When HDC is scanned from positive to negative saturations , the TS of a magnetic \nmaterial with uniaxial anisotropy demonstrates well -defined peaks at the anisotropy fields, HDC \n= ± 𝐻𝐾.64 But for a magnetic material comprising of randomly dispersed magnetic easy axes, \nthe TS shows broad maxima at the effective anisotropy fields, HDC = ±𝐻𝐾𝑒𝑓𝑓. Here, we show \nthe TS spectra as percentage change of the measured transverse susceptibility as, ∆𝜒𝑇\n𝜒𝑇(𝐻𝐷𝐶)=\n𝜒𝑇(𝐻𝐷𝐶)−𝜒𝑇(𝐻𝐷𝐶𝑚𝑎𝑥)\n𝜒𝑇(𝐻𝐷𝐶𝑚𝑎𝑥)×100% , where 𝜒𝑇(𝐻𝐷𝐶𝑚𝑎𝑥) is the value of 𝜒𝑇 at the maximum value of the \napplied DC magnetic field, 𝐻𝐷𝐶𝑚𝑎𝑥 which is chosen in such a way that 𝐻𝐷𝐶𝑚𝑎𝑥≫𝐻𝐷𝐶𝑠𝑎𝑡, where \n𝐻𝐷𝐶𝑠𝑎𝑡 is the saturation magnetic field. Fig. 3(e) shows the bipolar field scan ( +𝐻𝐷𝐶𝑚𝑎𝑥−\n𝐻𝐷𝐶𝑚𝑎𝑥 +𝐻𝐷𝐶𝑚𝑎𝑥) of ∆𝜒𝑇\n𝜒𝑇(𝐻𝐷𝐶) for MgO/CoFeCrGa  film measure d at T = 20 K for both IP ( HDC \nis parallel to the film surface) and OOP ( HDC is perpendicular to the film surface) \nconfigurations. For both the configurations, the TS shows maxima centering at 𝐻𝐷𝐶=±𝐻𝐾𝑒𝑓𝑓. \nHere, we define  𝐻𝐾𝑒𝑓𝑓= 𝐻𝐾𝐼𝑃 as the IP effective anisotropy field (for IP configuration) and \n𝐻𝐾𝑒𝑓𝑓= 𝐻𝐾𝑂𝑂𝑃 as the OOP effective anisotropy field (for OOP configuration). We found that   \n15 \n |𝐻𝐾𝑂𝑂𝑃|> |𝐻𝐾𝐼𝑃| at all the temperatures indicating IP easy axis of this film in the te mperature \nrange: 20 K ≤𝑇 ≤300 K. Furthermore, it is evident that the peaks at +𝐻𝐾𝐼𝑃(+𝐻𝐾𝑂𝑂𝑃) and \n−𝐻𝐾𝐼𝑃(−𝐻𝐾𝑂𝑂𝑃) are asymmetric with unequal peak heights  which is indicative of significant \nanisotropy dispersion in our MgO/CoFeCrGa  film for the both IP and OOP configurations. The \ntemperature variations of 𝐻𝐾𝐼𝑃 and 𝐻𝐾𝑂𝑂𝑃for our MgO/CoFeCrGa (95nm) film are shown in Fig. \n3(f). Clearly, both 𝐻𝐾𝐼𝑃 and 𝐻𝐾𝑂𝑂𝑃increase with decreasing temperature and 𝐻𝐾𝑂𝑂𝑃> 𝐻𝐾𝐼𝑃 \nthroughout the measured temperature range. Interestingly, with decreasing temperature, 𝐻𝐾𝑂𝑂𝑃 \nincreases more rapidly than 𝐻𝐾𝐼𝑃 which gives rise to large difference between 𝐻𝐾𝐼𝑃 and 𝐻𝐾𝑂𝑂𝑃 at \nlow temperatures.  Additionally, both 𝐻𝐾𝐼𝑃 and 𝐻𝐾𝑂𝑂𝑃 increases more rapidly below ≈ 200 K \ncompared to the temperature range  of 200 K ≤𝑇 ≤300 K. \n \n3.3. Thermal spin transport properties : ANE and LSSE  \nNext, we focus on the thermo -spin transport properties of our MgO/CoFeCrGa (95nm) film.  \nWe have performed anomalous Nernst effect ( ANE) and longitudinal spin Seebeck effect \n(LSSE) measurements on MgO/CoFeCrGa (95nm) and MgO/CoFeCrGa (95nm)/Pt(5nm) \nfilms, respectively. Figs. 4(a) and (b) demonstrate the schematic illustration s of our ANE and \nLSSE measurements . Both t he ANE and LSSE measurements on MgO/CoFeCrGa (95nm) and \nMgO/CoFeCrGa (95nm)/Pt films, respectively  were performed by sandwiching the film \nbetween two copper blocks  and applying a temperature gradient (along the + z-direction) that \ncreates a temperature difference, ∆𝑇 between th ose copper blocks  in presence of  an external \nDC magnetic field applied along the x-direction. The thermally generated Nernst and LSSE \nvoltage s generated along the y-direction w ere recorded us ing a Keithley 2182a nanovoltmeter  \nwhile scanning the DC magnetic field. According to the theory of thermally generated magnon -\ndriven interfacial spin pumping mechanism , simultaneous application of a  vertical ( z-axis) \ntemperature gradient ( 𝛁𝑻⃗⃗⃗⃗⃗ ) and an external transverse  dc magnetic field ( 𝝁𝟎𝑯⃗⃗⃗⃗⃗⃗⃗⃗ ) (x-axis) across   \n16 \n the MgO/CoFeCrGa (95nm)/Pt film gives rise to  transverse spin current pumping from the \nCoFeCrGa  layer into the Pt layer with the  interfacial spin current density :  𝑱𝑺⃗⃗⃗ = 𝐺↑↓\n2𝜋𝛾ℏ\n𝑀𝑆𝑉𝑎𝑘𝐵𝛁𝑻⃗⃗⃗⃗⃗  \nat the  CoFeCrGa /Pt interface,  where 𝐺↑↓, ℏ, 𝛾, 𝑀𝑆, 𝑉𝑎 and 𝑘𝐵 are the interfacial spin -mixing \nconductance, the reduced Planck’s constant (ℏ= ℎ\n2𝜋), the gyromagnetic ratio, the saturation \nmagnetization of CoFeCrGa , the magnon coherence volume and the Boltzmann constant, \nrespectively42,65,66. The magnetic coherence volume is expressed as : 𝑉𝑎= 2\n3𝜁(52⁄)(4𝜋𝐷\n𝑘𝐵𝑇)3/2\n ; \nwhere, 𝜁 is the Riemann Zeta function and 𝐷 is the spin -wave stif fness constant65,66. This \ntransverse spin current, 𝑱𝑺⃗⃗⃗  is then converted  into charge current, 𝑱𝑪⃗⃗⃗  = (2𝑒\nℏ)𝜃𝑆𝐻𝑃𝑡(𝑱𝑺⃗⃗⃗ × 𝝈𝑺⃗⃗⃗⃗⃗ ) \nalong the y-axis via the inverse spin Hall effect (ISHE), where e, 𝜃𝑆𝐻𝑃𝑡, and  𝝈𝑺⃗⃗⃗⃗⃗  are the electron \ncharge, the spin Hall angle of Pt, and the spin -polarization vector, respectively . The \ncorresponding voltage along the y-axis can be expressed as,42,67,68  \n 𝑉𝐿𝑆𝑆𝐸= 𝑅𝑦𝐿𝑦𝜆𝑃𝑡(2𝑒\nℏ)𝜃𝑆𝐻𝑃𝑡 𝐽𝑆tanh(𝑡𝑃𝑡\n2𝜆𝑃𝑡),    (1) \nwhere  𝑅𝑦,𝐿𝑦,𝜆𝑃𝑡,and 𝑡𝑃𝑡 are the electrical resistance between the voltage  leads, the \ndistance between the voltage  leads, the spin diffusion length of Pt, and the thickness  of Pt layer \n(= 5 nm ), respectively . Since CoFeCrGa  is a spin -gapless semiconductor with soft \nferromagnetic behavior,51,60 concomitant application of the temperature gradient (z-axis) and \ndc magnetic field (x-axis) also generates  a spin -polarized current in the CoFeCrGa  layer along \nthe y-axis due to ANE ,69 which  gives rise to an additional  contribution ( 𝑉𝐶𝑜𝐹𝑒𝐶𝑅𝐺𝑎𝐴𝑁𝐸) to the total \nvoltage signal measured across the Pt layer in the MgO/CoFeCrGa (95nm)/Pt heterostructure.    \n17 \n  \nFigure 4. (a) and (b) the schematic illustrations of our ANE and LSSE measurements , \nrespectively . (c) and (d) show the magnetic field dependence of the ANE voltage, 𝑉𝐴𝑁𝐸(𝐻) and \nISHE -induced in-plane voltage,  𝑉𝐼𝑆𝐻𝐸(𝐻) measured on the MgO/CoFeCrGa (95nm) and \nMgO/CoFeCrGa (95nm)/Pt films, respectively  for different values of the temperature \ndifference between the hot ( 𝑇ℎ𝑜𝑡) and cold ( 𝑇𝑐𝑜𝑙𝑑) copper blocks, ∆𝑇= (𝑇ℎ𝑜𝑡−𝑇𝑐𝑜𝑙𝑑) in the \nrange: +5 K≤ ∆𝑇 ≤+18 K at a fixed average sample temperature 𝑇= 𝑇ℎ𝑜𝑡+𝑇𝑐𝑜𝑙𝑑\n2 = 295 K.  \n(e) and (f) exhibit  the ∆𝑇dependence of the background -corrected ANE voltage, 𝑉𝐴𝑁𝐸(Δ𝑇)=\n [𝑉𝐴𝑁𝐸(+𝜇0𝐻𝑚𝑎𝑥,   Δ𝑇)−𝑉𝐴𝑁𝐸(  −𝜇0𝐻𝑚𝑎𝑥,   Δ𝑇)\n2] and the background -corrected (ANE+ LSSE ) \nvoltage, 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(Δ𝑇)= [𝑉𝐼𝑆𝐻𝐸(+𝜇0𝐻𝑚𝑎𝑥,   Δ𝑇)−𝑉𝐼𝑆𝐻𝐸(  −𝜇0𝐻𝑚𝑎𝑥,   Δ𝑇)\n2], respectively . \n \n  \n18 \n In presence of a transverse temperature gradient (𝛁𝑻⃗⃗⃗⃗⃗ ), the electric field generated by \nANE in a magnetic conductor/semiconductor with magnetization 𝑴⃗⃗⃗  can be expressed as,15  \n𝑬𝑨𝑵𝑬⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ∝ 𝑆𝐴𝑁𝐸(𝜇0𝑴⃗⃗⃗ ×𝛁𝑻⃗⃗⃗⃗⃗ )           (2)  \nwhere, 𝑆𝐴𝑁𝐸 is the anomalous Nernst coefficient. Furthermore, an additional voltage \ncontribution ( 𝑉𝑃𝑟𝑜𝑥𝐴𝑁𝐸) can appear due to the magnetic proximity effect (MPE) induced ANE in \nthe non -magnetic Pt layer.69,70 Note that, onl y a few layers of Pt close to the \nCoFeCrGa (95nm)/Pt interface gets magnetized (proximitized) due to the MPE, whereas the \nremaining layers remain unmagnetized. Hence both 𝑉𝐶𝑜𝐹𝑒𝐶𝑅𝐺𝑎𝐴𝑁𝐸 and 𝑉𝑃𝑟𝑜𝑥𝐴𝑁𝐸 are suppressed due to \nthe inclusion of t he 5 nm thick Pt layer on the top of CoFeCrGa  layer .41 Therefore, the resultant \nvoltage measured across the Pt layer of our MgO/CoFeCrGa (95nm)/Pt heterostructure can be \nexpressed as,71 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸= 𝑉𝐿𝑆𝑆𝐸+ 𝑉𝐶𝑜𝐹𝑒𝐶𝑅𝐺𝑎 ,   𝑆𝑢𝑝𝐴𝑁𝐸+ 𝑉𝑃𝑟𝑜𝑥,   𝑆𝑢𝑝𝐴𝑁𝐸; where , 𝑉𝐶𝑜𝐹𝑒𝐶𝑅𝐺𝑎 ,   𝑆𝑢𝑝𝐴𝑁𝐸 and \n𝑉𝑃𝑟𝑜𝑥,   𝑆𝑢𝑝𝐴𝑁𝐸 account for the suppressed ANE voltages due to the CoFeCrGa  layer and the MPE -\ninduced ANE voltage in the Pt layer, respectively. Previous studies show that the contribution \nfrom the MPE -induced ANE in the Pt layer is negligibly small for bilayers consisting of \nmagnetic semiconductors and Pt.41,69 Also, in our previous report ,71 we have shown that the \nMPE - induced LSSE contribution of the proximitized Pt layer is negligible as only a few layers \nof Pt close to the CoFeCrGa (95nm)/Pt interface are magnetized due to the  MPE41. Therefore, \nthe resultant voltage measured across the Pt layer of our MgO/CoFeCrGa (95nm)/Pt \nheterostructure can be expressed as: 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸= 𝑉𝐿𝑆𝑆𝐸+ 𝑉𝐶𝑜𝐹𝑒𝐶𝑟𝐺𝑎 ,   𝑆𝑢𝑝𝐴𝑁𝐸. Considering a \nparallel  circuit configuration of CoFeCrGa  and Pt layers, the suppressed ANE voltage (due to \nthe CoFeCrGa  layer) across the Pt layer of the MgO/CoFeCrGa (95nm)/Pt heterostructure can \nbe expressed as,41,69 \n𝑉𝐶𝑜𝐹𝑒𝐶𝑅𝐺𝑎 ,   𝑆𝑢𝑝𝐴𝑁𝐸= (𝐹\n1+𝐹)𝑉𝐶𝑜𝐹𝑒𝐶𝑟𝐺𝑎𝐴𝑁𝐸   (3)   \n19 \n where, 𝐹= 𝜌𝑃𝑡\n𝜌CoFeCrGa∙ 𝑡CoFeCrGa\n𝑡𝑃𝑡, 𝜌CoFeCrGa  (𝜌𝑃𝑡) is the electrical resistivity of the \nCoFeCrGa  (Pt) layer, and 𝑡CoFeCrGa  (𝑡𝑃𝑡) is the thickness of the CoFeCrGa  (Pt) layer, \nrespectively.  Therefore, the intrinsic LSSE voltage contribution can be disentangled from the \nANE contribution using the expressio n,41,71 \n   𝑉𝐿𝑆𝑆𝐸= 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸− (𝐹\n1+𝐹)𝑉𝐶𝑜𝐹𝑒𝐶𝑟𝐺𝑎𝐴𝑁𝐸   (4) \nFigs. 4(c) and (d) show the magnetic field dependence of the ANE voltage, 𝑉𝐴𝑁𝐸(𝐻) and ISHE -\ninduced in-plane voltage,  𝑉𝐼𝑆𝐻𝐸(𝐻) measured on the MgO/CoFeCrGa (95nm) and MgO/\nCoFeCrGa (95nm)/Pt films, respectively  for different values of the temperature difference \nbetween the hot ( 𝑇ℎ𝑜𝑡) and cold ( 𝑇𝑐𝑜𝑙𝑑) copper blocks, ∆𝑇= (𝑇ℎ𝑜𝑡−𝑇𝑐𝑜𝑙𝑑) in the range: \n+5 K≤ ∆𝑇 ≤+18 K at a fixed average sample temperature 𝑇= 𝑇ℎ𝑜𝑡+𝑇𝑐𝑜𝑙𝑑\n2 = 295 K. Clearly, \nboth 𝑉𝐴𝑁𝐸(𝐻) and 𝑉𝐼𝑆𝐻𝐸(𝐻) signals increase upon increasing  Δ𝑇. Figs. 4(e) and (f) exhibit  the \n∆𝑇dependence of the background -corrected ANE voltage, 𝑉𝐴𝑁𝐸(Δ𝑇)=\n [𝑉𝐴𝑁𝐸(+𝜇0𝐻𝑚𝑎𝑥,   Δ𝑇)−𝑉𝐴𝑁𝐸(  −𝜇0𝐻𝑚𝑎𝑥,   Δ𝑇)\n2] and the background -corrected (ANE+ LSSE ) \nvoltage, 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(Δ𝑇)= [𝑉𝐼𝑆𝐻𝐸(+𝜇0𝐻𝑚𝑎𝑥,   Δ𝑇)−𝑉𝐼𝑆𝐻𝐸(  −𝜇0𝐻𝑚𝑎𝑥,   Δ𝑇)\n2], respectively,  where \n𝜇0𝐻𝑚𝑎𝑥 (𝜇0𝐻𝑚𝑎𝑥≫𝜇0𝐻𝑠𝑎𝑡) is the maximum value of the applied magnetic field strength and \n𝜇0𝐻𝑠𝑎𝑡 = saturation magnetic field . Evidently , both 𝑉𝐴𝑁𝐸 and 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸  scale linearly with \nΔ𝑇 and |𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸|>|𝑉𝐴𝑁𝐸|, which confirm that the observed field dependen ces originate \nfrom the ANE and (ANE+LSSE), respectively41,71.  \n \n \n   \n20 \n  \nFigure 5. (a) and (b)  𝑉𝐴𝑁𝐸(𝐻) and 𝑉𝐼𝑆𝐻𝐸(𝐻) hysteresis loops measured at selected average \nsample temperatures  in the temperature range: 125 K≤ ∆𝑇 ≤295 K for a fixed value of Δ𝑇 \n= +1 5 K on MgO/CoFeCrGa (95nm) and MgO/CoFeCrGa (95nm)/Pt films, respectively . (c) \nThe 𝑇-dependence of the background -corrected ANE voltage, 𝑉𝐴𝑁𝐸(𝑇)=\n [𝑉𝐴𝑁𝐸(+𝜇0𝐻𝑚𝑎𝑥,   𝑇)−𝑉𝐴𝑁𝐸(  −𝜇0𝐻𝑚𝑎𝑥,   𝑇)\n2] and the background -corrected (ANE+ LSSE ) \nvoltage,  𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(𝑇)= [𝑉𝐼𝑆𝐻𝐸(+𝜇0𝐻𝑚𝑎𝑥,   𝑇)−𝑉𝐼𝑆𝐻𝐸(  −𝜇0𝐻𝑚𝑎𝑥,   𝑇)\n2] measured on MgO/\nCoFeCrGa (95nm) and MgO/CoFeCrGa (95nm)/Pt films, respectively . (d) Right y -scale: the \ntemperature dependence of the in trinsic LSSE voltage, 𝑉𝐿𝑆𝑆𝐸(𝑇) and the left y-scale: the \ntemperature dependence of [𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(𝑇)−𝑉𝐴𝑁𝐸(𝑇)]. \n \nIn Figs. 5(a) and (b), we show  𝑉𝐴𝑁𝐸(𝐻) and 𝑉𝐼𝑆𝐻𝐸(𝐻) hysteresis loops measured at \nselected average sample temperatures  in the temperature range: 125 K≤ ∆𝑇 ≤295 K for a \nfixed value of Δ𝑇 = +1 5 K on MgO/CoFeCrGa (95nm) and MgO/CoFeCrGa (95nm)/Pt \nfilms, respectively . Fig. 5(c) exhibits  the 𝑇-dependence of the background -corrected ANE \n  \n21 \n voltage, 𝑉𝐴𝑁𝐸(𝑇)= [𝑉𝐴𝑁𝐸(+𝜇0𝐻𝑚𝑎𝑥,   𝑇)−𝑉𝐴𝑁𝐸(  −𝜇0𝐻𝑚𝑎𝑥,   𝑇)\n2] and the background -corrected \n(ANE+ LSSE ) voltage, 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(𝑇)= [𝑉𝐼𝑆𝐻𝐸(+𝜇0𝐻𝑚𝑎𝑥,   𝑇)−𝑉𝐼𝑆𝐻𝐸(  −𝜇0𝐻𝑚𝑎𝑥,   𝑇)\n2] measured on \nMgO/CoFeCrGa (95nm) and MgO/CoFeCrGa (95nm)/Pt films, respectively . It is evident that \nboth |𝑉𝐴𝑁𝐸(𝑇)| and |𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(𝑇)| increase with decreasing temperature up to T = 200 K \nbelow which both of them decrease gradually with further reducing the temperature, resulting \nin a maximum around 200 K. Furthermore, |𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(𝑇)|> |𝑉𝐴𝑁𝐸(𝑇)|  throughout the \nmeasured temperature range, which confirms that both ANE and LSSE contribute towards the \nvoltage measured on the MgO/CoFeCrGa (95nm)/Pt heterostructure.  \n \nIn order to determine the temperature dependence of intrinsic LS SE voltage, we have \ndisentangled the LSSE  contribution from the ANE contribution using Eqn. 4 . The right y-scale \nof Fig. 5(d) shows the temperature dependence of the intrinsic LSSE voltage, 𝑉𝐿𝑆𝑆𝐸(𝑇) \nobtained by using by Eqn. 4  incorporating the correction factor:  (𝐹\n1+𝐹), whereas  the left y-scale \nshows the temperature dependence of the voltage difference [𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(𝑇)−𝑉𝐴𝑁𝐸(𝑇)] \nwithout incorporating the aforementioned correction factor, for comparison. A clear distinction \ncan be observed be tween 𝑉𝐿𝑆𝑆𝐸(𝑇) and [𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(𝑇)−𝑉𝐴𝑁𝐸(𝑇)] in terms of the absolute \nvalue as well the nature of the T-dependence, highlighting the importance of the correction \nfactor for accurately determining the intrinsic LSSE contribution. Evidently,  |𝑉𝐿𝑆𝑆𝐸(𝑇)| \nincreases with decreasing temperature and shows a broad maximum around 200 K below which \nit decreases gradually with further lowering the temperature, as shown in Fig. 5(d). To ensure \nthat the observed behavior of 𝑉𝐴𝑁𝐸(𝑇) and 𝑉𝐿𝑆𝑆𝐸(𝑇) are intrinsic to the MgO/\nCoFeCrGa (95nm) and MgO/CoFeCrGa (95nm)/Pt films, respectively, we repeated the same \nexperiments for two more  CoFeCrGa  films with different  thicknesses , namely, 𝑡CoFeCrGa= 50 \nand 200 nm .   \n22 \n  \nFigure 6. (a) and (b)  𝑉𝐴𝑁𝐸(𝐻) and 𝑉𝐼𝑆𝐻𝐸(𝐻) hysteresis loops measured on the MgO/\nCoFeCrGa (𝑡CoFeCrGa)  and MgO/CoFeCrGa (𝑡CoFeCrGa)/Pt  films  for \n𝑡CoFeCrGa (CoFeCrGa  film thickness )=50,95 and 200 nm at 295 K for Δ𝑇 = +1 5 K. (c) \n𝑉𝐴𝑁𝐸(𝜇0𝐻=1 T), 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(𝜇0𝐻=1 T) and 𝑉𝐿𝑆𝑆𝐸(𝜇0𝐻=1 T) at 295 K plotted as a \nfunction of 𝑡CoFeCrGa . (d), (e) and (f) show the comparison of  𝑉𝐴𝑁𝐸(𝑇), 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(𝑇) and \n𝑉𝐿𝑆𝑆𝐸(𝑇), respectively for different 𝑡CoFeCrGa. \n \nThe temperature dependent magnetometry and DC electrical transport pro perties of the \n𝑡CoFeCrGa= 50 and 200 nm films are displayed in the supplementary information ( Figures S1  \nand S3). Furthermore, similar to the 𝑡CoFeCrGa= 95 nm film, both 𝑉𝐴𝑁𝐸 and 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸 for \n𝑡CoFeCrGa= 50 and 200 nm films  scale linearly with Δ𝑇 and |𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸|>|𝑉𝐴𝑁𝐸|, as shown \nin Figure  S5. In Figure  S6, we demonstrate 𝑉𝐴𝑁𝐸(𝐻) and 𝑉𝐼𝑆𝐻𝐸(𝐻) hysteresis loops at \nselected average sample temperatures  in the range: 125 K≤ ∆𝑇 ≤295 K for a fixed value of \n  \n23 \n Δ𝑇 = +15 K for  𝑡CoFeCrGa= 50 and 200 nm films . In Figs. 6(a) and (b), we compare  𝑉𝐴𝑁𝐸(𝐻) \nand 𝑉𝐼𝑆𝐻𝐸(𝐻) hysteresis loops measured on the MgO/CoFeCrGa (𝑡CoFeCrGa) and MgO/\nCoFeCrGa (𝑡CoFeCrGa)/Pt  films  for 𝑡CoFeCrGa (CoFeCrGa  film thickness )=\n50,95 and 200 nm at 𝑇= 295 K for Δ𝑇 = +15 K. As shown in Figs. 6(c), 𝑉𝐴𝑁𝐸(𝜇0𝐻=1 T), \n𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(𝜇0𝐻=1 T) and 𝑉𝐿𝑆𝑆𝐸(𝜇0𝐻=1 T) at 295 K increase with increasing 𝑡CoFeCrGa . In \nFigs. 6(d), (e) and (f), we compare 𝑉𝐴𝑁𝐸(𝑇), 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(𝑇) and 𝑉𝐿𝑆𝑆𝐸(𝑇), respectively for \ndifferent 𝑡CoFeCrGa. Clearly, the values of all the three quantities: 𝑉𝐴𝑁𝐸, 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸 and 𝑉𝐿𝑆𝑆𝐸 \nare higher for thicker CoFeCrGa films at all temperatures.  Furthermore, |𝑉𝐴𝑁𝐸(𝑇)|, \n|𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(𝑇)| and |𝑉𝐿𝑆𝑆𝐸(𝑇)| exhibit the same behavior for all the three different \nCoFeCrGa  film thicknesses , i.e., all these quantities increase with decreasing temperature \nfrom 295 K and show a broad maximum around 200 K, which is followed by a gradual decrease \nwith further lowering  the temperature. These observations confirm that the observed behavior \nof 𝑉𝐴𝑁𝐸(𝑇) and 𝑉𝐿𝑆𝑆𝐸(𝑇) are intrinsic to the MgO/CoFeCrGa  and MgO/CoFeCrGa /Pt films, \nrespectively.  \n \n3.4. Mechanism of LSSE and ANE at low temperatures  \nSince the density of the thermally generated magnons -driven spin current is \nproportional to the effective temperature gradient across the CoFeCrGa film  through the \nexpression, |𝑱𝑺⃗⃗⃗ |=𝐺↑↓\n2𝜋𝛾ℏ\n𝑀𝑆𝑉𝑎𝑘𝐵|𝛁𝑻⃗⃗⃗⃗⃗ |, it is imperative to accurately determine the ef fective \ntemperature difference s between the top and bottom surfaces of the CoFeCrGa  film(∆𝑇𝑒𝑓𝑓). \nThe total temperature difference (Δ𝑇) across the MgO/CoFeCrGa /Pt heterostructure can be \nexpressed as a linear combination of temperature drops in the Pt layer , at the Pt/ CoFeCrGa  \ninterface, in the CoFeCrGa  layer, at the CoFeCrGa /MgO interface, across the GSGG substrate  \nas well as in the N-grease layers  (thickness ≈ 1 m) on both sides of the MgO/CoFeCrGa /Pt   \n24 \n heterostructure, and can be written  as,72 ∆𝑇= ∆𝑇𝑃𝑡+∆𝑇𝑃𝑡\nCoFeCrGa+∆𝑇CoFeCrGa+\n ∆𝑇CoFeCrGa\nMgO+∆𝑇MgO+2.∆𝑇N−Grease . Since the thermal resistance of Pt is very small \ncompared to the other contributions and the bulk contributions towards the measured ISHE \nvoltage dominate o ver the interfacial contributions when the thickness of the magnetic film \n(CoFeCrGa ) is high enough,72 the total temperature difference can be approximately written as,  \n∆𝑇= ∆𝑇CoFeCrGa+∆𝑇MgO+2.∆𝑇𝑁−𝐺𝑟𝑒𝑎𝑠𝑒 .  \n \nFigure 7. (a)-(c) Right  y-scale:  temperature dependence of ∆𝑇𝑒𝑓𝑓 for different 𝑡CoFeCrGa , left \ny-scale: temperature dependence of the modified LSSE coefficient, 𝑆𝐿𝑆𝑆𝐸𝑒𝑓𝑓(𝑇) for MgO/\nCoFeCrGa (𝑡CoFeCrGa)/Pt(5nm) films for 𝑡CoFeCrGa=50,95 and 200 nm, respectively, \nfitted with the expression 𝑆𝐿𝑆𝑆𝐸𝑒𝑓𝑓∝ (𝜃𝑆𝐻𝑃𝑡𝐺↑↓\n2𝜋𝑘𝐵\n𝐷3/2)𝑇𝑛.  (d) -(f) Temperature dependence of the \n  \n25 \n ANE coefficient, 𝑆𝑥𝑦𝐴𝑁𝐸(𝑇) for the MgO/CoFeCrGa (𝑡CoFeCrGa) films  for 𝑡CoFeCrGa=\n50,95 and 200 nm, respectively, fitted with Eqn. 6 . \n \nConsidering the 4 -slab model, the total thermal resistance between hot and cold plates \ncan be written as, 𝑅𝑇ℎ= 1\n𝐴(2𝑡𝑁−𝐺𝑟𝑒𝑎𝑠𝑒\n𝜅𝑁−𝐺𝑟𝑒𝑎𝑠𝑒+ 𝑡CoFeCrGa\n𝜅CoFeCrGa+𝑡MgO\n𝜅MgO), where, 𝑡N−Grease, 𝑡MgO and \n𝑡CoFeCrGa  are the thicknesses of the grease layers, MgO substrate  and the CoFeCrGa  layer , \nrespectively; 𝜅N−Grease,𝜅MgO and 𝜅CoFeCrGa  are the thermal conductivities of the grease layer s, \nMgO substrate  and the CoFeCrGa  layer ,  respectively, and 𝐴 is the c ross sectional area.  Since \nthe rate of heat flow across the entire heterostructure reaches a constant value in  the steady \nstate,  the effective temperature difference across the CoFeCrGa film can be written as,42 \n∆𝑇𝑒𝑓𝑓= ∆𝑇CoFeCrGa = 𝛥𝑇\n [1+𝜅𝐶𝑜𝐹𝑒𝐶𝑟𝐺𝑎\n𝑡𝐶𝑜𝐹𝑒𝐶𝑟𝐺𝑎(2𝑡𝑁−𝐺𝑟𝑒𝑎𝑠𝑒\n𝜅𝑁−𝐺𝑟𝑒𝑎𝑠𝑒 + 𝑡𝑀𝑔𝑂\n𝜅𝑀𝑔𝑂)]                                       (5) \nWe have measured the  temperature dependence of  thermal conductivity of bulk CoFeCrGa  \nusing the thermal transport option of the PPMS, as shown in the supplement ary information  \n(Figure S4 ). Using the reported values  of the thermal conductivities of the Apiezon N -grease ,73 \nand the MgO crystal74, we have determined the temperature dependence of ∆𝑇𝑒𝑓𝑓 for different \n𝑡CoFeCrGa  using Eqn. 5, as shown in Figs. 7(a)-(c). Here, we have ignored the interfacial \nthermal resistances between the N -grease and the hot/cold plates as well as between the sample \nand N -grease layers .14 \n \nUsing the T-dependence of ∆𝑇𝑒𝑓𝑓, we have estimated the T-dependence of the modified  \nLSSE coefficient , 𝑆𝐿𝑆𝑆𝐸𝑒𝑓𝑓(𝑇)= 𝑉𝐿𝑆𝑆𝐸(𝑇)\n∆𝑇𝑒𝑓𝑓(𝑇)𝑅𝑦(𝑇)×(𝐿𝑧\n𝐿𝑦) for MgO/CoFeCrGa (𝑡CoFeCrGa)/Pt(5nm) \nfilms for 𝑡CoFeCrGa=50,95 and 200 nm; where, 𝐿𝑦 (= 3 mm) is the distance  between the \nvoltage leads and 𝐿𝑧 = 𝑡CoFeCrGa  (see Figs. 7(a)-(c)). Note that we have measured the T-\ndependence of resistance (𝑅𝑦(𝑇)) between the voltage -leads placed on the Pt layer of the   \n26 \n MgO/CoFeCrGa (𝑡CoFeCrGa)/Pt heterostructure s using 4 -point probe configuration. Note that \nthe value of 𝑆𝐿𝑆𝑆𝐸𝑒𝑓𝑓(𝑇)for our MgO/CoFeCrGa (𝑡CoFeCrGa)/Pt heterostructures are ≈\n12.8,20.5 and 29.8 nV.K−1.Ω−1 at T = 295 K for 𝑡CoFeCrGa=50,95 and 200 nm, \nrespectively, which are higher than that of the half-metallic FM thin films of La 0.7Sr0.3MnO 3 \n(≈9 nV.K−1.Ω−1 at room temperature )43. As shown in Figs. 7(a)-(c), 𝑆𝐿𝑆𝑆𝐸𝑒𝑓𝑓(𝑇) for the \nMgO/CoFeCrGa (𝑡CoFeCrGa)/Pt(5nm) heterostructures for all the three CoFeCrGa film \nthicknesses increases as T decreases from room temperature and shows a peak around 225 K \nbelow which it decreases rapidly with further decrease in temperature. Since the saturation \nmagnetization, 𝑀𝑆≈ 𝑇−1 in the measured temperature range  (as shown in Fig. 3(a)) and, 𝑉𝑎∝\n𝑇−3/2, according to the theory of magnon -driven LSSE, |𝑱𝑺⃗⃗⃗ |∝ 𝐺↑↓\n2𝜋𝑘𝐵\n𝐷3/2𝑇5/2|𝛁𝑻⃗⃗⃗⃗⃗ |.42  \nConsidering tanh(𝑡𝑃𝑡\n2𝜆𝑃𝑡)≈1 for our  case and, 𝜆𝑃𝑡 ∝ 𝑇−1,75 according to the Eqn. 1 , the \nmodified LSSE coefficient  becomes 𝑆𝐿𝑆𝑆𝐸𝑒𝑓𝑓=𝑉𝐿𝑆𝑆𝐸\n∆𝑇𝑒𝑓𝑓𝑅𝑦𝐿𝑦∝ (𝜃𝑆𝐻𝑃𝑡𝐺↑↓\n2𝜋𝑘𝐵\n𝐷3/2)𝑇3/2.42 As shown in \nFigs. 7(a)-(c), 𝑆𝐿𝑆𝑆𝐸𝑒𝑓𝑓(𝑇) varies as 𝑇1.41±0.12, 𝑇1.48±0.08 and 𝑇1.49±0.1 for 𝑡CoFeCrGa=\n50,95 and 200 nm, respectively in the measured temperature range , which are in good \nagreement with the theory of thermally generated magnon -driven interfacial spin pumping \nmechanism42,65,66. \n \nNow, let us understand the origin of ANE in our MgO/CoFeCrGa (𝑡CoFeCrGa) films. The \ntransverse thermoelectric coefficient  (𝑆𝑥𝑦) is expressed  as, 𝑆𝑥𝑦= [𝛼𝑥𝑦− 𝑆𝑥𝑥𝜎𝑥𝑦\n𝜎𝑥𝑥], where  𝜎𝑥𝑥 \nand 𝜎𝑥𝑦 are the longitudinal  and transverse  electrical conductivit ies which are defined  as,9,18,22 \n𝜎𝑥𝑥= [ 𝜌𝑥𝑥 \n(𝜌𝑥𝑥)2 + (𝜌𝑥𝑦)2 ] and 𝜎𝑥𝑦= [ − 𝜌𝑥𝑦 \n(𝜌𝑥𝑥)2 + (𝜌𝑥𝑦)2 ], respectively . Also,  𝛼𝑥𝑦 and 𝑆𝑥𝑥 are the \ntransverse thermoelectric conductivity  and longitudinal Seebeck coefficient, which a ccording   \n27 \n to the Mott’s relations  can be expressed as , 9,15,76  𝛼𝑥𝑦= 𝜋2𝑘𝐵2𝑇\n3𝑒(𝜕𝜎𝑥𝑦\n𝜕𝐸)\n𝐸=𝐸𝐹and 𝑆𝑥𝑥=\n 𝜋2𝑘𝐵2𝑇\n3𝑒𝜎𝑥𝑥(𝜕𝜎𝑥𝑥\n𝜕𝐸)\n𝐸=𝐸𝐹, respectively, where 𝐸𝐹 is the Fermi energy . Since  ANE and anomalous Hall \nEffect (AHE) share the common physical origin  and the AHE follows the power law \nconnecting the anomalous Hall resistivity , 𝜌𝑥𝑦𝐴𝐻𝐸 with the longitudinal electrical resistivity, 𝜌𝑥𝑥 \nthrough the expression, 𝜌𝑥𝑦𝐴𝐻𝐸= 𝜆𝑀𝜌𝑥𝑥𝑛,9 where 𝜆 is the spin -orbit coupling constant  and 𝑛 is \na constant  exponent,  the anomalous Nernst coefficient  can be expressed  as,9,15 \n𝑆𝑥𝑦𝐴𝑁𝐸= 𝜌𝑥𝑥𝑛−1[𝜋2𝑘𝐵2𝑇\n3𝑒(𝜕𝜆\n𝜕𝐸)\n𝐸=𝐸𝐹−(𝑛−1)𝜆𝑆𝑥𝑥].    (6) \nWhen  n = 1, the extrinsic skew scattering is the predominant  mechanism for the \nanomalous Nernst /Hall transport, whereas  n = 2 indicates  the intrinsic Berry curvature or, the \nextrinsic side jump dominated anomalous Nernst/Hall transport25. Using the T-dependences of \nANE voltage, 𝑉𝐴𝑁𝐸(𝑇) and ∆𝑇𝑒𝑓𝑓, we have estimated the T-dependence of the ANE coefficient, \n𝑆𝑥𝑦𝐴𝑁𝐸(𝑇)= 𝑉𝐴𝑁𝐸(𝑇)\n∆𝑇𝑒𝑓𝑓(𝑇)×(𝐿𝑧\n𝐿𝑦) for the MgO/CoFeCrGa (𝑡CoFeCrGa) films , as shown in Figs. 7(d)-\n(f). Similar to the modified LSSE voltage, 𝑉𝐿𝑆𝑆𝐸𝑒𝑓𝑓(𝑇), 𝑆𝑥𝑦𝐴𝑁𝐸(𝑇) for the MgO/\nCoFeCrGa (𝑡CoFeCrGa) films for all the three CoFeCrGa film thicknesses also increases as T \ndecreases from room temperature and shows a maximum  around 225 K below which it \ndecreases rapidly with further decrease in temperature. Interestingly, 𝑆𝑥𝑦𝐴𝑁𝐸(𝑇) for the \nMgO/CoFeCrGa (200 nm) film increases slowly with decreasing temperature from the room \ntemperature  and the maximum around 225 K is much  broader in contrast to the films with \nlower thicknesses.  \n \nNote that the value s of 𝑆𝑥𝑦𝐴𝑁𝐸 for our MgO/CoFeCrGa (𝑡CoFeCrGa) films are ≈\n 1.28 ,1.86  and 4.9 μV.K−1 at T = 295 K and ≈1.75 ,2.63  and 5.1 μV.K−1 at 225 K , for \n𝑡CoFeCrGa=50,95 and 200 nm, respectively which are nearly two orders of magnitude higher   \n28 \n than that of the bulk polycrystalline sample of CoFeCrGa (≈0.018 μV.K−1 at 300 K)14 but, \ncomparable to that of the magnetic Weyl semimetal Co 2MnGa thin films  (≈2−3 μV.K−1 at \n300 K )77,78. We fitted the 𝑆𝑥𝑦𝐴𝑁𝐸(𝑇) data in the temperature range 125 K≤ 𝑇 ≤200 K for our \nMgO/CoFeCrGa (𝑡CoFeCrGa) films using Eqn.  6 considering 𝜆, (𝜕𝜆\n𝜕𝐸)\n𝐸=𝐸𝐹, and n as the fitting \nparameter s. The best fit was obtained for  𝑛 =0.61 ± 0.02,0.68 ± 0.06 and 0.87 ± 0.05, \nfor 𝑡CoFeCrGa=50,95 and 200 nm, respectively which implies  that the origin of  ANE in our \nMgO/CoFeCrGa (𝑡CoFeCrGa) films  is dominated by the asymmetric skew scattering of charge \ncarriers  below 200 K25. Note that, we have also observed skew -scattering dominated ANE in \nbulk polycrystalline sample of CoFeCrGa ,14 for which  𝑛≈0.78. \n \nNext, let us examine the temperature evolution of the anomalous off -diagonal \nthermoelectric conductivity, 𝛼𝑥𝑦𝐴𝑁𝐸(𝑇). To determine 𝛼𝑥𝑦𝐴𝑁𝐸(𝑇), we have performed the Hall \nmeasurements on the MgO/CoFeCrGa (𝑡CoFeCrGa) films . Figs. 8(a)-(c) present  the magnetic \nfield dependence of Hall resistivity 𝜌𝑥𝑦(𝐻) of our MgO/CoFeCrGa (𝑡CoFeCrGa) films for \n𝑡CoFeCrGa=50,95 and 200 nm, respectively recorded  at few selected temperatures in the \nrange: 125 K ≤𝑇 ≤295 K. By subtra cting the ordinary Hall effect (OHE) contribution from \n𝜌𝑥𝑦(𝐻), we determined  the T-dependence of  the anomalous Hall resistivity  𝜌𝑥𝑦𝐴𝐻𝐸(𝑇). The left -\ny scale s of Figs. 8(d)-(f) exhibit the T -dependence of the anomalous Hall conductivity, \n|𝜎𝑥𝑦𝐴𝐻𝐸|= [ 𝜌𝑥𝑦𝐴𝐻𝐸 \n(𝜌𝑥𝑥)2 + (𝜌𝑥𝑦𝐴𝐻𝐸)2 ] of our MgO/CoFeCrGa (𝑡CoFeCrGa) films for 𝑡CoFeCrGa=\n50,95 and 200 nm, respectively . Note that |𝜎𝑥𝑦𝐴𝐻𝐸(𝑇)| for our MgO/CoFeCrGa (𝑡CoFeCrGa) \nfilms increases almost linearly with decreasing temperature, unlike UCo 0.8Ru0.2Al for which \n|𝜎𝑥𝑦𝐴𝐻𝐸| is nearly temperature independent at low temperatures79. This implies that 𝜎𝑥𝑦𝐴𝐻𝐸 for our \nMgO/CoFeCrGa (95nm) film is strongly dependent on the scattering rate, which further   \n29 \n supports that the extrinsic mechanisms ( e.g., asymmetric skew scattering) dominate the \ntransverse thermoelectric response of our sample at low temperatures79. \n \n \nFigu re 8. (a)-(c) Magnetic field dependence of Hall resistivity 𝜌𝑥𝑦(𝐻) of our MgO/\nCoFeCrGa (𝑡CoFeCrGa) films for 𝑡CoFeCrGa=50,95 and 200 nm, respectively recorded  at few \nselected temperatures in the range: 125 K ≤𝑇 ≤295 K. (d)-(f) Left y -scale: the temperature \ndependence of the anomalous Hall conductivity, |𝜎𝑥𝑦𝐴𝐻𝐸| of our MgO/CoFeCrGa (𝑡CoFeCrGa) \nfilms for 𝑡CoFeCrGa=50,95 and 200 nm, respectively , right y -scale: corresponding \ntemperature variation s of transverse thermoelectric conductivity  𝛼𝑥𝑦𝐴𝑁𝐸. \n \n  \n30 \n The right y-scale of Figs. 8(d)-(f) illustrates the temperature variation of 𝛼𝑥𝑦𝐴𝑁𝐸 of our \nMgO/CoFeCrGa (𝑡CoFeCrGa) films for 𝑡CoFeCrGa=50,95 and 200 nm, respectively , which \nwas obtained by i ncorporating the T-dependences of 𝑆𝑥𝑥, 𝑆𝐴𝑁𝐸, 𝜌𝑥𝑥 and 𝜌𝑥𝑦𝐴𝐻𝐸 in the \nexpression ,20,21,80 𝛼𝑥𝑦𝐴𝑁𝐸= 𝑆𝑥𝑦𝐴𝑁𝐸𝜎𝑥𝑥+𝑆𝑥𝑥𝜎𝑥𝑦𝐴𝐻𝐸= [𝑆𝑥𝑦𝐴𝑁𝐸𝜌𝑥𝑥 − 𝑆𝑥𝑥𝜌𝑥𝑦𝐴𝐻𝐸\n(𝜌𝑥𝑥)2 + (𝜌𝑥𝑦𝐴𝐻𝐸)2 ]. It is evident that \n𝛼𝑥𝑦𝐴𝑁𝐸(𝑇) for all the films shows a maximum around 225 K, similar to  𝑆𝑥𝑦𝐴𝑁𝐸(𝑇). Note that \nsimilar to 𝑆𝑥𝑦𝐴𝑁𝐸(𝑇), 𝛼𝑥𝑦𝐴𝑁𝐸(𝑇) for the MgO/CoFeCrGa (200 nm) film increases slowly with \ndecreasing temperature from the room temperature  and the maximum around 225 K is much \nbroader in contrast to the films with lower thicknesses. The value s of 𝛼𝑥𝑦𝐴𝑁𝐸 at room temperature \n(295 K) for our  MgO/CoFeCrGa (𝑡CoFeCrGa) films are 0.55,0.77 and 1.4 A.m−1.K−1 \nfor𝑡CoFeCrGa=50,95 and 200 nm, respectively , which are much smaller than that of non-\ncentrosymmetric Kagome ferromagnet  UCo 0.8Ru0.2Al79 (≈15 A.m−1.K−1 at 40 K ), Co2MnGa  \nsingle crystal18 (≈7 A.m−1.K−1 at 300 K ) but closer to that of Co2MnGa  thin films78 (≈\n2 A.m−1.K−1 at 300 K) . \n \nNext, we focus on the origin of the maximum in both 𝑆𝐿𝑆𝑆𝐸𝑒𝑓𝑓(𝑇) and 𝑆𝑥𝑦𝐴𝑁𝐸(𝑇) centered \naround 225 K.  Note that the occurrence of maximum in both LSSE and ANE signals at the \nsame temperature has been observed in other ferromagnetic metallic films, e.g., mixed valent \nmanganites42, iron oxides71. The maximum in the temperature dependent LSSE signal in the \nmagnetically ordered state is commonly observed in different ferro - and ferrimagnets for \nexample , YIG, La0.7Ca0.3MnO 3 etc., which  originates as a consequence of the combined effects \nof boundary scattering and diffusive inelastic magnon -phonon or magnon -magnon  scattering \nprocesses  together with the reduction of magnon population at low temperatures33,42,81. In YIG, \nthe maximum in the LSSE signal is thickness dependent ; it shifts  from ≈ 70 K for bulk YIG \nslab to ≈ 2 00 K for 1 m YIG film.33 In ferromagnetic metals, extrinsic contributions arising   \n31 \n from electron -magnon scattering contributes significantly to the anomalous Nernst \nthermopower.45 In presence of a temperature gradient and external magnetic field, magnons \nare excited in the bulk of a ferromagnetic material and these thermally generated magnons \ntransfer  spin-angular mo menta to the itinerant electrons via electron -magnon  scattering as a \nresult of which the itinerant electrons of the ferromagnetic layer get spin polarized  and \ncontribute to the ANE.45 Since the observed ANE in our MgO/CoFeCrGa (𝑡CoFeCrGa) films has \ndominating contributi on from the extrinsic mechanism , the occurrence of maxima in 𝑆𝑥𝑦𝐴𝑁𝐸(𝑇) \naround 225 K and the subsequent decrease in 𝑆𝑥𝑦𝐴𝑁𝐸 in our MgO/CoFeCrGa (𝑡CoFeCrGa) films  \ncan also be attributed to the diffusive inelastic magnon scatterings and reduced magnon \npopulation at low temperatures.45 A decrease in the magnon population  at low temperatures \nalso reduces electron -magnon scattering which eventually diminishes the population of the \nspin-polarized itinerant electrons participating in the skew -scattering process.  \n \nIn case of LSSE, the magnon propagation length  (〈𝜉〉) of the ferr omagnetic material \nalso plays vital role in addition to the magnon population.  〈𝜉〉 signifies the critical length scale \nfor thermally -generated magnons to develop a spatial gradient of magnon accumulation inside \na ferro magnetic film  which is one of the crucial factors that governs spin angular momentum \ntransfer to the adjacent HM layer31,33,82. A decrease in 〈𝜉〉 also suppresses the LSSE signal. It \nwas theoretically shown that  〈𝜉〉 of a magnetic material with lattice constant 𝑎0 is related to the \neffective anisotropy constant ( 𝐾𝑒𝑓𝑓) and the Gilbert damping parameter ( 𝛼), through the \nrelation34,82 〈𝜉〉= 𝑎0\n2𝛼.√𝐽𝑒𝑥\n2𝐾𝑒𝑓𝑓, , where  𝐽𝑒𝑥 is the strength of the Heisenberg exchange \ninteraction between nearest neighbors . Since 𝐾𝑒𝑓𝑓= 1\n2𝑀𝑆𝐻𝐾𝑒𝑓𝑓, the aforementioned \nexpression can be written as, 〈𝜉〉=𝑎0\n2𝛼.√𝐽𝑒𝑥\n𝜇0𝑀𝑆𝐻𝐾𝑒𝑓𝑓. Thus,  〈𝜉〉 is inversely proportional to 𝛼 as   \n32 \n well as the square -root of (𝑀𝑆𝐻𝐾𝑒𝑓𝑓). This implies that the T-evolution of  〈𝜉〉 is related to that \nof 𝛼, 𝐻𝐾𝑒𝑓𝑓and 𝑀𝑆. As shown in Fig. 3(a), 𝑀𝑆 for our MgO/CoFeCrGa (95nm) film increases \nwith decreasing temperature.  Furthermore, 𝐻𝐾𝑒𝑓𝑓 of our MgO/CoFeCrGa (95nm) film for both \nIP and OOP configurations ( both 𝐻𝐾𝐼𝑃 and 𝐻𝐾𝑂𝑂𝑃) increases with decreasing temperature and \nthe increase is more rapid below ≈ 200 K compare d to the temperature range of 200 K ≤𝑇 ≤\n300 K, as indicated in Fig. 3(f). Notably, similar behavior of  𝐻𝐾𝑒𝑓𝑓 has been observed for the \nMgO/CoFeCrGa (200nm) film (see Figure S2 ). Therefore, both 𝐻𝐾𝑒𝑓𝑓and 𝑀𝑆 tends to suppress \n〈𝜉〉 (and hence , 𝑆𝐿𝑆𝑆𝐸𝑒𝑓𝑓) at low temperatures, especially below ≈ 200 K. To comprehend the role \nof 𝛼 in 〈𝜉〉 and hence, the LSSE signal at low temperatures,  we have investigated the spin -\ndynamic properties of our MgO/CoFeCrGa (95nm)and MgO/CoFeCrGa (95nm)/Pt(5nm) \nfilms  by employing the broadband ferromagnetic resonance (FMR) measurements . \n \n3.5. Magnetization dynamics and Gilbert damping  \nFigs. 9(a) and (b) display the field -derivative of microwave (MW) power absorption spectra \n(𝑑𝑃\n𝑑𝐻) as a function of the IP DC magnetic field for various frequencies in the range: 4 GHz ≤\n𝑓 ≤18 GHz  recorded at T = 250 K for the MgO/CoFeCrGa (95nm) and MgO/\nCoFeCrGa (95nm)/Pt(5nm) films, respectively . To extract the  resonance field  (𝐻𝑟𝑒𝑠) and \nlinewidth  (∆𝐻), we fitted the  𝑑𝑃\n𝑑𝐻 lineshapes with a linear combination of symmetric and \nantisymmetric Lorentzian function derivatives as,83 \n   𝑑𝑃\n𝑑𝐻= 𝑃𝑆𝑦𝑚∆𝐻\n2(𝐻𝑑𝑐−𝐻𝑟𝑒𝑠)\n[(𝐻𝑑𝑐−𝐻𝑟𝑒𝑠)2+(∆𝐻\n2)2\n]2+𝑃𝐴𝑠𝑦𝑚(∆𝐻\n2)2\n−(𝐻𝑑𝑐−𝐻𝑟𝑒𝑠)2\n[(𝐻𝑑𝑐−𝐻𝑟𝑒𝑠)2+(∆𝐻\n2)2\n]2+𝑃0               (7) \nwher e, 𝑃𝑆𝑦𝑚 and 𝑃𝐴𝑠𝑦𝑚  are the coefficients of the symmetric and antisymmetric Lorentzian \nderivatives, and 𝑃0 is a constant offset parameter. The fitted curves are represented by solid \nlines in Figs. 9(a) and (b) . To obtain  the temperature evolution of the damping parameter,  𝛼(𝑇)   \n33 \n for the MgO/CoFeCrGa (95nm) and MgO/CoFeCrGa (95nm)/Pt(5nm) films , we have fitted  \nthe ∆𝐻-f curves with the expression,84 ∆𝐻= ∆𝐻0+4𝜋𝛼\n𝛾𝜇0𝑓, where, ∆𝐻0 represents the \ninhomogeneous broadening , 𝛾\n2𝜋= 𝑔𝑒𝑓𝑓 𝜇𝐵\nℏ is the gyromagnetic ratio, 𝜇𝐵 is the Bohr magneton, \n𝑔𝑒𝑓𝑓 is the  Landé  g-factor . Figs. 9(c) shows the  ∆𝐻-f curves for the MgO/CoFeCrGa (95nm) \nfilm at different temperatures fitted with the aforementioned expression. Clearly, the slope of \nthe ∆𝐻-f curves increases with decreasing temperature which implies increase 𝛼 at low \ntemperatures.  In Fig. 9(d), we compare the  ∆𝐻-f curves for the MgO/CoFeCrGa(95nm) and \nMgO/CoFeCrGa (95nm)/Pt(5nm) films  recorded at T = 250 K. It is evident that  ∆𝐻 for \nMgO/CoFeCrGa (95nm)/Pt(5nm) is higher than that of MgO/CoFeCrGa (95nm)for all the \nfrequencies, which is because of the loss of spin angular momentum in the CoFeCrGa  film as \na result of spin pumping and can be expressed as,85 [ ∆𝐻CoFeCrGa /𝑃𝑡− ∆𝐻CoFeCrGa]=\n 𝐺𝑅↑↓(𝑔𝑒𝑓𝑓𝜇𝐵\n2𝛾𝑀𝑆𝑡CoFeCrGa)𝑓, where 𝐺𝑅↑↓ is the real  component of the interfacial spin mixing \nconductance  (𝐺↑↓). From the fits, we obtained  𝛼CoFeCrGa = (3.6±0.2)×10−2 and \n 𝛼CoFeCrGa /𝑃𝑡=(4.12±0.1)×10−2  at 250 K for theMgO/CoFeCrGa (95nm), and MgO/\nCoFeCrGa (95nm)/Pt(5nm) films , respectively . Clearly,  𝛼CoFeCrGa /𝑃𝑡> 𝛼CoFeCrGa  which is \ncaused by additional damping due to the spin pumping  effect85. In Fig. 9(e), we compare 𝛼(𝑇) \nfor theMgO/CoFeCrGa (95nm), and MgO/CoFeCrGa (95nm)/Pt(5nm) films . It is evident \nthat  𝛼CoFeCrGa /𝑃𝑡> 𝛼CoFeCrGa  at all the temperatures and both  𝛼CoFeCrGa /𝑃𝑡 and  𝛼CoFeCrGa  \nincrease with decrease temperature, especially below 225 K. Such  increase in 𝛼 and ∆𝐻 at low  \ntemperatures  can be primarily attributed to the impurity relaxation mechanisms86–88. Since \n〈𝜉〉∝ 1\n𝛼, an increase in 𝛼 at low temperatures gives rise to decrease in 〈𝜉〉, and hence, the LSSE \nsignal. The increase s in ∆𝐻0 at low  temperatures  for both MgO/CoFeCrGa (95nm) MgO/  \n34 \n CoFeCrGa (95nm)/Pt(5nm) films (see right y-scale of Fig. 9(f)) also support  the occurrence \nof impurity relaxation  at low temperatures89. \n \nFigure 9. (a) and (b) Field-derivative of microwave (MW) power absorption spectra (𝑑𝑃\n𝑑𝐻) as a \nfunction of the IP DC magnetic field for various frequencies in the range: 4 GHz ≤𝑓 ≤\n18 GHz recorded at 250 K for the MgO/CoFeCrGa (95nm) and MgO/CoFeCrGa (95nm)/\nPt(5nm) films, respectively  fitted with Eqn. 7 . (c) The ∆𝐻-f curves for the MgO/\nCoFeCrGa (95nm) film at different temperatures fitted with ∆𝐻= ∆𝐻0+4𝜋𝛼\n𝛾𝜇0𝑓. (d) The \ncomparison of the  ∆𝐻-f curves for the MgO/CoFeCrGa (95nm) and MgO/CoFeCrGa (95nm)/\nPt(5nm) films  recorded at 250 K. (e) Comparison of the temperature dependence of damping \nparameter 𝛼(𝑇) for theMgO/CoFeCrGa (95nm), and MgO/CoFeCrGa (95nm)/Pt(5nm) \nfilms . (f) Right y -scale: temperature dependence of  ∆𝐻0 for the MgO/CoFeCrGa (95nm) and \n  \n35 \n MgO/CoFeCrGa (95nm)/Pt films, left y-scale : temperature dependence of the real component \nof the spin mixing conductance  𝐺𝑅↑↓ for the MgO/CoFeCrGa (95nm)/Pt(5nm) film. \n \nTo have a quantitative understanding of the T-evolution of spin pumping efficiency in \nthe MgO/CoFeCrGa (95nm)/Pt(5nm) film, we estimated 𝐺𝑅↑↓ using the expression,90 𝐺𝑅↑↓=\n (2𝑒2\nℎ)(2𝜋𝑀𝑆𝑡CoFeCrGa\n𝑔𝑒𝑓𝑓𝜇𝐵)[ 𝛼CoFeCrGa /𝑃𝑡− 𝛼CoFeCrGa] where, 𝐺0=(2𝑒2\nℎ) is the conductance \nquantum, and found that  𝐺𝑅↑↓≈3.25 × 1014 Ω−1m−2 at 300 K which is close to  𝐺𝑅↑↓ =7.5 ×\n 1014 Ω−1m−2 in YIG/Pt91 and  𝐺𝑅↑↓ =5.7 × 1014 Ω−1m−2 in TmIG/Pt bilayers90. As shown \nin Fig. 9(d), 𝐺𝑅↑↓ for the MgO/CoFeCrGa (95nm)/Pt(5nm) film increases with decreasing \ntemperature, which is consistent with the phenomenological expression,92 𝐺𝑅↑↓∝(𝑇𝐶−𝑇), \nwhere 𝑇𝐶= Curie temperature.  Furthermore, to confirm the aforementioned behavior of the \ntemperature evolution of 𝛼, we have repeated the broadband FMR measurements on the \nMgO/CoFeCrGa (200nm)/Pt(5nm) film.  Figs. 10(a) display the magnetic fiel d dependence \nof the (𝑑𝑃\n𝑑𝐻) lineshapes in the range: 6 GHz ≤𝑓 ≤24 GHz recorded at T = 250 K for the \nMgO/CoFeCrGa (200nm)/Pt film, fitted with Eqn. 7 . To obtain the temperature evolution of \nthe damping parameter, 𝛼(𝑇) we have fitted the ∆𝐻-f curves at different temperatures in the \nrange  of 140 K ≤𝑇 ≤300 K with the expression,84 ∆𝐻= ∆𝐻0+4𝜋𝛼\n𝛾𝜇0𝑓, as shown in Fig. \n10(b). Evidently, the slope of the ∆𝐻-f curves increases with decreasing temperature which \nimplies increase 𝛼 at low temperatures. Moreover, in Fig. 10(c), we show the fitting of the f-\n𝐻𝑟𝑒𝑠 curves at T = 250 K using Kittel’s equation for magnetic thin films with IP magnetic \nfield,86 which is expressed as, 𝑓= 𝛾𝜇0\n2𝜋√𝐻𝑟𝑒𝑠(𝐻𝑟𝑒𝑠+𝑀𝑒𝑓𝑓), where  𝑀𝑒𝑓𝑓 is the effective \nmagnetization.    \n36 \n  \nFigure 10. (a) Field-derivative of (𝑑𝑃\n𝑑𝐻) as a function of the IP DC magnetic field for various \nfrequencies in the range: 6 GHz ≤𝑓 ≤24 GHz  recorded at T = 250 K for the MgO/\nCoFeCrGa (200nm)/Pt fitted with Eqn. 7 . (b) The ∆𝐻-f curves for the MgO/\nCoFeCrGa (200nm)/Pt film at different temperatures fitted with ∆𝐻= ∆𝐻0+4𝜋𝛼\n𝛾𝜇0𝑓. (c) \nFitting of the f vs. the resonance field, 𝐻𝑟𝑒𝑠 using the Kittel’s equation at T = 250 K for the \nMgO/CoFeCrGa (200nm)/Pt film. (d) Left y scale: temperature dependence of damping \nparameter 𝛼(𝑇) for theMgO/CoFeCrGa (200nm)/Pt, and r ight y-scale: temperature \ndependence of  ∆𝐻0 for the same . \n \nThe estimated value of 𝑔𝑒𝑓𝑓=(2.09 ±0.01) at 250 K  for the MgO/\nCoFeCrGa (200nm)/Pt(5nm) film, which is slightly higher than the free electron value ( 𝑔𝑒𝑓𝑓 \n= 2.002). Note that 𝑔𝑒𝑓𝑓=(2.046 ±0.01) and (2.048 ±0.02) for the MgO/\nCoFeCrGa (95nm) and MgO/CoFeCrGa (95nm)/Pt(5nm) films, respectively at 250 K . \nFinally, 𝛼(𝑇) for the MgO/CoFeCrGa (200nm)/Pt film is shown on the left y-axis of Fig. \n10(d). It is evident that 𝛼(𝑇) increases with decreasing temperature, especially below 225 K \nsimilar to what we have  observed for the MgO/CoFeCrGa (95nm) and MgO/\n  \n37 \n CoFeCrGa (95nm)/Pt(5nm) films. This observation furthe r confirms the contribution of 𝛼 \ntowards the observed decrease in the LSSE signal in the CoFeCrGa films below the \ntemperature range of 200 -225 K.  \n \n4. CONCLUSIONS  \nIn summary, we present a comprehensive investigation of the temperature ANE and intrinsic \nlongitudinal spin Seebeck effect (LSSE) in the quaternar y Heusler alloy based SGS thin films \nof CoFeCrGa  grown on MgO substrates. We found that the anomalous Nernst coefficient  for \nthe MgO/CoFeCrGa  (95 nm) film is ≈ 1.86 μV.K−1 at room temperature which is  much \nhigher than the bulk polycrystalline sample of CoFeCrGa (≈0.018 μV.K−1 at 300 K) but \ncomparable to that of the magnetic Weyl semimetal Co 2MnGa thin films ( ≈2−3 μV.K−1 at \n300 K ). Furthermore, the LSSE coefficient for our MgO/CoFeCrGa (95nm)/Pt(5nm) \nheterostructure is ≈20.5 nV.K−1.Ω−1 at 295 K which is twice larger  than that of the half-\nmetallic ferromagnet ic La0.7Sr0.3MnO 3 thin films (≈9 nV.K−1.Ω−1 at room temperature ). We \nhave show n that both ANE and LSSE coefficients follow identical temperatu re dependences \nand exhibit a maximum ≈225 K which is understood as the combined effects of inelastic \nmagnon scatterings and reduced magnon population at low temperatures . Our analys es not only \nindicate d that the extrinsic skew scattering is the dominating mechanism for ANE in these films \nbut also, provide d critical insight s into the functional  form of the observed temperature \ndependent LSSE  at low temperatures . Furthermore, by employing radio frequency transverse \nsusceptibility and broadband ferromagnetic r esonance in combination with the LSSE \nmeasurements, we have establish ed a correlation among the observed LSSE signal, magnetic \nanisotropy and Gilbert damping of the CoFeCrGa thin films which will be beneficial for \nfabricating tunable and highly efficient s pincaloritronic nanodevices.  We believe that our \nfindings will also attract the attention of materials science and spintronics community for   \n38 \n further exploration of different Heusler alloys based magnetic thin films and heterostructures \nco-exhibiting multip le thermo -spin effects with promising efficiencies.  \n \nACKNOWLEDGEMENTS  \nHS and MHP acknowledge support from  the US Department of Energy, Office of Basic Energy \nSciences, Division of Materials Science and Engineering under Award No. DE -FG02 -\n07ER46438 . HS thanks the Alexander von Humboldt foundation for a research award and also \nacknowledges a visiting professorship at IIT Bombay. D.A.A. acknowledges the support of the \nNational Science Foundation under Grant No. ECCS -1952957. DD and RC acknowledge the \nfinancial assistance received from DST Nanomission project (DST/NM/TUE/QM -11/2019).  \n \nSUPPORTING INFORMATION  \nMagnetometry, temperature dependence of electrical resistivity, magnetic field and \ntemperature dependences of transverse susceptibility, magnetic fiel d dependence of ANE and \nLSSE voltages for the MgO/CoFeCrGa  (200 nm) and MgO/CoFeCrGa  (50 nm) films.  \n \n \nDATA AVAILABILITY  \nThe data that support the findings of this study are available from the corresponding author \nupon reasonable request.  \n \n \n \n \n   \n39 \n REFERENCES  \n(1) Bauer, G. E. W.; Saitoh, E.; Van Wees, B. J. Spin Caloritronics. Nat. Mater.  2012 , 11 \n(5), 391 –399. \n(2) Uchida, K. -I. Transport Phenomena in Spin Caloritronics. Proc. Japan Acad. Ser. B  \n2021 , 97 (2), 69 –88. \n(3) Sakai, A.; Mizuta, Y. P.; Nugroho, A. A.; Sihombing, R.; Koretsune, T.; Suzuki, M. -\nT.; Takemori, N.; Ishii, R.; Nishio -Hamane, D.; Arita, R.; others. 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B  2015 , 92 (5), 54437.  \n(92) Uchida, K.; Qiu, Z.; Kikkawa, T.; Igu chi, R.; Saitoh, E. Spin Hall Magnetoresistance \nat High Temperatures. Appl. Phys. Lett.  2015 , 106 (5), 52405.  \n \n \n \n \n \n \n   \n50 \n Supplementary Information  \nLarge thermo -spin effects in Heusler alloy based spin -gapless \nsemiconductor thin film s \nAmit Chanda1*, Deepika Rani2, Derick DeT ellem1, Noha Alzahrani1, Dario A. Arena1, Sarath \nWitanachchi1, Ratnamala Chatterjee2, Manh -Huong Phan1 and Hari haran  Srikanth1* \n1 Department of Physics, University of South Florida, Tampa FL 33620  \n2 Physics Department, Indian Institute of Technology Delhi, New Delhi - 110016  \n*Corresponding authors: achanda@usf.edu ; sharihar@usf.edu  \n \n \n \n \n   \n51 \n  \nFigure S 1. (a) and (b) Magnetic field dependence of magnetization, 𝑀(𝐻) of our \nMgO/CoFeCrGa (200nm) and MgO/CoFeCrGa (50nm)/Pt films. respectively  measured at \nselected temperatures in the range: 125 K ≤𝑇 ≤300 K in presence of an in -plane sweeping \nmagnetic field , (c) and (d) temperature dependence of the saturation magnetization, MS for the \nsame films, respectively.  \n \n \n  \n52 \n  \nFigure S2. (a) Schematic illustration of the transverse susceptibilbity (TS) measurements. T he \nbipolar field scan s (+𝐻𝐷𝐶𝑚𝑎𝑥−𝐻𝐷𝐶𝑚𝑎𝑥 +𝐻𝐷𝐶𝑚𝑎𝑥) of the field dependence of TS,  ∆𝜒𝑇\n𝜒𝑇(𝐻𝐷𝐶) \nfor the MgO/CoFeCrGa (200nm)/Pt(5nm) film measured at T = 300 K and 20 K for the (b)  \nIP (HDC is parallel to the film surface) and (c) OOP ( HDC is perpendicular to the film surface) \nconfigurations . (d) Temperature variations of  the effective anisotropy fields:  𝐻𝐾𝐼𝑃 and 𝐻𝐾𝑂𝑂𝑃for \nthe MgO/CoFeCrGa (200nm)/Pt film. \n \n \n \n \n \n \n \n \n \n \n  \n53 \n  \nFigure S3.  Temperature  dependence of longitudinal resistivity, 𝜌𝑥𝑥(𝑇) for the  (a) MgO/\nCoFeCrGa (200nm), (b) MgO/CoFeCrGa (95nm) and (c) MgO/CoFeCrGa (200nm) films, \nrespectively  in the temperature range: 10 K ≤𝑇 ≤300 K. \n \n \n  \n54 \n  \nFigure S4.  Temperature variations of thermal conductivity of (a) Apiezon N -grease,1 (b) MgO \ncrystal2 and (c) bulk CoFeCrGa (measured) using the thermal transport option (TTO) of the \nPPMS. (d) Schematic illustration of  the heat flow through N -grease/MgO substrate/CoFeCrGa \nfilm/N -grease considering the 4 -slab model. (e) The temperture variation of the effective \ntemperature difference across the MgO/CoFeCrGa(95nm) film estimated from the expression,3 \n∆𝑇𝑒𝑓𝑓= ∆𝑇CoFeCrGa = 𝛥𝑇\n [1+𝜅𝐶𝑜𝐹𝑒𝐶𝑟𝐺 𝑎\n𝑡𝐶𝑜𝐹𝑒𝐶𝑟𝐺𝑎(2𝑡𝑁−𝐺𝑟𝑒𝑎𝑠𝑒\n𝜅𝑁−𝐺𝑟𝑒𝑎𝑠𝑒 + 𝑡𝑀𝑔𝑂\n𝜅𝑀𝑔𝑂)]                                        \n  \n55 \n  \nFigure S 5. (a) and ( b) The magnetic field dependence of the ANE voltage, 𝑉𝐴𝑁𝐸(𝐻) and ISHE -\ninduced in-plane voltage,  𝑉𝐼𝑆𝐻𝐸(𝐻) measured on the MgO/CoFeCrGa (200nm) and MgO/\nCoFeCrGa (200nm)/Pt films, respectively  for different values of the temperature difference \nbetween the hot ( 𝑇ℎ𝑜𝑡) and cold ( 𝑇𝑐𝑜𝑙𝑑) copper blocks, ∆𝑇= (𝑇ℎ𝑜𝑡−𝑇𝑐𝑜𝑙𝑑) in the range: \n+5 K≤ ∆𝑇 ≤+18 K at a fixed average sample temperature 𝑇= 𝑇ℎ𝑜𝑡+𝑇𝑐𝑜𝑙𝑑\n2 = 295 K.  (c) and \n(d) The  ∆𝑇dependence of the background -corrected ANE voltage, 𝑉𝐴𝑁𝐸(Δ𝑇)=\n [𝑉𝐴𝑁𝐸(+𝜇0𝐻𝑚𝑎𝑥,   Δ𝑇)−𝑉𝐴𝑁𝐸(  −𝜇0𝐻𝑚𝑎𝑥,   Δ𝑇)\n2] and the background -corrected (ANE+ LSSE ) \nvolta ge, 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(Δ𝑇)= [𝑉𝐼𝑆𝐻𝐸(+𝜇0𝐻𝑚𝑎𝑥,   Δ𝑇)−𝑉𝐼𝑆𝐻𝐸(  −𝜇0𝐻𝑚𝑎𝑥,   Δ𝑇)\n2], respectively.  \n  \n56 \n  \nFigure S6. (a) and (b)  𝑉𝐴𝑁𝐸(𝐻) hysteresis loops measured at selected average sample \ntemperatures  in the temperature range: 125 K≤ ∆𝑇 ≤295 K for a fixed value of Δ𝑇 = +1 5 \nK on the MgO/CoFeCrGa (200nm) and MgO/CoFeCrGa (500nm) films, respectively . (c) and \n(d) 𝑉𝐼𝑆𝐻𝐸(𝐻) hysteresis loops measured at selected average sample temperatures  in the \ntemperature range: 125 K≤ ∆𝑇 ≤295 K for a fixed value of Δ𝑇 = +1 5 K on the \nMgO/CoFeCrGa (200nm)/Pt and MgO/CoFeCrGa (50nm)/Pt films, respectively . \n \n \n \n \n \n \n \n \n  \n57 \n Reference s \n(1) Ashworth, T.; Loomer, J. E.; Kreitman, M. M. Thermal Conductivity of Nylons and \nApiezon Greases. In Advances in Cryogenic Engineering ; Springer, 1973; pp 271 –279. \n(2) Grimvall, G. Thermophysical Properties of Materials ; Elsevier, 1999.  \n(3) De, A.; Ghosh, A .; Mandal, R.; Ogale, S.; Nair, S. Temperature Dependence of the \nSpin Seebeck Effect in a Mixed Valent Manganite. Phys. Rev. Lett.  2020 , 124 (1), \n17203.  \n \n "    },    {        "title": "1907.11853v1.Two_improved_Gauss_Seidel_projection_methods_for_Landau_Lifshitz_Gilbert_equation.pdf",        "content": "Two improved Gauss-Seidel projection methods for\nLandau-Lifshitz-Gilbert equation\nPanchi Lia, Changjian Xiea, Rui Dua,b,\u0003, Jingrun Chena,b,\u0003, Xiao-Ping Wangc,\u0003\naSchool of Mathematical Sciences, Soochow University, Suzhou, 215006, China.\nbMathematical Center for Interdisciplinary Research, Soochow University, Suzhou, 215006, China.\ncDepartment of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay,\nKowloon, Hong Kong, China\nA B S T R A C T\nMicromagnetic simulation is an important tool to study various dynamic behaviors of\nmagnetic order in ferromagnetic materials. The underlying model is the Landau-Lifshitz-\nGilbert equation, where the magnetization dynamics is driven by the gyromagnetic torque\nterm and the Gilbert damping term. Numerically, considerable progress has been made in\nthe past decades. One of the most popular methods is the Gauss-Seidel projection method\ndeveloped by Xiao-Ping Wang, Carlos Garc\u0013 \u0010a-Cervera, and Weinan E in 2001. It \frst solves\na set of heat equations with constant coe\u000ecients and updates the gyromagnetic term in the\nGauss-Seidel manner, and then solves another set of heat equations with constant coe\u000ecients\nfor the damping term. Afterwards, a projection step is applied to preserve the length con-\nstraint in the pointwise sense. This method has been veri\fed to be unconditionally stable\nnumerically and successfully applied to study magnetization dynamics under various controls.\nIn this paper, we present two improved Gauss-Seidel projection methods with uncondi-\ntional stability. The \frst method updates the gyromagnetic term and the damping term\nsimultaneously and follows by a projection step. The second method introduces two sets of\napproximate solutions, where we update the gyromagnetic term and the damping term simul-\ntaneously for one set of approximate solutions and apply the projection step to the other set\nof approximate solutions in an alternating manner. Compared to the original Gauss-Seidel\nprojection method which has to solve heat equations 7 times at each time step, the improved\nmethods solve heat equations 5 times and 3 times, respectively. First-order accuracy in time\nand second-order accuracy in space are veri\fed by examples in both 1D and 3D. In addi-\ntion, unconditional stability with respect to both the grid size and the damping parameter is\ncon\frmed numerically. Application of both methods to a realistic material is also presented\nwith hysteresis loops and magnetization pro\fles. Compared with the original method, the\nrecorded running times suggest that savings of both methods are about 2 =7 and 4=7 for the\nsame accuracy requirement, respectively.\nKeywords: Landau-Lifshitz-Gilbert equation, Gauss-Seidel projection method, unconditional\nstability, micromagnetic simulation\n2000 MSC: 35Q99, 65Z05, 65M06\n1. Introduction\nIn ferromagnetic materials, the intrinsic magnetic order, known as magnetization M=\n(M1;M2;M3)T, is modeled by the following Landau-Lifshitz-Gilbert (LLG) equation [1, 2, 3]\n@M\n@t=\u0000\rM\u0002H\u0000\r\u000b\nMsM\u0002(M\u0002H) (1)\n\u0003Corresponding authors\ne-mail: LiPanchi1994@163.com (Panchi Li), 20184007005@stu.suda.edu.cn (Changjian Xie),\ndurui@suda.edu.cn (Rui Du), jingrunchen@suda.edu.cn (Jingrun Chen), mawang@ust.hk (Xiao-Ping Wang)\n1arXiv:1907.11853v1  [math.NA]  27 Jul 2019with\rthe gyromagnetic ratio and jMj=Msthe saturation magnetization. On the right-\nhand side of (1), the \frst term is the gyromagnetic term and the second term is the Gilbert\ndamping term with \u000bthe dimensionless damping coe\u000ecient [2]. Note that the gyromagnetic\nterm is a conservative term, whereas the damping term is a dissipative term. The local \feld\nH=\u0000\u000eF\n\u000eMis computed from the Landau-Lifshitz energy functional\nF[M] =1\n2Z\n\n\u001aA\nM2sjrMj2+ \b\u0012M\nMs\u0013\n\u00002\u00160He\u0001M\u001b\ndx+\u00160\n2Z\nR3jrUj2dx; (2)\nwhereAis the exchange constant,A\nM2sjrMjis the exchange interaction energy; \b\u0010\nM\nMs\u0011\nis the anisotropy energy, and for simplicity the material is assumed to be uniaxial with\n\b\u0010\nM\nMs\u0011\n=Ku\nM2s(M2\n2+M2\n3) withKuthe anisotropy constant; \u00002\u00160He\u0001Mis the Zeeman\nenergy due to the external \feld with \u00160the permeability of vacuum. \n is the volume occupied\nby the material. The last term in (2) is the energy resulting from the \feld induced by the\nmagnetization distribution inside the material. This stray \feld Hs=\u0000rUwhereU(x)\nsatis\fes\nU(x) =Z\n\nrN(x\u0000y)\u0001M(y)dy; (3)\nwhereN(x\u0000y) =\u00001\n4\u00191\njx\u0000yjis the Newtonian potential.\nFor convenience, we rescale the original LLG equation (1) by changes of variables t!\n(\u00160\rMs)\u00001tandx!LxwithLthe diameter of \n. De\fne m=M=Msandh=MsH. The\ndimensionless LLG equation reads as\n@m\n@t=\u0000m\u0002h\u0000\u000bm\u0002(m\u0002h); (4)\nwhere\nh=\u0000Q(m2e2+m3e3) +\u000f\u0001m+he+hs (5)\nwith dimensionless parameters Q=Ku=(\u00160M2\ns) and\u000f=A=(\u00160M2\nsL2). Here e2= (0;1;0),\ne3= (0;0;1). Neumann boundary condition is used\n@m\n@\u0017j@\n= 0; (6)\nwhere\u0017is the outward unit normal vector on @\n.\nThe LLG equation is a weakly nonlinear equation. In the absence of Gilbert damping,\n\u000b= 0, equation (4) is a degenerate equation of parabolic type and is related to the sympletic\n\row of harmonic maps [4]. In the large damping limit, \u000b!1 , equation (4) is related to\nthe heat \row for harmonic maps [5]. It is easy to check that jmj= 1 in the pointwise sense\nin the evolution. All these properties possesses interesting challenges for designing numerical\nmethods to solve the LLG equation. Meanwhile, micromagnetic simulation is an important\ntool to study magnetization dynamics of magnetic materials [3, 6]. Over the past decades,\nthere has been increasing progress on numerical methods for the LLG equation; see [7, 8, 9]\n2for reviews and references therein. Finite di\u000berence method and \fnite element method have\nbeen used for the spatial discretization.\nFor the temporal discretization, there are explicit schemes such as Runge-Kutta methods\n[10, 11]. Their stepsizes are subject to strong stability constraint. Another issue is that the\nlength of magnetization cannot be preserved and thus a projection step is needed. Implicit\nschemes [12, 13, 14] are unconditionally stable and usually can preserve the length of magne-\ntization automatically. The di\u000eculty of implicit schemes is how to solve a nonlinear system\nof equations at each step. Therefore, semi-implicit methods [15, 16, 17, 18, 19] provide a com-\npromise between stability and the di\u000ecult for solving the equation at each step. A projection\nstep is also needed to preserve the length of magnetization.\nAmong the semi-implicit schemes, the most popular one is the Gauss-Seidel projection\nmethod (GSPM) proposed by Wang, Garc\u0013 \u0010a-Cervera, and E [15, 18]. GSPM \frst solves a\nset of heat equations with constant coe\u000ecients and updates the gyromagnetic term in the\nGauss-Seidel manner, and then solves another set of heat equations with constant coe\u000ecients\nfor the damping term. Afterwards, a projection step is applied to preserve the length of mag-\nnetization. GSPM is \frst-order accurate in time and has been veri\fed to be unconditionally\nstable numerically.\nIn this paper, we present two improved Gauss-Seidel projection methods with uncondi-\ntional stability. The \frst method updates the gyromagnetic term and the damping term\nsimultaneously and follows by a projection step. The second method introduces two sets of\napproximate solutions, where we update the gyromagnetic term and the damping term simul-\ntaneously for one set of approximate solutions and apply the projection step to the other set\nof approximate solutions in an alternating manner. Compared to the original Gauss-Seidel\nprojection method, which solves heat equations 7 times at each time step, the improved\nmethods solve heat equations 5 times and 3 times, respectively. First-order accuracy in time\nand second-order accuracy in space are veri\fed by examples in both 1D and 3D. In addi-\ntion, unconditional stability with respect to both the grid size and the damping parameter is\ncon\frmed numerically. Application of both methods to a realistic material is also presented\nwith hysteresis loops and magnetization pro\fles. Compared with the original method, the\nrecorded running times suggest that savings of both methods are about 2 =7 and 4=7 for the\nsame accuracy requirement, respectively.\nThe rest of the paper is organized as follows. For completeness and comparison, we \frst\nintroduce GSPM in Section 2. Two improved GSPMs are presented in Section 3. Detailed\nnumerical tests are given in Section 4, including accuracy check and e\u000eciency check in both\n1D and 3D, unconditional stability with respect to both the grid size and the damping\nparameter, hysteresis loops, and magnetization pro\fles. Conclusions are drawn in Section 5.\n32. Gauss-Seidel projection method for Landau-Lifshitz-Gilbert equation\nBefore the introduction of the GSPM [15, 18], we \frst use the \fnite di\u000berence method\nfor spatial discretization. Figure 1 shows a schematic picture of spatial grids in 1D. Let\ni= 0;1;\u0001\u0001\u0001;M;M + 1,j= 0;1;\u0001\u0001\u0001;N;N + 1, andk= 0;1;\u0001\u0001\u0001;K;K + 1 be the indices of\ngrid points in 3D.\n0 1𝑥−1\n2𝑥1\n2𝑥𝑁−1\n2𝑥𝑁+1\n2𝑥3\n2𝑥𝑁−3\n2\nFig. 1. Spatial grids in 1D. Nodes x\u00001\n2andxN+1\n2are ghost points.\nSecond-order centered di\u000berence for \u0001 mreads as\n\u0001hmi;j;k=mi+1;j;k\u00002mi;j;k+mi\u00001;j;k\n\u0001x2\n+mi;j+1;k\u00002mi;j;k+mi;j\u00001;k\n\u0001y2\n+mi;j;k+1\u00002mi;j;k+mi;j;k\u00001\n\u0001z2; (7)\nwhere mi;j;k=m((i\u00001\n2)\u0001x;(j\u00001\n2)\u0001y;(k\u00001\n2)\u0001z). For the Neumann boundary condition,\na second-order approximation yields\nm0;j;k=m1;j;k;mM;j;k =mM+1;j;k; j = 1;\u0001\u0001\u0001;N;k = 1;\u0001\u0001\u0001;K;\nmi;0;k=mi;1;k;mi;N;k =mi;N+1;k; i= 1;\u0001\u0001\u0001;M;k = 1;\u0001\u0001\u0001;K;\nmi;j;0=mi;j;1;mi;j;K =mi;j;K +1; i= 1;\u0001\u0001\u0001;M;j = 1;\u0001\u0001\u0001;N:\nTo illustrate the main ideas, we \frst consider the following simpli\fed equation\nmt=\u0000m\u0002\u0001m\u0000\u000bm\u0002(m\u0002\u0001m);\nwhich can be rewritten as\nmt=\u0000m\u0002\u0001m\u0000\u000bm(m\u0001\u0001m) +\u000b\u0001m: (8)\nWe split (8) into two equations\nmt=\u0000m\u0002\u0001m; (9)\nmt=\u000b\u0001m: (10)\nHowever, (9) is still nonlinear. Therefore, we consider a fractional step scheme to solve\n(9)\nm\u0003\u0000mn\n\u0001t= \u0001hm\u0003\nmn+1=mn\u0000mn\u0002m\u0003\n4or\nmn+1=mn\u0000mn\u0002(I\u0000\u0001t\u0001h)\u00001mn;\nwhereIis the identity matrix. This scheme is subject to strong stability constraint, and thus\nthe implicit Gauss-Seidel scheme is introduced to overcome this issue. Let\ngn\ni= (I\u0000\u0001t\u0001h)\u00001mn\ni; i= 1;2;3: (11)\nWe then have 0\n@mn+1\n1\nmn+1\n2\nmn+1\n31\nA=0\n@mn\n1+ (gn\n2mn\n3\u0000gn\n3mn\n2)\nmn\n2+ (gn\n3mn+1\n1\u0000gn+1\n1mn\n3)\nmn\n3+ (gn+1\n1mn+1\n2\u0000gn+1\n2mn+1\n1)1\nA: (12)\nThis scheme solve (9) with unconditional stability. (10) is linear heat equation which can be\nsolved easily. However, the splitting scheme (9) - (10) cannot preserve jmj= 1, and thus a\nprojection step needs to be added.\nFor the full LLG equation (4), the GSPM works as follows. De\fne\nh=\u000f\u0001m+^f; (13)\nwhere ^f=\u0000Q(m2e2+m3e3) +he+hs.\nThe original GSPM [15] solves the equation (4) in three steps:\n\u000fImplicit Gauss-Seidel\ngn\ni= (I\u0000\u0001t\u000f\u0001h)\u00001(mn\ni+ \u0001t^fn\ni); i= 2;3;\ng\u0003\ni= (I\u0000\u0001t\u000f\u0001h)\u00001(m\u0003\ni+ \u0001t^fn\ni); i= 1;2; (14)\n0\n@m\u0003\n1\nm\u0003\n2\nm\u0003\n31\nA=0\n@mn\n1+ (gn\n2mn\n3\u0000gn\n3mn\n2)\nmn\n2+ (gn\n3m\u0003\n1\u0000g\u0003\n1mn\n3)\nmn\n3+ (g\u0003\n1m\u0003\n2\u0000g\u0003\n2m\u0003\n1)1\nA: (15)\n\u000fHeat \row without constraints\n^f\u0003=\u0000Q(m\u0003\n2e2+m\u0003\n3e3) +he+hn\ns; (16)\n0\n@m\u0003\u0003\n1\nm\u0003\u0003\n2\nm\u0003\u0003\n31\nA=0\n@m\u0003\n1+\u000b\u0001t(\u000f\u0001hm\u0003\u0003\n1+^f\u0003\n1)\nm\u0003\n2+\u000b\u0001t(\u000f\u0001hm\u0003\u0003\n2+^f\u0003\n2)\nm\u0003\n3+\u000b\u0001t(\u000f\u0001hm\u0003\u0003\n3+^f\u0003\n3)1\nA: (17)\n\u000fProjection onto S2\n0\n@mn+1\n1\nmn+1\n2\nmn+1\n31\nA=1\njm\u0003\u0003j0\n@m\u0003\u0003\n1\nm\u0003\u0003\n2\nm\u0003\u0003\n31\nA: (18)\nHere the numerical stability of the original GSPM [15] was founded to be independent of\ngridsizes but depend on the damping parameter \u000b. This issue was solved in [18] by replacing\n(14) and (16) with\ng\u0003\ni= (I\u0000\u0001t\u000f\u0001h)\u00001(m\u0003\ni+ \u0001t^f\u0003\ni); i= 1;2;\n5and\n^f\u0003=\u0000Q(m\u0003\n2e2+m\u0003\n3e3) +he+h\u0003\ns;\nrespectively. Update of the stray \feld is done using fast Fourier transform [15]. It is easy\nto see that the GSPM solves 7 linear systems of equations with constant coe\u000ecients and\nupdates the stray \feld using FFT 6 times at each step.\n3. Two improved Gauss-Seidel projection methods for Landau-Lifshitz-Gilbert\nequation\nBased on the description of the original GSPM in Section 2, we introduce two improved\nGSPMs for LLG equation. The \frst improvement updates both the gyromagnetic term and\nthe damping term simultaneously, termed as Scheme A. The second improvement introduces\ntwo sets of approximate solution with one set for implicit Gauss-Seidel step and the other set\nfor projection in an alternating manner, termed as Scheme B. Details are given in below.\n3.1. Scheme A\nThe main improvement of Scheme A over the original GSPM is the combination of (13)\n- (17), or (9) - (10).\n\u000fImplicit-Gauss-Seidel\ngn\ni= (I\u0000\u0001t\u000f\u0001h)\u00001(mn\ni+ \u0001t^fn\ni); i= 1;2;3;\ng\u0003\ni= (I\u0000\u0001t\u000f\u0001h)\u00001(m\u0003\ni+ \u0001t^f\u0003\ni); i= 1;2; (19)\n0\n@m\u0003\n1\nm\u0003\n2\nm\u0003\n31\nA=0\n@mn\n1\u0000(mn\n2gn\n3\u0000mn\n3gn\n2)\u0000\u000b(mn\n1gn\n1+mn\n2gn\n2+mn\n3gn\n3)mn\n1+\u000bgn\n1\nmn\n2\u0000(mn\n3g\u0003\n1\u0000m\u0003\n1gn\n3)\u0000\u000b(m\u0003\n1g\u0003\n1+mn\n2gn\n2+mn\n3gn\n3)mn\n2+\u000bgn\n2\nmn\n3\u0000(m\u0003\n1g\u0003\n2\u0000m\u0003\n2g\u0003\n1)\u0000\u000b(m\u0003\n1g\u0003\n1+m\u0003\n2g\u0003\n2+mn\n3gn\n3)mn\n3+\u000bgn\n31\nA:(20)\n\u000fProjection onto S2\n0\n@mn+1\n1\nmn+1\n2\nmn+1\n31\nA=1\njm\u0003j0\n@m\u0003\n1\nm\u0003\n2\nm\u0003\n31\nA: (21)\nIt is easy to see that Scheme A solves 5 linear systems of equations with constant coe\u000ecients\nand uses FFT 5 times at each step.\n3.2. Scheme B\nThe main improvement of Scheme B over Scheme A is the introduction of two sets of\napproximate solutions, one for (19) - (20) and the other for (21) and the update of these two\nsets of solutions in an alternating manner.\nGiven the initialized g0\ng0\ni= (I\u0000\u0001t\u000f\u0001h)\u00001(m0\ni+ \u0001t^f0\ni); i= 1;2;3; (22)\nScheme B works as follows\n6\u000fImplicit Gauss-Seidel\ngn+1\ni= (I\u0000\u0001t\u000f\u0001h)\u00001(m\u0003\ni+ \u0001t^f\u0003\ni); i= 1;2;3 (23)\nm\u0003\n1=mn\n1\u0000(mn\n2gn\n3\u0000mn\n3gn\n2)\u0000\u000b(mn\n1gn\n1+mn\n2gn\n2+mn\n3gn\n3)mn\n1+\n\u000b((mn\n1)2+ (mn\n2)2+ (mn\n3)2)gn\n1\nm\u0003\n2=mn\n2\u0000(mn\n3gn+1\n1\u0000m\u0003\n1gn\n3)\u0000\u000b(m\u0003\n1gn+1\n1+mn\n2gn\n2+mn\n3gn\n3)mn\n2+\n\u000b((m\u0003\n1)2+ (mn\n2)2+ (mn\n3)2)gn\n2\nm\u0003\n3=mn\n3\u0000(m\u0003\n1gn+1\n2\u0000m\u0003\n2gn+1\n1)\u0000\u000b(m\u0003\n1gn+1\n1+m\u0003\n2gn+1\n2+mn\n3gn\n3)mn\n3+\n\u000b((m\u0003\n1)2+ (m\u0003\n2)2+ (mn\n3)2)gn\n3 (24)\n\u000fProjection onto S2\n0\n@mn+1\n1\nmn+1\n2\nmn+1\n31\nA=1\njm\u0003j0\n@m\u0003\n1\nm\u0003\n2\nm\u0003\n31\nA: (25)\nHere one set of approximate solution fm\u0003gis updated in the implicit Gauss-Seidel step and\nthe other set of approximate solution fmn+1gis updated in the projection step. Note that\n(23) is de\fned only for fm\u0003gwhich can be used in two successive temporal steps, and thus\nonly 3 linear systems of equations with constant coe\u000ecients are solved at each step and 3\nFFT executions are used for the stray \feld. The length of magnetization can be preserved\nin the time evolution.\nThe computational cost of GSPM and its improvements comes from solving the linear\nsystems of equations with constant coe\u000ecients. To summarize, we list the number of linear\nsystems of equations to be solved and the number of FFT executions to be used at each step\nfor the original GSPM [18], Scheme A, and Scheme B in Table 1. The savings represent the\nratio between costs of two improved schemes over that of the original GSPM.\nGSPM Scheme Number of linear systems Saving Execution of FFT Saving\nOriginal 7 0 4 0\nScheme A 5 2=7 3 1=4\nScheme B 3 4=7 3 1=4\nTable 1. The number of linear systems of equations to be solved and the number of FFT\nexecutions to be used at each step for the original GSPM [18], Scheme A, and Scheme B. The\nsavings represent the ratio between costs of two improved schemes over that of the original\nGSPM.\n4. Numerical Experiments\nIn this section, we compare the original GSPM [15, 18], Scheme A, and Scheme B via a\nseries of examples in both 1D and 3D, including accuracy check and e\u000eciency check, uncon-\nditional stability with respect to both the grid size and the damping parameter, hysteresis\n7loops, and magnetization pro\fles. For convenience, we de\fne\nratio\u0000i=Time(GSPM)\u0000Time(Scheme i)\nTime(GSPM);\nfori= A and B, which quanti\fes the improved e\u000eciency of Scheme A and Scheme B over\nthe original GSPM [15, 18].\n4.1. Accuracy Test\nExample 4.1 (1D case). In 1D, we choose the exact solution over the unit interval \n =\n[0;1]\nme= (cos(\u0016x) sin(t);sin(\u0016x) sin(t);cos(t));\nwhich satis\fes\nmt=\u0000m\u0002mxx\u0000\u000bm\u0002(m\u0002mxx) +f\nwith \u0016x=x2(1\u0000x)2, and f=met+me\u0002mexx+\u000bme\u0002(me\u0002mexx). Parameters are\n\u000b= 0:00001 andT= 5:0e\u00002.\nWe \frst show the error kme\u0000mhk1withmhbeing the numerical solution with respect\nto the temporal stepsize \u0001tand the spatial stepsize \u0001x. As shown in Figure 2(a) and Fig-\nure 2(c), suggested by the least squares \ftting, both \frst-order accuracy in time and second-\norder accuracy in space are observed. Meanwhile, we record the CPU time as a function\nof accuracy (error) by varying the temporal stepsize and the spatial stepsize in Figure 2(b)\nand Figure 2(d), Table 2 and Table 3, respectively. In addition, from Table 2 and Table 3,\nthe saving of Scheme A over GSPM is about 2=7, which equals 1\u00005=7, and the saving of\nScheme B over GSPM is about 4=7, respectively. This observation is in good agreement with\nthe number of linear systems being solved at each step for these three methods, as shown in\nTable 1.\nXXXXXXXXXXCPU time\u0001tT/1250 T/2500 T/5000 T/10000 Reference\nGSPM 7.7882e-01 1.5445e+00 3.1041e+00 6.2196e+00 -\nScheme A 4.8340e-01 9.9000e-01 2.0527e+00 4.4917e+00 -\nScheme B 3.3010e-01 6.3969e-01 1.2281e+00 2.5510e+00 -\nratio-A 0.38 0.36 0.34 0.28 0.29(2/7)\nratio-B 0.58 0.59 0.60 0.59 0.57(4/7)\nTable 2. Recorded CPU time in 1D with respect to the approximation error when only \u0001tis\nvaried and \u0001x= 1=100.\nExample 4.2 (3D case). In 3D, we choose the exact solution over \n = [0;2]\u0002[0;1]\u0002[0;0:2]\nme= (cos(\u0016x\u0016y\u0016z) sin(t);sin(\u0016x\u0016y\u0016z) sin(t);cos(t));\nwhich satis\fes\nmt=\u0000m\u0002\u0001m\u0000\u000bm\u0002(m\u0002\u0001m) +f\n8log(∆t)-12.5 -12 -11.5 -11 -10.5 -10log(error)\n-12.5-12-11.5-11-10.5-10\nGSPM\nScheme A\nScheme B(a) Temporal accuracy\nlog(error)-12.5 -12 -11.5 -11 -10.5 -10log(time)\n-1.5-1-0.500.511.52\nGSPM\nScheme A\nScheme B (b) CPU time versus approximation error (\u0001 t)\nlog(∆x)-5.1 -5 -4.9 -4.8 -4.7 -4.6log(error)\n-16-15.9-15.8-15.7-15.6-15.5-15.4-15.3-15.2-15.1-15\nGSPM\nScheme A\nScheme B\n(c) Spatial accuracy\nlog(error)-16 -15.8 -15.6 -15.4 -15.2 -15log(time)\n77.588.599.5\nGSPM\nScheme A\nScheme B (d) CPU time versus approximation error (\u0001 x)\nFig. 2. Approximation error and CPU time in 1D. (a) Approximation error as a function of the\ntemporal step size; (b) CPU time as a function of the approximation error when \u0001tis varied\nand \u0001xis \fxed; (c) Approximation error as a function of the spatial step size; (d) CPU time\nas a function of the approximation error when \u0001xis varied and \u0001tis \fxed.\nwith \u0016x=x2(1\u0000x)2,\u0016y=y2(1\u0000y)2,\u0016z=z2(1\u0000z)2andf=met+me\u0002\u0001me+\u000bme\u0002(me\u0002\n\u0001me). Parameters are T= 1:0e\u000005and\u000b= 0:01.\nLike in the 1D case, we \frst show the error kme\u0000mhk1withmhbeing the numerical\nsolution with respect to the temporal stepsize \u0001tand the spatial stepsize \u0001x. As shown in\nFigure 3(a) and Figure 3(c), suggested by the least squares \ftting, both \frst-order accuracy in\ntime and second-order accuracy in space are observed. Meanwhile, we record the CPU time\nas a function of accuracy (error) by varying the temporal stepsize and the spatial stepsize in\nFigure 3(b) and Figure 3(d), Table 4 and Table 5, respectively. In addition, from Table 4 and\nTable 5, the saving of Scheme A over GSPM is about 2=7, and the saving of Scheme B over\nGSPM is about 4=7, respectively. This observation is in good agreement with the number of\nlinear systems being solved at each step for these three methods, as shown in Table 1.\nIt worths mentioning that all these three methods are tested to be unconditionally stable\nwith respect to the spatial gridsize and the temporal stepsize.\n9XXXXXXXXXXCPU time\u0001x1/100 1/120 1/140 1/160 Reference\nGSPM 3.3752e+03 5.2340e+03 9.0334e+03 1.0495e+04 -\nScheme A 2.4391e+03 3.7175e+03 6.5149e+03 8.0429e+03 -\nScheme B 1.4740e+03 2.2448e+03 3.9152e+03 4.8873e+03 -\nratio-A 0.28 0.29 0.28 0.23 0.29(2/7)\nratio-B 0.56 0.57 0.57 0.53 0.57(4/7)\nTable 3. Recorded CPU time in 1D with respect to the approximation error when only \u0001xis\nvaried and \u0001t= 1:0e\u00008.\nXXXXXXXXXXCPU time\u0001tT/10 T/20 T/40 T/80 Reference\nGSPM 3.5188e+01 6.8711e+01 1.4146e+02 2.9769e+02 -\nScheme A 2.3015e+01 4.3920e+01 8.6831e+01 1.7359e+02 -\nScheme B 1.3984e+01 2.6313e+01 5.1928e+01 1.0415e+02 -\nratio-A 0.35 0.36 0.39 0.42 0.29(2/7)\nratio-B 0.60 0.62 0.63 0.65 0.57(4/7)\nTable 4. Recorded CPU time in 3D with respect to the approximation error when only \u0001tis\nvaried and the spatial mesh is 128\u000264\u000210.\n4.2. Micromagnetic Simulations\nTo compare the performance of Scheme A and Scheme B with GSPM, we have carried out\nmicromagnetic simulations of the full LLG equation with realistic material parameters. In\nall our following simulations, we consider a thin \flm ferromagnet of size \n = 1 \u0016m\u00021\u0016m\u0002\n0:02\u0016m with the spatial gridsize 4 nm \u00024 nm\u00024 nm and the temporal stepsize \u0001 t= 1\npicosecond. The demagnetization \feld (stray \feld) is calculated via FFT [15, 18].\n4.2.1. Comparison of hysteresis loops\nThe hysteresis loop is calculated in the following way. First, a positive external \feld\nH0=\u00160His applied and the system is allowed to reach a stable state. Afterwards, the\nexternal \feld is reduced by a certain amount and the system is relaxed to a stable state\nagain. The process continues until the external \feld attains a negative \feld of strength H0.\nThen the external \feld starts to increase and the system relaxes until the initial applied\nexternal \feld H0is approached. In the hysteresis loop, we can monitor the magnetization\ndynamics and plot the average magnetization at the stable state as a function of the strength\nof the external \feld. The stopping criterion for a steady state is that the relative change of\nthe total energy is less than 10\u00007. The applied \feld is parallel to the xaxis. The initial state\nwe take is the uniform state and the damping parameter \u000b= 0:1.\nIn Figure 4, we compare the average magnetization in the hysteresis loop simulated by\nGSPM, Scheme A and Scheme B. Pro\fles of the average magnetization of these three methods\nare in quantitative agreements with approximately the same switch \feld 9 ( \u00060:4) mT.\n4.2.2. Comparison of magnetization pro\fles\nIt is tested that GSPM in [15] was unstable with a very small damping parameter \u000band\nwas resolved in [18]. This section is devoted to the unconditional stability of Scheme A and\n10log(∆t)-14 -13.5 -13 -12.5 -12 -11.5log(error)\n-14-13.5-13-12.5-12-11.5\nGSPM\nScheme A\nScheme B(a) Temporal accuracy\nlog(error)-14 -13.5 -13 -12.5 -12 -11.5log(time)\n2.533.544.555.56\nGSPM\nScheme A\nScheme B (b) CPU time versus approximation error (\u0001 t)\nThe spatial step size log( ∆x)-2.5 -2.4 -2.3 -2.2 -2.1 -2 -1.9 -1.8 -1.7log(error)\n-32.5-32-31.5-31\nGSPM\nScheme A\nScheme B\n(c) Spatial accuracy\nlog(error)-32.5 -32 -31.5 -31log(time)\n22.533.544.555.56\nGSPM\nScheme A\nScheme B (d) CPU time versus approximation error (\u0001 x)\nFig. 3. Approximation error and CPU time in 3D. (a) Approximation error as a function of the\ntemporal step size; (b) CPU time as a function of the approximation error when \u0001tis varied\nand \u0001x= \u0001y= \u0001zis \fxed; (c) Approximation error as a function of the spatial step size; (d)\nCPU time as a function of the approximation error when space is varied uniformly and \u0001tis\n\fxed.\nScheme B with respect to \u000b. We consider a thin \flm ferromagnet of size 1 \u0016m\u00021\u0016m\u00020:02\u0016m\nwith the spatial gridsize 4 nm \u00024 nm\u00024 nm and the temporal stepsize is 1 picosecond.\nFollowing [18], we consider the full LLG equation with \u000b= 0:1 and\u000b= 0:01 and without\nthe external \feld. The initial state is m0= (0;1;0) ifx2[0;Lx=5][[4Lx=5;Lx] and\nm0= (1;0;0) otherwise. The \fnal time is 10 ns. In Figures 5 to 7, we present a color plot\nof the angle between the in-plane magnetization and the xaxis, and an arrow plot of the\nin-plane magnetization for the original GSPM [15], Scheme A, and Scheme B, respectively.\nIn these \fgures, \u000b= 0:1 is presented in the top row and \u000b= 0:01 is presented in the bottom\nrow; a color plot of the angle between the in-palne magnetization and the xaxis is presented\nin the left column and an arrow plot of the in-plane magnetization is presented in the right\ncolumn.\n5. Conclusion\nIn this paper, based on the original Gauss-Seidel projection methods, we present two\nimproved Gauss-Seidel projection methods with the \frst-order accuracy in time and the\nsecond-order accuracy in space. The \frst method updates the gyromagnetic term and the\n11XXXXXXXXXXCPU time\u0001x1/6 1/8 1/10 1/12 Reference\nGSPM 2.1066e+01 9.2615e+01 1.9879e+02 3.7820e+02 -\nScheme A 1.5278e+01 6.5953e+01 1.4215e+02 2.6725e+02 -\nScheme B 8.9698e+00 3.8684e+01 8.4291e+01 1.5977e+02 -\nratio-A 0.27 0.29 0.28 0.29 0.29(2/7)\nratio-B 0.57 0.58 0.58 0.58 0.57(4/7)\nTable 5. Recorded CPU time in 3D with respect to the approximation error when only the\nspatial gridsize is varied with \u0001x= \u0001y= \u0001zand \u0001t= 1:0e\u000009.\n-50 -40 -30 -20 -10 0 10 20 30 40 50\n0 H (mT)-1-0.8-0.6-0.4-0.200.20.40.60.81M/Ms\n GSPM\n Scheme A\n Scheme B\nFig. 4. Comparison of hysteresis loops for GSPM, Scheme A and Scheme B. Pro\fles of the av-\nerage magnetization of these three methods are in quantitative agreements with approximately\nthe same switch \feld 9 (\u00060:4) mT . The applied \feld is parallel to the xaxis and the initial state\nis the uniform state.\ndamping term simultaneously and follows by a projection step, which requires to solve heat\nequations 5 times at each time step. The second method introduces two sets of approximate\nsolutions, where we update the gyromagnetic term and the damping term simultaneously for\none set of approximate solutions and apply the projection step to the other set of approximate\nsolutions in an alternating manner. Therefore, only 3 heat equations are needed to be solved\nat each step. Compared to the original Gauss-Seidel projection method, which solves heat\nequations 7 times at each step, savings of these two improved methods are about 2 =7 and\n4=7, which is veri\fed by both 1D and 3D examples for the same accuracy requirement. In\naddition, unconditional stability with respect to both the grid size and the damping parameter\nis con\frmed numerically. Application of both methods to a realistic material is also presented\nwith hysteresis loops and magnetization pro\fles.\nAcknowledgments\nThis work is supported in part by the grants NSFC 21602149 (J. Chen), NSFC 11501399\n(R. Du), the Hong Kong Research Grants Council (GRF grants 16302715, 16324416, 16303318\n12(a) Angle pro\fle ( \u000b= 0:1)\n0 0.2 0.4 0.6 0.8 1\nx (m)00.20.40.60.81y (m) (b) Magnetization pro\fle ( \u000b= 0:1)\n(c) Angle pro\fle ( \u000b= 0:01)\n0 0.2 0.4 0.6 0.8 1\nx (m)00.20.40.60.81y (m) (d) Magnetization pro\fle ( \u000b= 0:01)\nFig. 5. Simulation of the full Landau-Lifshitz-Gilbert equation using GSPM without any exter-\nnal \feld. The magnetization on the centered slice of the material in the xyplane is used. Top\nrow:\u000b= 0:1; Bottom row: \u000b= 0:01. Left column: a color plot of the angle between the in-plane\nmagnetization and the xaxis; Right column: an arrow plot of the in-plane magnetization.\n13(a) Angle pro\fle ( \u000b= 0:1)\n0 0.2 0.4 0.6 0.8 1\nx (m)00.20.40.60.81y (m) (b) Magnetization pro\fle ( \u000b= 0:1)\n(c) Angle pro\fle ( \u000b= 0:01)\n0 0.2 0.4 0.6 0.8 1\nx (m)00.20.40.60.81y (m) (d) Magnetization pro\fle ( \u000b= 0:01)\nFig. 6. Simulation of the full Landau-Lifshitz-Gilbert equation using Scheme A without any\nexternal \feld. The magnetization on the centered slice of the material in the xyplane is used.\nTop row: \u000b= 0:1; Bottom row: \u000b= 0:01. Left column: a color plot of the angle between the in-\nplane magnetization and the xaxis; Right column: an arrow plot of the in-plane magnetization.\n14(a) Angle pro\fle ( \u000b= 0:1)\n0 0.2 0.4 0.6 0.8 1\nx (m)00.20.40.60.81y (m) (b) Magnetization pro\fle ( \u000b= 0:1)\n(c) Angle pro\fle ( \u000b= 0:01)\n0 0.2 0.4 0.6 0.8 1\nx (m)00.20.40.60.81y (m) (d) Magnetization pro\fle ( \u000b= 0:01)\nFig. 7. Simulation of the full Landau-Lifshitz-Gilbert equation using Scheme B without any\nexternal \feld. The magnetization on the centered slice of the material in the xyplane is used.\nTop row: \u000b= 0:1; Bottom row: \u000b= 0:01. Left column: a color plot of the angle between the in-\nplane magnetization and the xaxis; Right column: an arrow plot of the in-plane magnetization.\n15and NSFC-RGC joint research grant N-HKUST620/15) (X.-P. Wang), and the Innovation\nProgram for postgraduates in Jiangsu province via grant KYCX19 1947 (C. Xie).\nReferences\n[1] L. Landau, E. Lifshitz, On the theory of the dispersion of magetic permeability in ferromagnetic bodies,\nPhys. Z. Sowjetunion 8 (1935) 153{169.\n[2] T. Gilbert, A lagrangian formulation of gyromagnetic equation of the magnetization \feld, Phys. Rev.\n100 (1955) 1243{1255.\n[3] W. F. B. Jr., Micromagnetics, Interscience Tracts on Physics and Astronomy, 1963.\n[4] P. Sulem, C. Sulem, C. Bardos, On the continuous limit limit for a system of classical spins, Comm.\nMath. Phys. 107 (1986) 431{454.\n[5] M. Struwe, On the evolution of harmonic maps in higher dimensions, J. Di\u000berential Geom. 28 (1988)\n485{502.\n[6] I. \u0014Zuti\u0013 c, J. Fabian, S. Das Sarma, Spintronics: Fundamentals and applications, Rev. Mod. Phys. 76\n(2004) 323{410.\n[7] M. Kruzik, A. Prohl, Recent developments in the modeling, analysis, and numerics of ferromagnetism,\nSIAM Rev. 48 (2006) 439{483.\n[8] I. Cimr\u0013 ak, A survey on the numerics and computations for the Landau-Lifshitz equation of micromag-\nnetism, Arch. Comput. Methods Eng. 15 (2008) 277{309.\n[9] C. J. Garc\u0013 \u0010a-Cervera, Numerical micromagnetics: a review, Bol. Soc. Esp. Mat. Apl. 39 (2007) 103{135.\n[10] A. Fran\u0018 cois, J. Pascal, Convergence of a \fnite element discretization for the Landau-Lifshitz equations\nin micromagnetism, Math. Models Methods Appl. Sci. 16 (2006) 299{316.\n[11] A. Romeo, G. Finocchio, M. Carpentieri, L. Torres, G. Consolo, B. Azzerboni, A numerical solution of\nthe magnetization reversal modeling in a permalloy thin \flm using \ffth order runge-kutta method with\nadaptive step size control, Physica B. 403 (2008) 1163{1194.\n[12] Y. H, H. N, Implicit solution of the Landau-Lifshitz-Gilbert equation by the Crank-Nicolson method,\nJ. Magn. Soc. Japan 28 (2004) 924{931.\n[13] S. Bartels, P. Andreas, Convergence of an implicit \fnite element method for the Landau-Lifshitz-Gilbert\nequation, SIAM J. Numer. Anal. 44 (2006) 1405{1419.\n[14] A. Fuwa, T. Ishiwata, M. Tsutsumi, Finite di\u000berence scheme for the Landau-Lifshitz equation, Japan\nJ. Indust. Appl. Math. 29 (2012) 83{110.\n[15] X. Wang, C. J. Garc\u0013 \u0010a-Cervera, W. E, A gauss-seidel projection method for micromagnetics simulations,\nJ. Comput. Phys. 171 (2001) 357{372.\n[16] W. E, X. Wang, Numerical methods for the Landau-Lisfshitz equation, SIAM J. Numer. Anal. 38 (2000)\n1647{1665.\n[17] J. Chen, C. Wang, C. Xie, Convergence analysis of a second-order semi-implicit projection method for\nLandau-Lifshitz equation, arXiv 1902.09740 (2019).\n[18] C. J. Garc\u0013 \u0010a-Cervera, W. E, Improved gauss-seidel projection method for micromagnetics simulations,\nIEEE Trans. Magn. 39 (2003) 1766{1770.\n[19] I. Cimr\u0013 ak, Error estimates for a semi-implicit numerical scheme solving the Landau-Lifshitz equation\nwith an exchange \feld, IMA J. Numer. Anal. (2005) 611{634.\n16"    },    {        "title": "2009.07620v1.Fast_convex_optimization_via_inertial_dynamics_combining_viscous_and_Hessian_driven_damping_with_time_rescaling.pdf",        "content": "Noname manuscript No.\n(will be inserted by the editor)\nFast convex optimization via inertial dynamics combining\nviscous and Hessian-driven damping with time rescaling\nHedy ATTOUCH \u0001A \u0010cha BALHAG \u0001Zaki\nCHBANI \u0001Hassan RIAHI\nthe date of receipt and acceptance should be inserted later\nAbstract In a Hilbert setting, we develop fast methods for convex unconstrained\noptimization. We rely on the asymptotic behavior of an inertial system combining\ngeometric damping with temporal scaling. The convex function to minimize enters\nthe dynamic via its gradient. The dynamic includes three coe\u000ecients varying with\ntime, one is a viscous damping coe\u000ecient, the second is attached to the Hessian-\ndriven damping, the third is a time scaling coe\u000ecient. We study the convergence\nrate of the values under general conditions involving the damping and the time\nscale coe\u000ecients. The obtained results are based on a new Lyapunov analysis and\nthey encompass known results on the subject. We pay particular attention to the\ncase of an asymptotically vanishing viscous damping, which is directly related to\nthe accelerated gradient method of Nesterov. The Hessian-driven damping signi\f-\ncantly reduces the oscillatory aspects. As a main result, we obtain an exponential\nrate of convergence of values without assuming the strong convexity of the objec-\ntive function. The temporal discretization of these dynamics opens the gate to a\nlarge class of inertial optimization algorithms.\nKeywords damped inertial gradient dynamics; fast convex optimization;\nHessian-driven damping; Nesterov accelerated gradient method; time rescaling\nMathematics Subject Classi\fcation (2010) 37N40, 46N10, 49M30, 65K05,\n65K10, 90B50, 90C25.\nHedy ATTOUCH\nIMAG, Univ. Montpellier, CNRS, Montpellier, France\nhedy.attouch@umontpellier.fr,\nSupported by COST Action: CA16228\nA \u0010cha BALHAG \u0001Zaki CHBANI\u0001Hassan RIAHI\nCadi Ayyad University\nS\u0013 emlalia Faculty of Sciences 40000 Marrakech, Morroco\naichabalhag@gmail.com \u0001chbaniz@uca.ac.ma \u0001h-riahi@uca.ac.maarXiv:2009.07620v1  [math.OC]  16 Sep 20202 Hedy ATTOUCH et al.\n1 Introduction\nThroughout the paper, His a real Hilbert space with inner product h\u0001;\u0001iand\ninduced normk\u0001k;andf:H!Ris a convex and di\u000berentiable function. We aim\nat developping fast numerical methods for solving the optimization problem\n(P) min\nx2Hf(x):\nWe denote by argminHfthe set of minimizers of the optimization problem ( P),\nwhich is assumed to be non-empty. Our work is part of the active research stream\nthat studies the close link between continuous dissipative dynamical systems and\noptimization algorithms. In general, the implicit temporal discretization of con-\ntinuous gradient-based dynamics provides proximal algorithms that bene\ft from\nsimilar asymptotic convergence properties, see [28] for a systematic study in the\ncase of \frst-order evolution systems, and [5,6,8,11,12,19,20,21] for some recent\nresults concerning second-order evolution equations. The main object of our study\nis the second-order in time di\u000berential equation\n(IGS)\r;\f;bx(t) +\r(t) _x(t) +\f(t)r2f(x(t)) _x(t) +b(t)rf(x(t)) = 0;\nwhere the coe\u000ecients \r;\f: [t0;+1[!R+take account of the viscous and Hessian-\ndriven damping, respectively, and b:R+!R+is a time scale parameter. We take\nfor granted the existence and uniqueness of the solution of the corresponding\nCauchy problem with initial conditions x(t0) =x02H, _x(t0) =v02H. Assuming\nthatrfis Lipschitz continuous on the bounded sets, and that the coe\u000ecients are\ncontinuously di\u000berentiable, the local existence follows from the nonautonomous\nversion of the Cauchy-Lipschitz theorem, see [24, Prop. 6.2.1]. The global existence\nthen follows from the energy estimates that will be established in the next section.\nEach of these damping and rescaling terms properly tuned, improves the rate of\nconvergence of the associated dynamics and algorithms. An original aspect of our\nwork is to combine them in the same dynamic. Let us recall some classical facts.\n1.1 Damped inertial dynamics and optimization\nThe continuous-time perspective gives a mechanical intuition of the behavior of the\ntrajectories, and a valuable tool to develop a Lyapunov analysis. A \frst important\nwork in this perspective is the heavy ball with friction method of B. Polyak [29]\n(HBF) x(t) +\r_x(t) +rf(x(t)) = 0:\nIt is a simpli\fed model for a heavy ball (whose mass has been normalized to one)\nsliding on the graph of the function fto be minimized, and which asymptoti-\ncally stops under the action of viscous friction, see [14] for further details. In this\nmodel, the viscous friction parameter \ris a \fxed positive parameter. Due to too\nmuch friction (at least asymptotically) involved in this process, replacing the \fxed\nviscous coe\u000ecient with a vanishing viscous coe\u000ecient ( i.e.which tends to zero as\nt!+1) gives Nesterov's famous accelerated gradient method [26] [27]. The other\ntwo basic ingredients that we will use, namely time rescaling, and Hessian-driven\ndamping have a natural interpretation (cinematic and geometric, respectively) inInertial dynamics with Hessian damping and time rescaling 3\nthis context. We will come back to these points later. Precisely, we seek to develop\nfast \frst-order methods based on the temporal discretization of damped inertial\ndynamics. By fast we mean that, for a general convex function f, and for each\ntrajectory of the system, the convergence rate of the values f(x(t))\u0000infHfwhich\nis obtained is optimal ( i.e.is achieved of nearly achieved in the worst case). The\nimportance of simple \frst-order methods, and in particular gradient-based and\nproximal algorithms, comes from the applicability of these algorithms to a wide\nrange of large-scale problems arising from machine learning and/or engineering.\n1.1.1 The viscous damping parameter \r(t).\nA signi\fcant number of recent studies have focused on the case \r(t) =\u000b\nt,\f= 0\n(without Hessian-driven damping), and b= 1 (without time rescaling), that is\n(AVD)\u000bx(t) +\u000b\nt_x(t) +rf(x(t)) = 0:\nThis dynamic involves an Asymptotically Vanishing Damping coe\u000ecient (hence\nthe terminology), a key property to obtain fast convergence for a general convex\nfunctionf. In [32], Su, Boyd and Cand\u0012 es showed that for \u000b= 3 the above system\ncan be seen as a continuous version of the accelerated gradient method of Nesterov\n[26,27] with f(x(t))\u0000minHf=O(1\nt2) ast!+1. The importance of the parame-\nter\u000bwas put to the fore by Attouch, Chbani, Peypouquet and Redont [9] and May\n[25]. They showed that, for \u000b>3, one can pass from capital Oestimates to small\no. Moreover, when \u000b>3, each trajectory converges weakly, and its limit belongs to\nargminf1. Recent research considered the case of a general damping coe\u000ecient \r(\u0001)\n(see [4,7]), thus providing a complete picture of the convergence rates for (AVD)\u000b:\nf(x(t))\u0000minHf=O(1=t2) when\u000b\u00153, andf(x(t))\u0000minHf=O\u0010\n1=t2\u000b\n3\u0011\nwhen\n\u000b\u00143, see [7,10] and Apidopoulos, Aujol and Dossal [3].\n1.1.2 The Hessian-driven damping parameter \f(t).\nThe inertial system\n(DIN)\r;\fx(t) +\r_x(t) +\fr2f(x(t)) _x(t) +rf(x(t)) = 0;\nwas introduced by Alvarez, Attouch, Bolte, and Redont in [2]. In line with (HBF),\nit contains a \fxed positive friction coe\u000ecient \r. As a main property, the introduc-\ntion of the Hessian-driven damping makes it possible to neutralize the transversal\noscillations likely to occur with (HBF), as observed in [2]. The need to take a\ngeometric damping adapted to fhad already been observed by Alvarez [1] who\nconsidered the inertial system\nx(t) +D_x(t) +rf(x(t)) = 0;\nwhereD:H!H is a linear positive de\fnite anisotropic operator. But still this\ndamping operator is \fxed. For a general convex function, the Hessian-driven damp-\ning in (DIN)\r;\fperforms a similar operation in a closed-loop adaptive way. (DIN)\nstands shortly for Dynamical Inertial Newton, and refers to the link with the\n1Recall that for \u000b= 3 the convergence of the trajectories is an open question4 Hedy ATTOUCH et al.\nLevenberg-Marquardt regularization of the continuous Newton method. Recent\nstudies have been devoted to the study of the inertial dynamic\nx(t) +\u000b\nt_x(t) +\fr2f(x(t)) _x(t) +rf(x(t)) = 0;\nwhich combines asymptotic vanishing damping with Hessian-driven damping [17].\n1.1.3 The time rescaling parameter b(t).\nIn the context of non-autonomous dissipative dynamic systems, reparameteriza-\ntion in time is a simple and universal means to accelerate the convergence of\ntrajectories. This is where the coe\u000ecient b(t) comes in as a factor of rf(x(t)).\nIn [11] [12], in the case of general coe\u000ecients \r(\u0001) andb(\u0001) without the Hessian\ndamping, the authors made in-depth study. In the case \r(t) =\u000b\nt, they proved that\nunder appropriate conditions on \u000bandb(\u0001),f(x(t))\u0000minHf=O(1\nt2b(t)). Hence a\nclear improvement of the convergence rate by taking b(t)!+1ast!+1.\n1.2 From damped inertial dynamics to proximal-gradient inertial algorithms\nLet's review some classical facts concerning the close link between continuous\ndissipative inertial dynamic systems and the corresponding algorithms obtained\nby temporal discretization. Let us insist on the fact that, when the temporal\nscalingb(t)!+1ast!+1, the transposition of the results to the discrete\ncase naturally leads to consider an implicit temporal discretization, i.e.inertial\nproximal algorithms. The reason is that, since b(t) is in front of the gradient, the\napplication of the gradient descent lemma would require taking a step size that\ntends to zero. On the other hand, the corresponding proximal algorithms involve\na proximal coe\u000ecient which tends to in\fnity (large step proximal algorithms).\n1.2.1 The case without the Hessian-driven damping\nThe implicit discretization of (IGS)\r;0;bgives the Inertial Proximal algorithm\n(IP)\u000bk;\u0015k(\nyk=xk+\u000bk(xk\u0000xk\u00001)\nxk+1= prox\u0015kf(yk)\nwhere\u000bkis non-negative and \u0015kis positive. Recall that for any \u0015>0, the proximity\noperator prox\u0015f:H!H is de\fned by the following formula: for every x2H\nprox\u0015f(x) := argmin\u00182H\u001a\nf(\u0018) +1\n2\u0015kx\u0000\u0018k2\u001b\n:\nEquivalently, prox\u0015fis the resolvent of index \u0015of the maximally monotone opera-\ntor@f. When passing to the implicit discrete case, we can take f:H!R[f+1g\na convex lower semicontinuous and proper function. Let us list some of the main\nresults concerning the convergence properties of the algorithm (IP)\u000bk;\u0015k:\n\u000f1. Case\u0015k\u0011\u0015>0 and\u000bk= 1\u0000\u000b\nk. When\u000b= 3, the (IP)1\u00003=k;\u0015algorithm\nhas a similar structure to the original Nesterov accelerated gradient algorithm[26],Inertial dynamics with Hessian damping and time rescaling 5\njust replace the gradient step with a proximal step. Passing from the gradient to the\nproximal step was carried out by G uler [22,23], then by Beck and Teboulle [18] for\nstructured optimization. A decisive step was taken by Attouch and Peypouquet in\n[16] proving that, when \u000b>3,f(xk)\u0000minHf=o\u00001\nk2\u0001. The subcritical case \u000b<3\nwas examined by Apidopoulos, Aujol, and Dossal [3] and Attouch, Chbani, and\nRiahi [10] with the rate of convergence rate of values f(xk)\u0000minHf=O\u0010\n1\nk2\u000b\n3\u0011\n.\n\u000f2. For a general \u000bk, the convergence properties of (IP)\u000bk;\u0015were analyzed\nby Attouch and Cabot [5], then by Attouch, Cabot, Chbani, and Riahi [6], in\nthe presence of perturbations. The convergence rates are then expressed using the\nsequence ( tk) which is linked to ( \u000bk) by the formula tk:= 1 +P+1\ni=kQi\nj=k\u000bj.\nUnder growth conditions on tk, it is proved that f(xk)\u0000minHf=O(1\nt2\nk). This\nlast results covers the special case \u000bk= 1\u0000\u000b\nkwhen\u000b\u00153.\n\u000f3. For a general \u0015k, Attouch, Chbani, and Riahi \frst considered in [11] the\ncase\u000bk= 1\u0000\u000b\nk. They proved that under a growth condition on \u0015k, we have\nthe estimate f(xk)\u0000minHf=O(1\nk2\u0015k). This result is an improvement of the one\ndiscussed previously in [16], because when \u0015k=k\u000ewith 0<\u000e<\u000b\u00003, we pass from\nO(1\nk2) toO(1\nk2+\u000e). Recently, in [13] the authors analyzed the algorithm (IP)\u000bk;\u0015kfor general \u000bkand\u0015k. By including the expression of tkpreviously used in [5,6],\nthey proved that f(xk)\u0000minHf=O\u00001=t2\nk\u0015k\u00001\u0001under certain conditions on \u0015k\nand\u000bk. They obtained f(xk)\u0000minHf=o\u00001=t2\nk\u0015k\u0001, which gives a global view of\nof the convergence rate with small o, encompassing [5,13].\n1.2.2 The case with the Hessian-driven damping\nRecent studies have been devoted to the inertial dynamic\nx(t) +\u000b\nt_x(t) +\fr2f(x(t)) _x(t) +rf(x(t)) = 0;\nwhich combines asymptotic vanishing viscous damping with Hessian-driven damp-\ning. The corresponding algorithms involve a correcting term in the Nesterov ac-\ncelerated gradient method which reduces the oscillatory aspects, see Attouch-\nPeypouquet-Redont [17], Attouch-Chbani-Fadili-Riahi [8], Shi-Du-Jordan-Su [30].\nThe case of monotone inclusions has been considered by Attouch and L\u0013 aszl\u0013 o [15].\n1.3 Contents\nThe paper is organized as follows. In section 2, we develop a new Lyapunov anal-\nysis for the continuous dynamic (IGS)\r;\f;b. In Theorem 1, we provide a system of\nconditions on the damping parameters \r(\u0001) and\f(\u0001), and on the temporal scaling\nparameter b(\u0001) giving fast convergence of the values. Then, in sections 3 and 4,\nwe present two di\u000berent types of growth conditions for the damping and tempo-\nral scaling parameters, respectively based on the functions \u0000 \randp\r, and which\nsatisfy the conditions of Theorem 1. In doing so, we encompass most existing re-\nsults and provide new results, including linear convergence rates without assuming\nstrong convexity. This will also allow us to explain the choice of certain coe\u000ecients\nin the associated algorithms, questions which have remained mysterious and only6 Hedy ATTOUCH et al.\njusti\fed by the simpli\fcation of often complicated calculations. In section 5, we\nspecialize our results to certain model situations and give numerical illustrations.\nFinally, we conclude the paper by highlighting its original aspects.\n2 Convergence rate of the values. General abstract result\nWe will establish a general result concerning the convergence rate of the values\nveri\fed by the solution trajectories x(\u0001) of the second-order evolution equation\n(IGS)\r;\f;b x(t) +\r(t) _x(t) +\f(t)r2f(x(t)) _x(t) +b(t)rf(x(t)) = 0:\nThe variable parameters \r(\u0001),\f(\u0001) andb(\u0001) take into account the damping, and\ntemporal rescaling e\u000bects. They are assumed to be continuously di\u000berentiable.\nTo analyze the asymptotic behavior of the solutions trajectories of the evolution\nsystem (IGS) \r;\f;b, we will use Lyapunov's analysis. It is a classic and powerful tool\nwhich consists in building an associated energy-like function which decreases along\nthe trajectories. The determination of such a Lyapunov function is in general a\ndelicate problem. Based on previous works, we know the global structure of such\na Lyapunov function. It is a weighted sum of the potential, kinetic and anchor\nfunctions. We will introduce coe\u000ecients in this function that are a priori unknown,\nand which will be identi\fed during the calculation to verify the property of decay.\nOur approach takes advantage of the technics recently developed in [4], [17], [12].\n2.1 The general case\nLetx(\u0001) be a solution trajectory of (IGS) \r;\f;b. Givenz2argminHf, we introduce\nthe Lyapunov function t7!E(t) de\fned by\nE(t) :=c(t)2b(t)(f(x(t))\u0000f(z))+\u0012(t)\u001b(t)2\n2kv(t)k2+\u0018(t)\n2kx(t)\u0000zk2;(1)\nwherev(t) :=x(t)\u0000z+1\n\u001b(t)(_x(t) +\f(t)rf(x(t))):\nThe four variable coe\u000ecients c(t);\u0012(t);\u001b(t);\u0018(t) will be adjusted during the calcu-\nlation. According to the classical derivation chain rule, we obtain\nd\ndtE(t) =d\ndt\u0010\nc2(t)b(t)\u0011\n(f(x(t))\u0000f(z))+c(t)2b(t)hrf(x(t));_x(t)i\n+1\n2d\ndt(\u0012(t)\u001b2(t))kv(t)k2+\u0012(t)\u001b2(t)h_v(t);v(t)i\n+1\n2_\u0018(t)kx(t)\u0000zk2+\u0018(t)h_x(t);x(t)\u0000zi:Inertial dynamics with Hessian damping and time rescaling 7\nFrom now, without ambiguity, to shorten formulas, we omit the variable t.\nAccording to the de\fnition of v, and the equation (IGS) \r;\f;b, we have\n_v= _x\u0000_\u001b\n\u001b2(_x+\frf(x))+1\n\u001bd\ndt(_x+\frf(x))\n= _x\u0000_\u001b\n\u001b2(_x+\frf(x))+1\n\u001b\u0010\nx+\fr2f(x) _x+_\frf(x)\u0011\n= _x\u0000_\u001b\n\u001b2(_x+\frf(x))+1\n\u001b\u0000\n\u0000\r_x\u0000brf(x) +_\frf(x)\u0001\n=\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u0013\n_x+\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\nrf(x):\nTherefore,\nh_v;vi=\u001c\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u0013\n_x+\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\nrf(x); x\u0000z+1\n\u001b(_x+\frf(x))\u001d\n=\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u0013\nh_x; x\u0000zi+\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u00131\n\u001bk_xk2\n+\u0012\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u0013\n\f\n\u001b+\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u00131\n\u001b\u0013\nhrf(x);_xi\n+\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\nhrf(x); x\u0000zi+\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\n\f\n\u001bkrf(x)k2:\nAccording to the de\fnition of v(t), after developing kv(t)k2, we get\nkvk2=kx\u0000zk2+1\n\u001b2\u0010\nk_xk2+\f2krf(x)k2\u0011\n+2\n\u001bh_x; x\u0000zi\n+2\f\n\u001bhrf(x); x\u0000zi+2\f\n\u001b2hrf(x);_xi:\nCollecting the above results, we obtain\nd\ndtE(t) =d\ndt\u0010\nc2b\u0011\n(f(x)\u0000f(z))+c2bhrf(x);_xi+1\n2_\u0018kx\u0000zk2+\u0018h_x;x\u0000zi\n+1\n2d\ndt(\u0012\u001b2)\u0010\nkx\u0000zk2+1\n\u001b2\u0010\nk_xk2+\f2krf(x)k2\u0011\n+2\n\u001bh_x; x\u0000zi\u0011\n+1\n2d\ndt(\u0012\u001b2)\u00102\f\n\u001bhrf(x); x\u0000zi+2\f\n\u001b2hrf(x);_xi\u0011\n+\u0012\u001b2\u0010\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u0013\nh_x; x\u0000zi+\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u00131\n\u001bk_xk2\u0011\n+\u0012\u001b2\u0012\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u0013\n\f\n\u001b+\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u00131\n\u001b\u0013\nhrf(x);_xi\n+\u0012\u001b2\u0010\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\nhrf(x); x\u0000zi+\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\n\f\n\u001bkrf(x)k2\u0011\n:\nIn the second member of the above formula, let us examine the terms that contain\nhrf(x); x\u0000zi. By grouping these terms, we obtain the following expression\n\u0010\f\n\u001bd\ndt(\u0012\u001b2) +\u0012\u001b2\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\u0011\nhrf(x); x\u0000zi:8 Hedy ATTOUCH et al.\nTo majorize it, we use the convex subgradient inequality hrf(x);x\u0000zi\u0015f(x)\u0000\nf(z);and we make a \frst hypothesis\f\n\u001bd\ndt(\u0012\u001b2)+\u0012\u001b2\u0010_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0011\n\u00140:Therefore,\nd\ndtE(t)\u0014\u0014\nd\ndt\u0010\nc2b\u0011\n+\f\n\u001bd\ndt(\u0012\u001b2) +\u0012\u001b2\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\u0015\n(f(x)\u0000f(z))\n+\u0014\nc2b+\f\n\u001b2d\ndt(\u0012\u001b2) +\u0012\u001b\u0012\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u0013\n\f+\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\u0013\u0015\nhrf(x);_xi\n+\u00141\n\u001bd\ndt(\u0012\u001b2) +\u0012\u001b2\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u0013\n+\u0018\u0015\nh_x; x\u0000zi\n+1\n2\u0014\nd\ndt(\u0012\u001b2) +_\u0018\u0015\nkx\u0000zk2+\u00141\n2\u001b2d\ndt(\u0012\u001b2) +\u0012\u001b\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u0013\u0015\nk_xk2\n+\u0014\n\f2\n2\u001b2d\ndt(\u0012\u001b2) +\u0012\u001b\f\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\u0015\nkrf(x)k2: (2)\nTo getd\ndtE(t)\u00140, we are led to make the following assumptions:\n(i)\f\n\u001bd\ndt(\u0012\u001b2) +\u0012\u001b2\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\n\u00140\n(ii)d\ndt\u0010\nc2b\u0011\n+\f\n\u001bd\ndt(\u0012\u001b2) +\u0012\u001b2\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\n\u00140;\n(iii)c2b+\f\n\u001b2d\ndt(\u0012\u001b2) +\u0012\u001b\u0012\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u0013\n\f+\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\u0013\n= 0;\n(iv)1\n\u001bd\ndt(\u0012\u001b2) +\u0012\u001b2\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u0013\n+\u0018= 0;\n(v)d\ndt(\u0012\u001b2) +_\u0018\u00140;\n(vi)1\n2\u001b2d\ndt(\u0012\u001b2) +\u0012\u001b\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u0013\n\u00140;\n(vii)\f2\n2\u001b2d\ndt(\u0012\u001b2) +\u0012\u001b\f\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\n\u00140:\nAfter simpli\fcation, we get the following equivalent system of conditions:\nA: Lyapunov system of inequalities involving c(t);\u0012(t);\u001b(t);\u0018(t):\n(i)d\ndt(\f\u0012\u001b)\u0000\u0012b\u001b\u00140\n(ii)d\ndt\u0000\nc2b+\f\u0012\u001b\u0001\n\u0000\u0012b\u001b\u00140;\n(iii)b(c2\u0000\u0012) +\f\u0012(\u001b\u0000\r) +d\ndt(\f\u0012) = 0;\n(iv)d\ndt(\u0012\u001b) +\u0012\u001b(\u001b\u0000\r)+\u0018= 0;\n(v)d\ndt(\u0012\u001b2+\u0018)\u00140;\n(vi)_\u0012+ 2(\u001b\u0000\r)\u0012\u00140;\n(vii)\f\u0010\n\f_\u0012+ 2\u0000_\f\u0000b\u0001\n\u0012\u0011\n\u00140:Inertial dynamics with Hessian damping and time rescaling 9\nLet's simplify this system by eliminating the variable \u0018. From (iv) we get\u0018=\n\u0000d\ndt(\u0012\u001b)\u0000\u0012\u001b(\u001b\u0000\r), that we replace in ( v), and recall that \u0018is prescribed to\nbe nonnegative. Now observe that the unkown function ccan also be eliminated.\nIndeed, it enters the above system via the variable bc2, which according to ( iii)\nis equal to bc2=b\u0012\u0000\f\u0012(\u001b\u0000\r)\u0000d\ndt(\f\u0012):Replacing in ( ii), which is the only\nother equation involving bc2, we obtain the equivalent system involving only the\nvariables\u0012(t);\u001b(t).\nB: Lyapunov system of inequalities involving the variables: \u0012(t);\u001b(t)\n(i)d\ndt(\f\u0012\u001b)\u0000\u0012b\u001b\u00140;\n(ii)d\ndt(b\u0012+\f\u0012\r)\u0000d2\ndt2(\f\u0012)\u0000\u0012b\u001b\u00140;\n(iii)b\u0012\u0000\f\u0012(\u001b\u0000\r)\u0000d\ndt(\f\u0012)\u00150;\n(iv)d\ndt(\u0012\u001b) +\u0012\u001b(\u001b\u0000\r)\u00140;\n(v)d\ndt\u0000\n\u0000d\ndt(\u0012\u001b) +\u0012\u001b\r\u0001\n\u00140;\n(vi)_\u0012+ 2(\u001b\u0000\r)\u0012\u00140;\n(vii)\f\u0010\n\f_\u0012+ 2\u0000_\f\u0000b\u0001\n\u0012\u0011\n\u00140:\nThen, the variables \u0018andcare obtained by using the formulas\n\u0018=\u0000d\ndt(\u0012\u001b)\u0000\u0012\u001b(\u001b\u0000\r)\nbc2=b\u0012\u0000\f\u0012(\u001b\u0000\r)\u0000d\ndt(\f\u0012):\nThus, under the above conditions, the function E(\u0001) is nonnegative and nonincreas-\ning. Therefore, for every t\u0015t0,E(t)\u0014E(t0);which implies that\nc2(t)b(t)f(x(t))\u0000min\nHf)\u0014E(t0):\nTherefore, as t!+1\nf(x(t))\u0000min\nHf=O\u00121\nc2(t)b(t)\u0013\n:\nMoreover, by integrating (2) we obtain the following integral estimates:\na) On the values:\nZ+1\nt0\u0010\n\u0012(t)b(t)\u001b(t)\u0000d\ndt\u0010\nc2(t)b(t) +\f(t)\u0012(t)\u001b(t)\u0011\u0011\u0012\nf(x(t))\u0000inf\nHf\u0013\ndt< +1;\nwhere we use the equality:\n\u0000\u0010\nd\ndt\u0000\nc2b\u0001+\f\n\u001bd\ndt(\u0012\u001b2) +\u0012\u001b2\u0010_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0011\u0011\n=\u0012b\u001b\u0000d\ndt\u0000\nc2b+\f\u0012\u001b\u0001\nand the fact that, according to ( ii), this quantity is nonnegative.10 Hedy ATTOUCH et al.\nb) On the norm of the gradients:\nZ+1\nt0q(t)krf(x(t))k2dt< +1:\nwhereqis the nonnegative weight function de\fned by\nq(t) :=\u0012(t)\f(t)\u0012_\u001b(t)\f(t)\n\u001b(t)+b(t)\u0000_\f(t)\u0013\n\u0000\f2(t)\n2\u001b2(t)d\ndt(\u0012\u001b2)(t)\n=b(t)\u0012(t)\f(t)\u00001\n2d\ndt(\u0012\f2)(t): (3)\nWe can now state the following Theorem, which summarizes the above results.\nTheorem 1 Letf:H!Rbe a convex di\u000berentiable function with argminHf6=;.\nLetx(\u0001)be a solution trajectory of\n(IGS)\r;\f;b x(t) +\r(t) _x(t) +\f(t)r2f(x(t)) _x(t) +b(t)rf(x(t)) = 0:\nSuppose that \r(\u0001),\f(\u0001), andb(\u0001), areC1functions on [t0;+1[such that there exists\nauxiliary functions c(t);\u0012(t);\u001b(t);\u0018(t)that satisfy the conditions (i)\u0000(vii)above. Set\nE(t) :=c(t)2b(t)(f(x(t))\u0000f(z))+\u0012(t)\u001b(t)2\n2kv(t)k2+\u0018(t)\n2kx(t)\u0000zk2;(4)\nwithz2argminHfandv(t) =x(t)\u0000z+1\n\u001b(t)(_x(t) +\f(t)rf(x(t))).\nThen,t7!E(t)is a nonincreasing function. As a consequence, for all t\u0015t0,\n(i)f(x(t))\u0000min\nHf\u0014E(t0)\nc2(t)b(t); (5)\n(ii)Z+1\nt0\u0010\n\u0012(t)b(t)\u001b(t)\u0000d\ndt\u0000\nc2b+\f\u0012\u001b\u0001(t)\u0011\u0012\nf(x(t))\u0000inf\nHf\u0013\ndt< +1; (6)\n(iii)Z+1\nt0\u0012\nb(t)\u0012(t)\f(t)\u00001\n2d\ndt\u0000\n\u0012\f2\u0001(t)\u0013\nkrf(x(t))k2dt< +1: (7)\n2.2 Solving system ( i)\u0000(vii)\nThe system of inequalities ( i)\u0000(vii) of Theorem 1 may seem complicated at \frst\nglance. Indeed, we will see that it simpli\fes notably in the classical situations.\nMoreover, it makes it possible to unify the existing results, and discover new\ninteresting cases. We will present two di\u000berent types of solutions to this system,\nrespectively based on the following functions:\np\r(t) = exp\u0012Zt\nt0\r(u)du\u0013\n; (8)\nand\n\u0000\r(t) =p\r(t)Z+1\ntdu\np\r(u): (9)\nThe use of \u0000 \rhas been considered in a series of articles that we will retrieve as a\nspecial case of our approach, see [4], [5], [7], [12]. Using p\rwill lead to new results,\nsee section 4.Inertial dynamics with Hessian damping and time rescaling 11\n3 Results based on the function \u0000 \r\nIn this section, we will systematically assume that condition ( H0) is satis\fed.\n(H0)Z+1\nt0ds\np(s)<+1:\nUnder (H0), the function \u0000 \r(\u0001) is well de\fned. It can be equally de\fned as the\nsolution of the linear non autonomous di\u000berential equation\n_\u0000\r(t)\u0000\r(t)\u0000\r(t) + 1 = 0; (10)\nwhich satis\fes the limit condition lim t!+1\u0000\r(t)\np\r(t)= 0.\n3.1 The case without the Hessian, i.e.\f\u00110\nThe dynamic writes\n(IGS)\r;0;b x(t) +\r(t) _x(t) +b(t)rf(x(t)) = 0:\nTo solve the system ( i)\u0000(vii) of Theorem 1, we choose\n\u0018\u00110; c(t) = \u0000\r(t); \u001b(t) =1\n\u0000\r(t); \u0012(t) =\u0000\r(t)2:\nAccording to (10), we can easily verify that conditions ( i);(iii)\u0000(vii) are satis\fed,\nand (ii) becomes\nd\ndt\u0010\n\u0000\r(t)2b(t)\u0011\n\u0000\u0000\r(t)b(t)\u00140:\nAfter dividing by \u0000 \r(t), and using (10), we obtain the condition\n0\u0015\u0000\r(t)_b(t)\u0000(3\u00002\r(t)\u0000\r(t))b(t):\nThis leads to the following result obtained by Attouch, Chbani and Riahi in [12].\nTheorem 2 [12, Theorem 2.1] Suppose that for all t\u0015t0\n\u0000\r(t)_b(t)\u0014b(t)(3\u00002\r(t)\u0000\r(t)); (11)\nwhere \u0000\ris de\fned from \rby(9). Letx: [t0;+1[!H be a solution trajectory of\n(IGS)\r;0;b. Givenz2argminHf, set\nE(t) := \u00002\n\r(t)b(t)(f(x(t))\u0000f(z))+1\n2kx(t)\u0000z+ \u0000\r(t) _x(t)k2: (12)\nThen,t7!E(t)is a nonincreasing function. As a consequence, as t!+1\nf(x(t))\u0000min\nHf=O\u00121\n\u0000\r(t)2b(t)\u0013\n: (13)\nPrecisely, for all t\u0015t0\nf(x(t))\u0000min\nHf\u0014C\n\u0000\r(t)2b(t); (14)\nwithC= \u0000\r(t0)2b(t0)(f(x(t0))\u0000minHf)+d(x(t0);argminf)2+\u0000\r(t0)2k_x(t0)k2:\nMoreover,\nZ+1\nt0\u0000\r(t)\u0010\nb(t)(3\u00002\r(t)\u0000\r(t))\u0000\u0000\r(t)_b(t)\u0011\n(f(x(t))\u0000min\nHf)dt< +1:\nRemark 1 Whenb\u00111, condition (11) reduces to \r(t)\u0000\r(t)\u00143\n2, introduced in [4].12 Hedy ATTOUCH et al.\n3.2 Combining Nesterov acceleration with Hessian damping\nLet us specialize our results in the case \f(t)>0, and\r(t) =\u000b\nt. We are in the case\nof a vanishing damping coe\u000ecient ( i.e.\r(t)!0 ast!+1). According to Su,\nBoyd and Cand\u0012 es [32], the case \u000b= 3 corresponds to a continuous version of the\naccelerated gradient method of Nesterov. Taking \u000b > 3 improves in many ways\nthe convergence properties of this dynamic, see section 1.1.1. Here, it is combined\nwith the Hessian-driven damping and temporal rescaling. This situation was \frst\nconsidered by Attouch, Chbani, Fadili and Riahi in [8]. Then the dynamic writes\n(IGS)\u000b=t;\f;b x(t) +\u000b\nt_x(t) +\f(t)r2f(x(t)) _x(t) +b(t)rf(x(t)) = 0:\nElementary calculus gives that ( H0) is satis\fed as soon as \u000b>1. In this case,\n\u0000\r(t) =t\n\u000b\u00001:\nAfter [8], let us introduce the following quantity which will simplify the formulas:\nw(t) :=b(t)\u0000_\f(t)\u0000\f(t)\nt: (15)\nThe following result will be obtained as a consequence of our general abstract The-\norem 1. Precisely, we will show that under an appropriate choice of the functions\nc(t);\u0012(t);\u001b(t);\u0018(t), the conditions ( i)\u0000(vii) of Theorem 1 are satis\fed.\nTheorem 3 [8, Theorem 1] Letx: [t0;+1[!H be a solution trajectory of\n(IGS)\u000b=t;\f;b x(t) +\u000b\nt_x(t) +\f(t)r2f(x(t)) _x(t) +b(t)rf(x(t)) = 0:\nSuppose that \u000b>1, and that the following growth conditions are satis\fed: for t\u0015t0\n(G2)b(t)>_\f(t) +\f(t)\nt;\n(G3)t_w(t)\u0014(\u000b\u00003)w(t):\nThen,w(t) :=b(t)\u0000_\f(t)\u0000\f(t)\ntis positive and\n(i)f(x(t))\u0000min\nHf=O\u00121\nt2w(t)\u0013\nast!+1;\n(ii)Z+1\nt0t\u0010\n(\u000b\u00003)w(t)\u0000t_w(t)\u0011\n(f(x(t))\u0000min\nHf)dt< +1;\n(iii)Z+1\nt0t2\f(t)w(t)krf(x(t))k2dt< +1:\nProof Take\u0012(t) = \u0000\r(t)2; \u001b(t) =1\n\u0000\r(t); \u0018(t)\u00110;and\nc(t)2=1\n(\u000b\u00001)2t\nb(t)\u0000\ntb(t)\u0000\f(t)\u0000t_\f(t)\u0001\n: (16)\nThis formula for c(t) will appear naturally during the calculation. Note that the\ncondition (G2) ensures that the second member of the above expression is positive,\nwhich makes sense to think of it as a square. Let us verify that the conditions ( i)\nand (iv);(v);(vi);(vii) are satis\fed. This is a direct consequence of the formula\n(10) and the condition ( G2):Inertial dynamics with Hessian damping and time rescaling 13\n(i)d\ndt(\f\u0012\u001b)\u0000\u0012b\u001b=d\ndt\f\u0000\u0000\u0000b=1\n\u000b\u00001\u0000d\ndt(t\f)\u0000tb\u0001=t\n\u000b\u00001\u0010\n_\f+\f\nt\u0000b\u0011\n\u00140.\n(iv)d\ndt(\u0012\u001b) +\u0012\u001b(\u001b\u0000\r)+\u0018=_\u0000 + \u0000\u00001\n\u0000\u0000\r\u0001=_\u0000 + 1\u0000\r\u0000 = 0.\n(v) Since\u0012\u001b2\u00111 and\u0018\u00111, we haved\ndt(\u0012\u001b2+\u0018) = 0.\n(vi)_\u0012+ 2(\u001b\u0000\r)\u0012= 2\u0000 _\u0000 + 2(\u0000\u0000\r\u00002) = 2\u0000( _\u0000 + 1\u0000\r\u0000) = 0.\n(vii)\f_\u0012+ 2\u0000_\f\u0000b\u0001\n\u0012= 2\u0000(\f_\u0000 + ( _\f\u0000b)\u0000) = 2\u00002(_\f\u0000b+\f\nt)\u00140:\nLet's go to the conditions ( ii) and (iii). The condition ( iii) gives the formula (16)\nforc(t). Then replacing c(t)2by this value in ( ii) gives the condition ( G3). Note\nthen thatb(t)c(t)2=1\n(\u000b\u00001)2t2!(t), which gives the convergence rate of the values\nf(x(t))\u0000min\nHf=O\u00121\nt2w(t)\u0013\n:\nLet us consider the integral estimate for the values. According to the de\fnition\n(16) forc2band the de\fnition of w, we have\n\u0012b\u001b\u0000d\ndt\u0010\nc2b+\f\u0012\u001b\u0011\n=1\n\u000b\u00001tb\u0000d\ndt\u00101\n\u000b\u00001t2w(t) +1\n\u000b\u00001t\f\u0011\n=t\n(\u000b\u00001)2\u0010\n(\u000b\u00001)b\u00002w\u0000t_w\u0000(\u000b\u00001)(_\f+\f\nt)\u0011\n=t\n(\u000b\u00001)2\u0010\n(\u000b\u00003)w\u0000t_w\u0011\n:\nAccording to Theorem 1 ( ii)\nZ+1\nt0t\u0010\n(\u000b\u00003)w(t)\u0000t_w(t)\u0011\n(f(x(t))\u0000min\nHf)dt< +1:\nMoreover, since \u0012\u001b2= 1, the formula giving the weighting coe\u000ecient q(t) in the\nintegral formula simpli\fes, and we get\nq(t) =\u0012(t)\u001b(t)\f(t) _\u001b(t)\f(t)\n\u001b2(t)+b(t)\n\u001b(t)\u0000_\f(t)\n\u001b(t)!\n=\f(t)\u0000\r(t)\u0010\n\u0000\f(t)_\u0000\r(t) +b(t)\u0000\r(t)\u0000_\f(t)\u0000\r(t)\u0011\n=\f(t)\u0000\r(t)2!(t):\nAccording to Theorem 1 ( iii)\nZ+1\nt0t2\f(t)w(t)krf(x(t))k2dt< +1\nwhich gives the announced convergence rates. u t\nRemark 2 Take\f= 0. Then, according to the de\fnition (15) of w, we havew=b,\nand the conditions of Theorem 3 reduce to\nt_b(t)\u0000(3\u0000\u000b)b(t)\u00140 fort2[t0;+1[:14 Hedy ATTOUCH et al.\nWe recover the condition introduced in [12, Corollary 3.4]. Under this condition,\neach solution trajectory xof\n(IGS)\u000b=t;0;b x(t) +\u000b\nt_x(t) +b(t)rf(x(t)) = 0;\nsatis\fes\nf(x(t))\u0000min\nHf=O\u00121\nt2b(t)\u0013\nast!+1:\n3.3 The case \r(t) =\u000b\nt,\fconstant\nDue to its practical importance, consider the case \r(t) =\u000b\nt,\f(t)\u0011\fwhere\fis a\n\fxed positive constant. In this case, the dynamic (IGS)\r;\f;bis written as follows\nx(t) +\u000b\nt_x(t) +\fr2f(x(t)) _x(t) +b(t)rf(x(t)) = 0: (17)\nThe set of conditions ( G2), (G3) boils down to: for t\u0015t0\n(G2)b(t)>\f\nt;\n(G3)t_w(t)\u0014(\u000b\u00003)w(t);\nwherew(t) =b(t)\u0000\f\nt. Therefore, b(\u0001) must satisfy the di\u000berential inequality\ntd\ndt\u0010\nb(t)\u0000\f\nt\u0011\n\u0014(\u000b\u00003)\u0012\nb(t)\u0000\f\nt\u0013\n:\nEquivalently\ntd\ndtb(t)\u0000(\u000b\u00003)b(t) +\f(\u000b\u00002)1\nt\u00140:\nLet us integrate this linear di\u000berential equation. Set b(t) =k(t)t\u000b\u00003wherek(\u0001) is\nan auxiliary function to determine. We obtain\nd\ndt\u0010\nk(t)\u0000\f\nt\u000b\u00002\u0011\n\u00140;\nwhich gives k(t) =\f\nt\u000b\u00002+d(t) withd(\u0001) nonincreasing. Finally, b(t) =\f\nt+d(t)t\u000b\u00003;\nwithd(\u0001) a nonincreasing function to be chosen arbitrarily. In summary, we get\nthe following result:\nProposition 1 Letx: [t0;+1[!H be a solution trajectory of\nx(t) +\u000b\nt_x(t) +\fr2f(x(t)) _x(t) +\u0010\f\nt+d(t)t\u000b\u00003\u0011\nrf(x(t)) = 0 (18)\nwhered(\u0001)is a nonincreasing positive function. Then, the following properties are sat-\nis\fed:\n(i)f(x(t))\u0000min\nHf=O\u00121\nt\u000b\u00001d(t)\u0013\nast!+1;\n(ii)Z+1\nt0\u0000_d(t)t\u000b\u00001(f(x(t))\u0000inf\nHf)dt< +1:\n(iii)Z+1\nt0t\u000b\u00001d(t)krf(x(t))k2dt< +1:\nProof According to the de\fnition of w(t) andb(t), we have the equalities\nt2w(t) =t2\u0010\nb(t)\u0000\f\nt\u0011\n=t2d(t)t\u000b\u00003=t\u000b\u00001d(t). Then apply Theorem 3. u tInertial dynamics with Hessian damping and time rescaling 15\n3.4 Particular cases\nAccording to Theorem 3 and Proposition 1, let us discuss the role and the impor-\ntance of the scaling coe\u000ecient b(t) in front of the gradient term.\na)The \frst inertial dynamic system based on the Nesterov method, and which\nincludes a damping term driven by the Hessian, was considered by Attouch, Pey-\npouquet, and Redont in [17]. This corresponds to b(t)\u00111, which gives:\nx(t) +\u000b\nt_x(t) +\fr2f(x(t)) _x(t) +rf(x(t)) = 0:\nIn this case, we have w(t) = 1\u0000\f\nt, and we immediately get that ( G2), (G3) are\nsatis\fed by taking \u000b>3 andt>\f . This corresponds to take d(t) =1\nt\u000b\u00003\u0000\f\nt\u000b\u00002,\nwhich is nonincreasing when t\u0015\u000b\u00002\n\u000b\u00003.\nCorollary 1 [17, Theorem 1.10, Proposition 1.11] Suppose that \u000b >3and\f >0.\nLetx: [t0;+1[!H be a solution trajectory of\nx(t) +\u000b\nt_x(t) +\fr2f(x(t)) _x(t) +rf(x(t)) = 0: (19)\nThen,\n(i)f(x(t))\u0000min\nHf=O\u00121\nt2\u0013\nast!+1;\n(ii)Z+1\nt0t(f(x(t))\u0000inf\nHf)dt< +1;\n(iii)Z+1\nt0t2krf(x(t))k2dt< +1:\nb)Another important situation is obtained by taking d(t) =1\nt\u000b\u00003. This is the\nlimiting case where the following two properties are satis\fed: d(\u0001) is nonincreasing,\nand the coe\u000ecient of rf(x(t)) is bounded. This o\u000bers the possibility of obtaining\nsimilar results for the explicit temporal discretized dynamics, that is to say the\ngradient algorithms. Precisely, we obtain the dynamic system considered by Shi,\nDu, Jordan, and Su in [30], and Attouch, Chbani, Fadili, and Riahi in [8].\nCorollary 2 [8, Theorem 3], [30, Theorem 5]\nSuppose that \u000b\u00153. Letx: [t0;+1[!H be a solution trajectory of\nx(t) +\u000b\nt_x(t) +\fr2f(x(t)) _x(t) +\u0010\n1 +\f\nt\u0011\nrf(x(t)) = 0 (20)\nThen, the conclusions of Theorem 3are satis\fed:\n(i)f(x(t))\u0000min\nHf=O\u00121\nt2\u0013\nast!+1;\n(ii)When\u000b>3;Z+1\nt0t(f(x(t))\u0000inf\nHf)dt< +1:\n(iii)Z+1\nt0t2krf(x(t))k2dt< +1:16 Hedy ATTOUCH et al.\nNote that (20) has a slight advantage over (19): the growth conditions are valid\nfort>0, while for (19) one has to take t>\f . Accordingly, the estimates involve\nthe quantity1\nt2instead of1\nt2(1\u0000\f\nt).\nc)Taked(t) =1\ntswiths >0. According to Proposition 1, for any solution\ntrajectory x: [t0;+1[!H of\nx(t) +\u000b\nt_x(t) +\fr2f(x(t)) _x(t) +\u0010\f\nt+t\u000b\u00003\u0000s\u0011\nrf(x(t)) = 0 (21)\nwe have:\n(i)f(x(t))\u0000min\nHf=O\u00121\nt\u000b\u00001\u0000s\u0013\nast!+1;\n(ii)Z+1\nt0t\u000b\u0000s\u00002(f(x(t))\u0000inf\nHf)dt< +1;Z+1\nt0t\u000b\u0000s\u00001krf(x(t))k2dt< +1:\n4 Results based on the function p\r\nIn this section, we examine another set of growth conditions for the damping\nand rescaling parameters that guarantee the existence of solutions to the system\n(i)\u0000(vii) of Theorem 1. In the following theorems, the Lyapunov analysis and the\nconvergence rates are formulated using the function p\r: [t0;+1[!R+de\fned by\np\r(t) := exp\u0012Zt\nt0\r(s)ds\u0013\n:\nIn Theorems 2 and 3, in line with the previous articles devoted to these questions\n(see [4], [7], [12]), the convergence rate of the values was formulated using the\nfunction\u0000\r(t) =p\r(t)R+1\nt1\np\r(s)ds. In fact, each of the two functions p\rand\u0000\r\ncaptures the properties of the viscous damping coe\u000ecient \r(\u0001), but their growths\nare signi\fcantly di\u000berent. To illustrate this, in the model case \r(t) =\u000b\nt,\u000b > 1,\nwe havep\r(t) =\u0000t\nt0\u0001\u000b, while\u0000\r(t) =t\n\u000b\u00001. Therefore, p\rgrows faster than \u0000\r\nast!+1, and we can expect to get better convergence rates when formulating\nthem using p\r. Moreover, p\rmakes sense and allows to analyze the case \u000b\u00141,\nwhile\u0000\rdoes not. Thus, we will see that the approach based on p\rprovides results\nthat cannot be captured by the approach based on \u0000\r. To illustrate this, we start\nwith a simple situation, then we consider the general case.\n4.1 A model situation\nConsider the system\n(IGS)\r;0;b x(t) +\r(t) _x(t) +b(t)rf(x(t)) = 0\nwith\r(t) =\r0(t) +1\np0(t)and p 0(t) = exp\u0012Zt\nt0\r0(s)ds\u0013\n:\nChoose\n\u0018\u00110; c(t) = p 0(t); \u001b(t) =1\np0(t); \u0012(t) = p 0(t)2:Inertial dynamics with Hessian damping and time rescaling 17\nAccording to _ p0(t) =\r0(t)p0(t), we can easily verify that the conditions ( i);(iii)\u0000\n(vii) of Theorem 1 are satis\fed, and ( ii) becomesd\ndt\u0000p0(t)2b(t)\u0001\n\u0000p0(t)b(t)\u00140:\nThen, a direct application of Theorem 1 gives the following result.\nTheorem 4 Suppose that for all t\u0015t0\np0(t)_b(t) +\u0010\n2\r0(t)p0(t)\u00001\u0011\nb(t)\u00140: (22)\nLetx: [t0;+1[!H be a solution trajectory of (IGS)\r;0;b. Then, ast!+1\nf(x(t))\u0000min\nHf=O\u00121\np0(t)2b(t)\u0013\n: (23)\nMoreover,R+1\nt0p0(t)\u0010\n1\u0000(2\r0(t)p0(t))\u0000p0(t)_b(t)\u0011\n(f(x(t))\u0000minHf)dt< +1:\nRemark 3 Let us rewrite the linear di\u000berential inequality (22) as follows:\n_b(t)\nb(t)\u00141\np0(t)\u00002_ p0(t)\np0(t):\nA solution corresponding to equality is b(t) = p 0(t)\u00002exphZt\nt0\u00101\np0(s)\u0011\ndsi\n.\nIn the case \r0(t)(t) =\u000b\nt, 0<\u000b< 1,t0= 1, we have p 0(t) =t\u000b;which gives\nb(t) =t\u00002\u000bexpht1\u0000\u000b\u00001\n1\u0000\u000bi\n:\nTherefore, for 0 <\u000b< 1;and for this choice of b, (23) gives\nf(x(t))\u0000min\nHf=O\u00101\nexph\nt1\u0000\u000b\n1\u0000\u000bi\u0011\n: (24)\nThus, we obtain an exponential convergence rate in a situation that cannot be\ncovered by the \u0000\rapproach.\n4.2 The general case, with the Hessian-driven damping\nTheorem 5 Letf:H!Rbe a convex function of class C1such that argminHf6=;.\nSuppose that \r(\u0001);\f(\u0001)areC1functions and b(\u0001)is aC2function which is nondecreasing.\nSuppose that randmare positive parameters which satisfy 0<r\u00141\n3and2r\u0014m\u0014\n1\u0000r. Suppose that the following growth conditions are satis\fed: for t\u0015t0\n(H1)\u00180(t)\u00150;\n(H2)\u00180(t)\u00002\u001b(t)\u0000(m+r)\r(t)\u0001\n\u00001\n2_\u00180(t) +\u0000\nm\u0000(1\u0000r)\u0001\n\r(t)\u001b2(t)\u00150;\n(H3)b(t)\u0000_\f(t) +\f(t)\u0000\n\u001b(t)\u0000\r(t)\u0001\n\u00150;\n(H4)d\ndt\u0010\n\u0012(w+\f\u001b)\u0011\n(t)\u0000\u0012(t)b(t)\u001b(t)\u00140:18 Hedy ATTOUCH et al.\nwhere\n\u00180(t) :=\u0000(1\u00002(r+m))\r(t) +\u001b(t)\u0001\n\u001b(t)\u0000_\u001b(t); (25)\n\u001b(t) :=m\r(t) +1\n3_b(t)\nb(t): (26)\nw(t) =b(t)\u0000_\f(t) +\f(t)\u001b(t) + (1\u00002r\u00002m)\r(t)\f(t): (27)\nThen, for each solution trajectory of x: [t0;+1[!H of(IGS)\r;\f;b, we have,\n(i)f(x(t))\u0000min\nHf=O \n1\np\r(t)2rw(t)b(t)\u00002\n3!\nast!+1 (28)\n(ii)Z+1\nt0p2r\n\r(t)\u0007(t)\u0012\nf(x(t))\u0000inf\nHf\u0013\ndt< +1; (29)\n(iii)Z+1\nt0\u0012\np2r\n\r(t)b1\n3(t)\f(t)\u0000d\ndt\u0000\np2r\n\rb\u00002\n3\f2\u0001(t)\u0013\nkrf(x(t))k2dt< +1:(30)\nHere\u0007(t) :=\u0010\n3\u001b(t)\u00002(r+m)\r(t)\u0011\nw(t)\u0000_w(t)\u00002(1\u0000r\u0000m)\r(t).\nProof According to Theorem 1, it su\u000eces to show that, under the hypothesis\n(H1)\u0000(H4);there exists c;\u0012;\u001b;\u0018 which satisfy the conditions ( i)\u0000(vii) of Theorem\n1. To perform the corresponding derivative calculation, let's start by establishing\nsome preliminary results.\n\u000flnp\r(t) =Rt\nt0\r(s)ds, which by derivation gives_p\r\np\r=\r, that is to say _ p\r=\rp\r:\n\u000fAccording to the de\fnition of \u001b,\nd\ndt\u0010\np2r\n\rb\u00002\n3\u0011\n= 2p2r\n\rb\u00002\n3\u0012\nr\r\u00001\n3_b\nb\u0013\n(31)\n= 2\u0012((r+m)\r\u0000\u001b): (32)\nLet us show that the following choice of the unknown parameters c;\u0012;\u001b;\u0018 satis\fes\nthe conditions ( i)\u0000(vii) of Theorem 1:\n\u0012:=p2r\n\rb\u00002\n3; \u001b :=m\r+1\n3_b\nb; \u0018 :=\u0012\u00180;\nand\nc2b:=\u0012w:=\u0012\u0010\nb\u0000_\f+\f\u001b+ (1\u00002r\u00002m)\r\f\u0011\n; (33)\nwhere\u00180has been de\fned in (25). We underline that under condition ( H3);\nc2b=\u0012\u0010\nb\u0000_\f+\f\u001b\u0000\r\f|{z}\n\u00150+2 (1\u0000r\u0000m)|{z}\n\u00150\r\f\u0011\n\u00150:\nAlso, according to (32), we have _\u0012= 2\u0012\u0000(r+m)\r\u0000\u001b\u0001\n:Inertial dynamics with Hessian damping and time rescaling 19\n(i)d\ndt(\f\u0012\u001b)\u0000\u0012b\u001b=_\f\u0012\u001b+\f(_\u0012\u001b+\u0012_\u001b)\u0000b\u0012\u001b\n=\u0012h\n_\f\u001b+ 2\u0010\n(r+m)\r\u0000\u001b\u0011\n\f\u001b+\f_\u001b\u0000b\u001bi\nbecause _\u0012= 2((r+m)\r\u0000\u001b)\n=\u0012\u0010\n\u0000\f\u00180\u0000\u001b(b\u0000_\f+\f\u001b\u0000\r\f)\u0011\n:\n(34)\nSincebis nondecreasing, then \u001b\u00150;so by (H1) and (H3);we get\nd\ndt(\f\u0012\u001b)\u0000\u0012b\u001b=\u0012\u0010\n\u0000\f\u00180\u0000\u001b(b\u0000_\f+\f\u001b\u0000\r\f)|{z}\n\u00150\u0011\n\u00140\n(ii) According to the derivation chain rule and ( H4), we conclude that\nd\ndt\u0010\nc2b\u0011\n+d\ndt(\f\u0012\u001b)\u0000\u0012b\u001b=d\ndt(\u0012w)+d\ndt(\f\u0012\u001b)\u0000\u0012b\u001b\n=d\ndt(\u0012w+\f\u0012\u001b)\u0000\u0012b\u001b\u00140:\n(iii)b(c2\u0000\u0012) +\f\u0012(\u001b\u0000\r) +d\ndt(\f\u0012) = 0 results from (33).\n(iv) According to the derivation chain rule, (31), and the de\fnition of \u001b\nd\ndt(\u0012\u001b) +\u0012\u001b(\u001b\u0000\r)+\u0018=_\u0012\u001b+\u0012_\u001b+\u0012\u001b(\u001b\u0000\r)+\u0018\n= 2\u0012\u001b\u0000(r+m)\r\u0000\u001b\u0001+\u0012_\u001b+\u0012\u001b(\u001b\u0000\r)+\u0018\n=\u0012\u0010\n_\u001b\u0000\u001b\u0000(1\u00002r\u00002m)\r+\u001b\u0001\u0011\n+\u0018:\nFor this quantity to be equal to zero, we therefore take \u0018=\u0012\u00180, where\u00180is\nde\fned in (26).\n(v) According to our choice \u0018=\u0012\u00180, we have\n(v)()d\ndt\u0010\n\u0012(\u001b2+\u00180)\u0011\n\u00140:\nLet's compute this quantity. According to the derivation chain rule and (32)\nd\ndt\u0010\n\u0012(\u001b2+\u00180)\u0011\n=\u0010\n\u0012_\u00180+_\u0012(\u001b2+\u00180) + 2\u0012_\u001b\u001b\u0011\n= 2\u0012\u00101\n2_\u00180+ (\u001b2+\u00180)((r+m)\r\u0000\u001b)+ _\u001b\u001b\u0011\n= 2\u0012\u00101\n2_\u00180+\u00180\u0000(r+m)\r\u00002\u001b\u0001+\u001b\u0000\n\u00180+\u001b\u0000(r+m)\r\u0000\u001b\u0001+ _\u001b\u0001\u0011\n:\nBy de\fnition of \u00180, we have\u00180+ _\u001b=\u0010\n(1\u00002(r+m))\r+\u001b\u0011\n\u001b. Therefore\nd\ndt\u0010\n\u0012(\u001b2+\u00180)\u0011\n= 2\u0012\u00101\n2_\u00180+\u00180((r+m)\r\u00002\u001b)+\r\u001b2(1\u0000(r+m))\u0011\n:\nSo, (v) is satis\fed under the condition\n1\n2_\u00180+\u00180\u0000(r+m)\r\u00002\u001b\u0001+\r\u001b2\u00001\u0000(r+m)\u0001\n\u00140;\nwhich is precisely ( H2).20 Hedy ATTOUCH et al.\n(vi) Let's compute\n_\u0012+ 2(\u001b\u0000\r)\u0012= 2\u0012\u0010\nr\r\u00001\n3_b\nb+ (m\u00001)\r+1\n3_b\nb\u0011\n= 2(r+m\u00001)\u0012\r:\nAccording to the assumption m\u00141\u0000r, this quantity is less or equal than zero.\nWe have (vii)()\f(\f_\u0012+ 2( _\f\u0000b)\u0012)\u00140:\nAccording to condition H3and the assumption m\u00141\u0000r;we conclude\n\f_\u0012+ 2( _\f\u0000b)\u0012= 2\u0012\u0000\n\f(r+m)\r\u0000\f\u001b\u0000b\u0001\n= 2\u0012h\n\u0000(b\u0000_\f+\f\u001b\u0000\r\u001b)|{z}\n\u00150\u0000\f\r(1\u0000r\u0000m)|{z}\n\u00150i\n\u00140:\nSo, (vii) is satis\fed\nAccording to Theorem 1, we obtain (28)-(29)-(30) which completes the proof. u t\n4.3 The case without the Hessian\nLet us specialize the previous results in the case \f= 0, i.e.without the Hessian:\n(IGS)\r;0;b x(t) +\r(t) _x(t) +b(t)rf(x(t)) = 0:\nTheorem 6 Suppose that the conditions (H1)and(H2)of Theorem 5are satis\fed.\nThen, for each solution trajectory x: [t0;+1[!H of(IGS)\r;0;b, we have, as t!+1\nf(x(t))\u0000min\nHf=O \n1\np\r(t)2rb(t)1\n3!\n: (35)\nMoreover, when m> 2r\nZ+1\nt0p\r(t)2rb(t)1\n3\r(t)\u0012\nf(x(t))\u0000inf\nHf\u0013\ndt< +1: (36)\nProof Conditions (H1) and (H2) in Theorem 5 remain unchanged since they are\nindependent of \f. We just need to verify ( H4), because (H3) is written b(t)\u00150\nand becomes obvious. Since \f= 0, we have (H4)()d\ndt\u0010\n\u0012b\u0011\n(t)\u0000\u0012(t)b(t)\u001b(t)\u00140:\nAccording to\nd\ndt\u0010\n\u0012b\u0011\n(t)\u0000\u0012(t)b(t)\u001b(t) =_\u0012(t)b(t) +\u0012(t)_b(t)\u0000\u0012(t)b(t)\u0012\nm\r+_b(t)\n3b(t)\u0013\n=b(t)1=3\u0014\nd\ndt\u0010\n\u0012(t)b(t)2=3\u0011\n\u0000m\r(t)\u0010\n\u0012(t)b(t)2=3\u0011\u0015\n=b(t)1=3\u0014\nd\ndt\u0010\np\r(t)2r\u0011\n\u0000m\r(t)\u0010\np\r(t)2r\u0011\u0015\n= (2r\u0000m)\r(t)b(t)1=3p\r(t)2r\u00140 since 2r\u0014m;\nwe conclude that ( H4) holds, which completes the proof. u t\nNext, we show that the condition ( H2) on the coe\u000ecients \r(\u0001) andb(\u0001) can be\nformulated in simpler form which is useful in practice.Inertial dynamics with Hessian damping and time rescaling 21\nTheorem 7 The conclusions of Theorem 6remain true when we replace (H2)by\n(H+\n2)\u001b(t)\u0000\n\u001b(t)\u0000(r+m)\r(t)\u0001\u00002\u001b(t) +\u00001\u00002(r+m)\u0001\n\r(t)\u0001+1\n2\u001b(t)\u00150,\nand assume moreover that b(\u0001)is log-concave, i.e.,d2\ndt2(ln(b(t)))\u00140.\nProof According to Theorem 6, it su\u000eces to show that ( H2) is satis\fed under the\nhypothesis (H+\n2). By de\fnition of \u001b, we have\n\u00002\u001b(t)\u0000(m+r)\r(t)\u0001=\u0010\n(m\u0000r)\r(t) +2\n3_b(t)\nb(t)\u0011\n:\nSo (H2) can be written equivalently as A\u00150, where\nA=:\u00180(t)\u0010\n(m\u0000r)\r(t) +2\n3_b(t)\nb(t)\u0011\n\u00001\n2_\u00180(t) +\u0010\nm+r\u00001\u0011\n\r(t)\u001b2(t): (37)\nA calculation similar to the one above gives\n\u00180(t) =\u0010\n(1\u00002r\u0000m)\r(t)\u0000m\r(t) +\u001b(t)\u0011\n\u001b(t)\u0000_\u001b(t);\n=\u0010\n(1\u00002r\u0000m)\r(t) +1\n3_b(t)\nb(t)\u0011\n\u001b(t)\u0000_\u001b(t): (38)\nIn (37), let's replace \u00180(\u0001) by its formulation (38), we obtain\nA=1\n2d2\ndt2\u001b(t)\u00001\n2d\ndt\u0014\n\u001b(t)\u0012\n(1\u00002r\u0000m)\r(t) +1\n3_b(t)\nb(t)\u0013\u0015\n\u0000_\u001b(t)\u0012\n(m\u0000r)\r(t) +2\n3_b(t)\nb(t)\u0013\n+\u0010\nm+r\u00001\u0011\n\r(t)\u001b2(t)\n+\u0010\n(m\u0000r)\r(t) +2\n3_b(t)\nb(t)\u0011\u0010\n(1\u00002r\u0000m)\r(t) +1\n3_b(t)\nb(t)\u0011\n\u001b(t):\nSet\nB:=\u0010\nm+r\u00001\u0011\n\r(t)\u001b2(t) +\u0010\n(m\u0000r)\r(t) +2\n3_b(t)\nb(t)\u0011\u0010\n(1\u00002r\u0000m)\r(t) +1\n3_b(t)\nb(t)\u0011\n\u001b(t);\nthen we have (by omitting the variable tto shorten the formulas)\nB=\u001bh\n(m+r\u00001)\r\u001b+\u0012\n(m\u0000r)\r(t) +2\n3_b\nb\u0013\u0012\n(1\u00002r\u0000m)\r+1\n3_b\nb\u0013i\n=\u001bh\n(m+r\u00001)\r\u001b+\u0012\n\u0000r\r+1\n3_b\nb+\u001b\u0013\u0012\n(1\u00002r)\r+2\n3_b\nb\u0000\u001b\u0013i\n=\u001bh\n(m+r\u00001)\r\u001b\u0000\u001b2+\r\u001b(\u0000m+ 1\u0000r)+\u001b2+\u0012\n\u0000r\r+1\n3_b\nb\u0013\u0012\n(1\u00002r)\r+2\n3_b\nb\u0013i\n=\u001b\u0012\n\u0000r\r+1\n3_b\nb\u0013\u0012\n(1\u00002r)\r+2\n3_b\nb\u0013\n:\nReplacingBinA, we obtain\nA=\u001b(t)\u0010\n\u001b(t)\u0000(m+r)\r(t)\u0011\u0010\n2\u001b(t) + (1\u00002(m+r))\r(t)\u0011\n+1\n2d2\ndt2\u001b(t) +C(t) (39)22 Hedy ATTOUCH et al.\nwhere\nC(t) :=\u0000_\u001b(t)\u0012\n(m\u0000r)\r(t) +2\n3_b(t)\nb(t)\u0013\n\u00001\n2d\ndt\u0014\n\u001b(t)\u0012\n(1\u00002r\u0000m)\r(t) +1\n3_b(t)\nb(t)\u0013\u0015\n:\nLet us show that C(t) is nonnegative. After replacing \u001b(t) by its value m\r(t)+1\n3_b(t)\nb(t),\nand developing, we get\nC(t) =\u0000m_\r(t)\r(t)(1\u00003r)\u00001\n6(4m\u00002r+ 1)_\r(t)_b(t)\nb(t)\n\u00001\n6d2\ndt2(ln(b(t)))\u0012\n(1 + 2(m\u00002r))\r(t) + 2_b(t)\nb(t)\u0013\n:\nBy assumption, m\u00002r\u00150, 1\u00003r\u00150,\r(\u0001) is nonincreasing, b(\u0001) is nondecreasing,\nandd2\ndt2(ln(b(t)))\u00140. We conclude that C(t)\u00150. According to (39), we obtain\nA\u0015\u001b(t)\u0010\n\u001b(t)\u0000(m+r)\r(t)\u0011\u0010\n2\u001b(t) + (1\u00002(m+r))\r(t)\u0011\n+1\n2d2\ndt2\u001b(t):\nThe condition (H+\n2) expresses that the second member of the above inequality is\nnonnegative. Therefore ( H+\n2) implies (H2), which gives the claim. u t\n4.4 Comparing the two approaches\nAs we have already underlined, Theorems 2 and 7 are based on the Lyapunov\nanalysis of the dynamic (IGS)\r;0;busing the functions \u0000\randp\r, respectively. As\nsuch, they lead to signi\fcantly di\u000berent growth conditions on the coe\u000ecients of\nthe dynamic. Precisely, using the following example, we will show that Theorem\n7 better captures the case where bhas an exponential growth. Take\nb(t) =e\u0016tqand\r(t) =\u000b\nt1\u0000qwith\u000b=\u0016q> 0; q2(0;1):\na)First, let us show that the condition ( H+\n2) of Theorem 7 is satis\fed. We have\n1\n2\u001b(t) +\u001b(t)\u0010\n\u001b(t)\u0000(m+r)\r(t)\u0011\u0010\n2\u001b(t) + (1\u00002(m+r))\r(t)\u0011\n= (\u0016q)3\u0012\nm+1\n3\u0013\u00121\n3\u0000r\u0013\u00125\n3\u00002r\u00131\nt3\u00003q+1\n2\u0016q\u0012\nm+1\n3\u0013\n(1\u0000q)(2\u0000q)1\nt3\u0000q\nwhich is nonnegative because of the hypothesis r\u00141\n3andq<1.\nb)Let us now examine the growth condition used in Theorem 2:\n\u0000(t)_b(t)\u0014b(t)\u0010\n3\u00002\r(t)\u0000(t)\u0011\nwhere\u0000(t) :=p(t)Z+1\ntds\np(s): (40)\nHerept) =e\u0016(tq\u0000tq\n0). Therefore \u0000(t) =e\u0016tqZ+1\nte\u0000\u0016sq\nds, which gives\n\u0000(t)_b(t)\u0000b(t)\u0010\n3\u00002\r(t)\u0000(t)\u0011\n= 3e\u0016tq\u0012\n\u0016qtq\u00001e\u0016tqZ+1\nte\u0000\u0016sq\nds\u00001\u0013\n:Inertial dynamics with Hessian damping and time rescaling 23\nLet us analyze the sign of the above quantity, which is the same as\nD(t) :=\u0016qtq\u00001e\u0016tqZ+1\nte\u0000\u0016sq\nds\u00001\n=\u0000\u0016qtq\u00001e\u0016tqZ+1\ntd\nds\u0010\ne\u0000\u0016sq\u00111\n\u0016qs1\u0000qds\u00001\nAfter integration by parts, we get\nD(t) :=\u00121\nq\u00001\u0013\n+1\u0000q\nqtq\u00001e\u0016tqZ+1\nte\u0000\u0016sq1\nsqds>\u00121\nq\u00001\u0013\n>0:\nTherefore, the condition (40) is not satis\fed.\n5 Illustration of the results\nLet us particularize our results in some important special cases, and compare them\nwith the existing litterature. We do not detail the proofs which result from the\ndirect applications of the previous theorems and the classical di\u000berential calculus.\n5.1 The case b(t) =p(t)3p0.\nRecall that p(t) = exp\u0012Zt\nt0\r(s)ds\u0013\n. We start with results in [7] concerning the\nrate of convergence of values in the case b(t) =c0p(t)3p0withp0\u00150 andc0\u00150.\nIn this case, the system (IGS) \r;0;bbecomes:\nx(t) +\r(t) _x(t) +c0exp\u0012\n3p0Zt\nt0\r(s)ds\u0013\nrf(x(t) = 0: (41)\nObserve that_b(t)\n3b(t)=p0\r(t) and\u00180(t) = (m+p0)\u0000(1\u00002r\u0000m+p0)\r2(t)\u0000_\r(t)\u0001.\nTherefore, conditions ( H1) and (H2) of Theorem 6 become after simpli\fcation:\n(H1) [(p0\u0000r) + (1\u0000r\u0000m)]\r2(t)\u0000_\r(t)\u00150;\n(H2) 2(p0\u0000r)(1 + 2(p0\u0000r))\r3(t)\u00002(1 + 3(p0\u0000r))\r(t)_\r(t) + \r(t)\u00150.\nSincem\u00141\u0000r, instead of (H1), it su\u000eces to verify\n(H+\n1)\u0000\np0\u0000r\u0001\n\r2(t)\u0000_\r(t)\u00150.\nTheorem 8 Let\r: [t0;+1)!R+be a nonincreasing and twice continuously di\u000ber-\nentiable function. Suppose that there exists r2(0;1\n3\u0003\nsuch that\n\r(t)\u00152\u0002min(0;p0\u0000r)\u00032\r3(t)on[t0;+1): (42)\nThen, for each solution trajectory x(\u0001)of(41), we have as t!+1\nf(x(t))\u0000min\nHf=O\u00121\np(t)2r+p0\u0013\n: (43)24 Hedy ATTOUCH et al.\nProof To prove the claim, we use Theorem 5 and distinguish two cases:\n?Supposep\u0000r\u00150, then (42) implies  \r(t)\u00150, and since \ris a nonincreasing,\nwe also have _ \r(t)\u00140; thus both conditions ( H+\n1) and (H2) are satis\fed.\n?Supposep\u0000r<0, then (42) becomes\n\r(t)\u0015(2p\u0000r)2\r3(t) on [t0;+1): (44)\nSince\r(\u0001) is a positive and nonincreasing, lim t!+1\r(t) =`exists and is equal to\nzero. Otherwise, by integrating (44) on [ t0;t] fort>t 0, we would have\n_\r(t)\u0000_\r(t0)\u00152(p\u0000r)2Zt\nt0\r(s)3ds\u00152(p\u0000r)2`3(t\u0000t0):\nThis in turn gives lim t!+1_\r(t) = +1, which implies lim t!+1\r(t) = +1, that is\na contradiction. Then, multiply (44) by _ \r(t). Since\r(\u0001) is nonincreasing, we obtain\n\r(t)_\r(t)\u00142(p\u0000r)2\r3(t)_\r(t)()1\n2d\ndt(_\r(t)2)\u0014(p\u0000r)2\n2d\ndt(\r4(t)):\nBy integrating this inequality from ttoT >t , we get\n_\r(T)2\u0000_\r(t)2\u0014(p\u0000r)2(\r4(T)\u0000\r4(t));\nLettingT!+1;and using lim T!+1\r(T) = 0, we obtain _ \r2(t)\u0015(p\u0000r)2\r4(t);\nwhich is equivalent to j_\r(t)j\u0015jp\u0000lj\r2(t):Since _\r(t)\u00140 andp < r , this gives\n\u0000_\r(t)\u0015(r\u0000p)\r2(t);8t>t 0, that is (H+\n1):We have\n[(p\u0000r) + (1\u0000r\u0000m)]\r2(t)\u0000_\r(t)\n=\u00002(p\u0000r)2\r3(t) + \r(t)|{z}\n\u00150 by (44)+2 (1\u00003r+ 3p)|{z}\n\u00150 sincep<r\r(t)\u0000(p\u0000r)\r2(t)\u0000_\r(t)|{z}\n\u00150 by (H+\n1)\u0001\n\u00150:\nTherefore, (H+\n1) and (H2) are satis\fed. Applying Theorem 5, we conclude. u t\nAs a particular case of Theorem 8, with p0= 0, we obtain the following result.\nTheorem 9 [7, Theorem 2.1] Let\r(\u0001)be a nonncreasing function of class C2, and\nx(\u0001)a solution trajectory of\nx(t) +\r(t) _x(t) +c0rf(x(t) = 0: (45)\nSuppose that\n(Hr;\r)9r>0such that\u00002r2\r3(t) + \r(t)\u00150fortlarge enough.\nThen,f(x(t))\u0000minHf=O\u0010\ne\u00002 min(r;1\n3)Rt\nt0\r(s)ds\u0011\nast!+1.\nRemark 4 The case\r(t) =1\nt(lnt)\u001a, for 0\u0014\u001a\u00141, was developed in [7]. In that case\ncondition (H3;\r) writes as\n2(lnt)2+ 3\u001alnt+\u001a(\u001a+ 1)\u00152r2(lnt)2(1\u0000\u001a);\nwhich is satis\fed for any r\u00141 and any t\u0015e.Inertial dynamics with Hessian damping and time rescaling 25\n{If\u001a= 1, thenp(t) = exp\u0012Zt\nt01\ns(lns)\u001ads\u0013\n= exp Zlnt\nlnt0du\nx!\n=lnt\nlnt0;\nand forr=1\n3, we getf(x(t))\u0000minHf=O\u0012\n1\n(lnt)2\n3\u0013\n:\n{If 0\u0014\u001a<1, thenp(t) = exp Zlnt\nlnt01\nu\u001adu!\n= exp\u0010\n1\n1\u0000\u001a\u0000(lnt)1\u0000\u001a\u0000(lnt0)1\u0000\u001a\u0001\u0011\n;\nand, forr=1\n3, we also get f(x(t))\u0000minHf=O\u0012\n1\nexp\u0010\n2\n3(1\u0000\u001a)(lnt)1\u0000\u001a\u0011\u0013\n:\n5.2 The case b(t) =c0tqand\r(t) =\u000b\nt.\nWhenb(t) =c0tqand\r(t) =\u000b\ntwhere\u000b > 0 andq\u00150, we \frst observe that\np(t) = exp\u0010Rt\nt0\r(s)ds\u0011\n=\u0000t\nt0\u0001\u000b:The second-order continuous system becomes:\nx(t) +\u000b\nt_x(t) +c0tqrf(x(t)) = 0: (46)\nApplying Theorem 8, we obtain the following new result.\nTheorem 10 Letx(\u0001)be a solution trajectory of (46) with\u000b>1andq\u00150. Suppose\nthat1<\u000b\u00143 +q. Then,\nf(x(t))\u0000min\nHf=O\u00121\nt2\u000b+q\n3\u0013\n;ast!+1: (47)\nRemark 5 Takingq= 0, a direct application of the above result covers the results\nobtained in [9,32] (case \u000b\u00153), and in [3,10], (case \u000b\u00143). It su\u000eces to take\n\r(t) =\u000b\ntandr=1\n\u000b. More precisely, we get :\n{if 0<\u000b\u00143 thenf(x(t))\u0000minHf=O(t\u00002\u000b\n3),\n{if\u000b>3 thenf(x(t))\u0000minHf=O(1\nt2).\n5.3 The case b(t) =e\u0016tqand\r(t) =\u000b\nt1\u0000q.\nSuppose that \u0016\u00150;0\u0014q\u00141 and\u000b>0. This will allow us to obtain the following\nexponential convergence rate of the values.\nTheorem 11 Letx: [t0;+1[\u0000!H be a solution trajectory of\nx(t) +\u000b\nt1\u0000q_x(t) +e\u0016tq\nrf(x(t)) = 0: (48)\nSuppose that \u000b\u0014\u0016q, then, ast!+1\nf(x(t))\u0000min\nHf=O\u0010\ne\u00002\u000b+\u0016q\n3tq\u0011\n:26 Hedy ATTOUCH et al.\nRemark 6 a) Forq=\u0016= 0, (48) reduces to the system initiated in [32], i.e.\nx(t) +\u000b\nt_x(t) +rf(x(t)) = 0:\nJust assuming \u000b>0, we obtain lim\nt!+1\u0012\nf(x(t))\u0000min\nHf\u0013\n= 0.\nb) Forq=1\n2we get\n{If\u000b\u0014\u0016, thenf(x(t))\u0000minHf=O\u0010\ne\u00002(2\u000b+\u0016)\n3)p\nt\u0011\n:\n{If\u000b\u0015\u0016, thenf(x(t))\u0000minHf=O\u0010\ne\u00002\u0016p\nt\u0011\n:\nc) Forq= 1, direct application of Theorem 11 gives:\nCorollary 3 (Linear convergence) Letx: [t0;+1[!H be a solution trajectory of\nx(t) +\u000b_x(t) +e\u0016trf(x(t)) = 0: (49)\nIf\u000b\u0014\u0016;thenf(x(t))\u0000minHf=O\u0010\ne\u00002\u000b+\u0016\n3t\u0011\n:\nLet us illustrate these results. Take f(x1;x2) :=1\n2\u0000\nx2\n1+x2\n2\u0001\n\u0000ln(x1x2), which is a\nstrongly convex function. Trajectories of\nx(t) +\u000b_x(t) +e\u0016trf(x(t)) +ce\u0017tr2f(x(t)) _x(t) = 0;\ncorresponding to di\u000berent values of the parameters \u000b,\u0016,\u0017, andc, are plotted in\nFigure 12. The parameter cshows the importance of the Hessian-damping.\nFig. 1 Evolution of f(x(t))\u0000minffor solutions of (49), (50), and f(x1;x2) =1\n2\u0000\nx2\n1+x2\n2\u0001\n\u0000\nln(x1x2).\n2From Scilab version 6.1.0 http://www.scilab.org as an open source softwareInertial dynamics with Hessian damping and time rescaling 27\n\r(t)\f(t)b(t) f(x(t))\u0000minf Reference\nCte 0 1 O\u0000\nt\u00001\u0001\n(1964) [29]\nCte Cte 1 O\u0000\nt\u00001\u0001\n(2002) [2]\n\u000b=t 0 1O\u0010\nt\u00002\n3\u000b\u0011\nif 0<\u000b\u00143\nO\u0000\nt\u00002\u0001\nif\u000b\u00153(2019) [10]\n(2014) [32]\n\u000b=t Cte 1O\u0000\nt\u00002\u0001\nif\u000b\u00153;\f> 0 (2016) [17]\n\r(t) 0b(t)O\u0012\u0010\np(t)R+1\nt(p(s))\u00001ds\u0011\u00002\n(b(t))\u00001\u0013\nwherep(t) := exp\u0010Rt\nt0\r(s)ds\u0011 (2019) [10]\n\u000b=t\f(t)b(t)O \u0012\nt2b(t)\u0000_\f(t)\u0000\f(t)\nt\u0013\u00001!\n(2020) [8]\nFig. 2 Convergence rate of f(x(t))\u0000minffor instances of Theorem 1 and general f.\n5.4 Numerical comparison\nFigure 2 summarizes our convergence results, according to the behavior of the\nparameters \r(t),\f(t),b(t). Let's comment on them and compare them, separately\nconsidering fto be strongly convex or not.\n5.4.1 Strongly convex case\nSuppose that fiss-strongly convex. Following Polyak's [29], the system\nx(t) + 2ps_x(t) +rf(x(t)) = 0 (50)\nprovides the linear convergence rate f(x(t))\u0000infHf\u0014Ce\u0000pst, see also [31, The-\norem 2.2]. In the presence of an additional Hessian-driven damping term\nx(t) + 2ps_x(t) +\fr2f(x(t)) _x(t) +rf(x(t)) = 0 (\f\u00150) (51)\na related linear rate of convergence can be found in [8, Theorem 7]. Let us insist\non the fact that, in Corollary 3, we obtain a linear convergence rate for a general\nconvex di\u000berentiable function f. In Figure 1, for the strongly convex function\nf(x1;x2) =1\n2\u0000\nx2\n1+x2\n2\u0001\n\u0000ln(x1x2);we can observe that some values of \u0016give a\nbetter speed of convergence of f(x(t))\u0000minf. We can also note that for \u0016correctly\nset, the system (49) provides a better linear convergence rate than the system (50).\n5.4.2 Non-strongly convex case\nWe illustrate our results on the following simple example of a non strongly convex\nminimization problem, with non unique solutions.\nmin\nR2f(x1;x2) =1\n2(x1+ 103x2)2: (52)\nFrom Figure 3 we get the following properties:\na) The convergence rate of the values is in accordance with Figure 2.\nb) The system (49) is best for its linear convergence of values.\nc) The Hessian-driven damping reduces the oscillations of the trajectories.28 Hedy ATTOUCH et al.\nFig. 3 Evolution of f(x(t))\u0000minffor systems in Figure 2, and f(x1;x2) =1\n2\u0000\nx2\n1+ 103x2\n2\u0001\n.\n6 Conclusion, perspectives\nOur study is one of the \frst works to simultaneously consider the combination of\nthree basic techniques for the design of fast converging inertial dynamics in con-\nvex optimization: general viscous damping (and especially asymptotic vanishing\ndamping in relation to the Nesterov accelerated gradient method), Hessian-driven\ndamping which has a spectacular e\u000bect on the reduction of the oscillatory aspects\n(especially for ill-conditionned minimization problems), and temporal rescaling.\nWe have introduced a system of equations-inequations whose solutions provide\nthe coe\u000ecients of a general Lyapunov functions for these dynamics. We have been\nable to encompass most of the existing results and \fnd new solutions for this sys-\ntem, thus providing new Lyapunov functions. Also, we have been able to explain\nthe mysterious coe\u000ecients which have been used in recent algorithmic develope-\nments, and which were just justi\fed until now by the simpli\fcation of complicated\ncalculations. Finally, by playing on fast rescaling methods, we have obtained linear\nconvergence results for general convex functions. This work provides a basis for\nthe development of corresponding algorithmic results.\nReferences\n1. F. \u0013Alvarez, On the minimizing property of a second-order dissipative system in Hilbert\nspaces, SIAM J. Control Optim., 38 (4) (2000), 1102-1119.\n2. F. \u0013Alvarez, H. Attouch, J. Bolte, P. Redont, A second-order gradient-like dissipative dy-\nnamical system with Hessian-driven damping, J. Math. Pures Appl., 81 (2002), 747{779.\n3. V. Apidopoulos, J.-F. Aujol, Ch. 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May, Asymptotic for a second order evolution equation with convex potential and\nvanishing damping term, Turkish Journal of Mathematics, 41 (3) (2016), 681{685.\n26. Y. Nesterov, A method of solving a convex programming problem with convergence rate\nO(1=k2), Soviet Mathematics Doklady, 27 (1983), 372{376.\n27. Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course. Springer\nScience+Business Media New York (2004).\n28. J. Peypouquet, S. Sorin, Evolution equations for maximal monotone operators: asymptotic\nanalysis in continuous and discrete time, J. Convex Anal, 17 (3-4) (2010), 1113{1163.\n29. B.T. Polyak, Some methods of speeding up the convergence of iteration methods, U.S.S.R.\nComput. Math. Math. Phys., 4 (1964), 1{17.\n30. B. Shi, S.S Du, M.I. Jordan, W.J. Su, Understanding the acceleration phenomenon via\nhigh-resolution di\u000berential equations, arXiv:submit/2440124[cs.LG] 21 Oct 2018.\n31. W. Siegel, Accelerated \frst-order methods: Di\u000berential equations and Lyapunov functions.\narXiv:1903.05671v1 [math.OC] (2019).\n32. W.J. Su, S. Boyd, E.J Cand\u0012 es, A di\u000berential equation for modeling Nesterov's accelerated\ngradient method: theory and insights,Neural Information Processing Systems, 27 (2014),\n2510{2518."    },    {        "title": "1604.02998v1.All_Optical_Study_of_Tunable_Ultrafast_Spin_Dynamics_in__Co_Pd__NiFe_Systems__The_Role_of_Spin_Twist_Structure_on_Gilbert_Damping.pdf",        "content": "All-Optical Study of Tunable Ultrafast Spin Dynamics in [Co/Pd]-NiFe Systems: The\nRole of Spin-Twist Structure on Gilbert Damping\nChandrima Banerjee,1Semanti Pal,1Martina Ahlberg,2T. N.\nAnh Nguyen,3, 4Johan \u0017Akerman,2, 4and Anjan Barman1,\u0003\n1Department of Condensed Matter Physics and Material Sciences,\nS. N. Bose National Centre for Basic Sciences, Block JD, Sec. III, Salt Lake, Kolkata 700 098, India\n2Department of Physics, University of Gothenburg, 412 96, Gothenburg, Sweden\n3Laboratory of Magnetism and Superconductivity,\nInstitute of Materials Science, Vietnam Academy of Science and Technology,\n18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam.\n4Department of Materials and Nano Physics, School of Information and Communication Technology,\nKTH Royal Institute of Technology, Electrum 229, SE-16440 Kista, Sweden\n(Dated: April 12, 2016)\nWe investigate optically induced ultrafast magnetization dynamics in [Co(0.5 nm)/Pd(1\nnm)] 5/NiFe( t) exchange-spring samples with tilted perpendicular magnetic anisotropy using a time-\nresolved magneto-optical Kerr e\u000bect magnetometer. The competition between the out-of-plane\nanisotropy of the hard layer, the in-plane anisotropy of the soft layer and the applied bias \feld reor-\nganizes the spins in the soft layer, which are modi\fed further with the variation in t. The spin-wave\nspectrum, the ultrafast demagnetization time, and the extracted damping coe\u000ecient all depend on\nthe spin distribution in the soft layer, while the latter two also depend on the spin-orbit coupling\nbetween the Co and Pd layers. The spin-wave spectra change from multimode to single-mode as t\nincreases. At the maximum \feld reached in this study, H=2.5 kOe, the damping shows a nonmono-\ntonic dependence on twith a minimum at t= 7.5 nm. For t<7.5 nm, intrinsic e\u000bects dominate,\nwhereas for t>7.5 nm, extrinsic e\u000bects govern the damping mechanisms.\nI. INTRODUCTION\nNonuniform magnetic structures, including exchange\nbias (ferromagnet/antiferromagnet)3,24and exchange-\nspring (ferromagnet/ferromagnet)5{8systems, have\nrecently been explored extensively on account of their\nintrinsic advantages for applications in both permanent\nmagnets and recording media. Exchange-spring (ES)\nmagnets are systems of exchanged-coupled hard and soft\nmagnetic layers that behave as a single magnet. Here,\nthe high saturation magnetization ( Ms) of the soft phase\nand the high anisotropy ( Hk) of the hard phase result in\na large increase in the maximum energy product. This\nmakes them useful as permanent magnets in energy ap-\nplications such as engines or generators in miniaturized\ndevices. On the other hand, for spintronic applications,\nthe soft phase is used to improve the writability of\nthe magnetic media, which in turn is stabilized by the\nmagnetic con\fguration of the hard layer. Consequently,\na wealth of research has been devoted to investigating\nthe static and dynamic magnetic properties, including\nthe switching behavior and exchange coupling strength,\nin ES systems.\nIn case of ES systems with tilted anisotropy, the hard\nand soft phases consist of materials with out-of-plane\n(OOP) and in-plane (IP) anisotropies, respectively. This\ncombination results in a canting of the magnetization\nof the soft layer with a wide and tunable range of tilt\nangles. The advantage of such a hybrid anisotropy sys-\ntem is that it is neither plagued by the poor writability\nand thermal instability of systems with IP anisotropy,\nnor does it lead to very high switching \felds, as in OOPsystems. As a result, these materials provide additional\ndegrees of freedom to control the magnetization dynam-\nics in magnetic nanostructures, and hint at potential\napplications in novel spintronic devices utilizing the\nspin-transfer torque (STT) e\u000bect|such as spin-torque\noscillators (STOs)25,26and STT-MRAMs.\nSo far, numerous studies have been performed on\nsuch systems where the exchange coupling between\nthe hard and soft layers has been tailored by varying\nthe layer thickness,12,13layer composition,19number\nof repeats,15and interfacial anisotropy.13The litera-\nture describes investigations of domain structure and\nother static magnetic properties for [Co/Pd]/Co,14\n[Co/Pd]/NiFe,12,14,19,21[Co/Pd]/CoFeB,14,15,20\n[Co/Pd]-Co-Pd-NiFe,13[Co/Ni]/NiFe,4and CoCrPt-\nNi11|these systems being studied with static mag-\nnetometry, magnetic force microscopy (MFM), and\nmicromagnetic simulations. The magnetization dy-\nnamics in such systems have also been measured using\nBrillouin light scattering (BLS)19,20and ferromagnetic\nresonance (FMR)21experiments, where the spin-wave\n(SW) modes have been investigated by varying the thick-\nness of the soft layer and changing the con\fguration of\nthe hard layer. In any process involving magnetization\ndynamics, the Gilbert damping constant ( \u000b) plays a key\nrole in optimizing writing speeds and controlling power\nconsumption. For example, in case of STT-MRAM\nand magnonic devices, low \u000bfacilitates a lower writing\ncurrent and the longer propagation of SWs, whereas a\nhigher\u000bis desirable for increasing the reversal rates and\nthe coherent reversal of magnetic elements, which are\nrequired for data storage devices.arXiv:1604.02998v1  [cond-mat.mtrl-sci]  11 Apr 20162\n46810121416350400450500\n  )sf( emit noitazitengameDt (nm)(d)(a) \n-202 600 1200 1800-2-10\n  Kerr rotation (a rb. unit)\nTime (ps)(b)0 10 20 30\n  Power (arb. unit)\nFrequency (GHz)(c)\n(b)\n-202 60012001800-2-10Kerrrotation(arb.unit)\nTime(ps)\nFigure 1. (color online) (a) Schematic of the two-color pump-\nprobe measurement of the time-resolved magnetization dy-\nnamics of exchange-spring systems. The bias \feld is applied\nwith a small angle to the normal of the sample plane. (b)\nTypical time-resolved Kerr rotation data revealing ultrafast\ndemagnetization, fast and slow relaxations, and precession\nof magnetization for the exchange-spring system with t=\n7.5 nm at H= 2.5 kOe. (c) FFT spectrum of the background-\nsubtracted time-resolved Kerr rotation. (d) Variation of de-\nmagnetization time with t.\nIn this paper, we present all-optical excitation and de-\ntection of magnetization dynamics in [Co(0.5 nm)/Pd(1\nnm)] 5/NiFe( t) tilted anisotropy ES systems, with varying\nsoft layer thickness ( t), using a time-resolved magneto-\noptical Kerr e\u000bect (TR-MOKE) magnetometer. The dy-\nnamical magnetic behavior of similar systems has previ-\nously been studied using BLS19and FMR21measure-\nments. However, a detailed study of the precessional\nmagnetization dynamics and relaxation processes in such\ncomposite hard/soft systems is yet to be carried out.\nThe advantage of implementing TR-MOKE is that here\nthe magnetization dynamics can be measured on di\u000ber-\nent time scales and the damping is measured directly\nin the time domain, and is therefore more reliable. We\ninvestigate the ultrafast magnetization dynamics over pi-\ncosecond and picosecond time scales. The ultrafast de-\nmagnetization is examined and found to change due to\nthe modi\fed spin structure in the soft layer for di\u000berent\ntvalues. The extracted SW spectra are strongly depen-\ndent on t. An extensive study of the damping coe\u000ecient\nreveals that the extrinsic contribution to the damping\nis more dominant in the higher thickness regime, while\nintrinsic mechanisms govern the behavior at lower thick-\nnesses.II. EXPERIMENTAL DETAILS\nA. Sample fabrication\nThe samples were fabricated using dc mag-\nnetron sputtering and have the following structure:\nTa(5nm)/Pd(3nm)/[Co(0.5nm)/Pd(1nm)] \u00025=Ni80Fe20(t)\n/Ta(5nm), where t= 4{20 nm. The chamber base pres-\nsure was below 3 \u000210\u00008Torr, while the Ar work\npressure was 2 and 5 mTorr for the Ta, NiFe and Co,\nPd layers, respectively. The samples were deposited\nat room temperature on naturally oxidized Si(100)\nsubstrates. The 5 nm Ta seed layer was used to induce\nfcc-(111) orientation in the Pd layer, which improves\nthe perpendicular magnetic anisotropy of the Co/Pd\nmultilayers; a Ta cap layer was used to avoid oxidation,\nwhich has been reported in previous studies.12{14The\nlayer thicknesses are determined from the deposition\ntime and calibrated deposition rates.\nB. Measurement technique\nTo investigate the precessional frequency and damp-\ning of these samples, the magnetization dynamics were\nmeasured by using an all-optical time-resolved magneto-\noptical Kerr e\u000bect (TR-MOKE) magnetometer2based on\na two-color optical pump-probe experiment. The mea-\nsurement geometry is shown in Fig. 1(a). The magne-\ntization dynamics were excited by laser pulses of wave-\nlength (\u0015) 400 nm (pulse width = 100 fs, repetition rate\n= 80 MHz) of about 16 mJ/cm2\ruence and probed by\nlaser pulses with \u0015= 800 nm (pulse width = 88 fs, rep-\netition rate = 80 MHz) of about 2 mJ/cm2\ruence. The\npump and probe beams are focused using the same micro-\nscope objective with N.A. of 0.65 in a collinear geometry.\nThe probe beam is tightly focused to a spot of about\n800 nm on the sample surface and, as a result, the pump\nbecomes slightly defocused in the same plane to a spot\nof about 1 \u0016m. The probe beam is carefully aligned at\nthe centre of the pump beam with slightly larger spot\nsize. Hence, the dynamic response is probed from a ho-\nmogeneously excited volume. The bias \feld was tilted\nat around 15\u000eto the sample normal (and its projection\nalong the sample normal is referred to as Hin this ar-\nticle) in order to have a \fnite demagnetizing \feld along\nthe direction of the pump beam. This \feld is eventually\nmodi\fed by the pump pulse which induces precessional\nmagnetization dynamics in the samples. The Kerr rota-\ntion of the probe beam, back-re\rected from the sample\nsurface, is measured by an optical bridge detector us-\ning phase sensitive detection techniques, as a function of\nthe time-delay between the pump and probe beams. Fig-\nure 1(b) presents typical time-resolved Kerr rotation data\nfrom the ES sample with t= 7.5 nm at a bias \feld H=\n2.5 kOe. The data shows a fast demagnetization within\n500 fs and a fast remagnetization within 8 ps, followed by\na slow remagnetization within 1800 ps. The precessional3\n(b) \n010 20 30 0 1 2\nPower (arb. unit)  Kerr Rotation(arb. unit)\n  \n  \n    \n  \n    \n  \nFrequency (GHz)4.5 nm\n5.5 nm\n7.5 nm\n8 nm\n15 nm\n  \nTime (ns)20 nm \n  \n  B\nA \n   \nNiFe  (t = 20 nm)  \nCo/Pd \n1 -1 Normalized Mz Co/Pd NiFe  (t = 6 nm)  \nCo/Pd NiFe ( t = 10 nm) (a) \nFigure 2. (color online) (a) Background-subtracted time-\nresolved Kerr rotation and the corresponding FFT spectra\nfor samples with di\u000berent tvalues at H= 2.5 kOe. The\nblack lines show the \ft according to Eq. 1. (b) Simulated\nstatic magnetic con\fgurations for samples with t= 20, 10,\nand 6 nm with a bias \feld H= 2.5 kOe in the experimental\ncon\fguration. The simulated samples are not to scale. The\ncolor map is shown at the bottom of the \fgure.\ndynamics appear as an oscillatory signal above the slowly\ndecaying part of the time-resolved Kerr rotation data.\nThis part was further analyzed and a fast Fourier trans-\nform (FFT) was performed to extract the corresponding\nSW modes, as presented in Fig. 1(c).III. RESULTS AND DISCUSSIONS\nIn order to closely observe the ultrafast demagnetiza-\ntion and fast remagnetization, we recorded the transient\nMOKE signals for delay times up to 30 ps at a resolution\nof 50 fs. In Fig. 1(d), the demagnetization times are plot-\nted as a function of t. We observe that the demagnetiza-\ntion is fastest in the thinnest NiFe layer ( t= 4 nm) and\nincreases sharply with the increase in t, becoming con-\nstant at 500 fs at t= 5 nm. At t= 10 nm, it decreases\ndrastically to 400 fs and remains constant for further in-\ncreases in t. For t<5 nm, the laser beam penetrates\nto the Co/Pd layer. In this regime, the large spin-orbit\ncoupling of Pd enhances the spin-\rip rate, resulting in a\nfaster demagnetization process. As tincreases, the top\nNiFe layer is primarily probed. Here, the spin con\fgura-\ntion across the NiFe layer, which is further a\u000bected by the\ncompetition between the in-plane and the out-of-plane\nanisotropies of the NiFe and [Co/Pd] layers, governs the\ndemagnetization process. Qualitatively, ultrafast demag-\nnetization can be understood by direct transfer of spin\nangular momentum between neighboring domains10,23.\nwhich may be explained as follows: For t>8 nm, the\nmagnetization orientation in the NiFe layer varies over a\nwide range of angles across the \flm thickness, where the\nmagnetization gradually rotates from nearly perpendicu-\nlar at the Co/Pd and NiFe interface to nearly parallel to\nthe surface plane in the topmost NiFe layer. Such a spin\nstructure across the NiFe layer thickness can be seen as a\nnetwork of several magnetic sublayers, where the spin ori-\nentation in each sublayer deviates from that of the neigh-\nboring sublayer. This canted spin structure accelerates\nthe spin-\rip scattering between the neighboring sublay-\ners and thus results in a shorter demagnetization time,\nsimilar to the work reported by Vodungbo et al.23On the\nother hand, for 5 nm <t<8 nm, the strong out-of-plane\nanisotropy of the Co/Pd layer forces the magnetization\nin the NiFe \flm towards the direction perpendicular to\nthe surface plane, giving rise to a uniform spin structure.\nThe strong coupling reduces the transfer rate of spin an-\ngular momentum and causes the demagnetization time\nto increase.\nTo investigate the variation of the precessional dynamics\nwith t, we further recorded the time-resolved data for a\nmaximum duration of 2 ns at a resolution of 10 ps. Fig-\nure 2(a) shows the background-subtracted time-resolved\nKerr rotation data for di\u000berent values of tatH=\n2.5 kOe and the corresponding fast Fourier transform\n(FFT) power spectra. Four distinct peaks are observed\nin the power spectrum of t= 20 nm, which reduces to\ntwo for t= 15 nm. This is probably due to the rela-\ntive decrease in the nonuniformity of the magnetization\nacross the NiFe thickness, which agrees with the varia-\ntion in the demagnetization time, as described earlier. To\ncon\frm this, we simulated the static magnetic con\fgura-\ntions of these samples in a \feld of H= 2:5 kOe using the\nLLG micromagnetic simulator.18Simulations were per-\nformed by discretizing the samples in arrays of cuboidal4\ncells with two-dimensional periodic boundary conditions\napplied within the sample plane. The simulations assume\nthe Co/Pd multilayer as an e\u000bective medium16with sat-\nuration magnetization Ms= 690 emu/cc, exchange sti\u000b-\nness constant A= 1.3\u0016erg/cm, and anisotropy constant\nKu1= 5.8 Merg/cc along the (001) direction, while the\nmaterial parameters used for the NiFe layer were Ms=\n800 emu/cc, A= 1.3\u0016erg/cm, and Ku1= 0.17The in-\nterlayer exchange between Co/Pd and NiFe is set to 1.3\n\u0016erg/cm and the gyromagnetic ratio \r= 18.1 MHz Oe\u00001\nis used for both layers. The Co/Pd layer was discretized\ninto cells of dimension 5 \u00025\u00022 nm3and the NiFe layer\nwas discretized into cells of dimension 5 \u00025\u00021 nm3.\nThe results are presented in Fig. 2(b) for t= 20, 10, and\n6 nm samples. The nonuniform spin structure is promi-\nnent in the NiFe layer of the t= 20 nm sample, which\nmodi\fes the SW spectrum of this sample, giving rise to\nthe new modes.17With the reduction of t, the spin struc-\nture in the NiFe layer gradually becomes more uniform,\nwhile at t= 7.5 nm it is completely uniform over the\nwhole thickness pro\fle. Hence, for low values of t, the\npower spectra shows a single peak due to the collective\nprecession of the whole stack. The variation in precession\nfrequency with tis plotted in Fig. 3(a). The frequency\nof the most intense mode shows a slow decrease down to\nt= 7.5 nm, below which it increases sharply down to the\nlowest thickness t= 4.5 nm, exhibiting precessional dy-\nnamics. This mode is basically the uniform mode of the\nsystem and follows Kittel's equation.9The variation in\nfrequency depicts the evolution of the e\u000bective anisotropy\nfrom OOP to IP with increasing t, which is in agreement\nwith previously reported results.21For lower t, the sys-\ntem displays an OOP easy axis, owing to the strong OOP\nanisotropy of the Co/Pd multilayer. This is manifested\nas a sharp increase in the frequency with decreasing t\nbelow 7.5 nm. For greater thicknesses, the e\u000bect of the\nperpendicular anisotropy of the Co/Pd multilayer gradu-\nally decreases and the e\u000bect of the in-plane NiFe becomes\nmore prominent. These two anisotropies cancel near t=\n7.5 nm, resulting in a minimum frequency, as shown in\nFig. 3(a).\nTo extract the damping coe\u000ecient, the time domain\ndata was \ftted with an exponentially damped harmonic\nfunction given by Eq. 1.\nM(t) =M(0)e\u0000t\n\u001csin(2\u0019ft\u0000\u001e) (1)\nwhere the relaxation time \u001cis related to the Gilbert\ndamping coe\u000ecient \u000bby the relation \u001c= 1/(2\u0019f\u000b).\nHere, fis the experimentally obtained precession fre-\nquency and \u001eis the initial phase of the oscillation. The\n\ftted data for various values of tis shown by the solid\nblack lines in Fig. 2. We did not extract a value of \u000bfort\n= 15 and 20 nm due to the occurrence of multimode os-\ncillations, which may lead to an erroneous estimate of the\ndamping. The extracted \u000bvalues are plotted against tin\nFig. 3(b) for two di\u000berent \feld values of 2.5 and 1.3 kOe.\nThe evolution of \u000bas a function of tdepends signi\fcantly\n5 6 7 8 9 100.0160.0200.0240.0280.032\n5 10 15 2041216\nt(nm) t(nm)(a) (b)Frequency(GHz)Figure 3. (color online) Evolution of (a) spin-wave frequency\nand (b) Gilbert damping constant as a function of tat 1.3 kOe\n(green circles) and 2.5 kOe (violet circles).\nonH, as can be seen from the \fgure. This is because of\nthe di\u000berent mechanisms responsible for determining the\ndamping in di\u000berent samples, as will be discussed later.\nAn interesting trend in the \u000bvs.tplot is observed\nforH= 2.5 kOe. For 10 nm \u0014t\u00147.5 nm,\u000bdecreases\nwith decreasing tand reaches a minimum value of about\n0.014 for t= 7.5 nm. Below this thickness, \u000bincreases\nmonotonically and reaches a value of about 0.022 for the\nlowest thickness. This variation of \u000bis somewhat cor-\nrelated with the variation of precession frequency with\nthickness. In the thinner regime, we probe both the\nNiFe layer and a fraction of the Co/Pd multilayer and\nthe relative contribution from the latter increases as t\ndecreases. The occurrence of a single mode oscillation\npoints towards a collective precession of the stack, which\nmay be considered a medium with e\u000bective magnetic pa-\nrameters consisting of both NiFe and Co/Pd layers. The\nvariation in damping may be related to the variation in\nthe anisotropy of the material. The competing IP and\nOOP anisotropies of the NiFe and Co/Pd layers lead\nto the appearance of a minimum in the damping. The\ndamping in this system may have multiple contributions,\nnamely (a) dephasing of the uniform mode in the spin-\ntwist structure1(b) interfacial d-dhybridization at the\nCo/Pd interface16, and (c) spin pumping into the Pd\nlayer.22The \frst is an extrinsic mechanism and is dom-\ninant in samples with higher NiFe thicknesses, while the\nother two mechanisms are intrinsic damping mechanisms.\nFort>7.5 nm, due to the nonuniformity of the spin\ndistribution, the dominant mode undergoes dynamic de-\nphasing and the damping thus increasescompared to the\nmagnetically uniform samples. With the increase in NiFe\nthickness, the nonuniformity of spin distribution and the\nconsequent mode dephasing across its thickness increases,\nleading to an increase in the damping value. Hence, in\nsamples with higher tvalues, dephasing is the dominant\nmechanism, while at lower tvalues|i.e., when the con-\ntribution from the Co/Pd multilayer is dominant|the\nspin-orbit coupling and spin pumping e\u000bects dominate.\nAt intermediate tvalues, the extrinsic and intrinsic ef-\nfects compete with each other, leading to a minimum\nin the damping. However, the damping increases mono-\ntonically with tin a lower \feld of H=1.3 kOe. For a\ndeeper understanding of this e\u000bect, we have measured \u000b5\n24681012140.0120.0160.0200.0240.0280.0324\n56789100.0140.0210.0280.0350.042(b) \n  \n5nm \n5.5nm \n6.5nm \n7nm/s61537F\nrequency (GHz)(a) \n  \n10nm \n8.5nm \n8nm \n7.5nm \n7nm/s61537F\nrequency (GHz)\nFigure 4. (color online) Dependence of Gilbert damping co-\ne\u000ecient on soft layer thickness ( t) for (a) 7{10 nm and (b)\n5{7 nm, respectively.\nas a function of precession frequency f. Figures 4(a){(b)\nshow the variation of \u000bwith f. Two di\u000berent regimes in\nthe thickness are presented in (a) and (b) to show the\nrate of variation more clearly. For 10 nm \u0014t\u00147 nm,\u000b\ndecreases strongly with the decrease in fand the rate of\nvariation remains nearly constant with t. This is the sig-\nnature of extrinsic damping generated by the nonuniform\nspin distribution. However, for t= 6.5 nm, the rate falls\ndrastically and for t\u00145.5 nm,\u000bbecomes nearly indepen-\ndent of t, which indicates that purely intrinsic damping is\noperating in this regime. This con\frms the competition\nbetween two di\u000berent types of damping mechanisms in\nthese samples.\nThe study demonstrates that various aspects of ul-\ntrafast magnetization dynamics|namely demagnetiza-\ntion time, precession frequency, number of modes, and\ndamping|are in\ruenced by the spin distribution in the\nsoft magnetic layer, as well as by the properties of the\nhard layer. By changing the thickness of the soft layer,\nthe relative contributions of these factors can be tuned\ne\u000bectively. This enables e\u000ecient control of the damp-\ning and other magnetic properties over a broad range,\nand will hence be very useful for potential applications\nin spintronic and magnonic devices.IV. CONCLUSION\nIn summary, we have employed the time-resolved\nMOKE technique to measure the evolution of ul-\ntrafast magnetization dynamics in exchange-coupled\n[Co/Pd] 5/NiFe( t) multilayers, with varying NiFe layer\nthicknesses, by applying an out-of-plane bias magnetic\n\feld. The coupling of a high-anisotropy multilayer with\na soft layer allows broad control over the spin struc-\nture, and consequently other dynamic magnetic prop-\nerties which are strongly dependent on t. The ultra-\nfast demagnetization displayed a strong variation with\nt. The reason for this was ascribed to the chiral-spin-\nstructure-dependent spin-\rip scattering in the top NiFe\nlayer, as well as to interfacial 3 d-4dhybridization of\nCo/Pd layer. The precessional dynamics showed mul-\ntiple spin-wave modes for t= 20 nm and 15 nm, whereas\na single spin-wave mode is observed for thinner NiFe lay-\ners following the change in the magnetization pro\fle with\ndecreasing t. The precession frequency and the damp-\ning show strong variation with the thickness of the NiFe\nlayer. The changes in frequency are understood in terms\nof the modi\fcation of the anisotropy of the system, while\nthe variation in damping originates from the competition\nbetween intrinsic and extrinsic mechanisms, which are\nsomewhat related to the anisotropy. The observed dy-\nnamics will be important for understanding the utiliza-\ntion of tilted anisotropy materials in devices such as spin-\ntransfer torque MRAM and spin-torque nano-oscillators.\nV. ACKNOWLEDGEMENTS\nWe acknowledge \fnancial support from the G oran\nGustafsson Foundation, the Swedish Research Coun-\ncil (VR), the Knut and Alice Wallenberg Foundation\n(KAW), and the Swedish Foundation for Strategic Re-\nsearch (SSF). This work was also supported by the Euro-\npean Research Council (ERC) under the European Com-\nmunity's Seventh Framework Programme (FP/2007{\n2013)/ERC Grant 307144 \"MUSTANG\". AB acknowl-\nedges the \fnancial support from the Department of Sci-\nence and Technology, Government of India (Grant no.\nSR/NM/NS-09/2011(G)) and S. N. Bose National Centre\nfor Basic Sciences, India (Grant no. SNB/AB/12-13/96).\nC.B. thanks CSIR for the senior research fellowship.\n\u0003abarman@bose.res.in\n1A. Barman and S. Barman. Dynamic dephasing of mag-\nnetization precession in arrays of thin magnetic elements.\nPhys. Rev. B , 79:144415, 2009.\n2A. Barman and A. Haldar. 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Weber, H. Nembach, S. Blomeier, B. Hillebrands,\nR. Kaltofen, J. Schumann, M. J. Carey, and J. Fassbender.\nAll-optical probe of magnetization dynamics in exchange\nbiased bilayers on the picosecond timescale. Eur. Phys. J.\nB, 45:243, 2005.\n25Y. Zhou, C. L. Zha, S. Bonetti, J. Persson, and\nJ.\u0017Akerman. Spin-torque oscillator with tilted \fxed layer\nmagnetization. Appl. Phys. Lett. , 92:262508, 2008.\n26Y. Zhou, C. L. Zha, S. Bonetti, J. Persson, and\nJ.\u0017Akerman. Microwave generation of tilted-polarizer spin\ntorque oscillator. J. Appl. Phys. , 105:07D116, 2009."    },    {        "title": "2106.04948v1.Grammage_of_cosmic_rays_in_the_proximity_of_supernova_remnants_embedded_in_a_partially_ionized_medium.pdf",        "content": "MNRAS 000, 000{000 (0000) Preprint 10 June 2021 Compiled using MNRAS L ATEX style \fle v3.0\nGrammage of cosmic rays in the proximity of supernova remnants\nembedded in a partially ionized medium\nS. Recchia1;2?, D. Galli3, L. Nava4, M. Padovani3, S. Gabici5, A. Marcowith6,\nV. Ptuskin7, G. Morlino3\n1Dipartimento di Fisica, Universit\u0013 a di Torino, via P. Giuria 1, 10125 Torino, Italy\n2Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy\n3INAF{Osservatorio Astro\fsico di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy\n4INAF-Osservatorio Astronomico di Brera, Via Bianchi 46, I-23807 Merate, Italy\n5Universit\u0013 e de Paris, CNRS, Astroparticule et Cosmologie, F-75006 Paris, France\n6Laboratoire Univers et particules de Montpellier, Universit\u0013 e Montpellier/CNRS, F-34095 Montpellier, France\n7Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radiowave Propagation, 108840, Troitsk, Moscow, Russia\nAccepted XXX. Received YYY; in original form ZZZ\nABSTRACT\nWe investigate the damping of Alfv\u0013 en waves generated by the cosmic ray resonant\nstreaming instability in the context of the cosmic ray escape and propagation in the\nproximity of supernova remnants. We consider ion-neutral damping, turbulent damp-\ning and non linear Landau damping in the warm ionized and warm neutral phases of\nthe interstellar medium. For the ion-neutral damping, up-to-date damping coe\u000ecients\nare used. We investigate in particular whether the self-con\fnement of cosmic rays\nnearby sources can appreciably a\u000bect the grammage. We show that the ion-neutral\ndamping and the turbulent damping e\u000bectively limit the residence time of cosmic rays\nin the source proximity, so that the grammage accumulated near sources is found to\nbe negligible. Contrary to previous results, this also happens in the most extreme\nscenario where ion-neutral damping is less e\u000bective, namely in a medium with only\nneutral helium and fully ionized hydrogen. Therefore, the standard picture, in which\nCR secondaries are produced during the whole time spent by cosmic rays throughout\nthe Galactic disk, need not to be deeply revisited.\nKey words:\n1 INTRODUCTION\nThe most popular hypothesis for the origin of Galactic cos-\nmic rays (CRs) invokes supernova remnants (SNRs) as the\nmain sources of such particles (see e.g. Blasi 2013; Gabici\net al. 2019). In this scenario, which in the last decades had\nbecome a paradigm, CR di\u000busion plays a central role. Di\u000bu-\nsion is the key ingredient at the base of the di\u000busive shock\nacceleration of particles at SNRs (e.g. Drury 1983). Di\u000bu-\nsion also a\u000bects the escape of CRs from the acceleration site\nand the subsequent propagation in the source region, with\nprominent implications for \r-ray observations (Aharonian\n& Atoyan 1996; Gabici et al. 2009; Casanova et al. 2010;\nOhira et al. 2011; Nava & Gabici 2013). Finally, di\u000busion\ndetermines the con\fnement time of CRs in the Galaxy, thus\na\u000becting the observed spectrum and the abundances of sec-\nondary spallation nuclei and of unstable isotopes (Ptuskin\n& Soutoul 1998; Wiedenbeck et al. 2007).\n?E-mail: sarah.recchia@unito.itThe di\u000busion of CRs is thought to be mostly due to the\nresonant scattering o\u000b plasma waves whose wavelength is\ncomparable to the particle's Larmor radius rL=\rmpc2=eB,\nwherempis the proton mass, Bis the magnetic \feld\nstrength and \rthe Lorentz factor (see e.g. Skilling 1975a).\nThe magneto-hydrodynamic (MHD) turbulence relevant for\nCR propagation is composed of incompressible Alfv\u0013 enic and\ncompressible (fast and slow) magnetosonic \ructuations (Cho\n& Lazarian 2002; Fornieri et al. 2021). MHD turbulence is\nubiquitous in the interstellar space and may be injected by\nastrophysical sources (see e.g. Mac Low & Klessen (2004))\nbut also by CRs themselves. The active role of CRs in pro-\nducing the waves responsible for their scattering has been\nwidely recognized (see e.g. Wentzel 1974; Skilling 1975b; Ce-\nsarsky 1980; Amato 2011). In fact, spatial gradients in the\nCR density, as those found in the source vicinity, lead to the\nexcitation of Alfv\u0013 en waves at the resonant scale (Ptuskin\net al. 2008). This process, called resonant streaming insta-\nbility , produces waves that propagate along magnetic \feld\nlines in the direction of decreasing CR density.\n©0000 The AuthorsarXiv:2106.04948v1  [astro-ph.HE]  9 Jun 20212S. Recchia et al.\nThe density of Alfv\u0013 en waves that scatter CRs is lim-\nited by several damping processes. The most relevant are:\n(i) ion-neutral damping in a partially ionized medium (Kul-\nsrud & Pearce 1969; Kulsrud & Cesarsky 1971; Zweibel &\nShull 1982); ( ii) turbulent damping, due to the interaction of\na wave with counter-propagating Alfv\u0013 en wave packets. Such\nwaves may be the result of a background turbulence injected\non large scales and cascading to the small scales (we indicate\nthis damping as FG, after Farmer & Goldreich 2004); ( iii)\nnon-linear Landau (NLL) damping, due to the interaction\nof background thermal ions with the beat of two interfering\nAlfv\u0013 en waves (see e.g. Felice & Kulsrud 2001; Wiener et al.\n2013). The relative importance of these e\u000bects depends sig-\nni\fcantly on the physical conditions and chemical composi-\ntion of the ambient medium. A few other collisionless and\ncollisional damping processes can impact magnetohydrody-\nnamical wave propagation in a partially ionized gas but they\nmostly a\u000bect high-wavenumber perturbations (Yan & Lazar-\nian 2004). Recently, it has been suggested that dust grains\nmay also contribute to the damping of Alfv\u0013 en waves (Squire\net al. 2021).\nIn this paper we investigate the escape of CRs from\nSNRs, and their subsequent self con\fnement in the source\nregion, as due to the interplay between the generation of\nAlfv\u0013 en waves by CR streaming instability, and the damp-\ning process mentioned above. Our main goal is to establish\nwhether the self-con\fnement of CRs nearby sources can ap-\npreciably a\u000bect the grammage accumulated by these parti-\ncles. In fact, if this is the case, a signi\fcant fraction of CR\nsecondaries would be produced in the vicinity of CR sources,\nand not during the time spent by CRs in the Galactic disk,\nas commonly assumed. This would constitute a profound\nmodi\fcation of the standard view of CR transport in the\nGalaxy (see, e.g. D'Angelo et al. 2016). In particular, we\nfocus on the CR propagation in partially ionized phases of\nthe interstellar medium (ISM), showing that the ion-neutral\nand FG damping can signi\fcantly a\u000bect the residence time\nof CRs nearby their sources. We \fnd that, for typical condi-\ntions, the grammage accumulated by CRs in the vicinity of\nsources is negligible compared to that accumulated during\nthe time spent in the Galaxy. Even in the case of a medium\nmade of fully ionized H and neutral He, the combination of\nion-neutral and turbulent damping can substantially a\u000bect\nthe con\fnement time1.\nThis paper is organized as follows: in Sec. 2 we describe\nthe damping of Alfv\u0013 en waves by ion-neutral collisions in vari-\nous partially ionized phases of the ISM, and by other damp-\ning mechanisms; in Sec. 3 we illustrate the equations and\nthe setup of our model of CR escape and propagation in the\nproximity of SNRs, the time dependent CR spectrum and\ndi\u000busion coe\u000ecient, the residence time of CRs in the source\nproximity and the implications on the grammage; in Sec. 4\nwe describe our results; and \fnally in Sec. 5 we draw our\nconclusions.\n1The case of a fully neutral (atomic or partially molecular)\nmedium and of a di\u000buse molecular medium (see, e.g. Brahimi\net al. 2020) are not treated here, since the \flling factor of such\nphases is small, but we report the ion-neutral damping rate for\nsuch media for the sake of completeness. The case of a fully ion-\nized medium has been extensively treated by Nava et al. (2019).2 DAMPING OF ALFV \u0013EN WAVES\n2.1 Ion-neutral damping\nThe Galaxy is composed, for most of its volume, by three\nISM phases, namely the warm neutral medium (WNM, \fll-\ning factor\u001825%), warm ionized medium (WIM, \flling fac-\ntor\u001825%) and hot ionized medium (HIM, \flling factor\n\u001850%, see e.g. Ferri\u0012 ere 2001; Ferri\u0012 ere 2019). The physi-\ncal characteristics of these phases are summarised in Ta-\nble 1 (from Jean et al. 2009, see also Ferri\u0012 ere 2001; Fer-\nri\u0012 ere 2019). The physical characteristics of the cold neu-\ntral medium (CNM) and the di\u000buse medium (DiM) are also\nlisted for completeness, while their \flling factor is .1% (Fer-\nri\u0012 ere 2001; Ferri\u0012 ere 2019). In the regions where neutrals are\npresent, like the WNM and the WIM, the rate of ion-neutral\ndamping depends on the amount and chemical species of the\ncolliding particles. In the WNM and WIM the ions are H+,\nwhile neutrals are He atoms (with a H/He ratio of \u001810%)\nand H atoms with a fraction that varies from phase to phase.\nThe main processes of momentum transfer (mt) be-\ntween ions and neutrals are elastic scattering by induced\ndipole, and charge exchange (ce). In the former case, domi-\nnant at low collision energies, the incoming ion is de\rected\nby the dipole electric \feld induced in the neutral species,\naccording to its polarizability (Langevin scattering); in the\nlatter case the incoming ion takes one or more electrons from\nthe neutral species, which becomes an ambient ion. The fric-\ntion force per unit volume Fiexerted on an ion iis thus the\nsum of Fi;mt+Fi;ce.\nWith the exception of collisions between an ion and a\nneutral of the same species, as in the important case of col-\nlisions of H+ions with H atoms (see Sec. A1), the two pro-\ncesses are well separated in energy. At low collision energies\nelastic scattering dominates, and the friction force is\nFi;mt=ninn\u0016inh\u001bmtviin(un\u0000ui); (1)\nwhereniandnnare the ion and neutral densities, uiand\nunare the ion and neutral velocities, \u0016inis the reduced\nmass of the colliding particles, \u001bmtis the momentum trans-\nfer (hereafter m.t.) cross section, and the brackets denote\nan average over the relative velocity of the colliding particles.\nAt high collision energies (above \u0018102eV), the dom-\ninant contribution to the transfer of momentum is charge\nexchange\nA++ B!A + B+: (2)\nIf the charge exchange rate coe\u000ecient is approximately in-\ndependent of temperature, and there is no net backward-\nforward asymmetry in the scattering process (two conditions\ngenerally well satis\fed), Draine (1986) has shown that the\nfriction force on the ions takes the form\nFi;ce=ninnh\u001bceviinm2\nnun\u0000m2\niui\nmn+mi; (3)\nwhere\u001bceis the charge exchange (hereafter c.e.) cross sec-\ntion, andmn(i)the mass of the neutral (ion).\nThe collisional rate coe\u000ecients h\u001bmtviinandh\u001bceviin\nare often estimated from the values given by Kulsrud & Ce-\nsarsky (1971) or Zweibel & Shull (1982) for H+{ H collisions\n(e.g. D'Angelo et al. 2016; Nava et al. 2016; Brahimi et al.\n2020). The rate coe\u000ecients for collisions between various\nMNRAS 000, 000{000 (0000)3\nTable 1. ISM phases and parameters adopted in this work. Tis the gas temperature, Bthe interstellar magnetic \feld, nthe total gas\ndensity,fthe ionisation fraction, \u001fthe helium fraction and Linjthe injection scale of the background magnetic turbulence.\nT(K)B(\u0016G)n(cm\u00003) neutral ion f \u001f L inj(pc)\nWIM 8000 5 0.35 H, He H+0.6\u00000.9 0\u00000.1\nHe H+1 0.1 50\nWNM 8000 5 0.35 H, He H+7\u000210\u00003\u00005\u000210\u000020\u00000.1 50\nCNM 80 5 35 H, He C+4\u000210\u00004\u000010\u000030.1 1-50\nDiM 50 5 300 H 2, He C+10\u000040.1 1-50\nHIM 1065\u00180:01 - H+1.0 0.0 100\nspecies of ions and neutrals adopted in this study are de-\nscribed in detail in Sec. A1. For elastic collisions, they have\nbeen taken from the compilation by Pinto & Galli (2008); for\ncharge exchange, they have been calculated from the most\nupdated available cross sections.\nIon-neutral collisions are one of the dominant damping\nprocesses for Alfv\u0013 en waves propagating in a partially ionized\nmedium (see Piddington 1956; Kulsrud & Pearce 1969). In\nthe case of elastic ion-neutral collisions, (Eq. 1), the disper-\nsion relation for Alfv\u0013 en waves in this case is\n!(!2\u0000!2\nk) +i\u0017in[(1 +\u000f)!2\u0000\u000f!2\nk] = 0; (4)\nwhere!is the frequency of the wave, !k=kvA;iis the\nwavevector in units of the Alfv\u0013 en speed of the ions\nvA;i=Bp4\u0019mini; (5)\n\u0017inis the ion-neutral collision frequency\n\u0017in=mn\nmi+mnh\u001bmtviinnn; (6)\nand\u000fis the ion-to-neutral mass density ratio\n\u000f=mini\nmnnn: (7)\nNotice that \u000fis a small quantity in the WNM and CNM but\nnot in the WIM2.\nThe dispersion relation Eq. (4) is a cubic equation for\nthe wave frequency !(with real and imaginary parts) as a\nfunction of the real wavenumber !k. Writing!=<(!)\u0000\ni\u0000in\nd, where \u0000in\nd>0 is the ion-neutral damping rate, and\nsubstituting in Eq. (4), one obtains (Zweibel & Shull 1982)\n!2\nk=2\u0000in\nd\n\u0017in\u00002\u0000in\nd[(1 +\u000f)\u0017in\u00002\u0000in\nd]2; (8)\nwhich implies 0 <\u0000in\nd<\u0017in=2. If\u000f\u001c1, then\n\u0000in\nd\u0019!2\nk\u0017in\n2[!2\nk+ (1 +\u000f)2\u00172\nin]: (9)\nAlfv\u0013 en waves resonantly excited by CR protons have fre-\nquency!k\u0019vA;i=rLThus, the frequency is related to the\n2To be precise, the dispersion relation Eq. (4) is valid only if\nthe friction force is proportional to the ion-neutral relative speed\nun\u0000ui, as in the case of momentum transfer by elastic collisions.\nHowever, we use the same relation also in the case of charge ex-\nchange, simply replacing h\u001bmtviinwithh\u001bceviin.kinetic energy of the CR proton E=\rmpc2as\n!k\u0019eBv A;i\nE: (10)\nThe e\u000bective Alfv\u0013 en velocity, vA=<(!)=k, felt by CRs de-\npends on the coupling between ions and neutrals. In general,\nthe following asymptotic behavior can be identi\fed:\n\u000fLow wavenumber, !k\u001c\u0017in:\nat large CR energy ions and neutrals are well coupled; the\ntotal density is n=nH+nHe+niand the Alfv\u0013 en speed\nrelevant for CRs resonant with the waves is\nvA;n=Bp4\u0019\u0016m pn; (11)\nwhere\u0016\u00181:4 is the mean molecular weight, and \u0000in\nd/E\u00002;\n\u000fHigh wavenumber, !k\u001d\u0017in:\nat small CR energy ions and neutrals are weakly coupled and\nion-neutral damping is most e\u000bective. The Alfv\u0013 en speed is\nthe one in the ions, vA;i, and \u0000in\nd\u0018const.\nNotice that if \u000f<1=8 there is a range of wavenumbers\nfor which the waves do not propagate in a partially ionized\nmedium (Zweibel & Shull 1982). This is marked as a shaded\nregion in Fig. 1-2. On the other hand, such non-propagation\nband is found in the absence of CRs propagating in the par-\ntially ionized medium. Recently it has been suggested (Re-\nville et al. 2021) that taking into account the presence of\nCRs may allow for the propagation of waves in that band.\nIntroducing the fraction of ionized gas fand the helium-\nto-hydrogen ratio \u001f,\nf=ni\nnH+ni; \u001f =nHe\nnH+ni; (12)\nEq. (6) becomes\n\u0017in=\u00141\u0000f\n1 + ~mih\u001bmtvii;H+4\u001f\n4 + ~mih\u001bmtvii;He\u0015n\n1 +\u001f:(13)\nwhere ~mi=mi=mp. In the following, the standard value\n\u001f= 0:1 is assumed, but the case \u001f= 0 is also considered for\nillustrative purposes and for a comparison with the results\nof D'Angelo et al. (2016), who neglect the contribution of\nhelium to ion-neutral damping.\n2.1.1 WIM and WNM\nIn this case H is partially ionized and the dominant ion is H+\n( ~mi= 1). Therefore \u000f=nH+=(nH+4nHe) =f=(1\u0000f+4\u001f),\nMNRAS 000, 000{000 (0000)4S. Recchia et al.\ni.e.\u000f= 0:005{0:05 and 0.75{9 for the WNM and the WIM,\nrespectively. The ion-neutral collision frequency is\n\u0017in=\u00141\u0000f\n2h\u001bmtviH+;H+4\u001f\n5h\u001bmtviH+;He\u0015n\n1 +\u001f:(14)\nFig. 1 shows the damping rate for waves resonant with CRs\nof energyE, as a function of the CR energy E. Notice the\nnon-propagation band found in the WNM ( \u000f<1=8).\n2.1.2 CNM and DiM\nIn this case H is neutral and the dominant ion is C+( ~mi=\n12), with fractional abundance nC+=nH\u0019(0:4{1)\u000210\u00003.\nTherefore\u000f= 12nC+=(nH+ 4nHe)\u0019(3{9)\u000210\u00003and\n\u0017in=\u00141\n13h\u001bmtviC+;H+\u001f\n4h\u001bmtviC+;He\u0015n\n1 +\u001f: (15)\nFig. 2 shows the damping rate for waves resonant with CRs\nof energyE, as a function of the CR energy E. Also in this\ncase non-propagation regions are found.\n2.2 Wave cascade and turbulent damping\nThe turbulent damping (FG) of self-generated Alfv\u0013 en waves\nis due to their interaction with a pre-existing background\nturbulence. Such turbulence may be injected by astrophys-\nical sources (see e.g. Mac Low & Klessen 2004) with a tur-\nbulent velocity vturband on scales, Linj, much larger than\nthe CR Larmor radius. For waves in resonance with particles\nwith a given energy E, the damping rate, that accounts for\nthe anisotropy of the turbulent cascade, has been derived\nby Farmer & Goldreich (2004); Yan & Lazarian (2004) and\nreads\n\u0000FG\nd=\u0012v3\nturb=Linj\nrLvA\u00131=2\n; (16)\nwherevAis the e\u000bective Alfv\u0013 en speed felt by CRs, as de\fned\nin Sec. 2.1. We take the turbulence as trans-Alfv\u0013 enic at the\ninjection scale, namely vturb=vA;n(at large scales waves\nare in the low wavenumber regime, where ions and neutrals\nare well coupled, as illustrated in Sec. 2.1). This is the\nlikely situation if the turbulence is mainly injected by old\nSNRs, with a forward shock becoming trans-sonic and trans-\nAlfv\u0013 enic. The FG damping rate is shown in Fig. 1 for the\nWIM and WNM.\nIn highly neutral media, such as the the WNM, CNM\nand DiM, the background turbulence responsible for the FG\ndamping can be damped by ion-neutral friction at a scale,\nlmin= 1=kmin(Xu et al. 2015, 2016; Lazarian 2016; Brahimi\net al. 2020). Correspondingly, there is a minimum particle\nenergy,Emin, such that rL(Emin) =lmin, below which the\nFG damping cannot a\u000bect the self-generated Alfv\u0013 en waves\n(Brahimi et al. 2020):\n1\nlmin=L1=2\ninj\u00122\u000f\u0017in\nvA;n\u00133=2r\n1 +vA;n\n2\u000f\u0017inLinj: (17)\nIn Fig. 1 the FG damping rate for the WNM is truncated\natEmin. In the WIM the cascade rate is found to be always\nlarger than the ion-neutral damping rate and there is no\nEmin.2.3 Non-linear Landau damping\nThe non-linear Landau (NLL) damping is caused by the in-\nteraction between the beat of two Alfv\u0013 en waves and the ther-\nmal (at temperature T) ions in the background medium. The\ndamping rate for resonant waves is given by (Kulsrud 1978;\nWiener et al. 2013)\n\u0000NLL\nd=1\n2s\n\u0019\n2\u0012kBT\nmp\u0013I(kres)\nrL; (18)\nwherekBis the Boltzmann constant and I(kres) is the wave\nenergy density (see Sec. 3 below for the de\fnition) at the\nresonant wavenumber kres= 1=rL.\n3 COSMIC RAY PROPAGATION IN THE PROXIMITY\nOF SNRS\nWe consider the escape of CRs from a SNR and the sub-\nsequent propagation in the source proximity. The propaga-\ntion region is assumed to be embedded in a turbulent mag-\nnetic \feld, with a large scale ordered component of strength\nB0. CRs are scattered by Alfv\u0013 en waves, which constitute a\nturbulent magnetic \feld background of relative amplitude\n\u000eB=B 0, wherekis the wavenumber. We only consider waves\nthat propagate along the uniform background \feld B0. In\nthe limit of \u000eB=B 0\u001c1, which is the one relevant for the\ncases treated in this paper, the CR di\u000busion along \feld lines\ncan be treated in the quasi-linear regime, with a di\u000busion\ncoe\u000ecient given by Berezinskii et al. (1990) and Kulsrud\n(2005)\nD(E) =4\u0019cr L(E)\n3I(kres)\f\f\f\f\nkres=1=rL=DB(E)\nI(kres); (19)\nwherecis the speed of light, I(kres) =\u000eB(kres)2=B2\n0is the\nwave energy density calculated at the resonant wavenumber\nkres= 1=rL, andDB(E) = (4\u0019=3)crL(E) is the Bohm di\u000bu-\nsion coe\u000ecient. We also assume that the dominant source of\nAlfv\u0013 enic turbulence is produced by the CR resonant stream-\ning instability.\nIn our model we adopt the \rux tube approximation for\nthe CR transport along B0(see, e.g., Ptuskin et al. 2008),\nand we neglect the di\u000busion across \feld lines, which is sup-\npressed in the \u000eB=B 0\u001c1 regime (see e.g. Drury 1983;\nCasse et al. 2002). Thus, we do not address the perpen-\ndicular evolution of the \rux tube (see, e.g., Nava & Gabici\n2013) and any possible CR feedback on it and, in general,\non the ISM dynamics (see, e.g., Schroer et al. 2020). Such\none-dimensional model for the CR propagation is applicable\nfor distances from the source below the coherence length, Lc,\nof the background magnetic turbulence, i.e. the scale below\nwhich the magnetic \rux tube is roughly preserved (see, e.g.,\nCasse et al. 2002).\nWhen particles di\u000buse away from the source at distances\nlarger than Lc, di\u000busion becomes 3-D and the CR density\ndrops quickly. In the Galactic disk, Lcis estimated observa-\ntionally and may range from few pc to \u0019100 pc, depending\non the ISM phase (see, e.g., Nava & Gabici 2013, and refer-\nences therein).\nWe follow the approach proposed by Nava et al. (2016,\n2019), and we determine: ( i) the escape time of CRs of a\nMNRAS 000, 000{000 (0000)5\nFigure 1. Damping rates \u0000in\ndand \u0000FG\nd(ion-neutral and turbulent) of Alfv\u0013 en waves in the WNM ( left-hand panel ) and WIM ( right-hand\npanel ) vs. CR energy E. Di\u000berent colors are used for di\u000berent values of the hydrogen ionization fraction f. Unless stated otherwise, a\nstandard\u001f= 0:1 He abundance is assumed. The considered parameters for the WNM and WIM are given in Table 1. In the left-hand\npanel the dotted lines refers to ion-neutral damping, while the dashed lines to the FG damping. The last are truncated to the minimum\nenergy,Emin, below which the background turbulence is damped by ion-neutral friction before reaching the scale relevant for damping\nself-generated waves resonant with particle of energy <E min(see Sec. 2.2). The shaded regions represent the range of non-propagation\nof Alfv\u0013 en waves (see Sec. 2.1). In the right-hand panel, the solid lines refer to ion-neutral damping, while the thin dotted lines refer to\nthe case where the contribution to damping from He is neglected ( \u001f= 0). The dashed lines refer to the FG damping, which in the case\nof WIM is found to depend little on the ionization fraction f.\nFigure 2. Ion-neutral damping rate \u0000in\ndof Alfv\u0013 en waves in the CNM (left-hand panel) and DiM (right-hand panel) as a function of the\nCR energy E. Di\u000berent colors are used for di\u000berent values of the C+abundance. The parameters adopted for the CNM and DiM are\ngiven in Table 1. The shaded regions represent the range of non-propagation of Alfv\u0013 en waves (see Sec. 2.1).\ngiven energy from the remnant; ( ii) the time dependent evo-\nlution of the CR cloud and of the self-generated di\u000busion\ncoe\u000ecient after escape; ( iii) the time spent by CRs in the\nsource vicinity and the corresponding grammage. We focus\non the warm ionized and warm neutral phases of the ISM.\nAs shown in the following sections, the results depend sig-\nni\fcantly on the considered ISM phase, and in particular on\nthe amount and type of neutral and ionized atoms in the\nbackground medium.3.1 Transport equations and initial/boundary conditions\nThe coupled CR and wave transport equations read (Nava\net al. 2016, 2019)\n@PCR\n@t+vA@PCR\n@z=@\n@z\u0012DB\nI@PCR\n@z\u0013\n(20)\nand\n@I\n@t+vA@I\n@z= 2(\u0000 CR\u0000\u0000d)I+Q: (21)\nMNRAS 000, 000{000 (0000)6S. Recchia et al.\nHerevAis an e\u000bective Alfv\u0013 en velocity that takes into ac-\ncount the coupling between ions and neutrals (see Sec. 2.1),\nwhilePCRis the partial pressure of CRs of momentum p\nnormalized to the magnetic \feld pressure\nPCR=4\u0019\n3cp4f(p)\nB2\n0=8\u0019: (22)\nThe term 2 \u0000 CRIgives the wave growth due to the CR\nstreaming instability and can be expressed as\n2\u0000CRI=\u0000vA@PCR\n@z: (23)\nThe term \u0000 dencompasses the relevant wave damping rates,\nwhich are described in detail in Sec. 2. The term Qrepre-\nsents the possible injection of turbulence from an external\nsource, other than the CR streaming. Here it is taken as\nQ= 2\u0000 dI0, whereI0is a parameter such that typical val-\nues of the Galactic CR di\u000busion coe\u000ecient are recovered at\nlarge distances from the source (see, e.g., Strong et al. 2007).\nIn this way, when the streaming instability is not relevant,\ndi\u000busion is regulated by the background turbulence. Notice\nthat in these equations we neglected adiabatic losses that\narises when Alfv\u0013 en speed varies in space (see, e.g., Brahimi\net al. 2020), since we are considering a homogeneous ISM\naround the SNR.\nThe coordinate zis taken along B0andz= 0 refers\nto the centre of the CR source. The equations are solved\nnumerically with a \fnite di\u000berence explicit method, using\nthe initial conditions\nPCR=\u001aP0\nCR ifz<R esc(E);\n0 ifz>R esc(E);\nand\nI=I0 everywhere: (24)\nHereResc(E) is the size of the region \flled by CRs at the\ntime of escape, and P0\nCRis the initial CR pressure inside this\nregion. The method used to determine the escape radius and\ntime for particles of energy Eis described in detail in Nava\net al. (2016, 2019). As for the initial condition for the waves,\nit is also possible to choose I\u001dI0forz<R esc(E) in order\nto mimic Bohm di\u000busion inside the source, as proposed by\nMalkov et al. (2013). However, as discussed by Nava et al.\n(2019), di\u000berent choices of Iinside the source have little im-\npact on the \fnal solution. As boundary conditions we impose\na symmetric CR distribution at z= 0, and\nPCR= 0; I =I0 atz&Lc: (25)\nThe 1-D model used in this paper is valid only up to a dis-\ntance from the source given by the coherence length Lcof\nthe magnetic \feld. At larger distances the di\u000busion becomes\n3-D and the CR density quickly drops, a behavior that can\nbe described in terms of a free escape boundary at Lc. The\nvalue ofLcis constrained from observations to be .100 pc\nbut its value is matter of debate and is likely phase- and\nposition-dependent in our galaxy. We take the free escape\nboundary at z&Lc, and we check that this assumption is\nnot signi\fcantly a\u000becting the results, for instance changing\nLcto 10Lc.4 RESULTS\nIn what follows we discuss the release of CRs of energy E\nfrom a SNR, giving an estimate of the age and radius of the\nsource at the moment of escape. Moreover, we investigate the\npropagation of runaway CRs in the source proximity, with\nparticular focus on the CR spectrum and self-generated dif-\nfusion coe\u000ecient. Finally, we estimate the residence time of\nCRs in the source proximity and we infer general implica-\ntions for the grammage accumulated in that region.\nWe treat these aspects in the cases of a WIM or a WNM\nsurrounding the remnant, with special emphasis on the role\nplayed by the ion-neutral damping of Alfv\u0013 en waves.\nIn our calculations we assume that: ( i) runaway CRs\nhave a total energy ECR= 1050erg, a power-law spectrum\nin energy between 1 GeV{5 PeV with spectral index \r=\n\u00002:0; and ( ii) the typical ISM di\u000busion coe\u000ecient is D(E) =\nD0(E=10 GeV)0:5withD0= 1028cm2s\u00001(see, e.g., Strong\net al. 2007).\n4.1 Escape of cosmic rays: source age and radius\nThe age and radius of escape of CRs of energy Eis estimated\nas follows (Nava et al. 2016, 2019): we de\fne the half-time,\nt1=2(E;R) of a CR cloud as the time after which half of the\nCRs of a given energy, initially con\fned within a region of\nsizeR, have escaped that region. The evolution of such CR\ncloud is studied by solving Eq. (20) and Eq. (21) with initial\nconditions given by Eq. (24) and boundary conditions given\nby Eq. (25).\nThe radius of a SNR expanding in a homogeneous\nmedium of density ncan be estimated as (Truelove & McKee\n1999)\nRSNR(t) = 0:5\u0012E51\nn\u00131=5\"\n1\u00000:09M5=6\nej\f\nE1=2\n51n1=3tkyr#2=5\nt2=5\nkyrpc;\n(26)\nwhereE51is the supernova explosion energy in units of\n1051erg,nis the total density of the ambient ISM in cm\u00003,\nMej\fis the mass of the supernova ejecta in solar masses,\nandtkyris the SNR age in kyr. This equation is valid\nin the adiabatic phase of the expansion, which starts af-\nter\u001986M5=6\nej\fE\u00001=2\n51n\u00001=3yr. Here we take E51= 1 and\nMej\f= 1:4, while the gas density depends on the medium\nand is reported in Table 1. At earlier times an approx-\nimate expression for the free expansion phase has to be\nused (Chevalier 1982). The adiabatic phase stops at roughly\n\u00191:4\u0002104E3=14\n51n\u00004=7yr, when the radiative phase starts\n(Cio\u000e et al. 1988). At this epoch we also assume that the\nacceleration of CRs becomes ine\u000bective and that all CRs are\ninstantly released (the validity of the assumption of instant\nrelease may depend on the conditions in the shock precursor,\nas discussed by Brahimi et al. 2020).\nThe escape time, tesc(E), of particles with energy Eis\nestimated as the time such that t1=2(E;R SNR) equals the\nage of a SNR of radius RSNR. This is the timescale over\nwhich waves can grow. The dependence of t1=2(E;R) on the\ninitial radius Rat \fxed energy E, and varying the back-\nground di\u000busion coe\u000ecient and the CR spectral index, has\nbeen extensively explored by Nava et al. (2016, 2019), and\nwe refer the reader to these works for a detailed discussion.\nHere we brie\ry summarize the most relevant points. At small\nMNRAS 000, 000{000 (0000)7\nRthe CR gradient is large and the ampli\fcation of waves is\nvery e\u000bective. In this regime, the NLL damping, that scales\nwith the wave energy density I(k), can play an important\nrole and may dominate over ion-neutral and FG damping\nat small enough radii. At intermediate R, ion-neutral damp-\ning dominates in most cases. In fact, as shown in Fig. 1,\nthe ion-neutral damping rate is larger than the FG rate at\nleast up to particle energies of \u001810 TeV, with the only ex-\nception of a WIM with fully ionized hydrogen and \u001810%\nof neutral helium. At large R, the CR gradient is reduced\nand the streaming instability is less e\u000bective. In this case\nthe expansion of the CR bubble is determined by the back-\nground turbulence and the test particle limit is recovered,\nwitht1=2/R2=D0.\nFig. 3 and Fig. 4 show the SNR age and radius at the\ntime of escape of CRs, as a function of the CR energy,in the\ncase of a WNM and WIM respectively. We con\frm the \fnd-\nings by Nava et al. (2016, 2019) and the qualitative results\nthat particles with higher energy escape earlier than the low\nenergy particles (see, e.g., Gabici 2011).\nIn the case of the WIM, Rescandtesctend to decrease\nwith a larger neutral density. The e\u000bect of ion-neutral damp-\ning is especially visible in the energy range \u00181{10 TeV,\nwhere the ion-neutral damping rate starts to decrease from\na roughly constant value at di\u000berent particle energies, de-\npending on the value of the ionized hydrogen fraction fand\non the helium-to-hydrogen ratio \u001f, as shown in Fig. 1. The\ndrop of \u0000in\ndcorresponds to an increase of tesc, and therefore\nto a better CR con\fnement.\nThe situation is more subtle in the case of the WNM.\nIn fact, here the con\fnement appears to be slightly more\ne\u000bective for a smaller value of f, a result opposite to what\nhappens in the WIM. This can be explained taking into ac-\ncount that, while the \u0000in\ndis practically the same at energies\nbelow\u001810 TeV for f\u00187\u000210\u00003\u00005\u000210\u00002, the e\u000bective\nAlfv\u0013 en speed felt by CRs, which at these energies is roughly\nthat of ions, as illustrated in Sec. 2, is a factor of \u00182{3 larger\nforf= 7\u000210\u00003. This re\rects on an increase by the same\namount of the FG damping rate (above the cut-o\u000b energy,\nsee Sec. 2.2), as shown in Fig. 1, but also of the growth rate\nterm, which is proportional to vA. On the other hand, at\nenergies in the range \u001810 GeV{1 TeV, the FG damping is\nalways subdominant compared to the ion-neutral damping,\nand the net e\u000bect of this change of vAis a slightly better\ncon\fnement at smaller fas a result of an enhanced wave\ngrowth rate.\nAlso in the case of a WIM the e\u000bective vAis a bit larger\nat smallerfbelow\u00181 TeV, but here the increase is only\nby a factor of\u00181:2 and cannot overcome the e\u000bect of ion-\nneutral damping.\n4.2 Cosmic ray spectra and di\u000busion coe\u000ecient\nThe spectrum of runaway CRs and the corresponding self-\ngenerated turbulence depend signi\fcantly on the distance\nfrom the source and on the time. In Fig. 5 and Fig. 6 we show\nthe spectrum of CRs that have already escaped to a distance\nof 50 pc from the centre of the remnant at di\u000berent times,\nfromt= 6\u0002103yr tot= 105yr, and for the WNM ( f=\n0:05,\u001f= 0:1) and the WIM ( f= 0:9,\u001f= 0:1), respectively.\nTo remove the e\u000bect of the simple 1-D geometry that we are\nadopting we multiply PCRbyR2\nesc(E)=R2\nSNR(t). For eachcase we also show the ratio between the self-generated and\nbackground di\u000busion coe\u000ecients.\nIn the same \fgures we also show the spectrum and di\u000bu-\nsion coe\u000ecient at the position of the shock of CRs that have\nescaped at earlier times (and are considered as decoupled\nfrom the accelerator) as well as the spectrum of particle still\ncon\fned in the accelerator, at di\u000berent ages of the remnant\nfromt= 6\u0002103yr tot= 3\u0002104yr. The spectrum of the\ncon\fned particles is estimated by assuming that the remnant\nprovides\u00191050erg in CRs with a /E\u00002spectrum that ex-\ntends from\u00181 GeV to\u00185 PeV, with an exponential cuto\u000b\nat the energy of the particles that escape at the considered\nage. The SNR is assumed to stop accelerating particles at\nthe onset of the radiative phase, which for the typical den-\nsity of the WIM and WNM takes place at trad\u00193\u0002104yr.\nThe corresponding shock radius is Rrad\u001922 pc. In all cases,\nat each time and location, the spectrum exhibit a sharp rise\nat low energies and a peak followed by a nearly power-law\nlike behaviour at higher energies.\nSince high energy particles escape earlier and di\u000buse\nfaster, they reach a given distance at earlier times, as shown\nfor the spectra at 50 pc, where the peak moves at lower\nenergies with time. At energies lower than the peak, particles\nhave not yet reached that position, giving the sharp rise. At\nlarger energies, particles have di\u000bused over a bigger distance,\ngiving a spectrum steeper than the spectrum released from\nthe SNR, here assumed to be /E\u00002.\nA similar behaviour is observed with the spectra at the\nshock. Here the position at which the spectra are shown\nvaries with time, since it is given by the shock radius at\na given age. This radius is always <50 pc for the cases\nillustrated here. Comparing the spectra at the shock and at\n50 pc for a given age, it is evident that the peak is at lower\nenergy in the former case. This again is the result of an\nearlier escape and faster di\u000busion of high energy particles.\nAs for theD=D 0ratio as a function of energy, it is possi-\nble to infer from the plots that, at a given time, the di\u000busion\ncoe\u000ecient is equal to its background value at low energies,\nwhere particles have not yet escaped or not yet reached the\nposition where Dis calculated, and at high energies, where\nthe turbulence produced by such particle is being damped.\nAt intermediate energies, and roughly corresponding to the\npeak in the spectrum, the di\u000busion coe\u000ecient is suppressed\ncompared to D0due to CR streaming instability. The ener-\ngies at which the suppression is evident move to lower values\nwith time.\nA more complex situation can be observed in some\ncases, for instance at t= 104yr at the shock and in the\nTeV range, for the WIM. Here Dstarts to deviate from D0\nat\u001850 GeV, then, above \u0018500 GeV,Dbegins to ap-\nproach again D0, but above\u00183 TeV the level of turbulence\nincreases again, with the result of a suppression of D. This\nis due to the energy dependence of the escape radius, which\nbecomes steeper above few TeV. This results in the fact that,\nabove that energy, the escape radius becomes small enough\nas to e\u000bectively excite the streaming instability. In fact the\ndensity of runaway CRs is /1=R2\nesc. The more e\u000ecient self-\ncon\fnement at this high energies also re\rects in a hardening\nof the spectrum (see also Nava et al. 2019 for a discussion).\nMNRAS 000, 000{000 (0000)8S. Recchia et al.\n102103104105\n101102103104 1 10 100\n-- Resc- tesctesc [yr]\nResc [pc]\nE [GeV]WNM\nf=0.007\nf=0.05\n104105106\n10110210310-210-1tres [yr]\nX [g/cm2]\nE [GeV]WNM\nf=0.007\nf=0.05 \nFigure 3. Case of WNM. Left panel: SNR age and radius at the time of escape of CRs, as a function of the CR energy. Right panel:\nResidence time of CRs in a region of \u0018100 pc around the source, as a function of the CR energy, and the corresponding grammage.\nDi\u000berent colors are used for di\u000berent values of the hydrogen ionization fraction f, while the helium-to-hydrogen ratio \u001f= 0:1. The\nparameters of the WNM are listed in Table 1.\n103104105\n101102103104 1 10 100\n-- Resc\n- tesctesc [yr]\nResc [pc]\nE [GeV]WIM\nf=0.6\nf=0.9\nf=0.9, χ =0.0\n104105106\n10110210310-210-1100tres [yr]\nX [g/cm2]\nE [GeV]WIM\nf=0.6\nf=0.9\nf=1\nf=1, χ=0\nf=1, χ=0, 20 pc\nFigure 4. Case of WIM. Left panel: SNR age and radius at the time of escape of CRs, as a function of the CR energy. Right panel:\nResidence time of CRs in a region of \u0018100 pc around the source, as a function of the CR energy, and the corresponding grammage. The\ncyan curve is computed by assuming that particles of all energies escape at SNR radius of 20 pc, as assumed by D'Angelo et al. (2016).\nDi\u000berent colors are used for di\u000berent values of the hydrogen ionization fraction fand helium-to-hydrogen ratio \u001f. The parameters of the\nWIM are listed in Table 1.\n4.3 Residence time and grammage\nThe grammage accumulated by CRs close to their sources,\nand its relevance compared to the grammage accumulated\nwhile di\u000busing in the whole Galaxy, is related to the resi-\ndence time in the proximity of the source. A formal deter-\nmination of the grammage should be done by solving the CR\ntransport equation for nuclei with the inclusion of spallation\ncontributions (see, e.g., Berezinskii et al. 1990; Ptuskin &\nSoutoul 1990).\nHere we adopt the following approach: we assume that\nN0particles of energy Eare instantly injected by a source\nlocated in a region of constant gas density \u001a=\u0016mpnand are\nsubject to free escape at a boundary located at a distance\nLcfrom the source. At a given time t, the total number\nof particles that are still in the region is Nin(t) while the\nnumber of escaped particles is Nesc=N0\u0000Nin(t). Particles\nthat cross the boundary at time thave accumulated the\ngrammageX(t) =\u001actand the average grammage gained byall escaped particles from t= 0 tot=1is given by\nhXi=1\nN0Z1\n0\u001actdNesc\ndtdt (27)\n=\u001ac\nN0\u0014Z1\n0Nin(t)dt\u0000lim\nt\u0000!1(tNin(t))\u0015\n=\u001ac\u001c res:\nThe residence time \u001cresis given by\n\u001cres=1\nN0\u0014Z1\n0Nin(t)dt\u0000lim\nt\u0000!1(tNin(t))\u0015\n: (28)\nThe number of particles with a given energy contained in\nthe region at a given time can be calculated from the CR\npressure (see Sec. 3):\nNin(t) =N0\nRescP0\nCRZLc\n0PCR(z;t;E )dz; (29)\nwhere we took into account that CRs on energy Eare ini-\ntially released from a region of size Resc(E).\nMNRAS 000, 000{000 (0000)9\nWNM \n10-210-1100\n10110210310450 pc6e+3\n1e+4 3e+4\n6e+4\n1e+5f(E)E2 x (Resc/Rsh)2 [eV/cm3]\nE [GeV]10-310-210-1100101102\n101102103104shock\n6e+3\n1e+43e+4f(E)E2 x (Resc/Rsh)2 [eV/cm3]\nE [GeV]\n 1\n101102103104D/D0\nE [GeV] 1\n101102103104D/D0\nE [GeV]\nFigure 5. Case of WNM ( f= 0:05,\u001f= 0:1): CR spectrum and ratio D=D 0as a function of energy, where D0(E) =\n1028(E=10 GeV)0:5cm2s\u00001. The di\u000berent colors refer to di\u000berent ages in yr (as marked). The left panels refer to a location at 50 pc\nfrom the center of the SNR, while the right panels refer to the shock position. In this case also the spectrum of CRs still con\fned in the\naccelerator is shown (dashed lines).\nFig. 3 and 4 show the residence time of CRs in a re-\ngion of\u0018100 pc around the remnant, for the WNM and\nthe WIM, respectively, and for di\u000berent values of the neu-\ntral fraction, and the corresponding grammage. Even in the\n(not very plausible) case of a fully ionized WIM ( f= 1:0,\n\u001f= 0:0), the grammage is nearly two orders of magnitudes\nsmaller than that accumulated in the disk (see e.g. Jones\net al. 2001; Gabici et al. 2019). This is due to the action of\nthe ion-neutral and FG damping. The presence of a \u001810%\nof neutral helium (which is less e\u000bective at wave damping\ncompared to hydrogen) noticeably reduces the CR residence\ntime compared to the case of a fully ionized background\nmedium, above few tens GeV.\nThese results are at odds with what previously sug-\ngested by D'Angelo et al. (2016), namely that ion-neutral\ndamping is totally negligible in the case of only neutral he-\nlium, since the c.e. cross section is very small. However, as\nshown in Sec. 2, not only c.e. but also m.t. has to be taken\ninto account in the ion-neutral damping. In fact, D'Angelo\net al. (2016) \fnd a grammage nearly a factor of ten larger\nthan what we \fnd. This is in part due to their assumption\nof ion-neutral damping of waves with neutral helium. In ad-dition, they assume a 20% acceleration e\u000eciency, while we\nassume 10%, which enhances the e\u000bect of streaming insta-\nbility, and they do not include the FG damping. Finally,\nD'Angelo et al. (2016) assume that CRs at all energies are\nreleased when the SNR radius is 20 pc, a scenario implying\na smaller release radius at low energies compared to our cal-\nculation, which translates in an enhancement of the stream-\ning instability (and the CR con\fnement) at energies below\n\u00191 TeV, as shown in Fig. 4.\nAny addition of neutral hydrogen compared to the\ncase of only neutral helium further reduces the con\fnement\ntime. Correspondingly, the CR grammage accumulated in\nthe source proximity results to be well below that inferred\nfrom observations. A similar result was found by Nava et al.\n(2019) for the HIM, where the residence time tends to be\nlarger than in a partially ionized medium, but the ISM den-\nsity is lower (\u00180:01 cm\u00003).\n5 CONCLUSIONS\nWe followed the method and the setup proposed by Nava\net al. (2016, 2019) to investigated the escape of CRs from\nMNRAS 000, 000{000 (0000)10 S. Recchia et al.\nWIM \n10-310-210-1100\n10110210310450 pc\n6e+31e+4\n3e+4\n6e+4\n1e+5f(E)E2 x (Resc/Rsh)2 [eV/cm3]\nE [GeV]10-310-210-1100101102\n101102103104shock\n6e+31e+43e+4f(E)E2 x (Resc/Rsh)2 [eV/cm3]\nE [GeV]\n 1\n101102103104D/D0\nE [GeV] 1\n101102103104D/D0\nE [GeV]\nFigure 6. Case of WIM ( f= 0:9,\u001f= 0:1): CR spectrum and ratio D=D 0as a function of energy, where D0(E) =\n1028(E=10 GeV)0:5cm2s\u00001. The di\u000berent colors refer to di\u000berent ages in yr (as marked). The left panels refer to a location at 50 pc\nfrom the center of the SNR, while the right panels refer to the shock position. In this case also the spectrum of CRs still con\fned in the\naccelerator is shown (dashed lines).\nSNRs embedded in a WNM and WIM, and the CR self con-\n\fnement in the source proximity. Our main objective was\nto determine whether, in realistic situations, the grammage\naccumulated by CRs in the source region could become com-\nparable to that inferred from observations. Ifthiswasthe\ncase, In this case, the standard picture in which CR secon-\ndaries are produced during the whole time spent by cosmic\nrays throughout the disk of the Galaxy, should be profoundly\nrevisited.\nWe con\frm the results found by Nava et al. (2016, 2019)\nthat CRs escaping from SNRs drive the excitation of Alfv\u0013 en\nwaves through the resonant CR streaming instability, which\nresults in a suppression of the di\u000busion coe\u000ecient and in the\nCR self-con\fnement in the source region. The SNR radius\nat which CRs of a given energy leave the source, Resc(E),\nresults to be a decreasing function of the energy, and regu-\nlates the CR streaming instability through a dilution factor\n/1=R2\nesc.\nWe found that the growth of self-generated Alfv\u0013 en\nwaves, and consequently the residence time of CRs in the\nsource region, is signi\fcantly limited by several damping\nprocesses, especially by the FG and ion-neutral damping.In particular, for the ion-neutral damping of Alfv\u0013 en waves\nwe have used up-to-date damping coe\u000ecients, based on ac-\ncurate experimental or theoretical determinations of the mo-\nmentum transfer and charge exchange cross sections between\nions and neutrals of di\u000berent species.\nIn the 1-D geometry adopted in our calculation a sup-\npression of the di\u000busion coe\u000ecient is found within a distance\nform the source of the order of the magnetic \feld coherence\nlengthLc\u001950\u0000100 pc, for a timescale that can be as large\nas\u0018105yr at\u001810 GeV. A smaller value for Lcwould make\nthe CR propagation to become 3-D closer to the remnant,\nthus reducing the residence time compared to our results.\nWe conclude that ion-neutral damping strongly limits\nthe CR grammage that can be accumulated in the source\nregion. Under the conditions typically met in the WIM and\nWNM of the Galactic disk, the CR source grammage is\nfound to be negligible compared to that inferred form ob-\nservation. A similar result was found for the HIM by Nava\net al. (2019), where the CR residence time is typically larger\nthan in a partially ionized medium but the ISM density is\nmore than a factor ten smaller. This makes alternative sce-\nnarios for the interpretation of quantities such as the B/C\nMNRAS 000, 000{000 (0000)11\nratio, in which an important contribution to the production\nof secondaries comes from the source region, less attractive.\nOur results on the residence time (and grammage) may\nchange with the inclusion of the contribution of other sources\nof turbulence to the CR con\fnement. In particular, it has\nrecently been suggested that another another CR-induced\ninstability, the so called non-resonant streaming instability ,\nwhich is mediated by the CR current and plays a crucial role\nin the acceleration of CRs (Bell 2004), may signi\fcantly en-\nhance the CR self-con\fnement in the source region above\n\u0018TeV energies (Schroer et al. 2020). Further investigations\nare thus needed in order to \frmly establish the importance\namount of CR grammage accumulated in the vicinity of\nsources by very-high energy particles.\nREFERENCES\nAharonian F. A., Atoyan A. 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The case of H+{ H collisions is special because the\nproton elastically scattered by the H atom is indistinguish-\nable from the recoiling proton produced by charge exchange\n(Krsti\u0013 c & Schultz 1999; Glassgold et al. 2005; Schultz et al.\n2008, 2016). Therefore, the elastic scattering and the charge\nexchange channels for collisions of H+with H, as for any col-\nlision between ions and their parent gas, cannot in general\nbe separated. Only at collision energies Ecm&1 eV can the\nMNRAS 000, 000{000 (0000)12 S. Recchia et al.\nforward elastic and the backward charge exchange be ap-\nproximately separated. In this limit, \u001bmt\u00192\u001bce(Dalgarno\n1958), a frequently used approximation (see e.g. Kulsrud &\nCesarsky 1971).\nFig. A1 shows the m.t. cross section for H+{ H colli-\nsions (Krsti\u0013 c & Schultz 1999; Glassgold et al. 2005), and\nthe corresponding rate coe\u000ecient. The \fgure also shows the\nm.t. cross sections and rate coe\u000ecients derived from either\ncharge exchange and elastic scattering in the distinguishable\nparticle approach (Schultz et al. 2008, 2016), and the exper-\nimental determination of the m. t. cross section obtained\nby Brennan & Morrow (1971) by measuring the attenuation\nof Alfv\u0013 en waves propagating in a partially-ionized hydrogen\nplasma atEcm\u00195 eV. Also shown in Fig. A1 are the rate co-\ne\u000ecients used by Kulsrud & Cesarsky (1971) and Zweibel &\nShull (1982), and frequently adopted also for other species\nby scaling the collision rate with the ratio mn=miof the\nneutral and ion masses (see e.g. (Zweibel & Shull 1982)). As\nshown in the following, this approximation is not generally\ncorrect, since the rate coe\u000ecients in the case of other species\nexhibit a di\u000berent dependence on the temperature compared\nto the H+{ H collisions case.\nFig. A2 shows the m.t. (Krsti\u0013 c & Schultz 1999, 2006)\nand c.e. (Loreau et al. 2018) cross sections for H+{ He col-\nlisions, and the corresponding rate coe\u000ecients. Charge ex-\nchange contributes to the transfer of momentum above col-\nlision energies\u0018103eV, corresponding to relative velocities\nof\u0018500 km s\u00001, much larger than typical thermal or Alfv\u0013 en\nspeeds in the WNM and WIM.\nA2 Collisions of C+with H and He atoms\nIn the CNM and DiM, ion-neutral damping is dominated by\nthe collisions between neutral hydrogen and ionized carbon.\nFig. A3 shows the cross sections and reaction rates for m.t.\nand c.e. in the case of collisions of C+ions with H atoms.\nFor collisions of C+ions with He atoms, no theoretical or\nexperimental are available. The m.t. transfer rate coe\u000ecient,\naccording to the Langevin theory, is h\u001bmtviC+;He= 1:33\u0002\n10\u00009cm3s\u00001(Pinto & Galli 2008). As in the case of H+{\nHe collisions, the large di\u000berence with the rate coe\u000ecients\nused in this work resides in the assumption that the rate\ncoe\u000ecient is the same for all species.\nFigure A1. Top panel: cross sections for collisions of H+with H\nvs. collision energy in the center-of-mass frame Ecm. Momentum\ntransfer cross section, from Krsti\u0013 c & Schultz (1999), Glassgold\net al. (2005), and Schultz et al. (2008); charge exchange cross\nsection, from Hodges & Breig (1991), and Schultz et al. (2008);\ndashed line: Langevin m.t. cross section. Experimental data for\nthe c.e. cross section: Newman et al. (1982) ( \flled triangles ). The\nempty circle shows the value of the m.t. cross section measured\nby Brennan & Morrow (1971) Bottom panel: Collisional rate co-\ne\u000ecient for momentum transfer and charge exchange. The \flled\nand empty circles show the rate coe\u000ecient adopted by Kulsrud\n& Cesarsky (1971) and Zweibel & Shull (1982), respectively.\nMNRAS 000, 000{000 (0000)13\nFigure A2. Top panel: cross sections for collisions of H+with\nHe vs. collision energy in the center-of-mass frame Ecm. Momen-\ntum transfer cross section, from Krsti\u0013 c & Schultz (1999, 2006);\nLangevin m.t. cross section ( dashed line ); charge exchange cross\nsection, from Loreau et al. (2018); and ionization cross section,\nfrom Shah & Gilbody (1985). Experimental data for the c.e. cross\nsection: Martin et al. (1981) ( empty triangles ), Shah & Gilbody\n(1985) ( \flled triangles ), Shah et al. (1989) ( empty squares ), and\nKusakabe et al. (2011) ( \flled squares ). Experimental data for\nthe ionization cross section: Shah & Gilbody (1985) ( empty cir-\ncles).Bottom panel: Collisional rate coe\u000ecient for Langevin m.t.\n(dashed line ), m.t. and c.e. ( solid lines ).\nFigure A3. Top panel: cross sections for collisions of C+with H\nvs. collision energy in the center-of-mass frame Ecm: m.t. cross\nsection computed by Flower & Pineau des Forets (1995) ( \flled\ntriangles ); \ft, from Pinto & Galli (2008) ( solid line ); Langevin\nm.t. cross section ( dashed line ). Experimental data for the c.e.\ncross section: Phaneuf et al. (1978) ( empty circles ), Go\u000be et al.\n(1979) ( empty squares ) and Stancil et al. (1998) ( \flled triangles );\nc.e. cross section recommended by Stancil et al. (1998) ( solid line ).\nBottom panel: Collisional rate coe\u000ecients: Langevin m.t. ( dashed\nline), m.t. and c.e.( solid lines ).\nMNRAS 000, 000{000 (0000)"    },    {        "title": "2209.07908v1.Pseudo_PT_symmetric_Dirac_equation___effect_of_a_new_mean_spin_angular_momentum_operator_on_Gilbert_damping.pdf",        "content": "arXiv:2209.07908v1  [quant-ph]  16 Sep 2022Pseudo- PTsymmetric Dirac equation : effect of a new mean\nspin angular momentum operator on Gilbert damping\nY. Bouguerra, S. Mehani, K. Bechane and M. Maamache\nLaboratoire de Physique Quantique et Syst` emes Dynamiques ,\nFacult´ e des Sciences, Universit´ e Ferhat Abbas S´ etif 1, S ´ etif 19000, Algeria\nP. -A. Hervieux\nInstitut de Physique et Chimie des Mat´ eriaux de Strasbourg ,\nCNRS and Universit´ e de Strasbourg BP 43, F-67034 Strasbour g, France\n(Dated: September 19, 2022)\nAbstract\nThe pseudo- PTsymmetric Dirac equation is proposed and analyzed by using a non-unitary\nFoldy-Wouthuysen transformations. A new spin operator PTsymmetric expectation value (called\nthe mean spin operator) for an electron interacting with a ti me-dependent electromagnetic field is\nobtained. Weshowthatspinmagnetization -whichisthequan tityusuallymeasuredexperimentally\n- is not described by the standard spin operator but by this ne w mean spin operator to properly\ndescribe magnetization dynamics in ferromagnetic materia ls and the corresponding equation of\nmotion is compatible with the phenomenological model of the Landau-Lifshitz-Gilbert equation\n(LLG).\nPACS numbers: Pseudo PT Symmetry; Non-Hermitian Dirac equation ; Foldy-Wouthuysen transformation;\nLandau-Lifshitz-Gilbert equation\n1In the field of micromagnetism ,which provides the physical framework for understanding\nand simulating ferromagnetic materials ,there is a fundamental unsolved problem which is\nthe microscopic origin of the intrinsic Gilbert damping. However, this d amping mechanism\nhas been introduced phenomenologically by T. L. Gilbert in 1955 for de scribing the spatial\nand temporal evolution of the magnetization (known as the LLG equ ation), a vector field\nwhich determines the properties of ferromagnetic materials on the sub-micron length scale\n[1]. Let us stress that this equation leads to the conservation of th e magnetization modulus.\nThis phenomenological model has since been validated by numerous e xperimental data and\nconstitutes the foundation of micromagnetism [2]. Moreover, magn etic damping plays a\ncrucial role in the operation of magnetic devices .The scattering theory can be used to\ncompute the Gilbert damping tensor [3].\nSpinisaquantumconcept [4]thatarisesnaturallyfromtheDiracth eoryandisassociated\nwith the operator ˆΣD≡1\n2iα∧α=\nσ0\n0σ\nwhereα=\n0σ\nσ0\nandσare the usual\n2×2 Pauli matrices [5]. For a classical version of the spin (see Supplementary\nMaterials). Usually, spin magnetization M(r,t) (the quantity which is experimentally\nmeasured) is defined as the expectation value of the spin angular mo mentum given by\nµB/angbracketleftBig\nΨD|ˆΣD|ΨD/angbracketrightBig\nwithµB≡e/planckover2pi1\n2mthe Bohr’s magneton ( e <0) and where ΨDisasolution\nof the Dirac equation. Indeed, in most magnetic materials the orbita l moment is quenched\nand therefore magnetism is only due to the spins [6].\nWhile for a free electron the spin angular momentum in the Heisenberg picture is not a\nconstant of motion/parenleftBigg\ndˆΣD\ndt/ne}ationslash= 0/parenrightBigg\n[5], there exists another spin operatorˆΣD\n,considered to\nbe a constant of motion ,(called the mean spin operator [7])\ndˆΣD\ndt= 0\n. In the presence\nof an electromagnetic field, which is relevant for exploring the micros copic origin of the\nLandau-Lifshitz-Gilbert (LLG) equation, a satisfactory result ha s not yet given.\nKnowingthatthespinorsintheDiractheoryconsistoffourcompon ents, itisimportantto\ncheck whether the Dirac equation yields physically reasonable result s in the non-relativistic\nexpansion case and to show that the Dirac equation reproduces th e two-component Pauli\nequation. We transform the Hamiltonian in such a way that all operat ors of the type αthat\ncouple the large to the small components will be removed. This can be achieved by a Foldy\nWouthuysen transformation [7–9] which is a non-relativistic expans ion of the Hamiltonian\n2into series of the particle ′s Compton wave lengths λC≡h\nmc.\nHickey and Moodera [10] have proposed that the spin-orbit interac tion, which arises from\nthe non-relativistic expansion of the Dirac equation, may be respon sible for the intrinsic\nferromagnetic line width. In their work, the term containing the curl of the electric field\nwhen coupled to Maxwell’s equations lead to a time-varying magnetic ind uction,and the\ntheoretical methods employed involve previously developed formalis ms in which an effective\nnon-Hermitian and time-dependent Hamiltonian is used. However, th e non-Hermiticity of\nthe Hamiltonian imposes new rules which are modified with respect to th ose of standard\nquantum mechanics. This fact was not explicitly taken into account b y the authors of [10]\nand therefore, their derivation of the intrinsic damping process is u nfortunately incorrect.\nMoreover, there is another fundamental issue which emerges fro m this work [10] concerning\nhow to properly perform the coupling between the classical Maxwell equations and the\nquantum evolution resulting from the non-relativistic limit of the Dirac equation. In what\nfollows, we show how to overcome this difficulty by using the well-known correspondence\nprinciple. Intheref. [11], themain goalwastodemonstratethatt hereisawaytoderivethe\nLLG equation coming from a non Hermitian quantum mechanics and to s park a discussion\nabout the connection between quantum and classical spin dynamics . Unfortunately, the\nquantum Heisenberg equation for a non-Hermitian Hamiltonian opera tor describing the\ndamping process is not compatible with the time-evolution operators for non-Hermitian\nHamiltonian operator.\nFromthe relativistic Diracequation, performing a Foldy-Wouthuyse n transformationand\nusing the Heisenberg equation of spin motion, Mondal et al [12–14] derive general relativistic\nexpressions for the Gilbert damping, but the term involving the cros s-product between the\nmagnetisationandthetime-derivativeofthemagneticfieldispurelyim aginary,andtherefore\nappears not to correspond to damping.\nIn the seminal work made by Dirac on relativistic quantum mechanics, the corresponding\nHamiltonian would be Hermitian. We stress that this property is very u seful to have in a\nphysical system, however we argue that the same features can b e achieved when starting\nfromnon-HermitianHamiltonian systems. These features canalso b eobtainedfromtheories\nbased on non-Hermitian Hamiltonians that have been considered in diff erent contexts. And\nwe distinguish three separate regimes: i) The PT-symmetric regime where the eigenval-\nues are real, ii) The spontaneously broken PTregime where the eigenvalues are complex\n3conjugate pairs and iii) The regime with complex, unrelated, eigenvalu es in which the PT\n-broken regime.\nOur objective here is to derive the LLG equation based on a non-Her mitian Dirac Hamil-\ntonian when compared to the most common standard approaches [ 12–14].\nTherefore, the Dirac equation in its fundamental representation is not unique to either\nHermitian quantum mechanics orquantum field theory. By relaxing the assumption of Her-\nmiticity andadoptinginstead the principles of P0T0-symmetry quantum mechanics outlined\nin the following paragraph, we will not make any modifications to the Dir ac equation. By\nP0T0-symmetry we mean reflection in space, with a simultaneous reversa l of time. The\nfundamental representation of the Dirac equation emerges comp letelyintact,identical in\nevery aspect to the Dirac equation derived from Hermitian theory. Before constructing the\nanalogous 4-d representation using the principles of P0T0quantum mechanics, let us briefly\nrecall the notion of P0T0-symmetry.\nThe Hermiticity of quantum Hamiltonians depends on the choice of the inner product\nof the states in the physical Hilbert space. This point was first point ed out by Bender et\nal [16, 17]. They showed that a wide class of Hamiltonians that respec tP0T0-symmetry\ncan exhibit entirely real spectra. Since then P0T0-symmetry has been a subject of intense\ninterest in the field of quantum mechanics.\nWhile any evidence of P0T0-symmetry has remained out of reach due to the hermitian\nnatureofthequantummechanics theory,opticshave providedafe rtilegroundforobservation\nof this property- P0T0-symmetry-since this field mainly relies on the presence of gain and\nloss.Note that even though HandP0T0commute, they do not continuously have identical\neigenvectors, as a result of the anti-linearity of the P0T0operator. If HandP0T0don’t have\nthe same eigenvectors, we say that the P0T0−symmetry is broken. The parity operator P0\neffects the momentum operator pand the position operator ras (P0:r→ −r,p→\n−p). This parity transformation has the following effect on the various vector potentials\nP0A(r,t)(P0)−1=−A(r,t) andP0Φ(r,t)(P0)−1= Φ(r,t), expressing thus their scalar\nand vector nature.\nThe anti-linear time reversal operator T0has the effect of changing the sign of the mo-\nmentum operator p,the pure imaginary complex quantity iand the time t(T0:r→r,\np→ −p, i→ −i, t→ −t). Since A(r,t) is generated by currents, which reverse s\nsignswhen the sense of time is reversed, it holds that T0A(r,t)(T0)−1=−A(r,t)\n4andT0Φ(r,t)(T0)−1= Φ(r,t).The two reflection operators commute with each other:\nP0T0=T0P0.\nTherefore, it is natural to introduce a modified Hilbert space, which is now endowed with\nP0T0-inner product, for the P0T0-symmetric nonself-adjoint theories. In such a Hilbert\nspace, the time evolution becomes unitary as the Hamiltonian is self- P0T0-adjoint and\nthe eigenfunctions form a complete set of orthonormal functions . But the norms of the\neigenfunctions have alternate signs even in the new Hilbert space en dowed with the P0T0-\ninnerproducts. Infact, anytheoryhavinganunbroken P0T0-symmetryitexistsasymmetry\nof the Hamiltonian associated with the fact that there are equal nu mbers of positive-norm\nand negative-norm states [17]:\n/an}b∇acketle{tψm,ψn/an}b∇acket∇i}htP0T0=/integraldisplay\ndx/bracketleftbig\nP0T0ψn(x)/bracketrightbig\nψm(x) =/integraldisplay\ndxψ∗\nn(−x)ψm(x)\n= (−1)nδmn (1)\nThe situation here is analogous to the problem that Dirac encounter ed in formulating the\nspinor wave equation in relativistic quantum theory [18] .\nThis again raises an obstacle in probabilistic interpretation in spite of t he systembeingin\nanunbroken P0T0phase. Afterwards, a new symmetry C, inherent to all P0T0-symmetric\nnon-Hermitian Hamiltonians, has been introduced [17]. Ccommutes with both HandP0T0\nand fixes the problem of negative norms of the eigenfunctions when the inner products are\ntaken with respect to CP0T0-adjoint.\nDoes aP0T0-symmetric Hamiltonian Hspecify a physical quantum theory in which the\nnorms of states are positive and time evolution is unitary? The answe r is that if Hhas an\nunbroken P0T0symmetry, then it has another symmetry represented by a linear o perator\nC.Thereforewecanconstruct atime-independent inner productwit hapositive-definite norm\nin terms of C.\nAnother possibility to explain the reality of the spectrum is making use of the\npseudo/quasi-Hermiticity transformations which do not alter the e igenvalue spectra. It was\nshown by Mostafazadeh [19] that P0T0-symmetric Hamiltonians are only aspecific class\nof the general families of thepseudo-Hermitian operators. A Hamiltonian is said to be\nη-pseudo-Hermitian if:\nH†=ηHη−1, (2)\n5whereηisametric operator . Theeigenvaluesofpseudo-HermitianHamiltoniansareeither\nreal or appear in complex conjugate pairs whilethe eigenfunctions satisfy bi-orthonormality\nrelations in the conventional Hilbert space. Due to this reason, suc h Hamiltonians do not\npossessacomplete set of orthogonal eigenfunctions in the conventional Hilb ert space and\nhence the probabilistic interpretation and unitarity of time evolution have not been satisfied\nby these pseudo-Hermitian Hamiltonians.\nHowever, like the case of P0T0-symmetric non-Hermitian systems, thepresence of the\nadditional operator ηin the pseudo-Hermitian theories allows usto define a new inner\nproduct in the fashion\n/an}b∇acketle{tφ|ψ/an}b∇acket∇i}htη=/an}b∇acketle{tηφ|ψ/an}b∇acket∇i}ht=/integraldisplay\n(ηφ(x))ψ(x)dx=/an}b∇acketle{tφ|ηψ/an}b∇acket∇i}ht. (3)\nLateranovel concept ofthepseudoparity-time (pseudo -P0T0)symmetry wasintroduced\nin [15] to connect the non-Hermitian Hamiltonian Hto its Hermitian conjugate H†\nH†=/parenleftbig\nP0T0/parenrightbig\nH/parenleftbig\nP0T0/parenrightbig−1(4)\nwhere in the expression of the inner product (3), the metric ηis replaced by P0T0. We now\nturn our attention to the main topic of interest, the spatial reflec tion and the time-reversal\ninvariance of the Dirac equation. The complete spatial reflection (p arity) transformation\nfor spinors and the complete time-inversion operator are denoted asP=γ0P0andT=\n−iα1α3T0=iγ1γ3T0such that\nPγ0P−1=γ0,PγiP−1=−γi\nTγ0T−1=γ0,TαT−1=−α (5)\nwhere the Hermitian matrices β=γ0andαsatisfy the ‘Dirac algebra’ {αi,αj}= 2δij;\n{αi,β}= 0.\nInwhat follows, we will denote by\nˆHD(t) =i/parenleftbig\ncα.(ˆp+ieA(r,t))−eΦ(r,t)+mc2β/parenrightbig\n(6)\nthe non-Hermitian DiracHamiltonian for a single electron in thepresence of a classical time-\ndependent external electromagnetic field defined by ( A(r,t),Φ(r,t)). The associated non-\nHermitian Dirac equation for a single electron in thepresence of a classical time-dependent\n6external electromagnetic field reads\ni/planckover2pi1∂ΨD(r,t)\n∂t=ˆHD(t)ΨD(r,t)\n=i/parenleftBig\nˆT+ˆV+mc2β/parenrightBig\nΨD(r,t), (7)\nwhere ΨD(r,t) = col(u(r,t),v(r,t)),is bispinors verifying the pseudo-othogonality relation\n/angbracketleftbig\nΨD|ΨD/angbracketrightbig\nPT=I,ˆV≡ −eΦ andˆT≡cα.ˆπ=cα.(ˆp+ieA(r,t)) is the kinetic energy which\nproduces a coupling between small and large components of the Dirac wavefun ction ΨD,\nα,βandγ5(see Eq.(15)) are the Dirac matrices [20]. The PTsymmetry condition\n(PT)ˆHD(t)(PT)−1=ˆH†D(t), (8)\nconnects the non-Hermitian Dirac Hamiltonian ˆHD(t) to its Hermitian conjugate\nˆH†D(t).This observation leads us to introduce a novel concept of the pseu do-parity-time\n(pseudo-PT) symmetry, where ( PT) is interpreted as a metric. Thus as the case of pseudo-\nhermiticity, the bispinor ΨD(r,t) = col(u(r,t),v(r,t)),verifiesthe pseudo-othogonality\nrelation/angbracketleftbig\nΨD|ΨD/angbracketrightbig\nPT=I. Note that, there is another situation which differs from that\ndescribed above ,also called pseudo- PTsymmetry which means that the system can have\na real eigenvalues whether or not the original system is PT- symmetric [21, 22].\nAs we are dealing with the non-relativistic expansion of the Dirac equa tion, the following\ndecomposition ΨD(r,t) = Π(r,t)×χD(us,t) [23] can be used, where χD(us,t) is the time-\ndependent bi-spinor representing the spin state oriented in the dir ection defined by usand\nΠ(r,t) thescalarpart of the wave function.\nWe argue that, in the non-relativistic limit, the operatorˆΣD\nis the one which must be\ninterpreted as the spin operator in the Pauli theory and used to de fine the magnetization as\nM(r,t)≡µB/angbracketleftbigg\nχD|ˆΣD\n|χD/angbracketrightbigg\nPT=µB/angbracketleftbigg\nχD|ˆU−1ˆΣFWˆU|χD/angbracketrightbigg\nPT=µB/an}b∇acketle{tχFW|ˆΣFW|χFW/an}b∇acket∇i}htPTwhere\nχFWis the spin part of the Dirac bi-spinor wave function ΨDin a non-relativistic expansion,\nobtained by using the Foldy-Wouthuysen (FW) transformation [7] andˆUis the associated\noperator (see the definition in the following). It worth mentioning th at,according to the\nabove definition ,expanding the spin operator (to a consistent order in h/mc) and using the\nDirac representation of the wave function is equivalent to expand ingthe wave function (also\nto a consistent order in h/mcusing the FW transformation) and keep ingthe original spin\noperator in the Dirac representation. In this work we have chosen to expand the mean value\noperator. The classical magnetization, M(r,t), is obtained by using the correspondence\n7principle. Indeed, we will show in what follows that the equation of mot ion of the mean\nspin operator for an electron interacting with a time-dependent ele ctromagnetic field leads\nto the LLG equation of motion revealing thus its microscopic origin.\nIna seminal work, FoldyandWouthuysen (FW)solved theproblem of finding a canonical\ntransformation thatallows to obtain a two-component theory in the low-energy limit (Pauli\napproximation) ,in the case of the Dirac equation coupled to an electromagnetic field [7 ].\nUnfortunately, contrarytothefree-electroncase, thesolutio ncannotbeexpressed inaclosed\nform. However, FW showed how to obtain successive approximation s of this transformation\nas a power series expansion in powers of the Compton wave length of the particle λC≡\nh\nmc. This procedure, generally restricted to the second-order in 1 /m, is presented in many\ntextbookson relativistic quantum mechanics [8, 20, 24, 25] and has been exte nded to fifth\norder in powers of 1 /m[26].\nIn what follows, the symbol [ ˆC,ˆD] ({ˆC,ˆD}) denotes the commutator (anticommutator)\nof the operators ˆCandˆD. We shall also use the following notations: ˆX≡X(r,t) and\nˆY≡Y(r,t).\nIn the Hermitian Dirac representation\nˆHhD(t) =/parenleftbig\ncα.(ˆp−eA(r,t))+eΦ(r,t)+mc2β/parenrightbig\n, (9)\nthe Heisenberg equation of motion for the spin operator reads as f ollows\ndˆΣD\ndt=i\n/planckover2pi1/bracketleftBig\nˆHhD,ˆΣD/bracketrightBig\n=−2c\n/planckover2pi1[α∧(ˆp−eA(r,t)]. (10)\nIt is well established that the expectation value onto/vextendsingle/vextendsingleΨD/angbracketrightbig\nof the above equation does\nnot lead to the LLG equation for the magnetization. However, as it w illbeshown in the\nfollowing, the latter can be obtained by using the non unitary FW tran sformation and the\nnew definitionof themagnetizationasanexpectation valueof means pinoperator. Thefirst-\nand second-order terms of the FW expansion in powers of 1 /mcorrespond, respectively, to\nthe precessional motion of the magnetization around an effective m agnetic field and its\ndamping.\nSince the Hamiltonian ˆHD(t) has a similar structure to the one in the Dirac case, by\nanalogy with the latter, we use for ˆU(t) the form eˆS(t)whereˆSis a non self-adjoint\n8operator. Therefore, the transformation ΨFW(r,t) =eˆS(t)ΨD(r,t)≡ˆU(t)ΨD(r,t) leads to a\nnew Hamiltonian\nˆHFW(t) =eˆS(t)/parenleftbigg\nˆHD(t)−i/planckover2pi1∂\n∂t/parenrightbigg\ne−ˆS(t), (11)\nwhereˆSis a non self-adjoint operator. More generally, any operator inthe FW\nrepresentation ,that is not explicitly time dependent, ˆOFWwill be transformed in the Dirac\nrepresentation as ˆOD(t) =ˆU−1(t)ˆOFWˆU(t).\nThe most natural extension oftheEhrenfest equation tonon-He rmitian pseudo- PTsym-\nmetric systems is by replacing a Hermitian ˆHhDwitha non-Hermitian one. The structure\nof the Ehrenfest equation does not change, having assumed that part of the action of T\n(i.e.T0) is to send t→ −t, i.e.;we show that it anticommute swith the operator ∂/∂t,\nconsequently, the operator PTcommuteswithi∂/∂t.Which immediately leads us to\ndeduce the Ehrenfest equation of motion for the diagonal matrix e lement of an operator\nˆOD(t)\nd\ndt/angbracketleftBig\nΨD|ˆOD|ΨD/angbracketrightBig\nPT=/angbracketleftBigg\nΨD|i\n/planckover2pi1/bracketleftBig\nˆHD,ˆOD/bracketrightBig\n+∂ˆOD\n∂t|ΨD/angbracketrightBigg\nPT. (12)\nIt’s straightforward to show that the equation of motion (12) lead s to\nd\ndt/angbracketleftBig\nΨFW|ˆOFW|ΨFW/angbracketrightBig\nPT=/angbracketleftbigg\nΨFW|i\n/planckover2pi1/bracketleftBig\nˆHFW,ˆOFW/bracketrightBig\n|ΨFW/angbracketrightbigg\nPT. (13)\nBy expanding ˆS=ˆS1m−1+ˆS2m−2+ˆS3m−3+...withˆS1=β\n2c(α.ˆπ),ˆS2=−i/planckover2pi1e\n4c3(α.E),\nˆS3=−β\n8c6/parenleftBig\n4c3\n3(α.ˆπ)(α.ˆπ)(α.ˆπ)+iec/planckover2pi12(α.∂tE)/parenrightBig\nandE=−∇Φ−∂A\n∂t[26], the mean spin\noperator is computed using the inverse FW transformation of the s pin operator asˆΣD\n=\ne−(ˆS1/m+ˆS2/m2+ˆS3/m3)ˆΣDe(ˆS1/m+ˆS2/m2+ˆS3/m3)and may be expanded in power series of (1 /m)\nleading toˆΣD\n=ˆΣD\n0+ˆΣD\n1m−1+ˆΣD\n2m−2+ˆΣD\n3m−3+...\nˆΣD\n0=ˆΣD,\nˆΣD\n1=−iβ\nc(α׈π),\nˆΣD\n2=1\n8c2/parenleftBig\n−4ie/planckover2pi1B+2e/planckover2pi1/parenleftBig\nˆΣD×B/parenrightBig\n−4/parenleftBig\nˆπ×/parenleftBig\nˆΣD׈π/parenrightBig/parenrightBig/parenrightBig\n−e/planckover2pi1\n2c3(α×E),\nˆΣD\n3=β\n48c3/bracketleftBig\n(α.ˆπ),/bracketleftBig\n(α.ˆπ),/bracketleftBig\n(α.ˆπ),ˆΣD/bracketrightBig/bracketrightBig/bracketrightBig\n+β\n6c3/bracketleftBig\n(α.ˆπ)(α.ˆπ)(α.ˆπ),ˆΣD/bracketrightBig\n−e/planckover2pi12β\n4c5(α×(∂tE)), (14)\n9whereB=∇ ×A.The free case which is investigated in [7] (may be obtained in closed\nform in this case) is recovered from the above formula by substitut ingE=B= 0 ,iβ→β\nandiˆπ→ˆpleading to\nˆΣD\n=ˆΣD−iβ(α∧ˆp)\nEp−/parenleftBig\nˆp∧/parenleftBig\nˆΣD∧ˆp/parenrightBig/parenrightBig\nEp(Ep+mc)\nwhereEp=/radicalbig\nm2c2+p2.\nThe pseudo- PTequation of motion (12) for the mean spin operator is\nd\ndt/angbracketleftbig\nΨD/vextendsingle/vextendsingleˆΣD/vextendsingle/vextendsingleΨD/angbracketrightbig\nPT\n=/angbracketleftbig\nΨD/vextendsingle/vextendsingleeβ\nm/parenleftBig\nˆΣD∧B/parenrightBig\n−e/planckover2pi1\n4m2c2/parenleftBig\nˆΣD∧∂tB/parenrightBig\n−ie\n2m2c2/parenleftBig\nˆΣD∧(E∧ˆπ)/parenrightBig\n+1\n4/planckover2pi1m2c/parenleftBig/bracketleftBig\n(α.π)ˆΣD(α.π),(α.π)/bracketrightBig\n−/bracketleftbig\n(α.π)(α.π)(α.π),ΣD/bracketrightbig/parenrightBig\n+ϑ(m−3)/vextendsingle/vextendsingleΨD/angbracketrightbig\nPT.(15)\n(In order) to check this result, we have applied to the above equat ion the direct FW trans-\nformation. It leads to\nd\ndt/angbracketleftbig\nΨD/vextendsingle/vextendsingleˆΣD/vextendsingle/vextendsingleΨD/angbracketrightbig\nPT=d\ndt/angbracketleftbig\nΨFW/vextendsingle/vextendsinglee(ˆS1/m+ˆS2/m2+ˆS3/m3)ˆΣD\ne−(ˆS1/m+ˆS2/m2+ˆS3/m3)/vextendsingle/vextendsingleΨFW/angbracketrightbig\nPT\n≡d\ndt/angbracketleftbig\nΨFW/vextendsingle/vextendsingleˆΣ/vextendsingle/vextendsingleΨFW/angbracketrightbig\nPT=/angbracketleftbig\nΨFW/vextendsingle/vextendsinglei\n/planckover2pi1/bracketleftbigg\nˆHFW,ˆΣFW/bracketrightbigg/vextendsingle/vextendsingleΨD/angbracketrightbig\nPT(16)\nwhereˆΣFW≡ˆΣand the expression of ˆHFWis given by\nˆHFW=iβmc2−ieˆΦ+iβˆπ2\n2m+iβˆπ4\n8m3c2+βe/planckover2pi12\n16m3c4{ˆπ,∂tE}\n−βe/planckover2pi1\n2mˆΣ./bracketleftbigg\nB−β/planckover2pi1\n4mc2/parenleftbigg\n(∇∧E)+2i\n/planckover2pi1E∧ˆπ/parenrightbigg\n+i/planckover2pi1\n8m2c4(∂tE∧ˆπ)+ ˆπ∧∂tE)/bracketrightbigg\n−βe/planckover2pi1\n8m3c2/braceleftBig\nˆπ2,ˆΣ.B/bracerightBig\n−iβ/parenleftbigge/planckover2pi1\n2m/parenrightbigg2B2\n2mc2+ie/planckover2pi12\n8m2c2∇.E+ϑ(m−4). (17)\nNow, the equation of motion for the mean spin operator (15) can be written in a simpler\nway being obtained as the pseudo- PTexpectation value a spin Dirac states |χD/an}b∇acket∇i}htassociated\ntoˆHD, one gets\nd\ndt/an}b∇acketle{tχD|ˆΣD\n|χD/an}b∇acket∇i}htPT\n=eβ\nm/an}b∇acketle{tχD|ˆΣD|χD/an}b∇acket∇i}htPT∧B−e/planckover2pi1\n4m2c2/an}b∇acketle{tχD|ˆΣD|χD/an}b∇acket∇i}htPT∧∂tB\n−e\n2mc2/an}b∇acketle{tχD|ˆΣD|χD/an}b∇acket∇i}htPT∧/parenleftbigg\nE∧/an}b∇acketle{tχD|i\nmˆπ|χD/an}b∇acket∇i}htPT/parenrightbigg\n+ϑ(m−3). (18)\n10The above expression has been obtained by using/angbracketleftbig\nΨD/vextendsingle/vextendsingleˆΣD/vextendsingle/vextendsingleΨD/angbracketrightbig\nPT=/an}b∇acketle{tχD|ˆΣD\n|χD/an}b∇acket∇i}htPT. It\nis of interest to mention that the non-diagonal terms which appear at thesecondline of\n(15) are due to the Zitterbewegung phenomenon [5]. They cancel o ut when they are pseudo\naveraged out in a Dirac states.\nLet us note that, it is more convenient to use the FW representatio n to find the evolution\nof spin, (indeed) from Eq.(7) whered\ndt/angbracketleftbig\nΨD/vextendsingle/vextendsingleˆΣD/vextendsingle/vextendsingleΨD/angbracketrightbig\nPT≡d\ndt/angbracketleftbig\nΨFW/vextendsingle/vextendsingleˆΣFW/vextendsingle/vextendsingleΨFW/angbracketrightbig\nPT, we get\nd\ndt/an}b∇acketle{tχFW|ˆΣ|χFW/an}b∇acket∇i}htPT=eβ\nm/an}b∇acketle{tχFW|ˆΣ∧/parenleftbigg\nB+1\n2c2/bracketleftbigg\nE∧iβ\nmˆπ/bracketrightbigg/parenrightbigg\n−e/planckover2pi1\n4m2c2/bracketleftBig\nˆΣ∧(∇∧E)/bracketrightBig\n|χFW/an}b∇acket∇i}htPT\n(19)\nConcerning the damping process, as previously explained, the only t erm of importance\nin the expression (15) is −e/planckover2pi1\n4m2c2/parenleftBig\nˆΣD∧∂tB/parenrightBig\nwhich has been obtained from the commutator\n[(α×E),(α.ˆπ)] coming from/bracketleftBig\nˆHD,ˆΣD/bracketrightBig\nand the partial derivative with respect to time\nof the mean spin angular momentum operator at second order in 1 /m[27] given in Eq.\n(14). In addition, classical Maxwell equations have been also employ ed. Similarly to the\nBreit Hamiltonian ,which is obtained from the classical Darwin Lagrangian (which also\noriginates from Maxwell equations) by using the correspondence p rinciple (CP) [25] ,we\nhere resort to the same procedure (in its inverse form, from quan tum to classical) for the\nMaxwell equations. According to this principle, the quantum counte rpartsˆf, ˆgof classical\nobservables f,gsatisfy/angbracketleftBig/bracketleftBig\nˆf,ˆg/bracketrightBig/angbracketrightBig\n=i/planckover2pi1{f,g}p,qwhere/angbracketleftBig/bracketleftBig\nˆf,ˆg/bracketrightBig/angbracketrightBig\nis the expectation value of\nthe commutator and the symbol {}p,qdenotes the Poisson bracket [28–31]. Let’s take for\ninstance the Maxwell-Faraday equation, we have\n∇∧E(r,t) =−∂B(r,t)\n∂t(20)\nwhich can be rewritten as\nǫijk{pi,Ej}p,qek=−∂B(r,t)\n∂t, (21)\nand using the CP one gets\nǫijk[ˆpi,Ej]\ni/planckover2pi1ek=−∂B(r,t)\n∂t≡ −∂tB. (22)\nConsequently, by using our definition of the magnetization M (r,t)≡\nµB/angbracketleftbigg\nχD|ˆΣD\n|χD/angbracketrightbigg\nPT≡µB/angbracketleftBig\nχFW|ˆΣ|χFW/angbracketrightBig\nPTand the CP, the equation of motion (18)\n11may be rewritten for the electron part as\ndM(r,t)\ndt=e\nmM(r,t)∧B(r,t)−e\n4m2c2M(r,t)∧∂tB(r,t)\n+e\n2mc2M(r,t)∧(E(r,t)∧v)+ϑ(m−3). (23)\nThe above equation constitutes the main result of this work.\nMoreover, if the electron is embedded in a magnetically polarizable med ium,defined\nby its magnetic polarizability χm,then∂M\n∂tgenerates a time-dependent magnetic induction\naccording to the relation ∂tB(r,t) =1\nχm∂M\n∂tand the equation (23) can be rewritten as\ndM(r,t)\ndt=−γM(r,t)∧Beff(r,t)−αG\nM/parenleftbigg\nM(r,t)∧∂M(r,t)\n∂t/parenrightbigg\n(24)\nwithγ=−e\nm>0 the gyromagnetic ratio for an isolated electron, Beff≡B−1\n2c2v∧E\nandαG≡eM\n4m2c2χm. The first term describes the precessional motion of the magnetiz ation\nvector around the direction of the effective magnetic field and the s econd term represents\nits damping ,characterized by the Gilbert’s constant αG.\nLet us stress that the first term in the right hand side of equation ( 24) can be retrieved\nfrom the non-relativistic expansion of the Bargmann-Michel-Telegd i’s equation [24, 32, 33]\nwhichrepresents the relativistic equation of motion of a classical magnetic dipole momen t\n[34]. However, the damping term cannot be obtained from this classic al description due to\nits quantum origin.\nIn summary, the mean spin angular momentum operator introduced for the first time\nby Foldy and Wouthuysen for the case of a free electron has been e xtended to the non\n-Hermitian or precisely to a pseudo PT-symmetric case of an electron interacting with\na time-dependent electromagnetic field. 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Jackson, Classical Electrodynamics, John Wiley ( 1998).\n[34] In order to be compatible with our description which doe s not include QED effects, the Land´ e\nfactor must be g= 2.\nSupplementary Materials\nIn terms of the conjugate variable ( q,p) the classical spin− →Sis desribed by [1, 2]\n\n\nSx=/radicalbig\nS2−p2cosq\nSy=/radicalbig\nS2−p2sinq\nSz=p(25)\nthe Poisson brackets {Si,Sj}=εijkSk(i,j,karex,yorz) are analogous to the same rela-\ntionships one has with spin components and commutators in quantum mechanics.\nSuppose we have the following Hamiltonian\nH=− →B.− →S (26)\nwhich is formally identical to the Hamiltonian for a spin 1 /2 system in a uniform magnetic\nfield. We can calculate the evolution of the vector components using the standard Hamil-\ntonian techniques and The motion of spin− →Son the sphere (phase space) with (conserved)\nradiusS=/vextendsingle/vextendsingle/vextendsingle− →S/vextendsingle/vextendsingle/vextendsinglegenerated by (26), can be obtained by regarding H(26) as classical hamil-\ntonian . It may be confirmed that Hamilton’s equation reproduce exa ctly what spin does in\na magnetic field i.e,− →·\nS=− →B∧− →S.\nThe two-level spin system can be written as a classical model if we em ploy the anticom-\nmuting Grassmann variables [3–6]− →ζwhich are transformed to the spin operator after the\n14quantization∧− →ζ=∧− →S/√\n2 . Unlike the classical spin defined in the equation ((25)) which\ndoes not tranformed into a spin operator after the quantization∧− →S/ne}ationslash=− →S.\n[1] M. V. Berry, in ”Fundamental Aspects of Quantum” (Edited by V. Gorini and A. Frigerio),\nPlenum, Nato ASI series vol. 144, 267-278 (1986)).\n[2] M. Maamache, exact solution and geometic Angle for the cl assical spin system, Phys. Scr. 54,\n21 (1996).\n[3] R. Casalbuoni, On the quantization of systems with antic ommuting variables, Nuovo Cimento\nA 33, 115 (1976).\n[4] F.A. Berezin and M.S. Marinov, Particle Spin Dynamics as the Grassmann Variant of Classical\nMechanics, Ann. Phys. (N.Y) 104, 336 (1977).\n[5] E. Gozzi and W. D. Thacker, Classical adiabatic holonomy in a Grassmannian system, Phys.\nRev. D 35, 2388 (1987).\n[6] M. Maamache and O. Cherbal, Evolution of Grassmannian in variant-angle coherent states and\nnonadiabatic Hannay’s angle, Eur. Phys. J. D 6, 145 (1999).\n15"    },    {        "title": "1703.03198v3.Material_developments_and_domain_wall_based_nanosecond_scale_switching_process_in_perpendicularly_magnetized_STT_MRAM_cells.pdf",        "content": "Material developments and domain wall based nanosecond-scale switching process in\nperpendicularly magnetized STT-MRAM cells\nThibaut Devolder\u0003and Joo-V on Kim\nCentre de Nanosciences et de Nanotechnologies, CNRS, Univ. Paris-Sud,\nUniversit ´e Paris-Saclay, C2N-Orsay, 91405 Orsay cedex, France\nJ. Swerts, S. Couet, S. Rao, W. Kim, S. Mertens, and G. Kar\nIMEC, Kapeldreef 75, B-3001 Leuven, Belgium\nV . Nikitin\nSAMSUNG Electronics Corporation, 601 McCarthy Blvd Milpitas, CA 95035, USA\nWe investigate the Gilbert damping and the magnetization switching of perpendicularly magnetized FeCoB-\nbased free layers embedded in magnetic tunnel junctions adequate for spin-torque operated magnetic memories.\nWe first study the influence of the boron content in MgO / FeCoB /Ta systems alloys on their Gilbert damping pa-\nrameter after crystallization annealing. Increasing the boron content from 20 to 30% increases the crystallization\ntemperature, thereby postponing the onset of elemental diffusion within the free layer. This reduction of the in-\nterdiffusion of the Ta atoms helps maintaining the Gilbert damping at a low level of 0.009 without any penalty on\nthe anisotropy and the magneto-transport properties up to the 400\u000eC annealing required in CMOS back-end of\nline processing. In addition, we show that dual MgO free layers of composition MgO/FeCoB/Ta/FeCoB/MgO\nhave a substantially lower damping than their MgO/FeCoB/Ta counterparts, reaching damping parameters as\nlow as 0.0039 for a 3 ˚A thick Tantalum spacer. This confirms that the dominant channel of damping is the\npresence of Ta impurities within the FeCoB alloy. On optimized tunnel junctions, we then study the duration of\nthe switching events induced by spin-transfer-torque. We focus on the sub-threshold thermally activated switch-\ning in optimal applied field conditions. From the electrical signatures of the switching, we infer that once the\nnucleation has occurred, the reversal proceeds by a domain wall sweeping though the device at a few 10 m/s.\nThe smaller the device, the faster its switching. We present an analytical model to account for our findings. The\ndomain wall velocity is predicted to scale linearly with the current for devices much larger than the wall width.\nThe wall velocity depends on the Bloch domain wall width, such that the devices with the lowest exchange\nstiffness will be the ones that host the domain walls with the slowest mobilities.\nI. INTRODUCTION\nTunnel magnetoresistance (TMR) and spin transfer torque\n(STT) – the fact that spin-polarized currents manipu-\nlate the magnetization of nanoscale magnets and in par-\nticular magnetic tunnel junction (MTJ) nanopillars – are\nthe basic phenomena underpinning an emerging technol-\nogy called Spin-Transfer-Torque Magnetic Random Access\nMemory (STT-MRAM)1, which combines high endurance,\nlow power requirement2,3, CMOS back-end-of-line (BEOL)\ncompatibility4and potentially large capacity5.\nThe core of an STT-MRAM stack is a magnetic tunnel\njunction composed6of an FeCoB/MgO/FeCoB central block.\nOne of the FeCoB layer is pinned to a high anisotropy syn-\nthetic ferrimagnet to create a fixed reference layer (RL) sys-\ntem while the second FeCoB acts as a free layer (FL). Histor-\nically, the FL is capped with (or deposited on) an amorphous\nmetal such as Ta4,7and more recently capped with a second\nMgO layer to benefit from a second interface anisotropy7–9\nin the so-called ’dual MgO’ configuration. So far, it is un-\nclear whether this benefit of anisotropy can be obtained with-\nout sacrificing the other important properties of the free layer,\nin particular the Gilbert damping.\nIn this paper, we will first tailor the Boron content inside\nthe FeCoB alloy to improve the properties of Ta / FeCoB /\nMgO ’single MgO’ free layers and their resilience to thermal\nannealing. The idea is to postpone the FeCoB crystalliza-tion till the very last stage of the BEOL annealing. Indeed\nmaintaining the amorphous state of FeCoB allows to mini-\nmize the interdiffusion of materials –in our case: tantalum–\nwithin the stack. This interdiffusion is otherwise detrimental\nto the Gilbert damping.\nWe then turn to dual MgO systems comprising a Ta spacer\nlayer in the midst of the FL. This spacer is empirically needed\nto allow proper crystallization and to effectively get perpen-\ndicular magnetic anisotropy (PMA)8,10–14. Unfortunately, the\npresence of heavy elements inside the FeCoB free layer is ex-\npected to alter its damping and to induce some loss of mag-\nnetic moment usually referred as the formation of magneti-\ncally dead layers. We study to what extend the Ta spacer in\nthe dual MgO free layers affects the damping and how this\ndamping compares with the one that can be obtained with sin-\ngle MgO free layers. Once optimized, damping factors as low\nas 0.0039 can be obtained a dual MgO free layer.\nBesides the material issues, the success of STT-MRAM\nalso relies on the capacity to engineer devices in accordance\nwith industry roadmaps concerning speed and miniaturiza-\ntion. To achieve fast switching and design devices accordingly\noptimized, one needs to elucidate the physical mechanism by\nwhich the magnetization switches by STT. Several categories\nof switching modes – macrospin15, domain-wall based16,\nbased on sub-volume nucleation17or based on the spin-wave\namplification18– have been proposed, but single-shot time-\nresolved experimental characterization of the switching patharXiv:1703.03198v3  [cond-mat.mtrl-sci]  4 Sep 20172\nare still scarce19–21. Here we study the nanosecond-scale spin-\ntorque-induced switching in perpendicularly magnetized tun-\nnel junctions with sizes from 50 to 300 nm. Our time-resolved\nexperiments argue for a reversal that happens by the motion\nof a single domain wall, which sweeps through the sample\nat a velocity set by the applied voltage. As a result, the\nswitching duration is proportional to the device length. We\nmodel our finding assuming a single wall moving in a uni-\nform material as a result of spin torque. The wall moves with\na time-averaged velocity that scales with the product of the\nwall width and the ferromagnetic resonance linewidth, such\nthat the devices with the lowest nucleation current densities\nwill be the ones that host the domain walls with the lowest\nmobilities.\nThe paper is split in first a material science part, followed\nby a study of the magnetization reversal dynamics. After a de-\nscription of the samples and the caracterization methods, sec-\ntion II C describes how to choose the optimal Boron content\nin an FeCoB-based free layer for STT-MRAM applications.\nSection II D discusses the benefits of ’dual MgO’ free layers\nwhen compared to ’single MgO’ free layers. Moving to the\nmagnetization switching section, the part III A gathers the de-\nscription of the main properties of the samples and the experi-\nmental methods used to characterize the STT-induced switch-\ning speed. Section III B describes the electrical signatures of\nthe switching mechanism at the nanosecond scale. The latter\nis modeled in section III C in an analytical framework meant\nto clarify the factors that govern the switching speed when the\nreversal involves domain wall motion.\nII. ADVANCED FREE LAYER DESIGNS\nA. Model systems under investigation\nOur objective is to study advanced free layer designs in\nfull STT-MRAM stacks. The stacks were deposited by phys-\nical vapor deposition in a Canon-Anelva EC7800 300 mm\ncluster tool. The MgO tunnel barriers were deposited by\nRF-magnetron sputtering. In dual MgO systems, the top\nMgO layer was fabricated by oxidation of a thin metallic Mg\nfilm. All stacks were post-deposition annealed in a TEL-MSL\nMRT5000 batch furnace in a 1 T perpendicular magnetic field\nfor 30 minutes. Further annealing at 400\u000eC were done in a\nrapid thermal annealing furnace in a N 2atmosphere for a pe-\nriod of 10 minutes.\nWe will focus on several kinds of free layers embod-\nied in state-of-the art bottom-pinned Magnetic Tunnel Junc-\ntions (MTJ) with various reference systems comprising ei-\nther [Co/Ni] and [Co/Pt] based hard layers22,23. Although we\nshall focus here on FLs deposited on [Co/Ni] based synthetic\nantiferromagnet (SAF) reference layers, we have conducted\nthe free layer development also on [Co/Pt] based reference\nlayers. While specific reference layer optimization leads to\nslightly different baseline TMR properties, we have found that\nthe free layer performances were not impacted provided the\nSAF structure is stable with the concerned heat treatment (not\nshown).The first category of samples are the so-called ’single-\nMgO’ free layers. We shall focus on samples with a free\nlayer consists of a 1.4 nm thick Fe 60Co20B20or a 1.6 nm\nthick Fe 52:5Co17:5B30layer sandwiched between the MgO\ntunnel oxide and a Ta (2 nm) metal cap. Note that these\nso-called ”boron 20%” and ”boron 30%” samples have dif-\nferent boron contents but have the same number of Fe+Co\natoms. A sacrificial4Mg layer is deposed before the Ta cap\nto avoid Ta and FeCoB mixing during the deposition, and\navoid the otherwise resulting formation of a dead layer. The\nMg thickness is calibrated so that the Mg is fully sputtered\naway upon cap deposition. This advanced capping method has\nproven to provide improved TMR ratios and lower RA prod-\nucts thanks to an improved surface roughness and a higher\nmagnetic moment4.\nThe second category of free layers are the so-called ’dual\nMgO’ free layers in which the FeCoB layer is sandwiched\nby the MgO tunnel oxide and an MgO cap which concur to\nimprove the magnetic anisotropy. The exact free layer com-\npositions are MgO (1.0 nm) / Fe 60Co20B20(1.1 nm) / spacer\n/ Fe 60Co20B20(0.9 nm) / MgO (0.5 nm). We study shall two\nspacers: a Mg/Ta(3 ˚A) spacer and a Mg/Ta(4 ˚A) spacer, both\ncomprising a sacrificial Mg layer.\nB. Experimental methods used for material quality assessment\nWe studied our samples by current-in-plane tunneling\n(CIPT), vibrating sample magnetometry (VSM) and Vector\nNetwork Ferromagnetic resonance (VNA-FMR)24in out-of-\nplane applied fields. CIPT was performed to extract the tun-\nnel magneto-resistance (TMR) and the resistance-area product\n(RA) of the junction. VSM measurements of the free layer\nminor loops have been used to extract the areal moments. We\nthen use VNA-FMR to identify selectively the properties of\neach subsystem. Our experimental method is explained in\nFig. 1, which gathers some VNAFMR spectra recorded on\noptimized MTJs. The first panel records the permeability of\na single MgO MTJ in the ffield-frequencygparameter space.\nWe systematically investigated a sufficiently large parameter\nspace to detect 4 different modes whose spectral characters\ncan be used to index them22. Three of the modes belong to the\nreference system that comprises 3 magnetic blocks coupled\nby interlayer exchange coupling through Ru and Ta spacers\nas usually done22,23; the properties of these 3 modes are inde-\npendent from the nature of the free layer. While we are not\npresently interested in analyzing the modes of the fixed sys-\ntem – thorough analyses can be found in ref.22,23– we empha-\nsize that it is necessary to detect all modes to unambiguously\nidentify the one belonging to the free layer, in order to study\nit separately. The free layer modes are the ones having V-\nshaped frequency versus field curves [Fig. 1(a)], whose slope\nchanges at the free layer coercivity. in each sample, the free\nlayer modes showed an asymmetric Lorentzian dispersion for\nthe real part of the permeability and a symmetric Lorentzian\ndispersion for the imaginary part [see the examples Fig. 1(b,\nc)]. As we found no signature of the two-layer nature of the\ndual MgO free layers, we modeled each free layer as a sin-3\nSingle MgOfree layerModes of the reference layers\nDual MgOTa spacer\u0000f2f=0.006\u0000f2f=0.016Contrastx 10Permeabilitymap\n↵=12@\u0000f@f=0.0039\nFIG. 1. (Color online). Examples of MTJ dynamical properties to\nillustrate the method of analysis. (a) Microwave permeability versus\nincreasing out-of-plane field and frequency for an MTJ with a sin-\ngle MgO free layer after an annealing of 300\u000eC. Note that the scale\nof the permeability was increased by a factor of 10 above 58 GHz\nfor a better contrast. The apparent vertical bars are the eigenmode\nfrequency jumps at the different switching fields of the MTJ. (b)\nReal and imaginary parts of the experimental (symbols) and modeled\n(lines) permeability for an out-of-plane field of 1.54 T for the same\nMTJ. The model is for an effective linewidth \u0001f=(2f) = 0:016,\nwhich includes both the Gilbert damping and a contribution from the\nsample inhomogeneity. (c) Same but for a dual MgO free layer based\non a 3 ˚A Ta spacer, modeled with \u0001f=(2f) = 0:006. (d) Cross\nsymbols: FMR half frequency linewidth versus FMR frequency for\na dual MgO free layer based on a 3 ˚A Ta spacer. The line is a guide\nto the eye corresponding to a Gilbert damping of 0.0039.\nglemacrospin, disregarding whether it was a single MgO or a\ndual MgO free layer.\nFMR frequency versus field fits [see one example in\nfig. 2(c)] were used to get the effective anisotropy fields\nHk\u0000Msof all free layers25. The curve slopes are \r0, where\n\r0= 230 kHz.m/A is the gyromagnetic factor \rmultiplied\nby the vacuum permeability \u00160. It was consistent with a spec-\ntroscopic splitting Land ´e factor ofg\u00192:08. Damping analy-\nsis was conducted as follows: the free layer composition can\nyield noticeable differences in the FMR linewidths [see for in-\nstance Fig. 1(b) and (c)]. To understand these differences, we\nsystematically separated the Gilbert damping contribution to\nthe linewidth from the contribution of the sample’s inhomo-\ngeneity using standard VNA-FMR modeling25. This is doneby plotting the half FMR linewidth \u0001f=2versus FMR fre-\nquencyfFMR [see one example in Fig. 1(d)]. The Gilbert\ndamping is the curve slope and the line broadening arising\nfrom the inhomogeneity of the effective field within the free\nlayer is the zero frequency intercept1\n2\r0\u0001fjf=0) of the curve.\nC. Boron content and Gilbert damping upon annealing of\nsingle MgO free layers\nDesigning advanced free layer in STT-MRAM stacks re-\nquires to minimize the Gilbert damping of the used raw ma-\nterial. In Ta/FeCoB/MgO ’single MgO’ free layers made of\namorphous FeCoB alloys or made of FeCoB that has been\njust crystallized, a damping of 0.008 to 0.011 can be found\ntypically19,25. (Note that lower values can be obtained but for\nthicknesses and anisotropies that are not adequate for spin-\ntorque application26). The damping of Ta/FeCoB/MgO sys-\ntems generally degrades substantially when further annealing\nthe already crystallized state27. Let us emphasize than even in\nthe best cases26, the damping of FeCoB based free layers are\nstill very substantially above the values of 0.002 or slightly\nless than can be obtained on FeCo of Fe bcc perfect single\ncrystals28,29.\nThere are thus potentially opportunities to improve the\ndamping of free layers by material engineering. We illustrate\nthis in fig. 2 in which we show that a simple increase of the\nBoron content is efficient to maintain the damping unaffected,\neven upon annealing at 400\u000eC in a single MgO free layer. In-\ndeed starting from Ta/FeCoB/MgO ’single MgO’ free layers\nsharing the same damping of 0.009 after annealing at 300\u000eC\n(not shown), an additional 100\u000eC yields\u000b= 0:015for the\nfree layer with 20% of boron, while the boron 30% free lay-\ners keep a damping of \u000b= 0:009[see fig. 2(d)]. Meanwhile\nthe anisotropies of these two free layers remain perpendicu-\nlar [fig. 2(c)] with \u00160(Hk\u0000Ms)being 0.27 and 0.17 T, re-\nspectively, after annealing at 400\u000eC. Let us comment on this\ndifference of damping.\nTwo mechanisms can yield to extra damping: spin-\npumping30and spin-flip impurity scattering of the conduc-\ntion electrons by a spin-orbit process31. Tantalum is known\nto be a poor spin-sink material as this early transition metal\nhas practically no delectrons and therefore its spin-pumping\ncontribution to the damping of an adjacent magnetic layer is\nweak30. We expect a spin pumping contribution to the damp-\ning of Ta (2 nm) / FeCoB (1.4 nm) / MgO ’single MgO’\nfree layers that compares with for instance that measured by\nMizukami et al. on Ta (3 nm) / Fe 20Ni80(3 nm) which was\nundetectable32since below 0.0001; we therefore expect that\nthe spin-pumping contribution to the total free layer damping\nis too negligible to account for the differences observed be-\ntween a free layer and the corresponding perfect single crys-\ntals. The main remaining contribution to the damping is the\nmagnon scattering by the paramagnetic impurities within the\nFeCoB material33. Indeed the Ta atoms within an FeCoB layer\nare paramagnetic impurities that contribute to the damping ac-\ncording to their concentration like any paramagnetic dopant;\nhowever the effect with Ta is particularly large34as Fe and Co4\n/s50/s50 /s50/s52 /s50/s54/s48 /s49\n/s49/s48/s50/s48/s51/s48\n/s49/s48 /s50/s48 /s51/s48/s49/s48 /s50/s48 /s51/s48\n/s48/s46/s51/s48/s46/s54\n/s40/s100/s41/s66/s111/s114/s111/s110/s32/s51/s48/s37\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41/s40/s99/s41\n/s40/s98/s41/s40/s97/s41/s66/s111/s114/s111/s110/s32/s50/s48/s37/s84/s114/s97/s110/s115/s118/s101/s114/s115/s101/s32/s112/s101/s114/s109/s101/s97/s98/s105/s108/s105/s116/s121\n/s32/s70/s77/s82/s32/s70/s114/s101/s113/s46/s32/s40/s71/s72/s122/s41/s32/s66/s32/s40/s84/s41\n/s72/s97/s108/s102/s32/s108/s105/s110/s101/s119/s105/s100/s116/s104/s32/s40/s71/s72/s122/s41\n/s32/s84\n/s97/s110/s110/s101/s97/s108/s32/s61/s32/s52/s48/s48/s176/s67\nFIG. 2. (Color online). Properties of single MgO free layers after\nannealing at 400\u000eC. (a) and (b): Real part (narrow lines) and imagi-\nnary part (bold lines) of the free layer permeability in a field of 0.7 T.\nThe lines are macrospin fits. (c) Ferromagnetic resonance frequency\nversus field curves. (d) Half linewidth versus FMR frequencies. The\nlines have slopes of \u000b= 0:009(red, B30%) and \u000b= 0:015(green,\nB20%)\natoms in direct contact with Ta atoms loose part of their mo-\nment and get an extra paramagnetic character, an effect usu-\nally referred as a ”magnetically dead layer”. Qualitatively, the\nTa atoms in the inner structure of the free layer degrade its\ndamping.\nAs the cap of Ta / FeCoB / MgO ’single MgO’ free lay-\ners contain many Ta atoms available for intermixing, a strong\ndegradation of the damping can be obtained in single MgO\nsystems when interdiffusion occurs. To prevent interdiffu-\nsion, we used the following strategy. Amorphous materials\n(including the glassy metals like FeCoB) are known to be ef-\nficient diffusion barriers, as they exhibit atom mobilities that\nare much smaller than their crystalline counterparts. To avoid\nthe diffusion of Ta atoms to the inner part of the FeCoB free\nlayer, a straightforward way is to maintain the FeCoB in an\namorphous state as long as possible during the annealing.\nIn metal-metalloid glasses, the crystallization temperature in-\ncreases with the metalloid content. In our FeCoB free lay-\ners, we find crystallization temperatures of 200, 300, 340 and\n375\u000eC for boron contents of respectively 10%, 20%, 25% and\n30%. Increasing the boron content in FeCo alloys is a way\nto conveniently increase the crystallization temperature and\nthus preserve a low damping. However since to obtain large\nTMR requires the FeCoB to be crystalline35,36, one should en-\ngineer the boron content such that the crystallization tempera-\nture matches with that used in the CMOS final BEOL anneal-\ning of 400\u000eC. In practice, we have found that this situation\nis better approached with a boron content of 30% than 0% to\n25%.D. Gilbert damping in single MgO and dual MgO free layers\nIn our search to further improve the free layers for STT-\nMRAM applications, we have compared the damping of op-\ntimized ’single MgO’ and optimized ’dual MgO’ free layers.\nFor a fair comparison, we first compare samples made from\nFeCoB with the same boron content of 20% and the same\n300\u000eC annealing treatement. From Fig. 1(b) and (c), there\nis a striking improvement of the FMR linewidths when pass-\ning from a single MgO to a dual MgO free layer. To discuss\nthis difference in linewidth, we have separated the Gilbert\ndamping contribution to the linewidth from the contribution\nof the sample’s inhomogeneity. We find that dual MgO sys-\ntems have systematically a substantially lower damping than\nsingle MgO free layers which confirms the trends indepen-\ndently observed by other authors9. Damping values as low\nas low as 0.0039\u00060:005were obtained in Ta 3 ˚A-spacer dual\nMgO stacks [Fig. 1(d)] after 300\u000eC annealing. Samples with\na thicker Ta spacer exhibit an increased damping (not shown).\nThis trend –lower damping in dual MgO systems –is main-\ntained after 400\u000eC annealing; for that annealing temperature,\nthe best damping are obtained for a slightly different internal\nconfiguration of the dual MgO free layer. Indeed a damping of\n0.0048 was obtained (not shown) in MgO / Fe 52:5Co17:5B30\n(1.4 nm) / Ta (0.2 nm) / Fe 52:5Co17:5B30(0.8 nm). This\nshould be compared with that the corresponding single MgO\nfree layer which had a damping of 0.009 for the same an-\nnealing condition [Fig. 2(d)]. This finding is consistent with\nthe results obtained on the single MgO free layer if we as-\nsume that the Ta impurities within an FeCoB layer contribute\nto the damping according to their concentration. Somehow,\nthe number of Tantalum atoms in the initial structure of the\nfree layer sets an upper bound for the maximum degradation\nof the damping upon its interdiffusion that can occur during\nthe annealing. Notably, the single MgO free layers contain\nmuch more Ta atoms (i.e. 2 nm compared to 0.2 to 0.4 nm)\navailable for intermixing: not only the initial number of Ta\nimpurities within the FeCoB layer directly after deposition is\nlarger in the case of single MgO free layer, but in addition a\nmuch stronger degradation of the damping can be obtained in\nsingle MgO systems when interdiffusion occurs, in line with\nour experimental findings. This interpretation – the dominant\nsource of damping is the Ta content – is further strengthened\nby the fact that the thickness of the Ta spacer strongly impacts\nthe damping in dual MgO free layers.\nLet us now study the spin-torque induced switching process\nin nanopillars processed from optimized MTJs.\nIII. SPIN-TORQUE INDUCED SWITCHING PROCESS\nA. Sample and methods for the switching experiments\nIn this section we use two kinds of perpendicularly magne-\ntized MTJ: a ’single MgO’ and a ’dual MgO’ free layer whose\nproperties are detailed respectively in ref.19and20. Note that\nthe devices are made from stacks that do not include all the\nlatest material improvement described in the previous sec-5\ntions and underwent only moderate annealing processes of\n300\u000eC. The ’single MgO’ free layer samples include a 1.4 nm\nFeCoB 20%free layer and a Co/Pt based reference synthetic\nantiferromagnet. Its most significant properties include19an\nareal moment of Mst\u00191:54mA, a damping of 0.01, an ef-\nfective anisotropy field of 0.38 T, a TMR of 150% . The ’dual\nMgO’ devices are made from tunnel junctions with a 2.2 nm\nthick FeCoB-based free layer and a hard reference system also\nbased on a well compensated [Co/Pt]-based synthetic antifer-\nromagnet. The perpendicular anisotropy of the (much thicker)\nfree layer is ensured by a dual MgO encapsulation and an iron-\nrich composition. After annealing, the free layer has an areal\nmoment of Mst\u00191:8mA and an effective perpendicular\nanisotropy field 0.33 T. Before pattering, standard ferromag-\nnetic resonance measurements indicated a Gilbert damping\nparameter of the free layer being \u000b= 0:008. Depending on\nthe size of the patterned device, the tunnel magnetoresistance\n(TMR) is 220 to 250%.\nBoth types of MTJs were etched into pillars of various size\nand shapes, including circles from sub-50 nm diameters to 250\nnm and elongated rectangles with aspect ratio of 2 and foot-\nprint up to 150\u0002300 nm. The MTJs are inserted in series\nbetween coplanar electrodes [Fig. 3(a)] using a device integra-\ntion scheme that minimizes the parasitic parallel capacitance\nso as to ensure an electrical bandwidth in the GHz range. The\njunction properties19,20are such that the quasi-static switching\nthresholds are typically 500 mV . Spin-wave spectroscopy ex-\nperiments similar to ref.37indicated that the main difference\nbetween the two sample series lies in the FL intralayer ex-\nchange stiffness. It is A= 8\u00009pJ/m in the 2.2 nm thick\ndual MgO free layers of the samples of ref.20and more usual\n(\u001920pJ/m) in the 1.4 nm thick ’single MgO’ free layers of\nthe samples of ref.19.\nFor switching experiments, the sample were characterized\nin a set-up whose essential features are described in Fig. 3(a):\na slow triangular voltage ramp is applied to the sample in se-\nries with a 50 \n oscilloscope. As the device impedance is\nmuch larger than the input impedance of the oscilloscope, we\ncan consider that the switching happens at an applied voltage\nthat is constant during the switching. We capture the elec-\ntrical signature of magnetization switching by measuring the\ncurrent delivered to the input of the oscilloscope [Fig. 3(b)].\nWhen averaging several switching events [as conducted in\nFig. 3(b)], the stochasticity of the switching voltage induces\nsome rounding of the electrical signature of the transition.\nHowever, the single shot switching events can also be cap-\ntured (Fig. 4-5). In that case, we define the time origins in\nthe switching as the time at which a perceivable change of the\nresistance suddenly happens (see the convention in Fig. 5).\nThis will be referred hereafter as the ”nucleation” instant.\nThis measurement procedure – slow voltage ramp and time-\nresolved current – entails that the studied reversal regime is\nthe sub-threshold thermally activated reversal switching. This\nsub-threshold thermally-activated switching regime is not di-\nrectly relevant to understand the switching dynamics in mem-\nory devices in which the switching will be forced by short\npulses of substantially higher voltage21. However elucidat-\ning the sub-threshold switching dynamics is of direct inter-\n50 ΩMTJVoltagebias\nOscilloscope50 Ω(a)(b)FIG. 3. (Color online). (a) Sketch of the experimental set-up. Mea-\nsurement procedure: the device is biased with a triangular kHz-rate\nvoltage (green) and the current (red) is monitored by a fast oscillo-\nscope connected in series. (b) The switching transitions are seen as\nabrupt changes of the current (red) followed by a change of the cur-\nrent slope. The resistance (blue) can be computed from the voltage-\nto-current ratio when the current is sufficiently non-zero. In this fig-\nure, the displayed currents and resistances are the averages over 1000\nevents for a 250 nm device with a dual MgO free layer of thickness\n2.2 nm and a weak exchange stiffness.\nest for the quantitative understanding of read disturb errors\nthat may happen at applied voltages much below the writing\npulses. Note finally that sending directly the current to the os-\ncilloscope has a drawback: the current decreases as the MTJ\narea such that the signal-to-noise ratio of our measurement\ndegrades substantially for small device areas (Fig. 5). As a\nresult, the comfortable signal-to-noise ratio allows for a very\nprecise determination of the onset of the reversal in large de-\nvices, but the precision degrades substantially to circa 500 ps\nfor the smallest (40 nm) investigated devices.\nB. Switching results\nIn samples whose (i) reference layers are sufficiently fixed\nto ensure the absence of back-hopping19and (ii) in which the\nstray field from the reference layer is rather uniform20, opti-\nmized compensation of the stray field of the reference layers\nleads to a STT-induced switching with a simple and abrupt\nelectrical signature [Fig. 4(a)]. If examined with a better time\nresolution, the switching event [Fig. 4(b)] appears to induce\na monotonic ramp-like evolution of the device conductance.\nFor a given MTJ stack, the switching voltage is practically in-\ndependent from the device size and shape in our interval of\ninvestigated sizes (not shown). This finding is consistent with\nthe consensual conclusion that the switching energy barrier\nis almost independent from the device area38,39for device ar-\neas above 50 nm. In spite of this quasi-independence of the\nswitching voltage and the device size, the switching duration\nwas found to strongly depend on device size (Fig. 5); we have\nfound that smaller devices switch faster, and the trend is that\nthe switching duration correlates linearly with the longest di-\nmension of the device. This is shown in Fig. 5: 40 nm devices\nswitch in typically 2 to 3 ns whereas devices that are 6 times6\n33PAP\nFIG. 4. (Color online). Single-shot time-resolved absolute value\nof the current during a spin-torque induced switching for parallel to\nantiparallel switching for a circular device of diameter 250 nm made\nwith a weak exchange stiffness, dual MgO 2.2 nm thick free layer.\n(a) Two microsecond long time trace, illustrating that the switching\nis complete, free of back-hopping phenomena, and occurs between\ntwo microwave quiet states. (b) 30 ns long time trace illustrating\nthe regular monotonic change of the device conductance during the\nswitching.\nlarger switch in 10 to 15 ns.\nSuch a reversal path can be interpreted this way: once a do-\nmain is nucleated at one edge of the device, the domain wall\nsweeps irreversibly through the system at a velocity set by\nthe applied voltage [sketch in Fig. 4(b)]. The average domain\nwall speed is then about 20 nm/ns for the low-exchange-free-\nlayers of ref.20. The other devices (not shown but described in\nref.19) based on a ’single MgO’ free layer with a more bulk-\nlike exchange switch with a substantially higher apparent do-\nmain wall velocity, reaching 40 m/s.\nC. Switching Model: domain wall-based dynamics\nTo model the switching, we assume that there is a domain\nwall (DW) which lies at a position qand moves along the\nlongest axis xof the device. The domain wall is assumed\nto be straight along the ydirection, as sketched in Fig. 4(b).\nWe describe the wall in the so-called 1D model40: the wall is\nassumed to be a rigid object of fixed width \u0019\u000epresenting a\ntilt\u001eof its magnetization in the device plane; by convention\n\u001e= 0is for a wall magnetization along x, i.e. a N ´eel wall.\n-10-5051015202501002000510m\nodelduration (ns)switchingd\nevice diam. (nm) \n90 nm \n60 nm 150 nm4\n0 nm80 nmCurrent (norm.)T\nime after nucleation (ns)250 nm  FIG. 5. (Color online). Single-shot time-resolved conductance traces\nfor parallel to antiparallel switching events occurring at at -0.5 V\nfor circular devices of various diameters. The curves are for the de-\nvices whose dual MgO free layer has a thickness of 2.2 nm and has a\nweak exchange stiffness. The curves have been vertically offset and\nvertically normalized to ease the comparison. The time origins and\nswitching durations are chosen at the perceivable onset and end of\nthe conductance change: they are defined by fitting the experimental\nconductance traces by 3 segments (see the sketch labelled ”model”).\nInset: duration of the switching events versus free layer diameter\n(symbols) and linear fit thereof with an inverse slope of 20 m/s.\nThe local current density at the domain wall position is\nwrittenj. The wall is subjected to an out-of-plane field Hz\nassumed to vary slowly in space at the scale of the DW width.\njis assumed to transfer p\u00191spin per electron to the DW by\na pure Slonczewski-like STT. We define\n\u001b=~\n2e\r0\n\u00160MSt(1)\nas the spin-transfer efficiency in unit such that \u001bjis a fre-\nquency. With typical FeCoB parameters, i.e. magnetization\nMs2[1:1;1:4]MA/m and free layer thickness t2[1:4;2:2]\nnm, we have \u001b2[0:018;0:036] Hz / (A/m2) where the low-\nest value corresponds to the largest areal moment Mst. With\nswitching current density of the order of 4\u00021010A/m2, this\nyields\u001bjdcbetween 0.72 and 1.4 GHz.\nFollowing ref.41, the wall position qand and wall tilt \u001eare7\nlinked by the two differential equations:\n_\u001e+\u000b\n\u000e_q=\r0Hz; (2)\n_q\n\u000e\u0000\u000b_\u001e=\u001bjdc+\r0HDW\n2sin(2\u001e) (3)\nin which\u0019\u000eis the width of a Bloch domain wall in an ultra-\nthin film, with \u000e2= 2A=(\u00160MsHeff\nk)whereAis the exchange\nstiffness. A wall parameter \u000e= 12 nm will be assumed for\nthe normal exchange 1.4 nm free layer from various estimates\nincluding ref.37for the exchange stiffness and ref.22for the\nanisotropy of the free layer. The domain wall stiffness field42\nHDWis the in-plane field that one would need to apply to have\nthe wall transformed from a Bloch wall to a N ´eel wall. As it\nexpresses the in-plane demagnetization field within the wall,\nit depends on the wall width \u0019\u000eand on the wall length when\nthe finite size of the device constrains the wall dimensions.\nUsing42, the domain wall stiffness field can be estimated\nto be at the most 20 mT in our devices. In circular devices,\nthe domain wall has to elongate upon its propagation38such\nthat the domain wall stiffness field HDWdepends in princi-\nple on the DW position. It should be maximal when the wall\nis along the diameter of the free layer. However we will see\nthatHDWis not the main determinant of the dynamics. In-\ndeed in the absence of stray field and current, the Walker field\nHWalker is proportional to the domain wall stiffness field times\nthe damping parameter, i.e. HWalker =\u000bH DW=2. As the sam-\nples required for STT switching are typically made of low\ndamped materials with \u000b < 0:01, the Walker field is very\nsmall and likely to be smaller than the stray fields emanating\nfrom either the reference layers or the applied field. This very\nsmall Walker field has implications: in practice as soon as\nthere is some field of some applied current, any domain wall\nin the free layer is bound to move in the Walker regime and\nto make the back-and-forth oscillatory movements that are in-\nherent to this regime. The DW oscillates at a generally fast\n(GHz) frequency43such that only the time-averaged velocity\nmatters to define how much it effectively advances.\nTo see quantitatively the effect of a constant current on the\ndomain wall dynamics, we assume that the sample is invari-\nant along the domain wall propagation direction (x) (like in\nan hypothetical stripe-shaped sample). Solving numerically\nEq. 2 and 3, we find that the Walker regime is maintained for\njdc6= 0(not shown). Two points are worth noticing:\nThe time-averaged domain wall velocity h_qivaries linearly\nwith the applied current density. When in the Walker regime,\nthe current effect can be understood from Eq. 3. Indeed the\nsin(2\u001e)term essentially averages out in a time integration as\n\u001eis periodic, and the term \u000b_\u001eis neligible, such that the time-\naveraged wall velocity reduces to:\nh_qi\u0019\u000e\u001bj dc (4)\nFor\u000e= 12 nm and\u001bjin the range of 1.4 GHz at the switching\nvoltage for the bulk-like exchange stiffness sample with free\nlayer thcikness 1.4 nm, the previous equation would predict atime-averaged domain wall velocity of 17 m/s (or nm/ns) dur-\ning the switching. More compact domain walls are expected\nfor the samples with a weaker exchange stiffness; the twice\nlower\u001bj\u00190:72GHz related to the larger thickness would\nreinforce this trend to a much a lower domain wall velocity (9\nm/s for our material parameters estimates). This expectation\ncompares qualitatively well with our experimental findings of\nslower walls in weakly exchanged materials (Fig. 5).\nWe wish to emphasize that Eq. 4 can be misleading regard-\ning the role of damping. Indeed a too quick look at Eq. 4 could\nlet people wrongly conclude the domain wall velocity is es-\nsentially set by the areal moment Mstand that the wall veloc-\nity under STT from a current perpendicular to the plane (CPP\ncurrent) is independent from the damping factor (see Eq. 1).\nHowever this is not the case as the switching current jdcis\na sweep-rate-dependent and temperature-determined fraction\n\u00112[1\n2;1]of the zero temperature instability current jc0of a\nmacrospin in the parallel state, which reads15,44:\njc0=\u000b4e\n~1 +p2\np\u00160MstHeff\nk\n2(5)\nwherep\u00191is an effective spin polarization.\nUsing Eq. 1, 4 and 5, the time-averaged wall velocity at the\npractical switching voltage is:\nh_qi\u0019\u000b\u000e\r 0Heff\nk\u0011 (6)\nThis expression indicates that the samples performing best\nin term of switching current (minimal damping and easy nu-\ncleation thanks to a small exchange) will host domain walls\nthat are inherently slow when pushed by the CPP current in\nthe Walker regime. The domain wall speed scales with the\ndomain wall width, which may be the reason why the low\nexchange stiffness samples host domain walls that are experi-\nmentally slower.\nTo summarize, once nucleated at the instability of the uni-\nformly magnetized state at jdc=\u0011jc0, the domain wall flows\nin a Walker regime through the device. The switching dura-\ntion varies thus simply with the inverse current:\n\u001cswitch =L\n\u000e\u001bj dc\u0019L\n\u000e\u00021\n\u000b\r0Heff\nk\u00021\n\u0011(7)\nLet us comment on this equation which is the main con-\nclusion of this section. The underlying simplifications are:\n(i) a rigid wall (ii) that does not sense the sample’s edges\n(iii) that moves at a speed equal to its average velocity in the\nWalker regime (iv) at a switching voltage that is independent\nfrom the sample geometry. Under these assumptions, the du-\nration of the switching scales with the length Lof the sam-\nple, as observed experimentally. It also scales with the inverse\nof the zero-field ferromagnetic resonance linewith 2\u000b\r0Heff\nk.\nThe practical switching voltage is below the zero temperature\nmacrospin switching voltage by a factor \u0011, which gathers the\neffect of the thermal activation and of the sweeping rate of the\napplied voltage45.\u0011\u00191=2for quasi-static experiments like\nreported here and \u0011!1for experiments in which the voltage\nrise timeVmax=_Vis short enough compared to the switching\nduration (Eq. 7).8\nIV . SUMMARY AND CONCLUSION\nIn summary, we have investigated the Gilbert damping of\nadvanced free layer designs: they comprise FeCoB alloys with\nvariable B contents from 20 to 30% and are organized in the\nsingle MgO or dual MgO free layer configuration fully em-\nbedded in functional STT-MRAM magnetic tunnel junctions.\nIncreasing the boron content increases the cristallization tem-\nperature, thereby postponing the onset of elemental diffusion\nwithin the free layer. This reduction of the interdiffusion of\nthe Ta atoms helps maintaining the Gilbert damping at a low\nlevel without any penalty on the anisotropy and the transport\nproperties. Thereby, increasing the Boron content to at least\n30% is beneficial for the thermal robustness of the MTJ up\nto the 400\u000erequired in CMOS back-end of line processing.\nIn addition, we have shown that dual MgO free layers have a\nsubstantially lower damping than their single MgO counter-\nparts, and that the damping increases as the thickness of the\nTa spacer within dual MgO free layers. This indicates that\nthe dominant source of extra damping is the presence of Ta\nimpurities within the FeCoB alloy. Using optimized MTJs,\nwe have studied the duration of the switching events as in-\nduced by spin-transfer-torque. Our experimental procedure –\ntime-resolving the switching with a high bandwidth but dur-\ning slow voltage sweep – ensures that we are investigating\nonly sub-threshold thermally activated switching events. In\noptimal conditions, the switching induces a ramp-like mono-\ntonic evolution of the device conductance that we interpret\nas the sweeping of a domain wall through the device. The\nswitching duration is roughly proportional to the device size:\nthe smaller the device, the faster it switches. We studied twoMTJ stacks and found domain wall velocities from 20 to 40\nm/s. A simple analytical model using a rigid wall approxima-\ntion can account for our main experimental findings. The do-\nmain wall velocity is predicted to scale linearly with the cur-\nrent for device sizes much larger than the domain wall widths.\nThe domain wall velocity depends on the material parame-\nters, such that the samples with the thinnest domain walls will\nbe the ones that host the domain walls with the lowest mo-\nbilities. Schematically, material optimization for low current\nSTT-induced switching (i.e. in practice: fast nucleation be-\ncause of low exchange stiffness Aand low damping \u000b) will\ncome together with slow STT-induced domain wall motion at\nleast in the range of device sizes in which the STT-induced re-\nversal proceeds through domain wall motion. 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Available:\nhttp://link.aps.org/doi/10.1103/PhysRevLett.92.088302"    },    {        "title": "1704.07006v1.Spin_injection_into_silicon_detected_by_broadband_ferromagnetic_resonance_spectroscopy.pdf",        "content": "1 \n Spin injection into silicon  \ndetected by broadband ferromagnetic resonance spectroscopy  \n \nRyo Ohshima,1 Stefan Klingler,2,3, Sergey Dushenko,1 Yuichiro Ando,1 Mathias Weiler,2,3 \nHans Huebl,2,3,4 Teruya Shinjo,1 Sebastian T. B. Goennenwein,2,3,4 and Masashi Shiraishi1* \n \n1Department of Electronic Science and Engineering, Kyoto Univ., 615 -8510 Kyoto, Japan.  \n2Walther -Meißner -Institut, Bayerische Akademie der Wissenschaften, 85748 Garching,  \nGermany.  \n3Physik -Department, Technische Universität München, 85748 Garching, Germany  \n4Nanosystems Initiative Munich , 80799 München, Germany  \n \n \nWe studied the spin injection in a NiFe(Py)/Si system using broadband \nferromagnetic resonance spectroscopy. The Gilbert damping parameter  of the Py layer \non top of the Si channel was determined as a function of the Si doping concentration  and \nPy layer thickness . For fixed Py thickness w e observe d an increase of the Gilbert damping \nparameter with decreasing resistivity of the Si channel . For a fixed Si doping \nconcentration we measured an increasing Gilbert damping parameter for decreasing Py \nlayer thickness.  No increase of the Gilbert damping param eter was found Py/Si samples \nwith an insulating interlayer . We attribute our observations to an enhanced spin injection \ninto the low -resistivity Si  by spin pumping . \n  2 \n Spin injection into semiconductors was relentlessly studied in recent years in hope to \nharness their long spin relaxation time, and gate tunability to realize spin metal -oxide -\nsemiconductor field -effect -transistors (MOSFET s). A central obstacle for a spin injection into \nsemiconductors was the conductance mismatch1 between ferromagnetic metals (used for the \nspin injection) and semiconductor channels. In an electrical spin injection method —widely \nused from the early years of the non -local spin transport studies —tunnel barriers between the \nsemiconductor and ferromagne t were formed to avoid the conductance mismatch problem2–5. \nUnfortunately, it complicated the production process of the devices , as high quality tunnel \nbarriers are not easy to grow , and presence of impurities, defects and pinholes take s a heavy \ntoll on th e spin injection efficiency and/or induces spurious effects. Meanwhile, the fabrication \nof electrical Si spin devices, like spin MOSFETs, with different resistivities is a time -\nconsuming process, which so far prevented systematic studies of the spin inject ion properties \n(such as spin lifetime, spin injection efficiency etc.). However,  such a systematic study is \nnecessary for further progress towards practical applications of spin MOSFETs.  \nIn 2002, a dynamical spin injection method, known as spin pumping, was introduced \nto the scene of spintronics research6,7. While the method was initially used in the metallic \nmultilayer sy stems, it was later  implemented to inject spin current s into semiconductors. In \ncontrast to electrical spin injection, spin pumping doe s not require the application of an electric \ncurrent across the ferromagnet/semiconductor interface. Devices that operate using spin \ncurrent s instead of charge current s can potentially reduce heat generation and power \nconsumption problems of modern electro nics. From a technological point of view, spin \npumping is also appealing because it does not require a tunnel barrier.  Spin injection —using \nspin pumping —into semiconductors from an adjacent ferromagnetic metal was achieved \ndespite the existence of conducti vity mismatch8-12. However, so far there was no systematic \nstudy of the spin pumping based spin injection in dependence on the resistivity of the Si channel.  3 \n In this letter, we focus on the study of spin injection  by spin pumping  in the NiFe(Py)/Si system  \nwith different resistivities  of the Si channel  using broadband ferromagnetic resonance (FMR) . \nThe broadband FMR method allows for a  precise determination of  the Gilbert damping \nparameter 𝛼, which increases in the presence of spin pumping, and  thus, spin injection . By \ntracking the change of the Gilbert damping parameter in various  Py/Si system s, we determine d \nthe spin pumping  efficiency  in the broad range of resistivities of the Si channel.   \nFor a first set of sample s 7nm-thick Py film s were  deposited  by electron beam \nevaporation  on top of various Si substrates  (1×1 cm2 in size) with resistivit ies in the range from \n10-3 to 103 ･cm (see Table 1 for the list of the prepared samples ). The oxidized surface of the \nSi substrates  was removed using 10% hydrofluoric acid (HF) prior to the Py evaporation. For \na second set of samples  Py films with  thickness es 𝑑Py between 5nm and 80nm  were deposited \non P -doped SOI  (silicon on insulator)  with the same technique . As a control  experiment , \nPy/AlO x and Py/ TiO x films were grown  on Si, P -doped SOI and SiO 2 substrates , as spin \npumping should be suppressed  in systems with an insulati ng barrier13 (see Table 2) . Both Al (3 \nnm, thermal deposition) and Ti  (2 nm, electron beam evaporation) were evaporated on the non -\ntreated substrates and left in the air for one day for oxidation of the surface (for the Al layer , \nthe process was repeated 3 times, with 1  nm of Al evaporated and oxidized at each step). After \noxidation, we evaporated 7nm thick  Py film s on the top of the tunnel barrier s. The properties \nof the prepared samples  are summarized in Table s 1 and 2 . \nA sketch of the broadband ferromagnetic res onance setup is shown in Fig. 1 (a). T he \nsamples were placed  face down on the center conductor  of a coplanar waveguide  (CPW), which \nwas located between the pole shoes of an electromagnet . A static magnetic field  |𝜇0𝐻|≤2.5 \nT was applied perpendicular to the surface of the samples to avoid extra damping due to two -\nmagnon scattering14. One end of the CPW was connected to a microwave source , where \nmicrowaves with frequenc y f < 40 GHz were generated . The other end of the CPW was 4 \n connected to a microwave diode and a lock -in amplifier to measure the rectified microwave \nvoltage  as a function of the applied magnetic field . All measurements were carried out at room \ntemperature.  \nThe microwave current in the  CPW generates an oscillating magnetic field around the \ncenter conductor  which results  in an oscillating torque on the s ample ’s magnetization. For \n𝜇0𝐻=𝜇0𝐻FMR  this torque results in a n absorption of microwave power . The resonance \ncondition i s given by the out -of-plane Kittel equation15,16: \n ℎ𝑓\n𝑔𝜇B=𝜇0𝐻FMR −𝜇0𝑀eff. (1)  \nHere,  ℎ is the Plan ck constant,  𝑔 is the Landé g-factor, 𝜇B is the Bohr magneton, 𝜇0 is \nthe vacuum permeability, and 𝑀eff is the effective saturation magnetization of Py.   \nWe use  the Gilbert damping model, which phenomenologically models the viscous \ndamping of the magnetic resonance. The linear relat ion between the full width at half maximum \n𝛥𝐻 of the resonance and the applied microwave frequency f is given by the Gilbert damping \nequation 17: \n 𝜇0𝛥𝐻=𝜇0𝛥𝐻0+2𝛼ℎ𝑓\n𝑔𝜇B. (2)  \nHere,  𝛥𝐻0 corresponds to  frequency independent scattering processes  and 𝛼 is the Gilbert  \ndamping parameter18,19: \n 𝛼=𝛼0+𝛼SP+𝛼EC. (3)  \nHere, 𝛼0 is the intrinsic Gilbert damping , 𝛼SP=𝑔𝜇𝐵𝑔r↑↓4𝜋𝑀S𝑑Py ⁄  is the damping due to \nspin pumping20, 𝑔r↑↓ is the real part of the spin mixing conductance,  and 𝛼EC=𝐶EC𝑑Py2 is the \neddy -current damping . The parameter  𝐶EC describes efficiency of  the eddy -current damping . \nTo realize a net spin injection via spin pumping, the following conditions should be \nfulfilled in the system: (i ) carriers should be present in the underlying channel, (ii) the spin \nrelaxation time in the channel should be small enough. The available carriers in the channel \ntransfer spin angular momentum away from the spin injection interface, allowing propagation 5 \n of the spin current. On the other hand, the long spin relaxation time in the channel leads to a \nlarge spin accumulation at the interface and generates a diffusive spin backflow in the direction \nopposite to the spin pumping current20 (see Figs. 1(b) and 1(c )). Thus, the spin backflow \neffectively cancels out the spin pumping current for long spin relaxation time, and the spin \npumping contribution to the Gilbert damping parameter should no longer be present in the \nsystem.  \nIn addition to spin pumping,  charge  currents can be induced in the Si channel  and Py  \nlayer due to  the Faraday ’s law  and Py magnetization precession, which results  in a Gilbert -like \ndamping contribution . These processes are  refer red to as radiative damping  and eddy -current \ndamping18. An e nhancement of the Gilbert damping parameter due to these processes  is \nexpected to be especially large for the Si channels with low resistivit ies and thick Py films , \nsince energy dissipation through eddy currents scales linearly  with the conductivity  of the  Si \nlayer  and quadratically with the Py layer thickness . Hence, both spin pumping and eddy current \ndamping are expected to be most efficien t for low -resistivity Si. I n contrast  to spin pumping , \nthe radiative damping  contribution doe s not  require a direct electrical contact between Py and \nSi, and is hence unaffected by the tunnel barrie r. \nFigure 1(d ) show s a typical FMR spectr um of a 7nm thick Py film on a phosphorous \nP-doped Si on insulator (SOI) substrate , where t he microwave frequency was fixed at 30 GHz  \nduring the sweep of the magnetic field. A single FMR  signal  was observed  (Fig. 1(d) red filled \ncircles) , from which 𝜇0𝐻FMR  and 𝜇0𝛥𝐻  were extracted  by a fit of the magnetic ac \nsusceptibility  (Fig. 1(d) black line) 16,21 (see Supplemental Material for additional fitting \nexamples).  An excellent agreement of the fit with the measurement is achieved.  \nFigures 2(a) and (b) show 𝐻FMR and 𝛥𝐻 versus the applied microwave frequency f \nfor the Py/P-doped SOI, Py/SOI and Py/SiO 2 samples. From the f itting of the frequency \ndependence of HFMR with Eq.(1) , 𝑔  and 𝜇0𝑀eff  of the Py /P-doped SOI ( Py/SOI) were 6 \n estimated to be 2.049 (2.051) and 0.732 T (0.724 T), and those of the Py /SiO 2 were estimated \nto be 2.038 and 0.935 T, respectively  (Supplemental Material  for fitting  and data from other \nsamples ). The difference in the 𝑔  and 𝑀eff  between the Py /Si and the Py/ SiO 2 sample  is \nattributed to the inter -diffusion of the Fe/Ni and Si at the  interface , which is  always present to \nsome extent during the growth at room temperature22,23. The Gilbert damping  𝛼 of the Py /P-\ndoped SOI , Py/SOI and Py/SiO 2 were estimated to be 1.25 ×10-2, 9.02 ×10-3 and 8.49 ×10-3, \nrespectively , from the linewidth vs. frequency evolution . The intrinsic Gilbert damping \nparameter 𝛼0  is determined from the linewidth evolution of the Py/SiO 2 and Py/quartz \nsamples to be 8.5×10−3 and 8.6×10−3, respectively, since no spin pumping contribution is \nexpected in these insulating materials ( 𝛼=𝛼0). From this we can see an increasing Gilbert \ndamping with decreasing resistivity. We additionally measured the samples with a n insulating \ntunnel barrier between the Si channel and the Py film. We found Gilbert damping parameters \nof 𝛼 = 8.8×10-3 for the Py/ AlO x/Si samples and 𝛼 = 7.5×10-3 for the Py/TiO x/Si samples, \nindependent of the Si resistivity.  The damping  values are in agreement with the intrinsic \ndamping extracted from the Py/SiO 2 sample , indicating that radiative damping is negligible in \nour samples.  \nFigure 3 (a) summarizes  the dependence of the Gilbert damping parameter 𝛼 on the \nresistivity of the Si channel (see Supplemental Material D for the g -factor, the effective \nsaturation magnetization and the frequency independent term ), including the measured control \nsamples.  The dashed lines  show  the intrinsic contributions 𝛼0  to the Gilbert damping \nparameter 𝛼 measured from the Py/SiO 2 and Py/quartz samples (red dashed), Py/AlO x/SiO 2 \n(blue dashed) and Py/TiO x/SiO 2 (green dashed). All  samples with Py on top of the conductive \nsubstrates without an additional tunnel barrier exhibited the Gilbert damping parameter 𝛼 \nlarger than the intrinsic contribution 𝛼0.  \nThe experimentally measured Gilbert damping parameter decreases logarithmically 7 \n with the resistivity. This result is in agreement with condition (i) for the spin pumping. In the \nSi channels with a small resistivity more carriers were available to transfer the injected angular \nmomentum, leading to an effective spin pumping. Additionally, both electron spin resonance25–\n28 and non -local 4 -terminal Hanle precession29,30 experiments showed, that the spin lifetime in \nSi is increas ing with increasing resistivity . Our samples with low resistivities have a large \ndoping concentration (see Table 1), leading to shorter spin relaxation time. In accordance with \nthe spin pumping condition (ii), the decrease of the spin relaxation time should lead to the \nincrease of the spi n pumping contribution, as now observe d experimentally. While the spin \npumping shows a logarithmic dependence on the resistivity of the channel, we note that an \nincreased Gilbert damping parameter  is observed even for the Si channel with high resistivit ies. \nWe comment on the Sb -doped sample, where the experimentally measured 𝛼SP was lower \nthan one expected from the logarithmic trend of the other samples. We speculate that this might \noriginate from the different doping profile, compared to the other samples . We note, that further \nstudies are necessary to separate the influence of the number of carriers in the channel and the \nspin relaxation time on the spin pumping process.  \nFinally, we show that the Py damping is increased in a broad range of Si resistivitie s \nand attribute this effect to the enhanced spin injection via spin pumping ( a discussion  of the \nspin mixing conductance  for various Si  resistivities is given in the Supplemental M aterial A). \nFigure 3(b) shows the Py thickness dependence of the 𝛼 for Py/P -doped SOI samples. The \nsolid line shows a fit of Eq.(3) to the measured data and a very good agreement of the spin \npumping theory with our measurements is achieved. From the fit we estimate 𝛼0=6.1×10−3, \n𝑔r↑↓=1.2×1019 m-2 and 𝐶EC=2.9×1011 m-2. Both the intrinsic damping and the real part \nof the spin mixing conductance are in good agreement with previous measurements30. The blue \ndashed line indicates 𝛼0+𝛼SP  and the green dashed line shows 𝛼0+𝛼EC . Dominant \ninfluence of spin pumping to the total damping is observed in samples with  small Py thickness , 8 \n while  eddy current contribution is dominant in samples with  thick Py  layer . For the 7 nm -thick \nPy sample, we find 𝛼SP = 3.6 ×10-3 and 𝛼EC = 1.4 ×10-5. Thus, the eddy -current damping in \nour 7 nm Py samples is negligibly small and cannot explain the increase of the damping with \ndecreasing resistivity. The Py thickness dependence of the Gilbert damping indicates  spin \npumping into the Si substrates.  \nIn conclusion , we studied spin pumping based spin injection from a Py layer into Si \nchannels with various resistivit ies using broadband ferromagnetic resonance . We determine d \nthe spin pumping contribution from the  change of the Gilbert damping parameter. The observed \nlogarithmic decrease of the Gilbert damping parameter with  increasing resistivity of the Si \nchannel is attribute  to the decrease in the number of carriers in the channel, and the increase in \nthe spin lifetime. De spite  the reduction of the spin pumping contribution to the Gilbert damping \nparameter with the increasing  resistivity  of the Si channel , we observe  spin pumping even for \nthe channels with high resistivity . We furthermore observe an increase of the Gilbert damping \nparameter for decreasing Py thickness which is in agreement with the spin pumping theory. \nOur results show that spin pumping can be potentially used in  a spin transistors, where low \ndoping concentration in the channel is necessary for the gate control of the device.  \n \nSupplement al Material  \n See Supplementary M aterial for a discussion  of the spin mixing conductance  for \nvarious Si resistivities and additional  fitting examples.  \n \nACKNOWLEGEMENTS  \nThis research was supported in part by a Gran t-in-Aid for Scientific Research from \nthe Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan, \nInnovative Area “Nano Spin Conversion Science ” (No. 2 6103003), Scientific Research (S) 9 \n “Semico nductor Spincurrentronics ” (No. 16H0633) and JSPS KAKENHI Grant  (No. \n16J00485).  R.O. acknowledges JSPS Research Fellowship. 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Shiraishi, Y. Suzuki, and K. Noguchi, Appl. Phys. \nExpress 4, 23003 (2011).  \n29 T. Tahara, Y. Ando, M. Kameno, H. Koike, K. Tanaka, S. Miwa, Y. Suzuki, T. Sasaki, T. \nOikawa, and M. Shiraishi, Phys. Rev. B 93, 214406 (2016).  \n30 Note, that the sample set with various Py thicknesses was grown in a different batch than \nthe samples with v arious Si doping. Hence, small deviations in the damping and the spin 12 \n mixing conductance are due to slightly different growth conditions . \n \nFigure 1: (a) Experimental setup for the broadband FMR measurement. The samples were \nplaced with the Py layer facing down on a coplanar waveguide. External magnetic field and \nmicrowave field from the waveguide induce the FMR of the Py and spins are injec ted into Si \nvia spin pumping. Schematic images of spin injection and dephasing in Si that have (b) long  \nand (c) short spin lifetimes. 𝜏1 and 𝜏2 are the spin lifetim e of Si in the case of (b) and (c), \nrespectively. Spin injection efficiency becomes large in the case of (c) because of a reduction \nof the backflow of spins. (d ) The derivative of the FMR signal of Py at 30 GHz microwave \nfrequency.  I is the microwave absorption intensity.  \n \nFigure 2: Frequency dependence of the (a) resonance field 𝐻FMR and (b) full width at half \nmaximum 𝛥𝐻 of the FMR spectra obtained from Py on top of P -doped SOI, SOI and  SiO 2. \nThe solid lines show fitting using  Eqs. (1) and (2) of 𝐻FMR and 𝛥𝐻, respectively.  \n \nFigure  3: (a) Si resistivity dependence of the Gilbert damping parameter  𝛼. The damping of \nthe samples with an insulating layer represents the intrinsic damping of the Py layer and is \nshown by the dashed line s. Red, blue and green coloration represents Py, Py/AlO x and Py/TiO x \nsamples, respectively. The damping of the Py/ P-doped SOI is an averaged value extracted from \nthe two Py/P-doped SOI samples fabricated at different times.  (b) Py thickness dependence of  \n𝛼. The solid line shows  a fit of Eq. (3) to the data . The b lue line shows 𝛼0+𝛼SP, whereas the \ngreen line shows 𝛼0+𝛼EC. \n \n \n 13 \n Fig. 1 R. Ohshima et al . \n \n \nFig. 2  R. Ohshima et al.  \n \n14 \n Fig. 3  R. Ohshima et al.  \n \n \nTable 1: Sample summary  \nName  Dopant  Doping \ndensity (cm-3) Structure  Resistivity \n(･cm) Gilbert damping \nparameter  Mixing \nconductance (m-2) \nPy/P -\ndoped SOI  P 6.5×1019 Py(7 nm)/Si(100 nm)/  \nSiO 2(200 nm)/Si  1.3×10−3 1.1×10−2 5.7×1018 \nPy/Sb -\ndoped Si  Sb 1×1019 Py(7 nm)/Si  5.0×10−3 9.3×10−3 2.3×1018 \nPy/N -\ndoped Si  N 1×1019 Py(7 nm)/Si  1.0×10−1 9.5×10−3 2.6×1018 \nPy/SOI  N/A 1×1015 Py(7 nm)/Si(100 nm)/  \nSiO 2(200 nm)/Si  4.5 9.0×10−3 1.3×1018 \nPy/P -\ndoped Si  P 1×1013 Py(7 nm)/Si  1.0×103 8.7×10−3 5.1×1017 \nPy/SiO 2 - - Py(7 nm)  \n/SiO 2(500 nm)/Si  - 8.5×10−3 - \nPy/Quartz  - - Py(7 nm) /Quartz  - 8.6×10−3 - \n \n \n \n \n \n15 \n Table 2: List of the samples for the control experiment  \nName  Dopant  Doping \ndensity (cm-3) Structure  Resistivity \n(･cm) Gilbert damping \nparameter  \nPy/AlO x/P-\ndoped SOI  P 6.5×1019 Py(7 nm)/AlO x(3 \nnm)/Si(100 nm)/  \nSiO 2(200 nm)/Si  1.3×10−3 8.6×10−3 \nPy/AlO x/P-\ndoped Si  P 1×1013 Py(7 nm)/AlO x(3 nm)/Si  1.0×103 8.5×10−3 \nPy/AlO x/ \nSiO 2 - - Py(7 nm)/ AlO x(3 nm)/  \nSiO 2(500 nm)/Si  - 8.8×10−3 \nPy/TiO x/P-\ndoped SOI  P 6.5×1019 Py(7 nm)/TiO x(2 \nnm)/Si(100 nm)/  \nSiO 2(200 nm)/Si  1.3×10−3 7.5×10−3 \nPy/TiO x/P-\ndoped Si  P 1×1013 Py(7 nm)/TiO x(2 nm)/Si  1.0×103 7.9×10−3 \nPy/TiO x/ \nSiO 2 - - Py(7 nm)/TiO x(2 nm)/  \nSiO 2(500 nm)/Si  - 7.8×10−3 \n \n "    },    {        "title": "1110.5112v3.CoB_Ni_Based_Multilayer_Nanowire_with_High_Speed_Domain_Wall_Motion_under_Low_Current_Control.pdf",        "content": "1\nCoB/Ni-Based Multilayer Nanowire with High-Speed Domain Wall Motion \nunder Low Current Control \nDuc-The Ngo*, Norihito Watanabe, and Hiroyuki Awanoj \nInformation Storage Materials Laboratory, Toyota Technological Institute, Nagoya 468-\n8511, Japan \n \n The spin-transfer torque motion of magnetic DWs in a CoB/Ni-based nanowire driven by \na low current density of (1.12±0.8)×10\n11 A m-2 has been observed indirectly by \nmagnetotransport measurements. A high DW velocity of 85±4 m/s at zero field was measured at the threshold current density. Upon increasing the current density to 2.6×10\n11 \nA m-2, the DW velocity increases to 197±16 m/s before decreasing quickly in the high-\ncurrent-density regime owing to nonadiabatic spin-transfer torque at a low damping factor and weak pinning. The addition of B atoms to the Co layers decreased the level of \nsaturation magnetization, Gilbert damping fact or, and density of pinning sites, making the \nCoB/Ni multilayer nanowire favora ble for practical applications.  \n \n \n \n*Present address: Department of Electrical and Computer Engineering, National University of Singapore, 4 \nEngineering Drive 3, Singapore 117576.  \nj E-mail address: awano@toyota-ti.ac.jp. 2\n \n1. Introduction \nWhen a spin-polarized electron current hits a magnetic moment, it exerts a torque on the \nmoment, transfers its angular momentum to the moment, and thereby affects the precession motion and switching of the mo ment. This phenomenon was theoretically \npredicted by Berger [1] and Slonczewski [2], and subsequently was named the spin-\ntransfer torque (STT). The motion of magne tic domain walls (DWs) caused by an \nelectrical current in magnetic nanostructures is also a consequence of the SST in which \nthe spin-polarized current switc hes the magnetic moments in the wall. This is nowadays \nwidely applied in spintronic technology such  as the DW logic gate [3,4] and racetrack \nmemory [5, 6]. Over the last 15 years, most studies have focused on a NiFe patterned \nfilm, a typical soft magnetic material with in-plane magnetic anisotropy and nearly zero \nmagnetocrystalline anisotropy, as it is cheap and highly stable and it is easy to fabricate and control its composition and properties. The motion of DWs with a very high velocity, up to ~200 m/s, has been demonstrated in a number of NiFe-based nanowire devices [3-6]. However, the motion in such in-plane anisotropy films was controllable only at a \nrelatively high current density (~10\n12 A m−2) owing to a wide DW and a low spin-torque \nefficiency (it should be noted that the threshold current density is expected to be proportional to the wall width [7]). A high control current consumes much energy and the \nheat released from electrical current would sometimes degrade the performance of such \ndevices. Therefore, deceasing of current dens ity is one of the most important technical \nissues at the moment.  3\nPerpendicularly magnetized thin films have recently been proposed to replace the \nin-plane NiFe film [8,9] to realize this goa l. In the perpendicular magnetic anisotropy \nfilms, formation of Bloch-type walls that are 1-2 orders of magnitudes thinner than Néel-\ntype walls in the in-plane films and a high spin-torque efficiency would lower the intrinsic current density by one or even two orders of magnitude. Many authors [10-13] \nhave presented the decrease in threshold current to (2-5)×10\n11 A m−2 using multilayer \nnanowires, e.g., Co/Pt, CoFe/Pt, and Co/Ni in which the perpendicular magnetic anisotropy was one of the keys to decrease the critical current density. Nonetheless, layer \nthickness in those multilayers was normally ~3-20 Ä and might be badly influenced by \nthe heat from electrical current. The perpendicular anisotropy is logically threatened to disappear owing to the diffusion of the layers under Joule heating of electrical current.  \nAmong the researchers, Yamanouchi et al. [14] were successful in establishing \nthe motion of magnetic DWs in a perpendicularly magnetized ferromagnetic semiconductor (Ga,Mn)As with a very low current density of about 10\n9 A m−2. However, \nthis material (and most ferromagnetic semiconductors) has a Curie temperature far below room temperature and therefore is not realistic for room-temperature devices. \nIn this article, we present the enhancement of the motion of the magnetic DWs in \nthe CoB/Ni multilayer nanowire at a low current density. The addition of B atoms to the Co layers decreased the density of the pinning sites in the film, enhanced the DW motion, \nand improved the stability of the multilayer  by preventing the diffusion between the \nCo/Ni interfaces.  \n 4\n2. Experimental Methods \nA multilayer film of Pt 5 nm/[CoB 0.6 nm/Ni 1.1 nm]4/CoB 0.6 nm/Pt 1 nm was \nfabricated by radio-frequency (RF) magnetron sputtering using Ar gas. The base vacuum \nof the deposition chamber was 3×10−8 Torr whereas the Ar pressure was maintained at 5 \nmTorr during the deposition process. The com position of the CoB ta rget was chosen as \nCo80B20 (at.%). The film was grown on a naturally oxidized Si substrate. A nanowire \nwith 300 nm width and 150 µm length was subsequently patterned by electron beam \nlithography and ion beam etching (Fig. 1). The nanowire was modified to have a planar \nHall shape for magnetotransport measurements. A square pad was made at one end of the \nwire as a source for DW nucleation [15], and the shape of the other end of the wire was modified to be triangular to prohibit the propagation of DW [16]. A Ti/Au electrode \npattern produced by photolithography was mounted to the wire for magnetotransport \nmeasurements. The magnetic properties of the film specimen were measured using an alternating gradient magnetometer (AGM).  \nThe magnetic DWs were nucleated in the square pad by an Oersted field \ngenerated from the 30 ns width, 6.5 MHz pulse current flowing in the Ti/Au electrode \ndeposited on the pad (Fig. 1, electrodes A-B). The motion of DWs in the nanowire was \nthen driven by a DC current (J\nDC, electrodes G-B in Fig. 1). Anomalous Hall effect \n[17,18] measurement (either electrodes C-D or E-F) was carried out to detect the \npropagation of DW in the wire. Hysteresis loop measurement on the continuous film \nspecimen (data not shown) using the AGM confirmed that the film exhibited a strong perpendicular magnetic anisotropy with a saturation magnetization of M\ns=5.6×105 A/m 5\n(about 15% lower than that of a Co/Ni-based film [13]) and a uniaxial anisotropy \nconstant of K u=3.57×105 J/m3 (~8% higher than that of a Co/Ni film [12, 13]). \n \n3. Results and Discussion \nThe time-resolved Hall effect voltage signal obtained from the nanowire at a driven \ncurrent (DC current) of 0.59 mA corresponding to a current density of  jDC=1.12×1011 A \nm−2 and an external field of +5 mT is illustrat ed in Fig. 2(a) and represents the movement \nof DW along the wire [17,18]. Initially, the wire was magnetically saturated, then \nmagnetization reversal was induced by the Oersted field generated from the pulse current \nwith nucleation of a tiny domain at the region between the wire and the square pad. The DC current sequentially forced the domain (with two walls on two sides) moving along \nthe wire, toward the Hall bar. When the domain moved into the Hall bar, the presence of \nthe domain in the cross bar induced a change in  the Hall voltage signal, as seen in Fig. \n2(a). The Hall signal switched to a low valu e when the domain (w ith the two walls) had \npassed the Hall bar. The progress of the Ha ll signal could be interpreted approximately \non the basis of a simple schematic shown in Fig. 2(b). Because of a periodic pulse, the \ndomains were nucleated and driven to the wire periodically and the Hall voltage signal \nappeared to be a periodic pulse. This result looks similar to the DW motion observed previously [17,18]. A square Hall-voltage hysteresis loop [inset of Fig. 2(a)] exhibits a \nsharp change in the magnetization, proving a fast propagation of a domain through the \nHall bar. The square aspect of the hysteresis loop indicated that a reversal occurred through DW nucleation followed by easy DW propagation. It should be noted that the \nOersted field released from the driven current was estimated to be about 30 mT, which 6\nwas much smaller than the coercive field of  the sample (see the Hall effect hysteresis \nloop in the inset of Fig. 2). Therefore, the influence of the Oersted field on the motion of \nthe wall along the wire (described in Fig. 2) could be minor, whereas the effect of the \nspin-polarized current is essentially considered.  \nThe current dependence of the variation of normalized Hall resistance, ΔRHall, is \nshown in Fig. 3. The normalized Hall resistance here was defined as the change in the \nHall voltage signal when the domain propagated through the Hall bar [Fig. 2(b)]. Therefore, the normalized Hall resistance became high (1) above a threshold current \ndensity of 1.12×10\n11 A m−2, whereas this value was low (0) below the threshold current \ndensity. This indicates that the motion of the magnetic DWs, denoted by a change in Hall resistance, could be induced when the density of the spin-polarized current is above \n1.12×10\n11 A m−2, confirming that it is possible to drive the DW motion in the 300 nm \nwidth CoB/Ni nanowire with a threshold current density of 1.12×1011 A m−2 by the STT \nmechanism. It is important that the threshold current density obtained here was reasonably lower than either ~(1-3)×10\n12 A m−2 in the NiFe-based devices [6,7] or ~(2-\n5)×1011 A m−2 in a similar multilayer Co/Ni wire [12,13], or lower than the current \ndensity in a spin-valve nanowire [19] reported recently. Moreover, it is shown in Fig. 2 \nthat the pulse like signal of Hall voltage is periodic and coherent with the nucleation \npulse, presuming the continuous propagation of a multidomain similar to a shift register \nwriting process. \nFrom the pulse like Hall signal, the velo city of DW moving in the Hall bar could \nbe derived [13] from Fig. 2(b): T 1 was the time when the front-edge wall of the domain \nstarted coming to the Hall bar and T 2 was the time that it passed the Hall bar (in L = 500 7\nnm). Therefore, the velocity of the front-edge wall could be referred as L/ Δt1 (Δt1 = T 2-\nT1). On the other hand, the velocity of the rear-edge wall could be attained from the time \ninterval Δt2 = T 4 – T 3. On the other hand, from the phase delay between the signals at the \nC-D and E-F Hall bars that reflected the time of flight of the wall between two Hall bars, \nthe velocity of the wall in the straight wire  (from C to E) was de termined. Figure 4 shows \nthe wall velocity in the straight wire area as a function of external magnetic field \nmeasured at the threshold current density. The field dependence here is consistent with the following expression [20]:  \nv(H) = µ\nH(H - H 0) + v(J),   (1) \nwhere µ H(J) is the DW mobility, J is the current density, and H 0 is the “dynamic coercive \nforce”. The term v(J) - µ HH0 can be referred as the velocity at zero field.  \nUsing this linear dependence, a zero-field wall velocity of 86±5 m/s was \ncalculated at the critical current density (1.12×1011 A m−2) with a mobility of 2640±170 \n(m s−1 T−1). This matches well with the velocity measured directly at H = 0 (85±4 m/s). It \nis interesting to note that the field-free wa ll velocity here was much higher than that of \nthe Co/Ni wire [12,13] or TbFeCo nanowire [18,21]. Therefore, this aspect is very \npromising for high-speed devices. As DW moved in the region of the Hall bars, the wall \nvelocity (as defined above) was found to be sli ghtly lower than that in the straight wire \narea but only in the error scale of the measurement. The non-zero DW velocity and linear \ndependence of the wall velocity on the external field can be attributed to the motion \ndriven by the nonadiabatic torque [20]. \nThe velocities of the front-edge and rear-edge walls were perfectly identical to \neach other and remained invariable at positions of two Hall bars. These suggest that i) the 8\neffect of the pinning on the motion of th e walls along the wire was predominantly \ngoverned by the material rather than the geometry of the Hall bars and  ii) no distortion of \nthe domain and the wall geometry as the domain length was conserved when they were \nlocated in the Hall bars. Usually, the distortion  of the domain in the Hall bar, denoted by \nthe small difference between the velocities of the front-edge wall (faster) and the rear-\nedge wall (slower), was only observed at a high applied field (over 90 mT), and can be \nimagined similarly to the distortion of a balloon, as reported elsewhere [22]. \nRegarding other interesting points, the time interval T 3 - T 1 [see Fig. 2(b)] \nexpresses the period necessary for the whole domain to reach the rear side of the Hall bar, \nallowing domain size to be estimated. Using this relationship, the average size of the domain was calculated to be 900±35 nm at  the critical current density of 1.12×10\n11 A m−2 \nand zero field. Under an external field, the domain size was slightly reduced to 630±25 \nnm at the field of 40 mT, which was similar to theoretical prediction [23]. It is supposed that the external field in this case acted oppositely to the nucleated field from the pulse current, and compressed the domain when it was nucleated. It should be noted that the domain length was conserved when the domain was located in two Hall bars. \nThe dependence of wall velocity at zero field on controlled current density is \nshown in Fig. 5. The velocity firstly incr eased with current density from 85 m/s at the \nthreshold current density to a maximum value of 197±16 m/s at a current density of \n2.63×10\n11 Am−2, then markedly dropped at higher current densities. The variation of wall \nvelocity with current density in this case can be explained qualitatively by referring to the \nmodel given in refs.#7 and 24. The model proposed by Tatara et al. [24] predicted that \nthe trend of the wall velocity variation (including a linear increase at low current and a 9\ndecrease with increasing current at high cu rrents) is a consequence of the nonadiabatic \ntorque driving when the damping factor is low and the pinning effect is weak. In the low \ncurrent regime, DW velocity was linearly dependent on current density, which is in \naccordance with the zero-field velocity de scribed in eq. (1) and somehow similar to a \nprevious experimental observation [20]. At a high current density (above 2.63×1011 A \nm−2), wall velocity appeared to decrease, indicating that the nonadiabatic parameter β was \nnot zero and not equal to the Gilbert damping fa ctor. This led to a deformation of the wall \nstructure above the Walker breakdown current density [20]. This dependence and linear \nfield-velocity function discussed in previous paragraphs indicated that the motion of the \nwalls in our device was mainly governed by the nonadiabatic term. \nIn an attempt to explain the decrease in the threshold current density, theoretical \nmodels [7,24] are employed, in which the threshold (or intrinsic critical) current density \ncould be referred to as follows [7]: \n21~s\nc\nBeMJgP\n ,  (2) \nwhere α and β are the Gilbert damping factor and nonadiabatic spin-transfer torque \nparameter, respectively;  is the DW width; Ms is the saturation magnetization; P is the \nspin polarization of the material; g is the gyromagnetic ratio,  e is the electron charge, and \nB is the Bohr magneton.  \nAs discussed in the previous paragraphs, the magnitude of saturation \nmagnetization of the studied CoB/Ni multilayer film was decreased by ~15%, whereas \nuniaxial anisotropy was slightly enhanced which subsequently led to a thinner DW. \nAdditionally, the substitution of B for Co is expected to decrease the Gilbert damping \nfactor of the film. Hence, such decreases in saturation magnetization magnitude and wall 10\nwidth could result in a decrease in the threshold critical current. Moreover, the addition of \nB atoms on the other hand weakens the pinning of the DW by decreasing the number of \npinning sites [25] as mentioned in previous paragraphs. This effect also enhances the \nvelocity of the walls. It should be noted that the Walker current at which the wall velocity \ndropped as seen in Fig. 5, is also expected to be a function of the intrinsic parameters of \nthe materials [26,27]. \nAdditionally, the addition of B to Co would make the devices more stable. B atoms with a small atomic radius would locate at the vacancies in the Co lattice and increase the \nclosed-package degree of the lattice, thus preventing the diffusion between the layers and \npreserving the magnetic properties of the film under heating caused by the electrical current applied. The addition of B atoms to th e Co lattice also decreases the difference in \nthe lattice constant between Co/Ni and result s in a smooth Co/Ni interface. It should be \nnoted that 20% addition of B to the Co layers (Co\n80B20) only gives rise to ~12% of \nresistivity in comparison with pure Co laye rs. From the technical point of view, these \nbenefits would to enhance the working stability of the devices.  \n \n4. Conclusions \nThe motion of magnetic DW in the CoB/Ni multilayer nanowire with a very low current \ndensity of (1.12±0.8)×1011 A m-2 and a high DW velocity of 85±4 m/s has been \nsuccessfully induced. DW velocity can be raised up to 197±16 m/s by increasing the \ncurrent density to 2.63×1011 A m−2. The variation of wall velocity was consistent with the \nnonadiabatic STT mechanism. These advantages were attributed to the presence of CoB \nlayers with a low Gilbert damping factor, a low saturation magnetization, and a low 11\ndensity of pinning sites. The addition of B also helps in preventing  the diffusion between \nCo and Ni layers and enhances the stability of the multilayer structure and the \nperformance of our device. Using a 30 ns pulse as a writing current, the device could \nperform shift-register writing of a multidomain state in the wire with an average domain size of 900±35 nm (without field) and a minimum size of 630±25 nm (with field). \n \nAcknowledgments \nThis work was completed with the financial support from the Toyota School Foundation. \nWe thank Professor T. Kato and Professo r S. Iwata (Nagoya University) for AGM \nmeasurements. 12\nReferences \n[1] L. Berger: Phys. Rev. B 54 (1996) 9353. \n[2] J. C. Slonczewski: J. Magn. Magn. Mater. 159 (1996) L1. \n[3] J. Jaworowicz, N. Vernier, J. Ferré, A. Maziewski, D. Stanescu, D. Ravelosona, A. \nS. Jacqueline, C. Chappert, B. Rodmacq, and B. Diény: Nanotechnology 20 (2009) \n215401. \n[4] L. Leem and J. S. Harris: J. Appl. Phys. 105 (2009) 07D102. \n[5] S. S. P. 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Koyama, K. Ueda , H. Tanigawa, S. Fukani, T. Suzuki, N. \nOhshima, N. Ishwata, Y. Nakatani, and T. Ono: Appl. Phys. Express 3 (2010) \n073004. \n[14] M. Yamanouchi, D. Chiba, F. Matsukura, and H. Ohno: Nature 428 (2004) 539. \n[15] D. McGrouther, S. McVitie, J. N. Chap man, and A. Gentils: Appl. Phys. Lett. 91 \n(2007) 022506.  \n[16] K. J. Kirk, J. N. Chapman, and C. D. W. Wilkinson: Appl. Phys. Lett. 71 (1997) \n539. \n[17] J. Heinen, D. Hinzke, O. Boulle, G. Malinowski, H. J. M. Swagten, B. Koopmans, \nC. Ulysse, G. Faini, and M. Kläui: Appl. Phys. Lett. 99 (2011) 242501. \n[18] D.-T. Ngo, K. Ikeda, and H. Awano: J. Appl. Phys. 111 (2012) 083921. \n[19] L. Y. Jang, S. Yoon, K. Lee, S. Lee, C. Nam, and B.-K. Cho: Nanotechnology 20 \n(2009) 125401. \n[20] G. S. D. Beach, C. Knutson, C. Nistor, M. Tsoi, and J. L. Erskine: Phys. Rev. Lett. \n97 (2006) 057203. \n[21] D.-T. Ngo, K. Ikeda, and H. Awano: Appl. Phys. Express 4 (2011) 093002. \n[22] D. Ravelosona, D. Lacour, J. A. Katine, B. D. Terris, and C. Chappert: Phys. Rev. \nLett. 95 (2005) 117203. \n[23] K.-J. Kim, K.-W. Moon, K.-S. Lee, and S.-B. Choe: Nanotechnology 22 (2011) \n025702. \n[24] G. Tatara, H. Kohno, and J. Shibata: J. Phys. Soc. Jpn. 77 (2008) 031003. \n[25] R. Lavrijsen, G. Malinowski, J. H. Franken, J. T. Kohlhepp, H. J. M. Swagten, B. \nKoopmans, M. Czapkiewicz, and T. Stobiecki: Appl. Phys. Lett. 96 (2010) 022501. 14\n[26] I. M. Miron, T. Moore, H. Szambolics, L. D. Buda-Prejbeanu, S. Auffret, B. \nRodmacq, S. Pizzini, J. Vogel, M. Bonfim, A. Schuhl, and G. Gaudin: Nat. Mater. \n10 (2011) 419. \n[27] O. Boulle, G. Malinowski, and M. Cläui: Mater. Sci. Eng. R 72 (2011) 159. 15\nFigure captions \nFig. 1 . Electron microscopy image of the CoB/Ni nanowire with Ti/Au electrodes for \nmagnetotransport measurements. The plus sign denotes the direction of applied field.  \nFig. 2 . (a) Time-resolved Hall voltage signal measured at a driven current of 0.59 mA \n(j\nDC=1.12×1011 A.m-2) and external field of 5 mT; (b) Interpretation of the Hall-voltage \npulse as domain motion progresses in the Hall bar. The inset shows the field dependence \nof Hall voltage.  \nFig. 3 . Hall resistance changes as a function of driven current density. The inset shows a \ntime-resolved Hall voltage signal measure at the critical current density and zero field.  \nFig. 4 . External field dependence of the velocity of the wall measured at the threshold \ncurrent density (j\nDC=1.12×1011 A.m-2). \n \nFig. 5 . Variation of wall velocity as a function of driven current density at zero external \nfield. 16\n \n \nFigure 1 17\n \n \nFigure 2 18\n \n \nFigure 3 19\n \n \nFigure 4 20\n \n \nFigure 5 "    },    {        "title": "1311.4070v1.Shear_viscosity_due_to_the_Landau_damping_from_quark_pion_interaction.pdf",        "content": "arXiv:1311.4070v1  [nucl-th]  16 Nov 2013Shear viscosity due to the Landau damping from quark-pion in teraction\nSabyasachi Ghosh1,2, Anirban Lahiri2, Sarbani Majumder2, Rajarshi Ray2, Sanjay K. Ghosh2\n1Instituto de Fisica Teorica, Universidade Estadual Paulis ta,\nRua Dr. Bento Teobaldo Ferraz, 271, 01140-070 Sao Paulo, SP, Brazil and\n2Center for Astroparticle Physics and Space Science, Bose In stitute,\nBlock EN, Sector V, Salt Lake, Kolkata 700091, India\nWe have calculated the shear viscosity coefficient ηof the strongly interacting matter in the\nrelaxation time approximation, where a quasi particle desc ription of quarks with its dynamical\nmass is considered from NJL model. Due to the thermodynamic s cattering of quarks with pseudo\nscalar type condensate (i.e. pion), a non zero Landau dampin g will be acquired by the propagating\nquarks. This Landau damping may be obtained from the Landau c ut contribution of the in-medium\nself-energy of quark-pion loop, which is evaluated in the fr amework of real-time thermal field theory.\nFrom the basic idea of the QCD asymptotic freedom\nat high temperatures and densities, a weakly interact-\ning quark gluon plasma (QGP) is naturally expected\nto be produced in the experiments of heavy ion colli-\nsion (HIC). However, the experimental data from RHIC,\nespecially the measured elliptic flow indicates that nu-\nclear matter as a strongly interacting liquid instead of a\nweakly interacting gas. The recent hydrodynamical cal-\nculations [1, 2] as well as some calculations of kinetic\ntransport theory [3, 4] conclude that the matter, pro-\nduced in HIC, must have very small shear viscosity. The\nshear viscosity of the fluid is generally quantified by the\nthe coefficient ηand it physically interprets the ability\nto transfer momentum over a distance of mean free path.\nHence the lower values of ηmeans the constituents of the\nmatter interact strongly to transfer the momentum eas-\nily. Whereasaweaklyinteractingsystemmust havelarge\nηbecause in this case the momentum transfer between\nthe constituents become strenuous.\nSeveral theoretical attempts [5–25] are taken to cal-\nculate theηof the strongly interacting matter at very\nhigh [5], intermediate [6, 7] and low temperature [8–16],\nwhere some special attentions are drawn on the small-\nness of its original value with respect to its lower bound\n(η=s\n4π, wheresisentropydensity), commonlyknownto\nas the KSS bound [26]. The most interesting fact, which\nhasbeen addedwiththe recenttheoreticalunderstanding\nofηfor strongly interacting matter, is that the η/smay\nreachaminimum inthe vicinityofaphasetransition[19–\n23] (see also [27]) like the liquid-gas phase transition of\ncertain materials e.g. Nitrogen, Helium or Water. These\ninvestigationsdemand a better understanding to zoom in\non the temperature ( T) dependence of ηof the strongly\ninteractingmatternearthephasetransition. Inspiringby\nthismotivation, inthisbriefreportwehaveaddressedthe\nη(T) due to forward and backward scattering of quark-\npion interaction.\nIn the relaxation time approximation, the ηof the\nquark [23] and pion [15, 16] medium (for µ= 0) can\nbe expressed as\nη=8β\n5/integraldisplayd3/vectork\n(2π)3/vectork4\nω2\nQnQ(1−nQ)\nΓQ(k)\n(U=l−k)(l)\n(k)\nQQ\n(A) (B)(k)\n(U=k−l)(l)\n(k)QQ\nQ\nFIG. 1: The diagram of quark (A)and pion (B)self-energy\nfor quark-pion and quark-anti quark loops respectively.\n+β\n5/integraldisplayd3/vectork\n(2π)3/vectork4\nω2πnπ(1+nπ)\nΓπ(1)\nwherenQ=1\neβωQ+1andnπ=1\neβωπ−1are respectively\nFermi-Dirac distribution of quark and Bose-Einstein dis-\ntribution of pion with ωQ=/radicalBig\n/vectork2+M2\nQandωπ=/radicalBig\n/vectork2+m2π. The Γ Qand Γ πare Landau damping of\nquark and pion respectively. Following the quasi parti-\ncle description of Nambu-Jona-Lasinio(NJL) model [28],\nthe dynamical quark mass MQis considered and it is\ngenerated due to quark condensate\n/angbracketleftψfψf/angbracketright=−MQ−mQ\n2G(2)\nwheremQis the current quark mass. In the medium,\nabove relation become (for µ= 0)\nMQ=mQ+4NfNcG/integraldisplayd3/vectork\n(2π)3MQ\nωQ(1−2nQ).(3)\nThis relation shows that the constituent quark mass\ntends to be the current quark mass at very high tem-\nperature where the non-zero quark condensate becomes\nsmall.\nThis Landau damping Γ Qand Γ πmay be estimated\nfrom the self-energy graphs of quark and pion at finite\ntemperature for quark-pion and quark-anti quark loops\nrespectively. These are respectively expressed as\nΓQ=−ImΣR(k0=/radicalBig\n/vectork2+M2\nQ,/vectork) (4)\nand\nΓπ=−1\nmπImΠR(k0=/radicalBig\n/vectork2+m2π,/vectork) (5)2\nwhere ΣRand ΠRare respectively retardedpart of quark\nand pion self-energy at finite temperature. Their dia-\ngrammatic representations are shown in Fig.1 (A)and\n(B)respectively. Following the real-time formalism of\nthermal field theory, the retarded part of in-medium\nquark self energy for quark-pion loop is given by [30]\nΣR(k0,/vectork) =/integraldisplayd3/vectorl\n(2π)31\n4ωl\nQωUπ[(1−nl\nQ)LQ\n1+nU\nπLQ\n3\nk0−ωl\nQ−ωUπ+iη\n+nl\nQLQ\n1+nU\nπLQ\n4\nk0−ωl\nQ+ωUπ+iη+−nl\nQLQ\n2−nU\nπLQ\n3\nk0+ωl\nQ−ωUπ+iη\n+nl\nQLQ\n2+(−1−nU\nπ)LQ\n4\nk0+ωl\nQ+ωUπ+iη]. (6)\nwhereLQ\ni,i= 1,..4 denote the values of LQ(l0,/vectorl) for\nl0=ωl\nQ,−ωl\nQ,k0−ωU\nπ,k0+ωU\nπrespectively with ωl\nQ=/radicalBig\n/vectork2+M2\nQandωU\nπ=/radicalBig\n(/vectork−/vectorl)2+m2π. Herenl\nQ(ωl\nQ)\nis Fermi-Dirac distribution function of quark whereas\nnU\nπ(ωU\nπ) denotes Bose-Einstein distribution function of π\nmeson.\nDuring extracting the imaginary part of ΣR(k0,/vectork) we\nwill get four delta functions associated with the four\nindividual terms of Eq. (6), which generate four dif-\nferent region in k0-axis where the ImΣR(k0,/vectork) will be\nnon-zero. From the non-zero values of ImΣR(k0,/vectork) the\nregion of discontinuities or branch cuts of ΣR(k0,/vectork)\ncan be identified. The regions coming from the 1st\nand 4th terms of (6) are respectively ( k0=−∞to\n−/radicalBig\n/vectork2+(mπ+MQ)2) and (k0=/radicalBig\n/vectork2+(mπ+MQ)2\nto∞). These are known as unitary cuts and different\nkind of forward and inverse decay processes are associ-\nated with these cut contributions [29, 30]. Similarly the\nregions (k0=−/radicalBig\n/vectork2+(mπ−MQ)2to 0) and (k0= 0 to/radicalBig\n/vectork2+(mπ−MQ)2) are coming from 2nd and 3rd terms\nrespectively. These purely medium dependent cuts are\nknown as Landau cuts and different kind of forward and\ninverse scattering processes are physically interpreted by\nthese cut contributions [29, 30]. So the 3rd term of\nImΣR(k0,/vectork) at the on-shell mass ( k0=/radicalBig\n/vectork2+M2\nQ,/vectork) of\nquark is responsible for the Landau damping Γ Qand it\nis given by [30]\nΓQ=−ImΣR(k0=/radicalBig\n/vectork2+M2\nQ,/vectork) = [/integraldisplayd3/vectorl\n(2π)3LQ\n2\n4ωl\nQωUπ\n(nl\nQ+nU\nπ)δ(k0+ωl\nQ−ωU\nπ)]k0=/radicalbig\n/vectork2+M2\nQ.(7)\nRearranging the statistical weight factor by\n(nl\nQ+nU\nπ) =nl\nQ(1+nU\nπ)+nU\nπ(1−nl\nQ),(8)\nwe can find thermalized πanduwith Bose enhanced\nprobability (1+ nU\nπ) and Pauli blocked probability (1 −nl\nQ) respectively. With the help of Eq. (8), the physi-\ncal significance of the Landau cut contribution may ex-\npressed as follows. During the propagation of uquark,\nit may absorb the thermalized ufrom the heat bath and\ncreate a thermalized πin the bath (indicated by the sec-\nond partof Eq.(8)). Again the thermalized πmaybe ab-\nsorbedbythe mediumandcreatethe thermalized ualong\nwith a propagating u, which is slightly off-equilibrium\nwith the medium (indicated by the first part of Eq. (8)).\nTo calculate LQ\nifrom quark-pion interaction, let us\nstart with free Lagrangian of quarks and demanding the\ninvariance properties of Lagrangian under chiral trans-\nformation,\nψ′\nf= exp(i/vector π·/vector τγ5\n2Fπ)ψf (9)\nwhere the chiral angle is associated with the pion field\n/vector πandFπis pion decay constant. Expanding up to first\norder of pion field, we obtain the quark-pion interaction\nterm [33, 34],\nLπQQ=−iMQ\nFπψf/vector π·/vector τγ5ψf\n=−iMQγ5\nFπ/parenleftbig\nud/parenrightbig/parenleftbigg\nπ0√\n2π+√\n2π−π0/parenrightbigg/parenleftbigg\nu\nd/parenrightbigg\n.(10)\nAs we are interested to calculate one-loop self-energy\n(ΣR) of any quark flavor, u(say) hence we have to con-\nsider two possible loops - uπ0anddπ+. Due to isospin\nsymmetry consideration in Lagrangian, we can evaluate\nanyone of the loops, say uπ0loop and then we have to\nmultiply it by a isospin factor\nIF= (1)2+(√\n2)2= 3. (11)\nFrom the interaction part,\nLπ0uu=−igπQQuγ5π0u,withgπQQ=MQ\nFπ(12)\nwe can calculate LQ\nab(l0,/vectorl) =−IFg2\nπQQ(l /−ml)ab, where\na,bare Dirac indices. For simplification we have taken\nthe scalar part only i.e. LQ(l0,/vectorl) =IFg2\nπQQml. We have\ntaken the parameters mQ= 0.0056 GeV,MQ= 0.4 GeV\n(forT= 0), three momentum cut-off Λ = 0 .588 GeV and\ncorresponding Tc= 0.222 GeV for µ= 0 [28].\nSimilar to Eq. (6), the pion self-energy ΠRfor quark-\nanti quark loop is also received similar kind of form only\nthe quantities nU\nπ,ωU\nπ=/radicalBig\n(/vectork−/vectorl)2+m2π,U=k−l\nandLQ\ni’s are changed to −nU\nQ,ωU\nQ=/radicalBig\n(/vectorl−/vectork)2+m2\nQ,\nU=−k+landLπ\ni’s respectively [31, 32]. As pion on-\nshell mass point ( k0=/radicalBig\n/vectork2+m2π,/vectork) will be inside the\nunitary cut region ( k0=/radicalBig\n/vectork2+(MQ+MQ)2to∞) of\nΠR, therefore\nΓπ=−ImΠR(k0=/radicalBig\n/vectork2+m2π,/vectork)\nmπ=−1\nmπ[/integraldisplayd3/vectorl\n(2π)3Lπ\n1\n4ωl\nQωU\nQ3\n00.10.20.30.40.50.6Γ (GeV)with folding\nwithout folding\nπcomponent\n0.15 0.2 0.25 0.3 0.35\nT (GeV)0102030τ (fm)k=0\nFIG. 2: Upper panel : The Tdependency of Γ Qwith (solid\nline) and without (dotted line) folding by Aπand Γ π(dashed\nline). Lower panel : The variation of corresponding collisi on\ntimeτwith temperature.\n010203040Aπ (GeV-2)T=0.300 GeV\nT=0.250 GeV\n00.050.10.150.20.250.30.350.4\nM (GeV)00.10.20.30.40.5ImΠ/mπ (GeV)k=0 \nFIG. 3: Lower panel shows Mdependency of the imaginary\npart of pion self-energy for Q¯Qloop, which is normalized after\ndividing by mπ. Upper panel shows invariant mass distribu-\ntion of pion spectral function due to its Q¯Qwidth. Dotted\nline indicates the position of pion pole.\n0.05 0.1 0.15 0.2 0.25 0.3\nT (GeV)00.010.020.030.04η (GeV3)\nπcomponent\nwithout folding\nwith folding\nRef. [23]\nRef. [12]\nRef. [14]\nFIG. 4: Temperature dependence of ηdue to Γ π(dashed\nline), Γ Qwithout (dotted line) and with (solid line) folding\nare separately shown. The results of Ref. [23](triangles) a re\nattached to compare with our results (solid line). [also the\nresults of hadronic domain by Ref. [12](stars), Ref. [14] (o pen\ncircles)].(1−nl\nQ−nU\nQ)δ(k0+ωl\nQ−ωU\nQ)]k0=√\n/vectork2+m2π(13)\nwhereLπ= 4IFg2\nπQQ[M2\nQ−l2−k·l] can be obtained\nfrom (12).\nIn Fig. (2), we can see the temperature dependency\nof Landau damping Γ (upper panel) and collision time\nτ=1\nΓ(lower panel) of quark (dotted line) and pion\n(dashed line) for their momentum /vectork= 0. Owing to\nthe on-shell condition, the Γ Qand Γ πare received the\nnon-zero values only in the temperature range where\nmπ>2MQ, which are clearly seen from dotted and\ndashed lines respectively. A correspondingnon-divergent\ncollisional times are also achieved by them in the same\ntemperature domain. Due to decay width of π→Q¯Q, it\nwill more realistic to consider the pion resonance of finite\nwidth in Eq. (7). The pion spectral function due to Q¯Q\nwidth may be defined as\nAπ(M) =1\nπIm/bracketleftBigg\n1\nM2−m2π+iImΠRvac(k0,/vectork)/bracketrightBigg\n(14)\nwhere ImΠR\nvac(k0,/vectork) is vacuum part of ImΠR(k0,/vectork) and\nM=/radicalBig\nk2\n0−/vectork2. The variation of ImΠR(M)/mπandAπ\nwithMfor two different temperatures are respectively\nshown in lower and upper panel of Fig. (3). Replacing\nmπof ΓQin (7) byMand then convoluting or folding it\nbyAπ(M), we have [31, 32]\nΓQ(mπ) =1\nNπ/integraldisplay\nΓQ(M)Aπ(M)dM2(15)\nwhereNπ=/integraltext\nAπ(M)dM2. One should notice that in\nthe narrow width approximation i.e. for ImΠR\nvac→0,\nEq. (15) is merged to (7). The Tdependency of Γ Qand\nits corresponding τafter folding are shown by solid line\nin the upper and lowerpanel of Fig. (2) respectively. Due\nto folding, Γ Qat lowTdomain (where mπ<2MQ) has\nacquired some non-zero values from its vanishing con-\ntributions and at the same time corresponding τrecover\nfromitsdivergenceuptothe approximatefreezeouttem-\nperature (T∼120−150GeV) of the strongly interacting\nmatter.\nBy using Γ Q(T,/vectork) from Eq. (7) and (15) in the quark\ncomponent (first term) of Eq. (1), we get the results of\nshear viscosity as function of T, which are respectively\ndescribed by dotted and solid line of Fig. (4). Being\nproportional to collisional time, the divergence of ηis\nremoved after folding in those temperature region, where\nmπ<2MQ. The contribution of ηdue to Γ π(T,/vectork) from\nEq. (13) is shown by dashed line in Fig. (4). After similar\nkind of folding as done in Eq. (15), an almost negligible\n(∼10−5 GeV3) contribution of ηfor pion component can\nbe obtained which is not included in final results.\nInlowtemperatureregion, ηisdecreasingwithincreas-\ning ofTwhich is analogous to the behavior of liquid\n(From our daily life experience, we see that the cooking\noil behaves like a less viscous medium when it is heated).4\nWhereas in high temperature domain, ηbecome an in-\ncreasing function of Tjust like a system of gas.\nThe magnitude of ηin our approach is very close to\nthe results of Sasaki and Redlich [23] (indicated by tri-\nangles) but underestimated with respect to the earlier\nestimation in NJL model by Zhuang et al. [22]. The\nLQCD calculation of η(η∼0.054−0.47 GeV3nearTc)\nby H. B. Meyer [7] is higher than all of these calculations.\nFrom the solid line in the lower panel of Fig. (2), we see\nthatτbelow theT∼160 MeV exceeds the typical value\nof time period ( ∼30−50 fm) during which a strongly\ninteracting matter survive in the labs of heavy ion colli-\nsions. Therefore the estimation of ηin low temperature\ndomain is quite higher than the standard calculations of\nηof hadronic matter [12, 14, 16]. The earlier calculations\nof NJL model [22, 23] also displayed these discrepancy in\nthe hadronic temperature domain.\nIn summary we have investigated the shear viscosity\nof strongly interacting matter in the relaxation time ap-proximation, where quarks with its dynamical mass may\nhave some non zero Landau damping because of its vari-\nous forward and inverse scattering with pions. This Lan-\ndau damping can be obtained from the thermal field the-\noretical calculation of quark self-energy for quark-pion\nloop. 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Ghosh and S. Sarkar, Eur. Phys. J. A 49, 97 (2013).\n[33] T. Frederico and G. A. Miller Phys. Rev. D 45, 4207\n(1992)\n[34] U. G. Meissner, Phys. Rep. 161, 213 (1988)."    },    {        "title": "1608.08326v3.Optimal_damping_ratios_of_multi_axial_perfectly_matched_layers_for_elastic_wave_modeling_in_general_anisotropic_media.pdf",        "content": "Optimal damping ratios of multi-axial perfectly matched layers for\nelastic-wave modeling in general anisotropic media\nKai Gaoa, Lianjie Huanga\naGeophysics Group, Los Alamos National Laboratory, Los Alamos, NM 87545\nAbstract\nThe conventional Perfectly Matched Layer (PML) is unstable for certain kinds of anisotropic media. This in-\nstability is intrinsic and independent of PML formulation or implementation. The Multi-axial PML (MPML)\nremoves such instability using a nonzero damping coe\u000ecient in the direction parallel with the interface be-\ntween a PML and the investigated domain. The damping ratio of MPML is the ratio between the damping\ncoe\u000ecients along the directions parallel with and perpendicular to the interface between a PML and the\ninvestigated domain. No quantitative approach is available for obtaining these damping ratios for general\nanisotropic media. We develop a quantitative approach to determining optimal damping ratios to not only\nstabilize PMLs, but also minimize the arti\fcial re\rections from MPMLs. Numerical tests based on \fnite-\ndi\u000berence method show that our new method can e\u000bectively provide a set of optimal MPML damping ratios\nfor elastic-wave propagation in 2D and 3D general anisotropic media.\nKey words: Anisotropic medium, elastic-wave propagation, Multi-axial Perfectly Matched Layers\n(MPML), damping ratio.\n1. Introduction\nElastic-wave modeling usually needs to absorb outgoing wave\felds at boundaries of an investigated\ndomain. Two main categories of boundary absorbers have been developed: one is called the Absorbing\nBoundary Condition (ABC) (e.g., Clayton and Engquist, 1977; Reynolds, 1978; Liao et al., 1984; Cerjan\net al., 1985; Higdon, 1986, 1987; Long and Liow, 1990; Peng and Toks oz, 1994), and the other is termed\nthe Perfectly Matched Layer (PML) (e.g., Berenger, 1994; Hastings et al., 1996; Collino and Tsogka, 2001).\nIn some literature, PML is considered as one of ABCs. However, there are fundamental di\u000berences in the\nconstruction of PML and its variants compared with traditional ABCs, we therefore di\u000berentiate them in\nnames. For a brief summary, please refer to Hastings et al. (1996) and other relevant references.\nThe PML approach was \frst introduced by Berenger (1994) for electromagnetic-wave modeling, and\nhas been widely used in elastic-wave modeling because of its simplicity and superior absorbing capability\n(e.g., Collino and Tsogka, 2001; Komatitsch and Tromp, 2003; Drossaert and Giannopoulos, 2007). Various\nimproved PML methods for elastic-wave modeling have been developed, such as non-splitting convolutional\nPML (CPML) to enahce absorbing capability for grazing incident waves (Komatitsch and Martin, 2007;\nPreprint submitted to Geophysical Journal International May 24, 2022arXiv:1608.08326v3  [physics.geo-ph]  21 Dec 2016Martin and Komatitsch, 2009), and CPML with auxiliary di\u000berential equation (ADE-PML) for modeling\nwith a high-order time accuracy formulation (Zhang and Shen, 2010; Martin et al., 2010). However, a well-\nknown problem of PML and its variants/improvements is that numerical modeling with PML is unstable in\ncertain kinds of anisotropic media for long-time wave propagation.\nTo address the instability problem of PML, B\u0013 ecache et al. (2003) analyzed the PML for 2D anisotropic\nmedia and found that, if there exists points where the ithcomponent of group velocity vhas an opposite\ndirection relative to the ithcomponent of wavenumber k, i.e.,viki<0 (no summation rules applied), then\nthexi-direction PML is unstable. The original version of this aforementioned condition was expressed with\nso-called \\slowness vector\" de\fned by B\u0013 ecache et al. (2003), but it can be recast in such form according\nto the de\fnition of the \\slowness vector\" in eq. (45) of (B\u0013 ecache et al., 2003). This PML instability is\nintrinsic and independent of PML/CPML formulations adopted for wave\feld modelings. To make PML\nstable, elasticity parameters of an anisotropic medium need to satisfy certain inequality relations (B\u0013 ecache\net al., 2003). These restrictions limit the applicability of PML for arbitrary anisotropic media.\nMeza-Fajardo and Papageorgiou (2008) presented an explanation for the instability of conventional PML.\nThey recast the elastic wave equations in PML to an autonomous system and found that the PML instability\nis caused by the fact that the PML coe\u000ecient matrix having one or more eigenvalues with positive imaginary\nparts. They showed that PML becomes stable when adding appropriate nonzero damping coe\u000ecients to PML\nin the direction parallel with the PML/non-PML interface. The ratio between the PML damping coe\u000ecients\nalong the directions parallel with and perpendicular to the PML/non-PML interface is called the damping\nratio. The resulting PML with nonzero damping ratios is termed the Multi-axial PML (MPML).\nA key step in the stability analysis of PML is to derive eigenvalue derivatives of the damped system\ncoe\u000ecient matrix. Meza-Fajardo and Papageorgiou (2008) derived expressions of the eigenvalue derivatives\nfor anisotropic media. However, these expressions are valid only for two-dimensional isotropic media and\nanisotropic media with up to hexagonal/orthotropic anisotropy, that is, C116= 0,C336= 0,C556= 0,\nC15=C35= 0, andC13can either be zero or nonzero depending on medium properties. Furthermore,\nalthough Meza-Fajardo and Papageorgiou (2008) showed that the nonzero damping ratios can stabilize PML,\nthey did not present a method to select the appropriate dampoing ratios. Adding these nonzero damping\ncoe\u000ecients makes the PML no longer \\perfectly matched\", and the larger are the ratios, the stronger the\narti\fcial re\rections become (Dmitriev and Lisitsa, 2011). This increase is linear. Therefore, it is necessary\nto \fnd a set of \\optimal\" damping ratios to not only ensure the stability of MPML, but to also eliminate\narti\fcial re\rections as much as possible.\nWe develop a new method to determine the optimal MPML damping ratios for general anisotropic media.\nWe show that, even for a two-dimensional anisotropic medium with nonzero C15andC35, the MPML stability\nanalysis is complicated, and new equations must be derived to calculate both the eigenvalues of the undamped\nsystem and the eigenvalue derivatives of the damped system. The resulting expressions are functions of all\nnonzeroCijcomponents as well as the wavenumber k. For 3D general anisotropic media, we \fnd that\n2such an analytic procedure becomes practically impossible because it requires de\fnite analytic expressions\nof eigenvalues and eigenvalue derivatives. In the 3D case, the dimension of the asymmetric system coe\u000ecient\nmatrix is up to 27 \u000227, and therefore a purely numerical approach should be employed. We present two\nalgorithms with slightly di\u000berent forms but essentially the same logic, to determine the optimal damping\nratios for 2D and 3D MPMLs. With these algorithms, it is possible to stabilize PML for any kind of\nanisotropic media without using a trial-and-error method. Our new algorithms enable us to use MPML\nfor \fnite-di\u000berence modeling of elastic-wave propagation in 2D and 3D general anisotropic media where all\nelastic parameters Cijmay be nonzero. These algorithms are also applicable to other elastic-wave modeling\nmethods such as spectral-element method (e.g., Komatitsch et al., 2000) and discontinuous Galerkin \fnite-\nelement method (e.g., de la Puente et al., 2007).\nOur paper is organized as follows. In the Methodology section, we derive the equations for the eigen-\nvalue derivatives for 2D and 3D general anisotropic media. We also present two algorithms to obtain the\noptimal MPML damping ratios. To validate our algorithms, we give six numerical examples in the Results\nsection, including three 2D anisotropic elastic-wave modeling examples and three 3D anisotropic elastic-wave\nmodeling examples, and show that our algorithms can give appropriate damping ratios for both 2D and 3D\nmodeling in general anisotropic media.\n2. Methodology\n2.1. Optimal damping ratios of 2D MPML\nIn this section, we concentrate our analysis on the x1x3-plane. This analysis is also valid for the x1x2-\nandx2x3-planes. We assume that C15andC35are generally nonzero for an anisotropic medium. The 2D\nelastic-wave equations in the stees-velocity form are given by (e.g., Carcione, 2007),\n\u001a@v\n@t=\u0003\u001b+f; (1)\n@\u001b\n@t=C\u0003Tv; (2)\nwhere \u001b= (\u001b11;\u001b33;\u001b13)Tis the stress wave\feld, v= (v1;v3)Tis the particle velocity wave\feld, fis the\nexternal force, \u001ais the mass density, Cis the elasticity tensor in Voigt notation de\fned as\nC=0\nBBB@C11C13C15\nC13C33C35\nC15C35C551\nCCCA; (3)\nand\u0003is the di\u000berential operator matrix de\fned as\n\u0003=0\n@@\n@x10@\n@x3\n0@\n@x3@\n@x11\nA: (4)\nIn the following analysis, we ignore the external force term fwithout loss of generality.\n3Using the convention in Meza-Fajardo and Papageorgiou (2008) for isotropic and VTI/HTI/othotropic\nmedia, the undamped system of eqs. (1){(2) can be also written as\n\u001a@\n@t2\n4v1\nv33\n5=2\n4@\n@x1\u001b11+@\n@x3\u001b13\n@\n@x1\u001b13+@\n@x3\u001b333\n5; (5)\n@\n@t2\n6664\u001b11\n\u001b33\n\u001b133\n7775=2\n6664C11C13C15C15\nC13C33C35C35\nC15C35C55C553\n77752\n6666664@\n@x1v1\n@\n@x3v3\n@\n@x1v3\n@\n@x3v13\n7777775: (6)\nEquivalently, the system of the above two equations can be written as\n@\n@tv=D1@\n@x1\u001b+D3@\n@x3\u001b; (7)\n@\n@t\u001b=C1@\n@x1v+C3@\n@x3v; (8)\nwhere\nv= (v1;v3)T; (9)\n\u001b= (\u001b11;\u001b33;\u001b13)T; (10)\nD1=\u001a\u000012\n41 0 0\n0 0 13\n5; (11)\nD3=\u001a\u000012\n40 0 1\n0 1 03\n5; (12)\nC1=2\n6664C11C15\nC13C35\nC15C553\n7775; (13)\nC3=2\n6664C15C13\nC35C33\nC55C553\n7775: (14)\nIn the conventional 2D PML, each \feld variable is split into two orthogonal components that are per-\npendicular to and parallel with the interface between the PML and the investigated domain, and the system\nof wave equations in PML can be written as (Meza-Fajardo and Papageorgiou, 2008)\n@\t\n@t=A\t; (15)\nwithA=A0+B, and\nA0=2\n6666664033 033@\n@x1C1@\n@x1C1\n033 033@\n@x3C3@\n@x3C3\n@\n@x1D1@\n@x1D1022 022\n@\n@x3D3@\n@x3D3022 0223\n7777775; (16)\n4B=2\n6666664\u0000d1I3033 032 032\n033\u0000d3I3032 032\n023 023\u0000d1I2022\n023 023 022\u0000d3I23\n7777775; (17)\nwhere 0mnis them\u0002nzero matrix, Imis them\u0002midentity matrix, and\n\t= (\u001b(1)\n11;\u001b(1)\n33;\u001b(1)\n13;\u001b(3)\n11;\u001b(3)\n33;\u001b(3)\n13;v(1)\n1;v(1)\n3;v(3)\n1;v(3)\n3)T(18)\nrepresents the split wave\feld variables in PML. In the damping matrix B,d1andd3represent the PML\ndamping coe\u000ecients along the x1- andx3-axis, respectively. The PML damping coe\u000ecients depend on the\nthickness of PML, the desired re\rection coe\u000ecient and the P-wave velocity at the PML/non-PML interface.\nUsually, they vary with the distance from a location inside the PML to the PML/non-PML interface according\nto the power of two or the power of three (e.g., Collino and Tsogka, 2001).\nTransforming system (15) into the wavenumber domain leads to\n@U\n@t=~AU; (19)\nwhere U=F[\t] is the Fourier transform of the split \fled variables \t,~A=~A0+B, and ~A0now is\n~A0=2\n6666664033 033ik1C1ik1C1\n033 033ik3C3ik3C3\nik1D1ik1D1022 022\nik3D3ik3D3022 0223\n7777775; (20)\nwherek1andk3are respectively the x1- andx3-components of wavenumber vector k.\nAs demonstrated by the stability theory of autonomous system in Meza-Fajardo and Papageorgiou (2008),\nfor a stable PML in an elastic medium, either isotropic or anisotropic, all the eigenvalues of the system matrix\n~Ashould have non-positive imaginary parts. In conventional PML, the outgoing wave\feld is damped only\nalong the direction perpendicular to the PML/non-PML interface, and the damping matrix Bfor PML in\nthex1andx3-directions can be respectively written as\nB1=2\n6666664\u0000d1I3033 032 032\n033 033 032 032\n023 023\u0000d1I2022\n023 023 022 0223\n7777775; (21)\nB3=2\n6666664033 033 032 032\n033\u0000d3I3032 032\n023 023 022 022\n023 023 022\u0000d3I23\n7777775; (22)\n5resulting in an unstable PML. Meza-Fajardo and Papageorgiou (2008) analyzed the derivatives of eigenvalues\nof~Awith respect to the damping parameters d1andd3, and showed that if an appropriate damping ratio\n\u00181or\u00183is added along the direction parallel with the PML/non-PML interface, i.e.,\nB1(\u00181) =2\n6666664\u0000d1I3 033 032 032\n033\u0000\u00181d1I33 032 032\n023 023\u0000d1I2 022\n023 023 022\u0000\u00181d1I223\n7777775; (23)\nB3(\u00183) =2\n6666664\u0000d3\u00183I33 033 032 032\n033\u0000d3I3 032 032\n023 023\u0000d3\u00183I22 022\n023 023 022\u0000d3I23\n7777775; (24)\nthen the PML becomes stable. The underlying principle of such stability comes from the fact that, after\nadding nonzero damping ratios \u00181and\u00183, the eigenvalue derivatives of ~Ahave negative values along all\nkdirections and consequently, all the relevant eigenvalues of ~A0have negative imaginary parts (other\neigenvalues are a pure zero), making autonomous system (15) stable.\nA key step for determing such damping ratios \u00181and\u00183is to compute the eigenvalue derivatives of ~A.\nMeza-Fajardo and Papageorgiou (2008) adopted the following procedure:\n1. Calculate eigenvalues eof undamped system coe\u000ecient matrix ~A0, of which six are a pure zero, and\nthe rest four are pure imaginary numbers as functions of elasticity coe\u000ecient Cand wavenumber k;\n2. Calculate the eigenvalue derivative of ~A=~A0+~B1(\u00181) or ~A=~A0+~B3(\u00183) with respect d1ord3at\nd1= 0 ord3= 0. These eigenvalue derivatives are functions of elasticity coe\u000ecient C, wavenumber k\nand damping ratio \u00181or\u00183;\n3. Choose an appropriate value to g ensure that the values of the eigenvalue derivatives are negative in the\nrange of (0;\u0019=2] (the direction of wavenumber k).\nBoth eigenvalues of ~A0and eigenvalue derivatives of ~Aare calculated analytically in the above procedure.\nSpecially, Step 2 involves implicit di\u000berentiation operation and solving roots for high-order polynomials, and\ncould not be accomplished numerically.\nWe adopt the above procedure for obtaining optimal damping ratios for 2D general anisotropic media\nwhereC15orC35may be nonzero. We \frst derive relevant expressions for the eigenvalues of ~A0and the\neigenvalue derivatives of ~A. Because our procedure is the same as that in Meza-Fajardo and Papageorgiou\n(2008), we only show the resulting equations. The four nonzero eigenvalues of ~A0(C;k) are\ne1(C;k) =\u0006ip\n2\u001aq\nP\u0000p\nQ; (25)\ne2(C;k) =\u0006ip\n2\u001aq\nP+p\nQ; (26)\n6P=\u001a[(C11+C55)k2\n1+ 2(C15+C35)k1k3+ (C33+C55)k2\n3]; (27)\nQ=\u001a2f[(C11+C55)k2\n1+ 2(C15+C35)k1k3+ (C33+C55)k2\n3]2\n+ 4[(C2\n15\u0000C11C55)k4\n1+ 2(C13C15\u0000C11C35)k3\n1k3\n+ (C2\n13\u0000C11C33\u00002C15C35+ 2C13C55)k2\n1k2\n3\n+ 2(\u0000C15C33+C13C35)k1k3\n3+ (C2\n35\u0000C33C55)]k4\n3g; (28)\nwhereCIJare components of the elasticity matrix and \u001ais the mass density. The two eigenvalues in e1or\ne2have the same length with di\u000berent signs, and we take only the negative ones, i.e.,\ne1(C;k) =\u0000ip\n2\u001aq\nP\u0000p\nQ; (29)\ne2(C;k) =\u0000ip\n2\u001aq\nP+p\nQ: (30)\nThe choice of the signs of e1ande2does not a\u000bect the following stability analysis and optimal damping\nratios.\nThe eigenvalue derivatives of ~Awith respect to the damping coe\u000ecient d1atd1= 0 can be written as\n\u001f(l)\n1(C;k;\u00181;el) = (2C15C35k2\n1k2\n3+ 3C15C33k1k3\n3\u00002C2\n35k4\n3\n+ 2C33C55k4\n3\u00002C2\n15k4\n1\u0018+ 2C15C35k2\n1k2\n3\u00181\n+C15C33k1k3\n3\u00181\u0000C2\n13k2\n1k2\n3(1 +\u00181)\n\u0000C13k1k3(C15k2\n1(1 + 3\u00181) +k3(2C55k1(1 +\u00181)\n+C35k3(3 +\u00181))) +e2\nl(k3(3C15k1(1 +\u00181)\n+ 3C35k1(1 +\u00181) +C33k3(2 +\u00181))\n+C55(k2\n3(2 +\u00181) +k2\n1(1 + 2\u00181)))\u001a+ 2e4\nl(1 +\u00181)\u001a2\n+C11k2\n1(2C55k2\n1\u00181+C33k2\n3(1 +\u00181) +C35k1k3(1 + 3\u00181) +e2\nl(1 + 2\u00181)\u001a))\n=(2((C2\n15\u0000C11C55)k4\n1+ 2(C13C15\u0000C11C35)k3\n1k3+ (C2\n13\u0000C11C33\u00002C15C35\n+ 2C13C55)k2\n1k2\n3+ 2(\u0000C15C33+C13C35)k1k3\n3+ (C2\n35\u0000C33C55)k4\n3)\n\u00003e2\nl((C11+C55)k2\n1+ 2(C15+C35)k1k3+ (C33+C55)k2\n3)\u001a\u00004e4\nl\u001a2); (31)\nwhere subscript \\ l\" is for the ltheigvenvalue, and elstands for the ltheigenvalue of ~A0. The eigenvalue\nderivatives of ~Awith respect to the damping coe\u000ecient d3atd3= 0 is given by\n\u001f(l)\n3(C;k;\u00183;el) = (\u00002C2\n15k4\n1+k2\n3(\u00002(C2\n35\u0000C33C55)k2\n3\u00183\n\u0000C2\n13k2\n1(1 +\u00183)\u0000C13k1(2C55k1(1 +\u00183)\n+C35k3(1 + 3\u00183))) +e2\nl(k3(3C35k1(1 +\u00183)\n+C33k3(1 + 2\u00183)) +C55(k2\n1(2 +\u00183) +k2\n3(1 + 2\u00183)))\u001a+ 2e4\nl(1 +\u00183)\u001a2\n+C15k1k3(\u0000C13k2\n1(3 +\u00183) +k3(2C35k1(1 +\u00183)\n7+C33k3(1 + 3\u00183)) + 3e2\nl(1 +\u00183)\u001a) +C11k2\n1(2C55k2\n1\n+k3(C33k3(1 +\u00183) +C35k1(3 +\u00183)) +e2\nl(2 +\u00183)\u001a))\n=(2((C2\n15\u0000C11C55)k4\n1+ 2(C13C15\u0000C11C35)k3\n1k3\n+ (C2\n13\u0000C11C33\u00002C15C35+ 2C13C55)k2\n1k2\n3+ 2(\u0000C15C33+C13C35)k1k3\n3\n+ (C2\n35\u0000C33C55)k4\n3)\u00003e2\nl((C11+C55)k2\n1\n+ 2(C15+C35)k1k3+ (C33+C55)k2\n3)\u001a\u00004e4\nl\u001a2): (32)\nThere exists a subtle trade-o\u000b between the PML stability and arti\fcail boundary re\rections for anisotropic\nmedia. On one hand, it is necessary to introduce nonzero damping ratios \u00181and\u00183to stabilize PML. On the\nother hand, adding these nonzero damping ratios to PML makes PML no longer perfectly matched, and the\nlarger are the damping ratios, the stronger the arti\fcial boundary re\rections become (Dmitriev and Lisitsa,\n2011). The original analysis of Meza-Fajardo and Papageorgiou (2008) only showed that a certain value of\n\u00181or\u00183can ensure that \u001f(l)\n1and\u001f(l)\n3are negative in all kdirections. However, it did not provide a method\nto determine how large the damping ratios \u00181and\u00183are adequate for an arbitrary anisotropic medium. We\ntherefore develop a procedure to determine the optimal damping ratios \u00181and\u00183to not only stabilize PMLs,\nbut to also minimize resulting arti\fcial boundary re\rections.\nWe employ the following procedure described in Algorithm 1 to obtain the optimal damping ratios \u00181\nand\u00183of MPML for 2D general anisotropic media.\nAlgorithm 1: Determine the optimal damping ratio \u0018i(i= 1;3) of MPML for 2D general anisotropic\nmedia\ninput :\u0018i= 0,\u000f=\u00000:005, \u0001\u0018= 0:001.\nfor\u00122(0;\u0019]do\n1) Calculate wavenumber k= (sin\u0012;cos\u0012)\n2) Calculate eigenvalues el(l= 1;2) of ~A0(C;k) using eqs. (25) and (26)\n3) Calculate eigenvalue derivatives \u001f(l)\ni(C;k;\u0018i;el)\n4)\u001fi;max= max(\u001f(1)\ni;\u001f(2)\ni) using eq. (31) or (32)\nif\u001fi;max>\u000fthen\n\u0018i=\u0018i+ \u0001\u0018\ngo to step 3\nend\nend\noutput:\u0018i\nWe apply the above procedure to both the x1- andx3-directions to obtain the optimal values of \u00181and\n\u00183.\n8Note that the eigenvalue derivatives along these two directions have di\u000berent expressions, although the\nexpressions for eigenvalues elare the same for both the x1- andx3-directions. Therefore, the optimal damping\nratios in the x1- andx3-directions might be di\u000berent from one another. In addition, the searching range for\nthe eigenvalue derivative should be (0 ;\u0019] instead of (0 ;\u0019=2]. We verify these \fndings in numerical examples\nin the next section.\nWe call the aforementioned procedure based on analytic expressions of eigenvalues and eigenvalue deriva-\ntives the analytic approach.\n2.2. Optimal damping ratios of 3D MPML\nElastic-wave equations (1){(2) are also valid for 3D general anisotropic media, but with\nv= (v1;v2;v3)T; (33)\n\u001b= (\u001b11;\u001b22;\u001b33;\u001b23;\u001b13;\u001b12)T; (34)\nC=2\n6666666666664C11C12C13C14C15C16\nC12C22C23C24C25C26\nC13C23C33C34C35C36\nC14C24C34C44C45C46\nC15C25C35C54C55C56\nC16C26C36C64C56C663\n7777777777775; (35)\n\u0003=2\n6664@\n@x10 0 0@\n@x3@\n@x2\n0@\n@x20@\n@x30@\n@x1\n0 0@\n@x3@\n@x2@\n@x103\n7775: (36)\nAnalogous to the 2D case, the 3D elastic-wave equations can be written using decomposed coe\u000ecient\nmatrices CiandDi(i= 1;2;3) as\n@\n@tv=D1@\n@x1\u001b+D2@\n@x2\u001b+D3@\n@x3\u001b; (37)\n@\n@t\u001b=C1@\n@x1v+C2@\n@x2v+C3@\n@x3v; (38)\nwhere\nC1=2\n6666666666664C11C16C15\nC12C26C25\nC13C36C35\nC14C46C45\nC15C56C55\nC16C66C563\n7777777777775; (39)\n9C2=2\n6666666666664C16C12C14\nC26C22C24\nC36C23C34\nC46C24C44\nC56C25C54\nC66C26C643\n7777777777775; (40)\nC3=2\n6666666666664C15C14C13\nC25C24C23\nC35C34C33\nC45C44C34\nC55C45C35\nC65C46C363\n7777777777775; (41)\nD1=2\n66641 0 0 0 0 0\n0 0 0 0 0 1\n0 0 0 0 1 03\n7775; (42)\nD2=2\n66640 0 0 0 0 1\n0 1 0 0 0 0\n0 0 0 1 0 03\n7775; (43)\nD3=2\n66640 0 0 0 1 0\n0 0 0 1 0 1\n0 0 1 0 0 03\n7775: (44)\nThe system of wave equations in PMLs for the 3D case can be expressed in the form of an autonomous\nsystem in the wavenumber domain as\n@U\n@t=~AU; (45)\nwhere\nU=F[\t]; (46)\n\t= (\u001b(1)\n11;\u001b(1)\n22;\u001b(1)\n33;\u001b(1)\n23;\u001b(1)\n13;\u001b(1)\n12;\u001b(2)\n11;\u001b(2)\n22;\u001b(2)\n33;\u001b(2)\n23;\u001b(2)\n13;\u001b(2)\n12;\n\u001b(3)\n11;\u001b(3)\n22;\u001b(3)\n33;\u001b(3)\n23;\u001b(3)\n13;\u001b(3)\n12;v(1)\n1;v(1)\n2;v(1)\n3;v(2)\n1;v(2)\n2;v(2)\n3;v(3)\n1;v(3)\n2;v(3)\n3)T; (47)\n~A=~A0+B; (48)\n10~A0=2\n6666666666664066 066 066ik1C1ik1C1ik1C1\n066 066 066ik2C2ik2C2ik2C2\n066 066 066ik3C3ik3C3ik3C3\nik1\u001a\u00001D1ik1\u001a\u00001D1ik1\u001a\u00001D1033 033 033\nik2\u001a\u00001D2ik2\u001a\u00001D2ik2\u001a\u00001D2033 033 033\nik3\u001a\u00001D3ik3\u001a\u00001D3ik3\u001a\u00001D3033 033 0333\n7777777777775; (49)\nB=2\n6666666666664\u0000d1I6066 066 063 063 063\n066\u0000d2I6066 063 063 063\n066 066\u0000d3I6063 063 063\n036 036 036\u0000d1I3033 033\n036 036 036 033\u0000d2I3033\n036 036 036 033 033\u0000d3I33\n7777777777775; (50)\nanddiis the damping coe\u000ecient along the xi-direction ( i= 1;2;3).\nTo stabilize the PML in the xi-direction, we need to employ nonzero damping ratios along the other two\ndirections perpendicular to xi. Therefore, for the x1-,x2- andx3-directions, we respectively set the damping\nmatrix to be\nB1(\u00181) =2\n6666666666664\u0000d1I6 066 066 063 063 063\n066\u0000d1\u00181I6 066 063 063 063\n066 066\u0000d1\u00181I6063 063 063\n036 036 036\u0000d1I3 033 033\n036 036 036 033\u0000d1\u00181I3 033\n036 036 036 033 033\u0000d1\u00181I33\n7777777777775; (51)\nB2(\u00182) =2\n6666666666664\u0000d2\u00182I6066 066 063 063 063\n066\u0000d2I6 066 063 063 063\n066 066\u0000d2\u00182I6 063 063 063\n036 036 036\u0000d2\u00182I3033 033\n036 036 036 033\u0000d2I3 033\n036 036 036 033 033\u0000d2\u00182I33\n7777777777775; (52)\nB3(\u00183) =2\n6666666666664\u0000d3\u00183I6 066 066 063 063 063\n066\u0000d3\u00183I6066 063 063 063\n066 066\u0000d3I6 063 063 063\n036 036 036\u0000d3\u00183I3 033 033\n036 036 036 033\u0000d3\u00183I3033\n036 036 036 033 033\u0000d3I33\n7777777777775: (53)\nIn the above 3D MPML damped matrices, we employ the same damping ratio along the two directions\n11parallel with the PML/non-PML interface. For instance, for damping along the x1-direction, we use the\nsame nonzero damping ratio \u00181for thex2- andx3-directions, so both the damping coe\u000ecients along the x2-\nandx3-directions in 3D MPML are \u00181d1. Using di\u000berent damping ratios along di\u000berent directions may also\nstabilize PML, but searching optimal values of damping ratios becomes even more complicated.\nWe develop a new approach to computing the eigenvalue derivatives of damped matrix ~Ausing eqs (51)-\n(53) for 3D general anisotropic media. Matrix ~A(as well as ~A0) has a dimension of 27 \u000227, resulting in\nan order of 27 of the characteristic polynomial of ~Aor~A0. Therefore, it is very di\u000ecult, if not impossible,\nto derive analytic expressions for the eigenvalues and eigenvalue derivatives, particularly for media with all\nCij6= 0.\nWe therefore adopt a numerical approach to solving for the eigenvalues and eigenvalue derivatives. Using\nthe de\fnitions of eigenvalues and eigenvectors of matrix ~A, we have\n(~A\u0000\u0015I27)P=0; (54)\nwhere\u0015is the eigenvalue of ~A, and the columns of Pare the eigenvectors of ~A. In addition, we have\nQT(~A\u0000\u0015I27) =0; (55)\nwhere the columns of QTare the left eigenvectors of ~A.\nDi\u000berentiating equation (54) with respect to damping parameter digives\n \n@~A\n@di\u0000@\u0015\n@diI27!\nP+ (~A\u0000\u0015I27)@P\n@di= 0: (56)\nMultiplying both sides of equation (56) with QTleads to\nQT \n@~A\n@di\u0000@\u0015\n@diI27!\nP+QT(~A\u0000\u0015I27)@P\n@di= 0; (57)\nwhich implies\nQT@~A\n@diP=QT@\u0015\n@diI27P: (58)\nTherefore,\n@\u0015\n@di=QT@~A\n@diP\nQTI27P: (59)\nBecause ~A=~A0+Bi(\u0018i) and ~A0is irrelevant to di, the eigenvalue derivative along xi-axis can be written\nas\n\u001fi(\u0018i) =QTRi(\u0018i)P\nQTP; (60)\n12where\nR1(\u00181) =2\n6666666666664\u0000I6066 066 063 063 063\n066\u0000\u00181I6066 063 063 063\n066 066\u0000\u00181I6063 063 063\n036 036 036\u0000I3033 033\n036 036 036 033\u0000\u00181I3033\n036 036 036 033 033\u0000\u00181I33\n7777777777775; (61)\nR2(\u00182) =2\n6666666666664\u0000\u00182I6066 066 063 063 063\n066\u0000I6066 063 063 063\n066 066\u0000\u00182I6063 063 063\n036 036 036\u0000\u00182I3033 033\n036 036 036 033\u0000I3033\n036 036 036 033 033\u0000\u00182I33\n7777777777775; (62)\nR3(\u00183) =2\n6666666666664\u0000\u00183I6066 066 063 063 063\n066\u0000\u00183I6066 063 063 063\n066 066\u0000I6063 063 063\n036 036 036\u0000\u00183I3033 033\n036 036 036 033\u0000\u00183I3033\n036 036 036 033 033\u0000I33\n7777777777775: (63)\nThese equations indicate that we only need to obtain the eigenvalues and the left and right eigenvectors\nof matrix ~A(C;k;\u0018i) for obtaining the optimal damping ratios of MPML in 3D general anisotropic media.\nThis can be achieved using a linear algebra library such as LAPACK, and the procedure is summarized in\nAlgorithm 2.\n13Algorithm 2: Determine the optimal damping ratio \u0018i(i= 1;2;3) for MPML in 3D general anisotropic\nmedia\ninput :\u0018i= 0,\u000f=\u00000:005, \u0001\u0018= 0:001.\nfor\u00122(0;\u0019]do\nfor\u001e2(0;\u0019]do\n1) Calculate wavenumber k= (cos\u001esin\u0012;sin\u001esin\u0012;cos\u0012)\n2) Calculate the left and right eigenvectors of ~A(C;k;\u0018i) using a numerical eigensolver\n3) Calculate the eigenvalue derivatives \u001f(l)\ni(l= 1;2;3) according to eq. (60)\n4)\u001fi;max= max(\u001f(1)\ni;\u001f(2)\ni;\u001f(3)\ni)\nif\u001fi;max>\u000fthen\n\u0018i=\u0018i+ \u0001\u0018\ngo to step 2\nend\nend\nend\noutput:\u0018i\nIn the above algorithm, it is not necessary to seek analytic forms of the left/right eigenvectors, which is\ngenerally impossible for matrix ~A. In our following numerical tests, we calculate the left/right eigenvectors\nwith the Intel Math Kernel Library wrapper for LAPACK. The above numerical approach is obviously\napplicable to the 2D case with trivial modi\fcations. Therefore, for 2D MPML, one can use either the\nanalytic approach or the numerical approach, yet for 3D MPML, one can use only the numerical approach.\n3. Results\nWe use three examples of 2D anisotropic media and three examples of 3D anisotropic media to validate\nthe e\u000bectiveness of our new algorithms for calculating optimal damping ratios in 2D and 3D MPMLs. In\nthe following, when presenting an elasticity matrix, we write only the upper triangle part of this matrix, but\nit should be clear that the elasticity matrix is essentially symmetric. We also assume that all the elasticity\nmatrices have units of GPa, and all the media have mass density values of 1000 kg/m3for convenience.\n3.1. MPML for 2D anisotropic media\nTo validate our new algorithm for determining the optimal damping ratios in MPML for 2D general\nanisotropic media, we consider a transversely isotropic medium with a horizontal symmetry axis (HTI\nmedium), a transversely isotropic medium with a tilted symmetry axis (TTI medium), and a transversely\nisotropic medium with a vertical symmetry axis (VTI medium) with serious qS triplication in both x1- and\nx3-directions.\n14-6000 -4000 -2000 0 2000 4000 6000\nx1component (m/s)\n-6000-4000-20000200040006000x3component (m/s)\nqP-wave\nqS-waveFigure 1: Wavefront curves in the 2D HTI medium with elasticity matrix (64).\nFor the HTI medium example, we use a well-known example with elasticity matrix (B\u0013 ecache et al., 2003;\nMeza-Fajardo and Papageorgiou, 2008):\nC=2\n66644 7:5 0\n20 0\n23\n7775: (64)\nNote that this medium is considered as an orthotropic medium in B\u0013 ecache et al. (2003) and Meza-Fajardo\nand Papageorgiou (2008). However, it could also be considered as an HTI medium on the x1x3-plane. The\nonly di\u000berences are that C11=C22andC44=C55for a 3D HTI medium, while there exists no such equality\nrestrictions for an orthotropic medium.\nThe wavefront curves of qP- and qS-waves in Fig. 1 show the anisotropy characteristics of this HTI\nmedium. We employ Algorithm 1 to determine the optimal damping ratios in MPML along the x1- and\nx3-directions, leading to\n\u00181= 0:108; \u0018 3= 0:259: (65)\nIn Meza-Fajardo and Papageorgiou (2008), the suggested values of damping ratios are \u00181= 0:30 and\n\u00183= 0:25 for this HTI medium. Their suggested value for damping ratio \u00181is much larger than the optimal\ndamping ratio given in eq. (65), while their suggested value for \u00183is similar to the optimal damping ratio.\nFigure 2 plots the values of eigenvalue derivatives of ~Aunder the optimal damping ratios in the x1-\nandx3-directions. In both panels, the blue curves represent the qP-wave eigenvalue derivatives, and the\nred curves are for the qS-wave eigenvalue derivatives. Clearly, the qS-wave gives rise to the large damping\nratios along both axes. Note that we set the threshold \u000f=\u00000:005, therefore, in both panels of Fig. 2, the\nmaximum values of the eigenvalue derivatives are \u00000:005.\nWe validate the e\u000bectiveness of our new MPML in numerical modeling of anisotropic elastic-wave pro-\n150 30 60 90 120 150\nWavenumber polar angle  (deg.)\n-1.5-1.2-0.9-0.6-0.30\nqPwave 1 with 1=0.108\nqSwave 1 with 1=0.108\n(a)\n0 30 60 90 120 150\nWavenumber polar angle  (deg.)\n-1.5-1.2-0.9-0.6-0.30\nqPwave 3 with 3=0.259\nqSwave 3 with 3=0.259\n (b)\nFigure 2: Eigenvalue derivatives of ~Aof MPML in the (a) x1- and (b)x3-directions under calculated optimal damping ratios\nin eq. (65) for the 2D HTI medium with elasticity matrix (64).\ngation. We use the rotated-staggered grid (RSG) \fnite-di\u000berence method (Saenger et al., 2000) to solve\nthe stress-velocity form elastic-wave equations (1){(2). The RSG \fnite-di\u000berence method has 16th-order\naccuracy in space with optimal \fnite-di\u000berence coe\u000ecients (Liu, 2014). We compute the wave\feld energy\ndecay curves of our wave\feld modelings to validate the e\u000bectiveness of MPML.\nIn our numerical modeling, the model is de\fned in a 400 \u0002400 grid, and a PML of 30-node thickness are\npadded around the model domain. The grid size is 10 m in both the x1- andx3-directions. A vertical force\nvector source is located at the center of the computational domain, and a Ricker wavelet with a 10 Hz central\nfrequency is used as the source time function. We simulate wave propagation for 20 s with a time interval of\n1 ms, which is smaller than what is required to satisfy the stability condition (about 1.54 ms). Figure 3 shows\nthe resulting wave\feld energy curve under the optimal damping ratios in eq. (65) , together with three others\nunder di\u000berent eigenvalue derivative threshold \u000fvalues, or equivalently, di\u000berent damping ratios. Figure 3\nshows that within the 20 s of wave propagation, the MPML with our calculated damping ratios \u00181= 0:108\nand\u00183= 0:259 is stable. These damping ratios are obtained under threshold value \u000f=\u00000:005, meaning\nthat the damping ratios have to ensure the eigenvalues derivatives \u001f1and\u001f3are not larger than \u00000:005 in\nthe entire range of wavenumber direction.\nWe test the behavior of MPML under threshold \u000f= 0:01, or equivalently, \u00181= 0:095 and\u00183= 0:248, and\nshow in Fig. 3 that the numerical modeling is stable. We further increase the threshold \u000fto be 0.05, and\nthe MPML becomes unstable quickly after about 2 s. Finally, the conventional PML, which is equivalent\nto MPML under \u00181=\u00183= 0, becomes unstable even earlier (before 1 s). These tests indicate that a small\npositive threshold \u000fmay still result in a stable MPML. However, there is no simple method to determine how\nlarge this positive \u000fto ensure the numerical stability. In this HTI medium case, \u000f= 0:01 results in stability\nwhile\u000f= 0:05 results in instability. Because a negative threshold resulting in stable MPML is consistent\nwith the stability theory presented by Meza-Fajardo and Papageorgiou (2008), we therefore should choose a\n160 5 10 15 20\nTime (s)1011\n109\n107\n105\n103\nWavefield energy |v|2=0.005,1=0.108,3=0.259\n=0.01,1=0.095,3=0.248\n=0.05,1=0.059,3=0.218\nConventional PMLFigure 3: Wave\feld energy decay curves under di\u000berent eigenvalue derivative thresholds for the 2D HTI medium with elasticity\nmatrix (64) within 20 s. Conventional PML can be considered as a special case of MPML with \u000f\u001d0 or equivalently \u00181=\u00183= 0.\nnegative threshold for the calculation of the optimal damping ratios to ensure that the resulting MPML is\nstable. This is also veri\fed in the hereinafter numerical examples.\nNext, we rotate the aforementioned HTI medium with respect to the x2-axis clockwise by \u0019=6 to obtain\na TTI medium represented by the following elasticity matrix:\nC=2\n66647:8125 7:6875 3:35585\n15:8125 3:57235\n2:18753\n7775; (66)\nwith unit GPa. The rotation can be accomplished by rotation matrix (e.g., Slawinski, 2010). The wavefront\ncurves in this TTI medium is shown in Fig. 4. Although this TTI medium is the rotation result of the HTI\nmedium in the previous numerical example, it is not obvious how to change the damping ratios accordingly.\nWe obtain the following optimal damping ratios of MPML under \u000f=\u00000:005 using Algorithm 1:\n\u00181= 0:157; \u0018 3= 0:226: (67)\nThe eigenvalue derivatives under this set of damping ratios are shown in Fig. 5. The eigenvalue derivative\ncurves are no longer symmetric with respect to \u0012=\u0019=2 (or has a period of \u0019=2) as those for the 2D HTI\nmedium (Fig. 2). Instead, they are periodic every \u0019angle, corresponding to the fact that there is always at\nleast one symmetric axis for whatever kind of 2D anisotropic medium in the axis plane. These curves also\nindicate that, for 2D general anisotropic medium (TTI medium in this example), it is necessary to determine\nthe values of eigenvalue derivatives within the range of (0 ;\u0019] instead of (0 ;\u0019=2]. Using only the range (0 ;\u0019=2]\ncan lead to a totally incorrect optimal value of \u00181, since the maximum value of \u001f1in the range (0 ;\u0019=2] is\nsmaller than that in the range ( \u0019=2;\u0019] for this TTI medium. In other words, even though \u001f1in (0;\u0019=2]\n17-6000 -4000 -2000 0 2000 4000 6000\nx1component (m/s)\n-6000-4000-20000200040006000x3component (m/s)\nqP-wave\nqS-waveFigure 4: Wavefront curves in the 2D HTI medium with elasticity matrix (66).\n0 30 60 90 120 150\nWavenumber polar angle  (deg.)\n-1.5-1.2-0.9-0.6-0.30\nqPwave 1 with 1=0.157\nqSwave 1 with 1=0.157\n0 30 60 90 120 150\nWavenumber polar angle  (deg.)\n-1.5-1.2-0.9-0.6-0.30\nqPwave 3 with 3=0.226\nqSwave 3 with 3=0.226\nFigure 5: Eigenvalue derivatives of ~Aof MPML in the (a) x1- and (b)x3-directions under calculated optimal damping ratios\nin eq. (67) for the 2D TTI medium with elasticity matrix (66).\nindicates a stable MPML, the MPML may still be unstable since \u001f1may be larger than zero in ( \u0019=2;\u0019].\nTherefore, for anisotropic media with symmetric axis not aligned with a coordinate axis, it is necessary to\nconsider the values of \u001fiin wavenumber direction \u00122(0;\u0019]. This statement is also true for 3D anisotropic\nmedia as shown in the hereinafter 3D numerical examples.\nFigure 6 displays the wave\feld energy decay curves for this TTI medium under the optimal damping\nratios, as well as under damping ratios calculated with positive eigenvalue derivative thresholds. In this\nexample, the wave\feld energy decays gradually within 20 s for both cases with \u000f=\u00000:005 and\u000f= 0:05.\nThe numerical modeling with \u000f= 0:15 becomes unstable. For comparison, in the previous HTI case,\n\u000f= 0:05 results in an unstable MPML. These results further demonstrate that a positive eigenvalue derivative\nthreshold should not be chosen to calculate the damping ratios, although a small positive \u000fmight result in\nstable MPML. In contrast, a negative \u000fcan always ensure the stability of MPML.\n180 5 10 15 20\nTime (s)1013\n1011\n109\n107\n105\n103\nWavefield energy |v|2=0.005,1=0.157,3=0.226\n=0.05,1=0.110,3=0.183\n=0.15,1=0.025,3=0.105\nConventional PMLFigure 6: Wave\feld energy decay curves under di\u000berent eigenvalue derivative thresholds in the 2D TTI medium with elasticity\nmatrix (66) within 20 s.\nOur next numerical example uses a VTI medium de\fned by\nC=2\n666410:4508 4:2623 0\n7:5410 0\n11:39343\n7775: (68)\nThe wavefront curves for this VTI medium are depicted in Fig. 7. The special feature of this VTI medium\nis that, in both the x1- andx3-directions, there exists serious qS-wave triplication phenomena. We obtain\nthe following optimal damping ratios using Algorithm 1:\n\u00181= 0:215; \u0018 3= 0:225: (69)\nThe corresponding eigenvalue derivatives in the x1- andx3-directions are displayed in Fig. 8. Again, it is the\nqS-wave that causes the damping ratios to be large to stabilize PML. Figure 9 depicts the wave\feld energy\ndecay curves under di\u000berent eigenvalue derivative thresholds. Similar with that of 2D TTI medium example,\na positive threshold 0.05 can still stabilize PML, yet a value of 0.15 makes the MPML unstable. We choose\na negative value of \u000fto ensure a stable MPML.\n3.2. MPML for 3D anisotropic media\nFor 3D anisotropic media, we need to determine the optimal MPML damping ratios along all three\ncoordinate directions. We use three di\u000berent anisotropic media (a quasi-VTI medium, a quasi-TTI medium\nand a triclinic medium) to demonstrate the determination of optimal MPML damping ratios.\n19-5000 -2500 0 2500 5000\nx1component (m/s)\n-5000-2500025005000x3component (m/s)\nqP-wave\nqS-waveFigure 7: Wavefront curves in the 2D VTI medium with elasticity matrix (68).\n0 30 60 90 120 150\nWavenumber polar angle  (deg.)\n-1.5-1.2-0.9-0.6-0.30\nqPwave 1 with 1=0.215\nqSwave 1 with 1=0.215\n0 30 60 90 120 150\nWavenumber polar angle  (deg.)\n-1.5-1.2-0.9-0.6-0.30\nqPwave 3 with 3=0.225\nqSwave 3 with 3=0.225\nFigure 8: Eigenvalue derivatives of ~Aof MPML in the (a) x1- and (b)x3-directions under calculated optimal damping ratios\nin eq. (69) for the 2D VTI medium with elasticity matrix (68).\n200 5 10 15 20\nTime (s)1020\n1015\n1010\n105\n100Wavefield energy |v|2=0.005,1=0.215,3=0.225\n=0.05,1=0.172,3=0.182\n=0.15,1=0.093,3=0.104\nConventional PMLFigure 9: Wave\feld energy decay curves under di\u000berent eigenvalue derivative thresholds in the 2D VTI medium with elasticity\nmatrix (68) within 20 s.\nWe \frst use a 3D anisotropic medium represented by the elasticity matrix\nC=2\n666666666666416:5 5 5 0 0 0\n16:5 5 0 0 0\n6:2 0 0 0\n4:96 0 0\n3:96 0\n5:963\n7777777777775: (70)\nThis elasticity matrix is modi\fed from the elasticity matrix of zinc (a VTI medium, or hexagonal anisotropic\nmedium) to increase the complexity of the resulting wavefronts and the characteristics of the eigenvalue\nderivatives along all three directions. This modi\fed elastic matrix still represents a physically feasible\nmedium since it is easy to verify that it satis\fes the following stability condition for anisotropic media\n(Slawinski, 2010):\ndet2\n6664C11\u0001\u0001\u0001C1n\n.........\nC1n\u0001\u0001\u0001Cnn3\n7775>0; (71)\nwheren= 1;2;\u0001\u0001\u0001;6. We call this anisotropic medium the quasi-VTI medium. Figure 10 shows the\nwavefront curves of this quasi-VTI medium on three axis planes.\nFor comparison, a standard 3D VTI medium can be expressed by its \fve independent elasticity constants\n21as (e.g., Slawinski, 2010)\nC=2\n6666666666664C11C12C13 0 0 0\nC11C13 0 0 0\nC33 0 0 0\nC44 0 0\nC44 0\nC11\u0000C12\n23\n7777777777775: (72)\nWe calculate the optimal damping ratios for MPML using Algorithm 2, and obtain\n\u00181= 0:088; \u0018 2= 0:131; \u0018 3= 0:041: (73)\nWe plot the eigenvalue derivatives in the polar angle range (0 ;\u0019] and azimuth angle range (0 ;\u0019] for three axis\ndirections in Fig. 11. Since the three symmetric axes of this VTI medium are aligned with three coordinate\naxes, the three eigenvalue derivatives are symmetric with respect to both \u0012=\u0019=2 and\u001e=\u0019=2 lines.\nWe conduct numerical wave\feld modeling to verify the stability of MPML under these optimal damping\nratios. The model is de\fned on a 400 \u0002400\u0002400 grid with a grid size of 10 m in all three directions. The\nthickness of PML layer is 25 grids. A vertical force vector is located at the center of the computational\ndomain, and the source time function is a Ricker wavelet with a central frequency of 10 Hz. The time step\nsize is 1 ms, which is smaller than the stability-required time step of 1.69 ms. A total of 15,000 time steps,\ni.e., 15 s, are simulated, and the wave\feld energy curve is shown in Fig. 12. The blue curve in Fig. 12 is\nfor the case with the optimal damping ratios in eq. (73). We also carry out a wave\feld modeling with the\neigenvalue derivative threshold \u000f= 0:015 and\u000f= 0:025. The MPMLs under these thresholds are unstable\naccording to the corresponding wave\feld energy variation curves in Fig. 12. As in the 2D MPML case, we\nshould always choose a negative \u000fto stabilize MPML for 3D anisotropic media.\nOur next numerical example uses a rotation version of the aforementioned quasi-VTI medium. We rotate\nthe quasi-VTI medium (70) with respect to the x1-axis by 30 degrees, the x2-axis by 50 degrees, and the\nx3-axis by 25 degrees, and the resulting elasticity matrix for this quasi-TTI medium is given by\nC=2\n666666666666415:7930 4:1757 4:9651 0:1582 0:6529\u00001:0343\n12:5979 4:1844 2:0903 0:8186\u00001:7513\n14:1587 1:9573 0:7643\u00000:1606\n3:7879\u00000:8909 0:8065\n5:0750 0:7668\n4:34233\n7777777777775: (74)\nThe corresponding wavefront curves are shown in Fig. 13.\nSimilar to the 3D quasi-VTI case, we obtain the following optimal damping ratios using Algorithm 2:\n\u00181= 0:089; \u0018 2= 0:051; \u0018 3= 0:080: (75)\n22-5000 -2500 0 2500 5000\nx1component (m/s)\n-5000-2500025005000x2component (m/s)\nqP-wave\nqS1-wave\nqS2-wave(a)\n-5000 -2500 0 2500 5000\nx1component (m/s)\n-5000-2500025005000x3component (m/s)\nqP-wave\nqS1-wave\nqS2-wave (b)\n-5000 -2500 0 2500 5000\nx2component (m/s)\n-5000-2500025005000x3component (m/s)\nqP-wave\nqS1-wave\nqS2-wave\n(c)\nFigure 10: Wavefront curves in the 3D quasi-VTI medium with elasticity matrix (70) on the (a) x1x2(b)x1x3and (c)x2x3\naxis plane. qS1 and qS2 represents the two qS-waves.\n2330 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1(a)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (b)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (c)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2\n(d)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (e)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (f)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1\n(g)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (h)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (i)\nFigure 11: Eigenvalue derivatives of ~Aof MPML in the (a)-(c) x1-, (d)-(f)x2-, and (g)-(i) x3-directions under calculated optimal\ndamping ratios in eq. (73) for the 3D quasi-VTI medium with elasticity matrix (70). (a), (d) and (g) represent qP-wave, (b),\n(e) and (h) represent qS1-wave, (c), (f) and (i) represent qS2-wave.\n240 5 10 15\nTime (s)1013\n1011\n109\n107\n105\nWavefield energy |v|2\n=0.005,1=0.089,2=0.131,3=0.041\n=0.015,1=0.068,2=0.112,3=0.020\n=0.025,1=0.058,2=0.102,3=0.009\nConventional PMLFigure 12: Wave\feld energy decay curve in the 3D VTI medium with elasticity matrix (70).\nThe eigenvalue derivatives under this set of damping ratios for three axis directions are shown in Fig. 14.\nObviously, for the quasi-TTI medium where the symmetric axes are not aligned with coordinate axes, the\neigenvalue derivatives of all three wave modes along any coordinate axis is no longer symmetric about any \u0012\nor\u001elines. Therefore, it is necessary to use the entire range of wavenumber polar angle \u0012and azimuth angle\n\u001e, i.e., (0;\u0019]\u0002(0;\u0019], to determine the optimal damping ratios.\nFigure 15 depicts the wave\feld energy decays under the optimal damping ratios in eq. (75) and damping\nratios with thresholds \u000f= 0:05 and\u000f= 0:075. For the case where \u000f= 0:05, the wave\feld energy does\nnot diverge immediately after maximum energy value occurred. Instead, the curve indicates a very slow\nenergy decay after about 1 s. In contrast, the optimal MPML with threshold \u000f=\u00000:005 shows a \\normal\"\nenergy decay. Therefore, although the MPML with \u000f= 0:05 does not show energy divergence within 15 s, it\nfails to e\u000bectively absorb the outgoing wave\feld, and we consider this as a \\quasi-divergence.\" Meanwhile,\nthe MPML with \u000f= 0:075 shows an energy divergence after about 4 s. These results further demonstrate\nthat the behavior of MPML with a positive eigenvalue derivative threshold is di\u000berent and unpredictable\nfor di\u000berent kinds of anisotropic media. Figure 15 also shows that the conventional PML gives an unstable\nresult.\nOur last 3D numerical example is based on a triclinic anisotropic medium represented by\nC=2\n666666666666410 3:5 2:5\u00005 0:1 0:3\n8 1:5 0:2\u00000:1\u00000:15\n6 1 0 :4 0:24\n5 0:35 0:525\n4\u00001\n33\n7777777777775; (76)\n25-5000 -2500 0 2500 5000\nx1component (m/s)\n-5000-2500025005000x2component (m/s)\nqP-wave\nqS1-wave\nqS2-wave(a)\n-5000 -2500 0 2500 5000\nx1component (m/s)\n-5000-2500025005000x3component (m/s)\nqP-wave\nqS1-wave\nqS2-wave (b)\n-5000 -2500 0 2500 5000\nx2component (m/s)\n-5000-2500025005000x3component (m/s)\nqP-wave\nqS1-wave\nqS2-wave\n(c)\nFigure 13: Wavefront curves in the 3D quasi-TTI medium with elasticity matrix (74) on the (a) x1x2(b)x1x3and (c)x2x3\naxis plane. qS1 and qS2 represents the two qS-waves. qS-wave wavefronts seem to be less complicated compared with those of\nthe 3D quasi-VTI medium (70) only because the qS-wave triplications are now out of axis planes after rotation.\n2630 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1(a)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (b)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (c)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1\n(d)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (e)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (f)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1\n(g)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (h)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (i)\nFigure 14: Eigenvalue derivatives of ~Aof MPML in the (a)-(c) x1-, (d)-(f)x2-, and (g)-(i) x3-directions under calculated optimal\ndamping ratios in eq. (75) for the 3D quasi-TTI medium with elasticity matrix (74). (a), (d) and (g) represent qP-wave, (b),\n(e) and (h) represent qS1-wave, (c), (f) and (i) represent qS2-wave.\n270 5 10 15\nTime (s)1024\n1019\n1014\n109\n104\nWavefield energy |v|2\n=0.005,1=0.089,2=0.051,3=0.080\n=0.05,1=0.034,2=0,3=0.024\n=0.075,1=0.009,2=0,3=0\nConventional PMLFigure 15: Wave\feld energy decay curve in the 3D quasi-TTI medium with elasticity matrix (74).\nwith unit GPa. The wavefront curves on three axis planes are shown in Fig. 16.\nWe solve for the optimal damping ratios for this triclinic anisotropic medium using Algorithm 2, and\nobtain the following optimal damping ratios with \u000f=\u00000:005:\n\u00181= 0:487; \u0018 2= 0:345; \u0018 3= 0:374: (77)\nThe damping ratios for this anisotropic medium are unexpectedly very large compared with those for the\nheretofore 2D and 3D examples. We seek the reasons of these large damping ratios from the eigenvalue\nderivatives shown in Fig. 17, and \fnd that it is the qS2-wave that leads to such large damping ratios to\nachieve a stable MPML. In fact, for the damping ratios in eq. (77), the corresponding eigenvalue derivatives\nof qP- and qS1-waves are far smaller than zero, yet the eigenvalue derivative of qS2-wave merely smaller\nthan zero (\u00000:005 under our threshold setting), leading to a set of relatively large damping ratios for this\n3D anisotropic medium.\nOur calculated optimal damping ratios result in a stable MPML, as indicated by the corresponding energy\ndecay curve shown in Fig. 18. The wave\feld energy decay curve with a threshold of \u000f= 0:1 displayed in\nFig. 18 is surprisingly almost identical with that of \u000f=\u00000:005. When using a threshold \u000f= 0:4, MPML\nbecome unstable, indicating that the positive threshold 0.4 is too large to make MPML stable. This veri\fes\nagain that, although a positive threshold might result in stable MPML, we should use a negative threshold\nto ensure a stable MPML for general anisotropic media. This is consistent with the stability condition\ndescribed in Meza-Fajardo and Papageorgiou (2008), and is perhaps the only practical method to stabilize\nPML using nonzero damping ratios.\n28-4000 -2000 0 2000 4000\nx1component (m/s)\n-3000-1500015003000x2component (m/s)\nqP-wave\nqS1-wave\nqS2-wave(a)\n-4000 -2000 0 2000 4000\nx1component (m/s)\n-3000-1500015003000x3component (m/s)\nqP-wave\nqS1-wave\nqS2-wave (b)\n-4000 -2000 0 2000 4000\nx2component (m/s)\n-3000-1500015003000x3component (m/s)\nqP-wave\nqS1-wave\nqS2-wave\n(c)\nFigure 16: Wavefront curves in the 3D triclinic anisotropic medium with elasticity matrix (74) on the (a) x1x2(b)x1x3and\n(c)x2x3axis plane. qS1 and qS2 represents the two qS-waves.\n2930 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5(a)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1.1-1-0.9-0.8-0.7-0.6-0.5-0.4 (b)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1.3-1.1-0.9-0.7-0.5-0.3-0.1 (c)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3\n(d)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3 (e)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1.3-1.1-0.9-0.7-0.5-0.3-0.1 (f)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3\n(g)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3 (h)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1.3-1.1-0.9-0.7-0.5-0.3-0.1 (i)\nFigure 17: Eigenvalue derivatives of ~Aof MPML in the (a)-(c) x1-, (d)-(f)x2-, and (g)-(i) x3-directions under calculated\noptimal damping ratios in eq. (77) for the 3D triclinic anisotropic medium with elasticity matrix (76). (a), (d) and (g) represent\nqP-wave, (b), (e) and (h) represent qS1-wave, (c), (f) and (i) represent qS2-wave.\n300 5 10 15\nTime (s)1015\n1012\n109\n106\n103\nWavefield energy |v|2\n=0.005,1=0.487,2=0.345,3=0.374\n=0.1,1=0.420,2=0.258,3=0.291\n=0.4,1=0.230,2=0.010,3=0.054\nConventional PMLFigure 18: Wave\feld energy decay curve in the 3D triclinic anisotropic medium with elasticity matrix (76). The blue curve\n(\u000f=\u00000:005) and red curve ( \u000f= 0:1) are almost identical.\n4. Conclusions\nA de\fnite analytic method for determining the optimal damping ratios of multi-axis perfectly matched\nlayers (MPML) is generally impossible for 3D general anisotropic media with possible all nonzero elasticity\nparameters. We have developed a new method to e\u000eciently determine the optimal damping ratios of MPML\nfor absorbing unwanted, outgoing propagating waves in 2D and 3D general anisotropic media. This numerical\napproach is very straightforward using the left and right eigenvectors of the damped system coe\u000ecient matrix.\nWe have used six numerical modeling examples of elastic-wave propagation in 2D and 3D anisotropic media\nto demonstrate that our new algorithm can e\u000bectively and correctly provide the optimal MPML damping\nratios for even very complex, general anisotropic media.\n5. Acknowledgments\nThis work was supported by U.S. Department of Energy through contract DE-AC52-06NA25396 to Los\nAlamos National Laboratory (LANL). The computation was performed using super-computers of LANL's\nInstitutional Computing Program.\nReferences\nB\u0013 ecache, E., Fauqueux, S., Joly, P., 2003. Stability of perfectly matched layers, group velocities and\nanisotropic waves. Journal of Computational Physics 188 (2), 399 { 433.\nURL http://www.sciencedirect.com/science/article/pii/S0021999103001840\n31Berenger, J.-P., 1994. A perfectly matched layer for the absorption of electromagnetic waves. Journal of\nComputational Physics 114 (2), 185 { 200.\nURL http://www.sciencedirect.com/science/article/pii/S0021999184711594\nCarcione, J. M., 2007. Wave \felds in real media: Wave propagation in anisotropic, anelastic, porous and\nelectromagnetic media (Second Edition). Elsevier, Amsterdam, Netherlands.\nCerjan, C., Koslo\u000b, D., Koslo\u000b, R., Reshef, M., 1985. A nonre\recting boundary condition for discrete acoustic\nand elastic wave equations. Geophysics 50 (4), 705{708.\nURL http://geophysics.geoscienceworld.org/content/50/4/705\nClayton, R., Engquist, B., 1977. Absorbing boundary conditions for acoustic and elastic wave equations.\nBulletin of the Seismological Society of America 67 (6), 1529{1540.\nURL http://bssa.geoscienceworld.org/content/67/6/1529\nCollino, F., Tsogka, C., 2001. Application of the perfectly matched absorbing layer model to the linear\nelastodynamic problem in anisotropic heterogeneous media. Geophysics 66 (1), 294{307.\nURL http://dx.doi.org/10.1190/1.1444908\nde la Puente, J., K aser, M., Dumbser, M., Igel, H., 2007. An arbitrary high-order discontinuous Galerkin\nmethod for elastic waves on unstructured meshes IV. Anisotropy. Geophysical Journal International\n169 (3), 1210{1228.\nURL http://dx.doi.org/10.1111/j.1365-246X.2007.03381.x\nDmitriev, M. N., Lisitsa, V. V., 2011. Application of M-PML re\rectionless boundary conditions to the\nnumerical simulation of wave propagation in anisotropic media. Part I: Re\rectivity. Numerical Analysis\nand Applications 4 (4), 271{280.\nURL http://dx.doi.org/10.1134/S199542391104001X\nDrossaert, F. H., Giannopoulos, A., 2007. A nonsplit complex frequency-shifted pml based on recursive\nintegration for fdtd modeling of elastic waves. Geophysics 72 (2), T9{T17.\nURL http://dx.doi.org/10.1190/1.2424888\nHastings, F. D., Schneider, J. B., Broschat, S. L., 1996. Application of the perfectly matched layer (PML) ab-\nsorbing boundary condition to elastic wave propagation. The Journal of the Acoustical Society of America\n100 (5).\nHigdon, R. L., Oct. 1986. Absorbing boundary conditions for di\u000berence approximations to the multi-\ndimensional wave equation. Math. Comput. 47 (176), 437{459.\nURL http://dx.doi.org/10.2307/2008166\n32Higdon, R. L., 1987. Numerical absorbing boundary conditions for the wave equation. Mathematics of\nComputation 49 (179), 65{90.\nURL http://www.jstor.org/stable/2008250\nKomatitsch, D., Barnes, C., Tromp, J., 2000. Simulation of anisotropic wave propagation based upon a\nspectral element method. Geophysics 65 (4), 1251{1260.\nURL http://dx.doi.org/10.1190/1.1444816\nKomatitsch, D., Martin, R., 2007. An unsplit convolutional perfectly matched layer improved at grazing\nincidence for the seismic wave equation. Geophysics 72 (5), SM155{SM167.\nURL http://dx.doi.org/10.1190/1.2757586\nKomatitsch, D., Tromp, J., 2003. A perfectly matched layer absorbing boundary condition for the second-\norder seismic wave equation. Geophysical Journal International 154 (1), 146{153.\nURL http://dx.doi.org/10.1046/j.1365-246X.2003.01950.x\nLiao, Z.-F., Huang, K.-L., Yang, B.-P., Yuan, Y.-F., 1984. A transmitting boundary for transient wave\nanalyses. Science China: Mathematics 27 (10), 1063.\nURL http://math.scichina.com:8081/sciAe/EN/abstract/article_379434.shtml\nLiu, Y., 2014. Optimal staggered-grid \fnite-di\u000berence schemes based on least-squares for wave equation\nmodelling. Geophysical Journal International 197, 1033{1047.\nURL http://gji.oxfordjournals.org/content/early/2014/02/20/gji.ggu032.abstract\nLong, L. T., Liow, J. S., 1990. A transparent boundary for \fnite-di\u000berence wave simulation. Geophysics\n55 (2), 201{208.\nURL http://geophysics.geoscienceworld.org/content/55/2/201\nMartin, R., Komatitsch, D., 2009. An unsplit convolutional perfectly matched layer technique improved at\ngrazing incidence for the viscoelastic wave equation. Geophysical Journal International 179 (1), 333{344.\nURL http://dx.doi.org/10.1111/j.1365-246X.2009.04278.x\nMartin, R., Komatitsch, D., Gedney, S. D., Bruthiaux, E., 2010. A high-order time and space formulation\nof the unsplit perfectly matched layer for the seismic wave equation using auxiliary di\u000berential equations\n(ADE-PML). Computer Modeling in Engineering & Sciences 56 (1), 17{40.\nMeza-Fajardo, K. C., Papageorgiou, A. S., 2008. A nonconvolutional, split-\feld, perfectly matched layer for\nwave propagation in isotropic and anisotropic elastic media: Stability analysis. Bulletin of the Seismological\nSociety of America 98 (4), 1811{1836.\nURL http://www.bssaonline.org/content/98/4/1811.abstract\n33Peng, C., Toks oz, M. N., 1994. An optimal absorbing boundary condition for \fnite di\u000berence modeling of\nacoustic and elastic wave propagation. The Journal of the Acoustical Society of America 95 (2), 733{745.\nURL http://scitation.aip.org/content/asa/journal/jasa/95/2/10.1121/1.408384\nReynolds, A. C., 1978. Boundary conditions for the numerical solution of wave propagation problems. Geo-\nphysics 43 (6), 1099{1110.\nURL http://geophysics.geoscienceworld.org/content/43/6/1099\nSaenger, E. H., Gold, N., Shapiro, S. A., 2000. Modeling the propagation of elastic waves using a modi\fed\n\fnite-di\u000berence grid. Wave Motion 31 (1), 77 { 92.\nURL http://www.sciencedirect.com/science/article/pii/S0165212599000232\nSlawinski, M. A., 2010. Waves and Rays in Elastic Continua. World Scienti\fc.\nURL http://www.worldscientific.com/worldscibooks/10.1142/7486#t=aboutBook\nZhang, W., Shen, Y., 2010. Unsplit complex frequency-shifted PML implementation using auxiliary di\u000ber-\nential equations for seismic wave modeling. Geophysics 75 (4), T141{T154.\nURL http://dx.doi.org/10.1190/1.3463431\n34"    },    {        "title": "1507.01124v1.Comments_on_turbulence_theory_by_Qian_and_by_Edwards_and_McComb.pdf",        "content": "arXiv:1507.01124v1  [physics.flu-dyn]  4 Jul 2015Comments on turbulence theory by Qian and by Edwards and\nMcComb\nR. V. R. Pandya∗\nDepartment of Mechanical Engineering,\nUniversity of Puerto Rico at Mayaguez, PR 00682, USA\nAbstract\nWe reexamine Liouville equation based turbulence theories proposed by Qian [Phys. Fluids 26,\n2098 (1983)] and Edwards and McComb [J. Phys. A: Math. Gen. 2, 157 (1969)], which are\ncompatible with Kolmogorov spectrum. These theories obtai ned identical equation for spectral\ndensityq(k)and different results for damping coefficient. Qian proposed v ariational approach and\nEdwards and McComb proposed maximal entropy principle to ob tain equation for the damping co-\nefficient. We show that assumptions used in these theories to o btain damping coefficient correspond\nto unphysical conditions.\n∗rvrptur(AT)yahoo.com\n1I. INTRODUCTION\nEdwards [2] proposed turbulence theory based on Liouville e quation for joint probability\ndistribution function of Fourier modes uα(k,t)of velocity field governed by forced Navier-\nStokes equation. Following Edwards, a few more turbulence t heories [3, 4, 10] were proposed\nto solve Liouville equation and were reviewed by Leslie [7] a nd McComb [8]. Edwards theory\nseeks Fokker-Planck model representation for Liouville eq uation and obtains closed set of\nequations for spectral density q(k)and damping coefficient (total viscosity) ω(k)for sta-\ntionary, isotropic turbulence. The theory failed to be cons istent with Kolmogorov spectrum\n[1, 5] and the failure was attributed to equation for ω(k)[8]. As a modification to Edwards’\ntheory, Edwards and McComb [3] proposed principle of maxima l entropy to derive an equa-\ntion forω(k)compatible with Kolmogorov spectrum. Within the Fokker-Pl anck framework\nof Edwards, Qian [10] proposed variational approach and obt ained different equation for\ndamping coefficient consistent with Kolmogorov spectrum. In stead of using uα(k,t), Qian\nused real dynamical modal variables Xiand their governing equations which were suggested\nby Kraichnan [6] and later utilized by Herring [4] for his sel f-consistent turbulence theory.\nHerring’s theory also uses Liouville equation and obtains e quation for q(k)identical to equa-\ntion obtained by Edwards. In this paper, we reexamine Kolmog orov spectrum compatible\ntheories proposed by Qian [10] and Edwards and McComb [3] to o btain damping coefficient.\nWe show in the next two sections that assumptions made in thes e theories correspond to\nunphysical conditions.\nII. VARIATIONAL APPROACH BY QIAN\nFor discussion purpose, hereafter we refer to Qian’s variat ional approach as Q83. We\nuse (Q83; #) to represent equation number (#) in Q83 paper [10 ]. Qian based his theory\non governing equation for real dynamical modal variables Xifor stationary, homogeneous,\nisotropic turbulence, written as\ndXi\ndt=−(νi−ν′\ni)Xi+/summationdisplay\nj,mAijmXjXm,(Q83;8) (1)\nwhereν′\niXirepresents external driving force. Einstein summation con vention of repeated in-\ndices is not utilized in Eq. (1) and in this section. It should be noted that the real dynamical\n2modal variables and their equations were first suggested by K raichnan within the context of\nhydromagnetic turbulence [6]. Qian’s theory [10] seeks Lan gevin model representation for\nisotropic turbulence in the form\nd\ndtXi≃ −ηiXi+fi, ηi=ζi+(ν−ν′\ni) (Q83;12) (2)\nby using\n/summationdisplay\nj,mAijmXjXm∼=−ζiXi+fi(Q83;11) (3)\nwhere−ζiXiis dynamical damping term and fiis white noise type forcing term. Qian pro-\nposed variational approach to obtain damping coefficient ηi. The approach yields equation\nforηiby minimizing a function I(ηi), written as\nI=/summationdisplay\ni/angbracketleftBigg\n/summationdisplay\nj,mAijmXjXm−(−ζiXi)\n2/angbracketrightBigg\n(Q83;29) (4)\nand using\n∂I\n∂ηi= 0.(Q83;28) (5)\nHere/angbracketleft /angbracketrightrepresents ensemble average. Qian considered variation in Iunder constraint φi=\nconstant , whereφiis related to /angbracketleftX2\ni/angbracketrightby\n/angbracketleftBig\nX2\ni/angbracketrightBig\n=φi/parenleftBigg\n1−νi−ν′\ni\nηi/parenrightBigg\n.(Q83;25) (6)\nAlso,φiis proportional to spectral density q(k)[10].\nWe now show that for I=I(ηi)and under the constraint of φi=constant ,\n∂I\n∂ηi/negationslash= 0 (7)\nwithin the framework of Langevin model considered by Qian. C onsequently, the use of Eq.\n(5) to obtain ηiis in error. For stationary turbulence, solution of Eq. (2) s uggests\n2ηi/angbracketleftBig\nX2\ni/angbracketrightBig\n=Fi (8)\nin which correlation of white noise forcing term\n/angbracketleftfi(t)fi(t′)/angbracketright=Fiδ(t−t′) (9)\n3is utilized. Here δ(t−t′)is Dirac delta function. Using Eqs. (3) , (4) and (9), functio nI(ηi)\ncan be written as\nI=/summationdisplay\ni/angbracketleftBigg\n/summationdisplay\nj,mAijmXjXm−(−ζiXi)\n2/angbracketrightBigg\n=/summationdisplay\ni/angbracketleftfifi/angbracketright=δ(0)/summationdisplay\niFi. (10)\nFurther, using Eqs (6), (8), (10) and for φi=constant , we can write\n∂I\n∂ηi=δ(0)/summationdisplay\nj∂Fj\n∂ηi=δ(0)/summationdisplay\nj∂2ηj/angbracketleftBig\nX2\nj/angbracketrightBig\n∂ηi= 2δ(0)φi. (11)\nThis Eq. (11) suggests that for all i\n∂I\n∂ηi/negationslash= 0 (12)\nasφi/negationslash= 0. In view of this, use of Eq. (5), i.e.∂I\n∂ηi= 0, in Q83 to obtain ηiis in error and\ncorresponds to unphysical condition φi= 0,∀ifor stationary turbulence.\nNow we suggest possible modification within the framework of Q83. Consider a function\nV, written as\nV=/summationdisplay\nj∂I\n∂ηj=/summationdisplay\nj2δ(0)φj, (13)\nwhich satisfies an exact condition\n∂V\n∂ηi= 0. (14)\nThis condition along with Eqs. (4) can be used, instead of Eq. (5), to obtain equation for\nηi.\nIII. MAXIMAL ENTROPY PRINCIPLE BY EDWARDS AND MCCOMB\nFor discussion purpose, hereafter we refer to Edwards and Mc Comb’s theory as EM69.\nWe use (M90; #) to represent equation number (#) in McComb’s b ook [8]. Edwards and\nMcComb [3] considered stationary, homogeneous, isotropic , turbulence inside a cubic box of\nsideL. Their Liouville equation based theory uses equation for Fo urier modes uα(k,t)of\nthe velocity field uα(x,t)governed by forced Navier-Stokes equation, written as\n/parenleftBigg∂\n∂t+νk2/parenrightBigg\nuα(k,t) =Mαβγ(k)/summationdisplay\njuβ(j,t)uγ(k−j,t)+fα(k,t),(M90;4.81) (15)\n4wherefαrepresents external driving force, kis wavevector and k2=|k|2. The Einstein\nsummation convention for repeated Greek indices is utilize d while writing Eq. (15) and\nin this section. Edwards and McComb [3] theory seeks Fokker- Planck model equation for\nLiouville equation. The model equation contains two model p arameters, namely r(k)and\ns(k), which account for contribution of nonlinear term in Eq. (15 ). The dynamical damping\ncoefficient r(k)is related to damping coefficient ω(k)by\nω(k) =νk2+r(k).(M90;6.80) (16)\nThe coefficient s(k)accounts for correlation of white noise forcing in the Lange vin equa-\ntion for Fokker-Planck equation. Within the framework of Ed wards and McComb and for\nstationary turbulence, ω(k)ands(k)are related by\n2ω(k)q(k) =d(k) (M90;6.84) (17)\nwhere\nd(k) =W(k)+s(k) (M90;6.79) (18)\nandW(k)accounts for correlation of forcing term fα(k,t). The spectral density q(k)is\ndefined by\n/parenleftbigg2π\nL/parenrightbigg3\n/angbracketleftuα(k)uβ(−k)/angbracketright=Dαβ(k)q(k),(M90;6.85) (19)\nwhereDαβ(k) =δαβ−kαkβ\n|k|2. EM69 considered entropy function S=S[q(k),ω(k)]to obtain\nequation for ω(k)by maximizing S, corresponding to the condition\nδS\nδω(k)+/summationdisplay\nj/bracketleftBiggδS\nδq(j)/bracketrightBiggδq(j)\nδω(k)= 0.(M90;7.88) (20)\nHere\nδq(j)\nδω(k)=−d(k)δ(k−j)\n2ω2(k)+1\n2ω(j)δd(j)\nδω(k).(M90;7.89) (21)\nandδ(k−j) = 1 whenk=jotherwise δ(k−j) = 0 . Edwards and McComb realized\nthe difficulty in obtaining the second term on the right-hand s ide (rhs) of Eq. (21). After\nneglecting the second term, an approximate equation\nδq(j)\nδω(k)=−d(k)δ(k−j)\n2ω2(k)(22)\n5was used for further calculation in EM69 [8]. This Eq. (22) su ggests thatδq(j)\nδω(k)= 0,∀j/negationslash=k.\nAs a consequence, EM69 used following approximate equation\nδS\nδω(k)−/bracketleftBiggd(k)\n2ω2(k)/bracketrightBiggδS\nδq(k)= 0 ( M90;7.90) (23)\nto obtain equation for ω(k).\nWe now show that the neglect of the second term on the rhs of Eq. (21) corresponds to\nunphysical condition. Consequently, the use of Eq. (23) to o btainω(k)is in error. Since\nEM69 is proposed for stationary turbulence,\n/parenleftbigg2π\nL/parenrightbigg3/summationdisplay\nj1\n2/angbracketleftuα(j)uα(−j)/angbracketright=/summationdisplay\njq(j) =Constant. (24)\nand from which we can write exact equation\n/summationdisplay\njδq(j)\nδω(k)=δq(k)\nδω(k)+/summationdisplay\nj,j/negationslash=kδq(j)\nδω(k)= 0. (25)\nSubstituting approximate Eq. (22) of EM89 into Eq. (25) and u sing Eq. (17), we obtain\nd(k)\n2ω2(k)=q(k)\nω(k)= 0 (26)\nand which is not correct for all k. In view of this, approximation used in EM89 to obtain\nω(k)corresponds to unphysical condition q(k) = 0 and does not comply with conservation\nof energy Eq. (24) for stationary turbulence where q(k)/negationslash= 0,∀k.\nIt should be noted that, within the framework of EM69, the unp hysical behavior can be\navoided if\nδS\nδω(k)= 0 (27)\nalong with q(k) =constant, ∀kis used instead of Eq. (20). This means that S=S(ω(k))\nand second term on the rhs of Eq. (20) is equal to zero and is neg lected. This kind of neglect\nby Qian in Q83 for function I(ηi)was considered mathematically incorrect by McComb [8].\nIn our view, Eq. (27) can be considered as valid equation whic h seeks to optimize Entropy\nwhen energy of turbulence remains constant as q(k) =constant, ∀k.\nIV. CONCLUDING REMARKS\nWithin the Eulerian framework, a very few renormalized pert urbation theories of turbu-\nlence are consistent with Kolmogorov spectrum [7, 8]. In thi s paper, we have reexamined\n6two such theories proposed by Qian [10] and Edwards and McCom b [3] and have revealed\nhidden unphysical conditions in these theories. We have sug gested possible modifications,\nEqs. (13), (14) and (27), to these theories but have not explo red their usefulness in obtain-\ning damping coefficient consistent with Kolmogorov spectrum . This will be explored in a\nbroader context of our future work on turbulence theory deve lopment within the framework\nof Kraichnan’s direct interaction approximation [9].\n[1] G. K. Batchelor. The Theory of Homogeneous Turbulence . Cambridge University Press, Cam-\nbridge, UK, 1959.\n[2] S. F. Edwards. The statistical dynamics of homogeneous t urbulence. J. Fluid Mech. , 18:239–\n273, 1964.\n[3] S. F. Edwards and W. D. McComb. Statistical mechanics far from equilibrium. J. Phys. A ,\n2:157–171, 1969.\n[4] J. R. Herring. Self-Consistent-Field approach to turbu lence theory. Phys. Fluids , 8:2219–2225,\n1965.\n[5] A. N. Kolmogorov. The local structure of turbulence in in compressible viscous fluid for large\nReynolds numbers. Dokl. Akad. Nauk SSSR , 30(11):301–305, 1941.\n[6] R. H. Kraichnan. Irreversible statistical mechanics of incompressible hydromagnetic turbu-\nlence. Phys. Rev. , 109:1407–1422, 1958.\n[7] D. C. Leslie. Developments in the Theory of Turbulence . Clarendon Press, Oxford, 1973.\n[8] W. D. McComb. The Physics of Fluid Turbulence . Oxford University Press, New York, NY,\n1990.\n[9] R. V. R. Pandya. Development of Eulerian theory of turbul ence within Kraichnan’s direct\ninteraction approximation framework. arXiv:1407.1828 , pages 1–18, 2014.\n[10] J. Qian. Variational approach to the closure problem of turbulence theory. Phys. Fluids ,\n26:2098–2104, 1983.\n7"    },    {        "title": "0811.4118v1.The_quantum_mechanical_basis_of_an_extended_Landau_Lifshitz_Gilbert_equation_for_a_current_carrying_ferromagnetic_wire.pdf",        "content": "arXiv:0811.4118v1  [cond-mat.mtrl-sci]  25 Nov 2008The quantum-mechanical basis of an extended\nLandau-Lifshitz-Gilbert equation for a\ncurrent-carrying ferromagnetic wire\nD.M. Edwards1and O. Wessely1,2\n1 Department of Mathematics, Imperial College, London SW7 2BZ, U nited Kingdom\n2 Department of Mathematics, City University,London EC1V 0HB, Un ited Kingdom\nE-mail:d.edwards@imperial.ac.uk\nAbstract. An extended Landau-Lifshitz-Gilbert (LLG) equation is introduced to\ndescribe the dynamics of inhomogeneous magnetization in a current -carrying wire.\nThe coefficients of all the terms in this equation are calculated quant um-mechanically\nfor a simple model which includes impurity scattering. This is done by co mparing\nthe energies and lifetimes of a spin wave calculated from the LLG equa tion and from\nthe explicit model. Two terms are of particular importance since they describe non-\nadiabatic spin-transfer torque and damping processes which do no t rely on spin-orbit\ncoupling. It is shown that these terms may have a significant influenc e on the velocity\nof a current-driven domain wall and they become dominant in the cas e of a narrow\nwall.\nPACS numbers:An extended Landau-Lifshitz-Gilbert equation 2\n1. Introduction\nThe effect of passing an electric current down a ferromagnetic wire is of great current\ninterest. If the magnetization is inhomogeneous it experiences a sp in-transfer torque\ndue to the current [1, 2, 3, 4]. The effect is described phenomenolo gically by adding\nterms to the standard LLG equation [5, 6]. The leading term in the spin -transfer\ntorque is an adiabatic one arising from that component of the spin po larization of the\ncurrent which is in the direction of the local magnetization. However , in considering\nthe current-induced motion of a domain wall, Li and Zhang [3, 4] foun d that below a\nvery large critical current the adiabatic term only deforms the wall and does not lead\nto continuous motion. To achieve this effect they introduced [7] a ph enomenological\nnon-adiabatic term associated with the same spin non-conserving p rocesses responsible\nfor Gilbert damping. Subsequently Kohno et al[8] derived a torque of the Zhang-Li\nformquantum-mechanically using amodel ofspin-dependent scatt ering fromimpurities.\nThis may arise from spin-orbit coupling on the impurities. More recent ly Wessely et\nal[9] introduced two further non-adiabatic terms in the LLG equation in order to\ndescribe their numerical calculations of spin-transfer torques in a domain wall. These\nquantum-mechanical calculationsusingtheKeldyshformalismwerem adeintheballistic\nlimit without impurities and with spin conserved. Other terms in the LLG equation,\ninvolving mixed space and time derivatives, have been considered by S obolevet al[12],\nTserkovnyak et al[10], Skadsen et al[11] and Thorwart and Egger [13].\nThe object of this paper is to give a unified treatment of all these te rms in the LLG\nequation and to obtain explicit expressions for their coefficients by q uantum-mechanical\ncalculations for a simple one-band model with and without impurity sca ttering. The\nstrategy adopted is to consider a uniformly magnetized wire and to c alculate the effect\nof a current onthe energy andlifetime of a long wavelength spin wave propagating along\nthe wire. It is shown in section 2 that coefficients of spin-transfer t orque terms in the\nLLG equation are directly related to qandq3terms in the energy and inverse lifetime\nof a spin wave of wave-vector q. The Gilbert damping parameter is the coefficient of the\nωterm in the inverse lifetime, where ωis the spin-wave frequency. It corresponds to the\ndamping of a q= 0 spin wave while higher order terms ωqandωq2relate to damping\nof spin waves with finite wave-vector q. The relation between the qterm in the spin\nwave energy and the adiabatic spin-transfer torque has been not iced previously [2, 14].\nWe find that the qterm in the spin wave lifetime relates to the Zhang-Li non-adiabatic\nspin transfer torque. Our result for the coefficient of the Zhang- Li term is essentially the\nsame as that obtained by Kohno et al[8] and Duine et al[15] but our derivation appears\nsimpler. The q3terms in the spin wave energy and lifetime are related to the additiona l\nnon-adiabatic torques we introduced into the LLG equation [9], toge ther with an extra\none arising from spin non-conserving scattering. Explicit expressio ns for the coefficients\nof these terms are obtained in section 3. In section 4 we discuss brie fly the importance\nof the additional terms in our extended LLG equation for current- driven motion of a\ndomain wall. Some conclusions are summarized in section 5.An extended Landau-Lifshitz-Gilbert equation 3\n2. The LLG equation and spin waves\nWe write our extended LLG equation in the dimensionless form\n∂s\n∂t+αs×∂s\n∂t+α1s×∂2s\n∂z∂t−α′\n1s×/parenleftbigg\ns×∂2s\n∂z∂t/parenrightbigg\n−α′\n2s×∂3s\n∂z2∂t−α2s×/parenleftbigg\ns×∂3s\n∂z2∂t/parenrightbigg\n=s×∂2s\n∂z2−bexts×ez−a∂s\n∂z−fs×∂s\n∂z\n+a1/braceleftBigg\ns×/parenleftbigg\ns×∂3s\n∂z3/parenrightbigg\n+/bracketleftBigg\ns·∂2s\n∂z2−1\n2/parenleftbigg∂s\n∂z/parenrightbigg2/bracketrightBigg\n∂s\n∂z/bracerightBigg\n−f1s×/bracketleftbigg\ns×∂\n∂z/parenleftbigg\ns×∂2s\n∂z2/parenrightbigg/bracketrightbigg\n+g1s×∂3s\n∂z3. (1)\nHeres(z,t) is a unit vector in the direction of the local spin polarisation, time tis\nmeasured in units of ( γµ0ms)−1and the coordinate zalong the wire is in units of the\nexchange length lex= (2A/µ0m2\ns)1/2. The quantities appearing here are the gyroscopic\nratioγ= 2µB//planckover2pi1,thepermeabilityoffreespace µ0andtwopropertiesoftheferromagnetic\nmaterial, namely the saturation magnetisation msand the exchange stiffness constant\nA.ezis a unit vector in the zdirection along the wire. The equation expresses the\nrate of change of spin angular momentum as the sum of various torq ue terms, of which\ntheα1,α′\n1,a,f,a1,f1andg1terms are proportional to the electric current flowing.\nThe second term in the equation is the standard Gilbert term, with da mping factor\nα, while the α′\n1andα′\n2terms introduce corrections for spin fluctuations of finite wave-\nvector. Skadsem et al[11] point out the existence of the α′\n2term but do not consider\nit further. It was earlier introduced by Sobolev et al[12] within a microscopic context\nbased on the Heisenberg model. The α1andα2terms are found to renormalise the spin\nwave frequency, but for the model considered in section 3 we find t hatα1is identically\nzero. We shall argue that this result is model-independent. Tserko vnyaket al[10] and\nThorwart and Egger [13] find non-zero values of α1which differ from each other by a\nfactor 2; they attribute this to their use of Stoner-like and s−dmodels, respectively.\nThorwartandEgger[13]alsofindthe α′\n1termandtheyinvestigatetheeffectof α1andα′\n1\nterms on domain wall motion. Their results are difficult to assess beca use the constant\n|s|= 1 is not maintained during the motion. In eq.(1) we have omitted term s involving\nthe second order time derivatives, whose existence was pointed ou t by Thorwart and\nEgger [13]; one of these is discussed briefly in section 3.2.\nThe first term on the right-hand side of eq. (1) is due to exchange s tiffness and the\nnext term arises from an external magnetic field Bextezwith dimensionless coefficient\nbext=Bext/µ0ms. The third term is the adiabatic spin transfer torque whose coefficie nt\nais simple and well-known. In fact [3, 4]\na=1\n2/planckover2pi1JP\neµ0m2\nslex(2)An extended Landau-Lifshitz-Gilbert equation 4\nwhereJis the charge current density and eis the electron charge (a negative quantity).\nThe spin polarisation factor P= (J↑−J↓)/(J↑+J↓), where J↑, J↓are the current\ndensities for majority and minority spin in the ferromagnet ( J=J↑+J↓). Eq.(2) is\nvalid for both ballistic and diffusive conduction . The fourth term on th e right-hand\nside of eq.(1) is the Zhang-Li torque which is often characterised [8 ] by a parameter\nβ=f/a. The next term is the E1term of eq.(7) in ref. [9]. It is a non-adiabatic torque\nwhich is coplanar with s(z) ifs(z) lies everywhere in a plane. As shown in ref. [9] it\nis thezderivative of a spin current , which is characteristic of a torque occ urring from\nspin-conserving processes. In fact this term takes the form\na1∂\n∂z/bracketleftBigg\ns×/parenleftbigg\ns×∂2s\n∂z2/parenrightbigg\n−1\n2s/parenleftbigg∂s\n∂z/parenrightbigg2/bracketrightBigg\n. (3)\nThef1term may be written in the form\n−f1/parenleftbigg\ns·∂s\n∂z×∂2s\n∂z2/parenrightbigg\ns+f1∂\n∂z/parenleftbigg\ns×∂2s\n∂z2/parenrightbigg\n. (4)\nIfs(z) lies in a plane, the case considered in ref. [9], the first term vanishes and we\nrecover the F1term of eq.(9) in ref. [9]. Its derivative form indicate that it arises fr om\nspin-conserving processes so we conclude that the coefficient f1is of that origin. This\nis not true of the last term in eq.(1) and we associate the coefficient g1with spin non-\nconserving processes. For a spin wave solution of the LLG equation , where we work\nonly to first order in deviations from a state of uniform magnetisatio n, the last three\nterms of eq.(1) may be replaced by the simpler ones\n−a1∂3s\n∂z3+(f1+g1)s×∂3s\n∂z3. (5)\nApart fromadditional terms, eq.(1) looksslightly different fromeq.( 7) ofref. [9] because\nwe use the spin polarisation unit vector srather than the magnetisation vector mand\ns=−m. Furthermore the dimensionless coefficients will take different nume rical values\nbecause we have used different dimensionless variables zandtto avoid introducing the\ndomain wall width which was specific to ref. [9]. The torques due to an isotropy fields\nwere also specific to the domain wall problem and have been omitted in e q.(1).\nWe suppose that the wire is magnetised uniformly in the zdirection and consider a\nspin wave as a small transverse oscillation of the spin polarisation abo ut the equilibrium\nstate or, when a current flows, the steady state. Thus we look fo r a solution of eq.(1)\nof the form\ns=/parenleftbig\ncei(qz−ωt),dei(qz−ωt),−1/parenrightbig\n(6)\nwhere the coefficients of the xandycomponents satisfy c≪1,d≪1. This represents\na spin wave of wave-vector qand angular frequency ωpropagating along the zaxis.\nWhen (6) is substituted into eq.(1) the transverse components yie ld, to first order in c\nandd, the equations\n−iλc+µd= 0, µc+iλd= 0 (7)An extended Landau-Lifshitz-Gilbert equation 5\nwhere\nλ=ω−aq+a1q3−α2ωq2+iα′\n1qω\nµ=−iαω+bext+q2+ifq+i(f1+g1)q3+α1ωq−iωq2α′\n2. (8)\nOn eliminating canddfrom eq.(7) we obtain λ2=µ2. To obtain a positive real part\nfor the spin wave frequency, we take λ=µ. Hence\nω/parenleftbig\n1−α1q−α2q2/parenrightbig\n=bext+aq+q2−a1q3\n+i/bracketleftbig\nω/parenleftbig\n−α−α′\n1q−α′\n2q2/parenrightbig\n+fq+(f1+g1)q3/bracketrightbig\n. (9)\nThus the spin wave frequency is given by\nω=ω1−iω2 (10)\nwhere\nω1≃/parenleftbig\n1−α1q−α2q2/parenrightbig−1/parenleftbig\nbext+aq+q2−a1q3/parenrightbig\nω2≃/parenleftbig\n1−α1q−α2q2/parenrightbig−1/bracketleftbig\nω1/parenleftbig\nα+α′\n1q+α′\n2q2/parenrightbig\n−fq−(f1+g1)q3/bracketrightbig\n.(11)\nHere we have neglected terms of second order in α,α′\n1,α′\n2,f,f1andg1, the coefficients\nwhich appear in the spin wave damping. This form for the real and imag inary parts\nof the spin wave frequency is convenient for comparing with the qua ntum-mechanical\nresults of the next section. In this way we shall obtain explicit expre ssions for all the\ncoefficients in the phenomenological LLG equation. Coefficients of od d powers of qare\nproportional to the current flowing whereas terms in even powers ofqare present in the\nequilibrium state with zero current.\n3. Spin wave energy and lifetimes in a simple model\nAs a simple model of an itinerant electron ferromagnet we consider t he one-band\nHubbard model\nH0=−t/summationdisplay\nijσc†\niσcjσ+U/summationdisplay\nini↑ni↓−µBBext/summationdisplay\ni(ni↑−ni↓), (12)\nwherec†\niσcreates an electron on site iwith spin σandniσ=c†\niσciσ. We consider a simple\ncubic lattice and the intersite hopping described by the first term is r estricted to nearest\nneighbours. The second term describes an on-site interaction bet ween electrons with\neffective interaction parameter U; the last term is due to an external magnetic field. It\nis convenient to introduce a Bloch representation, with\nc†\nkσ=1√\nN/summationdisplay\niek·Ric†\niσ, nkσ=c†\nkσckσ, (13)\nǫk=−t/summationdisplay\nieik·ρi=−2t(coskxa0+coskya0+coskza0). (14)An extended Landau-Lifshitz-Gilbert equation 6\nThe sum in eq.(13) is over all lattice cites Riwhereas in eq.(14) ρi=\n(±a0,0,0),(0,±a0,0),(0,0,±a0) are the nearest neighbour lattice sites. Then\nH0=/summationdisplay\nkσǫknkσ+U/summationdisplay\nini↑ni↓−µBBext/summationdisplay\nk(nk↑−nk↓). (15)\nTo discuss scattering of spin waves by dilute impurities we assume tha t the effect of\nthe scattering from different impurity sites adds incoherently; hen ce we may consider\ninitially a single scattering center at the origin, We therefore introdu ce at this site a\nperturbing potential u+vl·σ, wherel= (sinθcosφ,sinφsinθ,cosθ) is a unit vector\nwhose direction will finally be averaged over. uis the part of the impurity potential\nwhich is indepndent of the spin σand the spin dependent potential vl·σis intended to\nsimulateaspin-orbit L·σinteractionontheimpurity. Itbreaksspinrotationalsymmetry\nin the simplest possible way. Clearly spin-orbit coupling can only be trea ted correctly\nfor a degenerate band such as a d-band, where on-site orbital angular momentum L\noccurs naturally. The present model is equivalent to that used by K ohnoet al[8] and\nDuineet al[15]. In Bloch representation the impurity potential becomes V=V1+V2\nwith\nV1=v↑1\nN/summationdisplay\nk1k2c†\nk1↑ck2↑+v↓1\nN/summationdisplay\nk1k2c†\nk1↓ck2↓\nV2=ve−iφsinθ1\nN/summationdisplay\nk1k2c†\nk1↑ck2↓+veiφsinθ1\nN/summationdisplay\nk1k2c†\nk1↓ck2↑ (16)\nandv↑=u+vcosθ,v↓=u−vcosθ. To avoid confusion we note that the spin\ndependenceoftheimpuritypotentialwhichoccursinthemany-bod yHamiltonian H0+V\nis not due to exchange, as would arise in an approximate self consiste nt field treatment\n(e.g. Hartree-Fock) of the interaction Uin a ferromagnet.\n3.1. Spin wave energy and wave function\nIn this section we neglect the perturbation due to impurities and det ermine expressions\nfor the energy and wave function of a long-wave length spin wave in t he presence of an\nelectric current. The presence of impurities is recognised implicitly sin ce the electric\ncurrent is characterised by a perturbed one-electron distributio n function fkσwhich\nmight be obtained by solving a Boltzman equation with a collision term. We consider\na spin wave of wave-vector qpropagating along the zaxis, which is the direction\nof current flow. Lengths and times used in this section and the next , except when\nspecified, correspond to actual physical quantities, unlike the dim ensionless variables\nused in section 2.\nWe first consider the spin wave with zero electric current and treat it, within\nthe random phase approximation (RPA), as an excitation from the H artree-Fock (HF)\nground state of the Hamiltonian (15). The HF one electron energies are given by\nEkσ=ǫk+U/angbracketleftn−σ/angbracketright−µBσBext (17)An extended Landau-Lifshitz-Gilbert equation 7\nwhereσ= 1,−1 for↑and↓respectively, and /angbracketleftn−σ/angbracketrightis the number of −σspin electrons\nper site. In a self-consistent ferromagnetic state at T= 0,/angbracketleftnσ/angbracketright=N−1/summationtext\nkfkσand\nn=/summationtext\nσ/angbracketleftnσ/angbracketright, where, fkσ=θ(EF−Ekσ),nis the number of electrons per atom, and\nEFis the Fermi energy. Nis the number of lattice sites and θ(E) is the unit step\nfunction. The spin bands Ekσgiven by eq.(17) are shifted relative to each other by an\nenergy ∆+2 µBBextwhere ∆ = U/angbracketleftn↑−n↓/angbracketrightis the exchange splitting. The ground state\nis given by |0/angbracketright=/producttext\nkσc†\nkσ|/angbracketrightwhere|/angbracketrightis the vacuum state and the product extends over\nall states kσsuch that fkσ= 1. Within the RPA, the wave function for a spin wave of\nwave-vector q, excited from the HF ground state, takes the form\n|q/angbracketright=Nq/summationdisplay\nkAkc†\nk+q↓ck↑|0/angbracketright (18)\nwhereNqis a normalisation factor. The energy of this state may be written\nEq=Egr+/planckover2pi1ωq=Egr+2µBBext+/planckover2pi1ω′\nq (19)\nwhereEgris the energy of the HF ground state and /planckover2pi1ωqis the spin wave excitation\nenergy. On substituting (18) in the Schr¨ odinger equation ( H0−Eq)|q/angbracketright= 0 and\nmultiplying on the left by /angbracketleft0|c†\nk′↑ck′−q↓, we find\nAk′/parenleftbig\nǫk′+q−ǫk′+∆−/planckover2pi1ω′\nq/parenrightbig\n=U\nN/summationdisplay\nkAkfk↑(1−fk+q↓). (20)\nHence we may take\nAk= ∆/parenleftbig\nǫk+q−ǫk+∆−/planckover2pi1ω′\nq/parenrightbig−1(21)\nand, for small q,/planckover2pi1ω′\nqsatisfies the equation\n1 =U\nN/summationdisplay\nkfk↑−fk+q↓\nǫk+q−ǫk+∆−/planckover2pi1ω′q. (22)\nThisistheequationforthepolesofthewell-knownRPAdynamicalsus ceptibility χ(q,ω)\n[16]. The spin wave pole is the one for which /planckover2pi1ω′\nq→0 asq→0.\nTo generalise the above considerations to a current-carrying sta te we proceed as\nfollows. We re-interpret the state |0/angbracketrightsuch that /angbracketleft0|...|0/angbracketrightcorresponds to a suitable\nensemble average with a modified one-electron distribution fkσ. When a current flows\nin thezdirection we may consider the ↑and↓spin Fermi surfaces as shifted by small\ndisplacement δ↑ˆkz,δ↓ˆkzwhereˆkzis a unit vector in the zdirection. Thus\nfkσ=θ(EF−Ek+δσˆkz,σ)\n≃θ(EF−Ekσ)−δσδ(EF−Ekσ)∂ǫk\n∂kz(23)\nand the charge current density carried by spin σelectrons is\nJσ=e\n/planckover2pi1Na3\n0/summationdisplay\nk∂ǫk\n∂kzfkσ=−eδσ\n/planckover2pi1Na3\n0/summationdisplay\nk/parenleftbigg∂ǫk\n∂kz/parenrightbigg2\nδ(EF−Ekσ)\n=−eδσ\n/planckover2pi1a3\n0/angbracketleftBigg/parenleftbigg∂ǫk\n∂kz/parenrightbigg2/angbracketrightBigg\nσρσ(EF) (24)An extended Landau-Lifshitz-Gilbert equation 8\nwhere/angbracketleft(∂ǫk/∂kz)2/angbracketrightσis an average over the σspin Fermi surface and ρσ(EF) is the\ndensity of σspin states per atom at the Fermi energy. We shall also encounter the\nfollowing related quantities;\nKσ=1\nN∆2a3\n0/summationdisplay\nk∂ǫk\n∂kz∂2ǫk\n∂k2zfkσ\n=/planckover2pi1Jσ\n∆2e/angbracketleftBigg/parenleftbigg∂ǫk\n∂kz/parenrightbigg2∂2ǫk\n∂k2z/angbracketrightBigg\nσ/slashbigg/angbracketleftBigg/parenleftbigg∂ǫk\n∂kz/parenrightbigg2/angbracketrightBigg\nσ(25)\nLσ=1\nN∆3a3\n0/summationdisplay\nk/parenleftbigg∂ǫk\n∂kz/parenrightbigg3\nfkσ\n=/planckover2pi1Jσ\n∆3e/angbracketleftBigg/parenleftbigg∂ǫk\n∂kz/parenrightbigg4/angbracketrightBigg\nσ/slashbigg/angbracketleftBigg/parenleftbigg∂ǫk\n∂kz/parenrightbigg2/angbracketrightBigg\nσ. (26)\nTo derive eqs.(25) and (26), δσhas been eliminated using eq.(24).\nTo solve eqn.(22) for /planckover2pi1ω′\nqwe expand the right-hand side of the equation in powers\nof (ǫk+q−ǫk−/planckover2pi1ω′\nq)/∆ and make the further expansions\nǫk+q−ǫk=q∂ǫk\n∂kz+1\n2q2∂2ǫk\n∂k2z+1\n6q3∂3ǫk\n∂k3z... (27)\n/planckover2pi1ω′\nq=Bq+Dq2+Eq3+... (28)\nin powers of q. We retain all terms up to q3except those involving B2; the coefficients\nBandEare proportional to the current and we keep only terms linear in the current.\nHence we find a solution of eq.(22) in the form (28) with\nB=1\nN↑−N↓/summationdisplay\nk(fk↑−fk↓)∂ǫk\n∂kz=Na3\n0\nN↑−N↓/planckover2pi1\ne(J↑−J↓) (29)\nD=1\nN↑−N↓/bracketleftBigg\n1\n2/summationdisplay\nk(fk↑+fk↓)∂2ǫk\n∂k2z−1\n∆/summationdisplay\nk(fk↑−fk↓)/parenleftbigg∂ǫk\n∂kz/parenrightbigg2/bracketrightBigg\n(30)\nE=−a2\n0B\n6\n+B\n(N↑−N↓)∆/bracketleftBigg/summationdisplay\nk(fk↑+fk↓)∂2ǫk\n∂k2\nz−3\n∆/summationdisplay\nk(fk↑−fk↓)/parenleftbigg∂ǫk\n∂kz/parenrightbigg2/bracketrightBigg\n−Ua3\n0/summationdisplay\nσ(Kσ−σLσ). (31)\nHereNσis the total number of σspin electrons so that Nσ=N/angbracketleftnσ/angbracketright.\nIn the absence of spin-orbit coupling the expression for Bin terms of spin current\nis a general exact result even in the presence of disorder, as show n in Appendix A. The\ncoefficient Dis the standard RPA spin-wave stiffness constant (e.g ref. [16]). W e note\nthat, in the limit ∆ → ∞, E takes the simple form −a2\n0B/6.\nOn restoring the correct dimensions (as indicated after eq.(1)) to the expression\nforω1in eq.(11) we may determine the coefficients aanda1by comparing with theAn extended Landau-Lifshitz-Gilbert equation 9\nequation\n/planckover2pi1ωq= 2µBBext+Bq+Dq2+Eq3. (32)\nFrom the coefficient of qwe have\na+α1bext=B/(2µBµ0mslex). (33)\naandBare both determined directly from the spin current JPindependently of a\nparticular model (see appendix A) so that bextshould not enter their relationship. We\nconclude quite generally that α1= 0. In this case we find that on combining eqs.(33)\nand (29), and noting that ms=−µB(N↑−N↓)/Na3\n0, eq.(2) is obtained as expected.\nIn section 3.2 we show explicitly for the present model that α1= 0. This conflicts\nwith the results of refs.[10] and [13]. From the coefficients of q2in eqs.(11) and (32) we\nfind 1+α2bext=D/(4µBA/ms). Thus an external field slightly disturbs the standard\nrelationA=Dms/4µB. However in the spirit of the LLG equation we take Aand\nms, which enter the units of length and time used in eq.(1), to be consta nts of the\nferromagnetic material in zero external field. The coefficients of q3in eqs.(11) and (32)\nyield the relation (taking α1= 0),\n−a1+α2a=E//parenleftbig\n2µBµ0msl3\nex/parenrightbig\n. (34)\nWe defer calculation of α2until section 3.2 and the result is given in eq.(44). Combining\nthis with eqs.(34) and (31) we find\n2µBµ0msl3\nexa1=a2\n0B\n6−2BD\n∆+Ua3\n0/summationdisplay\nσ(Kσ−σLσ). (35)\nWe have thus derived an explicit expression , for a simple model, for th e coefficient a1\nof a non-adiabatic spin torque term which appears in the LLG equatio n (1). We have\nneglected the effect of disorder due to impurities . In the absence o f spin-orbit coupling\nthe expression for the adiabatic torque coefficient a, given by eq.(2), is exact even in\npresence of impurities. In the next section we shall calculated furt her non-adiabatic\ntorque terms, with coefficients f1andg1, as well as damping coefficients α,α′\n1andα′\n2.\nIn the present model all these depend on impurity scattering for t heir existence.\n3.2. Spin wave lifetime\nThe solutions of eq.(22) are shown schematically in figure 1. They inclu de the spin\nwave dispersion curve and the continuum of Stoner excitations c†\nk+q↓ck↑|0/angbracketrightwith energies\nEk+q↓−Ek↑. The Zeeman gap2 µBBextinthe spin wave energy at q= 0 doesnot appear\nbecause we have plotted /planckover2pi1ω′\nqrather than /planckover2pi1ωq(see eq.(19)). Within the present RPA the\nspin wave in a pure metal has infinite lifetime outside the continuum and cannot decay\ninto Stoner excitations owing to conservation of the momentum q. However, when the\nperturbation V1due to impurities is introduced (see eqn.(16)), crystal momentum is no\nlonger conserved and such decay processes can occur. These ar e shown schematically\nby the dotted arrow in figure 1. If the bottom of the ↓spin band lies above the Fermi\nlevel there is a gap in the Stoner spectrum and for a low energy (sma llq) spin waveAn extended Landau-Lifshitz-Gilbert equation 10\nFigure 1. Spin-flip excitations from the ferromagnetic ground state. The do tted\narrow shows the mechanism of decay of a spin wave into Stoner excit ations which is\nenabled by the impurity potential V1.\nsuch processes cannot occur. However the spin-flip potential V2enables the spin wave\nto decay into single particle excitations c†\nk+qσckσ|0/angbracketrightabout each Fermi surface and these\ndo not have an energy gap.\nThe lifetime τ−1\nqof a spin wave of wave-vector qis thus given simply by the “golden\nrule” in the form\nτ−1\nq=2π\n/planckover2pi1Nimp(T1+T2) (36)\nwhereNinpis the number of impurity sites and\nT1=/summationdisplay\nkp/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig\n0/vextendsingle/vextendsingle/vextendsinglec†\nk↑cp↓V1/vextendsingle/vextendsingle/vextendsingleq/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle2\nfk↑(1−fp↓)δ(/planckover2pi1ωq−Ep↓+Ek↑)\nT2=/summationdisplay\nkpσ/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig\n0/vextendsingle/vextendsingle/vextendsinglec†\nkσcpσV2/vextendsingle/vextendsingle/vextendsingleq/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle2\nfkσ(1−fpσ)δ(/planckover2pi1ωq−ǫp+ǫk). (37)\nWe first consider T1and, using eqns.(16) and (18), we find\n/angbracketleftBig\n0/vextendsingle/vextendsingle/vextendsinglec†\nk↑cp↓V1/vextendsingle/vextendsingle/vextendsingleq/angbracketrightBig\n=Nq\nNfk↑(1−fp↓)[Akv↓(1−fp↓)−Ap−qv↑fp−q↑]\n=Nq\nNfk↑(1−fp↓)(Akv↓−Ap−qv↑) (38)\nfor small q. The last line follows from two considerations. Firstly, because of th eδ-\nfunction in eq.(37) we can consider the states k↑andp↓to be close to their respective\nFermi surfaces. Secondly the ↓spin Fermi surface lies within the ↑Fermi surface and q\nis small. Hence\nT1=N2\nq\nN2/summationdisplay\nkpfk↑(1−fp↓)δ(/planckover2pi1ωq−Ep↓+Ek↑)(Akv↓−Ap−qv↑)2.(39)An extended Landau-Lifshitz-Gilbert equation 11\nTo evaluate this expression in the case when a current flows we use t he distribution\nfunction fkσgiven by eq.(23). Thus, neglecting a term proportional to the squa re of the\ncurrent, we have\nT1=N2\nq\nN2/summationdisplay\nkpδ(/planckover2pi1ωq−Ep↓+Ek↑)(Akv↓−Ap−qv↑)2\n×/bracketleftbigg\nθ(EF−Ek↑)θ(Ep↓−EF)−δ↑θ(Ep↓−EF)δ(EF−Ek↑)∂ǫk\n∂kz\n+δ↓θ(EF−Ek↑)δ(EF−Ep↓)∂ǫp\n∂pz/bracketrightbigg\n. (40)\nWe wish to expand this expression, and a similar one for T2, in powers of qtoO(q3) so\nthat we can compare with the phenomenological expression (eq.(11 )) for the imaginary\npartofthespinwave frequency, which isgiven by τ−1\nq/2. Itisstraight-forwardtoexpand\nthe second factor in the above sum by using eqs.(21) and (28). We s hall show that the\ncontribution to T1of the first term in square brackets in eq.(40) leads to a contributio n\nproportional to spin wave frequency ωq. Together with a similar contribution to T2it\nyields the Gilbert damping factor αas well as the coefficients α′\n1,α′\n2of the terms in\neq.(11) which give the qdependence of the damping. The remaining terms in eq.(40)\nyield the spin-transfer torque coefficients f,f1andg1.\nThe normalisation factor N2\nqwhich appear in eq.(40) leads naturally to the factor\n(1−α1q−α2q2)−1which appears in eq.(11). From eq.(18) it is given by\n1 =/angbracketleftq|q/angbracketright=N2\nq\nN/summationdisplay\nk/parenleftbig\nA2\nkfk↑−A2\nk−qfk↓/parenrightbig\n. (41)\nBy expanding A2\nk−qin powers of q, and using eq.(23), we find to O(q2) that\nN−2\nq= (N↑−N↓)\n×/braceleftBigg\n1+q2\n∆2(N↑−N↓)/summationdisplay\nk/parenleftbigg∂ǫk\n∂kz/parenrightbigg2\n[θ(EF−Ek↑)−θ(EF−Ek↓)]/bracerightBigg\n.(42)\nWe deduce that\nα1= 0 (43)\nand\nα2=−1\nl2\nex∆2(N↑−N↓)/summationdisplay\nk/parenleftbigg∂ǫk\n∂kz/parenrightbigg2\n[θ(EF−Ek↑)−θ(EF−Ek↓)].(44)\nThe result α1= 0, which was predicted on general grounds in section 3.1 and in\nAppendix 1, arises here through the absence of a qterm, proportional to current,\nin the spin wave normalisation factor. In the derivation of eq.(42) th is occurs due\nto a cancellation involving the Bqterms in the spin energy, which appears in Ak.\nWithout this cancellation we would have α1= 2B/lex∆ which is of the form obtained\nby Tserkovnyak et al[10] and Thorwart and Egger [13].An extended Landau-Lifshitz-Gilbert equation 12\nWe now return to the programme for calculating the LLG coefficients\nα,α′\n1,α′\n2,f,f1,g1which was outlined after eq.(40). We have seen that the qdependence\nofN2\nqcorresponds to the prefactor in eq.(11). Hence to determine the coefficients listed\nabove we can take N2\nq=N2\n0= (N↑−N↓)−1inT1andT2when we expand terms in\npowers of qto substitute in eq.(36) and compare with eq.(11). We first consider the\ncaseq= 0 in order to determine the Gilbert damping factor α. Thus only the first term\nin square brackets in eq.(40) contributes, since ∂ǫk/∂kzis an odd function kz, and\nT1(q= 0) =4v2cos2θ\nN↑−N↓\n×N−2/summationdisplay\nkpδ(/planckover2pi1ω0−Ep↓+Ek↑)θ(EF−Ek↑)θ(Ep↓−EF) (45)\nwherecos2θis an average over the angle appearing in the impurity potential V(eq.(16))\nand we shall assume cosθ= 0. The summations in eq.(45) may be replaced by energy\nintegrals involving the density of states of per atom ρσ(ǫ) of the states Ekσ. Then, to\norder (/planckover2pi1ω0)2,\nT1(q= 0) =/bracketleftBigg\n4v2cos2θ\nN↑−N↓/bracketrightBigg/bracketleftbigg\n/planckover2pi1ω0ρ↑ρ↓+1\n2(/planckover2pi1ω0)2/parenleftbig\nρ↑ρ′\n↓−ρ′\n↑ρ↓/parenrightbig/bracketrightbigg\n(46)\nwhereρσ(ǫ) and its derivative ρ′\nσ(ǫ) are evaluated at ǫ=EF. Similarly\nT2(q= 0) =/bracketleftBigg\nv2sin2θ\nN↑−N↓/bracketrightBigg\n/planckover2pi1ω0/parenleftbig\nρ2\n↑+ρ2\n↓/parenrightbig\n(47)\nand noω2\n0terms appear. We have included the ω2\n0term in eq.(46) merely because it\ncorresponds to a term s×/parenleftBig\ns×∂2s\n∂t2/parenrightBig\nin the LLG equation whose existence was noted by\nThorwald and Egger [13]. We shall not pursue terms with second-ord er time derivatives\nany further. Since the imaginary part of the spin wave frequency is given by τ−1\nq/2 it\nfollows from eqs.(11), (36), (46) and (47) that\nα=πcv2\n/angbracketleftn↑−n↓/angbracketright/bracketleftBig\n4cos2θρ↑ρ↓+sin2θ/parenleftbig\nρ2\n↑+ρ2\n↓/parenrightbig/bracketrightBig\n, (48)\nwherec=Nimp/Nis the concentration of impurities, in agreement with Khono et al[8]\nand Duine et al[15]. If the direction of the spin quantisation axis of the impurities is\ndistributed randomly cos2θ= 1/3,sin2θ= 2/3 so that αis proportional to ( ρ↑+ρ↓)2.\nTo investigate the qdependence of Gilbert damping, and thus evaluate α′\n1andα′\n2\nin eq.(11), the second factor in the summation of eq.(40) must be ex panded in powers\nofq. All the terms which contribute to the sum are of separable form g(k)h(p). The\ncontribution to T1of interest here , proportional to ωq, again arises from the first term\nin square brackets in eq.(40), and similarly for T2. The summations required in eq.(40)\nare of the form\n/summationdisplay\nkpδ(/planckover2pi1ωq−Ep↓+Ek↑)θ(EF−Ek↑)θ(Ep↓−EF)g(k)h(p)\n=/angbracketleftg(k)/angbracketright↑/angbracketlefth(k)/angbracketright↓ρ↑ρ↓/planckover2pi1ωq (49)An extended Landau-Lifshitz-Gilbert equation 13\nwhere/angbracketleftg(k)/angbracketrightσ=N−1/summationtext\nkg(k)δ(EF−Ekσ) is an average over the Fermi surface, as used\npreviously in section 3.1. After some algebra we find\nα′\n1= 2Bα/∆lex (50)\nα′\n2=πc\n/angbracketleftn↑−n↓/angbracketrightl2\nex∆2/braceleftbigg\nρ↑ρ↓/parenleftBig\nu2+5v2cos2θ/parenrightBig/summationdisplay\nσ/angbracketleftBigg/parenleftbigg∂ǫk\n∂kz/parenrightbigg2/angbracketrightBigg\nσ\n−2ρ↑ρ↓∆v2cos2θ/summationdisplay\nσσ/angbracketleftbigg∂2ǫk\n∂k2z/angbracketrightbigg\nσ\n−v2sin2θ/bracketleftBigg\n∆/summationdisplay\nσσρ2\nσ/angbracketleftbigg∂2ǫk\n∂k2z/angbracketrightbigg\nσ−3/summationdisplay\nσρ2\nσ/angbracketleftBigg/parenleftbigg∂ǫk\n∂kz/parenrightbigg2/angbracketrightBigg\nσ/bracketrightBigg/bracerightbigg\n+2Dα\n∆l2ex. (51)\nWe note that, unlike αandα′\n1, the coefficient α′\n2is non-zero even when the spin-\ndependent part of the impurity potential, v, is zero. In this case the damping of a\nspin wave of frequency ωand small wave-vector qis proportional to ρ↑ρ↓u2ωq2. In\nzero external field ω∼q2so that the damping is of order q4. This damping due to\nspin-independent potential scattering by impurities was analysed in detail by Yamada\nand Shimizu [17]. One of the Fermi surface averages in eq.(51) is easily evaluated using\neqs.(14) and (17). Thus/angbracketleftbigg∂2ǫk\n∂k2\nz/angbracketrightbigg\nσ=−a2\n0\n3/angbracketleftǫk/angbracketrightσ=−a2\n0\n3(Ef−U/angbracketleftn−σ/angbracketright+σµBBext). (52)\nIn the spirit of the LLG equation we should take Bext= 0 in evaluating the coefficients\nα′\n2.\nWe now turn to the evaluation of the non-adiabatic spin-transfer t orque coefficients\nf,f1andg1. These arise from the second and third terms in square brackets in eq.(40),\nand in a similar expression for T2. The summations involved in these terms differ from\nthose in eq.(49) since one θ-function is replaced by a δ-function. This leads to the\nomission of the frequency factor /planckover2pi1ωq. The Fermi surface shifts δσare elininated in\nfavour of currents Jσby using eq.(24).\nBy comparing the coefficient of qin the expansion of eq.(36) with that in eq.(11)\nwe find the coefficient of the Zhang-Li torque in the form\nf=πcv2\nµ0m2s∆lex/planckover2pi1\ne/bracketleftBig\n2cos2θ(ρ↑J↓−ρ↓J↑)+sin2θ(ρ↓J↓−ρ↑J↑)/bracketrightBig\n.(53)\nThis is in agreement with Khono et al[8] and Duine et al[15]. In the “isotropic”\nimpurity case, with cos2θ= 1/3,sin2θ= 2/3, it follows from eqs.(53), (48) and (2) that\nβ=f\na=α2\nU(ρ↑+ρ↓). (54)\nIn the limit of a very weak itinerant forromagnet ρσ→ρ, the paramagnetic density of\nstates, and Uρ→1 by the Stoner criterion. Thus in this limit β=α. Tserkovnyak etAn extended Landau-Lifshitz-Gilbert equation 14\nal[10] reached a similar conclusion. For a parabolic band it is straightfor ward to show\nfrom Stoner theory that β/α >1 and may be as large as 1.5.\nAs discussed in section 2 the coefficient f1is associated with spin-conserving\nprocesses, and hence involves the spin independent potential u. The coefficient g1\nis associated with spin non-conserving processes and involves v. By comparing the\ncoefficient of q3in the expansion of eq.(36) with that in eq.(11) we deduce that\nf1=πc\n2µ0m2sl3exu2(K1+2L1+M1) (55)\nand\ng1=1\nl2ex/parenleftbigg3D\n∆−a2\n0\n6/parenrightbigg\nf\n+πcv2\n2µ0m2sl3ex/bracketleftBig\ncos2θ(5K1+6L1−M1)+sin2θ(3K2+4L2)/bracketrightBig\n. (56)\nHere\nK1=K↓ρ↑+K↑ρ↓, K2=K↓ρ↓+K↑ρ↑\nL1=L↓ρ↑−L↑ρ↓, L2=L↓ρ↓−L↑ρ↑\nM1=/planckover2pi1\ne∆3/summationdisplay\nσ/bracketleftBigg\n2σ/angbracketleftBigg/parenleftbigg∂ǫk\n∂kz/parenrightbigg2/angbracketrightBigg\n−σ+∆/angbracketleftbigg∂2ǫk\n∂k2z/angbracketrightbigg\n−σ/bracketrightBigg\nJσρ−σ. (57)\nThis complete the derivation of expressions for all the LLG coefficien ts of eq.(1) within\nthe present impurity model\n4. The extended LLG equation applied to current-driven doma in wall\nmotion\nIn a previous paper [9] we introduced the a1andf1terms of the extended LLG equation\n(cf. eqns.(1), (3) and (4)) in order to describe numerically-calcula ted spin-transfer\ntorques acting on a domain wall when it is traversed by an electric cur rent. In that\nwork the origin of the small f1term for a pure ferromagnetic metal was specific to\nthe domain wall problem; it was shown to be associated with those elec tronic states at\nthe bulk Fermi surface which decay exponentially as they enter the wall. The analytic\nderivation of f1in section 3 (see eqn.(55)) is based on impurity scattering in the bulk\nferromagnet and applies generally to any slowly-varying magnetizat ion configuration.\nFor a ferromagnetic alloy such as permalloy both mechanisms should c ontribute in the\ndomain wall situation but the impurity contribution would be expected to dominate.\nTo describe a domain wall we must add to the right-hand side of eqn.(1 ) anisotropy\nterms of the form\n−(s·ey)s×ey+b−1(s·ez)s×ez, (58)\nwhereeyis a unit vector perpendicular to the plane of the wire. The first term\ncorresponds to easy-plane shape anisotropy for a wire whose widt h is large compared\nwith its thickness and the second term arises from a uniaxial field Hualong the wire,An extended Landau-Lifshitz-Gilbert equation 15\nso thatb=ms/Hu. The solution of eqn.(1), with the additional terms (58), for a\nstationary N´ eel wall in the plane of the wire, with zero external fie ld and zero current,\nis\ns= (sech(z/b1/2),0,−tanh(z/b1/2)). (59)\nAs pointed out in ref. [9] there is no solution of the LLG equation of th e form\ns=F(z−vWt), corresponding to a uniformly moving domain wall, when the f1term\nis included. It is likely that the wall velocity oscillates about an average value, as\npredicted by Tatara and Kohno [18, 19] for purely adiabatic torque above the critical\ncurrent density for domain wall motion. However, we may estimate t he average velocity\nvWusing the method of ref. [9]. The procedure is to substitute the ap proximate form\ns=F(z−vWt) in the extended LLG equation (1), with the terms (58) added, tak e the\nscalar product with F×F′and integrate with respect to zover the range ( −∞,∞).\nThe boundary conditions appropriate to the wall are s→ ∓ezasz→ ±∞. Hence for\nbext= 0 we find the dimensionless wall velocity to be\nvW=f/integraltext∞\n−∞(F×F′)2dz+f1/integraltext∞\n−∞(F×F′′)2dz+g1/integraltext∞\n−∞(F′′)2dz\nα/integraltext∞\n−∞(F×F′)2dz+α′\n2/integraltext∞\n−∞(F′′)2dz.(60)\nTo estimate the integrals we take F(z) to have the form of the stationary wall s(z)\n(eqn.(59)) and, with the physical dimensions of velocity restored, the wall velocity is\ngiven approximately by\nvW=v0β\nα1+f1(3fb)−1\n1+α′\n2(αb)−1(61)\nwherev0=µBPJ/(mse). We have neglected g1here because, like fandα, it depends\non spin-orbit coupling but is a factor ( a0/lex)2smaller than f(cf. eqns.(53) and (56)).\nf1andα′\n2are important because they do not depend on spin-orbit coupling.\nIt is interesting to compare vWwith the wall velocity observed in permalloy\nnanowires by Hayashi et al[20]. We first note that v0is the velocity which one obtains\nvery simply from spin angular momentum conservation if the current -driven wall moves\nuniformly without any distortion such as tilting out of the easy plane a nd contraction\n[21]. This is never the case, even if f1= 0,α′\n2= 0, unless β=α. For a permalloy\nnanowire, with µ0ms= 1 T,v0= 110Pm/s forJ= 1.5·108A/cm2. Thus, from the\nstandard theory with f1= 0,α′\n2= 0,vW= 110Pβ/αm/s for this current density. In\nfact Hayashi et al[20] measure a velocity of 110 m/s which implies β > αsince the spin\npolarization Pis certainly less than 1. They suggest that βcannot exceed αand that\nsome additional mechanism other than spin-transfer torque is ope rating. However in the\ndiscussion following eqn.(54) we pointed out that in the model calculat ions it is possible\nto haveβ > α. Even if this is not the case in permalloy we can still have vW> v0if the\nlast factor in eqn.(61) is greater than 1 when f1andα′\n2are non-zero. We can estimate\ntermsinthisfactorusingtheobservationfromref. [20],that lW=lexb1/2= 23nm, where\nlWis the width of the wall. From eqns.(53) and (55) we find f1/(fb)∼(u/v)2(kFlW)−2,\nwherekFis a Fermi wave-vector. In permalloy we have Fe impurities in Ni so tha t inAn extended Landau-Lifshitz-Gilbert equation 16\nthe impurity potential u+v·σwe estimate u∼1 eV and v∼0.005 eV. The value for v\nis estimated by noting that the potential v·σis intended to model spin-orbit coupling\nof the form ξL·σwithξ/lessorsimilar0.1 eV and /angbracketleftLz/angbracketrightFe∼0.05,Lzbeing the component of orbital\nangular momentum in the direction of the magnetization [22]. Hence u/v∼200 and\nkFlW∼200 so that f1/(fb)∼1.α′\n2/(αb) is expected to be of similar magnitude. We\nconclude that the α′\n2andf1terms in the LLG equation (1) can be important in domain\nwall motion and should be included in micromagnetic simulations such as O OMMF\n[23]. For narrower domain walls these terms may be larger than the Gilb ert damping α\nand non-adiabatic spin- transfer torque fterms which are routinely included. Reliable\nestimates of their coefficients are urgently required using realistic m ultiband models of\nthe ferromagnetic metal or alloy.\n5. Conclusions\nThe coefficients of all the terms in an extended LLG equation for a cu rrent-carrying\nferromagnetic wire have been calculated for a simple model. Two of th ese (f1and\nα′\n2) are of particular interest since they do not rely on spin-orbit coup ling and may\nsometimes dominate the usual damping and non-adiabatic spin-tran sfer torque terms.\nOne term ( α1) which has been introduced by previous authors is shown rigorously to\nbe zero, independent of any particular model. Solutions of the exte nded LLG equation\nfor domain wall motion have not yet been found but the average velo city of the wall is\nestimated. It is pointed out that the f1andα′\n2terms are very important for narrow\nwalls and should be included in micrmagnetic simulations such as OOMMF. I t is shown\nthat there is no theoretical reason why the wall velocity should not exceed the simplest\nspin-transfer estimate v0, as is found to be the case in experiments on permalloy by\nHayashiet al[20]\nAcknowledgments\nWe are grateful to the EPSRC for financial support through the S pin@RT consortium\nand to other members of this consortium for encouragement and s timulation.\nAppendix A.\nThe simple single-band impurity model used in the main text is useful fo r obtaining\nexplicit expressions for all the coefficients in the LLG equation (1). H ere we wish to\nshow that some of these results are valid for a completely general s ystem. We suppose\nthe ferromagnetic material is described by the many-body Hamilton ian\nH=H1+Hint+Hext (A.1)\nwhereH1is a one-electron Hamiltonian of the form\nH1=Hk+Hso+V. (A.2)An extended Landau-Lifshitz-Gilbert equation 17\nHereHkis the total electron kinetic energy, Hsois the spin-orbit interaction, Vis a\npotential term, Hintis the coulomb interaction between electrons and Hextis due to an\nexternal magnetic field Bextin thezdirection. Thus\nHext=−2µBSz\n0Bext (A.3)\nwhereS0\nzis thezcomponent of total spin. Both HsoandVcan contain disorder. Since\nwe are interested in the energy and lifetime of a long-wavelength spin wave we consider\nthe spin wave pole, for small q, of the dynamical susceptibility.\nχ(q,ω) =/integraldisplay\ndt/angbracketleft/angbracketleftS−\nq(t),S+\n−q/angbracketright/angbracketrighte−iω−t(A.4)\n(ω−=ω−iǫ) whereS±\nq=Sx\nq±iSy\nqare Fourier components of the total transverse spin\ndensity. Here\n/angbracketleft/angbracketleftS−\nq(t),S+\n−q/angbracketright/angbracketright=i\n/planckover2pi1/angbracketleft/bracketleftbig\nS−\nq(t),S+\n−q/bracketrightbig\n/angbracketrightθ(t). (A.5)\nIngeneral we shall take theaverage /angbracketleft/angbracketrightina steady statein which a charge current density\nJis flowing in the qdirection. Following the general method of Edwards and Fisher\n[24] we use equations of motion to find that\nχ(q,ω) =−2/angbracketleftSz\n0/angbracketright\n/planckover2pi1(ω−bext)+1\n/planckover2pi12(ω−bext)2/braceleftbig\nχc(q,ω)−/angbracketleft/bracketleftbig\nC−\nq,S+\n−q/bracketrightbig\n/angbracketright/bracerightbig\n(A.6)\nwhere/planckover2pi1bext= 2µBBext,C−\nq= [S−\nq,H1] and\nχc(q,ω) =/integraldisplay\ndt/angbracketleft/angbracketleftC−\nq(t),C+\n−q/angbracketright/angbracketrighte−iωt. (A.7)\nFor small qandω,χis dominated by the spin wave pole, so that\nχ(q,ω) =−2/angbracketleftSz\n0/angbracketright\n/planckover2pi1(ω−bext−ωq)(A.8)\nwherebext+ωqis the spin wave frequency, in general complex corresponding to a fi nite\nlifetime. Following ref. [24] we compare (A.6) and (A.8) in the limit ωq≪ω−bextto\nobtain the general result\nωq=−1\n2/angbracketleftSz\n0/angbracketright/planckover2pi1/braceleftbigg\nlim\nω→bextχc(q,ω)−/angbracketleft/bracketleftbig\nC−\nq,S+\n−q/bracketrightbig\n/angbracketright/bracerightbigg\n. (A.9)\nEdwards and Fisher [24] were concerned with Reωqwhereas Kambersky [25] derived the\nabove expression for Imωqfor the case q= 0, and zero current flow. His interest was\nGilbert damping in ferromagnetic resonance. Essentially the same re sult was obtained\nearlier in connection with electron spin resonance, by Mori and Kawa saki [26], see also\nOshikawa and Affleck [27]. Since S−\nqcommutes with the potential term V, even in the\npresence of disorder, we have\nC−\nq=/bracketleftbig\nS−\nq,H1/bracketrightbig\n=/bracketleftbig\nS−\nq,Hk/bracketrightbig\n+/bracketleftbig\nS−\nq,Hso/bracketrightbig\n. (A.10)\nFor simplicity we now neglect spin-orbit coupling so that\nC−\nq=/bracketleftbig\nS−\nq,Hk/bracketrightbig\n=/planckover2pi1qJ−\nq (A.11)An extended Landau-Lifshitz-Gilbert equation 18\nwhere the last equation defines the spin current operator J−\nq. For a general system, with\nthenthelectron at position rnwith spin σnand momentum pn,\nS−\nq=/summationdisplay\nneiq·rnσ−\nn, Hk=/summationdisplay\nnp2\nn/2m. (A.12)\nHence, from eqns.(A.11) and (A.12),\n/angbracketleft/bracketleftbig\nC−\nq,S+\n−q/bracketrightbig\n/angbracketright=N/planckover2pi12q2\n2m+2/planckover2pi1/summationdisplay\nn/angbracketleftσz\nnvn/angbracketright·q (A.13)\nwhereNis the total number of electrons and vn=pn/mis the electron velocity, so\nthate/summationtext\nn/angbracketleftσz\nnvn/angbracketrightis the total spin current. Hence from eq.(A.9), we find\nωq=/planckover2pi1q2\n2/angbracketleftSz\n0/angbracketright/bracketleftbiggN\n2m−lim\nω→bextχJ(0,ω)/bracketrightbigg\n+Bq\n/planckover2pi1(A.14)\nwith\nB=/planckover2pi1µBPJ/em s. (A.15)\nThis expression for Bhas been obtained by Bazaliy et al[2] and Fern´ andez-Rossier et\nal[14] for simple parabolic band, s−dand Hubbard models. The derivation here is\ncompletelygeneralforanyferromagnet,eveninthepresenceof disorderduetoimpurities\nor defects, as long as spin-orbit coupling is neglected. Eqs.(2) and ( A.14) are both valid\nfor arbitrary bext, so that in eq. (33) we must have α1= 0.\nReferences\n[1] Berger L 1978 J.Appl.Phys. 492156\n[2] Bazaliy Y B, Jones B A and Zhang S C 1998 Phys. Rev. B 57R3213\n[3] Li Z and Zhang S 2004 Phys. Rev. B 70024417\n[4] Li Z and Zhang S 2004 Phys. Rev. Lett. 92207203\n[5] Landau L D, Lifshitz E M and Pitaevski L P 1980 Statistical Physics , part 2 (Oxford: Pergamon)\n[6] Gilbert T L 1955 Phys. 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Microsc. 54i69\n[20] Hayashi M, Thomas L, Rettner C, Moriya R, Bazaliy Y B and Parkin S S P 2007 Phys. Rev. Lett.\n98037204\n[21] Barnes S E and Maekawa S 2005 Phys. Rev. Lett. 95107204\n[22] Daalderop G H O, Kelly P J and Schuurmans M F H 1990 Phys. Rev. B 4111919\n[23] Donahue M and Porter D http://math.nist.gov/oommfAn extended Landau-Lifshitz-Gilbert equation 19\n[24] Edwards D M and Fisher B 1971 J. Physique 32C1 697\n[25] Kambersk´ y V 1976 Czech. J. Phys. B 261366\n[26] Mori H and Kawasaki K 1962 Prog. Theor. Phys. 27529\n[27] Oshikawa M and Affleck I 2002 Phys. Rev. B 65134410"    },    {        "title": "1206.4819v2.Fast_domain_wall_propagation_in_uniaxial_nanowires_with_transverse_fields.pdf",        "content": "arXiv:1206.4819v2  [cond-mat.mtrl-sci]  18 Aug 2013Fast domain wall propagation in uniaxial nanowires with tra nsverse fields\nArseni Goussev1,2, Ross G. Lund3, JM Robbins3, Valeriy Slastikov3, Charles Sonnenberg3\n1Department of Mathematics and Information Sciences,\nNorthumbria University, Newcastle Upon Tyne, NE1 8ST, UK\n2Max Planck Institute for the Physics of Complex Systems,\nN¨ othnitzer Straße 38, D-01187 Dresden, Germany\n3School of Mathematics, University of Bristol, University W alk, Bristol BS8 1TW, United Kingdom\n(Dated: May 14, 2018)\nUnder a magnetic field along its axis, domain wall motion in a u niaxial nanowire is much slower\nthan in the fully anisotropic case, typically by several ord ers of magnitude (the square of the di-\nmensionless Gilbert damping parameter). However, with the addition of a magnetic field transverse\nto the wire, this behaviour is dramatically reversed; up to a critical field strength, analogous to the\nWalker breakdown field, domain walls in a uniaxial wire propa gate faster than in a fully anisotropic\nwire (without transverse field). Beyond this critical field s trength, precessional motion sets in, and\nthe mean velocity decreases. Our results are based on leadin g-order analytic calculations of the\nvelocity and critical field as well as numerical solutions of the Landau-Lifshitz-Gilbert equation.\nPACS numbers: 75.75.-c, 75.78.Fg\nIntroduction\nThe dynamics of magnetic domain walls in ferromag-\nnetic nanowiresunder external magnetic fields [1–12] and\nspin-polarised currents [12–20] is a central problem in\nmicromagnetics and spintronics, both as a basic physical\nphenomenonaswellasacornerstoneofmagneticmemory\nand logic technology [3, 16–18]. From the point of view\nof applications, it is desirable to maximise the domain\nwall velocity in order to optimise switching and response\ntimes.\nPartly because of fabrication techniques, attention has\nbeen focused on nanowires with large cross-sectional as-\npect ratio, typically of rectangular cross-section. In this\ncase, even if the bulk material is isotropic (e.g., permal-\nloy), the domain geometry induces a fully anisotropic\nmagnetic permeability tensor, with easy axis along the\nwire and hard axis along its shortest dimension [21, 22].\nNanowires with uniaxial permeability, characteristic of\nmoresymmetricalcross-sectionalgeometries(e.g., square\nor circular), have been less studied [23–25]. Here we in-\nvestigate domain wall (DW) motion in uniaxial wires in\nthe presence of transverse fields. We show that the DW\nvelocityinuniaxialwiresdependsstronglyonthelongitu-\ndinal applied field H1, increasing with H1up to a certain\ncritical field and thereafter falling off as precessional mo-\ntion sets in. We employ a systematic asymptotic expan-\nsion scheme, which differs from alternative approaches\nbased on approximate dynamics for the DW centre and\norientation; a detailed account of this scheme, also in-\ncluding anisotropy and current-induced torques, will be\ngiven separately [29] .\nWe employ a continuum description of the magnetisa-\ntion. For a thin nanowire, this is provided by the one-\ndimensionalLandau-Lifshitz-Gilbert(LLG)equation[22,26–28], which we write in the non-dimensionalised form\n˙M=γM×H−αM×(M×H). (1)\nHereM(x,t) is a unit-vector field specifying the orien-\ntation of the magnetisation, which we shall also write\nin polar form M= (cosΘ ,sinΘcosΦ ,sinΘsinΦ). The\neffective magnetic field, H(m), is given by\nH=Am′′+K1m1ˆx−K2m2ˆy+Ha.(2)\nHereAis the exchange constant, K1is the easy-axis\nanisotropy, K2>0 is the hard-axis anisotropy, Hais\nthe applied magnetic field (taken to be constant), γis\nthe gyromagnetic ration, and αis the Gilbert damping\nparameter. For convenience we choose units for length,\ntime and energy so that A=K1=γ= 1.Domains cor-\nrespond to locally uniform configurations in which Mis\naligned along one of the local minima, denoted m+and\nm−, of the potential energy\nU(m) =−1\n2(m2\n1−K2m2\n2)−m·Ha.(3)\nTwo distinct domains separated by a DW are described\nby the boundary conditions M(±∞,t) =m±.\nFor purely longitudinal fields Ha=H1ˆxand forH1\nbelowtheWalkerbreakdownfield HW=αK2/2,theDW\npropagates as a travelling wave [1], the so-called Walker\nsolution Θ( x,t) =θW(x−VWt), Φ(x,t) =φW, whereθW\nandφWare given by\nθW(ξ) = 2tan−1(e−ξ/γ),sin2φW=H1/HW.(4)\nThe width of the DW, γ, is given by γ= (1 +\nK2cos2φW)−1/2, and the velocity is given by\nVW=−γ(α+1/α)H1. (5)\nForH1> HW, the DW undergoes non-uniform preces-\nsion and translation, with mean velocity decreasing with2\nH1[1, 5, 6, 9]. The effects of additional transverse fields\nhave been examined recently [7, 11].\nIf the cross-sectional geometry is sufficiently symmet-\nrical (e.g., square or circular), the permeability tensor\nbecomes uniaxial, so that K2= 0 [21, 22]. The dynamics\nin this case is strikingly different. The LLG equation has\nan exact solution, Θ( x,t) =θ0(x−VPt), Φ(x,t) =−H1t,\nin which the DW propagates with velocity\nVP=−αH1 (6)\nand precesses about the easy axis with angular velocity\n−H1[23, 24]. The precessing solution persists for all H1\n– there is no breakdown field – but becomes unstable for\nH1/greaterorsimilar1/2 [25].\nForH1< HW, the ratio VW/VPisgiven by γ(α−2+1).\nFor typical values of α(0.01 – 0.1), the uniaxial velocity\nVPis less than the fully anisotropic velocity VWby sev-\neral orders of magnitude. As we show below, applying a\ntransverse field H2>0 to a uniaxial wire dramatically\nchanges its response to an applied longitudinal field H1.\nThe transverse field, analogous to hard-axis anisotropy,\ninhibits precession and facilitates fast DW propagation.\nForH1lessthanan H2-dependentcriticalfield H1c, given\nin the linear regime by (29) below, there appears a trav-\nelling wave, while for H1> H1c, there appears an oscil-\nlating solution, as in the Walker case. The DW velocity\nof travelling wave exceeds that of oscillating solution.\nVelocity of travelling wave\nWe first obtain a general identity, of independent in-\nterest, which relates the velocity of a travelling wave\nM(x,t) =m(x−Vt) (assuming one exists) to the change\nin potential energy across the profile (for zero transverse\nfield, this coincides with results of [1] and [10]). Noting\nthat˙M=−Vm′, we take the squareof (1) and integrate\nover the length of the wire to obtain\nV2||m′||2= (1+α2)||m×H||2. (7)\nHere we use the notation\n||u||2=/angbracketleftu,u/angbracketright,/angbracketleftu,v/angbracketright=/integraldisplay∞\n−∞u·vdx(8)\nfor theL2-norm and inner product of vector fields (anal-\nogous notation for scalar fields is used below). Next, we\ntake the inner product of (1) with Hto obtain\nV/angbracketleftm′,H/angbracketright=−α||m×H||2. (9)\nNoting that m′·H=/parenleftbig1\n2m′·m′−U(m)/parenrightbig′, we combine\n(7) and (9) to obtain\nV=1\n2(α+1/α)||m′||−2(U(m−)−U(m+)).(10)The identity (10) has a simple physical interpretation;\nthe velocity is proportional to the potential energy dif-\nference across the wire, and inversely proportional to the\nexchange energy of the profile.\nFrom now on, we consider the uniaxial case K2= 0\nandappliedfieldwithlongitudinalandtransversecompo-\nnentsH1,H2>0 (by symmetry, we can assume H3= 0)\nwith|Ha|<1. An immediate consequence of (10) is\nthat, in the uniaxial case, the velocity must vanish as\nH1goes to zero. For when H1= 0, the local minima\nm±are related by reflection through the 23-plane, and\nU(m+) =U(m−).\nSmall transverse field\nIn order to understand travelling wave and oscillating\nsolutions as well as the transition between them, we first\ncarry out an asymptotic analysis in which both H1and\nH2are regarded as small, writing H1=ǫh1,H2=ǫh2\nand rescaling time as τ=ǫt(a systematic treatment in-\ncludingcurrent-inducedtorqueswillbegivenin[29]). We\nseek a solution of the LLG equation (1) of the following\nasymptotic form:\nΘ(x,t) =θ0(x,τ)+ǫθ1(x,τ)+..., (11)\nΦ(x,t) =φ0(x,τ)+ǫφ1(x,τ)+... (12)\nIt is straightforward to check that the boundary condi-\ntions, namely that mapproach distinct minima of Uas\nx→ ±∞, imply that\nm(±∞,τ) = (±1,ǫh2,0)+O(ǫ2).(13)\nThe leading-order equations for Θ and Φ become\nθ0,xx−1\n2(1+φ2\n0,x)sin2θ0= 0, (14)\n/parenleftbig\nsin2θ0φ0,x/parenrightbig\nx= 0. (15)\nThe only physical (finite-energy) solutions of (14) and\n(15) consistent with the boundary conditions (13) are of\nthe form\nφ0(x,τ) =φ0(τ) (16)\nθ0(x,τ) = 2arctanexp( −(x−x∗(τ))),(17)\nwhereφ0andx∗respectivelydescribetheDWorientation\nand centre, and are functions of τalone. It is convenient\nto introduce a travelling coordinate ξ=x−x∗(τ) and\nrewrite the ansatz (11)–(12) as\nΘ(x,t) =θ0(ξ,τ)+ǫθ1(ξ,τ)+..., (18)\nΦ(x,t) =φ0(ξ,τ)+ǫφ1(ξ,τ)+... (19)\nTo obtain equations for φ0(τ) andx∗(τ) we must pro-\nceed to the next order. It is convenient to introduce new3\nvariables at order ǫwhich, in light of the boundary con-\nditions (13), vanish at x=±∞, as follows:\nΘ1:=θ1−h2cosφ0cosθ0, (20)\nu:=φ1sinθ0+h2sinφ0. (21)\nThese satisfy the linear inhomogeneous equations\nLΘ1=f, (22)\nLu=g. (23)\nHereLis the self-adjoint Schr¨ odinger operator given by\nL=−∂2\n∂ξ2+W(ξ), (24)\nwhere\nW=θ′′′\n0\nθ′\n0= 1−2sech2ξ, (25)\nandf(ξ,τ) andg(ξ,τ) are given by\nf= (1+α2)−1sinθ0(−α˙x∗−˙φ0)−h1sinθ0,\ng= (1+α2)−1sinθ0(˙x∗−α˙φ0)+2h2sin2θ0sinφ0.\n(26)\nTheDWposition x∗andorientation φ0aredetermined\nfrom the solvability conditions for (22) – (23). According\nto the Fredholm alternative, given a self-adjoint opera-\ntorLonL2(R), a necessary condition for the equation\nLΘ1=fto have a solution Θ 1is thatfbe orthogonal to\nthe kernel of L. If this is the case, a sufficient condition is\nthatthespectrumof Lisisolatedawayfrom0. From(24)\nand (25) it is clear that θ′\n0belongs to the kernel of L, and\nsince the eigenvalues of a one-dimensional Schr¨ odinger\noperator are nondegenerate, it follows that θ′\n0spans the\nkernel of L. Moreover, since W(ξ)→1 asξ→ ±∞, it\nfollowsthat the spectrum of Lis discrete near0. (In fact,\nWis a special case of the exactly solvable P¨ oschl-Teller\npotential, but we won’t make use of this fact.) Requiring\nfandgin (22) and (23) to be orthogonal to θ′\n0and not-\ning that /angbracketleftθ′\n0,θ′\n0/angbracketright= 2,/angbracketleftθ′\n0,sinθ0/angbracketright=−2,/angbracketleftθ′\n0,1/angbracketright=−π, and\n/angbracketleftθ′\n0,cosθ0/angbracketright= 0, we obtain the following system of ODEs\nforφ0andx∗:\n˙φ0=−h1−απ\n2h2sinφ0, (27)\n˙x∗=−αh1+π\n2h2sinφ0. (28)\nTravelling wave solutions appear provided (27) has\nfixed points; this occurs for h1below a critical field h1,c\ngiven by\nh1,c=απh2\n2, (29)The velocity and orientation of the travelling wave are\ngiven by\n˙x∗=−/parenleftbigg\nα+1\nα/parenrightbigg\nh1, (30)\nsinφ0=−h1\nh1,c. (31)\nThere are two possible solutions for φ0∈[0,2π), only\none of which is stable. Oscillating solutions appear for\nh1> h1c, and are given by\nh1tan1\n2φ0=−h1,c−/radicalBig\nh2\n1−h2\n1,ctan/parenleftBig\n1\n2/radicalBig\nh2\n1−h2\n1,cτ/parenrightBig\n(32)\nwith the period T= 2π//radicalBig\nh2\n1−h2\n1,c. The mean preces-\nsional and translational velocities are obtained by aver-\naging over a period, with result\n/angbracketleftBig\n˙φ0/angbracketrightBig\n=−sgn(h1)/radicalBig\nh2\n1−h2\n1,c, (33)\n/angbracketleft˙x∗/angbracketright=−/parenleftbigg\nα+1\nα/parenrightbigg\nh1+1\nαsgn(h1)/radicalBig\nh2\n1−h2\n1,c.(34)\nNote that for h1=h1,c, (34) coincides with the travelling\nwave velocity (30), whereas for h1≫h1,c, (34) reduces\nto the velocity of the precessing solution given by (6).\nThe behaviour is similar in many respects to the\nWalker case (i.e., K2/negationslash= 0 and H2= 0). Here, the trans-\nverse field rather than hard-axis anisotropy serves to ar-\nrest the precession of the DW (provided the longitudinal\nfield is not too strong). There are differences as well;\nin the transverse-field case there is just one stable trav-\nelling wave, whereas in the Walker case there are two.\nAlso, in the transverse-field case the asymptotic value of\nthe magnetisation has a transverse component, whereas\nin the Walker case it has none.\nModerate transverse field\nWe can extend the travelling wave analysis to the\nregime where H2is no longer regarded as small. We\ncontinue to regard H1as small, writing H1=ǫh1and\nV=ǫv, and expand the travelling wave ansatz Θ( x,t) =\nθ(x−Vt), Φ(x,t) =φ(x−Vt) to first order in ǫ, writing\nθ=θ0+ǫθ1,φ=φ0+ǫφ1. Substituting into the LLG\nequation, we obtain the O(ǫ0) equations\nθ′\n0= (H2−sinθ0), φ0= 0, (35)\nwith boundary conditions sin θ0±=H2,θ0+> π/2 and\nθ0−< π/2. Thus, for H2=O(ǫ0), azimuthal symmetry\nisbrokenatleadingorder,andthestaticprofileisparallel\nto the transversefield (the alternativesolution with φ0=\nπis unstable). The solution of (35) is given by\ntanθ0\n2=κ\nH2tanh/bracketleftbigg\ntanh−1/parenleftbiggH2−1\nκ/parenrightbigg\n−κ\n2ξ/bracketrightbigg\n+1\nH2,\n(36)4\nwhereκ=/radicalbig\n1−H2\n2.\nAt order ǫwe obtain the linear inhomogeneous equa-\ntions\nLθ1=α\n1+α2vθ′\n0−h1sinθ0, (37)\nMφ1=1\n1+α2v(cosθ0)′, (38)\nwhere\nL=−d2\ndξ2+θ′′′\n0\nθ′\n0, M=−d\ndξsin2θ0d\ndξ+H2sinθ0.(39)\nHereθ0is given by (36), and θ1,φ1are requiredto vanish\nasξ→ ±∞. As above, the Fredholm alternative implies\nthat the right-hand side of (37) must be orthogonal to θ′\n0\nin order for a solution to exist. Calculation yields\nV=−/parenleftbigg\nα+1\nα/parenrightbigg\n(1−(H2/κ)cos−1H2)−1H1.(40)\nForH2= 0, this coincides with (30); thus, (40) gives\nH2-nonlinear corrections to the velocity. Moreover, it is\nstraightforward to show that (40) is consistent with the\ngeneral identity (10). Finally, one can also show that\nMhas trivial kernel with spectrum bounded away from\nzero, so that (38) is automatically solvable.\nItisinterestingtocomparetheDWvelocitywithtrans-\nverse field to the Walker case. From (5) and (40),\nVW/V=γ(1−(H2/κ)cos−1H2)<1.(41)\nThus, to leading orderin H1, the DWvelocity in a uniax-\nial wire with transverse field exceeds the Walker velocity.\nNumerical results below establish that this continues to\nhold asH1approaches the critical field H1c.\nNumerical results\nToverifyouranalyticalresults, wesolvetheLLGequa-\ntion (1) using a finite-difference scheme on a domain\n−L≤x≤LwhereL= 100 (the DW has width of\norder 1). Neumann boundary conditions, m′= 0, are\nmaintained at the endpoints. The damping parameter α\nis taken to be 0 .1 throughout. As initial condition we\ntake the stationary profile, with θ0given by (36) and\nφ0= 0. After an initial transient period, during which\nthe asymptotic values of matx→ ±Lconverge to m±,\na stable solution emerges, in which the DW propagates\nwith a characteristic mean velocity V. (For convenience,\nwehavetaken H1<0, sothat Vis positive.) In Figure1,\nnumerically computed values of Vare plotted as a func-\ntion of|H1|for three fixed values of the transverse field:\nH2= 0.2,H2= 0.1, and the limiting case H2= 0, where\nthe dynamics is given by the precessing solution. There\nis good quantitative agreement with the analytic results\nfor small transverse fields, (30), for |H1|< H1,c, and(34), for |H1|> H1,c, In Figure 2, the analytic expres-\nsions for the velocity for small and moderate transverse\nfields are compared to numerical results for H2= 0.2 and\n|H1| ≪H1c. The moderate-field expression (40), which\ndepends nonlinearly in H2, gives excellent agreement for\nsmalldrivingfields. Fornonzero H2, the velocityexhibits\na peak at a critical field |H1c|, which depends on H2.\n0 0.1 0.2 0.3 0.400.050.10.150.20.250.30.35\n|H1|V\n  \nH2= 0.2\nH2= 0.1\nH2= 0\nFIG. 1: Average DW velocity Vas a function of the driving\nfield|H1|for three values of the transverse field H2. The an-\nalytic formulas (solid curves) (30), for |H1|< H1,c, and (34),\nfor|H1|> H1,c, are plotted against numerically computed\nvalues (open circles). For H2= 0, the analytic formula is\nexact.\n00.005 0.01 0.015 0.02 0.02500.050.10.150.20.250.3\n|H1|V\n  \nSmall −H2theory\nModerate −H2theory\nNumerics\nFIG. 2: DW velocity Vas a function of the driving field\n|H1|forH2= 0.2. The expressions for small-transverse field\n(30) (red curve) and moderate-transverse field (40) (light b lue\ncurve) are plotted against numerically computed values (op en\ncircles).\nFigure 3 shows the dependence of the critical field |H1,c|5\nonH2, in close agreement with the analytic result (29).\n0 0.05 0.1 0.15 0.200.0050.010.0150.020.0250.030.035\nH2|H1,c|\nFIG. 3: The critical driving field |H1,c|as a function of the\ntransverse field H2. A linear fit (blue curve) through the nu-\nmerically computed data (blue diamonds) is plotted alongsi de\nthe analytical result (29) (red curve).\nFIG. 4: The magnetization distribution, θ(x,t) andφ(x,t),\nfor two values of the driving field: H1=−0.01 in figures\n(a) and (b), and H1=−0.05 in figures (c) and (d). The\ntransverse field is taken as H2= 0.1 throughout.\nAs in the Walker case, the properties of the propagat-\ning solution are qualitatively different for driving fields\n|H1|below and above the critical field. This is confirmed\nin Figure 4, which shows contour plots of the magne-\ntization in the ( x,t)-plane. Figs. 4(a) and 4(b), where\nH1=−0.01, exemplify the case |H1|<|H1c|. The mag-\nnetisation evolves as a fixed profile translating rigidly\nwith velocity V. For|H1|>|H1c|, as exemplified by\nFigs. 4(c) and 4(d), where H1=−0.05. the magnetiza-\ntion profile exhibits a non-uniform precession as it prop-agates along the nanowire, with mean velocity in good\nagreement with (34).\nSummary\nWe haveestablished, both analyticallyin leading-order\nasymptotics and numerically, the existence of travelling\nwaveandoscillatingsolutionsoftheLLGequationinuni-\naxial wires in applied fields with longitudinal and trans-\nverse components. We have obtained analytic expres-\nsions for the velocity, (30) and (40), and for the critical\nlongitudinal field, (29), above which the travelling wave\nsolution ceases to exist. 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Sch¨ afer, Magnetic Domains: The\nAnalysis of Magnetic Microstructures (Springer, Berlin,\n1998).\n[23] Z. Z. Sun and J. Schliemann, Phys. Rev. Lett. 104,\n037206 (2010).\n[24] A. Goussev, J.M. Robbins, V. Slastikov, Phys. Rev. Lett .\n104, 147202 (2010).\n[25] Y. Gou, A. Goussev, J. M. Robbins, V. Slastikov, Phys.\nRev. B84, 104445 (2011)[26] L. D. Landau and E. M. Lifshitz, Phys. Zeitsch. Sowietu-\nnion8, 153 (1935).\n[27] T. L. Gilbert, Phys. Rev. 100, 1243 (1955); IEEE Trans.\nMag.40, 3443 (2004).\n[28] A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev, Phys.\nRep.194, 117 (1990).\n[29] A. Goussev, R. Lund, J.M. Robbins, C. Sonnenberg,\nV. Slastikov, in preparation."    },    {        "title": "2210.00366v1.Nonlinear_features_of_the_superconductor__ferromagnet__superconductor___varphi_0__Josephson_junction_in_ferromagnetic_resonance_region.pdf",        "content": "Nonlinear features of the superconductor{ferromagnet{superconductor '0Josephson\njunction in ferromagnetic resonance region\nAliasghar Janalizadeh1, Ilhom R. Rahmonov2;3;4, Sara A.\nAbdelmoneim5, Yury M. Shukrinov2;3;4, and Mohammad R. Kolahchi1\n1Department of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), P.O. Box 45137-66731, Zanjan, Iran\n2BLTP, JINR, Dubna, Moscow Region, 141980, Russia\n3Dubna State University, Dubna, 141980, Russia\n4Moscow Institute of Physics and Technology, Dolgoprudny, 141700, Moscow Region, Russia\n5Physics department, Meno\fya University, Faculty of Science, 32511, Shebin Elkom,Egypt\n(Dated: October 4, 2022)\nWe demonstrate the manifestations of the nonlinear features in magnetic dynamics and IV-\ncharacteristics of the '0Josephson junction in the ferromagnetic resonance region. We show that\nat small values of system parameters, namely, damping, spin-orbit interaction, and Josephson to\nmagnetic energy ratio, the magnetic dynamics is reduced to the dynamics of the scalar Du\u000eng os-\ncillator, driven by the Josephson oscillations. The role of increasing superconducting current in the\nresonance region is clari\fed. Shifting of the ferromagnetic resonant frequency and the reversal of\nits damping dependence due to nonlinearity are demonstrated by the full Landau-Lifshitz-Gilbert-\nJosephson system of equations, and in its di\u000berent approximations. Finally, we demonstrate the\nnegative di\u000berential resistance in the IV{characteristics, and its correlation with the foldover e\u000bect.\nI. I. INTRODUCTION\nThe coupling of superconducting phase di\u000berence with\nmagnetic moment of ferromagnet in the '0junction leads\nto a number of unique features important for supercon-\nducting spintronics, and modern information technology\n[1{5]. It allows to control the magnetization preces-\nsion by superconducting current and a\u000bects the current{\nvoltage (IV) characteristics by magnetic dynamics in the\nferromagnet, in particular, to create a DC component in\nthe superconducting current [6{8]. A remarkable mani-\nfestation of such coupling is the possibility to stimulate\na magnetization reversal in the ferromagnetic layer by\napplying current pulse through the '0-junction [3, 9{13].\nThere are two features of our Josephson junction that\ncome into play in our study. One is the broken inver-\nsion symmetry in the weak link of the Josephson junc-\ntion, when the link is magnetic, which introduces an ex-\ntra phase in the current|-phase relation, preventing it\nfrom being antisymmetric. Such Josephson junctions are\nnamed'0junctions [1], and examples exist such as MnSi\nand FeGe. Second is the nonlinear property of the system\nthat makes for an anomalous resonance behavior [14].\nWe couple such a Josephson junction to the model\nthat describes the magnetodynamics in thin \flms or\nheterostructure, to form the Landau-Lifshitz-Gilbert-\nJosephson model (LLGJ)[14{16]. It is shown that for\na particular set of parameters, the coupled equations\nreduce to the dynamics of a Du\u000eng oscillator [14].\nThe cubic nonlinearity in this oscillator has applications\nin describing several e\u000bects in other models too [17].\nOne being the resonance e\u000bects in the antiferromagnetic\nbimeron in response to an alternating current, which has\napplications in the detection of weak signals [15, 18, 19].\nThe Gilbert damping term is added phenomenologi-\ncally to the Landau|-Lifshitz model, to reproduce the\ndamping of the precessing magnetic moment. Gilbertdamping is important in modeling other resonance fea-\ntures too, as its temperature dependence a\u000bects them\n[20, 21], and in return in the superconducting correla-\ntions that a\u000bect it [22]. The magnetization precession\nin the ultra thin Co20Fe60B20layer stimulated by mi-\ncrowave voltage under a large angle, needs modeling by\nDu\u000eng oscillator too. This gets help from the so called\nfoldover features, again due to nonlinearity [16, 23, 24].\nThe consequences of the nonlinear nature of the cou-\npled set of LLGJ system of equations in the weak cou-\npling regime was demonstrated recently in Ref. [14]. We\nshowed in this regime, where the Josephson energy is\nsmall compared to the magnetic energy, the '0Joseph-\nson junction is equivalently described by a scalar non-\nlinear Du\u000eng equation. An anomalous dependence of\nthe ferromagnetic resonant frequency (FMR) with the\nincrease of the Gilbert damping was found. We showed\nthat the damped precession of the magnetic moment is\ndynamically driven by the Josephson supercurrent, and\nthe resonance behavior is given by the Du\u000eng spring.\nThe obtained results were based on the numerical simu-\nlations. The role of dc superconducting current, and the\nstate with negative di\u000berential resistance (NDR) in IV-\ncharacteristic were not clari\fed. Also, the e\u000bects of the\nJosephson to magnetic energy ratio and the spin-orbit\ncoupling (SOC) were not investigated at that time.\nIn the present paper, we study the nonlinear aspects\nof the magnetic dynamics and IV-characteristics of the\n'0Josephson junction in the ferromagnetic resonance re-\ngion. We compare description of the anomalous damp-\ning dependence (ADD) exhibited by full LLGJ system\nof equations with approximated equations and demon-\nstrate the Du\u000eng oscillator features in the small param-\neter regime. E\u000bects of the Josephson to magnetic energy\nratio, and the spin-orbit coupling on the ADD, referred\nto earlier as the \u000b-e\u000bect [14] are demonstrated. By de-\nriving the formula which couples the dc superconduct-arXiv:2210.00366v1  [cond-mat.supr-con]  1 Oct 20222\ning current and maximal amplitude of magnetization we\ndiscuss the correlation of superconducting current and\nthe negative di\u000berential resistance in the resonance re-\ngion. Finally, we discuss the experimentally important\nfeatures by emphasizing the details of the magnetization\ndynamics and the IV-characteristics of the '0junction.\nWe have shown that in the limit of small system pa-\nrameters; that is, the Josephson to magnetic energy ra-\ntioG, the damping \u000b, and the spin-orbit coupling r, the\ndynamics is given by the Du\u000eng spring [14]. We focus\non the shift in resonance and the e\u000bects of nonlinear in-\nteractions. We give semi-analytic models to explain our\nresults in various limits.\nThe paper is organized as follows. In Section II we\noutline the theoretical model and discuss the methods\nof calculations. The ferromagnetic resonance and ef-\nfects of system parameters on the anomalous damping\ndependence are considered in Subsection A of Section\nIII. In Subsection B we present analytical description of\nthe dynamics and IV-characteristics of the '0junction\nat small system parameters. Manifestation of the nega-\ntive di\u000berential resistance in IV-characteristics through\nthe foldover e\u000bect is discussed. We compare the de-\nscription of the anomalous damping dependence by full\nLLGJ system of equation with approximated equation,\nand show how the Du\u000eng oscillator captures the non-\nlinearities in the small parameter regime in Subsection\nC. We present results on the critical damping and de-\nrive the formula which couples the dc superconducting\ncurrent and maximal amplitude of magnetization in the\nferromagnetic layer. Finally, in Section IV we concludes\nthe paper.\nII. II. MODELS AND METHOD\nThe following section is closely related to our work\nin [13]. The '0junction [6, 12, 25] that we study is shown\nin Fig.1. The current-phase relation in varphi 0junction\nhas the form Is=Icsin ('\u0000'0), where'0=rMy=M0,\nMydenotes the component of magnetic moment in ^ ydi-\nrection,M0is the modulus of the magnetization. The\nphysics of'0Josephson juncton is determined by system\nof equations which consists of Landau-Lifshits-Gilbert\n(LLG), resistively capacitively shunted junction (RCSJ)\nmodel expression with current-phase relation ( Is) de-\nscribed above, and Josephson relation between phase dif-\nference and voltage.\nThe dynamics of the magnetic moment Mis described\nby the LLG equation [26]\ndM\ndt=\u0000\rM\u0002Heff+\u000b\nM0\u0012\nM\u0002dM\ndt\u0013\n; (1)\nwhere Mis the magnetization vector, \ris the gyromag-\nnetic relation, Heffis the e\u000bective magnetic \feld, \u000bis\nGilbert damping parameter, M0=jMj.\nFigure 1. Schematic view of SFS '0Josephson junction. The\nexternal current applied along x direction, ferromagnetic easy\naxis is along z direction.\nIn order to \fnd the expression for the e\u000bective mag-\nnetic \feld we have used the model developed in Ref.[6],\nwhere it is assumed that the gradient of the spin-orbit\npotential is along the easy axis of magnetization taken to\nbe along ^z. In this case the total energy of the system\ncan be written as\nEtot=\u0000\b0\n2\u0019'I+Es(';' 0) +EM('0); (2)\nwhere'is the phase di\u000berence between the supercon-\nductors across the junction, Iis the external current,\nEs(';' 0) =EJ[1\u0000cos ('\u0000'0)], andEJ= \b 0Ic=2\u0019\nis the Josephson energy. Here \b 0is the \rux quantum,\nIcis the critical current, r=l\u001dso=\u001dFl= 4hL=~\u001dF,L\nis the length of Flayer,his the exchange \feld of the\nFlayer,EM=\u0000KVM2\nz=(2M2\n0), the parameter \u001dso=\u001dF\ncharacterizes a relative strength of spin-orbit interaction,\nKis the anisotropic constant, and Vis the volume of the\nferromagnetic ( F) layer.\nThe e\u000bective \feld for LLG equation is determined by\nHe\u000b=\u00001\nV@Etot\n@M\n=\nF\n\r\u0014\nGrsin\u0012\n'\u0000rMy\nM0\u0013\nby+Mz\nM0bz\u0015\n(3)\nwhere \n F=\rK=M 0is frequency of ferromagnetic reso-\nnance andG=EJ=(KV) determines the ratio of Joseph-\nson energy to magnetic one.\nIn order to describe the full dynamics '0junction the\nLLG equations should be supplemented by the equation\nfor phase di\u000berence ', i.e. equation of RCSJ model for\nbias current and Josephson relation for voltage. Accord-\ning to the extended RCSJ model, which takes into ac-\ncount derivative of '0phase shift, the current \rowing\nthrough the system in underdamped case is determined\nby\nI=~C\n2ed2'\ndt2+~\n2eR\u0014d'\ndt\u0000r\nM0dMy\ndt\u0015\n(4)\n+Icsin\u0012\n'\u0000r\nM0My\u0013\n:\nwhereIis the bias current, CandRare the capacitance\nand resistance of Josephson junction respectively. The3\nJosephson relation for voltage is given by :\n~\n2ed'\ndt=V: (5)\nWe note that in the framework of RCSJ{model the\ndisplacement current is proportional to the \frst deriva-\ntive of voltage (or second derivative of phase di\u000berence).\nFrom the other hand, the magnetization dynamics plays\nrole of the external force and \frst order derivative of '0\nis a source of external current for JJ. This was demon-\nstrated in Ref.[25, 27] where the authors included the \frst\nderivative of '0as the source of the electromotive force.\nVoltage is determined by the phase di\u000berence, and does\nnot depend on '0. From this point of view, in the frame-\nwork of RCSJ model the external current source cannot\nmodify the expression for displacement current. That's\nwhy we do not include the second derivative of varphi 0\nin our model.\nUsing (1), (3), (4) and (5) we can write the system of\nequations, in normalised variables, which describes the\ndynamics of '0junction\n_mx=!F\n1 +\u000b2f\u0000mymz+Grm zsin('\u0000rmy)\n\u0000\u000b[mxm2\nz+Grm xmysin('\u0000rmy)]g;\n_my=!F\n1 +\u000b2fmxmz\n\u0000\u000b[mym2\nz\u0000Gr(m2\nz+m2\nx) sin('\u0000rmy)]g;\n_mz=!F\n1 +\u000b2f\u0000Grm xsin('\u0000rmy)\n\u0000\u000b[Grm ymzsin('\u0000rmy)\u0000mz(m2\nx+m2\ny)]g;\n_V=1\n\fc[I\u0000V+r_my\u0000sin('\u0000rmy)];\n_'=V(6)\nwheremx;y;z =Mx;y;z=M0and satisfy the constraintP\ni=x;y;zm2\ni(t) = 1,\fc= 2eIcCR2=~is McCumber pa-\nrameter. In order to use the same time scale in the\nLLG and RCSJ equations in this system of equations\nwe have normalized time to the !\u00001\nc, where!c=2eIcR\n~,\nand!F= \n F=!cis the normalized frequency of ferro-\nmagnetic resonance \n F=\rK=M 0. Bias current is nor-\nmalized to the critical current Icand voltage V{ to the\nVc=IcR. The system of equations (6), is solved numer-\nically using the fourth-order Runge-Kutta method(see\nRef.[14]).\nIII. III. RESULTS AND DISCUSSION\nA. A. E\u000bect of system parameters on the\nanomalous damping dependence\nADD of the FMR frequency with increasing \u000bwas dis-\ncussed in Ref. [14]. It was found that the resonance\ncurves demonstrate features of Du\u000eng oscillator, re-\n\recting the nonlinear nature of Landau-Lifshitz-Gilbert-\n 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14\n 0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8mymax\nValpha=0.01\nalpha=0.02\nalpha=0.03\nalpha=0.04\nalpha=0.05\nalpha=0.06\nalpha=0.07\nalpha=0.08\nalpha=0.09\nalpha=0.1\n 0 0.1\n 0.5Figure 2. Maximal amplitude of magnetization\nmy\u0000component at each values of bias current and voltage\nalong IV-characteristics of the '0junction in the ferromag-\nnetic resonance region for various \u000b. Inset enlarges the main\nmaximum. Parameters: \fc= 25,G=0.05,r=0.05, !F= 0:5.\nJosephson (LLGJ) system of equations. There is a criti-\ncal damping value at which anomalous dependence comes\ninto play. This critical value depends on the system pa-\nrameters. Here we present the details of such transforma-\ntion from usual to anomalous dependence with variation\nin spin-orbit coupling and ratio of Josephson to magnetic\nenergy.\nTo investigate the e\u000bect of damping, we calculate\nthe maximal amplitude of magnetization component my\ntaken at each value of the bias current based on the\nLLGJ system of equations (6). In Fig.2 we show the\nvoltage dependence of maximal amplitude mmax\nyin the\nferromagnetic resonance region at di\u000berent damping pa-\nrameter and small values of Josephson to magnetic en-\nergy ratio G=0.05 and spin-orbit coupling r= 0:05. We\nfound that the ferromagnetic resonance curves demon-\nstrate the di\u000berent forms. An increase in damping shows\na nonuniform change in the resonant frequency: it is ap-\nproaching the !Finstead of moving away with increase\nin\u000b. We stress that this happens at small Gandr. We\nconsider that such behavior can be explained by the non-\nlinear nature of the LLGJ system of equations. There is\na manifestation of subharmonics of the FMR in Fig.2 at\n!= 1=2;1=3;1=4.\nWe usually expect the resonance peak to move away\nfrom resonance as the \u000bincreases. Figure 2 shows that\nthis normal e\u000bect is accompanied with an anomalous be-\nhaviour as can be seen in the inset to this \fgure, where\nthe resonance peak approaches !Fas\u000bincreases [14].\nThe manifestation of FMR in IV-characteristics of the\n'0junction at three values of damping parameter is\ndemonstrated in Fig. 3. The strong deviation of the\nIV-curve is observing at \u000b= 0:01, which is characteristic4\nFigure 3. Part of the IV characteristic of the '0junction\natG= 0:05;r= 0:05 and di\u000berent values of Gilbert damp-\ning. The numbers show \u000bvalue. Inset shows the total IV-\ncharacteristic and arrow indicates the resonance region\nvalue for many magnetic materials. This fact indicates\nthat ADD can be observed experimentally by measuring\nIV-characteristics in wide interval of the damping param-\neter.\nInteresting features of ADD appear by a variation of\nspin-orbit coupling. As it was demonstrated in Ref.[28],\nan increase in SOC leads to the essential change in IV-\ncharacteristics and magnetization precession in the fer-\nromagnetic resonance region. The nonlinearity is going\nstronger and the state with negative di\u000berential resis-\ntance appears at large SOC.\nFigure 4(a) demonstrates results of numerical simu-\nlations ofmmax\nydependence on \u000bat di\u000berent values of\nSOC parameter r. It shows two speci\fc features of ADD.\nFirst, with an increase in r, the critical value of Vpeakis\ndecreasing (the curve moves away from !F). The sec-\nond important feature is an increasing of \u000bcritwhich is\ndemonstrated in this \fgure by arrows.\nAnother model parameter which a\u000bects the phe-\nnomenon discussed in the present paper is the ratio G\nof Josephson to magnetic energies. Figure 4(b) demon-\nstrates the results of numerical simulations of mmax\nyde-\npendence on \u000bat di\u000berent values of G.\nSimilar to the e\u000bect of r, increasing Galso causes the\nvalue of\u000bcritto increase. By changing the volume of the\nferromagnetic layer, the ferromagnetic energy and con-\nsequently the value of G can be changed [6]. For small\nG, i.e. a situation where the magnetic energy is much\nlarger than the Josephson energy, the magnetic layer re-\nceives less energy, and its amplitude decreases in the y\ndirection, and also the maximum value of the oscillation\nfrequency is closer to the magnetic frequency, !F.\nVpeakα\n0.46 0.47 0.48 0.49 0.500.050.10.150.2\nαcrit\n1 23\nVpeakα\n0.46 0.47 0.48 0.49 0.500.050.10.150.2\nαcrit\n1 23Figure 4. (a) Demonstration of ADD at di\u000berent values of\nSOC parameter ratG= 0:05. Numbers indicate: 1 -\nr= 0:05; 2 -r= 0:1; 3 -r= 0:5; Arrows show critical \u000b\nvalue, corresponded to the reversal in the \u000bdependence (b)\nDemonstration of ADD at di\u000berent values of the Josephson\nto magnetic energy ratio Gatr= 0:05. Numbers indicate: 1\n-G= 0:01; 2 -G= 0:1; 3 -G= 1.\nB. B. Dynamics and IV-characteristics of the '0\njunction at small system parameters\nAs it was discussed in Refs.[6, 29, 30], in case of\nG;r;\u000b<< 1 andmz\u00191, \frst three equations of the sys-\ntem (6) can be simpli\fed. Taking into account '=!Jt\nand neglecting quadratic terms of mxandmy, we get\n(\n_mx=!F[\u0000my+Grsin(!Jt)\u0000\u000bmx]\n_my=!F[mx\u0000\u000bmy];(7)\nThis system of equations can be written as the second\norder di\u000berential equation with respect to my\nmy+ 2\u000b!F_my+!2\nFmy=!2\nFGrsin!Jt: (8)5\nCorresponding solution for myhas the form\nmy(t) =!+\u0000!\u0000\nrsin(!Jt)\u0000\r++\r\u0000\nrcos(!Jt);(9)\nwhere\n!\u0006=Gr2!F\n2!J\u0006!F\n\n\u0006; (10)\nand\n\r\u0006=Gr2!F\n2\u000b!J\n\n\u0006: (11)\nwith \n \u0006= (!J\u0006!F)2+ (\u000b!J)2(see Ref.[6] and corre-\nsponded Erratum[31]).\nWhen the Josephson frequency !Jis approaching the\nferromagnetic one !F,mydemonstrates the damped fer-\nromagnetic resonance. Di\u000berential resistance in the res-\nonance region is decreasing and it is manifested in the\nIV{characteristic as a resonance branch [7].\nTaking into account rmy<<1, we rewrite expression\nfor superconducting current as\nIs(t) = sin(!Jt\u0000rmy(t))\n= sin(!Jt)\u0000rmycos(!Jt) (12)\nUsing solution (9) we can obtain\nIs(t) = sin!Jt\u0000!+\u0000!\u0000\n2sin 2!Jt\n+\r++\r\u0000\n2cos 2!Jt+I0(\u000b) (13)\nwhere\nI0=\r++\r\u0000\n2: (14)\nThis superconducting current explains the appearance\nof the resonance branch in the IV{characteristic. The\ngenerated current I0can be expressed through the am-\nplitude ofmyand SOI parameter r\nI0=r\n2mmax\ny(!J); (15)\nwithmmax\ny(!J) being the frequency response of my.\nAt small model parameters \u000b<<Gr<< 1 of SFS'0\nJosephson junction, the states with a negative di\u000berential\nresistance appear in the IV-characteristics in the FMR\nregion. Due to the nonlinearity, the resonance peak is\nasymmetric. Increasing of nonlinearity leads to the bista-\nbility (foldover e\u000bect). A natural questions appears if the\nstates with a negative di\u000berential resistance are the origin\nof the foldover and ADD. In order to clarify this question,\nwe show in Fig.5 a part of IV{characteristics of '0junc-\ntion together with IV-characteristic of SIS junction in\nthe ferromagnetic resonance region and numerically cal-\nculated superconducting current through this junction.\nThe total IV{characteristics are demonstrated in the in-\nset to this \fgure.\nFigure 5. IV-characteristic of '0and SIS junctions and calcu-\nlated average superconducting current through the '0junc-\ntion.\nWe see the correlation of the foldover e\u000bect in super-\nconducting current (blue) with the NDR part of IV-curve.\nThe peak in the superconducting current and minimum\nof IV-curve are at the same voltage value. So, both ef-\nfects re\rect the nonlinear features of the ferromagnetic\nresonance in '0junction. But in contrast to the foldover\nand ADD e\u000bects, which start manifesting themselves at\nrelatively small deviations from linear case, the nonlin-\nearity in case of the NDR plays a more essential role.\nWe note that in the resonance region for considered\nlimit of model parameters the myamplitude are coupled\nto the value of superconducting current (see Eq.(15)). We\nstress the importance of the performed analysis demon-\nstrating the analytical coupling of time independent su-\nperconducting current and magnetization, re\recting the\nDu\u000eng oscillator features of the '0junction.\nAs it is well known, the states with negative di\u000beren-\ntial resistance appear in IV-characteristics of Josephson\nstructures in di\u000berent physical situations. In particular,\nthe nonlinear superconducting structures being driven\nfar from equilibrium exhibit NDR states [32]. The NDR\nstates plays an essential role in the applications related\nto the THz radiation emission [33]. A detailed expla-\nnation for di\u000berent types of negative di\u000berential resis-\ntance (NDR) in Josephson junction (i.e., N-shaped and\nS-shaped) is introduced in[34]. The authors emphasize\nthat the nonlinear behavior of the Josephson junction\nplays a key role in the NDR feature. In our case, the\nNDR states appear as a result of system's nonlinearity\nat small values of '0junction parameters, such as SOC,\nratio of Josephson to magnetic energy and Gilbert damp-\ning. We demonstrate these e\u000bects here by by presented\nresults of detailed investigation of the NDR state at dif-\nferent system parameters and discuss possibility of their\ncontrol near the ferromagnetic resonance.\nFigure 6 shows the e\u000bect of the spin-orbit coupling on\nthe IV-characteristic at G= 0:05 and\u000b= 0:01. We\nsee the NDR feature which is getting more pronounced\nwith increase in r. A further increase in rleads to the\njump down in voltage and then practically linear growth6\nIV\n0.49 0.495 0.5 0.505 0.510.460.470.480.490.09\n0.15\nIV\n0 0.5 100.20.40.60.811.2r =0.15\nFigure 6. Enlarged parts of the IV-curves in the resonance\nregion at di\u000berent values of SOC parameter rat\u000b= 0:01\nandG= 0:05. The numbers indicate the increasing of rfrom\n0.09 to 0.15 with an increment 0.01. Inset demonstrates total\nIV-characteristic ar r= 0:15\nof IV-characteristic.\nAn interesting question is concerning the e\u000bect of\nGilbert damping. Results of IV-characteristics simula-\ntions in the resonance region at a certain range of the\ndamping parameter \u000batG= 0:05 andr= 0:13 are\nshown in Fig.7(a). In this case, the most pronounced\ncharacteristic appears at \u000b= 0:01. At chosen set of pa-\nrametersG= 0:05 andr= 0:13, the range of \u000bwith\npronounced NDR features is 0 :016\u000b<0:014.\nThe maximal amplitude mmax\nyas a function of volt-\nage is shown in the Fig.7(b). Based on the results, pre-\nsented in Fig.7(a) and Fig.7(b), we came to the important\nconclusion that the foldover e\u000bect (bistability) and NDR\nstate have strong correlations and have the same origin\nrelated to the nonlinearity at small system parameters.\nBut anomalous damping dependence does not show a\none-to-one correlation with either negative di\u000berential re-\nsistance or foldover e\u000bect. The resonance peak positions\nofmmax\nyin bias current Ipeak, and in voltage Vpeakas the\nfunctions of \u000bare demonstrated in Fig.7(c). According\nto our results, we can divide \u000binterval into two regions\n(see Fig.7(c)). Region I introduces the values of \u000bwhere\nthe NDR feature is present, while in the Region II it dis-\nappears. In the Region II the foldover e\u000bect (bistability)\ndisappears as well, but ADD is realized.\nC. C. Du\u000eng oscillator features of '0junction and\ncritical damping\nThe system of equations (6) is nonlinear and very com-\nplex, so in order to provide analytical study of dynamics\nof the'0junction we need to derive an approximated\nequation for some limited values of model parameters.\nIn Ref. [14] was shown that the resonance curves demon-\nstrate features of Du\u000eng oscillator, re\recting the nonlin-\near nature of Landau-Lifshitz-Gilbert-Josephson (LLGJ)\nIV\n0.495 0.5 0.505 0.510.4650.470.4750.480.4850.49\n0.01(a)\n0.018\nIV\n0 0.5 100.20.40.60.811.2\nα= 0.01\nαIpeak, Vpeak\n0.01 0.012 0.014 0.016 0.018 0.020.470.480.490.5Ipeak\nVpeak(c)\nΙΙΙFigure 7. (a) Enlarged parts of the IV-curves at di\u000berent\nvalues of\u000b; (b) Voltage dependence of mmax\nyat di\u000berent \u000b;\n(c)\u000b-dependence of the resonance curve maximum in current\n(Ipeak) and voltage ( Vpeak). The numbers indicate the value\nof\u000bfrom 0.01 to 0.018 in (a) and from 0.01 to 0.02 in (b)\nby an increment 0.001. Results are obtained at r= 0:13, and\nG= 0:05.\nsystem of equations. In this section, we present analyt-\nical approach to describe the nonlinear dynamics of '0-\njunction and compare analytical results followed from ap-\nproximated Du\u000eng equation with numerical simulations\nof total system of equations (6). We show that in the\nlimit of\u000b << G;r << 1, we arrive to the Du\u000eng os-\ncillator. We start by \frst three equations of the (6) for7\nmagnetization components\n_mx\n!F=\u0000mymz+Grm zsin('\u0000rmy)\u0000\u000bmxm2\nz\n_my\n!F=mxmz\u0000\u000bmym2\nz (16)\n_mz\n!F=\u0000Grm xsin('\u0000rmy) +\u000bmz(m2\nx+m2\ny)\nSimplifying this system of equations by the same pro-\ncedure as it was done in Ref.[14] we can write equation\nformyas\nmy+2\u0018\u000b_my+\u00182(1+\u000b2)my\u0000\u00182(1+\u000b2\u0000\u000b4)m3\ny=\u00182\rsin':\n(17)\nFinally, by neglecting the \u000b2and\u000b4terms which are\nmuch smaller than 1, we come to well known Du\u000eng\nequation,\nmy+ 2!F\u000b_my+!2\nFmy\u0000!2\nFm3\ny=!2\nFGrsin': (18)\nIn the small parameter range, this Du\u000eng equation\ncan describe the dynamics of my. We will have the full\ndynamics, once we consider the coupling with the Joseph-\nson equation,\n'+1\n\fc[ _'\u0000r_my+ sin('\u0000rmy)] =1\n\fcI: (19)\nThe system of equations, (18) and (19) can replace the\nLLGJ equations in the limit of G;r<< 1 andG;r>>\u000b .\nTaking into account '=!Jtwe can right analytically\nobtained frequency response for equation (18)\n(mmax\ny)2=\u0000\nGr\u00012\n\u0002\n!2\u00001 +3\n4(mmaxy)2\u00032+\u0000\n2\u000b!\u00012\n(20)\nwhere!=!J=!F. From Eq. (20) we get\n(mmax\ny)6+8\n3(!2\u00001)(mmax\ny)4\n+\u00124\n3\u00132\u0014\n(!2\u00001)2+\u0000\n2\u000b!\u00012\u0015\n(mmax\ny)2\n\u0000\u00124\n3Gr\u00132\n= 0: (21)\nThis equation allows to determine analytically fre-\nquency dependence of the mmax\nyamplitude. To \fnd it\nwe solve the equation (21) by the Newton method. Re-\nsults of analytical calculations (blue dots) corresponded\nto (21) and numerical one (red doted line) corresponded\nto the full system of equation (6) are demonstrated in\nFig.8.\nFigure 8. Numerically (curve 1) and analytically (curve 2)\ncalculated amplitude dependence of my.\nFigure 9. Numerically calculated superconducting current for\nSFS junction (plot 1) and analytical I0(plot 2) and super-\nconducting current for SIS junction (plot 3).\nWe can see that they are close to each other which\nproves the correctness of the chosen approximation.\nBoth curves demonstrate an asymmetric resonance peak,\nwhich is common for Du\u000eng oscillator. When a role of\nthe cubic term is getting larger, we observe a bistability\nof the resonance curve, which is usually called a foldover\ne\u000bect. Note that the foldover e\u000bect can be also achieved\nby the damping decreasing; i.e., by the decreasing of dis-\nsipative term in (18), we can increase the in\ruence of the\ncubic term in this equation.\nThe comparison of analytically and numerically cal-\nculated superconducting current as a function of the\nJosephson frequency is demonstrated in Fig. 9. We note\nthat in our normalization V=!J. We can see the man-\nifestation of the asymmetric resonance peak in the fre-\nquency dependence of superconducting current. So, the\napproximated system of equations 7 re\rects one of the\nmain feature of Du\u000eng oscillator.\nFigure (10) compares anomalous damping dependence\nof the resonance peak of mmax\ny(V) calculated numeri-\ncally according to the full LLGJ system of equations (6)\nwith calculated numerically according to the generalized\nDu\u000eng model (equations (17, 19)). We see that in the\ndamping parameter interval [0.001 { 0.2] the coincidence8\nFigure 10. The \u000b-dependence of the resonance maximum of\nmmax\ny(V) in the damping parameter interval [0.001 { 0.12].\nGreen squares show results calculated numerically according\nto the full system of equations (6), blue circles show results\ncalculated numerically according to the generalized Du\u000eng\nand Josephson equations (17,19). The dashed line connects\nthe symbols to guide eyes. Solid line show analytical \u000b-\ndependence calculated according to the Eq. (22). All calcu-\nlation have been done at \fc= 25, G=0.05, r=0.05, !F= 0:5.\nof the dependences is enough good.\nUsing equation (18) with '=!Jt, we can \fnd (see\nSupplementary materials ??) a relation between posi-\ntion of the resonance peak in mmax\ny(V) dependence and\ndamping\n!peak=s\n1\u00003\u000b2\n2+1\n2r\n(1\u0000\u000b2)2\u000012(Gr\n4\u000b)2(22)\nwhere!peak=!J;peak\n!Fdetermines the position of the res-\nonance peak.\nEquation (22) allows to \fnd the formula for critical\ndamping\u000bcritwhich is an important parameter deter-\nmining the reversal point in damping dependence of the\nresonance peak of mmax\ny(V) .\nTaking into account equation (22) we can write equa-\ntion with respect of Gr=(4\u000b) (See supplementary mate-\nrials??).\n9\u0012Gr\n4\u000bcrit\u00134\n+ 3\u000b2\ncrit(10\u000b2\ncrit\u00001)\u0012Gr\n4\u000bcrit\u00132\n(23)\n\u00002\u000b4\ncrit(\u000b2\ncrit\u00001)2= 0\nUsing approximation 10 \u000b2\ncrit<<1 and\u000b2\ncrit<<1 it\ngives (see Supplementary Materials)\n\u000bcrit\u00191\n2sr\n3\n2Gr (24)\nFigure 11. Numerical calculations according to Eq. (6)\n(squares), analytical according to Eq. (23)(solid line) and\napproximated analytical according to Eq. (24) (dashed line).\nTable 1: A comparison between the numerical and an-\nalytical values of \u000bcrit:at di\u000berent values of Gandr.\nG r Gr\u000bcrit:;numerics \u000bcrit:;analytics\n0.01 0.05 0.0005 0.0100 0.0123\n0.05 0.05 0.0025 0.0300 0.0276\n0.05 0.10 0.0050 0.0400 0.0391\n0.05 0.30 0.0150 0.0700 0.0677\n0.05 0.50 0.0250 0.0900 0.0874\n0.10 0.05 0.0050 0.0391 0.0391\n0.60 0.05 0.0300 0.0950 0.0958\n0.70 0.05 0.0350 0.1000 0.1035\n1.00 0.05 0.0500 0.1200 0.1237\nFigure 11 presents comparison of numerical and ana-\nlytical results \u000bcritversusGr.\nAs we see, it shows a good agreement of numerical\nand analytical results of calculations at small product of\nJosephson to magnetic energy ratio and spin-orbit inter-\naction.\nIV. IV. CONCLUSIONS\nThe understanding of the nonlinear features of\nmagnetization dynamics in superconductor-ferromagnet-\nsuperconductor Josephson junction and their manifesta-\ntion in the IV-characteristics has implications for super-\nconductor spintronics, and modern information technol-\nogy. In'0junctions the nonlinear features can a\u000bect the\ncontrol of magnetization precession by superconducting\ncurrent and external electromagnetic radiation [28].\nHere, using numerical and analytic approaches, we\nhave demonstrated that at small system parameters,9\nnamely, the damping, spin-orbit interaction and Joseph-\nson to magnetic energy ratio in '0junction, magnetic dy-\nnamics is reduced to the dynamics of the scalar Du\u000eng\noscillator, driven by the Josephson oscillations. We have\nclari\fed the role of increasing superconducting current\nin the resonance region leading to the foldover e\u000bect in\nthe ferromagnet magnetization. We have demonstrated\nthe parameter dependence of the anomalous ferromag-\nnetic resonant shifting with anomalous damping depen-\ndence due to nonlinearity of the full LLGJ equation and\nin its di\u000berent approximations. We have derived the an-\nalytical expression for critical damping value. Also, we\ndemonstrated appearance of negative di\u000berential resis-\ntance in the IV-characteristics and the correlation with\noccurrence of the foldover e\u000bect in the magnetization of\nferromagnet.\nWe have stressed that the manifestation of negative\ndi\u000berential resistance is related to the nonlinear features\nof the system[34, 35]. It was demonstrated that in the\nsmall model parameters case the equation for magnetic\nsubsystem takes form of Du\u000eng equation where nonlin-\nearity manifest itself as the cubic term. We have shown\nthat the appearance of negative di\u000berential resistance in\nthe I-V curve is related to the appearance of foldover inthemmax\ny-Vcurve.\nWe believe that the experimentally measured IV-\ncharacteristics of '0junction with manifestations dis-\ncussed in detail in the present paper, would allow close\ninvestigations of its nonlinear features important for su-\nperconductor electronics and spintronics.\nV. SUPPLEMENTARY\nIn supplementary material are presented the details of\ncalculations for Eq.22 and Eq.24.\nVI. FUNDING\nNumerical simulations were funded by Project No. 18-\n71-10095 of the Russian Science Foundation. The pre-\nsented results concerning the calculations of DC super-\nconducting current in the section V are supported by the\nRussian Science Foundation in the framework of project\n22-42-04408. A.J. and M.R.K. are grateful to IASBS for\n\fnancial support.\n[1] Buzdin, A. Physical Review Letters 2008 ,101 (10),\n107005.\n[2] Linder, J., Robinson, J. W. 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P.\nIEEE Transactions on Applied Superconductivity 2014 ,\n24(6), 1{7.\n[35] Nagel, J., Speer, D., Gaber, T., Sterck, A., Eichhorn, R.,\nReimann, P., Ilin, K., Siegel, M., Koelle, D., Kleiner, R.\nPhysical Review Letters 2008 ,100, 217001."    },    {        "title": "1002.4958v1.Correlation_Effects_in_the_Stochastic_Landau_Lifshitz_Gilbert_Equation.pdf",        "content": "arXiv:1002.4958v1  [cond-mat.mes-hall]  26 Feb 2010Correlation Effects in Stochastic Ferromagnetic Systems\nThomas Bose and Steffen Trimper\nInstitute of Physics, Martin-Luther-University, D-06099 Halle, Germany∗\n(Dated: June 16, 2018)\nAbstract\nWe analyze the Landau-Lifshitz-Gilbert equation when the p recession motion of the magnetic\nmoments is additionally subjected to an uniaxial anisotrop y and is driven by a multiplicative cou-\npled stochastic field with a finite correlation time τ. The mean value for the spin wave components\noffers that the spin-wave dispersion relation and its damping is strongly influenced by the deter-\nministic Gilbert damping parameter α, the strength of the stochastic forces Dand its temporal\nrangeτ. The spin-spin-correlation function can be calculated in t he low correlation time limit by\nderiving an evolution equation for the joint probability fu nction. The stability analysis enables us\nto find the phase diagram within the α−Dplane for different values of τwhere damped spin wave\nsolutions are stable. Even for zero deterministic Gilbert d amping the magnons offer a finite life-\ntime. We detect a parameter range wherethe deterministic an d the stochastic damping mechanism\nare able to compensate each other leading to undamped spin-w aves. The onset is characterized by\na critical value of the correlation time. An enhancement of τleads to an increase of the oscillations\nof the correlation function.\nPACS numbers: 75.10.Hk, 05.40.-a, 75.30.Ds,72.70.+m,76.60.Es\n∗thomas.bose@physik.uni-halle.de; steffen.trimper@physik.uni-halle.de\n1I. INTRODUCTION\nMagnetism can be generally characterized and analyzed on different length and time scales.\nThe description of fluctuations of the magnetization, the occurre nce of damped spin waves\nand the influence of additional stochastic forces are successfully performed on a mesoscopic\nscale where the spin variables are represented by a continuous spa tio-temporal variable [1].\nIn this case a well established approach isbased uponthe Landau-L ifshitz equation [2] which\ndescribes the precession motion of the magnetization in an effective magnetic field. This\nfield consists of a superposition of an external field and internal fie lds, produced by the in-\nteracting magnetic moments. The latter one is strongly influenced b y the isotropic exchange\ninteraction and the magnetocrystalline anisotropy, for a recent r eview see [3]. The studies\nusing this frame are concentrated on different dynamical aspects as the switching behav-\nior of magnetic nanoparticles which can be controlled by external tim e-dependent magnetic\nfields [4] and spin-polarized electric currents [5, 6]. Such a current- induced spin transfer\nallows the manipulation of magnetic nanodevices. Recently, it has bee n demonstrated that\nan electric current, flowing through a magnetic bilayer, can induce a coupling between the\nlayers [7]. Likewise, such a current can also cause the motion of magn etic domain walls in\na nanowire [8]. Another aspect is the dynamical response of ferrom agnetic nanoparticles\nas probed by ferromagnetic resonance, studied in [9]. In describing all this more complex\nbehavior of magnetic systems, the Landau-Lifshitz equation has t o be extended by the in-\nclusion of dissipative processes. A damping term is introduced pheno menologically in such a\nmanner, that the magnitude of the magnetization /vectorSis preserved at any time. Furthermore,\nthe magnetization should align with the effective field in the long time limit. A realization\nis given by [2]\n∂S\n∂t=−γ[S×Beff]−ε[S×(S×Beff)]. (1)\nThe quantities γandεare the gyromagnetic ratio and the damping parameter, respectiv ely.\nAn alternative equation for the magnetization dynamics had been pr oposed by Gilbert [10].\nThe Gilbert equation yields an implicit form of the evolution of the magne tization. A com-\nbination of both equations, called Landau-Lifshitz-Gilbert equation (LLG) will be used as\nthe basic relation for our studies, see Eq. (2). The origin of the dam ping term as a non-\nrelativistic expansion of the Dirac equation has been discussed in [11 ] and a generalization\nof the LLG for conducting ferromagnetics is offered in [12]. The form of the damping seems\n2to be quite general as it has been demonstrated in [13] using symmet ry arguments for fer-\nroelectric systems.\nAs a new aspect let us focus on the influence of stochastic fields. Th e interplay between\ncurrent and magnetic fluctuations and dissipation has been studied recently in [14]. Via the\nspin-transfer torque, spin-current noise causes a significant en hancement of the magnetiza-\ntion fluctuations. Such a spin polarized current may transfer mome ntum to a magnet which\nleads to a spin-torque phenomenon. The shot noise associated with the current gives rise to\na stochastic force [15]. In our paper we discuss the interplay betwe en different dissipation\nmechanism, namely the inherent deterministic damping in Eq. (1) and t he stochastic mag-\nnetic field originated for instance by defect configurations giving ris e to a different coupling\nstrength between the magnetic moments. Assuming further, tha t the stochastic magnetic\nfield is characterized by a finite correlation time, the system offers m emory effects which\nmight lead to a decoherent spin precession. To that aim we analyze a f erromagnet in the\nclassical limit, i.e., the magnetic order is referred to single magnetic at oms which occupy\nequivalent crystal positions, and the mean values of their spins exh ibit a parallel orientation.\nThe last one is caused by the isotropic exchange interaction which will be here supplemented\nby a magneto-crystalline anisotropy that defines the direction of t he preferred orientation.\nEspecially, we discuss the influence of an uniaxial anisotropy. The co upling between differ-\nent dissipation mechanisms, mentioned above, leads to pronounced correlations, which are\ndiscussed below. Due to the multiplicative coupling of the stochastic fi eld and the finite\ncorrelation time the calculation of the spin-spin correlation function is more complicated.\nTo that aim we have to derive an equivalent evolution equation for the joint probability\ndistribution function. Within the small correlation time limit this approa ch can be fulfilled\nin an analytical manner. Our analysis is related to a recent paper [16] in which likewise the\nstochastic dynamics of the magnetization in ferromagnetic nanopa rticles has been studied.\nFurther, we refer also to a recent paper [17] where the mean first passage time and the\nrelaxation of magnetic moments has been analyzed. Different to tho se papers our approach\nis concentrated on the correlation effects in stochastic system wit h colored noise.\nOur paper is organized as follows: In Sec.II we discuss the LLG and ch aracterize the ad-\nditional stochastic field. The equations for the single and the two pa rticle joint probability\ndistribution are derived in Sec.III. Using these functions we obtain t he mean value of the\nspin wave variable and the spin-spin correlation function. The phase diagram, based on the\n3stability analysis, is presented in Sec.IV. In Sec.V we finish with some co nclusions.\nII. MODEL\nIn order to develop a stochastic model for the spin dynamics in ferr omagnetic systems let\nus first consider the deterministic part of the equation of motion. W e focus on a description\nbased upon the level of Landau-Lifshitz phenomenology [2], for a r ecent review see [3]. To\nfollow this line we consider a high spin systems in a ferromagnet sufficien tly below the\nCurie temperature. In that regime the dynamics of the magnet are dominated by transverse\nfluctuationsofthespatio-temporalvaryinglocalmagnetization. Theweakexcitations, called\nspin waves or magnons, are determined by a dispersion relation, the wavelength of which\nshould be large compared to the lattice constant a, i.e., the relation q·a≪1 is presumed to\nbe satisfied, where qis the wavenumber. In this limit the direction of the spin varies slowly\nwhile its magnitude |S|=msremains constant in time. A proper description for such a\nsituation is achieved by applying the Landau-Lifshitz-Gilbert equatio n (LLG) [4, 10, 18].\nThe spin variable is represented by S=msˆ n, whereˆ n(r,t) is a continuous variable which\ncharacterizes the local orientation of the magnetic moment. The e volution equation for that\nlocal orientation reads\n∂ˆ n\n∂t=−γ\n1+α2ˆ n×[Beff+α[ˆ n×Beff]]. (2)\nThe quantities γandαare the gyromagnetic ratio and the dimensionless Gilbert damping\nparameter, respectively, where αis related to εintroduced in Eq. (1). Beffis the effective\nmagnetic field that drives the motion of the spin density. Generally, it consists of an internal\npart originated by the interaction of the spins and an external field . This effective field is\nrelated to the Hamiltonian of the system by functional variation with respect to ˆ n\nBeff=−m−1\nsδH\nδˆ n. (3)\nIn absence of an external field the Hamiltonian can be expressed as [19, 20]\nH=/integraldisplay\nd3r{wex+wan},with\nwex=1\n2msκ(∇ˆ n)2andwan=1\n2msΓ sin2θ.(4)\nThereby, the constants κand Γ denote the exchange energy density and the magneto-\ncrystalline anisotropy energy density. To be more precise, κ∝Ja2,Jbeing the coupling\n4strength that measures theinteraction between nearest neighb ors inthe isotropic Heisenberg\nmodel [21]. Once again ais thelattice constant. Notice that the formof theexchange ener gy\nin the Hamiltonian (4) arises from the Heisenberg model in the classica l limit. The quantity\nθrepresents the angle between ˆ nand the anisotropy axis ˆν= (0,0,1), where ˆνpoints\nin the direction of the easy axis in the ground state in the case of zer o applied external\nfield. Thus, the constant Γ >0 characterizes anisotropy as a consequence of relativistic\ninteractions (spin-orbital and dipole-dipole ones [20]). In deriving Eq . (4) we have used\nˆ n2= 1. Although it is more conventional to introduce the angular coord inates (θ,Φ) [2, 4],\nwe find it more appropriate to use Cartesian coordinates. To proce ed, we divide the vector\nˆ ninto a static and a dynamic part designated by µandϕ, respectively. In the linearized\nspin wave approach let us make the ansatz\nˆ n(r,t) =µ(r)+ϕ(r,t) =µˆν+ϕ, µ= const., (5)\nwhereˆ n2= 1 is still valid. The effective field can now be obtained from Eqs. (3) an d (4).\nThis yields\nBeff=κ∇2ϕ−Γϕ′;ϕ′= (ϕ1,ϕ2,0). (6)\nEq. (2) together with Eqs. (3) and (4) represent the determinist ic model for a classical\nferromagnet. In order to extent the model let us supplement the effective magnetic field in\nEq. (6) by a stochastic component yielding an effective random field Beff=Beff+η(t). The\nstochastic process η(t) is assumed to be Gaussian distributed with zero mean and obeying\na colored correlation function\n˜χij(t,t′) =∝an}b∇acketle{tηi(t)ηj(t′)∝an}b∇acket∇i}ht=˜Dij\n˜τijexp/bracketleftbigg\n−|t−t′|\n˜τij/bracketrightbigg\n. (7)\nHere,˜Dijand ˜τijare the noise strength and the finite correlation time of the noise η.\nDue to the coupling of the effective field to the spin orientation ˆ nthe stochastic process\nis a multiplicative one. Microscopically, such a random process might be originated by\na fluctuating coupling strength for instance. The situation associa ted with our model is\nillustrated in Fig. 1 and can be understood as follows: The stochastic vector fieldη(t) is able\nto change the orientation of the localized moment at different times. Therefore, fixed phase\nrelations between adjacent spins might be destroyed. Moreover, theη(tk) are interrelated\ndue to the finite correlation time τ. The anisotropy axis defines the preferred orientation of\nthe mean value of magnetization. Due to the inclusion of η(t) the deterministic Eq. (2) is\n5xyz\nanisotropy axis ˆν\nexchange ∝Jaη(t1)η(t2)η(t3)random field at\ndifferent times ti\nFIG. 1. Part of a ferromagnetic domain influenced by stochast ic forces for the example of cubic\nsymmetry with lattice constant a. The black spin in the center only interacts with its nearest\nneighbors (green), where Jis a measure for the exchange integral.\ntransformed into the stochastic LLG. Using Eq. (5) it follows\n∂ϕ\n∂t=−γ\n1+α2(µ+ϕ)×[Beff+α[(µ+ϕ)×Beff]]. (8)\nThe random magnetic field is defined by\nBeff=κ∇2ϕ−Γϕ′+η(t), (9)\nwhereϕ′is given in Eq. (6). With regard to the following procedure we suppose the random\nfield to be solely generated dynamically, i.e., ˆ n×η(t) =ϕ×η(t). So far, the dynamics\nof our model (Eqs. (8) and (9)) are reflected by a nonlinear, stoc hastic partial differential\nequation (PDE). Using Fourier transformation, i.e., ψ(q,t) =F{ϕ(r,t)}and introducing\nthe following dimensionless quantities\nβ= (l0q)2+1, l2\n0=κ\nΓ, ω=γΓ,¯t=ωt ,λ(t) =η(t)\nΓ,(10)\nthe components ψi(q,t) fulfill the equation\nd\ndtψi(q,t) = Ωi(ψ(q,t))+Λ ij(ψ(q,t))λj(t). (11)\n6The quantity l0is the characteristic magnetic length [22]. The vector Ωand the matrix Λ\nare given by\nΩ=ξµβ\n−(αµψ1+ψ2)\nψ1−αµψ2\n0\n, ξ=1\n1+α2, (12)\nand\nΛ =ξ\nαµψ3ψ3−(ψ2+αµψ1)\n−ψ3αµψ3ψ1−αµψ2\nψ2−ψ1 0\n. (13)\nFor convenience we have substituted ¯t→tagain. The statistical properties of λ(t) are\nexpressed as ∝an}b∇acketle{tλ(t)∝an}b∇acket∇i}ht= 0 and\nχkl(t,t′) =∝an}b∇acketle{tλk(t)λl(t′)∝an}b∇acket∇i}ht=Dkl\nτklδklexp/bracketleftbigg\n−|t−t′|\nτkl/bracketrightbigg\nτkl→0− −− →2Dklδklδ(t−t′).(14)\nIncidentally, in the limit τ→0 the usual white noise properties are recovered. We empha-\nsize that although we regard the long-wavelength limit ( a·q≪1), wave vectors for which\nl0·q≫1 (in Eq. (10)) can also occur [22]. But this case is not discussed in the present\npaper and will be the content of future work. Whereas, in what follo ws we restrict our\nconsiderations to the case q→0 so that, actually, l0·q≪1 is fulfilled. Hence, we can set\nβ= 1 approximately in Eq. (10). Due to the anisotropy the spin wave dis persion relation\noffers a gap at q= 0. Owing to this fact ψis studied at zero wave vector. For this situation\nthe assumption of a space-independent stochastic force ηi(t), compare Eq. (7), is reasonable.\nFor non-zero wave vector the noise field should be a spatiotempora l fieldηi((r,t). Because\nour model is based on a short range interaction we expect that the corresponding noise\ncorrelation function is δ-correlated, i.e. instead of (14) we have\nχkl(r,t;r′,t′) =Dkl\nτklδklexp/bracketleftbigg\n−|t−t′|\nτkl/bracketrightbigg\n2Mδ(r−r′),\nwhereMis the strength of the spatial correlation. Using this relation we are able to study\nalso the case of small qwhich satisfies l0·q≪1. In the present paper we concentrate on\nthe case of zero wave vector q= 0.\n7III. CORRELATION FUNCTIONS\nIn the present section let us discuss the statistical behavior of th e basic Eqs. (11)-(14). They\ndescribe a non-stationary, non-Markovian process attributed t o the finite correlation time.\nDue to their common origin both characteristics can not be analyzed separately. In the\nlimitτ→0, Eq. (11) defines a Markovian process which provides also station arity by an\nappropriate choice of initial conditions [23]. However, the present s tudy is focused on the\neffectofnonzerocorrelationtimes. Tothatpurposeweneedapro perprobabilitydistribution\nfunction which reflects the stochastic process defined by Eqs. (1 1)-(14). In deriving the\nrelevant joint probability distribution function we follow the line given in [24], where the\ndetailedcalculationshadbeencarriedout, seealsothereferences citedtherein. Inparticular,\nit has been underlined in those papers that in order to calculate corr elation functions of type\n∝an}b∇acketle{tψi(t)ψj(t′)∝an}b∇acket∇i}hta single probability distribution function P(ψ,t) is not sufficient. Instead of\nthat one needs a joint probability distribution of the form P(ψ,t;ψ′,t′). Before proceeding\nlet us shortly summarize the main steps to get the joint probability dis tribution function.\nTo simplify the calculation we assume τkl=τδklandDkl=Dδkl. Notice that our system\nhas no ergodic properties what would directly allow us to relate the st ochastic interferences\nwith temperature fluctuations by means of a fluctuation-dissipatio n theorem. Based on\nEq. (11) the appropriate joint probability distribution is defined by [2 4, 25], for a more\ngeneral discussion compare also [26]:\nP(ψ,t;ψ′,t′) =∝an}b∇acketle{tδ(ψ(t)−ψ)δ(ψ(t′)−ψ′)∝an}b∇acket∇i}ht. (15)\nHere the average is performed over all realizations of the stochas tic process. In defining the\njoint probability distribution function we follow the convention to indic ate the stochastic\nprocess by the function ψ(t) whereas the quantity without arguments ψstands for the\nspecial values of the stochastic variable. These values are even re lalized with the probaility\nP(ψ,t;ψ′,t′). The equation of motion for this probability distribution reads acco rding to\n8[24]\n∂\n∂tP(ψ,t;ψ′,t′)\n=−∂\n∂ψit/integraldisplay\n0χjk(t,t1)/angbracketleftigg/bracketleftbiggδψi(t)\nδλk(t1)/bracketrightbigg\nψ(t)=ψ·δ(ψ(t)−ψ)δ(ψ(t′)−ψ′)/angbracketrightigg\ndt1\n−∂\n∂ψ′it′/integraldisplay\n0χjk(t,t1)/angbracketleftigg/bracketleftbiggδψi(t′)\nδλk(t1)/bracketrightbigg\nψ(t′)=ψ′·δ(ψ(t)−ψ)δ(ψ(t′)−ψ′)/angbracketrightigg\ndt1,(16)\nwhere Novikov’s theorem [27] has been applied. Expressions for the response functions\nδψi(t)/δλk(t1) andδψi(t′)/δλk(t1) can be found by formal integration of Eq. (11) and\niterating the formal solution. After a tedious but straightforwar d calculation including the\ncomputation of the response functions to lowest order in ( t−t1) and (t′−t1) and the\nevaluation of several correlation integrals referring to χklfrom Eq. (14), Eq. (16) can be\nrewritten in the limit of small correlation time τas\n∂\n∂tPs(ψ,t;ψ′,t′) =/braceleftbig\nL0(ψ,τ)\n+exp[−(t−t′)/τ]D∂\n∂ψiΛik(ψ)∂\n∂ψ′\nnΛnk(ψ′)/bracerightbigg\nPs(ψ,t;ψ′,t′).(17)\nThereby, transient terms and terms of the form ∝τexp[−(t−t′)/τ] (these terms would lead\nto terms of order τ2in Eq. (22)) have been neglected. The result is valid in the stationary\ncase characterized by t→ ∞andt′→ ∞but finites=t−t′. In Eq. (17) L0is the operator\nappearing in the equation for the single probability density. Following [2 4, 28] the operator\nreads\nL0(ψ,τ) =−∂\n∂ψiΩi(ψ)+∂\n∂ψiΛik(ψ)∂\n∂ψn/braceleftigg\nD/bracketleftbig\nΛnk(ψ)−τMnk(ψ)/bracketrightbig\n+D2τ/bracketleftbigg\nKnkm(ψ)∂\n∂ψlΛlm(ψ)+1\n2Λnm(ψ)∂\n∂ψlKlkm(ψ)/bracketrightbigg/bracerightigg\n,(18)\nwith\nMnk= Ωr∂Λnk\n∂ψr−Λrk∂Ωn\n∂ψr\nKnlk= Λrk∂Λnl\n∂ψr−∂Λnk\n∂ψrΛrl.(19)\nThe equation of motion for the expectation value ∝an}b∇acketle{tψi∝an}b∇acket∇i}htscan be evaluated from the single\nprobability distribution in the stationary state\n∂\n∂tPs(ψ,t) =L0Ps(ψ,t). (20)\n9One finds\nd\ndt∝an}b∇acketle{tψi(t)∝an}b∇acket∇i}hts=∝an}b∇acketle{tΩi∝an}b∇acket∇i}hts+D/angbracketleftbigg∂Λik\n∂ψn/parenleftbig\nΛnk−τMnk/parenrightbig/angbracketrightbigg\ns−D2τ/braceleftigg/angbracketleftbigg∂\n∂ψr/parenleftbigg∂Λik\n∂ψnKnkm/parenrightbigg\nΛrm/angbracketrightbigg\ns\n+1\n2/angbracketleftbigg∂\n∂ψr/parenleftbigg∂Λik\n∂ψnΛnm/parenrightbigg\nKrkm/angbracketrightbigg\ns/bracerightigg\n.(21)\nThe knowledge of the evolution equation of the joint probability distr ibutionP(ψ,t;ψ′,t′)\ndue to Eqs. (17) and (18) allows us to get the corresponding equat ion for the correlation\nfunctions. Following again [24], it results\nd\ndt∝an}b∇acketle{tψi(t)ψj(t′)∝an}b∇acket∇i}hts=∝an}b∇acketle{tΩi(ψ(t))ψj(t′)∝an}b∇acket∇i}hts+D/angbracketleftbigg/bracketleftbigg∂Λik\n∂ψn/parenleftbig\nΛnk−τMnk/parenrightbig/bracketrightbigg\ntψj(t′)/angbracketrightbigg\ns\n−D2τ/braceleftigg/angbracketleftbigg/bracketleftbigg∂\n∂ψr/parenleftbigg∂Λik\n∂ψnKnkm/parenrightbigg\nΛrm/bracketrightbigg\ntψj(t′)/angbracketrightbigg\ns\n+1\n2/angbracketleftbigg/bracketleftbigg∂\n∂ψr/parenleftbigg∂Λik\n∂ψnΛnm/parenrightbigg\nKrkm/bracketrightbigg\ntψj(t′)/angbracketrightbigg\ns/bracerightigg\n+Dexp/bracketleftbigg\n−t−t′\nτ/bracketrightbigg\n∝an}b∇acketle{tΛik(ψ(t))Λjk(ψ(t′))∝an}b∇acket∇i}hts,(22)\nwhere the symbol [ ...]tdenotes the quantity [ ...] at timet. As mentioned above the result\nis valid for t, t′→ ∞whiles=t−t′>0 remains finite. The quantities MnkandKklmare\ndefined in Eq. (19). The components Ω iand Λ ijare given in Eqs. (12) and (13). Performing\nthe summation over double-indices according to Eqs. (21) and (22) we obtain the evolution\nequations for the mean value and the correlation function\nd\ndt∝an}b∇acketle{tψi(t)∝an}b∇acket∇i}hts=Gik∝an}b∇acketle{tψk(t)∝an}b∇acket∇i}hts, (23)\nand\nd\ndsCij(s) =d\nds∝an}b∇acketle{tψi(t′+s)ψj(t′)∝an}b∇acket∇i}hts=Gik∝an}b∇acketle{tψk(t′+s)ψj(t′)∝an}b∇acket∇i}hts\n+Dexp/bracketleftig\n−s\nτ/bracketrightig\n∝an}b∇acketle{tΛik(ψ(t′+s))Λjk(ψ(t′))∝an}b∇acket∇i}hts.(24)\nNotice, that in the steady state one gets Cij(t,t′) =Cij(s) withs=t−t′. The matrix\ncomponents of Gikare given by\nGik=\n−A1A20\n−A2−A10\n0 0 −A3\n, (25)\n10where\nA1=−D2τ(6µ2α2−1)ξ4+2µ2αDτξ3−D(µ2α2−2)ξ2+µ2αξ\nA2=1\n2µαD2τ/parenleftbig\n11−3µ2α2/parenrightbig\nξ4+µDτ/parenleftbig\nµ2α2−1/parenrightbig\nξ3+3µDαξ2−µξ\nA3= +D2τ/parenleftbig\n3µ2α2+1/parenrightbig\nξ4−4µ2αDτξ3+2Dξ2,(26)\nandξis defined in Eq. (12). At this point let us stress that in the case t′= 0 the term\n∝exp[−(t−t′)/τ] on the rhs. in Eqs. (22) and (24), respectively, would vanish in the steady\nstate, i.e.\n∝an}b∇acketle{tψi(t′+s)ψj(t′)∝an}b∇acket∇i}hts∝ne}ationslash=∝an}b∇acketle{tψi(s)ψj(0)∝an}b∇acket∇i}hts.\nTheoccurrenceofsuchatermisastrongindicationforthenon-st ationarityofourmodel. An\nexplicit calculation shows, that in general this inequality holds for non -stationary processes\n[23].\nIV. RESULTS\nThe solution of Eq. (23) can be found by standard Greens function methods and Laplace\ntransformation. As the result we find\n∝an}b∇acketle{tψ(t)∝an}b∇acket∇i}hts=\ne−A1tcos(A2t)e−A1tsin(A2t) 0\n−e−A1tsin(A2t)e−A1tcos(A2t) 0\n0 0 e−A3t\n·∝an}b∇acketle{tψ0∝an}b∇acket∇i}hts, (27)\nwhere∝an}b∇acketle{tψ0∝an}b∇acket∇i}hts=∝an}b∇acketle{tψ(t= 0)∝an}b∇acket∇i}htsare the initial conditions. The parameters A1,A3andA2defined\nin Eqs. (26) play the roles of the magnon lifetime and the frequency o f the spin wave at\nzero wave vector, respectively. As can be seen in Eq. (26) all of th ese three parameters are\naffected by the correlation time τand the strength Dof the random force. Moreover, the\nGilbert damping parameter αinfluences the system as well. The solution of Eq. (24) for\nthe correlation function in case of t′= 0 is formal identical to that of Eq. (27). The more\ngeneral situation t′∝ne}ationslash= 0 allows no simple analytic solution and hence the behavior of the\ncorrelation function C(s) is studied numerically. In order to analyze the mean values and\nthe correlation function let us first examine the parameter range w here physical accessible\nsolutions exist. In the following we assume ∝an}b∇acketle{tψ1(0)∝an}b∇acket∇i}ht=∝an}b∇acketle{tψ2(0)∝an}b∇acket∇i}ht=∝an}b∇acketle{tψ0∝an}b∇acket∇i}htand∝an}b∇acketle{tψ3(0)∝an}b∇acket∇i}ht= 0, since\nthe solutions for ψ1(t) andψ2(t) on the one hand and ψ3(t) on the other hand are decoupled\n11in Eq. (27). Therefore, spin wave solutions only exists for non-zer o averages ∝an}b∇acketle{tψ1(t)∝an}b∇acket∇i}htand\n∝an}b∇acketle{tψ2(t)∝an}b∇acket∇i}ht. The existence of such non-trivial solutions are determined in depe ndence on the\nnoise parameters Dandτand the deterministic damping parameter α. Notice, that the\ndimensionless quantity D=˜D/Γ, i.e.,Dis the ratio between the strength of the correlation\nfunction (Eq. (7)) and the anisotropy field in the original units. The stability of spin wave\nsolutions is guaranteed for positive parameters A1andA3. According to Eqs. (26) the\nphase diagrams are depicted in Fig. 2 within the α−Dplane for different values of the\ncorrelation time τ. The separatrix between stable and unstable regions is determined by\nthe condition A1= 0. The second condition A3= 0 is irrelevant due to the imposed initial\nconditions. As the result of the stability analysis the phase space dia gram is subdivided into\nfour regions where region IV does not exist in case of τ= 0, see Fig. 2(a). For generality,\nwe take into account both positive and negative values of Dindicating correlations and\nanti-correlations of the stochastic field. Damped spin waves are ob served in the areas I and\nIV, whereas the sectors II and III reveal non-accessible solutio ns. In those regions the spin\nwave amplitude, proportional to exp[ −A1t], tends to infinity which should not be realized,\ncompare Figs. 2(b)-2(d). Actually, a reasonable behavior is obser ved in regions I and IV. As\nvisible from Fig. 2 damped spin waves will always emerge for D>0 even in the limit of zero\ndamping parameter αand vanishing correlation time τ. This behavior is shown in Fig. 3,\nwhere theevolution of ∝an}b∇acketle{tψ1(t)∝an}b∇acket∇i}htis depicted for different valuesof α. As canbeseen in Fig.2(a)\nthesolutionfor D<0isunlimitedandconsequently, itshouldbeexcludedfurther. Contr ary\nto this situation, additional solutions will be developed in region IV in ca se ofτ >0 and\nsimultaneously α= 0, see Figs. 2(b)-2(d). Thereby the size of area IV grows with inc reasing\nτ. Likewise, the extent of region I decreases for an enhanced τ. However, in the limit of\nD= 0 and consequently for τ= 0, too, only damped spin waves are observed. Immediately\non the separations line undamped periodic solutions will evolve, compa re the sub-figures in\nFig. 2. This remarkable effect can be traced back to the interplay be tween the deterministic\ndamping and the stochastic forces. Both damping mechanism are co mpensated mutually\nwhich reminds of a kind of resonance phenomenon. The difference to conventional resonance\nbehavior consists of the compensation of the inherent determinist ic Gilbert damping and the\nstochastic one originated from the random field. This statement is e mphasized by the fact\nthat undamped periodic solutions do not develop in the absence of st ochastic interferences,\ni.e.,D= 0. The situation might be interpreted physically as follows: the requ ired energy\n12(a)τ= 0 (b)τ= 0.1\n(c)τ= 1 (d)τ= 10\nFIG. 2.α−Dplane for fixed magnetization µ= 0.9 and different values of τ.\nthat enables the system to sustain the deterministic damping mecha nisms is delivered by\nthe stochastic influences due to the interaction with the environme nt. To be more precise, in\ngeneral, the Gilbert damping enforces the coherent alignment of th e spin density along the\nprecession axis. Contrary, the random field supports the dephas ing of the orientation of the\nclassical spins. Surprisingly, the model predicts the existence of a critical value τ=τc≥0\n13FIG. 3. Evolution of the mean value ∝an}b∇acketle{tψ1(t)∝an}b∇acket∇i}ht, withµ= 0.9,D= 0.1 andτ= 0.αvaries from 0\n(dash-dotted line), 0 .05 (solid line), 0 .5 (dotted line) and 1 (dashed line).\ndepending on αandDwhich determines the onset of undamped periodic solutions. Notice,\nthat negative values of τcare excluded. The critical value is\nτc=−[µ2(α3−Dα2+α)+2D](1+α2)2\n2Dµ2(α3−3Dα2+α)+D2. (28)\nHence, this result could imply the possibility of the cancellation of both damping processes.\nExamples according to the damped and the periodic case are displaye d in Fig. 4. An increas-\ningτfavors the damping process as it is visible in Fig. 4(a). Based on estima tions obtained\nfor ferromagnetic materials [29] and references therein, the Gilbe rt damping parameter can\nrange between 0 .04<α<0.22 in thin magnetic films, whereas the bulk value for Co takes\nαb≈0.005. The phase space diagram in Fig. 2 offers periodic solutions only fo r values of\nαlarger than those known from experiments. Therefore such perio dic solutions seem to be\nhard to see experimentally. We proceed further by analyzing the be havior of the correlation\nfunction by numerical computation of the solution of Eq. (24) with E qs. (25) and (26). As\ninitial values we choose Cik(t=t′,t′) =Cik(s= 0) =C0for every combination i,k={1,2,3}.\nThe results are depicted in Figs. 5 and 6. Inspecting Figs. 5(a)-5(c ) one recognizes that an\nenhancement of the correlation time τleads to an increase of the oscillations within the\ncorrelation functions C1k,k={1,2,3}. Moreover, Fig. 5(d) reveals that the oscillatory\n14(a) (b)\nFIG. 4. Evolution of the mean values ∝an}b∇acketle{tψ1,2(t)∝an}b∇acket∇i}ht, withµ= 0.9. (a):D= 0.1,α= 0.005 andτvaries\nfrom 10 (solid line), 1 (dotted line) and 0 (dash-dotted line ). (b):D= 2,α= 1 andτ=τc≈1.79\n(Eq. (28)). The solid line represents ∝an}b∇acketle{tψ1∝an}b∇acket∇i}htand the dash-dotted line is ∝an}b∇acketle{tψ2∝an}b∇acket∇i}ht.\nbehavior of C31seems to be suppressed. Obviously, the decay of the correlation f unction is\nenhanced if τgrowths up. The pure periodic case for τ=τc, corresponding to Fig. 4(b),\nis depicted in Fig. 6. Exemplary, C12andC31are illustrated. The behavior of the latter is\nsimilar to the damped case, displayed in Fig. 5(d), unless slight oscillatio ns occur. However,\nif one compares the form of C12in Fig. 5(b) and Fig. 6 the differences are obvious. The am-\nplitude of the correlation function for the undamped case grows to the fourfold magnitude\nin comparison with C0, whereas the damped correlation function approaches zero. Fur ther,\na periodic behavior is shown in Fig. 6, and therefore the correlation w ill oscillate about zero\nbut never vanish for all s=t−t′>0.\nV. CONCLUSIONS\nIn this paper we have analyzed the dynamics of a classical spin model with uniaxial\nanisotropy. Aside from the deterministic damping due to the Landau -Lifshitz-Gilbert\nequation the system is subjected to an additional dissipation proce ss by the inclusion of a\nstochastic field with colored noise. Both dissipation processes are a ble to compete leading to\n15(a) (b)\n(c) (d)\nFIG. 5. Correlation functions Cik(s) forµ= 0.9,D= 0.1 andα= 0.005.τtakes 0 (dotted line),\n1 (solid line) and 10 (dash-dotted line).\na more complex behavior. To study this one we derive an equation for the joint probability\ndistribution which allows us to find the corresponding spin-spin-corr elation function. This\nprogram can be fulfilled analytically and numerically in the spin wave appr oach and the\nsmall correlation time limit. Based on the mean value for the spin wave c omponent and\n16FIG. 6. Correlation functions Cik(s) forτ=τc≈1.79 (Eq. (28)), µ= 0.9,D= 2 andα= 1. The\ndotted line represents C12and the solid line is C31.\nthe correlation function we discuss the stability of the system in ter ms of the stochastic\nparameters, namely the strength of the correlated noise Dand the finite correlation time\nτ, as well as the deterministic Gilbert damping parameter α. The phase diagram in the\nα−Dplane offers that the system develops stable and unstable spin wave solutions due to\nthe interplay between the stochastic and the deterministic damping mechanism. So stable\nsolutions evolve for arbitrary positive Dand moderate values of the Gilbert damping α.\nFurther, we find that also the finite correlation time of the stochas tic field influences the\nevolution of the spin waves. In particular, the model reveals for fix edDandαa critical\nvalueτcwhich characterizes the occurrence of undamped spin waves. The different situa-\ntions are depicted in Fig. 2. Moreover, the correlation time τaffects the damped spin wave\nwhich can be observed in regions I and IV in the phase diagram. If the parameters Dand\nαchanges within these regions, an increasing τleads to an enhancement of the spin wave\ndamping, cf. Fig. 4(a). The influence of τon the correlation functions is similar as shown\nin Figs. 5(a)-5(c). The study could be extended by the inclusion of fi nite wave vectors and\nusing an approach beyond the spin wave approximation.\n17ACKNOWLEDGMENTS\nOne of us (T.B.) is grateful to the Research Network ’Nanostructu red Materials’, which is\nsupported by the Saxony-Anhalt State, Germany.\n18[1] L. D. Landau, E. Lifshitz, and L. Pitaevskii, Electrodynamics of continuous media (Pergamon\nPress, Oxford, 1989).\n[2] L. Landau and E. Lifshitz, Zeitschr. d. Sowj. 8, 153 (1935).\n[3] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halpe rin, Rev. Mod. Phys. 77, 1375\n(2005).\n[4] A. Sukhov and J. Berakdar, J. Phys. - Cond. 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Daniel and M. Lakshmanan, Physica A 120, 125 (1983).\n[19] M. Lakshmanan and K. Nakamura, Phys. Rev. Lett. 53, 2497 (1984).\n[20] V. G. Bar’Yakhtar, M. V. Chetkin, B. A. Ivanov, andS. N. G adetskii, Dynamics of Topological\nMagnetic Solitons: Experiment and Theory (Springer Tracts i n Modern Physics) (Springer,\n1994).\n[21] M. Lakshmanan, T. W. Ruijgrok, and C. J. Thompson, Physi ca A84, 577 (1976).\n[22] A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev, Phys. Rep .194, 117 (1990).\n[23] A. Hernandez-Machado and M. San Miguel, J. Math. Phys. 25, 1066 (1984).\n[24] A. Hernandez-Machado, J. M. Sancho, M. San Miguel, and L . Pesquera, Zeitschr. f. Phys. B\n52, 335 (1983).\n19[25] N. G. van Kampen, Braz. J. Phys. 28, 90 (1998).\n[26] N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amster-\ndam, 1981).\n[27] E. A. Novikov, Sov. Phys. JETP 20, 1290 (1965).\n[28] H. Dekker, Phys. Lett. A 90, 26 (1982).\n[29] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev . Lett.88, 117601 (2002).\n20"    },    {        "title": "2310.18878v1.Asymptotic_profiles_for_the_Cauchy_problem_of_damped_beam_equation_with_two_variable_coefficients_and_derivative_nonlinearity.pdf",        "content": "arXiv:2310.18878v1  [math.AP]  29 Oct 2023ASYMPTOTIC PROFILES FOR THE CAUCHY PROBLEM OF\nDAMPED BEAM EQUATION WITH TWO VARIABLE\nCOEFFICIENTS AND DERIVATIVE NONLINEARITY\nMOHAMED ALI HAMZA1, YUTA WAKASUGI2∗AND SHUJI YOSHIKAWA3\n1Basic Sciences Department, Deanship of Preparatory Year an d Supporting\nStudies, P. O. Box 1982, Imam Abdulrahman Bin Faisal Univers ity, Dammam,\nKSA.\n2Laboratory of Mathematics, Graduate School of Advanced Sci ence and\nEngineering, Hiroshima University, Higashi-Hiroshima, 7 39-8527, Japan\n3Division of Mathematical Sciences, Faculty of Science and T echnology, Oita\nUniversity, Oita, 870-1192, Japan\nAbstract. In this article we investigate the asymptotic profile of solu tions for\nthe Cauchy problem of the nonlinear damped beam equation wit h two variable\ncoefficients:\n∂2\ntu+b(t)∂tu−a(t)∂2\nxu+∂4\nxu=∂x(N(∂xu)).\nIn the authors’ previous article [ 17], the asymptotic profile of solutions for\nlinearized problem ( N≡0) was classified depending on the assumptions for\nthe coefficients a(t) andb(t) and proved the asymptotic behavior in effective\ndamping cases. We heregive the conditions ofthe coefficients andthe nonlinear\nterm in order that the solution behaves as the solution for th e heat equation:\nb(t)∂tu−a(t)∂2\nxu= 0 asymptotically as t→ ∞.\n1.Introduction\nWe study the Cauchy problem of nonlinear damped beam equation\n/braceleftBigg\n∂2\ntu+b(t)∂tu−a(t)∂2\nxu+∂4\nxu=∂x(N(∂xu)), t∈(0,∞),x∈R,\nu(0,x) =u0(x), ∂tu(0,x) =u1(x), x ∈R,(1.1)\nwhereu=u(t,x)isareal-valuedunknown, a(t) andb(t)aregivenpositivefunctions\noft,N(∂xu) denotes the nonlinear function, and u0andu1are given initial data.\nE-mail address :mahamza@iau.edu.sa, wakasugi@hiroshima-u.ac.jp,\nyoshikawa@oita-u.ac.jp .\nDate: October 31, 2023.\nKey words and phrases. Nonlinear damped beam equations; asymptotic behavior; glo bal ex-\nistence; variable coefficients.\n2010 Mathematics Subject Classification. 35G25; 35B40; 35A 01\n∗Corresponding author\n12 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA\nBefore giving more precise assumptions for a(t),b(t) andNand our result,\nwe first mention the physical background and the mathematical mo tivations of\nthe problem. The equation ( 1) corresponds to the so-called Falk model under\nisothermal assumption with the damping term. The Falk model is one o f the\nmodels for a thermoelastic deformation with austenite-martensite phase transitions\non shape memory alloys:\n\n\n∂2\ntu+∂4\nxu=∂x/braceleftbig\n(θ−θc)∂xu−(∂xu)3+(∂xu)5/bracerightbig\n,\n∂tθ−∂2\nxθ=θ∂xu∂t∂xu, t ∈(0,∞),x∈R,\nu(0,x) =u0(x), ∂tu(0,x) =u1(x), θ(0,x) =θ0(x), x ∈R,\nwhereuandθare the displacement and the absolute temperature, respectively ,\nandθcis a positive constant representing the critical temperature for t he phase\ntransition. If we assume the temperature arecontrollableand unif ormly distributed\nwith respect to the space, that is, θis given function uniform in xsuch asθ−θc=\na(t) and setN(ε) =ε5−ε3, then the problem ( 1) is surely derived. Our interest\ndirects to the behavior of solution around the initial temperature θ0is closed to\nthe critical temperature θc. Indeed, the Lyapunov stability for the solution of\nthe Falk model is shown in [ 11], which claims that the temperature tends to the\nfunction uniformly distributed in xin the bounded domain case. For more precise\ninformation of the Falk model of shape memory alloys, we refer the r eader to\nChapter 5 in [ 2]. We are also motivated by the extensible beam equation proposed\nby Woinovsky-Krieger [ 16]:\n∂2\ntu−/parenleftbigg/integraldisplay\nR|∂xu|2dx/parenrightbigg\n∂2\nxu+∂4\nxu= 0.\nIn [1], the model with the damping term was proposed and the stability res ult was\nshown. The problem ( 1) corresponds to the nonlinear generalization for the equa-\ntion. As the observation similar to the Kirchhoff equation, the lineariz ed problem\nsubstituting the given function a(t) into the nonlocal term was also studied by e.g.\n[4], [10], [17] and [6].\nNext, let us explain the mathematical background of our problem. I t is well-\nknown that the solution of the Cauchy problem for the damped wave equation\n∂2\ntu+∂tu−∂2\nxu= 0 behaves as the solution for the heat equation ∂tu−∂2\nxu= 0\nasymptotically as t→ ∞(see e.g. [ 7]). Roughly speaking, this implies that ∂2\ntu\ndecays faster than ∂tuast→ ∞. From the same observation, the solution for\nthe beam equation ∂2\ntu+∂tu−∂2\nxu+∂4\nxu= 0 behaves as the solution for the\nheat equation ∂tu−∂2\nxu= 0 asymptotically as t→ ∞, because∂4\nxudecays faster\nthan∂2\nxu. The above observation induces the investigation of the solution fo r the\nequation with time variable coefficient: ∂2\ntu+b(t)∂tu−∂2\nxu= 0 withb(t)∼(1+t)β.\nThe precise analysis implies that the solution behaves as the solution f or the heat\nequationb(t)∂tu−∂2\nxu= 0 whenβ <−1, and on the other hand, that the solution\nbehaves as the solution for the wave equation ∂2\ntu−∂2\nxu= 0 when −1< β <1\n(see e.g. [ 8], [9], [14] and [15]). Correspondingly, the authors in [ 17] studied the\nasymptotic behavior of the solution for the linearized problem of ( 1)\n∂2\ntu+b(t)∂tu−a(t)∂2\nxu+∂4\nxu= 0.NONLINEAR DAMPED BEAM EQUATION 3\nAs in Figure 1, we divide the two-dimensional regions Ω jfor (α,β) (j= 1,2,3,4,5)\nby\nΩ1:=/braceleftbig\n(α,β)∈R2| −1<β <min{α+1,2α+1}/bracerightbig\n,\nΩ2:=/braceleftbig\n(α,β)∈R2|max{−1,2α+1}<β <1/bracerightbig\n,\nΩ3:=/braceleftbig\n(α,β)∈R2|β <−1<α/bracerightbig\n,\nΩ4:=/braceleftbig\n(α,β)∈R2|β <−1,α<−1/bracerightbig\n,\nΩ5:=/braceleftbig\n(α,β)∈R2|max{1,α+1}<β/bracerightbig\n.\nαβ\n01\n−1−1\n2\nβ=−1β=α+1\nβ= 2α+1Ω1Ω2\nΩ3Ω4Ω5\nFigure 1.\nBy a scaling argument, the result gives the conjecture for the clas sification of\nthe asymptotic behavior of the solution:\n(1) In (α,β)∈Ω1,u(t) behaves as the solution for b(t)ut−a(t)uxx= 0.\n(2) In (α,β)∈Ω2,u(t) behaves as the solution for b(t)ut+uxxxx= 0.\n(3) In (α,β)∈Ω3,u(t) behaves as the solution for utt−a(t)uxx= 0.\n(4) In (α,β)∈Ω4,u(t) behaves as the solution for utt+uxxxx= 0.\n(5) In (α,β)∈Ω5,u(t) behaves as the solution for over damping case.\nAs a partial answer, the authors proved the effective damping cas es (1) and (2) in\n[17]. Here we shall give the result for the nonlinear problem ( 1) in the case (1).\nFrom now on, we shall give our main result. To state it precisely, we pu t the\nfollowing assumptions.\nAssumption (A) The coefficients a(t) andb(t) are smooth positive functions\nsatisfying\nC−1(1+t)α≤a(t)≤C(1+t)α, C−1(1+t)β≤b(t)≤C(1+t)β\nand\n|a′(t)| ≤C(1+t)α−1,|b′(t)| ≤C(1+t)β−1\nwith some constant C≥1 and the parameters α,β∈R. Moreover, we assume\n(α,β)∈Ω1:=/braceleftbig\n(α,β)∈R2| −1<β <min{α+1,2α+1}/bracerightbig\n.\nAssumption (N) The function N(∂xu) is a linear combination of ( ∂xu)2andp-th\norder terms with p≥3. More precisely, the function Nhas the form\nN(z) =µz2+˜N(z),4 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA\nwith someµ∈R, where ˜N∈C2(R) satisfies\n˜N(j)(0) = 0 and |˜N(j)(z)−˜N(j)(w)| ≤C(|z|+|w|)p−1−j|z−w|(z,w∈R)\nholds with some p≥3 forj= 0,1,2.\nA typical example of our nonlinearity is\n∂x(N(∂xu)) =∂x(∂xu)2+∂x/parenleftbig\n|∂xu|p−1∂xu/parenrightbig\nwithp≥3.\nRemark 1.1. The assumption (A) implies\n−β+1\nα−β+1<2<p.\nThis means that the nonlinearity is supercritical (see the a rgument in Section 4.1).\nWe further prepare the following notations. Let G=G(t,x) be the Gaussian,\nthat is,\nG(t,x) =1√\n4πtexp/parenleftbigg\n−x2\n4t/parenrightbigg\n.\nWe define\nr(t) =a(t)\nb(t)andR(t) =/integraldisplayt\n0r(τ)dτ.\nRemark that\nC−1(1+t)α−β+1≤R(t)≤C(1+t)α−β+1\nholds with some constant C≥1, thanks to the assumption (A).\nTheorem 1.2. Under the assumptions (A) and (N), there exists a constant ε0>0\nsuch that if (u0,u1)∈/parenleftbig\nH2,1(R)∩H3,0(R)/parenrightbig\n×/parenleftbig\nH0,1(R)∩H1,0(R)/parenrightbig\nand\n/bardblu0/bardblH2,1∩H3,0+/bardblu1/bardblH0,1∩H1,0≤ε0,\nthen there exists a unique solution\nu∈C([0,∞);H2,1(R)∩H3,0(R))∩C1([0,∞);H0,1(R)∩H1,0(R)).(1.2)\nMoreover, the solution uhas the asymptotic behavior\n/bardblu(t,·)−m∗G(R(t),·)/bardblL2≤C(R(t)+1)−1\n4−λ\n2(/bardblu0/bardblH2,1∩H3,0+/bardblu1/bardblH0,1∩H1,0)\nwith some constants C >0,m∗∈R, andλ>0.\nRemark 1.3. From the proof, λis taken to be arbitrary so that\n0<λ<min/braceleftbigg1\n2,2(β+1)\nα−β+1,2α−β+1\nα−β+1/bracerightbigg\n.\nThe proof is based on the method by Gallay and Raugel [ 5] using the self-similar\ntransformation and the standard energy method.\nThis paper is organized as follows. In Section 2, we rewrite the proble m through\nthe self-similar transformation. Thereafter, we show several en ergy estimates in\nSection 3, and give a priori estimates through the estimates for th e nonlinear terms\nin Section 4. In the appendix, we also give a lemma for energy identities which is\nfrequently used in the proof and the proof of local-in-time existenc e of solution for\nthe readers’ convenience.NONLINEAR DAMPED BEAM EQUATION 5\nAt the end of this section we prepare notation and several definitio ns used\nthroughout this paper. We denote by Ca positive constant, which may change\nfrom line to line. The symbol a(t)∼b(t) means that C−1b(t)≤a(t)≤Cb(t) holds\nfor some constant C≥1.Lp=Lp(R) stands for the usual Lebesgue space, and\nHk,m=Hk,m(R) fork∈Z≥0andm∈Ris the weighted Sobolev space defined by\nHk,m(R) =/braceleftBigg\nf∈L2(R);/bardblf/bardblHk,m=k/summationdisplay\nℓ=0/bardbl(1+|x|)m∂ℓ\nxf/bardblL2<∞/bracerightBigg\n.\n2.Scaling variables\nThelocalexistenceofthe solutioninthe class( 1.2) isstandard(seeAppendix B).\nThus, it suffices to show the a priori estimate. To prove it, following t he argument\nof Gallay and Raugel [ 5], we introduce the scaling variables\ns= log(R(t)+1), y=x/radicalbig\nR(t)+1,\nand definev=v(s,y) andw=w(s,y) by\nu(t,x) =1/radicalbig\nR(t)+1v/parenleftBigg\nlog(R(t)+1),x/radicalbig\nR(t)+1/parenrightBigg\n,\nut(t,x) =R′(t)\n(R(t)+1)3/2w/parenleftBigg\nlog(R(t)+1),x/radicalbig\nR(t)+1/parenrightBigg\n.\nThen, by a straightforward computation, the Cauchy problem ( 1) is rewritten as\n\n\nvs−y\n2vy−1\n2v=w,\nr2e−s\na/parenleftbigg\nws−y\n2wy−3\n2w/parenrightbigg\n+/parenleftbigg\n1+r′\na/parenrightbigg\nw=vyy−e−s\navyyyy+es\na∂y/parenleftbig\nN/parenleftbig\ne−svy/parenrightbig/parenrightbig\n,\nv(0,y) =v0(y), w(0,y) =w0(y),\n(2.1)\nwherev0(y) =u0(y) andw0(y) =1\nr(0)u1(y). Here, we also note that the functions\na,b,r,r′appearing the above precisely mean such as a(t(s)) =a(R−1(es−1)).\nRemark 2.1. WhenN(z) =|z|p−1z, the nonlinearity has a bound\n/vextendsingle/vextendsingle/vextendsingle/vextendsinglees\na(t(s))∂y/parenleftbig\nN(e−svy)/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Ce(−β+1\nα−β+1−p)s|vy|p−1|vyy|.\nSince−β+1\nα−β+1<2, the assumption (N) implies that the nonlinearity can be tre ated\nas remainder.\nTo investigate the asymptotic behavior of the solution of ( 2), we define\nm(s) :=/integraldisplay\nRv(s,y)dy.\nBy the first equation of ( 2) and the integration by parts, we have\nms(s) =d\ndsm(s) =/integraldisplay\nRvsdy=/integraldisplay\nR/parenleftbiggy\n2vy+1\n2v+w/parenrightbigg\ndy=/integraldisplay\nRwdy.6 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA\nWe also define\nφ(y) :=G(1,y) =1√\n4πexp/parenleftbigg\n−y2\n4/parenrightbigg\n,\nψ(y) :=φyy(y).\nUsing them, we decompose ( v,w) as\nv(s,y) =m(s)φ(y)+f(s,y),\nw(s,y) =ms(s)φ(y)+m(s)ψ(y)+g(s,y),(2.2)\nand we expect that the functions fandgdefined above can be treated as remain-\nders.\nBy a direct calculation, we obtain the following equation for m(s):\nLemma 2.2. We have\nr2e−s\na(mss−ms) =−/parenleftbigg\n1+r′\na/parenrightbigg\nms.\nFrom the above lemma and the straightforward computation, we ca n see that\n(f,g) satisfies the following equations.\n\n\nfs−y\n2fy−1\n2f=g,\nr2e−s\na/parenleftbigg\ngs−y\n2gy−3\n2g/parenrightbigg\n+/parenleftbigg\n1+r′\na/parenrightbigg\ng=fyy−e−s\nafyyyy+es\na∂y/parenleftbig\nN/parenleftbig\ne−svy/parenrightbig/parenrightbig\n+h,\n(2.3)\nwhere\nh=−r2e−s\na/parenleftbigg\n2msψ−y\n2mψy−3\n2mψ/parenrightbigg\n−r′\namψ−e−s\namψyy.(2.4)\nThey satisfy\n/integraldisplay\nRf(s,y)dy=/integraldisplay\nRg(s,y)dy=/integraldisplay\nRh(s,y)dy= 0. (2.5)\nTherefore, our goal is to give energy estimates of the solutions ( f,g) to the equation\n(2) under the condition ( 2).\n3.Energy estimates\nIn this section, we give energy estimates of ( f,g) defined by ( 2). We first prepare\nthe following general lemma for energy identities.\nLemma 3.1. Letl,m∈R,n∈N∪ {0}, and letc1(s),c2(s),c4(s)be smooth\nfunctions defined on [0,∞). We consider a system for two functions f=f(s,y)\nandg=g(s,y)given by\n\n\nfs−y\n2fy−lf=g,\nc1(s)/parenleftBig\ngs−y\n2gy−mg/parenrightBig\n+c2(s)g+g=fyy−c4(s)fyyyy+h(s,y)∈(0,∞)×R,\n(3.1)NONLINEAR DAMPED BEAM EQUATION 7\nwhereh=h(s,y)is a given smooth function belonging to C([0,∞);H0,n(R)). We\ndefine the energies\nE1(s) =1\n2/integraldisplay\nRy2n/parenleftbig\nf2\ny+c4(s)f2\nyy+c1(s)g2/parenrightbig\ndy,\nE2(s) =/integraldisplay\nRy2n/parenleftbigg1\n2f2+c1(s)fg/parenrightbigg\ndy.\nThen, we have\nd\ndsE1(s) =−/integraldisplay\nRy2ng2dy+/parenleftbigg\n−2n−1\n4+l/parenrightbigg/integraldisplay\nRy2nf2\nydy+/parenleftbigg\n−2n−3\n4+l/parenrightbigg\nc4(s)/integraldisplay\nRy2nf2\nyydy\n+/parenleftbigg\n−2n+1\n4+m/parenrightbigg\nc1(s)/integraldisplay\nRy2ng2dy−c2(s)/integraldisplay\nRy2ng2dy\n−2n/integraldisplay\nRy2n−1fygdy−2n(2n−1)c4(s)/integraldisplay\nRy2n−2fyygdy−4nc4(s)/integraldisplay\nRy2n−1fyygydy\n+c′\n4(s)\n2/integraldisplay\nRy2nf2\nyydy+c′\n1(s)\n2/integraldisplay\nRy2ng2dy+/integraldisplay\nRy2nghdy\nand\nd\ndsE2(s) =−/integraldisplay\nRy2nf2\nydy−c4(s)/integraldisplay\nRy2nf2\nyydy+/parenleftbigg\n−2n+1\n4+l/parenrightbigg/integraldisplay\nRy2nf2dy\n+c1(s)/integraldisplay\nRy2ng2dy+/parenleftbigg\n−2n+1\n2+l+m/parenrightbigg\nc1(s)/integraldisplay\nRy2nfgdy−c2(s)/integraldisplay\nRy2nfgdy\n−2n/integraldisplay\nRy2n−1ffydy−4nc4(s)/integraldisplay\nRy2n−1fyfyydy−2n(2n−1)c4(s)/integraldisplay\nRy2n−2ffyydy\n+c′\n1(s)/integraldisplay\nRy2nfgdy+/integraldisplay\nRy2nfhdy.\nThe casen= 0 is given by [ 17, Lemma 3.1]. We will prove a slightly more\ngeneral version of this lemma in Appendix A.\nTo bring out the decay property of the solutions ( f,g) to (2) from the condition\n(2), we define the auxiliary functions\nF(s,y) :=/integraldisplayy\n−∞f(s,z)dz, G(s,y) :=/integraldisplayy\n−∞g(s,z)dz, H(s,y) :=/integraldisplayy\n−∞h(s,z)dz.\nThen, bythefollowingHardyinequality, theconditions( 2)andf(s),g(s)∈H0,1(R)\nensureF(s),G(s)∈L2(R).\nLemma 3.2 (Hardy-type inequality [ 13, Lemma 3.9]) .Letf=f(y)∈H0,1(R)\nand satisfy/integraltext\nRf(y)dy= 0. LetF(y) =/integraltexty\n−∞f(z)dz. Then, we have\n/integraldisplay\nRF(y)2dy≤4/integraldisplay\nRy2f(y)2dy.8 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA\nFrom (2),FandGsatisfy the following system.\n\n\nFs−y\n2Fy=G,\nr2e−s\na/parenleftBig\nGs−y\n2Gy−G/parenrightBig\n+/parenleftbigg\n1+r′\na/parenrightbigg\nG=Fyy−e−s\naFyyyy+es\naN/parenleftbig\ne−svy/parenrightbig\n+H.\nWe define the energies of ( F,G) by\nE01(s) :=1\n2/integraldisplay\nRFy(s,y)2dy+e−s\n2a/integraldisplay\nRFyy(s,y)2dy+r2e−s\n2a/integraldisplay\nRG(s,y)2dy,\nE02(s) :=1\n2/integraldisplay\nRF(s,y)2dy+r2e−s\na/integraldisplay\nRF(s,y)G(s,y)dy.\nLemma 3.3. We have\nd\ndsE01(s)+/integraldisplay\nRG2dy=1\n2E01(s)−a′\n2ra2/integraldisplay\nRF2\nyydy−ra′\n2a2/integraldisplay\nRG2dy\n+/integraldisplay\nRGes\naN/parenleftbig\ne−svy/parenrightbig\ndy+/integraldisplay\nRGHdy,\nd\ndsE02(s)+1\n2E02(s)+2E01(s) = 2r2e−s\na/integraldisplay\nRG2dy+/parenleftbiggr′\na−ra′\na2/parenrightbigg/integraldisplay\nRFGdy\n+/integraldisplay\nRFes\naN/parenleftbig\ne−svy/parenrightbig\ndy+/integraldisplay\nRFHdy.\nProof.We apply Lemma 3.1asf=F,g=Gandh=es\naN(e−svy)+Hwithl= 0,\nm= 1,n= 0,c1(s) =r2e−s/a,c2(s) =r′/a, andc4(s) =e−s/a. Noting that\nd\ndsr(t(s)) =d\ndsr(R−1(es−1)) =r′(t(s))\nr(t(s))es,\nwe first have\nc′\n1(s) =1\na2/parenleftbigg\n2rdr\ndsae−s−r2ae−s−r2da\ndse−s/parenrightbigg\n=2r′\na−r2e−s\na−ra′\na2,(3.2)\nc′\n4(s) =1\na2/parenleftbigg\n−ae−s−da\ndse−s/parenrightbigg\n=−e−s\na−a′\nra2. (3.3)\nThus, we obtain\nd\ndsE01(s) =−/integraldisplay\nRG2dy+1\n4/integraldisplay\nRF2\nydy+3e−s\n4a/integraldisplay\nRF2\nyydy\n+3r2e−s\n4a/integraldisplay\nRG2dy−r′\na/integraldisplay\nRG2dy−1\n2/parenleftbigge−s\na+a′\nra2/parenrightbigg/integraldisplay\nRF2\nyydy\n+1\n2/parenleftbigg2r′\na−r2e−s\na−ra′\na2/parenrightbigg/integraldisplay\nRG2dy\n+/integraldisplay\nRGes\naN/parenleftbig\ne−svy/parenrightbig\ndy+/integraldisplay\nRGHdy\n=−/integraldisplay\nRG2dy+1\n2E01(s)−a′\n2ra2/integraldisplay\nRF2\nyydy−ra′\n2a2/integraldisplay\nRG2dy\n+/integraldisplay\nRGes\naN/parenleftbig\ne−svy/parenrightbig\ndy+/integraldisplay\nRGHdy.NONLINEAR DAMPED BEAM EQUATION 9\nNext, we compute the derivative of E02(s). By Lemma 3.1, we obtain\nd\ndsE02(s) =−/integraldisplay\nRF2\nydy−e−s\na/integraldisplay\nRF2\nyydy−1\n4/integraldisplay\nRF2dy\n+r2e−s\na/integraldisplay\nRG2dy+r2e−s\n2a/integraldisplay\nRFGdy−r′\na/integraldisplay\nRFGdy\n+/integraldisplay\nRFes\naN/parenleftbig\ne−svy/parenrightbig\ndy+/integraldisplay\nRFHdy\n+/parenleftbigg2r′\na−r2e−s\na−ra′\na2/parenrightbigg/integraldisplay\nRFGdy\n=−1\n2E02(s)−2E01(s)\n+2r2e−s\na/integraldisplay\nRG2dy+/parenleftbiggr′\na−ra′\na2/parenrightbigg/integraldisplay\nRFGdy\n+/integraldisplay\nRFes\naN/parenleftbig\ne−svy/parenrightbig\ndy+/integraldisplay\nRFHdy.\nThis completes the proof. /square\nNext, forn= 0,1, we define the energies of ( f,g) by\nE(n)\n11(s) :=1\n2/integraldisplay\nRy2nfy(s,y)2dy+e−s\n2a/integraldisplay\nRy2nfyy(s,y)2dy+r2e−s\n2a/integraldisplay\nRy2ng(s,y)2dy,\nE(n)\n12(s) :=1\n2/integraldisplay\nRy2nf(s,y)2dy+r2e−s\na/integraldisplay\nRy2nf(s,y)g(s,y)dy.\nLemma 3.4. Forn= 0,1, we have\nd\ndsE(n)\n11(s)+/integraldisplay\nRy2ng2dy=3−2n\n2E(n)\n11(s)−2n/integraldisplay\nRy2n−1fygdy\n−2n(2n−1)e−s\na/integraldisplay\nRy2n−2fyygdy−4ne−s\na/integraldisplay\nRy2n−1fyygydy\n−a′\n2ra2/integraldisplay\nRy2nf2\nyydy−ra′\n2a2/integraldisplay\nRy2ng2dy\n+/integraldisplay\nRy2ng/parenleftbigges\na∂y(N(e−svy))+h/parenrightbigg\ndy\nand\nd\ndsE(n)\n12(s)+1\n2E(n)\n12(s)+2E(n)\n11(s) = (1−n)E(n)\n12(s)+2r2e−s\na/integraldisplay\nRy2ng2dy−2n/integraldisplay\nRy2n−1ffydy\n−4ne−s\na/integraldisplay\nRy2n−1fyfyydy−2n(2n−1)e−s\na/integraldisplay\nRy2n−2ffyydy\n+/parenleftbiggr′\na−ra′\na2/parenrightbigg/integraldisplay\nRy2nfgdy+/integraldisplay\nRy2nf/parenleftbigges\na∂y(N(e−svy))+h/parenrightbigg\ndy.10 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA\nProof.Forn= 0,1,weapplyLemma 3.1asf=f,g=gandh=es\na∂yN(e−svy)+h\nwithl=1\n2,m=3\n2,c1(s) =r2e−s/a,c2(s) =r′/a, andc4(s) =e−s/a. Using ( 3)\nand (3), we have\nd\ndsE(n)\n11(s) =−/integraldisplay\nRy2ng2dy+3−2n\n4/integraldisplay\nRy2nf2\nydy+5−2n\n4e−s\na/integraldisplay\nRy2nf2\nyydy\n+5−2n\n4r2e−s\na/integraldisplay\nRy2ng2dy−r′\na/integraldisplay\nRy2ng2dy−2n/integraldisplay\nRy2n−1fygdy\n−2n(2n−1)e−s\na/integraldisplay\nRy2n−2fyygdy−4ne−s\na/integraldisplay\nRy2n−1fyygydy\n−1\n2/parenleftbigge−s\na+a′\nra2/parenrightbigg/integraldisplay\nRy2nf2\nyydy+1\n2/parenleftbigg2r′\na−r2e−s\na−ra′\na2/parenrightbigg/integraldisplay\nRy2ng2dy\n+/integraldisplay\nRy2ng/parenleftbigges\na∂y(N(e−svy))+h/parenrightbigg\ndy\n=−/integraldisplay\nRy2ng2dy+3−2n\n2E(n)\n11(s)\n−2n/integraldisplay\nRy2n−1fygdy−2n(2n−1)e−s\na/integraldisplay\nRy2n−2fyygdy−4ne−s\na/integraldisplay\nRy2n−1fyygydy\n−a′\n2ra2/integraldisplay\nRy2nf2\nyydy−ra′\n2a2/integraldisplay\nRy2ng2dy\n+/integraldisplay\nRy2ng/parenleftbigges\na∂y(N(e−svy))+h/parenrightbigg\ndy.\nSimilarly, we calculate\nd\ndsE(n)\n12(s) =−/integraldisplay\nRy2nf2\nydy−e−s\na/integraldisplay\nRy2nf2\nyydy+1−2n\n4/integraldisplay\nRy2nf2dy\n+r2e−s\na/integraldisplay\nRy2ng2dy+3−2n\n2r2e−s\na/integraldisplay\nRy2nfgdy−r′\na/integraldisplay\nRy2nfgdy\n−2n/integraldisplay\nRy2n−1ffydy−4ne−s\na/integraldisplay\nRy2n−1fyfyydy−2n(2n−1)e−s\na/integraldisplay\nRy2n−2ffyydy\n+/parenleftbigg2r′\na−r2e−s\na−ra′\na2/parenrightbigg/integraldisplay\nRy2nfgdy+/integraldisplay\nRy2nf/parenleftbigges\na∂y(N(e−svy))+h/parenrightbigg\ndy\n=1−2n\n2E(n)\n12(s)−2E(n)\n11(s)+2r2e−s\na/integraldisplay\nRy2ng2dy\n−2n/integraldisplay\nRy2n−1ffydy−4ne−s\na/integraldisplay\nRy2n−1fyfyydy−2n(2n−1)e−s\na/integraldisplay\nRy2n−2ffyydy\n+/parenleftbiggr′\na−ra′\na2/parenrightbigg/integraldisplay\nRy2nfgdy+/integraldisplay\nRy2nf/parenleftbigges\na∂y(N(e−svy))+h/parenrightbigg\ndy.\nThis completes the proof. /squareNONLINEAR DAMPED BEAM EQUATION 11\nFinally, to control the bad term −4e−s\na/integraldisplay\nRyfyygydyind\ndsE(1)\n11(s), we consider\nthe energies\nE21(s) =1\n2/integraldisplay\nR/parenleftbigg\nf2\nyy+e−s\naf2\nyyy+r2e−s\nag2\ny/parenrightbigg\ndy,\nE22(s) =/integraldisplay\nR/parenleftbigg1\n2f2\ny+r2e−s\nafygy/parenrightbigg\ndy.\nSince (fy,gy) satisfies the equations\n\n\n(fy)s−y\n2(fy)y−fy=gy,\nr2e−s\na/parenleftBig\n(gy)s−y\n2(gy)y−2gy/parenrightBig\n+gy+r′\nagy= (fy)yy−e−s\na(fy)yyyy+es\na∂2\nyN(e−svy))+hy,\nwe have the following lemma.\nLemma 3.5. We have\nd\ndsE21(s)+/integraldisplay\nRg2\nydy=5\n2E21(s)−a′\n2ra2/integraldisplay\nRf2\nyyydy−ra′\n2a2/integraldisplay\nRg2\nydy\n+/integraldisplay\nRgy/parenleftbigges\na∂2\ny(N(e−svy))+hy/parenrightbigg\ndy\nand\nd\ndsE22(s)+2E21(s) =3\n2E22(s)+2r2e−s\na/integraldisplay\nRg2\nydy+/parenleftbiggr′\na−ra′\na2/parenrightbigg/integraldisplay\nRfygydy\n+/integraldisplay\nRfy/parenleftbigges\na∂2\ny(N(e−svy))+hy/parenrightbigg\ndy.\nProof.Applying Lemma 3.1asf=fy,g=gyandh=es\na∂yyN(e−svy)+hywith\nl= 1,m= 2,n= 0,c1(s) =r2e−s\na,c2(s) =r′\na, andc4(s) =e−s\na, and also using ( 3)\nand (3), we have\nd\ndsE21(s) =−/integraldisplay\nRg2\nydy+5\n4/integraldisplay\nRf2\nyydy+7\n4e−s\na/integraldisplay\nRf2\nyyydy\n+7\n4r2e−s\na/integraldisplay\nRg2\nydy−r′\na/integraldisplay\nRg2\nydy\n−1\n2/parenleftbigge−s\na+a′\nra2/parenrightbigg/integraldisplay\nRf2\nyyydy+1\n2/parenleftbigg2r′\na−r2e−s\na−ra′\na2/parenrightbigg/integraldisplay\nRg2\nydy\n+/integraldisplay\nRgy/parenleftbigges\na∂2\ny(N(e−svy))+hy/parenrightbigg\ndy\n=−/integraldisplay\nRg2\nydy+5\n2E21(s)\n−a′\n2ra2/integraldisplay\nRf2\nyyydy−ra′\n2a2/integraldisplay\nRg2\nydy\n+/integraldisplay\nRgy/parenleftbigges\na∂2\ny(N(e−svy))+hy/parenrightbigg\ndy.12 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA\nSimilarly, we have\nd\ndsE22(s) =−/integraldisplay\nRf2\nyydy−e−s\na/integraldisplay\nRf2\nyyydy+3\n4/integraldisplay\nRf2\nydy\n+r2e−s\na/integraldisplay\nRg2\nydy+5\n2r2e−s\na/integraldisplay\nRfygydy−r′\na/integraldisplay\nRfygydy\n+/parenleftbigg2r′\na−r2e−s\na−ra′\na2/parenrightbigg/integraldisplay\nRfygydy+/integraldisplay\nRfy/parenleftbigges\na∂2\ny(N(e−svy))+hy/parenrightbigg\ndy\n=3\n2E22(s)−2E21(s)\n+2r2e−s\na/integraldisplay\nRg2\nydy+/parenleftbiggr′\na−ra′\na2/parenrightbigg/integraldisplay\nRfygydy\n+/integraldisplay\nRfy/parenleftbigges\na∂2\ny(N(e−svy))+hy/parenrightbigg\ndy.\nThis completes the proof. /square\nFinally, we define\nEm1(s) :=1\n2r2e−s\nams(s)2,\nEm2(s) :=1\n2m(s)2+r2e−s\nam(s)ms(s).\nThen, we have the following energy identities.\nLemma 3.6. We have\nd\ndsEm1(s)+1\n2Em1(s)+ms(s)2=/parenleftbigg3r2e−s\n4a−ra′\n2a2/parenrightbigg\nms(s)2\nand\nd\ndsEm2(s) = 2Em1(s)+/parenleftbiggr′\na−ra′\na2/parenrightbigg\nm(s)ms(s).\nProof.By (3) and Lemma 2.2, we have\nd\ndsEm1(s) =1\n2d\nds/parenleftbiggr2e−s\na/parenrightbigg\nm2\ns+r2e−s\namsmss\n=1\n2/parenleftbigg2r′\na−r2e−s\na−ra′\na2/parenrightbigg\nm2\ns\n+r2e−s\nam2\ns−/parenleftbigg\n1+r′\na/parenrightbigg\nm2\ns\n=1\n2r2e−s\nam2\ns−1\n2ra′\na2m2\ns−m2\ns\n=−1\n2Em1(s)−m2\ns+/parenleftbigg3r2e−s\n4a−ra′\n2a2/parenrightbigg\nm2\ns.NONLINEAR DAMPED BEAM EQUATION 13\nSimilarly, we have\nd\ndsEm2(s) =mms+d\nds/parenleftbiggr2e−s\na/parenrightbigg\nmms+r2e−s\nam2\ns+r2e−s\nammss\n=mms+/parenleftbigg2r′\na−r2e−s\na−ra′\na2/parenrightbigg\nmms\n+r2e−s\nam2\ns+r2e−s\namms−/parenleftbigg\n1+r′\na/parenrightbigg\nmms\n=r2e−s\nam2\ns+/parenleftbiggr′\na−ra′\na2/parenrightbigg\nmms\n= 2Em1(s)+/parenleftbiggr′\na−ra′\na2/parenrightbigg\nmms.\n/square\n3.1.Energy estimates. Now, we combine the energy identities in the previous\nsubsection to obtain the energyestimates. First, we prepare the following estimates\nfor remainders.\nLemma 3.7. Set\nδ:= min/braceleftbiggβ+1\nα−β+1,2α−β+1\nα−β+1/bracerightbigg\n,\nwhich is positive if (α,β)∈Ω1. Then, we have\nr2e−s\na∼e−β+1\nα−β+1s≤Ce−δs,e−s\na∼e−2α−β+1\nα−β+1s≤Ce−δs,\nand\n/vextendsingle/vextendsingle/vextendsingle/vextendsinglea′\nra2/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Ce−s\na,/vextendsingle/vextendsingle/vextendsingle/vextendsinglera′\na2/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Cr2e−s\na,/vextendsingle/vextendsingle/vextendsingle/vextendsingler′\na/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Cr2e−s\na.\nDefine\nE0(s) :=E01(s)+c0E02(s),\nwherec0>0 is a sufficiently large constant determined later.\nLemma 3.8. There exists a constant c0>0satisfying the following: For any\nη>0, there exists s0>0such that for any s≥s0, we have\nE0(s)≥C/parenleftbigg/integraldisplay\nRF2\nydy+e−s\n2a/integraldisplay\nRF2\nyydy+r2e−s\n2a/integraldisplay\nRG2dy+/integraldisplay\nRF2dy/parenrightbigg\nand\nd\ndsE0(s)+1\n2E0(s)+1\n4/integraldisplay\nRG2dy\n≤ηE0(s)+C(η)/parenleftbigg\n/bardblH(s)/bardbl2\nL2+e2s\na2/bardblN(e−svy)/bardbl2\nL2/parenrightbigg\n.14 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA\nProof.Letη >0 be arbitrary. Lemmas 3.3and3.7, and the Schwarz inequality\nimply\nd\ndsE01(s)+/integraldisplay\nRG2dy≤1\n2E01(s)+C1e−s\na/integraldisplay\nRF2\nyydy+C1r2e−s\na/integraldisplay\nRG2dy\n+1\n2/integraldisplay\nRG2dy+C/parenleftbigg\n/bardblH(s)/bardbl2\nL2+e2s\na2/bardblN(e−svy)/bardbl2\nL2/parenrightbigg\n≤/parenleftbigg1\n2+2C1/parenrightbigg\nE01(s)+1\n2/integraldisplay\nRG2dy\n+C/parenleftbigg\n/bardblH(s)/bardbl2\nL2+e2s\na2/bardblN(e−svy)/bardbl2\nL2/parenrightbigg\nwith someC1>0 and\nd\ndsE02(s)+1\n2E02(s)+2E01(s)≤C2(η1)e−δs/integraldisplay\nRG2dy+η1/integraldisplay\nRF2dy\n+C(η1)/parenleftbigg\n/bardblH(s)/bardbl2\nL2+e2s\na2/bardblN(e−svy)/bardbl2\nL2/parenrightbigg\n,\nwith someC2(η1)>0, whereη1is an arbitrary small positive number determined\nlater. We take c0sufficiently large so that 2 c0−1\n2−2C1≥1\n2. Then, letting s0\nsufficiently large so that c0C2(η1)e−δs≤1\n4holds for any s≥s0, we conclude\nd\ndsE0(s)+1\n2E0(s)+1\n4/integraldisplay\nRG2dy\n≤2c0η1/integraldisplay\nRF2dy+C/parenleftbigg\n/bardblH(s)/bardbl2\nL2+e2s\na2/bardblN(e−svy)/bardbl2\nL2/parenrightbigg\n. (3.4)\nOn the other hand, we remark that\nr2e−s\na/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRF(s,y)G(s,y)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1\n4/integraldisplay\nRF(s,y)2dy+C/parenleftbiggr2e−s\na/parenrightbigg2/integraldisplay\nRG(s,y)2dy\n≤1\n4/integraldisplay\nRF(s,y)2dy+Ce−δsr2e−s\n2a/integraldisplay\nRG(s,y)2dy.\nFrom this, retaking s0larger if needed, we have for s≥s0,\nE0(s)≥C/parenleftbigg/integraldisplay\nRF2\nydy+e−s\n2a/integraldisplay\nRF2\nyydy+r2e−s\n2a/integraldisplay\nRG2dy+/integraldisplay\nRF2dy/parenrightbigg\n,\nwhich shows the first assertion. In particular, it gives/integraltext\nRF2\nydy≤CE0(s) Applying\nthis to the right-hand side of ( 3.1) and taking η1so thatη= 2c0Cη1, we have the\ndesired estimate. /square\nNext, forn= 0,1, we define\nE(n)\n1(s) :=E(n)\n11(s)+c(n)\n1E(n)\n12(s),\nwherec(0)\n1andc(1)\n1are sufficiently large constants determined later. The following\ntwo lemmas are the estimates for E(0)\n1(s) andE(1)\n1(s), respectively.NONLINEAR DAMPED BEAM EQUATION 15\nLemma 3.9. There exist positive constants c(0)\n1ands(0)\n1such that for any s≥s(0)\n1,\nwe have\nE(0)\n1(s)≥C/parenleftbigg/integraldisplay\nRf2\nydy+e−s\n2a/integraldisplay\nRf2\nyydy+r2e−s\n2a/integraldisplay\nRg2dy+/integraldisplay\nRf2dy/parenrightbigg\nand\nd\ndsE(0)\n1(s)+1\n2E(0)\n1(s)+1\n4/integraldisplay\nRg2dy\n≤CE0(s)+C/parenleftbigg\n/bardblh(s)/bardbl2\nL2+e2s\na2/vextenddouble/vextenddouble∂yN/parenleftbig\ne−svy/parenrightbig/vextenddouble/vextenddouble2\nL2/parenrightbigg\n.\nProof.By Lemmas 3.4and3.7, and the Schwarz inequality, we have\nd\ndsE(0)\n11(s)+1\n2E(0)\n11(s)+/integraldisplay\nRg2dy\n≤2E(0)\n11(s)+Ce−s\na/integraldisplay\nRf2\nyydy+Cr2e−s\na/integraldisplay\nRg2dy\n+1\n2/integraldisplay\nRg2dy+C/parenleftbigg\n/bardblh(s)/bardbl2\nL2+e2s\na2/bardbl∂y/parenleftbig\nN/parenleftbig\ne−svy/parenrightbig/parenrightbig\n/bardbl2\nL2/parenrightbigg\n,\nwhich implies\nd\ndsE(0)\n11(s)+1\n2E(0)\n11(s)+1\n2/integraldisplay\nRg2dy\n≤(2+C1)E(0)\n11(s)+C/parenleftbigg\n/bardblh(s)/bardbl2\nL2+e2s\na2/bardbl∂y/parenleftbig\nN/parenleftbig\ne−svy/parenrightbig/parenrightbig\n/bardbl2\nL2/parenrightbigg\nwith some constant C1>0. In a similar way, we also obtain\nd\ndsE(0)\n12(s)+1\n2E(0)\n12(s)+2E(0)\n11(s)\n≤C/integraldisplay\nRf2dy+C2e−δs/integraldisplay\nRg2dy\n+C/parenleftbigg\n/bardblh(s)/bardbl2\nL2+e2s\na2/bardbl∂y/parenleftbig\nN/parenleftbig\ne−svy/parenrightbig/parenrightbig\n/bardbl2\nL2/parenrightbigg\nwith some constant C2>0. Therefore, taking c(0)\n1ands(0)\n1sufficiently large so that\n2c(0)\n1−(2+C1)≥1\n2andc(0)\n1C2e−δs≤1\n4holds for any s≥s(0)\n1, we conclude\nd\ndsE(0)\n1(s)+1\n2E(0)\n1(s)+1\n4/integraldisplay\nRg2dy\n≤C/integraldisplay\nRf2dy+C/parenleftbigg\n/bardblh(s)/bardbl2\nL2+e2s\na2/vextenddouble/vextenddouble∂yN/parenleftbig\ne−svy/parenrightbig/vextenddouble/vextenddouble2\nL2/parenrightbigg\n.\nFinally,by/integraltext\nRf2dy=/integraltext\nRF2\nydy≤CE0, theproofofthesecondassertioniscomplete.\nThe first assertion is proved in the same way as the previous lemma an d we omit\nthe detail. /square\nLemma 3.10. There exists a constant c(1)\n1>0satisfying the following: for any\nη′>0, there exists a constant s(1)\n1>0such that for any s≥s(1)\n1, we have\nE(1)\n1(s)≥C/parenleftbigg/integraldisplay\nRy2f2\nydy+e−s\n2a/integraldisplay\nRy2f2\nyydy+r2e−s\n2a/integraldisplay\nRy2g2dy+/integraldisplay\nRy2f2dy/parenrightbigg16 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA\nand\nd\ndsE(1)\n1(s)+1\n2E(1)\n1(s)+1\n4/integraldisplay\nRy2g2dy\n≤η′E(1)\n1(s)+CE(0)\n1(s)−4e−s\na/integraldisplay\nRyfyygydy+Ce−δs/integraldisplay\nRg2dy\n+C(η′)/parenleftbigg\n/bardblyh(s)/bardbl2\nL2+e2s\na2/bardbly∂y(N(e−svy))/bardbl2\nL2/parenrightbigg\n.\nProof.Letη′>0 be arbitrary. By Lemmas 3.4and3.7, and the Schwarzinequality,\nwe have\nd\ndsE(1)\n11(s)+1\n2E(1)\n11(s)+/integraldisplay\nRy2g2dy\n≤E(1)\n11(s)+Ce−s\na/integraldisplay\nRy2f2\nyydy+Ce−s\na/integraldisplay\nRf2\nyydy+Cr2e−s\na/integraldisplay\nRy2g2dy\n+1\n2/integraldisplay\nRy2g2dy+C/integraldisplay\nRf2\nydy+Ce−δs/integraldisplay\nRg2dy\n−4e−s\na/integraldisplay\nRyfyygydy+C/parenleftbigg\n/bardblyh(s)/bardbl2\nL2+e2s\na2/bardbly∂y(N(e−svy))/bardbl2\nL2/parenrightbigg\n,\nwhich implies\nd\ndsE(1)\n11(s)+1\n2E(1)\n11(s)+1\n2/integraldisplay\nRy2g2dy\n≤(1+C′\n1)E(1)\n11(s)+CE(0)\n1(s)+Ce−δs/integraldisplay\nRg2dy\n−4e−s\na/integraldisplay\nRyfyygydy+C/parenleftbigg\n/bardblyh(s)/bardbl2\nL2+e2s\na2/bardbly∂y(N(e−svy))/bardbl2\nL2/parenrightbigg\nwith some constant C′\n1>0. Next, for E(1)\n12(s), Lemmas 3.4and3.7and the Schwarz\ninequality imply\nd\ndsE(1)\n12(s)+1\n2E(1)\n12(s)+2E(1)\n11(s)\n=2r2e−s\na/integraldisplay\nRy2g2dy−2/integraldisplay\nRyffydy−4e−s\na/integraldisplay\nRyfyfyydy−2e−s\na/integraldisplay\nRffyydy\n+/parenleftbiggr′\na−ra′\na2/parenrightbigg/integraldisplay\nRy2fgdy+/integraldisplay\nRy2f/parenleftbigges\na∂y(N(e−svy))+h/parenrightbigg\ndy\n≤η′\n1/integraldisplay\nRy2f2dy+E(1)\n11(s)+C′\n2(η′\n1)e−δs/integraldisplay\nRy2g2dy\n+C/integraldisplay\nRf2\nydy+C/parenleftbigge−s\na/parenrightbigg2/integraldisplay\nRf2\nyydy+C/integraldisplay\nRf2dy\n+C/parenleftbigg\n/bardblyh(s)/bardbl2\nL2+e2s\na2/bardbly∂y(N(e−svy))/bardbl2\nL2/parenrightbigg\n≤η′\n1/integraldisplay\nRy2f2dy+E(1)\n11(s)+C′\n2(η′\n1)e−δs/integraldisplay\nRy2g2dy+CE(0)\n1(s)\n+C/parenleftbigg\n/bardblyh(s)/bardbl2\nL2+e2s\na2/bardbly∂y(N(e−svy))/bardbl2\nL2/parenrightbiggNONLINEAR DAMPED BEAM EQUATION 17\nfor arbitrary small η′\n1>0 determined later and some constant C′\n2(η′\n1)>0. There-\nfore, taking c(1)\n1ands(1)\n1so thatc(1)\n1−(1+C′\n1)≥1\n2andc(1)\n1C′\n2(η′\n1)e−δs≤1\n4holds\nfor anys≥s(1)\n1, we conclude\nd\ndsE(1)\n1+1\n2E(1)\n1+1\n4/integraldisplay\nRy2g2dy\n≤η′\n1c(1)\n1/integraldisplay\nRy2f2dy−4e−s\na/integraldisplay\nRyfyygydy+C/integraldisplay\nRf2\nydy+Ce−δs/integraldisplay\nRg2dy\n+CE(0)\n1(s)+C/parenleftbigg\n/bardblyh(s)/bardbl2\nL2+e2s\na2/bardbly∂y(N(e−svy))/bardbl2\nL2/parenrightbigg\n.\nTakingη′\n1so that the first term of the right-hand side is bounded by η′E(1)\n1(s) and\nusing/integraltext\nRf2\nydy≤CE(0)\n1(s), we complete the proof of the second assertion. The first\nassertion is proved in the same way as before and we omit the detail. /square\nNext, we define\nE2(s) :=E21(s)+c2E22(s),\nwherec2is a sufficiently large constant determined later.\nLemma 3.11. There exist positive constants c2ands2such that for any s≥s2,\nwe have\nE2(s)≥C/parenleftbigg/integraldisplay\nRf2\nyydy+e−s\n2a/integraldisplay\nRf2\nyyydy+r2e−s\n2a/integraldisplay\nRg2\nydy+/integraldisplay\nRf2\nydy/parenrightbigg\nand\nd\ndsE2(s)+1\n2E2(s)+1\n4/integraldisplay\nRg2\nydy\n≤CE(0)\n1(s)+C/parenleftbigg\n/bardbl∂yh(s)/bardbl2\nL2+e2s\na2/bardbl∂2\ny(N(e−svy))/bardbl2\nL2/parenrightbigg\n.\nProof.By Lemmas 3.5and3.7and the Schwarz inequality, we have\nd\ndsE21(s)+/integraldisplay\nRg2\nydy≤5\n2E21(s)+C1e−s\na/integraldisplay\nRf2\nyyydy+C1r2e−s\na/integraldisplay\nRg2\nydy\n+1\n2/integraldisplay\nRg2\nydy+C/parenleftbigg\n/bardblhy(s)/bardbl2\nL2+e2s\na2/bardbl∂2\nyN(e−svy)/bardbl2\nL2/parenrightbigg\n≤/parenleftbigg5\n2+2C1/parenrightbigg\nE21(s)+1\n2/integraldisplay\nRg2\nydy\n+C/parenleftbigg\n/bardblhy(s)/bardbl2\nL2+e2s\na2/bardbl∂2\nyN(e−svy)/bardbl2\nL2/parenrightbigg\n.18 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA\nwith someC1>0 and\nd\ndsE22(s)+1\n2E22(s)+2E21(s)\n≤2E22(s)+Cr2e−s\na/integraldisplay\nRg2\nydy+Cr2e−s\na/integraldisplay\nRf2\nydy\n+C/integraldisplay\nRf2\nydy+C/parenleftbigg\n/bardblhy(s)/bardbl2\nL2+e2s\na2/bardbl∂2\nyN(e−svy)/bardbl2\nL2/parenrightbigg\n≤C2e−δs/integraldisplay\nRg2\nydy+C/integraldisplay\nRf2\nydy\n+C/parenleftbigg\n/bardblhy(s)/bardbl2\nL2+e2s\na2/bardbl∂2\nyN(e−svy)/bardbl2\nL2/parenrightbigg\nwith someC2>0. We take c2sufficiently large so that 2 c2−5\n2−2C1≥1\n2. Then,\nlettings2sufficiently large so that c2C2e−δs≤1\n4holds for any s≥s2, we conclude\nd\ndsE2(s)+1\n2E2(s)+1\n4/integraldisplay\nRg2\nydy\n≤C/integraldisplay\nRf2\nydy+C/parenleftbigg\n/bardblhy(s)/bardbl2\nL2+e2s\na2/bardbl∂2\nyN(e−svy)/bardbl2\nL2/parenrightbigg\nfor anys≥s2. This and/integraltext\nRf2\nydy≤CE(0)\n1(s) complete the proof of the second\nassertion. The first assertion is proved in the same way as before a nd we omit the\ndetail. /square\nFinally, let us combine the estimates in Lemmas 3.8–3.11. Fix\nλ∈/parenleftbigg\n0,min/braceleftbigg1\n2,2(β+1)\nα−β+1,2α−β+1\nα−β+1/bracerightbigg/parenrightbigg\nand lets′\n∗= max{s0,s(0)\n1,s(1)\n1,s2}. We first note that the Schwarz inequality and\nLemma3.7imply\n−4e−s\na/integraldisplay\nRyfyygydy≤η′′E(1)\n1(s)+C(η′′)e−δs/integraldisplay\nRg2\nydy\nfor anyη′′>0. We take η,η′\n1in Lemmas 3.8and3.10andη′′above so that\n1\n2−η≤λandη′+η′′≤1\n2−λ. Then, we take ˜ c0≫˜c(0)\n1≫˜c(1)\n1≫1 and define\nE(s) = ˜c0E0(s)+˜c(0)\n1E(0)\n1(s)+˜c(1)\n1E(1)\n1(s)+E2(s)+Em1(s),\nG(s) =/integraldisplay\nR/parenleftBig\n˜c0G2+˜c(0)\n1g2+˜c(1)\n1y2g2+g2\ny/parenrightBig\ndy,\n/tildewideE(s) =E(s)+Em2(s).NONLINEAR DAMPED BEAM EQUATION 19\nThen, adding the estimates in Lemmas 3.6and3.8–3.11, we conclude that\nd\ndsE(s)+λE(s)+1\n4G(s)+ms(s)2\n≤Ce−δs/integraldisplay\nRg2dy+Ce−δs/integraldisplay\nRg2\nydy+/parenleftbigg3r2e−s\n4a−ra′\n2a2/parenrightbigg\nms(s)2\n+C/parenleftbig\n/bardblH(s)/bardbl2\nL2+/bardblh(s)/bardbl2\nH0,1+/bardblhy(s)/bardbl2\nL2/parenrightbig\n+C/parenleftbigges\na/parenrightbigg2/parenleftBig\n/bardblN(e−svy)/bardbl2\nL2+/vextenddouble/vextenddouble∂yN/parenleftbig\ne−svy/parenrightbig/vextenddouble/vextenddouble2\nH0,1+/bardbl∂2\ny(N(e−svy))/bardbl2\nL2/parenrightBig\nholds fors≥s′\n∗. Moreover, Lemma 3.7leads to\n/parenleftbigg3r2e−s\n4a−ra′\n2a2/parenrightbigg\n≤Ce−δs.\nTherefore, we finally reach the following energy estimate.\nProposition 3.12. There exist constants s∗>0andC >0such that for any\ns≥s∗, we have\nd\ndsE(s)+λE(s)+1\n8/parenleftbig\nG(s)+m2\ns/parenrightbig\n≤C/parenleftbig\n/bardblH(s)/bardbl2\nL2+/bardblh(s)/bardbl2\nH0,1+/bardblhy(s)/bardbl2\nL2/parenrightbig\n+C/parenleftbigges\na/parenrightbigg2/parenleftBig\n/bardblN(e−svy)/bardbl2\nL2+/vextenddouble/vextenddouble∂yN/parenleftbig\ne−svy/parenrightbig/vextenddouble/vextenddouble2\nH0,1+/bardbl∂2\ny(N(e−svy))/bardbl2\nL2/parenrightBig\n.\n4.Estimates of remainder terms and the proof of a priori estima te\nIn this section, we give estimates of the right-hand side of Proposit ion3.12, and\ncomplete the a priori estimate, which ensures the existence of the global solution.\n4.1.Estimates of remainder terms. First, by the Hardy-type inequality in\nLemma3.2, we have\n/bardblH(s)/bardbl2\nL2≤4/bardblyh(s)/bardbl2\nL2,/bardblN(e−svy)/bardbl2\nL2≤4/bardbly∂yN(e−svy)/bardbl2\nL2.\nHence, it suffices to estimate\n/bardblh(s)/bardbl2\nH0,1,/bardblhy(s)/bardbl2\nL2,/parenleftbigges\na/parenrightbigg2/vextenddouble/vextenddouble∂yN/parenleftbig\ne−svy/parenrightbig/vextenddouble/vextenddouble2\nH0,1,/parenleftbigges\na/parenrightbigg2\n/bardbl∂2\ny(N(e−svy))/bardbl2\nL2.\nFirst, from the definition of h(see (2)) and Lemma 3.7, we easily obtain\n/bardblh(s)/bardbl2\nH0,1+/bardblhy(s)/bardbl2\nL2≤Ce−2δs/parenleftbig\nm(s)2+ms(s)2/parenrightbig\n≤e−2δs/tildewideE(s).\nNext, we estimate the nonlinear term. By Assumption (N), we see th at\n∂yN(e−svy) = 2µe−2svyvyy+˜N′(e−svy)e−svyy,\n∂2\nyN(e−svy) = 2µe−2s(v2\nyy+vyvyyy)+˜N′′(e−svy)e−2sv2\nyy+˜N′(e−svy)e−svyyy.\nTherefore, by |˜N′(z)| ≤C|z|p−1, the Sobolev embedding theorem, and\n/bardblf/bardbl2\nH2,1≤C/parenleftbigge−s\na/parenrightbigg−1/parenleftbig\nE(0)\n1(s)+E(1)\n1(s)/parenrightbig\n≤Ce2α−β+1\nα−β+1s/tildewideE(s),20 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA\nwe have\n/parenleftbigges\na/parenrightbigg2/vextenddouble/vextenddouble∂yN/parenleftbig\ne−svy/parenrightbig/vextenddouble/vextenddouble2\nH0,1\n≤C/parenleftbigges\na/parenrightbigg2\ne−4s/bardblvyvyy/bardbl2\nH0,1+C/parenleftbigges\na/parenrightbigg2\ne−2ps/bardbl|vy|p−1vyy/bardbl2\nH0,1\n≤Ce−2(1+α\nα−β+1)s/bardblvy/bardbl2\nL∞/bardblvyy/bardbl2\nH0,1+Ce−2(p−2)se−2(1+α\nα−β+1)s/bardblvy/bardbl2(p−1)\nL∞/bardblvyy/bardbl2\nH0,1\n≤Ce−2(2α−β+1)\nα−β+1s/bardblvy/bardbl2\nH1/bardblvyy/bardbl2\nH0,1+Ce−2(p−2)se−2(2α−β+1)\nα−β+1s/bardblvy/bardbl2(p−1)\nH1/bardblvyy/bardbl2\nH0,1\n≤C/parenleftBig\ne−2(2α−β+1)\nα−β+1s(/bardblf/bardbl2\nH2,0+m(s)2)+e−2(p−2)se−2(2α−β+1)\nα−β+1s(/bardblf/bardbl2\nH2,0+m(s)2)p−1/parenrightBig\n×(/bardblf/bardbl2\nH2,1+m(s)2)\n≤Ce−2α−β+1\nα−β+1s/tildewideE(s)2+e−[2(p−2)+2α−β+1\nα−β+1]s/tildewideE(s)p.\nSimilarly, by |˜N′′(z)| ≤C|z|p−2, the Sobolev embedding theorem, and\n/bardblf/bardbl2\nH3,0≤C/parenleftbigge−s\na/parenrightbigg−1/parenleftbig\nE(0)\n1(s)+E21(s)/parenrightbig\n≤Ce2α−β+1\nα−β+1s/tildewideE(s),\nwe obtain\n/parenleftbigges\na/parenrightbigg2/vextenddouble/vextenddouble∂2\nyN/parenleftbig\ne−svy/parenrightbig/vextenddouble/vextenddouble2\nL2\n≤C/parenleftbigges\na/parenrightbigg2\ne−4s/parenleftbig\n/bardblv2\nyy/bardbl2\nL2+/bardblvyvyyy/bardbl2\nL2/parenrightbig\n+C/parenleftbigges\na/parenrightbigg2\ne−2ps/parenleftbig\n/bardbl|vy|p−2v2\nyy/bardbl2\nL2+/bardbl|vy|p−1vyyy/bardbl2\nL2/parenrightbig\n≤Ce−2(1+α\nα−β+1)s(/bardblvyy/bardbl2\nL∞/bardblvyy/bardbl2\nL2+/bardblvy/bardbl2\nL∞/bardblvyyy/bardbl2\nL2)\n+Ce−2(p−2)se−2(1+α\nα−β+1)s(/bardblvy/bardbl2(p−2)\nL∞/bardblvyy/bardbl2\nL∞/bardblvyy/bardbl2\nL2+/bardblvy/bardbl2(p−1)\nL∞/bardblvyyy/bardbl2\nL2)\n≤Ce−2(2α−β+1)\nα−β+1)s(/bardblvyy/bardbl2\nH1,0/bardblvyy/bardbl2\nL2+/bardblvy/bardbl2\nH1,0/bardblvyyy/bardbl2\nL2)\n+Ce−2(p−2)se−2(2α−β+1)\nα−β+1)s(/bardblvy/bardbl2(p−2)\nH1,0/bardblvyy/bardbl2\nL∞/bardblvyy/bardbl2\nL2+/bardblvy/bardbl2(p−1)\nH1,0/bardblvyyy/bardbl2\nL2)\n≤C/parenleftBig\ne−2(2α−β+1)\nα−β+1)s(/bardblf/bardbl2\nH2,0+m(s)2)+e−2(p−2)se−2(2α−β+1)\nα−β+1)s(/bardblf/bardbl2\nH2,0+m(s)2)p−1/parenrightBig\n×(/bardblf/bardbl2\nH3,0+m(s)2)\n≤Ce−2α−β+1\nα−β+1s/tildewideE(s)2+e−[2(p−2)+2α−β+1\nα−β+1]s/tildewideE(s)p.\n4.2.Proof of a priori estimate. Combining the energy estimates obtained in\nProposition 3.12with the estimates of remainder terms given in the previous sub-\nsection, we deduceNONLINEAR DAMPED BEAM EQUATION 21\nd\ndsE(s)+λE(s)+1\n8/parenleftbig\nG(s)+ms(s)2/parenrightbig\n≤Ce−2δs/tildewideE(s)+Ce−2α−β+1\nα−β+1s/tildewideE(s)2+Ce−[2(p−2)+2α−β+1\nα−β+1]s/tildewideE(s)p(4.1)\nFrom Lemmas 3.6and3.7, we see that\nd\ndsEm2(s) = 2Em1(s)+/parenleftbiggr′\na−ra′\na2/parenrightbigg\nm(s)ms(s)\n≤Ce−δsms(s)2+1\n16ms(s)2+Ce−2δsm(s)2.\nTherefore, there exists constants sm≥s∗andc>0 such that for any s≥sm, we\nhave\nd\nds/tildewideE(s)+λ/tildewideE(s)+c(G(s)+ms(s)2)\n≤C3e−2δs/tildewideE(s)+C3/parenleftBig\ne−2α−β+1\nα−β+1s/tildewideE(s)2+e−[2(p−2)+2α−β+1\nα−β+1]s/tildewideE(s)p/parenrightBig\n(4.2)\nwith someC3>0. Define\nΛ(s) = exp/parenleftbigg\n−C3/integraldisplays\nsme−2δσdσ/parenrightbigg\n.\nNote that\nΛ(s) = exp/parenleftbiggC3\n2δ/parenleftbig\ne−2δs−e−2δsm/parenrightbig/parenrightbigg\n∼1 and Λ( sm) = 1.\nMultiplying ( 4.2) by Λ(s), we deduce\nd\nds/bracketleftBig\nΛ(s)/tildewideE(s)/bracketrightBig\n+λΛ(s)E(s)+cΛ(s)/parenleftbig\nG(s)+ms(s)2/parenrightbig\n≤C3Λ(s)/parenleftBig\ne−2α−β+1\nα−β+1s/tildewideE(s)2+e−[2(p−2)+2α−β+1\nα−β+1]s/tildewideE(s)p/parenrightBig\n.\nIntegrating the above over [ sm,s], we have\nΛ(s)/tildewideE(s)≤/tildewideE(sm)+C3/integraldisplays\nsmΛ(σ)/parenleftBig\ne−2α−β+1\nα−β+1s/tildewideE(σ)2+e−[2(p−2)+2α−β+1\nα−β+1]s/tildewideE(σ)p/parenrightBig\ndσ\nFinally, we put\n/tildewideEmax(s) = max\nσ∈[sm,s]/tildewideE(σ)\nfors≥sm. Then, the above estimate implies\n/tildewideEmax(s)≤C0/tildewideE(sm)+C′\n0/parenleftBig\n/tildewideEmax(s)2+/tildewideEmax(s)p/parenrightBig\nwith some constants C0,C′\n0>0, where we have used δ >0 andp >−β+1\nα−β+1(see\nRemark 1.1). Thus, we conclude the a priori estimate\n/tildewideEmax(s)≤2C0/tildewideE(sm) (4.3)\nfor alls≥sm, provided that /tildewideE(sm) is sufficiently small. From the local existence\nresult (Proposition B.2), we see that, for sufficiently small initial dat a, the local\nsolution uniquely exists over [0 ,sm], and it satisfies /tildewideE(sm)≤C(/bardblu0/bardblH2,1∩H3,0+\n/bardblu1/bardblH0,1∩H1,0) (for the detail, see the proof of Proposition B.2 (vi) ). Thus, /tildewideE(sm)22 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA\ncan be controlled by the norm of initial data. This and Proposition B.2 ( iii) (blow-\nupalternative)indicatetheexistenceoftheglobalsolutioniftheinit ialdata(u0,u1)\nis sufficiently small.\nIt remains to prove the asymptotic estimate. To this end, we go bac k to the\nestimate ( 4.2). By virtue of the a priori estimate ( 4.2), we have\nd\ndsE(s)+λE(s)+1\n8/parenleftbig\nG(s)+ms(s)2/parenrightbig\n≤Ce−min{2δ,2α−β+1\nα−β+1,2(p−2)+2α−β+1\nα−β+1}s/tildewideE(sm)\n=Ce−min{2(β+1)\nα−β+1,2α−β+1\nα−β+1}s/tildewideE(sm),\nwhere we have also used /tildewideE(sm), which can be assumed without loss of generality.\nNow, recall\nλ∈/parenleftbigg\n0,min/braceleftbigg1\n2,2(β+1)\nα−β+1,2α−β+1\nα−β+1/bracerightbigg/parenrightbigg\n,\nand multiply the above estimate by eλs. Then, we obtain\nd\nds/bracketleftbig\neλsE(s)/bracketrightbig\n+1\n8eλs/parenleftbig\nG(s)+ms(s)2/parenrightbig\n≤Ceλ−min{2(β+1)\nα−β+1,2α−β+1\nα−β+1}s/tildewideE(sm).\nIntegrating this over [ sm,s] implies\neλsE(s)+1\n8/integraldisplays\nsmeλσ/parenleftbig\nG(σ)+ms(σ)2/parenrightbig\ndσ≤C/tildewideE(sm).\nTherefore, we have\nE(s)≤Ce−λs/tildewideE(sm) (4.4)\nfor alls≥sm. Moreover, we deduce\n/integraldisplays\nsmeλσms(σ)2dσ≤C/tildewideE(sm).\nThis shows, for any s≥s′≥sm,\n|m(s)−m(s′)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplays\ns′ms(σ)dσ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/parenleftbigg/integraldisplays\ns′e−λσdσ/parenrightbigg1/2/parenleftbigg/integraldisplays\ns′eλσms(σ)2dσ/parenrightbigg1/2\n≤/parenleftbigg1\nλ(e−λs′−e−λs)/parenrightbigg1/2\nC/tildewideEm(sm)1/2\n→0 (s′,s→ ∞).\nThis means that the limit m∗= lims→∞m(s) exists and satisfies\n|m∗−m(s)|2≤C/tildewideE(sm)e−λs\nfor alls≥sm. Consequently, by the above estimate and ( 4.2), we have\n/bardblv(s)−m∗ϕ/bardbl2\nL2=/bardblm(s)ϕ+f(s)−m∗ϕ/bardbl2\nL2\n≤C/parenleftbig\n|m∗−m(s)|2/bardblϕ/bardbl2\nL2+/bardblf(s)/bardbl2\nL2/parenrightbig\n≤Ce−λs/tildewideE(sm)\n≤Ce−λs/parenleftbig\n/bardblu0/bardbl2\nH2,1∩H3,0+/bardblu1/bardblH0,1∩H1,0/parenrightbig2NONLINEAR DAMPED BEAM EQUATION 23\nfors≥sm, which implies\n/bardblu(t)−m∗G(R(t))/bardbl2\nL2≤C(R(t)+1)−1\n2−λ/parenleftbig\n/bardblu0/bardbl2\nH2,1∩H3,0+/bardblu1/bardblH0,1∩H1,0/parenrightbig2\nfort≥tm:=R−1(es−1). This completes the proof of the asymptotic estimate.\nAppendix A.A general lemma for the energy identity\nIn this appendix, we give a proof of Lemma 3.1. Actually, we give a slightly\nmore general version of it and prove the following lemma. If we take k=1\n2and\nc3(s)≡1, then we have Lemma 3.1.\nLemma A.1. Letk,l,m∈R,n∈N∪ {0}, and letcj=cj(s) (j= 1,2,3,4)\nbe smooth functions defined on [0,∞). We consider a system for two functions\nf=f(s,y)andg=g(s,y)given by\n/braceleftBigg\nfs−kyfy−lf=g,\nc1(s)(gs−kygy−mg)+c2(s)g+g=c3(s)fyy−c4(s)fyyyy+h(s,y)∈(0,∞)×R,\n(A.1)\nwhereh=h(s,y)is a given smooth function belonging to C([0,∞);H0,n(R)). We\ndefine the energies\nE1(s) =1\n2/integraldisplay\nRy2n/parenleftbig\nc3(s)f2\ny+c4(s)f2\nyy+c1(s)g2/parenrightbig\ndy,\nE2(s) =/integraldisplay\nRy2n/parenleftbigg1\n2f2+c1(s)fg/parenrightbigg\ndy.\nThen, we have\nd\ndsE1(s) =−/integraldisplay\nRy2ng2dy+/parenleftbigg\n−2n−1\n2k+l/parenrightbigg\nc3(s)/integraldisplay\nRy2nf2\nydy+/parenleftbigg\n−2n−3\n2k+l/parenrightbigg\nc4(s)/integraldisplay\nRy2nf2\nyydy\n+/parenleftbigg\n−2n+1\n2k+m/parenrightbigg\nc1(s)/integraldisplay\nRy2ng2dy−c2(s)/integraldisplay\nRy2ng2dy\n−2nc3(s)/integraldisplay\nRy2n−1fygdy−2n(2n−1)c4(s)/integraldisplay\nRy2n−2fyygdy−4nc4(s)/integraldisplay\nRy2n−1fyygydy\n+c′\n3(s)\n2/integraldisplay\nRy2nf2\nydy+c′\n4(s)\n2/integraldisplay\nRy2nf2\nyydy+c′\n1(s)\n2/integraldisplay\nRy2ng2dy+/integraldisplay\nRy2nghdy24 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA\nand\nd\ndsE2(s) =−c3(s)/integraldisplay\nRy2nf2\nydy−c4(s)/integraldisplay\nRy2nf2\nyydy+/parenleftbigg\n−2n+1\n2k+l/parenrightbigg/integraldisplay\nRy2nf2dy\n+c1(s)/integraldisplay\nRy2ng2dy+(−(2n+1)k+l+m)c1(s)/integraldisplay\nRy2nfgdy−c2(s)/integraldisplay\nRy2nfgdy\n−2nc3(s)/integraldisplay\nRy2n−1ffydy−4nc4(s)/integraldisplay\nRy2n−1fyfyydy−2n(2n−1)c4(s)/integraldisplay\nRy2n−2ffyydy\n+c′\n1(s)/integraldisplay\nRy2nfgdy+/integraldisplay\nRy2nfhdy.\nProof of Lemma A.1.We calculate\nd\ndsE1(s) =d\nds/bracketleftbigg1\n2/integraldisplay\nRy2n/parenleftbig\nc3(s)f2\ny+c4(s)f2\nyy+c1(s)g2/parenrightbig\ndy/bracketrightbigg\n=c3(s)/integraldisplay\nRy2nfyfysdy+c′\n3(s)\n2/integraldisplay\nRy2nf2\nydy\n+c4(s)/integraldisplay\nRy2nfyyfyysdy+c′\n4(s)\n2/integraldisplay\nRy2nf2\nyydy\n+c1(s)/integraldisplay\nRy2nggsdy+c′\n1(s)\n2/integraldisplay\nRy2ng2dy.\nUsing the equation ( A.1), we rewrite the above identity as\nd\ndsE1(s) =c3(s)/integraldisplay\nRy2nfy(kyfy+lf+g)ydy+c′\n3(s)\n2/integraldisplay\nRy2nf2\nydy\n+c4(s)/integraldisplay\nRy2nfyy(kyfy+lf+g)yydy+c′\n4(s)\n2/integraldisplay\nRy2nf2\nyydy\n+c1(s)/integraldisplay\nRy2ng(kygy+mg)dy−c2(s)/integraldisplay\nRy2ng2dy−/integraldisplay\nRy2ng2dy\n+c3(s)/integraldisplay\nRy2ngfyydy−c4(s)/integraldisplay\nRy2ngfyyyydy+/integraldisplay\nRy2nghdy\n+c′\n1(s)\n2/integraldisplay\nRy2ng2dy.NONLINEAR DAMPED BEAM EQUATION 25\nBy noting the relations\ny2nfy(yfy)y=/parenleftbiggy2n+1\n2f2\ny/parenrightbigg\ny−2n−1\n2y2nf2\ny,\ny2nfyy(yfy)yy=/parenleftbiggy2n+1\n2f2\nyy/parenrightbigg\ny−2n−3\n2y2nf2\nyy,\ny2ng(ygy) =/parenleftbiggy2n+1\n2g2/parenrightbigg\ny−2n+1\n2y2ng2,\ny2ngfyy=/parenleftbig\ny2ngfy/parenrightbig\ny−y2nfygy−2ny2n−1fyg,\ny2ngfyyyy=/parenleftbig\ny2ngfyyy/parenrightbig\ny−/parenleftbig\n(y2ng)yfyy/parenrightbig\ny\n+/parenleftbig\n2n(2n−1)y2n−2g+4ny2n−1gy+y2ngyy/parenrightbig\nfyy,\nwe have\nd\ndsE1(s) =/parenleftbigg\n−2n−1\n2k+l/parenrightbigg\nc3(s)/integraldisplay\nRy2nf2\nydy+c3(s)/integraldisplay\nRy2nfygydy+c′\n3(s)\n2/integraldisplay\nRy2nf2\nydy\n+/parenleftbigg\n−2n−3\n2k+l/parenrightbigg\nc4(s)/integraldisplay\nRy2nf2\nyydy+c4(s)/integraldisplay\nRy2nfyygyydy+c′\n4(s)\n2/integraldisplay\nRy2nf2\nyydy\n+/parenleftbigg\n−2n+1\n2k+m/parenrightbigg\nc1(s)/integraldisplay\nRy2ng2dy−c2(s)/integraldisplay\nRy2ng2dy−/integraldisplay\nRy2ng2dy\n−c3(s)/integraldisplay\nRy2nfygydy−2nc3(s)/integraldisplay\nRy2n−1fygdy\n−2n(2n−1)c4(s)/integraldisplay\nRy2n−2fyygdy−4nc4(s)/integraldisplay\nRy2n−1fyygydy−c4(s)/integraldisplay\nRy2nfyygyydy\n+/integraldisplay\nRy2nghdy+c′\n1(s)\n2/integraldisplay\nRy2ng2dy.\nThus, we conclude\nd\ndsE1(s) =−/integraldisplay\nRy2ng2dy+/parenleftbigg\n−2n−1\n2k+l/parenrightbigg\nc3(s)/integraldisplay\nRy2nf2\nydy+/parenleftbigg\n−2n−3\n2k+l/parenrightbigg\nc4(s)/integraldisplay\nRy2nf2\nyydy\n+/parenleftbigg\n−2n+1\n2k+m/parenrightbigg\nc1(s)/integraldisplay\nRy2ng2dy−c2(s)/integraldisplay\nRy2ng2dy\n−2nc3(s)/integraldisplay\nRy2n−1fygdy−2n(2n−1)c4(s)/integraldisplay\nRy2n−2fyygdy−4nc4(s)/integraldisplay\nRy2n−1fyygydy\n+c′\n3(s)\n2/integraldisplay\nRy2nf2\nydy+c′\n4(s)\n2/integraldisplay\nRy2nf2\nyydy+c′\n1(s)\n2/integraldisplay\nRy2ng2dy+/integraldisplay\nRy2nghdy.26 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA\nThis gives the desired identity for E1(s). Next, we compute\nd\ndsE2(s) =d\nds/bracketleftbigg/integraldisplay\nRy2n/parenleftbigg1\n2f2+c1(s)fg/parenrightbigg\ndy/bracketrightbigg\n=/integraldisplay\nRy2nffsdy+c1(s)/integraldisplay\nRy2nfsgdy+c1(s)/integraldisplay\nRy2nfgsdy+c′\n1(s)/integraldisplay\nRy2nfgdy.\nUsing the equation ( A.1), we rewrite the above identity as\nd\ndsE2(s) =/integraldisplay\nRy2nf(kyfy+lf+g)dy+c1(s)/integraldisplay\nRy2n(kyfy+lf+g)gdy\n+c1(s)/integraldisplay\nRy2nf(kygy+mg)dy−c2(s)/integraldisplay\nRy2nfgdy−/integraldisplay\nRy2nfgdy\n+c3(s)/integraldisplay\nRy2nffyydy−c4(s)/integraldisplay\nRy2nffyyyydy\n+/integraldisplay\nRy2nfhdy+c′\n1(s)/integraldisplay\nRy2nfgdy.\nBy noting the relations\ny2nf(yfy) =/parenleftbiggy2n+1\n2f2/parenrightbigg\ny−2n+1\n2y2nf2,\ny2nf(ygy) =/parenleftbig\ny2n+1fg/parenrightbig\ny−y2n+1fyg−(2n+1)y2nfg,\ny2nffyy=/parenleftbig\ny2nffy/parenrightbig\ny−y2nf2\ny−2ny2n−1ffy,\ny2nffyyyy=/parenleftbig\ny2nffyyy/parenrightbig\ny−/parenleftbig\n(y2nf)yfyy/parenrightbig\ny\n+(2n(2n−1)y2n−2f+4ny2n−1fy+y2nfyy)fyy,\nwe have\nd\ndsE2(s) =/parenleftbigg\n−2n+1\n2k+l/parenrightbigg/integraldisplay\nRy2nf2dy+/integraldisplay\nRy2nfgdy\n+kc1(s)/integraldisplay\nRy2n+1fygdy+lc1(s)/integraldisplay\nRy2nfgdy+c1(s)/integraldisplay\nRy2ng2dy\n−kc1(s)/integraldisplay\nRy2n+1fygdy+(−(2n+1)k+m)c1(s)/integraldisplay\nRy2nfgdy\n−c2(s)/integraldisplay\nRy2nfgdy−/integraldisplay\nRy2nfgdy\n−c3(s)/integraldisplay\nRy2nf2\nydy−2nc3(s)/integraldisplay\nRy2n−1ffydy\n−2n(2n−1)c4(s)/integraldisplay\nRy2n−2ffyydy−4nc4(s)/integraldisplay\nRy2n−1fyfyydy−c4(s)/integraldisplay\nRy2nf2\nyydy\n+/integraldisplay\nRy2nfhdy+c′\n1(s)/integraldisplay\nRy2nfgdy.NONLINEAR DAMPED BEAM EQUATION 27\nThus, we conclude\nd\ndsE2(s) =−c3(s)/integraldisplay\nRy2nf2\nydy−c4(s)/integraldisplay\nRy2nf2\nyydy+/parenleftbigg\n−2n+1\n2k+l/parenrightbigg/integraldisplay\nRy2nf2dy\n+c1(s)/integraldisplay\nRy2ng2dy+(−(2n+1)k+l+m)c1(s)/integraldisplay\nRy2nfgdy−c2(s)/integraldisplay\nRy2nfgdy\n−2nc3(s)/integraldisplay\nRy2n−1ffydy−4nc4(s)/integraldisplay\nRy2n−1fyfyydy−2n(2n−1)c4(s)/integraldisplay\nRy2n−2ffyydy\n+c′\n1(s)/integraldisplay\nRy2nfgdy+/integraldisplay\nRy2nfhdy.\nThis completes the proof. /square\nAppendix B.Local existence\nWe discuss the local existence and basic properties of solutions to ( 1). Let\nX=H3,0(R)×H1,0(R) and\nU:=/parenleftbigg\nu\n∂tu/parenrightbigg\n, U0:=/parenleftbigg\nu0\nu1/parenrightbigg\n.\nLetD(A) =H5,0(R)×H3,0(R) and define\nA=/parenleftbigg\n0 1\n−∂4\nx0/parenrightbigg\n,T(t) = exp(tA).\nWe also define\nK(σ;U(s)) =/parenleftbigg\n0\n−b(σ)∂tu(s)+a(σ)∂2\nxu(s)+∂xN(∂xu(s))/parenrightbigg\n,\nnamely,σandsdenote the variables for the coefficients a(t),b(t) and the unknown\nu, respectively.\nNow, we introduce the definition of the strong solution and the mild so lution.\nDefinition B.1. LetI= [0,T]with someT >0orI= [0,∞). We say that a\nfunctionu(orU=t(u,∂tu)) is a strong solution to (1)onIif\n\n\nU∈C(I;D(A))∩C1(I;X),\nd\ndtU(t) =AU(t)+K(t;U(t))onI,\nU(0) =U0.\nAlso, we say that a function u(orU=t(u,∂tu)) is a mild solution to (1)onIif\n\n\nU∈C(I;X),\nU(t) =T(t)U0+/integraldisplayt\n0T(t−s)K(s;U(s))dsinC(I;X).\nProposition B.2. (i) (Local existence) For anyU0∈X, there exists T >0such\nthat there exists a mild solution to (1)on[0,T].\n(ii) (Uniqueness) LetT >0. IfUandVare mild solutions in C([0,T];X)with the\nsame initial condition U(0) =V(0) =U0, thenU=V.28 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA\n(iii) (Blow-up alternative) LetTmax=Tmax(U0)be\nTmax= sup{T∈(0,∞];∃U∈C([0,T];X) :a mild solution to (1)}.\nIfTmax<∞, thenlimt→Tmax−0/bardblU(t)/bardblX=∞.\n(iv) (Continuous dependence on the initial data) LetU0∈Xand{U(j)\n0}∞\nj=1a\nsequence in Xsatisfying limj→∞/bardblU(j)\n0−U0/bardblX= 0. LetUandU(j)be the corre-\nsponding mild solutions to the initial data U0andU(j)\n0, respectively. Then, for any\nfixedT∈(0,Tmax(U0)), we haveTmax(U(j)\n0)>Tfor sufficiently large jand\nlim\nj→∞sup\nt∈[0,T]/bardblU(j)(t)−U(t)/bardblX= 0.\n(v) (Regularity) LetT >0. IfU0∈D(A), then the mild solution in (i)on[0,T]\nbecomes a strong solution on [0,T].\n(vi) (Small data almost global existence) For anyT >0, there exists ε0>0such\nthat if/bardblU0/bardblX< ε0, then the corresponding mild solution Ucan be extended to\n[0,T].\n(vii) (Boundedness of weighted norm) LetT >0andY:=H2,1(R)×H0,1(R). If\nU0∈X∩Y, then the corresponding mild solution Uon[0,T]belongs toC([0,T];X∩\nY).\nProof.LetT0>0 be fixed. Then, a(t),b(t) are positive, and they and their first\nderivativesarebounded by someconstant CT0>0on [0,T0]. LetT∈(0,T0]. Then,\nfor anyU=t(u,v)∈Xandt∈[0,T], we have\n/bardblK(t;U)/bardblX=/vextenddouble/vextenddouble−b(t)∂tu+a(t)∂2\nxu+∂xN(∂xu)/vextenddouble/vextenddouble\nH1\n≤CT0/parenleftbig\n/bardbl∂tu/bardblH1+/bardbl∂2\nxu/bardblH1/parenrightbig\n+C/parenleftBig\n/bardbl∂xu/bardblW1,∞+/bardbl∂xu/bardblp−1\nW1,∞/parenrightBig\n/bardbl∂2\nxu/bardblH1<∞,\nthat is,K(t;·) :X→X. Moreover, for M >0 andU=t(u,v),W=t(w,z)∈\nBM={U∈X;/bardblU/bardblX≤M}, we calculate\n/bardblK(t;U)−K(t;W)/bardblX≤CT0(/bardblv−z/bardblH1+/bardblu−w/bardblH3)\n+C(/bardblu/bardblW2,∞+/bardblw/bardblW2,∞)/bardblu−w/bardblH3\n+C(|u/bardblW2,∞+/bardblw/bardblW2,∞)p−1/bardblu−w/bardblH3\n≤CT0,M/bardblU−W/bardblX.\nTherefore,K(t;·) islocally Lipschitzcontinuousin X. Therefore, fromthe proofs of\n[3, Lemmas 4.3.2, Proposition 4.3.3], there exist T >0 and a unique mild solution\nuonI= [0,T]. Also, [ 3, Theorem 4.3.4] shows the property (iii). Moreover, by\n[3, Proposition 4.3.7], the continuous dependence on the initial data. T his proves\n(i)–(iv).\nNext, we prove (iv) along with the argument of [ 3, Lemma 4.3.9]. Take U0∈\nD(A) andT∈(0,Tmax). Leth >0,t∈[0,T−h], andM:= sups∈[0,T]/bardblU(s)/bardblX.NONLINEAR DAMPED BEAM EQUATION 29\nConsider\nU(t+h)−U(t) =T(h)T(t)U0−T(t)U0\n+/integraldisplayt\n0T(s){K(t+h−s;U(t+h−s))−K(t−s;U(t−s)))}ds\n+/integraldisplayh\n0T(t+s)K(h−s;U(h−s))ds\n=:J1+J2+J3.\nForJ1,J2, we estimate\n/bardblJ1/bardblX≤ /bardblT(h)U0−U0/bardblX=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayh\n0T(s)AU0ds/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nX≤h/bardblAU0/bardblX,\n/bardblJ3/bardblX≤hsup\ns∈[0,T]/bardblK(s;U(s))/bardblX.\nForJ2, using the Lipschitz continuity of a,b: [0,T]→RandK(s:·) :X→X, we\ncan show\n/bardblJ2/bardblX≤/integraldisplayt\n0/bardblK(t+h−s;U(t+h−s))−K(t+h−s;U(t−s))/bardblXds\n+/integraldisplayt\n0/bardblK(t+h−s;U(t−s))−K(t−s;U(t−s))/bardblXds\n≤CT0,Mh+CT0,M/integraldisplayt\n0/bardblU(s+h)−U(s)/bardblXds.\nThen, the Gronwall inequality implies\n/bardblU(t+h)−U(t)/bardblX≤CT0,Mh,\nthat is,U: [0,T]→Xis Lipschitz continuous. This further leads to\n/bardblK(t;U(t))−K(s;U(s))/bardblX≤ /bardblK(t;U(t))−K(s;U(t))/bardblX+/bardblK(s;U(t))−K(s;U(s))/bardblX\n≤CT0,M|t−s|,\ni.e.,K(·;U(·)) : [0,T]→Xis Lipschitz continuous, and hence, K(·;U(·))∈\nW1,1((0,T);X). This enables us to apply [ 3, Lemma 4.16] and ubecomes a strong\nsolution. This proves (v).\nNext, we prove (vi). Let T >0 be arbitrary fixed, I:= [0,T], and\nCT,a,b:=/integraldisplayT\n0(|a(s)|+|b(s)|)ds.\nLetε >0 be sufficiently small so that 2(1 + CT,a,b)ε <1 and let Bε={U∈\nC([0,T];X); supt∈[0,T]/bardblU(t)/bardblX≤2(1 +CT,a,b)ε}. Define a map Φ : C(I:X)→\nC(I;X) by\nΦ[U](t) :=T(t)U0+/integraldisplayt\n0T(t−s)K(s;U(s))ds.30 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA\nThen, forU0satisfying /bardblU0/bardblX≤εandU=t(u,v)∈ Bε, we see that\n/bardblΦ[U](t)/bardblX≤ /bardblT(t)U0/bardblX+/integraldisplayt\n0/bardblT(t−s)K(s;U(s))/bardblXds\n≤ /bardblU0/bardblX+/integraldisplayt\n0/parenleftbig\n|b(s)|/bardblv(s)/bardblH1+|a(s)|/bardbl∂2\nxu(s)/bardblH1/parenrightbig\nds\n+/integraldisplayt\n0/bardbl∂xN(∂xu(s))/bardblH1ds\n≤(1+CT,a,b)ε+TCN(2(1+CT,a,b)ε)2,\nwhereCN>0 is a constant depending only on the nonlinearity N. Similarly, we\nhave, forU,V∈ Bε,\n/bardblΦ[U](t)−Φ[V](t)/bardblX≤/integraldisplayt\n0/bardblK(s;U(s))−K(s;V(s))/bardblXds\n≤T˜CN(2(1+CT,a,b)ε) sup\ns∈[0,T]/bardblU(s)−V(s)/bardblX,\n˜CN>0 is a constant depending only on the nonlinearity N. Therefore, taking ε\nfurther small so that\nTCN(2(1+CT,a,b)ε)≤1, T˜CN(2(1+CT,a,b)ε)≤1\n2,\nwe see that Φ is a contraction mapping on Bε. This and the uniqueness of mild\nsolution imply that the mild solution obtained in (i) can be extended to [0 ,T].\nFinally, we prove (vii). Let T >0,I= [0,T],U0∈Y, and letUbe the corre-\nspondingmildsolutionon[0 ,T]totheinitialdata U0. WeputM:= supt∈I/bardblU(t)/bardblX.\nIn order to justify the following energy method, we take a sequenc e{U(j)\n0}∞\nj=1from\n[C∞\n0(R)]2such that lim j→∞U(j)\n0=U0inX∩Y. Then, the corresponding strong\nsolutionU(j)∈C(I;D(A))∩C1(I;X) to the data U(j)\n0satisfies lim j→∞U(j)=U\ninC(I;X) by the continuous dependence on the initial data. In particular, t aking\nsufficiently large j, we may suppose that supj∈N,t∈I/bardblU(j)(t)/bardblX≤2M.\nLet\nχ∈C∞\n0(R),0≤χ≤1, χ(x) =/braceleftBigg\n1 (|x| ≤1),\n0 (|x| ≥2),\nχn(x) :=χ/parenleftBigx\nn/parenrightBig\n(n∈N).\nBy suppχn⊂[−2n,2n], we easily see that\n|∂x(x2χn(x)2)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle2xχn(x)2+2x2\nnχ′/parenleftBigx\nn/parenrightBig\nχn(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C|x|χn(x),\n|∂2\nx(x2χn(x)2)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle2χn(x)2+4x\nnχ′/parenleftBigx\nn/parenrightBig\nχn(x)+2x2\nn2/parenleftbigg/parenleftBig\nχ′/parenleftBigx\nn/parenrightBig/parenrightBig2\n+χ′′/parenleftBigx\nn/parenrightBig\nχn(x)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤CNONLINEAR DAMPED BEAM EQUATION 31\nwith some constant C >0. Denote U=t(u,∂tu),U(j)=t(u(j),∂tu(j)), and\nconsider\nEn(t;u) :=/integraldisplay\nRx2χn(x)2/parenleftbig\n|∂tu(t,x)|2+a(t)|∂xu(x)|2+|∂2\nxu(t,x)|2+|u(t,x)|2/parenrightbig\ndx,\nE(t;u) :=/integraldisplay\nRx2/parenleftbig\n|∂tu(t,x)|2+a(t)|∂xu(x)|2+|∂2\nxu(t,x)|2+|u(t,x)|2/parenrightbig\ndx.\nNote thatEn(t;u(j)) is finite thanks to χn. Differentiating it, we have\nd\ndtEn(t;u(j)) = 2/integraldisplay\nRx2χn(x)2/parenleftBig\n∂tu(j)∂2\ntu(j)+a(t)∂xu(j)∂t∂xu(j)+∂2\nxu(j)∂t∂2\nxu(j)/parenrightBig\ndx\n+2/integraldisplay\nRx2χn(x)2u(j)∂tu(j)dx+/integraldisplay\nRx2χn(x)2a′(t)|∂xu(j)|2dx.\nBy the integration by parts and using the equation ( 1), the right-hand side can be\nwritten as\n2/integraldisplay\nRx2χn(x)2∂tu(j)/parenleftBig\n−b(t)∂tu(j)+∂xN(∂xu(j))/parenrightBig\ndx\n−2/integraldisplay\nR∂x(x2χn(x)2)a(t)∂xu(j)∂tu(j)dx\n+4/integraldisplay\nR∂x(x2χn(x)2)a(t)∂3\nxu(j)∂tu(j)dx+2/integraldisplay\nR∂2\nx(x2χn(x)2)a(t)∂2\nxu(j)∂tu(j)dx\n+2/integraldisplay\nRx2χn(x)2u(j)∂tu(j)dx+/integraldisplay\nRx2χn(x)2a′(t)|∂xu(j)|2dx.\nThe above quantity can be further estimated by\nC(2M)2+CT,a,b,MEn(t;u(j))\nwith some constants C,CT,a,b,M>0. Hence, the Gronwall inequality implies\nEn(t;u(j))≤˜CT,a,b,M,\nwhere the constant ˜CT,a,b,Mis independent of nandj. Lettingj→ ∞first and\nusing the continuous dependence on the initial data, we have\n/integraldisplay\nRx2χn(x)2/parenleftbig\n|∂tu(t,x)|2+a(t)|∂xu(x)|2+|∂2\nxu(t,x)|2+|u(t,x)|2/parenrightbig\ndx≤˜CT,a,b,M.\nThen, letting n→ ∞, we conclude\n/integraldisplay\nRx2/parenleftbig\n|∂tu(t,x)|2+a(t)|∂xu(x)|2+|∂2\nxu(t,x)|2+|u(t,x)|2/parenrightbig\ndx≤˜CT,a,b,M,\nwhich shows U(t)∈Yfor anyt∈[0,T]. The continuity of /bardblU(t)/bardblYintfollows\nfrom the estimate\n|En(t;u(j))−En(s;u(j))| ≤/integraldisplayt\ns/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndσEn(σ;u(j))/vextendsingle/vextendsingle/vextendsingle/vextendsingledσ≤CT,a,b,M(t−s)\nfors<tand taking the limits j→ ∞andn→ ∞.\n/square32 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA\nAcknowledgements\nThis work was supported by JSPS KAKENHI Grant Numbers JP18H01 132,\nJP20K14346.\nReferences\n[1]J. M. Ball, Stability theory for an extensible beam, J. Differential Equations, 14(1973),\n399–418.\n[2]M. Brokate, J. 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Differential\nEquations 272(2021), 938–957."    },    {        "title": "2006.16510v1.Negative_Gilbert_damping_in_cavity_optomagnonics.pdf",        "content": "Negative Gilbert damping in cavity optomagnonics\nYunshan Cao\u0003and Peng Yany\nSchool of Electronic Science and Engineering and State Key Laboratory of Electronic Thin Films and Integrated Devices,\nUniversity of Electronic Science and Technology of China, Chengdu 610054, China\nExceptional point (EP) associated with the parity-time ( PT) symmetry breaking is receiving considerable\nrecent attention by the broad physics community. By introducing balanced gain and loss, it has been realized in\nphotonic, acoustic, and electronic structures. However, the observation of magnonic EP remains elusive. The\nmajor challenge is to experimentally generate the negative Gilbert damping, which was thought to be highly\nunlikely but is demanded by the PT symmetry. In this work, we study the magneto-optical interaction of\ncircularly-polarized lasers with a submicron magnet placed in an optical cavity. We show that the o \u000b-resonant\ncoupling between the driving laser and cavity photon in the far-blue detuning can induce the magnetic gain (or\nnegative damping) exactly of the Gilbert type. A hyperbolic-tangent function ansatz is found to well describe\nthe time-resolved spin switching as the intrinsic magnetization dissipation is overcome. When the optically\npumped magnet interacts with a purely lossy one, we observe a phase transition from the imbalanced to passive\nPTsymmetries by varying the detuning coe \u000ecient. Our findings provide a feasible way to manipulate the sign\nof the magnetic damping parameter and to realize the EP in cavity optomagnonics.\nIntroduction. —One of the most fundamental principles in\nquantum mechanics is that a physical observable should be\ndescribed by a Hermitian operator to guarantee real eigenval-\nues [1]. However, Bender and Boettcher [2] reported a class\nof non-Hermitian Hamiltonians that allow entirely real spec-\ntrum as long as the combined parity ( P) and time (T)-reversal\nsymmetries are respected. By tuning system parameters, both\nthe eigenvalues and eigenstates of the PT-symmetric Hamil-\ntonian simultaneously coalesce [3, 4], giving rise to a non-\nHermitian degeneracy called exceptional point (EP). The na-\nture around the EP that is accompanied by a phase transi-\ntion can trigger many intriguing phenomena, such as unidi-\nrectional invisibility [5, 6], loss-induced laser suppression and\nrevival [7] and optical transparency [8], laser mode selection\n[9], and EP enhanced sensing [10–13]. Over the past decades,\nthe experimental observation of EPs has been realized in a\nbroad field of photonics [14–17], acoustics [18, 19], and elec-\ntronics [20–22]. Very recently, the concept of PTsymmetry\nis attracting significant attention in spintronics and magnonics\n[23–31]. The simplest way to obtain a PT-symmetric system\nconsists in coupling two identical subsystems, one with gain\nand the other with equal amount of loss. The composite sys-\ntem isPT symmetric because space reflection interchanges\nthe subsystems, and time reversal interchanges gain and loss.\nIndeed, aPT-symmetric magnetic structure composed of two\nidentical ferromagnets with balanced gain and loss was first\nproposed by Lee et al. [23] and subsequently investigated by\nYang et al. [27]. One recent breakthrough was made by Liu\net al. [32] who reported EP in passive PT-symmetric devices\nin the form of a trilayer structure with two magnetic layers\nof di\u000berent (positive) Gilbert damping. However, the exper-\nimental observation of genuine PT symmetry for magnons\n(the quanta of spin waves)—as elementary excitations in or-\ndered magnets—is still elusive. The di \u000eculty lies in that the\nGilbert damping can hardly be tuned to be negative [33, 34].\nThe past ten years have witnessed the development and\napplication of spin cavitronics, allowing cavity photons res-\nonantly coupled to magnons with the same microwave fre-quency [35–46]. One recent trend beyond microwaves is the\nrealization of the parametric coupling between optical lasers\nand magnons, that would generate interesting new opportuni-\nties. Tantalizing physics indeed has been demonstrated, such\nas nonreciprocal Brillouin light scattering [47], microwave-\nto-optical converting [48, 49], optical cooling of magnons\n[50], etc. In these studies, considerable interests have been\ndrawn to the scalar properties of magnons, e.g., magnon num-\nber (population), temperature, and chemical potential, which\nis successful to describe the small-angle spin precession. In\ncontrast, their vectorial behavior, i.e., the full time-evolution\nof the magnetic moment driven by optical lasers, remains\nlargely unexplored, with few exceptions [51]. It has been\nshown that a ferromagnetic-to-antiferromangetic phase tran-\nsition may emerge in the vicinity of the magnonic EP [27]. In\nsuch case, the magnetic moment would significantly deviate\nfrom its equilibrium direction, and a vectorial field descrip-\ntion becomes more relevant than a scalar one.\nS\nz\nx y\nCircularly polarized \nlaser beamOptical cavity(a) (b)\n(c)ωlas > ωcav\nωlas‘ωm\nRed-detuningBlue-detuning\nωlas < ωcav\nωlas‘\nωm\nFIG. 1: (a) Schematic illustration of a macrospin Sinteracting\nwith three orthogonally propagating circularly-polarized lasers (red\nbeams) in an optical cavity. O \u000b-resonant coupling between the driv-\ning laser (!las) and the cavity photon ( !cav) mediated by magnons\n(!m\u001c!cav) in the blue (b) and red (c) detuning regimes.\nIn this Letter, we propose to realize the negative GilbertarXiv:2006.16510v1  [cond-mat.mtrl-sci]  30 Jun 20202\ndamping by considering the optomagnonic interaction be-\ntween three orthogonally propagating circularly-polarized\nlasers and a submicron magnet placed in an optical cavity [see\nFig. 1(a)]. By solving the coupled equations of motion and\nintegrating the photon’s degree of freedom, we derive the an-\nalytical formula of the optical torque acting on the macrospin.\nIn the far-blue detuning, we find that the optical torque exactly\ntakes the Gilbert form \u0000\u000bopt\nS˙S\u0002Swith\u000bopt>0 (see below).\nThe total Gilbert damping becomes negative when the intrin-\nsic dissipation is overcome. In such case, a hyperbolic-tangent\nfunction ansatz is found to well describe the time-resolved\nspin switching. We further study the optically pumped spin\ninteracting with a purely lossy one, and observe a phase transi-\ntion from the imbalanced to passive PTsymmetries by vary-\ning the detuning parameter.\nModel. —The proposed setup is schematically plotted in\nFig. 1(a). Three circularly-polarized laser beams propagat-\ning respectively along x;y;zdirections drive the parametric\ncoupling with a macrospin S=(ˆSx;ˆSy;ˆSz) inside the optical\ncavity. The Hamiltonian reads\nH=\u0000~!0ˆSz\u0000~X\nj=x;y;z\u0010\n\u0001j\u0000gjˆSj\u0011\nˆcy\njˆcj+Hdr; (1)\nwhere!0=\rB0is the Larmor frequency around the exter-\nnal magnetic field B0pointing to the negative z-direction with\n\rbeing the gyromagnetic ratio, \u0001j=!las;j\u0000!cavis the de-\ntuning between the laser frequency !las;jand the cavity reso-\nnant frequency !cav, and ˆ cy\nj(ˆcj) is the creation (annihilation)\noperator of the optical cavity photons, with j=x;y;z. The\ncoupling strength gjbetween the spin and optical photon orig-\ninates from the Faraday-induced modification of the electro-\nmagnetic energy in ferromagnets [52]. The last term describes\nthe interaction between the driving laser and the cavity pho-\ntonHdr=i~P\nj(Ajˆcy\nj\u0000h:c:), where Aj=(2\u0014jPj=~!las;j)1=2is\nthe field amplitude, with \u0014jthe laser loss rate and Pjbeing the\ndriving power.\nThe Heisenberg-Langevin equations of motion for coupled\nphotons and spins are expressed as ( o\u0011hˆoi),\n˙cj=(i\u0001j\u0000\u0014j)cj\u0000igjSjcj+Aj; (2a)\n˙Sx=!0Sy+gynySz\u0000gznzSy; (2b)\n˙Sy=\u0000!0Sx\u0000gxnxSz+gznzSx; (2c)\n˙Sz=\u0000gynySx+gxnxSy; (2d)\nwhere nj=hˆcy\njˆcjiis the average photon number in the cav-\nity. Because the spin dynamics usually is much slower than\noptical photons, one can expand the cavity photon operator as\ncj(t)\u0019cj0(t)+cj1(t)+\u0001\u0001\u0001, in orders of ˙Sj. Equation (2a) then\ncan be recast in series\n0=(i\u0001j\u0000\u0014j)cj0\u0000igjSjcj0+Aj; (3a)\n˙cj0=(i\u0001j\u0000\u0014j)(cj0+cj1)\u0000igjSj(cj0+cj1)+Aj;(3b)\nby keeping up to the first-order terms. We can therefore derivethe formula of photon number in the cavity\nnj(t)\u0019 jcj0j2+2Re[ c\u0003\nj0cj1]\n=A2\nj\n(\u0001j\u0000gjSj)2+\u00142\nj\u00004\u0014jA2\njgj(\u0001j\u0000gjSj)\nh\n(\u0001j\u0000gjSj)2+\u00142\nji3˙Sj:(4)\nSubstituting (4) into Eqs. (2b)-(2d), we obtain\n˙S=\u0000\rS\u0002Be\u000b+\u000b\nS(˙S\u0002S)\u0000\fopt\u0002S; (5)\nwhere the e \u000bective magnetic field Be\u000b=\u0000B0ez+Boptin-\ncludes both the external magnetic field and the optically in-\nduced magnetic field\nBopt=X\nj\r\u00001gjA2\nj\n(\u0001j\u0000gjSj)2+\u00142\njej; (6)\nwhich is the zeroth-order of ˙Sj. The second term in the right\nhand side of (5) is the intrinsic Gilbert damping torque, with\nS=jSjthe total spin number and \u000b > 0 being the intrinsic\nGilbert damping constant. The last term in (5) represents the\noptical torque with the anisotropic e \u000bective field\n\fopt=X\nj4\u0014jA2\njg2\nj(\u0001j\u0000gjSj)\nh\n(\u0001j\u0000gjSj)2+\u00142\nji3˙Sjej; (7)\nwhich is linear with the first-order time-derivative of Sj. Be-\nlow, we show that the anisotropic nature of (7) can be smeared\nout under proper conditions.\nNegative Gilbert damping. —To obtain the optical torque of\nexactly the Gilbert form, we make two assumptions: (i) the\nthree laser beams are identical, i.e., Aj=A;gj=g;\u0014j=\u0014;\nand\u0001j= \u0001; (ii) the optomagnonic coupling works in the far\ndetuning regime, i.e., j\u0011j\u001d1 with\u0011= \u0001=(gS), which allows\nus to drop the gjSjterms in Eq. (7). The optically induced\ne\u000bective fields then take the simple form\nBopt=\r\u00001gA2\n\u00012+\u00142X\njej; (8)\nand\n\fopt=\u000bopt\nS˙S;with\u000bopt=4\u0014A2g2S\u0001\n(\u00012+\u00142)3(9)\nbeing the laser-induced magnetic gain or loss that depends the\nsign of the detuning \u0001. Based on the above results, we finally\nobtain the optically modulated spin dynamics\n˙S=\u0000\rS\u0002Be\u000b+\u000be\u000b\nS(˙S\u0002S); (10)\nwith\u000be\u000b=\u0000\u000bopt+\u000b. One can observe that a negative ef-\nfective Gilbert constant ( \u000be\u000b<0) emerges in the far-blue de-\ntuning regime, i.e., 1 < \u0011 < \u0011 c. In case of the red detun-\ning (\u0011 < 0), we have \u000bopt<0, which indicates the enhance-\nment of the magnetic attenuation. In the deep-blue detuning3\nηηηPTηC\nηC=7.11η P (W)P (μWĎ\nr (m)αopt /α Bopt (μT)αeff =0\nBopt =333 μT(a)\n(b)(c)\n(d)\nηPTηCBopt =440 μTαeff =-α\n×\nFIG. 2: Optically induced magnetic gain (a) and magnetic field (b)\nvs. the optical detuning parameter \u0011. (c)\u0011PT(orange) and \u0011C(green)\nas a function of the driving laser power. (d) Radius dependence of\nthe laser power at the compensation point \u0011C=7:11.\nregime (\u0011>\u0011 c), driving lasers can still generate the magnetic\ngain (\u000bopt>0) but cannot compensate the intrinsic dissipa-\ntion, i.e., 0 < \u000b opt< \u000b. Here\u0011cis the critical detuning pa-\nrameter at which the e \u000bective Gilbert damping vanishes. The\nphysics can be understood from the diagram plotted in Figs.\n1(b) and 1(c): In the blue detuning regime ( !las> ! cav), mi-\ncrowave magnons are emitted in the non-resonant interaction\nbetween the driving laser and the cavity photon, representing\na magnetic gain. On the contrary, they are absorbed in the red\ndetuning (!las< ! cav), manifesting a magnon absorption or\ncooling. Below we discuss practical materials and parameters\nto realize this proposal.\nMaterials realizations. —For a ferromagnetic insulator like\nyttrium ion garnet (YIG), the intrinsic Gilbert constant \u000btyp-\nically ranges 10\u00003\u001810\u00005[53–55]. We take \u000b=10\u00004\nin the following calculations. The magneto-optical coupling\nstrength is determined by the Faraday rotation coe \u000ecient\u0012F\nof the materials gS'c\u0012F=p\u000fr, with cthe speed of light and\n\u000frthe relative permittivity (for YIG, we choose \u000fr=15 [56]\nand\u0012F=188\u000e=cm [57]). We thus have gS=2\u0019\u00191 GHz. The\noptical cavity is set at the resonant frequency !cav=2\u0019=100\nTHz with the loss rate \u0014=2\u0019=1 GHz. For a YIG sphere of\nradius r=10 nm and spin density \u001as\u00191028m\u00003, we esti-\nmate the total spin number S=\u001asr3\u0019104and the coupling\nstrength g=2\u0019\u00190:1 MHz. Materials parameters are summa-\nrized in Table I. Because g\u001c\u0014, all interesting physics occurs\nin the weak coupling regime. A negative \u000be\u000bis demanded for\nrealizing thePTsymmetry in magnetic system. Considering\nthe driving laser with a fixed power P=1\u0016W, the e \u000bective\nTABLE I: Parameters for optical cavity and YIG.\n!cav=2\u0019 \u0014= 2\u0019 ! 0=2\u0019 gS=2\u0019 r\u000b\n100 THz 1 GHz 10 GHz 1 GHz 10 nm 10\u00004Gilbert-type magnetic gain is \u000be\u000b=\u0000\u000bat\u0011PT'6:16, and\nthe critical gain-loss point \u000be\u000b=0 occurs at\u0011C'7:11, indeed\nsatisfying the large-detuning condition j\u0011j\u001d1 in deriving (9).\nFigure 2(a) shows the monotonically decreasing dependence\nof the optically induced magnetic gain \u000bopton the detuning pa-\nrameter\u0011. The\u0011-dependence of the optical field is plotted in\nFig. 2(b), showing that it monotonically decreases with the in-\ncreasing of the detuning, too. Enhancing the laser power will\npush the two critical points \u0011Cand\u0011PTinto the deep detuning\nregion, as demonstrated in Fig. 2(c). For a magnetic sphere\nof larger volume (1 \u0016m)3\u0018(1 mm)3that contains a total spin\nnumber S=1010\u00181019with the reduced magneto-optical\ncoupling strength g=2\u0019=10\u00001\u001810\u000010Hz, the required laser\npower then should be 6 \u001815 orders of magnitude higher than\nthe nm-scale sphere case, as shown in Fig. 2(d).\nTime-resolved spin flipping. —To justify the approximation\nadopted in deriving the Gilbert-type magnetic gain, we di-\nrectly simulate the time evolution of the unit spin components\n(sj\u0011Sj=S) based on both Eq. (5) and Eq. (10). Numer-\nical results are, respectively, plotted in Figs. 3(a) and 3(b)\nfor the same detuning parameter \u0011=1:8 (corresponding to\nan e\u000bective magnetic gain \u000be\u000b=\u00000:0453) and!0=2\u0019=10\nGHz. Both figures show that the very presence of the negative\nGilbert damping can flip the spin in a precessional manner,\nwith similar switching curves. The fast Fourier transforma-\ntion (FFT) analysis of the spatiotemporal oscillation of sxalso\nconfirms this point (see the insets). Although the analytical\nform of sz(t) by solving (5) generally is unknown [58, 59], we\nfind an ansatz that can well describe the time-resolved spin\nswitching\nsj sj\nszη=1.8 (αeff =-0.0453 )\n(a) (c)\n(d) (b)szsysx\nττ τ\nηEq. (5)\nEq. (10)τ0=98.9 (a) fitting\n(b) fitting\nTheoryτ0=107.8’’\nτ0=100.4’τp=22.1τp=22.4’’τp=14.9’ tanh(-      )τ-τ0τp\nτ0’\nτ0’’\nTheory\nτp’\nτp’’\nTheory46810121401020 9.779.71\nFrequency (GHz)Frequency (GHz)FFT of s x FFT of s x46810121401020\nFIG. 3: Time evolution of unit spin components ( sx;sy;sz) at de-\ntuning\u0011=1:8 based on Eq. (5) (a) and Eq. (10) (b). Insets\nshow the FFT spectrum of sx. (c) Theoretical fittings of szusing\nthe hyperbolic-tangent ansatz (11) (dashed curves). The solid green\ncurve is the analytical formula without any fitting. (d) Numerical re-\nsults of the \u0011-dependence of the two characteristic times \u001c0and\u001cp,\ncomparing with formula (12) (solid curves).4\nsz(\u001c)'tanh \n\u0000\u001c\u0000\u001c0\n\u001cp!\n; (11)\nwhich is reminiscent of the Walker solution for modeling the\nprofile of 180\u000emagnetic domain wall [60] by replacing the\ntime coordinate \u001cwith the space coordinate x. Here\u001c0is\nthe switching time, \u001cprepresents the life-time of uniform\nmagnons, and \u001c=!0t. From perturbation theory, we derive\nthe analytical form of these two parameters\n\u001cp=\u00001+\u000b2\ne\u000b\n\u000be\u000b;and\u001c0=\u001cptanh\u00001vt\n1\u00004B2\nopt\nB2\ne\u000b:(12)\nFigure 3(c) shows the time evolution of sz. Symbols repre-\nsent the numerical results, dashed curves label the theoretical\nfittings of ansatz (11), and the solid curve is the analytical for-\nmula without fitting. The fitted switching time \u001c0\n0=100:4\n(\u001c00\n0=107:8) and magnon life-time \u001c0\np=14:9 (\u001c00\np=22:4)\nfrom from Eq. (5) [Eq. (10)] compare well with the analyt-\nical formula (12) which gives \u001c0=98:9 and\u001cp=22:1. We\nfurther show that the analytical ansatz agrees excellently with\nnumerical results in a broad range of detuning parameters, as\nplotted in Fig. 3(d).\nPhase transition in spin dimers. —We have shown that un-\nder proper conditions, one can realize the Gilbert-type mag-\nnetic gain which is essential for observing PT-symmetry in\npurely magnetic structures. Next, we consider the optically\npumped spin Sinteracting with a lossy one S0, as shown in\nFig. 4(a). The coupled spin dynamics is described by the\nLandau-Lifshitz-Gilbert equation\n˙s=\u0000\rs\u0002Be\u000b+!exs\u0002s0+\u000be\u000b˙s\u0002s; (13a)\n˙s0=\u0000\rs0\u0002B0\ne\u000b+!exs0\u0002s+\u000b˙s0\u0002s0; (13b)\nwhere s(0)\u0011S(0)=Sis the unit spin vector. Since the optically\ninduced magnetic field is the same order of magnitude with\nthe geomagnetic field (much smaller than B0), it can be safely\nignored. Spin s0is exchange coupled to the optically pumped\nspins, and su \u000bers an intrinsic Gilbert damping. If \u000be\u000b=\u0000\u000b,\nthe two-spin system satisfies the PT-symmetry: Eqs. (13) are\ninvariant in the combined operation of the parity P(s$s0\nandBe\u000b$B0\ne\u000b) and the time reversal T(t!\u0000 t,s!\u0000s,\ns0!\u0000s0,Be\u000b!\u0000Be\u000b, and B0\ne\u000b!\u0000B0\ne\u000b).\nAssuming a harmonic time-dependence for the small-angle\nspin precession sx;y(t)=sx;yei!twithjsx;yj \u001c 1, one can\nsolve the eigenspectrum of Eqs. (13). By tuning the spin-\nspin coupling strength !ex, we observe a transition from exact\nPT phase to the broken PT phase, separated by the EP at\n!c\nex=2\u0019=1 MHz for\u0011=\u0011PT=6:16, as shown in Figs. 4(b)\nand 4(c). Interestingly, the unequal gain and loss, i.e., \u000be\u000b<0\nand\u000be\u000b,\u0000\u000b, leads to an imbalanced parity-time ( IPT )-\nsymmetry. In this region ( \u0011>\u0011 IPT=5:66), the eigenfrequen-\ncies have di \u000berent real parts but share the identical imaginary\none, as plotted in Fig. 4(d). A passive parity-time ( PPT )-\nsymmetry is further identified when \u000be\u000b>0. In such case\nRe[(ω-ω0)/2π] (MHz)(b)(a)\n(d)\n(c)\nωex /2π (MHz)Im[(ω-ω0)/2π] (MHz)\nηη=ηPT\nηPTηPPT ηIPTωex \n2πc\n=1 MHz(e)\nωexs s’\nωex \n2π=1.5 MHzFIG. 4: (a) Spin dimmer consisting of an optically pumped spin sand\na purely lossy one s0. Evolution of eigenfrequencies vs. the exchange\ncoupling (b,c) at the detuning \u0011PT=6:16, and vs. the detuning pa-\nrameter (d,e) at the exchange coupling !ex=2\u0019=1:5 MHz.\n(\u0011 > \u0011 PPT=7:11), the imaginary part of both branches is\nsmaller than their intrinsic damping [see Fig. 4(e)].\nDiscussion. —In the above derivation, we focus on the case\nthat the intrinsic Gilbert damping is isotropic. Our approach\ncan also be generalized to treat the case when the intrinsic\ndamping is anisotrpic [61, 62]. The three propagating lasers\nthen should be accordingly adjusted to match the tensor form\nof the intrinsic magnetic damping, by modulating the driving\npower or the frequency of each beam, for instance. The red-\ndetuning region is appealing to cool magnons to the subtle\nquantum domain. Inspired by PT-symmetric optics [19], we\nenvision a giant enhancement of the magnonic gain and an\nultralow-threshold magnon lasing in a two-cavity system with\nbalanced optical gain and loss, which is an open question for\nfuture study. While the magnonic passive PTsymmetry has\nbeen observed by Liu et al. [32], the exact and imbalanced\nPTphases are still waiting for the experimental discovery.\nConclusion. —To summarize, we have proposed an opto-\nmagnonic method to generate the negative Gilbert damp-\ning in ferromagnets, by studying the parametric dynamics\nof a macrospin coupled with three orthogonally propagating\ncircularly-polarized lasers in an optical cavity. We analyti-\ncally derived the formula of the optical torque on the spin\nand identified the condition for the magnetic gain exactly in\nthe Gilbert form. 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Bailey\nDept. of Applied Physics & Applied Mathematics,\nColumbia University, New York NY 10027, USA\n(Dated: November 21, 2021)\nA wavenumber-dependentdissipative term to magnetization dynamics, mirroring the conservative\nterm associated with exchange, has been proposed recently f or ferromagnetic metals. We present\nmeasurements ofwavenumber-( k-)dependentGilbert dampinginthree metallic ferromagnet s, NiFe,\nCo, and CoFeB, using perpendicular spin wave resonance up to 26 GHz. In the thinnest films\naccessible, where classical eddy-current damping is negli gible, size effects of Gilbert damping for the\nlowest and first excited modes support the existence of a k2term. The new term is clearly separable\nfrom interfacial damping typically attributed to spin pump ing. Higher-order modes in thicker films\ndo not show evidence of enhanced damping, attributed to a com plicating role of conductivity and\ninhomogeneous broadening. Our extracted magnitude of the k2term, ∆α∗\nkE= ∆α∗\n0+A∗\nkk2where\nA∗\nk=0.08-0.1 nm2in the three materials, is an order of magnitude lower than th at identified in prior\nexperiments on patterned elements.\nThe dynamical behavior of magnetization for ferro-\nmagnets (FMs) can be described by the Landau-Lifshitz-\nGilbert (LLG) equation[1]:\n˙ m=−µ0|γ|m×Heff+αm×˙ m (1)\nwhereµ0is the vacuum permeability, m=M/Msis the\nreduced magnetization unit vector, Heffis the effective\nmagnetic field, γis the gyromagnetic ratio, and αis the\nGilbert damping parameter. The LLG equation can be\nequivalently formulated, for small-angle motion, in terms\nof a single complex effective field along the equilibrium\ndirection, as ˜Heff=Heff-iαω/|γ|; damping torque is in-\ncluded in the imaginary part of ˜Heff.\nFor all novel spin-transport related terms to the LLG\nidentified so far[2–7], each real (conservative) effective\nfield term is mirrored by an imaginary (dissipative)\ncounterpart. In spin-transfer torque, there exist both\nconventional[2, 3] and field-like[8] terms in the dynamics.\nIn spin-orbit torques (spin Hall[4] and Rashba[6] effect)\ndampinglike and fieldlike components have been theoret-\nically predicted[9] and most terms have been experimen-\ntally identified[5, 6]. For pumped spin current[7], theory\npredictsrealandimaginaryspinmixingconductances[10]\ng↑↓\nrandg↑↓\niwhich introduce imaginary and real effective\nfields, respectively.\nIt is well known that the exchange interaction, respon-\nsible for ferromagnetism, contributes a real effective field\n(fieldlike torque) quadratic in wavenumber kfor spin\nwaves[11]. It isthennaturaltoaskwhetheracorrespond-\ning imaginary effective field might exist, contributing a\ndampinglike torque to spin waves. Theoretically such an\ninteraction has been predicted due to the intralayer spin-\ncurrent transport in a spin wave[12–15], reflected as an\nadditional term in Eq. (1):\n˙ m=···−(|γ|σ⊥/Ms)m×∇2˙ m (2)\nwhereσ⊥is the transverse spin conductivity. This term\nrepresents a continuum analog of the well-established in-terlayer spin pumping effect[7, 16, 17]. For spin wave\nresonance (SWR) with well-defined wavenumber k, Eq.\n(2) generates an additional Gilbert damping ∆ α(k) =\n(|γ|σ⊥/Ms)k2. In this context, Gilbert damping refers\nto an intrinsic relaxation mechanism in which the field-\nswept resonance linewidth is proportional to frequency.\nRemarkably, the possible existence of such a term has\nnot been addressed in prior SWR measurements. Previ-\nous studies of ferromagneticresonance (FMR) linewidths\nof spin waves[18–21] were typically operated at fixed fre-\nquency, not allowing separation of intrinsic (Gilbert) and\nextrinsic linewidths. Experiments have been carried out\non thick FM films, susceptible to a large eddy current\ndamping contribution[22]. Any wavenumber-dependent\nlinewidth broadening in these systems has been at-\ntributed to eddy currents or inhomogeneous broadening,\nnot intrinsic torques which appear in the LLG equation.\nIn this Manuscript, wepresent a study of wavenumber-\ndependent Gilbert damping in the commonly applied\nferromagnetic films Ni 79Fe21(Py), Co, and CoFeB. A\nbroad range of film thicknesses (25-200 nm) has been\nstudied in order to exclude eddy-current effects. We\nobserve a thickness-dependent difference in the Gilbert\ndamping for uniform and first excited spin wave modes\nwhich is explained well by the intralayer spin pump-\ning model[14]. Corrections for interfacial damping, or\nconventional spin pumping, have been applied and are\nfound to be small. The measurements show that the\nwavenumber-dependent damping, as identified in contin-\nuousfilms, isinreasonableagreementwith thetransverse\nspin relaxation lengths measured in Ref. [23], but an or-\nder of magnitude smaller than identified in experiments\non sub-micron patterned Py elements[24].\nTwo different types of thin-film heterostructures were\ninvestigated in this study. Films were deposited by\nUHV sputtering with conditions given in Ref. [23, 25].\nMultilayers with the structure Si/SiO 2(substrate)/Ta(5\nnm)/Cu(5 nm)/ FM(tFM)/Cu(5 nm)/Ta(5 nm), where2\nFM= Py, Co and CoFeB and tFM= 25-200 nm, were\ndesigned to separate the effects of eddy-current damp-\ning and the intralayer damping mechanism proposed in\nEq. (2). The minimum thickness investigated here is\nour detection threshold for the first SWR mode, 25 nm.\nA second type of heterostructure focused on much thin-\nner Py films, with the structure Si/SiO 2(substrate)/Ta(5\nnm)/Cu(5 nm)/Py( tPy)/Cu(5 nm)/ X(5 nm),tPy= 3-30\nnm. Here the cap layer X= Ta or SiO 2was changed,\nfor two series of this type, in order to isolate the effect\nof interfacial damping (spin pumping) from Cu/Ta inter-\nfaces.\nTo study the Gilbert damping behavior of finite-\nwavenumber spin waves in the samples, we have\nexcited perpendicular standing spin wave resonance\n(PSSWR)[26] using a coplanar waveguide from 3 to 26\nGHz. The spin-wave mode dispersion is given by the\nKittel equation ω(k)/|γ|=µ0(Hres−Ms+Hex(k)); the\neffective field from exchange, µ0Hex(k) = (2Aex/Ms)k2\nwithAexas the exchange stiffness, gives a precise mea-\nsurement of the wavenumber excited ((Fig. 1 inset)).\nPSSWR modes are indexed by the number of nodes p,\nwithk=pπ/tFMin the limit of unpinned surface spins.\nThe full-width half-maximum linewidth, ∆ H1/2, is fit-\nted using µ0∆H1/2(ω) =µ0∆H0+ 2αω/|γ|to extract\nthe Gilbert damping α. Forp= 1 modes we fix µ0∆H0\nas the values extracted from the corresponding p= 0\nmodes for ( tFM≤40 nm), because frequency ranges are\nreduced due to large exchange fields. In unconstrained\nfits for films of this thickness, the inhomogeneous broad-\neningµ0∆H0of thep= 1 modes does not exhibit a\ndiscernible trend with 1 /t2\nFM(ork2)[19–21], justifying\nthis approximation[27].\nTo fit our data, we have solved Maxwell’s equations\nand the LLG equation (Eq. 1), including novel torques\nsuch as those given in Eq. (2), according to the method\nof Rado[28]. The model (designated ’EM+LLG’) is de-\nscribed in the Supplemental Information. Values calcu-\nlated using the EM+LLG model are shown with curves\nin Fig. 1 and dashed lines in Fig. 4. Comparison with\nsuch a model has been necessary since in our first type of\nsample series, tFM= 25-200 nm, eddy-current damping\nis negligible for thinner films (25 nm), the Akk2contri-\nbution is negligible for thicker films (200 nm), but the\ntwo effects coexist for the intermediate region.\nIn Fig. 1(a-c)we comparethe measuredGilbert damp-\ning for the uniform ( p= 0,αu) and first excited ( p= 1,\nαs)spinwavemodes. Thedominantthickness-dependent\ncontribution to Gilbert damping of the uniform modes of\nPy, Co, and CoFeB is clearly due to eddy currents which\narequadraticinthickness. Notethateddy-currentdamp-\ning is negligible for the thinnest films investigated (25\nnm), but quite significant for the thickest films (200 nm).\nThis term sums with the bulk Gilbert damping α0[29].\nThe simulation of αu, shown by black curves in Fig. 1,\nmatches closely with the analytical expression for bulkand eddy-current damping only[30] of αu=αu0+αE0,\nwhereαE0=µ2\n0γMst2\nFM/12ρcdenotes the eddy-current\ndamping for uniform modes. Fittings of αuyield resis-\ntivitiesρc= 16.7, 26.4 and 36.4 µΩ·cm for Py, Co and\nCoFeB, respectively.\nUnlike the uniform-mode damping, the 1st SWR\n(×10 -3 )\nuu\ns\nu\ns(a) (b) \n(c) Co \nCoFeB Py μ0HB (T). . .p=0 p=1\nPy 75nm  μ 0Hex \nx50\ns\n(×10 -3 )\nkus-=\nFIG. 1. Thickness dependence of αuandαsfor (a) Py, (b)\nCo and (c) CoFeB thin films. Curves are calculated from\na combined solution of Maxwell’s equations and the LLG\n(EM+LLG). For αuthe values of µ0Ms,α(Table I), effective\nspin mixing conductance (Supplemental Information Sectio n\nC) g-factor (2.12 for Py and CoFeB and 2.15 for Co) and ρc\n(from analytical fitting) are used. For αsthe values of ∆ α∗\nkE\nand ∆α∗\nk0(Table I) are also included in the simulation. Inset:\n10 GHz FMR spectra of p= 0 and p= 1 modes in Py 75 nm\nfilm.\nmode damping αsis found to exhibit a minimum as a\nfunction of thickness. For decreasing thicknesses below\n75 nm,αsis increased. This behavior indicates an addi-\ntional source of Gilbert damping for the 1st SWR modes.\nIn CoFeB the increased αsis less visible in Fig. 1(c) due\nto fluctuations in damping for samples of different thick-\nness, but is evident in the difference, αs−αu, plotted in\nFig. 2.\nIn order to isolate this new damping mechanism, we\nplot in Fig. 2 the increased damping for the 1st SWR\nmode, ∆ αk=αs−αu, side-by-side with exchange field\nµ0Hexas a function of ( π/tFM)2taken as the wavenum-\nberk2. When π/tFMis large, a linear k2dependence\nof ∆αkin all three ferromagnets mirrors the linear de-\npendence of µ0Hexonk2. This parallel behavior reflects\nthe wavenumber-dependent imaginary and real effective\nfields acting on magnetization, respectively. To quantify\nthe quadratic wavenumber term in ∆ αk, we also show\nthe eddy-current-corrected values ∆ αkE=∆αk−∆αEin\nFig. 2(a). Here ∆ αE=αE1−αE0denotes the differ-\nence in eddy current damping between p= 1 and p= 0\nmodes according to the theory of Ref. [30], for weak sur-\nface pinning, where αE1≈0.23αE0(See Supplemental\nInformation for more details). We then fit this eddy-3(×10 -3 )\nPy \nCo \nCoFeB \nPy 150 nm (a)\n(b)\n(π/t FM )2 (×10 16  m -2 )Py \nCo \nCoFeB kk\nFIG. 2. Imaginary (damping, a) and real (exchange, b) ef-\nfective fields as a function of k2for Py, Co and CoFeB. (a)\nAdditional SWR damping ∆ αk(circle) and eddy-current cor-\nrected value ∆ αkE(cross) as a function of ( π/tFM)2. Solid\nlines are guides to eye and dashed lines are fits to Eq. (3). (b)\nExchange field µ0Hexas a function of ( π/tFM)2((pπ/tFM)2,\np=0-6, for Py 150 nm). Lines are fits to µ0Hex= (2A/Ms)k2.\ncurrent-corrected value to a linearization of Eq. (2), as:\n∆αkE= ∆αk0+Akk2(3)\nwithAk=|γ|σ⊥/Msand ∆αk0a constant offset. The\nvalues of Akestimated this way are 0 .128±0.022 nm2,\n0.100±0.011 nm2and 0.100±0.018 nm2for Py, Co and\nCoFeB.\nRecently, Kapelrud et al.[31] have predicted that\ninterface-localized (e.g. spin-pumping) damping terms\nwill also be increased in SWR, with interfacial terms\nforp≥1 modes a factor of two greater than those for\nthep= 0 mode. Using the second series of thinner\nPy films, we have applied corrections for the interfacial\nterm to our data, and find that these effects introduce\nonly a minor ( ∼20%) correction to the estimate of\nAk. Thep= 0 mode damping associated with the\nCu/Ta interface has been measured from the increase\nin damping upon replacement of SiO 2with Ta at the\ntop surface (Fig. 3, inset). Here Cu/SiO 2is taken as\na reference with zero interfacial damping; insulating\nlayers have been shown to have no spin pumping\ncontribution[32]. We find the damping enhancement to\nbe inversely proportional to tFM, indicating an interfa-\ncial damping term quantified as spin pumping into Ta[7]\nwith ∆αsp=γ¯h(g↑↓/S)/4πMstFM. Using the values\nin Table I yields the effective spin mixing conductance\nasg↑↓\nPy/Cu/Ta/S=2.5 nm−2, roughly a factor of three\nsmaller than that contributed by Cu/Pt interfaces[17].\nUsing the fitted g↑↓\nFM/Cu/Ta/S, we calculate andcorrect for the additional spin pumping contribution to\ndamping of the p= 1 mode, 2∆ αsp(from top and bot-\ntom interfaces). The corrected values for the 1st SWR\ndamping enhancement, ∆ α∗\nkE= ∆αkE−2∆αsp, are\nplotted for Py(25-200nm)in Fig. 3. These correctionsdo\nnot change the result significantly. We fit the k2depen-\ndence of ∆ α∗\nkEto Eq. (3) to extract the corrected values\nA∗\nkand ∆α∗\nk0. The fitted value, A∗\nk= 0.105±0.021 nm2\nfor Py, is slightly smaller than the uncorrected value\nAk. Other extracted interfacial-corrected values A∗\nkare\nlisted in Table I. Note that the correction of wavenumber\nby finite surface anisotropy will only introduce a small\ncorrection of AkandA∗\nkwithin errorbars. We also\nshow the EM+LLG numerical simulation results for\nthe uniform modes and the first SWR modes in Fig. 1\n(solid curves). Those curves coincide with the analytical\nexpressions of eddy-current damping plus k2damping\n(not shown) and fit the experimental data points nicely.\nThe negative offsets ∆ α∗\nk0between uniform modes\n(π/t FM )2 (×10 16  m -2 )(×10 -3 )*(= )\n**u0 u0 \ntFM  (nm)\nFIG. 3. Interfacial damping correction for Py. Main panel:\n∆αkEand ∆α∗\nkEas a function of ( π/tFM)2. Dashed lines\nare fits to k2-dependent equation as Eq. (3); ∆ α∗\nk0are ex-\ntracted from ∆ α∗\nkEfits.Inset:size effect of uniform-modes\nGilbert damping in Py/Cu/Ta and Py/Cu/SiO 2samples (cir-\ncles). The dashed curve is the theoretical reproduction of\nPy/Cu/SiO 2usingαu0+ ∆αsp(tFM). The shadow is the\nsame reproduction using αu0+ ∆αsp(tFM) +A∗\nkk2where\nthe error of shadow is from A∗\nk. Here kis determined by\nAexk2= 2Ks/tFM.\nand spin wave modes for Py and CoFeB are attributed\nto resistivitylike intrinsic damping[33]: because ˙ mis\naveraged through the whole film for uniform modes\nand maximized at the interfaces for unpinned boundary\ncondition, the SWR mode experiences a lower resistivity\nnear low-resistivity Cu and thus a reduced value of\ndamping. For Co a transition state between resistivity-\nlike and conductivitylike mechanisms[34] corresponds to\nnegligible ∆ α∗\nk0as observed in this work.\nIn addition to the thickness-dependent comparison of\np= 0 and p= 1 modes, we have also measured Gilbert4\ndamping for a series of higher-order modes in a thick\nPy (150 nm) film. Eddy-current damping ( αE∼0.003)\nis the dominant mode-dependent contribution in this\nfilm. The wavenumber kfor the mode p= 6 is roughly\nequal to that for the first SWR, p= 1, in the 25 nm\nfilm. Resonance positions are plotted with the dashed\nlines in Fig. 2(b), as a function of k, and are in good\nagreement with those found from the p= 1 data. In Fig.\n4 we plot the mode-related Gilbert damping αpup to\np= 6, which gradually decreases as pincreases. We have\nagain conducted full numerical simulations using the\nEM+LLG method with ( A∗\nk= 0.105 nm2) or without\n(A∗\nk= 0) the intralayer spin pumping term, shown in\nred and black crosses, respectively. Neither scenario fits\nthe data closely; an increase at p= 3 is closer to the\nmodel including the k2mechanism, but experimental α\natp= 6 falls well below either calculation.\nWe believe there are two possibilities why the α∝p2\ndamping term is not evident in this configuration. First,\nthe effective exchange field increases with p, resulting\nin a weaker (perpendicular) resonance field at the same\nfrequency. When the perpendicular biasing field at\nresonance is close to the saturation field, the spins\nnear the boundary are not fully saturated, which might\nproduce an inhomogeneous linewidth broadening at\nlower frequencies and mask small Gilbert contributions\nfrom wavenumber effect. From the data in Fig. 4 inset\nthe high- pSWR modes is more affected by this inhomo-\ngeneous broadening and complicate the extraction of k2\ndamping. Second, high- pmodes in thick films are close\nto the anomalous conductivity regime, kλM∼1, where\nλMis the electronic mean free path. The Rado-type\nmodel such as that applied in Fig. 4 is no longer valid\nin this limit[35], beyond which Gilbert damping has\nbeen shown to decrease significantly in Ni and Co[36].\nBased on published ρλMproducts for Py[37] and our\nexperimental value of ρc= 16.7µΩ·cm, we find λM∼8\nnm andkλM∼1 for the p= 6 mode in Py 150 nm. For\nthe 1st SWR mode in Py 25 nm, on the other hand,\neddy currents are negligible and the anomalous behavior\nis likely suppressed due to surface scattering, which\nreducesλM.\nAn important conclusion of our work is that the\nintralayer spin pumping, as measured classically through\nPSSWR, is indeed present but more than 10 times\nsmaller than estimated in single nanoscale ellipses[24].\nThe advantages of the PSSWR measurements presented\nin this manuscript are that the one-dimensional mode\nprofile is well-defined, two-magnon effects are reduced,\nif not absent[39], and there are no lithographic edges to\ncomplicate the analysis. The lower estimates of A∗\nkfrom\nPSSWR aresensible, basedonphysicalparametersofPy,\nCo, and CoFeB. The polarization of continuum-pumped\nspins in a nearly uniformly magnetized film, like that\nof pumped spin current in a parallel-magnetized F/N/F\nstructure, is transverse to the magnetization[14]. Fromthe measured transverse spin conductance σ⊥we extract\nthat the relaxation lengths of pumping intralayer spin\ncurrent are 0.8-1.9 nm for the three ferromagnets[27], in\ngood agreement with the small transverse spin coherence\nlengths found in these same ferromagneticmetals[23, 40].\nFinally, we show that the magnitude of the intralayer\nmT p=1, tFM =25-200 nm \np=0-6, t FM =150 nm \n(pπ/t FM )2 (×1016  m -2 )\np*\n*\nFIG.4. Mode-dependentdamping αpfor Py(150nm), 0 ≤p≤\n6. Crosses are EM+LLG calculated values with and without\nthe wavenumber-dependent damping term. Inset: Inhomoge-\nneous broadening ∆ H0vs 0≤p≤6, 150nm film. Larger,\nk-dependent values are evident, compared with those in the\nthickness series ( tFM=25-200 nm).\nspin pumping identified here is consistent with the\ndamping size effect notattributable to interlayer spin\npumping, in layers without obvious spin sinks. For the\np= 0 mode, a small but finite wavenumber is set by the\nsurface anisotropy through[30, 41] Aexk2= 2Ks/tFM.\nThe damping enhancement due to intralayer spin\npumping will, like the interlayer spin pumping, be\ninverse in thickness, leading to an ’interfacial’ term as\nα= 2Ks(A∗\nk/Aex)t−1\nFM. This contribution is indicated\nby the grey shadow in Fig. 3 insetand provides a\ngood account of the additional size effect in the SiO 2-\ncapped film. Here we use Ks=0.11 mJ/m2extracted\nby fitting the thickness-dependent magnetization to\nµ0Meff=µ0Ms−4Ks/MstFM. While alternate\ncontributions to the observed damping size effect for the\nSiO2-capped film cannot be ruled out, the data in Fig.\n3insetplace an upper bound on A∗\nk.\nIn summary, we have identified a wavenumber-\ndependent, Gilbert-type damping contribution to spin\nwaves in nearly uniformly magnetized, continuous\nfilms of the metallic ferromagnets Py, Co and CoFeB\nusing classical spin wave resonance. The term varies\nquadratically with wavenumber, ∆ α∼A∗\nkk2, with the\nmagnitude, A∗\nk∼0.08-0.10 nm2, amounting to ∼20% of\nthe bulk damping in the first excited mode of a 25 nm\nfilm of Py or Co, roughly an order of magnitude smaller\nthan previously identified in patterned elements. The\nmeasurements quantify this texture-related contribution5\nto magnetization dynamics in the limit of nearly homo-\ngeneous magnetization.\nµ0Ms(T)α0Aex(J/m) A∗\nk(nm2)∆α∗\n0\nPy 1.00 0.0073 1.2×10−110.11±0.02 -0.0008\nCo 1.47 0.0070 3.1×10−110.08±0.01 -0.0002\nCoFeB 1.53 0.0051 1.8×10−110.09±0.02 -0.0011\nTABLE I. Fit parameters extracted from resonance fields and\nlinewidths of uniform and 1st SWR modes. Values of A∗\nk\nand ∆α∗\n0for Co and CoFeB are calculated using the spin\nmixing conductances measured in FM/Cu/Pt[17]. See the\nSupplemental Material for details.\n[1] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n[2] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1\n(1996).\n[3] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[4] J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).\n[5] S. O. Valenzuela and M. Tinkham, Nature442, 176\n(2006).\n[6] I. M. Miron, G. Gaudin, S. Auffret, B. Rodmacq, A.\nSchuhl, S. Pizzini, J. Vogel and P. Gambardella, Nature\nMater.9, 230 (2010).\n[7] Y. Tserkovnyak, A. Brataas and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n[8] S. Zhang, P. M. Levy and A. Fert, Phys. Rev. Lett. 88,\n236601 (2002).\n[9] P. M. Haney, H. W. Lee, K. J. Lee, A. Manchon and M.\nD. Stiles, Phys. Rev. B 87, 174411 (2013).\n[10] M. Zwierzycki, Y. Tserkovnyak, P. J. Kelly, A. Brataas\nand G. E. W. Bauer, Phys. Rev. B 71, 064420 (2005).\n[11] C. Kittel, Phys. Rev. 81, 869 (1951).\n[12] E. M. Hankiewicz, G. Vignale and Y. Tserkovnyak, Phys.\nRev. B78, 020404(R) (2008).\n[13] J. Foros, A. Brataas, Y. Tserkovnyak and G. E. W.\nBauer,Phys. Rev. B 78, 140402(R) (2008).\n[14] Y. Tserkovnyak, E. M. Hankiewicz and G. Vignale, Phys.\nRev. B79, 094415 (2009).[15] S. Zhang and Steven S.-L. Zhang, Phys. Rev. Lett. 102,\n086601 (2009).\n[16] S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B\n66, 104413 (2002).\n[17] A. Ghosh, J. F. Sierra, S. Auffret, U. Ebels and W. E.\nBailey,Appl. Phys. Lett. 98, 052508 (2011).\n[18] P. E. Wigen, Phys. Rev. 133, A1557 (1964).\n[19] T. G. Phillips and H. M. Rosenberg, Phys. Lett. 8, 298\n(1964).\n[20] G. C. Bailey, J. Appl. Phys. 41, 5232 (1970).\n[21] F. Schreiber and Z. Frait, Phys. Rev. B 54, 6473 (1996).\n[22] P. Pincus, Phys. Rev. 118, 658 (1960).\n[23] A. Ghosh, S. Auffret, U. Ebels and W. E. Bailey, Phys.\nRev. Lett. 109, 127202 (2012).\n[24] H. T. Nembach, J. M. Shaw, C. T. Boone and T. J. Silva,\nPhys. Rev. Lett. 110, 117201 (2013).\n[25] Y. Li, Y. Lu and W. E. Bailey, J. Appl. Phys. 113,\n17B506 (2013).\n[26] M. H. Seavey and P. E. Tannenwald, Phys. Rev. Lett. 1,\n168 (1958).\n[27] See the Supplemental Information for details.\n[28] W. S. Ament and G. T. Rado, Phys. Rev. 97, 1558\n(1955).\n[29] C. Scheck, L. Cheng and W. E. Bailey, Appl. Phys. Lett.\n88, 252510 (2006).\n[30] M. Jirsa, phys. stat. sol. (b) 113, 679 (1982).\n[31] A. Kapelrud and A. Brataas, Phys. Rev. Lett. 111,\n097602 (2013).\n[32] O. Mosendz, J. E. Pearson, F. Y. Fradin, S. D. Bader\nand A. Hoffmann, Appl. Phys. Lett. 96, 022502 (2009).\n[33] K. Gilmore, Y. U. Idzerda and M. D. Stiles, Phys. Rev.\nLett.99, 027204 (2007).\n[34] S. M. Bhagat andP.Lubitz, Phys. Rev. B 10, 179(1974).\n[35] G. T. Rado, J. Appl. Phys. 29, 330 (1958).\n[36] V. Korenman and R. E. Prange, Phys. Rev. B 6, 2769\n(1972).\n[37] B. A. Gurney, V. S. Speriosu, J.-P. Nozieres, H. Lefakis ,\nD. R. Wilhoit and O. U. Need, Phys. Rev. Lett. 71, 4023\n(1993).\n[38] J. N. Lloyd and S. M. Bhagat, Solid State Commun. 8,\n2009 (1970).\n[39] R. D. McMichael, M. D. Stiles, P. J. Chen and W. F.\nEgelhoff Jr., J. Appl. Phys. 83, 7037 (1998).\n[40] J. Zhang, P. M. Levy, S. F. Zhang and V. Antropov,\nPhys. Rev. Lett. 93, 256602 (2004).\n[41] R. F. Soohoo, Phys. Rev. 131, 594 (1963)."    },    {        "title": "1608.08043v1.Sub_micrometer_yttrium_iron_garnet_LPE_films_with_low_ferromagnetic_resonance_losses.pdf",        "content": "Sub-micrometer yttrium iron garnet LPE \flms with low ferromagnetic\nresonance losses\nCarsten Dubs,1Oleksii Surzhenko,1Ralf Linke,1Andreas Danilewsky,2Uwe Br uckner,3and Jan Dellith3\n1)INNOVENT e.V., Technologieentwicklung, Pr ussingstr. 27B, 07745 Jena, Germany\n2)Kristallographie, Albert-Ludwigs-Universit at Freiburg, Hermann-Herder-Str. 5, 79104 Freiburg,\nGermany\n3)Leibniz-Institut f ur Photonische Technologien (IPHT), Albert-Einstein-Str. 9, 07745 Jena,\nGermany\n(Dated: 30 August 2016)\nUsing liquid phase epitaxy (LPE) technique (111) yttrium iron garnet (YIG) \flms with thicknesses of \u0019100 nm\nand surface roughnesses as low as 0.3 nm have been grown as a basic material for spin-wave propagation\nexperiments in microstructured waveguides. The continuously strained \flms exhibit nearly perfect crys-\ntallinity without signi\fcant mosaicity and with e\u000bective lattice mis\fts of \u0001 a?=as\u001910\u00004and below. The\n\flm/substrate interface is extremely sharp without broad interdi\u000busion layer formation. All LPE \flms ex-\nhibit a nearly bulk-like saturation magnetization of (1800 \u000620) Gs and an `easy cone' anisotropy type with\nextremely small in-plane coercive \felds <0.2 Oe. There is a rather weak in-plane magnetic anisotropy with a\npronounced six-fold symmetry observed for saturation \feld <1.5 Oe. No signi\fcant out-of-plane anisotropy is\nobserved, but a weak dependence of the e\u000bective magnetization on the lattice mis\ft is detected. The narrowest\nferromagnetic resonance linewidth is determined to be 1.4 Oe @ 6.5 GHz which is the lowest values reported\nso far for YIG \flms of 100 nm thicknesses and below. The Gilbert damping coe\u000ecient for investigated LPE\n\flms is estimated to be close to 1 \u000210\u00004.\nPACS numbers: 81.15.Lm, 75.50.Gg, 76.50.+g\nI. INTRODUCTION\nMagnonics is an increasingly growing new branch\nof spin-wave physics, speci\fcally addressing the use of\nmagnons for information transport and processing1{4.\nSingle crystalline yttrium iron garnet (YIG), which is a\nferrimagnetic insulator with the smallest known magnetic\nrelaxation parameter5, appears to be a superior candi-\ndate for this purpose6{8. As bulk or as thick \flm mate-\nrial, which is commonly grown by liquid phase epitaxy\n(LPE)9, it has a very low damping coe\u000ecient and allows\nmagnons to propagate over distances exceeding several\ncentimeters6. However YIG functional layers for practi-\ncal magnonics should be nanometer-thin with extremely\nsmooth surfaces in order to achieve optimum e\u000eciency\nin data processing and dramatic reduction in energy con-\nsumption of sophisticated spin-wave devices. Therefore,\nhigh-quality thin and ultra-thin YIG \flms were grown us-\ning di\u000berent growth techniques such as LPE, pulsed laser\ndeposition (PLD) and rf-magnetron sputtering to inves-\ntigate diverse spin-wave e\u000bects and to design YIG waveg-\nuides as well as nanostructures for spin wave excitation,\nmanipulation and detection in prospective magnonic cir-\ncuits.\nFrom previous reports about sub-micrometer YIG\n\flms with thicknesses between 100 and 20 nm10{15avail-\nable microwave and magnetic key parameters were taken\nand summarized in Table I. Thus, ferromagnetic reso-\nnance (FMR) data were included which have been ex-\ntracted from measurements of the absorption curves or\nabsorption derivative curves versus sweeping magnetic\nin-plane \feld Hat a \fxed frequency for vs. sweep-ing rf-exciting \feld hrfwith an applied in-plane static\nmagnetic bias \feld. The reported FMR linewidths \u0001 H\nand converted peak to peak linewidths \u0001 Hp\u0000pof the\n\feld derivative values (\u0001 H=p\n3\u0001Hp\u0000p), which will\nbe given during the further paper as full-width at half-\nmaximum \u0001 HFWHM , varied between 3 Oe and 13 Oe.\nThe Gilbert damping coe\u000ecient \u000bwere found in the\nrange from 2\u000210\u00004to 8\u000210\u00004. Only the lowest given \u000b\nvalue of 0:9\u000210\u00004was obtained for a very short \ft range\nof about 4 GHz without any given data in the low fre-\nquency range below 10 GHz13and is therefore not really\ncomparable with the other reported values. From this\ncompilation it is obvious that neither \u0001 HFWHM nor\u000bis\nsigni\fcantly in\ruenced by the YIG \flm thickness down\nto 20 nm. The di\u000berences are probably resulted from\nadditional ferromagnetic losses due to contributions of\nhomogeneous and/or inhomogeneous broadening by mi-\ncrostructural imperfections or magnetic inhomogeneities.\nIn this report we present microstructural, magnetic\nand FMR properties of LPE-grown 100 nm thin YIG and\nLanthanum substituted (La:YIG) \flms with low ferro-\nmagnetic resonance losses. Film thicknesses were deter-\nmined by X-ray re\rectometry (XRR) and surface rough-\nness by atomic force microscopy (AFM) measurements.\nCrystalline perfection and compositional homogeneity\nwere investigated by high-resolution X-ray di\u000braction\n(HR-XRD) and X-ray photoelectron spectroscopy (XPS)\nas well as by secondary ion mass spectroscopy (SIMS).\nStatic and dynamic (microwave) magnetic characteriza-\ntions were carried out by vibrating sample magnetometry\n(VSM) and by Vector Network Analysis (VNA), respec-\ntively.arXiv:1608.08043v1  [cond-mat.mtrl-sci]  29 Aug 20162\nTABLE I: Key parameters reported for thin/ultrathin YIG \flms on (111) GGG substrates\nGrowth method Thick- RMS- 4 \u0019MsaHca\u0001Haf0 \u0001H0a\u000b\n(Reference) ness roughness FWHM FWHM \u000210\u00004\n(nm) (nm) (kGs) (Oe) (Oe) (GHz) (Oe)\nLPE10100 - 1.81 - 3.0 7 1.6 2.8\nLPE (this study) 83{113 0.3{0.8 1.78{1.82 \u00140.2 1.4{1.6 6.5 0.5{0.7 1.2{1.7\nPLDb 1179 0.2 1.72 <2 3.0 10 1.4 2.2\nPLD1223 - 1.60 <1 3.5c9.6 3.5{7c2{4\nSputtering1322 0.13 1.78 0.4 12c16.5 6.4c0.9\nSputtering1420 0.2 - 0.4 13c9.7 7c8\nPLD1520 0.2{0.3 2.10 0.2 3.3c6 2.4c2.3\naMeasurements at RT with the in-plane external magnetic \feld H\nbYIG \flms grown on the (100) GGG substrates\ncPeak-to-peak value \u0001 Hp\u0000pof the derivative of FMR absorption transformed into \u0001 HFWHM = \u0001Hp\u0000p\u0002p\n3\nII. RESULTS\nA. Microstructural properties\nSelected microstructural and magnetic properties of\nliquid phase epitaxial grown YIG (sample A-C) and\nLa:YIG (sample D) \flms are given in Table II. The con-\nsistent magnetic as well as microwave properties obtained\nfor \flms deposited during di\u000berent growth runs demon-\nstrate a high reproducibility of the LPE growth tech-\nnique. Fig. 1a shows XRR plots of \flms with thicknesses\nof about 100 nm which are smaller than the previously re-\nported thinnest LPE YIG \flms16{18. The smallest root-\nmean-square (RMS) surface roughness of about 0.25 nm\nobtained for the sample B in Fig. 1b is nearly compa-\nrable with epi-polished GGG substrate quality of \u00190.15\nnm and with the best PLD and sputtered YIG \flms (see\ne.g. Table I). Besides, \flms with slightly rougher surfaces\n(see Table II)) were obtained as a result of additional\ndendritic aftergrowth and/or due to plateau formation,\nso called \\mesas\", if any solution droplet adheres to the\nsample surface.\nHR-XRD studies of our thin epitaxial LPE \flms have\nbeen found to be di\u000ecult because of the nearly super-\nimposed di\u000braction pattern of YIG \flm and GGG sub-\nstrate. Although the angle distances between \flm and\nsubstrate Bragg re\rections were above the resolution\nlimit of our HR-XRD equipment, the di\u000braction inten-\nsity of the \flm re\rection was very low and results only\nin a broadening of the GGG Bragg re\rection. Fig. 2a\nshows a!-scan (rocking curve) with a Gaussian-like \ft-\nted GGG substrate 444 re\rection and a second \ftted\npeak at the right shoulder which corresponds to the YIG\n444 \flm re\rection. This indicates a tensile stressed YIG\n\flm because of the smaller \flm lattice parameter com-\npared to the commercially available Czochralski-grown\nGGG substrate ( as=1.2382 nm). For La:YIG \flms we ob-\nserved a perfect pseudo-Voigt \ftted substrate peak with-\nout any additional shoulder (not shown) which indicates\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s49/s48/s49/s49/s48/s51/s49/s48/s53/s49/s48/s55\n/s100/s32/s61/s32/s57/s55/s32/s110/s109/s100/s32/s61/s32/s56/s51/s32/s110/s109/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s99/s112/s115/s41\n/s80/s111/s115/s105/s116/s105/s111/s110/s32 /s32 /s40/s67/s117/s45/s75/s76\n/s50/s44/s51/s41/s83/s97/s109/s112/s108/s101/s32/s67\n/s83/s97/s109/s112/s108/s101/s32/s68(a)\n(b)\nFIG. 1: (a) XRR plots of sub-micrometer-thick YIG\nLPE \flms. (b) 5\u00025\u0016m2AFM surface topography of\nsample B with RMS roughness of 0.25 nm.\na perfect lattice match between substrate and LPE \flm.\nThis is in remarkable contrast to YIG \flms deposited\nby various gas phase techniques such as PLD and rf-3\nTABLE II: YIG/La:YIG \flm properties grown on (111) GGG substrates by LPE technology\nThick- RMS- Relative lattice VSMaFMRa\nSample ness roughness mis\ft \u0001 a?=as 4\u0019MsHc 4\u0019Me\u000b \u0001HFWHMb\u0001H0\u000b\n(nm) (nm) \u000210\u00004(kGs) (Oe) (kGs) (Oe) (Oe) \u000210\u00004\nA 113 0.8 4.7 1.82 0.10 1.637 1.4 0.5 1.4\nB 106 0.3 1.8 1.78 0.20 1.658 1.5 0.7 1.2\nC 83 0.6 0.3 1.82 0.16 1.672 1.4 0.5 1.6\nDc97 0.8 0.0 1.78 0.18 1.712 1.6 0.7 1.7\nAccuracy \u00061 \u00060:1 \u00060:3 \u00060:04 \u00060:03 \u00060:010 \u00060:1 \u00060:1 \u00060:1\naVSM and FMR measurements at room temperature with applied in-plane magnetic \feld\nbFMR linewidth value at frequency f=6.5 GHz\ncLa:YIG LPE \flm\nsputtering11{15,19,20on GGG substrates. For those \flms\nthe YIG re\rection has always been detected at consider-\nably lower Bragg angles compared to the GGG substrate\nindicating a signi\fcant distortion of the cubic YIG garnet\ncell with signi\fcantly enlarged lattice parameters (com-\npressive stress)19,21.\nThe relative e\u000bective mis\ft \u0001 a?=as= (as\u0000a?\nYIG)=as\nobtained from strained \flm lattice parameter in growth\ndirectiona?\nYIGand the substrate lattice parameter ascan\nbe used as a measure for epitaxial induced in-plane ten-\nsion or strain. Due to YIG Poisson's ratio of \u0017P= 0:29\npseudomorphously grown, fully strained YIG \flms with\nan ideal YIG bulklattice parameter aYIG= 1:2375 nm22\nshould have a relative e\u000bective mis\ft of \u0001 a?=as=\n+11\u000210\u00004(tensile stress). In the case of our sub-\nmicrometer YIG \flms \u0001 a?=ashas been determined to\nbe in the range between zero and +5 \u000210\u00004(see Ta-\nble II) compared to PLD-grown YIG \flms with up to\n\u0001a?=as=\u0000100\u000210\u00004(see e.g. Ref.12). Hence, our\nLPE \flms are under tension but not to the extent which\nwe expected for nominally pure YIG material without\nadditional lattice expansion by lattice defects or impu-\nrities. To \fnd the reason for this, high-resolution re-\nciprocal space map (HR-RSM) and XPS investigations\nwere performed. Fig. 2b shows a HR-RSM plot around\nthe symmetrical 444 Bragg re\rection with symmetrical\ndi\u000bracted intensity for the GGG substrate and asym-\nmetric di\u000bracted intensity toward higher scattering an-\ngles along Qz(2\u0012-!-Scan) which we attribute to the\nYIG 444 \flm re\rection. Broadening of the \flm re\rec-\ntion along Qzis due to the \fnite coherence lenght of\nthe sub-micrometer thin \flm in growth direction and\nother broadening mechanisms as for example heteroge-\nneous strain. The extension of the \flm re\rection up\nto the substrate peak position suggests that the \flm is\ncontinuously strained due to an existing compositional\nand/or strain gradient. No peak broadening along the\nQxdirection (!-scan) indicates single crystalline perfec-\ntion parallel to the \flm plane without signi\fcant mosaic-\nity due to tilts of epitaxial regions with respect to one\nanother.\nTo evaluate the compositional homogeneity along the\n25.34 25.36 25.38 25.401000200030004000\n \nYIG 444\nFWHM=0.011□Intensity (cps)\nAngle ω (deg) Sample B\n Gaussian fit GGG\n Gaussian fit YIG\n Cumulative fit\nFWHM=0.0058□GGG 444(a)\n/s45/s48/s46/s48/s48/s50 /s45/s48/s46/s48/s48/s49 /s48/s46/s48/s48/s48 /s48/s46/s48/s48/s49 /s48/s46/s48/s48/s50/s53/s46/s53/s57/s48/s53/s46/s53/s57/s50/s53/s46/s53/s57/s52/s53/s46/s53/s57/s54/s53/s46/s53/s57/s56/s53/s46/s54/s48/s48/s53/s46/s54/s48/s50\n/s81\n/s88/s32/s40/s49/s47/s110/s109/s41/s81\n/s90/s32/s40/s49/s47/s110/s109/s41\n/s49/s50/s52/s56/s49/s54/s51/s50/s54/s52/s49/s50/s56/s50/s53/s54/s53/s49/s50/s49/s48/s50/s52/s50/s48/s52/s56\n/s52/s52/s52\n/s83/s97/s109/s112/s108/s101/s32/s65\n(b)\nFIG. 2: (a) HR-XRD !-scan around substrate/\flm 444\nBragg re\rection of sample B. By \ftting procedures the\nYIG \flm peak has been extracted. (b) HR-RSM scans\naround substrate/\flm 444 reciprocal point reveal\nasymmetric di\u000bracted intensities towards higher Q z\nvalues for sample A.\ngrowth direction of the \flms and to detect expected\nimpurities (e.g. Pb from solvent) depth pro\fle analy-4\n0 500 1000 1500 2000 2500 30000246810\nsubstrate\n Intensity (a.u.)\nEtch time (s) Y 3p3\n Fe 2p\n O 1s\n Gd 3d5\n Ga 2p3\n Pb 4f< 5 nm\nYIG/GGG\ninterfaceYIG film\n(a)\n0 250 500 750 1000 1250 15001234\n  \n GasubstrateLa:YIG film La:YIG/GGG\n     interfaceIntensity 139La, 69Ga (a.u.)\nEtch time (s) La< 11 nm\nx 100\n(b)\nFIG. 3: (a) XPS depth pro\fle of sample B reveals a\nvery narrow interface between \flm and substrate. The\nPb 4f signal could not be detected within the detection\nlimit of about 0.1 at-%. (b) SIMS depth pro\fle analysis\ndetects the139La signal of the \flm as well as the69Ga\nsignal of the substrate (sample D) and their changes at\nthe \flm/substrate interface.\nses were carried out by XPS. Fig. 3a shows a homo-\ngeneous distribution of the YIG matrix elements along\nthe \flm growth direction and a sharp transition at the\n\flm/substrate interface. The obtained width of the tran-\nsition layer for sample B is below 5 nm. But the obtained\ndepth pro\fle consists of a convolution of the true concen-\ntration pro\fle with the depth resolution of the XPS sys-\ntem under the concrete measuring conditions and should\nbe narrower. Therefore, these pro\fles demonstrate that\nno broad interdi\u000busion layer is formed by element in-\ntermixing at the interface at an early state of epitaxial\ngrowth or by di\u000busion of substrate ions into the epitaxial\nlayer and vice versa during the subsequent growth pro-\ncess.\nWhereas XPS surface analysis of the very \frst atomic\nlayers (not shown) gives a Pb content of about 0.2 at-%,\nno Pb signal could be observed during the depth pro\fleanalyses within the detection limit of 0.1 at-%23. There-\nfore, it is assumed that the Pb signal corresponds to\na surface contamination of condensed PbO vapor from\nhigh temperature solution and this contamination is com-\npletely removed by the \frst argon-ion etching step. For\nYIG \flms grown in La 2O3containing solution no La sig-\nnal could be detected by XPS that give indicates that\nthe La content must be below 0.5 at-%24. In order to\nimprove the detection capability additional qualitative\nSIMS measurements were carried out. Due to the result-\ning sputtering e\u000bect and by time-dependent detection of\nthe sputtered sample ions one obtains depth pro\fles of\nthe \flm elements as shown for139La in Fig. 3b. Here, the\ncounts of two separate measurements taken under identi-\ncal measuring conditions at neighboring sample positions\nwere added up in order to enhance the statistical signi\f-\ncance. It is clearly visible that the lanthanum signal de-\ncreases at the \flm/substrate interface whereas substrate\nsignals like69:7Ga simultaneously increase.\nB. Static magnetic measurements\nThe vibrating sample magnetometry was used to mea-\nsure the net magnetic moment mof the YIG/GGG sam-\nples at room temperature. As a thickness of GGG sub-\nstrates\u00195000 times exceeded these of the studied YIG\n\flms, a proper calculation of the YIG parameters re-\nquired us (i) to extract the GGG contribution that lin-\nearly increased with the external \feld Hand (ii) to prefer\nthe in-plane sample orientation that ensured considerably\nlower \felds Hsfor the YIG \flms to attain the saturation.\nFig. 4a presents a typical dependence of the total mag-\nnetic moment mvs the in-plane magnetic \feld Hand\nillustrates the method allowing us to separate the m'\ncomponents produced by the YIG \flm and the GGG\nsubstrate. Being subsequently normalized to the \flm\nvolume, the YIG component loops yield the following\nmaterial parameters { a saturation magnetization Ms, a\ncoercivityHcand a saturation \feld Hs, i.e. the \feld (av-\neraged over ascending H\"and descending H#branches\nof hysteresis loops) where the YIG \flm magnetization\napproaches 0.9\u0002Ms. In order to estimate the in-plane\nanisotropy, we have repeated this procedure for the sam-\nples rotated around the h111iaxis perpendicular to \flm\nsurfaces. Fig. 4b demonstrates such results as polar\nsemi-log plots vs the azimuthal angle '. A saturation\nmagnetization Msin Fig. 4b seems independent of '.\nThe obtained 4 \u0019Msvalues cluster around 1800 Gs usu-\nally reported25for bulk YIG single crystals. Within an\nexperimental error (mostly de\fned by the YIG volume\nuncertainity of\u00062%), the same is valid for the 4 \u0019Msval-\nues in other LPE \flms listed in Table II. The obtained\ncoercivity ( Hc\u00140:2 Oe) in studied LPE \flms is among\nthe best values reported for gas phase epitaxial \flms (see\nTable I). No distinct in\ruence of the crystallographic ori-\nentation on the Hcvalues is also registered. In contrast,\nthe azimuthal dependence of the saturation \feld Hsob-5\n-150 -100 -50 0 50 100 150-800-600-400-2000200400600800\n-2 0 2-2000200m (µemu)\nH (Oe) YIG+GGG\n YIG\n GGG  m (µemu)\n H (Oe)\n(a)\n/s48/s46/s49/s49/s54/s48 /s49/s50/s48\n/s49/s56/s48\n/s50/s52/s48 /s51/s48/s48/s48/s46/s49\n/s49/s32/s72\n/s67/s32/s44/s32/s72\n/s83/s32/s40/s79/s101/s41/s32/s32/s32/s32/s32/s32/s32/s32/s52 /s77\n/s83/s32/s40/s107/s71/s115/s41\n/s32/s32/s52 /s77\n/s115\n/s32/s72\n/s115\n/s32/s72\n/s99\n(b)\nFIG. 4: (a) The net VSM magnetic moment mof the\nsample D as well as its components induced by the YIG\n\flm and the GGG substrate vs the in-plane magnetic\n\feldHparallel to theh110idirection. (b) Azimuthal\nangle dependencies for the VSM loop parameters of\nsample D, i.e. a saturation magnetization Ms, a\nsaturation \feld Hsand a coercivity Hc. TheHssix-fold\nsymmetry with the mimima along the h112i`easy axes'\nand the maxima along the h110i`hard axes' indicates\nthe cubic magnetocrystalline anisotropy.\nviously reveals the six-fold symmetry which matches the\ncrystallographic symmetry of YIGs. The Hsmaxima co-\nincide with the in-plane h110iprojections of the hard\nmagnetization axes, whereas the Hsminima correspond\nto theh112icrystallographic directions. The h112i`easy\naxes' orientation suggests an `easy cone' anisotropy after\nUbizskii26. He has also demonstrated27that relatively\nsmall in-plane magnetic \felds lead to single-domain YIG\n\flms, although a deviation of magnetization vector from\nthe \flm plane still remains due to \fnite values of the\ncubic anisotropy constants.\nIn conclusion, as the demagnetizing factor at the out-\nof-plane YIG \flm orientation is 1, the out-of-plane satu-\nration \feld has to be close to the in-plane 4 \u0019Msvalues.\nThis fact is qualitatively con\frmed by our out-of-plane\nmeasurements. Unfortunately, the GGG component ofthe total VSM signal at \felds H?\u00191:8 kOe is much\nlarger than magnetic moments of YIG \flms with a thick-\nness of\u0019100 nm (see, for instance, Fig. 4a) and, hence,\na reasonable accuracy of \u00060.5 % at the GGG signal elim-\nination inevitably results in too large errors for the YIG\nparameters. One may conclude that the out-of-plane con-\n\fguration may provide reliable results when the ratio of\nYIG to GGG thickness exceeds, at least, 10\u00003.\nC. FMR absorption\nFMR absorption spectra for each of studied YIG \flms\nwere recorded at several values ( H\u00145 kOe) of the in-\nplane magnetic \feld. The inset in Fig. 5 shows such a\nspectrum at H= 1:6 kOe that looks like the Lorentz\nfunction with a linewidth \u0001 fFWHM\u00194 MHz centered\nnear the FMR frequency f\u00196:5 GHz. Since the FMR\nlinewidth is mostly expressed in units of magnetic \feld,\nwe, at \frst, used the centers fof measured spectra and\nthe corresponding in-plane \felds Hto estimate the gy-\nromagnetic ratio \rand the e\u000bective magnetization Me\u000b\nin the Kittel formula28\nf=\rp\nH(H+ 4\u0019Me\u000b): (1)\nThen, the best \ftting pair of \randMe\u000ballowed us (i) to\nconvert every frequency spectrum into the magnetic \feld\nscale, (ii) to \ft rescaled spectra with the Lorentz function\nand (iii) to evaluate, thereby, the corresponding linewidth\n\u0001HFWHM . The selected results of the described proce-\ndure { namely, 4 \u0019Me\u000band \u0001HFWHM at the reference\nfrequencyf= 6:5 GHz { are listed in Table II, while\nthe whole summary of the obtained \u0001 HFWHM values is\n0 5 10 15012\n012\n6.48 6.49 6.500.0000.0050.0100.015\n  \n  A: 1.4 x10-4; 0.5 Oe\n  B: 1.2 x10-4; 0.7 Oe\n  C: 1.6 x10-4; 0.5 Oe\n  D: 1.7 x10-4; 0.7 Oe\n ∆HFWHM (Oe)\nf (GHz)Sample: α ; ∆H0 \n f (GHz)1-S21\nFIG. 5: Frequency dependence of FMR absorption\nlinewidth \u0001 HFWHM for YIG LPE \flms A{D at various\nvalues of the in-plane magnetic \feld ( H\u00145 kOe).\nStraight lines are linear \fts that the Gilbert damping\nfactors\u000bare obtained from. Inset shows an example of\nFMR absorption spectrum measured for the sample A\natH= 1:6 kOe.6\n0 5 10 15 200.00.51.01.52.02.5\n α = 1.2×10-4\nα = 0.7×10-4\nα = 0.4×10-4 d = 106 nm\n d = 410 nm\n d = 3.0 µm\n d = 300 µm∆HFWHM (Oe)\nf (GHz)α = 0.5×10-4\nFIG. 6: Frequency dependencies of the FMR linewidth\n\u0001HFWHM for YIG LPE \flms of various thickness dand\nthe YIG sphere with diameter d= 300\u0016m. The Gilbert\ndamping factors \u000bare calculated from slopes of the best\nlinear \fts according to Eq. (2).\npresented in Fig. 5 vs the FMR frequency. The plots in\nFig. 5 are known10{14to provide data about the Gilbert\ndamping coe\u000ecient \u000band the inhomogeneous contribu-\ntion \u0001H0to the FMR linewidth that are mutually related\nby\n\u0001HFWHM = \u0001H0+2\u000bf\n\r(2)\nAs the FMR performance of thin YIG \flms strongly\ndepends on the working frequency of future magnonic\napplications, we have included various quality parame-\nters in Table II, viz. i) the Gilbert damping coe\u000ecient\n\u000bwhich is mostly responsible for the FMR losses at\nhigh magnetic \felds ( H\u001d4\u0019Me\u000b), ii) the inhomoge-\nneous contribution \u0001 H0that dominates at small \felds\n(H\u001c4\u0019Me\u000b) as well as iii) the FMR linewidth at the\nreference frequency f=6.5 GHz which approximately cor-\nresponds to the case H\u00194\u0019Me\u000b. The latter is estimated\ndown to \u0001HFWHM =1.4 Oe that is to our knowledge the\nnarrowest value reported so far for YIG \flms with a thick-\nness of about 100 nm and smaller. The Gilbert damping\ncoe\u000ecients are estimated to be close to \u000b\u00191\u000210\u00004\nwhich is comparable to the best values reported so far\n(compare with Table I). The zero frequency term \u0001 H0is\nfound almost the same for all YIG \flms including the La\nsubstituted one. The obtained value \u0001 H0\u00190:5\u00000:7 Oe\nappears as well appreciably lower than that for gas phase\nepitaxial \flms (see Table I).\nIn summary, optimized LPE growth and post-\nprocessing conditions improve FMR linewidths and\nGilbert damping coe\u000ecients (compare this study and\nRef.29 with Ref.10). However, the improved values are\nstill far from these in bulk YIGs and relatively thick YIG\n\flms (see Fig. 6) due to the decreasing volume to inter-\nface ratio in sub-micrometer \flms. For example, imper-\nfections at the \flm interface of thin \flms should have astronger in\ruence on the magnetic losses in contrast to\nthe dominating volume properties of perfect thick \flms.\nIt requires us to undertake further attempts to mini-\nmize the FMR performance deterioration with a decrease\nof the YIG \flm thickness. These attempts will be fo-\ncused on avoiding the most probable sources of FMR\nlosses such as contributions due to homogeneous broad-\nening (interface roughness, homogeneously distributed\ndefects and impurities) and inhomogeneous broadening\n(geometric and magnetic mosaicity, single surface de-\nfects) and, thus, on approaching the \\target\" parameters\nof \u0001HFWHM = 0:3 Oe at 6.5 GHz and \u000b= 0:4\u000210\u00004\nreported by R oschmann and Tolksdorf30for bulk discs\nmade of single YIG crystals.\nIII. OUTLOOK AND CONCLUSIONS\nBesides the e\u000borts to avoid growth defects as well as\ninterface roughness and to reduce impurity incorpora-\ntion during the LPE deposition process further high-\nresolution investigations are necessary to gain more in-\nsight into the YIG microstructure and to identify the\nproperties which play an essential role for its FMR per-\nformance. Therefore, in future studies we will carry out\nHR-RSM scans with asymmetrical re\rections to deter-\nmine in-plane and axial strain, respectively, the Time-\nof-Flight (ToF) SIMS analysis technique using element\nstandards to precisely quantify the La substitution con-\ncentration as well as to detect impurity elements from the\nhigh-temperature solutions in our sub-micrometer LPE\n\flms. Furthermore, angular dependent measurements of\nthe resonance \feld and of the FMR linewidth will be in-\ntended to determine the in\ruence of uniaxial magnetic\nanisotropies on the ferromagnetic resonance losses.\nIn conclusion, liquid phase epitaxy has the potential to\nprovide sub-micrometer YIG \flms with outstanding crys-\ntalline and magnetic properties to meet the requirements\nfor future magnon spintronics with ultra-low e\u000bective\nlosses if a drastic miniaturization down to the nanometer\nscale is possible. First sub-100 nm lateral sized structures\nhave presently been prepared31which could be the next\nstep to LPE-based microscaled spintronic circuits. The\ndevelopment of YIG LPE \flms with thicknesses below\n100 nm is now in progress and remains a big challenge\nfor the classical thick-\flm LPE technique.\nIV. METHODS\nA. Sample fabrication\nYIG \flms were grown from PbO-B 2O3based high-\ntemperature solutions resistively-heated in a platinum\ncrucible at about 900\u000eC using standard dipping LPE\ntechnique. During di\u000berent growth runs nominally pure\nYIG \flms were grown on one-inch (111) gadolinium gal-\nlium garnet (GGG) substrates to check the reproducibil-7\nity of the sub-micrometer liquid phase epitaxial growth.\nFor La substituted \flms La 2O3was added to the al-\nready used high-temperature solution. To remove solu-\ntion remnants from the sample surfaces the holder had\nto be stored in a hot acidic solution after room tem-\nperature cooling. Afterwards the reverse side layer was\nremoved by mechanical polishing from the double-side\ngrown samples. Chips of di\u000berent sizes were prepared by\na diamond wire saw and sample surfaces were cleaned\nusing ethanol, distilled water and acetone. The LPE \flm\nthickness was determined by X-ray re\rectometry using a\nPANanalytical/X-Pert Pro system.\nB. Microstructural properties\nThe root-mean-square surface roughness was deter-\nmined by AFM measurements for each sample at three\ndi\u000berent regions over 25 \u0016m2ranges using a Park Scien-\nti\fc Instruments, M5. HR-XRD studies were performed\nby a \fve-crystal di\u000braction spectrometer of Seifert (3003\nPTS HR) equipped with a four-fold Ge 440 asymmetric\nmonochromator using CuK \u000bradiation. The resolution\nlimit was 1\u000210\u00004deg. GGG substrate lattice param-\neters were obtained by the Bond method. Depth pro-\n\fle analyses were carried out by an Axis UltraDLDXPS\nsystem (Kratos Analytical Ltd.) using a mono-atomic\nargon-ion etching technique. Qualitative SIMS (Hiden\nAnalytical) measurements were carried out. Here, a \flm\narea of 500\u0002500\u0016m2is irradiated by 5 keV oxygen ions.\nC. Magnetic properties\nThe vibrating sample magnetometer (MicroSense\nLLC, EZ-9) was used to register the in-plane hystere-\nsis loops of the YIG/GGG samples at room tempera-\nture. The external magnetic \feld Hwas controlled with\nan error of\u00140.01 Oe. To estimate the magnetization of\nthe YIG \flms we removed a contribution of the GGG\nsubstrates from the total VSM signal. To monitor the\nin-plane anisotropy as a function of the crystallographic\norientation, the hysteresis loops at the azimuthal angles\n0\u000e\u0014'\u0014360\u000ewere measured with an angular step of\n3\u000e. The FMR absorption spectra were registered with a\nvector network analyzer (Rohde & Schwarz GmbH, ZVA\n67) attached to a broadband stripline. The sample was\ndisposed face-down over a stripline and the transmission\nsignals (S21&S12) were recorded. During the measure-\nments, a frequency of microwave signals with the input\npower of\u000010 dBm (0.1 mW) was swept across the res-\nonance frequency, while the in-plane magnetic \feld H\nwas constant and measured with an accuracy of 1 Oe.\nEach recorded spectrum was \ftted by the Lorentz func-\ntion and allowed us to de\fne the resonance frequency\nand the FMR linewidth \u0001 HFWHM corresponding to the\napplied \feld H.1V. V. Kruglyak and R. J. Hicken, \\Magnonics: Experiment to\nprove the concept,\" J. Magn. Magn. Mater. 306, 191{194 (2006),\ncond-mat/0511290.\n2S. Neusser, B. Botters, and D. Grundler, \\Localization, con\fne-\nment, and \feld-controlled propagation of spin waves in Ni 80Fe20\nantidot lattices,\" Phys. Rev. B 78, 054406 (2008).\n3V. V. Kruglyak, S. O. Demokritov, and D. Grundler, \\Magnon-\nics,\" J. Phys. D: Appl. Phys. 43, 264001 (2010).\n4R. L. Stamps, S. Breitkreutz, J. \u0017Akerman, A. V. Chumak,\nY. Otani, G. E. W. Bauer, J.-U. Thiele, M. Bowen, S. A.\nMajetich, M. Kl aui, I. Lucian Prejbeanu, B. Dieny, N. M.\nDempsey, and B. Hillebrands, \\The 2014 Magnetism Roadmap,\"\nJ. Phys. D: Appl. Phys. 47, 333001 (2014), arXiv:1410.6404\n[cond-mat.mtrl-sci].\n5R. C. LeCraw, E. G. Spencer, and C. S. Porter, \\Ferromagnetic\nResonance Line Width in Yttrium Iron Garnet Single Crystals,\"\nPhys. Rev. 110, 1311{1313 (1958).\n6A. A. Serga, A. V. Chumak, and B. Hillebrands, \\YIG magnon-\nics,\" J. Phys. D: Appl. Phys. 43, 264002 (2010).\n7A. V. Chumak, A. A. Serga, and B. Hillebrands, \\Magnon tran-\nsistor for all-magnon data processing,\" Nat. Commun. 5, 4700\n(2014).\n8A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands,\n\\Magnon spintronics,\" Nat. Phys. 11, 453{461 (2015).\n9E. A. Giess, J. D. Kuptsis, and E. A. D. White, \\Liquid phase\nepitaxial growth of magnetic garnet \flms by isothermal dipping\nin a horizontal plane with axial rotation,\" J. Cryst. Growth 16,\n36{42 (1972).\n10P. Pirro, T. Br acher, A. V. Chumak, B. L agel, C. Dubs,\nO. Surzhenko, P. G ornert, B. Leven, and B. Hillebrands, \\Spin-\nwave excitation and propagation in microstructured waveguides\nof yttrium iron garnet/Pt bilayers,\" Appl. Phys. Lett. 104,\n012402 (2014), arXiv:1311.6305 [cond-mat.mes-hall].\n11M. C. Onbasli, A. Kehlberger, D. H. Kim, G. Jakob, M. Kl aui,\nA. V. Chumak, B. Hillebrands, and C. A. Ross, \\Pulsed laser\ndeposition of epitaxial yttrium iron garnet \flms with low Gilbert\ndamping and bulk-like magnetization,\" APL Mater. 2, 106102\n(2014).\n12B. M. Howe, S. Emori, H.-M. Jeon, T. Oxhol, J. G. Jones, K. Ma-\nhalingam, Y. Zhuang, N. X. Sun, and G. J. Brown, \\Pseudomor-\nphic yttrium iron garnet thin \flms with low damping and inho-\nmogeneous linewidth broadening,\" IEEE Magn. Lett. 8, 3500504\n(2015).\n13H. Chang, P. Li, W. Zhang, T. Liu, A. Ho\u000bmann, L. Deng,\nand M. Wu, \\Nanometer-thick yttrium iron garnet \flms with\nextremely low damping,\" IEEE Magn. Lett. 5, 6700104 (2014).\n14H. Wang, Understanding of Pure Spin Transport in a Broad\nRange of Y3Fe5O12-based Heterostructures , Ph.D. thesis, The\nOhio State University (2015).\n15O. d'Allivy Kelly, A. Anane, R. Bernard, J. Ben Youssef,\nC. Hahn, A. H. Molpeceres, C. Carr\u0013 et\u0013 ero, E. Jacquet, C. Der-\nanlot, P. Bortolotti, R. Lebourgeois, J.-C. Mage, G. de Loubens,\nO. Klein, V. Cros, and A. Fert, \\Inverse spin Hall e\u000bect in\nnanometer-thick yttrium iron garnet/Pt system,\" Appl. Phys.\nLett. 103, 082408 (2013), arXiv:1308.0192 [cond-mat.mtrl-sci].\n16V. Castel, N. Vlietstra, B. J. van Wees, and J. B. Youssef, \\Fre-\nquency and power dependence of spin-current emission by spin\npumping in a thin-\flm YIG/Pt system,\" Phys. Rev. B 86, 134419\n(2012), arXiv:1206.6671 [cond-mat.mtrl-sci].\n17C. Hahn, G. de Loubens, O. Klein, M. Viret, V. V. Naletov, and\nJ. Ben Youssef, \\Comparative measurements of inverse spin Hall\ne\u000bects and magnetoresistance in YIG/Pt and YIG/Ta,\" Phys.\nRev. B 87, 174417 (2013), arXiv:1302.4416 [cond-mat.mes-hall].\n18L. J. Cornelissen, J. Liu, R. A. Duine, J. B. Youssef, and B. J.\nvan Wees, \\Long-distance transport of magnon spin information\nin a magnetic insulator at room temperature,\" Nat. Phys. 11,\n1022{1026 (2015), arXiv:1505.06325 [cond-mat.mes-hall].\n19S. A. Manuilov, R. Fors, S. I. Khartsev, and A. M. Grishin, \\Sub-\nmicron Y 3Fe5O12Film Magnetostatic Wave Band Pass Filters,\"\nJ. Appl. Phys. 105, 033917{033917 (2009).8\n20Y. Sun, Y.-Y. Song, H. Chang, M. Kabatek, M. Jantz, W. Schnei-\nder, M. Wu, H. Schultheiss, and A. Ho\u000bmann, \\Growth and\nferromagnetic resonance properties of nanometer-thick yttrium\niron garnet \flms,\" Appl. Phys. Lett. 101, 152405 (2012).\n21S. A. Manuilov, S. I. Khartsev, and A. M. Grishin, \\Pulsed laser\ndeposited Y 3Fe5O12\flms: Nature of magnetic anisotropy I,\" J.\nAppl. Phys. 106, 123917{123917 (2009).\n22R. Hergt, H. Pfei\u000ber, P. G ornert, M. Wendt, B. Keszei, and\nJ. Vandlik, \\Kinetic Segregation of Lead Impurities in Garnet\nLPE Films,\" Phys. Stat. Sol. (a) 104, 769{776 (1987).\n230.1 at-% Pb corresponds to xPb\u00190:02 formular units in stoi-\nchiometric Y 3\u0000xPbxFe5O12.\n240.5 at-% La corresponds to yLa\u00190:1 formular units in stoichio-\nmetric Y 3\u0000yLayFe5O12.\n25G. Winkler, \\Magnetic garnets,\" in Vieweg tracts in pure and\napplied physics; Volume 5 (Friedrich Vieweg & Sohn Verlag,\nBraunschweig, Wiesbaden, 1981) Chap. 2, pp. 75{79.\n26S. B. Ubizskii, \\Orientational states of magnetization in epitaxial\n(111)-oriented iron garnet \flms,\" J. Magn. Magn. Mater. 195,\n575{582 (1999).\n27S. B. Ubizskii, \\Magnetization reversal modelling for (111)-\noriented epitaxial \flms of iron garnets with mixed anisotropy,\"\nJ. Magn. Magn. Mater. 219, 127{141 (2000).\n28C. Kittel, \\On the Theory of Ferromagnetic Resonance Absorp-\ntion,\" Phys. Rev. 73, 155{161 (1948).\n29V. Lauer, D. A. Bozhko, T. Br acher, P. Pirro, V. I. Vasyuchka,\nA. A. Serga, M. B. Jung\reisch, M. Agrawal, Y. V. Kobljanskyj,\nG. A. Melkov, C. Dubs, B. Hillebrands, and A. V. Chumak,\n\\Spin-transfer torque based damping control of parametrically\nexcited spin waves in a magnetic insulator,\" Appl. Phys. Lett.\n108, 012402 (2016), arXiv:1508.07517 [cond-mat.mes-hall].\n30P. R oschmann and W. Tolksdorf, \\Epitaxial growth and anneal-\ning control of FMR properties of thick homogeneous Ga substi-\ntuted yttrium iron garnet \flms,\" Mat. Res. Bull. 18, 449{459\n(1983).31T. L ober, A. V. Chumak, and B. Hillebrands, Unpublished re-\nsults.\nV. ACKNOWLEDGEMENTS\nWe acknowledge the partial \fnancial support by\nDeutsche Forschungsgemeinschaft (DU 1427/2-1). We\nthank M. Frigge for EPMA analysis, Ch. Schmidt for\nXRR measurements and R. Meyer and B. Wenzel for\ntechnical support.\nVI. AUTHOR CONTRIBUTIONS STATEMENT\nC.D. conceived the experiments, prepared all samples\nand analyzed the data. O.S. performed VSM and FMR\nmeasurements and analyzed the data. R.L. performed\nthe XPS experiments. J.D. and U.B. performed the SIMS\nexperiments. A.D. conducted the XRD experiments and\nanalyzed the data. C.D. and O.S. wrote the manuscript.\nAll authors contributed to scienti\fc discussions and the\nmanuscript review.\nVII. ADDITIONAL INFORMATION\nA. Competing \fnancial interests\nThe authors declare no competing \fnancial interests."    },    {        "title": "1612.07020v2.Spin_Pumping__Dissipation__and_Direct_and_Alternating_Inverse_Spin_Hall_Effects_in_Magnetic_Insulator_Normal_Metal_Bilayers.pdf",        "content": "Spin Pumping, Dissipation, and Direct and Alternating Inverse Spin Hall E\u000bects in\nMagnetic Insulator-Normal Metal Bilayers\nAndr\u0013 e Kapelrud and Arne Brataas\nDepartment of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\nWe theoretically consider the spin-wave mode- and wavelength-dependent enhancement of the\nGilbert damping in magnetic insulatornormal metal bilayers due to spin pumping as well as the\nenhancement's relation to direct and alternating inverse spin Hall voltages in the normal metal. In\nthe long-wavelength limit, including long-range dipole interactions, the ratio of the enhancement\nfor transverse volume modes to that of the macrospin mode is equal to two. With an out-of-\nplane magnetization, this ratio decreases with both an increasing surface anisotropic energy and\nmode number. If the surface anisotropy induces a surface state, the enhancement can be an order of\nmagnitude larger than for to the macrospin. With an in-plane magnetization, the induced dissipation\nenhancement can be understood by mapping the anisotropy parameter to the out-of-plane case\nwith anisotropy. For shorter wavelengths, we compute the enhancement numerically and \fnd good\nagreement with the analytical results in the applicable limits. We also compute the induced direct-\nand alternating-current inverse spin Hall voltages and relate these to the magnetic energy stored\nin the ferromagnet. Because the magnitude of the direct spin Hall voltage is a measure of spin\ndissipation, it is directly proportional to the enhancement of Gilbert damping. The alternating spin\nHall voltage exhibits a similar in-plane wave-number dependence, and we demonstrate that it is\ngreatest for surface-localized modes.\nPACS numbers: 76.50.+g, 75.30.Ds, 75.70.-i, 75.76.+j, 75.78.-n\nI. INTRODUCTION\nIn magnonics, one goal is to utilize spin-based sys-\ntems for interconnects and logic circuits1. In previous\ndecades, the focus was to gain control over these systems\nby exploiting long-range dipole interactions in combina-\ntion with geometrical shaping. However, the complex\nnature of the nonlinear magnetization dynamics persis-\ntently represents a challenge in using geometrical shaping\nalone to realize a variety of desired properties1.\nIn magnonic systems, a unique class of materials con-\nsists of magnetic insulators. Magnetic insulators are elec-\ntrically insulating, but localized magnetic moments cou-\nple to form a long-range order. The prime example is\nYttrium Iron Garnet (YIG). YIG is a complex crystal2\nin the Garnet family, where the Fe2+and Fe3+ions at\ndi\u000berent sites in the unit cell contribute to an overall fer-\nrimagnetic ordering. What di\u000berentiates YIG from other\nferromagnetic (ferrimagnetic) systems is its extremely\nlow intrinsic damping. The Gilbert damping parame-\nter measured in YIG crystals is typically two orders of\nmagnitude smaller than that measured in conventional\nmetallic ferromagnets (Fe, Co, Ni, and alloys thereof).\nThe recent discovery that the spin waves in mag-\nnetic insulators strongly couple to spin currents in ad-\njacent normal metals has re-invigorated the \feld of\nmagnonics3{12. Although there are no mobile charge car-\nriers in magnetic insulators, spin currents \row via spin\nwaves and can be transferred to itinerant spin currents in\nnormal metals via spin transfer and spin pumping13,14.\nThese interfacial e\u000bects open new doors with respect to\nlocal excitation and detection of spin waves in magnonic\nstructures. Another key element is that we can transfer\nknowledge from conventional spintronics to magnonics,opening possibilities for novel physics and technologies.\nTraditionally, spin-wave excitation schemes have focused\non the phenomenon of resonance or the use of \u001frsted\n\felds from microstrip antennas.\nA cornerstone for utilizing these systems is to estab-\nlish a good understanding of how the itinerant elec-\ntrons in normal metals couple across interfaces with\nspin-wave dynamics in magnetic insulators. Good mod-\nels for adressing uniform (macrospin) magnetization\nthat agrees well with experiments have been previously\ndeveloped13{15. We recently demonstrated that for long-\nwavelength magnons the enhanced Gilbert damping for\nthe transverse volume modes is twice that of the uniform\nmode, and for surface modes, the enhancement can be\nmore than ten times stronger. These results are con-\nsistent with the theory of current-induced excitations\nof the magnetization dynamics16because spin pump-\ning and spin transfer are related by Onsager reciprocity\nrelations17. Moreover, mode- and wave-vector-dependent\nspin pumping and spin Hall voltages have been clearly\nobserved experimentally4.\nIn this paper, we extend our previous \fndings18in the\nfollowing four aspects. i) We compute the in\ruence of\nthe spin back\row on the enhanced spin dissipation. ii)\nWe also compute the induced direct and alternating in-\nverse spin Hall voltages. We then relate these voltages to\nthe enhanced Gilbert damping and the relevant energies\nfor the magnetization dynamics. The induced voltages\ngive additional information about the spin-pumping pro-\ncess, which can also be directly measured. iii) We also\nprovide additional information on the e\u000bects of interfa-\ncial pinning of di\u000berent types in various \feld geometries.\niv) Finally, we explain in more detail how the numerical\nanalysis is conducted for a greater number of in-planearXiv:1612.07020v2  [cond-mat.mes-hall]  6 Apr 20172\nwave numbers.\nIt was discovered19{23and later quantitatively\nexplained13,15,24,25that if a dynamic ferromagnetic mate-\nrial is put in contact with a normal metal, the magnetiza-\ntion dynamics will exert a torque on the spins of electrons\nin the immediate vicinity of the magnet. This e\u000bect is\nknown as spin pumping (SP)13,15,25. As the electrons are\ncarried away from the ferromagnet-normal metal inter-\nface, the electrons spin with respect to each other, caus-\ning an overall loss of angular momentum. The inverse\ne\u000bect, in which a spin-polarized current can a\u000bect the\nmagnetization of a ferromagnet, is called spin-transfer\ntorque (STT)26{28.\nThe discovery that a precessing magnetization in mag-\nnetic insulators3, such as YIG, also pumps spins into an\nadjacent metal layer was made possible by the fact that\nthe mixing conductance in YIG-normal metal systems is\nof such a size that the extra dissipation of the magneti-\nzation due to the spin pumping is of the same order of\nmagnitude as the intrinsic Gilbert damping. A conse-\nquence of this e\u000bect is that the dissipation of the magne-\ntization dynamics is enhanced relative to that of a system\nin which the normal metal contact is removed.\nThis paper is organized in the following manner. Sec-\ntion II presents the equation of motion for the magne-\ntization dynamics and the currents in the normal metal\nand the appropriate boundary conditions, both for gen-\neral nonlinear excitations and in the fully linear response\nregime. In Section III, we derive approximate solutions\nto the linearized problem, demonstrating how the mag-\nnetization dissipation is enhanced by the presence of an\nadjacent metal layer. Section IV presents our numerical\nmethod and results. Finally, we summarize our \fndings\nin Section V.\nII. EQUATIONS OF MOTION\nThe equation of motion for the magnetization is given\nby the Landau-Lifshitz-Gilbert equation29(presented\nhere in CGS units)\n@M\n@t=\u0000\rM\u0002He\u000b+\u000b\nMsM\u0002@M\n@t; (1)\nwhere\r=jg\u0016B=~jis the magnitude of the gyromagnetic\nratio;g\u00192 is the Land\u0013 e g-factor for the localized elec-\ntrons in the ferromagnetic insulator (FI); and \u000bis the di-\nmensionless Gilbert damping parameter. In equilibrium,\nthe magnitude of the magnetization is assumed to be\nclose to the saturation magnetization Ms. The magneti-\nzation is directed along the z-axis in equilibrium. Out of\nequilibrium, we assume that we have a small transverse\ndynamic magnetization component, such that\nM=M(r;t) =Ms+m(r;t) =Ms^z+m(r;t);(2)\nwherejmj\u001cMsandm\u0001^z= 0. Furthermore, we assume\nthat the dynamic magnetization can be described by a\nx\nhzx\nyz\nfq(a)\nd\nL2\n-L2NM\nFI\nSUBx (b)\nFIG. 1. a) The coordinate system. ^\u0018is the \flm normal\nand ^\u0010is the spin-wave propagation direction. \u0018\u0011\u0010form a\nright-handed coordinate system. The ^zaxis is the direc-\ntion of the magnetization in equilibrium, such that xyis the\nmagnetization-precession plane. b) The \flm stack is in the\nnormal direction.\nplane wave traveling along the in-plane \u0010-axis. In the\n(\u0018;\u0011;\u0010 ) coordinate system (see Figure 1), we have\nm(r;t) =m(\u0018;\u0010;t ) =mQ(\u0018)ei(!t\u0000Q\u0010); (3)\nwhere!is the harmonic angular frequency, Qis the in-\nplane wave number, and mQ(\u0018) =XQ(\u0018)^x+YQ(\u0018)^y,\nwhereXQandYQare complex functions. Note that m\nis independent of the \u0011coordinate due to translational\ninvariance.\nHe\u000bis the e\u000bective \feld, given as the functional deriva-\ntive of the free energy29,30\nHe\u000b(r;t) =\u0000\u000eU[M(r;t)]\n\u000eM(r;t)=Hi+2A\nM2sr2M(r;t)+\n+ 4\u0019ZL\n2\n\u0000L\n2d\u00180bGxy(\u0018\u0000\u00180)m(\u00180;\u0010;t);(4)\nwhere Hiis the internal \feld, which is composed of\nthe applied external \feld and the static demagnetization\n\feld. The direction of Hide\fnes the z-axis (see Fig-\nure 1). The second term of Eq. (4) is the \feld, Hex,\ninduced by to the exchange interaction (assuming cu-\nbic symmetry), where Ais the exchange sti\u000bness pa-\nrameter. The last term is the dynamic \feld, hd(r;t),\ninduced by dipole-dipole interactions, where bGxyis the\nupper 2\u00022 part of the dipole{dipole tensorial Green's\nfunctionbG\u0018\u0011\u0010in the magnetostatic approximation31ro-\ntated to the xyzcoordinate system (see Appendix A for\ncoordinate-transformation matrices).32\nThe e\u000bect of the dipolar interaction on the spin-wave\nspectrum depends on the orientation of the internal \feld\nwith respect to both the interface normals of the thin\n\flm, ^\u0018, and the in-plane spin-wave propagation direc-\ntion, ^\u0010. Traditionally, the three main con\fgurations are\nthe out-of-plane con\fguration ( \u0012= 0), in the forward\nvolume magnetostatic wave (FVMSW) geometry (see\nFig. 2a); the in-plane and parallel-to- ^\u0010con\fguration, in3\nthebackward volume magnetostatic wave (BVMSW) ge-\nometry (see Fig. 2b); and the in-plane and perpendicular-\nto-^\u0010con\fguration, in the magnetostatic surface wave\n(MSSW) geometry (see Fig. 2c).1,32{36Here, the term\n\\forward volume modes\" denotes modes that have posi-\ntive group velocities for all values of QL, whereas back-\nward volume modes can have negative group velocities\nin the range of QL, where both exchange and dipolar\ninteractions are signi\fcant. Volume modes are modes in\nwhich mQ(\u0018) is distributed across the thickness of the\nentire \flm, whereas the surface modes are localized more\nclosely near an interface.\nA. Spin-Pumping Torque\nWe consider a ferromagnetic insulator (FI) in contact\nwith a normal metal (NM) (see Figure 1). If the magneti-\nzation in the FI close to the interface is precessing around\nthe e\u000bective \feld, electron spins in the NM re\rected at\nthe interface will start to precess due to the local ex-\nchange coupling to the magnetization in the FI. The re-\n\rected electrons carry the angular momentum away from\nthe interface, where the spin information can get lost\nthrough dephasing of the spins within a typical spin di\u000bu-\nsion length lsf. This loss of angular momentum manifests\nitself as an increased local damping of the magnetization\ndynamics in the FI. The magnetization dissipation due\nto the spin-pumping e\u000bect can be taken into account by\nadding the local dissipation torque15\n\u001csp=\r~2g?\n2e2M2s\u000e(\u0018\u0000L\n2)M(r;t)\u0002@M(r;t)\n@t;(5)\nto the right-hand side (rhs) of Eq. (1). Here, g?is the\nreal part of the spin-mixing conductance per area, and e\nis the electron charge. We neglect the contribution from\nthe imaginary part of the mixing conductance, because\nthis has been shown to be signi\fcantly smaller than that\nof the real part, in addition to a\u000becting only the gyro-\nmagnetic ratio.15The spin-current density pumped from\nthe magnetization layer is thus given by\nj(s)\nsp=\u0000~2g?\n2e2M2s\u0014\nM(r;t)\u0002@M(r;t)\n@t\u0015\n\u0018=L=2;(6)\nin units of erg. Next, we will see how the spin pumping\na\u000bects the boundary conditions.\nB. Spin-Pumping Boundary Conditions\nFollowing the procedure of Rado and Weertman37, we\nintegrate Eq.(1) with the linear expansion of Eq. (2) over\na small pill-box volume straddling one of the interfaces\nof the FI. Upon letting the pill box thickness tend to\nzero, only the surface torques of the equation survive.\nAccounting for the direction of the outward normal ofthe lid on the di\u000berent top and bottom interfaces, we\narrive at the exchange-pumping boundary condition\n\u00142A\nM2sM\u0002@M\n@\u0018+~2\n2e2M2sg?M\u0002@m\n@t\u0015\n\u0018=\u0006L=2= 0:(7)\nThere is no spin current pumped at the interface to the\ninsulating substrate; thus, a similar derivation results in\na boundary condition that gives an unpinned magnetiza-\ntion,\n@M(r;t)\n@\u0018\f\f\f\f\n\u0018=\u0000L=2= 0: (8)\nIn the next section, we will generalize the bound-\nary conditions of Eqs. (7) by also considering possible\nsurface-anisotropy energies.\nIncluding surface anisotropy:\nIn the presence of surface anisotropy at an interface\nwith an easy-axis (EA) pointing along the direction ^n,\nthe surface free energy is\nUs[M(r;t)] =Z\ndV Ks\"\n1\u0000\u0012M(r;t)\u0001^n\nMs\u00132#\n\u000e(\u0018\u0000\u0018i);\n(9)\nwhereKsis the surface-anisotropy energy density at the\ninterface, which is assumed to be constant; ^nis the direc-\ntion of the anisotropy easy axis; and\u0018iis the transverse\ncoordinate of the interface. The contribution from the\nEA surface-anisotropy energy to the e\u000bective \feld is de-\ntermined by\nHs=\u0000\u000eUs[M(r;t)]\n\u000eM(r;t)=2Ks\nM2s(M\u0001^n)\u000e(\u0018\u0000\u0018j)^n:\nHowever, if we have an easy-plane (EP) surface\nanisotropy with, ^nbeing the direction of the hard axis,\nthe e\u000bective \feld is the same as that for the EA case,\nexcept for a change of sign of Ks. We unify both cases\nby de\fning Ks>0 to imply that we have an EA surface\nanisotropy with its easy axis along ^n, whereasKs<0\nimplies that we have an EP surface anisotropy with its\nhard axis along ^n.\nFollowing the approach from Section II B, the total\nboundary condition, including exchange, pumping and\nsurface anisotropy, becomes\n\u0014\n\u00062A\nM2sM\u0002@M\n@\u0018\u00002Ks\nM2s(M\u0001^n) (M\u0002^n) +\n+~2\n2e2M2sg?M\u0002@M\n@t\u0015\n\u0018=\u0006L=2= 0;(10)\nwhere the positive (negative) sign in front of the exchange\nterm indicates that the bulk FI is located below (above)\nthe interface coordinate.4\nx,z\nz\n(a)\nx\nz,zq (b)\nx\nz\nzfq (c)\nFIG. 2. Laboratory \feld con\fgurations, i.e., directions of ^z(green arrow) in relation to \flm normal ^\u0018and the spin-wave\npropagation direction ^\u0010, resulting in the di\u000berent geometries: a) FVMSW geometry; b) BVMSW geometry; c) MSSW geometry.\nC. Linearization\nWe linearize the equation of motion using Eq. (2) with\nrespect to the dynamic magnetization m. The linearized\nequation of motion for the bulk magnetization Eq. (1)\nbecomes32\n\u001a\ni!\n!M\u0012\n\u000b\u00001\n1\u000b\u0013\n+11\u0014!H\n!M+ 8\u0019\r2A\n!2\nM\u0012\nQ2\u0000d2\nd\u00182\u0013\u0015\u001b\n\u0001\n\u0001mQ(\u0018) =ZL\n2\n\u0000L\n2d\u00180bGxy(\u0018\u0000\u00180)mQ(\u00180);(11)\nwhere!H\u0011\rHi,!M\u00114\u0019\rMs, and 11 =\u00001 0\n0 1\u0001\n.\nNext, we linearize the boundary conditions of Eq. (10).\nWe choose the anisotropy axis to be perpendicular to the\n\flm plane, ^n=^\u0018, which in the xyzcoordinate system\nis given by ^\u0018xyz= (sin\u0012;0;cos\u0012), where\u0012is the angle\nbetween the z-axis and the \flm normal (see Fig. 1). The\n\fnite surface anisotropy forces the magnetization to be\neither perpendicular or coplanar with the \flm surface so\nthat\u0012= 0;\u0019=2;\u0019. Linearizing to 1st order in the dy-\nnamic magnetization, we arrive at the linearized bound-\nary conditions for the top interface\n\u0012\nL@\n@\u0018+i!\n!M\u001a+dcos(2\u0012)\u0013\nmQ;x(\u0018)\f\f\f\f\n\u0018=L\n2= 0;(12a)\n\u0012\nL@\n@\u0018+i!\n!M\u001a+dcos2(\u0012)\u0013\nmQ;y(\u0018)\f\f\f\f\n\u0018=L\n2= 0;(12b)\nwhered\u0011LKs=Ais the dimensionless surface-\npinning parameter that relates the exchange to the\nsurface anisotropy and the \flm thickness and \u001a\u0011\n!ML~2g?=4Ae2is a dimensionless constant relating the\nexchange sti\u000bness and the spin-mixing conductance.\nD. Spin Accumulation in NM and Spin Back\row\nThe pumped spin current induces a spin accumulation,\n\u0016(s)=\u0016(s)^s, in the normal metal. Here, ^sis the spin-\npolarization axis, and \u0016(s)= (\u0016\"\u0000\u0016#)=2 is half of thedi\u000berence between chemical potentials for spin-up and\nspin-down electrons in the NM.\nAs the spin accumulation is a direct consequence of\nthe spin dynamics in the FI (see Eq. (6)), the spin ac-\ncumulation cannot change faster than the magnetization\ndynamics at the interface. Thus, assuming that spin-\rip\nprocesses in the NM are must faster than the typical pre-\ncession frequency of the magnetization in the FI25, we can\nneglect precession of the spin accumulation around the\napplied \feld and any decay in the NM. With this assump-\ntion, the spin-di\u000busion equation@\u0016(s)\n@t=Dr2\u0016(s)\u0000\u0016(s)\n\u001csf,\nwhereDis the spin-di\u000busion constant, and \u001csfis the\nmaterial-speci\fc average spin-\rip relaxation time, be-\ncomes\n\u0016(s)\u0019l2\nsfr2\u0016(s); (13)\nwherelsf\u0011p\u001csfDis the average spin-\rip relaxation\nlength.\nThe spin accumulation results in a back\rowing spin-\ncurrent density, given by\nj(s)\nbf(L=2) =~g?\ne2M2sh\nM(r;t)\u0002\u0010\nM(r;t)\u0002\u0016(s)(r;t)\u0011i\n\u0018=L=2;\n(14)\nwhere the positive sign indicates \row from the NM into\nthe FI. This spin current creates an additional spin-\ntransfer torque on the magnetization at the interface\n\u001cbf=\u0000\r~g?\ne2M2s\u000e\u0010\n\u0018\u0000L\n2\u0011\nM(r;t)\u0002\u0010\nM(r;t)\u0002\u0016(s)\u0011\n:\n(15)\nBecause the spin accumulation is a direct result of the\npumped spin current, it must have the same orientation\nas the M(r;t)\u0002@tM(r;t) term in Eq. (5). That term is\ncomprised of two orthogonal components: the 1st-order\ntermMs^z\u0002_min thexyplane, and the 2nd-order term\nm\u0002_moriented along ^z. Because the magnetization is\na real quantity, care must be taken when evaluating the\n2nd-order term. Using Eq. (3), the 2nd-order pumped5\nspin current is proportional to\nRefmg\u0002@tRefmg\f\f\f\n\u0018=L=2=e\u00002Imf!gtRef!g\u0002\n\u0002^zh\nImXQReYQ\u0000ReXQImYQi\n;(16)\nwhich is a decaying direct-current (DC) term. This is in\ncontrast to the 1st-order term, which is an alternating-\ncurrent (AC) term. Thus, we write the spin accumula-\ntion as\n\u0016(s)=\u0016(s)\nAC(^z\u0002^mt) +\u0016(s)\nDC^z; (17)\nwhere we have used the shorthand notation mt=_m(\u0018=\nL=2), such that ^mt=mt=jmtj, which in general is not\nparallel to mbut guaranteed to lie in the xyplane. In-\nserting Eq. (17) into Eq. (13) gives one equation each for\nthe AC and DC components of the spin accumulation,\n@2\u0016(s)\nj\n@\u00182=l\u00002\nsf;j\u0016(s)\nj; (18)\nwherejdenotes either the AC or DC case and lsf,DC =lsf\nwhilelsf,AC =lsf(1+l2\nsfQ2)\u00001=2because mt/exp(i(!t\u0000\nQ\u0010)). Eq. (18) can be solved by demanding spin-current\nconservation at the NM boundaries: at the free surface of\nthe NM, there can be no crossing spin current; thus, the \u0018\ncomponent of the spin-current density must vanish there,\n@\u0018\u0016(s)\njj\u0018=L=2+d= 0. Similarly, by applying conservation\nof angular momentum at the FI-NM interface, the net\nspin-current density crossing the interface, due to spin\npumping and back\row, must equal the spin current in\nthe NM layer, giving\n\u0014\n\u0000~2g?\n2e2M2sM\u0002@M\n@t+~g?\ne2M2sM\u0002\u0010\nM\u0002\u0016(s)\u0011\u0015\n\u0018=L=2\n=\u0000~\u001b\n2e2@\u0018\u0016(s)j\u0018=L=2;(19)\nwhere\u001bis the conductivity of the NM. Using these\nboundary conditions, we recover the solutions (see,\ne.g.,25,38)\n\u0016(s)\nj=\u0016(s)\nj;0sinh\u0010\nl\u00001\nsf;j\u0002\n\u0018\u0000(L=2 +d)\u0003\u0011\nsinh\u0010\n\u0000d\nlsf;j\u0011; (20)\nwhere\u0016(s)\nj;0is time dependent, and depends on the \u0010co-\nordinate only in the AC case. We \fnd that the AC and\nDC spin accumulations \u0016(s)\nj;0are given by\n\u0016(s)\nAC;0=\u0000~\n2mt\nMs\u0014\n1 +\u001b\n2g?lsf;ACcoth\u0012d\nlsf;AC\u0013\u0015\u00001\n;\n(21)\n\u0016(s)\nDC;0=\u0000lsf~\n\u001bM2s~g?tanh\u0012d\nlsf\u0013\n^z\u0001[m\u0002_m]\u0018=L=2;\n(22)TABLE I. Typical values for the parameters used in the\ncalculations.6,7,11,39{41\nParameter Value Unit\nA 3:66\u000110\u00007erg cm\u00001\n\u000b 3\u000110\u00004{\nKs 0:05 erg cm\u00002\ng? 8:18\u00011022cm\u00001s\u00001\n\r 1:76\u0001107G\u00001s\u00001\n4\u0019M s 1750 G\n\u001b 8:45\u00011016s\u00001\nd 50 nm\nlsf 7.7 nm\n\u0002 0.1 {\nwhere ~g?is a renormalized mixing conductance, which is\ngiven by\n~g?=g?(\n1\u0000\u0014\n1 +\u001b\n2g?lsf;ACcoth\u0012d\nlsf;AC\u0013\u0015\u00001)\n:\n(23)\nThis scaling of g?occurring in the DC spin accumulation\noriginates from the second-order spin back\row due to the\nAC spin accumulation that is generated in the normal\nmetal.\nAdding both the spin-pumping and the back\row\ntorques to Eq. (1) and repeating the linearization pro-\ncedure from Sec. II C, we \fnd that the AC spin accumu-\nlation renormalizes the pure spin-mixing conductance.\nThus, the addition of the back\row torque can be ac-\ncounted for by replacing g?with ~g?in the boundary con-\nditions of Eqs. (12), making the boundary conditions Q-\ndependent in the process. Using the values from Table I,\nwhich are based on typical values for a YIG-Pt bilayer\nsystem, we obtain ~ g?=g?\u00180:4 forQL\u001c1, whereas\n~g?=g?!1 for large values of QL. Thus, AC back\row is\nsigni\fcant for long-wavelength modes and should be con-\nsidered when estimating g?from the linewidth broaden-\ning in ferromagnetic resonance (FMR) experiments.11\nInverse Spin Hall E\u000bect\nThe inverse spin Hall e\u000bect (ISHE) converts a spin\ncurrent in the NM to an electric potential through the\nspin-orbit coupling in the NM. For a spin current in\nthe^\u0018direction, the ISHE electric \feld in the NM layer\nisEISHE =\u0000e\u00001\u0002h(@\u0018\u0016(s))\u0002^\u0018i\u0018, where \u0002 is the di-\nmensionless spin-Hall angle, and h\u0001i\u0018is a spatial average\nacross the NM layer, i.e., for \u00182(L=2;L=2 +d). Using\nthe previously calculated spin accumulation, we \fnd that6\nthe AC electric \feld is\nEAC\nISHE =\u0000\u0002~\n2deMs\u0014\n1 +\u001b\n2g?lsf;ACcoth\u0012d\nlsf;AC\u0013\u0015\u00001\n\u0002\n\u0002\u0002\n\u0000^\u0011(\u0000mt;ycos\u0012cos\u001e+mt;xsin\u001e)+\n+^\u0010(\u0000mt;xcos\u001e\u0000mt;ycos\u0012sin\u001e)\u0003\n;(24)\nwhere\nmt;i=\u0000[Im!Remi+Re!Immi]\u0018=L=2; (25)\nandi=x;y. For BVMSW ( \u0012=\u0019=2;\u001e= 0) modes,\nthe AC \feld points along ^\u0010, whereas for MSSW ( \u0012=\n\u001e=\u0019=2) modes, it points along ^\u0011(i.e., in plane, but\ntransverse to \u0010; see Fig. 1). Notice that for both BVMSW\nand MSSW mode geometries, only the xcomponent of\nmtcontributes to the \feld. In contrast, for FVMSW\n(\u0012= 0) modes, the \feld points somewhere in the \u0011\u0010\nplane, depending on the ratio of mt;xtomt;y.\nSimilarly to the AC \feld, the DC ISHE electric \feld is\ngiven by\nEDC\nISHE = \u0002\u0016(s)\nDC;0\ndesin\u0012(^\u0011cos\u001e\u0000^\u0010sin\u001e);(26)\nwhich is perpendicular to the AC electric \feld and zero\nfor the FVMSW mode geometry.\nThe total time-averaged energy in the ferromagnet\nEtotal(see Morgenthaler42) is given by\nhEtotaliT=Z\nferriteRe\u0014\n\u0000i\u0019!\u0003\n!M(m\u0002m\u0003)^z\u0015\ndV; (27)\nwhere the integral is taken over the volume of the ferro-\nmagnet.\nBecause the DC ISHE \feld is in-plane, the voltage\nmeasured per unit distance along the \feld direction,\n^\u0003=^\u0011cos\u001e\u0000^\u0010sin\u001e, can be used to construct an esti-\nmate of the mode e\u000eciency. Taking the one-period time\naverage of Eq. (26) using Eq. (22) and normalizing it by\nEq. (27) divided by the in-plane surface area, A, we \fnd\nan amplitude-independent measure of the DC ISHE:\n\u000fDC=he^\u0003\u0001EDC\nISHEiT\nhEtotaliT=A=\u00002\r\u0002lsf~\nd\u001bMs~g?tanh\u0012d\nlsf\u0013\nsin\u0012\u0002\n\u0002Reh\n\u0000i!\u0003\n!M(m\u0002m\u0003)^zi\n\u0018=L=2RL=2\n\u0000L=2Reh\n\u0000i!\u0003\n!M(m\u0002m\u0003)^zi\nd\u0018;(28)\ngiven in units of cm, and where f\u0001g\u0003denotes complex\nconjugation.\nSimilarly, the AC ISHE electric \feld, being time-\nvarying, will contribute a power density that, when nor-\nmalized by the power density in the ferromagnet, be-comes\n\u000fAC=h\u001b\u0000\nEAC\nISHE\u00012iT\nRef!g\n2\u0019ALhEtotaliT=\u0019\u001b\nRef!g\u0012\u0002~\n2deMs\u00132\n\u0002\n\u0002\u0014\n1 +\u001b\n2g?lsf;ACcoth\u0012d\nlsf;AC\u0013\u0015\u00002\n\u0002\n\u0002jmt;xj2+ cos2\u0012jmt;yj2\n1\nLRL=2\n\u0000L=2Reh\n\u0000i!\u0003\n!M(m\u0002m\u0003)^zi\nd\u0018:(29)\nTo be able to calculate explicit realizations of the mode-\ndependent equations Eqs. (28) and (29), one will need to\n\frst calculate the dispersion relation and mode pro\fles\nin the ferromagnet.\nIII. SPIN-PUMPING THEORY FOR\nTRAVELLING SPIN WAVES\nBecause, the linearized boundary conditions (see\nEqs. (12)) explicitly depend on the eigenfrequency !, we\ncannot apply the method of expansion in the set of pure\nexchange spin waves, as was performed by Kalinikos and\nSlavin32. Instead, we analyze and solve the system di-\nrectly for small values of QL, whereas the dipole-dipole\nregime ofQL\u00181 is explored using numerical computa-\ntions in Sec. IV.\nA. Long-Wavelength Magnetostatic Modes\nWhenQL\u001c1 Eq. (11) is simpli\fed to\n( \nsin2\u00120\n0 0!\n+i!\n!M \n\u000b\u00001\n1\u000b!\n+\n+11\u0014!H\n!M\u00008\u0019\r2A\n!2\nMd2\nd\u00182\u0015\u001b\n\u0001mQ(\u0018) = 0;(30)\nwhere the 1st-order matrix term describe the dipole-\ninduced shape anisotropy and stems from bGxy(see32).\nWe make the ansatz that the magnetization vector in\nEq. (3) is composed of plane waves, e.g., mQ(\u0018)/eik\u0018.\nInserting this ansatz into Eq. (30) produces the disper-\nsion relation\n\u0010!\n!M\u00112\n=\u0010!H\n!M+\u00152\nexk2+i\u000b!\n!M\u0011\n\u0002\n\u0002\u0010!H\n!M+\u00152\nexk2+ sin2\u0012+i\u000b!\n!M\u0011\n;(31)\nwhere\u0015ex\u0011p\n8\u0019\r2A=!2\nMis the exchange length . Keep-\ning only terms to \frst order in the small parameter \u000b,\nwe arrive at\n!(k)\n!M=\u0006r\u0010!H\n!M+\u00152exk2\u0011\u0010!H\n!M+\u00152exk2+ sin2\u0012\u0011\n+\n+i\u000b\u0010!H\n!M+\u00152\nexk2+sin2\u0012\n2\u0011\n: (32)7\nThe boundary conditions in Eq. (12) depend explicitly on\n!andkand give another equation k=k(!) to be solved\nsimultaneously with Eq. (32). However, in the absence\nof spin pumping, i.e., when the spin-mixing conductance\nvanishesg?!0, it is su\u000ecient to insert the constant k\nsolutions from the boundary conditions into Eq. (32) to\n\fnd the eigenfrequencies.\nDi\u000berent wave vectors can give the same eigenfre-\nquency. It turns out that this is possible when !(k) =\n!(i\u0014), which has a non-trivial solution relating \u0014tok:\n\u00152\nex\u00142= sin2\u0012+\u00152\nexk2+ 2!H\n!M\u0006i2\u000b!(k)=!M:(33)\nWith these \fndings, a general form of the magnetiza-\ntion is\nmQ(\u0018) = \n1\nr(k)!h\nC1cos\u0000\nk(\u0018+L\n2)\u0001\n+C2sin\u0000\nk(\u0018+L\n2)\u0001i\n+\n+ \n1\nr(i\u0014)!h\nC3cosh\u0000\n\u0014(\u0018+L\n2)\u0001\n+C4sinh\u0000\n\u0014(\u0018+L\n2)\u0001i\n;\n(34)\nwherefCigare complex coe\u000ecients to be determined\nfrom the boundary conditions, and where \u0014=\u0014(k) is\ngiven by Eq. (33). The ratio between the transverse com-\nponents of the magnetization, r(k) =YQ=XQ, is deter-\nmined from the bulk equation of motion (see Eq. (30))\nand is in linearized form\nr(k) =\u0000\u000bsin2\u0012\u00062ir\u0010\n!H\n!M+\u00152exk2\u0011\u0010\n!H\n!M+\u00152exk2+ sin2\u0012\u0011\n2\u0010\n!H\n!M+\u00152exk2\u0011 ;\n(35)\nimplying elliptical polarization of mQwhen\u00126= 0.\nInserting Eq. (34) into Eq. (8) only leads to a solution\nwhenk= 0, such that C2=C4= 0 in the general case.\nBy solving Eq. (12b) for C3, we \fnd\nC3\nC1=\u0000!H\n!M+\u00152\nexk2+ sin2\u0012+i\u000b!\n!M\n!H\n!M\u0000\u00152ex\u00142+ sin2\u0012+i\u000b!\n!M\u0002\n\u0002(i!\n!M~\u001a+dcos2\u0012) cos(kL)\u0000kLsin(kL)\n(i!\n!M~\u001a+dcos2\u0012) cosh(\u0014L) +\u0014Lsinh(\u0014L);(36)\nwhere ~\u001a\u0011\u001ajg?!~g?is the pumping parameter altered by\nthe AC spin back\row from the NM (see Section II D). C1\nis chosen to be the free parameter that parameterizes the\ndynamic magnetization amplitude, which can be deter-\nmined given a particular excitation scheme. Lineariza-\ntion of Eq. (36) with respect to \u000bis straightforward, but\nthe expression is lengthy; we will therefore not show it\nhere.\nInserting the ansatz with C2=C4= 0 andC3given\nby Eq. (36) into Eq. (12a) gives the second equation for\nkand!(the \frst is Eq. (32)). In the general case, thenumber of terms in this equation is very large; thus, we\ndescribe it as\nf(k;!;\u000b; ~\u001a) = 0; (37)\ni.e., an equation that depends on the wave vector k, fre-\nquency!, Gilbert damping constant \u000band spin-pumping\nparameter ~\u001a.\nBecause both the bulk and interface-induced dissipa-\ntion are weak, \u000b\u001c1, ~\u001a\u001c1, the wavevector is only\nslightly perturbed with respect to a system without dis-\nsipation, i.e., k!k+\u000ekwhere\u0015ex\u000ek\u001c1. It is therefore\nsu\u000ecient to expand fup to 1storder in these small quan-\ntities:\nf(k;!; 0;0) + (~\u001a)@f\n@~\u001a\f\f\f\f\n0+\u000b@f\n@\u000b\f\f\f\f\n0+\n+ (\u0015ex\u000ek)@f\n@(\u0015ex\u000ek)\f\f\f\f\n0\u00190;(38)\nwhere the sub-index 0 means evaluation in a system with-\nout dissipation, i.e., when ( \u000b;~\u001a;\u000ek) = (0;0;0). By solv-\ning the system of equations in the absence of dissipation,\nf(k;!; 0;0) = 0, the dissipation-induced change in the\nwave vector \u000ekis given by\n\u000ek\u0019\u0000~\u001a@f\n@~\u001a\f\f\f\n0+\u000b@f\n@\u000b\f\f\f\n0\n\u0015ex@f\n@(\u0015ex\u000ek)\f\f\f\n0: (39)\nIn turn, this change in the wave vector should be in-\nserted into the dispersion relation of Eq. (31) to \fnd\nthe dissipation. Inspecting Eq. (31), we note that \u000ek-\ninduced additional terms proportional to !are of the\nform (k+\u000ek)2\u0000k2\u00192k\u000ekwhich renormalize the Gilbert-\ndamping term i\u000b!\n!M. Thus, in Eq. (39), there are terms\nproportional to the frequency in both terms in the numer-\nator. We extract these terms /i!\n!Mby di\u000berentiating\nwith respect to !and de\fne the renormalization of the\nGilbert damping, i.e., \u000b!\u000b+ \u0001\u000b, from spin pumping\nas\n\u0001\u000b=i2\u0015exk!M@!\u0000\n\u0015ex\u000ekj\u000b=0\u0001\ni2\u0015exk!M@!\u0000\n\u0015ex\u000ekj~\u001a=0\u0001\n\u00001; (40)\nwhere@!represents the derivative with respect to !and\nkis the solution to the 0th-order equation. Note that in\nperforming a further local analysis around some point k0\nin thek-space of Eq. (37), a series expansion of faround\nk0must be performed before evaluating Eqs. (39) and\n(40).\nEq. (40) is generally valid, except when d= 0 and\nkL!0, which we discuss below. In the following sec-\ntion, we will determine explicit solutions of the 0th-order\nequation for some key cases, and mapping out the spin-\nwave dispersion relations and dissipation in the process.8\nB. No Surface Anisotropy ( d= 0)\nLet us \frst investigate the case of a vanishing sur-\nface anisotropy. In this case, the 0th-order expansion\nof Eq. (37) has a simple form and is independent of the\nmagnetization angle \u0012. The equation to determine kis\ngiven by\nkLtan(kL) = 0; (41)\nwith solutions k=n\u0019=L , wheren2Z. Similarly, the\nexpression for \u000ekis greatly simpli\fed, \u000ekn=i!\n!M~\u001a\nn\u0019\u0015ex\nL,\nn6= 0, such that the mode-dependent Gilbert damping\nis\n\u0001\u000bn= 2~\u001a\u0012\u0015ex\nL\u00132\n; n6= 0: (42)\nFor the macrospin mode, when n= 0, the linear ex-\npansion in \u000ekbecomes insu\u000ecient. This is because\nkLtan(kL)\u0018(kL)2forkL!0; thus, we must expand\nthe function fto second order in the deviation \u000ekaround\nkL= 0. Ford= 0, we \fnd that the boundary condi-\ntion becomes \u000ek2L2=i!\n!M~\u001a\u00152\nex, and when inserted into\nEq. (31), it immediately gives\n\u0001\u000b0= ~\u001a\u0012\u0015ex\nL\u00132\n=1\n2\u0001\u000bn; (43)\nwhich is the macrospin renormalization factor found in\nRef. 15. Using a di\u000berent approach, our results in this\nsection reproduce our previous result that the renormal-\nization of the Gilbert damping for standing waves is\ntwice the renormalization of the Gilbert damping of the\nmacrospin.18Next, we will obtain analytical results be-\nyond the description in Ref. 18 for the enhancement of\nthe Gilbert damping in the presence of surface anisotropy.\nC. Including Surface Anisotropy ( d6= 0)\nIn the presence of surface anisotropy, the out-of-plane\nand in-plane \feld con\fgurations must be treated sepa-\nrately. This distinction is because the boundary condi-\ntion Eq. (37) has di\u000berent forms for the two con\fgura-\ntions in this scenario.\n1. Out-of-plane Magnetization\nWhen the magnetization is out of plane, i.e., \u0012= 0, the\nspin-wave excitations are circular and have a high degree\nof symmetry. A simpli\fcation in this geometry is that\nthe coe\u000ecient C3= 0. In the absence of dissipation,\nthe boundary condition Eq. (37) determining the wave\nvectors becomes\nkLtan(kL) =d: (44)\nLet us consider the e\u000bects of the two di\u000berent\nanisotropies in this geometry.\n0510152025300.00.51.01.52.02.5\nLKsADaEA,nDa0\nn=0n=5FIG. 3. The ratio of enhanced Gilbert damping \u0001 \u000bEA,n=\u0001\u000b0\nin a system with easy-axis surface anisotropy versus the en-\nhanced Gilbert damping of macrospin modes in systems with\nno surface anisotropy as a function of surface-anisotropy en-\nergy.nrefers to the mode number, where n= 0 is the\nuniform-like mode. The dashed line represents the ratio\n\u0001\u000bn=\u0001\u000b0in the case of no surface anisotropy (see Eq. (42)).\na. Easy-Axis Surface Anisotropy ( d > 0):When\nd\u00181 or larger, the solutions of Eq. (44) are displaced\nfrom the zeroes of tan( kL), i.e., the solutions we found in\nthe case of no surface anisotropy, and towards the upper\npoles located at kuL= (2n+1)\u0019=2, wheren= 0;1;2;:::.\nWe therefore expand fin Eq. (37) (and thus also in\nEq. (44)) into a Laurent series around the poles from\nthe \frst negative order up to the \frst positive order in\nkLto solve the boundary condition for kL, giving\nkL\u0019\u0015ex\nL3(1 +d) + 2(kuL)2\u0000p\n12(kuL)2+ 9(1 +d)2\n2kuL:\n(45)\nUsing this result and the Laurent-series expansion for\nfin Eq. (39) and Eq. (40), we \fnd the Gilbert-damping\nrenormalization term ( \u000b!\u000b+ \u0001\u000b(oop)\nEA,n) and the ratio\nbetween the modes\n\u0001\u000b(oop)\nEA,n\n\u0001\u000b0\u00193\u0000\n3(1 +d) + 2(kuL)2\u0000p\n12(kuL)2+ 9(1 +d)2\u0001\n\u0002\n\u0002\u0000p\n4(kuL)2+ 3(1 +d)2\u0000p\n3(1 +d)\u0001\n2(kuL)2p\n4(kuL)2+ 3(1 +d)2:\n(46)\nThis ratio is plotted in Figure 3 for n\u00145. We see that\nthe ratio vanishes for large values of d. For small values\nof the anisotropy energy d, the approximate ratio exceeds\nthe exact result of the ratio we found in the limiting case\nof no surface anisotropy (see Eq. (42)). For moderate\nvalues ofd\u00185, the expansion around the upper poles\nis su\u000ecient, but only for the \frst few modes. This im-\nplies that moderate-strength easy-axis surface anisotropy\nquenches spin pumping for the lowest excited modes but\ndoes not a\u000bect modes with higher transverse exchange\nenergy.9\n05101520253001234\nLÈKsÈADaEP,nDa0\nn=1n=5\nFIG. 4. Plot of \u0001 \u000b(oop)\nEP,n=\u0001\u000b0. The dashed line represents\nthe ratio \u0001 \u000bn=\u0001\u000b0in the case of no surface anisotropy (see\nEq. (42)).\nb. Easy-Plane Surface Anisotropy ( d < 0):Easy-\nplane surface anisotropy is represented by a negative sur-\nface anisotropy din Eq. (44). In this case, the boundary\ncondition must be treated separately for the uniform-\nlike (n= 0) mode and the higher excitations. When\njdj>1, we can obtain a solution by expanding along the\nimaginary axis of kL. This corresponds to expressing the\nboundary condition in the form \u0000ikLtanh(ikL) =\u0000jdj,\nwith the asymptotic behavior kL\u0019 \u0000ijdj. Using the\nasymptotic form of the boundary condition in Eqs. (39)\nand calculating the renormalization of the Gilbert damp-\ning using Eq. (40), we \fnd that the renormalization is\n\u000b!\u000b+ \u0001\u000b(oop)\nEP,0, where\n\u0001\u000b(oop)\nEP,0\n\u0001\u000b0= 2jdj: (47)\nThus, the Gilbert damping of the lowest mode is much\nenhanced by increasing surface anisotropy. The surface-\nanisotropy mode is localized at the surface because it\ndecays from the spin-active interface and into the \flm.\nBecause the e\u000bective volume of the mode is reduced,\nspin pumping more strongly causes dissipation out of the\nmode and into the normal metal.\nFor the higher modes ( n > 0), the negative term on\nthe rhs of Eq. (44) forces the kLsolutions closer to thenegative, lower poles of tan( kL), located at k(l)\nnL= (2n\u0000\n1)\u0019=2, wheren= 1;2;3;:::. We repeat the procedure\nused for the EA case by expanding finto a Laurent series\naround these lower poles, arriving at\nkL\u00193(1\u0000jdj) + 2(k(l)\nnL)2+q\n12(k(l)\nnL)2+ 9(1\u0000jdj)2\n2k(l)\nnL:\n(48)\nUsing this relation and the new lower-pole Laurent ex-\npansion for f, Eqs. (39) and (40) give us the renormal-\nization of the Gilbert damping ( \u000b!\u000b+ \u0001\u000b(oop)\nEP,n) and\nthe ratio\n\u0001\u000b(oop)\nEP,n\n\u0001\u000b0\u00193\u0000\n3(1\u0000jdj) + 2(kuL)2+p\n12(kuL)2+ 9(1\u0000jdj)2\u0001\n\u0002\n\u0002\u0000p\n4(kuL)2+ 3(1\u0000jdj)2+p\n3(1\u0000jdj)\u0001\n2(kuL)2p\n4(kuL)2+ 3(1\u0000jdj)2:\n(49)\nThis ratio is plotted in Figure 4 from n= 1 up ton= 5.\nWe see that the ratio vanishes for large values of jdj.\nSimilar to the case of EA surface anisotropy, the approx-\nimation breaks down for large nand/or small values of\njdj.\nWhereas the n= 0 mode exhibits a strong spin-\npumping enhanced dissipation in this \feld con\fguration,\nthe DC ISHE \feld vanishes when \u0012= 0 (see Eq. (26)).\nThis is one of the reasons why this con\fguration is sel-\ndom used in experiments. However, this con\fguration\ncan lead to a signi\fcant AC ISHE, and a similar AC sig-\nnal was recently detected12. Because of the strong dissi-\npation enhancement, the EP surface anisotropy induced\nlocalized mode in perpendicular magnetization geometry\ncould be important in future experimental work.\n2. In-plane Magnetization\nWe will now complete the discussion of the spin-\npumping enhanced Gilbert damping by treating the case\nin which the magnetization is in plane ( \u0012=\u0019=2). For\nsuch systems, the coe\u000ecient C36= 0, and the 0th-order\nexpansion of Eq. (37) becomes\nkLtankL=\u0000d\u0000\n(\u0015exk)2+!H\n!M\u0001q\n1 + (\u0015exk)2+ 2!H\n!Mq\n1 + (\u0015exk)2+ 2!H\n!M\u0000\n1 + 2(\u0015exk)2+ 2!H\n!M\u0001\n\u0000d\u0015ex\nL\u0000\n1 + (\u0015exk)2+!H\n!M\u0001\ncoth\u0010\nL\n\u0015exq\n1 + (\u0015exk)2+ 2!H\n!M\u0011:\n(50)\nFor typical \flm thicknesses, of some hundred nanome-\nters, we have L=\u0015 ex\u001d1 and (\u0015exk)2\u001c1 for the lowest\neigenmodes. Thus, we take the asymptotic coth \u00181 and\nneglect the ( \u0015exk)2terms, ridding the rhs of Eq. (50) ofanykdependence. Eq (50) now becomes similar to the\nout-of-plane case\nkLtan(kL) =de\u000b; (51)10\nwhere\nde\u000b=\u0000d!H\n!Mq\n1 + 2!H\n!M\n\u0000\n1 + 2!H\n!M\u00013=2\u0000d\u0015ex\nL\u0000\n1 +!H\n!M\u0001: (52)\nde\u000bis positive if d<0 and negative for d>0 up to a crit-\nical valued\u0015ex=L=\u0015exKs=A=\u0000\n1 + 2!H\n!M\u00013=2=\u0000\n1 +!H\n!M\u0001\n,\nwhere the denominator becomes zero. For negative d,\njde\u000bj<jdj, whereas for positive d,jde\u000bjis initially smaller\nthan that ofjdjbut quickly approaches the critical value.\nWith the value Ksfrom Tab. I, we have jde\u000bj<jdj, in-\ndependent of the sign of d.\nWith this relation, we can calculate an approximate\nGilbert damping renormalization in both the EA and\nEP cases using the EP and EA relations, respectively,\nobtained in the out-of-plane con\fguration. Thus,\n\u0001\u000bip\nEA,0\u0019\u0001\u000boop\nEP,0jd!deff= 2jde\u000bj; (53)\n\u0001\u000bip\nEA,n\u0019\u0001\u000boop\nEP,njd!deff; (54)\n\u0001\u000bip\nEP,n\u0019\u0001\u000boop\nEA,njd!deff: (55)\nTo summarize this section regarding the enhancement\nof Gilbert damping, we see that the enhancement can be\nvery strong for the surface modes because their e\u000bective\nsizes are smaller than the thickness of the \flm. For all\nother modes, the enhancement decreases with increasing\nmagnitude of the surface-anisotropy energy.\nIV. NUMERICAL CALCULATIONS\nThe \frst step in the numerical method is to approxi-\nmate the equation of motion of Eq. (11) into by \fnite-size\nmatrix eigenvalue problem. We discretize the transverse\ncoordinate\u0018on the interval [\u0000L=2;L=2] intoNpoints la-\nbeled byj= 1;2;:::;N , and characterize the transverse\ndiscrete solutions of the dynamic magnetization vectors\nmQby (mx;j;my;j) of size 2N.\nWe approximate the 2nd-order derivative arising from\nthe exchange interaction using a nth-order central dif-\nference method. For the n\u00002 discretized points next to\nthe boundaries, we also use nth-order methods, using for-\nward (backward) di\u000berence schemes for the lower (upper)\n\flm boundary. This strategy avoids the introduction of\n\\ghost\" points outside the interval [ \u0000L=2;L=2] to satisfy\nthe boundary conditions.\nThus, the total operator acting on the magnetiza-\ntion on the left-hand side of Eq. (11) becomes a sparse\n2N\u00022Nmatrix operator. On the right-hand side of\nEq. (11), we also represent the convolution integral as\na 2N\u00022Ndense matrix operator, where each row is\nweighted according to the extended integration formu-\nlas for closed integrals to nth order43. The four N\u0002N\nsub-blocks of this integration operator correspond to the\nfour tensor elements of bGxy. In the \fnal discrete form,\nwe obtained a 2 N\u00022N !-dependent matrix.Next, the 4 boundary conditions (at the left and right\nboundaries for the two components, mxandmy) are used\nto reduce the number of equations to 2 N\u00004. This is per-\nformed by algebraically solving the discretized boundary\nconditions with respect to the boundary points, i.e., by\ndetermining miwherei2f1;N;N + 1;2Ngin terms of\nthe magnetizations at the interior points.\nFinally, each (2 N\u00004)\u0002(2N\u00004) matrix is separated\ninto two parts: a term independent of the frequency !\nand a term proportional to !. The dipole interaction\ncauses the eigenvalue problem to be non-Hermitian and\ntherefore computationally more demanding than a gen-\neralized eigenvalue problem. We \fnd the dispersion rela-\ntion and magnetization vectors by solving this eigenvalue\nproblem. The resulting eigenvectors are used to \fnd the\nmagnetization at the boundary by back-substitution into\nthe equations for the boundary conditions.\nWe are interested in \fnding the mode and wave-vector\ndependence of the spin-pumping enhanced Gilbert damp-\ning. To obtain this information numerically, we perform\ntwo independent calculations of the (complex) eigenval-\nues. First, we calculate the complex eigenvalues !dwhen\nthere is no spin pumping, but dissipation occurs via the\nconventional bulk Gilbert damping. Second, we calcu-\nlate the complex eigenvalues !spwhen spin pumping is\nactive at the FI-NM interface but there is no bulk Gilbert\ndamping. A mode- and wave-vector-dependent measure\nof the e\u000bective enhanced Gilbert damping enhancement\nis then given by\n\u0001\u000b=\u000bIm!sp\nIm!d: (56)\nTo ensure that we treat the same modes in the two in-\ndependent calculations, we check the convergence of the\nrelative di\u000berence in the real part of the eigenvalues. Ta-\nble I lists the values for the di\u000berent system parameters\nthat are used throughout this section.\nLet us \frst discuss the renormalization of the Gilbert\ndamping when there is no surface anisotropy. We will\npresent the numerical results for the three main geome-\ntries described in Sec. I and compare the results to the\nanalytical results of Sec. III A.\nA. FVMSW ( \u0012= 0)\nFigure 5 shows the wave-vector dependent renormal-\nization of the Gilbert damping \u0001 \u000bdue to spin pumping\nat the FI-NM interface in the FVMSW geometry. In\nthis geometry, waves travelling along \u0006^\u0010have the same\nsymmetry; thus, each line is doubly degenerate and cor-\nresponds to two waves of \u0006!. The \\spikes\" in the \fgure\nare due to degeneracies, i.e., mode crossings, and upon\ninspection, these spikes can be observed in the dispersion\nrelation.11\n0.010.1110100QL0.050.100.150.200.250.30DaH10-2L\n0.11100.60.70.80.91.0Re8wwM<-L2xL2m®\nHxL¤\nFIG. 5. \u0001\u000bversus wave vector for the FVMSW geometry\nof the four smallest eigenvalues. Top inset: Magnitudes of\neigenvectors (in arbitrary units) across the \flm at QL= 10.\nBottom inset: dispersion relation in the dipole-dipole active\nregime.\n1. Easy-Axis Surface Anisotropy ( ^\u0018easy axis)\n0.010.1110100QL0.20.40.60.81.01.2DaH10-3L\n0.11100.60.70.80.91.0Re8wwM<\n-L2xL2m®\nHxL¤\nFIG. 6. \u0001\u000bEAversus wave vector for the FVMSW geometry\nshowing the four smallest eigenvalues. The horizontal dashed\nlines indicate solutions of Eq. (46). Left inset: Magnitudes of\neigenvectors (in arbitrary units) across the \flm at QL= 5.\nRight inset: Dispersion relation in the dipole-dipole active\nregime.\nFigure 6 shows \u0001 \u000bEAfor the FVMSW geometry with\nan EA surface anisotropy at the spin-active interface. As\npredicted in Sec. III C 1 a, all modes exhibit a decreased\n\u0001\u000bcompared with those in Eqs. (43) and (42). For small\nQLand the chosen value of Ks(see Tab. I), the 1stfour\nmodes match the analytical result of Eq. (46), which is\nconsistent with the plot in Figure 3. For even higher ex-\ncited modes, the e\u000bect of the EA surface anisotropy be-\ncomes weaker due to the increase in transverse exchange\nenergy. These modes (not shown in the \fgure) approach\nthe value of \u0001 \u000bn.2. Easy-Plane Surface Anisotropy ( ^\u0018hard axis)\naLDaH10-3L45\n0.010.11100.00.10.20.3\nQL0.11100.50.60.70.80.91.0Re8wwM<bL\n-L2xL2m®\nHxL¤cL\nFIG. 7. a) \u0001 \u000bEPversus wave vector for the FVMSW geom-\netry, showing the four smallest eigenvalues. The dashed lines\nrepresent the analytic solutions from Sec. III C 1 b. b) Disper-\nsion relation in the dipole-dipole active regime. c) Magnitude\nof eigenvectors (in arbitrary units) across the \flm at QL= 5.\nFigure 7 shows \u0001 \u000bEPfor the FVMSW geometry with\nan EP surface anisotropy. We see that the mode corre-\nsponding to n= 0 has been promoted to a surface mode\nwith a large \u0001 \u000b, which for small values of QLmatches\nEq. (47). For the higher excited modes, we observe a\ndecrease in \u0001 \u000bcompared to the case with no surface\nanisotropy.\nB. BVMSW ( \u0012=\u0019=2and\u001e= 0)\n0.010.1110100QL0.20.40.60.81.0DaH10-3L\n0.11100.70.80.91.01.11.2Re8wwM<\n-L2xL2m®\nHxL¤\nFIG. 8. \u0001\u000bversus wave vector for the BVMSW geometry\n(\u0012=\u0019=2 and\u001e= 0) withKs= 0, plotted for the four small-\nest eigenvalues. Left inset: magnitudes of normalized eigen-\nvectors across the \flm at QL= 5. Right inset: dispersion\nrelation in the dipole-dipole active regime.\nFigure 8 shows the QL-dependent renormalization of\nthe Gilbert damping due to spin pumping at the FI-\nNM interface in the BVMSW geometry. We see that\nthe enhancement \u0001 \u000bagrees with the analytic limits in12\nEqs. (43) and (42) for small values of QL. For large val-\nues ofQL, we are in the strong exchange regime, in which\nthe in-plane exchange energy becomes large compared to\nall other energy contributions. This in-plane exchange\nsti\u000bness e\u000bectively quenches the coupling to the normal\nmetal layer, causing \u0001 \u000b!0 for large values of QL.\nAlthough Figure 8 only appears to show the three \frst\neigenvalues and eigenvectors, it actually contains dou-\nble this amount. Because ^zis parallel to the wave-\npropagation direction ^\u0010in this geometry, there is no\nchange in dipolar energies, regardless of whether the wave\ntravels in the + ^\u0010direction or in the \u0000^\u0010direction; thus,\nthe Gilbert damping is enhanced equally in both wave\ndirections. A slight o\u000bset from this con\fguration, taking\neither\u0012 < \u0019= 2 or\u001e6= 0, would result in a splitting of\neach line in Figure 8 into two distinct lines.\nIncluding Surface Anisotropy\nFigure 9 shows both the EA and the EP surface-\nanisotropy calculations in the BVMSW geometry. In the\ncase of an EA surface anisotropy, the mode correspond-\ning ton= 0 gets promoted to a surface mode, similarly\nto the case in which there is EP surface anisotropy in the\nFVMSW geometry. The increase in \u0001 \u000bis much smaller\nfor the same magnitude of Ks, as explained in detail in\nSec. III C. The higher modes, corresponding to n > 0,\nexhibit increased quenching of the Gilbert damping en-\nhancement. In the case of EP surface anisotropy, all\nmodes exhibit quenched Gilbert damping enhancement.\nC. MSSW ( \u0012=\u001e=\u0019=2)\nFigure 10 shows the QL-dependent renormalization of\nthe Gilbert damping due to spin pumping at the FI-NM\ninterface in the MSSW geometry. The computed eigen-\nvalues agree with Eqs. (43) and (42) for small values of\nQL. We see in the inset of Figure 10 that in this geom-\netry, the macrospin-like mode behaves as predicted by\nDamon and Eshbach3433, cutting through the dispersion\nrelations of the higher excited modes for increasing val-\nues ofQLin the dipole-dipole regime. A prominent fea-\nture of this geometry is the manner in which the modes\nwith di\u000berent signs of Ref!gbehave di\u000berently due to\nthe dipole-dipole interaction. This is because the inter-\nnal \feld direction ( ^z) is not parallel to the direction of\ntravel ( ^\u0010) of the spin wave. Hence, changing the sign of\n!is equivalent to inverting the externally applied \feld,\nchanging the xyzcoordinate system in Figure 1 from a\nright-handed coordinate system to a left-handed system.\nIn the middle of the dipole regime, the lack of symme-\ntry with respect to propagation direction has di\u000berent\ne\u000bects on the eigenvectors; e.g., in the dipole-dipole ac-\ntive region the modes with positive or negative Ref!g\nexperience an increased or decreased magnitude of the\ndynamic magnetization, depending on the value of QL,as shown in Figure 10e & f. This magnitude di\u000berence\ncreates di\u000berent renormalizations of the Gilbert damp-\ning, as the plot of \u0001 \u000b(\u0006)in Figure 10b & c shows.\nIncluding Surface Anisotropy\nFigure 11 shows \u0001 \u000bcomputed for modes in the MSSW\ngeometry with EA and EP surface anisotropies. We can\nclearly see that for small QLan exponentially localized\nmode exists in the EA case, and as predicted in Sec. III C,\nall the lowest-energy modes have spin pumping quenched\nby EP surface anisotropy. This is similar to the corre-\nsponding case in the BVMSW geometry.\nD. AC and DC ISHE\nFigure 12 shows the DC and AC ISHE measures for\nthe BVMSW geometry corresponding to the data repre-\nsented in Figure 8. In this geometry, the angular term,\nsin\u0012, in Eq. (28) is to equal one, ensuring that the DC\nmeasure is nonzero. This is not the case for all geometries\nbecause the DC electric \feld vanishes in the FVMSW ge-\nometry. The mode-dependent DC ISHE measure exhibits\nthe sameQL-dependence as the spectrum of the Gilbert\ndamping enhancement in all geometries where sin \u00126= 0.\nWe have already presented the renormalization of the\nGilbert damping in the most general cases above. There-\nfore, we restrict ourselves to presenting the simple case of\nthe BVMSW geometry with no surface anisotropy here.\nThe AC ISHE measure plotted in Figure 12 exhibits\na similarQLdependence to the Gilbert damping renor-\nmalization (and hence the DC ISHE measure), but with a\nslight variation in the spectrum towards higher values of\nQL. Note that because Eq. (24) is non-zero for all values\nof\u0012, the AC e\u000bect should be detectable in the FVMSW\ngeometry. By comparing the computed renormalization\nof the Gilbert damping for the di\u000berent geometries in\nthe previous subsections, we see that the strong renor-\nmalization of the n= 0 induced surface mode that oc-\ncurs in the FVMSW geometry with easy-plane surface\nanisotropy (see Sec. IV A 2 and Fig. 7) can have a pro-\nportionally strong AC ISHE signal in the normal metal.\nV. CONCLUSION\nIn conclusion, we have presented analytical and numer-\nical results for the spin-pumping-induced Gilbert damp-\ning and direct- and alternating terms of the inverse spin-\nHall e\u000bect. In addition to the measures of the magnitudes\nof the DC and AC ISHE, the e\u000bective Gilbert damp-\ning constants strongly depend on the modes through the\nwave numbers of the excited eigenvectors.\nIn the long-wavelength limit with no substantial sur-\nface anisotropy, the spectrum is comprised of standing-\nwave volume modes and a uniform-like (macrospin)13\n0.010.1110100QL0.51.01.52.0aLDaH10-3L\n0.11100.70.80.91.01.11.2Re8wwM<\n-L2xL2m®\nHxL¤\n0.0010.010.1110100QL0.20.40.60.81.0bLDaH10-3L\n0.11100.70.80.91.01.11.2Re8wwM<\n-L2xL2m®\nHxL¤\nFIG. 9. a) Dispersion relation versus wave vector for the BVMSW geometry ( \u0012=\u0019=2,\u001e= 0) for the four lowest eigenvalues in\nthe case of EA surface anisotropy. b) Dispersion relation in the case of EP surface anisotropy. In both \fgures, the horizontal\ndashed lines mark the value of \u0001 \u000bnin the case of no surface anisotropy.\n0.010.11101000.00.20.40.60.8Da+H10-3LaL\n0.010.11101000.00.20.40.60.81.0\nQLDa-H10-3LbL\n0.010.11101000.91.1.1ÈReHwwMLÈcL\nQL\n-L2xL20.0.51.1.5m®\nHxL¤dL\n-L2xL20.0.51.m®\nHxL¤eL\nFIG. 10. Gilbert damping renormalization in the MSSW geometry. Subplots a) and b) show Gilbert damping renormalization\n\u0001\u000bfor modes with positive (negative) Ref!g. The horizontal dashed lines represent the analytical values \u0001 \u000b0and \u0001\u000bnfor\nsmallQL. c) Dispersion relation versus wave vector for the MSSW geometry ( \u0012=\u001e=\u0019=2) for the four smallest eigenvalues,\ncolored pairwise in \u0006!. Subplot d (e) shows the magnitude of normalized eigenvectors (in arbitrary units) at QL= 3 across\nthe \flm modes with positive (negative) Ref!g.\n0.010.11101000.0.51.1.5Da+H10-3LaL\n0.010.11101000.0.51.1.5\nQLDa-H10-3LbL\n0.010.11101000.00.20.40.60.8Da+H10-3LcL\n0.010.11101000.00.20.40.60.8\nQLDa-H10-3LdL\nFIG. 11. a) and b) Gilbert damping renormalization from spin pumping in the MSSW geometry ( \u0012=\u001e=\u0019=2) for modes with\npositive (negative) Ref!gin the case of EA surface anisotropy. The four smallest eigenvalues are colored pairwise in \u0006!across\nthe plots. c) and d) show the Gilbert damping renormalization in the case of EP surface anisotropy.\nmode. These results are consistent with our previous\n\fndings18: in the long-wavelength limit, the ratio be-\ntween the enhanced Gilbert damping for the higher vol-ume modes and that of the macrospin mode is equal\nto two. When there is signi\fcant surface anisotropy,\nthe uniform mode can be altered to become a pure lo-14\n0.010.11101000.00.10.20.30.40.50.6\nQLeACH10-4LaL\n0.010.11101000.00.51.01.5\nQLeDCH10-9cmLbL\nFIG. 12. ISHE as a function of in-plane wave vector in the\nBVMSW geometry with Ks= 0. a) AC ISHE measure of\nEq. (28); b) DC ISHE measure of Eq. (28).\ncalized surface mode (in the out-of-plane geometry and\nwith EP surface anisotropy), a blend between a uniform\nmode and a localized mode (in-plane geometries and EA\nsurface anisotropy), or quenched uniform modes (out-of-\nplane \feld con\fguration and EA surface anisotropy, or\nin-plane \feld con\fguration and EP surface anisotropy).\nThe e\u000bective Gilbert damping is strongly enhanced for\nthe surface modes but decreases with increasing surface-\nanisotropy energies for all the other modes.\nThe presented measures for both the AC and DC in-\nverse spin-Hall e\u000bects are strongly correlated with the\nspin-pumping renormalization of the Gilbert damping,\nwith the DC e\u000bect exhibiting the same QLdependency,\nwhereas the AC e\u000bect exhibits a slighthly di\u000berent vari-\nation for higher values of QL. Because the AC e\u000bect\nis nonzero in both in-plane and out-of-plane geometries\nand because both EP and EA surface anisotropies in-\nduce surface-localized waves at the spin-active interface,\nthe AC ISHE can be potentially large for these modes.\nACKNOWLEDGMENTS\nWe acknowledge support from EU-FET grant no.\n612759 (\\InSpin\"), ERC AdG grant no. 669442 (\\In-sulatronics\"), and the Research Council of Norway grant\nno. 239926.\nAppendix A: Coordinate transforms\nThe transformation for vectors from \u0018\u0011\u0010toxyzcoor-\ndinates (see Fig. 1) is given by an a\u000ene transformation\nmatrixT, so that\nf(xyz)=T\u0001f(\u0018\u0011\u0010);\nfor some arbitrary vector f. Tensor{vector products are\ntransformed by inserting a unity tensor I=T\u00001Tbe-\ntween the tensor and vector and by left multiplication by\nthe tensor T, such that the tensor transforms as TbGT\u00001\nfor some tensor bGwritten in the \u0018\u0011\u0010basis.\nTis given by the concatenated rotation matrices T=\nR2\u0001R1, whereR1is a rotation \u001earound the \u0018-axis,\nandR2is a rotation \u0012\u0000\u0019\n2around the new \u0011-axis/y-axis.\nHence,\nR1=0\nB@1 0 0\n0 cos\u001e\u0000sin\u001e\n0 sin\u001ecos\u001e1\nCA; (A1)\nR2=0\nB@sin\u00120\u0000cos\u0012\n0 1 0\ncos\u00120 sin\u00121\nCA; (A2)\nsuch that\nT=0\nB@sin\u0012\u0000cos\u0012sin\u001e\u0000cos\u0012cos\u001e\n0 cos\u001e\u0000sin\u001e\ncos\u0012sin\u0012sin\u001e sin\u0012cos\u001e1\nCA: (A3)\nThis transformation matrix consists of orthogonal trans-\nformations; thus, the inverse transformation, which\ntransforms xyz!\u0018\u0011\u0010, is just the transpose, T\u00001=TT.\n1A. Serga, A. Chumak, and B. 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(Cambridge University Press,\n2007)."    },    {        "title": "1805.01776v2.Superparamagnetic_Relaxation_Driven_by_Colored_Noise.pdf",        "content": "arXiv:1805.01776v2  [cond-mat.stat-mech]  7 May 2018Superparamagnetic Relaxation Driven by Colored Noise\nJ. G. McHugh,1R. W. Chantrell,1I. Klik,2and C. R. Chang2\n1Department of Physics, The University of York, York, YO10 5D D, UK\n2Department of Physics, National Taiwan University, Taipei , Taiwan\nA theoretical investigation of magnetic relaxation proces ses in single domain particles driven\nby colored noise is presented. Two approaches are considere d; the Landau-Lifshitz-Miyazaki-Seki\nequation, which is a Langevin dynamics model based on the int roduction of an Ornstein-Uhlenbeck\ncorrelated noise into the Landau-Lifshitz-Gilbert equati on and a Generalized Master Equation ap-\nproach whereby the ordinary Master Equation is modified thro ugh the introduction of an explicit\nmemory kernel. It is found that colored noise is likely to bec ome important for high anisotropy\nmaterials where the characteristic system time, in this cas e the inverse Larmor precession frequency,\nbecomes comparable to the correlation time. When the escape time is much longer than the corre-\nlation time, the relaxation profile of the spin has a similar e xponential form to the ordinary LLG\nequation, while for low barrier heights and intermediate da mping, for which the correlation time is\na sizable fraction of the escape time, an unusual bi-exponen tial decay is predicted as a characteristic\nof colored noise. At very high damping and correlation times , the time profile of the spins exhibits\na more complicated, noisy trajectory.\nI. INTRODUCTION\nThermally-activated magnetization reversal over an\nanisotropic energy barrier is the driving force for switch-\ning in magnetic materials. Theoretical understanding\nwasfirst developed by N´ eel1basedon the transition state\ntheory (TST) leading to an Arrhenius-like relaxation\ntime proportional to exp( EB/kBT) whereEBis the en-\nergy barrier, kBthe Boltzmann constant and Tthe tem-\nperature. Brown2provided further insight through the\nconstruction ofthe Langevinequation for the problem by\nthe introduction of white-noise fields into the Landau-\nLifshitz equation with Gilbert damping, leading to the\nstochastic Landau-Lifshitz-Gilbert (LLG) equation, An\nexpression for the relaxation time of thermally-driven\nescape over the energy barrier is then found through\nthe lowest eigenvalue of the correspondingFokker-Planck\nequation(FPE)governingthetime-evolutionoftheprob-\nability density function of the magnetization orientation.\nThe routeto the Arrhenius-likerelaxationtime expres-\nsion is one of two directions leading from the Langevin\nequation. The second, Langevin Dynamics (LD) ap-\nproach is the direct numerical solution of the Langevin\nequation3–6. There is a natural separation of timescales,\nwith LD used for high frequency applications such as\nmagnetic recording and the Arrhenius-like relaxation\ntimeusedforslowdynamicbehaviorarisingfromthermal\nactivationoverenergybarriers. The twoapproacheshave\nbeen compared by Kalmykov et. al.,7who calculated es-\ncape times for both cases giving excellent agreement for\nthe variation of escape time with damping constant and\ndemonstrating the importance of starting the LD calcu-\nlations from the correct thermal equilibrium distribution\nwithin the energy minimum.The LLG equation for a single spin takes the well-\nknown form\ndS\ndt=−γ\n1+α2/parenleftbig\nS×H+αS×(S×H)/parenrightbig\n,(1)\nwhereαis the phenomenological damping constant, γ=\n1.7611T−1s−1andSisaunitvectorinthedirectionofthe\nspin,S=µ/µs. The local magnetic field, H, is derived\nfrom the first derivative of the spin Hamiltonian Hwith\nrespect to the spin degree of freedom,\nH=−1\nµs∂H\n∂S. (2)\nThermal fluctuations are necessary to incorporate the\ndeviations of a particular spin from the average tra-\njectory. This is done via the formal inclusion of ran-\ndom fields in the LLG equation. In order to realize the\nFluctuation-Dissipation theorem for this system, these\nthermal fields must also be proportionalto the same phe-\nnomenological damping constant, αthat occurs in the\ndamping. The moments of the thermal field are then\ngiven by\n∝angbracketleftHth,i(t)∝angbracketright= 0 (3)\n∝angbracketleftHth,i(t)Hth,j(t′)∝angbracketright=2αkBT\nγµsδ(t−t′)δij(4)\nwherei,jlabel the spin components.\nIn all numerical simulations, we interpret the stochas-\ntic equation in the Stratonovich sense and employ the\nHeun method An implicit assumption of this approach is\nthe presence of white noise, which exists in the zero cor-\nrelation time limit for some physical noise process with a2\nwell-defined correlation time. Such a colored noise may\nbe implemented for a magnetic system through the use\nof the Landau-Lifshitz-Miyazaki-Seki pair of Langevin\nequations, which take the form\ndS\ndt=γS×/parenleftbig\nH+η/parenrightbig\n, (5)\ndη\ndt=−1\nτc(η−χS)+R, (6)\nwhereτcis the correlation time and χis a spin-bath cou-\npling which is related to the phenomenological damping\nparameter as α=γχτcin the limit of small correlation\ntimes. The autocorrelation of the white noise field, R, is\ngiven by\n∝angbracketleftRi(t)Rj(t′)∝angbracketright=2χkBT\nτcµsδijδ(t−t′).(7)\nThis pair of Langevin equations leads to a frequency-\ndependent damping of the spin together with an expo-\nnentially correlated noise term in the spin-only space,\n∝angbracketleftˆηi(t)ˆηj(t′)∝angbracketright=χkBT\nµse−(t−t′)\nτcδij=χkBT\nµsK(t−t′)δij\n(8)\nwhereK(t) = exp−(t−t′)\nτcis the exponential memory ker-\nnel. For completeness, additional background on the\nLLMS Langevin equation and colored noise is included\nin Appendix A.\nAn alternative approach to the Langevin equation is\nthe discrete orientation approximation, whereby, in the\nlimit oflarge barriers, the detailed dynamics are replaced\nby phenomenological rate equations describing transi-\ntions between the minima of the magnetic potential. We\nmay augment this description by the introduction of a\nmemory kernel into the rates, thus replacing the master\nequation description with a generalized master equation\nwhich explicitly incorporates the retardation effect into\nthe rate equations.\nHere we investigate the introduction of colored noise\ninto the calculation of escape rates. This leads to sig-\nnificant effects for materials with large magnetocrys-\ntalline anisotropy energies, including the prediction of\nbi-exponential behavior at intermediate damping, when\nthe characteristic time of the relaxation process becomes\ncomparable to the heat bath correlationtime. The paper\nis organized as follows. We first outline thermally acti-\nvated escape times for single nanoparticles, followed by\nanintroductionofcolorednoiseintotheLangevinformal-\nism via the LLMS equations. We then derive the relax-\nation profile from the non-Markovian generalized exten-\nsion of the rate equation, followed by a systematic inves-\ntigationoftheeffects ofthebarrierheightandcorrelation\ntimes on the relaxation profile from LLMS simulations.A. Thermally-Assisted Magnetization Reversal\nWe will investigate here the effect that colored noise\nhas on the dynamics of the thermal escape problem for a\nmagnetic nanoparticle. The spin Hamiltonian of the sys-\ntem contains both an applied field and anisotropy term,\ntaking the form\nH=−KVS2\nz−µs/vectorH·S, (9)\nwhereKis the anisotropy constant and Vis the particle\nvolume. For the escape problem we have a spin energy\npotential of the form\nV(θ,φ) =σβ−1/parenleftbig\nsin2θ−2h(cosψcosθ(10)\n+sinψsinθcosφ)/parenrightbig\n,\nwhereθ,φare respectively the polar and azimuthal\ncomponents of the spin in spherical coordinates, σ=\nKV/k BTis the reduced barrier height parameter, h=\nH/2σis the reduced field, β= (kBT)−1andψis\nthe angle between the easy-axis and the applied field.\nThis potential has a bistable character under the condi-\ntion than the critical applied field value, h < h c(ψ) =\n((cos2/3ψ+sin2/3ψ)−3/213, in which case there are local\nand global minima in the north and south polar regions,\nwith an equatorial saddle point between them. We are\nthen interested in the calculation of the characteristic es-\ncape time of a spin initialized in one such minimum.\nFor the special case of aligned field and easy axis, for\nwhichψ= 0 the potential is\nV(θ) =σβ−1/parenleftbig\nsin2θ−2hcosθ/parenrightbig\n. (11)\nIn this case the escape time takes the Arrhenius form,\nwhere the barrier energy, EB, is proportional to the\nanisotropy energy, leading to an escape time\nτ∝f−1\n0eKV/k BT(12)\nwheref0is the attempt frequency, the frequency of Lar-\nmor gyromagnetic precession at the bottom of the well.\nWeinvestigatetheescapetimeinthecoloredandwhite\nnoise cases through repeated numerical integration of the\nLangevin equations for a spin initialized in a potential\nminimum. An important consideration for such simula-\ntions is the choice of initial and switching condition for\nthe spin. We will initialize the spins with the Boltz-\nmann distribution at the bottom of the well in order to\navoidinconsistenciesat lowdamping, while the switching\ncondition is chosen such that Sz<−0.5, with the spin\ninitialized in the positive z-direction, so that the spin is\nsufficiently deep in the well such that it has escaped.\nII. LLMS ESCAPE TIMES & COLORED NOISE\nA. System time τsvs.τccharacteristic bath time.\nFor the uniaxial escape problem the external field in\nthe LLMS will consist of an external applied part and an3\nanisotropy contribution\nH=Ha+H0 (13)\nthe magnitude of the anisotropic contribution depends\non the orientation of the spin and is given by Ha=\n2ku\nµs/vectorSz·/vectorz=Hk/vectorSz·/vectorzwhere/vectorzis the direction of easy\nmagnetization and kuis the anisotropy energy. To gain\nintuition into the relevant timescales for the relaxation\nproblem, we will assume the uniaxial case in the follow-\ning, where the external field is applied along the same\ndirection as the easy axis, such that both fields only have\ncomponents in the z-direction.\nWe note that the anisotropic field contribution varies\nwith the projection of the spin on to the easy-axis as\nHa=Hk(S·z)/vectorz= (Hkcosθ)/vectorz, (14)\nThelargest field magnitude and consequently the fastest\ntimescale of the problem is set by the value for which\nthe anisotropic field contribution is at its largest, which\nis when the spin and the easy-axis precisely coalign. For\nany other orientation, the field will be smaller and the\ntimescale of oscillation hence slower. We may then take\nthe spin-only Langevin equation,\ndS\ndt=γS(t)×/parenleftbig\n(Hkcos(θ))/vectorz+¯η−χ/integraldisplayt\n−∞dt′K(t−t′)dS(t′)\ndt′/parenrightbig\n,\n(15)\nand proceed to scale this equation by the maximum\nanisotropy field value. Defining the system time for the\nspin asτs= (γHk)−1then\ndS\ndt=1\nτsS(t)×/parenleftbig\ncos(θ)/vectorz+H−1\nk¯η\n−H−1\nkχ/integraldisplayt\n−∞dt′K(t−t′)dS(t′)\ndt′/parenrightbig\n.(16)\nWe may scale the time variable in the Langevin equa-\ntionsothatthesystemtimeisremovedbytaking ζ=τst.\nThen we have\ndS\ndζ=S(ζ)×cos(θ)/vectorz+S(ζ)×/parenleftbigg\nH−1\nk¯η(ζ)\n+H−1\nkχ/integraldisplayζ′\n−∞dζ′e−(ζ−ζ′)τs\nτcdS(ζ′)\ndζ′/parenrightbigg\n(17)\nThe autocorrelation of the noise is similarly transformed\nto become\n∝angbracketleft¯η(ζ)¯η(ζ′)∝angbracketright=τs\nτc¯De−ζ−ζ′)τs\nτc=¯D\nτe−(ζ−ζ′)/τ(18)\nwhereτs/τc=τand¯D=D/τs=χτkBT/µs. We\ncan then write the coupling as ¯ χ=χ/Hk, and ab-\nsorb theHkfactor into the diffusion constant for the\nthermal field. Since the thermal fields are given by\n¯η(ζ) =√\n2D\nτ/integraltextζ\n−∞K(ζ−ζ′)Γ(ζ′), the diffusion constant\nbecomes\n¯D=χτkBT\nµsH2\nk=¯χτkBT\n2ku=¯χτ\n2σ(19)11.21.41.61.82\n0.01 0.1 1τ/τLLG\nγ Hk τcσ = 2\nσ = 5\nσ = 7.5\nFIG. 1. Escape time, normalized to the uncorrelated LLG\nescape time vs correlation time, from LLMS simulations for a\nCo nanoparticle with α= 0.05 and different reduced barrier\nheights, σ,\nwhereσ=ku/kBT. The final expression for the\nLangevin equation is then\ndS(ζ)\ndt=S(ζ)×/parenleftbig\ncos(θ)/vectorz+¯η−¯χ/integraldisplayζ\n−∞dζ′K(ζ−ζ′)dS(ζ′)\ndζ′/parenrightbig\n.\n(20)\nIn the case that τ≪1 andτc≪τs, the memory\nkernels appearing in the noise and damping terms are\nreduced to delta functions and the white noise behav-\nior is restored. Additionally the bath coupling and the\nstrength of the thermal fluctuations are reduced by the\nanisotropy field, so that in the event of a very large\nanisotropy the precessional dynamics of the spin dom-\ninate the thermal and damping parts. We then conclude\nthat the condition τc/greaterorsimilar(γHk)−1dictates whether the\neffect of correlations are relevant in the system dynamics\nin the high barrier limit.\nThis prediction is borne out in numerical simulations\nof the LLMS equation. Figure 1 depicts the escape time\ncalculated using the LLMS model for a Co nanoparti-\ncle of volume V= 8×10−27m3, with anisotropy en-\nergyKV= 1.12×1021J, and a magnetic moment µs=\n1.12×10−20J/T.wherethe correlationtime is normalized\nby the inverse of the Larmor precession frequency, and\nthe escape time in the LLMS is normalized by the escape\ntime calculated from the Markovian LLG equation. The\nescape rate departs from the LLG escape rate only once\nthe correlation time is some significant fraction of the\nLarmor time, and for increasing barrier height the corre-\nlation time must be a larger fraction of the gyromagnetic\nprecession before the escape rate departs from the LLG\nprediction.\nFigure 2 shows a comparison of the escape time for the\nCo nanoparticle and a SmCo 5nanoparticle of the same4\n110\n1.0E-15 1.0E-14 1.0E-13 1.0E-12 1.0E-11 1.0E-10τ/τLLG\nτcSmCo\nCo\nFIG. 2. Comparison of simulation results for systems with pa -\nrameters chosen tobe similar toSmCo 5and Co nanoparticles,\nrespectively, for large reduced barriers σ= 13.5, and a fixed\nα= 0.05. The higher anisotropy SmCo 5exhibits departure\nfrom LLG behavior at smaller correlation times.\nvolume. The SmCo 5material parameters are taken to be\nµs= 6.4×10−18J/T, and anisotropy KV= 2.16×10−16,\na much higher anisotropy energy density than Co. This\nhigher anisotropy gives the nanoparticle a faster system\ntime, which causes the LLMS to depart from the LLG\nfor smaller bath correlation times, τc, on the order of\n50−100fsfor the SmCo 5particle, while it is approxi-\nmately 1psfor the Co nanoparticle. The fact that the\nsystem time is inversely proportional to the magnitude\nof the anisotropy field is exhibited in the simulations by\nthe difference between LLMS and LLG escape rates at\nsmaller values of the bath correlation time for the mate-\nrial with higher magnetic anisotropy.\nB. Arrhenius Behavior\nCrucially, it is found that the Arrhenius behavior of\nthe escape rate is recovered from LLMS simulations in\nthe limit of large barrier height. In figure 3 we show the\ntemperature- dependence of the escape time vs reduced\nbarrier height.\nIn the high damping case, we see that the escape rates\nbegin to convergeas the temperature tends towardszero.\nAs the escape time between the wells becomes much\nlonger than the bath correlation time, the detailed dy-\nnamics of the spin within the well becomes less relevant.\nAt low damping, the LLMS and LLG appear not to\nconverge even at the larger barrier heights considered\nhere. We attribute this difference to the difference in\ndamping regimes and the physically distinct mechanisms\ninvolved in the escape process between the two regimes.\nEscape at high damping is mediated by thermal fluctu-0.010.1110100100010000100000\n0246810121416τkr γ Hk\nσLLMS, h=0.2, α = 0.01\nLLG, h=0.2, α = 0.01 Ψ = 1/4\n0.010.1110100100010000\n0246810121416τkr γ Hk\nσLLMS, h=0.3, α = 1, Ψ = 1/4\nLLG, h=0.3, α = 0.01, Ψ = 1/4\n0.010.1110100100010000\n0246810121416τkr γ Hk\nσLLMS, h=0.2, α = 1\nLLG, h=0.2, α = 1 Ψ = 1/4\n0.010.11101001000\n0246810121416τkr γ Hk\nσLLMS, h=0.3, α = 1, Ψ = 1/4\nLLG, h=0.3, α = 1, Ψ = 1/4\nFIG. 3. Escape time, τγHkvs reduced barrier height, σ,\nfrom LLMS and LLG simulations, for different values of the\napplied field h=µsH/σand damping, α, with a fixed angle\nof Ψ = π/4 between the applied field and the easy axis of\nmagnetization. 1:Low damping, h= 0.2,2:Low damping,\nh= 0.3.3:High damping, h= 0.2,4:High damping,\nh= 0.3.\nations, which liberate the bound spin. In the limit of\nvanishing temperature the infrequency of thermal oscil-\nlations of sufficient energy dominate the escape behavior\nand the escape rates converge.\nIn contrast, the energy-controlled diffusion regime is\ncharacterized by the almost-free precessional motion of\nthe spin in the well. In the highly correlated case, the5\nsimple damping is replaced with a frequency-dependent\ndamping, an effect which increases the overall effective\ndamping. In the limit T→0, this inhibits the escape\nratebetweenthewellsbydecreasingtherateatwhichthe\nspin is able to attain a trajectory with sufficient escape\nenergy.\nIII. RATE EQUATIONS FOR\nTHERMALLY-ACTIVATED MAGNETIZATION\nREVERSAL\nA. Master Equation\nThe master equation is a phenomenological set of first-\norder differential rate equations for a multi-level system,\nwhich takes the form\ndni\ndt= Γij(t)nj(t), (21)\nwhereniis a probability vector representing the proba-\nbility that the system is in one of a discrete set of states,\nandi,jlabel those discrete states, while the matrix of\ncoefficients Γ i,jdictates the transition rate from state i\nto the state jof the system.\nThe dynamics of the thermally-assisted escape prob-\nlem in a magnetic system may be approximated by such\na master equation under the condition that the energy\nbarrier is large compared to the thermal energy, σ >1,\nbut not too large such that it would inhibit inter-well\ntransitions. This approximation to the Langevin dynam-\nics is called the discrete orientation approximation . The\nspin orientations are assumed to be restricted only to the\n2 minima of the potential energy dictated by the spin\nHamiltonian. The time evolution of the occupation of\neach state follows from Eq. 21, where i,j= 1,2. The\ntransition matrix elements follow from the applied field,\nanisotropy and temperature. In particular, we will as-\nsume a fixed applied field, such that the transition rates\nare constant in time and the matrix takes the form\nΓij=/parenleftbigg\n−κ12κ21\nκ12−κ21/parenrightbigg\n, (22)\nIn the uniaxial case these rates are given by κ1→2=\nκ12=f0exp(−σ(1+h)2) andκ2→1=κ21=\nf0exp(−σ(1−h)2), whereσandhare the reduced bar-\nrier height and applied field, respectively. The time evo-\nlution of the population of the state n1is then explicitly\ngiven by\ndn1\ndt=−κ12n1+κ21n2= (κ12+κ21)n1+κ21.(23)\nThe time-evolution of the magnetization follows from\nthe individual rates for the two wells, where the magne-\ntization is given by m(t) =n1(t)−n2(t) and is subject\nto the normalization condition n1(t) +n2(t) = 1. The\ndifferential equation for the magnetization is then\ndm\ndt=−Γ1m(t)−Γ2, (24)where Γ 1=κ12+κ21and Γ 2=κ12−κ21. This is the\nsame form as the rate for the individual wells, Eq. 23.\nFor an initial magnetization m0=n1(t= 0)−n2(t=\n0), the magnetization as a function of time is a simple\nexponential,\nm(t) =e−Γ1t(Γ1m0+Γ2)\nΓ1−Γ2\nΓ1,(25)\nwhich tends to the value\n−Γ1\nΓ2=κ21−κ12\nκ12+κ21. (26)\nIn the long-time limit, the steady state magnetization\ncorresponding to the difference in the transition rates\nbetween the wells, if κ2→1> κ1→2, the transition rate\ninto well 1 is greater than the rate out, and we have a\npositive magnetization, as expected.\nB. Generalized Master Equation\nThe non-Markovian extension of the master equation\nformalismiswhat iscalledageneralizedmasterequation.\nUnder this model, the set of i×jrates represented in the\ntransition matrix in Eq. 21 are promoted to a set of i×j\nmemory kernels for the transitions between the wells i,j,\nreplacing the set of first-order differential equations with\na set of integro-differential equations for the population\nof each well,\ndni\ndt=/integraldisplay∞\n0Mij(t−τ)n(τ)dτ. (27)\nWe will consider the simplified case\nMij(t) =e−t/Θ\nΘAij=K(t)Γij, (28)\nwhere Γ ijare the same constant transition rates consid-\nered in the Markovian master equations, now modified\nby a simple exponential kernel over the recent popula-\ntion of the well. The integro-differential expression for\nthe magnetization then becomes\ndm\ndt=−Γ1/integraldisplay∞\n0K(t−τ)m(τ)dτ−Γ2/integraldisplay∞\n0K(t−τ)dτ.\n(29)\nWhere we note that for the exponential kernel, K(t) =\ne−t/Θ\nΘ, the uncorrelated form of the master equation\nis recovered in the limit of vanishing correlation time,\nlimΘ→0K(t) =δ(t).\nThe Laplace transform of this equation is\nωm(ω)−m0=−Γ1K(ω)m(ω)−Γ2\nωK(ω),(30)\nwhereK(ω) =L(K(t)) is the Laplace transform of the\nmemory kernel,\nK(ω) =Θ−1\nω+Θ−1=1\n1+Θω, (31)6\nwe then have\nm(ω) =−Γ2\nωK(ω)+m0\nω+Γ1K(ω). (32)\nAfter inserting the expression for the Laplace transform\nof the kernel we find\nm(ω) =−Γ2\nω+m0(1+Θω)\nΘω2+ω+Γ1. (33)\nFinally we solve for the time-dependence of the mag-\nnetization by taking the inverse Laplace transform,\nm(t) =L−1[(1+Θω)\nΘω2+ω+Γ1] =φ(t)(Γ1m0+Γ2)\nΓ1−Γ2\nΓ1,\n(34)\nwe note that this bears a strong resemblance to the\nMarkovian expression, Eq. 25, with the exponential be-\ning replaced by the function φ(t), which is\nφ(t) =1\n2β/parenleftBig\n(β−1)e−t(1+β)/2Θ+(β+1)e−t(1−β)/2Θ/parenrightBig\n,\n(35)\nwhereβ=√1−4Γ1Θ. In the limit t→ ∞, the value of\nthe magnetization again tends to−Γ2\nΓ1. To see that this\nagrees with the uncorrelated solution for small correla-\ntion times, we may expand βin Θ for small Θ, hence\nβ= 1−2Γ1Θ, inserting into the magnetization it be-\ncomes\nm(t) =β−1\n2βe−t/2ΘeΓ1t+(β+1)\n2βe−Γ1t.(36)\nAs Θ→0,β→1, and only the second term in the\nexpression for the magnetization remains, m(t) =e−Γ1t,\nso the small correlation time limit of the spin evolution\nagrees with the non Markovian master equation.\n00.10.20.30.40.50.60.70.80.91\n00.511.522.53m(t)\nΓ1tR < 0.01\nR=0.1\nR = 0.24\nFIG. 4. m(t) vst, forR= 0,0.1,0.2, under the initial condi-\ntionm= 1, with transition rates κ12= 1,κ21= 0Finally, we note that the solution for the magnetiza-\ntion breaks down into two regimes. First, we note that\nthe expression for βdepends only on the product of the\ncorrelation time, Θ, and the rate Γ 1, and not on their\nspecific individual values. We may then discuss the be-\nhavior of the model in terms of only the ratio parameter\nR= Γ1Θ = Θ/Γ−1\n1, which gives the ratio of the well\ncorrelation time to the escape time. Rewriting the Eq.35\nfor the spin vs time,\nm(t) =(Γ1m0+Γ2)\nΓ1/parenleftBig\n(e−t/2Θ([eβt/2Θ(37)\n−e−βt/2Θ]/2β+[e−βt/2Θ+eβt/2Θ]/2)/parenrightBig\n−Γ2\nΓ1,\nwhich may be simplified in terms of hyperbolic trigono-\nmetric functions,\nm(t) =e−t/2Θ/parenleftBigsinh(βt/2Θ)\nβ+cosh(βt/2Θ)/parenrightBig\n.(38)\nFor smaller R <1\n4, we have a real value of β=√1−4R, and the time-dependence of the spin corre-\nsponds to Eq. 38. In Figure 4, we plot the time-evolution\nfor values of R <1\n4. Once the correlation time is some\nsizable fraction of the escape time, the behavior begins\nto depart from the simple exponential behaviorpredicted\nin the Markovian system. At early times the magnetiza-\ntion decays more slowly than the exponential decay and\nat later times it decays more quickly, while the timescale\nover which the decay occurs (Γ 1) remains the same. The\neffect of the increasing correlation time between the pop-\nulations of the wells is then to shift the process to differ-\nent, lower frequencies.\nIn the case that R >1\n4, we have an imaginary argu-\nment to sinh and cosh, we then have an expression for\nm(t)\nm(t) =e−t/2Θ(sin(bt/2Θ)\nb+cos(bt/2Θ)) (39)\nwhereb=√\n4R−1. We note that the solutions take\nthe form of damped oscillations which tends toward the\nequilibrium value of the magnetization. However, these\nsolutions are unphysical as the occupation in individual\nwells maybecome lessthan 0 for these values. This is not\nsurprising, as for longer correlation times the generalized\nmaster equation will overestimate the population in each\nwell and generate a time evolution which will continue to\nreduce the population of a well, even when that well is\npresentlyempty. Itisalsounclearwhatitwouldmeanfor\nthe correlation time of the well population to exceed or\nbe on the order of the overall escape time, as this would\nimplythatthetimescaleoverwhichthespinpopulationis\ncorrelatedexceeds the overallescape time for the system,\nwhich is itself determined by changes in the individual\nwell populations.7\nIV. COMPARISON\nWe may now directly compare the magnetic relaxation\nprofiles calculated from explicit numerical integration of\nEqs. 5, 6 at various barrier heights, damping and cor-\nrelation times, to the biexponential decay predicted by\nthe generalized master equation. In all of the present\nsimulations we again use simulation parameters compa-\nrable to the Co nanoparticle of volume V= 8×10−27m3,\nanisotropy energy density K= 4.2×105J/m3giving an\nanisotropy energy KV= 1.12×1021J, and a magnetic\nmomentµs= 1.12×10−20J/T, while no external applied\nfield is assumed, Hext= 0.\n00.20.40.60.81\n0e+002e-094e-096e-098e-091e-08M(t)/Mr\nt (s)σ = 2, α = 0.01\ne-t/τ\n00.20.40.60.81\n0e+002e-094e-096e-098e-091e-08M(t)/Mr\nt (s)σ = 6, α = 0.5\ne-t/τ\nFIG. 5. Spin relaxation profiles from LLG simulations for\nTOP:σ= 2,α= 0.01, giving an exponential decay with\ncharacteristic escape time τ= 5×10−9sandBOTTOM :\nσ= 6,α= 0.5,τ= 4.5×10−9s\n.\nThe spins are initialized in the equilibrium Boltz-\nmann distribution in one of the minima of the po-\ntential energy, according to the distribution P(θ)∝\nsin(θ)exp(−ku/kBTsin2(θ)). To ensure that the noise\nis equilibrated with the spin at the correct temperature,\nthe noise is initially set to ηi,j,k= 0, and is then evolved\nin the presence of the equilibrium distribution in the well\nuntiltheycomeintothermalequilibrium. Theinitialcon-\ndition of the noise is important, as, for example, a choice\nofη(t= 0) = 0, will result in a field which quickly alignswith the spins in the potential minimum and give an un-\nphysical increase in the well population from equilibrium\nat short times.\nThe time-evolution of the magnetization, M(t) =\n∝angbracketleftSz(i)∝angbracketrightis then plotted, normalized by the initial rema-\nnent magnetization inside of the well, Mr=M(0).\n00.20.40.60.81\n0e+002e-094e-096e-098e-091e-08M(t)/Mr\nt (s)τc = 1, σ = 2, α = 0.01\ne-t/τ\n00.20.40.60.81\n0e+002e-094e-096e-098e-091e-08M(t)/Mr\nt (s)τc = 1, σ = 6, α = 0.5\ne-t/τ\nFIG. 6. Exponential behavior from LLMS simulations, for\nTOP:τc= 1,σ= 2.α= 0.01 we have an exponential decay\nwith escape time τ= 5.5×10−9, andBOTTOM :τc= 1,\nσ= 6.α= 0.5τ= 5.3×0−9s For low damping and large\nbarrier heights, the correlation time is much smaller than t he\nescape time.\nIn Figure 5, we depict the numerical calculation of the\nrelaxation profile from the LLG. This gives rise to an ex-\nponential behavior with a single relaxation time, which\nis directly comparable to the exponential decay of the\nmaster equation. In general, the relaxation profile from\nthe LLG may be non-exponential, with both the inte-\ngral relaxation time and the decay profile depending on\nthe higher-order eigenvalues of the Fokker-Planck opera-\ntor and the equilibrium correlation functions of the spin,\nτi\nint=/summationtext\nkτi\nkλk. However, the relaxation is dominated\nby the first eigenvalue in the high-barrier limit and for\nsmall applied fields , for σ>1, with good agreement be-\ntween the LLG and exponential decay for σas low as 2,\nas is shown in Figure 5.\nFigure 6 shows the relaxation from LLMS simulations\nof the Co nanoparticle, where the correlation time is cho-8\n0.10.20.30.40.50.60.70.80.91\n0e+005e-111e-101e-102e-103e-10M(t)/Mr\nt (s)τc = 1, σ = 2, α = 0.5\ne-t/τ\nFIG. 7. Biexponential behavior from LLMS simulations for\nτ= 1,σ= 2,α= 0.5 andτ= 1.48x10−10\nsen to be of the order of the inverse Larmor precession\ntime such that τc≈(γHk)−1. In both the cases of low\ndamping and higher barriers, we see that the ordinary\nexponential behavior of the LLG is retained. In this case\nthe escape time is much larger than the correlation time\nofthenoise,andtherelaxationaldynamicsareunaffected\nby the intra-well dynamics of the spin which occur on a\nmuch faster timescale than the relaxation, τc/τ≈0.01\nfor both simulations.\nIn the intermediate-to-high damping and high damp-\ning regimes, the behavior of the magnetization becomes\nmuchmoreinterestinganddepartsfromthe LLG. In par-\nticular,forarelativelysmallbarrierof σ= 2,α= 0.5and\nacorrelationtimeagainoftheorderoftheinverseLarmor\nfrequency. In this case the ratio of the escape to the cor-\nrelationtime is τc/τ= 9.4×10−12s/1.48×10−10s≈0.06.\nThe influence of the spin correlation is now visible in the\nrelaxation profile of the escape, as shown in Figure 7,\nwhich is similar to the biexponential deviation predicted\nby the generalized master equation, with the relaxation\nproceeding more slowly at earlier times and speeding up\nat later times.\nFinally, for very long correlationtimes and high damp-\ning, the correlation time remains a sizable fraction of\nthe escape time. However the biexponential behavior\nis no longer evident as shown in figure 8. The decay\nremains approximately exponential with a highly noisy\npath, a possible indication that the precise decay profile\nis extremely dependent on the initial conditions for such\nstrong coupling between the spin and bath.\nV. CONCLUSIONS\nWe have investigated thermal relaxation in magnetic\nnanoparticles introducing colored noise. Two models00.20.40.60.81\n0e+002e-104e-106e-108e-101e-09M(t)/Mr\nt (s)τc = 5, σ = 2, α = 0.5\ne-t/τ\n00.20.40.60.81\n0e+001e-102e-103e-104e-105e-10M(t)/Mr\nt (s)τc = 5, σ = 2, α = 5\ne-t/τ\nFIG. 8. LLMS simulations at high damping and long corre-\nlation times. The behavior continues to depart from a purely\nexponential decay, but now exhibits a noisy, more compli-\ncated time-dependence. TOP:τc= 5,σ= 2 ,α= 0.5 and\nτ= 4.5×10−10,BOTTOM :τc= 5,σ= 2 ,α= 5 and\nτ= 1.8×10−10\n.\nare considered. The first is an approach based on the\nnumerical solution of the Landau-Lifshitz-Miyazaki-Seki\n(LLMS) model, which replaces the white noise approx-\nimation associated with the use of LLB-equation based\nmodels. Due to computational requirements the LLMS\napproach is useful for relatively short timescales, conse-\nquently a second approach is derived based on a general-\nizedmasterequationapproachinvolvingtheintroduction\nof a memory kernel. We find that the importance of col-\nored noise is determined by the ratio of the correlation\ntimeτcto the characteristic system time τs= (γHk)−1,\nwhich is essentially the Larmor precession time. Con-\nsequently correlated noise should become important for\nmaterials with large magnetic anisotropy such as SmCo 5\nwhere the characteristic time approaches femtoseconds.\nBoth models, the LLMS-based approach and the mas-\nter equation, although derived for different timescales,\nexhibit an unusual bi-exponential decay of the magneti-\nzation, which represents an interesting signature of the\npresence of colored noise.9\nAppendix A: Colored Noise\nIn this appendix we present some relevant background\nmaterial on the LLMS equation and colored noise.\n1. Landau-Lifshitz-Miyazaki-Seki\nThe LLMS equations constitute an implementation of\na colored noise in a system with a thermalization con-\ndition represented through the Fluctuation-Dissipation\ntheorem. We reproduce here the original derivation by\nMiyazaki and Seki10, of the spin-only expression of the\nLLMS, which allows us to compare the LLMS thermal\nfluctuations directly to the Ornstein-Uhlenbeck. The\ntime evolution of the LLMS noise term is similar to the\nOU, with an additional term which couples explicitly to\nthe spin,\ndη\ndt=−1\nτc/parenleftBig\nη(t)−χS(t)/parenrightBig\n+R. (A1)\nTakingD=χkBT\nµs, then the autocorrelation of the field\nRis∝angbracketleftR(t)R(t′)∝angbracketright= 2D\nτcδ(t−t′), and proceeding to solve\nas a first-order linear differential equation in the same\nmanner as the OU noise, we have\nη(t) =χ\nτc/integraldisplayt\n−∞dt′K(t−t′)S(t′) (A2)\n+/radicalbigg\n2D\nτc/integraldisplayt\n−∞dt′K(t−t′)Γ(t).\nAfter integrating the first term by parts, we have\nη(t) =/radicalbigg\n2D\nτc/integraldisplayt\n−∞dt′K(t−t′)Γ(t) (A3)\n−χ/integraldisplayt\n−∞dt′K(t−t′)dS(t′)\ndt′,\nand by inserting this into the precessional equation for\nthe spin, we get the spin-only form for the LLMS equa-\ntion,\ndS\ndt=γS(t)×/parenleftBig\nH+¯η−χ/integraldisplayt\n−∞dt′K(t−t′)dS(t′)\ndt′/parenrightBig\n,(A4)\nwhere we now label the thermal fluctuations by ¯η(t),\n¯η(t) =/radicalbigg\n2D\nτc/integraldisplayt\n−∞dt′K(t−t′)Γ(t′).(A5)\nThe autocorrelation of this thermal field is\n∝angbracketleft¯η(t)¯η(t′)∝angbracketright=DK(t−t′) (A6)\n=χkBT\nµsK(t−t′) =β−1\nµsχK(t−t′),\nRecognizing χK(t−t′) as the damping term, we see that\nthis is a representation of the Fluctuation-Dissipationtheorem for the colored noise, where the additional fac-\ntor ofµsarises from the spin normalization. Taking the\nzero correlation time limit,\nlim\nτc→0∝angbracketleft¯η(t)¯η(t′)∝angbracketright= 2Dτcδ(t−t′).(A7)\n00.20.40.60.81\n00.511.522.533.5P(θ)\nθLLMS, σ = 1\nAnalytical\n00.20.40.60.81\n00.511.522.533.5P(θ)\nθLLMS, σ = 10\nAnalytical\nFIG. 9. P(θ) vsθ, from numerical simulations of the LLMS\nequation for TOP:σ= 1 and BOTTOM :σ= 10, with\nτcγHk= 2.\n.\nWe note that the LLMS thus derived from the physi-\ncal consideration of the spin-field interaction is not im-\nmediately comparable with the typical expression for the\nOrnstein-Uhlenbeck colored noise, owing to the fact that\nthe 1/τcterm has been implicitly absorbed in the white\nnoise term. If we rescale the driving noise such that\nQ(t) =τcR(t), we then have a pair of Langevin equa-\ntions\ndS\ndt=γ(S×(H+η)), (A8)\nwhile the noise evolves as,\ndη\ndt=−1\nτc/parenleftBig\nη(t)−χS(t)+Q/parenrightBig\n. (A9)\nThe autocorrelation of the white noise is\n∝angbracketleftQ(t)Q(t′)∝angbracketright=2χτckBT\nµsδ(t−t′) = 2Dδ(t−t′),(A10)10\nwithD=χτckBT\nµs, while the limit of the autocorrelation\nof the thermal term in the spin-only expression is now,\nlim\nτc→0∝angbracketleft¯Q(t)¯Q(t′)∝angbracketright=D\nτcδ(t−t′),(A11)\nwhich is directly comparable to the Ornstein-Uhlenbeck\nform of the colored noise. The expression of the LLMS\nin terms of the bath variable Qhas the additional benefit\nthat/bracketleftbig\nQ/bracketrightbig\n=Tand so we can interpret Qas the thermal\nmagnetic field contribution to the evolution of the bath\nfield.\nFinally, we may see that the limit of the LLMS equa-\ntion for vanishing correlation time is the LLG equation.\nFor small correlation times we can then take the Taylor\nexpansion about the time tint′, so that the damping\nterm becomes,\n/integraldisplayt\n−∞K(t−t′)dS(t′)\ndt′dt′=/bracketleftBig/integraldisplayt\n−∞K(t′)dt′/bracketrightBigdS(t)\ndt+...\n(A12)\nHence the spin and memory kernel decouple in the small\ncorrelation time limit, and the Langevin equation be-\ncomes\ndS\ndt=γS(t)×/parenleftBig\nH+¯η−/bracketleftBig\nχ/integraldisplayt\n−∞dt′K(t−t′)/bracketrightBigdS(t)\ndt/parenrightBig\n,\n(A13)\nAfter performing the integration over t′, the damping is\nχ/integraldisplayt\n−∞dt′e−(t−t′)/τc=χτc. (A14)\nand by direct comparison of the damping terms in this\nexpression and in Gilbert’s equation we have the rela-\ntionship of the phenomenological damping to the LLMS\nparameters α=χγτc. We note also that this expression\ncan be seen if we identify the driving white noise in the\nbath field of the LLMS with the thermal magnetic fieldsof the LLG.\n∝angbracketleftQ(t)Q(t′)∝angbracketright=2χτckBT\nµsδ(t−t′)\n=2αkBT\nγµsδ(t−t′)\n=∝angbracketleftHth(t)Hth(t′)∝angbracketright(A15)\nunder the assumption that α=γχτc.\n2. Thermalization\nAs a quantitative evaluation of the LLMS model and\nour implementation thereof, we compare the equilibrium\nbehavior to the appropriate analytical Boltzmann distri-\nbution, which the Markovian LLG equation also satis-\nfies. We simulate a single spin under the influence of\nanisotropy only. The Boltzmann distribution for such a\nsystem is\nP(θ)∝sinθexp(−kusin2θ\nkBT) (A16)\nwhereθis the angle between the spin and the easy-\naxis and the factor of sin θarises from normalizing the\nprobability distribution on the sphere. WE initialize the\nspin along the easy-axis direction, then allow the spin to\nevolve for 108steps after equilibration and evaluate the\nprobability distribution by recording the number of steps\nthe spin spends at each angle to the easy-axis.\nIn Figure 9, we compare the numerical results to the\nanalytical expression for both the LLMS model and the\nstandard LLG augmented by Ornstein-Uhlenbeck fields\nof the type generated by the Langevin equation in Eq. 4.\nThe simulations using the LLMS model agree with the\nanticipated Boltzmann distribution at equilibrium, while\nthe LLG with Ornstein-Uhlenbeck fails to reproduce the\ncorrect distribution. This is because, as we have argued,\nthis does not comprise a correct implementation of the\nFluctuation-Dissipation theorem, with deviations corre-\nsponding to the missing high-frequency components of\nthe damping.\n1L. N´ eel, Ann. G´ eophys. C.N.R.S. 5, 99 (1949).\n2W. F. Brown, Jr., Phys. Rev. 130, 1677 (1963).\n3A. Lyberatos, D.V. Berkov and R.W. Chantrell, J Phys:\nCondens. Matter 5, 8911 (1993)\n4A. Lyberatos and R. W. Chantrell, J. Appl. Phys. 73, 6501\n(1993).\n5J. L. Garc´ ıa-Palacios and F. J. L´ azaro, Phys. Rev. B 58,\n14937 (1998).\n6D. V. Berkov, IEEE Trans. Magn. 38, 2489 (2002).7Y. P. Kalmykov, W. T. Coffey, U. Atxitia, O. Chubykalo-\nFesenko, P. M. D´ ejardin, and R. W. Chantrell, Phys. Rev.\nB82, 024412 (2010).\n8R. Street and J. C. Woolley, Proc. Phys. Soc. A 62, 562\n(1949).\n9U Atxitia, O. Chubykalo-Fesenko, R. W. Chantrell, U\nNowak and A. Rebei, Phys. Rev. Lett. 102, 057203 (2009).\n10K. Miyazaki and K. Seki, J. Appl. Phys. 112, 121301\n(2012).11\n11U. Atxitia and O. Chubykalo-Fesenkoo, Phys. Rev. B 84,\n144414 .(2011)\n12P. H¨ anggi, P. Jung, Adv. Chem. Phys. 89, 239 (1995).\n13U. Nowak, Annual Reviews of Computational Physics IX,\npg. 105-151 , World Scientific (2001).\n14U. Nowak, R. W. Chantrell, and E. C. Kennedy, Phys.\nRev. Lett. 84, 163 (2000).15W. T. Coffey and Y. P. Kalmykov, J. Appl. Phys. 112,\n121301 (2012).\n16R. F. L. Evans, W. J. Fan, P. Chureemart, T. A. Ostler,\nM. O. A. Ellis and R. W. Chantrell, J. Phys.: Condens.\nMatter26, 103202 (2014).\n17I. M. Sokolov, Phys. Rev. E 66, 041101 (2002).\n18I. M. Sokolov, Phys. Rev. E 63, 056111 (2001)."    },    {        "title": "1510.03949v1.The_General_Solution_to_Vlasov_Equation_and_Linear_Landau_Damping.pdf",        "content": "The General Solution to Vlasov Equation and Linear Landau Damping  \n \nDeng Zhou \nInstitute of Plasma Physics, Chinese Academy of Sciences  \nHefei 230031, P . R. China  \n \nABSTRACT  \n  \nA general solution to linearized Vlasov equation for an electron electrostatic wave  in a \nhomogeneous unmagnetized plasma is derived. The quasi -linear diffusion coefficien t \nresulting from this solution is a continuous function of 𝜔 at 𝐼𝑚(𝜔)=0 in contrast to  \nthat derived from the traditional Vlasov treatment. The general solution is also equivalent to the Landau’s treatment of the plasma normal oscillations, and hence leads to the well-known Landau damping.  \n        Linear e lectron plasma waves in a collisionless plasma can be obtained by solving the \nlinearized Vlasov equation together with the Poisson equation, which was first treated by Vlasov\n1\n. The dispersion relation is given by  \n𝐷(𝜔,𝑘)=1+𝑒2\n𝑚𝑘𝜖0∫𝜕𝑓0/𝜕𝑣\n𝜔−𝑘𝑣+∞\n−∞𝑑𝑣=0  (1)  \nwhere 𝜔 and 𝑘 are respectively the frequency and the wave number ，other symbols are \nobvious . 𝐷(𝜔,𝑘) is usually called the plasma dielectric function.  From Eq. (1), we can get \nthe usual Langmuir wave. However, it is inadequate to treat all the effects of thermal particl es. We notice from Eq. (1) that there is a singularity at 𝑣=𝜔/𝑘 in the integra l and \n𝐷(𝜔,𝑘) is not a continuous function of 𝜔 at 𝐼𝑚(𝜔)=0.  \n   To overcome these insufficiencies, Landau solved the Vlasov equation as an initial -value \nproblem and introduc ed the deformed integration route in the  plasma dielectric function \nto make sure that 𝐷(𝜔,𝑘)  is an analytical function in the whole complex plane of 𝜔. \nLandau’s treatment leads to the famous landau damping\n2, which was later demonstrated in \nexperiments3. The introduction on the two  kinetic treatment s of plasma waves can be \nfound in most plasma textbooks, such as  Ref.[4]. An overview by Ryutov summarized the \nstudies of Landau damping by the end of twentieth century5. The Landau damping derived \nby Landau is a pure achievement of applied mathematics. On it ’s physical explanation \ndifferent people have different points of view ( see, for example, Drummond6 and \nreferences therein ).  \n   On a weak nonlinear level, a quasi- linear approach is adopted to describe  the \ninteraction between waves and particles7,8. The averaged background particle distribution \nmay experience diffusion in the velocity space  due to the wave -particle interaction . Taking \nthe one dimensional electrostatic case as an example, one gets  distribution evolution \nequation \n      𝜕𝑓0\n𝜕𝑡=𝜕\n𝜕𝑣𝒟𝜕𝑓0\n𝜕𝑣  (2)  \nwith the quasi -linear diffusion coefficient   \n       𝒟∝𝛾\n(Ω−𝑘𝑣)2+𝛾2      (3)  \nwhere Ω(𝛾) is the real (imaginary) component of 𝜔. Eq. (3) is derived from the Vlasov’s \nsolution for unstable modes, i. e. 𝛾>0.  When the resonance broadening disappears one meets a difficulty in extending (3) to the 𝛾<0 cases. We note that in the limit 𝛾→0, \nEq. (3) reduces to  \n      𝒟 ∝𝑠𝑖𝑔𝑛 (𝛾)𝜋𝛿(Ω−𝑘𝑣)  (4)  \nwhere  𝑠𝑖𝑔𝑛 (𝛾) denotes sign of 𝛾. The resonance broadening disappears and a \ndiscontinuity appears at 𝛾=0. To overcome  this problem, one has to turn to Landau’ s \ntreatment to get the distribution function. It is difficult to get an explicit function of \ndistribution since an inverse Laplace transform is involved. To circumvent this problem9, \nHellinger et al  recently extended the linear solution to Vlasov equation for the 𝛾<0 case \nby adding a term involving a complex Dirac delta function. However, these authors didn’t \nrealize  that the extended solution is the real solution to Vlasov equation and is related to \nlinear Landau damping, and moreover Dirac delta function is not a we ll-defined function in \ncomplex plane.  \n   In the following , we derive a general solution to Vlasov equation for the electrostatic \nLangmuir waves and from this solution the quasi -linear coefficient is derived and shown to \nbe continuous from growing to damped modes. We also show that the general solution is \nequivalent to the Landau treatment and leads to the famous landau damping.  \nWe consider the one dimentional electrostatic oscillation in a homogeneous \nunmagnetized plasma, the linearized Vlasov equation is  \n𝜕𝑓1\n𝜕𝑡+𝑣𝜕𝑓1\n𝜕𝑥=𝑒𝐸1\n𝑚𝜕𝑓0\n𝜕𝑣   (5)  \nThe perturbation of electric field and particle distribution is written in the normal mode form  \n    𝐸\n1=𝐸�𝑘𝑒−𝑖(𝜔𝑡−𝑘𝑥)      (6)  \n     𝑓 1=𝑓̂𝑘𝑒−𝑖(𝜔𝑡−𝑘𝑥)       (7)  \nEq. ( 6) is then reduced to  \n    −𝑖𝜔𝑓̂𝑘+𝑖𝑘𝑣𝑓̂𝑘=𝑒𝐸�𝑘\n𝑚𝜕𝑓0\n𝜕𝑣      (8)  \nThe general form of solution to Eq. (8) is  \n  𝑓̂𝑘=−𝑖𝑒𝐸�𝑘\n𝑚𝑘𝜕𝑓0\n𝜕𝑣�1\n𝑣−𝜔𝑘⁄+𝑆𝜔,𝑘(𝑣)𝛥(𝑣,𝜔𝑘⁄)�    (9)  \nwhere 𝑆𝜔,𝑘(𝑣) is any function of 𝜔, 𝑘 and 𝑣, and  \n    Δ (𝑣,𝜔𝑘⁄)=𝛿(𝑣−Ω𝑘⁄)+(−𝑖𝛾/𝑘)𝛿′(𝑣−Ω𝑘⁄)+(−𝑖𝛾/𝑘)2\n2!𝛿′′(𝑣−Ω𝑘⁄)+⋯   \n(10) \nSetting 𝑆𝜔,𝑘(𝑣)=0, we recover the Vlasov treatment and get the dispersion relation Eq. \n(1). To prove that Eq. (9) is the solution of Eq. (8), one needs to demonstrate  \n  (𝑣−𝜔𝑘⁄)Δ(𝑣,𝜔𝑘⁄)=0      (11)  \nThis relation  is obvious for 𝛾=0. For 𝛾≠0, we have the following integrals for any \nsmooth function 𝐹(𝑣) \n  ∫𝐹(𝑣)+∞\n−∞(𝑣−𝜔𝑘⁄)δ(𝑣,Ω𝑘⁄)𝑑𝑣=−(𝑖𝛾/𝑘)𝐹(Ω𝑘⁄)     (12a)  \n \n  ∫𝐹(𝑣)+∞\n−∞(𝑣−𝜔𝑘⁄)(−𝑖𝛾/𝑘)𝛿′(𝑣−Ω𝑘⁄)𝑑𝑣=(𝑖𝛾/𝑘)𝐹(Ω𝑘⁄)−(𝑖𝛾/𝑘)2𝐹′(Ω𝑘⁄)    \n(12b)  \n     \n  \n∫1\n𝑛!𝐹(𝑣)+∞\n−∞(𝑣−𝜔𝑘⁄)(−𝑖𝛾/𝑘)𝑛𝛿(𝑛)(𝑣−Ω𝑘⁄)𝑑𝑣=\n(𝑖𝛾/𝑘)𝑛\n(𝑛−1)!𝐹(𝑛−1)(Ω𝑘⁄)−(𝑖𝛾/𝑘)𝑛+1\n𝑛!𝐹(𝑛)(Ω𝑘⁄)  (12c)  Combining (10) (12a -c), we obtain  \n   ∫𝐹(𝑣)+∞\n−∞(𝑣−𝜔𝑘⁄)Δ(𝑣,𝜔𝑘⁄)𝑑𝑣=0   (13)  \nSince 𝐹(𝑣) can be any smooth function, then from the basic theorem of function theory, \nEq. (11) is verified.  \n  The form  of  Δ (𝑣,𝜔𝑘⁄) is superficially the Taylor expansion of the complex Dirac \ndelta function 𝛿(𝑣−𝜔𝑘⁄). However, it is impossible to define a proper complex Dirac \ndelta function 𝛿(𝑧) which is analytical at 𝑧=0 since we know that a complex function \nanalytical at whole complex plane is only a constant.  \nNow we need to determine 𝑆𝜔,𝑘(𝑣). Without loss of generality, we set 𝑆𝜔,𝑘(𝑣)=0 \nfor 𝛾>0 and derive the value for 𝛾≤0 through the continuity of density perturbation \nat 𝛾=0. \nThe density perturbation is given by  \n    𝑛�𝑘=−𝑖∫𝑒𝐸�𝑘\n𝑚𝑘𝜕𝑓0\n𝜕𝑣�1\n𝑣−𝜔𝑘⁄+𝑆𝜔,𝑘(𝑣)𝛥(𝑣,𝜔𝑘⁄)�𝑑𝑣+∞\n−∞   (14)  \nThe integration route is from −∞ to +∞ in 𝑣-space. In Eq. (14), the principal value of \nintegral is taken if a singularity lies on  the integration route.  \n  For 𝛾>0, we get from (14)  \n    𝑛�𝑘=𝑖∫𝑒𝐸�𝑘\n𝑚𝑘𝜕𝑓0\n𝜕𝑣1\n𝑣−𝜔𝑘⁄𝑑𝑣𝐶𝑅+2𝜋𝑒𝐸�𝑘\n𝑚𝑘\u0000𝜕𝑓0\n𝜕𝑣�\n𝑣=𝜔𝑘⁄       (15)  \nwhere 𝐶𝑅 denotes the integration route of the semi -circle on the upper half complex \nplane from +∞ to −∞, as indi cated in Fig. 1. For 𝛾<0, the density perturbation is given \nby \n   𝑛�𝑘=𝑖∫𝑒𝐸�𝑘\n𝑚𝑘𝜕𝑓0\n𝜕𝑣1\n𝑣−𝜔𝑘⁄𝑑𝑣𝐶𝑅−𝑖𝑆𝑒𝐸�𝑘\n𝑚𝑘�\u0000𝜕𝑓0\n𝜕𝑣�\n𝑣=𝜔\n𝑘+(𝑖𝛾)\u0000𝜕2𝑓0\n𝜕𝑣2�\n𝑣=𝜔\n𝑘+(𝑖𝛾)2\n2!\u0000𝜕3𝑓0\n𝜕𝑣3�\n𝑣=𝜔\n𝑘+∙∙∙�   (16) \nOne obtains 𝑆=2𝜋𝑖 for 𝛾<0 from (15) and (16) by the requirement of 𝑛�𝑘(𝛾→0+)=\n𝑛�𝑘(𝛾→0−). The same procedure yields 𝑆=𝜋𝑖 for 𝛾=0. In summary, we get the \ngeneral solution to Eq. (8) given by Eq. (9) and (10) with \n          𝑆 𝜔,𝑘(𝑣)=�0          𝛾 >0\n 𝜋𝑖        𝛾 =0         \n2𝜋𝑖      𝛾 <0               \u0000    (17)  \nIn the weak non- linear level the averaged background particle distribution changes slowly \ndue to the wave -particle interaction, which is described through a quasi -linear approach. \nKeeping to the second order of perturba tion in Vlasov equation and taking a spa ce \naverag ing in large space scale, one obtains the distribution evolution  \n   𝜕𝑓0\n𝜕𝑡=𝑅𝑒〈𝑒𝐸1∗\n𝑚𝜕𝑓1\n𝜕𝑣〉     (18)  \nwhere 𝑅𝑒 ( 𝐼𝑚 ) denotes the real ( imaginary ), bracket denotes sp ace averaging, and the \nsuper script star denotes complex conjugate.    \n  Inserting (6) (7) and (9) into (18), we get the  diffusion equation, Eq. (2), for  distribution \n𝑓0 with the diffusion coefficient  \n    𝒟 =�𝑒\n𝑚�2\n�𝐸�𝑘�2𝐼𝑚�1\n𝑘𝑣−𝜔+𝑆𝜔,𝑘(𝑣)𝛥(𝑣,𝜔𝑘⁄)\n𝑘� \n      = �𝑒\n𝑚�2\n�𝐸�𝑘�2�𝛾\n(Ω−𝑘𝑣)2+𝛾2+𝑆\n𝑖𝑘�𝛿(𝑣−Ω𝑘⁄)−�𝛾\n𝑘�2\n𝛿′′(𝑣−Ω𝑘⁄)+⋯��    (19)  \nFor simplicity we  have kept only one wave number while the real diffusion is contributed \nfrom all the excited modes.  \n   It is obvious that the diffusion coefficient given by (19) is continuous as 𝛾→0±, which \nis  \n    𝒟 (𝛾→0+)=𝒟(𝛾→0−)∝𝜋𝛿(𝑘𝑣−Ω)     (20)   The general solution, Eq. (9), can lead to linear Landau damping. Substituting the \nperturbation of the electric field and the distribution into the one dimensional Poisson \nequation \n    𝜖0𝜕𝐸1\n𝜕𝑥=−𝑒∫𝑓1+∞\n−∞𝑑𝑣    (21) \nwe obtain the dispersion relation \n   1−𝜔𝑝𝑒2\n𝑛0𝑘2∫𝜕𝑓0\n𝜕𝑣�1\n𝑣−𝜔𝑘⁄+𝑆𝜔,𝑘(𝑣)𝛥(𝑣,𝜔𝑘⁄)�𝑑𝑣+∞\n−∞   (22)  \nwhere 𝜔𝑝𝑒=�𝑛0𝑒2\n𝜖0𝑚 is the usual Langmuir  frequency and 𝑛0 is the background plasma \ndensity.  \n   Substituting (10) in (22) yields  \n      1−𝜔𝑝𝑒2\n𝑛0𝑘2∫𝜕𝑓0\n𝜕𝑣1\n𝑣−𝜔𝑘⁄𝑑𝑣+∞\n−∞−𝑆𝜔𝑝𝑒2\n𝑛0𝑘2\u0000𝜕𝑓0\n𝜕𝑣�\n𝑣=𝜔\n𝑘=0        (23)  \nThis is exactly the plasma dielectric  function one obtains from Landau’s treatment by \nadopting a modified Landau integra tion contour in complex 𝑣-plane. Hence, the general \nsolution to Vlasov equation is equivalent to the Landau’s solution in the treatment of plasma normal modes and leads to the well- known Landau damping.  \nIn a recent work\n10, Wesson reexamined the problem of Landau damping. He solved \nVlasov equation separately  for 𝛾>0 and 𝛾<0 using real variables . The two cases are \nrelated through the continuity of density perturbation at 𝛾→0. Our present treatment is \nequivalent to Wesson’s since we also solve Vlasov equation separately for 𝛾>0 and \n𝛾<0 and also use the continuity condition to rel ate the two cases. To show this, we  take \nthe case 𝛾<0  as an example. Like Wesson’s  treatment, the density perturbation is \nseparated into two parts, 𝑛1𝑏  and 𝑛1𝑤. 𝑛1𝑏 is the contribution from the basic thermal \ndistribution, which is treated as a Dirac delta function, i. e. 𝑓0=𝑛0𝛿(𝑣). Hence \n𝑛1𝑏=−𝑖�𝑒𝐸�𝑘\n𝑚𝑘𝜕𝑓0\n𝜕𝑣1\n𝑣−𝜔𝑘⁄𝑑𝑣+∞\n−∞ \n                       = −𝑖∫𝑒𝐸�𝑘\n𝑚𝑘𝜕𝑓0\n𝜕𝑣𝑣−Ω𝑘⁄+𝑖γ𝑘⁄\n(𝑣−Ω𝑘⁄)2+(γ𝑘⁄)2𝑑𝑣+∞\n−∞ \n                       = −𝑖𝑒𝐸�𝑘\n𝑚𝑘�∫𝜕𝑓0\n𝜕𝑣1\n𝑣−Ω𝑘⁄𝑑𝑣+∞\n−∞+∫𝜕𝑓0\n𝜕𝑣𝑖γ𝑘⁄\n(𝑣−Ω𝑘⁄)2𝑑𝑣+∞\n−∞�     \n =−𝑖𝑒𝐸�𝑘\n𝑚𝑘�𝑛0\n(Ω𝑘⁄)2−2𝑖𝑛0γ𝑘⁄\n(Ω𝑘⁄)3�              (24)  \nwhere we have neglected the term (γ𝑘⁄)2 in the denominate of the second line since  \n𝛾≪Ω. 𝑛1𝑤 is contribut ed from the particles in resonance with the wav e. Taking  𝜕𝑓0\n𝜕𝑣 to be \na constant, we have  \n           \n𝑛1𝑤=−𝑖�𝑒𝐸�𝑘\n𝑚𝑘𝜕𝑓0\n𝜕𝑣�1\n𝑣−𝜔𝑘⁄+𝑆𝛥(𝑣,𝜔𝑘⁄)�𝑑𝑣+∞\n−∞ \n                             = −𝑖𝑒𝐸�𝑘\n𝑚𝑘\u0000𝜕𝑓0\n𝜕𝑣�\n𝑣=Ω\n𝑘∫�𝑣−Ω𝑘⁄+𝑖γ𝑘⁄\n(𝑣−Ω𝑘⁄)2+(γ𝑘⁄)2+𝑆𝛥(𝑣,𝜔𝑘⁄)�𝑑𝑣+∞\n−∞   (25) \nChanging to the coordinate system travelling with the wave velocity, one obtains  \n 𝑛1𝑤=−𝑖𝑒𝐸�𝑘\n𝑚𝑘\u0000𝜕𝑓0\n𝜕𝑣�\n𝑣=𝜔\n𝑘∫�𝑣+𝑖γ𝑘⁄\n𝑣2+(γ𝑘⁄)2+𝑆𝛥(𝑣,𝜔𝑘⁄)�𝑑𝑣+∞\n−∞ \n                 = −𝑖𝑒𝐸�𝑘\n𝑚𝑘�𝑖𝜋\u0000𝜕𝑓0\n𝜕𝑣�\n𝑣=𝜔\n𝑘−𝛾2𝜋\u0000𝜕2𝑓0\n𝜕𝑣2�\n𝑣=𝜔\n𝑘+⋯�           (26)  \nSubstitution of (24) (26) into the Poisson equation (21) yields respectively from the real and the imaginary parts  \n         𝛾 =𝜋Ω3\n2𝑛0𝑘2\u0000𝜕𝑓0\n𝜕𝑣�\n𝑣=𝜔\n𝑘          (27)  \nand  \n         Ω =𝜔𝑝𝑒=�𝑛0𝑒2\n𝜖0𝑚          (28)  \nThey are exactly the same as  those  given by Wesson.  \n   In summary, we have presented the general form of solution to Vlasov equation for the \nelectrostatic  plasma wave , shown by Eqs. (9) (10) and (17). From this solution, one can get \nthe quasi -linear diffusion coefficients  valid for both da mping and growing modes. The \nsolution can also lead to the well -known Landau damping and it is equivalent  to Wesson ’s \nrecent explanation of Landau damping using real variables.  \n ACKNOWLEDGEMENT  \n This work is supported by National  Natural Science Foundation of China under Grant  No. \n11175213.  \n      \n1A. Vlasov, J. Phys. (USSR) 9, 25 (1945). \n2L. D. Landau, J. Phys. (USSR) 10, 25 (1946).  \n3J. Malmberg and C. Wharton, Phys. Rev. Lett. 17, 175 (1966).  \n4R. Goldston and P . Rutherford, Intro duction to Plasma Physics (Institute of Physics \nPublishing, 1995).  \n5D. Ryutov, Plasma phys. Contr. Fusion 41,  A1 (1999) . \n6W. Drummond, Phys. Plasmas  11, 552 (2004).  \n7A. Vedenov, E. Velikhov, and R. Sagdeev, Nucl. Fusion 1, 82 (1961).  \n8W. Drummond and D. Pines, Nucl. Fusion Suppl. 3, 1049 (1962).  \n9P . Hellinger and P . Travnicek, Phys. Plasmas 19, 062307 (2012). \n10J. Wesson, Phys. Plasmas 22, 0 22519 (2015 ). \n         \n \n      \n \n   \n \nFig. 1 The  integration route used in calculation of density perturbation.  \n \n  \n"    },    {        "title": "1906.10326v2.Conductivity_Like_Gilbert_Damping_due_to_Intraband_Scattering_in_Epitaxial_Iron.pdf",        "content": "1 \n Conductivity -Like Gilbert Damping due to Intraband Scattering in Epitaxial Iron  \n Behrouz Khodadadi1, Anish Rai2,3, Arjun Sapkota2,3, Abhishek Srivastava2,3, Bhuwan Nepal2,3, \nYoungmin  Lim1, David A. Smith1, Claudia Mewes2,3, Sujan Budhathoki2,3, Adam J. Hauser2,3, \nMin Gao4, Jie-Fang Li4, Dwight D. Viehland4, Zijian Jiang1, Jean J. Heremans1, Prasanna  V. \nBalachandran5,6, Tim Mewes2,3, Satoru Emori1* \n1 Department of Physics, Virginia Tech , VA 24061, U.S.A  \n2Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA  \n 3 Center for Materials for Information Technology (MINT), University of Alabama, Tuscaloosa, \nAL 35487, U .S.A. \n4  Department  of Material Science and Engineering, Virginia Tech ,  \n Blacksburg, VA 24061, U.S.A . \n5  Department of Material Science and Engineering, University of Virginia,  \n Charlottesville, VA 22904, U.S.A . \n6  Department of Mechanical and Aerospace Engineering , Univer sity of Virginia,  \n Charlottesville, VA 22904, U.S.A.  \n*email: semori@vt.edu  \n \nConfirming  the or igin of Gilbert damping by experiment has remained  a challenge  for \nmany decades , even for simple  ferromagnetic metals . In this Letter, we experimentally \nidentify  Gilbert damping that increases with decreasing electronic scattering  in epitaxial \nthin films of pure Fe . This observation of  conductivity -like damping, which cannot be  \naccounted for by  classical eddy current loss , is in excellent quantitative agreement with \ntheoretical  predictions of Gilbert damping due to intraband scatte ring. Our results  resolve 2 \n the longstanding question about a fundamental damping mechanism  and offer  hints for \nengineering low -loss magnetic metals for cryogenic spintronic s and quantum devices.    \n \nDamping determines how fast the magnetization relaxes towards the effective magnetic \nfield and plays a  central  role in many aspects  of magnetization dynamics  [1,2] . The magnitude of \nviscous Gilbert damping governs the  threshold current for spin -torque magnetic switching  and \nauto-oscillations  [3,4] , mobility of magnetic domain walls  [5,6] , and  decay leng ths of diffusive \nspin waves and superfluid -like spin current s [7,8] . To enable spintronic technologies with low \npower dissipation , there is currently  much interest in  minimizing Gilbert damping in thin films of \nmagnetic m aterials  [9–13], especially ferromagnetic metals  [14–23] that are compatible with \nconventional device fabrication schemes . Despite the fundamental and technological importance \nof Gilbert damping, its physical mechanisms  in various magnetic materials have yet to be \nconfirmed by experiment .   \nGilbert damping is generally attributed to spin-orbit coupling that ultimately dissipates  \nthe energy of the magnetic system  to the lattice  [1,2] . Kambersky’s torque correlation model  [24] \nqualitatively captures  the temperature dependence of damping in some experiments  [25–28] by \npartitioning Gilbert damping into two mechanisms due to spin -orbit coupling, namely interband \nand intraband scattering mechanisms, each with a distinct dependence  on the elect ronic \nmomentum scattering tim e e. For the interband scattering mechanism  where magnetization \ndynamics can excite electron -hole pairs across dif ferent bands, the resulting Gilbert damping is \n“resistivity -like” as its magnitude scales with e-1, i.e., increased  electronic scattering results in \nhigher damping  [29,30] . By contrast, the intraband scattering mechanism  is typically understood \nthrough the breathing Fermi surface mode l [31], where electron -hole pairs are excited in the 3 \n same band , yielding  “conductivity -like” Gilbert damping  that scales with  e, i.e., reduced \nelectronic scattering  results in  higher damping.  \nConductivity -like Gilbert damping was reported  experimentally more than  40 years ago \nin bulk crystals of pure Ni and Co at low temperatures , but surprisingly not in pure Fe [25]. The  \napparent absence  of co nductivity -like damping in Fe has been  at odds with many theoretical \npredictions that intraband scattering should dominate at low temperatures  [32–38], although \nsome theoretical studies have suggested that intraband scattering may be  absent alt ogether in \npure metals  [39,40] . To date, no experimental work has conclusively addressed  the role of \nintraband scattering in  pure Fe1. There thus remains  a significant gap in the fundamental \nunderstanding of damping in one of the simplest ferromagnetic metals.  Intrinsic conductivity -\nlike Gilbert damping in Fe is also technologically relevant,  since  minimizing damping in \nferromagnetic metals at low temperatures is crucial for cryogenic superconducting spintronic \nmemories  [41,42]  and quantum information  transduction schemes  [43,44] .  \nIn this Letter, we experimentally demonstrate the presence of conductivity -like Gilbert \ndamping due to intr aband scattering  in epitaxial thin films of body -centered -cubic (BCC) Fe. By \ncombining broadband ferromagnetic resonance (FMR) measurements with characterization of \nstructural and transport properties of these model -system thin films, we show that conductivity -\nlike Gilbert damping dominates at lo w temperatures in epitaxial Fe . These experimental  results \n                                                           \n1 Ref.  [36] includes experimental data that suggest the presence of conductivity -like Gilbert damping in an ultrathin \nFe film, although no detailed information is given about the sample and t he experimental results deviate \nconsiderably from the calculations. An earlier study by Rudd et al. also suggests an increase in Gilbert damping with \ndecreasing temperature  [27], but quantification of the Gilbert damping parameter in this experiment is difficult.  \n 4 \n agree remarkably well with the magnitude of Gilbert damping derived from first -principles \ncalculations  [32,33,36] , thereby  providing evidence for intraband scatterin g as a key mechanism \nfor Gilbert damping in pure BCC Fe. Our experiment  thus resolves the longstanding  question  \nregarding the origin of damping in the prototypical ferromagnetic metal . Our results also confirm  \nthat – somewhat counterintuitively – disorder can partially suppress intrinsic damping at low \ntemperatures in ferromagnetic metals, such that optimally disordered films  may be well suited \nfor cryogenic spintronic and quantum applications  [41–44].   \nEpitaxial BCC Fe thin films were sputter deposited on (001) -oriented MgAl 2O4 (MAO) \nand MgO single crystal substrates. The choices of substrates were inspired by the recent \nexperiment by Lee et al. [20], where epitaxial growth is enabled with t he [100] axis of a B CC \nFe-rich alloy oriented 45o with respect to  the [100] axis of MAO or MgO. MAO with a lattice \nparameter of a MAO /(2√2) = 0.2858  nm exhibits a lattice mismatch of less than 0.4% with Fe (a Fe \n≈ 0.287 nm) , whereas the lattice mismatch between MgO ( aMgO/√2 = 0.2978  nm) and Fe is of the \norder  4%. Here , we focus on 25 -nm-thick Fe films that were grown  simultaneously on MAO and \nMgO by confocal DC magnetron sputtering  [45].  In the Supplemental Material  [45], we report \non additional films depos ited by off -axis magnetron sputtering.  \nWe verified the crystalline quality of the epitaxial Fe films by X -ray diffraction, as s hown \nin Fig. 1( a-c). Only (00X )-type peaks of the substrate and film are found in each 2θ-ω scan, \nconsistent with the single -phase  epitaxial growth of the Fe films. The 2θ-ω scans reveal a larger \namplitude of film peak for MAO/Fe, suggesting higher crystalline quality  than that of MgO/Fe. \nPronounced Laue oscillations, indicative of atomically smooth film interfaces, are o bserved \naround  the film peak of MAO/Fe, whereas they are absent for MgO/Fe. The high crystalline \nquality of MAO/Fe is also evidenced  by its narrow film -peak rocking curve with a FWHM of 5 \n only 0.02o, comparable to the rocking curve F WHM of the substrate2. By contrast, the film -peak \nrocking curve of MgO/Fe has a FWHM of  1o, which indicates substantial mosaic  spread in the \nfilm due to the large  lattice mismatch with the MgO substrate.  \nResults of 2θ -ω scans for different film thicknesses  [45] suggest  that the 25 -nm-thick Fe \nfilm may be  coherently strained to the MAO substrate , consistent with  the smooth interfaces and \nminimal mosaic spread of MAO/Fe . By contrast, i t is likely that 25 -nm-thick Fe on MgO is \nrelaxed to accommodate the large film-substrate lattice mismatch. Static magnetometry provides \nfurther evidence that Fe is strained  on MAO a nd relaxed  on MgO  [45]. Since  strained MAO/Fe \nand relaxed MgO/Fe exhibit  distinct  crystalline quality,  as evidenced by an approximately 50 \ntimes narrower rocking FWHM for  MAO/Fe , we have two model systems that enable \nexperimental investigation of the impact of structural disorder on Gilbert damping.   \nThe residual electrical resistivity also reflects  the structural quality  of metal s. As shown \nin Fig. 1(d ), the residual resistivity is 20 % lower for MAO/Fe compared to MgO/Fe, which \ncorroborates the lower defect density in MAO/Fe.  The resistivity increases  by nearly an order of \nmagnitude with increasing  temperature, reaching  1.1×10-7  m for both samples  at room \ntemperature , consistent with behavior expected for pure metal  thin film s.  \nWe now examine how the difference in crystalline quality correlates with  magnetic \ndamping in MAO/Fe and MgO /Fe. Broadband FMR measurements were performed at room \ntemperature up to 65 GHz with a custom spectrometer that empl oys a coplanar waveguide \n(center conductor width  0.4 mm ) and an electromagnet (maximum field < 2 T) . For each \nmeasurement at a fixed excitat ion frequency, an external bias magnetic field was swept parallel \nto the film plane along the [110] axis of Fe , unless otherwise noted. I n the Supplemental \n                                                           \n2 The angular resolu tion of the diffractometer is 0.0068o.  6 \n Material  [45], we show similar results with the  field applied along the [110] and [100] axes of \nFe; Gilbert damping is essentially isotropic within the film plane for our epitaxial Fe films , in \ncontrast to a recent report of anisotropic damping in ultrathin epitaxial Fe  [22]. \nFigure 2  shows that the peak -to-peak FMR linewidth Hpp scales linearly with frequency \nf, enabling a precise  determination of  the measured  Gilbert damping parameter  𝛼𝑚𝑒𝑎𝑠  from the \nstandard equation,  \n𝜇0∆𝐻𝑝𝑝=𝜇0∆𝐻0+2\n√3𝛼𝑚𝑒𝑎𝑠\n𝛾′𝑓,    (1) \nwhere  Hpp,0 is the zero -frequency linewidth and 𝛾′=𝛾/2𝜋≈29.5 GHz/T is the reduced \ngyromagnetic ratio . Despite the difference  in crystalline  quality , we find essentially the same \nmeasured  Gilbert damp ing parameter of  𝛼𝑚𝑒𝑎𝑠  ≈ 2.3×10-3 for MAO/Fe and MgO/Fe. We note \nthat t his value of 𝛼𝑚𝑒𝑎𝑠  is comparable to the lowest damping parameters reported for epitaxial Fe \nat room temperature  [15–17]. Our results indicate  that Gilbert  damping at room temperature is \ninsensitive to the strain state or structural disorder  in epitaxial Fe.3  \n The measured damping parameter 𝛼𝑚𝑒𝑎𝑠  from  in-plane FMR can generally include a \ncontribution from non-Gilbert  relaxation , namely two -magnon scattering  driven by defects  [46–\n49]. However,  two-magnon scattering is suppressed when the film is magnetized out-of-\nplane  [19,48] . To isolate any two -magnon scattering contribution to d amping, we performed out-\nof-plane  FMR measurements under  a sufficiently large magnetic field (>4 T) for complete \nsaturation of  the Fe film, using a custom W-band shorted waveguide  combined with a \n                                                           \n3 However, the crystallographic texture of Fe has significant impact on damping; for example, non -epitaxial Fe films \ndeposited directly on amorphous SiO 2 substrates exhibit an order of magnitude wider linewidths, due to much more \npronounced non -Gilbert damping (e.g., two -magnon scattering), compared to (001) -oriented epitaxial Fe films.  \n 7 \n superconducting magnet. As shown in Fig. 2, the out -of-plane and in -plane FMR data yield  the \nsame slope  and hence 𝛼𝑚𝑒𝑎𝑠  (Eq. 1)  to within <  8%. This finding indicates  that two -magnon \nscattering is negligible and that frequency -dependent magnetic relaxation is dominated by \nGilbert damping in epitaxial Fe examined here.  \nThe insensitivity of Gilbert damping to disorder found in Fig. 2 can be explained by the \ndominance of the interband (resistivity -like) mechanis m at room temperature, with phonon \nscattering dominating over defect scattering. Indeed, since MAO/Fe and MgO/Fe have the same \nroom -temperature resistivity  (Fig. 1(d )), any contributions to Gilbert damping from  electronic \nscattering should be identical for  both samples at room temperature. Moreover, according to our \ndensity functional theory calculations  [45], the density of states of BCC Fe at the Fermi energy, \nD(EF), does not depend significantly on  the strain state of the crystal. Therefore, i n light of the \nrecent reports  that Gilbert damping is proportional to  D(EF) [18,50,51] , the different strain states \nof MAO/Fe and MgO/Fe are not expe cted to cause a significant  difference in Gilbert damping.   \n However, since MAO/Fe and MgO/Fe exhibit distinct resistivities (electronic scattering \ntimes e) at low temperatures, one might expect  to observe distinct temperature dependence in \nGilbert damping for these two samples. To this end, we performed variable -temperature FMR \nmeasurements using  a coplanar -waveguide -based spectrometer (maximum frequency 40 GHz, \nfield < 2 T)  equipped with  a clos ed-cycle cryostat4. Figure 3(a,b) shows that meas is enhanced \nfor both samples at lower temperatures. Notably, this damping enhancement with decreasing \ntemperature is significantl y greater for MAO/Fe . Thus, at low temperatures, we find a \n                                                           \n4 The W -band spectrometer for out -of-plane FMR (Fig. 2) could not be  cooled below room temperature  due to its \nlarge thermal mass , limiting us to in -plane FMR measurements at low temperatures.  8 \n conductivity -like damping increase that is evidently more pronounced in epitaxial Fe with less \nstructural disorder.  \nWhile this increased damping at low temperatures is reminiscent of intrinsic Gilbert \ndamping from intraband scattering  [31–38], we first consider other possible contributions. One \npossibility is two -magnon scattering  [46–49], which we have ruled out at room temperature (Fig. \n2) but could  be present in our low -temperature in-plane FMR measurements . From Fig. 3(a,b), \nthe zero -frequency linewidth   H0  (Eq. 1 ) – typically attributed to magnetic inhomogeneity – is \nshown to increase  along with meas at low temperatures  [45], which might  point to  the emergence \nof two -magnon scattering  [48,49] . However, our mean -field model calculations (see \nSupplemental Material  [45]) shows that H0  correlates with  meas due to  interactions among \ndifferent regions  of the inhomogeneous film  [52]. The increase of H0 at low temperatures is \ntherefore readily accounted for  by increased  Gilbert damping , rather than two -magnon scattering .  \nWe are also not aware of  any mechanism that enhance s two-magnon scattering with \ndecreasing temperature, particularly given that  the saturation magnetization (i.e., dipolar \ninteractions) is constant across  the measured temperature range  [45]. Moreover, the isotropic in -\nplane damping found in our study is inconsistent with typically anisotropic two-magnon \nscattering tied to the crystal symmetry of  epitaxial films  [46,47] , and the film thickness in our \nstudy (e.g., 25 nm) rules out t wo-magnon scattering of interfacial origin  [49]. As such, we \nconclude  that two -magnon scattering does not play any essential  role in our experimental \nobservations.  \n Another  possible contribution  is dissipation due to classical eddy current s, which \nincrease s proportionally with the increasing conductivity  𝜎 at lower temperatures . We estimate \nthe eddy current contribution to the measured Gilbert damping with [15,53]   9 \n 𝛼𝑒𝑑𝑑𝑦 =𝜎\n12𝛾𝜇02𝑀𝑠𝑡𝐹2,    (2) \nwhere  𝜇0𝑀𝑠≈2.0 T is the saturation magnetization  and tF is the film thickness . We find that  \neddy curr ent damping accounts  for only ≈20% (≈ 30%) of  the total measured  damping of  \nMAO/Fe (MgO/Fe)  even at the lowest measured temperature  (Fig. 3(c)) . Furthermore, a s shown \nin the Supplemental Material  [45], thinner MAO/Fe film s, e.g., tF = 11 nm , with negligible eddy \nstill exhibit a significant increase in damping with decreasing temperature. Our results thus \nindicate a substantial contribution to conductivity -like Gilbert damping that is not accounted for \nby classica l eddy current damping.  \n For further discussion , we subtract the eddy -current damping from the measured damping \nto denote the Gilbert damping parameter attributed to intrinsic spin-orbit coupling as \n𝛼𝑠𝑜= 𝛼𝑚𝑒𝑎𝑠 − 𝛼𝑒𝑑𝑑𝑦. To correlate electronic transport and magnetic damping across the entire \nmeas ured temperature range, we perform a phenomenological fit of the temperature dependence \nof Gilbert damping with  [26]  \n𝛼𝑠𝑜=𝑐𝜎(𝑇)\n𝜎(300  𝐾)+𝑑𝜌(𝑇)\n𝜌(300  𝐾),   (3) \nwhere  the conductivity -like (intraband) and resistivity -like (interband) terms are scaled by \nadjustable parameters c and d, respectively. As shown in Fig.  4(a),(b), t his simple \nphenomenological model  using the experimental transport results (Fig. 1(d))  agrees remarkably \nwell with the temperature dependence of Gilbert damping for both MAO/Fe and MgO/Fe.  \nOur fi nding s that Gilbert damping can  be phenomenologically partit ioned  into two \ndistinct contributions (Eq. 3 ) are in line with  Kambersky’s  torque correlation model . We \ncompare our experimental resul ts to first-principles calculations by Gilmore et al. [32,33]  that \nrelate  electronic momentum scattering rate e-1 and Gilbert damping through Kambersky’s torque \ncorrelation model. We use the  experimentally measured resistivity ρ (Fig. 1(d)) to convert the 10 \n temperature to e-1 by assuming the constant conversion factor ρ e = 1.30×10-21  m s [33]. To \naccount for the difference in electronic scattering time for the minority spin  and majority spin \n, we take the calculated curve from Gilmore et al. with / = 4 [33], which is close to the \nratio of D(EF) of the spin-split bands  for BCC Fe , e.g., derived from our density functional \ntheory calculations  [45]. For explicit comparison with Refs.  [32,33] , the Gilbert damping \nparameter in Fig. 4(c) is converted to the magnetic relaxation rate 𝜆= 𝛾𝛼𝑠𝑜𝜇0𝑀𝑠. The \ncalculated prediction is in excellent  quantitative agreement with our experimental results for both \nstrained MAO/Fe and relaxed MgO/Fe  (Fig. 4(c)) , providing additional experimental evidence \nthat intraband scattering predominately contribute s to Gilber t damping at low temperatures.   \n We also compare our experimental results to a more recent first -principles calculation \nstudy by Mankovsky et al., which  utilizes the linear response formalism  [36]. This approach  \ndoes not  rely on a phenomenological electronic scattering rate and instead  allows for explicitly \nincorporating thermal  effects  and structural disorder . Figure 4(d ) shows the calculated \ntemperature dependence of the Gilbert damping parameter for BCC Fe with a small density of \ndefects, i.e., 0.1% vacancies , adapted from Ref.  [36]. We again find good quantitative agreement \nbetween the ca lculations and our experimental results for MAO/Fe. On the other hand, the \nGilbert damping parameter s at low temperatures  for relaxed MgO/Fe are significantly below the \ncalculated values . This  is consistent with  the reduction  of intraband scattering  due to enhanced \nelectronic scattering (enhanced e-1) from defects in relaxed MgO/Fe .  \n Indeed, significant defect -mediated electronic scattering may explain  the absence of \nconductivity -like Gilbert damping for crystalline Fe in prior  experiments. For example, Ref.  [25] \nreports an upper limit of  only a two -fold increase of the estimated Gilbert damping parameter \nfrom T = 300 K to 4 K . This relatively small damping enhancement is similar to that for MgO/Fe 11 \n in our study  (Fig. 4(b)) , suggesting that intraband scattering may have been suppressed in Fe in \nRef. [25] due to a similar degree of structural disorder  to MgO/Fe. We therefore conclude that \nconductivity -like Gilbert damping from  intraband scattering is highly sensitive to disorder in \nferromagnetic metals.  \nMore generally , the presence of defects  in all real metals – evidenced by finite residual \nresistivity  – ensures that the Gilbert damping parameter is finite even in the zero -temperature \nlimit . This circumvents the theoretical deficiency of Kambersky’s  torque correlation model \nwhere Gilbert damping would diverge in a perfectly clean ferromagnetic metal at T  0 [39,40] . \nWe also remark that a fully quantum mechanical many -body theory of magnetization dynami cs \nyields finite Gilbert damping even in the clean, T = 0 limit  [54].  \n In summary, we have demonstrated the dominance of conductiv ity-like Gilbert damping \ndue to intraband scattering at low temperatures in high-quality epitaxial Fe . Our experimental \nresults  also validate  the longstanding theoretical prediction of  intraband scattering as an essential  \nmechanism for Gilbert damping in pure ferromagnetic metals  [32–38], thereby advancing the \nfundamental understanding of magnetic relaxation in real materials . Moreover, we have \nconfirmed that, at low temperatures, a ma gnetic metal with imperfect crystallinity can exhibit \nlower Gilbert damping (sp in decoherence) than its cleaner  counterpart. This somewhat \ncounterintuitive finding suggests that  magnetic thin film s with optimal structural or chemical \ndisorder may be useful for cryogenic spintronic memories  [41,42]  and spin-wave -driven  \nquan tum information systems  [43,44] .   \n \n \n 12 \n Acknowledgements  \nThis research was funded in part by 4 -VA, a collaborative partnership for advancing the  \nCommonwealth of Vir ginia , as well as by the ICTAS Junior Faculty Award . A. Sapkota and C. \nMewes would like to acknowledge support by NSF-CAREER  Award No. 1452670 , and A. \nSrivastava would like to acknowledge support by NASA Award No. CAN80NSSC18M0023 . \nWe thank M. D. Stiles , B. K. Nikolic , and F. Mahfouzi  for helpful discussions on theoretical \nmodels for computing Gilbert damping , as well as R. D. McMichael for his input on the mean -\nfield modeling of interactions in inhomogeneous ferromagnetic films.    \n \n \n1.  B. 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B \n96, 214421 (2017).  \n \n  19 \n  \nFigure 1. (a,b) 2θ -ω X-ray diffraction scans of MAO/Fe and MgO/Fe (a) over a wide angle range \nand (b) near the BCC Fe (002) film peak. (c) Rocking curve scans about the film peak. (d) \nTemperature dependence of resistivity plotted on a log -log scale.  \n \n \nFigure 2. Frequency dependence of FMR linewidth Hpp for MAO/Fe and MgO/Fe at room \ntemperature. Linewidths measured under in -plane field are shown as open symbols, whereas \nthose measured under out -of-plane (OP) field are shown as filled symbols .  \n62 64 66 68log(intensity) [a.u.]\n2q [deg.]\n30 40 50 60 70log(intensity) [a.u.]\n2q [deg.]\n-1.0 -0.5 0.0 0.5 1.0intensity [a.u.]\nw002 [deg.]\n10 10010-810-7r [ m]\nT [K] MgO/Fe\n MAO/Fe\nMAO (004)MgO (002)\nFe (002)MAO/Fe\nMgO /Fe\n( 4)(a) (b) (c) (d)MAO/Fe\nMgO /Fe\n(a)\n0 20 40 60 80 100 120024681012 MAO/Fe (OP)\nMgO/Fe\nMAO/Fem0Hpp [mT]\nf [GHz]20 \n  \nFigure 3. (a,b) Frequency dependence of FMR linewidth for MA O/Fe and MgO/Fe at (a) T = 100 \nK and (b) T = 10 K. (c) Temperature dependence of measured  Gilbert damping parameter meas \nand estimated eddy -current damping parameter eddy. \n \n \n0 50 100 150 200 250 3000246810\n meas MAO/Fe  \n eddy estimate\n meas MgO/Fe\n eddy estimatemeas, eddy [10-3]\nT [K]\n0 10 20 30 40051015\nMAO/Fe\nMgO/Fem0Hpp (mT)\nf (GHz)T = 100 K\n0 10 20 30 40051015m0Hpp (mT)\nf (GHz)T = 10 K(c)(a) (b)21 \n   \nFigure 4. (a,b) Temperature dependence of the spin-orbit -induced Gilbert damping parameter \nso, fit phenomenologically with the experimentally measured resistivity for (a) MAO/Fe and (b) \nMgO/Fe. The dashed and dotted curves indicate the conductivity -like and resistivity -like \ncontributions, respectively; the solid curve represents the fit curve for the total spin -orbit -induced \nGilbert damping parameter. (c,d) Comparison of our experimental results with calculated Gilbert \ndamping parameters by (c) Gilmore et al. [32,33]  and (d) Mankovsky et al. [36]. \n \n \n0 100 200 30002468\nr-liker-likeso [10-3]\nT [K]s-likeMAO/Fe\n0 100 200 30002468\nMgO/Fe\nr-likes-likeso [10-3]\nT [K](a) (b)\n0 100 200 30002468\n MAO/Fe\n MgO/Fe\n calculated [Mankovsky]so [10-3]\nT [K]\n0 50 1000123\nr-like MAO/Fe\n MgO/Fe\n calculated [Gilmore]l [109 s-1]\ne-1 [1012 s-1]s-like0.0 0.5 1.0\n02468\nso [10-3]r [10-7 m]\n(c)\n(d)"    },    {        "title": "2103.05871v3.Anisotropic_superconducting_spin_transport_at_magnetic_interfaces.pdf",        "content": "Anisotropic superconducting spin transport at magnetic interfaces\nYuya Ominato1, Ai Yamakage2, and Mamoru Matsuo1;3;4;5\n1Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China\n2Department of Physics, Nagoya University, Nagoya 464-8602, Japan\n3CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n4Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan and\n5RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n(Dated: October 18, 2022)\nWe present a theoretical investigation of anisotropic superconducting spin transport at a magnetic\ninterface between a p-wave superconductor and a ferromagnetic insulator. Our formulation describes\nthe ferromagnetic resonance modulations due to spin current generation depending on spin-triplet\nCooper pair, including the frequency shift and enhanced Gilbert damping, in a uni\fed manner. We\n\fnd that the Cooper pair symmetry is detectable from the qualitative behavior of the ferromagnetic\nresonance modulation. Our theory paves the way toward anisotropic superconducting spintronics.\nIntroduction.| Use of spin-triplet Cooper pairs as car-\nriers for spin currents in the emergent \feld of super-\nconducting spintronics is challenging1,2. Previous stud-\nies have demonstrated spin transport mediated by spin-\ntriplet Cooper pairs that formed at the s-wave supercon-\nductor (SC)/ferromagnet interfaces of Josephson junc-\ntions. The spin-singlet pairs in SCs are converted into\nspin-triplet pairs in half-metallic CrO 23. However, pre-\nvious studies on spin-triplet pairs at magnetic interfaces\nhave been limited to cases induced by the proximity ef-\nfect.\nOne promising candidate material system for investi-\ngation of spin-triplet currents to enable more active use\nof spin-triplet pairs is the p-wave SC/ferromagnetic in-\nsulator (FI) bilayer thin \flm system4,5. Tunneling of the\nspins is driven by the magnetization dynamics excited\nby ferromagnetic resonance (FMR) in the ferromagnetic\nmaterial via interfacial exchange coupling between the\nmagnetization in the FI and the electron spins in the\np-wave SC, and a spin-triplet current is expected to be\ngenerated. Furthermore, as a backaction of spin injec-\ntion, both the FMR frequency and the Gilbert damping\nof the FI should be modulated6{8. Although similar sce-\nnarios have already been studied vigorously in s-wave\nSC/ferromagnet systems, most previous studies have fo-\ncused on the Gilbert damping modulation due to spin\ninjection9{22. To gain an in-depth understanding of the\nspin-triplet transport mechanisms, the FMR modulation\nprocesses, including both the frequency shift and the en-\nhanced Gilbert damping, should be formulated micro-\nscopically in a systematic manner.\nDetermination of the pairing symmetry of the spin-\ntripletp-wave SCs within the same framework is also\ndesirable. Despite many years of research based on sev-\neral experimental techniques that detect the pairing sym-\nmetry, including nuclear magnetic resonance23, polar-\nized neutron scattering24{26, and muon-spin resonance\ntechniques27, there are few established candidate systems\nfor spin-triplet SCs28{32. The FMR modulation has been\nobserved in various nanoscale magnetic multilayers. Ac-\ncordingly, the technique is widely used to investigate a\n(c) FMR modulation due to the coupling between\n     spin-triplet Cooper pair and magnetization\nH\nH0H0+ δH\n(b) Spin-triplet Cooper pair\n(i) Chiral p-wave (ii) Helical p-wave\nα+ δα\nH\nH0αH0H0+ δHFISC\nFIxz\nY, yθ(a) System\nθZ\nXS-HFISC\nFIG. 1. Mechanism of FMR modulation due to anisotropic\nsuperconducting spin transport at magnetic interfaces. (a)\nPrecession axis located on the x-zplane, where the angle\nbetween the precession axis and the zaxis is\u0012(where 0\u0014\u0012\u0014\n\u0019=2). (b) Two types of spin-triplet Cooper pairs considered\nin this work. (c) FMR signal modulation in the SC/FI bilayer\nsystem compared with the signal in the FI monolayer.\nspin transport property in a variety of nanoscale thin\n\flm systems because it is highly sensitive. Thus one can\nexpect that the FMR measurements in p-wave SC/FI bi-\nlayer systems provide useful information about pairing\nsymmetry.\nIn this Letter, we investigate anisotropic superconduct-\ning spin transport at the magnetic interfaces of hybrid\nsystems composed of p-wave SC/FI thin \flms theoret-\nically, as illustrated in Fig. 1(a). The two-dimensional\nbulk SC is placed on the FI, where the FMR occurs. The\nprecession axis is rotated by an angle \u0012from the direc-\ntion perpendicular to the interface. Here, we use two\ncoordinate systems: ( x;y;z ) and (X;Y;Z ). Thezaxis is\nperpendicular to the interface and the xandyaxes arearXiv:2103.05871v3  [cond-mat.supr-con]  15 Oct 20222\nalong the interface. The ( X;Y;Z ) coordinate is obtained\nby rotating the angle \u0012around the yaxis, so that the\nprecession axis and the Zaxis are parallel. Figure 1(b)\nshows a schematic image of the spin-triplet Cooper pairs\nfor the chiral and helical p-wave SCs considered in this\nwork. Figure 1(c) shows a schematic image of the FMR\nsignal in the FI monolayer and the SC/FI bilayer. The\nFMR frequency and linewidth in the SC/FI bilayer are\nboth modulated because of the spin transfer occurring at\nthe interface.\nUsing the nonequilibrium Green's function method,\nwe formulate the FMR modulations due to the back\naction of the spin-triplet transport process systemati-\ncally. The main advantage of using the nonequilibrium\nGreen's function is dealing with both a spectral function\nand a nonequilibrium distribution function. Indeed, the\ninterface spin current is given by the expression using\nthe nonequilibrium distribution function, which shows\nthat the interface spin current by the spin pumping and\nthe enhanced Gilbert damping are proportional to each\nother. Furthermore, as an advantage of \feld theoretical\ntreatment, the frequency shift and the enhanced Gilbert\ndamping are both described in a uni\fed manner. Addi-\ntionally, it is shown that the symmetry of the spin-triplet\npairs can be extracted from the FMR modulations. The\nresults presented here o\u000ber a pathway toward develop-\nment of anisotropic superconducting spintronics.\nModel Hamiltonian.| The FMR modulation due to the\nSC adjacent to the FI is calculated microscopically using\nthe spin tunneling Hamiltonian method9{11,33{38. The\ne\u000bect of the SC on the FI is treated as a perturbation\nand suppression of ferromagnetism with the onset of su-\nperconductivity is assumed to be negligible, which is con-\nsistent with the results of spin pumping experiments in\nmagnetic multilayer thin \flms. The details of the model\nHamiltonians and the formulations are described in the\nSupplemental Material39. In the main text, we focus on\ngiving an overview of the model Hamiltonians and the\nformulations.\nThe total Hamiltonian H(t) comprises three terms\nH(t) =HFI(t) +HSC+Hex: (1)\nThe \frst term HFI(t) describes the bulk FI,\nHFI(t) =X\nk~!kby\nkbk\u0000h+\nac(t)by\nk=0\u0000h\u0000\nac(t)bk=0;(2)\nwhereby\nkandbkdenote the creation and annihilation op-\nerators of magnons with the wave vector k= (kx;ky;kz),\nrespectively. We assume the parabolic dispersion ~!k=\nDk2\u0000~\rH, where\r(<0) is the electron gyromagnetic ra-\ntio. The coupling between the microwave radiation and\nthe magnons is given by h\u0006\nac(t) = ~\rhacp\nSN=2e\u0007i!t,\nwherehacand!are the amplitude and the frequency of\nthe microwave radiation, respectively. Sis the magni-\ntude of the localized spin and Nis the number of sites\nin the FI. Note that the precession axis for the localized\nspin is \fxed along the Zaxis [see Fig. 1(a)].The second term HSCdescribes the two-dimensional\nbulk SCs,\nHSC=1\n2X\nkcy\nkHBdGck; (3)\nwhere we use the four-component notations\ncy\nk= (cy\nk\";cy\nk#;c\u0000k\";c\u0000k#); (4)\nck= (ck\";ck#;cy\n\u0000k\";cy\n\u0000k#)T: (5)\nHere,cy\nksandcksdenote creation and annihilation op-\nerators, respectively, of electrons with the wave vector\nk= (kx;ky) and thezcomponent of the spin s=\";#.\nThe Bogoliubov-de Gennes Hamiltonian HBdGis a 4\u00024\nmatrix given by\nHBdG=\u0012\n\u0018k\u001b0\u0001k\n\u0000\u0001\u0003\n\u0000k\u0000\u0018k\u001b0\u0013\n; (6)\nwhere\u0018krepresents the energy of the electrons as mea-\nsured from their chemical potential, \u001b0is a 2\u00022 unit\nmatrix, and the pairing potential \u0001 kis also a 2\u00022 ma-\ntrix. We consider three pairing potential types, including\nthe spin-singlet s-wave pairing \u0001 k= \u0001i\u001byand two spin-\ntripletp-wave pairings \u0001 k= (dk\u0001\u001b)i\u001by, where their d\nvectors are given by\ndk=(\n\u0001(0;0;ei\u001ek) : Chiral p\u0000wave\n\u0001(\u0000sin\u001ek;cos\u001ek;0) : Helical p\u0000wave(7)\nwhere\u001ek= arctan(ky=kx) is an azimuth angle. The\nphenomenological form of the gap function is assumed\n\u0001 = 1:76kBTctanh\u0010\n1:74p\nTc=T\u00001\u0011\n; (8)\nwithTcthe superconducting transition temperature. By\ndiagonalizing HBdG, the quasiparticle energy is given by\nEk=p\n\u00182\nk+ \u00012for all SCs considered here. There-\nfore, one cannot distinguish them by the energy spectrum\nalone, and they are simple models suitable for studying\nthe di\u000berence of the magnetic responses due to the pair-\ning symmetry40.\nThe third term Hexrepresents the proximity exchange\ncoupling that occurs at the interface, which describes the\nspin transfer between the SC and the FI10,33,\nHex=X\nq;k\u0000\nJq;k\u001b+\nqS\u0000\nk+ h:c:\u0001\n; (9)\nwhereJq;kis the matrix element for the spin transfer pro-\ncesses,\u001b\u0006\nq= (\u001bX\nq\u0006i\u001bY\nq)=2 represent the spin-\rip opera-\ntors for the electron spins in the SCs, and S\u0000\n\u0000k=p\n2Sby\nk\nandS+\nk=p\n2Sbkrepresent the Fourier component of the\nlocalized spin in the FI. Note that the precession axis is\nalong theZaxis, so that the Zcomponent of the spin\nis injected into the SC when the FMR occurs. Using3\nthe creation and annihilation operators of electrons and\nmagnons,Hexis written as\nHex=X\nq;k;k0;s;s0\u0010p\n2SJq;k\u001b+\nss0cy\nk0sck0+qs0by\n\u0000k+ h:c:\u0011\n:\n(10)\nFrom the above expression, one can see that Hexde-\nscribes electron scattering processes with magnon emis-\nsion and absorption.\nModulation of FMR.| The FMR modulation can be\nread from the retarded component of the magnon Green's\nfunction33, which is given by\nGR\nk(!) =2S=~\n!\u0000!k+i\u000b!\u0000(2S=~)\u0006R\nk(!); (11)\nwhere the Gilbert damping constant \u000bis introduced\nphenomenologically41{43. In the second-order perturba-\ntion calculation with respect to the matrix element Jq;k,\nthe self-energy caused by proximity exchange coupling is\ngiven by\n\u0006R\nk(!) =\u0000X\nqjJq;kj2\u001fR\nq(!); (12)\nwhere the dynamic spin susceptibility of the SCs is de-\n\fned as\n\u001fR\nq(!) :=Z\ndtei(!+i0)ti\n~\u0012(t)h[\u001b+\nq(t);\u001b\u0000\n\u0000q(0)]i:(13)\nThe pole of GR\nk(!) indicates the FMR modulation, i.e.,\nthe shift of resonance frequency and the enhancement of\nthe Gilbert damping. By solving the equation\n!\u0000!k=0\u0000(2S=~)Re\u0006R\nk=0(!) = 0; (14)\nat a \fxed microwave frequency !, one obtains the mag-\nnetic \feld at which the FMR occurs. The imaginary part\nof the self-energy gives the enhancement of the Gilbert\ndamping. Consequently, the frequency shift and the en-\nhanced Gilbert damping are given by\n\u000eH=2S\n\r~Re\u0006R\nk=0(!); \u000e\u000b =\u00002S\n~!Im\u0006R\nk=0(!):(15)\nFrom the above equations and Eq. (12), one can see that\nthe FMR modulation provides information about both\nthe interface coupling properties and the dynamic spin\nsusceptibility of the SCs.\nThe form of matrix element Jq;k=0depends on the\ndetails of the interface. In this work, we assume the\ninterface with uncorrelated roughness. jJq;k=0j2is given\nby\njJq;k=0j2=J2\n1\nN\u000eq;0+J2\n2l2\nNA; (16)\nwhere the \frst and second terms describe averaged uni-\nform contribution and uncorrelated roughness contribu-\ntion, respectively39.J1andJ2correspond to the meanvalue and variance, respectively. Ais the area of the in-\nterface, which is equal to the system size of the SC. lis\nan atomic scale length. Using Eq. (16), the self-energy\nfor the uniform magnon mode is given by\n\u0006R\nk=0(!) =\u0000J2\n1\nN\u001fR\nuni(!)\u0000J2\n2l2\nNA\u001fR\nloc(!); (17)\nwhere the uniform and local spin susceptibilities are de-\n\fned as\n\u001fR\nuni(!) := lim\njqj!0\u001fR\nq(!); \u001fR\nloc(!) :=X\nq\u001fR\nq(!):(18)\nThe self-energy \u0006R\nk=0(!) consists of two terms originating\nfrom the uniform and roughness contributions, so that\nboth\u001fR\nuni(!) and\u001fR\nloc(!) contribute to \u000eHand\u000e\u000b.\nHere, we discuss the FI thickness dependence on the\nFMR modulation44. From Eqs. (15), and (17), one can\nsee that the FMR modulation is inversely proportional\nto the FI thickness ( /A=N ) because\u001fR\nuni(!)/Aand\n\u001fR\nloc(!)/A2. This is consistent with the experiments on\nthe spin pumping in Y 3Fe5O12=Pt heterostructures45. In\norder to observe the FMR modulation experimentally, it\nis necessary to prepare a sample that is su\u000eciently thin,\ne.g., typically, the thickness of several tens of nanometers.\nNumerical results.| In the following, we consider a \rat\ninterface where J2= 0, so that the behavior of the FMR\nmodulation is determined by \u001fR\nuni(!). The roughness\ncontribution proportional to \u001fR\nloc(!) is discussed later.\nFigure 2 shows the frequency shift \u000eHand the enhanced\nGilbert damping \u000e\u000bas a function of temperature and fre-\nquency. Here, we set \u0012= 0 and \u0000=kBTc= 0:05, where \u0000\nis a constant level broadening of the quasiparticle intro-\nduced phenomenologically39.\nFirst, we explain the qualitative properties of \u000eHand\n\u000e\u000bfor the chiral p-wave SC. In the low frequency re-\ngion, where ~!=kBTc\u00141,\u000eHis \fnite and remains al-\nmost independent of !near the zero temperature and\n\u000e\u000bdecreases and becomes exponentially small with the\ndecrease of the temperature. In the high frequency re-\ngion, where ~!=kBTc\u00151, a resonance peak occurs at\n~!= 2\u0001 for both \u000eHand\u000e\u000b. The qualitative proper-\nties of\u000eHand\u000e\u000bfor the helical p-wave SC are the same\nas those of the chiral p-wave SC.\nNext, we explain the qualitative properties of \u000eHand\n\u000e\u000bfor thes-wave SC. In the low frequency region, where\n~!=kBTc\u00141, both\u000eHand\u000e\u000bdecrease and become\nexponentially small with the decrease of the temperature.\nIn the high frequency region, where ~!=kBTc\u00151, both\n\u000eHand\u000e\u000bvanish.\nThep-wave SCs show two characteristic properties\nthat thes-wave SC does not show: a \fnite \u000eHatT= 0\nand a resonance peak of \u000eHand\u000e\u000b. These properties\ncan be understood by the analogy between SCs and band\ninsulators as follows. The uniform dynamic spin suscepti-\nbility consists of contributions from intraband transitions\nwithin particle (hole) bands and interband transitions\nbetween particles and holes. In the low temperature or4\n(a) (b)Γ/kBTc=0.05 Chiral p-wave\n(c) (d)Γ/kBTc=0.05 Helical p-wave\n(e) (f)Γ/kBTc=0.05 s-waveT/T\nc\nhω/kBTcδα/δα1\n0.00.51.0 0\n2\n410\n5\n0\nT/T\nc\nhω/kBTcδH/δH1\n0.00.51.0 0\n2\n410\n5\n0\nT/T\nc\nhω/kBTcδα/δα1\n0.00.51.0 0\n2\n410\n5\n0\nT/T\nc\nhω/kBTcδH/δH1\n0.00.51.0 0\n2\n410\n5\n0\nT/T\nc\nhω/kBTcδH/δH1\n0.00.51.0 0\n2\n42\n1\n0\nT/T\nc\nhω/kBTcδα/δα1\n0.00.51.0 0\n2\n410\n5\n0\nFIG. 2. The frequency shift \u000eHand the enhanced Gilbert\ndamping\u000e\u000bas a function of temperature and frequency nor-\nmalized by the characteristic values \u000eH1=\u0000SJ2\n1DF=(N\r~)\nand\u000e\u000b1=SJ2\n1DF=(NkBTc) in the normal state. DF(/A)\nis the density of states at the Fermi level in the normal state.\nWe set\u0012= 0 and \u0000=kBTc= 0:05. The sign of \u000eHcorresponds\nto the sign of Re \u001fR\nuni(!), which can be positive and negative\nat low and high frequencies, respectively. In contrast, \u000e\u000bis\npositive at any frequency.\nhigh frequency region, the intraband contribution is neg-\nligible and the interband contribution is dominant. In\nthe case of the s-wave SC, the interband transitions are\nforbidden because the Hamiltonian and the spin operator\ncommute. As a result, there is no spin response in the\nlow-temperature or high-frequency regions. In contrast,\nthe Hamiltonian for the p-wave SCs and the spin operator\ndo not commute. Therefore, \u000eHhas a \fnite value near-\nzero temperature due to the interband contribution. In\naddition, a resonance peak occurs when ~!= 2\u0001 because\nthe density of states diverges at the band edge E=\u0006\u0001.\nA detailed proof of the above statement is given in the\nSupplemental Material39.\nThe angle dependences of \u000eHand\u000e\u000bare distinct for\nchiral and helical p-wave SCs, as shown in Fig. 3. In\nboth cases, we set ~!=kBTc= 3:0 as the typical values\nat high frequencies, where the main contribution of the\nuniform spin susceptibility is the interband transitions.\nIn the chiral p-wave SC,\u000eHand\u000e\u000btend to decrease and\nare halved at a \fxed temperature when \u0012increases from\n0 to\u0019=2. Conversely, in the helical p-wave SC, the qual-\n0.4\nT/Tc1.0 0.2 0.08\n4\n0\n0.8 0.6δH/δH1\n2\n-210\n60.4\nT/Tc1.0 0.2 0.08\n4\n0\n0.8 0.6δH/δH1\n2\n-210\n6\nHelical p-waveChiral p-wave\nθ=0\nπ/4\nπ/2\nθ=0π/4π/2\n0.4\nT/Tc1.0 0.2 0.02\n0\n0.8 0.6δα/δα1\n13θ=0\nπ/4\nπ/2\nθ=0π/4π/20.4\nT/Tc1.0 0.2 0.02\n0\n0.8 0.6δα/δα1\n13Γ/kBTc=0.05, hω/kBTc=3.0\n(a) (b)\n(c) (d)Γ/kBTc=0.05, hω/kBTc=3.0FIG. 3. Frequency shift and the enhanced Gilbert damping\nas a function of temperature at angles of \u0012= 0;\u0019=4;\u0019=2. The\nupper and lower panels show the characteristics for the chiral\nand helical p-wave SCs, respectively.\nitative behavior shows the opposite trend. \u000eHand\u000e\u000b\nboth tend to increase and become 1 :5 times larger at a\n\fxed temperature when \u0012increases from 0 to \u0019=2. In\nfact, the angle dependences are approximately obtained\nto be/1 +cos2\u0012and 1 +(sin2\u0012)=2 for chiral and helical\np-wave SCs, respectively39. Therefore, the spin con\fg-\nuration of the Cooper pair can be detected from the \u0012\ndependence data for the FMR modulation.\nThe FMR modulation properties of the three SCs are\nsummarized in Table I. All SCs considered here can be\ndistinguished based on three properties: the frequency\nshift in the low temperature limit, the presence of their\nresonance peak, and their \u0012dependence. For the s-wave\nSC,\u000eHbecomes exponentially small in T!0, while for\nthep-wave SCs, \u000eHis \fnite in T!0. For the s-wave\nSC,\u000eHand\u000e\u000bshow no resonance and no \u0012dependence,\nwhile for the chiral and helical p-wave SCs, both \u000eHand\n\u000e\u000bexhibit a resonance at ~!= 2\u0001 and a \u0012dependence.\nIn addition, these two p-wave SCs can be distinguished\nfrom their\u0012dependences of \u000eHand\u000e\u000b, which are char-\nacterized by @\u0012(\u000eH) and@\u0012(\u000e\u000b), respectively. Here, it\nshould be emphasized that the pairing symmetry can be\ncharacterized by the sign of @\u0012(\u000eH) and@\u0012(\u000e\u000b). These\nproperties are summarized in the Table I.\nSpin-triplet current generation.| The relationship be-\ntween the enhanced Gilbert damping discussed above5\nand the spin-triplet current generation must also be dis-\ncussed. The enhancement of the Gilbert damping is\nknown to originate from the spin current generation at\nthe magnetic interface6,33. The interface spin current in-\nduced by FMRhISiSPis given by39\nhISiSP=N(~\rhac)2\n2\u000b\u0002\n\u0000ImGR\nk=0(!)\u0003\n\u000e\u000b: (19)\nOne can see that hISiSPand\u000e\u000bare proportional to each\nother. In our setup, the enhanced Gilbert damping \u000e\u000b\nwill lead to the generation of both the Cooper pair spin-\ntriplet current and the quasiparticle spin current. Since\nthe angular dependence of \u000e\u000bre\rects the direction of\nthe Cooper pair spins, it is expected that the spin-triplet\ncurrent can be controlled by varying the magnetization\ndirection of the FI.\nDiscussion.| We have considered a \rat SC/FI inter-\nface. In the presence of roughness, the correction term\nproportional to \u001fR\nloc(!) contributes to the FMR mod-\nulation, as shown in Eq. (17). In the rough limit,\nJ2\n1\u001cJ2\n2,\u001fR\nloc(!) dominates to make the FMR modu-\nlation isotropic, due to the angle average by summation\noverq. Namely, the anisotropy peculiar to p-wave SC\nis smeared by the roughness. The detailed behavior of\n\u001fR\nloc(!) is shown in the Supplemental Material39. This\nresult implies that it is crucial to control the interface\nroughness. In principle, the roughness of the interface\ncan be observed using transmission electron microscopy\nof interfaces46{48and it is possible to detect whether the\ninterface of the sample is \rat or rough. More detailed\nspectroscopy can be obtained from the FMR modulation\nby using a \rat interface.\nOur results show that the pairing symmetry can be\ndetected by the sign of @\u0012(\u000eH) and@\u0012(\u000e\u000b) around the\nin-plane magnetic \feld ( \u0012\u0018\u0019=2), where the vortices are\nnegligible. When the external magnetic \feld has a large\nout-of-plane component, the vortex formation may cause\nproblems in observing the angular dependence. The qual-\nitative behavior is expected to change when the out-of-\nplane magnetic \feld approaches the upper critical \feld\n(H\u0018Hc2\u00181T). This is because the coherence length\nof the Cooper pair and the distance between the vor-\ntices can become comparable. Indeed, it has been exper-\nimentally reported that the vortex formation suppresses\nthe characteristic properties in the spin pumping into\nSCs20. Therefore, the out-of-plane magnetic \feld should\nbe as small as possible when FMR measurements are per-\nformed for H\u0018Hc2.\nTABLE I. FMR modulation properties for the \rat SC/FI in-\nterface where J16= 0 andJ2= 0.\nPairing symmetry s Chiral Helical\n\u000eHin the limit of T!0 0 \fnite \fnite\nResonance peak of \u000eH,\u000e\u000b { X X\n@\u0012(\u000eH),@\u0012(\u000e\u000b) 0 negative positiveRecent experiments have reported that UTe 2is a can-\ndidate material for spin-triplet p-wave SCs31, which has\nattracted a great deal of attention. Various experi-\nments, including spectroscopic measurements, are now\nin progress to investigate the pairing symmetry of UTe 2,\nand indicated that the superconducting transition tem-\nperature is about 1K \u001830 GHz. Therefore, the resonance\ncondition ~!= 2\u0001 shown above is accessible to recent\nbroadband FMR measurements.\nIn addition, experiments on spin pumping into d-wave\nSCs have recently been reported49and a theoretical in-\nvestigation of the enhancement of the Gilbert damp-\ning in ad-wave SC/FI bilayer system has recently been\npresented50. Thus anisotropic superconducting spintron-\nics can be expected to develop as a new research direc-\ntion.\nWe should emphasize two important aspects of the\nFMR method presented here: the spectroscopic probe\nmethod for the p-wave SC thin \flms and the versa-\ntile spin injection method. First, the FMR measure-\nment procedure can provide a new spin-sensitive mea-\nsurement method that will complement other measure-\nment methods to enable a breakthrough in the discovery\nof spin-triplet SCs. Second, the FMR method represents\na promising way to generate spin-triplet currents in p-\nwave SC thin \flms.\nConclusions.| We have investigated the anisotropic\nsuperconducting spin transport at magnetic interfaces\ncomposed of a p-wave SC and an FI based on a micro-\nscopic model Hamiltonian. The FMR signal in these p-\nwave SC/FI bilayer systems is modulated via spin trans-\nfer at the interface, which generates spin-triplet currents.\nWe have shown that the pairing symmetry of the SCs\ncan be extracted from the FMR modulation character-\nistics. Our approach provides a unique way to explore\nanisotropic superconducting spintronics, which will be\nuseful for application to emerging device technologies.\nNote added.| After the submission of this manuscript,\nwe became aware of a closely related work, where a way\nto convert spin-triplet currents to magnon spin currents\nin SC/FI bilayer systems is discussed51.\nWe thank R. Ohshima, M. Shiraishi, H. Chudo, G.\nOkano, K. Yamanoi, and Y. Nozaki for helpful discus-\nsions. This work was supported by the Priority Pro-\ngram of the Chinese Academy of Sciences under Grant\nNo. XDB28000000, and by JSPS KAKENHI under\nGrants Nos. JP20K03835, JP20H04635, JP20H01863,\nJP21H04565, and JP21H01800.6\nSUPPLEMENTAL MATERIAL\nI. MODEL HAMILTONIAN\nIn this section, we describe the derivation and details of the model Hamiltonian used in the main text.\nA. Ferromagnetic Heisenberg model\nThe ferromagnetic Heisenberg model with the transverse AC magnetic \feld due to the microwave radiation is given\nby\nHFI(t) =\u0000JX\nhi;jiSi\u0001Sj+~\rHX\njSZ\nj\u0000~\rhacX\nj\u0000\nSX\njcos!t\u0000SY\njsin!t\u0001\n; (S.1)\nwhereJ >0 is the exchange coupling constant, hi;jirepresents summation over all nearest-neighbor sites, Sjis the\nlocalized spin at site jin the ferromagnetic insulator (FI), \r(<0) is the gyromagnetic ratio, His a static magnetic\n\feld,hacis an amplitude of an transverse oscillating magnetic \feld due to the microwave radiation with a frequency\n!. The rotated coordinates ( X;Y;Z ) are shown in Fig. 1(a).\nIt is convenient to introduce the boson creation and annihilation operators in order to formulate the problem in\nterms of the quantum \feld theory. In the current problem, we perturbatively treat the excitation of the FI. In this\ncase, the Holstein-Primako\u000b transformation is useful, where the localized spin can be described using boson creation\nand annihilation operators bj;by\njin Hilbert space constrained to 2 S+ 1 dimensions. The spin operators are written as\nS+\nj=SX\nj+iSY\nj=\u0010\n2S\u0000by\njbj\u00111=2\nbj; (S.2)\nS\u0000\nj=SX\nj\u0000iSY\nj=by\nj\u0010\n2S\u0000by\njbj\u00111=2\n; (S.3)\nSZ\nj=S\u0000by\njbj; (S.4)\nwhere we require [ bi;by\nj] =\u000ei;j;in order that the S+\nj,S\u0000\nj, andSZ\njsatisfy the commutation relation of angular\nmomentum. The deviation of SZ\njfrom its ground-state value Sis quanti\fed by the boson particle number.\nWe consider low-energy excitation in the FI, where the deviation of SZ\njfrom the ground state is small hby\njbji=S\u001c1.\nThe ladder operators S\u0006\njare approximated as\nS+\nj\u0019(2S)1=2bj; (S.5)\nS\u0000\nj\u0019(2S)1=2by\nj; (S.6)\nwhich is called spin-wave approximation. Here, we de\fne the magnon operators\nbk=1p\nNX\nje\u0000ik\u0001rjbj; (S.7)\nby\nk=1p\nNX\njeik\u0001rjby\nj; (S.8)\nwhereNis the number of sites and k= (kx;ky;kz). The inverse transformation is then given by\nbj=1p\nNX\nkeik\u0001rjbk; (S.9)\nby\nj=1p\nNX\nke\u0000ik\u0001rjby\nk: (S.10)\nThe magnon operators satisfy [ bk;by\nk0] =\u000ek;k0and describe the quantized collective excitations. Using the spin-wave\napproximation and the magnon operators, the Hamiltonian HFI(t) is written as\nHFI(t)\u0019X\nk~!kby\nkbk\u0000h+\nac(t)by\nk=0\u0000h\u0000\nac(t)bk=0; (S.11)7\nwhere ~!k=Dk2\u0000~\rHwithD= 2JSa2and the lattice constant a,h\u0006\nac(t) =~\rhacp\nSN=2e\u0007i!t, and constant\nterms are omitted.\nB. BCS Hamiltonian\nWe derive a mean-\feld Hamiltonian, which describes a bulk superconductor (SC), and we diagonalize the mean-\feld\nHamiltonian with the Bogoliubov transformation. At the end of this section, the spin density operators of the SC are\nwritten in terms of the Bogoliubov quasiparticle creation and annihilation operators.\nWe start with the e\u000bective Hamiltonian in momentum space\nHSC=X\nk;s\u0018kcy\nkscks+1\n2X\nk;k0;s1;s2;s3;s4Vs1;s2;s3;s4(k;k0)cy\n\u0000ks1cy\nks2ck0s3c\u0000k0s4; (S.12)\nwhere\u0018kis the band energy measured relative to the chemical potential, and cy\nksandcksare the creation and\nannihilation operators of electrons with the wave vector k= (kx;ky) and thezcomponent of the spin s=\";#. The\nmatrix elements satisfy\nVs1;s2;s3;s4(k;k0) =\u0000Vs2;s1;s3;s4(\u0000k;k0); (S.13)\nVs1;s2;s3;s4(k;k0) =\u0000Vs1;s2;s4;s3(k;\u0000k0); (S.14)\nbecause of the anticommutation relation of fermions, and\nVs1;s2;s3;s4(k;k0) =V\u0003\ns4;s3;s2;s1(k0;k); (S.15)\nbecause of the Hermitianity of the Hamiltonian. We consider a mean-\feld, which is called a pair potential\n\u0001k;ss0=\u0000X\nk0;s3;s4Vs0;s;s 3;s4(k;k0)hck0s3c\u0000k0s4i; (S.16)\nand its conjugate\n\u0001\u0003\n\u0000k;ss0=X\nk0;s1;s2Vs1;s2;s0;s(k0;k)hcy\n\u0000k0s1cy\nk0s2i: (S.17)\nHere, we consider a mean-\feld approximation where the interaction term is replaced as follows\ncy\n\u0000ks1cy\nks2ck0s3c\u0000k0s4!cy\n\u0000ks1cy\nks2hck0s3c\u0000k0s4i+hcy\n\u0000ks1cy\nks2ick0s3c\u0000k0s4\u0000hcy\n\u0000ks1cy\nks2ihck0s3c\u0000k0s4i; (S.18)\nso that the interaction term is rewritten as\nX\nk;k0;s1;s2;s3;s4Vs1;s2;s3;s4(k;k0)cy\n\u0000ks1cy\nks2ck0s3c\u0000k0s4!X\nk;s1;s2h\n\u0001k;s1s2cy\nks1cy\n\u0000ks2\u0000\u0001\u0003\n\u0000k;s1s2c\u0000ks1cks2i\n; (S.19)\nwhere an constant term is omitted. Consequently, we derive a mean-\feld Hamiltonian\nHSC=X\nk;s\u0018kcy\nkscks+1\n2X\nk;s1;s2\u0002\n\u0001k;s1s2cy\nks1cy\n\u0000ks2\u0000\u0001\u0003\n\u0000k;s1s2c\u0000ks1cks2\u0003\n: (S.20)\nUsing a four-component notation\ncy\nk= (cy\nk\";cy\nk#;c\u0000k\";c\u0000k#); (S.21)\nck= (ck\";ck#;cy\n\u0000k\";cy\n\u0000k#)T; (S.22)\nthe mean-\feld Hamiltonian is written as\nHSC=1\n2X\nkcy\nkHBdGck: (S.23)8\nHBdGis the 4\u00024 matrix\nHBdG= \n\u0018k\u001b0\u0001k\n\u0000\u0001\u0003\n\u0000k\u0000\u0018k\u001b0!\n; (S.24)\nwhere\u001b0is the 2\u00022 unit matrix and \u0001 kis the 2\u00022 matrix given as\n\u0001k= \n\u0001k;\"\"\u0001k;\"#\n\u0001k;#\"\u0001k;##!\n: (S.25)\nIn principle, the pair potential is obtained by solving the gap equation self-consistently for an explicit form of the\nmatrix elements Vs1;s2;s3;s4(k;k0). In this work, we do not solve the gap equation, but instead assume an explicit\nform of the pair potential and perform calculations using a phenomenological gap function. For the singlet pairing,\nthe pair potential is given by\n\u0001k= ki\u001by; (S.26)\nwith an even function  k= \u0000k. For ans-wave SC, the pair potential is given by\n\u0001k= \u0001 \n0 1\n\u00001 0!\n: (S.27)\nFor the triplet pairing, the pair potential is given by\n\u0001k= [dk\u0001\u001b]i\u001by; (S.28)\nwith an odd vectorial function dk=\u0000d\u0000k. For a chiral p-wave SC and a helical p-wave SC,dkis given by\ndk=(\n\u0001(0;0;ei\u001ek) : chiral p\u0000wave\n\u0001(\u0000sin\u001ek;cos\u001ek;0) : helical p\u0000wave(S.29)\nwith\u001ek= arctan(ky=kx), so that the pair potential is given by\n\u0001k=8\n>>>><\n>>>>:\u0001 \n0ei\u001ek\nei\u001ek0!\n: chiralp\u0000wave\n\u0001 \nie\u0000i\u001ek0\n0iei\u001ek!\n: helicalp\u0000wave(S.30)\nThe phenomenological gap function is given by\n\u0001 = 1:76kBTctanh\u0010\n1:74p\nTc=T\u00001\u0011\n: (S.31)\nThe Bogoliubov transformation to diagonalize HBdGis given by\nUk= \nukvk\nv\u0003\n\u0000ku\u0003\n\u0000k!\n; (S.32)\nUy\nk= \nuk\u0000vk\n\u0000v\u0003\n\u0000ku\u0003\n\u0000k!\n; (S.33)\nwith the 2\u00022 matricesukandvkgiven by\nuk=s\n1\n2\u0012\n1 +\u0018k\nEk\u0013\n\u001b0; (S.34)\nvk=\u0000s\n1\n2\u0012\n1\u0000\u0018k\nEk\u0013\u0001k\n\u0001; (S.35)9\nwhereEkis the eigenenergy\nEk=q\n\u00182\nk+ \u00012: (S.36)\nUsing the Bogoliubov transformation Uk, the 4\u00024 matrixHBdGis diagonalized as\nUy\nkHBdGUk=0\nBBB@Ek0 0 0\n0Ek0 0\n0 0\u0000Ek0\n0 0 0\u0000Ek1\nCCCA: (S.37)\nThe excitation of HSCis described by the creation and annihilation operators of the Bogoliubov quasiparticles \r(y)\nk\n\ry\nk= (\ry\nk\";\ry\nk#;\r\u0000k\";\r\u0000k#); (S.38)\n\rk= (\rk\";\rk#;\ry\n\u0000k\";\ry\n\u0000k#)T; (S.39)\nwhere they are obtained by the Bogoliubov transformation\n\rk=Uy\nkck; (S.40)\n\ry\nk=cy\nkUk: (S.41)\nThe spin density operators \u001ba(r) (a=x;y;z ) is de\fned as\n\u001ba(r) :=1\nAX\nk;k0;s;s0e\u0000i(k\u0000k0)\u0001r\u001ba\nss0cy\nksck0s0; (S.42)\nwhereAis the area of the system. \u001ba(r) (a=x;y;z ) is expanded in Fourier series\n\u001ba(r) =1\nAX\nqeiq\u0001r\u001ba\nq; (S.43)\nand the Fourier coe\u000ecient is given by\n\u001ba\nq=Z\ndre\u0000iq\u0001r\u001ba(r) =X\nk;s;s0\u001ba\nss0cy\nksck+qs0: (S.44)\nUsing the Bogoliubov transformation Uk, the above expression is rewritten as\n\u001ba\nq=X\nk;s;s0\"\u0010\nsa(1)\nk;k+q\u0011\ns;s0\ry\nks\rk+qs0+\u0010\nsa(2)\nk;k+q\u0011\ns;s0\r\u0000ks\ry\n\u0000k\u0000qs0+\u0010\nsa(3)\nk;k+q\u0011\ns;s0\ry\nks\ry\n\u0000k\u0000qs0+\u0010\nsa(4)\nk;k+q\u0011\ns;s0\r\u0000ks\rk+qs0#\n;\n(S.45)\nwith the 2\u00022 matricessa(i)\nk;k+qgiven by\nsa(1)\nk;k+q=uy\nk\u001bauk+q; (S.46)\nsa(2)\nk;k+q=vy\nk\u001bavk+q; (S.47)\nsa(3)\nk;k+q=uy\nk\u001bavk+q; (S.48)\nsa(4)\nk;k+q=vy\nk\u001bauk+q: (S.49)\nThe \frst and second terms describe the intraband transition from particle-to-particle and from hole-to-hole, respec-\ntively. The third and fourth terms describe the interband transition from hole-to-particle and from particle-to-hole,\nrespectively.10\nC. Proximity exchange coupling at interface\nWe start with a model for the proximity exchange coupling given by\nHex=Z\ndrX\njJ(r;rj)\u001b(r)\u0001Sj: (S.50)\nWe rewrite the above expression in the real space into the expression in the wave space. The proximity exchange\ncoupling is rewritten as\nHex=Z\ndrX\njJ(r;rj)1\nAp\nNX\nq;kei(q\u0001r+k\u0001rj)\u0000\n\u001b+\nqS\u0000\nk+\u001b\u0000\nqS+\nk\u0001\n+Z\ndrX\njJ(r;rj)\u001bZ(r)SZ\nj; (S.51)\nwhere the Fourier series are given by\n\u001b(r) =1\nAX\nqeiq\u0001r\u001bq; (S.52)\nSj=1p\nNX\nkeik\u0001rjSk; (S.53)\nwith the area of the SC, A, and the number of sites in the FI, N, and the ladder operators are given by\n\u001b\u0006=1\n2(\u001bX\u0006i\u001bY); (S.54)\nS\u0006=SX\u0006iSY: (S.55)\nThe matrix element is given by\nJq;k=1\nAp\nNZ\ndrX\njJ(r;rj)ei(q\u0001r+k\u0001rj): (S.56)\nConsequently, the exchange coupling which we use in the main text is derived as\nHex=X\nq;k\u0000\nJq;k\u001b+\nqS\u0000\nk+J\u0003\nq;k\u001b\u0000\n\u0000qS+\n\u0000k\u0001\n; (S.57)\nwhere we use a relation J\u0000q;\u0000k=J\u0003\nq;k, and we omit the last term\nZ\ndrX\njJ(r;rj)\u001bZ(r)SZ\nj; (S.58)\nin order to focus on the spin transfer at the interface. For the uniform magnon mode jkj= 0, the matrix element is\ngiven by\nJq;k=0=1\nAp\nNZ\ndrX\njJ(r;rj)eiq\u0001r: (S.59)\nII. TIME DEPENDENT QUANTUM AVERAGE\nIn this section, we show that the ferromagnetic resonance (FMR) frequency and linewidth are read from the\nmagnon Green's function. We consider the Hamiltonian H(t) composed of the unperturbed Hamiltonian H0and the\nperturbation V(t)\nH(t) =H0+V(t): (S.60)\nThe time-dependent quantum average of a physical quantity Ois calculated as\nhO(t)i=hSy(t;\u00001)~O(t)S(t;\u00001)i; (S.61)11\nwhere ~O(t) is the interaction picture and the S matrix S(t;t0) is given by\nS(t;t0) =Texp Zt\nt0dt0~V(t0)\ni~!\n: (S.62)\nThe time-dependent quantum average hO(t)iis written as\nhO(t)i=hOieq+\u000ehO(t)i; (S.63)\nwherehOieq= Tr (\u001aeqO) is the equilibrium value and \u000ehO(t)iis deviation from the equilibrium. When the perturbation\nis written as V(t) =\u0000AF(t), the \frst order perturbation calculation gives\n\u000ehO(t)i=\u0000Zt\n\u00001dt01\ni~h[~O(t);~A(t0)]iF(t0)\n=\u0000Z1\n\u00001dt0GR(t0)F(t\u0000t0); (S.64)\nwhere we de\fne the retarded Green's function\nGR(t) =1\ni~\u0012(t)h[~O(t);~A(0)]i: (S.65)\nWhen the external force is written as F(t) =Fe\u0000i(!+i0)t,\u000ehO(t)iis written as\n\u000ehO(t)i=\u0000Fe\u0000i(!+i0)tZ1\n\u00001dt0ei(!+i0)t0GR(t0)\n=\u0000Fe\u0000i!tGR(!): (S.66)\nUsing the above formula, the dynamics of \u000ehS+\nk=0(t)iis written as\n\u000ehS+\nk=0(t)i=\u0000~\rhacp\nN\n2e\u0000i!tGR\nk=0(!); (S.67)\nwhereGR\nk(!) is the Fourier transform of the retarded component of the magnon Green's function GR\nk(t). They are\nde\fned as\nGR\nk(t) :=1\ni~\u0012(t)h[S+\nk(t);S\u0000\n\u0000k(0)]i; (S.68)\nGR\nk(!) :=Z1\n\u00001dt0ei(!+i0)t0GR\nk(t0): (S.69)\nFrom Eq. (S.67), one can see that the FMR frequency and linewidth are read from GR\nk(!).\nIII. MAGNON GREEN'S FUNCTION\nIn this section, we perform perturbative calculation for the magnon Green's function. We treat the proximity\nexchange coupling as a perturbation. The Hamiltonian is written as\nH=H0+V; (S.70)\nwhereH0is the unperturbed Hamiltonian\nH0=X\nk~!kby\nkbk+X\nk;sEk\ry\nks\rks; (S.71)\nandVis the perturbation\nV=X\nq;k\u0000\nJq;k\u001b+\nqS\u0000\nk+ h:c:\u0001\n: (S.72)12\n(e) Vertex(b) Keldysh contour\n(d) Self-energytime(a) Magnon Green’s function (c) Dyson equation\nk/uni2032 +qs/uni2032 −k\n−k/uni2032 −qs/uni2032 /uni03C3+\nqS−\nk= + + +−k −k −k−k/uni2032 s k/uni2032 s k/uni2032 s\n−k/uni2032 −qs/uni2032 k/uni2032 +qs/uni2032 −k/uni2032 sGk(/uni03C4,/uni03C4/uni2032 )= = + /uni03A3\n/uni03A3 = + + +\nk/uni2032 +qs/uni2032 −k/uni2032 −qs/uni2032 −k/uni2032 s k/uni2032 s k/uni2032 s\n−k/uni2032 −qs/uni2032 k/uni2032 +qs/uni2032 −k/uni2032 s\nBogoliubov quasiparticle: Magnon:\nFIG. 4. (a) The Feynman diagram for the magnon Green's function. (b) Keldysh contour to perform perturbative calculations.\n(c) The Feynman diagram for the Dyson equation. (d) The self-energy within the second-order perturbation is given by the\ndynamic spin susceptibility of the SCs. (e) The Feynman diagrams for the vertex \u001b+\nqS\u0000\nk, which represent scattering of a\nBogoliubov quasiparticle with magnon emission. The solid and wavy lines represent a Bogoliubov quasiparticle and a magnon,\nrespectively.\nWe de\fne the magnon Green's function\nGk(\u001c;\u001c0) :=1\ni~hTCS+\nk(\u001c)S\u0000\n\u0000k(\u001c0)i; (S.73)\nwhereTCis the time-ordering operator on the Keldysh contour (see Figs. 4(a) and (b)). To perform the perturbative\ncalculation, we introduce interaction picture. The perturbation is written as\n~V(t) =X\nq;k\u0010\nJq;k~\u001b+\nq(t)~S\u0000\nk(t) + h:c:\u0011\n: (S.74)\nThe magnon Green's function is given by\nGk(\u001c;\u001c0) =1\ni~hTCSC~S+\nk(\u001c)~S\u0000\n\u0000k(\u001c0)iconn; (S.75)\nwhereh\u0001\u0001\u0001i connmeans the connected diagrams and the S matrix is given by\nSC=TCexp Z\nCd\u001c~V(\u001c)\ni~!\n: (S.76)\nThe above expressions lead to the Dyson equation (see Fig. 4(c))\nGk(\u001c;\u001c0) =G(0)\nk(\u001c;\u001c0) +Z\nCd\u001c1Z\nCd\u001c2G(0)\nk(\u001c;\u001c1)\u0006k(\u001c1;\u001c2)Gk(\u001c2;\u001c0); (S.77)\nwhereG(0)\nk(\u001c;\u001c0) is the unperturbed magnon Green's function\nG(0)\nk(\u001c;\u001c0) =1\ni~hTC~S+\nk(\u001c)~S\u0000\n\u0000k(\u001c0)i; (S.78)\nand \u0006 k(\u001c1;\u001c2) is the self-energy. Within the second-order perturbation, the self-energy is given by (see Fig. 4(d))\n\u0006k(\u001c;\u001c0) =1\ni~X\nqjJq;kj2hTC~\u001b+\nq(\u001c)~\u001b\u0000\n\u0000q(\u001c0)i: (S.79)\nThe Feynman diagram for the vertex is shown in Fig. 4(e). Substituting the ladder operators expressed in terms of13\n\r(y)\nks, the self-energy is written as\n\u0006k(\u001c;\u001c0) =\u0000i~X\nqjJq;kj2X\nk0;s;s0\"\u0012\f\f\f(s+(1)\nk0;k0+q)s;s0\f\f\f2\n\u0000(s+(1)\nk0;k0+q)s;s0(s\u0000(2)\n\u0000k0\u0000q;\u0000k0)\u0003\ns0;s\u0013\ngk0;s(\u001c0;\u001c)gk0+q;s0(\u001c;\u001c0)\n+\u0012\f\f\f(s+(2)\nk0;k0+q)s;s0\f\f\f2\n\u0000(s+(2)\nk0;k0+q)s;s0(s\u0000(1)\n\u0000k0\u0000q;\u0000k0)\u0003\ns0;s\u0013\ng\u0000k0;s(\u001c;\u001c0)g\u0000k0\u0000q;s0(\u001c0;\u001c)\n\u0000\u0012\f\f\f(s+(3)\nk0;k0+q)s;s0\f\f\f2\n\u0000(s+(3)\nk0;k0+q)s;s0(s\u0000(3)\n\u0000k0\u0000q;\u0000k0)\u0003\ns0;s\u0013\ngk0;s(\u001c0;\u001c)g\u0000k0\u0000q;s0(\u001c0;\u001c)\n\u0000\u0012\f\f\f(s+(4)\nk0;k0+q)s;s0\f\f\f2\n\u0000(s+(4)\nk0;k0+q)s;s0(s\u0000(4)\n\u0000k0\u0000q;\u0000k0)\u0003\ns0;s\u0013\ng\u0000k0;s(\u001c;\u001c0)gk0+q;s0(\u001c;\u001c0)#\n;\n(S.80)\nwhere the quasiparticle Green's function is de\fned as\ngk;s(\u001c;\u001c0) :=1\ni~hTC~\rks(\u001c)~\ry\nks(\u001c0)i: (S.81)\nThe \frst and second terms give the intraband contribution, and the third and fourth terms give the interband\ncontribution. Evaluating the Dyson equation, the retarded component of the magnon Green's function is given by\nGR\nk(!) =1\nh\nG(0)R\nk(!)i\u00001\n\u0000\u0006R\nk(!); (S.82)\nwhere the unperturbed Green's function is written as\nG(0)R\nk(!) =2S=~\n!\u0000!k+i\u000b!: (S.83)\nHere, we introduce the phenomenological dimensionless damping parameter \u000b. Using Eq.(S.83), the retarded Green's\nfunction is written as\nGR\nk(!) =2S=~\n!\u0000!k+i\u000b!\u0000(2S=~)\u0006R\nk(!): (S.84)\nFrom the above expression, the frequency shift at a \fxed !is given by\n\u000eH=2S\n\r~Re\u0006R\nk(!); (S.85)\nand the enhanced Gilbert damping is given by\n\u000e\u000b=\u00002S\n~!Im\u0006R\nk(!): (S.86)\nThe Fourier transform of the self-energy is given as\n\u0006R\nk(!) =Z\ndtei(!+i0)t\u0006R\nk(t) =\u0000X\nqjJq;kj2\u001fR\nq(!); (S.87)\nwhere the dynamic spin susceptibility of the SC is de\fned as\n\u001fR\nq(!) :=Z\ndtei(!+i0)ti\n~\u0012(t)h[~\u001b+\nq(t);~\u001b\u0000\n\u0000q(0)]i: (S.88)\nEvaluating the self-energy Eq. (S.87), one can obtain the information of the FMR modulation, \u000eHand\u000e\u000b. Using the\nsystem's symmetry, the dynamic spin susceptibility \u001fR\nq(!) can be written as\n\u001fR\nq(!) = cos2\u0012\u001fxx\nq(!) +\u001fyy\nq(!) + sin2\u0012\u001fzz\nq(!); (S.89)\nwhich means that both \u000eHand\u000e\u000bshow a dependence on \u0012when the dynamic spin susceptibility is anisotropic.14\nIV. SPIN CURRENT AT THE INTERFACE\nIn this section, we derive the general expression of spin current at the interface. We treat the tunneling Hamiltonian\nas a perturbation and the other terms as the unperturbed Hamiltonian\nH(t) =H0(t) +Hex; (S.90)\nH0(t) =HFI(t) +HSC: (S.91)\nThe operator of spin current \rowing from the SC to the FI at the interface is de\fned by\nIS:=\u0000~\n2_\u001bZ\ntot=\u0000~\n21\ni~[\u001bZ\ntot;Hex] =i\n2[\u001bZ\ntot;Hex]; (S.92)\nwhere\u001bZ\ntotis given by\n\u001bZ\ntot=Z\ndr\u001bZ(r): (S.93)\nCalculating the commutation relation, we obtain the following expression\nIS=iX\nq;k\u0000\nJq;k\u001b+\nqS\u0000\nk\u0000h:c:\u0001\n: (S.94)\nThe time-dependent quantum average of ISis written as\nhIS(t)i= Re2\n42iX\nq;kJq;kh\u001b+\nq(t)S\u0000\nk(t)i3\n5; (S.95)\nwhereh\u0001\u0001\u0001i = Tr[\u001a0\u0001\u0001\u0001] denotes the statistical average with an initial density matrix \u001a0. In order to develop the\nperturbation expansion, we introduce the interaction picture\nhIS(\u001c1;\u001c2)i= Re2\n42iX\nq;kJq;khTCSC~\u001b+\nq(\u001c1)~S\u0000\nk(\u001c2)i3\n5: (S.96)\nSCand ~O(t) are given by\nSC=TCexp Z\nCd\u001c~Hex(\u001c)\ni~!\n; (S.97)\nand\n~O(t) =Uy\n0(t;t0)OU0(t;t0); (S.98)\nwhere\nU0(t;t0) =Texp\u0012Zt\nt0dt0H0(t0)\ni~\u0013\n: (S.99)\nExpandingSCas\nSC\u00191 +Z\nCd\u001cTC~Hex(\u001c)\ni~; (S.100)\nthe spin current is given by\nhIS(\u001c1;\u001c2)i=X\nq;kjJq;kj2Re\"\n2\n~Z\nCd\u001chTC~\u001b+\nq(\u001c1)~\u001b\u0000\n\u0000q(\u001c)ihTC~S+\n\u0000k(\u001c)~S\u0000\nk(\u001c2)i#\n: (S.101)15\nUsing the contour ordered Green's functions\n\u001fq(\u001c1;\u001c) =\u00001\ni~hTC~\u001b+\nq(\u001c1)~\u001b\u0000\n\u0000q(\u001c)i; (S.102)\nGk(\u001c;\u001c2) =1\ni~hTC~S+\nk(\u001c)~S\u0000\n\u0000k(\u001c2)i; (S.103)\nthe above equation is rewritten as\nhIS(\u001c1;\u001c2)i=X\nq;kjJq;kj2Re\"\n2~Z\nCd\u001c\u001fq(\u001c1;\u001c)G\u0000k(\u001c;\u001c2)#\n: (S.104)\nWe put\u001c2on the forward contour and \u001c1on the backward contour to describe spin transfer at the interface in\nappropriate time order. Assuming a steady state, the spin current is written as\nhISi= 2~X\nq;kjJq;kj2Re\"Z1\n\u00001d!0\n2\u0019\u0010\n\u001fR\nq(!0)G<\n\u0000k(!0) +\u001f<\nq(!0)GA\n\u0000k(!0)\u0011#\n: (S.105)\nWe introduce the distribution functions as\n\u001f<\nq(!) =fSC\nq(!)\u0002\n2iIm\u001fR\nq(!)\u0003\n; (S.106)\nG<\nk(!) =fFI\nk(!)\u0002\n2iImGR\nk(!)\u0003\n: (S.107)\nThe formula of the spin current at the interface is derived as\nhISi= 4~X\nq;kjJq;kj2Z1\n\u00001d!0\n2\u0019Im\u001fR\nq(!0)\u0002\n\u0000ImGR\n\u0000k(!0)\u0003\u0002\nfFI\n\u0000k(!0)\u0000fSC\nq(!0)\u0003\n: (S.108)\nWhen both the SC and the FI are in equilibrium, the di\u000berence of the distribution functions is zero (i.e. fFI\n\u0000k(!0)\u0000\nfSC\nq(!0) = 0), so that no spin current is generated. Under the microwave irradiation, the distribution function of\nthe FI deviates from equilibrium, which generates the interface spin current. Performing a second-order perturbation\ncalculation, the deviation of the distribution function of the FI, \u000efFI\n\u0000k(!0), is given by\n\u000efFI\n\u0000k(!0) =2\u0019NS (\rhac=2)2\n\u000b!0\u000ek;0\u000e(!0\u0000!): (S.109)\nConsequently, the interface spin current is written as\nhISiSP= 4~X\nq;kjJq;kj2Z1\n\u00001d!0\n2\u0019Im\u001fR\nq(!0)\u0002\n\u0000ImGR\n\u0000k(!0)\u0003\n\u000efFI\n\u0000k(!0): (S.110)\nFinally, one can show that the spin current is proportional to the enhanced Gilbert damping\nhISiSP= 4~NS(\rhac=2)2\n\u000b!\u0002\n\u0000ImGR\nk=0(!)\u0003X\nqjJq;k=0j2Im\u001fR\nq(!);\n=N(~\rhac)2\n2\u000b\u0002\n\u0000ImGR\nk=0(!)\u0003\n\u000e\u000b: (S.111)\nV. MODEL FOR INTERFACE CONFIGURATIONS\nIn order to calculate Eq. (S.87), one needs to set up an explicit expression for jJq;k=0j2. We consider an interface\nwith uncorrelated roughness. To model this interface, we assume that J(r;rj) satis\fes\nDX\njJ(r;rj)E\nave=J1; (S.112)\nDX\nj;j0J(r;rj)J(r0;rj0)E\nave=J2\n1+J2\n2l2\u000e(r\u0000r0); (S.113)16\nwhereh\u0001\u0001\u0001i avemeans interface con\fguration average. The spatially averaged J(r;rj) is given by a constant J1as\nshown in Eq. (S.112). Equation (S.113) means that the interface roughness is uncorrelated and J2\n2l2is a variance.\nJ1andJ2are coupling constants with dimension of energy, and are independent of the system size. lis introduced\nbecause the Hamiltonian of the SCs is treated as a continuum model. Performing the interface con\fguration average,\nand using Eq. (S.112) and (S.113), one can obtain the expression for jJq;k=0j2in the main text.\nVI. DYNAMIC SPIN SUSCEPTIBILITY OF SC\nEvaluating the retarded component of the self-energy Eq. (S.80), the dynamic spin susceptibility of the SC is given\nby\n\u001fR\nq(!) =\u0000Z1\n\u00001dEf(E)X\n\u0015;k(\nM\u0015;\u0015(a)\nk;k+q\u0014\n\u00001\n\u0019ImgR\n\u0015;k(E)gR\n\u0015;k+q(E+~!)\u00001\n\u0019ImgR\n\u0015;k+q(E)gA\n\u0015;k(E\u0000~!)\u0015\n+M\u0015;\u0000\u0015(a)\nk;k+q\u0014\n\u00001\n\u0019ImgR\n\u0015;k(E)gR\n\u0000\u0015;k+q(E+~!)\u00001\n\u0019ImgR\n\u0000\u0015;k+q(E)gA\n\u0015;k(E\u0000~!)\u0015)\n;\n(S.114)\nwhereM\u0015;\u00150(a)\nk;k+qwitha=s;c;andhare given by\nM\u0015;\u00150(s)\nk;k+q=(\u0018k+\u0015Ek)(\u0018k+q+\u00150Ek+q)\n4\u0015Ek\u00150Ek+q+\u00012\n4\u0015Ek\u00150Ek+q; (S.115)\nM\u0015;\u00150(c)\nk;k+q=(\u0018k+\u0015Ek)(\u0018k+q+\u00150Ek+q)\n4\u0015Ek\u00150Ek+q\u0000\u00012e\u0000i(\u001ek\u0000\u001ek+q)\n4\u0015Ek\u00150Ek+qcos2\u0012; (S.116)\nM\u0015;\u00150(h)\nk;k+q=(\u0018k+\u0015Ek)(\u0018k+q+\u00150Ek+q)\n4\u0015Ek\u00150Ek+q\u0000\u00012sin\u001eksin\u001ek+q\n4\u0015Ek\u00150Ek+qsin2\u0012: (S.117)\n\u0015;\u00150=\u0006give a sign, and a=s;c;andhcorrespond to matrix elements for s-wave, chiral p-wave, and helical p-wave\nSCs, respectively. In Eq. (S.114), the terms multiplied by M\u0015;\u0015(a)\nk;k+qdescribe the intraband transition processes, i.e.,\ntransition processes from particle to particle and from hole to hole, and the terms multiplied by M\u0015;\u0000\u0015(a)\nk;k+qdescribe\nthe interband transition processes, i.e., transition processes from particle to hole and vice versa. The retarded and\nadvanced Green's functions of the quasiparticles gR=A\n\u0015;k(E) are given by\ngR\n\u0015;k(E) =1\nE\u0000\u0015Ek+i\u0000; (S.118)\ngA\n\u0015;k(E) =1\nE\u0000\u0015Ek\u0000i\u0000; (S.119)\nwhere \u0000 is a constant level broadening introduced phenomenologically. \u0000 is introduced to incorporate the intraband\ncontribution in the calculation of the uniform spin susceptibility. The details are explained in the next section.\nThe sum over kis replaced by the integral near the Fermi energy\nX\nkF(k)!DFZ1\n0dEDs(E)X\n\u0011=\u0006F\u0011(E); (S.120)\nX\nkF(k) sin2\u001ek!DFZ1\n0dEDs(E)X\n\u0011=\u00061\n2F\u0011(E); (S.121)\nwhereDFis the density of states near the Fermi energy in the normal state and Ds(E) is the density of states of\nquasiparticles\nDs(E) =jEjp\nE2\u0000\u00012\u0012(jEj\u0000\u0001): (S.122)\nF\u0011(E) means to assign \u0011p\nE2\u0000\u00012to\u0018contained in F(k).17\nVII. UNIFORM SPIN SUSCEPTIBILITY\nIn this section, we explain three properties related to the calculation of the uniform spin susceptibility. First, the\nmatrix element's properties are explained, which is essential to understand the qualitative di\u000berence between spin-\nsinglets-wave and spin-triplet p-wave SCs. Second, the reason to introduce the constant level broadening \u0000. Third,\nthe analytical expression for the uniform spin susceptibility of the p-wave SCs is given.\nPerforming the angular integral and replacing the sum over kby theEintegral, the matrix elements are replaced\nby\nM\u0015;\u00150(s)\nk;k!1 +\u0015\u00150\n4\u0015\u00150; (S.123)\nM\u0015;\u00150(c)\nk;k!(1 +\u0015\u00150)E2\u0000(1 + cos2\u0012)\u00012\n4\u0015\u00150E2; (S.124)\nM\u0015;\u00150(h)\nk;k!(1 +\u0015\u00150)E2\u0000(1 +1\n2sin2\u0012)\u00012\n4\u0015\u00150E2: (S.125)\nHere, the \frst-order terms in \u0018kare omitted because they vanish in the Eintegral. From the above expressions, the\nintraband matrix elements become \fnite for all SCs considered here, while the interband matrix elements vanish in\nthes-wave SC and becomes \fnite in the p-wave SCs. The above properties of the intraband and interband matrix\nelements can be understood using the commutation relation between the Hamiltonian and the spin operators. We\nintroduce the BdG form of the spin operators \u001ba\nBdG(a=x;y;z ) as below\n\u001ba\nBdG= \n\u001ba0\n0\u0000(\u001ba)T!\n: (S.126)\nThe commutation relation of HBdGand\u001ba\nBdGis given by\n[HBdG;\u001ba\nBdG] = 0 :s\u0000wave; (S.127)\n[HBdG;\u001ba\nBdG]6= 0 :p\u0000wave: (S.128)\nEquation (S.127) means that both the Hamiltonian and the spin operator are diagonalized simultaneously, so that the\nmatrix elements of the spin operator between a particle and a hole with the same wave-number vanish. This is because\nthes-wave SC is spin singlet. Therefore, the interband matrix elements vanishes in the s-wave SC. In contrast, in\nthep-wave SCs, the commutation relation between the Hamiltonian and the spin operator is \fnite as shown in Eq.\n(S.128), so that the matrix elements of the spin operator between a particle and a hole with the same wave-number\nis \fnite. This is because the p-wave SCs are spin triplet. As a result, the interband matrix elements are \fnite.\nHere, we explain the reason to introduce the constant level broadening \u0000 for gR=A\n\u0015;k(E). The intraband and interband\ntransitions are schematically shown in Fig. 5. The quasiparticles are scattered due to the magnon emission or\nabsorption. The scattering process conserves the wave-number. Consequently, in the case of the intraband transition,\nthe transition process is forbidden when \u0000 = 0. In order to incorporate the intraband processes, one needs to introduce\n\u0000, otherwise the intraband contribution vanishes, which can be directly shown by calculating Eq. (S.114).\nWhen \u0000 = 0, the uniform spin susceptibility for the chiral p-wave SCs is given by\nRe\u001fR\nuni(!) =2DFZ1\n\u0001dEEp\nE2\u0000\u00012(1 + cos2\u0012)\u00012\n4E2(f(E)\u0000f(\u0000E))\u00121\n2E+~!+1\n2E\u0000~!\u0013\n; (S.129)\nand\nIm\u001fR\nuni(!) =2\u0019DFj~!=2jp\n(~!=2)2\u0000\u00012(1 + cos2\u0012)\u00012\n(~!)2(f(\u0000~!=2)\u0000f(~!=2)): (S.130)\nFrom the above expressions, one can show that both the real part and imaginary part of the uniform spin susceptibility\ndiverge at ~!= 2\u0001, leading a resonance peak. The expressions for the helical p-wave SC can be obtained by replacing\ncos2\u0012with1\n2sin2\u0012. Therefore, \u0012dependence of \u001fR\nuni(!) explained in the main text is obtained from the above\nexpressions.18\nΓintra\ninter\nk+Ek\n−EkE E\nSpectral\nfunction\n2/uni0394/uni210F/uni03C9\n/uni210F/uni03C9Γ\nFIG. 5. Schematic image of intraband transition and interband transitions. The intraband transition gives contribution to the\nuniform spin susceptibility when the excitation energy is comparable to or smaller than the level broadening, ~!.\u0000. The\ninterband contribution is dominant when the excitation energy is comparable to the superconucting gap, ~!\u00192\u0001.\nVIII. LOCAL SPIN SUSCEPTIBILITY\nPerforming the angular integral and replacing the sum over k;qby theE;E0integral, the matrix elements are\nreplaced by\nM\u0015\u00150(s)\nk;q!1\n4+\u00012\n4\u0015E\u00150E0; (S.131)\nM\u0015\u00150(c)\nk;q!1\n4; (S.132)\nM\u0015\u00150(h)\nk;q!1\n4: (S.133)\nThe matrix elements for the chiral and helical p-wave SCs are identical. From the above expressions, one can see\nthat the interband contribution in the s-wave SC is suppressed. Unlike the uniform spin susceptibility, the intraband\ncontribution for the local spin susceptibility is \fnite even when \u0000 = 0. This is because the transition processes\nconsidered here leads to momentum transfer and the intraband transition is not forbidden. Therefore, we calculate\nthe local spin susceptibility at \u0000 = 0. The local spin susceptibility for the s-wave SC is given by\n\u001fR\nloc(!) =\u0000D2\nFZ1\n\u00001dEZ1\n\u00001dE0Ds(E)Ds(E0)\u0012\n1 +\u00012\nEE0\u0013f(E)\u0000f(E0)\nE\u0000E0+~!+i0; (S.134)\nand the local spin susceptibility for the p-wave SCs is given by\n\u001fR\nloc(!) =\u0000D2\nFZ1\n\u00001dEZ1\n\u00001dE0Ds(E)Ds(E0)f(E)\u0000f(E0)\nE\u0000E0+~!+i0: (S.135)\nIX. FMR MODULATION: ROUGH INTERFACE\nIn this section, we show the numerical results and summarize the characteristic properties of the FMR modulation\nfor the rough interface limit. In the following calculations, we set J1= 0 and assume that only \u001fR\nloc(!) contributes to\n\u000eHand\u000e\u000b.\nFigures 6 show (a) \u000eHand (b)\u000e\u000bfor the chiral and helical p-wave SCs as a function of frequency and temperature.\n\u000eHis \fnite inT!0 and has a resonance peak at ~!= 2\u0001.\u000e\u000bexhibits a coherence peak just below the transition\ntemperature in the su\u000eciently low frequency region, where ~!=kBTc\u001c1.\u000e\u000bdrops abruptly at ~!= 2\u0001.\u000e\u000bis\nalmost independent of both frequency and temperature when ~!>2\u0001.\nFigures 6 show (c) \u000eHand (d)\u000e\u000bfor thes-wave SC as a function of frequency and temperature. In the low\nfrequency region, where ~!=kBTc\u00141,\u000eHat a \fxed frequency decreases by about thirty percent with the decrease of\nthe temperature, and \u000eHis \fnite inT!0. As the frequency increases, \u000eHis almost independent of the temperature.\n\u000e\u000bshows a coherence peak just below the transition temperature in the su\u000eciently low frequency, where ~!=kBTc\u001c1.19\nThe coherence peak in the s-wave SC is larger than the corresponding coherence peak in the p-wave SCs. \u000e\u000bhas a\nkink structure at ~!= 2\u0001.\nNote that the cuto\u000b energy Ecwas introduced here to cause the integral for Re \u001fR\nloc(!) to converge. Although\nRe\u001fR\nloc(!) is approximately proportional to Ec, the qualitative properties explained above are independent of Ec.\nThe FMR modulation properties of the three SCs are summarized in Table II. In the case of the rough interface\nlimit, the pairing symmetry can be detected from either the absence or the existence of the resonance peak of \u000eH. The\npairing symmetry may also be detected from the properties of \u000e\u000b, the height of the coherence peak, and the structure\nat~!= 2\u0001. When compared with the resonance peak for \u000eH, however, the properties of \u000e\u000bare too ambiguous to\nallow the pairing symmetry to be distinguished clearly.\n(c) (d)s-wave(a) (b)Chiral & Helical p-wave\nT/T\nchω/kBTcδH/δH2\n0.00.51.00\n2\n410\n8\n6\nT/T\nchω/kBTcδα/δα2\n0.00.51.00\n2\n42\n1\n0\nT/T\nchω/kBTcδH/δH2\n0.00.51.00\n2\n410\n8\n6\nT/T\nchω/kBTcδα/δα2\n0.00.51.00\n2\n42\n1\n0\nFIG. 6. (a) The frequency shift and (b) the enhanced Gilbert damping as a function of both frequency and temperature for the\np-wave SCs. (c) The frequency shift and (d) the enhanced Gilbert damping as a function of both frequency and temperature\nfor thes-wave SC. The terms \u000eH2and\u000e\u000b2are given by \u000eH2=\u00002\u0019SJ2\n2l2D2\nFkBTc=(NA\r ~) and\u000e\u000b2= 2\u0019SJ2\n2l2D2\nF=(NA),\nwhere they are characteristic values in the normal state. The cuto\u000b energy is set to be Ec=kBTc= 10.\nTABLE II. 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Through strict derivation, we\nseparate the damping coefficients for two finite open systems, i.e., the bi-plate system and the sensor\ncore system, into base damping and diffusion damping. This elucidates the relationship between\nthe free damping in the infinite gas volume and the proximity damping in the constrained volume,\nunifies them into one microscopic picture, and allows us to point out three pathways of energy\ndissipation in the bi-plate gap. We also provide the conditions that need to be met to achieve this\nseparation. In applications, for space gravitational wave detection, our results for the residual gas\ndamping coefficient for the 4TM torsion balance experiment is the closest one to the experimental\nand simulation data compared to previous models. For the LISA mission, our estimation for residual\ngas acceleration noise at the sensitive axis is consistent with the simulation result, within about 5%\ndifference. In addition, in the test of the gravitational inverse-square law, our results suggest that\nthe constraint on the distance between TM and the conducting membrane can be reduced by about\n28%.\nI. INTRODUCTION\nEinstein’s general relativity, which still stands as the\nmost successful theory of gravitation, has a far-reaching\ninfluence on the vision of nature as geometry. With\nthe establishment of the Dicke framework [1] based on\nEinstein’s equivalence principle (EP) in the 1960s, ex-\nperimental gravity or gravitational experiments can be\nroughly divided into two classes, including the tests\nof foundations of gravitational theory such as EP and\nthe precision measurements of spacetime curvature ef-\nfects of the so-called metric theories of gravity [2, 3].\nBoth classes of gravitational experiments heavily rely on\nthe establishments of high-precision inertial references\nand the measurements of the relative free-falling mo-\ntions or geodesic deviations between them, and these\ninclude, for example, the MICROSCOPE mission [4–\n6] for the test of the weak EP, ground-based or space-\nborne gravitational wave (GW) antennas like the LIGO-\nVIRGO collaboration [7–9], the LISA [10, 11] and LISA-\nlike missions [12, 13], satellite gravity recovery missions\nsuch as GRACE/GRACE-FO [14, 15] and GOCE [16],\nand ground-based experiments with high-precision tor-\nsion balances [17–26] et al. In present days, isolated\ntest masses (TMs) coupled with high-precision readout\nsystems, such as capacity sensors [27], laser interferome-\nters [28], or SQUIDs [29], are the main technical imple-\nmentation methods for high-precision inertial reference\n∗dongjq@lzu.edu.cn\n†huangl@lzu.edu.cnsystems. The TM, as the key unit, is generally suspended\n(electrically or magnetically in space, or through a wire\nfor ground-based experiments) inside a very stable envi-\nronment with extremely weak couplings to external dis-\nturbances, which ensures the free-falling state of certain\ndegrees of freedom of the TM in the gravitational field.\nWhile, the unavoidable fluctuations of the physical fields\ncoupled to the TM, especially the random collisions from\nresidual gas molecules, will limit the acceleration noise of\nTM and therefore the precisions or sensitivities of the in-\nertial reference [30]. Therefore, the accurate assessments\nof numerous environmental noises become a crucial task\nin the high-precision gravity-related experiments.\nAmong the various stray forces [30], those associated\nwith residual gases include damping force noise [31, 32],\nradiometer effect, and outgassing effect [33, 34]. Damp-\ning force noise refers to the Brownian motion of TMs due\nto gas collisions, which typically becomes one of the lim-\nits in high-precision measurements [31, 32, 35–40]. The\ndamping force noise can be obtained from the damp-\ning coefficient through the fluctuation-dissipation theo-\nrem [31, 41–44]\nSf(ω) = 4 kBTRe\u0012\n−∂F(ω)\n∂v(ω)\u0013\n= 4kBTRe[Z(ω)],(1)\nwhere Sf(ω) is the power spectral density of the fluctua-\ntion force on the TM, kBis the Boltzmann constant, Tis\nthe temperature, Fis the damping force, vis the TM’s\nvelocity, Z(ω) is the mechanical impedance of the sys-\ntem. In high-precision gravity-related experiments, to re-\nduce residual gas noise, the environment pressure is gen-\nerally less than 10−4Pa [19, 34, 45], and the environmentarXiv:2401.04808v1  [gr-qc]  9 Jan 20242\naround the TM is in a free molecular flow regime, where\nthe mean free path is much larger than the distance from\nthe TM to the surrounding walls, and collisions between\nmolecules rarely occur. The collisions between molecules\nand the surfaces are inelastic, following the Knudsen hy-\npothesis [46–49]. So the collisions can be treated as in-\ndependent impulses, the resulting fluctuation force noise\nhas a frequency independent spectrum [32], so residual\ngas leads to an impedance Z(ω) =β, with βreferred to\nhere as the gas damping coefficient.\nThe residual gas damping coefficient is dependent on\nthe specific environments surrounding TMs. In high-\nprecision gravity-related experimental setups with TMs,\ntwo fundamental structures frequently appear, i.e., bi-\nplate and sensor core [50, 51]. The bi-plate structure\ncomprises a movable TM and a fixed parallel plate. The\nsensor core is a cuboid nested structure within gravita-\ntional reference sensors (GRSs)/inertial sensors, with the\nTM in the middle and surrounded by an electrode hous-\ning. And they constitute the finite open systems. In\nGW detections, the damping caused by the motion of the\nTM is called proximity-enhanced damping [32, 38, 39],\nwhile in other fields like MEMS/NEMS, it is known as\nsqueeze-film damping [52, 53]. In traditional squeeze-film\ndamping studies, when the pressure is at one atmospheric\npressure or less, the gas is treated as a continuous viscous\nfluid, and damping forces are calculated using the viscos-\nity coefficient or effective viscosity coefficient [54–56]. As\nthe pressure decreases until collisions between molecules\nrarely occur, i.e. the free molecular flow regime, the envi-\nronment around TM and the choice of surface boundary\nconditions and analysis methods significantly influence\nthe damping calculations. Commonly employed meth-\nods include the free molecular model [57–61], isothermal\npiston model [32, 52], shot noise model [32], and non-\nisothermal models [38, 62].\nIn traditional analyses, the damping coefficient βin the\nfinite open systems consists of two parts [32, 37, 39, 52],\ni.e., the free damping/kinetic damping β∞within an in-\nfinite gas volume, and proximity damping ∆ βwithin a\nconstrained volume. And the total damping coefficient\nisβ=β∞+ ∆β, corresponding to the force noise power\nspectral density, Sf(ω) =S∞\nf+ ∆Sf(ω). However, for\nthis linear separation, there is no consistent understand-\ning of the relationship between the two dampings. For\nexample, Ref. [32] gives a simple explanation: there is\na continuum of behavior between the free damping limit\nand proximity damping, which is confirmed by Monte\nCarlo simulations. Reference [52] compares the magni-\ntudes of two dampings, and kinetic damping is eventu-\nally ignored. And Ref. [39] considers only the proximity\ndamping on the front side, with shear free damping on\nthe lateral side. Therefore, a quantitative and rigorous\nanalysis for the separation of the damping and the cor-\nresponding condition is needed.\nTo address the above issues, some researchers have\nstudied from the perspective of non-isothermal thermo-\ndynamics. Ref. [62] considers the impact of gas tem-perature increase during the compression process, intro-\nduces the thermal-squeeze film damping, and highlights\nthe influence of molecular degrees of freedom and surface\nroughness on damping. However, their use of adiabatic\nconditions contradicts the Knudsen hypothesis. Subse-\nquently, Ref. [38] introduces isothermal conditions, and\nemploys effective temperature to compensate for the ex-\ncess internal energy in the non-equilibrium state com-\npared to the equilibrium state. While effective tempera-\nture is a macroscopic approximation, and high-precision\nmeasurements demand a more microscopically accurate\nexplanation for residual gas noise.\nIn this paper, we introduce finite open models, and pre-\ncisely decompose the damping coefficient into base damp-\ning and diffusion damping for both bi-plate and sensor\ncore finite open systems. This clarifies the relationship\nbetween the free damping and the proximity damping,\nand unifies them into one microscopic picture. In ap-\nplications, the finite open models effectively correct the\nconstraints on gap spacing in the test of the gravitational\ninverse-square law (ISL), and emerge as the theoretical\nmodel most aligned with the simulation and measure-\nment results of the residual gas damping in GRSs for the\nLISA mission and other LISA-like space GW detection\nmissions.\nThe rest of this paper is organized as follows: Section II\ncalculates the changes in molecular number density. The\nseparation of damping coefficients is carried out for both\nbi-plate and sensor core finite open systems in Sec. III.\nSection IV gives separation and sources of damping from\nan energy perspective. The finite open models are ap-\nplied to experiments in the test of the ISL and the GW\ndetections in Sec. V. Discussions and conclusions are pro-\nvided in Sec. VI. Appendices include detailed derivations\nand expressions for some physical quantities.\nII. CHANGES IN MOLECULAR NUMBER\nDENSITY IN THE BI-PLATE FINITE OPEN\nSYSTEM\nThe distinction between the bi-plate finite open system\nand the infinite volume lies in: as the TM moves, the\nmolecular number density within the gap changes. When\ncalculating the momentum exchange between molecules\nand the TM, the number density is required. This will\nlead to the unique physical processes. The change in\nnumber density is addressed in the following discussion.\nIn Fig. 1, as the TM moves, the gap volume V, the\ntotal number of molecules Nand the number density\nnundergo continuous changes over time. Considering\nthe total differential of the molecular number density\nn(t) =N(t)/V(t), and taking the derivative with respect\nto time, we have\nd∆n\ndt=1\nV0d∆N\ndt−N0\nV2\n0d∆V\ndt, (2)\nwhere ∆ nis the change in molecular number density,3\nGapTM\nFixed plate\n0+∆ℎL \nW \nFIG. 1. Schematic side view of the bi-plate finite open system.\nThe TM moves downward parallel to the x-axis, with a gap\nspacing denoted as h. Gas molecules can exchange with the\nenvironment through the gap boundary. The pure light-gray\ngas molecules are from the fixed plate surface or the environ-\nment, while the gas molecules represented by light-gray and\nblack desorb from the TM surface. The color black repre-\nsents more or less momentum or energy carried compared to\nthe case when the TM is stationary. The effective molecular\nnumber density on the windward and leeward sides is denoted\nasn′\nWandn′\nL, respectively.\n∆N(t) and ∆ V(t) are the changes in the molecular num-\nber and gap volume relative to the equilibrium point.\nConsidering the diffusion process for the molecules to es-\ncape into the environment, as shown by the arrows in\nthe left of Fig. 1, we deal with the first and second terms\non the right side of Eq. (2) in Appendix A, then under\ncondition ∆ h≪h0, Eq. (2) is rearranged as\nd∆n\ndt=−∆n\nτ0−n0\nh0dh\ndt, (3)\nwhere τ0is diffusion time for one molecule to escape from\nthe gap and h(t) =h0+Bcos(ωt) is gap spacing in the\nfrequency domain, and the general solution to Eq. (3) is\n∆n=n0Bωτ 0\nh0p\n1 +ω2τ2\n0cos(ωt+φ) +Ce−t\nτ0,(4)\nwhere φ= (arctan(1 /ωτ0)−π)∈(−π,−π/2) is the phase\nangle, and Cis the integration constant. The first term in\nEq. (4) is the steady-state solution, and the second term\nis the transient solution, which diminishes over time. In\nthe most high-precision gravity-related experiments, the\nproduct of detection frequency bands and diffusion times\nisωτ0∈(10−8,10−2) [7, 10–16, 18–23, 63, 64], and thus\nφ≈ −π/2. The steady-state part of Eq. (4) is approxi-\nmated as:\n∆n≈n0Bωτ 0\nh0cos(ωt−π\n2). (5)\nThe system’s finite openness can be characterized by the\ndiffusion time τ0. A small degree of finite openness means\nthat the number of exchanges between the gap and the\nenvironment is small per unit time, equivalent to a largediffusion time τ0, i.e., a long relaxation time for the sys-\ntem, and vice versa. When the gap boundary is closed,\nthe gap spacing change ∆ his in phase opposition with\nthe density change ∆ nclosed, this is consistent with our\ngeneral understanding. As the gap boundary opens, τ0\ndecreases, the phase φof ∆nchanges −π→ −π/2. Inter-\nestingly, when ∆ hcontinues to increase beyond the equi-\nlibrium position, ∆ nopenalso increases, which may seem\ncounterintuitive. The explanation is that when the vTM\nis very small, the difference in number density between\nthe gap and the environment is small. However, when\nthevTMis very large, due to the reaction time required\nfor diffusion, the number density in the gap will differ\nsignificantly from that in the environment. Furthermore,\nconsidering the direction of motion, ∆ nopenshould be in\nphase opposition to vTM.\nIII. SEPARATION OF DAMPING\nCOEFFICIENTS\nThe Knudsen hypothesis posits that molecules, upon\nimpacting a surface, undergo adsorption, remain for a\nbrief period, and forget their velocity and direction be-\nfore impact. During desorption, the speed of emitted\nmolecules follows the Maxwell-Boltzmann distribution,\nand the emission angles obey cosine law [40, 65–72]. Ac-\ncording to the cosine law, the probability for a molecule\nto leave the surface within the solid angle d ωis expressed\nas dP= dωcosθ/π, where θdenotes the angle with the\nsurface normal direction.\nThe cosine law is initially discovered in experiments\nusing glass surfaces [48, 49], and subsequent studies on\nvarious materials have also verified the same emission\nrule [35, 40, 65–72]. Especially, Ref. [40] provides a condi-\ntion for non-elastic collisions in a vacuum ( p0<10−4Pa)\nwith gold-plated surfaces. In realistic applications in the\nfield of MEMS/NEMS, various models considering dif-\nferent boundary conditions are available [53, 60, 73, 74],\nincluding isothermal and adiabatic, rough and smooth\nsurfaces, diffuse and specular scattering, etc., which are\noften selected based on experimental data.\nDespite some deviations observed in individual experi-\nments [65, 75], it is assumed that the cosine law remains\nvalid for surfaces of general materials. The cosine law and\nthe parallel arrangement of the bi-plate ensure that the\ngas within the gap adheres to the Maxwell-Boltzmann\nvelocity distribution [76–78]. Given the extremely short\nadsorption time at room temperature (on the order of\npicoseconds or nanoseconds) [32], much shorter than the\nmotion time of molecules in the gap (on the order of\nmicroseconds), it is reasonable to assume a negligible ad-\nsorption time.\nNext, we will proceed to separate the damping coeffi-\ncients in the bi-plate and sensor core finite open systems,\nrespectively.4\nA. Bi-plate finite open system\nAs shown in Fig. 1, assuming the TM is moving down-\nward, the frequency of collisions with molecules increases\non the windward side and decreases on the leeward side,\nimpacting the force experienced by the TM. We will an-\nalyze this effect separately for surfaces perpendicular ( x\nsurface) and parallel ( yorzsurface) to the direction of\nmotion.\nDamping force of incident molecules on xsurface. —\nConsider a surface element d Axperpendicular to the x-\naxis. Assuming the TM has motion solely parallel to the\nx-axis, the number of molecular per unit time colliding\nwith the element d Axon the windward side, with a ve-\nlocity component vxin the range vx∼vx+ dvxis given\nby [77–79]\ndNvx= (n0+ ∆n)(vx−vTM)f(vx)dvxdAx,(6)\nwhere fis the Maxwell-Boltzmann velocity distribution\nfunction, i.e. f(vx) = (m/2πkBT)1/2exp\u0000\n−mv2\nx/2kBT\u0001\n.\nThe molecules of vx> v TMwill collide with d Ax, so\nvx∈(vTM,∞). Taking into account momentum conser-\nvation, the force exerted per unit time by collisions on\nthe windward side is\nFin\n⊥,W= (n0+ ∆n)AxmZ∞\nvTM(vx−vTM)2f(vx)dvx,(7)\nwhere mis the molecular mass. Similarly, the force ex-\nerted per unit time by collisions on the leeward side is\nFin\n⊥,L=−n0AxmZvTM\n−∞(vx−vTM)2f(vx)dvx. (8)\nNote that the leeward side is in the external environ-\nment, and the number density remains n0. From the\nEq. (5), it can be seen that ∆ n=−n0τ0vTM/h0. Tak-\ning into account that in gravity-related experiments, vTM\nis very small, only the first-order terms are retained in\nthe subsequent derivation. The total damping force from\nmolecules incident on surfaces perpendicular to the di-\nrection of motion is then approximately given by\nFin\n⊥≈ −2Axn0\u00122mkBT\nπ\u00131/2\nvTM−Axn0τ0kBT\n2h0vTM.\n(9)\nDamping force of molecules desorbed from the xsur-\nface.— Since molecules lose memory after colliding with\nthe surface, the incident and outgoing processes are in-\ndependent. Therefore, the force exerted by desorbed\nmolecules on the TM can be calculated separately. The\ncosine law ensures that the incident distribution is the\nsame as the outgoing distribution when taking the TM\nas the stationary coordinate system [76–78]. Due to the\nidentical distributions, the outgoing process is equivalent\nto the incident process, and Eq. (6) can be used directly.\nAs the TM moves, the number of incident molecules will\nchange, so will the number of outgoing molecules. In thisscenario, we refer to the outgoing number as the effective\nincident number.\nFrom Eq. (6), after integrating the velocity vx, only the\nvariables nandTremain. Under isothermal conditions,\nTremains constant, so the only variable that can change\nisn=n0+ ∆n. Therefore, the difference between the\nreal incident number and the effective incident number\nlies in the effective number densities n′. By setting the\noriginal incident number equal to the outgoing number\nper unit time, we can write the following identity:\n−n′\nWZ0\n−∞vxf(vx)dvx= (n0+∆n)Z∞\nvTM(vx−vTM)f(vx)dvx,\n(10)\nConsidering that as vTM→0, Erfhp\nm/(2kBT)vTMi\n→\n0, and exp[ −mv2\nTM/(2kBT)]→1, Eq. (10) can be solved\nto obtain the effective number density on the windward\nside\nn′\nW= (n0+ ∆n)\"\n−vTM\u0012πm\n2kBT\u00131/2\n+ 1#\n. (11)\nSimilarly on the leeward side,\nn′\nL=n0\"\nvTM\u0012πm\n2kBT\u00131/2\n+ 1#\n. (12)\nThus, the total damping force generated by molecules\nemitted parallel to the direction of motion is:\nFout\n⊥=−n0Ax\u0012πmk BT\n2\u00131/2\nvTM−Axn0τ0kBT\n2h0vTM.\n(13)\nDamping forces on yorzsurface. — Since the three di-\nrections of the Maxwell-Boltzmann velocity distribution\nare completely independent, we can first use the vydis-\ntribution to get the numberR\nvydNy,vyhitting the side,\nand then use this number to calculate the damping coeffi-\ncient in the x-axis. Since the zdirection is perpendicular\nto the direction of motion of the TM, it does not need to\nbe calculated, and only the velocity distribution of vxin\nthexdirection is considered. Such analysis also applies\nto surface z.\nSince vy/zis perpendicular to the direction of motion,\nthe motion of the TM does not affect the velocity distri-\nbution and integration limits during calculation. From\nthis, we can get the force per unit area on the side\ndFy/z,v x=m(vx−vTM)f(vx)dvxdAy/zZ\nvy/zdNy/z,v y/z.\n(14)\nThe outgoing molecule velocities cancel each other out in\nall directions, so the net force exerted in the xdirection\nis zero. Thus, we obtain the damping force on a single\nlateral surface:\nF∥,y/z=−n0Ay/z\u0012mkBT\n2π\u00131/2\nvTM. (15)5\nFinally, combining Eq. (9), (13), and (15), we can get the\ntotal damping force and then the damping coefficient\nβB=(4Ax+πAx+ 2Ay+ 2Az)p0\u0012m\n2πkBT\u00131/2\n+Axp0τ0\nh0,(16)\nwhere p0=n0kBTis the pressure at equilibrium in the\ngap.\nB. Sensor core finite open system\nAs depicted in Fig. 2, the sensor core used in GRSs for\nspace GW detection missions such as LISA and similar\nprojects [10–13] is characterized by a cuboid nested struc-\nture. This structure is also employed in global gravity\nfield recovery missions albeit with variations in size [14–\n16, 27, 63, 64, 80], and serves to monitor the real-time\nposition of the TM by connecting to the front-end circuit\nand apply electrostatic forces to control the TM’s behav-\nior. The sensor core forms a closed gap structure between\nthe TM and the electrode housing. However, for a gap on\none side, molecules can exchange with other gaps, form-\ning a finite open structure. In Fig. 2, we decompose it\ninto six bi-plate finite open systems, Gx1,Gx2,Gy1,Gy2,\nGz1,Gz2, and handle them separately.\n\n Electrode HousingTest Mass\n2 12\n1\nFIG. 2. Schematic side view of the sensor core in GRS. The\nsensor core consists of a TM in the middle and an electrode\nhousing surrounding it, with a gap between them. The gap\ncan be divided into six bi-plate gaps, i.e., bottom gaps Gx1,\nGx2, and lateral gaps Gz1,Gz2,Gy1,Gy2, and the last two\nare not shown.\nDamping force of incident molecules on xsurface. —\nIn a manner analogous to the previous subsection III A,\nthe force exerted by all molecules hitting the windward\nface per unit time remains the same as in Eq. (7). Thechange in the number density in gap Gx1is opposite to\nthe change in number density in gap Gx2, i.e., ∆ nx2=\n−∆nx1=−∆n. Consequently, the force exerted by all\nmolecules hitting the leeward face per unit time becomes\nFin\n⊥,L=−(n0−∆n)AxmZvTM\n−∞(vx−vTM)2f(vx)dvx.\n(17)\nThe total damping force of molecules incident perpen-\ndicular to the direction of motion is then approximately\ngiven by\nFin\n⊥≈ −2Axn0\u00122mkBT\nπ\u00131/2\nvTM−Axn0τ0kBT\nh0vTM.\n(18)\nDamping force of molecules desorbed from xsurface. —\nThe effective molecular number density on the windward\nface is the same as in Eq. (11), while on the leeward side,\nthe effective number density becomes\nn′\nL= (n0−∆n)\"\nvTM\u0012πm\n2kBT\u00131/2\n+ 1#\n, (19)\nThe total damping force generated by molecules emitted\nperpendicular to the direction of motion is given by\nFout\n⊥=−n0Ax\u0012πmk BT\n2\u00131/2\nvTM−Axn0τ0kBT\nh0vTM.\n(20)\nDamping forces on yorzsurface. — As the TM moves\nup and down, the number of molecules in the gaps Gx1\nandGx2changes due to diffusion, causing changes in the\nnumber ∆ Nsideand number density of molecules in the\nlateral gap ( Gy1,Gy2,Gz1,Gz2). Molecules diffuse into\ngapGx1while diffusing out of gap Gx2, resulting in a den-\nsity gradient in the x-axis direction on the lateral gaps.\nHowever, according to Eq. (14), the molecular density on\nthe side is linearly superposed, allowing for non-uniform\ndensity distribution on the side. The side force can be\nreexpressed as\nF∥,y/z=−Ay/zn0\u0012mkBT\n2π\u00131/2\nvTM\n−∆Nside,y/z\nhy/z\u0012mkBT\n2π\u00131/2\nvTM.(21)\nCondsidering the number density changes in the gaps\nGx1andGx2are ∆ nx1and ∆ nx2, and the instanta-\nneous volumes are Vx1=Ax(h0+Bcos(ωt)) and Vx2=\nAx(h0−Bcos(ωt)), respectively. Thus, we obtain\n∆Nside=−(∆nx1Vx1+ ∆nx2Vx2)\n=4n0Bτ0Ax\nh0cos(ωt)vTM,(22)6\nSubstituting Eq. (22) into Eq. (21),\nF∥,y/z≈ −Ay/zn0\u0012mkBT\n2π\u00131/2\nvTM. (23)\nUnlike the gaps Gx1andGx2, the fluctuation of number\ndensity does not affect the lateral damping force.\nCombining Eqs. (18), (20), and (23), we can obtain the\ntotal damping force and then the damping coefficient\nβS=(4Ax+πAx+ 2Ay+ 2Az)p0\u0012m\n2πkBT\u00131/2\n+2Axp0τ0\nh0.(24)\nThe first term in Eqs. (16) and (24) is consistent with\nthe free damping in the infinite gas volume [36], while\nthe second term is consistent with the proximity damping\nin the isothermal piston model [32, 52]. The first term\ncan be regarded as a basic damping that always exists\nregardless of the environment, whereas the second term\nrepresents damping induced by molecular diffusion. We\nrefer to them as base damping and diffusion damping,\nrespectively.\nThe models described above are denoted as finite open\nmodels. Finally, we achieve the natural separation of\ndamping coefficients within the finite open systems.\nIV. SEPARATION AND SOURCES OF\nDAMPING FROM AN ENERGY PERSPECTIVE\nAs shown in Fig. 1, when the molecules desorb from\nthe moving TM surface, in addition to the velocity given\nby the Maxwell-Boltzmann distribution, there is also a\nbackground translational velocity of the TM. In other\nwords, the TM only affects the energy of the molecules\nin translational degrees of freedom.\nUsing subscripts u for gas molecules desorbed from the\nTM surface, and d for those desorbed from the fixed plate\nbelow, with the total gas molecules indexed as i, the total\nenergy can be expressed as\nE=1\n2X\nimv2\ni=1\n2X\ndmv2\nd+1\n2X\numv2\nu\n≈1\n2X\nimv′2\ni+1\n2X\numv2\nTM+X\numv′\nu·vTM,(25)\nwhere v′\niandv′\nu=vu−vTMrepresent the velocities\nof molecules in the center-of-mass frame assuming that\nthe TM does not move. The first term in Eq. (25),P\nimv′2\ni/2≈P\ndmv2\nd/2 +P\numv′2\nu/2, assumes that the\nnumber Nuof molecules desorbed from the TM surface\nand the number Nddesorbed from the bottom fixed plate\nare approximately equal, Nu≈Nd. Molecules desorb\nfrom the TM surface and emit to the fixed plate, and\nthen back to the TM surface, this process takes a very\nshort time, ∆ t∼10−5s, while the TM’s velocity is veryslow, vTM<10−8m/s. In other words, during this time,\nNd(t0+ ∆t)≈Nu(t0), so this approximation is reason-\nable. The total number N(t) =Nu+Ndchanges due to\nthe diffusion of gas molecules.\nLetv′\nuxbe the projection of v′\nuin the direction of\nthe TM motion, its probability density distribution is\n2f(v′\nux). Therefore, the third term can be re-expressed\nas\nX\numuv′\nu·vTM=Nu\u00122mkBT\nπ\u00131/2\nvTM. (26)\nConsidering the Knudsen hypothesis, the Eq. (25) can be\nre-expressed as\nE(t) =3\n2N(t)kBT+1\n4N(t)mv2\nTM+N(t)\u0012mkBT\n2π\u00131/2\nvTM.\n(27)\nThe first term is consistent with the isothermal internal\nenergy of the gas in an equilibrium state, and the second\nand third terms are related to the translational velocity\nof the TM, representing additional energy introduced by\nnon-equilibrium processes. This result is different from\nthe K¨ onig’s theorem [81], which states that the energy of\nthe system can be separated into the translational kinetic\nenergy of the center-of-mass and the relative kinetic en-\nergy within the center-of-mass system. In the Maxwell-\nBoltzmann distribution, the overall translational veloc-\nity of the gas is zero. In other words, when we refer to\ntemperature, this means that the system is discussed in\nthe center-of-mass system. Thus, it is inaccurate to use\nisothermal gas or effective temperature assumptions in\nthis system. In the free molecular flow regime, molecules\ndo not collide, thus energy cannot be evenly distributed\namong the degrees of freedom [82], except for the trans-\nlational degrees of freedom related to the TM.\nAccording to analytical mechanics, the damping co-\nefficient means the dissipation of energy, and there are\nthree pathways of energy dissipation in the bi-plate gap.\nWhen molecules are incident on the TM surfaces, the in-\ncident damping forces, i.e. Eqs. (9) and (18) correspond\nto the energy dissipation of the TM. In Fig. 1, due to\nthe motion of the TM, the molecules desorbed from the\nTM will dissipate more kinetic energy of the TM to the\nfixed plate. This part of the energy results in the addi-\ntion of the last two terms in Eq. (27). For the bi-plate\nsize at the bottom gaps of the sensor core for LISA mis-\nsion, only about 9% molecules desorbed from the TM\n(see Appendix B) leave the gap directly. Similarly, the\nratio from the environment directly to TM is the same.\nThe ratio is proportional to the energy dissipated, so the\nmajority of the dissipated energy flows to the fixed plate\nand the TM, with a very small amount flowing to the\nenvironment.\nCompared to base damping, diffusion damping ac-\ncounts for changes in number density resulting from the\ndiffusion process. Since the change in number density is7\nin counter-phase with TM’s velocity, it is equivalent to\nincreasing the number of incident and outgoing molecules\nper unit time, which in turn amplifies damping force or\nenergy dissipation. Consequently, the dissipation flow\nfrom the gap to the environment as noticed in Ref. [32]\nis not a leading term to the diffusion damping of the\nenergy dissipation. Instead, as mentioned above, this en-\nergy is primarily dissipated into the TM and the fixed\nplate. In isothermal piston models, the utilization of the\nideal gas state equation implies that compression is a\nquasi-static process, wherein only the diffusion damping\nresulting from changes in number density is considered.\nAs a result, its internal energy corresponds solely to the\nfirst term in Eq. 27, excluding the last two terms associ-\nated with the damping of the non-equilibrium part.\nV. APPLICATIONS\nThe investigation of residual gas noise holds significant\nimportance in gravity-related experiments. Residual gas\ndamping noise typically determines the limit of sensitiv-\nities in many high-precision gravity-related experiments.\nIn the subsequent sections, we apply the sensor core and\nbi-plate finite open models to space gravitational wave\ndetection and the test of the gravitational inverse-square\nlaw, respectively.\nA. Space gravitational wave detection\nThe discovery of GWs has positioned it as a new mes-\nsenger for observing the universe. LISA is a spaceborne\nGW detection mission proposed by the European Space\nAgency [10, 11, 83]. To investigate the residual gas damp-\ning noise used in such missions, the University of Trento\n(UTN) employs a torsion balance to measure the decay\ncurves in the GRSs [31].\nThe 4TMs torsion balance experiment setup is shown\nin Fig. 3. The experimental temperature is 293 K, and\nthe molecular mass utilized in the simulation is 30 u. By\nmeasuring the amplitude decay time during free pendu-\nlum motion, the torsional damping coefficients can be ob-\ntained. The experimental results indicate that the resid-\nual gas damping coefficient is much larger than the theo-\nretical prediction of infinite volume damping. However, it\nis in good agreement with UTN’s simulation predictions,\nas shown in Fig. 4.\nSince the torsion balance measures the torsional damp-\ning coefficient, the translational damping coefficient βSof\nthe finite open model is needed to convert to the 4TM ro-\ntational damping coefficient β4TM\nrot=r2βtran+βrot. The\nGRS\nMirror\nS2TMTM\nEH\nEH(a)\n(b)(c)\nFIG. 3. Schematic drawing of the 4TMs torsion balance ex-\nperiment setup (not to scale). (a) A cross-shaped structure is\nsuspended by a wire, with four hollow cubic TMs attached at\neach end. The side length of each TM is s= 46 mm, and the\npendulum arm length is r= 0.1065 m. (b) The gap spacings\nbetween the GRS’s TM and the electrode housing (EH) in the\nx,y, and zaxes are 4 .0 mm, 2 .9 mm, and 3 .5 mm, respec-\ntively. (c) The gap spacings between the S2’s TM and the\nEH in the x,y, and zaxes are 8 .0 mm, 6 .0 mm, and 8 .0 mm,\nrespectively [31].\ntotal rotational damping coefficient is given by\nβ4TM\nrot=r2p0s2\"\n2τside(hx)\nhx+\u001232m\nπkBT\u00131/2\u0010\n1 +π\n8\u0011#\n+1\n6p0s4\u0014τside(hx)\nhx+τside(hz)\nhz\u0015\n+p0s4\u00122m\nπkBT\u00131/2\u0010\n1 +π\n12\u0011\n.\n(28)\nIn Fig. 4, the result of the finite open model demon-\nstrates a 62% improvement compared to the constrained\nvolume model [39], and exhibits an 8% difference from\nthe UTN’s simulation data. The latter utilizes the dif-\nfusion time under bi-plate structures, but the electrode\nhousing alters the diffusion conditions at the bi-plate\nboundaries, subsequently changing the diffusion time un-\nder the bi-plates. This alteration increases the diffusion\ntime. Finally, we substitute the simulation result into\nthe finite open model, with τside(hx) = 3 .22×10−4s and\nτside(hz) = 3 .02×10−4s, while τcuboid = 1.31×10−4s.\nThe 8% discrepancy may come from the gradient of\nmolecular number density in the lateral gap ( Gy1,Gy2,\nGz1,Gz2) in the x-axis. This might lead to the deviation,\ni.e., the N0/τ0term in Eq. (A1) is not accurate.\nThe results in Fig. 4 demonstrate the effectiveness of\nthe finite open model. Ultimately, we apply the finite\nopen model to the analysis of the residual gas damping\nnoise in the GRS’s sensitive axis for the LISA space GW\ndetection mission [31] and other LISA-like space GW de-\ntection missions, such as Taiji [13], TianQin [12]. This\nnoise largely determines the detection sensitivity of GRSs\nin the primary frequency range. For the 2 kg TM moving8\n0 1 2 3 4010203040Experimental data fit of UTN\nSimulation of UTN\nOur model\nConstrained volume model\nInfinite volume model\nFIG. 4. Comparison of different rotational damping coeffi-\ncients in the 4TM torsion balance experiment. The black\ndashed line (+) and black dashed line ( ×) represent the ex-\nperimental measured values and simulation values from the\nUTN, respectively [31]. The green solid line (+) represents\nthe theoretical values from Eq. (28), i.e., our finite open model\nin this paper, and the short dashed gray line ( ×) represents\nthe theoretical values from the constrained volume model [39].\nThe short dashed gray line represents the theoretical result of\ndamping coefficients in an infinite volume [36].\nalong the sensitive axis ( x-axis), we can use Eqs. (1) and\n(28) to estimate the residual gas acceleration noise, that\nis\nS1/2\na=1.23×10−15m s−2Hz−1/2\n×\u0010p\n10−6Pa\u00111/2\u0010m\n30 u\u00111/4\u0012T\n293 K\u00131/4\n.(29)\nUnder the same conditions, compared to Ref. [39], our\nresult is closer to the simulation result S1/2\na= 1.3×\n10−15m s−2Hz−1/2from Ref. [31], with only a 5% dif-\nference.\nThe validity of the finite open model also corrects the\nprevious view on the diffusion time in the sensor core.\nRef. [31] states that the diffusion time is the average time\nfor molecules to diffuse from gap Gx1toGx2. However, as\nshown in Eq. (22), we found that the molecular number\nchanges from Gx1andGx2have a negligible second-order\neffect on the y/zsurfaces, and the diffusion time can be\napproximated as the time to diffuse just from the bottom\ngap ( Gx1orGx2) to the lateral gap ( Gy1,Gy2,Gz1,Gz2).\nB. Test of the gravitational inverse-square law\nTo reconcile general relativity and the standard model,\nstring theory or M-theory predicts deviations from the\nISL at short distances [84, 85]. Huazhong University of\nScience and Technology (HUST) designs a torsion bal-\nance and a eightfold azimuthal symmetric attractor ex-\n\n\nConducting membranePendulumVacuum feedthrough\nFiber\nClamp111\n1\n2\n22\n2FIG. 5. Schematic drawing of the test of the ISL experiment\nsetup (not to scale). An I-shaped pendulum is suspended from\nthe bottom of the torsion balance through tungsten wire, and\nthe pendulum faces the position of the attractor. The middle\npart of the pendulum is a glass block (M), and two glass\nbases ( G1,G2) are symmetrically installed at both ends. Two\nglass substrates ( Gt1,Gt2), two tungsten TMs ( Wt1,Wt2),\nand two gravitational compensation pieces ( Wtc 1,Wtc 2) are\nglued to the glass bases opposite the attractor. A special\nglass clamp is installed on the top of the middle glass block,\nit can fix the suspended tungsten wire to the center of the\npendulum. On the left side of the conducting membrane,\nthere is an attractor not shown here. The attractor consists\nof 8 tungsten source masses and 8 tungsten compensation\nmasses, arranged alternately on a rotatable glass disk [19].\nperimental platform, and tests the ISL at short distances\nby dually modulating the signal of interest and the grav-\nity calibration signal [18–23].\nThe experimental setup of the test of the ISL is illus-\ntrated in Fig. 5. To facilitate the experimental design of\nthe gap spacing between the pendulum and the conduct-\ning membrane, it is necessary to assess the influence of\nresidual gas damping noise with varying gap spacings. To\nobtain the constraints on gap spacing hcunder different\ngas pressures p0[38], the gas damping noise is equated\nto the torsion balance’s internal damping thermal noise.\nThe internal thermal noise of the torsion balance is\nSth(ω) = 4 kBTk/(ωQ)≈3.61×10−30N2m2Hz−1.\nHere, Qis the quality factor of the torsion balance, ω\nis the resonant frequency, and kis the spring constant.\nAccording to the bi-plate finite open model, the fluctu-\nation torque noise power spectral density Srot,Mof the\nmiddle glass block, and Srot,G,Wt of the glass blocks ( G1,\nG2), TMs ( Wt1,Wt2) and gravitational compensation\npieces ( Wtc 1,Wtc 2) at both ends can be obtained in\nAppendix C. From this, the relation\np0=f(hc) =Sth\n(Srot,M+Srot,G,Wt )/p0, (30)9\nis obtained, and consequently\nhc=f−1(p0). (31)\n02040608010020406080100Non-isothermal model\nOur model\nFIG. 6. Comparison of gap spacing constraints under dif-\nferent pressures between the finite open model and the non-\nisothermal model. The shaded area below the green solid line\nrepresents the constraints given by Eq. (31), i.e., our bi-plate\nfinite open model in this paper, and the shaded area below\nthe gray dashed line represents the constraints given by the\nnon-isothermal model [38].\nGiven temperature T= 297 K, molecular mass 19 .1 u,\nand thermal adjustment coefficient σ= 1, the results\nare shown in Fig. 6. Due to the very small gap spacing,\n∼100µm, the diffusion damping is magnified, and the\nbase damping is almost negligible. In the range of 10 to\n100µPa, the constraints on the gap spacing hcobtained\nby the finite open model are lower by about 28% com-\npared to the non-isothermal model. This suggests that\nthe additional temperature effect in the non-isothermal\nmodel is almost negligible. These results are crucial for\nreducing the test length scale λin ISL experiments, pro-\nviding valuable insights into the constraints imposed by\nresidual gas damping noise on experimental parameters.\nVI. DISCUSSIONS AND CONCLUSIONS\nWe separate the damping coefficients for two finite\nopen systems, i.e., the bi-plate system and the sensor\ncore system, into base damping and diffusion damping\nthrough rigorous derivation. This effectively elucidates\nthe relationship between the free damping and the prox-\nimity damping. This separation needs to meet the con-\ndition ωτ0→0, which holds for most high-precision\ngravity-related experiments. In our derivation, the key\nlies in how to deal with the change in molecular number\ndensity. The change is so small compared to the overall\ndensity that it can be almost ignored [52], as evidenced in\nthe LISA’s GRSs (∆ n/n 0<10−11). However, detailedderivation shows that the results corresponding to such\na small amount can not be ignored, which renders high-\nprecision measurements truly distinctive. We address the\nimpact of changes in number density within the bottom\ngaps on the lateral gaps, which also reveals the possibil-\nities for analyzing cross-talk effects in residual gas noise\nbetween the sensitive axis and other axes.\nThe diffusion time τis used to quantitatively charac-\nterize the degree of finite openness in the system, which\nis different from the constrained volume. It is general\nand can account for the effects of different shapes of the\nstructure around the TM. However, in a sensor core with\nlateral walls, an increase in constrained volume does not\nnecessarily equate to a reduction in diffusion time. Inter-\nestingly, as τ→0, i.e., the system is completely open, the\ndiffusion damping disappears, and the damping becomes\nconsistent with the free damping in an infinite volume.\nConversely, as τ→ ∞ , i.e., the system is completely\nclosed, in contrast to the zero damping of the isother-\nmal piston model [32, 52], there still exists base damping\nindependent of gap spacing. This result is reflected in\nthe lower limit of the power spectral density shown in\nRef. [32]. In brief, the concept of finite openness can\ncharacterize damping more comprehensively and accu-\nrately.\nIn terms of the physical picture, the previous analyses\nof damping in an infinite volume and a constrained vol-\nume are disjointed. The finite open models presented in\nthis paper provide a complete microscopic picture that\nincludes both types of damping. This allows us to point\nout three pathways of energy dissipation of the TM from\nthe microscopic level in the bi-plate gap, that is: energy\nabsorbed by the TM, assimilated by the fixed plate, and a\nvery small portion directly diffused into the environment.\nIn practical applications, for space GW detection, our\nmodel predicts a damping coefficient for the 4TM torsion\nbalance experiment 62% higher than previous analyses,\nand is the closest to the experimental and simulation\ndata to the best of our knowledge. For the LISA mis-\nsion, our theoretical estimation of the residual gas accel-\neration noise along the sensitive axis aligns most closely\nwith simulation results, differing only by 5%. These re-\nsults validate the effectiveness of the finite open models.\nAnd these results are also applicable to other LISA-like\nspace GW detection missions, such as Taiji [13], Tian-\nQin [12]. In the test of the ISL, our model reduces the\nconstraints on the gap spacing between the TM and the\nconducting membrane by about 28%, compared with the\nnon-isothermal model. And this result may further help\nto reduce the test length scale in ISL experiments.\nFurthermore, our theory can also be applied to other\nhigh-precision measurements, such as ultra-sensitive\nspace accelerometers [27, 80, 86, 87], measurements of\nthe gravitational constant G[24–26], MEMS/NEMS ac-\ncelerometers [88, 89] and Casimir force measurements [90,\n91].10\nACKNOWLEDGMENTS\nWe thank Profs. Yun-Kau Lau and Lamberto Rondoni\nfor their inspiring discussions and advice, as well as Drs.\nZuolei Wang and Da Fan for their help. This work is sup-\nported by the National Key Research and Development\nProgram of China under Grant No. 2020YFC2200601,\nthe National Natural Science Foundation of China under\nGrants No. 12175090, No. 12305045 and No. 12247101,\nand the 111 Project under Grant No. B20063.\nAppendix A: The Two Terms in the Equation (2)\nThe first term. — Suppose the average diffusion time\nfor one molecule to escape from the gap is τ(h), and N/τ\nis the number of escaping molecules per unit time, then\nd∆N=−N(t)\nτ(h)dt+N0\nτ0dt, (A1)\nwhere ∆ Nis the net number of diffusing molecules, N0\nandτ0are at the equilibrium point. Equation (A1) is\nalso known as the Fick rule [38]. Considering gap spacing\nchange ∆ h≪h0, we can have the approximation τ(h)≈\nτ0. In this case, dividing Eq. (A1) by d tandV0gives\n1\nV0d∆N\ndt=−n(t)\nτ0+n0\nτ0, (A2)\nwhere n0=N0/V0. In addition, considering ∆ V(t)≪\nV0, the approximation N(t)/V0≈n(t) is made.\nThe second term. — Assuming the TM vibrates near\nthe equilibrium point, in the frequency domain, the dis-\nplacement relative to the equilibrium point is ∆ h(t) =\nBcos(ωt), and the instantaneous volume of the gap is\nV(t) =Ax(h0+Bcos(ωt)) = Axh(t), where Axis the\nsurface area of the TM perpendicular to the x-axis. Thus,\nthe second term on the right side of Eq. (2) can be written\nas\n−N0\nV2\n0d∆V\ndt=−n0d\ndt\u0012Axh\nV0\u0013\n=−n0\nh0dh\ndt. (A3)\nAppendix B: The proportion of Molecules Escaping\nDirectly from the Test Mass Surface to the\nEnvironment\nMolecules undergo an average of Ncollisions inside the\ngap before leaving. The last time the molecules escape\ndirectly from the TM surface, the proportion is 1 /2, so\nthe proportion of molecules escaping directly from the\nTM surface to the environment is approximately given\nby\nPout≈1\n21\nN\n2≈0.09, (B1)where we use the conclusion from Ref. [32]. For the size\nof sensor core for LISA mission, Nis approximately given\nby\nN ≈R2\nh2lnh\n1 +\u0000R\nh\u00012i≈11, (B2)\nwhere the radius Ris obtained by assuming that the area\nof the square is consistent with the area of the equiva-\nlent circle. The proportion of molecules escaping directly\nfrom the TM surface to the environment is 0.09, which\nis a small fraction.\nAppendix C: Fluctuating Force Power Spectral\nDensity in ISL Experiment\nAccording to the theoretical model of the bi-plate finite\nopen system, the fluctuating torque noise Srot,Mfor the\nmiddle glass block can be obtained as follows:\nSrot,M=s3\nM,xsM,z\n6\u001232mkBT\nπ\u00131/2\u0010\n1 +π\n4\u0011\np0\n+ \ns3\nM,xsM,z\n2+s3\nM,xsM,y\n6+s3\nM,ysM,x\n6!\n×\u00128mkBT\nπ\u00131/2\np0.(C1)\nThe torque noise Srot,G,Wt for the glass blocks ( G1,\nG2), the TMs ( Wt1,Wt2), and gravitational compensa-\ntion pieces ( Wtc 1,Wtc 2) at both ends is given by:\nSrot,G,Wt = 8kBTDl2p0, (C2)\nwhere\nD=(4sG,xsG,z+πsG,xsG,z+ 2sG,ysG,z+ 2sG,xsG,y\n−sM,zsM,y)\u0012m\n2πkBT\u00131/2\n+sWt,xsWt,zτ0,Wt\nhc\n+sWtc,x sWtc,z τ0,Wtc\nhc+ (sGt,y+sWt,y−sWtc,y ),\n(C3)\nwhere, the arm length from the suspension point to the\ncenter of the TM is l= (sG,x+sM,x)/2 = 38 .0605 mm,\nand the dimensions are given as sM,x×sM,y×sM,z=\n61.491×8.000×12.000 mm3,sG,x×sG,y×sG,z=\n14.630×19.756×27.138 mm3,sGt,x×sGt,y×sGt,z=\n14.630×0.486×12.003 mm3,sWt,x×sWt,y×sWt,z =\n14.630×0.200×12.003 mm3,sWtc,x×sWtc,y×sWtc,z =\n14.630×0.606×15.139 mm3. The diffusion time τ0,Wtand\nτ0,Wtc useτcuboid in Appendix D. The parameters here\nmay differ somewhat from those in Refs. [19, 37, 38]. The\nparameters in this paper are for reference only.\nConsidering that the base damping here is almost neg-\nligible, theoretically, when the diffusion time used in the11\nnon-isothermal model is the same as the diffusion time\nformula used in the bi-plate finite open model, the ratio of\nthe power spectral density of fluctuating forces between\nthe two models is approximately 7:6. When the damp-\ning noise is equal to the intrinsic damping thermal noise,\nunder the same pressure, the ratio of the gap spacing hc\nconstraints between the two models is approximately 7:6.\nIn other words, the latter is about 1/7 lower than the for-\nmer. As long as the base damping can be neglected, this\nresult will not change with variations in experimental pa-\nrameters.Appendix D: Diffusion Time for Bi-plate Finite\nOpen Systems\nReference [39] provides theoretical result for the diffu-\nsion time τcuboid of square bi-plates:\nτx=bp\n72πkBT/m1\nς\u0000√\n1 +ς2−ς\u0001\n×2 arctan(1 /ς) + 2ςlnς−ςln(1 + ς2)\n2ς−2√\n1 +ς2+ ln\u0000\n1 +√\n1 +ς2\u0001\n−lnς,(D1)\nwhere ς=h/2b,aandbare the side lengths of the square\nplates along the xandyaxes, respectively. Similarly, τy\ncan be calculated, and the total diffusion time is given\nby:\nτcuboid =a\na+bτx+b\na+bτy. (D2)\nAdditionally, Ref. [39] also provides diffusion time τcircle\nof circular bi-plate.\nThe bi-plate model has no special requirements on the\nshape of the plate. For example, it can be a ring, a tri-\nangle, or a more complex shape. However the diffusion\ntime needs to be calculated through numerical simula-\ntion. However, in sensor cores, the gap boundary is dif-\nferent. 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Watson Research Center, Yorktown Heights, NY 10598 , USA\n(Dated: November 18, 2018)\nFerromagnetic resonance has been used to study the magnetic properties and magnetization dy-\nnamics of polycrystalline Fe 1−xVxalloy films with 0 ≤x <0.7. Films were produced by co-\nsputtering from separate Fe and V targets, leading to a compo sition gradient across a Si substrate.\nFMR studies were conducted at room temperature with a broadb and coplanar waveguide at fre-\nquencies up to 50 GHz using the flip-chip method. The effective demagnetization field 4 πMeffand\nthe Gilbert damping parameter αhave been determined as a function of V concentration. The\nresults are compared to those of epitaxial FeV films.\nI. INTRODUCTION\nA decade ago, it was predicted that a spin polar-\nized current from a relatively thick ferromagnet (FM)\ncould be used to switch the magnetization of a thin FM\n[1]. Since then, this effect, known as spin-transfer, has\nbeen demonstrated in spin-valves [2] and magnetic tun-\nnel junctions [3]. In a macrospin model with collinear\nlayer magnetizations, there is a threshold current den-\nsityJcfor an instability necessary for current-induced\nmagnetization switching of the thin FM layer [1, 4]:\nJc=2eαMstf(Hk+2πMs)\n/planckover2pi1η, (1)\nwhereαisthedampingconstant. tfandMsarethethick-\nness and the magnetization density of the free layer, re-\nspectively. Hkis the in-plane uniaxial anisotropy field. η\nis the currentspin-polarization. In orderfor spin-transfer\nto be used in high density memory devices Jcmust be re-\nduced. From Eq. 1 it is seen that this can be achieved by\nemploying materials with low Msandαin spin-transfer\ndevicesor, equivalently materialswith lowGilbert damp-\ning coefficients, G = αMs(gµB//planckover2pi1).\nVery recently, an experimental study of epitaxial FeV\nalloy thin films demonstrated a record low Gilbert damp-\ning coefficient [5]. This material is therefore of interest\nfor spin transfer devices. However, such devices are gen-\nerally composed of polycrystalline layers. Therefore it is\nof interest to examine polycrystalline FeV films to assess\ntheir characteristics and device potential.\nIn this paper, we present a FMR study of thin poly-\ncrystalline Fe 1−xVxalloy films with 0 ≤x <0.7 grown\nby co-sputtering. The FeV layers were embedded be-\ntween two Ta |Cu layers, resulting in the layer structure\n||5 Ta|10 Cu|FeV|5 Cu|10 Ta||, where the numbers are\nlayer thickness in nm. FeV polycrystalline films were\nprepared by dc magnetron sputtering at room tempera-\nture from two separate sources, oriented at a 45oangle\n(Fig. 1a). The substrate, cut from a Silicon wafer with\n100 nm thermal oxide, was 64 mm long and about 5 mm\nwide. The Fe and V deposition rates were found to vary/s49 /s50 /s51 /s52/s45/s50/s45/s49/s48/s40/s98/s41\n/s32/s86/s32/s116/s97/s114/s103/s101/s116\n/s52/s53/s111\n/s32\n/s32/s115/s117/s98/s115/s116/s114/s97/s116/s101/s70/s101/s32/s116/s97/s114/s103/s101/s116/s40/s97/s41\n/s72\n/s114/s101/s115/s65/s98/s115/s111/s114/s112/s116/s105/s111/s110/s32/s108/s105/s110/s101/s32/s32/s40/s97/s117/s41\n/s72\n/s97/s112/s112/s32/s32/s40/s107/s79/s101/s41/s49/s52/s32/s71/s72/s122\n/s32/s32/s32\n/s32/s120 /s61/s48/s46/s51/s55/s120 /s61/s48/s46/s53/s50\n/s120 /s61/s48/s46/s49/s57/s72\n/s115/s117/s98/s115/s116/s114/s97/s116/s101/s32/s104/s111/s108/s100/s101/s114\n/s97/s120/s105/s115\nFIG. 1: a) The co-sputtering setup. b) Typical absorp-\ntion curves at 14 GHz for a selection of ||Ta|Cu|7.5 nm\nFe1−xVx|Cu|Ta||films, with x=0.19, 0.37 and 0.52. The res-\nonance field Hresand the linewidth ∆ Hare indicated.\nlinearly across the wafer. The Fe and V rates were then\nadjusted to produce a film in which xvaries from 0.37\nto 0.66 across the long axis of the wafer. The base pres-\nsure in the UHV chamber was 5 ×10−8Torr and the\nAr pressure was set to 3.5 mTorr. The FeV was 7.5 nm\nin thickness, varying by less than 0.3 % across the sub-\nstrate. An Fe 1−xVxfilm 3 nm thick was also fabricated.\nTo produce films with x <0.30 the rate of the V source\nwas decreased. Finally, pure Fe films with a thickness\ngradient ranging from 7 nm to 13.3 nm were deposited.\nTheFMRmeasurementswerecarriedoutatroomtem-\nperature using a coplanar wave-guide (CPW) and the\nflip-chip method. Details of the experimental setup and\nofthe CPWstructuralcharacteristicscan be found in [6].\nAdc magnetic field, up to 10 kOe, wasapplied in the film\nplane, perpendiculartotheacmagneticfield. Absorption\nlines at frequencies from 2 to 50 GHz were measured by\nmonitoring the relative change in the transmitted signal\nas a function of the applied magnetic field.2\n/s48/s46/s48 /s48/s46/s51 /s48/s46/s54/s48/s53/s49/s48/s49/s53/s50/s48\n/s48/s46/s48 /s48/s46/s51 /s48/s46/s54/s50/s46/s48/s53/s50/s46/s49/s48/s50/s46/s49/s53/s49/s50/s51\n/s120\n/s32/s52 /s77\n/s101/s102/s102/s32/s32/s40/s107/s71/s41/s55/s46/s53/s32/s110/s109/s32/s70/s101\n/s49/s45 /s120/s86\n/s120\n/s51/s32/s110/s109/s32/s70/s101\n/s48/s46/s54/s51/s86\n/s48/s46/s51/s55\n/s49/s50/s46/s57/s32/s110/s109/s32/s70/s101\n/s120/s32\n/s32/s103/s45/s102/s97/s99/s116/s111/s114/s40/s98/s41/s40/s99/s41/s32\n/s32\n/s32/s72\n/s114/s101/s115/s32/s32/s40/s107/s79/s101/s41\n/s49/s49/s32/s71/s72/s122/s40/s97/s41\nFIG. 2: a) The resonance field at 11 GHz versus xand b) the\neffective demagnetization field versus x. The solid line is a\nguide to the eye. c) The Land´ e gfactor as a function of x.\nThe dotted line shows the g-factor value of bulk Fe.\nII. RESULTS\nTypical absorption lines at 14 GHz of selected FeV al-\nloyfilmsareshowninFig. 1b. Thelinesarelorentzianfor\nmost frequencies. At a fixed frequency, the FMR absorp-\ntion decreaseswith increasingV content. The FMR peak\nof a film 7.5 nm thick with x= 0.66 is about 100 times\nsmallerthanthatofapureFeofthesamethickness. This\nis accompaniedby a shift of Hrestowardshigher field val-\nues (Fig. 2a). The effective demagnetization field 4 πMeff\nand the Land´ eg-factorgwere determined by fitting the\nfrequency dependence of the resonance field Hresto the\nKittel formula [7]:\nf2=/parenleftBiggµB\nh/parenrightBig2\nHres(Hres+4πMeff),(2)\nwhere the effective demagnetization field is:\n4πMeff= 4πMs−H⊥. (3)\nNote that in the absence of a perpendicular anisotropy\nfieldH⊥, the effective field would be directly related to\nMs. The dependence of 4 πMeffon V concentration is\nshown in Fig. 2c. As xincreases the effective demagneti-\nzation field decreases dramatically, going from about 16\nkG forx= 0 to 1.1 kG for x= 0.66. Note that the effec-\ntive demagnetization field of the 7.5 nm Fe film is about\n25 % lower than that of bulk Fe (21.5 kG). The 12.9\nnm Fe film exhibits a larger 4 πMeff, which is, however,\nstill lower than 4 πMsof the bulk material. Similarly,\nthe 4πMeffof an Fe 0.63V0.37film is thickness dependent:\ndecreasing with decreasing layer thickness.\nThe Land´ eg-factor increases monotonically with in-\ncreasing V concentration (Fig. 2b). The minimum g-\nfactor is measured for the Fe film: g= 2.11±0.01,\nwhich is slightly larger than the value of bulk material\n(g= 2.09). Note that gof a Fe film 12.9 nm thick is\nequal to that of Fe bulk. However, the g-factor of the/s48 /s50/s48 /s52/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56\n/s48/s46/s48/s48/s46/s51/s48/s46/s54\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s48/s46/s53/s49/s46/s48/s49/s46/s53 /s40/s99/s41/s120 /s32/s61/s32/s48/s46/s53/s50/s120/s32 /s61/s32/s48/s46/s52/s51/s120/s32 /s61/s32/s48/s40/s98/s41/s32/s40/s49/s48/s45/s50\n/s41/s76/s105/s110/s101/s119/s105/s100/s116/s104/s32/s32/s40/s107/s79/s101/s41/s32\n/s32/s72 /s32/s32/s40/s107/s79/s101/s41\n/s72\n/s48/s72\n/s48/s72\n/s49/s52/s32/s32\n/s32/s40/s97/s41\n/s120/s32 /s32/s102/s32/s32/s40/s71/s72/s122/s41\nFIG. 3: a)Frequency dependence of the linewidth for 7.5 nm\nFe1−xVxalloy film with x=0, 0.43 and 0.52. The solid lines\nare the best linear fit of the experimental data. b) ∆ H14, the\nlinewidth at 14 GHz , and ∆ H0are shown as a function of x.\nc) The magnetic damping parameter versus V concentration.\nFe0.63V0.37, does not appear to be thickness dependent:\nthe 3 nm Fe 0.63V0.37layer has about the same gvalue\nthan the 7.5 nm Fe 0.63V0.37layer.\nThe half-power linewidth, ∆ H, was studied as a func-\ntion of the frequency and of the V concentration. Fig.\n3b shows the dependence of the FMR linewidth on xat\n14 GHz. The general trend is that ∆ Hincreases with\nx. However, there are two regimes. For x >0.4, the\nlinewidth depends strongly on x, increasing by a factor\n5 whenxis increased from 0.4 to 0.66. The dependence\nof the linewidth on xis more moderate for the films with\nx <0.4: it increases by about 30 %. For all samples,\nthe linewidth scales linearly with the frequency. A least\nsquare fit of ∆ H(f) gives ∆ H0, the intercept at zero\nfrequency, and the Gilbert damping parameter αwhich\nis proportional to the slope: d∆H/df= (2h/gµB)α[8].\n∆H0is typically associated with an extrinsic contribu-\ntion to the linewidth and related to magnetic and struc-\ntural inhomogeneities in the layer. For two samples with\nthe highest Vanadium concentration, x= 0.60, 0.66, the\nlinewith is dominated by inhomogeneous broadening and\nit wasnot possible to extract α. Asxincreases, ∆ H0and\nαincreases. The damping parameter and ∆ H0remain\npractically unchanged for x≤0.4 and when x >0.4,\nboth the intercept and the slope of ∆ Hversusfincrease\nrapidly.\nIII. DISCUSSION\nSeveral factors can contribute to the dependence of\n4πMeffon the V concentration. First, the decrease of\nthe effective demagnetizationfield canbe associatedwith\nthe reduction of the alloy magnetization density Mssince\nthe Fe content is reduced. In addition, a neutron scat-\ntering study showed that V acquires a magnetic moment3\nantiparallel to the Fe, and that the Fe atom moment de-\ncreases with increasing V concentration [9]. The Curie\ntemperature of Fe 1−xVxdepends on x. In fact, Tcfor\nx=0.65 is near room temperature [10]. It is important to\nmention that a factor that can further decrease 4 πMeff\nis an out-of-plane uniaxial anisotropy field H⊥(Eq. 3).\nIn thin films, the perpendicular anisotropy field is com-\nmonly expressed as H⊥= 2K⊥/(Mst), where K⊥>0\nis the anisotropy constant and tthe ferromagnetic film\nthickness [11]. In this simple picture, it is assumed that\nK⊥is nearly constant over the thickness range of our\nfilms. This anisotropy can be associated with strain\ndue to the lattice mismatch between the FeV alloy and\nthe adjacent Cu layers and/or with an interface contri-\nbution to the magnetic anisotropy. For Fe films with\nt= 7.5 and 12.9 nm, a linear fit of 4 πMeffversus 1/t\ngives 4πMs= 20.2 kG and K⊥= 2.5 erg/cm2. The\nvalue extracted for 4 πMsis in the range of the value of\nFebulk. AsimilaranalysisconductedonFe 0.63V0.37films\nof thickness t=3 and 7.5 nm gives 4 πMs= 12.2 kG and\nK⊥= 0.1 erg/cm2. The result suggests that the surface\nanisotropy constant decreases with increasing x.\nIV. SUMMARY\nThe effective demagnetization field of the polycrys-\ntalline Fe 1−xVxalloy films decreases with increasing xandalmostvanishesfor x≈0.7. AFMRstudyonepitax-\nial films haveshown a similar xdependence of 4 πMeff[5].\nUsing the value of Mscalculated in the analysis above,\nwe estimate the Gilbert damping constant of a 7.5 nm Fe\nlayer and 7.5 nm Fe 0.63V0.37alloy film to be G Fe= 239\nMHz and G FeV= 145 MHz respectively. The decrease of\nthe effective demagnetization field of Fe 1−xVxwith in-\ncreasing xis accompanied by a decrease of the Gilbert\ndamping constant. A similar xdependence of G was\nobserved in epitaxial films [5]. The authors explained\nthe decrease of G by the reduced influence of spin-orbit\ncoupling in lighter ferromagnets. Note that the Gilbert\ndampingofourfilmsislargerthanwhatwasfoundforthe\nepitaxial films (G=57 MHz for epitaxial Fe 8 nm thick).\nWe note that the Fe 0.63V0.37alloy film, which has\n4πMsapproximatly the same as that of Permalloy, has a\nmagnetic damping constant of the same order than that\nof Py layer in a similar layer structure [12]. Hence, with\ntheir low Msandα, polycrystalline FeV alloy films are\npromising materials to be integrated in spin-tranfer de-\nvices.\n[1] J. C. Slonczewski, J. Magn. Magn. Mater. 159(1-2), L1\n(1996) ; L. Berger, Phys. Rev. B 54(12), 9353 (1996).\n[2] see, for example, J. A. Katine et al., Phys. Rev. Lett. 84,\n3149 (2000) ; B. Oezyilmaz et al., Phys. Rev. Lett. 91,\n067203 (2003).\n[3] see, for example, G. D. Fuchs et al., J. Appl. Phys. 85\n(7), 1205 (2004).\n[4] J. Z. Sun, Phys. Rev. B 62(1), 570 (2000).\n[5] C. Scheck et al., Phys. Rev. Lett. 98, 117601 (2007).\n[6] J-M. L. Beaujour et al., Europhys. J. B, DOI:\n10.1140/epjb/e2007-00071-1 (2007).\n[7] C. Kittel in Introduction to Solid State Physics, Ed. 7,\np.505.[8] see, for example, D. L. Mills and S. M. Rezende in Spin\nDynamics in Confined Magnetic Structures II (Eds. B.\nHillebrands andK.Ounadjela), pp.27-58, (Springer, Hei-\ndelberg 2002).\n[9] I. Mirebeau, G. Parette, and J. W. Cable. J. Phys. F:\nMet. Phys. 17, 191 (1987).\n[10] Y. Kakehashi, Phys. Rev. B 32(5), 3035 (1985).\n[11] Y. K. Kim and T. J. Silva, Appl. Phys. Lett. 68, 2885\n(1996).\n[12] S. Mizukami et al., J. Magn. Magn. Mater. 239, 42\n(2002)."    },    {        "title": "2009.10832v2.Sharp_exponential_decay_rates_for_anisotropically_damped_waves.pdf",        "content": "SHARP EXPONENTIAL DECAY RATES FOR ANISOTROPICALLY\nDAMPED WAVES\nBLAKE KEELER AND PERRY KLEINHENZ\nAbstract. In this article, we study energy decay of the damped wave equation on com-\npact Riemannian manifolds where the damping coe\u000ecient is anisotropic and modeled by a\npseudodi\u000berential operator of order zero. We prove that the energy of solutions decays at\nan exponential rate if and only if the damping coe\u000ecient satis\fes an anisotropic analogue of\nthe classical geometric control condition, along with a unique continuation hypothesis. Fur-\nthermore, we compute an explicit formula for the optimal decay rate in terms of the spectral\nabscissa and the long-time averages of the principal symbol of the damping over geodesics, in\nanalogy to the work of Lebeau for the isotropic case. We also construct genuinely anisotropic\ndampings which satisfy our hypotheses on the \rat torus.\n1.introduction\nLetpM;gqbe a smooth, compact manifold without boundary and let \u0001 gbe the as-\nsociated Laplace-Beltrami operator (taken with the convention that \u0001 g¤0). Suppose\nW:L2pMqÑL2pMqis bounded and nonnegative. We consider the generalized damped\nwave equation given by#\nB2\ntu\u0001\u0001gu\u00002WBtu\u00100\npu;Btuq|t\u00100\u0010pu0;u1q;(1.1)\nforpu0;u1qTPH:\u0010H1pMq`L2pMq, where His taken with the natural norm\n}pu0;u1qT}2\nH\u0010}p1\u0001\u0001gq1\n2u0}2\nL2pMq\u0000}u1}2\nL2pMq:\nWe study the asymptotic properties of the energy of solutions to (1.1) as tÑ8 . Here, the\nenergy is de\fned by\nEpu;tq\u00101\n2»\nM|rgupt;xq|2\u0000|Btupt;xq|2dvgpxq; (1.2)\nwheredvgis the Riemannian volume form on M:It is straightforward to compute that\nd\ndtEpu;tq\u0010\u0001 2RexWBtu;Btuy¤0; (1.3)\nwherex\u0004;\u0004ydenotes the inner product on L2pM;gq:Thus, the assumption that Wis a non-\nnegative operator guarantees that the energy of solutions to (1.1) experiences dissipation,\nbut (1.3) does not indicate how quickly the energy decays as tÑ8 . The most straightfor-\nward type of decay is uniform stabilization, i.e. when there exists a constant C¡0 and a\nreal-valued function tÞÑrptqwithrptqÑ0 astÑ8 such that\nEpu;tq¤CrptqEpu;0q:\nDate : March 22, 2022.\n1arXiv:2009.10832v2  [math.AP]  21 Mar 20222 B. KEELER AND P. KLEINHENZ\nIn the case where Wacts via multiplication by a bounded, nonnegative function b, a great\ndeal is known about energy decay rates. Perhaps the most well known result states that\nsolutions to (1.1) experience uniform stabilization with an exponential rate if and only if W\nsatis\fes the geometric control condition (GCC) [RT75,Ral69]. The GCC is satis\fed if there\nexists some T¡0 such that every geodesic with length at least Tintersects the set where bis\nbounded below by some positive constant. Many other works have proved weaker decay rates\nin the setting where the GCC is not satis\fed (c.f. [Leb96] [Bur98] [BZ16] [Chr07] [Chr10]\n[BC15] [LR05] [BH07]). With more restrictive assumptions on WandM, one can obtain\nsharp decay rates (c.f. [AL14] [Sta17] [LL17] [Kle19a] [DK20] [Kle19b], [Sun22] [DJN19a]\n[Jin20]).\nA distinct shortcoming of the multiplicative case is that the damping force is sensitive only\nto positional information and not to the direction in which the solution propagates. For this\nreason, one can classify multiplicative damping as an isotropic force, but many physical sys-\ntems which experience anisotropic damping forces are studied in materials science, physics,\nand engineering [KKBH16,Cra08,JSC11]. However, a general analysis of the damped wave\nequation in the anisotropic case has not yet been done. This article aims to address this gap\nin the literature by studying the case where the anisotropic damping force is modeled by a\npseudodi\u000berential operator.\nIt is common in analysis of the generalized damped wave equation (1.1) to assume that W\ntakes the form of a square, i.e. W\u0010B\u0006Bfor some bounded operator B(c.f. [AL14]). This\nguarantees that Wis nonnegative and enables the use of certain techniques from spectral\ntheory. We allow for a slightly more general assumption here, namely that Wtakes the form\nW\u0010N°\nj\u00101B\u0006\njBjfor some \fnite collection tBjuN\nj\u00101\t0\nc`pMq, where \t0\nclpMqdenotes the space\nof classical pseudodi\u000berential operators on Mof order zero with polyhomogeneous symbols.\nThe corresponding space of symbols is denoted S0\nc`pT\u0006Mq. We note that allowing Wto take\nthe form of a sum of squares is indeed a generalization, since it is not generically possible\nto write°N\nj\u00101B\u0006\njBjasB\u0006Bfor someBP\t0\nc`pMq, since the pseudodi\u000berential calculus only\nallows for the computation of square roots modulo a smoothing remainder. We denote by\nwPS0\nc`pT\u0006Mqthe principal symbol of W, taken to be positively \fber-homogeneous of degree\n0 outside a small neighborhood of the zero section in T\u0006M:That is,wpx;s\u0018q\u0010wpx;\u0018qfor\nalls¡0 and all|\u0018|¥cfor somec¡0 which can be chosen to be arbitrarily small. This\nhomogeneity allows us to treat was a function on the co-sphere bundle\nS\u0006M:\u0010tpx;\u0018qPT\u0006M:|\u0018|g\u00101\n2u;\nwhere the choice of1\n2is made for the sake of convenience in later arguments.\nWe now state the required assumptions for the main theorem. The \frst is an anisotropic\nanalogue of the classical geometric control condition.\nAssumption 1 (Anisotropic Geometric Control Condition ).Let'tdenote the lift of\nthe geodesic \row to T\u0006M. Assume that there exists a compact neighborhood Kof the zero\nsection inT\u0006Mand constants T0;c¡0such that for every px0;\u00180qPT\u0006MzK,\n1\nTT»\n0wp'tpx0;\u00180qqdt¥c;forT¥T0:SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 3\nThat is, the long-time averages of wover geodesics are uniformly bounded below. In this\ncase, we say Wsatis\fes the anisotropic geometric control condition (AGCC).\nNote that in the case of multiplicative damping, Assumption 1 is equivalent to classical\ngeometric control condition in [RT75].\nThe second key assumption requires that the kernel of Wcontain no nontrivial eigenfunc-\ntions of \u0001 g.\nAssumption 2. IfvPL2pMqsatis\fes\u0001\u0001gv\u0010\u00152vwith\u0015\u00180, thenWv\u00180:\nIn the case where W\u0010bpxq, Assumption 2 is satis\fed when bis supported on any open set,\nsince eigenfunctions of \u0001 gcannot vanish on open sets by the unique continuation principle\n(c.f. [RT75]). It is for this reason that we sometimes refer to Assumption 2 as a \\unique\ncontinuation hypothesis.\"\nWith these assumptions stated, we then have the following equivalence.\nTheorem 1. All solutions uto(1.1) withWP\t0\nc`pMqsatisfy\nEpu;tq¤Ce\u0001\ftEpu;0q (1.4)\nfor someC; \f¡0and for all t¥0if and only if Wsatis\fes Assumptions 1 and 2.\nIn other words, solutions experience uniform stabilization at an exponential rate if and only\nifWsatis\fes Assumptions 1 and 2.\nThe existing literature on anisotropic damping coe\u000ecients is quite limited. In the con-\ntext of pseudodi\u000berential W, Sj ostrand [Sj o00] studied the asymptotic distribution of eigen-\nvalues of the stationary damped wave equation. Christianson, Schenck, Vasy, and Wun-\nsch [CSVW14] showed that a polynomial resolvent estimate for a related complex absorbing\npotential problem gives another polynomial resolvent estimate of the same order for the\nstationary damped wave equation. However, these results do not consider anisotropic damp-\ning in a time-dependent setting and so do not provide energy decay results. Theorem 1\naddresses this gap in the literature by providing conditions which guarantee exponential\nuniform stabilization, in analogy to the classical result of Rauch and Taylor [RT75].\nSince Theorem 1 only claims the existence of some exponential decay rate \f, a natural\nquestion is to determine the optimal rate of decay for a given damping coe\u000ecient. Given a\n\fxedWP\t0\nc`pMq, we de\fne the best exponential decay rate as in [Leb96] via\n\u000b:\u0010supt\fPR:DC¡0 such that Epu;tq¤Ce\u0001\ftEpu;0q @uwhich solve (1.1) u:(1.5)\nOur next result shows that \u000bcan be expressed in terms two fundamental quantities: the\nspectral abscissa, and the long-time averages of the principal symbol of Wover geodesics.\nThe spectral abscissa is de\fned with respect to\nAW:\u0010\u0002\n0 Id\n\u0001g\u00012W\n;\nwhich is the in\fnitesimal generator of the solution semigroup for (1.1). For each R¡0, we\nset\nDpRq\u0010suptRep\u0015q:|\u0015|¡R; \u0015PSpecpAWqu:\nWe then de\fne the spectral abscissa as\nD0\u0010lim\nRÑ0\u0000DpRq: (1.6)4 B. KEELER AND P. KLEINHENZ\nWe also de\fne for tPRthe time-average of the damping along geodesics\nLptq\u0010 inf\npx;\u0018qPS\u0006M1\ntt»\n0wp'spx;\u0018qqds;\nand the long-time limit\nL8\u0010lim\ntÑ8Lptq: (1.7)\nWe can then characterize \u000bas follows.\nTheorem 2. The best exponential decay rate for solutions to (1.1) withWP\t0\nc`pMqis\n\u000b\u00102 mint\u0001D0;L8u;\nwhereD0andL8are de\fned by (1.6) and(1.7) , respectively.\nRemark 1.1. It is of considerable note that the formula for the optimal decay rate here\nis an exact analogy of the multiplicative case studied by Lebeau (c.f. [Leb96, Theorem 2]).\nWhile the broad structure of our proof is similar, there are portions of the analysis which\ndiverge greatly, particularly in Section 3 where we investigate the action of pseudodi\u000berential\noperators on Gaussian beams.\nRemark 1.2. Theorem 2 is signi\fcantly stronger than Theorem 1, although this is not\nimmediately obvious. The main portion of this article is dedicated to the proof of Theorem 2.\nWe then show that Theorem 2 implies Theorem 1 in Section 6.\nTheorems 1 and 2 \ft into a broad range of existing results which attempt to reproduce\nthe equivalence of the GCC and exponential decay under modi\fed hypotheses on the damp-\ning. It is not uncommon for such statements to be somewhat inconclusive. For example,\nwhen the damping is allowed to be time-dependent [LRLTT17] showed that for time peri-\nodic damping, the GCC is indeed equivalent to exponential decay, but it is not currently\nknown if this result is true for non-periodic damping. In the setting where the damping\nis allowed to take negative values (commonly called \\inde\fnite damping\"), the state of the\nart is similarly mixed. If Mis an open domain in RnwithC2boundary, [LRZ02] proves\nan exponential decay rate provided that the damping is positive in a neighborhood of BM\n(which implies the GCC) and inf xPMWpxqis not too negative. However, it is currently not\nknown if an appropriate generalization of the GCC is equivalent to exponential stability in\nthe inde\fnite case. The limitations of these results illustrate that seemingly simple changes\nto hypotheses on the damping coe\u000ecient can create substantial barriers to reproducing the\nclassical equivalence theorem. So, the fact that Theorem 1 provides a direct analogy of the\nGCC for pseudodi\u000berential damping which is equivalent to exponential decay is somewhat\nexceptional. Note also that these other generalizations do not possess an analogy of Theo-\nrem 2. Although [LRZ02] and [LRLTT17] both provide a rate for the exponential decay, it\nis not shown to be sharp.\nOur \fnal result concerns Assumption 2, which is necessary in order to obtain Theorem 1.\nTo see this, suppose that vsatis\fes\u0001\u0001gv\u0010\u00152vwith\u0015\u00180 andWv\u00100. Then, the function\nupt;xq\u0010eit\u0015vpxq;\nsolves (1.1), but has energy Epu;tq\u0010\u00152}v}2\nL2pMqfor allt. As previously mentioned, when W\nis a multiplication operator supported on any open set, unique continuation results guaranteeSHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 5\nthatWdoes not annihilate any eigenfunctions of \u0001 g, making Assumption 2 unnecessary.\nHowever, in the pseudodi\u000berential setting, verifying this assumption is more di\u000ecult.\nA special case in which Assumption 2 is easy to check is when Wis constructed from\nfunctions of \u0001 g. Suppose W\u0010B\u0006BwithB\u0010fp\u0001\u0001gq, wheref:RÑRsatis\fes a\n\\symbol-type\" estimate of the form\n|Bk\nsf|¤Cp1\u0000|s|q\u0001k\nfor anyk:The functional calculus of Strichartz [Str72] shows that Wis pseudodi\u000berential\nwhen constructed in this way. The calculus also immediately implies that Assumption 2\nholds as long as fdoes not vanish on the spectrum of \u0001\u0001g, since for any eigenfunction v\nwith eigenvalue \u0015, we haveWv\u0010fp\u0015q2v. However, damping coe\u000ecients constructed in\nthis fashion are somewhat uninteresting in the sense that the principal symbol is a function\nof|\u0018|2\ng, and therefore independent of direction. Thus, examples of this type are not truly\nanisotropic. In general, it is not obvious that one can always construct nontrivial anisotropic\nexamples satisfying Assumption 2, although we expect that a rich class of examples do indeed\nexist. The following theorem demonstrates that one can always produce such examples when\npM;gqis real analytic.\nTheorem 3. IfpM;gqis compact and real analytic, then there exists WP\t0\nclpMqof the\nformW\u0010°N\nj\u00101B\u0006\njBj,such that for each xPM, the principal symbol of Wvanishes on an\nopen cone in T\u0006\nxMand for any van eigenfunction of \u0001g;Wv\u00180:\nThe fact that the principal symbol vanishes in an open cone of directions at each point implies\nthat theWin this theorem is not built from functions of \u0001 g, excluding the somewhat trivial\ncase discussed previously. Using the machinery developed in the proof of Theorem 3, we are\nable to produce explicit examples on the \rat 2-torus of operators WP\t0\nclwhich satisfy both\nAssumptions 1 and 2. This construction is presented in Section 7.\nRemark 1.3. As mentioned previously, Assumption 2 follows directly from the geometric\ncontrol condition in the multiplicative case. One might hope that this could be generalized\nto the scenario where Wis pseudodi\u000berential, but this problem is exceedingly di\u000ecult in\ngeneral. The only result of this type known to the authors is that of [DJN19b], which uti-\nlized the fractal uncertainty principle to show that when Mis an Anosov surface, vis an\neigenfunction of the Laplacian, and the principal symbol of Wis not identically zero, then\nthere is a quantitative lower bound on the size of Wv. So for Anosov surfaces, Assumption 1\nimplies Assumption 2. But even in this specialized case, the proof involves highly sophisti-\ncated techniques. An analysis of the general case is an open problem, and we suspect that\nit would be a signi\fcant undertaking.\n1.1.Outline of the article. The majority of this article is devoted to the proof of Theo-\nrem 2, which spans Sections 2, 3 4, and 5. The primary tool is microlocal defect measures.\nWe begin in Section 2 by analyzing the behavior of defect measures associated to sequences\nof solutions to (1.1) when propagated by the Hamiltonian \row for P\u0010B2\nt\u0001\u0001g. The formula\nwe produce follows largely from direct computations and measure-theoretic arguments, in\nanalogy to [Leb96,Kle17]. In Section 3, we perform a detailed study of the action of certain\npseudodi\u000berential operators on coherent states, which is a critical component of constructing\nquasimodes for (1.1). This analysis is a key point where the pseudodi\u000berential case becomes\nsigni\fcantly more di\u000ecult than the multiplicative setting. In Section 4, we then combine\nthe results of Sections 2 and 3 to produce quasimodes for the damped wave equation whose6 B. KEELER AND P. KLEINHENZ\nenergy is strongly localized near a \fxed geodesic. The analysis of these quasimodes allows\nus to prove that \u000b¤2 mint\u0001D0;L8u. The proof of Theorem 2 is completed in Section 5,\nwhere we prove the lower bound \u000b¥2 mint\u0001D0;L8u:This section follows in close analogy\nto [Leb96], and so we omit some of the more technical details.\nIn Section 6, we show that Theorem 2 implies Theorem 1. This follows directly from\nspectral theory analysis.\nFinally, in Section 7, we restrict to the case of real analytic manifolds to produce some\nexamples. We provide a fairly generic condition on pseudodi\u000berential operators which guar-\nantees that they satisfy Assumption 2. We also show that one can always produce examples\nwhich fall into this category, thus proving Theorem 3. We then conclude by constructing\nsome explicit examples on the \rat torus which satisfy both Assumptions 1 and 2.\n1.2.Acknowledgements. The authors would like to thank J. Wunsch and A. Vasy for\ntheir frequent helpful comments throughout the course of this project. Wunsch's suggestions\nregarding the analytic wave front set argument in Section 7 were particularly useful. We are\nalso grateful to M. Taylor for helping us better understand the proof of his original result\nwith J. Rauch on exponential energy decay in [RT75]. We also wish to thank H. Christianson,\nY. Canzani and J. Galkowski for their comments on an earlier version of this paper. Finally,\nBK was supported in part by NSF Grant DMS-1900519 through his advisor Y. Canzani.\n2.Propagation of the microlocal defect measure\nIn this section, we compute the propagation of microlocal defect measures associated to a\nsequence of solutions of (1.1) with WP\t0\nc`pMq. We begin by noting some general facts\nabout microlocal defect measures. We will not prove these here, and we direct the reader to\nthe seminal article of G\u0013 erard [G\u0013 er91] for details and proofs. Here we consider defect measures\nas measures on S\u0006pR\u0002Mq\u0010tpt;x;\u001c;\u0018qPT\u0006pR\u0002Mq:\u001c2\u0000|\u0018|2\ng\u00101\n2u;the co-sphere bundle\nofR\u0002Mtreated as a Riemannian manifold with the the product metric. For any sequence\ntukuHmpR\u0002Mqconverging weakly to 0, there exists a sub-sequence tukjuand a positive\nRadon measure \u0017onS\u0006pR\u0002Mqsuch that for any AP\t2m\nc`pR\u0002Mqwith compact support\nint, we have\nlim\njÑ8xAukj;ukjy\u0010»\nS\u0006pR\u0002Mqad\u0017;\nwherex\u0004;\u0004ydenotes the standard inner product on L2pR\u0002Mq, which is the natural pairing\nbetweenH\u00011pR\u0002MqandH1pR\u0002Mq:Iftukuhas a defect measure without the need for\npassing to a subsequence, we say that tukuispure.\nFrom this point on, we specialize to the case where tukuH1pR\u0002Mqis a pure sequence\nof solutions to the damped wave equation converging weakly to zero, with associated defect\nmeasure\u0017. A key piece of the proof of Theorem 2 is the propagation of this defect measure\nunder the Hamiltonian \row on T\u0006pR\u0002Mqgenerated by ppt;x;\u001c;\u0018q\u0010|\u0018|2\ng\u0001\u001c2, the principal\nsymbol ofP\u0010B2\nt\u0001\u0001g. We denote this \row by \b s, and it can be written as\n\bspt;x;\u001c;\u0018q\u0010pt\u00012s\u001c;\u001c;'spx;\u0018qq;\nwhere we recall that 'sis the geodesic \row on T\u0006M. More precisely, the propagation of the\ndefect measure refers to the behavior of \u0017under the pushforward p\bsq\u0006.\nWe now show that there exists a smooth function sÞÑGsPC8pS\u0006pR\u0002Mqqsuch that\np\bsq\u0006\u0017\u0010G\u0001s\u0017, and thatGscan be de\fned as the solution to a certain di\u000berential equation.SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 7\nLemma 2.1. For any \fxed pt;x;\u001c;\u0018qPS\u0006pR\u0002Mq, de\fneGspt;x;\u001c;\u0018qas the solution to\nthe initial value problem\n#\nG0pt;x;\u001c;\u0018q\u00101;\nBsGspt;x;\u001c;\u0018q\u0010tp;Gsupt;x;\u001c;\u0018q\u00004\u001cwpx;\u0018qGspt;x;\u001c;\u0018q;(2.1)\nwherepandware the principal symbols of P\u0010 B2\nt\u0001\u0001gandW, respectively. Then,\np\bsq\u0006\u0017\u0010G\u0001s\u0017. Equivalently, for any bPC8pS\u0006pR\u0002Mqqwhich is compactly supported\ninpt;xq,»\nS\u0006pR\u0002Mqb\u0005\bsd\u0017\u0010»\nS\u0006pR\u0002MqbG\u0001sd\u0017: (2.2)\nRemark 2.2. Note that the content of Lemma 2.1 is analogous to that of [Kle17, Proposition\n8], and the proof goes through in a very similar fashion for the case of pseudodi\u000berential\ndamping.\nProof. First, observe that in order to prove (2.2), it is su\u000ecient to show that for all sPR\nand anybPC8pS\u0006pR\u0002Mqqwith compact support in pt;xq\n»\nS\u0006pR\u0002Mqpb\u0005\bsqGsd\u0017\u0010»\nS\u0006pR\u0002Mqbd\u0017:\nFurthermore, since G0\u00101, this is equivalent to showing that\nBs»\nS\u0006pR\u0002Mqpb\u0005\bsqGsd\u0017\u00100:\nBy direct computation, we see that\nBsrpb\u0005\bsqGss\u0010tp;b\u0005\bsuGs\u0000pb\u0005\bsqBsGs;\nsince \bsis the Hamiltonian \row generated by p:Using algebraic properties of the Poisson\nbracket, we have tp;b\u0005\bsuGs\u0010tp;pb\u0005\bsqGsu\u0001tp;Gsupb\u0005\bsq:Therefore,\nBs»\nS\u0006pR\u0002Mqpb\u0005\bsqGsd\u0017\u0010»\nS\u0006pR\u0002Mqtp;pb\u0005\bsqGsu\u0001tp;Gsupb\u0005\bsq\u0000pb\u0005\bsqBsGsd\u0017: (2.3)\nTo rewrite the \frst term on the right-hand side above, let us extend pb\u0005\bsqGsto a function\nonT\u0006pR\u0002Mqwhich is \fber-homogeneous of degree 1 outside a small neighborhood of the\nzero section. This can be accomplished by choosing some \u001fPC8pRqwhich vanishes in a\nneighborhood of zero and is equal to one outside a slightly larger neighborhood. Then, for\nany \fxedsPR;\n\u001fp|\u0018|qpb\u0005\bsqGsPS1\nc`pT\u0006pR\u0002Mqq;\nand hence\nB:\u0010Opp\u001fp|\u0018|qpb\u0005\bsqGsqP\t1\nc`pMq:\nThus,rB;PsP\t2\nc`pMq, and hence\nlim\nkÑ8xrP;Bsuk;uky \u0010»\nS\u0006pR\u0002Mq1\nitp;pb\u0005\bsqGsud\u00178 B. KEELER AND P. KLEINHENZ\nby the de\fnition of the defect measure. On the other hand, since each uksolves the damped\nwave equation, we also have\nxrP;Bsuk;uky\u0010xBuk;Puky\u0001xPuk;Buky\n\u0010xBuk;\u00012WBtuky\u0000x2WBtuk;Buky\n\u0010x2pBtWB\u0000BWBtquk;uky:\nTaking the limit of both sides as kÑ8 gives\nlim\nkÑ8xrP;Bsuk;uky\u0010»\nS\u0006pR\u0002Mq4i\u001cwpb\u0005\bsqGsd\u0017:\nTherefore,»\nS\u0006pR\u0002Mqtp;pb\u0005\bsqGsud\u0017\u0010\u0001»\nS\u0006pR\u0002Mq4\u001cwpb\u0005\bsqGsd\u0017:\nCombining the above with (2.3), we obtain\nBs»\nS\u0006pR\u0002Mqpb\u0005\bsqGsd\u0017\u0010»\nS\u0006pR\u0002Mqpb\u0005\bsqpBsGs\u0001tp;Gsu\u00014\u001cwGsqd\u0017;\nwhich is clearly zero if Gssatis\fes (2.1). \u0003\nAnother important observation about the defect measure \u0017is its support is closely related\nto the characteristic set of P, de\fned as\nCharpPq\u0010tpt;x;\u001c;\u0018qPT\u0006pR\u0002Mq:ppt;x;\u001c;\u0018q\u00100u:\nThe following is a well-known result, but we provide a short proof for the bene\ft of the\nreader.\nLemma 2.3. Giventukuand\u0017as above, the support of \u0017is contained in the intersection\nCharpPqXS\u0006pR\u0002Mq.\nProof. First, let\u001fPC8pR\u0002Mqbe supported in a neighborhood of Char pPqand identically\none on a slightly smaller neighborhood. Then, since Pis elliptic on the support of 1 \u0001\u001f,\nthere exist a parametrix QP\t\u00012\nc`pR\u0002Mqsuch that\npId\u0001Opp\u001fqquk\u0010QPuk\u0000Ruk\u0010Ruk\nfor some smoothing operator Rand allk. Now \fx an interval Iand let PC8\n0pIq. SinceR\nis smoothing,\n} ptqpI\u0001Opp\u001fqquk}H1pR\u0002Mq\u0010} ptqRuk}H1pI\u0002Mq¤}uk}L2pI\u0002Mq:\nBy assumption, ukconverges weakly to zero in H1pR\u0002Mq, and therefore its restriction to\nI\u0002Mconverges weakly to zero in H1pI\u0002Mq. Thus, there exists a subsequence tukjuwhich\ncoverges strongly to zero in L2pI\u0002Mq, sinceI\u0002Mis compact. Therefore, }ukj}L2pI\u0002MqÑ0,\nwhich implies that  ptqpId\u0001Opp\u001fqqukjÑ0 strongly in H1pI\u0002Mq. Now consider the\noperator\npB2\nt\u0000\u0001gqr ptqpI\u0001Opp\u001fqqsP \t2\nc`pR\u0002Mq;SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 9\nwhich is supported in I\u0002M:Since extracting a subsequence does not change the defect\nmeasure, we must have\nxpB2\nt\u0000\u0001gq\u0010\n ptqpId\u0001Opp\u001fqqukj\u0018\n;ukjqyÑ\u00011\n2»\nS\u0006pR\u0002Mq ptqp1\u0001\u001fpt;x;\u001c;\u0018qqd\u0017; (2.4)\nrecalling that \u001c2\u0000|\u0018|2\ng\u00101\n2onS\u0006pR\u0002Mq:However, we also have that\n}pB2\nt\u0000\u0001gqr ptqpId\u0001Opp\u001fqqsukj}H\u00011pI\u0002Mq\n¤C} ptqpId\u0001Opp\u001fqqukj}H1pI\u0002Mq:\nRecalling that  ptqpI\u0001Opp\u001fqqukjÑ0 strongly in H1pI\u0002Mqand noting that tukjuis\nbounded in H1pI\u0002Mq, we must have\nxpB2\nt\u0000\u0001gq\u0010\n ptqpId\u0001Opp\u001fqqukj\u0018\n;ukjqyÑ 0:\nCombining this with (2.4), we see that the support of \u0017must be disjoint from that of\n ptqp1\u0001\u001fpt;x;\u001c;\u0018qq. Since was arbitrary, and the argument holds for any \u001fwith the\nappropriate support properties, we must have that the support of \u0017is contained in Char pPqX\nS\u0006pR\u0002Mq: \u0003\nObserve that p\u00100 exactly when \u001c\u0010 \b|\u0018|g, and therefore Char pPqXS\u0006pR\u0002Mqis\ncomprised of the two connected components\nS\b\u0010t\u001c\u0010\t1{2uXS\u0006pR\u0002Mq:\nIt is helpful to de\fne \u0017\bandG\b\nsto be the restrictions of \u0017andGstoS\b, respectively. The\nde\fnition of Gsgiven in (2.1) then implies that\nBsG\b\ns\u0010tp;G\b\nsu\t2wG\b\ns: (2.5)\nNow, sincewdepends only on px;\u0018qand\u001cis constant on S\u0000, we may treat G\b\nsas functions on\nS\u0006M. For the purposes of this article, it su\u000eces to consider only G\u0000\ns. It follows immediately\nfrom Lemma 2.1 that G\u0000\nsgives the propagation of \u0017\u0000onS\u0000under the \row \b si.e.p\bsq\u0006\u0017\u0000\u0010\nG\u0000\n\u0001s\u0017\u0000:\nAs in [Kle17], we claim that G\u0000\nscan be realized as the solution of a much simpler di\u000berential\nequation by observing that it is a cocycle map. That is, for any px;\u0018qPS\u0006Mand anyr;sPR,\nwe haveG\u0000\ns\u0000rpx;\u0018q\u0010G\u0000\nrp'spx;\u0018qqG\u0000\nspx;\u0018q:To see this, note that by (2.5) and properties of\nthe Poisson bracket,\nBs\u0000\npG\u0000\nr\u0005'sqG\u0000\ns\b\n\u0010pG\u0000\nr\u0005'sq\u0000\ntp;G\u0000\nsu\u00012wG\u0000\ns\b\n\u0000tp;G\u0000\nr\u0005'suG\u0000\ns\n\u0010tp;pG\u0000\nr\u0005'sqG\u0000\nsu\u00012wpG\u0000\nr\u0005'sqG\u0000\ns:\nSo,pG\u0000\nr\u0005'sqG\u0000\nsandG\u0000\ns\u0000rboth satisfy the same initial value problem, and must be equal.\nUsing the cocycle property and the fact that G\u0000\n0\u00111, we have\nBsG\u0000\ns\u0010lim\nhÑ0G\u0000\ns\u0000h\u0001G\u0000\ns\nh\u0010lim\nhÑ0pG\u0000\nh\u0005'sqG\u0000\ns\u0001G\u0000\ns\nh\n\u0010G\u0000\nslim\nhÑ0G\u0000\nh\u0005's\u0001G\u0000\n0\u0005's\nh\u0010G\u0000\nsBr\u0000\nG\u0000\nr\u0005's\b\u0007\u0007\nr\u00100:10 B. KEELER AND P. KLEINHENZ\nSinceG\u0000\n0\u00111, we have that tp;G\u0000\nru|r\u00100\u00100. This along with the fact that 'sis independent\nofrgivesBrpG\u0000\nr\u0005'sq\u0007\u0007\nr\u00100\u0010\u00012w\u0005's:Thus,G\u0000\nscan be realized as the solution of the initial\nvalue problem#\nG\u0000\n0px;\u0018q\u00101\nBsG\u0000\nspx;\u0018q\u0010\u0001 2wp'spx;\u0018qqG\u0000\nspx;\u0018q;\nwhich has solution\nG\u0000\nspx;\u0018q\u0010exp\u0004\n\u0005\u0001s»\n02wp'rpx;\u0018qqdr\f\n\r: (2.6)\nThus, the propagation of the defect measure exhibits exponential decay in proportion to the\namount of time geodesics spend in the region where wpx;\u0018qis positive.\n3.Pseudodifferential Operator Acting on Coherent States\nA key component of the proof of Theorem 2 is to build quasimodes for (1.1) using Guassian\nbeams, which are strongly localized along a given geodesic. In this section, we obtain precise\nestimates for pseudo-di\u000berential operators acting on slightly simpler objects, namely coherent\nstates. A coherent state on Rnis a sequence of smooth functions thkutaking the form\nhkpxq\u0010kn\n4eikxx\u0001x0;\u00180yeik\n2xApx\u0001x0q;px\u0001x0qybpxq (3.1)\nfor some \fxed px0;\u00180q PS\u0006Rn, wherebPC8\ncpRnqandAPCn\u0002nhas positive de\fnite\nimaginary part. Heuristically, one thinks of hkas being strongly microlicalized near px0;\u00180q.\nThe objective of this section is to show that if a symbol aPSm\nc`pR2nqvanishes to some \fnite\norder atpx0;\u00180q, then}Oppaqhk}L2pRnqsatis\fes a bound which depends on the symbol order\nmand on the order of vanishing.\nRemark 3.1. For the purposes of this section only, we use the more standard convention\nthatS\u0006Rn\u0010tpx;\u0018qPR2n:|\u0018|\u00101ufor the sake of convenience, but this does not alter the\nanalysis in any way.\nProposition 3.2. Fixpx0;\u00180qPS\u0006Rn,bPC8\ncpRnq, and a matrix APCn\u0002n;with positive\nde\fnite imaginary part. Then, for any k¥1;lethkbe given by (3.1) . LetaPSm\nc`pR2nqhave\ncompact support in xand a polyhomogeneous expansion given by\na\u0012¸\nj¥0am\u0001j;\nwhere each am\u0001jPSm\u0001j\nc`pR2nqsatis\fesam\u0001jpx;s\u0018q \u0010sm\u0001jam\u0001jpx;\u0018qfor alls¡0and\n|\u0018|¥c¡0for some small c. Suppose there exists an `PNsuch thatam\u0001jvanishes to order\n`\u00012jatpx0;\u00180qfor allj¤`\n2. Then, for each \"¡0, there exists a C\"¡0so that\n}Oppaqhk}L2pRnq¤C\"km\u0001`\n2\u0000\": (3.2)\nProof. By the polyhomogeneity of a, for anyN0¥0 there exists rN0PSm\u0001N0\nc` such that\na\u0010N0\u00011¸\nj\u00100am\u0001j\u0000rN0: (3.3)\nWe begin with the following lemma, which handles the remainder term in this expansion.SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 11\nLemma 3.3. LetrPS\u0001s\nc`pR2nqwiths¥0:Then, there exists a C¡0such that\n}Opprqhk}L2pRnq¤Ck\u0001s\n2: (3.4)\nProof. By the quantization formula, we have\n}Opprqhk}2\nL2pRnq\u0010»\nRn\u0007\u0007\u0007\u0007\u0007\u0007»\nR2neixx\u0001y;\u0018yrpx;\u0018qhkpyqdyd\u0018\u0007\u0007\u0007\u0007\u0007\u00072\ndx:\nAssume without loss of generality that x0\u00100. We then change variables via xÞÑk\u00011\n2x,\nyÞÑk\u00011\n2y, and\u0018ÞÑk1\n2\u0018:Recalling the de\fnition of hk, we obtain\n}Opprqhk}2\nL2pRnq\u0010»\nRn\u0007\u0007\u0007\u0007\u0007\u0007»\nR2neixx\u0001y;\u0018yrpk\u00011\n2x;k1\n2\u0018qeik1\n2xx;\u00180yei\n2xAy;yybpk\u00011\n2yqdyd\u0018\u0007\u0007\u0007\u0007\u0007\u00072\ndx: (3.5)\nFor notational convenience, we de\fne gApyq\u0010ei\n2xAy;yyand let\u001cs:C8pRnqÑC8pRnqdenote\ndilation by s¡0. That is, \u001csfpyq \u0010fpsyq:Then, using ^ to denote the standard Fourier\ntransform, we de\fne\nFkp\u0018q:\u0010»\nRne\u0001ixy;\u0018ygApyqbpk\u00011{2yqdy\u0010kn{2rpgA\u0006\u001ck1\n2pbsp\u0018q: (3.6)\nThus, we can rewrite (3.5) as\n}Opprqhk}2\nL2pRnq\u0010»\nRn\u0007\u0007\u0007\u0007\u0007\u0007»\nRneixx;\u0018yrpk\u00011\n2x;k1\n2\u0018qFkp\u0018\u0001k1\n2\u00180qd\u0018\u0007\u0007\u0007\u0007\u0007\u00072\ndx: (3.7)\nWe claim that for any NPNand any multi-index \f;there exists a constant CN;\f¡0 so\nthat \u0007\u0007\u0007B\f\n\u0018Fkp\u0018q\u0007\u0007\u0007¤CN;\fp1\u0000|\u0018|q\u0001Nfor allkPN: (3.8)\nTo see this, consider the case where |\f|\u00100 and note that\n\u0007\u0007\u0007p1\u0000|\u0018|qNkn{2rpgA\u0006\u001ck1\n2pbsp\u0018q\u0007\u0007\u0007¤CNkn{2»\nRnp1\u0000|\u0018\u0001\u0011|qNp1\u0000|\u0011|qN|pgAp\u0018\u0001\u0011q||pbpk1\n2\u0011q|d\u0011\n¤CNkn{2»\nRnp1\u0000|\u0011|qN|pbpk1\n2\u0011q|d\u0011;\nwhere the last inequality follows from the fact that pgAis Schwartz class, since Ahas positive\nde\fnite imaginary part. Now, observe that\nkn{2»\nRnp1\u0000|\u0011|qN|pbpk1\n2\u0011qd\u0011|¤kn{2»\nRnp1\u0000k1\n2|\u0011|qN\u0000n\u00001\np1\u0000k1\n2|\u0011|qn\u00001|pbpk1\n2\u0011q|d\u0011¤CNkn{2»\nRnp1\u0000k1\n2|\u0011|q\u0001n\u00011d\u0011;\nfor some new CN¡0, sincepbis Schwartz-class. Changing variables via \u0011ÞÑk\u00011\n2\u0011, we obtain\nthat \u0007\u0007\u0007p1\u0000|\u0018|qNkn{2rpgA\u0006\u001ck1\n2pbsp\u0018q\u0007\u0007\u0007¤CNkn{212 B. KEELER AND P. KLEINHENZ\nafter potentially increasing CN:Dividing through by kn{2p1\u0000|\u0018|qNcompletes the proof of\n(3.8) for|\f|\u00100. To obtain the estimate when |\f|\u00180;simply repeat the above proof with\npgAreplaced byB\f\n\u0018pgA:\nNow, in order to estimate (3.7) we introduce a smooth cuto\u000b function \u001fwhich is identically\none in a neighborhood of x\u00100. We then write\n}Opprqhk}2\nL2pRnq\u0010I\u0000II;\nwhereIis de\fned by\nI\u0010»\nRn\u0007\u0007\u0007\u0007\u0007\u0007»\nRneixx;\u0018y\u001fpxqrpk\u00011\n2x;k1\n2\u0018qFkp\u0018\u0001k1\n2\u00180qd\u0018\u0007\u0007\u0007\u0007\u0007\u00072\ndx;\nandIIis de\fned analogously with \u001fpxqreplaced by 1 \u0001\u001fpxq. To estimate I, we note that\nwhen|\u0018|¤1,\n|rpk\u00011\n2x;k1\n2\u0018qFkp\u0018\u0001k1\n2\u00180q|¤CNp1\u0000|\u0018\u0001k1\n2\u00180|q\u0001N¤C1\nNk\u0001N{2;\nfor someCN; C1\nN¡0 and anyN, by (3.8) and the fact that rhas nonpositive order and is\ntherefore uniformly bounded. Thus,\nI\u0010»\nRn\u0007\u0007\u0007\u0007\u0007\u0007\u0007»\n|\u0018|¥1eixx;\u0018y\u001fpxqrpk\u00011\n2x;k1\n2\u0018qFkp\u0018\u0001k1\n2\u00180qd\u0018\u0007\u0007\u0007\u0007\u0007\u0007\u00072\ndx\u0000Opk\u00018q: (3.9)\nNow, when|\u0018|¥1,\n|rpk\u00011\n2x;k1\n2\u0018q|¤Cp1\u0000k1\n2|\u0018|q\u0001s¤Ck\u0001s\n2:\nCombining this with (3.9), we have\nI¤Ck\u0001s}Fk}2\nL1pRnq¤C1k\u0001s; (3.10)\nwhere the \fnal inequality follows from (3.8).\nNow, consider II. Since 1\u0001\u001fvanishes in a neighborhood of x\u00100, we may integrate by\nparts arbitrarily many times in \u0018using the operatorxx;r\u0018y\ni|x|2, which preserves eixx;\u0018y. That is,\nfor any\u0017¥0, we have\nII\u0010»\nRn\u0007\u0007\u0007\u0007\u0007\u0007»\nRneixx;\u0018yp1\u0001\u001fpxqq\u0002ixx;r\u0018y\n|x|2\n\u0017\u0001\nrpk\u00011\n2x;k1\n2\u0018qFkp\u0018\u0001k1\n2\u00180q\t\nd\u0018\u0007\u0007\u0007\u0007\u0007\u00072\ndx:\nBy (3.8) and the fact that rPS\u0001s\nc`, we have for any multi-index \fand anyN,\n\u0007\u0007\u0007B\f\n\u0018\u0001\nrpk\u00011\n2x;k1\n2\u0018qFkp\u0018\u0001k1\n2\u00180q\t\u0007\u0007\u0007¤¸\n|\r|¤|\f|C\r;Nk|\r|\n2p1\u0000k1\n2|\u0018|q\u0001s\u0001|\r|p1\u0000|\u0018\u0001k1\n2\u00180|q\u0001N:\nIn the region where |\u0018| ¥1, the above is bounded by CNk\u0001s\n2p1\u0000|\u0018\u0001k1\n2\u00180|q\u0001Nfor some\nCN¡0. Alternatively, when |\u0018| ¤ 1, we have a bound of the form CNk\u0001N{2, since\n1\u0000|\u0018\u0001k1\n2\u00180|¥Ck1\n2. Combining these facts, we have\n»\nRn\u0007\u0007\u0007\u0007\u0007\u0007»\nRneixx;\u0018yp1\u0001\u001fpxqq\u0002ixx;r\u0018y\n|x|2\n\u0017\u0001\nrpk\u00011\n2x;k1\n2\u0018qFkp\u0018\u0001k1\n2\u00180q\t\nd\u0018\u0007\u0007\u0007\u0007\u0007\u00072\ndx¤CNk\u0001s;SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 13\nfor someCN¡0, provided \u0017¡n\u00001\n2so that the integral in xis convergent. Therefore,\nII¤CNk\u0001s:\nCombining this with (3.10) and taking square roots of both sides completes the proof.\n\u0003\nWe now return to the proof of Proposition 3.2. We aim to estimate each of the terms in\nthe sum in (3.3) separately. By de\fnition, for each j¤N0\u00011,\n}Oppam\u0001jqhk}2\nL2pRnq\u0010»\nRn\u0007\u0007\u0007\u0007»\nR2neixx\u0001y;\u0018yam\u0001jpx;\u0018qhkpyqdyd\u0018\u0007\u0007\u0007\u00072\ndx:\nAs before, we change variables via xÞÑk\u00011\n2x;yÞÑk\u00011\n2y;and\u0018ÞÑk1\n2\u0018:This gives\n}Oppam\u0001jqhk}2\nL2pRnq\u0010»\nRn\u0007\u0007\u0007\u0007»\nR2neixx\u0001y;\u0018yam\u0001jpk\u00011\n2x;k1\n2\u0018qFkp\u0018\u0001k1\n2\u0018q\u0007\u0007\u0007\u00072\ndx;\nwhereFkis given by (3.6). Now, let \u001fpx;\u0018q\u0010\u001fp\u0018qbe a smooth function which is identically\none for|\u0018|¤1\n2and zero outside |\u0018|¤1. Then, for any 0 \u000b1{2 and anyk¡0, de\fne\n\u001fk;\u000bp\u0018q\u0010\u001fpk\u0001\u000bp\u0018\u0001k1\n2\u00180qq;\nso that\u001fk;\u000bis identically one on the ball of radius1\n2k\u000bcentered at k1\n2\u00180and zero outside\nthe corresponding ball of radius k\u000b. Since\u001fk;\u000bis supported on |\u0018|¡2 for su\u000eciently large\nk, the homogeneity of am\u0001jimplies\n\u001fk;\u000bp\u0018qam\u0001jpk\u00011\n2x;k1\n2\u0018q\u0010km\u0001j\n2\u001fk;\u000bp\u0018qam\u0001jpk\u00011\n2x;\u0018q: (3.11)\nRecall that am\u0001jvanishes to order `j:\u0010`\u00012jatpx0\u00100;\u00180qfor allj¤`\n2, so by Taylor\nexpansion, there exists a collection of fm\u0001j;\r; gm\u0001j;\rPSm\u0001j\nc`pR2nqsuch that\nam\u0001jpx;\u0018q\u0010¸\n|\r|\u0010`jx\rfm\u0001j;\rpx;\u0018q\u0000\u0002\u0018\n|\u0018|\u0001\u00180\n\r\ngm\u0001j;\rpx;\u0018q:\nCombining this with (3.11), we have\n\u001fk;\u000bp\u0018qam\u0001jpk\u00011\n2x;k1\n2\u0018q\u0010km\u0001j\n2¸\n|\r|\u0010`j\u0002\nk\u0001`j\n2x\rfm\u0001j;\rpk\u00011\n2x;\u0018q\u0000\u0002\u0018\n|\u0018|\u0001\u00180\n\r\ngm\u0001j;\rpk\u00011\n2x;\u0018q\n:\nThen, we de\fne\nAj;1pxq\u0010km\u0001j\u0001`j\n2¸\n|\r|\u0010`j»\nRneixx;\u0018y\u001fk;\u000bp\u0018qx`jfm\u0001j;\rpk\u00011\n2x;\u0018qFkp\u0018\u0001k1\n2\u00180qd\u0018;(3.12)\nAj;2pxq\u0010km\u0001j\n2¸\n|\r|\u0010`j»\nRneixx;\u0018y\u001fk;\u000bp\u0018q\u0002\u0018\n|\u0018|\u0001\u00180\n\r\ngm\u0001j;\rpk\u00011\n2x;\u0018qFkp\u0018\u0001k1\n2\u00180qd\u0018: (3.13)\nand\nRjpxq\u0010»\nRneixx;\u0018yp1\u0001\u001fk;\u000bp\u0018qqam\u0001jpk\u00011\n2;k1\n2\u0018qFkp\u0018\u0001k1\n2\u0018qd\u0018; (3.14)\nso that\nOppam\u0001jqhk\u0010Aj;1\u0000Aj;2\u0000Rj:14 B. KEELER AND P. KLEINHENZ\nWe claim that Rjis negligible for large k. SinceFkp\u0018\u0001k1\n2\u00180qis a Gaussian centered at k1\n2\u00180\nand 1\u0001\u001fk;\u000bis supported at least k\u000baway from that center, we are able to show that Rjis\ncontrolled by an arbitrarily negative power of k.\nLemma 3.4. For anyN1PNthere exists CN1¡0such that\n}Rj}L2pRnq¤CN1k\u0001N1: (3.15)\nProof. To begin note that for any multi-index \f\nB\f\n\u0018\u001fk;\u000bp\u0018q\u0010k\u0001\u000b|\f|pB\f\n\u0018\u001fqpk\u0001\u000bp\u0018\u0001k1\n2\u00180qq: (3.16)\nCombining (3.16) with (3.8) shows that for any NPNand any multi-index \f, there exists\nCN;\f¡0 such that\nB\f\n\u0018rp1\u0001\u001fk;\u000bqFkp\u0018qs¤CN;\fk\u0001\u000b|\f|1suppp1\u0001\u001fk;\u000bqp\u0018qp1\u0000|\u0018\u0001k1\n2\u00180|q\u0001N;\nwhere for any set ERn,1Edenotes the indicator function of E.\nNow, when|x| ¥1, for\u0017¥0 we may integrate by parts in (3.14) as in the proof of\nLemma 3.3 to obtain\nRjpxq\u0010»\nRneixx;\u0018y\u0002ixx;r\u0018y\n|x|2\n\u0017\u0011\np1\u0001\u001fk;\u000bp\u0018qqam\u0001jpk\u00011\n2x;k1\n2\u0018qFkp\u0018q\u0019\nd\u0018:\nSinceam\u0001jPSm\u0001j\nc`pR2nq, we have that for any multi-index \f,\n|B\f\n\u0018am\u0001jpk\u00011\n2x;k1\n2\u0018q|¤Ck|\f|\n2p1\u0000k1\n2|\u0018|qm\u0001j\u0001|\f|¤Ck|\f|\n2p1\u0000k1\n2|\u0018|qm\nThus, for any NPN, there exists a constant CNsuch that whenever |x|¥1,\n|Rjpxq|¤CNsup\n|\f|¤\u00171\n|x|\u0017»\nRn1suppp1\u0001\u001fk;\u000bqp\u0018qp1\u0000|\u0018\u0001k1\n2\u00180|q\u0001Nk|\f|\n2p1\u0000k1\n2|\u0018|qmd\u0018:\nRecall that|\u00180|\u00101, and so by the triangle inequality\n1\u0000k1\n2|\u0018|¤1\u0000k\u0000k1\n2|\u0018\u0001k1\n2\u00180|¤Ckp1\u0000|\u0018\u0001k1\n2\u00180|q:\nThus,\n|Rjpxq|¤CNsup\n|\f|¤\u00171\n|x|\u0017»\nRn1suppp1\u0001\u001fk;\u000bqp\u0018qp1\u0000|\u0018\u0001k1\n2\u00180|q\u0001N\u0000mkm\u0000|\f|\n2d\u0018: (3.17)\nUsing polar coordinates \u0018\u0001k1\n2\u00180\u0010r!withrPR\u0000; !PSn\u00011, we compute\n»\nRn1suppp1\u0001\u001fk;\u000bqp\u0018qp1\u0000|\u0018\u0001k1\n2\u00180|q\u0001N\u0000md\u0018¤»\nSn\u000118»\nk\u000b{2p1\u0000rq\u0001N\u0000mrn\u00011drd!\n¤Ck\u000bpm\u0000n\u0001Nq:\nCombining this with (3.17), we have\n|Rjpxq|¤CN|x|\u0001\u0017k\u000bpm\u0000n\u0001Nq\u0000m\u0000|\f|\n2;if|x|¥1:\nSince\u0017andNwere both arbitrary, given any N1¥0 we can choose \u0017¥n\u00001\n2andN\nsu\u000eciently large so that\n|Rjpxq|¤CN1k\u0001N1|x|\u0001n\u00001\n2;if|x|¥1:SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 15\nBy an analogous argument when |x|¤1, except without integration by parts, we have\n|Rjpxq|¤CN1k\u0001N1if|x|¤1:\nCombining these inequalities and taking the L2norm completes the proof of (3.15). \u0003\nIt remains to estimate Aj;1andAj;2. It is here that we take advantage of the compatibility\nof the vanishing of am\u0001jwith the particular form of the coherent state hk. We \frst consider\nAj;1.\nLemma 3.5. For anyj¥0, there exists Cj¡0such that\n}Aj;1}L2pRnq¤Cjkm\u0001j\u0001`j\n2: (3.18)\nProof. Note on the support of \u001fk;\u000b\nk1\n2\u0001k\u000b¤|\u0018|¤k1\n2\u0000k\u000b:\nAlso, recall that fm\u0001j;\rPSm\u0001j\nc`pR2nq, so|B\f\n\u0018fm\u0001j;\rpx;\u0018q|¤C\f|\u0018|m\u0001j\u0001|\f|. Therefore,\nsup\nxPRn|B\f\n\u0018fm\u0001j;\rpx;\u0018q|¤C\fkm\u0001j\u0001|\f|\n2;for all\u0018Psupp\u001fk;\u000b: (3.19)\nNow, when|x|¥1, we may integrate by parts as before to obtain that for any \u0017¥0 and\nanyNsu\u000eciently large,\n|Aj;1pxq|¤km\u0001j\u0001`j\n2¸\n|\r|\u0010`j\u0007\u0007\u0007\u0007»\nRneixx;\u0018y\u001fk;\u000bp\u0018qx\rfm\u0001j;\rpk\u00011\n2x;\u0018qFkp\u0018\u0001k1\n2\u00180qd\u0018\u0007\u0007\u0007\u0007\n¤km\u0001j\u0001`j\n2|x|`j»\nRn\u0007\u0007\u0007\u0007\u0002ixx;r\u0018y\n|x|2\n\u0017\u0011\n\u001fk;\u000bp\u0018qfm\u0001j;\rpk\u00011\n2x;\u0018qFkp\u0018\u0001k1\n2\u00180q\u0019\u0007\u0007\u0007\u0007d\u0018\n¤C\u0017;Nkm\u0001j\u0001`j\n2|x|`j\u0001\u0017»\nRnp1\u0000|\u0018\u0001k1\n2\u00180|q\u0001Nd\u0018\n¤C1\n\u0017;Nkm\u0001j\u0001`j\n2|x|`j\u0001\u0017;\nwhere the second-to-last inequality follows from (3.8), (3.16) and (3.19). When |x|¤1, by\nthe same argument with \u0017\u00100, we obtain\n|Aj;1pxq|¤Ckm\u0001j\u0001`j\n2:\nThus, for each \u0017¥0, there exists a constant C\u0017¡0 so that\n|Aj;1pxq|¤C\u0017km\u0001j\u0001`\n2p1\u0000|x|q`j\u0001\u0017for allxPRn:\nChoosing\u0017so that`j\u0001\u0017¤\u0001n\u00001\n2and taking the L2norm gives the desired inequality. \u0003\nFinally, we turn our attention to Aj;2. The estimation of this term is the most subtle of\nthe three, and requires some very technical analysis of the relationship between the vanishing\nfactor\u0001\n\u0018\n|\u0018|\u0001\u00180\t\nand the structure of the support of \u001fk;\u000b.\nLemma 3.6. For anyj¥0;there exists Cj¡0such that\n}Aj;2}L2pRnq¤Cjkm\u0001j\u0000p\u000b\u00011\n2q`j: (3.20)16 B. KEELER AND P. KLEINHENZ\nProof. As before, we \frst consider the case where |x|¥1 and the case |x|¤1 will follow\nfrom an analogous argument. When |x|¥1, we may again use integration by parts to see\nthat for any \u0017¥0 and anyNsu\u000eciently large,\n|Aj;2pxq|¤km\u0001j\n2¸\n|\r|\u0010`j\u0007\u0007\u0007\u0007\u0007\u0007»\nRneixx;\u0018y\u0002ixx;r\u0018y\n|x|2\n\u0017\u0012\n\u001fk;\u000bp\u0018q\u0002\u0018\n|\u0018|\u0001\u00180\n\r\ngm\u0001j;\rpk\u00011\n2x;\u0018qFkp\u0018\u0001k1\n2\u00180q\u001a\nd\u0018\u0007\u0007\u0007\u0007\u0007\u0007:\nNote that on supp \u001fk;\u000b; k1\n2\u0001k\u000b¤|\u0018|¤k1\n2\u0000k\u000b, and so|B\f\n\u0018gm\u0001j;\r|¤Ckm\u0001j\u0001|\f|\n2¤Ckm\u0001j\n2.\nCombining this with (3.8) gives\n|Aj;2pxq|¤Ckm\u0001j|x|\u0001\u0017¸\n\f¤\u0017¸\n|\r|\u0010`j»\nRn\u0007\u0007\u0007\u0007B\f\n\u0018\u0002\n\u001fk;\u000bp\u0018q\u0002\u0018\n|\u0018|\u0001\u00180\n\r\n\u0007\u0007\u0007\u0007p1\u0000|\u0018\u0001k1\n2\u00180|q\u0001Nd\u0018;(3.21)\nwhen|x|¥1. Therefore, it is su\u000ecient to show that for any multi-indices \f; \r with|\f|¤\u0017\nand|\r|\u0010`j, there exists C¡0 such that\n\u0007\u0007\u0007\u0007B\f\n\u0018\u0002\n\u001fk;\u000bp\u0018q\u0002\u0018\n|\u0018|\u0001\u00180\n\r\n\u0007\u0007\u0007\u0007¤Ckp\u000b\u00011\n2q`j: (3.22)\nTo show this, it is convenient to choose coordinates on Rnso that\u00180\u0010p1;0;:::; 0q. Writing\n\u0018\u0010p\u00181;\u00182;:::;\u0018nq, we have that on the support of \u001fk;\u000b;\n|\u0018\u0001k1{2\u00180|\u0010b\np\u00181\u0001k1{2q2\u0000\u00182\n2\u0000\u0004\u0004\u0004\u0000\u00182\nn¤k\u000b;\nby the de\fnition of \u001fk;\u000b. Thus,\n|\u0018r|¤k\u000bfori\u00181 and|\u0018|¥k1{2\u0001k\u000b: (3.23)\nAlso, note that \u00181¡0 on supp\u001fk;\u000bforklarge enough, and so we can write\n|\u0018|\u0001\u00181\u0010p|\u0018|\u0000\u00181q\u00011p\u00182\n2\u0000\u0004\u0004\u0004\u0000\u00182\nnq:\nCombining these facts, we have that for large k,\n|\u0018|\u0001\u00181\u0010\u00182\n2\u0000\u0004\u0004\u0004\u0000\u00182\nn\n|\u0018|\u0000\u00181¤pn\u00011qk2\u000b\nk1{2\u0001k\u000b¤Ck2\u000b\u00011\n2:\nWe can now show (3.22) for \f\u00100. Recalling that \r\u0010p\r1;\r2;:::;\rnqPNnis a multi-index\nwith|\r|\u0010\r1\u0000\u0004\u0004\u0004\u0000\rn\u0010`j;we make use of the above inequality and (3.23) to obtain\n\u0007\u0007\u0007\u0007\u0002\u0018\n|\u0018|\u0001\u00180\n\r\u0007\u0007\u0007\u0007\u0010|\u00181\u0001|\u0018||\r1|\u00182|\r2\u0004\u0004\u0004|\u0018n|\rn\n|\u0018|`j\n¤Ck2\u000b\r1\nk\r1\n2\u0004k\u000b\r2\u0004\u0004\u0004k\u000b\rn\nk`j\n2\n\u0010Ckp\u000b\u00011\n2q`j\u0000p\u000b\u00011\n2q\r1\n¤Ckp\u000b\u00011\n2q`j;\nwhich proves (3.22) in the case where |\f| \u00100. To handle the case where \f\u00180 we let\n\f1;\f2;:::;\fnbe multi-indices and expand via the product rule to obtain\u0007\u0007\u0007\u0007B\f\n\u0018\u0002\u0018\n|\u0018|\u0001\u00180\n\r\u0007\u0007\u0007\u0007¤¸\n|\f1\u0000\f2\u0000\u0004\u0004\u0004\u0000\fn|\u0010\fC\fj\u0007\u0007\u0007\u0007B\f1\n\u0018\u0002\u00181\u0001|\u0018|\n|\u0018|\n\r1\nB\f2\n\u0018\u0002\u00182\n|\u0018|\n\r2\n\u0004\u0004\u0004B\fn\n\u0018\u0002\u0018n\n|\u0018|\n\rn\u0007\u0007\u0007\u0007:(3.24)SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 17\nSince\u00181\u0001|\u0018|\n|\u0018|and\u0018j\n|\u0018|are homogeneous of degree zero, we have that for any multi-index \u0012,\n\u0007\u0007\u0007\u0007B\u0012\n\u0018\u0002\u00181\u0001|\u0018|\n|\u0018|\n\u0007\u0007\u0007\u0007¤C\n|\u0018||\u0012|;and\u0007\u0007\u0007\u0007B\u0012\n\u0018\u0002\u0018r\n|\u0018|\n\u0007\u0007\u0007\u0007¤C\n|\u0018||\u0012|ifr\u00181:\nFurthermore, we recall that |\u0018|¥k1\n2\u0001k\u000b¥Ck1\n2on supp\u001fk;\u000b;and so\n\u0007\u0007\u0007\u0007\u001fk;\u000bp\u0018qB\u0012\n\u0018\u0002\u00181\u0001|\u0018|\n|\u0018|\n\u0007\u0007\u0007\u0007¤Ck\u0001|\u0012|\n2;and\u0007\u0007\u0007\u0007\u001fk;\u000bp\u0018qB\u0012\n\u0018\u0002\u0018r\n|\u0018|\n\u0007\u0007\u0007\u0007¤Ck\u0001|\u0012|\n2; r\u00181: (3.25)\nNow consider B\fr\n\u0018\u0001\n\u0018r\n|\u0018|\t\rr\n. Expanding via the product rule, we can rewrite this as a linear\ncombination of terms of the form\u0002\u0018r\n|\u0018|\ntr\nB\u000e1\n\u0018\u0002\u0018r\n|\u0018|\n\u0004\u0004\u0004B\u000eq\n\u0018\u0002\u0018r\n|\u0018|\n;\nwhere each \u000eiis a multi-index with |\u000ei|¥1,\u000e1\u0000\u0004\u0004\u0004\u0000\u000eq\u0010\fr;andtr\u0000q\u0010\rr:That is,\nthere aretrfactors of\u0018r\n|\u0018|which do not have any derivatives, and the remaining \rr\u0001trfactors\neach have at least 1 derivative applied to them. Note then that tr¥maxp0;\rr\u0001|\fr|q:By\n(3.23) each factor with no derivatives is bounded by k\u000b\u00011\n2. The homogeneous estimate (3.25)\ncontrols the factors with derivatives, giving\u0007\u0007\u0007\u0007\u0007\u001fk;\u000bp\u0018q\u0002\u0018r\n|\u0018|\ntr\nB\u000e1\n\u0018\u0002\u0018r\n|\u0018|\n\u0004\u0004\u0004B\u000eq\n\u0018\u0002\u0018r\n|\u0018|\n\u0007\u0007\u0007\u0007\u0007¤kp\u000b\u00011\n2qtrk\u0001|\u000e1|\n2\u0004\u0004\u0004k\u0001|\u000eq|\n2¤Ckp\u000b\u00011\n2qtr\u0001|\fr|\n2:\nThus, by the triangle inequality, we have\n\u0007\u0007\u0007\u0007\u001fk;\u000bp\u0018qB\fr\n\u0018\u0002\u0018r\n|\u0018|\n\rr\u0007\u0007\u0007\u0007¤Ckp\u000b\u00011\n2qtr\u0001|\fr|\n2:\nWhen\rr\u0001|\fr|¡0, we have\nkp\u000b\u00011\n2qtrk\u0001|\fr|\n2¤kp\u000b\u00011\n2qp\rr\u0001|\fr|qk\u0001|\fr|\n2¤Ckp\u000b\u00011\n2q\rr;\nsincetr¥\rr\u0001|\fr|¡0 and\u000b\u00011\n20. On the other hand, when \rr¤|\fr|, we still have\ntr¥0 and so\nkp\u000b\u00011\n2qtrk\u0001|\fr|\n2¤Ck\u0001|\fr|\n2¤Ck\u0001\rr\n2¤Ckp\u000b\u00011\n2q\rr;\nsince\u000b¡0:Thus, there exists a C\f¡0 so that\n\u0007\u0007\u0007\u0007\u001fk;\u000bp\u0018qB\fr\n\u0018\u0002\u0018r\n|\u0018|\n\rr\u0007\u0007\u0007\u0007¤C\fkp\u000b\u00011\n2q\rr: (3.26)\nAn analogous argument shows\n\u0007\u0007\u0007\u0007\u001fk;\u000bp\u0018qB\f1\n\u0018\u0002\u00181\u0001|\u0018|\n|\u0018|\n\r1\u0007\u0007\u0007\u0007¤C\fkp\u000b\u00011\n2q\r1; (3.27)\nfor some potentially di\u000berent C\f¡0:Combining (3.26) and (3.27) with (3.16) and (3.24)\nyields\n\u0007\u0007\u0007\u0007B\f\n\u0018\u0002\n\u001fk;\u000bp\u0018q\u0002\u0018\n|\u0018|\u0001\u00180\n\r\n\u0007\u0007\u0007\u0007¤C\fkp\u000b\u00011\n2qp\r1\u0000\r2\u0000\u0004\u0004\u0004\u0000\rnq\u0010C\fkp\u000b\u00011\n2q|\r|\u0010C\fkp\u000b\u00011\n2q`j:\nWe have therefore proved (3.22).18 B. KEELER AND P. KLEINHENZ\nCombining (3.21) and (3.22), we have that for any \u0017¥0 and anyNlarge enough, there\nexistsC\u0017;C1\n\u0017¡0 so that\n|Aj;2pxq|¤C\u0017|x|\u0001\u0017km\u0001j\n2\u0000p\u000b\u00011\n2q`j¸\n|\f1|¤\u0017¸\n|\r|\u0010`jkm\u0001j\u0001|\f1|\n2»\nRnp1\u0000|\u0018\u0001k1\n2\u00180|q\u0001Nd\u0018¤C1\n\u0017|x|\u0001\u0017km\u0001j\u0000p\u000b\u00011\n2q`j:\nWe then have\n|Aj;2pxq|¤C\u0017|x|\u0001\u0017km\u0001j\u0000p\u000b\u00011\n2q`j;for all|x|¥1; (3.28)\nfor someC\u0017¡0.\nTo estimate Aj;2pxqwhen|x|¤1, we repeat the above argument without integrating by\nparts. From this, we obtain\n|Aj;2pxq|¤Ckm\u0001j\u0000p\u000b\u00011\n2q`j;for all|x|¤1: (3.29)\nChoosing\u0017¡n\u00011\n2, we can combine (3.28) with (3.29), then take L2norms to obtain (3.20)\nas desired. \u0003\nRecalling the constructions of Rj;Aj;1;andAj;2, we combine Lemmas 3.4, 3.5, and 3.6 to\nobtain that for each 0 ¤j¤N0\u00011,\n}Oppam\u0001jqhk}L2pRnq¤}Aj;1}L2pRnq\u0000}Aj;2}L2pRnq\u0000}Rj}L2pRnq¤Cjkm\u0001j\u0000p\u000b\u00011\n2q`j;(3.30)\nfor someCj¡0. Since`j\u0010`\u00012j, (3.4) and (3.30) imply that for any N0¥m, there exists\na collection of constants tCjuN0\nj\u00100so that\n||Oppaqhk||L2¤N0\u00011¸\nj\u00100||Oppam\u0001jqhk||L2\u0000||OpprN0qhk||L2\n¤N0\u00011¸\nj\u00100Cjkm\u0001j\u0000p\u000b\u00011\n2q`j\u0000CN0km\u0001N0\n2\n\u0010N0\u00011¸\nj\u00100Cjkm\u0001`\n2\u0000p`\u00012jq\u000b\u0000CN0km\u0001N0\n2\n¤Ckm\u0001`\n2\u0000`\u000b\u0000CN0km\u0001N0\n2;\nfor someC¡0:ChoosingN0large enough and \u000bsmall enough completes the proof of\nProposition 3.2. \u0003\n4.The upper bound for \u000b\nIn this section, we show that \u000b¤2 mint\u0001D0;L8u, whereD0andL8are de\fned as\nin Section 1. That \u00012D0is an upper bound is straightforward to show. To do so, let\n\u0015jPSpecpAWqzt0u. Thus there exists u\u0010 pu0;u1q \u00180 such that AWu\u0010\u0015ju, where we\nrecall\nAW\u0010\u0002\n0 Id\n\u0001g\u00012W\n:\nIt is then immediate that upxq \u0010et\u0015ju0pxqsolves the damped wave equation with initial\ndatapu0;u1q, and\nEpu;tq\u0010e2tRep\u0015jqEpu;0q:SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 19\nSinceEpu;0q\u00180, we have that \u000b¤\u00012Rep\u0015jqfor allj. Furthermore, by the de\fnition of D0,\nthere must either exist some \u0015j0with Rep\u0015j0q\u0010D0, or a sequence of \u0015jwith Rep\u0015jqÑD0.\nIn either case, we must have \u000b¤\u00012D0.\nShowing that 2 L8is also an upper bound is more complicated. Our technique for this\nis inspired by the method of Gaussian beams introduced by Ralston [Ral69, Ral82]. Using\nGaussian beams, one can produce quasimodes for the wave equation with energy strongly\nlocalized near a single geodesic. Intuitively, solutions to (1.1) should decay only when they\ninteract with the damping coe\u000ecient. Motivated by this, we modify the Gaussian beam\nconstruction using the propagation of the defect measure derived in Section 2, in analogy\nto [Kle17]. From this, we obtain solutions whose energy decays at a rate proportional to the\nintegral of the symbol of Walong the chosen geodesic.\nTo begin, we recall Ralston's original Gaussian beam construction on Rnwith a Riemann-\nian metricg. LetAptqbe ann\u0002nsymmetric matrix-valued function with positive de\fnite\nimaginary part. Let tÞÑpxt;\u0018tqdenote a geodesic trajectory and set\n px;tq\u0010x\u0018t;x\u0001xty\u00001\n2xAptqpx\u0001xtq;x\u0001xty:\nLetbPC8pR\u0002Rnq. Then, we de\fne\nukpx;tq\u0010k\u00011\u0000n{4bpt;xqeik px;tq: (4.1)\nThe work of [Ral82] guarantees that there exist appropriate choices of bandAptqso thatuk\nis a quasimode of the undamped wave equation with positive energy, which is concentrated\nalong the geodesic pxt;\u0018tq. We summarize some notable facts from [Ral82] in the following\nLemma.\nLemma 4.1 [Ral82] .FixT¡0andpx0;\u00180qPS\u0006M. For'tpx0;\u00180q\u0010pxt;\u0018tq;there exists a\nbPC8pR\u0002Rnqand ann\u0002nsymmetric matrix-valued function tÞÑAptqso that for any\nk¥1;for theukde\fned in (4.1)\nsup\ntPr0;Ts}B2\ntukp\u0004;tq\u0001\u0001gukp\u0004;tq}L2pRnq¤Ck\u00011\n2: (4.2)\nFurthermore, for all tPr0;Ts,\nlim\nkÑ8Epuk;tq¡0; (4.3)\nand the limit is always \fnite and independent of t.\nRemark 4.2. By (4.3), we may assume without loss of generality that lim\nkÑ8Epuk;tq\u00101 for\nalltPr0;Ts:\nRemark 4.3. Using coordinate charts and a partition of unity, we can extend this con-\nstruction to the case of manifolds, which results in a sequence tuku C8pR\u0000\u0002Mqsuch\nthat lim kÑ8Epuk;tq\u00101 and the appropriate analogue of (4.2) holds.\nNext, we modify tukuusing the propagation of the defect measure from Section 2 to\nproduce a sequence of quasimodes for the damped wave equation. Recall that G\u0000\nt:S\u0006MÑC\nis given by G\u0000\ntpx0;\u00180q\u0010exp\u0001\n\u0001³t\n02wpxs;\u0018sqds\t\nwherepxs;\u0018sq\u0010'spx0;\u00180q.\nRecall also the time averaging function tÞÑLptq\nLptq\u00101\ntinf\npx0;\u00180qPS\u0006Mt»\n0wpxs;\u0018sqds:20 B. KEELER AND P. KLEINHENZ\nNote thatLptqcan be rewritten in terms of G\u0000\nt\nLptq\u0010\u00011\ntsup\npx;\u0018qPS\u0006Mln\u0000\nG\u0000\ntpx;\u0018q\b\n:\nMotivated by the form of G\u0000\nt, we \fxpx0;\u00180qPS\u0006Mand set\nvkpt;xq\u0010G\u0000\ntpx0;\u00180qukpt;xq:\nThat is, we modify the quasimode for the free wave equation so that it decays exponentially\nat a rate proportional to integral of wpxt;\u0018tqalong the geodesic it is concentrated on. We\nnow show that for any \"¡0; vkis anOpk\u00011\n2\u0000\"qquasimode of (1.1).\nProposition 4.4. Givenpx0;\u00180qPS\u0006M, letukpt;xqbe as speci\fed in Remark 4.3 and set\nvkpt;xq\u0010G\u0000\ntpx0;\u00180qukpt;xq. For anyT¡0and\"¡0, there exists a constant C\";T¡0so\nthat\nsup\ntPr0;Ts}pB2\nt\u0001\u0001g\u00002WBtqvkpt;\u0004q}L2pMq¤C\";Tk\u00011\n2\u0000\": (4.4)\nProof. By direct computation\npB2\nt\u0001\u0001g\u00002WBtqvk\u0010G\u0000\ntpB2\nt\u0001\u0001gquk\u00002BtG\u0000\ntBtuk\u0000pB2\ntG\u0000\ntquk\u00002WBtpG\u0000\ntukq\n\u0010G\u0000\ntpB2\nt\u0001\u0001gquk\u00012wpxt;\u0018tqG\u0000\ntBtuk\u0001Btpwpxt;\u0018tqG\u0000\ntquk\n\u00002WG\u0000\ntBtuk\u00012wpxt;\u0018tqWG\u0000\ntuk\n\u0010G\u0000\ntpB2\nt\u0001\u0001gquk\u00002pW\u0001wpxt;\u0018tqqG\u0000\ntBtuk\n\u0000\u0000\nwpxt;\u0018tq2\u00012wpxt;\u0018tqW\u0001Btwpxt;\u0018tq\b\nG\u0000\ntuk:\nBy the construction of ukand the boundedness of G\u0000\nt, we have\nsup\ntPr0;Ts}G\u0000\ntpB2\nt\u0001\u0001gqupt;\u0004q}L2pMq¤Opk\u00011\n2q:\nSinceWis order zero, and therefore bounded on L2pMq, we obtain\nsup\ntPr0;Ts\u0007\u0007\u0007\u0007pwpxt;\u0018tq2\u00012wpxt;\u0018tqW\u0001Btwpxt;\u0018tqqG\u0000\ntuk\u0007\u0007\u0007\u0007\nL2pMq¤Csup\ntPr0;Ts||ukpt;\u0004q||L2\u0010Opk\u00011q;\nwhere the \fnal equality follows from the fact that³\nRnkn\n4e\u0001k|y|2dyis uniformly bounded in k.\nTo estimate W\u0001wpxt;\u0018tqqG\u0000\ntBtukpt;\u0004qwe will apply Proposition 3.2 with m\u00100 and`\u00101.\nTo do so, note W\u0001wpxt;\u0018tqis a pseudo with appropriate vanishing properties. Furthermore\nBtukpx;tq\u0010k\u00011\u0000n\n4Btbpt;xqeik px;tq\u0000ikn\n4bpt;xqBt px;tqeik px;tq;\nand for \fxed t, both of these terms take the form of a coherent state hkas de\fned in\nProposition 3.2 (the fact that the \frst has an extra factor of k\u00011is irrelevant, as it only\nimproves the estimate). Since all quantities depend on tin aC8fashion, for any \"¡0\nsup\ntPr0;Ts\u0007\u0007\u0007\u00072pW\u0001wpxt;\u0018tqqG\u0000\ntBtukpt;\u0004q\u0007\u0007\u0007\u0007\nL2pMq¤Cpk\u00011{2\u0000\"q: (4.5)\nBy the triangle inequality, we obtain (4.4), which completes the proof. \u0003\nThe next step in the proof of the upper bound for \u000bis given apx0;\u00180qproduce a sequence\nofexact solutions to (1.1) whose energy approaches |G\u0000\ntpx0;\u00180q|2.SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 21\nProposition 4.5. Given any T¡0, any\"¡0, and anypx0;\u00180q PS\u0006M, there exists an\nexact solution uof the generalized damped wave equation (1.1) with\n|Epu;0q\u00011|\"\nand\u0007\u0007Epu;Tq\u0001|G\u0000\nTpx0;\u00180q|2\u0007\u0007\": (4.6)\nProof. Letukandvkbe as de\fned previously. Then, de\fne !kas the unique solution of the\ndamped wave equation with initial conditions !kpx;0q\u0010vkpx;0qandBt!kpx;0q\u0010Btvkpx;0q.\nIt is immediate that\nEp!k;0q\u0010Epvk;0q\u0010Epuk;0qÑ1;askÑ8:\nTo see (4.6), \frst note by the triangle inequality\n|Ep!k;tq1\n2\u0001Epvk;tq1\n2|¤Ep!k\u0001vk;tq1\n2: (4.7)\nThus, it su\u000eces to prove that lim kÑ8Epvk;tq\u0010|G\u0000\ntpx0;\u00180q|2and that lim\nkÑ8Ep!k\u0001vk;tq\u00100.\nTo see that lim kÑ8Epvk;tq\u0010|G\u0000\ntpx0;\u00180q|2, note that by the de\fnition of vkand properties\nofG\u0000\nt\nEpvk;tq\u00101\n2»\nM|G\u0000\ntpx0;\u00180qBtukpx;tq\u0001wpxt;\u0018tqG\u0000\ntpx0;\u00180qukpx;tq|2\n\u0000|G\u0000\ntpx0;\u00180qrgukpx;tq|2dvgpxq:\nNow sincewpxt;\u0018tqandG\u0000\ntare bounded\n»\nM\u0007\u0007wpxt;\u0018tqG\u0000\ntpx0;\u00180qukpx;tq\u0007\u00072dvgpxq¤C||ukpt;\u0004q||2\nL2¤C1k\u00012;\nfor someC; C1¡0. Thus,\nlim\nkÑ8Epvk;tq\u0010lim\nkÑ81\n2»\nM\u0007\u0007G\u0000\ntpx0;\u00180qBtukpx;tq\u0007\u00072\u0000\u0007\u0007G\u0000\ntpx0;\u00180qrgukpx;tq\u0007\u00072dvgpxq\n\u0010\u0007\u0007G\u0000\ntpx0;\u00180q\u0007\u00072lim\nkÑ8Epuk;tq\n\u0010\u0007\u0007G\u0000\ntpx0;\u00180q\u0007\u00072; (4.8)\nwhere in the \fnal equality we used that lim\nkÑ8Epuk;tq\u00101.\nTo controlEp!k\u0001vk;tq, letfk\u0010pB2\nt\u0001\u0001\u00002WBtqvk:Then\npB2\nt\u0001\u0001\u00002WBtqpvk\u0001!kq\u0010fk:\nBy Proposition 4.4, for any \";T¡0 there exists a C\";T¡0 such that\nsup\ntPr0;Ts}fkpt;\u0004q}L2pMq¤C\";Tk\u00011\n2\u0000\": (4.9)22 B. KEELER AND P. KLEINHENZ\nBy direct computation\nBtEp!k\u0001vk;tq\u0010»\nMpB2\nt\u0001\u0001gqp!k\u0001vkqBtp!k\u0001vkq\u0000pB2\nt\u0001\u0001gqp!k\u0001vkqBtp!k\u0001vkqdvgpxq\n\u00102Re»\nMrfk\u00012WBtp!k\u0001vkqsBtp!k\u0001vkqdvgpxq\n\u00102Re»\nMfk\u0004Btp!k\u0001vkqdvgpxq\u00014RexWBtp!k\u0001vkq;Btp!k\u0001vkqyL2pMq:\nNote that the second term on the right-hand side above is nonpositive, since Wis a non-\nnegative operator. Now, using (4.9) and that ||Btpvk\u0001!kqpt;\u0004q||L2is uniformly bounded for\nkPNandtPr0;Ts, there exists C1\n\";T¡0 such that\nsup\ntPr0;Ts\u0007\u0007\u0007\u0007\u0007\u00072Re»\nMfkBtp!k\u0001vkqdvgpxq\u0007\u0007\u0007\u0007\u0007\u0007¤2}fkpt;\u0004q}L2}Btp!k\u0001vkqpt;\u0004q}L2¤C1\n\";Tk\u00011\n2\u0000\":\nThus, for any \"¡0\nsup\ntPr0;Ts|BtEp!k\u0001vk;tq|¤C1\n\";Tk\u00011\n2\u0000\":\nIntegrating in tgives\nsup\ntPr0;TsEpvk\u0001!k;tq¤C1\n\";TTk\u00011\n2\u0000\":\nCombining this with (4.7) and (4.8) yields (4.6). \u0003\nFor the penultimate step in the proof of the upper bound for \u000b, we will show that tÞÑtLptq\nis superadditive. That is, for r;t¥0,pt\u0000rqLpt\u0000rq¥tLptq\u0000rLprq. To see this observe\npt\u0000rqLpt\u0000rq\u0010 inf\npx0;\u00180qPS\u0006M»r\u0000t\n0wpxs;\u0018sqds\n\u0010 inf\npx0;\u00180qPS\u0006M\u0002»t\n0wpxs;\u0018sqds\u0000»t\u0000r\ntwpxs;\u0018sqds\n¥ inf\npx0;\u00180qPS\u0006M»t\n0wpxs;\u0018sqds\u0000 inf\npx0;\u00180qPS\u0006M»t\u0000r\ntwpxs;\u0018sqds\n\u0010 inf\npx0;\u00180qPS\u0006M»t\n0wpxs;\u0018sqds\u0000 inf\npx0;\u00180qPS\u0006M»r\n0wpxs;\u0018sqds\n\u0010tLptq\u0000rLprq:\nNow by Fekete's lemma, L8:\u0010lim\ntÑ8Lptq\u0010 sup\ntPr0;8qLptq, and thusLptq¤L8for allt. That the\nsupremum is not in\fnite follows from the fact that wpx;\u0018qis uniformly bounded on T\u0006M:\nWe are now ready to show that \u000b¤2L8. Assume for the sake of contradiction that\n\u000b\u00102L8\u00003\u0011for some\u0011¡0. Then since 2 pL8\u0000\u0011q\u000b, there exists a C¡0 such that for\nallt¥0 and all solutions uof (1.1),\nEpu;tq¤CEpu;0qe\u00012tpL8\u0000\u0011q: (4.10)SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 23\nFor the next step, it is convenient to remove the factor of C:To accomplish this, choose\nT¡0 large enough so that max pC;1qeT\u0011. Then\nCe\u00012TpL8\u0000\u0011qe\u0001Tp2L8\u0000\u0011q:\nSinceLptq¤L8for allt, we obtain\nCe\u00012TpL8\u0000\u0011qe\u00012TL8\u0001T\u0011¤e\u00012TLpTq\u0001T\u0011: (4.11)\nNow, we recall that\n\u0001TLpTq\u0010 sup\npx;\u0018qPS\u0006MlnG\u0000\nTpx;\u0018q:\nThus, there exists a point px0;\u00180qPS\u0006Msuch that ln G\u0000\nTpx0;\u00180q¡\u0001TLpTq\u00011\n2T\u0011:Therefore,\ne\u00012TLpTq\u0001T\u0011|G\u0000\nTpx0;\u00180q|2:\nSo by (4.11) there exists a \u000e¡0 such that\nCe\u00012TpL8\u0000\u0011q|G\u0000\nTpx0;\u00180q|2\u0001\u000e:\nNow, by Proposition 4.5, there exists an exact solution uof (1.1) such that\n1¡Epu;0q\u0001\u000e\n2andEpu;Tq¡|G\u0000\nTpx0;\u00180q|2\u0001\u000e\n2:\nThus,\nEpu;Tq¡Epu;Tq\u0002\nEpu;0q\u0001\u000e\n2\n¡Epu;0q\u0002\n|G\u0000\nTpx0;\u00180q|2\u0001\u000e\n2\n\u0001\u000e\n2Epu;Tq\n¡Epu;0q\u0002\n|G\u0000\nTpx0;\u00180q|2\u0001\u000e\n2\n\u0001\u000e\n2Epu;0q\n\u0010Epu;0q\u0000\n|G\u0000\nTpx0;\u00180q|2\u0001\u000e\b\n:\nTherefore,\nEpu;Tq¡Epu;0qp|G\u0000\nTpx0;\u00180q|2\u0001\u000eq¡CEpu;0qe\u00012TpL8\u0000\u0011q;\nbut this contradicts (4.10). Thus, we must have \u000b¤2L8:Combining this with the discussion\nat the beginning of this section, we have proved the upper bound\n\u000b¤2 mint\u0001D0;L8u:\nWe complete the proof of Theorem 2 in the next section by proving the corresponding lower\nbound for\u000b:\n5.The lower bound for \u000b\nIn this section, we prove that the best exponential decay rate satis\fes\n\u000b¥2 mint\u0001D0;L8u; (5.1)\nwhich is the \fnal component of the proof of Theorem 2. In contrast to the proof of the\nupper bound, this section proceeds in direct analogy to the work of Lebeau, and so we omit\nmany of the details which can be found in [Leb96, Kle17]. While the proofs presented here\nare not new, we include them to introduce notation that is used later in Section 6, where we\nuse Theorem 2 to prove Theorem 1.24 B. KEELER AND P. KLEINHENZ\nWe begin with the following energy inequality, which for the multiplicative case is presented\nas Lemma 1 in [Leb96].\nLemma 5.1. For everyT¡0and every\"¡0, there exists a constant cp\";Tq¡0so that\nfor every solution uof(1.1) ,\nEpu;Tq¤p 1\u0000\"qe\u00012TLpTqEpu;0q\u0000cp\";Tq}pu0;u1q}2\nL2ÀH\u00011; (5.2)\nThis inequality is proved using straightforward properties of the propagation of the defect\nmeasure, so the proof from [Leb96] goes through with no modi\fcation. To obtain the desired\nlower bound on \u000bwe must further control the }pu0;u1q}2\nL2ÀH\u00011on the right hand side.\nGiven Lemma 5.1, we proceed by introducing the adjoint A\u0006\nW\u0010\u0002\n0\u0001Id\n\u0001\u0001g\u00012W\nof the\nsemigroup generator AW. Note that the spectrum of A\u0006\nWis the conjugate of the spectrum of\nAW. Thus, we denote by E\u0006\n\u0015jthe generalized eigenspace of A\u0006\nWwith associated eigenvalue\n\u0015j. Recall that H\u0010H1pMq`L2pMq, equipped with the natural norm. It is also useful to\nintroduce the 9Hseminorm de\fned for elements of Hby\n}pu0;u1qT}2\n9H\u0010}ru0}2\nL2\u0000}u1}2\nL2:\nFor eachN¥1, de\fne the subspace\nHN\u0010#\n'PH:x'; yH\u00100;@ Pà\n|\u0015j|¤NE\u0006\n\u0015j+\n:\nOur \frst observation is that HNis invariant under the action of the semigroup etAW. To\ndemonstrate this, let t kube a basis of the \fnite dimensional spaceÀ\n|\u0015j|¤NE\u0006\n\u0015jDpA\u0006\nWq:Now,\nsinceE\u0006\n\u0015jis invariant under A\u0006\nW, we can express each A\u0006\nW las a \fnite linear combination of\nthet ku. Thus, for each `and any'PHN, we have\nBtxetAW'; `yH\u0007\u0007\nt\u00100\u0010xetAW';A\u0006\nW `yH\u0007\u0007\nt\u00100\u0010¸\nc`;kx'; kyH\u00100;\nby the de\fnition of HN. Repeating this argument, we see that Bj\ntxetAW'; `yH\u0007\u0007\nt\u00100\u00100 for\nallj:Observing that xetAW'; `yHis an analytic function of t, we havexetAW'; `yH\u00100\nfor alltPR:ThereforeetAW'PHN.\nNow, de\fne H1\u0010L2`H\u00011and let\u0012Ndenote the norm of the embedding of HNin\nH1, which is well-de\fned since Mis compact. Since Wis bounded on L2, it is compact\nas an operator from L2ÑH\u00011. Therefore A\u0006\nW:HÑH1is a compact perturbation of\nthe skew-adjoint operator\u0002\n0\u0001Id\n\u0001P 0\n. Thus, the family tE\u0006\n\u0015ju8\nj\u00100is total in H, and so\nlimNÑ8\u0012N\u00100 (c.f. [GK69, Ch. 5, Theorem 10.1]).\nWe can now proceed with the proof of (5.1). Assume that 2 min t\u0001D0;L8u¡0, otherwise\nthe statement is trivial. Choose \u0011¡0 small enough so that \f\u00102 mint\u0001D0;L8u\u0001\u0011¡0\nand takeTlarge enough so that 4 |L8\u0001LpTq|\u0011ande\u0011T\n2¡3. Then, by Lemma 5.1 with\n\"\u00101, there exists a constant cp1;Tqsuch that for every solution uof (1.1)\nEpu;tq¤2e\u00012TLpTqEpu;0q\u0000cp1;Tq}pu0;u1q}2\nH1: (5.3)SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 25\nNext, choose Nlarge enough so that cp1;Tq\u00122\nN¤e\u00012TLpTq. Then, for solutions uof (1.1)\nwith initial data pu0;u1qTPHN\nEpu;Tq¤3e\u00012TLpTqEpu;0q:\nSinceHNis invariant under evolution by etAW\nEpu;kTq¤3ke\u00012kTLpTqEpu;0q;@kPN:\nThen, we can use the fact that 4 |L8\u0001LpTq|\u0011and\u0011T\n2¡ln 3 to obtain that\nEpu;kTq¤3ke\u00012kTpL8\u0001\u0011{4qEpu;0q\n¤\u0001\neln 3\u0001\u0011T\n2\tk\ne\u00012kTL8Epu;0q\n¤e\u0001kT\fEpu;0q;\nwhere the \fnal inequality follows from the fact that \f¤2L8\u0001\u00112L8by de\fnition. Since\nthe energy is nondecreasing, it follows that\nEpu;tq¤Ce\u0001\ftEpu;0q @t¥0; (5.4)\nfor some constant C¡0:\nTo extend (5.4) to all solutions of (1.1), let \u0005 denote the orthogonal projection from H\nontoÀ\n|\u0015j|¤NE\u0015j:Then for any v\u0010 pu0;u1qTPH, there is an orthogonal decomposition of\nthe formv\u0010\u0005v\u0000pId\u0001\u0005qv. SinceE\u0015jandE\u0006\n\u0015kare orthogonal for \u0015j\u0018\u0015k, we have that\npId\u0001\u0005qvPHN;and henceHK\nN\u0010À\n|\u0015j|¤NE\u0015j. SinceE\u0015jis invariant under etAWandHK\nNis\n\fnite dimensional, we have that there exists a C¡0 so that for all solutions uof (1.1) with\ninitial data in HK\nN,\nEpu;tq¤Ce2D0Epu;0q¤Ce\u0001\ftEpu;0q;@t¥0: (5.5)\nFinally, since \u0005 and Id \u0001\u0005 are continuous with respect to the 9Hseminorm, for some C¡0\nEp\u0005u;0q\u0000EppId\u0001\u0005qu;0q¤CEpu;0q:\nTherefore, using the decomposition \u0005 \u0000pId\u0001\u0005qon the initial data of any solution uwe can\napply (5.4) and (5.5) to obtain\nEpu;tq¤Ce\u0001\ftEpu;0q;@t¥0; (5.6)\nfor some possibly larger C¡0:By de\fnition of the best possible decay rate, \u000b¥\f\u0010\n2 mint\u0001D0;L8u\u0001\u0011. Since\u0011can be taken arbitrarily small, this proves (5.1). Combining\nthis with the upper bound obtained in Section 4 completes the proof of Theorem 2.\n6.Proof of Theorem 1\nIn this section we show that Theorem 2 implies Theorem 1. First, we will assume both\nAssumptions 1 and 2 are satis\fed. We will show this implies \u000b¡0, which is equivalent to\nexponential energy decay. Note that Assumption 1 immediately implies that L8¥c¡0.\nThus, we only need to show that D00. For this, we introduce the quantity\nD8:\u0010lim\nRÑ8suptRep\u0015q:|\u0015|¡R; \u0015PSpecAWu:\nWe claim that D8¤\u0001L8. To show this, \frst recall E\u0015jandHNfrom Section 5. Let ube a\nsolution to (1.1) with initial data pu0;u1qTPE\u0015jwith|\u0015j|¡N. Thenu\u0010etAWpu0;u1qT\u001026 B. KEELER AND P. KLEINHENZ\net\u0015jpu0;u1qT. Note that E\u0015jHNwhenever|\u0015j| ¡N. Combining this with the proof of\n(5.4)\ne2Rep\u0015jqtEpu;0q\u0010Epu;tq¤Ce\u0001\ftEpu;0q;\nfor every 0\f2L8. Hence, 2Re p\u0015jq¤\u0001\fwhenever|\u0015j|¥N, and so Rep\u0015jq¤\u0001L8\nfor such\u0015j. It immediately follows that D8¤\u0001L80.\nBy the abstract spectral theory arguments in the proof of [AL14, Lemma 4.2], the spectrum\nofAWconsists only of isolated eigenvalues and Re p\u0015q¤0 for all\u0015PSpecpAWq. Thus, in\norder to have D0\u00100, eitherD8\u00100 or there exists a nonzero eigenvalue of AWon the\nimaginary axis. Since we have already shown D80, we need only rule out nonzero\nimaginary eigenvalues. Suppose i\u0015PSpecpAWqwith\u0015PRand corresponding eigenvector\npv0;v1qT:Thenv1\u0010\u0015v0, and\n\u0001gv0\u0000\u00152v0\u00012i\u0015Wv 0\u00100: (6.1)\nTaking the L2inner product of both sides with v0and then taking the imaginary part gives\n\u00012\u0015xWv 0;v0y\u00100:\nIf\u0015\u00100, the equation is trivially satis\fed. However, if \u0015\u00180, thenxWv 0;v0y\u00100. Recalling\nthatW\u0010°B\u0006\njBjfor some collection of operators Bj, we must have Wv 0\u00100. Then by\n(6.1)v0is an eigenfunction of \u0001 gwith eigenvalue \u0001\u00152andv0PkerW. But by Assumption 2,\nthis is impossible. Thus, the only possible eigenvalue of AWon the imaginary axis is zero\nand we cannot have D0\u00100:Combining this with the fact that L8¡0, we have shown\nthat Assumptions 1 and 2 imply \u000b¡0, which in turn demonstrates that solutions to (1.1)\nexperience exponential energy decay.\nWe now prove the reverse implication in Theorem 1. For this, we assume that (1.4) holds\nwith some \f¡0 for all solutions uand we want to see that Assumptions 1 and 2 hold.\nBy de\fnition, \u000b¥\f¡0, and hence both \u0001D0andL8are strictly positive. Because\nL8¥\u000b{2¡0 Assumption 1 holds. Similarly, since D00, there cannot be any eigenvalues\nofAWon the imaginary axis except possibly at zero. Now suppose that vPL2satis\fes\n\u0001\u0001gv\u0010\u00152vwith\u0015\u00180 andWv\u00100:Thenpv;i\u0015vqTis an eigenvector of AWwith eigenvalue\ni\u0015\u00180, which is a contradiction. Thus, Assumption 2 must also hold, which completes the\nproof of Theorem 1.\n7.A Class of Examples on Analytic Manifolds\nOne of the key hypotheses of Theorem 1 was that the damping coe\u000ecient Wmust not\nannihilate any eigenfunctions of \u0001 gassociated with nonzero eigenvalues. In the case where W\nis a multiplication operator whose support satis\fes the classical geometric control condition,\nthis is always satis\fed by the unique continuation properties of elliptic operators [RT75].\nHowever, when the damping is pseudodi\u000berential it is much more di\u000ecult to check this\nhypothesis.\nIn this section, we produce a collection of operators on real analytic manifolds which\nsatisfy Assumption 2 and are not multiplication operators. We also give an example of\nan explicit pseudodi\u000berential damping coe\u000ecient on T2which satis\fes Assumptions 1 and\n2. The primary tool in this discussion is the analytic wavefront set, and so we begin by\nproviding some background de\fnitions for the reader's convenience. More details can be\nfound in [H or83, §8.4-8.6].SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 27\nGiven a set XRnand a distribution uPD1pXq, ifuis real analytic on an open\nneighborhood of x0we write that uPCanearx0PX:In analogy with the relationship\nbetween the standard wavefront set and C8singularities, one can resolve Casingularities\nby de\fning the analytic wavefront set, written WFApuqand de\fned as follows.\nDe\fnition 7.1. We say that a point px0;\u00180qPT\u0006Xz0is not inWFApuq, if there exists an\nopen neighborhood Uofx0, a conic neighborhood \u0000of\u00180and a bounded sequence uNPE1pXq,\nwhich are equal to uonU, and each satisfy\n|puNp\u0018q|¤C\u0002N\u00001\n|\u0018|\nN\n; (7.1)\nfor all\u0018P\u0000:\nBy [H or83, Prop. 8.4.2], we have that uPCanearx0if and only if WFApuqcontains no\npoints of the form px0;\u0018qwith\u0018\u00180:\nWe also introduce a set, which we can be thought of as the analytically invertible directions\nofudenoted by \u0000 Apuq. Its complement is commonly called the (analytic) characteristic set\nofu[H or83].\nDe\fnition 7.2. We say that \u00180PRnz0is in \u0000Apuqif there exists a complex conic neighbor-\nhoodVof\u00180and a function \b, which is holomorphic in t\u0018PV:|\u0018|¡cufor somec¡0;\nsatisfying \bpu\u00101inVXRnand there exists C;N¡0such that\n|\bp\u0010q|¤C|\u0010|N;\nfor\u0010PV:\nThe \fnal preliminary we require is the notion of the normal set of a closed region F\ncontained within a manifold M. For the purposes of this de\fnition, we only require that M\nbeC2.\nDe\fnition 7.3. LetFbe a closed region in a C2manifoldM:The exterior normal set,\nNepFq, is de\fned as the set of all px0;\u00180q PT\u0006Mz0such thatx0PFand such that there\nexists a real valued function fPC2pMqwithdfpx0q\u0010\u00180\u00180and\nfpxq¤fpx0q; xPF:\nThe interior normal set of Fis then de\fned by NipFq\u0010tpx;\u0018q:px;\u0001\u0018qPNepFquand the\nfull normal set is de\fned as NpFq\u0010NepFqNipFq. We write NpFqto denote the closure\nof the normal set of F.\nNote that the projection of NepFqontoMis dense inBFbut might not be equal to BF[H or83,\nProp. 8.5.8].\nWith these de\fnitions in hand, we are able to describe a class of pseudodi\u000berential oper-\nators which do not annihilate any eigenfunctions of \u0001 g.\nLemma 7.4. LetpM;gqbe a compact, real analytic manifold of dimension n. Suppose\n\u001f;r\u001fPC8\ncpMqare cuto\u000b functions supported entirely within a single coordinate patch, with\nr\u001f\u00111on an open neighborhood of the support of \u001f. Let PC8pRnqbe homogeneous\nof degree 0 outside a compact neighborhood of the origin, and de\fne BP\t0\nclpMqin local\ncoordinates by Bu\u0010r\u001fOpp p\u0018qq\u001fu. Letq denote the inverse Fourier transform of  and\n\u00192:Rn\u0002RnÑRndenote the natural projection onto the \fber variables \u0018, if\n\u00192pNpsupp\u001fqqX\u0000Apq q\u0018H;28 B. KEELER AND P. KLEINHENZ\nthen for any eigenfunction uof\u0001g, we haveBu\u00180:\nProof. We proceed by contradiction, so assume Bu\u00100 for some eigenfunction uof \u0001g.\nThusWFApBuq \u0010 H and we aim to show there exists some px0;\u00180q PWFApBuq. First,\nby [H or83, Thm 8.5.6'], we have\nNpsupp\u001fuqWFAp\u001fuq:\nSinceuis an eigenfunction, it cannot vanish identically on any open set. We claim that this\nimplies\nBpsupp\u001fqBp supp\u001fuq: (7.2)\nTo see this, suppose xPBpsupp\u001fqand letVbe any open neighborhood of x. Since\u001fpxq\u00100,\nwe have that \u001fpxqupxq\u00100, so it is enough to show that \u001fuis not identically zero on all of V.\nWithout loss of generality, we may assume that Vlies entirely within the same coordinate\npatch containing supp \u001f. Sincexis a boundary point of the support, \u001fdoes not vanish\nidentically on V. By continuity, this implies the existence of a smaller open neighborhood\nrVV(not containing x) where\u001fis never zero. Since uis an eigenfunction, it cannot vanish\nidentically on rV, and hence \u001fuis not identically zero on rVV, which proves (7.2).\nNow we want to show Npsupp\u001fq Npsupp\u001fuq. Takepx0;\u00180q PNpsupp\u001fq;notex0\nmaximizes a function fon supp\u001fwithdfpx0q\u00180, sox0PBpsupp\u001fqBp supp\u001fuq. That\nisx0is not an interior point. Furthermore, since supp \u001fsupp\u001fuandfis maximized at\nx0in supp\u001fit must also be maximized at x0when restricted to the smaller set supp \u001fu:\nThereforeNpsupp\u001fqNpsupp\u001fuqandNpsupp\u001fqNpsupp\u001fuq.\nHence,\nNpsupp\u001fqWFAp\u001fuq: (7.3)\nSince the cuto\u000b function \u001fis supported in a single coordinate patch, we can treat \u001fu\nand Opp q\u001fuas functions on Rn:Now, observe that q \u0006\u001fu\u0010Opp q\u001fu, where\u0006denotes\nstandard convolution. This, along with [H or83, Thm 8.6.15] gives\nWFAp\u001fuqWFApOpp q\u001fuqYpRn\u0002\u0000Apq qcq: (7.4)\nApplying (7.3), we obtain\nNpsupp\u001fqWFApOpp q\u001fuqYpRn\u0002\u0000Apq qcq;\nand therefore,\nNpsupp\u001fqXpRn\u0002\u0000Apq qqWFApOpp q\u001fuq:\nBy hypothesis, there exists a point\npx0;\u00180qPNpsupp\u001fqXpRn\u0002\u0000Apq qqWFApOpp q\u001fuq:\nIn particular, x0Psupp\u001f, and since r\u001f\u00111 on a neighborhood of supp \u001f, we see thatpx0;\u00180q\nmust also lie inside WFApr\u001fOpp q\u001fuq\u0010WFApBuq. This contradicts the assumption that\nBu\u00100, and thus the proposition is proved.\n\u0003\nRemark 7.5. It is worth noting that the argument of this lemma works for when \u0001 gis\nreplaced by P, an elliptic second order pseudodi\u000berential operator, as long as P's eigenfunc-\ntions do not vanish identically on open sets.\nGiven Proposition 7.4, the proof of Theorem 3 is straightforward.SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 29\nProof of Theorem 3. Given a real analytic manifold pM;gq;take\u001f;r\u001fas in the statement of\nProposition 7.4. Let px0;\u00180qPNepsupp\u001fqbe an arbitrary exterior normal. Then, take any\n PC8pRnqwhich is identically one in a conic neighborhood of \u00180, zero on the complement\nof a slightly larger conic neighborhood, and homogeneous of degree 0 away from the origin.\nThen \u0000Apq qcontains\u00180because \u00111 on a conic neighborhood of \u00180, and so one may take\n\b\u00111 in the de\fnition of \u0000 A:Proposition 7.4 then guarantees that B\u0010r\u001fOpp q\u001fdoes not\nannihilate any eigenfunctions of \u0001 g;and thus neither does W\u0010B\u0006B:One can repeat this\nprocess in any \fnite number of coordinate patches to show that there exists W\u0010°N\nj\u00101B\u0006\njBj\nwith the same property. \u0003\nWe now construct a pseudodi\u000berential damping coe\u000ecient on T2which satis\fes Assump-\ntions 1 and 2.\nExample 7.6. LetT2\u0010R2{Z2denote the two-dimensional torus equipped with the \rat\nmetric, and let \u0001 be the associated Laplace-Beltrami operator. Let \u000e¡0 and let\u001f1P\nC8\ncpT2qbe supported in the vertical strip tpxp1q;xp2qq PT2:1\n2\u0001\u000e¤xp1q¤1\n2\u0000\u000euand\nequal to one on a smaller vertical strip. De\fne r\u001f1in a similar way, but with r\u001f1\u00111\non the support of \u001f1. Analogously, let \u001f2PC8\ncpT2qbe supported in the horizontal strip\ntpxp1q;xp2qqPT2:1\n2\u0001\u000e¤xp2q¤1\n2\u0000\u000euand equal to one on a smaller horizontal strip, and\nde\fner\u001f2similarly with r\u001f2\u00111 on the support of \u001f2.\nNow, let\"¡0 and let 1PC8pS1qbe supported in the set\n\u00022\"\u0010\u0001\n\u0001\u0019\n4\u00012\";\u0019\n4\u00002\"\t\nY\u00023\u0019\n4\u00012\";5\u0019\n4\u00002\"\nand equal to one on the smaller set \u0002 \". Similarly, let  2PC8pS1qbe nonzero on \u0002 2\"\u0000\u0019\n2\nand equal to one on \u0002 \"\u0000\u0019\n2. Choose\fPC8\ncpRqto be supported in r1\n4;8qand equal to one\nonr1\n2;8q. Then de\fne symbols bjPS0\nclpT\u0006T2qby\nbjp\u0018q\u0010 jp\u0012q\fprq; j\u00101;2;\nwhere\u0018\u0010 pr;\u0012qin standard polar coordinates on T\u0006\nxT2:Figure 1 illustrates the cone of\ndirections in T\u0006\nx0T2in whichb1is supported at some arbitrary x0Psupp\u001f1:Now de\fne\nBj\u0010r\u001fjOppbjq\u001fj, and set the damping coe\u000ecient Wto be\nW\u0010B\u0006\n1B1\u0000B\u0006\n2B2:\nTo see kerWcontains no nontrivial eigenfunctions of \u0001 we apply Proposition 7.4. Note\nNpsupp\u001f1qcontains all points of the form px;\u0018qwithxPBpsupp\u001f1qand\u0018\u0010pr;\u0012q, where\u0012\u0010\n0 or\u0012\u0010\u0019. Sinceb1is constant in a conic neighborhood of both of these cotangent directions,\nthe hypotheses of Proposition 7.4 are satis\fed. Thus, ker B1contains no eigenfunctions of the\nLaplacian. An analogous argument holds for B2, and since B\u0006\n1B1andB\u0006\n2B2are nonnegative\noperators,Wcannot annihilate any eigenfunctions of the Laplacian.\nTo show exponential energy decay with Was the damping coe\u000ecient, we must also demon-\nstrate that Wsatis\fes the AGCC. For this, it is convenient to observe that the AGCC is\nequivalent to the existence of some T0¡0 andc¡0 such that every trajectory tÞÑ'tpx0;\u00180q\nencounters the set\nWc\u0010tpx;\u0018qPT\u0006T2:wpx;\u0018q¥c¡0u30 B. KEELER AND P. KLEINHENZ\nFigure 1. The cone of directions containing the support of b1px0;\u0004qinT\u0006\nx0T2.\nin timeT¤T0:Recall that the geodesics on T2are the projections of straight lines in R2\nunder the quotient map. Thus, the geodesic \row on S\u0006T2is given by\npx;\u0018qÞÑppx\u0000t\u0018qmodZ2;\u0018q:\nGiven an arbitrary point px0;\u00180qPS\u0006T2, we will show that p\rptq;\r1ptqq\u0010ppx0\u0000t\u00180qmodZ2;\u00180q\nmust intersect Wcin some \fxed time T0¡0. Let us write \u00180PS1aspcos\u00120;sin\u00120q, and\nconsider the case where \u00120lies in \u0002 \". Suppose \frst that\n\u00120P\u0001\n\u0001\u0019\n4\u0001\";\u0019\n4\u0000\"\t\n;\nwhich implies b1p\u00180q\u00180. Then, if x0\u0010pxp1q\n0;xp2q\n0q, the horizontal coordinate of \rptqis given\nby\npxp1q\n0\u0000tcos\u00120qmodZ;\nwhich must reach1\n2in some time less than1\ncos\u00120¤1\ncosp\u0019{4\u0000\"q. Therefore,p\rptq;\r1ptqqintersects\nthe region where b1is strictly positive in time less than1\ncosp\u0019{4\u0000\"q. The same argument holds\nif instead\u00120Pp3\u0019\n4\u0001\";5\u0019\n4\u0000\"q, and so whenever \u00120P\u0002\", we have that there exists a c¡0\nsuch thatp\rptq;\r1ptqqintersectstb1px;\u0018q¥?cuin \fnite time. Analogously, if \u00120P\u0002\"\u0000\u0019\n2,\nthen the vertical component of \rptq, given bypxp2q\n0\u0000tsin\u00120qmodZ, must equal1\n2in some\ntime less than1\nsinp\u0019{4\u0001\"q. Therefore, p\rptq;\r1ptqqintersectstb2px;\u0018q ¥?cuin \fnite time.\nSince\nT2\u0002\u0001\n\u0002\"Yp\u0002\"\u0000\u0019\n2q\t\n\u0010S\u0006T2;\nand sincewpx;\u0018q \u0010b2\n1px;\u0018q\u0000b2\n2px;\u0018q, we have that for every px0;\u00180q PS\u0006T2, the curve\n'tpx0;\u00180qintersects Wcin some \fxed time T0¡0. We have therefore shown that Was\nde\fned here satis\fes both Assumptions 1 and 2. Thus by Theorem 1, all solutions to the\ndamped wave equation on T2with damping coe\u000ecient Wexperience exponential energy\ndecay.\nRemark 7.7. In the previous example, one may notice that on the intersection of the\nvertical and horizontal strips, the principal symbol of the damping coe\u000ecient is supportedSHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 31\nin all directions \u0018PT\u0006T2z0:So in this region, Wbehaves very much like a multiplication\noperator for frequencies away from zero. A natural question is whether or not there must\nalways be a point of \\full microsupport\" if the hypotheses of Theorem 1 are to be satis\fed.\nIn fact, there need not be such a point. To see this, we can modify our example above as\nfollows.\nDe\fne\u001f1;r\u001f1; \u001f2;r\u001f2andb1in a similar fashion to the previous example, but now de\fne\nb2to be supported only in the directions with angle \u0012Pp\u0019\n4\u00012\";3\u0019\n4\u00002\"qand identically one\nonp\u0019\n4\u0001\";3\u0019\n4\u0000\"q. Next, we introduce another horizontal strip, disjoint from the \frst, with\na corresponding pair of cuto\u000b functions \u001f3;r\u001f3. Then, de\fne  3PC8pS1qto be supported\ninp5\u0019\n4\u00012\";7\u0019\n4\u00002\"qand equal to one on p5\u0019\n4\u0001\";7\u0019\n4\u0000\"q, and letb3p\u0018q\u0010 3p\u0012q\fprq, where\n\u0018\u0010pr;\u0012qas before. This is illustrated in Figure 2. Then, if we de\fne B3\u0010r\u001f3Oppb3q\u001f3and\nsetW\u0010°3\nj\u00101B\u0006\njBj;we can apply arguments similar to those above to see that Assumptions\n1 and 2 are still satis\fed, but there does not exist any point xPT2wherewpx;\u0018qis supported\nin all directions.\nFigure 2. The cones containing the supports of b2px0;\u0004qandb3px1;\u0004q.\nReferences\n[AL14] N. Anantharaman and M. L\u0013 eautaud. Sharp polynomial decay rates for the damped wave equation\non the torus. Anal. PDE , 7(1):159{214, 2014. doi:10.2140/apde.2014.7.159. With an appendix\nby St\u0013 ephane Nonnenmacher.\n[BC15] N. Burq and H. Christianson. Imperfect geometric control and overdamping for the damped\nwave equation. Communications in Mathematical Physics , 336(1):101{130, 2015.\n[BH07] N. Burq and M. Hitrik. Energy decay for damped wave equations on partially rectangular\ndomains. Mathematical Research Letters , 14(1):35{47, 2007.\n[Bur98] N. Burq. 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Christianson, E. Schenck, A. Vasy, and J. Wunsch. From resolvent estimates to damped\nwaves. J. Anal. Math. , 121(1):143{162, 2014.\n[DJN19a] S. Dyatlov, L. Jin, and S. Nonnenmacher. Control of eigenfunctions on surfaces of variable\ncurvature. arXiv:1906.08923 , 2019.\n[DJN19b] S. Dyatlov, L. Jin, and S. Nonnenmacher. Control of eigenfunctions on surfaces of variable\ncurvature. arXiv preprint arXiv:1906.08923 , 2019.\n[DK20] K. Datchev and P. Kleinhenz. Sharp polynomial decay rates for the damped wave equation with\nh older-like damping. Proc. Amer. Math. Soc. , 2020. doi:10.1090/proc/15018.\n[G\u0013 er91] P. G\u0013 erard. Microlocal defect measures. Communications in Partial Di\u000berential\nEquations , 16(11):1761{1794, 1991, https://doi.org/10.1080/03605309108820822.\ndoi:10.1080/03605309108820822.\n[GK69] I. C. Gohberg and M. G. Kre\u0015 \u0010n. Introduction to the theory of linear nonselfadjoint operators .\nTranslations of Mathematical Monographs, Vol. 18. American Mathematical Society, Providence,\nR.I., 1969. Translated from the Russian by A. Feinstein.\n[H or83] L. H ormander. The Analysis of Linear Partial Di\u000berential Operators I . Berlin: spring-verlag,\n1983. doi:10.1007/978-3-642-61497-2.\n[Jin20] L. Jin. Damped wave equations on compact hyperbolic surfaces. Communications in Mathemat-\nical Physics , 373(3):771{794, 2020.\n[JSC11] S. V. Joubert, M. Y. Shatalov, and C. E. Coetzee. Analysing manufacturing imperfections in\na spherical vibratory gyroscope. In 2011 4th IEEE International Workshop on Advances in\nSensors and Interfaces (IWASI) , pages 165{170. IEEE, 2011.\n[KKBH16] D. Krattiger, R. Khajehtourian, C. L. Bacquet, and M. I. Hussein. Anisotropic dissipation in\nlattice metamaterials. AIP Advances , 6(12):121802, 2016.\n[Kle17] G. Klein. Best exponential decay rate of energy for the vectorial damped wave equation. SIAM\nJournal on Control and Optimization , 56, 07 2017. doi:10.1137/17M1142636.\n[Kle19a] P. Kleinhenz. Stabilization Rates for the Damped Wave Equation with H older-Regular Damping.\nCommun. Math. Phys. , 369(3):1187{1205, 2019.\n[Kle19b] P. Kleinhenz. Decay rates for the damped wave equation with \fnite regularity damping. arXiv\npreprint arXiv:1910.06372 , 2019.\n[Leb96] G. Lebeau. Equation des ondes amorties. In Algebraic and Geometric Methods in Mathematical\nPhysics: Proceedings of the Kaciveli Summer School, Crimea, Ukraine, 1993 , pages 73{109.\nSpringer Netherlands, Dordrecht, 1996.\n[LL17] M. L\u0013 eautaud and N. Lerner. Energy decay for a locally undamped wave equation. Annales de\nla facult\u0013 e des sciences de Toulouse S\u0013 er.6 , 26(1):157{205, 2017.\n[LR05] Z. Liu and B. Rao. Characterization of polynomial decay rate for the solution of linear evolution\nequation. Zeitschrift f ur angewandte Mathematik und Physik ZAMP , 56(4):630{644, 2005.\n[LRLTT17] J. Le Rousseau, G. Lebeau, P. Terpolilli, and E. Tr\u0013 elat. Geometric control condition for the wave\nequation with a time-dependent observation domain. Analysis & PDE , 10(4):983{1015, 2017.\n[LRZ02] K. Liu, B. Rao, and X. Zhang. Stabilization of the wave equations with potential and inde\fnite\ndamping. Journal of mathematical analysis and applications , 269(2):747{769, 2002.\n[Ral69] J. Ralston. Solutions of the wave equation with localized energy. Communications on Pure and\nApplied Mathematics , 22(6):807{823, 1969.\n[Ral82] J. Ralston. Gaussian beams and the propagation of singularities. Studies in Partial Di\u000berential\nEquations, MAA Studies in Mathematics , 23:206{248, 1982.\n[RT75] J. Rauch and M. Taylor. Exponential decay of solutions to hyperbolic equations in bounded\ndomains. Indiana Univ. Math. J. , 24(1):79{86, 1975.\n[Sj o00] J. Sj ostrand. Asymptotic distribution of eigenfrequencies for damped wave equations. Publica-\ntions of the Research Institute for Mathematical Sciences , 36(5):573{611, 2000.\n[Sta17] R. Stahn. Optimal decay rate for the wave equation on a square with constant damping on a\nstrip. Zeitschrift f ur angewandte Mathematik und Physik , 68(2):36, 2017.\n[Str72] R. Strichartz. A functional calculus for elliptic pseudo-di\u000berential operators. American Journal\nof Mathematics , 94(3):711{722, 1972.SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 33\n[Sun22] C. Sun. Sharp decay rate for the damped wave equation with convex-shaped damping. Interna-\ntional Mathematics Research Notices , 2022.\nEmail address :bkeeler@live.unc.edu\nDepartment of Mathematics and Statistics, McGill University, 805 Rue Sherbrooke Ouest,\nMontr \u0013eal, QC H3A 0B9\nEmail address :pkleinhe@gmail.com\nDepartment of Mathematics, Michigan State University, 619 Cedar River Rd, East Lans-\ning, MI 48823"    },    {        "title": "1208.1462v1.Observation_of_Coherent_Helimagnons_and_Gilbert_damping_in_an_Itinerant_Magnet.pdf",        "content": "arXiv:1208.1462v1  [cond-mat.str-el]  7 Aug 2012ObservationofCoherent HelimagnonsandGilbertdamping in anItinerant Magnet\nJ. D. Koralek1,∗,†, D. Meier2,∗,†, J. P. Hinton1,2, A. Bauer3, S. A. Parameswaran2,\nA. Vishwanath2, R. Ramesh1,2R. W. Schoenlein1, C. Pfleiderer3, J. Orenstein1,2\n1Materials Science Division, Lawrence Berkeley National La boratory, Berkeley, California 94720, USA\n2Department of Physics, University of California, Berkeley , California 94720, USA and\n3Physik Department E21, Technische Universit¨ at M¨ unchen, D-85748 Garching, Germany\n(Dated: DatedAugust 30, 2018)\nWe study the magnetic excitations of itinerant helimagnets by applying time-resolved optical spectroscopy\nto Fe0.8Co0.2Si. Optically excited oscillations of the magnetization in the helical state are found to disperse\nto lower frequency as the applied magnetic field is increased ; the fingerprint of collective modes unique to\nhelimagnets,knownashelimagnons. Theuseoftime-resolve dspectroscopyallowsustoaddressthefundamen-\ntal magnetic relaxation processes by directly measuring th e Gilbert damping, revealing the versatility of spin\ndynamics inchiralmagnets.\nTheconceptofchiralitypervadesallofscience,havingpro -\nfound implications in physics, chemistry and biology alike .\nIn solids, relativistic spin-orbit coupling can give rise t o the\nDzyaloshinskii-Moriya (DM) interaction,2,3imparting a ten-\ndency for the electron spins to form helical textures with a\nwell-definedhandednessincrystalslackinginversionsymm e-\ntry. Helical spinorderisespeciallyinterestingwhenthem ag-\nnetismarisesfromthesameelectronsresponsibleforcondu c-\ntion as is the case in doped FeSi which displays unconven-\ntional magnetoresitence,4,5helimagnetism,6and the recently\ndiscovered Skyrmion lattice.7,8The excitations of helimag-\nnets have been studied over the past 30 years, culminating\nrecently in a comprehensive theory of spin excitations call ed\nhelimagnons.9,10Signatures of helimagnons have been ob-\nserved in neutron scattering11and microwave absorption,12\nyet little is known about their magnetodynamics and relax-\nation phenomenaon the sub-picosecondtimescales on which\nmagnetic interactions occur. Understanding the dynamics,\nhowever,isofgreatimportanceregardingspintransfertor que\neffects in chiral magnets, and related proposed spintronic s\napplications.13–15\nIn this work we study the dynamics of collective spin ex-\ncitationsin the itineranthelimagnetFe 0.8Co0.2Si. Ouroptical\npump-probemeasurementsidentifyanomalousmodesatzero\nwavevector ( q=0) which we identify unmistakably as heli-\nmagnons. These helimagnons manifest as strongly damped\nmagnetization oscillations that follow a characteristic s caling\nrelation with respect to temperature and magnetic field. The\nsub-picosecond time resolution of our technique enables de -\nterminationof the intrinsic Gilbert dampingparameterwhi ch\nis foundto be oneorderof magnitudelargerthan in localized\nsystems, revealing the versatility of the spin-lattice int erac-\ntions available in the emergent class of DM-driven helimag-\nnets.\nDespite being a non-magnetic insulator, FeSi is trans-\nformed into an itinerant magnet upon doping with cobalt.4,16\nWe have chosen Fe 0.8Co0.2Si for our study because it can\neasily be prepared in high quality single crystals17with a\nreasonably high magnetic ordering temperature TN, and its\nexotic equilibrium properties are well characterized, ope n-\ning the door for non-equilibrium dynamical studies. Small-\nangle neutron scattering8was used to determine the phase\ndiagram and has revealed helimagnetic spin textures belowTN=30 K that emerge from the interplay between the fer-\nromagnetic exchange and DM interactions. In zero magnetic\nfield the spins form a proper helix with a spatial period of\n≈350˚A,18whereasfinitefieldscantthespinsalongthehelix\nwavevector, kh, (see Fig. 1(c))inducinga conicalstate witha\nnet magnetization. Sufficiently high fields, H≥Hc, suppress\nthe conical order in favor of field alignment of all spins. In\ntheexperimentsreportedhere,femtosecondpulsesoflinea rly\npolarized 1.5 eV photons from a Ti:Sapphire oscillator were\nused to excite a (100) oriented single crystal of Fe 0.8Co0.2Si\nat near normal incidence. The changes induced in the sam-\nple by the pump pulse were probed by monitoringthe reflec-\ntion and Kerr rotation of time-delayed probe pulses from the\nsame laser. In order to minimize laser heating of the sam-\nple the laser repetition rate was reduced to 20 MHz with an\nelectro-optic pulse picker, ensuring that thermal equilib rium\nwasreachedbetweensuccessivepumppulses. Signaltonoise\nwas improved by modulation of the pump beam at 100 KHz\nandsynchronouslock-indetectionofthereflectedprobe. Ke rr\nrotation was measured using a Wollaston prism and balanced\nphotodiode. All temperature and field scans presented in thi s\nwork were performed from low to high TandH||(100)after\nzero-fieldcooling.\nFig. 1 shows the transient reflectivity, ΔR/R, as a function\nof temperature and magnetic field. At high temperature we\nobserve a typical bolometric response from transient heati ng\nof the sample by the pump pulse (Fig 1 (a)).19This is char-\nacterized by a rapid increase in reflectivity, followed by tw o-\ncomponent decay on the fs and ps timescales, corresponding\nto the thermalizationtimes between differentdegreesof fr ee-\ndom (electron, spin, lattice, etc.).20As the sample is cooled\nbelowTN, the small thermal signal is beset by a much larger\nnegative reflectivity transient (Fig. 1 (b)) with a decay tim e\nof roughly τR≈175 ps at low temperature (Fig. 3 (b)). A\nnatural explanation for this is that the pump pulse weakens\nthe magnetic order below TN, which in turn causes a change\nin reflectivity via the resulting shift of spectral weight to low\nenergy.21Thetemperaturedependenceof the peak ΔR/Rval-\nues is plotted in Fig. 1 (c) for several applied fields, showin g\nonlyweakfielddependence.\nToaccessthemagnetizationdynamicsmoredirectlywean-\nalyze the polarizationstate of the probe pulses, which rota tes\nby an angle θKupon reflection from the sample surface, in2\nFIG.1: Timedependence ofthepump-inducedtransientreflec tivityΔR/Rinthe(a)paramagneticand(b)helimagneticstates. Thetem perature\ndependence of the maximum ΔR/Ris plottedin(c)for several applied magnetic fields.\nproportion to the component of the magnetization along the\nlight trajectory. The change in Kerr rotation induced by the\noptical pump, ΔθK, is shown in Fig. 2 as a function of tem-\nperature and field. The upper panels show temperature scans\nat fixedmagneticfield,while afieldscan atfixedtemperature\nisshowninpanel(d). Weobservethat ΔθKchangessignas H\nisreversed(notshown),andgoestozeroas Hgoesto zeroor\nas temperatureis raised above TN. Oscillationsof the magne-\ntization are clearly visible in the raw data below 25 K in the\nhelimagneticphase.\nIn order to analyze the magnetization dynamics, we use a\nsimple phenomenological function that separates the oscil la-\ntory and non-oscillatorycomponentsseen in the data. It con -\nsists ofadecayingsinusoidaloscillation,\nΔθK=e−t\nτK[A+Bsin(ωt)] (1)\nwitha timedependentfrequency,\nω(t)=2πf0/bracketleftBig\n1+0.8/parenleftBig\ne−t\nτK/parenrightBig/bracketrightBig\n(2)\nwhich decays to a final value ω0. We emphasize that there is\nonly a single decay time τKdescribing the magneto dynam-\nics, and it is directly related to the Gilbert damping parame -\nterα=(2πf0τK)−1. This function produces excellent fits to\nthe data as illustrated in Fig. 3 (a), allowing accurate extr ac-\ntion of the oscillation frequencies and decay times shown in\nFigs. 3 (b)-(d). The oscillation frequency is reduced as ei-\nther field or temperature is increased, while the decay time\nτKis roughly constant and equal to τRbelow 20 K. As the\ntemperature is raised towards the phase transition, the rel ax-\nation time τKdiverges, which can be understood in terms of\na diverging magnetic correlation length due to the presence\nof a critical point. The similarity between the decay times τR\nandτKwithin the ordered phase reflects strongly correlated\ncharge and spin degrees of freedom, and supports the notion\nthatΔR/Risdeterminedbythemagneticorder.\nThemagneticoscillationfrequencyreaches f0≈4.8GHzat\nlowtemperature,whichcorrespondstoaLarmorprecessiono f\nspinssubjectedtoafieldof170mT,whichisroughlythecrit-\nical field Hcrequiredto destroy the spin helix. This, togetherwith the fact that the oscillation frequencyis nonzero only in\nthehelicalstate,suggeststhattheoscillationsarecomin gfrom\nexcitations unique to the helical structure. It is well know n\nthat magnetization oscillations can be optically induced b y\nultrafast generation of coherent magnons,24–26however, or-\ndinary magnons cannot explain our data as their frequency\nwouldincreasewith H,oppositetowhatisseenin Fig. 3(c).\nBased on these observations, we propose the following in-\nterpretation of our results: In the helical magnetic phase, the\npump photons weaken the magnetic order through the ultra-\nfast demagnetization process.27As described above, this re-\nduction in magnetic order gives rise to a decrease in the re-\nflectivity at 1.5 eV which is nearly field independent. As a\nmagnetic field is applied the spins become canted along the\nhelix wavevector,giving rise to a macroscopic magnetizati on\nwhichweobserveinKerrrotationviaitscomponentalongthe\nprobelight trajectory. The demagnetizationfromthe pumpi s\nresponsible for the initial peak seen in the ΔθKtime traces,\nand is captured by the exponential component of our fitting\nfunction (green curve in Fig 3 (a)). The pump photons also\nlaunch a coherentspin wave, giving rise to the oscillations in\nΔθK(red curve in Fig. 3 (a)). The form of the oscillatory\ncomponent goes like sin (ωt)rather than [1−cos(ωt)], sug-\ngesting impulsive stimulated Raman scattering as the mecha -\nnism of excitation.25The anomalousfield dependenceshown\nin Fig. 3 (c) leads to the unambiguous conclusion that the\noptically excited spin waves are the fundamental modes of\nhelimagnetstermedhelimagnons.10Specifically,theoptically\naccessible helimagnon mode consists of the constituents of\nthe spin helix precessing in-phase about their local effect ive\nfield. Since this local effective field is reduced during the u l-\ntrafast demagnetizationprocess, the oscillation frequen cyde-\ncreases as a function of time delay as the field recovers, ne-\ncessitatingthetimedependentfrequencyinEq. 1. Theabili ty\nto resolve helimagnons with femtosecond time resolution at\nq=0isuniquetoouropticalprobe,andcomplimentsneutron\nscatteringwhichisrestrictedtomappinghelimagnonbands at\nhigherq. This regionof reciprocalspace is particularlyinter-\nestinginthecaseofhelimagnetsastheperiodicityintrodu ced\nbythehelicalspintexturegeneratesbandsthatarecentere dat\nq=±khand therefore have finite frequency modes at q=03\nFIG. 2: (a),(b),(c) Time dependence of the pump-induced cha nge in Kerr rotation, ΔθK, as a function of temperature for several applied\nmagneticfields. (d) ΔθKasafunctionofmagneticfieldforseveraltemperatures. Cur vesareoffsetforclarity. Alsoshownisaschematicphase\ndiagram, adapted from Reference 8,withredarrows illustra tingthe temperature and fieldscans usedin(a)-(d).\neven in the absence of a gap. This is in contrast to ordinary\nmagnonsinwhichthebandsaregenerallycenteredat q=0so\nthattheassociatedmodehaszerofrequency. Wenotethatour\nobservationsare in agreementwith previouswork on the col-\nlectivemodesofskyrmions28whichcoexistwithhelimagnons\nin theA-Phase(see Fig. 3).12Theappearanceofthese modes\nis not expected in our data as their corresponding oscillati on\nperiodsexceedtheobserveddampingtimein Fe 0.8Co0.2Si.\nInordertoquantitativelytestthehelimagnoninterpretat ion\nwetaketheexpressionforthe q=0helimagnonfrequencyin\nanexternalmagneticfield,\nf0=gµBHc/radicalbig\n1+cos2θ (3)\nwheregistheeffectiveelectron g-factor,µBistheBohrmag-\nneton,andπ\n2−θistheconicalanglei.e. theamountthespins\nare canted away from khby the applied field H. Ignoringde-\nmagnetization effects of the spin waves themselves, we can\nwrite sinθ=H\nHc, whereHcis the critical field at which the\nspinsall alignwiththe field andthe helimagnonceasesto ex-\nist asa well-definedmode. Thenweobtain,9\nf0=gµBHc/radicalBigg\n1−1\n2/parenleftbiggH\nHc/parenrightbigg2\n(4)\nwhich expresses the magnon frequency as a function of ap-\nplied field. This expression fits the data remarkably well as\nshown in Fig. 3 (c), capturingthe decrease in frequencywith\nincreasing Hwhich is unique to helimagnons. However, due\ntothefactthattheoscillationperiodexceedsthedampingt imeforfieldsabove75mT,itisnotpossibletoextractthevalueo f\nthecritical field Hcinthis system. The solidline in Fig. 3(d)\nisafittotheform f0∝/radicalBig\n1−T\nTNwhichgives TNasafunction\nofHin reasonableagreementwithpublisheddata.8\nThe Gilbert damping parameter can be directly obtained\nfrom the measured decay times through the relation α=\n(2πf0τK)−1, which gives a value of α≈0.4 for the heli-\nmagnetic phase of Fe 0.8Co0.2Si. This is an order of magni-\ntude larger than what was seen in insulating Cu 2OSeO3,12\nwhere helimagnetism arises from localized rather than itin -\nerant spins. The contrast in dynamics between these systems\nis critical in the context of potential spintronic applicat ions\nbasedonhelimagnetismwherethereisatradeoffbetweenfas t\nswitching which requires large damping, and stability whic h\nreliesonlowdamping.\nIn summary, this work demonstrates ultrafast coherent op-\ntical excitation of spin waves in an itinerant DM-driven spi n\nsystem and reveals the underlying spin dynamics. We iden-\ntifytheseexcitationsashelimagnonsthroughtheiranomal ous\nfield dependence and explain our observations with a com-\nprehensive model. Our experiments directly yield the intri n-\nsic Gilbertdampingparameter,revealingastrikingdiffer ence\nin spin relaxationphenomenabetweenitinerant andlocaliz ed\nhelimagnets. The results elucidate the dynamicsof collect ive\nmodes common to the actively studied B20 transition metal\ncompounds that codetermine their performance in potential\nspinbasedapplications.\nAcknowledgments: The work in Berkeley was supported\nby the Director, Office of Science, Office of Basic Energy4\nFIG.3: (a) Exemplary ΔθKoscillation data (blue circles) and fit (black line) using th e model described inthe text. The fitis decomposed into\nan exponential term (green curve) and an oscillatory term (r ed curve). The fitting function uses a single time constant τKfor all terms which\nis plotted in panel (b) as a function of temperature and field. For comparison we also plot the decay time of the reflectivity ,τR, averaged over\nall fields. The solid lines are guides to the eye. Panels (c) an d (d) show the reduced magnetization oscillation frequency for field scans and\ntemperature scans respectively, andsolidlines are fitstot he data as described inthe maintext.\nSciences,MaterialsSciencesandEngineeringDivision,of the\nU.S. Department of Energy under Contract No. DE-AC02-\n05CH11231. C.P. and A.B. acknowledge support through\nDFGTRR80(FromElectronicCorrelationstoFunctionality) ,\nDFG FOR960 (Quantum Phase Transitions), and ERC AdG\n(291079, TOPFIT). A.B. acknowledges financial support\nthrough the TUM graduate school. D.M. acknowledges sup-portfromtheAlexandervonHumboldtfoundationandS.A.P.\nacknowledgessupportfrom the SimonsFoundation. C.P. and\nA.B. also thank S. Mayr, W. M¨ unzer, and A. Neubauer for\nassistance.\n∗Theseauthorscontributedequallytothiswork.\n†Email address: jdkoralek@lbl.gov and meier@berkeley.edu\n2I. E.Dzyaloshinskii, Sov. Phys.JETP 5, 1259 (1957).\n3T. Moriya, Phys. Rev. 120, 91(1960).\n4N. Manyala etal., Nature404, 581 (2000).\n5N. 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Kapeliovich and T. L. Perelman, Sov. Phy s.\nJETP39, 375 (1975).\n21F.P.Mena etal.,Phys. Rev. B 73, 085205 (2006).\n22S.A.Brazovskii,S.G.Dmitriev,Sov.Phys.JETP 42,497(1976).\n23M. Janoschek, M. Garst, A. Bauer, P. Krautscheid, R. Georgii ,\nP.B¨ oni, andC.Pfleiderer,arXiv:1205.4780v1(2012).\n24M. van Kampen etal.,Phys. Rev. Lett. 88, 227201 (2002).\n25A. M.Kalashnikova et al.,Phys.Rev. B 78, 104301 (2008).\n26D. Talbayev et al.,Phys. Rev. Lett. 101, 097603 (2008).\n27A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. Phys. 82,\n2731 (2010).\n28M. Mochizuk, Phys. Rev. Lett. 108, 017601 (2012)."    },    {        "title": "0708.3323v1.Enhancement_of_the_Gilbert_damping_constant_due_to_spin_pumping_in_noncollinear_ferromagnet_nonmagnet_ferromagnet_trilayer_systems.pdf",        "content": "arXiv:0708.3323v1  [cond-mat.mes-hall]  24 Aug 2007Enhancement of the Gilbert damping constant due to spin pump ing in non-collinear\nferromagnet / non-magnet / ferromagnet trilayer systems\nTomohiro Taniguchi1,2, Hiroshi Imamura2\n1Institute for Materials Research, Tohoku University, Send ai 980-8577,\n2Nanotechnology Research Institute, National Institute of Advanced Industrial Science and Technology,\n1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan\n(Dated: October 29, 2018)\nWe analyzed the enhancement of the Gilbert damping constant due to spin pumping in non-\ncollinear ferromagnet / non-magnet / ferromagnet trilayer systems. We show that the Gilbert\ndamping constant depends both on the precession angle of the magnetization of the free layer and\non the direction of the magntization of the fixed layer. We find the condition to be satisfied to\nrealize strong enhancement of the Gilbert damping constant .\nPACS numbers: 72.25.Mk, 75.70.Cn, 76.50.+g, 76.60.Es\nThere is currently great interest in the dynamics of\nmagnetic multilayers because of their potential applica-\ntions in non-volatile magnetic random access memory\n(MRAM) and microwave devices. In the field of MRAM,\nmuch effort has been devoted to decreasing power con-\nsumption through the use of current-induced magnetiza-\ntion reversal (CIMR) [1, 2, 3, 4, 5, 6, 7]. Experimentally,\nCIMR is observed as the current perpendicular to plane-\ntype giant magnetoresistivity (CPP-GMR) of a nano pil-\nlar, in which the spin-polarized current injected from the\nfixed layer exerts a torque on the magnetization of the\nfree layer. The torque induced by the spin current is\nutilized to generate microwaves.\nThe dynamics of the magnetization Min a ferromag-\nnet under an effective magnetic field Beffis described by\nthe Landau-Lifshitz-Gilbert (LLG) equation\ndM\ndt=−γM×Beff+α0M\n|M|×dM\ndt,(1)\nwhereγandα0are the gyromagnetic ratio and the\nGilbert damping constant intrinsic to the ferromagnet,\nrespectively. The Gilbert damping constant is an im-\nportant parameter for spin electronics since the critical\ncurrent density of CIMR is proportional to the Gilbert\ndamping constant [8, 9] and fast-switching time magne-\ntization reversal is achieved for a large Gilbert damp-\ning constant [10]. Several mechanisms intrinsic to ferro-\nmagnetic materials, such as phonon drag [11] and spin-\norbit coupling [12], have been proposed to account for\nthe origin of the Gilbert damping constant. In addition\nto these intrinsic mechanisms, Mizukami et al.[13, 14]\nand Tserkovnyak et al.[15, 16] showed that the Gilbert\ndampingconstantinanon-magnet(N) /ferromagnet(F)\n/ non-magnet(N) trilayersystem is enhanced due to spin\npumping. Tserkovnyak et al.[17] also studied spin pump-\ning in a collinear F/N/F trilayer system and showed that\nenhancement of the Gilbert damping constant depends\non the precession angle of the magnetization of the free\nlayer.\nOn the other hand, several groups who studied CIMR\nin a non-collinear F/N/F trilayer system in which theFIG. 1: (Color online) The F/N/F trilayer system is schemat-\nically shown. The magnetization of the F 1layer (m1) pre-\ncesses around the z-axis with angle θand angular velocity ω.\nThe magnetization of the F 2layer (m2) is fixed with tilted\nangleρ. The precession of the magnetization in the F 1layer\npumpsspin current Ipump\nsintotheNandF 2layer, andcreates\nthe spin accumulation µNin the N layer. The spin accumu-\nlation induces the backflow spin current Iback(i)\ns(i= 1,2).\nmagnetization of the free layer is aligned to be perpen-\ndicular to that of the fixed layer have reported the reduc-\ntion of the critical current density [5, 6, 7]. Therefore, it\nis intriguing to ask how the Gilbert damping constant is\naffected by spin pumping in non-collinear F/N/F trilayer\nsystems.\nIn this paper, we analyze the enhancement of the\nGilbert damping constant due to spin pumping in non-\ncollinear F/N/F trilayer systems such as that shown in\nFig. 1. Following Refs. [15, 16, 17, 18], we calculate the\nspin current induced by the precession of the magnetiza-\ntion of the free layer and the enhancement of the Gilbert\ndamping constant. We show that the Gilbert damping\nconstant depends not only on the precession angle θof\nthe magnetization of a free layer but also on the angle ρ\nbetweenthemagnetizationsofthefixedlayerandthepre-\ncession axis. The Gilbert damping constant is strongly\nenhanced if angles θandρsatisfy the condition θ=ρor\nθ=π−ρ.\nThe system we consider is schematically shown in Fig.\n1. A non-magnetic layer is sandwiched between two fer-\nromagnetic layers, F 1and F 2. We introduce the unit2\nvectormito represent the direction of the magnetiza-\ntion of the i-th ferromagnetic layer. The equilibrium\ndirection of the magnetization m1of the left free fer-\nromagnetic layer F 1is taken to exist along the z-axis.\nWhen an oscillatingmagnetic field is applied, the magne-\ntization of the F 1layer precesses around the z-axis with\nangleθ. The precession of the vector m1is expressed\nasm1= (sinθcosωt,sinθsinωt,cosθ), whereωis the\nangular velocity of the magnetization. The direction of\nthe magnetization of the F 2layer,m2, is assumed to be\nfixed and the angle between m2and thez-axis is repre-\nsented byρ. The collinear alignment discussed in Ref.\n[17] corresponds to the case of ρ= 0,π.\nBefore studying spin pumping in non-collinear sys-\ntems, we shall give a brief review of the theory of\nspin pumping in a collinear F/N/F trilayer system [17].\nSpin pumping is the inverse process of CIMR where the\nspin current induces the precession of the magnetization.\nContrary to CIMR, spin pumping is the generation of\nthe spin current induced by the precession of the mag-\nnetization. The spin current due to the precession of the\nmagnetization in the F 1layer is given by\nIpump\ns=/planckover2pi1\n4πg↑↓m1×dm1\ndt, (2)\nwhereg↑↓is a mixing conductance [18, 19] and /planckover2pi1is the\nDirac constant. Spins are pumped from the F 1layer\ninto the N layer and the spin accumulation µNis cre-\nated in the N layer. Spins also accumulate in the F 1\nand F 2layers. In the ferromagnetic layers the trans-\nverse component of the spin accumulation is assumed to\nbe absorbed within the spin coherence length defined as\nλtra=π/|k↑\nFi−k↓\nFi|, wherek↑,↓\nFiis the spin-dependent\nFermi wave number of the i-th ferromagnet. For fer-\nromagnetic metals such as Fe, Co and Ni, the spin co-\nherence length is a few angstroms [20]. Hence, the spin\naccumulation in the i-th ferromagnetic layer is aligned to\nbe parallel to the magnetization, i.e., µFi=µFimi. The\nlongitudinal component of the spin accumulation decays\non the scale of spin diffusion length, λFi\nsd, which is of the\norder of 10 nm for typical ferromagnetic metals [21].\nThe difference in the spin accumulation of ferromag-\nnetic and non-magnetic layers, ∆ µi=µN−µFimi(i=\n1,2), induces a backflow spin current, Iback(i)\ns, flowing\ninto both the F 1and F 2layers. The backflow spin cur-\nrentIback(i)\nsis obtained using circuit theory [18] as\nIback(i)\ns=1\n4π/braceleftbigg2g↑↑g↓↓\ng↑↑+g↓↓(mi·∆µi)mi\n+g↑↓mi×(∆µi×mi)/bracerightbig\n,(3)\nwhereg↑↑andg↓↓are the spin-up and spin-down con-\nductances, respectively. The total spin current flowing\nout of the F 1layer is given by Iexch\ns=Ipump\ns−Iback(1)\ns\n[17]. The spin accumulation µFiin the F ilayer is ob-\ntained by solving the diffusion equation. We assume\nthat spin-flip scattering in the N layer is so weak thatwe can neglect the spatial variation of the spin current\nwithin the N layer, Iexch\ns=Iback(2)\ns. The torque τ1\nacting on the magnetization of the F 1layer is given by\nτ1=Iexch\ns−(m1·Iexch\ns)m1=m1×(Iexch\ns×m1). For\nthe collinear system, we have\nτ1=g↑↓\n8π/parenleftbigg\n1−νsin2θ\n1−ν2cos2θ/parenrightbigg\nm1×dm1\ndt,(4)\nwhereν= (g↑↓−g∗)/(g↑↓+g∗) is the dimensionless\nparameter introduced in Ref. [17]. The Gilbert damping\nconstant in the LLG equation is enhanced due to the\ntorqueτ1asα0→α0+α′with\nα′=gLµBg↑↓\n8πM1dF1S/parenleftbigg\n1−νsin2θ\n1−ν2cos2θ/parenrightbigg\n,(5)\nwheregLis the Land´ e g-factor,µBis the Bohr magneton,\ndF1is the thickness of the F 1layer andSis the cross-\nsection of the F 1layer.\nNext, we move on to the non-collinear F/N/F trilayer\nsystem with ρ=π/2, in which the magnetization of the\nF2layer is aligned to be perpendicular to the z-axis. Fol-\nlowing a similar procedure, the LLG equation for the\nmagnetization M1in the F 1layer is expressed as\ndM1\ndt=−γeffM1×Beff+γeff\nγ(α0+α′)M1\n|M1|×dM1\ndt,(6)\nwhereγeffandα′are the effective gyromagnetic ratio\nand the enhancement of the Gilbert damping constant,\nrespectively. The effective gyromagnetic ratio is given by\nγeff=γ/parenleftbigg\n1−gLµBg↑↓νcotθcosψsinωt\n8πMdF1Sǫ/parenrightbigg−1\n,(7)\nwhere cosψ= sinθcosωt=m1·m2and\nǫ= 1−ν2cos2ψ−ν(cot2θcos2ψ−sin2ψ+sin2ωt).(8)\nThe enhancement of the Gilbert damping constant is ex-\npressed as\nα′=gLµBg↑↓\n8πMdF1S/parenleftbigg\n1−νcot2θcos2ψ\nǫ/parenrightbigg\n.(9)\nItshouldbenotedthat, fornon-collinearsystems, both\nthegyromagneticratioandtheGilbert dampingconstant\nare modified by spin pumping, contrary to what occurs\nin collinear systems. The modification of the gyromag-\nnetic ratio and the Gilbert damping constant due to spin\npumping can be explained by considering the pumping\nspincurrentandthe backflowspincurrent[SeeFigs. 2(a)\nand 2(b)]. The direction of the magnetic moment car-\nried by the pumping spin current Ipump\nsis parallel to\nthe torque of the Gilbert damping for both collinear and\nnon-collinear systems. The Gilbert damping constant is\nenhanced by the pumping spin current Ipump\ns. On the\notherhand, the directionofthe magneticmoment carried3\nFIG. 2: (Color online) (a) Top view of Fig. 1. The dotted\ncircle in F 1represents the precession of magnetization M1\nand the arrow pointing to the center of this circle represent s\nthe torque of the Gilbert damping. The arrows in Ipump\nsand\nIback(1)\nsrepresent the magnetic moment of spin currents. (b)\nThe back flow Iback(1)\nshas components aligned with the di-\nrection of the precession and the Gilbert damping.\nby the backflow spin current Iback(1)\nsdepends on the di-\nrection of the magnetization of the F 2layer. As shown in\nEq. (3), the backflowspin current in the F 2layerIback(2)\ns\nhas a projection on m2. Since we assume that the spin\ncurrent is constant within the N layer, the backflow spin\ncurrent in the F 1layerIback(1)\nsalso has a projection on\nm2. For the collinear system, both Ipump\nsandIback(1)\ns\nare perpendicular to the precession torque because m2\nis parallel to the precession axis. However, for the non-\ncollinear system, the vector Iback(1)\nshas a projection on\nthe precession torque, as shown in Fig. 2(b). Therefore,\nthe angular momentum injected by Iback(1)\nsmodifies the\ngyromagnetic ratio as well as the Gilbert damping in the\nnon-collinear system.\nLet us estimate the effective gyromagnetic ratio using\nrealistic parameters. According to Ref. [17], the con-\nductancesg↑↓andg∗for a Py/Cu interface are given\nbyg↑↓/S= 15[nm−2] andν≃0.33, respectively. The\nLand´ eg-factor is taken to be gL= 2.1, magnetization is\n4πM= 8000[Oe] and thickness dF1= 5[nm]. Substitut-\ning these parameters into Eqs. (7) and (8), one can see\nthat|γeff/γ−1| ≃0.001. Therefore, the LLG equation\ncan be rewritten as\ndM1\ndt≃ −γM1×Beff+(α0+α′)M1\n|M1|×dM1\ndt.(10)\nThe estimated value of α′is of the order of 0.001. How-\never, we cannot neglect α′since it is of the same order\nas the intrinsic Gilbert damping constant α0[22, 23].\nExperimentally, the Gilbert damping constant is mea-\nsuredasthe width ofthe ferromagneticresonance(FMR)\nabsorptionspectrum. LetusassumethattheF 1layerhas\nno anisotropy and that an external field Bext=B0ˆzisapplied along the z-axis. We also assume that the small-\nangle precession of the magnetization around the z-axis\nis excited by the oscillating magnetic field B1applied in\nthexy-plane. The FMR absorption spectrum is obtained\nas follows [24]:\nP=1\nT/integraldisplayT\n0dtαγMΩ2B2\n1\n(γB0−Ω)2+(αγB0)2,(11)\nwhere Ω is the angular velocity of the oscillating mag-\nnetic field, T= 2π/Ω andα=α0+α′. Sinceαis very\nsmall, the absorption spectrum can be approximately ex-\npressedasP∝α0+∝an}bracketle{tα′∝an}bracketri}htandthehighestpointofthepeak\nproportional to ∝an}bracketle{t1/(α0+α′)∝an}bracketri}ht, where∝an}bracketle{tα′∝an}bracketri}htrepresents the\ntime-averaged value of the enhancement of the Gilbert\ndamping constant. In Fig. 3(a), the time-averaged value\n∝an}bracketle{tα′∝an}bracketri}htfor a non-collinear system in which ρ=π/2 is plot-\nted by the solid line as a function of the precession an-\ngleθ. The dotted line represents the enhancement of\nthe Gilbert damping constant α′for the collinear system\ngiven by Eq. (5). The time-averaged value of the en-\nhancement of the Gilbert damping constant ∝an}bracketle{tα′∝an}bracketri}httakes\nits maximum value at θ= 0,πfor the collinear system\n(ρ= 0,π). Contrary to the collinear system, ∝an}bracketle{tα′∝an}bracketri}htof the\nnon-collinear system in which ρ=π/2 takes its maxi-\nmum value at θ=π/2.\nAs shown in Fig. 2(b), the backflow spin current\ngives a negative contribution to the enhancement of the\nGilbert damping constant. This contribution is given by\nthe projection of the vector Iback(1)\nsonto the direction\nof the torque of the Gilbert damping, which is repre-\nsented by the vector m1×˙m1. Therefore, the condition\nto realize the maximum value of the enhancement of the\nGilbert damping is satisfied if the projection of Iback(1)\ns\nontom1×˙m1takes the minimum value; i.e., θ=ρor\nθ=π−ρ.\nWe can extend the above analysis to the non-collinear\nsystemwith anarbitraryvalue of ρ. After performingthe\nappropriate algebra, one can easily show that the LLG\nequation for the magnetization of the F 1layer is given\nby Eq. (6) with\nγeff=γ/bracketleftBigg\n1−gLµBg↑↓νsinρsinωt(cotθcos˜ψ−cscθcosρ)\n8πMdS˜ǫ/bracketrightBigg−1\n(12)\nα′=gLµBg↑↓\n8πMdS/braceleftBigg\n1−ν(cotθcos˜ψ−cscθcosρ)2\n˜ǫ/bracerightBigg\n,\n(13)\nwhere cos ˜ψ= sinθsinρcosωt+ cosθcosρ=m1·m2\nand\n˜ǫ=1−ν2cos2˜ψ\n−ν{(cotθcos˜ψ−cscθcosρ)2−sin2˜ψ+sin2ρsin2ωt}.\n(14)\nSubstituting the realistic parameters into Eqs. (12) and\n(14), we can show that the effective gyromagnetic ratio4\n\u0013\n\u0003\u0013\u0011\u0013\u0013\u0015\u001a\u0003\u0013\u0011\u0013\u0013\u0016\u0015\u0003\u0013\u0011\u0013\u0013\u0016\u001a\n\u000bD\f\n\u000bE\f\u0013\u0011\u0013\u0013\u0016\u001a\n\u0013\u0011\u0013\u0013\u0016\u0015\n\u0013\u0011\u0013\u0013\u0015\u001a/c50/c0f/c12 /c50\n/c51/c1c/c41\n/c1e\n\u0013 /c50/c0f/c12 /c50\n/c52\u0013/c50/c0f/c12/c50\n/c51/c1c/c41\n/c1e\nFIG. 3: (Color online) (a) The time-averaged value of the en-\nhancement of the Gilbert damping constant α′is plotted as a\nfunction of the precession angle θ. The solid line corresponds\nto the collinear system derived from Eq. (9). The dashed\nline corresponds to the non-collinear system derived from E q.\n(5). (b) The time-averaged value of the enhancement of the\nGilbert damping constant α′of the non-collinear system is\nplotted as a function of the precession angle θand the an-\ngleρbetween the magnetizations of the fixed layer and the\nprecession axis.γeffcan be replaced by γin Eq. (6) and that the LLG\nequation reduces to Eq. (10). Figure 3(b) shows the\ntime-averaged value of the enhancement of the Gilbert\ndamping constant ∝an}bracketle{tα′∝an}bracketri}htof Eq. (13). Again, the Gilbert\ndamping constant is strongly enhanced if angles θandρ\nsatisfy the condition that θ=ρorθ=π−ρ.\nIn summary, we have examined the effect of spin\npumping on the dynamics of the magnetization of mag-\nnetic multilayers and calculated the enhancement of the\nGilbertdampingconstantofnon-collinearF/N/Ftrilayer\nsystems due to spin pumping. The enhancement of the\nGilbert damping constant depends not only on the pre-\ncession angle θof the magnetization of a free layer but\nalso on the angle ρbetween the magnetizations of the\nfixed layerand the precession axis, as shown in Fig. 3(b).\nWe have shown that the θ- andρ-dependence of the en-\nhancement of the Gilbert damping constant can be ex-\nplained by analyzing the backflow spin current. The con-\ndition to be satisfied to realizestrongenhancement of the\nGilbert damping constant is θ=ρorθ=π−ρ.\nThe authors would like to acknowledge the valuable\ndiscussions we had with Y. Tserkovnyak, S. Yakata, Y.\nAndo, S. Maekawa, S. Takahashi and J. Ieda. This work\nwas supported by CREST and by a NEDO Grant.\n[1] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1\n(1996).\n[2] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[3] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Em-\nley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph,\nNature425, 380 (2003).\n[4] A. Deac, K. J. 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Lett. 88, 117601 (2002).\n[16] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B66, 224403 (2002).\n[17] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B67, 140404(R) (2003).\n[18] A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Eur.\nPhys. J. B 22, 99 (2001).\n[19] A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Phys.\nRev. Lett. 84, 2481 (2000).\n[20] M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407\n(2002).\n[21] J. Bass and W. P. Jr., J. Phys.: Condens. Matter 19,\n183201 (2007).\n[22] J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers,\nand D. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000).\n[23] F. Schreiber, J. Pflaum, Th. M¨ uhge, and J. Pelzl, Solid\nState Commun. 93, 965 (1995).\n[24] S. V. Vonsovskii, ed., FERROMAGNETIC RESO-\nNANCE (Israel Program for Scientific Translations Ltd.,\nJersalem, 1964)."    },    {        "title": "1405.2347v1.Magnetization_dynamics_and_damping_due_to_electron_phonon_scattering_in_a_ferrimagnetic_exchange_model.pdf",        "content": "Magnetization Dynamics and Damping due to Electron-Phonon Scattering in a\nFerrimagnetic Exchange Model\nAlexander Baral,\u0003Svenja Vollmar, and Hans Christian Schneidery\nPhysics Department and Research Center OPTIMAS,\nKaiserslautern University, P. O. Box 3049, 67663 Kaiserslautern, Germany\n(Dated: June 4, 2018)\nWe present a microscopic calculation of magnetization damping for a magnetic \\toy model.\"\nThe magnetic system consists of itinerant carriers coupled antiferromagnetically to a dispersionless\nband of localized spins, and the magnetization damping is due to coupling of the itinerant carriers\nto a phonon bath in the presence of spin-orbit coupling. Using a mean-\feld approximation for\nthe kinetic exchange model and assuming the spin-orbit coupling to be of the Rashba form, we\nderive Boltzmann scattering integrals for the distributions and spin coherences in the case of an\nantiferromagnetic exchange splitting, including a careful analysis of the connection between lifetime\nbroadening and the magnetic gap. For the Elliott-Yafet type itinerant spin dynamics we extract\ndephasing and magnetization times T1andT2from initial conditions corresponding to a tilt of the\nmagnetization vector, and draw a comparison to phenomenological equations such as the Landau-\nLifshitz (LL) or the Gilbert damping. We also analyze magnetization precession and damping for\nthis system including an anisotropy \feld and \fnd a carrier mediated dephasing of the localized spin\nvia the mean-\feld coupling.\nPACS numbers: 75.78.-n, 72.25.Rb, 76.20.+q\nI. INTRODUCTION\nThere are two widely-known phenomenological ap-\nproaches to describe the damping of a precessing mag-\nnetization in an excited ferromagnet: one introduced\noriginally by Landau and Lifshitz1and one introduced\nby Gilbert,2which are applied to a variety of prob-\nlems3involving the damping of precessing magnetic mo-\nments. Magnetization damping contributions and its in-\nverse processes, i.e., spin torques, in particular in thin\n\flms and nanostructures, are an extremely active \feld,\nwhere currently the focus is on the determination of novel\nphysical processes/mechanisms. Apart from these ques-\ntions there is still a debate whether the Landau-Lifshitz\nor the Gilbert damping is the correct one for \\intrin-\nsic\" damping, i.e., neglecting interlayer coupling, inter-\nface contributions, domain structures and/or eddy cur-\nrents. This intrinsic damping is believed to be caused\nby a combination of spin-orbit coupling and scattering\nmechanisms such as exchange scattering between s and d\nelectrons and/or electron-phonon scattering.4{6Without\nreference to the microscopic mechanism, di\u000berent macro-\nscopic analyses, based, for example, on irreversible ther-\nmodynamics or near equilibrium Langevin theory, prefer\none or the other description.7,8However, material param-\neters of typical ferromagnetic heterostructures are such\nthat one is usually \frmly in the small damping regime so\nthat several ferromagnetic resonance (FMR) experiments\nwere not able to detect a noticeable di\u000berence between\nLandau Lifshitz and Gilbert magnetization damping. A\nrecent analysis that related the Gilbert term directly to\nthe spin-orbit interaction arising from the Dirac equa-\ntion does not seem to have conclusively solved this dis-\ncussion.9\nThe dephasing term in the Landau-Lifshitz form isalso used in models based on classical spins coupled\nto a bath, which have been successfully applied to\nout-of-equilibrium magnetization dynamics and magnetic\nswitching scenarios.10The most fundamental of these\nare the stochastic Landau-Lifshitz equations,10{13from\nwhich the Landau-Lifshitz Bloch equations,14,15can be\nderived via a Fokker-Planck equation.\nQuantum-mechanical treatments of the equilibrium\nmagnetization in bulk ferromagnets at \fnite temper-\natures are extremely involved. The calculation of\nnon-equilibrium magnetization phenomena and damp-\ning for quantum spin systems in more than one dimen-\nsion, which include both magnetism and carrier-phonon\nand/or carrier-impurity interactions, at present have to\nemploy simpli\fed models. For instance, there have been\nmicroscopic calculations of Gilbert damping parameters\nbased on Kohn-Sham wave functions for metallic ferro-\nmagnets16,17and Kohn-Luttinger p-dHamiltonians for\nmagnetic semiconductors.18While the former approach\nuses spin density-functional theory, the latter approach\ntreats the anti-ferromagnetic kinetic-exchange coupling\nbetween itinerant p-like holes and localized magnetic\nmoments originating from impurity d-electrons within a\nmean-\feld theory. In both cases, a constant spin and\nband-independent lifetime for the itinerant carriers is\nused as an input, and a Gilbert damping constant is ex-\ntracted by comparing the quantum mechanical result for\n!!0 with the classical formulation. There have also\nbeen investigations, which extract the Gilbert damping\nfor magnetic semiconductors from a microscopic calcula-\ntion of carrier dynamics including Boltzmann-type scat-\ntering integrals.19,20Such a kinetic approach, which is of\na similar type as the one we present in this paper, avoids\nthe introduction of electronic lifetimes because the scat-\ntering is calculated dynamically.arXiv:1405.2347v1  [cond-mat.mtrl-sci]  9 May 20142\nThe present paper takes up the question how the spin\ndynamics in the framework of the macroscopic Gilbert\nor Landau-Lifshitz damping compare to a microscopic\nmodel of relaxation processes in the framework of a rel-\natively simple model. We analyze a mean-\feld kinetic\nexchange model including spin-orbit coupling for the itin-\nerant carriers. Thus the magnetic mean-\feld dynamics is\ncombined with a microscopic description of damping pro-\nvided by the electron-phonon coupling. This interaction\ntransfers energy and angular momentum from the itin-\nerant carriers to the lattice. The electron-phonon scat-\ntering is responsible both for the lifetimes of the itiner-\nant carriers and the magnetization dephasing. The lat-\nter occurs because of spin-orbit coupling in the states\nthat are connected by electron-phonon scattering. To be\nmore speci\fc, we choose an anti-ferromagnetic coupling\nat the mean-\feld level between itinerant electrons and\na dispersion-less band of localized spins for the magnetic\nsystem. To keep the analysis simple we use as a model for\nthe spin-orbit coupled itinerant carrier states a two-band\nRashba model. As such it is a single-band version of the\nmulti-band Hamiltonians used for III-Mn-V ferromag-\nnetic semiconductors.18,21{24The model analyzed here\nalso captures some properties of two-sublattice ferrimag-\nnets, which are nowadays investigated because of their\nmagnetic switching dynamics.25,26The present paper is\nset apart from studies of spin dynamics in similar mod-\nels with more complicated itinerant band structures19,20\nby a detailed comparison of the phenomenological damp-\ning expressions with a microscopic calculation as well as\na careful analysis of the restrictions placed by the size\nof the magnetic gap on the single-particle broadening in\nBoltzmann scattering.\nThis paper is organized as follows. As an extended\nintroduction, we review in Sec. II some basic facts con-\ncerning the Landau-Lifshitz and Gilbert damping terms\non the one hand and the Bloch equations on the other.\nIn Sec. III we point out how these di\u000berent descriptions\nare related in special cases. We then introduce a micro-\nscopic model for the dephasing due to electron-phonon\ninteraction in Sec. IV, and present numerical solutions\nfor two di\u000berent scenarios in Secs. V and VI. The \frst\nscenario is the dephasing between two spin subsystems\n(Sec. V), and the second scenario is a relaxation process\nof the magnetization toward an easy-axis (Sec. VI). A\nbrief conclusion is given at the end.\nII. PHENOMENOLOGIC DESCRIPTIONS OF\nDEPHASING AND RELAXATION\nWe summarize here some results pertaining to a single-\ndomain ferromagnet, and set up our notation. In equilib-\nrium we assume the magnetization to be oriented along\nits easy axis or a magnetic \feld ~H, which we take to\nbe thezaxis in the following. If the magnetization\nis tilted out of equilibrium, it starts to precess. As\nillustrated in Fig. 1 one distinguishes the longitudinal\nFIG. 1. Illustration of non-equilibrium spin-dynamics in pres-\nence of a magnetic \feld without relaxation (a) and within\nrelaxation (b).\ncomponent Mk, inzdirection, and the transverse part\nM?\u0011q\nM2\u0000M2\nk, precessing in the x-yplane with the\nLarmor frequency !L.\nIn connection with the interaction processes that re-\nturn the system to equilibrium, the decay of the trans-\nverse component is called dephasing. There are three\nphenomenological equations used to describe spin de-\nphasing processes:\n1. The Bloch(-Bloembergen) equations27,28\n@\n@tMk(t) =\u0000Mk(t)\u0000Meq\nT1(1)\n@\n@tM?(t) =\u0000M?(t)\nT2(2)\ndescribe an exponential decay towards the equilib-\nrium magnetization Meqinzdirection. The trans-\nverse component decays with a time constant T2,\nwhereas the longitudinal component approaches its\nequilibrium amplitude with T1. These time con-\nstants may be \ft independently to experimental\nresults or microscopic calculations.\n2. Landau-Lifshitz damping1with parameter \u0015\n@\n@t~M(t) =\u0000\r~M\u0002~H\u0000\u0015~M\nM\u0002\u0000~M\u0002~H\u0001\n(3)\nwhere\ris the gyromagnetic ratio. The \frst term\nmodels the precession with a frequency !L=\rj~Hj,\nwhereas the second term is solely responsible for\ndamping.\n3. Gilbert damping2with the dimensionless Gilbert\ndamping parameter \u000b\n@\n@t~M(t) =\u0000\rG~M\u0002~H+\u000b\u0010~M\nM\u0002@t~M\u0011\n(4)\nIt is generally accepted that \u000bis independent of\nthe static magnetic \felds ~Hsuch as anisotropy\n\felds,18,29and thus depends only on the material\nand the microscopic interaction processes.3\nThe Landau-Lifshitz and Gilbert forms of damping are\nmathematically equivalent2,7,30with\n\u000b=\u0015\n\r(5)\n\rG=\r(1 +\u000b2) (6)\nbut there are important di\u000berences. In particular, an in-\ncrease of\u000blowers the precession frequency in the dynam-\nics with Gilbert damping, while the damping parameter\n\u0015in the Landau-Lifshitz equation has no impact on the\nprecession. In contrast to the Bloch equations, Landau-\nLifshitz and Gilbert spin-dynamics always conserve the\nlengthj~Mjof the magnetization vector.\nAn argument by Pines and Slichter,31shows that there\nare two di\u000berent regimes for Bloch-type spin dynamics\ndepending on the relation between the Larmor period and\nthe correlation time. As long as the correlation time is\nmuch longer than the Larmor period, the system \\knows\"\nthe direction of the \feld during the scattering process.\nStated di\u000berently, the scattering process \\sees\" the mag-\nnetic gap in the bandstructure. Thus, transverse and\nlongitudinal spin components are distinguishable and the\nBloch decay times T1andT2can di\u000ber. If the correlation\ntime is considerably shorter than the Larmor period, this\ndistinction is not possible, with the consequence that T1\nmust be equal to T2. Within the microscopic approach,\npresented in Sec. IV D, this consideration shows up again,\nalbeit for the energy conserving \u000efunctions resulting from\na Markov approximation.\nThe regime of short correlation times has already been\ninvestigated in the framework of a microscopic calcula-\ntion by Wu and coworkers.32They analyze the case of\na moderate external magnetic \feld applied to a non-\nmagnetic n-type GaAs quantum well and include di\u000ber-\nent scattering mechanisms (electron-electron Coulomb,\nelectron-phonon, electron-impurity). They argue that\nthe momentum relaxation rate is the crucial time scale\nin this scenario, which turns out to be much larger than\nthe Larmor frequency. Their numerical results con\frm\nthe identity T1=T2expected from the Pines-Slichter\nargument.\nIII. RELATION BETWEEN\nLANDAU-LIFSHITZ, GILBERT AND BLOCH\nWe highlight here a connection between the Bloch\nequations (1, 2) and the Landau-Lifshitz equation (3).\nTo this end we assume a small initial tilt of the mag-\nnetization and describe the subsequent dynamics of the\nmagnetization in the form\n~M(t) =0\n@\u000eM?(t) cos(!Lt)\n\u000eM?(t) sin(!Lt)\nMeq\u0000\u000eMk(t)1\nA (7)\nwhere\u000eM?and\u000eMjjdescribe deviations from equilib-\nrium. Putting this into eq. (3) one gets a coupled set ofequations.\n@\n@t\u000eM?(t) =\u0000\u0015HMeq\u0000\u000eMk(t)\nj~M(t)j\u000eM?(t) (8)\n@\n@t\u000eMk(t) =\u0000\u0015H1\nj~M(t)j\u000eM2\n?(t) (9)\nEq. (8) is simpli\fed for a small deviation from equilib-\nrium, i.e.,\u000eM(t)\u001cMeqandj~M(t)j\u0019Meq:\n\u000eM?(t) =Cexp(\u0000\u0015Ht) (10)\n\u000eMk(t) =C2\n2Meqexp(\u00002\u0015Ht) (11)\nwhereCis an integration constant. For small excitations\nthe deviations decay exponentially and Bloch decay times\nT1andT2result, which are related by\n2T1=T2=1\n\u0015H: (12)\nOnly this ratio of the Bloch times is compatible with a\nconstant length of the magnetization vector at low exci-\ntations. By combining Eqs. (12) and (5) one can connect\nthe Gilbert parameter \u000band the dephasing time T2\n\u000b=1\nT2!L: (13)\nIf the conditions for the above approximations apply, the\nGilbert damping parameter \u000bcan be determined by \ft-\nting the dephasing time T2and the Larmor frequency !L\nto computed or measured spin dynamics. This dimen-\nsionless quantity is well suited to compare the dephasing\nthat results from di\u000berent relaxation processes.\nFigure 2 shows the typical magnetization dynamics\nthat results from (3), i.e., Landau-Lifshitz damping. As\nan illustration of a small excitation we choose in Fig. 2(a)\nan angle of 10\u000efor the initial tilt of the magnetization,\nwhich results in an exponential decay with 2 T1=T2.\nFrom the form of Eq. (3) it is clear that this behavior\npersists even for large !Land\u0015. Obviously the Landau-\nLifshitz and Gilbert damping terms describe a scenario\nwith relatively long correlation times (i.e., small scat-\ntering rates), because only in this regime both decay\ntimes can di\u000ber. The microscopic formalism in Sec. IV\nworks in the same regime and will be compared with\nthe phenomenological results. For an excitation angle\nof 90\u000e, the Landau-Lifshitz dynamics shown in Fig. 2(b)\nbecome non-exponential, so that no well-de\fned Bloch\ndecay times T1,T2exist.\nIV. MICROSCOPIC MODEL\nIn this section we describe a microscopic model that in-\ncludes magnetism at the mean-\feld level, spin-orbit cou-\npling as well as the microscopic coupling to a phonon\nbath treated at the level of Boltzmann scattering inte-\ngrals. We then compare the microscopic dynamics to4\n0 5000.51δM⊥/Meq\ntime (ps)0 5000.51\ntime (ps)δM/bardbl/Meq0 5000.010.02δM/bardbl/Meq\ntime (ps)0 5000.10.2\ntime (ps)δM⊥/Meq\n  \nT1= 5.02 ps(a)\n(b)T2= 10.04 ps\nFIG. 2. Dynamics of \u000eM?and\u000eMkcomputed using to\nLandau-Lifshitz damping ( !L= 1 ps\u00001,H= 106A\nm\u0019\n1:26\u0001104Oe,\u0015= 10\u00007m\nA ps). (a) An angle of 10\u000eleads to\nexponential an exponential decay with well de\fned T1andT2\ntimes. (b). For an angle of 90\u000e, the decay (solid line) is not\nexponential as comparison with the exponential \ft (dashed\nline) clearly shows.\nthe Bloch equations (1), (2), as well as the Landau-\nLifshitz (3) and Gilbert damping terms (4). The mag-\nnetic properties of the model are de\fned by an anti-\nferromagnetic coupling between localized magnetic im-\npurities and itinerant carriers. As a prototypical spin-\norbit coupling we consider an e\u000bectively two-dimensional\nmodel with a Rashba spin-orbit coupling. The reason\nfor the choice of a model with a two-dimensional wave\nvector space is not an investigation of magnetization dy-\nnamics with reduced dimensionality, but rather a reduc-\ntion in the dimension of the integrals that have to be\nsolved numerically in the Boltzmann scattering terms.\nSince we treat the exchange between the localized and\nitinerant states in a mean-\feld approximation, our two-\ndimensional model still has a \\magnetic ground state\"\nand presents a framework, for which qualitatively dif-\nferent approaches can be compared. We do not aim at\nquantitative predictions for, say, magnetic semiconduc-\ntors or ferrimagnets with two sublattices. Finally, we\ninclude a standard interaction hamiltonian between the\nitinerant carriers and acoustic phonons. The correspond-\ning hamiltonian reads\n^H=^Hmf+^Hso+^He\u0000ph+^Haniso: (14)\nOnly in Sec. VI an additional \feld ^Haniso is included,\nwhich is intended to model a small anisotropy.A. Exchange interaction between itinerant carriers\nand localized spins\nThe \\magnetic part\" of the model is described by the\nHamiltonian\n^Hmf=X\n~k\u0016~2k2\n2m\u0003^cy\n~k\u0016^c~k\u0016+J^~ s\u0001^~S: (15)\nwhich we consider in the mean-\feld limit. The \frst term\nrepresents itinerant carriers with a k-dependent disper-\nsion relation. In the following we assume s-like wave\nfunctions and parabolic energy dispersions. The e\u000bective\nmass is chosen to be m\u0003= 0:5me, wheremeis the free\nelectron mass, and the ^ c(y)\n~k\u0016operators create and annihi-\nlate carriers in the state j~k;\u0016iwhere\u0016labels the itinerant\nbands, as shown in Fig. 3(a).\nThe second term describes the coupling between itiner-\nant spins~ sand localized spins ~Svia an antiferromagnetic\nexchange interaction\n^~ s=1\n2X\n~kX\n\u0016\u00160h~k;\u00160j^~ \u001bj~k;\u0016i^cy\n~k\u0016^c~k\u00160 (16)\n^~S=1\n2X\n\u0017\u00170h\u00170j^~ \u001bj\u0017iX\n~K^Cy\n~K\u0017^C~K\u00170 (17)\nHere, we have assumed that the wave functions of the lo-\ncalized spins form dispersionless bands, i.e., we have im-\nplicitly introduced a virtual-crystal approximation. Due\nto the assumption of strong localization there is no or-\nbital overlap between these electrons, which are therefore\nconsidered to have momentum independent eigenstates\nj\u0017iand a \rat dispersion, as illustrated in Fig. 3(a). The\ncomponents of the vector ^~ \u001bare the Pauli matrices ^ \u001biwith\ni=x;y;z , and ^C(y)\n~K\u0017are the creation and annihilation op-\nerators for a localized spin state.\nWe do notinclude interactions among localized or itin-\nerant spins, such as exchange scattering. For simplicity,\nwe assume both itinerant and localized electrons to have\na spin 1=2 and therefore \u0016and\u0017to run over two spin-\nprojection quantum numbers \u00061=2. In the following we\nchosse an antiferromagnetic ( J > 0) exchange constant\nJ= 500 meV, which leads to the schematic band struc-\nture shown in Fig. 3(b).\nIn the mean \feld approximation used here, the itiner-\nant carriers feel an e\u000bective magnetic \feld ^Hloc\n~Hloc=\u0000J\u0016B\u0016\ng~S (18)\ncaused by localized moments and vice versa. Here \u0016B\nis the Bohr magneton and g= 2 is the g-factor of the\nelectron. The permeability \u0016is assumed to be the vac-\nuum permeability \u00160. This time-dependent magnetic\n\feld~Hloc(t) de\fnes the preferred direction in the itiner-\nant sub-system and therefore determines the longitudinal\nand transverse component of the itinerant spin at each\ntime.5\nr#k (a) (b) E(k) \nk \n\u0010\nk \n\u000e\u0010,k\n\u000e,kE(k) \nEF \nFIG. 3. Sketch of the band-structure with localized (\rat\ndispersions) and itinerant (parabolic dispersions) electrons.\nAbove the Curie-Temperature TCthe spin-eigenstates are de-\ngenerate (a), whereas below TCa gap between the spin states\nexists.\nB. Rashba spin-orbit interaction\nThe Rashba spin-orbit coupling is given by the Hamil-\ntonian\n^Hso=\u000bR(^\u001bxky\u0000^\u001bykx) (19)\nA Rashba coe\u000ecient of \u000bR= 10 meV nm typical for semi-\nconductors is chosen in the following calculations. This\nvalue, which is close to the experimental one for the\nInSb/InAlSb material system,33is small compared to the\nexchange interactions, but it allows the exchange of an-\ngular momentum with the lattice.\nC. Coherent dynamics\nFrom the above contributions (15) and (19) to the\nHamiltonian we derive the equations of motion contain-\ning the coherent dynamics due to the exchange interac-\ntion and Rashba spin-orbit coupling as well as the inco-\nherent electron-phonon scattering. We \frst focus on the\ncoherent contributions. In principle, one has the choice\nto work in a basis with a \fxed spin-quantization axis or\nto use single-particle states that diagonalize the mean-\n\feld (plus Rashba) Hamiltonian. Since we intend to use\na Boltzmann scattering integral in Sec. IV D we need to\napply a Markov approximation, which only works if one\ndeals with diagonalized eigenenergies. In our case this is\nthe single-particle basis that diagonalizes the entire one-\nparticle contribution of the Hamiltonian ^Hmf+^Hso. In\nmatrix representation this one-particle contribution for\nthe itinerant carriers reads:\n^Hmf+^Hso= \n~2k2\n2m\u0003+ \u0001loc\nz(\u0001loc\n++R~k)\u0003\n\u0001loc\n++R~k~2k2\n2m\u0003\u0000\u0001loc\nz!\n(20)\nwhere we have de\fned \u0001loc\ni=J1\n2h^SiiandR~k=\n\u0000i\u000bRkexp(i'k) with'k= arctan(ky=kx). The eigenen-\nergies are\n\u000f\u0006\n~k=~2k2\n2m\u0003\u0007q\nj\u0001loczj2+jR~k+ \u0001loc\n+j2: (21)and the eigenstates\nj~k;+i=\u0012\n1\n\u0018~k\u0013\n;j~k;\u0000i=\u0012\u0000\u0018\u0003\n~k\n1\u0013\n(22)\nwhere\n\u0018~k=\u0001loc\n++R~k\n\u0001locz+q\nj~\u0001locj2+jR~kj2(23)\nIn this basis the coherent part of the equation of mo-\ntion for the itinerant density matrix \u001a\u0016\u00160\n~k\u0011 h^cy\n~k\u0016^c~k\u00160i\nreads\n@\n@t\u001a\u0016\u00160\n~k\f\f\f\ncoh=i\n~\u0000\n\u000f\u0016\n~k\u0000\u000f\u00160\n~k\u0001\n\u001a\u0016\u00160\n~k: (24)\nNo mean-\feld or Rashba terms appear explicitly in these\nequations of motion since their contributions are now hid-\nden in the time-dependent eigenstates and eigenenergies.\nSince we are interested in dephasing and precessional\ndynamics, we assume a comparatively small spin-orbit\ncoupling, that can dissipate angular momentum into the\nlattice, but does not have a decisive e\u000bect on the band-\nstructure. Therefore we use the spin-mixing only in the\ntransition matrix elements of the electron-phonon scat-\nteringM~k0\u00160\n~k\u0016(31). For all other purposes we set R~k= 0.\nIn particular, the energy-dispersion \u000f\u0006\n~kis assumed to be\nuna\u000bected by the spin-orbit interaction and therefore it\nis spherically symmetric.\nWith this approximation the itinerant eigenstates are\nalways exactly aligned with the e\u000bective \feld of the local-\nized moments ~Hloc(t). Since this e\u000bective \feld changes\nwith time, the diagonalization and a transformation of\nthe spin-density matrix in \\spin space\" has to be re-\npeated at each time-step. This e\u000bort makes it easier\nto identify the longitudinal and transverse spin compo-\nnents with the elements of the single-particle density\nmatrix: The o\u000b-diagonal entries of the density matrix\n\u001a\u0006\u0007\n~k, which precess with the k-independent Larmor fre-\nquency!L= 2\u0001loc=~, always describe the dynamics of\nthe transverse spin-component. The longitudinal compo-\nnent, which does not precess, is represented by the diag-\nonal entries \u001a\u0006\u0006\n~k. Since both components change their\nspatial orientation continuously, we call this the rotating\nframe. The components of the spin vector in the rotating\nframe are\nh^ski=1\n2X\n~k\u0000\n\u001a++\n~k\u0000\u001a\u0000\u0000\n~k\u0001\n(25)\nh^s?i=X\n~k\f\f\u001a+\u0000\n~k\f\f (26)\nThe components in the \fxed frame are obtained from\nEq. (16)\nh^~ si=1\n2X\n~kX\n\u0016\u00160h~k;\u00160j^~ \u001bj~k;\u0016i\u001a\u0016\u00160\n~k(27)6\nIn this form, the time-dependent states carry the infor-\nmation how the spatial components are described by the\ndensity matrix at each time step. No time-independent\n\\longitudinal\" and \\transverse\" directions can be identi-\n\fed in the \fxed frame.\nIn a similar fashion, the diagonalized single-particle\nstates of the localized spin system are obtained. The\neigenenergies are\nE\u0006=\u0007\f\f~\u0001itin\f\f (28)\nwhere \u0001itin\ni=J1\n2h^siiis the localized energy shift caused\nby the itinerant spin component si. The eigenstates are\nagain always aligned with the itinerant magnetic mo-\nment. In this basis the equation of motion of the localized\nspin-density matrix \u001a\u0017\u00170\nloc\u0011P\n~Kh^Cy\n~K\u0017^C~K\u00170iis simply\n@\n@t\u001a\u0017\u00170\nloc=i\n~(E\u0017\u0000E\u00170)\u001a\u0017\u00170\nloc (29)\nand does not contain explicit exchange contributions.\nEqs. (25), (26), and (27) apply in turn to the components\nhSkiandhS?iof the localized spin and its spin-density\nmatrix\u001a\u0017\u00170\nloc.\nD. Electron-phonon Boltzmann scattering with\nspin splitting\nRelaxation is introduced into the model by the interac-\ntion of the itinerant carriers with a phonon bath, which\nplays the role of an energy and angular momentum sink\nfor these carriers. Our goal here is to present a derivation\nof the Boltzmann scattering contributions using stan-\ndard methods, see, e.g., Refs. 34 and 36. However, we\nemphasize that describing interaction as a Boltzmann-\nlike instantaneous, energy conserving scattering process\nis limited by the existence of the magnetic gap. Since we\nkeep the spin mixing due to Rashba spin-orbit coupling\nonly in the Boltzmann scattering integrals, the resulting\ndynamical equations describe an Elliott-Yafet type spin\nrelaxation.\nThe electron-phonon interaction Hamiltonian reads34\n^He\u0000ph=X\n~ q~!ph\nq^by\n~ q^b~ q\n+X\n~k~k0X\n\u0016\u00160\u0000\nM~k0\u00160\n~k\u0016^cy\n~k\u0016^b~k\u0000~k0^c~k0\u00160+ h.c.\u0001(30)\nwhere ^b(y)\n~ qare the bosonic operators, that create or an-\nnihilate acoustic phonons with momentum ~ qand linear\ndispersion!ph(q) =cphj~ qj. The sound velocity is taken\nto becph= 40 nm/ps and we use an e\u000bectively two-\ndimensional transition matrix element35\nM~k0\u00160\n~k\u0016=Dq\nj~k\u0000~k0jh~k;\u0016j~k0;\u00160i (31)\nwhere the deformation potential is chosen to be D=\n60 meVnm1=2. The scalar-product between the initialstatej~k0;\u00160iand the \fnal state j~k;\u0016iof an electronic\ntransition takes the spin-mixing due to Rashba spin-orbit\ncoupling into account.\nThe derivation of Boltzmann scattering integrals for\nthe itinerant spin-density matrix (24) leads to a memory\nintegral of the following shape\n@\n@t\u001aj(t)\f\f\f\ninc=1\n~X\nj0Zt\n\u00001ei(\u0001Ejj0+i\r)(t\u0000t0)Fjj0[\u001a(t0)]dt0;\n(32)\nregardless whether one uses Green's function36or\nequation-of-motion techniques.34Since we go through a\nstandard derivation here, we highlight only the impor-\ntant parts for the present case and do not write the equa-\ntions out completely. In particular, for scattering process\nj0=j\u00160;~k0i!j=j\u0016;~ki, we useFjj0[\u001a(t0)] as an abbre-\nviation for a product of dynamical electronic spin-density\nmatrix elements \u001a, evaluated at time t0<t, and equilib-\nrium phononic distributions. The corresponding energy\ndi\u000berence is denoted by \u0001 Ejj0=Ej\u0000Ej0\u0006~!ph(j~k\u0000~k0j),\nand\rdescribes the decay of the exponential function due\nto dissipation and/or higher order correlation functions.\nIn general, the integral has to be evaluated numerically\nand contains memory e\u000bects. To apply the Markov ap-\nproximation one needs to compare two time scales: the\n\\memory depth\" 1 =\r, i.e., the time scale on which the\nexp[\u0000\r(t\u0000t0)] factor essentially cuts o\u000b the integral, and\nthe typical time scale on which the Fterm changes. In\nthis paper we deal with relaxation processes not too far\naway from equilibrium, so that the typical time scale of\nthe components of the spin-density matrix contained in\nFis set by the Bloch times T1andT2. We can thus\napproximate Fjj0[\u001a(t0)] byFjj0[\u001a(t)], for all transitions\nlabeled byjandj0, if the memory depth is shorter than\nthe Bloch time(s), or\n\r\u001d1\nT1: (33)\nProvided condition (33) holds, the integral (32) can\nbe done using the Markov approximation Fjj0[\u001a(t0)]'\nFjj0[\u001a(t)]\n@\n@t\u001aj(t)\f\f\f\nincoh=i\n~X\nj0Fjj0[\u001a(t)]1\n\u0001Ejj0+i~\r:(34)\nAs it is customary, we neglect in the following the real\npart of the complex energy denominator, which results in\nshifts of the single-particle energies. While these shifts\nmay play an important role in non-Markovian problems\nwith discrete energy levels37, the imaginary parts yield\nthe relaxation contributions that are important for the\npresent paper\n@\n@t\u001aj(t)\f\f\f\nincoh=X\nj0Fjj0[\u001a(t)]~\r\n(\u0001Ejj0)2+ (~\r)2(35)\nAll transitions are thus weighted by a Lorentzian peaked\nat resonant transitions (\u0001 Ejj0= 0) with a broadening7\nof~\rthat may be interpreted as an energy uncertainty.\nFor relaxation processes in a system with a spin-splitting\n(due to internal \felds and/or spin-orbit coupling), this\nbroadening must not be so large as to blur the distinc-\ntion between the split bands. Consequently, only if the\nbroadening \ris smaller than the magnetic splitting, i.e,,\nif\r\u001c!L, it is possible to distinguish between longi-\ntudinal and transverse components of the spin-density\nmatrix. With Eq. (33) the inequality \r\u001c!Lyields the\ncondition\n!L\u001d1\nT1(36)\nfor the Larmor frequencies and Bloch times, for which\nit is permissible to replace the Lorentzian by an energy\nconserving \u000efunction\n~\r\n(\u0001Ejj0)2+ (~\r)2\r!0\u0000!\u0019\u000e(\u0001Ejj0): (37)This reduces the numerical e\u000bort very considerably, be-\ncause it allows one to eliminate an integration from the\nscattering term, and the energy conserving \u000efunction is\ntherefore often used without explicitly checking its valid-\nity.\nThe considerations leading to the connection between\nEqs. (36) and (37) are a microscopic version of an argu-\nment due to Pines and Slichter,31according to which T1\nandT2can di\u000ber only for correlation times that long in\ncomparison to a Larmor period. The microscopic Boltz-\nmann scattering terms, which contain the energy con-\nserving\u000efunctions and will be used in the following,\ndo not apply in a regime outside of condition (36). If\n!L'1=T1, a \fnite broadening \rhas to be taken into\naccount. Together the full equation of motion in the\nregime (36) for the itinerant-carrier spin density matrix\nthus reads\n@\n@t\u001a\u0016\u00160\n~k=i\n~\u0000\n\u000f\u0016\nk\u0000\u000f\u00160\nk\u0001\n\u001a\u0016\u00160\n~k\n+\u0019\n~X\nk0X\n\u00161\u00162\u00163M~k\u0016\n~k0\u00161M~k0\u00162\n~k\u00163\u000e\u0000\n\u0001E~k0\u00162~k\u00163\u0001h\u0000\n1 +Nph\nj~k0\u0000~kj\u0001\n\u001a\u00161\u00162\n~k0\u0000\n\u000e\u00163\u00160\u0000\u001a\u00163\u00160\n~k\u0001\n\u0000Nph\nj~k0\u0000~kj\u001a\u00163\u00160\n~k\u0000\n\u000e\u00161\u00162\u0000\u001a\u00161\u00162\n~k0\u0001i\n\u0000\u0019\n~X\n~k0X\n\u00161\u00162\u00163M~k\u0016\n~k0\u00161M~k0\u00162\n~k\u00163\u000e\u0000\n\u0001E~k\u00163~k0\u00162\u0001h\u0000\n1 +Nph\nj~k\u0000~k0j\u0001\n\u001a\u00163\u00160\n~k\u0000\n\u000e\u00161\u00162\u0000\u001a\u00161\u00162\n~k0\u0001\n\u0000Nph\nj~k\u0000~k0j\u001a\u00161\u00162\n~k0\u0000\n\u000e\u00163\u00160\u0000\u001a\u00163\u00160\n~k\u0001i\n+\u0019\n~X\n~k0X\n\u00161\u00162\u00163M~k0\u00161\n~k\u00160M~k\u00163\n~k0\u00162\u000e\u0000\n\u0001E~k0\u00162~k\u00163\u0001h\u0000\n1 +Nph\nj~k0\u0000~kj\u0001\n\u001a\u00162\u00161\n~k0\u0000\n\u000e\u0016\u00163\u0000\u001a\u0016\u00163\n~k\u0001\n\u0000Nph\nj~k0\u0000~kj\u001a\u0016\u00163\n~k\u0000\n\u000e\u00162\u00161\u0000\u001a\u00162\u00161\n~k0\u0001i\n\u0000\u0019\n~X\n~k0X\n\u00161\u00162\u00163M~k0\u00161\n~k\u00160Mk\u00163\n~k0\u00162\u000e\u0000\n\u0001E~k\u00163~k0\u00162\u0001h\u0000\n1 +Nph\nj~k\u0000~k0j\u0001\n\u001a\u0016\u00163\n~k\u0000\n\u000e\u00162\u00161\u0000\u001a\u00162\u00161\n~k0\u0001\n\u0000Nph\nj~k\u0000~k0j\u001a\u00162\u00161\n~k0\u0000\n\u000e\u0016\u00163\u0000\u001a\u0016\u00163\n~k\u0001i\n(38)\nHere, \u0001E~k\u0016~k0\u00160=\u000f\u0016\nk\u0000\u000f\u00160\nk0\u0000~!ph\nj~k\u0000~k0j, andNph\nqis the\noccupation function of a thermalized phonon bath, given\nby a Bose-Einstein distribution\nNph\nq=1\ne\f~!ph\u00001(39)\nwhere\f= 1=(kBTph). The numerical results for the mi-\ncroscopic dynamics in the following sections are obtained\nby numerically solving the equations of motion (29) and\n(38). In the numerical calculations the spin-density ma-\ntrix is transformed to the single-particle basis of the in-\nstantaneous, diagonalized eigenstates (22).\nV. DEPHASING BETWEEN LOCALIZED AND\nITINERANT SPINS: NUMERICAL RESULTS\nSince we are interested in this paper in a comparison\nof the model described above with Landau-Lifshitz and\nGilbert damping, we investigate magnetization dynam-\nics with an initial spin-density matrix that correspondsto a tilting of the spins out of their equilibrium posi-\ntion without changing the kinetic energy of the carriers,\nbecause we need initial conditions that lead to generic\nmagnetization dephasing without carrier heating and the\ncorresponding demagnetization dynamics.\nA. Initial state and spin dynamics\nThus we take as the equilibrium initial state the\nsteady-state that is reached for the coupled spins inter-\nacting with the phonon bath at low temperature ( Tph=\n1 K), as shown in Fig. 4(a). In this equilibrium state\nthe spin density matrix is characterized by shifted Fermi\nfunctions for the distributions \u001a\u0016\u0016\n~k=f(\u000f\u0016\nk\u0000EF;Tph)\nand vanishing coherences \u001a+\u0000\n~k= 0. In particular, the\nsteady-state calculation determines the equilibrium mag-\nnetic gap.\nWe then change the itinerant density matrix to that\ncorresponding to an itinerant spin tilted by \f= 10\u000e\nout of equilibrium, see Fig. 4(b). This initial condition8\nFIG. 4. (a) Localized spin h~Siand itinerant spin h~ siin ther-\nmal equilibrium. (b) Itinerant spins tilted out of equilibrium\nby an angle \f.\n0 1 2 300.51\nk(nm−1)Occupations  \n0 1 2 3−0.500.5\nk(nm−1)Coherences\n  ρ|k|− −\nρ|k|+ +\nℜ(ρ|k|− +)\nℑ(ρ|k|− +)\nFIG. 5. Initial spin density-matrix (occupations and coher-\nences) for itinerant electrons corresponding to a tilt \f= 50\u000e.\nThe deviation from equilibrium occurs only between the Fermi\nwave-vectors of the \\+\" and \\ \u0000\" bands.\nachieves a tilting of the spins without heating and avoids\ngeneric de- and remagnetization dynamics. Microscop-\nically the tilted spin corresponds to the spin density-\nmatrix shown in Fig. 5. The perturbation for distribu-\ntions and coherences exists only between the two Fermi\nwave vectors for the \u0016= + and\u0016=\u0000bands. For smaller\ntilt angles the deviation is much less pronounced.\nFrom this initial condition, both spins start to precess\naround the instantaneous direction, along which the ex-\nchange interaction tries to align them. This direction\nis determined for the itinerant carriers by the localized\nspins, and vice versa. A return back into equilibrium re-\nquires the scattering of itinerant electrons with phonons.\nIf we switch o\u000b spin mixing, the dynamics shown in\nFig. 6(a) result: No angular momentum is exchanged\nwith the phonon bath, the excited system cannot relax\ninto equilibrium and the precession goes on inde\fnitely.\nFig. 6(b) shows the same result including spin-mixed itin-\nerant states. Now angular momentum can be transferred\nfrom the itinerant sub-system into the lattice and the to-\ntal spinh~Si+h~ sichanges. In the presence of spin-orbit\ncoupling, electron-phonon scattering, which is by itself\nspin-diagonal, can return the spin system into equilib-\nrium, characterized by aligned spins and vanishing trans-\nverse components. Since we consider here a small Rashba\n(a) (b) \n5&:P; 5&:P; \nO&:P; O&:P; FIG. 6. Non-equilibrium dynamics of localized (red) and itin-\nerant (blue) spins including electron-phonon scattering with-\nout spin-mixing (a) and within spin-mixing (b).\n0 1 2 30.120.130.14\ntime (ps)|s|0 1 2 3−0.0400.04\ntime (ps)sx\n0 1 2 3−0.0400.04\ntime (ps)sy\n0 1 2 3−0.14−0.13−0.12\ntime (ps)sz\nFIG. 7. Relaxation dynamics of itinerant spins in the \fxed\nframe.\ncoupling, the interaction with the phonon bath removes\nenergy much faster than angular momentum. The car-\nrier temperature therefore stays practically equal to the\nphonon temperature Tphduring the entire relaxation pro-\ncess, and no heat-induced demagnetization processes oc-\ncur. The \fnal magnetization is, however, not necessarily\noriented in the zdirection of the \fxed frame, because\nin the results discussed in the this section there is no\nexternal \feld or anisotropy to induce such an alignment.\nWe plot the resulting dynamics of the itinerant spins\nduring the dephasing process in Fig. 7, which shows that9\n00.511.522.53−0.133−0.132−0.131\ntime(ps)s/bardbl\n00.511.522.5300.010.020.03\ntime(ps)s⊥\nFIG. 8. Relaxation dynamics of itinerant spins in the rotating\nframe. An exponential \ft determines the Bloch decay times\ntoT1= 0:5 ps andT2= 1:0 ps.\nall itinerant spatial components precess, and no spatially\n\fxed component can be considered to be longitudinal.\nFirst, the localized spin turns away from the zdirec-\ntion due to the tilted itinerant spin and subsequently\nboth spin-systems precess around each other because the\nquantization axis of each system changes continuously\ndue to the mutual interaction. During the entire relax-\nation process the absolute value of the itinerant spin is\nconserved to better than 1%. Fig. 8 shows the same\ndynamics in the rotating frame, where the longitudinal\nand transverse dynamics can be seen clearly. Both itin-\nerant components show exponential dynamics, which are\ntherefore well described by decay times T1andT2. If well-\nde\fned decay times exist, one expects a ratio T2=T1= 2\nas long as the length of the spin is conserved. The \ft for\nour numerical results indeed gives 2 T1'T2.\nFigure 9 plots the T1andT2values extracted from the\ndynamics as a function of the strength of the electron-\nphonon coupling, or deformation potential, D. The de-\npendence of the decay times on Dcan be \ft extremely\nwell by a 1=D2relation, which demonstrates the propor-\ntionality of the Bloch decay times T1;2/1=D2. Further,\nthe ratioT2=T1stays equal to 2 for all coupling strengths.\nAs discussed in Sec. IV D about the Markov approxima-\ntion, our microscopic description cannot reach regimes\nwhereT1andT2are indistinguishable and therefore\nequal. However, these results show that for small tilting\nangles, not even a pronounced electron-phonon coupling\nleads to a noticeable deviation from the T2= 2T1be-\nhavior. Because this relation between T1andT2holds,\nthe dynamics in Fig. 8 can be equally well described by\nan Landau-Lifshitz or Gilbert damping term. By \ftting\nthe dephasing time T2\u00191 ps and the Larmor-frequency\n!L\u0019281 ps\u00001, equation (13) yields the corresponding\nGilbert damping parameter \u000biso\u00193:6\u000110\u00003.\n204060801001201401.822.2\nD(meV√nm)T2/T1\n  2040608010012014010−2100102\nD(meV√nm)T1,2(ps)\n  FIG. 9. Top: Longitudinal (black squares) and the transverse\n(blue circles) Bloch decay times vs. electron-phonon coupling\nstrengthD. Two \ft curves /1=D2show thatT1;2/1=D2.\nBottom:The ratio of both decay times remains almost con-\nstantT2=T1\u00192 over the entire range of coupling.\n369  \n0 0.5 1279279.5280280.5281\n1/T2(ps−1)ω(ps−1)\n  \nLL\nGilbert\nMicroscopic\nFIG. 10. Precession frequency with respect to the damping\nstrength in terms of the decay rate 1 =T2.\nB. Precession-frequency shift due to dephasing\nIn the Landau-Lifshitz equation the contributions de-\nscribing, respectively, the precession and the damping are\ncompletely independent, so that the damping constant \u0015\nhas no impact on the precession. By contrast, an increase\nof\u000bin the Gilbert equation does not only increase the\ndephasing rate, it lowers the precession frequency as well.\nIn this section we investigate the change of the preces-\nsion frequency in the microscopic calculation and com-\npare it with the macroscopic descriptions. To this end,\nwe use the o\u000b-diagonal components of the itinerant den-\nsity matrix \u001a+\u0000(t) =P\n~k\u001a+\u0000\n~k(t), which describe the dy-\nnamics of the transverse spin components in the rotating\nframe. The modulus of its Fourier transform j\u001a+\u0000(!)j\nshows a distinct peak, which is exactly at the precession\nfrequency.\nTo compare the precession frequency for di\u000berent\ndamping parameters, we use the dependence on T2, be-\ncause all dephasing parameters can be related to T2for10\nsmall excitations. Fig. 10 plots the Larmor frequency vs.\nthe transverse relaxation rate 1 =T2. The precession pa-\nrameters of the Landau-Lifshitz and the Gilbert equation\nare chosen such that the Larmor frequency in the un-\ndamped limiting case is equal to that of the microscopic\nsimulation. In order to stay within the bounds set by con-\ndition (36), we do not extend the plot in Fig. 10 to higher\ndephasing rates. Fig. 10 shows that the microscopic cal-\nculation yields a reduction of the precession frequency\nwith the damping rate. Although the Gilbert dynamics\nalso show such a reduction, it occurs only at shorter T2.\nAs mentioned above, for the Landau-Lifshitz damping,\nthe frequency is independent of the damping parame-\nter. Even though the change of precession frequency in\nthe microscopic calculation is small, the Landau-Lifshitz\ndamping completely fails to include this e\u000bect. While\nGilbert damping does show a reduction of precession fre-\nquency, it is not at all close to the microscopic calcula-\ntion on the frequency scale considered here. Both phe-\nnomenological damping expressions thus do not repro-\nduce the dependence of the precession frequency on T2.\nEven though the numerical di\u000berences are small, these\ndi\u000berences already occur in the small-excitation regime,\nand may perhaps be detectable.\nC. Dephasing at larger excitation angles\nThe phenomenological Landau-Lifshitz and Gilbert\ndamping contributions describe an exponential decay\nonly for small excitation angles, as studied in the pre-\nvious section. In this section we investigate the e\u000bect of\nlarger excitation angles ( >10\u000e) on the spin dynamics in\nthe microscopic calculation. Apart from this the initial\ncondition of the dynamics is the same as before, in par-\nticular, the itinerant spin is tilted such that the absolute\nvalue of the spin is unchanged.\nFigure 11 shows the time development of the skand\ns?components of the itinerant spin in the rotating frame\nfor an initial tilt angle \f= 140\u000e. While the transverse\ncomponent s?in the rotating frame can be well described\nby an exponential decay, the longitudinal component sk\nshows a di\u000berent behavior. It initially decreases with a\ntime constant of less than 1 ps, but does not reach its\nequilibrium value. Instead, the eventual return to equi-\nlibrium takes place on a much longer timescale, during\nwhich the s?component is already vanishingly small.\nThe long-time dynamics are therefore purely collinear.\nFor the short-time dynamics, the transverse component\ncan be \ft well by an exponential decay, even for large ex-\ncitation angles. This behavior is di\u000berent from Landau-\nLifshitz and Gilbert dynamics, cf. Fig. 2, which both ex-\nhibit non-exponential decay of the transverse spin com-\nponent.\nIn Fig. 12 the dependence of T2on the excitation an-\ngle is shown. From small \fup to almost 180\u000e, the decay\ntime decreases by more than 50%. This dependence is\nexclusively due to the \\excitation condition,\" which in-\n0 1 2 3 4−0.100.1\ntime (ps)s/bardbl\n0 1 2 3 400.050.1\ntime (ps)s⊥FIG. 11. Dynamics of the longitudinal and transverse itiner-\nant spin components in the rotating frame (solid lines) for a\ntilt angle of \f= 140\u000e, together with exponential \fts toward\nequilibrium (dashed lines). The longitudinal equilibrium po-\nlarization is shown as a dotted line.\nvolves only spin degrees of freedom (\\tilt angle\"), but no\nchange of temperature. Although one can \ft such a T2\ntime to the transverse decay, the overall behavior with\nits two stages is, in our view, qualitatively di\u000berent from\nthe typical Bloch relaxation/dephasing picture.\nTo highlight the similarities and di\u000berences from the\nBloch relaxation/dephasing we plot in Fig. 13 the mod-\nulus of the itinerant spin vector j~ sjin the rotating\nframe, whose transverse and longitudinal components\nwere shown in Fig. 11. Over the 2 ps, during which the\ntransverse spin in the rotating frame essentially decays,\nthe modulus of the spin vector undergoes a fast initial\ndecrease and a partial recovery. The initial length of ~ s\nis recovered only over a much larger time scale of several\nhundred picoseconds (not shown). Thus the dynamics\ncan be seen to di\u000ber from a Landau-Lifshitz or Gilbert-\nlike scenario because the spin does not precess toward\nequilibrium with a constant length. Additionally they\ndi\u000ber from Bloch-like dynamics because there is a com-\nbination of the fast and slow dynamics that cannot be\ndescribed by a single set of T1andT2times. We stress\nthat the microscopic dynamics at larger excitation angles\nshow a precessional motion of the magnetization with-\nout heating and a slow remagnetization. This scenario is\nsomewhat in between typical small angle-relaxation, for\nwhich the modulus of the magnetization is constant and\nwhich is well described by Gilbert and Landau-Lifshitz\ndamping, and collinear de/remagnetization dynamics.\nVI. EFFECT OF ANISOTROPY\nSo far we have been concerned with the question\nhow phenomenological equations describe dephasing pro-\ncesses between itinerant and localized spins, where the11\n0 50 100 1500.40.60.81\nβ(◦)T2(ps)\nFIG. 12.T2time extracted from exponential \ft to s?dynam-\nics in rotating frame for di\u000berent initial tilting angles \f.\n0 0.5 1 1.5 20.040.060.080.10.120.14\ntime (ps)|s|\n  \n10°\n50°\n90°\n140°\nFIG. 13. Dynamics of the modulus j~ sjof the itinerant spin\nfor di\u000berent initial tilt angles \f. Note the slightly di\u000berent\ntime scale compared to Fig. 11.\nmagnetic properties of the system were determined by a\nmean-\feld exchange interaction only. Oftentimes, phe-\nnomenological models of spin dynamics are used to de-\nscribe dephasing processes toward an \\easy axis\" deter-\nmined by anisotropy \felds.29\nIn order to capture in a simple fashion the e\u000bects of\nanisotropy on the spin dynamics in our model, we sim-\nply assume the existence of an e\u000bective anisotropy \feld\n~Haniso, which enters the Hamiltonian via\n^Haniso =\u0000g\u0016B\u0016^~ s\u0001~Haniso (40)\nand only acts on the itinerant carriers. Its strength is\nassumed to be small in comparison to the \feld of the\nlocalized moments ~Hloc. This additional \feld ~Haniso has\nto be taken into account in the diagonalization of the\ncoherent dynamics as well, see section IV C.\nFor the investigation of the dynamics with anisotropy,\nwe choose a slightly di\u000berent initial condition, which is\nshown in Fig. 14. In thermal equilibrium, both spins\nare now aligned, with opposite directions, along the\nanisotropy \feld ~Haniso, which is assumed to point in the\nzdirection. At t= 0 they are both rigidly tilted by an\n5&:P; \nO&:P; U T \nV E *_lgqm FIG. 14. Dynamics of the localized spin ~Sand itinerant spin\n~ s. Att= 0, the equilibrium con\fguration of both spins is\ntilted (\f= 10\u000e) with respect to an anisotropy \feld ~Haniso.\nThe anisotropy \feld is only experienced by the itinerant sub-\nsystem.\n01002003004005006000.490.4950.5\ntime(ps)Sz(t)\n010020030040050060000.050.1\ntime(ps)/radicalBig\nS2x(t)+S2y(t)\n  \nFIG. 15. Relaxation dynamics of the localized spin toward the\nanisotropy direction for longitudinal component Szand the\ntransverse componentp\nS2x+S2y. An exponential \ft yields\nBloch decay times of Taniso\n1 = 67:8 ps andTaniso\n2 = 134:0 ps.\nangle\f= 10\u000ewith respect to the anisotropy \feld.\nFigure 14 shows the time evolution of both spins in the\n\fxed frame, with zaxis in the direction of the anisotropy\n\feld for the same material parameters as in the previous\nsections and an anisotropy \feld ~Haniso =\u0000108A\nm\u0001~ ez.\nThe dynamics of the entire spin-system are somewhat\ndi\u000berent now, as the itinerant spin precesses around the\ncombined \feld of the anisotropy and the localized mo-\nments. The localized spin precesses around the itinerant\nspin, whose direction keeps changing as well.\nFigure 15 contains the dynamics of the components\nof the localized spin in the rotating frame. Both com-\nponents show an exponential behavior that allows us to\nextract well de\fned Bloch-times Taniso\n1 andTaniso\n2. Again\nwe \fnd the ratio of 2 Taniso\n1\u0019Taniso\n2, because the abso-\nlute value of the localized spin does not change, as it is\nnot coupled to the phonon bath.\nIn Fig. 16 the Larmor-frequency !aniso\nL, which is the\nprecession frequency due to the anisotropy \feld, and the\nBloch decay times Taniso\n2 are plotted vs. the strength of\nthe anisotropy \feld ~Haniso. The Gilbert damping pa-12\n0 5 10 150510\nHaniso(107A/m)ωaniso\nL (ps−1)\n0 5 10 1505001000\nHaniso(107A/m)Taniso\n2 (ps)\n0 5 10 1501020\nHaniso(107A/m)αaniso (10−4)\nFIG. 16. Larmor frequency !aniso\nL and Bloch decay time Taniso\n2\nextracted from the spin dynamics vs. anisotropy \feld Haniso,\nas well as the corresponding damping parameter \u000baniso.\nrameter\u000baniso for the dephasing dynamics computed via\nEq. (13) is also presented in this \fgure.\nThe plot reveals a decrease of the dephasing time Taniso\n2\nand a almost linear increase of the Larmor frequency\n!aniso\nL with the strength of the anisotropy \feld Haniso.\nThe Gilbert damping parameter \u000baniso shows only a neg-\nligible dependence on the anisotropy \feld Haniso. This\ncon\frms the statement that, in contrast to the dephas-\ning rates, the Gilbert damping parameter is independent\nof the applied magnetic \feld. In the investigated range\nwe \fnd an almost constant value of \u000baniso'9\u000210\u00004.\nThe Gilbert damping parameter \u000baniso for the de-\nphasing toward the anisotropy \feld is about 4 times\nsmaller than \u000biso, which describes the dephasing between\nboth spins. This disparity in the damping e\u000eciency\n(\u000baniso< \u000b iso) is obviously due to a fundamental di\u000ber-\nence in the dephasing mechanism. In the anisotropy case\nthe localized spin dephases toward the zdirection with-\nout being involved in scattering processes with itinerant\ncarriers or phonons. The dynamics of the localized spins\nis purely precessional due to the time-dependent mag-\nnetic moment of the itinerant carriers ~Hitin(t). Thus,\nonly this varying magnetic \feld, that turns out to be\nslightly tilted against the localized spins during the en-\ntire relaxation causes the dephasing, in presence of the\ncoupling between itinerant carriers and a phonon bath,\nwhich acts as a sink for energy and angular momentum.\nThe relaxation of the localized moments thus occurs only\nindirectly as a carrier-meditated relaxation via their cou-\npling to the time dependent mean-\feld of the itinerant\nspin.\nNext, we investigate the dependence of the Gilbert pa-\nrameter\u000baniso on the bath coupling. Fig. 17 shows that\n0 50 100 150 20000.0040.0080.012\nD(meV√nm)αaniso\n  FIG. 17. Damping parameter \u000baniso vs. coupling constant D\n(black diamonds). The red line is a quadratic \ft, indicative\nof\u000baniso/D2.\n\u000baniso increases quadratically with the electron-phonon\ncoupling strength D.\nSince Fig. 9 establishes that the spin-dephasing rate\n1=T2for the fast dynamics discussed in the previous sec-\ntions, is proportional to D2, we \fnd\u000baniso/1=T2. We\nbrie\ry compare these trends to two earlier calculations\nof Gilbert damping that employ p-dmodels and assume\nphenomenological Bloch-type rates 1 =T2for the dephas-\ning of the itinerant hole spins toward the \feld of the\nlocalized moments. In contrast to the present paper, the\nlocalized spins experience the anisotropy \felds. Chovan\nand Perakis38derive a Gilbert equation for the dephasing\nof the localized spins toward the anisotropy axis, assum-\ning that the hole spin follows the \feld ~Hlocof the localized\nspins almost adiabatically. Tserkovnyak et al.39extract\na Gilbert parameter from spin susceptibilities. The re-\nsulting dependence of the Gilbert parameter \u000baniso on\n1=T2in both approaches is in qualitative accordance and\nexhibits two di\u000berent regimes. In the the low spin-\rip\nregime, where 1 =T2is small in comparison to the p-dex-\nchange interaction a linear increase of \u000baniso with 1=T2\nis found, as is the case in our calculations with micro-\nscopic dephasing terms. If the relaxation rate is larger\nthan thep-ddynamics,\u000baniso decreases again. Due to\nthe restriction (36) of the Boltzmann scattering integral\nto low spin-\rip rates, the present Markovian calculations\ncannot be pushed into this regime.\nEven though the anisotropy \feld ~Haniso is not cou-\npled to the localized spin ~Sdirectly, both spins precess\naround the zdirection with frequency !aniso\nL. In analogy\nto Sec. V B we study now the in\ruence of the damping\nprocess on the precession of the localized spin around\nthe anisotropy axis and compare it to the behavior of\nLandau-Lifshitz and Gilbert dynamics. Fig. 18 reveals a\nsimilar behavior of the precession frequency as a function\nof the damping rate 1 =Taniso\n2 as in the isotropic case. The\nmicroscopic calculation predicts a distinct drop of the\nLarmor frequency !aniso\nL for a range of dephasing rates\nwhere the precession frequency is unchanged according\nto the Gilbert and Landau-Lifshitz damping models. Al-\nthough Gilbert damping eventually leads to a change in\nprecession frequency for larger damping, this result shows\na qualitative di\u000berence between the microscopic and the13\n0 0.02 0.04 0.06 0.087.127.167.27.24\n1/Taniso\n2(ps−1)ωaniso(ps−1)\n  \nGilbert\nLL\nMicroscopic\nFIG. 18. Precession frequency of the localized spin around\nthe anisotropy \feld vs. Bloch decay time 1 =Taniso\n2.\nphenomenological calculations.\nVII. CONCLUSION AND OUTLOOK\nIn this paper, we investigated a microscopic descrip-\ntion of dephasing processes due to spin-orbit coupling\nand electron-phonon scattering in a mean-\feld kinetic\nexchange model. We \frst analyzed how spin-dependent\ncarrier dynamics can be described by Boltzmann scat-\ntering integrals, which leads to Elliott-Yafet type relax-\nation processes. This is only possible for dephasing rates\nsmall compared to the Larmor frequency, see Eq. (36).\nThe microscopic calculation always yielded Bloch times\n2T1=T2for low excitation angles as it should be due\nto the conservation of the absolute value of the mag-\nnetization. A small decrease of the e\u000bective precession\nfrequency occurs with increasing damping rate, which is\na fundamental di\u000berence to the Landau-Lifshitz descrip-\ntion and exceeds the change predicted by the Gilbert\nequation in this regime.We modeled two dephasing scenarios. First, a relax-\nation process between both spin sub systems was studied.\nHere, the di\u000berent spins precess around the mean-\feld of\nthe other system. In particular, for large excitation an-\ngles we found a decrease of the magnetization during the\nprecessional motion without heating and a slow remag-\nnetization. This scenario is somewhat in between typi-\ncal small angle-relaxation, for which the modulus of the\nmagnetization is constant and which is well described\nby Gilbert and Landau-Lifshitz damping, and collinear\nde/remagnetization dynamics. Also, we \fnd important\ndeviations from a pure Bloch-like behavior.\nThe second scenario deals with the relaxation of the\nmagnetization toward a magnetic anisotropy \feld expe-\nrienced by the itinerant carrier spins for small excitation\nangles. The resulting Gilbert parameter \u000baniso is inde-\npendent of the static anisotropy \feld. The relaxation of\nthe localized moments occurs only indirectly as a carrier-\nmeditated relaxation via their coupling to the time de-\npendent mean-\feld of the itinerant spin.\nTo draw a meaningful comparison with Landau-\nLifshitz and Gilbert dynamics we restricted ourselves\nthroughout the entire paper to a regime where the elec-\ntronic temperature is equal to the lattice temperature Tph\nat all times. In general our microscopic theory is also ca-\npable of modeling heat induced de- and remagnetization\nprocesses. We intend to compare microscopic simulations\nof hot electron dynamics in this model, including scat-\ntering processes between both types of spin, with phe-\nnomenological approaches such as the Landau-Lifshitz-\nBloch (LLB) equation or the self-consistent Bloch equa-\ntion (SCB)40.\nWe \fnally mention that we derived relation (13) con-\nnecting the Bloch dephasing time T2and the Gilbert\ndamping parameter \u000b. 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Phys. 113, 163911 (2013)."    },    {        "title": "2305.13564v1.Current_driven_motion_of_magnetic_topological_defects_in_ferromagnetic_superconductors.pdf",        "content": "Current-driven motion of magnetic topological defects in ferromagnetic\nsuperconductors\nSe Kwon Kim1,∗and Suk Bum Chung2, 3,†\n1Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea\n2Department of Physics and Natural Science Research Institute,\nUniversity of Seoul, Seoul 02504, Republic of Korea\n3School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea\n(Dated: May 24, 2023)\nRecent years have seen a number of instances where magnetism and superconductivity intrinsically\ncoexist. Our focus is on the case where spin-triplet superconductivity arises out of ferromagnetism,\nand we make a hydrodynamic analysis of the effect of a charge supercurrent on magnetic topological\ndefects like domain walls and merons. We find that the emergent electromagnetic field that arises out\nof the superconducting order parameter provides a description for not only the physical quantities\nsuch as the local energy flux density and the interaction between current and defects but also\nthe energy dissipation through magnetic dynamics of the Gilbert damping, which becomes more\nprominent compared to the normal state as superconductivity attenuates the energy dissipation\nthrough the charge sector. In particular, we reveal that the current-induced dynamics of domain\nwalls and merons in the presence of the Gilbert damping give rise to the nonsingular 4 πand 2 π\nphase slips, respectively, revealing the intertwined dynamics of spin and charge degrees of freedom\nin ferromagnetic superconductors.\nI. INTRODUCTION\nWhile magnetism has traditionally been regarded as\ninimical to superconductivity, recent years have seen ob-\nservation of ferromagnetism and superconductivity co-\nexisting or cooperating in varieties of materials which\nincludes uranium heavy-fermion compounds [1–3] and\ntwo-dimensional moir´ e materials such as twisted bilayer\ngraphene [4–6]. It has been known that such coexistence\ncan be naturally accommodated by the Cooper pairing\nof spin-polarized electrons [7]. In such cases, it is natural\nto question what effect, if any, ferromagnetism may have\non superconductivity and vice versa.\nIt is well established in magnetism and spintronics\nthat the current-induced motions of spin textures such\nas domain walls in magnetic metals give rise to the spin\nand energy dissipation into the baths of quasiparticles or\nphonons, commonly known as the Gilbert damping [8, 9].\nThe conservation of energy dictates that the dissipated\nenergy should be externally supplied by the input power.\nIn the case of normal metals, however, resistivity-induced\nenergy dissipation is present regardless of the presence\nor the absence of any spin textures. Hence the Gilbert\ndamping gives rise to only an additional term in the en-\nergy dissipation and, in this sense, its presence can be\ndifficult to confirm solely through charge transport.\nCharge transport detection of the Gilbert damping in\nferromagnetic superconductors may be more straightfor-\nward despite involving a feature unconventional for su-\nperconductors. To maintain a steady-state motion of spin\ntextures in the presence of the Gilbert damping, a fer-\nromagnetic superconductor needs the finite input power\n∗sekwonkim@kaist.ac.kr\n†sbchung0@uos.ac.kr\n(a)(b)\nFIG. 1. (a) The illustration of the mutually orthogonal unit\nvectors ˆs,ˆu, and ˆvthat describe the directional degrees of\nfreedom of the order parameter of a ferromagnetic supercon-\nductor. (b) The configuration of the triad {ˆs,ˆu,ˆv}for a do-\nmain wall in a ferromagnetic superconductor with easy-axis\nspin anisotropy along the zdirection.\nthat goes out of the superconductor solely in the form of\nthe Gilbert damping. This indicates voltage arising in-\nside the superconductor in the direction of the current by\nthe dynamics of spin textures. The mechanisms by which\na superconductor acquires a finite voltage difference be-\ntween two points is referred to as phase slips [10, 11].\nIn conventional superconductors, these phase slips gen-\nerally accompany the singularities, i.e.the vanishing of\nthe order parameter at a certain time during the phase\nslips.\nIn this paper, we show that this is not necessar-\nily the case for ferromagnetic superconductors by us-\ning the concrete example of the current-induced mo-\ntions of two types of magnetic defects, domain walls and\nmerons, which are schematically illustrated in Fig. 1(b)\nand Fig. 3, respectively. To this end, we begin by ex-\namining the order parameter of the spin-polarized super-\nconductor and show how the Cooper pair spin rotation\naround the spin polarization direction is actually equiv-\nalent to the twisting of the overall phase. This gives rise\nto a channel for the interaction between ferromagnetism\nand superconductivity, namely the coupling of CooperarXiv:2305.13564v1  [cond-mat.supr-con]  23 May 20232\npairs to the effective gauge field arising from spin tex-\nture [7, 12, 13].\nWe then proceed to show how such formalism can be\nused to obtain the current-induced motion of topologi-\ncal spin defects such as a domain wall and a meron in\npresence of a background superflow. First, for a domain\nwall, we show that the current-induced motion of do-\nmain walls in the presence of the Gilbert damping ac-\ncompanies the precessional dynamics of the local spin\npolarization and this in turn gives rise to the nonsingu-\nlar 4πphase slips through the generation of an emergent\nelectric field. The induced phase slip opens a channel\nthrough which the ferromagnetic superconductor can ac-\nquire input power, which is shown to be dissipated by the\nspin dynamics entirely via the Gilbert damping. Also,\na current-induced motion of a meron is shown to give\nrise to the nonsingular 2 πphase slips perpendicular to\nits motion, engendering a channel for the input power\nthat is dissipated via the Gilbert damping. The gener-\nation of the 2 πphase slips can be understood from the\nemergent electromagnetic field associated with the meron\ndynamics. For ferromagnetic metals, the emergent elec-\ntromagnetic fields associated with spin textures and their\ndynamics have been discussed theoretically [14–17] and\nconfirmed experimentally [18–21]. However, their man-\nifestations in the dynamics of magnetic defects in fer-\nromagnetic superconductors and the resultant nonsingu-\nlar phase slips have not been discussed yet. Our work\nreveals that the current-induced dynamics of magnetic\ndefects exemplify the intertwined dynamics of spin and\ncharge degrees of freedom in ferromagnetic superconduc-\ntors, where the emergent electromagnetic fields play cru-\ncial roles.\nThe paper is organized as follows. The general formal-\nism for the order parameter and its dynamics of ferro-\nmagnetic superconductors is developed phenomenologi-\ncally in Sec. II. The current-induced dynamics of a do-\nmain wall and its relation to the nonsingular 4 πphase\nslips are discussed in Sec. III. Section IV concerns the\ncurrent-induced dynamics of a meron and its relation to\nthe nonsingular 2 πphase slips. We conclude the paper\nin Sec. V with discussions.\nII. GENERAL FORMALISM\nA. Order parameter\nThe order parameter of a fully spin-polarized triplet su-\nperconductor provides a starting point for understanding\nhow superconductivity and magnetism are intertwined\nthrough the emergent gauge field. In the d-vector for-\nmalism defined by i(d·σσy)s,s′≡∆s,s′, it is given\nby [7, 12, 13]\nd=√ρ\n2eiϕ(ˆu+iˆv) =rρ\n2eiϕ(ˆu+iˆv)√\n2≡rρ\n2ˆd,(1)where ρ= 2d∗·dis the number density of the Cooper\npairs, ˆuandˆvare perpendicular unit vectors, and ˆd∗·ˆd=\n1; the simplest example would be ˆu=ˆx,ˆv=ˆywhich\ngives ∆ s,s′= 0 except for ∆ ↑↑(see Appendix A for the\ndetails). There is an ambiguity here in defining ϕas\nthe above order parameter remains invariant under the\nfollowing simultaneous change of ϕandˆuandˆv:\neiϕ(ˆu+iˆv) =ei(ϕ+δϕ)\u0002\ne−iδϕ(ˆu+iˆv)\u0003\n≡ei(ϕ+δϕ)(ˆu′+iˆv′),\n(2)\nwhere ˆu′,ˆv′are obtained by rotating ˆu,ˆvby +δϕaround\nˆu׈v. As the spin density in units of ℏcan be written\nas\ns= 2id×d∗=ρˆu׈v≡ρˆs, (3)\nEq. (2) denotes the U(1) ϕ+sorder parameter redundancy\n[7, 12], i.e.the invariance of the order parameter when\nthe angle of the spin rotation around ˆsequals the change\ninϕ. Such redundancy implies the existence of an effec-\ntive gauge field arising from the spin degrees of freedom.\nSee Fig. 1(a) for the illustration of the three mutually\northogonal unit vectors ˆs,ˆu, and ˆv, which are depicted\nby red, green, and blue arrows, respectively.\nFor deriving the vector potential and magnetostatics\nof this effective gauge field, the above order parameter\nsuffices. From the spin rotation angle around ˆsdefined\nasα, the effective vector gauge can be written as [14]\nai≡ℏ\nq∂iα=ℏ\nqˆs·(ˆu×∂iˆu) ; (4)\nhence the emergent gauge is a direct consequence of the\nU(1) ϕ+sorder parameter redundancy of Eq. (2). Indeed,\nthe emergent gauge field of spatial curvature in the chi-\nral superconductor has been attributed to the analogous\norder parameter redundancy there [22–25]. Here, while\nwe have kept the charge, q, generic, q=−2e <0 holds in\nsuperconductors. From this emergent vector potential, it\nis straightforward to obtain the emergent magnetic field,\nbi=ϵijk∂jak=−ℏϵijk\n2qˆs·(∂jˆs×∂kˆs) ; (5)\nnote that this is in the same form as the well-known\nMermin-Ho relation between the orbital angular momen-\ntum texture and superfluid velocity in the 3He-A super-\nfluid [26, 27]. Yet this discussion does not include any\ndynamics, for which we shall adopt a two-step approach\nof first formulating the simplest free energy for the order\nparameter [Eq. (2)] and then use its Lagrangian to obtain\nthe equations of motion.\nB. Free energy\nGiven that we seek results relevant to wide-ranging su-\nperconductors whose common attributes may not extend\nbeyond the spin-polarized Cooper pairing [1–6] we will\nconsider for our free energy the simplest minimal model3\nthat includes the spin anisotropy and the Zeeman cou-\npling:\nF[d] =Z\ndVF′\n0[d] +Z\ndVU\n2\u0000\n2|d|2−ρ0\u00012,(6)\nwhere\nF′\n0=A′\nd\n2|(∇−iq\nℏA)d|2+ρ\u0014A′\ns\n2|∇ˆs|2−D\n2(ˆs·ˆz)2−Hsz+qV\u0015\n,\nwhere A′\nsrepresents the excess spin stiffness; note that,\nin contrast to previous analysis [12], our treatment will\nencompass both the easy-axis anisotropy D > 0 and the\neasy-plane anisotropy D < 0. As we will focus on the\ncases where the fluctuation of the condensate density ρ≡\n2|d|2is strongly suppressed, it is convenient to separatelygroup together terms dependent on ρfluctuations [12, 13]\nF=Z\ndV ρF0+Z\ndV\u0014A′\nd\n16ρ(∇ρ)2+U\n2(ρ−ρ0)2\u0015\n,\nwhere\nF0=A′\nd\n2|(∇−iq\nℏA)ˆd|2+A′\ns\n2|∇ˆs|2−D\n2(ˆs·ˆz)2−Hsz+qV\nis the free energy density per unit density. The gauge\ntransformation is implemented as\nA7→A+∇Λ,d7→eiqΛ/ℏd.\nThe free energy can be recast into the following form:\nF=Z\ndV ρ\u001aAc\n2h\n∂iϕ−q\nℏAi−ˆs·(ˆu×∂iˆu)i2\n+As\n2(∂iˆs)2−D\n2(ˆs·ˆz)2−Hsz+qV\u001b\n+Z\ndV\u0014Ac\n8ρ(∇ρ)2+U\n2(ρ−ρ0)2\u0015\n,\n(7)\nwhere Ac=A′\ndandAs=A′\ns+A′\nd/2. Here, Acand\nAsrepresent the charge stiffness and the spin stiffness,\nrespectively. The similar expression without the sec-\nond and the third terms can be found in Eq. (2.3) and\nEq. (2.5) of Ref. [28].\nAccordingly, the charge supercurrent density is modi-\nfied to\nJi=−δF\nδAi=q\nℏρAc(∂iϕ−q\nℏAi−q\nℏai), (8)\nwith the velocity field is given by\nvi=Ji\nqρ=Ac\nℏ(∂iϕ−q\nℏAi−q\nℏai). (9)\nIt satisfies the following equation (by assuming a nonsin-\ngular θ):\n∇×v=−qAc\nℏ2(B+b). (10)\nThe free energy formalism provides a convenient\nspringboard for extending our analysis to dynamics as\nwell. In particular, such analysis helps us understand\nhow emergent electric field would arise, in analogy with\nthe standard electrodynamics. This is accomplished by\nconsidering the Langrangian for this minimal model.\nC. Equations of motion\nFor the dynamics analysis, we now obtain the classical\nequation of motion for both the charge and the spin com-ponent of the order parameter through considering the\nLagrangian of the spin-polarized superconductor. This\ncan be written as\nL=Z\ndV2iℏd∗·∂td−F\n=−Z\ndVℏρ[∂tϕ−ˆs·(ˆu×∂tˆu)]−F . (11)\nThe first term of the above Lagrangian arises from 2 iℏd∗\nbeing the conjugate variable to d; the detailed derivation\nof its relation to ρ, ϕ,ˆscan be found in Appendix B.\nThe low-energy dynamics of the order parameter can be\ndescribed by the three Euler-Lagrange equations for ϕ, ρ,\nandˆs.\nThe equations for ρandϕare basically analogous to\nthose of the conventional superconductors. The equation\nof motion for the density ρ,\n˙ρ=−1\nℏ∂in\nρAch\n∂iϕ−q\nℏAi−ˆs·(ˆu×∂iˆu)io\n=−1\nq∂iJi,\n=−∇·(ρv) (12)\nis obtained from δL/δϕ = 0 and is none other than the\ncontinuity equation for the Cooper pair density. Simi-\nlarly, the equation of motion for the phase ϕ\n−ℏ[∂tϕ−ˆs·(ˆu×∂tˆu)] =F0+U(ρ−ρ0)−Ac\n4ρ∇2ρ(13)\nobtained from δL/δρ = 0 (where only terms constant or\nlinear in ρare retained) comes out to be the Josephson\nrelation. We, however, want to obtain a hydrodynamic\nequation of motion for Cooper pairs, for which purpose\nwe take the spatial derivative of the Josephson relation:4\n−ℏh\n∂t∂iϕ−q\nℏ(Ei+ei)−q\nℏ∂t(Ai+ai)i\n=ℏ2\nAcv·∂iv+∂i\u0014\nρ′\ne+U(ρ−ρ0)−Ac\n4ρ∇2ρ\u0015\n,\nwhere\nei=−ℏ\nqˆs·(∂iˆs×∂tˆs) (14)\nis the emergent electric field and\nρ′\ne=As(∂iˆs)2\n2−Ds2\nz\n2−Hsz (15)\nis the magnetic energy density (per unit density). By using\nv·∂iv=1\n2∂i(v2) = (v·∇)vi+ϵijkvj(∇×v)k= (v·∇)vi−qAc\nℏ2ϵijkvj(Bk+bk)\nand defining the material derivative Dt≡∂t+v·∇and the effective mass of a Cooper pair, m≡ℏ2/Ac, we obtain\nmDtv=q(E+e) +qv×(B+b)−∂i\u0014\nρ′\ne+U(ρ−ρ0)−Ac\n4ρ∇2ρ\u0015\n. (16)\nThe novelty in the ferromagnetic superconductor is the\nequation of motion for the spin direction ˆsthat is derived\nfrom δL/δˆs= 0 (see Appendix C for details):\nℏρ∂tˆs=−ℏJi\nq∂iˆs+∂i[ρAs(ˆs×∂iˆs)]+ρD(ˆs·ˆz)ˆs׈z+ρHˆs׈z,\n(17)\nwhich is identical to the Landau-Lifshitz equation [29]\naugmented by the adiabatic spin-transfer torque [9, 30,\n31]. This can be also written as\nℏρDtˆs=∂i[ρAs(ˆs×∂iˆs)] +ρD(ˆs·ˆz)ˆs׈z+ρHˆs׈z.\nBy using ˙ ρ=−∂iJi/q, we can obtain the spin continuity\nequation:\n∂t(ℏρˆs) =−∂iJs\ni+ρD(ˆs·ˆz)ˆs׈z+ρHˆs׈z,(18)\nwhere\nJs\ni=ℏJi\nqˆs−ρAs(ˆs×∂iˆs),\nis the spin current density. The first term and the second\nterm on the right-hand side are longitudinal spin currents\nproportional to the charge current and the transverse\nspin current that is carried by a spin texture, respec-\ntively.\nA complete set of equations describing the hydrody-\nnamics of a ferromagnetic superconductor in the absence\nof external fields ( E= 0 and B= 0) can now be given;\nthe analogous equations have been written down for a\nspinor BEC [32]. It is convenient to measure energy in\nthe unit of the anisotropy energy absolute value |D|andlength in the unit derived from the combination of |D|\nwith the spin stiffness As,i.e.\nl=s\nAs\n|D|, ϵ=|D|.\nAlso, we will use ˜ ρ≡ρ/ρ0, Uρ/D ≡η. This gives us the\ndimensionless equations\n−Dt˜ρ= ˜ρ(∇·v),\n˜m(∇×v) =−b,\n˜mDtv=e+v×b\n−∂i[ρe+η(˜ρ−1)− ∇2˜ρ/4],\n˜ρDtˆs=∂i[˜ρ(ˆs×∂iˆs)] + ˜ρν(ˆs·ˆz)ˆs׈z+ ˜ρhˆs׈z,\n(19)\nwhere ν≡sgn(D), ˜m≡As/Acthe dimensionless mass\nwhich is on the order of unity, h≡H/D the dimension-\nless external field, and\nρe=1\n2(∂iˆs)2−ν1\n2(ˆs·ˆz)2−hsz\nthe dimensionless magnetic energy density; for zero ex-\ncess spin stiffness ˜ m= 1/2. The emergent electromag-\nnetic fields are now re-defined as\nei=−ˆs·(∂iˆs×∂tˆs), b i=−ϵijk\n2ˆs·(∂jˆs×∂kˆs),\nwhere the charge qis absorbed into the fields.5\nD. Gilbert damping\nDue to the inevitable nonconservation of spin angu-\nlar momentum in solids, it is reasonable to expect the\ndamping of spin dynamics and the associated energy dis-\nsipation, which are not included in the hydrodynamics\nequations [Eq. (19)], to play an important role in spin\ndynamics of the ferromagnetic superconductors as in any\nother solid-state systems. The spin sinks can be quasi-\nparticles, phonons, and any other excitations that can\npossess angular momentum [33–36]. The spin dissipation\ncan be treated phenomenologically with the addition of\nthe Gilbert damping term α˜ρˆs×∂tˆs[8] to the spin equa-\ntion of motion in Eq. (19),\n˜ρDtˆs+α˜ρˆs×∂tˆs=∂i[˜ρ(ˆs×∂iˆs)]+ ˜ρν(ˆs·ˆz)ˆs׈z+ ˜ρhˆs׈z.\n(20)\nIn the incompressible limit η→ ∞ where ˜ ρis uniform and\nconstant, this gives us the energy continuity equation,\n∂tρe+∇·je=−v·e−α(∂tˆs)2, (21)\nwhere\nje=−∂tˆs·∇ˆs (22)\nis the magnetic energy flux density (per unit density).\nThis continuity equation, which has been previously\nnoted in literature [37, 38], can be derived by taking the\nproduct of both sides of Eq. (20) with ˆs×∂tˆs. One can\nnote that the first term on the right-hand side of Eq. (21)\n(−v·e) is the power dissipated (supplied) by the su-\nperflow vflowing parallel (antiparallel) to the direction\nof the emergent electric field e, while the second term\n(−α(∂tˆs)2) is the energy dissipation through the Gilbert\ndamping.\nEquation (21) implies that, in the incompressible limit,\nthe energy dissipated by the Gilbert damping is equal to\nthe work done by the emergent electric field when spin\ntexture is transported without any distortion. This is\nbecause the total magnetic energy should be unchanged\nin this process and hence the left-hand side of Eq. (21)\nintegrated over the whole system should be zero.\nIII. DOMAIN WALL\nFor the supercurrent-driven motion of topological de-\nfects, we first consider an easy-axis ferromagnetic super-\nconductor with spin-anisotropy sign ν= sgn( D) = +1.\nA domain wall is a generically stable topological defect\nbetween two different ground states for easy-axis spin\nsystems and intrinsically has no skyrmion density, i.e.\nno emergent magnetic field. Therefore for the rest of this\nsection, we will take the Cooper pair velocity to be ir-\nrotational, i.e.∇×v= 0 [Eq. (19)]. In addition, we\nalso set the applied magnetic field to be zero and hence\nh= 0.A. Deriving dynamics from a static solution\nThe solution for a domain-wall motion in the back-\nground of the constant and uniform background super-\nflow can be straightforwardly constructed from the static\ndomain wall solution in absence of any background su-\nperflow for the incompressible limit η→ ∞ . For this\ncase, the absence of the emergent magnetic field allows\nus to consider only the spin equation of motion [40]\n(∂t+v·∇+αˆs×∂t)ˆs=∂i[(ˆs×∂iˆs)] + (ˆs·ˆz)ˆs׈z; (23)\nfrom the hydrodynamic equations Eq. (19); others are\neither irrelevant due to b= 0 or merely provides the\nconstraint ˜ ρ= 1 in the incompressible limit. Given the\nintrinsically quasi-one-dimensional nature of the domain\nwall, we can set the boundary condition\nˆs(x→ ±∞ ) =±ˆz,\nfor any domain-wall configuration. For the static domain\nwall at x= 0 in absence of background superflow, the\nsolution is given by the following Walker ansatz [41]:\nˆs0= (ˆxcosφ0+ˆysinφ0)sech x+Qˆztanhx, (24)\nwhere Q=±1 represents the domain wall type, satisfies\nthe static domain-wall equation\n0 =∂x[(ˆs0×∂xˆs0)] + ( ˆs0·ˆz)ˆs0׈z, (25)\nderived from Eq. (23), for an arbitrary domain-wall angle\nφ0; it is important to note here that Eq. (25) is sufficient\nas Eq. (24) gives us v= 0 and b= 0 everywhere.\nFrom Eq. (25), it can be shown that the general solu-\ntion can be obtained by applying to the static solution\nboth giving boost in the spatial direction and precession\naround the easy-axis:\nˆs= (ˆxcos Ω t+ˆysin Ωt)sech( x−V t) +Qˆztanh( x−V t).\n(26)\nIn deriving the domain-wall velocity ˆxVand the preces-\nsion rate Ω, it is convenient to note that Eq. (26) also\nsatisfies Eq. (25) as the latter equation involves no time\nderivatives. Also, as Eq. (26) is obtained from boost and\nprecession,\n∂tˆs= (−V ∂x+ˆzΩ×)ˆs.\nThe velocity Vand the precession rate Ω therefore can\nbe obtained from\n[v∂x+ (1 + αˆs×)(−V ∂x+ˆzΩ×)]ˆs= 0, (27)\nwhere we set the background superflow to be perpendic-\nular to the domain wall without any loss of generality,\nv=vˆx. By taking the scalar product of the above equa-\ntion with ˆzandˆz׈swe obtain\nv−V=QαΩ,Ω =−QαV6\n(a)<latexit sha1_base64=\"mvV1IC9/vAkApJWzfWyUx3QadJw=\">AAAB8XicbVDLSgNBEJz1GeMr6tHLYBDiJexKQL0FvXiMYB6YLGF20kmGzM4uM71iWPIXXjwo4tW/8ebfOEn2oIkFDUVVN91dQSyFQdf9dlZW19Y3NnNb+e2d3b39wsFhw0SJ5lDnkYx0K2AGpFBQR4ESWrEGFgYSmsHoZuo3H0EbEal7HMfgh2ygRF9whlZ66CA8YVpiZ5NuoeiW3RnoMvEyUiQZat3CV6cX8SQEhVwyY9qeG6OfMo2CS5jkO4mBmPERG0DbUsVCMH46u3hCT63So/1I21JIZ+rviZSFxozDwHaGDIdm0ZuK/3ntBPuXfipUnCAoPl/UTyTFiE7fpz2hgaMcW8K4FvZWyodMM442pLwNwVt8eZk0zstepXx1VylWr7M4cuSYnJAS8cgFqZJbUiN1wokiz+SVvDnGeXHenY9564qTzRyRP3A+fwBSnJC1</latexit>\n(b)<latexit sha1_base64=\"Lyp4rKMmNObzzWYxIHNaXwltWlk=\">AAAB8XicbVDLSgNBEJz1GeMr6tHLYBDiJexKQL0FvXiMYB6YLGF20kmGzM4uM71iWPIXXjwo4tW/8ebfOEn2oIkFDUVVN91dQSyFQdf9dlZW19Y3NnNb+e2d3b39wsFhw0SJ5lDnkYx0K2AGpFBQR4ESWrEGFgYSmsHoZuo3H0EbEal7HMfgh2ygRF9whlZ66CA8YVoKzibdQtEtuzPQZeJlpEgy1LqFr04v4kkICrlkxrQ9N0Y/ZRoFlzDJdxIDMeMjNoC2pYqFYPx0dvGEnlqlR/uRtqWQztTfEykLjRmHge0MGQ7NojcV//PaCfYv/VSoOEFQfL6on0iKEZ2+T3tCA0c5toRxLeytlA+ZZhxtSHkbgrf48jJpnJe9SvnqrlKsXmdx5MgxOSEl4pELUiW3pEbqhBNFnskreXOM8+K8Ox/z1hUnmzkif+B8/gBUIpC2</latexit>\n(c)<latexit sha1_base64=\"eT4ibbwfOOWkpDCTAUUKdbMXNmE=\">AAAB8XicbVDLSgNBEJz1GeMr6tHLYBDiJexKQL0FvXiMYB6YLGF20kmGzM4uM71iWPIXXjwo4tW/8ebfOEn2oIkFDUVVN91dQSyFQdf9dlZW19Y3NnNb+e2d3b39wsFhw0SJ5lDnkYx0K2AGpFBQR4ESWrEGFgYSmsHoZuo3H0EbEal7HMfgh2ygRF9whlZ66CA8YVriZ5NuoeiW3RnoMvEyUiQZat3CV6cX8SQEhVwyY9qeG6OfMo2CS5jkO4mBmPERG0DbUsVCMH46u3hCT63So/1I21JIZ+rviZSFxozDwHaGDIdm0ZuK/3ntBPuXfipUnCAoPl/UTyTFiE7fpz2hgaMcW8K4FvZWyodMM442pLwNwVt8eZk0zstepXx1VylWr7M4cuSYnJAS8cgFqZJbUiN1wokiz+SVvDnGeXHenY9564qTzRyRP3A+fwBVqJC3</latexit>\nFIG. 2. (a) A series of snapshots of a precessing domain wall moving to the right, where xandtare the spatial and the\ntemporal coordinates, respectively. The red arrows, blue arrows, and green arrows represent ˆs,ˆu, and ˆv, respectively. The\ndomain-wall position is denoted by the gray dot. The domain-wall angle, which is the azimuthal angle of ˆsat the center of the\ndomain wall, changes from φ0= 0 to φ0=−2πgradually with increasing time from bottom to top. On the left end, ˆu(green\narrow) rotates by −2πabout ˆs(red arrow), whereas on the right end, ˆurotates by 2 πabout ˆs. This process of a domain-wall\nprecession can be considered as a nonsingular 4 πphase slip since these opposite 2 πrotations of ˆuaround ˆsat the left and\nthe right ends induces a finite voltage across the wire. (b) Mapping of the instantaneous configuration of the two vectors ˆs\n(red arrows) and ˆu(green arrows) onto the unit sphere with the normal vector identified with ˆs. The yellow line represents\na spatial dimension of the system. (c) Collection of the mapping of the configuration of ˆsandˆuonto the unit sphere with ˆs\nidentified with the normal vector for all the snapshots shown in (a). Note that the unit tangent vector field ˆuis not uniquely\ndetermined at the north and the south poles as dictated by the Poincar´ e-Hopf theorem [39]. Rather, ˆurotates once around\nˆscounterclockwise (counterclockwise) at the north (south) pole as the domain wall completes one cycle of rotation, which is\nconsistent with the Euler number 2 of the sphere. See the main text for further detailed discussions.\nrespectively, giving us\nV=1\n1 +α2v ,Ω =−Qα\n1 +α2v; (28)\nNote that in absence of the Gilbert damping, α= 0, there\nwould have been no precession and the domain wall would\nhave remained static with respect to the background su-\nperflow. See Fig. 2 for the illustration of the domain-wall\ndynamics with precession.\nFrom the above solution, it is straightforward to con-\nfirm that all work done by the emergent electric field is\ndissipated through the Gilbert damping. The work done\nby the emergent electric field is\n−v·e=vjˆs·(∂jˆs×∂tˆs) =QvΩ[1−(ˆs·ˆz)2],(29)\nwhich gives the total energy input of\nW=Z\ndx(−v·e) = 2 QvΩ. (30)\nThe energy dissipation rate per unit density is given by\nα(∂tˆs)2=α(V2+ Ω2)[1−(ˆs·ˆz)2]. (31)\nThis energy dissipation through the spin dynamics and\nthe work rate done by the emergent electric field on the\nsuperflow are the same since\nQvΩ =α\n1 +α2v2(32)\nand\nα(V2+ Ω2) =α\n1 +α2v2. (33)B. 4πphase slips from a domain-wall dynamics\nDue to the U(1) ϕ+sorder parameter redundancy, the\nenergy dissipation from the damping-induced precession\nin the domain-wall motion can be regarded as an equiva-\nlent of 4 πphase slips. As shown in Fig. 2, ˆuandˆvrotate\naround ˆsby±2πby adiabatically following the dynamics\nof the local spin direction ˆsin one cycle of precession. To\nsee this, note that if we adopt the condition ˆv·ˆz= 0 in\ndefining ˆv, Eq. (26) will give us ˆv=−ˆxsin Ωt+ˆycos Ω t.\nBut given the U(1) ϕ+sredundancy, this is equivalent to\nthe±2πphase twist on the left and the right end, re-\nspectively.\nThe voltage arising from this precession can be under-\nstood either as arising from the emergent electric field, or\nequivalently, arising from the constant rate of 4 πphase\nslips. When φincreases at the rate ˙ φ= Ω, the emergent\nscalar potential at the two ends of the wire, x=±∞, is\ngiven by\n˜Ve=−ˆs·(ˆu×∂tˆu) =(\n−QΩ at x=−∞,\nQΩ at x=∞,(34)\nwhich is exactly the Josephson voltage for the 4 Qπphase\nslip occurring at the rate of Ω /2π. Then, to maintain the\nfinite superflow, there must be work given by\nW=v[˜Ve(x=∞)−˜Ve(x=−∞)] = 2 QvΩ,(35)\nby external reservoirs on the system, matching the work7\n[Eq. (30)]. Figure 2(a) shows the time evolution of the\ntriad{ˆs,ˆu,ˆv}associated with the domain-wall that both\nmoves and precesses. Note that ˆu(green arrow) at the\nleft end ( x→ −∞ ) and the right end ( x→ ∞ ) ro-\ntates clockwise and counterclockwise, respectively, about\nˆs(red arrow), engendering the nonsingular phase slips\nacross the xdirection. In a nutshell, the domain-wall an-\ngular dynamics produces the nonsingular 4 πphase slips\nthat give rise to a finite voltage difference between the\ntwo ends of the superconducting wire, which constitutes\nour first main result.\nThere is a topological reason why one precession of a\nmagnetic domain wall induces a 4 πphase slip, which can\nbe derived from the Poincar´ e-Hopf theorem or Poincar´ e-\nBrower theorem [39]. For concrete discussion on this, we\nwill consider in the following the case with Q= 1 as\nshown in Fig. 2(b) and (c). At a given time, the instan-\ntaneous configuration of ˆsandˆucan be mapped onto a\nline connecting the north pole [ ˆs(x→ ∞ )] and the south\npole [ ˆs(x→ −∞ )] on the unit sphere by identifying ˆs\nwith the surface normal as shown in Fig. 2(b). When we\nconsider the collection of the configuration of {ˆs,ˆu}onto\nthe unit sphere during one complete precession of the do-\nmain wall, i.e., φ0→φ0+2π,ˆucan be regarded as a unit\ntangent vector field on the sphere since it is perpendic-\nular to ˆs, i.e., the surface normal as shown in Fig. 2(c).\nHere, note that ˆuis not uniquely determined at the north\npole and the south pole, which is consistent with the well-\nknown topological property of the sphere that the unit\ntangent vector field cannot be defined without a singu-\nlarity on it. Instead of being uniquely determined, ˆu\nrotates by 2 πaround the north pole and rotates by −2π\naround the south pole as the domain wall completes one\ncycle of precession, which gives rise to a 4 πphase slip\nacross the wire as discussed above [Eq. (34)]. This can\nbe understood by applying the Poincar´ e-Hopf theorem to\nthe unit tangent vector field ˆuon the sphere. The Euler\nnumber of the unit sphere is 2, meaning that the sum of\nthe indices of the isolated singularities of the unit tan-\ngent vector field on the sphere must be 2. In our case,\nthe indices of the north pole and the south pole associ-\nated with ˆuare both 1, adding up to 2, agreeing with the\nEuler number of the unit sphere.\nIV. MERON\nWe now consider ferromagnetic superconductors with\neasy-plane spin anisotropy ( ν= sgn( D) =−1). For easy-\nplane spin systems, a meron, with its one-half skyrmion\ncharge, is a generically stable topological defect [42–\n45]. Therefore, we consider the rotational Cooper pair\nvelocity in absence of the applied magnetic field i.e.\n∇×v=−b/˜mwith h= 0. See Fig. 3 for the schematic\nillustration of a meron.\n(a)<latexit sha1_base64=\"mvV1IC9/vAkApJWzfWyUx3QadJw=\">AAAB8XicbVDLSgNBEJz1GeMr6tHLYBDiJexKQL0FvXiMYB6YLGF20kmGzM4uM71iWPIXXjwo4tW/8ebfOEn2oIkFDUVVN91dQSyFQdf9dlZW19Y3NnNb+e2d3b39wsFhw0SJ5lDnkYx0K2AGpFBQR4ESWrEGFgYSmsHoZuo3H0EbEal7HMfgh2ygRF9whlZ66CA8YVpiZ5NuoeiW3RnoMvEyUiQZat3CV6cX8SQEhVwyY9qeG6OfMo2CS5jkO4mBmPERG0DbUsVCMH46u3hCT63So/1I21JIZ+rviZSFxozDwHaGDIdm0ZuK/3ntBPuXfipUnCAoPl/UTyTFiE7fpz2hgaMcW8K4FvZWyodMM442pLwNwVt8eZk0zstepXx1VylWr7M4cuSYnJAS8cgFqZJbUiN1wokiz+SVvDnGeXHenY9564qTzRyRP3A+fwBSnJC1</latexit>\n(b)<latexit sha1_base64=\"Lyp4rKMmNObzzWYxIHNaXwltWlk=\">AAAB8XicbVDLSgNBEJz1GeMr6tHLYBDiJexKQL0FvXiMYB6YLGF20kmGzM4uM71iWPIXXjwo4tW/8ebfOEn2oIkFDUVVN91dQSyFQdf9dlZW19Y3NnNb+e2d3b39wsFhw0SJ5lDnkYx0K2AGpFBQR4ESWrEGFgYSmsHoZuo3H0EbEal7HMfgh2ygRF9whlZ66CA8YVoKzibdQtEtuzPQZeJlpEgy1LqFr04v4kkICrlkxrQ9N0Y/ZRoFlzDJdxIDMeMjNoC2pYqFYPx0dvGEnlqlR/uRtqWQztTfEykLjRmHge0MGQ7NojcV//PaCfYv/VSoOEFQfL6on0iKEZ2+T3tCA0c5toRxLeytlA+ZZhxtSHkbgrf48jJpnJe9SvnqrlKsXmdx5MgxOSEl4pELUiW3pEbqhBNFnskreXOM8+K8Ox/z1hUnmzkif+B8/gBUIpC2</latexit>\n(c)<latexit sha1_base64=\"eT4ibbwfOOWkpDCTAUUKdbMXNmE=\">AAAB8XicbVDLSgNBEJz1GeMr6tHLYBDiJexKQL0FvXiMYB6YLGF20kmGzM4uM71iWPIXXjwo4tW/8ebfOEn2oIkFDUVVN91dQSyFQdf9dlZW19Y3NnNb+e2d3b39wsFhw0SJ5lDnkYx0K2AGpFBQR4ESWrEGFgYSmsHoZuo3H0EbEal7HMfgh2ygRF9whlZ66CA8YVriZ5NuoeiW3RnoMvEyUiQZat3CV6cX8SQEhVwyY9qeG6OfMo2CS5jkO4mBmPERG0DbUsVCMH46u3hCT63So/1I21JIZ+rviZSFxozDwHaGDIdm0ZuK/3ntBPuXfipUnCAoPl/UTyTFiE7fpz2hgaMcW8K4FvZWyodMM442pLwNwVt8eZk0zstepXx1VylWr7M4cuSYnJAS8cgFqZJbUiN1wokiz+SVvDnGeXHenY9564qTzRyRP3A+fwBVqJC3</latexit>\nFIG. 3. (a) An illustration of the meron with polarity p= 1\nand vorticity n= 1, which is a nonsingular topological de-\nfect of ferromagnetic superconductors with easy-plane spin\nanisotropy. The red, the green, and the blue arrows repre-\nsentˆs,ˆu, and ˆv, respectively. The meron core is denoted by\nthe gray dot. The local spin direction ˆsrotates by 2 πcoun-\nterclockwise about the zaxis when we follow the infinitely\ndistant trajectory encircling the meron core counterclockwise\n(and thus the vorticity n= 1). Note that ˆuandˆvalso change\nspatially to keep their orthonormality to ˆs; they rotate by 2 π\nclockwise about the local spin direction ˆswhen we enclose\nthe meron center counterclockwise. (b) Mapping of the con-\nfiguration of ˆs(red arrows) and ˆu(green arrows) along the\ninfinitely distant circle in (a) onto the equator with the surface\nnormal identified with ˆs. (c) Mapping of the configuration of\nˆsand ˆuof the entire system onto the northern hemisphere\nwith ˆsidentified with the surface normal. Note that ˆuis a\nwell-defined unit tangent vector field on the northern hemi-\nsphere without any singularity. Since the Euler number of\nthe hemisphere is 1, if the unit tangent vector field defined on\nthe northern hemisphere has no singularity, it should rotate\naround the surface normal by 2 πalong the equator according\nto the Poincar´ e-Hopf theorem [39], which is exactly what ˆu\n(unit tangent vector) does around ˆs(surface normal).\nA. Static solution\nAnalogous to the case of the domain wall in the previ-\nous section, a straightforward construction of the meron\nmotion solution here in the background of the constant\nand uniform background superflow is possible from the\nstatic meron solution in absence of any background su-\nperflow for the incompressible limit, η→ ∞ [32, 46].\nThe static solution can be obtained from the following8\ntwo equations;\n˜m(∇×v) =−b,\n(v·∇)ˆs=∂i[(ˆs×∂iˆs)]−(ˆs·ˆz)ˆs׈z.(36)\nIt is important to note here that while we are dealing\nwith a static configuration we still have v̸= 0 due to\nthe intrinsic emergent magnetic flux of the meron. In\naddition, this solution would possess the axial symmetry,\ni.e.\nˆs0= (sin θcosφ,sinθsinφ,cosθ), (37)\nwith θ=θ(r) and φ=nχ+ Φ where ( r, χ) are polar\ncoordinates for the two-dimensional system, and follow\nthe universal boundary conditions for merons are given\nby\nθ(r= 0) = (1 −p)π\n2, θ(r→ ∞ ) =π\n2;\np=±1 here is the polarity, which is the z-component of\nthe local spin direction ˆsat the meron center, and n∈Z\nthe vorticity, which counts how many times ˆswinds in\nthe easy plane along the closed trajectory encircling the\nmeron center. For our purpose, obtaining the differential\nequation for θ(r) is sufficient for showing Eq. (37) to be\nthe solution of Eq. (36). We start by noting that the\nemergent magnetic field is aligned entirely along the z-\naxis and\nbz=−ˆs0·(∂xˆs0×∂yˆs0) =−nsinθ\nrdθ\ndr(38)\nis function only for θ, which gives us the well-known re-\nsult\nZ\ndxdyb z=Z\nrdχdr\u0012\n−nsinθ\nrdθ\ndr\u0013\n=−2πpn (39)\nfor the total emergent magnetic flux. With this emergent\nmagnetic field, the first equation of Eq. (36) requires the\ncirculating velocity v=ˆφv(r) around the meron with\n1\nrd(rv)\ndr=nsinθ\n˜mrdθ\ndr.\nInserting this relation into the second equation of\nEq. (36) gives us\nsinθ(1−cosθ)1\n˜mr2=1\nrd\ndr\u0012\nrdθ\ndr\u0013\n+cos θsinθ\u0012\n1−1\nr2\u0013\n,\nfrom which θ(r) can be obtained numerically.\nTo find explicit expressions for ˆuandˆv, note that the\nfollowing three unit vectors form an orthonormal triad:\nˆs0=ˆer≡(sinθcosφ,sinθsinφ,cosθ),\nˆeθ≡(cosθcosφ,cosθsinφ,−sinθ),\nˆeφ≡(−sinφ,cosφ,0),which gives us\nˆu0(r, χ) = cos φ(χ)ˆeθ(r, χ)−sinφ(χ)ˆeφ(r, χ),(40)\nˆv0(r, χ) = sin φ(χ)ˆeθ(r, χ) + cos φ(χ)ˆeφ(r, χ).(41)\nThe local configuration of the triad ( ˆs0,ˆu0,ˆv0) for a\nmeron with p= 1 and n= 1 is shown in Fig. 3. Note\nthat it is nonsingular, differing from a conventional vor-\ntex of a s-wave superconductor [10]. An analogous non-\nsingular topological defects that give rise to 4 πnonsin-\ngular phase slips has been discussed by Anderson and\nToulouse [47] and has been termed the skyrmion solu-\ntion in the more recent literature [28, 32]. By contrast,\nour solution {ˆs0,ˆu0,ˆv0}[Eqs. (37,40,41)] represents an\nexplicit solution for the nonsingular topological defect\nin the easy-plane case that gives rise to 2 πnonsingular\nphase slip, as can be seen from Eqs (36) and (39): it\nharbors the emergent magnetic flux −2πpnand thus its\nmotion gives rise to the emergent electric field, i.e., phase\nslips perpendicular to its motion.\nThe non-trivial emergent gauge field ai, and thus the\nquantized non-zero emergent magnetic flux, of our non-\nsingular topological defect can be understood from the\nrotation of ˆuaround ˆsas stated in Eq. (4). There exists\na topological constraint dictating that a meron texture\nofˆsshould trap a quantized non-zero emergent magnetic\nflux, which can be understood by invoking the Poincar´ e-\nHopf theorem [27, 39] as follows. For the given meron\nconfiguration with p= 1, let us consider the mapping of\ntwo unit vectors ˆsandˆuonto the northern hemisphere\nsuch that ˆsis identified with the surface normal. Then,\nˆubecomes a unit tangent vector field on the hemisphere\nsince it is perpendicular to ˆs, i.e., the surface normal as\nshown in Fig. 3(c). The Euler number of the hemisphere\nis 1, and thus the Poincar´ e-Hopf theorem dictates that, if\nthe unit tangent vector field is nonsingular on the north-\nern hemisphere, it should rotate exactly one time about\nthe surface normal while traversing the equator, which is\nexactly what ˆudoes around ˆsin Fig. 3(b). Therefore, the\none-time rotation of ˆuaround ˆsalong the closed loop that\ncontains, but is also infinitely far from, the meron core\nas shown in Fig. 3(a), which gives rise to the quantized\nemergent magnetic flux, can be regarded as the physical\nmanifestation of the topological constraint that should\nbe satisfied by the nonsingular unit tangent vector field\ndefined on the hemisphere.\nB. Dynamics with a background superflow\nAnalogous to the domain wall motion, it is straight-\nforward to work out the spin equation of motion for the\nmeron motion driven by a uniform constant background\nsuperflow v0if we assume the meron motion to be rigid,\ni.e.ˆs(r, t) =ˆs0(r−Vt) (the same holds for ˆuandˆv). The\nCooper pair velocity would then be given by v=vm+v0,\nwhere vmis the Cooper pair velocity around a static\nmeron. We can therefore employ the collective coordi-\nnate approach to describe the dynamics of a meron, and9\nFIG. 4. Snapshots of a meron moving in the ydirection in the increasing time from (a) to (d), where the meron core is depicted\nby a gray dot. At x→ −∞ ,ˆu(green arrow) rotates about ˆs(red arrow) counterclockwise, whereas at x→ ∞ ,ˆurotates about\nˆsclockwise, inducing phase slips and thereby generating a finite voltage in the xdirection.\nuse the fact that ˆs0(r−Vt) should satisfy the original\nspin equation of motion of Eq. (20) with, as we are in\nthe incompressible limit, the constant ˜ ρwhile ˆs0(r) also\nsatisfies the equation for the static meron of Eq. (36).\nSubtraction between the two equations, together with\n∂tˆs0(r−Vt) =−V·∇ˆs0(r−Vt) gives us\n−(1 +αˆs0×)V·∇ˆs0=−v0·∇ˆs0.\nTaking the scalar product with s0×∂is0on both sides\ngives us\n−Vj[s0·(∂iˆs0×∂jˆs0)]−αVj(∂iˆs0·∂jˆs0)\n=−v0,j[s0·(∂iˆs0×∂jˆs0)].(42)\nStrictly speaking, this result shows that whereas we have\nan exact rigid motion solution with V=v0in absence\nof damping, no rigid motion solution can be exact in\npresence of damping. Yet to the zeroth order in v0and\nalso in the spirit of collective coordinate, we can average\nout the effect of the spin texture, i.e.defining\nGij=Z\ndxdys0·(∂iˆs0×∂jˆs0), D ij=Z\ndxdy∂ iˆs0·∂jˆs0,\nknown respectively as the gyrotropic coefficients and the\ndissipation coefficients [48–50] ( Dij=DδijandGxy=\n−Gyx≡G= 2πpnfor a meron). By integrating Eq. (42),\nwe have\n(αD+Gˆz×)V=Gˆz×v0, (43)\nwhich yields the following solution for the velocity V:\nV=G\nG2+α2D2(G+αDˆz×)v0. (44)\nTo consider a concrete example, we will hereafter restrict\nthe discussion to the case where the background super-\nflow flows in the xdirection: v0= (v0,0). In this case,we have\n\u0012\nVx\nVy\u0013\n=G\nG2+α2D2\u0012\nGv0\n−αDv 0\u0013\n. (45)\nNote that the presence of damping gives rise to the com-\nponent of the meron velocity transverse to the uniform\nsuperflow in proportion to the skyrmion number G/2πof\nthe meron, Vy=−α[GD/(G2+α2D2)]v0, exhibiting the\nso-called skyrmion Hall effect [16, 51–53]. This transverse\nmotion of the meron with respect to the superflow gives\nrise to the finite voltage in the direction of the superflow\nvia the 2 πphase slips, which we turn our attention now.\nC. 2πphase slips from a meron motion\nAgain analogous to the domain wall motion, the\nU(1) ϕ+sorder parameter redundancy allows the energy\ndissipation due to the meron motion as equivalent to the\n2πphase slips. This can be seen from the emergent elec-\ntric field\ne=−ˆs·(∇ˆs×∂tˆs)\n=ˆs0·(∇ˆs0×V·∇ˆs0) =−V×b, (46)\nwhich is in the same form as the Josephson electric field\narising from the vortex motion [10]. The same can natu-\nrally be said about the input power density required for\ndriving the uniform constant background superflow\n−v0·e=v0,xˆs·(∂xˆs×∂tˆs),\n=−v0,xVyˆs·(∂xˆs×∂yˆs).\nIt can be checked explicitly that the total energy rate for\ndriving the superflow\n−Z\ndxdyv0·ˆe=−v0,xVyG=αG2D\nG2+α2D2v2\n0,10\nis equal to the energy dissipation rate\nZ\ndxdyα (∂tˆs)2=Z\ndxdyαV iVj(∂iˆs0·∂jˆs0)\n=αDV2=αG2D\nG2+α2D2v2\n0.\nThis energy must come from external reservoirs\nthrough the boundary of the system, meaning that there\nshould be a development of a finite voltage across the\nsystem in the xdirection. This can be explicitly seen\nfrom\n˜Ve=−ˆs·(ˆu×∂tˆu) = (1 −cosθ)∂tφ . (47)\nTo see how a finite voltage is generated by the motion\nof a vortex, let us assume that a vortex moves in the y\ndirection, V=Vyˆy. Then for a given point at large x,\nZ∞\n−∞dtˆs·(ˆu×∂tˆu) =−Z∞\n−∞dt∂tφ=nπ , forx→+∞.\n(48)\nThe vector ˆu(x→ ∞ ) rotates by nπaround the ˆs. Also,\nZ∞\n−∞dtˆs·(ˆu×∂tˆu) =−Z∞\n−∞dt∂tφ=−nπ , forx→ −∞ .\n(49)\nThe vector ˆu(x→ −∞ ) rotates by −nπaround the\nˆs. This indicates that the motion of a meron in the\nydirection induces a nonsingular 2 πphase slips across\nthexdirection. To keep the superflow in the xdirec-\ntion constant, we need to counteract the effect of these\nphase slips. The corresponding work on the system is\ndissipated to external baths such as quasiparticles or\nphonons through the Gilbert damping. Figure 4 shows\nthe schematic illustration of the process. As the vor-\ntex moves in the positive ydirection from Fig. 4(a) to\nFig. 4(d), ˆuatx→ ∞ andx→ −∞ rotates about the\nlocal spin direction ˆscounterclockwise and clockwise, re-\nspectively, producing phase slips in the xdirection. This\nis our second main result: The dynamics of a meron en-\ngenders the nonsingular 2 πphase slips perpendicular to\nits motion through the generation of the emergent elec-\ntric field, showcasing the intertwined dynamics of spin\nand charge degrees of freedom of ferromagnetic super-\nconductors.\nV. DISCUSSION\nWithin the phenomenological framework for the dy-\nnamics of the order parameter of ferromagnetic super-\nconductors, we have shown that the current-induced dy-\nnamics of magnetic defects in the presence of the spin\ndissipation, i.e., the Gilbert damping, give rise to nonsin-\ngular phase slips via the emergent electromagnetic fields.\nThe input power, which is the product of the applied\ncurrent that drives the magnetic defects and the voltage\ngenerated by the phase slip, is shown to be equivalent tothe dissipated energy in the form the Gilbert damping\ninto the baths of quasiparticles or phonons. Our work on\nthe dynamics of magnetic defects showcases the intrinsic\ninterplay of spin and charge dynamics of ferromagnetic\nsuperconductors.\nA few remarks are on order about the limitations of\nour work. First, we did not include the effects of the\nnon-adiabatic spin-transfer torque by a supercurrent on\nthe dynamics of magnetic defects, which is expected to be\npresent on general grounds whenever the Gilbert damp-\ning is present [9, 40, 54, 55]. While we believe that the\ninclusion of the non-adiabatic spin-transfer torque in our\nmodel would not qualitatively change the relations that\nwe have found between the dynamics of magnetic defects\nand nonsingular phase slips, it will certainly enrich the\nphysics of the interplay of spin and charge dynamics in\nferromagnetic superconductors. Secondly, in this work,\na ferromagnetic superconductor is assumed to be a fully\nspin-polarized triplet superconductor as in Ref. [12], by\nleaving the generalization to a partially spin-polarized\ncase as future work. Thirdly, our analysis shows that\nthe assumption of rigid motion is not exact for merons\nin presence of damping. The coupling between current\nand magnons bound to meron cores may be a relevant\ntopic for future study. Lastly, the dynamics of magnetic\ndefects has been discussed in the incompressible limit,\nwhere the dynamics of the order-parameter amplitude\nis frozen. Releasing this assumption would allow us to\nstudy the interplay of spin dynamics and longitudinal\norder-parameter dynamics, which is beyond the scope of\nthe current work.\nACKNOWLEDGMENTS\nWe thank Mike Stone, Grigori Volovik, Daniel\nAgterberg and Jim Sauls for useful discussions.\nS.K.K. was supported by Brain Pool Plus Program\nthrough the National Research Foundation of Korea\nfunded by the Ministry of Science and ICT (NRF-\n2020H1D3A2A03099291) and by the National Research\nFoundation of Korea funded by the Korea Government\nvia the SRC Center for Quantum Coherence in Con-\ndensed Matter (NRF-RS-2023-00207732). S.B.C. was\nsupported by the National Research Foundation of Korea\n(NRF) grants funded by the Korea government (MSIT)\n(NRF-2023R1A2C1006144, NRF-2020R1A2C1007554,\nand NRF-2018R1A6A1A06024977).11\nAppendix A: Details of the order parameter\nThe multicomponent superconducting gap is given by\nˆ∆ =\u0012\n∆↑↑∆↑↓\n∆↓↑∆↓↓\u0013\n≡\u0012\n−˜dx+i˜dy˜dz\n˜dz˜dx+i˜dy\u0013\n=i(d·σ)σy. (A1)\nThen,\nˆ∆ˆ∆†=\u0012\n∆↑↑∆↑↓\n∆↓↑∆↓↓\u0013\u0012∆∗\n↑↑∆∗\n↓↑\n∆∗\n↑↓∆∗\n↓↓\u0013\n=\u0012|∆↑↑|2+|∆↑↓|2∆↑↑∆∗\n↓↑+ ∆↑↓∆∗\n↓↓\n∆↓↑∆∗\n↑↑+ ∆↓↓∆∗\n↑↓|∆↓↓|2+|∆↓↑|2\u0013\n=\u0012\n−˜dx+i˜dy˜dz\n˜dz˜dx+i˜dy\u0013\u0012−˜d∗\nx−i˜d∗\ny˜d∗\nz\n˜d∗\nz˜d∗\nx−i˜d∗\ny\u0013\n=\u0012|˜dx|2+|˜dy|2+|˜dz|2+i(˜dx˜d∗\ny−˜d∗\nx˜dy) ( −˜dx˜d∗\nz+i˜dy˜d∗\nz) + c.c.\n(−˜d∗\nx−i˜d∗\ny)˜dz) + c.c. |˜dx|2+|˜dy|2+|˜dz|2−i(˜dx˜d∗\ny−˜d∗\nx˜dy)\u0013\n=|d|2σ0+i(d×d∗)·σ, (A2)\nwhere d= (˜dx,˜dy,˜dz). The total number density of the Cooper pairs is given by\n|∆↑↑|2+|∆↑↓|2+|∆↓↑|2+|∆↓↓|2= Tr[ ˆ∆ˆ∆†] = 2d·d∗= 2|d|2. (A3)\nThe expected spin angular momentum is given in units of ℏby\nTr[ˆ∆ˆ∆†σ] = Tr[ i(d×d∗)·σσ] = 2id×d∗. (A4)\nFor a fully spin-polarized triplet superconductor, we can use\nd=deiϕ(ˆu+iˆv), (A5)\nwith ˆu⊥ˆv. Then, we have d·d∗= 2d2. Therefore, d=√ρ/2. The spin polarization is then given by\ns= 2id×d∗= 4d2(ˆu׈v)\n≡4d2ˆs=ρˆs. (A6)\nTo see if the result makes sense, let us consider ˆs=ˆzwith ˆu=ˆxandˆv=ˆy. Then, d=deiϕ(ˆx+iˆy). Then,\n˜dx=deiϕand˜dy=deiϕi. Then, ∆ ↑↑=−2d,∆↓↓= ∆↑↓= ∆↓↑= 0. The condensate density (of the Cooper pairs) is\ngiven by ρ=|∆↑↑|2= 4d2and the spin density is given by s= 4d2ˆs=ρˆs.\nAppendix B: Kinetic term of the Lagrangian\nThe kinetic term of the Lagrangian density is given by\nLK= 2iℏd∗·∂td\n=iℏ\n2√ρe−iϕ(ˆu−iˆv)·∂t\u0002√ρeiϕ(ˆu+iˆv)\u0003\n=iℏ\n2√ρe−iϕ(ˆu−iˆv)·\u0014∂tρ\n2√ρeiϕ(ˆu+iˆv) + (i∂tϕ)√ρeiϕ(ˆu+iˆv) +√ρeiϕ(∂tˆu+i∂tˆv)\u0015\n=iℏ\n2√ρ(ˆu−iˆv)·\u0014∂tρ\n2√ρ(ˆu+iˆv) + (i∂tϕ)√ρ(ˆu+iˆv) +√ρ(∂tˆu+i∂tˆv)\u0015\n=iℏ\n2\u0014∂tρ\n22 + (i∂tϕ)ρ2 +ρ(iˆu·∂tˆv−iˆv·∂tˆu)\u0015\n=−ℏρ(∂tϕ+ˆu·∂tˆv)\n=−ℏρ[∂tϕ−ˆs·(ˆu×∂tˆu)]. (B1)12\nThe factor of 2 in front is to ensure the commutation relation, [ d∗\ni(r), dj(r′)] = 2 iℏδ(r−r′). Then, the total Lagrangian\ndensity is given by\nL=−ℏρ[∂tϕ−ˆs·(ˆu×∂tˆu)]\n−ρ\u001aAc\n2h\n∂iϕ+q\nℏAi+ˆs·(ˆu×∂iˆu)i2\n+As\n2(∂iˆs)2−D\n2(ˆs·ˆz)2−Hsz+qV\u001b\n−\u001aAc\n16(∇ρ)2+U\n2(ρ−ρ0)2\u001b\n.(B2)\nFrom this, we can read the emergent scalar potential\nVe=−ℏ\nqˆs·(ˆu×∂tˆu), (B3)\nand the emergent vector potential\nai=ℏ\nqˆs·(ˆu×∂iˆu). (B4)\nAppendix C: Equations of motion\nThe dynamics of the order parameter can be uniquely characterized by the dynamics of three variables, the con-\ndensate number density of the Cooper pairs ρ, the phase of the order parameter ϕ, and the spin direction ˆs.\nFirst, the equation of motion for ϕcan be obtained by\nδL\nδρ= 0,\n⇒ −ℏ[∂tϕ−ˆs·(ˆu×∂tˆu)] =δF\nδρ,\n⇒ −ℏ[∂tϕ−ˆs·(ˆu×∂tˆu)] =\u001aAc\n2h\n∂iϕ−q\nℏAi−ˆs·(ˆu×∂iˆu)i2\n+As\n2(∂iˆs)2−D\n2(ˆs·ˆz)2−Hsz+qV\u001b\n−Ac\n8∇2ρ+U(ρ−ρ0).\n(C1)\nSecond, the equation of motion for ρcan be obtained by\nd\ndt\u0012δL\nδ˙ϕ\u0013\n−δL\nδϕ= 0,\n⇒ −ℏ˙ρ=−δF\nδϕ,\n⇒ −ℏ˙ρ=∂in\nρAch\n∂iϕ−q\nℏAi−ˆs·(ˆu×∂iˆu)io\n,\n⇒˙ρ=−1\nq∂iJi, (C2)\nwhere Jiis the charge current density. This is nothing but the continuity equation.\nThird, to obtain the equation of motion for ˆs, by considering infinitesimal variations of the three vectors that\nmaintain the orthonormality conditions,\nˆs=ˆs0+δˆs=ˆs0+aˆu0+bˆv0,\nˆu=ˆu0−aˆs0,\nˆv=ˆv0−bˆs0, (C3)\nwe observe that, to zeroth order in aandb,\nδ\nδˆs(ˆs·(ˆu×∂tˆu)) = ˆu0∂\n∂a(ˆs·(ˆu×∂tˆu)) +ˆv0∂\n∂b(ˆs·(ˆu×∂tˆu))\n=−ˆu0(ˆs0·(ˆu0×∂tˆs0)) +ˆv0(ˆv0·(ˆu0×∂tˆu0))\n=ˆs0×∂tˆs0. (C4)13\nAlso, by the analogous steps,\nδ\nδˆs(ˆs·(ˆu×∂iˆu)) =ˆs×∂iˆs. (C5)\nThen, from the Lagrangian, we obtain\nδL\nδˆs= 0,\n⇒ℏρˆs×∂tˆs=δF\nδˆs,\n⇒ℏρ∂tˆs=−ℏJi\nq∂iˆs+∂i[ρAs(ˆs×∂iˆs)] +ρD(ˆs·ˆz)ˆs׈z+ρHˆs׈z. (C6)\nBy using ˙ ρ=−(∂iJi)/q, the last equation can be recast into\n∂t(ℏρˆs) =−∂i\u0014ℏJi\nqˆs−[ρAs(ˆs×∂iˆs)]\u0015\n+ρD(ˆs·ˆz)ˆs׈z+ρHˆs׈z. (C7)\nThe left-hand side is the spin density, s=ℏρˆs. 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Rashba spin-orbit coupling-induced aniso tropy of these phenomena is thoroughly\nanalysed. For the case of two partly filled spin subbands, a re markable relation between the\nanisotropic STT, GD, and ESR is established. In the absence o f magnetic field and other torques\non magnetization, this relation corresponds to a current-i nduced motion of a magnetic texture with\nthe classical drift velocity of conduction electrons. Fina lly, we compute spin susceptibility of the\nsystem and generalize the notion of spin-polarized current .\nPossibility to efficiently manipulate magnetic order by\nmeansofelectriccurrenthasgainedalotofattentionover\nthe past decades1,2. Potential applications include race\ntrackmemory3,4, spin torquemagnetization switching5,6,\nskyrmion-based technology7,8, and other promising con-\ncepts. Spintronic logic and memory devices based on\ncurrent-driven magnetization dynamics are believed to\nachieve high speed, low volatility, outstanding durabil-\nity, and low material costs with promises to outperform\ncharge-trapping solid-state memory devices9.\nIn the light of recent detection of fast domain wall\n(DW) motion in magnetic films10,11and predictions of\nevenhigherDWvelocitiesinantiferromagnets12, current-\ninduced dynamics of domain walls, skyrmions, and other\nmagnetic textures remain an important research subject\nin the field of spintronics. Such dynamics is mainly de-\ntermined by the interplay of the two phenomena: Gilbert\ndamping (GD) and spin torques13–16.\nIn the absence of spin-orbit coupling (SOC), spin\ntorques emerge only in the systems with nonuniform\nmagnetization profiles and are most often referred to as\nspin-transfer torques (STT). At the same time, the clas-\nsification of spin torques usually gets more complicated\nif coupling between spin and orbital degrees of freedom\nbecomes pronounced. Moreover, the debate on the mi-\ncroscopic origin of spin torques in the latter case remains\nongoing17,18. Below, we regard STT, in the continuum\nlimit, as a contribution to the total torque on magnetiza-\ntion that is linearwith respect to both the electricfield E\nand the first spatial derivatives of the unit vector of mag-\nnetization direction n. We note that, in the absence of\nSOC, physics of STT is well understood15,16.\nIn a similar fashion, Gilbert damping may be gener-\nally associated with the terms of the Landau-Lifshitz-\nGilbert (LLG) equation that are odd under time reversal\nand linear with respect to the time derivative of n. In\nthe most simplistic approach, GD is modeled by a sin-\ngle phenomenological term αn×∂tnthat corresponds to\n“isotropic” damping.\nHowever, it has been known for quite a while that GD\nmay exhibit anisotropic behaviour19–27. Or, to be more\nprecise, that the scalar damping constant α, in general,should be replaced by a damping matrix with the com-\nponents depending on the orientation of n. These two\nmanifestations of anisotropy may be referred to as rota-\ntional and orientational anisotropy, respectively22. Ex-\nperimental observation of the orientational anisotropy\nof Gilbert damping has been reported very recently in\na metal ferromagnet (FM)/semiconductor interface of\nFe/GaAs(001)28and in epitaxial CoFe films29. The au-\nthors of Ref. [28] argued that the measured anisotropy\nrooted in the interplay of interfacial Rashba and Dressel-\nhaus spin-orbit interaction.\nGiven the equal importance of GD and STT in the\ncontext of current-induced magnetization dynamics and\nthe significant progressmade in the understanding of the\nanisotropic nature of Gilbert damping, we find it surpris-\ning that the anisotropyof spin-transfer torques has so far\nonly been addressed phenomenologically24,30.\nIn the present paper, we consider a 2D Rashba FM\nwithspin-independent electronscattering. Amicroscopic\nanalysis, performed for an arbitrarymagnetization direc-\ntion, allows us to quantify the rotational as well as the\norientationalanisotropyofboth STT and GD induced by\nRashba SOC. Our results indicate that, for a Rashba FM\nsystem, spin-transfer torques TSTTand Gilbert damp-\ningTGDentering the LLG equation\n∂tn=γn×Heff+TSTT+TGD+... (1)\nnaturally acquire the following forms:\nTSTT=ξ0∂vn−ξ/bardbl[n×∂vn/bardbl]−ξ⊥[n×∂vn⊥],(2a)\nTGD=ξ0∂tn−ξ/bardbl[n×∂tn/bardbl]−ξ⊥[n×∂tn⊥],(2b)\nwhereξi=ξi(n), the operator ∂v= (vd·∇) is expressed\nvia the classical electron drift velocity vd=eE/planckover2pi1τ/m,\nandn/bardbl/⊥stands for the in-plane/perpendicular-to-the-\nplane component of the vector field n:\nn=n/bardbl+n⊥,n⊥=eznz=ezcosθ.(3)\nFor convenience, we have included the term ξ0∂tninto\nthe definition of TGD. This term, being even under time\nreversal, leads to a renormalization of spin in the LLG2\nequation16and does not contribute to damping. In what\nfollows, we refer to such renormalization as effective spin\nrenormalization (ESR).\nThe rotational and orientational anisotropy arising in\nEqs. (2) appear to be a natural consequence of the fact\nthat the Rashba spin-orbit interaction singles out the di-\nrection perpendicular to the electron 2D plane. The ori-\nentationalanisotropyofthedimensionlessfunctions ξi(n)\nis determined by all space symmetries of the system and,\nfor a general Rashba FM, may turn out to be rather\ncomplex. However, for the particular interface model of\ntheC∞vsymmetry class, which we consider below, one\nsimply finds ξi=ξi(n2\nz).\nBefore we proceed, let us describe at least two impor-\ntant outcomes of Eqs. (2). First, according to the usual\nconvention, STT consist of two contributions: the adi-\nabatic torque ∝(js·∇)nand the nonadiabatic torque\n∝n×(js·∇)n, wherejsdenotes a spin-polarized cur-\nrent. For vanishing SOC, the adiabatictorque has aclear\nphysical meaning. As far as spins of conduction elec-\ntrons adiabatically follow local magnetization direction,\nthe corresponding change of their angular momentum is\ntransferred to the magnetic texture. Since ↑and↓spins\npoint in the opposite directions along n, the transfer rate\nis proportional to ( js·∇)n, where js=j↑−j↓. In\nthe presence of SOC, however, conduction spins are no\nlonger aligned with the direction of nand, thus, the en-\ntire concept of spin-polarized current becomes somewhat\nvague. For the particular Rashba model, our results re-\nveal an important relation between the adiabatic torque\nand ESR, providing steps toward better understanding\nof the former for systems with SOC.\nAnother remarkable property of Eqs. (2) is a simple\nand exact relation between the nonadiabatic torque and\nGD, which has an important implication for current-\ninduced motion of magnetic textures (e.g., domain walls\nor skyrmions). Indeed, by transforming Eq. (1) into the\nmoving reference frame31r′=r−vdt, one immedi-\nately observes that both components of the nonadiabatic\ntorqueareexactlycancelledbythe correspondingGilbert\ndamping terms. Therefore, if the effect of other driving\ntorques on the motion of a magnetic texture is negligible,\nthen its terminal velocity, in the moving reference frame,\nshall vanish for mediate currents32,33(in the absence of\nmagnetic field). This implies that, in the laboratory ref-\nerence frame, the texture moves with the universal elec-\ntron drift velocity vd. Certainly, in the presence of, e.g.,\nspin-orbit torques, which can assist motion of domain\nwalls and skyrmions10,34, the resulting dynamics might\ndiffer. In any case, the analysisofsuch dynamics can still\nbe performed in the moving reference frame, where the\neffect of the nonadiabatic spin-transfer torque is conve-\nniently absent.\nHaving outlined our main results, we skip further dis-\ncussion until Sec VII. The rest of the paper is organized\nas follows. In Sec. I we introduce the model and use\nan expansion in spatial gradients to reduce the analysis\nto a study of a homogeneous system. Self-energy andKubo formulas are addressed in Sec. II. A general re-\nlation between STT, GD, and ESR (in the considered\nmodel) is obtained in Sec. III, while in Sec. IV we estab-\nlish the exact vector structures of these quantities. Some\nanalytical insight into our general results is provided in\nSec. V and Sec. VI. An extensive Discussion of Sec VII\nis followed by Conclusions (and seven Appendices).\nI. MODEL\nA. Generalized torque in s-dmodel\nIn whatfollows, weadoptthe ideologyofthe s-dmodel\nby performing a decomposition of a FM into a system of\nlocalized spins Siand a system of noninteracting con-\nduction electrons. Despite being rather simplistic, this\napproach has proven to describe very well the key prop-\nerties of current-induced magnetization dynamics in fer-\nromagnetic systems35–38.\nIf the value of |Si|=Scan be assumed sufficiently\nlarge, then it is natural to treat the localized spins clas-\nsically by means of the unit vector n(ri) =Si/S, which\npoints in the opposite to local magnetization direction.\nIn this case, the s-d-like local exchange interaction be-\ntween the localized spins and conduction electrons is\ngiven, in the continuum limit, by\nHsd=JsdSn(r,t)·σ, (4)\nwithJsdquantifying the strength of the exchange and\nPauli matrices σrepresenting the spins of conduction\nelectrons.\nIt is known16that interaction of the form of Eq. (4),\nleads to the following LLG equation for the dynamics of\nthe vector n:\n∂tn=γn×Heff+JsdA\n/planckover2pi1[s(r,t)×n(r,t)],(5)\nwhereγisthebaregyromagneticratio, Heffdescribesthe\neffective magnetic field, Adenotes the areaof the magnet\nunit cell, and s(r,t) stands for the nonequilibrium spin\ndensity of conduction electrons39. The second term on\nthe right hand side of Eq. (5) represents the generalized\ntorque on magnetization\nT=JsdA\n/planckover2pi1[s(r,t)×n(r,t)]. (6)\nAssuming slow dynamics of n(r,t) on the scale of elec-\ntron scattering time and smoothness of magnetization\nprofile on the scale of electron mean free path, one may\nexpand the generalized torque in time and space gradi-\nents ofn. In this paper, we consider twoparticularterms\nof such expansion,\nT=TSTT+TGD+..., (7)\nignoring all other contributions (such as, e.g., spin-orbit\ntorques). In Eq. (7) and below, we identify spin-transfer3\ntorquesTSTTas a double response of Tto the electric\nfieldEandtothespatialgradientsof n, whiletheGilbert\ndamping vector TGD(which also includes the ESR term)\nis defined as a response to the time derivative of n,\nTSTT\nα=/summationdisplay\nβγδTSTT\nαβγδEβ∇γnδ, (8a)\nTGD\nα=/summationdisplay\nδTGD\nαδ∂tnδ. (8b)\nMicroscopic analysis of the tensors TSTTandTGDis the\nmain subject of the present work.\nB. Single particle problem\nAccording to Eqs. (8), the vectors TSTTandTGDrep-\nresent linear response to the time derivative of magne-\ntization direction and to the time derivative of vector\npotential, respectively. Hence, computation of both vec-\ntors can be performed with the help of Kubo formulas\nthat make use of Green’s functions of the correspond-\ning time-independent problem. We choose the latter to\noriginate in the 2D Rashba model40with the effective\ns-d-type term of Eq. (4),\nH=p2/2m+αR[p×σ]z+JsdSn(r)·σ,(9)\nwhereαRcharacterizes the strength of Rashba coupling\nandmis the effective electron mass.\nThe Hamiltonian of Eq. (9) should be supplemented\nwith a momentum relaxation mechanism since both STT\nandGDtensors,similarlytotheconductivitytensor,con-\ntain essentially dissipative components. We assume that\nmomentum relaxation in the system is provided by scat-\ntering on a spin-independent Gaussian white-noise dis-\norder potential Vdis(r). Thus, the full Hamiltonian of a\nsingle conduction electron reads\nHdis=H+Vdis(r), (10)\nwhere the disorder potential is characterized by the zero\naverage∝an}b∇acketle{tVdis(r)∝an}b∇acket∇i}ht= 0 and the pair correlator\n∝an}b∇acketle{tVdis(r)Vdis(r′)∝an}b∇acket∇i}ht= (/planckover2pi12/mτ)δ(r−r′).(11)\nThe angular brackets in Eq. (11) stand for the averaging\nover the disorder realizations, τis the mean scattering\ntime measured in the inverse energy units.\nOne can readily observe from Eq. (6) that the general-\nized torque Tcan be understood as a spatial density of a\ndisorder-averagedmeanvalueoftheoperator( JsdA//planckover2pi1)ˆT,\nwhere we refer to\nˆT=σ×n(r), (12)\nas the dimensionless torque operator.C. Expansion in spatial gradients\nComputation of STT involves the expansion of the\nHamiltonian Hof Eq. (9) and the corresponding Green’s\nfunction\nGR,A= (ε−H±i0)−1(13)\nin the first spatial gradients of nup to the linear terms.\nWe obtain the latter utilizing the Taylor expansion\nn(r) =n(r∗)+/summationdisplay\nγ(r−r∗)γ∇γn(r∗),(14)\nat some particular point r∗.\nWith the help of Eq. (14), Hcan be, then, approxi-\nmated as\nH=H+JsdS/summationdisplay\nγ(r−r∗)γ∇γn(r∗)·σ,(15)\nwhere the Hamiltonian\nH=p2/2m+αR[p×σ]z+JsdSn(r∗)·σ(16)\ndescribes the homogeneouselectronic system with a fixed\ndirection of magnetization set by n(r∗).\nSimilarly, we approximate the Green’s function GR,A,\nemploying the Dyson series\nGR,A(r,r′) =GR,A(r−r′)+JsdS/integraldisplay\nd2r′′GR,A(r−r′′)\n×/bracketleftig/summationdisplay\nγ(r′′−r∗)γ∇γn(r∗)·σ/bracketrightig\nGR,A(r′′−r′) (17)\nand the Green’s function\nGR,A= (ε−H±i0)−1(18)\nthat corresponds to the homogeneous system. Note that,\nin Eq. (17), we kept only the terms that are linear in the\ngradients of n, as prescribed.\nD. Spectrum of the homogeneous system\nThe spectrum of Hincorporates two spectral branches\nε±(p) =p2/2m±/radicalig\n∆2\nsd+(αRp)2−2ςαR∆sdpsinθsinϕ,\n(19)\nwhere the angle θstands for the polar angle of nwith\nrespecttothe zaxis[seealsoEq.(3)], while ϕisthe angle\nbetween the momentum pand the in-plane component\nof the vector n:ϕ=φp−φn. We have also introduced\nthe notations\n∆sd=|Jsd|S, ς = signJsd, (20)\nwhere ∆ sdhas a meaning of half of the exchange\ninteraction-induced splitting (in the absence of SOC).4\nFIG. 1. Guide for an eye: spectrum of the homogeneous sys-\ntem of conduction electrons with a fixed direction of magne-\ntization. Note that the actual spectrum is not isotropic, an d\nthe two subbands may even touch each other. We restrict the\nanalysis to the case of ε >∆sd. For the latter, both subbands\nare always partly filled.\nIf the chemical potential εexceeds the value of ∆ sd,\nboth subbands are always partly filled41. Below, we fo-\ncus solely on the latter case, which is schematically illus-\ntrated in Fig. 1. Note that the spectrum is not isotropic.\nMoreover, for finite values of sin θ, separation of the\ntwo subbands diminishes and they may even touch each\nother.\nIn what follows, we also find it convenient to intro-\nduce the energy scale ∆ so=|αR|√\n2mε, which is equal\nto half of the spin-orbit coupling-induced splitting of the\nbranches (for vanishing ∆ sd).\nE. Roots of dispersion relation\nNow let us analyze the roots of the dispersion of\nEq. (19). Using, for example, Ref. [42], one can show\nthat, under the assumption ε >∆sd, the quartic func-\ntion (ε+(p)−ε)(ε−(p)−ε) of the absolute value of mo-\nmentum palways has four real roots: two positive and\ntwonegative. The former twodefine the angle-dependent\nFermi momenta p±corresponding to ε±branches. The\nfour roots are distinct in all cases, except one. Namely,\nwhenn⊥= 0 (i.e., when sin θ= 1) and ∆ so= ∆sd, the\nsubbands touch each other. We will not consider this\nparticular case.\nUsing the notation p±,negfor the negative roots, we\nhave\np−> p+>0> p+,neg> p−,neg, (21)\nwhere\np∓=1\n2/parenleftbigg√\n2u±/radicalig\n−2u−2q−r/radicalbig\n2/u/parenrightbigg\n,(22a)\np±,neg=1\n2/parenleftbigg\n−√\n2u±/radicalig\n−2u−2q+r/radicalbig\n2/u/parenrightbigg\n,(22b)\nu >0 is the largest root of the resolvent cubic\nu3+qu2−(s−q2/4)u−r2/8, (23)while the parameters q,s, andrare given by\nq=−4m(ε+mα2\nR), s= (2m)2(ε2−∆2\nsd),(24a)\nr= 8m2αRς∆sdsinθsinϕ. (24b)\nIt is straightforward to see, from Eqs. (24), that the\ndependence on the momentum angle enters Eq. (23) only\nvia the parameter r2. As a result, the quantity umay\nonly depend on sin2ϕand other parameters of the model\nthat areϕindependent. This will play an important role\nbelow.\nForαR= 0 (vanishing SOC), ∆ sd= 0 (nonmagnetic\nlimit), or n=n⊥(perpendicular-to-the-plane magneti-\nzation) situation with the rootsbecomes less complex. In\nthese cases, ( ε+(p)−ε)(ε−(p)−ε) is biquadratic (with\nrespect to p) andp±=−p±,neg, as one can also see di-\nrectly from Eqs. (22). Furthermore, the Fermi momenta\np±, then, are angle independent, while their values yield\nthe relations\np2\n±= 2m[ε∓∆sd],forαR= 0,(25a)\np2\n±= 2m/bracketleftbig\nε+mα2\nR∓λ(0)/bracketrightbig\n,for ∆sd= 0,(25b)\np2\n±= 2m/bracketleftbig\nε+mα2\nR∓λ(∆sd)/bracketrightbig\n,forn=n⊥,(25c)\nwhereλ(Υ) =/radicalbig\nΥ2+2εmα2\nR+m2α4\nR.\nII. DISORDER AVERAGING\nHaving analysed the spectrum of the “clean” homoge-\nneous system, we can proceed with the inclusion of the\ndisorder. In what follows, we assume ε0τ≫1, where\nε0is the difference between the Fermi energy εand the\nclosest band edge. We start with a calculation of the\nself-energy in the first Born approximation.\nA. Self-energy\nAccording to Eq. (11), the self-energy is defined as\nΣR,A(r) = (/planckover2pi12/mτ)GR,A(r,r), (26)\nwith the Green’s function GR,Aof Eq. (13). It should be\nexplicitly pronounced that ΣR,A(r) may have a spatial\ndependenceoriginatinginthespatialdependenceof n(r).\nHowever, as we are about to see, the first spatial gradi-\nents of magnetization do not affect the self-energy in the\nmodel under consideration.\nDisregarding the “real” part of the self-energy that\nshould be included in the renormalized value of the\nchemical potential, we focus only on the calculation of\nImΣ(r) =−i[ΣR(r)−ΣA(r)]/2. By substituting the\nexpansion of Eq. (17) into Eq. (26), switching to mo-\nmentum representation, and symmetrizing the result we\nobtain\nImΣ(r) = Σ(0)+/summationdisplay\nγδ/braceleftig\n(r−r∗)γΣ(1)\nδ+Σ(2)\nγδ/bracerightig\n∇γnδ(r∗),\n(27)5\nwith\nΣ(0)=1\n2imτ/integraldisplayd2p\n(2π)2/parenleftbig\nGR−GA/parenrightbig\n,(28a)\nΣ(1)\nδ=ς∆sd\n2imτ/integraldisplayd2p\n(2π)2/parenleftig\nGRσδGR−GAσδGA/parenrightig\n,(28b)\nΣ(2)\nγδ=ς∆sd/planckover2pi1\n4mτ/integraldisplayd2p\n(2π)2/parenleftig\nGRσδGRvγGR−\nGRvγGRσδGR+h.c./parenrightig\n,(28c)\nwhere “h.c.” denotes Hermitian conjugate, GR,Ais the\nGreen’s function of Eq. (18) in momentum representa-\ntion,\nGR,A=ε−p2/2m+αR[p×σ]z+ς∆sdn(r∗)·σ\n(ε−ε+(p)±i0)(ε−ε−(p)±i0),(29)\nandv=∂H/∂pis the velocity operator. In Eqs. (28),\nΣ(0)defines the scattering time (for uniform magneti-\nzation), Σ(1)corresponds to the renormalization of the\ngradient term on the right hand side of Eq. (15), while\nΣ(2)determinesthe possible dependence ofthe scattering\ntime on the first spatial gradients of magnetization.\nTo proceed, we take advantage of the additional sym-\nmetrization of the integrands with respect to the trans-\nformation43ϕ→π−ϕand observe that, in the first\nBorn approximation, integration over the absolute value\nof momentum, in Eqs. (28), is reduced to a calculation\nof residues at p=p±. Using Eqs. (22), we, then, get\nΣ(0)=−1\n2τ/integraldisplay2π\n0dϕ\n2π/bracketleftbig\n1+rW1+rW2n(r∗)·σ\n+W3n/bardbl(r∗)·σsinϕ/bracketrightbig\n,(30)\nwhereWi=Wi/parenleftbig\nr2,u(r2)/parenrightbig\nare some functions of the pa-\nrameterr2andϕ-independent parameters of the model.\nSincer∝sinϕand, obviously, all integrals of the form/integraltext2π\n0W(sin2ϕ)sinϕdϕvanish for arbitrary function W,\nwe obtain a particularly simple result for the constant\npart of the self-energy,\nΣ(0)=−1/2τ. (31)\nSimilar, but more lengthy, analysis shows that each\ncomponent of Σ(1)and Σ(2)is equal to zero. Therefore,\nthereexistsnorenormalizationofthegradienttermofthe\nHamiltonian Has well as no scattering time dependence\non the first magnetization gradients. The self-energy, in\nthe first Born approximation, is found as\nΣR,A(r) =∓i/2τ. (32)\nB. Kubo formula for STT\nAs was outlined in Sec. IB, the generalized torque\nT(r0) of Eq. (6), at a certain position r0in space, is de-\nfined as a disorder-averaged mean value of the operatorFIG.2. Diagrammatic representationoftheSTTtensor TSTT\nαβγδ\nof Eq. (34). Solid lines correspond to the disorder-average d\nGreen’s functions gR,A. Vertex corrections (impurity ladders)\nare represented by green fillings.\n(JsdA//planckover2pi1)δr0ˆT, whereδr0=δ(r−r0). At zero tempera-\nture, thelinearresponse44ofTα(r0)tothezerofrequency\nelectric field Eis given by the standard Kubo expression\ne/planckover2pi1\n2πJsdA\n/planckover2pi1/angbracketleftig\nTr/bracketleftig\nGAδr0ˆTαGRv/bracketrightig\nE/angbracketrightig\n,(33)\nwherev=∂H/∂pis the velocity operator, Tr stands for\nthe operator trace, and angular brackets represent the\ndisorder averaging.\nFrom Eq. (33), we can further deduce the Kubo for-\nmula for spin-transfer torques. In order to do that, we\nsubstitutetheexpansionofEq.(17)intoEq.(33)andcol-\nlect all terms proportional to ∇γnδ(r∗). Then we switch\ntomomentum representationandperformspatialaverag-\ning of torque on the scale of transport mean free path in\nthe vicinity of r=r0. In the noncrossingapproximation,\nthis leads to the general formula for the STT tensor,\nTSTT\nαβγδ=e∆2\nsdA\n2π/planckover2pi1S/integraldisplayd2p\n(2π)2\n×itr/bracketleftig\ngAσδgAvγgAˆTvc\nαgRvvc\nβ−h.c./bracketrightig\n,(34)\nwhere the superscript “vc” marks the vertices corrected\nwith the impurity ladders, the notation tr refers to the\nmatrix trace, and\ngR,A=∝an}b∇acketle{tGR,A∝an}b∇acket∇i}ht= (ε−H±i/2τ)−1(35)\nis the disorder-averaged Green’s function of the homoge-\nneous system. In Eq. (35), we have used the result for\nthe self-energy obtained in Sec. IIA.\nThe expression of Eq. (34) is represented diagrammat-\nically in Fig. 2. We note that similar diagrams have been\nused in Ref. [45] to compute STT in a 3D FM, in the\nabsence of SOC, and in Ref. [46] to study STT for the\nmodel of massive Dirac fermions.\nC. Kubo formula for GD and ESR\nSimilarly, from the zero frequency linear response44of\nTα(r0) to the time derivative of n,\nJsdS/planckover2pi1\n2πJsdA\n/planckover2pi1/angbracketleftig\nTr/bracketleftig\nGAδr0ˆTαGRσ/bracketrightig\n∂tn/angbracketrightig\n,(36)6\nonemayderivethe formulaforthe GDtensorofEq.(8b),\nTGD\nαδ=∆2\nsdA\n2π/planckover2pi12S/integraldisplayd2p\n(2π)2tr/bracketleftig\ngAˆTvc\nαgRσδ/bracketrightig\n,(37)\nwhere, according to the definition of TGD, spatial depen-\ndence of nis completely disregarded.\nNote that n,∇γnδ, and∂tnin Eqs. (8), (34), and (37)\nare all taken at r=r0. From now on, we consistently\nomit the argument of all these functions.\nD. Relation between TGDand vertex corrections\nto the torque operator ˆT\nVertex corrected torque operator that enters both\nEqs. (34) and (37) can be expressed with the help of\nvertex corrected Pauli matrices. One can infer the latter\nfrom the “matrix of one dressing” M, whose elements\nMij=1\n2mτ/integraldisplayd2p\n(2π)2tr/bracketleftig\ngAσigRσj/bracketrightig\n(38)\narethe coordinates(in the basis {σx,σy,σz}) ofthe oper-\natorσidressed with a single impurity line. We note that,\nin the model considered, vertex corrected Pauli matrices\nσvc\niappear to have zero trace if ε >∆sd. This is a direct\nconsequence of the fact that the self-energy in Eq. (32) is\nscalar. Hence, {σx,σy,σz}is, indeed, a proper basis for\nthe operators σvc\ni.\nMatrixrepresentationofthe operator ˆT=σ×n, with\nrespect to this basis, is defined as\nˆTi=/summationdisplay\njUijσj, U=\n0nz−ny\n−nz0nx\nny−nx0\n.(39)\nSince, obviously,\nˆTvc\ni=/summationdisplay\njUijσvc\nj, (40)\nwe can see, from Eq. (38), that the geometric series\nT=U(M+M2+···) =UM(I−M)−1,(41)\nprovides the matrix representation of vertex corrections\nto the torque operator. Moreover, from Eq. (37), it is\nevident that the GD tensor is, in fact, determined by the\nsame matrix T,\nTGD\nαδ=∆2\nsdAmτ\nπ/planckover2pi12STαδ. (42)\nE. Crossing diagrams\nIt has been demonstrated recently that the diagrams\nwith two crossing impurity lines may contribute to such\nquantitiesasthe anomalousHall effect47–49, the spinHalleffect50, and the Kerr effect51in the same leading or-\nder with respect to the small parameter ( ε0τ)−1, as the\nconventionalnoncrossingapproximationdoes. Scattering\nmechanisms associated with these diagrams, in general,\nshould affect spin torques and damping as well.\nIn the presentstudy we, however,completely disregard\nthe crossing diagrams, as being significantly more diffi-\ncult to calculate. At the same time, preliminary anal-\nysis shows that the related additional contributions to\nSTT, GD, and ESR are parametrically different from the\npresent resultsand that, for ε≫∆sd, they are negligible.\nIII. RELATION BETWEEN STT, GD, AND ESR\nA. Symmetrization of STT diagrams\nCalculation of spin-transfer torques can be performed\nwith the help of Eq. (34) directly. Such brute-force cal-\nculation has been originally performed by us. We have,\nhowever, subsequently found a shortcut that makes it\npossible not only to obtain the same results in a much\nmore concise manner but also to establish a general re-\nlation between TSTTandTGDtensors. This alternative\napproach takes a reformulation of the result of Eq. (34)\nin a more symmetric form.\nWe apply the identity gAvγgA=∂gA/∂pγin Eq. (34)\nand perform integration by parts. Then, we take a half-\nsum of the result obtained and the original expression of\nEq. (34). This leads to the formula\nTSTT\nαβγδ=δTSTT\nαβγδ+e∆2\nsdA\n2π/planckover2pi1S/integraldisplayd2p\n(2π)2\n×i\n2tr/bracketleftbigg\n−gAσδgAˆTvc\nαgR∂vvc\nβ\n∂pγ−h.c./bracketrightbigg\n,(43)\nwhere the first term on the right-hand side\nδTSTT\nαβγδ=e∆2\nsdA\n2π/planckover2pi1S/integraldisplayd2p\n(2π)2\ni\n2tr/bracketleftig\ngAσδgAvγgAˆTvc\nαgRvvc\nβ−gAvγgAσδgAˆTvc\nαgRvvc\nβ\n−gAσδgAˆTvc\nαgRvγgRvvc\nβ−h.c./bracketrightig\n.(44)\nis illustrated schematically in Fig. 3 by a group of en-\ncircled diagrams. The remaining two diagrams in Fig. 3\ncorrespond to the second term on the right-hand side of\nEq.(43). We will seebelowthat, in fact, the entiretensor\nδTSTTdoes vanish.\nB. Relation between TSTTand vertex corrections\nto the torque operator ˆT\nAs was argued in Ref. 52 on the basis of perturbative\nexpansions, the velocity operator v=p/m−αR[ez×σ],7\nFIG. 3. Another diagrammatic representation of the STT tens orTSTT\nαβγδ, given by Eq. (43). Six diagrams encircled by the\ndashed line define the δTSTT\nαβγδtensor of Eq. (44) that vanishes for any direction of nprovided ε >∆sd. Solid lines correspond\nto the disorder-averaged Green’s functions gR,A. Vertex corrections (impurity ladders) are represented by green fillings.\ncorrectedbyanimpurityladder,hasaparticularlysimple\nform in the present model,\nvvc=p/m. (45)\nA formal proof of this statement that does not refer to\nany perturbative expansion is presented in Appendix A.\nInterestingly, Eq. (45) also allows to make a spin-orbit\ntorque (SOT) calculation extremely concise. We provide\na brief discussion of this matter in the same Appendix A.\nIt is important that the momentum operator p, as well\nasvvc, commutes with the Green’s function gR,A. In Ap-\npendix B, we demonstrate that this is sufficient for the\nentire tensor δTSTTto vanish. As a result, TSTTis de-\ntermined by the second term on the right hand side of\nEq. (43) alone. Computation of the this term is facili-\ntated by the relation\n∂vvc\nβ/∂pγ=δβγ/m, (46)\nwhereδq1q2is Kronecker delta. With the help of the\nabove, the STT tensor of Eq. (43) readily simplifies to\nTSTT\nαβγδ=δβγe∆2\nsdA\n2π/planckover2pi1Sm/integraldisplayd2p\n(2π)2\n×i\n2tr/bracketleftig\n−gAσδgAˆTvc\nαgR−h.c./bracketrightig\n,(47)\nsince, as we have mentioned, δTSTT= 0.\nEmploying the Hilbert’s identity for the Green’s func-\ntions of Eq. (35),\ngA−gR=gR(i/τ)gA, (48)\nwe can further reduce44Eq. (47) to the formula\nTSTT\nαβγδ=δβγe∆2\nsdAτ\n2π/planckover2pi1Sm/integraldisplayd2p\n(2π)2tr/bracketleftig\ngAˆTvc\nαgRσδ/bracketrightig\n,(49)which resembles very closely the formula of Eq. (37) for\nthe GD tensor. The result of Eq. (49) can also be ex-\npressed in terms of the matrix Tas\nTSTT\nαβγδ=δβγe∆2\nsdAτ2\nπ/planckover2pi1STαδ, (50)\nwhere we have again used the argumentationof Sec. IID.\nC. Relation between TSTTandTGD\nIt can now be seen that both TSTTandTGDvectors\nturn out to be fully defined by the matrix of vertex cor-\nrectionsTto the torque operator. Moreover, comparison\nof Eq. (42) and Eq. (50) reveals a remarkable direct con-\nnection between the STT and GD tensors,\nTSTT\nαβγδ=δβγe/planckover2pi1τ\nmTGD\nαδ, (51)\nwhich is one of the central results of the paper.\nAccordingtothe definitionsofEqs.(8), the established\nrelation between the two tensors indicates that all quan-\ntities of interest (STT, GD, and ESR) may be related to\nthe action of a single linear operator Ξ,\nTSTT= Ξ[∂vn],TGD= Ξ[∂tn],(52)\non one of the vectors, ∂vnor∂tn. We remind here the\nshort-handed notations for the directional spatial deriva-\ntive53∂v= (vd·∇) and for the classical drift velocity of\nconduction electrons vd=eE/planckover2pi1τ/m.\nThe matrix of the operator Ξ coincides with the ma-\ntrixTGD, being also proportional to the matrix T[see\nEqs. (8b), and (42)]. In the next section we obtain the\ngeneral form of the latter and then use it to derive the\nexact vector forms of TSTTandTGD.8\nIV. VECTOR FORMS\nA. Matrix gauge transformation\nIn order to establish the structure of the operator Ξ,\nit should be first noted that the constraint n2≡1 is\nresponsibleforanessentialfreedominthedefinition of T.\nFor an arbitrary operator of differentiation ∂, we have\n1\n2∂n2=/summationdisplay\nδnδ∂nδ= 0. (53)\nTherefore, the left hand sides of\nTSTT\nα=∆2\nsdAmτ\nπ/planckover2pi12S/summationdisplay\nδTαδ∂vnδ,(54a)\nTGD\nα=∆2\nsdAmτ\nπ/planckover2pi12S/summationdisplay\nδTαδ∂tnδ,(54b)\nremain invariant under the addition of the matrix row\nR= (nx,ny,nz), with an arbitrary coefficient, to any of\ntherowsofthematrix T. Inotherwords,thetransforma-\ntionT → T Xdoes not change TSTTandTGD, provided\nTX=T+XR, (55)\nwith any matrix column X= (X1,X2,X3)T.\nB. Vector structure of TSTTandTGD\nThe matrix Tis defined in Eq. (41) with the help of\nthe matrix M. The latter is determined by the disorder-\naveraged Green’s function which, in momentum repre-\nsentation, takes the form\ngR,A=ε±i/2τ−p2/2m+αR[p×σ]z+ς∆sdn·σ\n(ε−ε+(p)±i/2τ)(ε−ε−(p)±i/2τ).\n(56)\nUsing Eq. (56), one can prove that M, in general, is\nexpressed as a linear combination of six matrices,\nI, P, U, U2, P UP, P U2P, (57)\nwhereUis introducedin Eq.(39) and P= diag(1 ,1,0)is\na diagonal matrix. In Appendix C, we demonstrate how\nthe components of this decomposition can be calculated\nforn∝ne}ationslash=n⊥.\nThen, in Appendix D, we show that any power of M\nretains the same structure. It immediately follows that\nthe matrix T=U(M+M2+···) can be represented as\nT=c1U+c2UP+c3U2+c4U3+c5UP UP+c6UP U2P,\n(58)\nwhereciare some dimensionless scalar functions.\nThe representation of Eq. (58) can be substantially\nsimplified with the use of the matrix gauge transforma-\ntion described in the previous section. Namely, by takingadvantage of the directly verifiable relations\nU2=RTR−I, U3=−U, (59a)\nUP UP= (I−P)RTR−n2\nzI, (59b)\nUP U2P=UPRTR−UP+n2\nzU(I−P) (59c)\nwe find that the choice of the gauge\n/tildewideX=−[c3I+c5(I−P)+c6UP]RT,(60)\nfor the transformation T → T /tildewideX≡/tildewideT, leads to\n/tildewideT=t0I+t/bardblUP+t⊥U(I−P),(61)\nor, more explicitly, to\n/tildewideT=\nt0nzt/bardbl−nyt⊥\n−nzt/bardblt0nxt⊥\nnyt/bardbl−nxt/bardblt0\n, (62)\nwhere the quantities tiare related to the matrix Tby\nmeans of the relations\nt0=−c3−c5n2\nz, (63a)\nt/bardbl=c1+c2−(c4+c6), (63b)\nt⊥=c1−c4+c6n2\nz. (63c)\nReplacing Twith/tildewideTin Eqs. (54),\nTSTT\nα=∆2\nsdAmτ\nπ/planckover2pi12S/summationdisplay\nδ/tildewideTαδ∂vnδ,(64a)\nTGD\nα=∆2\nsdAmτ\nπ/planckover2pi12S/summationdisplay\nδ/tildewideTαδ∂tnδ,(64b)\nwe observe that the operator Ξ in Eq. (52) is represented\nby three dimensionless quantities ξ0,ξ/bardbl,ξ⊥, such that\nξi=∆2\nsdAmτ\nπ/planckover2pi12Sti, (65)\nwhile the vector structure of TSTTandTGDis, indeed,\nprovided by the formulas\nTSTT=ξ0∂vn−ξ/bardbl[n×∂vn/bardbl]−ξ⊥[n×∂vn⊥],\nTGD=ξ0∂tn−ξ/bardbl[n×∂tn/bardbl]−ξ⊥[n×∂tn⊥],\nannouncedin the introductorypart. With someremarks,\nthey remain valid for n=n⊥as well. We consider this\nspecific case separately, in Sec. VB.\nIn the next section, we derive closed-form results for\nξ0,ξ/bardbl, andξ⊥, in two particular regimes. Afterwards, we\nfind asymptotic expansions of these functions in either\nsmallαRor in small ∆ sd. All the obtained results are\ncollected in Table I and represented in Fig. 4 alongside\nwith the corresponding numerical curves.9\nV. CLOSED-FORMS\nThe analysis of TSTTandTGDtensors, as has been\npointed out, reduces to integration in Eq. (38) and sub-\nsequent matrix arithmetics. Unfortunately, for arbitrary\ndirection of magnetization, the results cannot be ex-\npressed in terms of elementary functions. For example,\nforn⊥= 0, Eq. (38) already involves elliptic integrals.\nThe complexity is caused, primarily, by the angle de-\npendence of the dispersion relation roots p±,p±,negof\nEqs. (22). Additional complications arise due to the fact\nthat all four roots are distinct.\nOn the other hand, if the parameter rdefined in\nEq. (24b) vanishes, then the angle dependence of p±,\np±,negis absent and, furthermore, p±=−p±,neg(see\nalso Sec. IE). In this case, angle integration in Eq. (38)\nis trivial, while integration over the absolute value pof\nmomentum can be replaced with an integration over p2.\nFor such integrals, we can extend the integration contour\nto−∞and close it through the upper half-plane. Then\nthe value of the integral is given by a sum of residues at\nthep2\n±poles of Eqs. (25) that acquire finite imaginary\nparts due to a ε→ε+i/2τshift.\nHence, computation of the matrix Mis straightfor-\nward when αR= 0, ∆ sd= 0, orn=n⊥. In this section,\nwe calculate ξ0,ξ/bardbl, andξ⊥, for the first and third cases.\nIn the next section, we use the first two cases as reference\npoints for perturbative analysis of these functions.\nA. Vanishing spin-orbit coupling\nWewillstudythecaseof αR= 0first. Inthe absenceof\nSOC, conservation of spin brings a technical difficulty to\nthe calculation of T. Namely, at zero frequency and zero\nmomentum, the matrix of disorder-averaged advanced-\nretarded spin-spin correlators M(I− M)−1that enters\nEq. (41) cannot be finite. Indeed, using the formulas of\nAppendix C with αR= 0, one finds\nM=I−2ςτ∆sd\n1+(2τ∆sd)2U(I−2ςτ∆sdU),(66)\nso thatI− Mis proportional to U. But det U= 0\nand, therefore, M(I− M)−1=∞. Physically, this di-\nvergence is caused by the absence of linear response of\nelectron spins polarized along nto time-dependent ho-\nmogeneous perturbations of Jsd(cf. Sec. 8.3 in Ref. 54).\nNevertheless, even in the limit of zero momentum and\nzero frequency, STT, GD, and ESR remain finite, since\nthe series\nT=UM+UM2+UM3+... (67)\nactually converges.\nThe sum in Eq. (67) is most easily calculated in the\ndiagonal representation of U,\nU=VUdiagV†, U diag= diag(i,−i,0),(68)which is defined by the unitary matrix\nV=\niny−nxnz√\n2(n2x+n2y)−iny+nxnz√\n2(n2x+n2y)nx\n−inx+nynz√\n2(n2x+n2y)inx−nynz√\n2(n2x+n2y)ny√\nn2x+n2y√\n2√\nn2x+n2y√\n2nz\n.(69)\nIntroducing MU=V†MVand making use of the rela-\ntionUdiag=UdiagP, to take care of the potential diver-\ngence, we can rewrite Eq. (67) as\nT=VUdiag(PMU+PM2\nU+PM3\nU+...)V†,(70)\nwhere, according to Eqs. (66) and (68),\nPMk\nU= diag/parenleftig\n[1+2iςτ∆sd]−k,[1−2iςτ∆sd]−k,0/parenrightig\n.\n(71)\nSummation in Eq. (70) is trivially performed, leading to\nT=−ς\n2τ∆sdVU2\ndiagV†=−ς\n2τ∆sdU2=\nς\n2τ∆sd/parenleftbig\nI−RTR/parenrightbig\n=/tildewideT −/tildewideXR,(72)\nwhere/tildewideT= (ς/2τ∆sd)Irepresents the gauge of Eq. (61)\nand we have used the first identity of Eq. (59a).\nThe above result clearly corresponds to t0=ς/2τ∆sd\nandt/bardbl=t⊥= 0, or\nξ0=ς∆sdAm\n2π/planckover2pi12S, ξ /bardbl=ξ⊥= 0. (73)\nHence, Gilbert damping and the nonadiabatic spin-\ntransfertorqueareboth absentwhen αR= 0, asit should\nbe in the model with no SOC, spin-dependent disorder,\nor other sources of spin relaxation.\nThe parameter ξ0defines the effective spin renormal-\nization(duetoconductionelectrons)intheLLGequation\nas16ξ0=−δSeff/S. In fact, for αR= 0, the effective spin\nrenormalization coincides with actual spin renormaliza-\ntion. Indeed, without SOC, all electrons are polarized\nalong±n, and, for the calculation of the total electron\nspin in a unit cell,\nδS=δS↑−δS↓=ς\n2(N+−N−) =\nςA\n8π2/planckover2pi12\n/integraldisplay\nε+(p)≤εpdpdφ p−/integraldisplay\nε−(p)≤εpdpdφ p\n,(74)\none may use ε±(p)≤ε⇔p2≤2m(ε∓∆sd) to obtain\nδS=−ς∆sdAm\n2π/planckover2pi12. (75)\nThus,δS=−ξ0S=δSeffin this case.\nIn Appendix E, we compute spin susceptibility of the\nsystem for αR∝ne}ationslash= 0 and demonstrate that the spin renor-\nmalization does not depend on the SOC strength. At the10\nsametime, the effectivespinrenormalizationdoes. More-\nover, the identity δSeff=δSis, in fact, a very specific\ncase. It holds either for vanishing spin-orbit interaction,\noratsomeparticularvalueof∆ so≈∆sd, asonecanlearn\nfrom Table I and Fig. 4 (we recall that ∆ so=|αR|√\n2mε\ncharacterizes the SOC-induced splitting of the spectral\nbranches).\nB. Perpendicular-to-the-plane magnetization\nNow we turn to the n=n⊥regime. The formulas of\nAppendix C arenot applicable in this case. Nevertheless,\none can perform the integration in Eq. (38) directly, uti-\nlizing the expression for the Green’s function of Eq. (56)\nwith sinθ= 0 (and n=ezcosθ). It follows that\nM=/bracketleftbig\n1+4τ2(∆2\nsd+∆2\nso)/bracketrightbig−1/parenleftig/bracketleftbig\n1+2(τ∆so)2/bracketrightbig\nP\n+/bracketleftbig\n1+4(τ∆sd)2/bracketrightbig\n(I−P)−2ςτ∆sdP UP/parenrightig\n(76)\nand, after some arithmetic,\nT=ς\n2τ∆sd/bracketleftigg\n1−/parenleftbig\nτ∆2\nso/parenrightbig2\n∆2\nsd+τ2(2∆2\nsd+∆2so)2/bracketrightigg\nP\n+1\n2/bracketleftigg\n∆2\nso/bracketleftbig\n1+2τ2(2∆2\nsd+∆2\nso)/bracketrightbig\n∆2\nsd+τ2(2∆2\nsd+∆2so)2/bracketrightigg\nP UP.(77)\nSubstitution of this result into Eqs. (54) shows that, in\nthis case, both TSTTandTGDare represented as linear\ncombinations of two vector forms: ∂n/bardblandn⊥×∂n/bardbl.\nSincen=n⊥and, thus, ∂n⊥= 0, the coefficients in\nfront of these forms should be recognized as t0andt/bardbl,\nrespectively. With the help of Eq. (75), we, therefore,\nfind\nξ0=−δS\nS/bracketleftigg\n1−/parenleftbig\nτ∆2\nso/parenrightbig2\n∆2\nsd+τ2(2∆2\nsd+∆2so)2/bracketrightigg\n,(78a)\nξ/bardbl=/vextendsingle/vextendsingle/vextendsingleδS\nS/vextendsingle/vextendsingle/vextendsingleτ∆sd/bracketleftigg\n∆2\nso/bracketleftbig\n1+2τ2(2∆2\nsd+∆2\nso)/bracketrightbig\n∆2\nsd+τ2(2∆2\nsd+∆2so)2/bracketrightigg\n.(78b)\nFor a fixed n=n⊥, however, one cannot directly de-\nfineξ⊥. Indeed, the latter function, in this case, is a\nprefactor in front of the vanishing vector form n×∂n⊥\nand, in principle, can be even taken arbitrary. The only\nway to assign a clear meaning to ξ⊥, here, is to consider\nitsasymptoticbehaviouratsmallvaluesofsin θ. Namely,\none should expand the integrands in Eq. (38) up to sin2θ\nand, after the integration, compute the coefficients of the\ndecomposition of Eq. (58) with the same accuracy. Ap-\nplication of a sin θ→0 limit in Eq. (63c), afterwards,\nwill lead to\nξ⊥=/vextendsingle/vextendsingle/vextendsingleδS\nS/vextendsingle/vextendsingle/vextendsingleτ∆sd/bracketleftigg\n1\n2∆2\nso/bracketleftbig\n1+(2τ∆sd)2/bracketrightbig\n∆2\nsd+τ2(2∆2\nsd+∆2so)2/bracketrightigg\n.(79)One may use Eqs. (78b) and (79) to evaluate the\nstrengthoftherotationalanisotropyofGDandthenona-\ndiabatic STT, given n≈n⊥. We see, for example, that,\nfor small sin θ, the ratio\nξ/bardbl/ξ⊥= 2+∆2\nso\n∆2\nsd+(1/2τ)2+O(sin2θ),(80)\nexceeds 2, making the rotational anisotropy considerable\neven if SOC is weak. At the same time, for strong spin-\norbit coupling, ξ/bardblcan potentially be orders of magnitude\nlarger than ξ⊥(see also Fig. 4).\nFor the perpendicular-to-the-plane magnetization, GD\nwas analyzed previously in Ref. [55] under an additional\nassumption of large chemical potential. Our result for\nthe Gilbert damping coefficient ξ/bardbl, given by Eq. (78b),\ncoincides with the expression on the right hand side of\nEq. (25) of Ref. [55], to an overall factor that we were\nunable to identify (most likely, it is equal to 4). The\nτ→ ∞limit of the same expression was derived recently\nin Ref. [56] (with another overall factor). This paper also\nmentions the role of the diagonal terms of the GD tensor\non ESR.\nA separate study of the nonadiabatic STT (also lim-\nited to the n=n⊥case) was reported in Ref. [57]. As\nwe have shown above, this torque should be fully de-\ntermined by the very same function ξ/bardblas is GD. The\nauthors, however, ignored vertex corrections, and, as it\nseems, overlooked this fact. In any case, their results\ndiffer from those of Eq. (78b).\nVI. ASYMPTOTIC EXPANSIONS\nWe proceed with a calculation of the ξiexpansions\nin either small αRor small ∆ sd. To perform such cal-\nculation, one should expand the integrands in Eq. (38)\nor, alternatively, in Eqs. (C2), with respect to the corre-\nspondingvariable. Thentheresultcanbeintegratedover\nthe poles, provided by Eqs. (25a) and (25b), respectively\n(whereεshould be replaced with ε+i/2τ).\nA. Weak spin-orbit coupling\nKeeping the notation of Sec. VA for the matrices M\nandTin the absence of SOC, below we use the symbols\nδMandδTtorepresenttherespectivecontributionspro-\nvided by finite αR.\nSinceδM ∝ne}ationslash= 0, the result of matrix inversion in\nT+δT=U(M+δM)(I−M−δM)−1(81)\nis finite, making the analysis straightforward yet rather\ncumbersome. Retaining only proportionalto α2\nRterms in\nδM(see Appendix F for explicit formulas), we obtain\nδT=δc2P+δc3U+δc4U2+..., (82)11\nξ0/(−δS\nS) orδSeff/δS ξ/bardbl/(|δS\nS|τ∆sd) ξ⊥/(|δS\nS|τ∆sd)\nαR= 0 1 0 0\nO(∆2\nso)1+2(τ∆so)2\n1+(2τ∆sd)21−n2\nz\n1+n2z(∆so/∆sd)2\n1+(2τ∆sd)2/bracketleftbigg\n(2τ∆sd)2+2\n1+n2z/bracketrightbigg(∆so/∆sd)2\n1+(2τ∆sd)21+(2nzτ∆sd)2\n1+n2z\n∆sd→0/parenleftbigg∆sd\n∆so/parenrightbigg2/bracketleftbigg\n4n2\nz+1+n2\nz\n2(τ∆so)2/bracketrightbigg\n2+1\n(τ∆so)21\n2(τ∆so)2\nn=n⊥1−(τ∆2\nso)2\n∆2\nsd+τ2(2∆2\nsd+∆2so)2∆2\nso/bracketleftbig\n1+2τ2(2∆2\nsd+∆2\nso)/bracketrightbig\n∆2\nsd+τ2(2∆2\nsd+∆2so)21\n2∆2\nso/bracketleftbig\n1+(2τ∆sd)2/bracketrightbig\n∆2\nsd+τ2(2∆2\nsd+∆2so)2\nTABLE I. Closed-form results and asymptotic expansions for the dimensionless functions ξ0,ξ/bardbl, andξ⊥that define anisotropic\nspin-transfer torques, Gilbert damping, and effective spin renormalization. The results are expressed in terms of the e nergy\nscales ∆ sd=|Jsd|Sand ∆ so=|αR|√\n2εmthat describe, respectively, the exchange and spin-orbit- induced splitting. The second\nrow shows the expansion up to the second order in ∆ so. The third row provides the leading order terms of the expans ion with\nrespect to small ∆ sd. Spin renormalization is defined in Eq. (75) by δS=−JsdSAm/2π/planckover2pi12.\nwhere dots represent terms that do not contribute to the\nδ/tildewideTgauge in the α2\nRorder and\nδc2=∆2\nso\n2∆2\nsd1\n1+n2z, (83a)\nδc3=−τ∆2\nso\nς∆sd/bracketleftig\n1+(2τ∆sd)2/bracketrightig1−n2\nz\n1+n2z,(83b)\nδc4=−∆2\nso\n2∆2\nsd1+(2nzτ∆sd)2\n1+(2τ∆sd)21\n1+n2z.(83c)\nThen, utilizingEqs.(63)with cireplacedby δci,wearrive\nat the second-orderexpansionsin small SOC strength for\nthe functions ξi. Those are collected in the second row\nof Table I.\nWe may again use the obtained results to quantify the\nrotational anisotropy of GD and the nonadiabatic STT\nby computing the ratio\nξ/bardbl/ξ⊥= 2+1−n2\nz\nn2z+1/(2τ∆sd)2+O(∆2\nso).(84)\nFor weak spin-orbit coupling, the rotational anisotropy\nis minimal when magnetization is perpendicular to the\nplane and increases for the magnetization approaching\nthe in-plane direction.\nWe also note that the asymptotic expansions up to the\norderα2\nRallowustoestimatetheorientationalanisotropy\nofξi. Employing the notation ξi=ξi(n2\nz), we find\nξ0(0)−ξ0(1) =2(τ∆so)2\n1+(2τ∆sd)2, (85a)\nξ/bardbl(0)−ξ/bardbl(1) =1\n1+(2τ∆sd)2∆2\nso\n∆2\nsd, (85b)\nξ⊥(0)−ξ⊥(1) =1−(2τ∆sd)2\n1+(2τ∆sd)2∆2\nso\n2∆2\nsd,(85c)\nfor weak SOC. Clearly, ξ0andξ/bardblare both maximal for\nn⊥= 0. On the other hand, the expression on the righthand side of Eq. (85c) can change sign, depending on the\nvalue of τ∆sd. Therefore, the orientational anisotopy of\nξ⊥in a “clean” system ( τ∆sd≫1) differs from that in a\n“dirty” one (Fig. 4 corresponds to the case of a “clean”\nsystem).\nInterestingly, at αR= 0 the matrix function δTturns\nout to be discontinuous. Namely, its elements have fi-\nnite limits for αR→0. This discontinuity has, however,\nno physical consequences, since the matrix δTitself is\nnot gauge invariant. In the δ/tildewideTgauge, the discontinuity\nis removed and, thus, it does not affect the physically\nrelevant quantities ξ0,ξ/bardbl, andξ⊥. This property demon-\nstrates the importance of full analysis of all components\nof the STT and GD tensors.\nB. Weak exchange interaction\nUp to the linear order in ∆ sd, we have\nM=I+2(τ∆so)2P\n1+4(τ∆so)2−2ςτ∆sdU+4(τ∆so)2P UP\n[1+4(τ∆so)2]2.\n(86)\nThis corresponds to the following coefficients of the de-\ncomposition of Eq. (58),\nc1=1\n4(τ∆so)2, c2= 1+1\n4(τ∆so)2, (87a)\nc3=−ςτ∆sd\n4(τ∆so)4, c4= 0, (87b)\nc5=−ςτ∆sd/bracketleftbig\n1+8(τ∆so)2/bracketrightbig\n4(τ∆so)4, c6= 0.(87c)\nSubstituting the latter expressions into Eqs. (63), one\nobtains the leading-order contributions to ξiin the limit\nof small ∆ sd. The respective results are presented in the\nthird row of Table I. Using them, we can find yet another\nexpression for the ratio\nξ/bardbl/ξ⊥= 2+(2τ∆so)2+O(∆2\nsd).(88)12\nFIG. 4. Dimensionless functions ξ0,ξ/bardbl, andξ⊥that define anisotropic spin-transfer torques, Gilbert dam ping, and effective spin\nrenormalization as functions of the spin-orbit coupling st rengthαRfor four different polar angles of magnetization ( nz= cosθ).\nThe notations coincide with those of Table I. We use the dimen sionless combinations ετ= 50,τ∆sd= 10. Since for θ= 0 it is\nimpossible to compute ξ⊥numerically, only analytical result is shown. The O/parenleftbig\n1/∆4\nso/parenrightbig\nexpansion is addressed in Appendix G.\nRemarkably, the rotational anisotropy of GD and the\nnonadiabatic STT, ξ/bardbl/ξ⊥= 2, persists to both limits\n∆sd≪∆so≪1/τand ∆ so≪∆sd≪1/τ,(89)\nin which the Fermi surfaces defined in Eq. (19) are not\nonly essentially isotropic but, at the same time, do get\nstrongly broadened by the disorder (the broadening 1 /τexceeds the splitting of the subbands).\nIt is also interesting to mention that, for small values\nof ∆sd, the nonadiabatic spin-transfer torque dominates\nover the adiabatic one: ξ/bardbl,⊥/ξ0∝1/∆sd. This agrees\nwiththeintuitivelogicthat, foraweakexchangebetween\nconduction and localized spins, the former would rather\nnot adiabatically follow the direction of the latter.13\nVII. DISCUSSION\nA. Role of vertex corrections\nWe would like to begin this final section by stressing\nthat it is the accurate consideration of vertex corrections\nthat is responsible for the established vector structures\nof anisotropic STT, GD, and ESR, as well as for the\nrelation between them. Practically none of this would be\nseen from an uncontrolled analysis that ignores vertex\ncorrections.\nFor example, if one does not apply the disorder dress-\ning to the current vertex v, the relation of Eq. (50) will\nno longer be valid. Instead, the STT tensor, in this case,\nwill contain 18 additional nonzero components of differ-\nent symmetries, which one might by mistake interpret as\nphysical torques.\nB. Renormalization of spin\nIn Sec. VA, we have demonstrated that, in the limit\nof vanishing SOC, the ESR factor δSeff=−ξ0Sdoes\ncoincide with the actual total electron spin in a unit cell\nδS=−JsdSAm/2π/planckover2pi12. On the other hand, this equality\nbreaksdown forfinite αR, and the ratio δSeff/δSstarts to\ndepend on all of the parameters of the system, including\nscattering time (see Table I and Fig. 4).\nForlargevaluesofspin-orbit-inducedsplitting∆ so, the\nquantity ξ0(which determines ESR) understandably de-\ncays due to the effective randomization of the electron\nspin direction induced by SOC. What is, however, rather\ninteresting, is that, for relatively small values of αR, the\nESRfactor δSeffexceedsδS,reachingthemaximumvalue\nat ∆so≈∆sd. We do not have an intuitive explanation\nfor such behaviour.\nC. LLG equation\nIt is instructive to compare the microscopic LLG\nEq.(1)toitsconventionalphenomenologicalcounterpart.\nIn the absence of spin-orbit, thermal, and other torques\nthat we do not consider in this study, the latter equation\nreads\n∂tn=γn×Heff+(js·∇)n\n−α[n×∂tn]−β[n×(js·∇)n],(90)\nwhere the vector quantity jsis interpreted as the phe-\nnomenological spin-polarized current, while the param-\netersαandβdefine Gilbert damping and the nonadi-\nabatic spin-transfer torque, respectively. The latter is\nalsocommonlyreferredtoasthe β-torque. Theadiabatic\nspin-transfertorqueisrepresentedbytheterm( js·∇)n,\nwhileHeffstands for effective field contributions.\nFirst, taking into account Eqs. (2), we can rewrite the\nmicroscopic LLG Eq. (1) in a form which is similar tothat of Eq. (90),\n∂tn= ¯γn×Heff+(js·∇)n\n−α/bardbl[n×∂tn/bardbl]−β/bardbl[n×(js·∇)n/bardbl]\n−α⊥[n×∂tn⊥]−β⊥[n×(js·∇)n⊥],(91)\nwhere\njs=vdξ0\n1−ξ0=−vdδSeff\nS+δSeff,(92a)\nα/bardbl,⊥=ξ/bardbl,⊥\n1−ξ0, β/bardbl,⊥=ξ/bardbl,⊥\nξ0,¯γ=γ\n1−ξ0(92b)\nand each of the quantities js,α/bardbl,⊥,β/bardbl,⊥, ¯γdepend on\nthe orientation of the vector n. For the particular 2D\nRashba FM model system considered in this paper,\njs=js(n2\nz), α /bardbl,⊥=α/bardbl,⊥(n2\nz),(93a)\nβ/bardbl,⊥=β/bardbl,⊥(n2\nz),¯γ= ¯γ(n2\nz). (93b)\nWe see that the microscopic LLG Eq. (91) is essentially\nanisotropic, in contrast with the phenomenological LLG\nEq. (90). Namely, the coefficients αandβgot split into\ntwo components each. Moreover, the new coefficients\nα/bardbl,⊥andβ/bardbl,⊥as well as the other parametersof the LLG\nequation became dependent on the direction of magneti-\nzation. We note that the splitting of the GD coefficient α\nhas been reported, for a Rashba FM, in Ref. [58].\nNext, let us comment on the microscopic definiton of\nthe spin-polarized current formulated in Eq. (92a). Nor-\nmally, ifspins ofconductionelectrons(travellingwith the\ncharacteristic velocity v) adiabatically follow the direc-\ntion ofn, one assumes js=−vδS/(S+δS), where δS\nis a contribution from conduction electrons to the total\nspin of the system. In this case, Eq. (90) can be simply\nviewedasamanifestationofthetotalangularmomentum\nconservation (for n×Heff= 0),\n(S+δS)∂tn+δS(v·∇)n= 0. (94)\nwhere−δS(v·∇)nis the rate of angular momentum\ntransfer from conduction to total spin.\nThe definition of the vector quantity js, given by\nEq. (92a), provides a perfect generalization of the above\nlogic for a system with finite Rashba SOC. Indeed, con-\nduction spins no longer follow the direction of n(due to,\ne.g., nonzero damping). Nevertheless, −δSeff(vd·∇)n\nstill has a meaning of the rate of “angular momentum\ntransfer” from the effective conduction spin δSeffto the\ntotalS+δSeff. Importantly, it was a fully controllable\naccurate microscopic treatment of the problem that led\nus to Eq. (92a). (We identified the drift velocity vd\nas a “proportionality coefficient” between the STT and\nGD tensors and observedthat the adiabatic spin-transfer\ntorque and ESR are described by the same quantity ξ0.)\nFinally, for the sake of historical integrity, let us also\nmention that the equalities α/bardbl=β/bardblandα⊥=β⊥, in\nthis system, are equivalent59to the relation\nδSeff=−S/2, (95)\nwhich appears to be rather unphysical.14\nD. Material derivative and moving reference frame\nIn the presence of the anisotropic STT and GD of\nEqs. (2), it is natural to analyse the microscopic LLG\nEq. (1) in such a frame, where the effect of the nonadia-\nbaticspin-transfertorqueisabsent. Namely, inthe frame\nthat moves with the classical drift velocity of conduction\nelectrons vd. One may use a nice analogy to continuum\nmechanics as an illustration of this fact.\nIndeed, despite the essentially anisotropic character of\nbothTSTTandTGD, their sum is conveniently expressed\nin the LLG Eq.(1) viathe operatorofmaterialderivative\nDt=∂t+(vd·∇) as\n(1−ξ0)Dtn=γn×Heff+(vd·∇)n−ξ/bardbl/bracketleftbig\nn×Dtn/bardbl/bracketrightbig\n−ξ⊥[n×Dtn⊥]+...,(96)\nwhere we have moved the term ξ0Dtnto the left hand\nside and added ( vd·∇)nto both sides. By considering\nconduction electrons as a “fluid” flowing with the drift\nvelocity vd, one may interpret the material derivatives\nof Eq. (96) as the change rates of components of nthat\nare associated with the electronic “fluid parcels”. Thus,\nin the moving (“flowing”) frame, r′=r−vdt, the ma-\nterial derivatives Dtare automatically replaced31by the\nordinary time derivatives ∂t.\nIn other words, in the movingreferenceframe, Eq. (96)\ntakes the form of the LLG equation\n(1−ξ0)∂tn=γn×Heff+(vd·∇)n−ξ/bardbl/bracketleftbig\nn×∂tn/bardbl/bracketrightbig\n−ξ⊥[n×∂tn⊥]+...(97)\nthat comprises the analogue of the adiabatic torque\n(vd·∇)n, two components of damping, and (repre-\nsentedherebydots)allotherpossibletorques. As longas\nthe latterareabsent, the dynamicsofamagnetictexture,\ngoverned by such equation (under mediate currents and\nin the absence of magnetic field), is likely to be a motion\nwith zero terminal velocity (as it is32,33, in the isotropic\ncase, for domain walls). For a general situation, current-\ninduced magnetic dynamics can differ significantly. Nev-\nertheless, it should still be more convenient to perform\nthe analysis once the effect of the nonadiabatic STT has\nbeen accounted for by switching to the “flowing” frame.\nInterestingly, any “propagating” texture of the form\nn(r,t)=ζ(r−vdt)=ζr(t)nullifiesthesum TSTT+TGD.\nHence, for such textures, the LLG Eq. (1) reads\ndζr/dt=γζr×Heff+..., (98)\nwherercan be regardedas aparameter. Ifone takesinto\naccountonlyspin-transfertorquesandfieldlike spin-orbit\ntorque, solutions of this equation will have an oscillatory\ncharacter. Note that Eq. (98) is different from the LLG\nequation\n0 =γζr×Heff (99)\nthat describes the uniform motion of the ground state in\nthe presence of the Galilean invariance [the case α=β\nin Eq. (90)]13,15,16,60.E. Response to electric current\nSo far, we have computed spin-transfer torques as a\nlinear response of the system to the external electric field\nE. In experiment, however, it is not the electric field\nbut rather the electric current jwhich is externally ap-\nplied. To relate spin torques to the latter, one should\ncompute the conductivity tensor ˆ σand, afterwards, use\nthe identity\nE= ˆσ−1j (100)\nto replace Ewithj. Importantly, the conductivity ten-\nsor has to be computed up to the linear order in first\nmagnetization gradients ∇αnβ.\nF. Relation to Edelstein effect\nIt is worth noting that some of our results can be inde-\npendently benchmarked. As it was suggested in Ref. 36,\nthereexistsaconnectionbetweensomeparticularpairsof\nquantities in the model of Eq. (9), as, e.g., between the\nDzyaloshinskii-Moriya interaction strength and the ex-\nchange stiffness, or between spin-orbit torques and spin-\ntransfer torques. The latter relation is relevant to our\nstudy.\nA general interpretation of the approach described in\nRef. 36 would be the following. Suppose there exists\na quantity F(αR) which, for the model with αR= 0,\ndepends on the gradients of n, such that\nF(0) =F(∇xn,∇yn). (101)\nThen, up to the linear order with respect to αR, one\nwould obtain61\nF(αR) =F(0)+αR/bracketleftbigg∂\n∂αRF(/tildewide∇xn,/tildewide∇yn)/bracketrightbigg\nαR=0,(102)\nwhere\n/tildewide∇in=∇in+2mαR\n/planckover2pi1[n×[ez×ei]].(103)\nLet us now choose three functions Fi(αR) to be the\ncomponents of the vector TSTT. Using the expression\nfor the quantity ξ0in the limit αR= 0 (see Table I), we\ncan write\nTSTT=eA\n2π/planckover2pi1Jsdτ(E·∇)n. (104)\nFrom Eq. (102) we, then, find another contribution to\nthe generalized torque in the ∝αRorder\nTSOT=2mαR\n/planckover2pi1eA\n2π/planckover2pi1Jsdτ[n×[ez×E]],(105)\nwhichis preciselythe expressionfor the Edelsteineffect62\ninaformofafieldliketorqueonmagnetization. Inasimi-\nlarway,vanishingofthefunctions ξ/bardblandξ⊥whenαR= 015\ncan be translated into the absence52of the antidamping\nSOT in the model of Eq. (9).\nTheresultofEq.(105)coincideswith thedirectderiva-\ntion of SOT, for the model of Eq. (9), that has been re-\nported previously52. A more compact and accurate form\nof this derivation is also presented in Appendix A. Such\nindependent consistency check adds to the credibility of\nour results.\nCONCLUSIONS\nWe have presented a thorough microscopic analysis of\nSTT, GD, and ESR, for the particular 2D FM system\nwith Rashba spin-orbit coupling and spin-independent\nGaussian white-noise disorder. Assuming arbitrary di-\nrection of magnetization, we have established the ex-\nact relation between these effects. We have intro-\nduced the notion of the matrix gauge transformation\nfor magnetization-dependent phenomena and used it to\nexpress spin-transfer torques, Gilbert damping, and ef-\nfective spin renormalization in terms of meaningful vec-\ntor forms. The latter allowed us to quantify the SOC-\ninduced anisotropy of the former. We have analysed,\nboth analytically and numerically, three dimensionless\nfunctions that fully define anisotropic STT, GD, and\nESR. We have also generalized the concept of spin-\npolarized current, computed spin susceptibility of the\nsystem, and obtained a number of other results.\nIt would be an interesting challenge to observe the\nanisotropy of STT experimentally. It might be possi-\nble to do this by measuring current-induced corrections\nto the magnon spectrum asymmetry that is normally as-\nsociated with the Dzyaloshinskii-Moriya interaction. We\nalso believe that, to some extent, the anisotropy of STT\nand GD might explain the differences in dynamics of do-\nmainwalls(andskyrmions)with differentcharacteristics.\nACKNOWLEDGMENTS\nWe would like to thank Jairo Sinova for pointing out a\nnumber of flaws in the original version of the manuscript.\nWe are also grateful to Artem Abanov, Arne Brataas,\nSergey Brener, Ivan Dmitriev, Rembert Duine, Olena\nGomonay, Andrew Kent, Alessandro Principi, Alireza\nQaiumzadeh, and Yaroslav Tserkovnyak for helpful dis-\ncussions. This research was supported by the JTC-\nFLAGERAProjectGRANSPORTandbytheDutch Sci-\nence Foundation NWO/FOM 13PR3118. M.T. acknowl-\nedges the support from the Russian Science Foundation\nunder Project 17-12-01359.Appendix A: Vertex corrections to velocity\noperator; spin-orbit torque\nIn order to compute vertex corrections to the velocity\noperator v=p/m−αR[ez×σ], we first apply a single\nimpurity line to the scalar part of the latter,\n(p/m)1×dr=1\nmτ/integraldisplayd2p\n(2π)2gR(p/m)gA.(A1)\nDuetothefactthatthemomentumoperator pcommutes\nwith the Green’s functions gR,A, the above relation can\nbe equivalently written as\n(p/m)1×dr=i\nm/integraldisplayd2p\n(2π)2(p/m)/parenleftbig\ngR−gA/parenrightbig\n,(A2)\nwhere we have used the Hilbert’s identity of Eq. (48).\nThe subsequent analysis follows the route of Sec. IIA.\nIntegration over the absolute value of momentum in\nEq. (A2) is performed by computing residues at p=p±.\nSymmetrization of the obtained result, with respect to\nthe transformation43ϕ→π−ϕ, leads to\n(p/m)1×dr=/integraldisplay2π\n0dϕ\n2π/parenleftig\nαR(1+rW4)[ez×σ]\n+(αR+rW5)/braceleftbig\nn/bardbl[σ×n]z−/parenleftbig\nn/bardbl·σ/parenrightbig\n[ez×n]/bracerightbig\ncos2ϕ\n+(W6+W7n·σ)[ez×n]sinϕ/parenrightig\n,(A3)\nwhereWi=Wi/parenleftbig\nr2,u(r2)/parenrightbig\nare some functions of the pa-\nrameterr2andϕ-independent parameters of the model.\nAgain, all terms that contain Wivanish identically after\nintegration over the angle and we conclude that\n(p/m)1×dr=αR[ez×σ]. (A4)\nNext, we observe that the corrected by an impurity\nladder velocity operator vvccan be recast in the form\nvvc=/braceleftbig\np/m−αR[ez×σ]/bracerightbigvc=\np/m+/braceleftbig\n(p/m)1×dr−αR[ez×σ]/bracerightbigvc.(A5)\nAccording to Eq. (A4), expression inside the brackets on\nthe second line vanishes, leading us to the desired result,\nvvc=p/m, (A6)\nwhich coincides with Eq. (45) of the main text. Note\nthat, since the momentum operator commutes with the\nGreen’s functions, Eq. (A6) determines both advanced-\nretardedand retarded-advancedvertexcorrectionsto the\nvelocity operator.\nOne immediate consequence of Eqs. (A4) and (A6) is\na trivial form of spin-orbit torque in the considered inter-\nface Rashba model. Indeed, it was conjectured in Ref. 52\nthat the antidamping SOT, in this model, is identically16\nabsent, while the field-like SOT is entirely isotropic. To\nprove the conjecture, we use the Kubo formula for SOT\nTSOT=eJsdA\n2π/planckover2pi12/integraldisplayd2p\n(2π)2tr/braceleftig\nˆTgR(vvc·E)gA/bracerightig\n.(A7)\nSubstituting vvc=p/mand using Eq. (A1), we immedi-\nately find\nTSOT=eJsdAmτ\n2π/planckover2pi12tr/braceleftig\nˆT/parenleftig\n(p/m)1×dr·E/parenrightig/bracerightig\n,(A8)\nFinally, with the help of Eqs. (12) and (A4), we obtain\nthe expression for spin-orbit torque,\nTSOT=eJsdAmτα R\n2π/planckover2pi12tr/braceleftbig\n[σ×n]([ez×σ]·E)/bracerightbig\n=\neJsdAmτα R\nπ/planckover2pi12[n×[ez×E]],(A9)\nwhich coincides with that of Eq. (105), as expected.\nAppendix B: Vanishing of δTSTT\nWe will now prove that the absence of the spin compo-\nnent inthe vertexcorrectedvelocityoperator vvcnullifies\nthe contribution δTSTTto the STT tensor of Eq. (44).\nUsing cyclic permutations under the matrix trace and\nthe fact that vvc=p/mcommutes with any function of\nmomentum, one can rewrite Eq. (44) as\nδTSTT\nαβγδ=−e∆2\nsdA\n2π/planckover2pi1S/integraldisplayd2p\n(2π)2pβτ\n2mtr[Λ1+Λ2] (B1)\nwith\nΛ1=/parenleftig\nvγgAˆTvc\nαgRσδ−σδgAˆTvc\nαgRvγ/parenrightiggRgA\niτ,(B2a)\nΛ2=/parenleftig\nσδgAvγgAˆTvc\nα−vγgAσδgAˆTvc\nα/parenrightiggRgA\niτ\n−/parenleftig\nˆTvc\nαgRvγgRσδ−ˆTvc\nαgRσδgRvγ/parenrightiggRgA\niτ.(B2b)\nIn Eq. (B2a), we employ the Hilbert’s indentity of\nEq. (48) to replace the factor gRgA/iτwithgR−gAand\nagain use cyclic permutations to obtain\nΛ1=ˆTvc\nαgRσδgRvγgA−ˆTvc\nαgRvγgRσδgA\n−ˆTvc\nαgRσδgAvγgA+ˆTvc\nαgRvγgAσδgA.(B3)\nA similar procedure is performed to simplify the expres-\nsion for Λ 2. We note, however, that terms with only re-\ntarded or only advanced Green’s functions, in Eq. (B2b),\nshould be disregarded44. Hence, gRgA/iτis replaced\nwithgRin the first line of Eq. (B2b) and with −gAin\nthe second line. After moving the torque operator to the\nfirst place in each term,\nΛ2=ˆTvc\nαgRσδgAvγgA−ˆTvc\nαgRvγgAσδgA\n+ˆTvc\nαgRvγgRσδgA−ˆTvc\nαgRσδgRvγgA,(B4)\nwe conclude that Λ 1+Λ2= 0 and, therefore, δTSTT= 0\nas well.Appendix C: Structure of M\nUsing Green’s function of Eq. (56) we compute the\nmatrix trace in Eq. (38) and further symmetrize the in-\ntegrandswith respect to the transformation43ϕ→π−ϕ.\nThis results in the decomposition\nM=γ1I+γ2P+γ3U+γ4U2+γ5P UP+γ6P U2P(C1)\nwhere the coefficients are given in the integral form,\nγ1= 2/bracketleftig/parenleftig\n∆2\nsd+|ε+i/2τ|2/parenrightig\nI0−2(ε+δso)I1+I2/bracketrightig\n,\n(C2a)\nγ2=−4/bracketleftigg\n2δson2\nz\n1−n2zI1+/parenleftbig\n1+n2\nz/parenrightbig\nς∆sd/radicalbig\n1−n2zJ1−1+n2\nz\n1−n2zJ2/bracketrightigg\n,\n(C2b)\nγ3=−2\nτ/bracketleftigg\nς∆sdI0−1/radicalbig\n1−n2zJ1/bracketrightigg\n, (C2c)\nγ4= 4ς∆sd/bracketleftigg\nς∆sdI0−1/radicalbig\n1−n2zJ1/bracketrightigg\n,(C2d)\nγ5=−2\nτ/radicalbig\n1−n2zJ1, (C2e)\nγ6=−4/bracketleftigg\n2δso\n1−n2zI1+ς∆sd/radicalbig\n1−n2zJ1−2\n1−n2zJ2/bracketrightigg\n,(C2f)\nwithδso=mα2\nRand\nIk=/integraldisplayd2p\n(2π)2(2mτ)−1/parenleftbig\np2/2m/parenrightbigk\n|ε−ε+(p)+i/2τ|2|ε−ε−(p)+i/2τ|2,\n(C3a)\nJk=/integraldisplayd2p\n(2π)2(2mτ)−1(αRpsinϕ)k\n|ε−ε+(p)+i/2τ|2|ε−ε−(p)+i/2τ|2.\n(C3b)\nSomeofEqs.(C2)formallybecomeinvalidwhen n=n⊥.\nHowever,structureof MandTintherespectivecasewas\nanalysed directly in Sec. VB.\nAppendix D: Structure of Mk\nWe have already demonstrated that\nM ∈spanL,L={I, P, U, U2, P UP, P U2P},(D1)\nLet us now prove that any natural power of Mbelongs\nto the same linear span,\nMk∈spanL,∀k∈N. (D2)\nTheoperationofmatrixproduct, byitself, isnotclosed\non spanL. Moreover, 14 of 36 elements of L×Ldo not17\nbelong to span L. On the other hand, a combination of\ntwo such elements (matrices P UandUP),\nP U+UP={P,U}=U+P UP, (D3)\nobviously does. Similarly, the remaining 12 “unsuit-\nable” elements of L × Ldo form 6 pairs, such that\nthe corresponding anticommutators (namely, {P,U2},\n{P UP,U},{P U2P,U},{P UP,U2},{P U2P,U2}, and\n{P UP,P U2P}) belong to span L.\nIn general, the following statement holds: operation of\nmatrix anticommutation sends elements of L × Lto a\nlinear span of L,\n{,}:L×L → spanL. (D4)\nTaking into account the fact that anticommutator is a\nbilinear map, we deduce from Eq. (D4):\n{,}: spanL×spanL →spanL.(D5)\nFinally, since for arbitrary kwe have\nMk=1\n2{M,Mk−1}, (D6)\nthe desired result, Mk∈spanL, is proven by induction.\nAppendix E: Spin susceptibility in the presence of\nSOC\nIn this Appendix, the total spin δSof conduction elec-\ntronsin a unit cell ofthe area Ais computed fora general\ncase ofαR∝ne}ationslash= 0. We use the following standard definition:\nδS=A\n2πi/integraldisplay\ndǫf(ǫ)/integraldisplayd2p\n(2π/planckover2pi1)2tr/bracketleftigσ\n2/parenleftbig\nGA−GR/parenrightbig/bracketrightig\n,(E1)\nwherefstands for the Fermi-Dirac distribution,\nf(ǫ) = (1+exp[( ǫ−ε)/T])−1, (E2)\nandGA,Rrefers to the momentum-dependent Green’s\nfunction of Eq. (29). We will first consider the in-plane\ncomponent of δS.\nMatrix trace calculation followed by an integration\noverǫ, in Eq. (E1), gives\nδSx=A/integraldisplayd2p\n(2π/planckover2pi1)2ς∆sdnx−αRpy\nε+(p)−ε−(p)(f+−f−),(E3a)\nδSy=A/integraldisplayd2p\n(2π/planckover2pi1)2ς∆sdny+αRpx\nε+(p)−ε−(p)(f+−f−),(E3b)\nwheref±=f(ε±(p)). It is convenient to introduce the\nquantity δS+=δSx+iδSy. For the latter, we find\nδS+=A\n4αR/integraldisplayd2p\n(2π/planckover2pi1)2(f+−f−)\n×/parenleftbigg\ni∂\n∂px−∂\n∂py/parenrightbigg\n[ε+(p)−ε−(p)],(E4)where we took advantage of the fact that the fractions\nin Eqs. (E3) can be expressed as the derivatives with\nrespect to the components of momentum. In the zero-\ntemperaturelimit, onecanuseGreen’stheoremtoreduce\nthe double integrals in Eq. (E4) to the integrals over the\nclosed curves C±={p|ε±(p) =ε},\nδS+=δS+\n++δS−\n+, (E5a)\nδS±\n+=±A\n4αR/integraldisplay\nC±dpx+idpy\n(2π/planckover2pi1)2[ε+(p)−ε−(p)].(E5b)\nNext, we follow the approach used by K.-W. Kim et al.\nin Ref. 41. Using the variable w=px+ipyand the\nrelationε±(p) =p2/2m±[ε+(p)−ε−(p)]/2, we find\nδS±\n+=A\n16π2/planckover2pi12αR/integraldisplay\nC±dw/parenleftbigg\n2ε−w∗w\nm/parenrightbigg\n,(E6)\nwherew∗w=p2andC±={w|ε±(w,w∗) =ε}are now\nregarded as contours in the complex w-plane. Since the\ncontours are closed, Eq. (E6) is further simplifed to\nδS±\n+=−A\n16π2/planckover2pi12mαR/integraldisplay\nC±dww∗w. (E7)\nIn order to perform integration in Eq. (E7), we solve\nthe equation ε±(w,w∗) =εforw∗and express the result\nas a function of w∈C±,\nw∗=2m\nw2/parenleftig\nw/bracketleftbig\nε+mα2\nR/bracketrightbig\n−imαRς∆sdn+±√\nR/parenrightig\n,(E8)\nwheren+=nx+inyandRis a cubic function of w.\nDifferent signs in front of the square root in Eq. (E8)\ncorrespond to two different functions w∗=w∗\n±(w) of\nw∈C±, respectively. We do not specify which sign\ncorrespondsto which function. Such ambiguity, however,\ndoes not affect the final result for δS+. Indeed, it can\nbe proven41that all three zeroes of Rare of the form\nwk=irkn+with real rk. Then, from the general relation\n[ε−ε+(w,w∗)][ε−ε−(w,w∗)] =−R\n+/parenleftbiggw∗w\n2m−/bracketleftbig\nε+mα2\nR/bracketrightbig\n+imαRς∆sdn+\nw/parenrightbigg2\n,(E9)\nwe learn that\n[ε−ε+(wk,w∗\nk)][ε−ε−(wk,w∗\nk)]≥0 (E10)\nand, thus, ε−(wk,w∗\nk)< ε⇒ε+(wk,w∗\nk)≤ε. Hence,\nall the singularities of w∗\n−that lie inside the contour C−\nare, in fact, located inside or, at most, on the contour\nC+(note that C+is inside C−). Disregarding the case63\nwk∈C±and using Cauchy integral theorem, we can\nshrink64C−in Eq. (E7) to obtain\nδS+=−A\n16π2/planckover2pi12mαR/integraldisplay\nC+dw/parenleftbig\nw∗\n++w∗\n−/parenrightbig\nw,(E11)18\nso that the terms ±√\nR, in Eq. (E8), do not contribute\ntoδS+. The only remaining singularity of the integrand\nis located at the origin and, by the residue theorem,\nδS+=−ς∆sdAm\n2π/planckover2pi12n+orδS/bardbl=−ς∆sdAm\n2π/planckover2pi12n/bardbl,(E12)\nwhich completes the computation of the in-plane compo-\nnent ofδS.\nIn order to calculate δSz, it is useful to introduce the\n“magnetization”vector M=ς∆sdn. Intermsof M,one\ncan straightforwardlyestablish the “thermodynamic” re-\nlationδSi=∂Ω/∂Mi, where Ω has a meaning of the\nelectronic grand potential in a unit cell,\nΩ =−TA\n2πi/integraldisplay\ndǫg(ǫ)/integraldisplayd2p\n(2π/planckover2pi1)2tr/bracketleftbig\nGA−GR/bracketrightbig\n,(E13a)\ng(ǫ) = log(1+exp[( ε−ǫ)/T]). (E13b)\nWe further note that, according to Eq. (E12), δSxand\nδSydo not depend on Mz. Therefore, equating the sec-\nond derivatives, we find\n∂δSz\n∂Mα=∂2Ω\n∂Mα∂Mz=∂δSα\n∂Mz= 0,(E14)\nwhereα=x,y. As a result, δSzdoes not depend on Mx\nandMyand, thus, can be computed for Mx=My= 0\n(or, equivalently, for nx=ny= 0).\nFrom Eq. (E1) we obtain\nδSz=A/integraldisplayd2p\n(2π/planckover2pi1)2ς∆sdnz\nε+(p)−ε−(p)(f+−f−),(E15)which, for nx=ny= 0, can be integrated over the mo-\nmentum angle with the result\nδSz=Aς∆sdnz\n4π/planckover2pi12∞/integraldisplay\n0pdpf+−f−/radicalig\n∆2\nsd+(αRp)2.(E16)\nAtzerotemperature, theintegrationdomaininEq.(E16)\nis reduced to a finite interval p+< p < p −, wherep±are\ngiven by Eq. (25c). After some algebraic practice, we\nfinally arrive at\nδSz=Aς∆sdnz\n4π/planckover2pi12α2\nR/radicalig\n∆2\nsd+(αRp)2/vextendsingle/vextendsingle/vextendsingle/vextendsinglep−\np+=−ς∆sdAm\n2π/planckover2pi12nz.\n(E17)\nCombining the results of Eqs. (E12) and (E17) into a\nsingle vector form\nδS=−ς∆sdAm\n2π/planckover2pi12n, (E18)\nweseethat, onaverage,evenforfinite valuesofspin-orbit\ncoupling strength αR, spins of conduction electrons, in\ntheequilibrium, arealignedwiththe localmagnetization.\nMoreover, the spin susceptibility tensor is fully isotropic\nand is expressed by a single scalar parameter\nδS=−|δS|=−ς∆sdAm\n2π/planckover2pi12, (E19)\nwhich coincides with that given by Eq. (75) of the main\ntext.\nAppendix F: Expansion of Mup toα2\nR\nExpansion of Eqs. (C2) up to α2\nR= ∆2\nso/2εmprovides us with the coefficients\nδγ1=−/bracketleftbigg2τ∆so\n1+(2τ∆sd)2/bracketrightbigg2/bracketleftbig\n1+(2nzτ∆sd)2/bracketrightbig\n, δγ 2= 2/bracketleftbiggτ∆so\n1+(2τ∆sd)2/bracketrightbigg2/bracketleftbig\n1−(1+2n2\nz)(2τ∆sd)2/bracketrightbig\n,(F1a)\nδγ3=/bracketleftbigg4τ∆so\n1+(2τ∆sd)2/bracketrightbigg21+(2nzτ∆sd)2\n1+(2τ∆sd)2ςτ∆sd, δγ 4=−2/bracketleftbigg4τ2∆so∆sd\n1+(2τ∆sd)2/bracketrightbigg21+(2nzτ∆sd)2\n1+(2τ∆sd)2, (F1b)\nδγ5=−2/bracketleftbigg2τ∆so\n1+(2τ∆sd)2/bracketrightbigg2\nςτ∆sd, δγ 6=−/bracketleftbigg4τ2∆so∆sd\n1+(2τ∆sd)2/bracketrightbigg2\n(F1c)\nof the decomposition that we refer to in Sec. VIA: δM=δγ1I+δγ2P+δγ3U+δγ4U2+δγ5P UP+δγ6P U2P.\nAppendix G: O(1/∆4\nso)expansion of ξi(limit of strong SOC)\nThe quantities ξiare shown in the plots of Fig. 4 as functions of the spin-orbit coupling strength αR(while keeping\nbothmandεconstant). Therefore, the right “tails” of the curves can be pro perly fit using the asymptotic expansion\nwith respect to the parameter 1 /∆so. Such expansion can be obtained indirectly, from the expansion in sm all ∆sd.\nBelow, for consistency with the results of Sec. VIB, we list all the co ntributions to ξithat do not exceed the fourth19\norder in 1 /∆so,\nξ0=−δS\nS/bracketleftigg/parenleftbigg∆sd\n∆so/parenrightbigg2/bracketleftigg\n4n2\nz+1+n2\nz\n2(τ∆so)2/bracketrightigg\n+6/parenleftbigg∆sd\n∆so/parenrightbigg4/bracketleftbig\n1−3n2\nz/bracketrightbig\nn2\nz/bracketrightigg\n, (G1a)\nξ/bardbl=/vextendsingle/vextendsingle/vextendsingleδS\nS/vextendsingle/vextendsingle/vextendsingleτ∆sd/bracketleftigg\n2+1\n(τ∆so)2−/parenleftbigg∆sd\n∆so/parenrightbigg2/bracketleftigg\n4n2\nz−1−7n2\nz\n(τ∆so)2/bracketrightigg\n−4/parenleftbigg∆sd\n∆so/parenrightbigg4/bracketleftbig\n1−3n2\nz/bracketrightbig\nn2\nz/bracketrightigg\n, (G1b)\nξ⊥=/vextendsingle/vextendsingle/vextendsingleδS\nS/vextendsingle/vextendsingle/vextendsingleτ∆sd/bracketleftigg\n1\n2(τ∆so)2+/parenleftbigg∆sd\n∆so/parenrightbigg2/bracketleftigg\n2n2\nz+1−5n2\nz\n2(τ∆so)2/bracketrightigg\n+2/parenleftbigg∆sd\n∆so/parenrightbigg4/bracketleftbig\n1−5n2\nz/bracketrightbig\nn2\nz/bracketrightigg\n. 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Together with similar con-\ntributions from retarded-advanced (or advanced-retarded )\ncorrelators, they should either vanish or be included in\nrenormalization of the parameters of the model (for large\nvalues of the parameter ε0τ). For example, amore accurate\ncomputation of the GD tensor that takes into account con-\ntributions from p= 0 includes the following self-consistent\nrenormalization of the exchange splitting (at αR= 0):\n∆′\nsd= ∆sd−(4πτ∆′\nsd)−1/bracketleftBig\nlog(ε+∆′\nsd)−log(ε−∆′\nsd)/bracketrightBig\n.\nFermi sea contributions to STT, GD, and ESR are dis-\nregarded throughout the paper as well. For the present\nmodel, they vanish when ε >∆sd.\n45H. Kohno, G. Tatara, and J. Shibata,\nJ. Phys. Soc. Jpn. 75, 113706 (2006).\n46A. Sakai and H. Kohno, Phys. Rev. B 89, 165307 (2014).\n47I. A. Ado, I. A. Dmitriev, P. M. Ostrovsky, and M. Titov,\nEurophys. Lett. 111, 37004 (2015).\n48I. A. Ado, I. A. Dmitriev, P. M. Ostrovsky, and M. Titov,\nPhys. Rev. 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Lett. 95, 107204 (2005).\n61According to the definition of Eq. (9), the spin-orbit cou-\npling term has an opposite sign as compared to that used\nin Ref. 36.\n62V. M. Edelstein, Solid State Commun. 73, 233 (1990).\n63The conditions wk∈C±can only be fulfilled for some\nparticular values of ε. SinceδSis a continuous function\nofε, one may just ignore such values.\n64See Ref. 41 for important details on branch cuts."    },    {        "title": "1203.0607v1.Scaling_of_intrinsic_Gilbert_damping_with_spin_orbital_coupling_strength.pdf",        "content": "arXiv:1203.0607v1  [cond-mat.mtrl-sci]  3 Mar 2012Scaling of intrinsic Gilbert damping with spin-orbital cou pling strength\nP. He1,4, X. Ma2, J. W. Zhang4, H. B. Zhao2,3, G. L¨ upke2, Z. Shi4, and S. M. Zhou1,4\n1Surface Physics State Laboratory and Department of Physics , Fudan University, Shanghai 200433, China\n2Department of Applied Science, College of William and Mary, Williamsburg, Virginia 23185\n3Key Laboratory of Micro and Nano Photonic Structures (Minis try of Education) and\nDepartment of Optical Science and Engineering, Fudan Unive rsity, Shanghai 200433, China and\n4Shanghai Key Laboratory of Special Artificial Microstructu re Materials and\nTechnology &Physics Department, Tongji University, Shanghai 200092, C hina\n(Dated: November 6, 2018)\nWe have experimentally and theoretically investigated the dependence of the intrinsic Gilbert\ndamping parameter α0on the spin-orbital coupling strength ξby using L1 0ordered FePd 1−xPtx\nternary alloy films with perpendicular magnetic anisotropy . With the time-resolved magneto-optical\nKerr effect, α0is found to increase by more than a factor of ten when xvaries from 0 to 1.0. Since\nchanges of other leading parameters are found to be neglecte d, theα0has for the first time been\nproven to be proportional to ξ2.\nPACS numbers: 75.78.Jp; 75.50.Vv; 75.70.Tj; 76.50.+g\nMagnetization dynamics has currently become one\nof the most popular topic in modern magnetism due\nto its crucial importance in information storage. Real\nspace trajectory of magnetization processional switching\ntriggered by magnetic field pulses, fs laser pulses, and\nspin-polarized current1–6, can be well described by\nthe phenomenological Landau-Lifshitz-Gilbert (LLG)\nequation that incorporates the Gilbert damping term7\nwhich controls the dissipation of magnetic energy\ntowards the thermal bath. The time interval from the\nnon-equilibrium magnetization to the equilibrium state\nis governed by the Gilbert parameter α. It has very\nrecently been shown that the laser-induced ultrafast\ndemagnetization is also controlled by the α8.\nThe intrinsic Gilbert damping α0has been exten-\nsively studied in theory9–15, and in general believed\nto arise from the spin orbital coupling (SOC). In the\nSOC torque-correlation model proposed by Kambersk´ y,\ncontributions of intraband and interband transitions are\nthought toplay adominant rolein the α0at lowand high\ntemperatures Tand are predicted to be proportional to\nξ3(ξ=the SOC strength) and ξ2, respectively10,14. Up\nto date, however, no experiments have been reported\nto demonstrate the quantitative relationship between\nα0andξalthough many experimental attempts have\nbeen made to study the α0in various metallic and alloy\nfilms16–23. It is hard to rule out effects other than the\nSOC because α0is also strongly related to parameters\nsuch as the electron scattering time and density of state\nD(EF) at Fermi surface EF21,23,24which in turn change\namong various metals and alloys. In order to rigorously\naddress the ξdependence of α0in experiments, it is\ntherefore essential to find magnetic alloys in which the\nξcan be solely adjusted while other parameters almost\nkeep fixed.\nIn this Letter, we elucidate the ξdependence of α0by\nusing L1 0FePd1−xPtx(=FePdPt) ternary alloy films.\nHere, only ξis modulated artificially by the Pt/Pd\nconcentration ratio because heavier atoms are expectedto have a larger ξ27–29and parameters other than ξare\ntheoretically shown to be almost fixed. Experimental\nresults have shown that α0is proportional to ξ2. It is\ntherefore the first time to have given the experimental\nevidence of the ξ2scaling law. This work will also facili-\ntate exploration of new magnetic alloys with reasonably\nlarge perpendicular magnetic anisotropy (PMA) and low\nα.\nL10FePdPt ternary alloy films with 0 ≤x≤1.0 were\ndeposited on single crystal MgO (001) substrates by\nmagnetron sputtering. The FePdPt composite target\nwas formed by putting small Pt and Pd pieces on an\nFe target. During deposition, the substrates were kept\nat 500◦C. After deposition, the samples were annealed\nin situ at the same temperature for 2 hours. The base\npressure of the deposition system was 1 ×10−5Pa and\nthe Ar pressure was 0.35 Pa. Film thickness was deter-\nmined by X-ray reflectivity (XRR) to be 12 ±1 nm. In\norder to measure the Gilbert damping parameter α25,26,\ntime-resolved magneto-optical Kerr effect (TRMOKE)\nmeasurements were performed in a pump-probe setup\nusing a pulsed Ti:sapphire laser in the wavelength of\n400 nm (800 nm) for pump (probe) pulses with a pulse\nduration of 200 fs and a repetition rate of 250 kHz. An\nintense pump pulse beam with a fluence of 0.16 mJ/cm2\nwas normally incident to excite the sample, and the\ntransient Kerr signal was detected by a probe pulse\nbeam which is time-delayed with respect to the pump.\nThe intensity ratio of the pump to probe pulses was\nset to be about 6:1, and their respective focused spot\ndiameters were 1 mm and 0.7 mm. A variable magnetic\nfieldHup to 5 T was applied at an angle of 45 degrees\nwith respect to the film normal using a superconducting\nmagnet. TRMOKE measurements were performed at\n200 K and other measurements were performed at room\ntemperature.\nMicrostructural analysis was accomplished with the\naid of X-ray diffraction (XRD). Figures 1(a)-1(c) show\nthe XRD patterns for L1 0FePdPt films with x= 1,2\n/s50/s52 /s51/s50 /s52/s48 /s52/s56 /s53/s54 /s45/s51/s48 /s45/s50/s48 /s45/s49/s48 /s48 /s49/s48 /s50/s48 /s51/s48/s70/s101/s80/s116\n/s32/s32\n/s40/s97/s41\n/s70/s101/s80/s116/s40/s100/s41\n/s32/s32\n/s32/s79/s85/s84\n/s32/s73/s78\n/s40/s98/s41 /s70/s101/s80/s100\n/s48/s46/s53/s80/s116\n/s48/s46/s53\n/s32\n/s70/s101/s80/s100\n/s48/s46/s53/s80/s116\n/s48/s46/s53/s40/s101/s41\n/s32\n/s70/s101/s80/s100\n/s48/s46/s55/s53/s80/s116\n/s48/s46/s50/s53/s40/s99/s41\n/s50 /s32/s40/s100/s101/s103/s114/s101/s101/s41\n/s32/s73/s110/s116/s101/s114/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s117/s110/s105/s116/s115/s41\n/s109/s32/s40/s97/s114/s98/s46/s117/s110/s105/s116/s115/s41\n/s40/s102/s41\n/s70/s101/s80/s100\n/s48/s46/s55/s53/s80/s116\n/s48/s46/s50/s53\n/s70/s105/s101/s108/s100/s32/s40/s107/s79/s101/s41\n/s32\nFIG. 1: XRD patterns(a, b, c), out-of-plane and in-plane\nhysteresis loops (d,e,f)for L1 0FePd1−xPtxfilms with x= 1\n(a,d),x= 0.5 (b,e) and x= 0.25 (c,f).\nx= 0.5, andx= 0.25, respectively. The films are\nof the L1 0ordered structure in the presence of (001)\nsuperlattice peak. The chemical ordering degree Scan\nbe calculated with the intensity of the (001) and (002)\npeaks and found to be 0 .7±0.1 for all samples. Since\nno other diffraction peaks exist except for (001) and\n(002) ones, all samples are of L1 0single phase with c\naxis perpendicular to the film plane. Here, c= 3.694˚A.\nMagnetization hysteresis loops were measured by vibrat-\ning sample magnetometer. Figures 1(d)-1(f) display the\ncorresponding out-of-plane and in-plane magnetization\nhysteresis loops. As shown in Fig.1(d), for x= 1\n(L10FePt) the out-of-plane hysteresis loop is almost\nsquare-shaped with coercivity HC= 3.8 kOe, indicating\nthe establishment of high PMA. With decreasing x, the\nHCdecreases. For x= 0.25 in Fig. 1(f), HCapproaches\nzero and the out-of-plane and in-plane loops almost\noverlap with each other, indicating a weak PMA. Ap-\nparently, the PMA increases with increasing x. Similar\nphenomena have been reported elsewhere28,29.\nFigure 2(b) displays the typical TRMOKE results\nfor L1 0FePdPt films with x= 0.25 under θH= 45oas\nshown in Fig.2(a). For the time delay longer than 5.0\nps, damped oscillatory Kerr signals are clearly seen due\nto the magnetization precession. The precession period\nbecomes short significantly with increasing H. In order\nto extract the precession frequency, the Kerr signal was\nfitted by following exponentially damped sine function,\na+bexp(−t/t0) +Aexp(−t/τ)sin(2πft+ϕ), where\nparameters A,τ,fandϕare the amplitude, relaxation\ntime, frequency, and phase of damped magnetization\nprecession, respectively30. Here,a,b, andt0correspond\nto the background signal owing to the slow recovery\nprocess. The experimental data are well fitted by the\nabove equation, as shown in Fig.2(b).\nFigure 3(a) shows that for all samples studied\nhere, the extracted precession frequency fincreases\nmonotonically as Hincreases. Moreover, fshows an\nFIG. 2: Schematic illustration of the TRMOKE geometry (a)\nand TRMOKE results for x= 0.25 under various magnetic\nfields (b). Here θH= 45◦. Curves are shifted for clarity. The\nsolid lines are fit results.\nincreasing tendency with increasing xat fixed H. For\nx= 1 (L1 0FePt),fis in a very high frequency range of\n180-260 GHz due to the high PMA. Figure 3(b) shows\nthat the relaxation time τdisplays a decreasing trend\nwith increasing H. Moreover, τincreases by two orders\nof magnitude when Pd atoms are replaced by Pt ones.\nIn particular, we observed the short relaxation time\nτ= 3 ps for x= 1 (L1 0FePt). When the oscillation\nperiod is longer than the relaxation time for low Hthe\nprecession cannot be excited for x= 131.\nWithα≪1.0, one can obtain the follow-\ning dispersion equation, 2 πf=γ√H1H2, where\nH1=Hcos(θH−θ) +HKcos2θandH2=\nHcos(θH−θ)+HKcos2θ, whereHK= 2KU/MS−4πMS\nwith uniaxial anisotropy constant KU. The equilibrium\nmagnetization angle θis calculated from the following\nequation sin2 θ= (2H/HK)sin(θH−θ), which is derived\nby taking the minimum of the total free energy. The\nmeasured Hdependence of fcan be well fitted, as shown\nin Fig.3(a). With the measured MSof 1100 emu/cm3,\ntheKUcan be calculated. The gfactor is equal to 2.16\nforx= 1, 0.7, and 0.5, and to 2.10 and 2.03 for x= 0.25\nand 0.15, respectively. A small fraction of the orbital3\n/s48/s56/s48/s49/s54/s48/s50/s52/s48/s51/s50/s48/s52/s48/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50/s48/s46/s48/s51/s48/s46/s50/s48/s46/s52/s120/s61/s49/s32/s32/s32/s32 /s120/s61/s48/s46/s55/s32/s32\n/s120/s61/s48/s46/s53/s32 /s120/s61/s48/s46/s50/s53/s32/s32\n/s120/s61/s48/s46/s49/s53/s102/s32/s40/s71/s72/s122/s41/s40/s97/s41\n/s40/s98/s41/s32/s40/s110/s115/s41\n/s72/s32/s40/s107/s79/s101/s41\nFIG. 3: Uniform magnetization precession frequency f(a)\nand relaxation time τ(b) as a function of Hfor all samples\nstudied here. Solid lines refer to fit results.\n/s48/s46/s48/s48/s46/s49/s48/s46/s50/s49/s50/s51/s52\n/s32/s77/s101/s97/s115/s117/s114/s101/s100\n/s32/s67/s97/s108/s99/s117/s108/s97/s116/s101/s100/s40/s98/s41\n/s32/s32/s40/s97/s41/s75\n/s85 /s32/s40/s49/s48/s55\n/s32/s101/s114/s103/s47/s99/s109/s51\n/s41\n/s32\n/s48/s46/s48/s48 /s48/s46/s50/s53 /s48/s46/s53/s48 /s48/s46/s55/s53 /s49/s46/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s40/s99/s41\n/s32/s32 /s32/s40/s101/s86/s41\n/s120\nFIG. 4: Measured KU(a), measured (solid box) and calcu-\nlated (solid circles) α0(b),ξcalculated in this work (solid\ncircles) and elsewhere38(open ones) (c) as a function of x.\nThe lines serve as a visual guide in (b) and refer to the fit\nresults in (c).\nangular momentum is therefore restored by the SOC10\nand close to results reported elsewhere32.\nThe measured Hdependence of τcan be well fitted\nbyτ= 2/αγ(H1+H2) with the fitted values of g\nandHKforα≪1.0. Here, the Gilbert damping αis\nan adjustable parameter. As shown in Fig.3(b), the\nexperimental and fitted data coincide with each other\nat highHand exhibit significant deviation from each\nother at low H. It is therefore illustrated that the\nextrinsic magnetic relaxation contributes to the αat\nlowHand becomes weak at high H. This is because\nthe extrinsic magnetic relaxation may arise from the\ninhomogeneous PMA distribution and the interfacial\neffect and is greatly suppressed under high H33–35. The\nintrinsic α0therefore plays a dominant role at high H,\nthat is to say, α0is fitted in Fig. 3(b).\nTo determine the SOC strength ξand intrinsicdamping parameter α0in L1 0ordered FePd 1−xPtx\nternary alloys, we perform spin dependent first prin-\nciples calculations based on linear muffin-tin orbital\ndensity functional theorem, where the lattice constants\narea= 3.86˚Aandc= 3.79˚Afor L1 0ordered FePt. The\nD(EF) is 2.55, 2.47, 2.43, and 2.39 per atom per eV for\nxvarying from 0, 0.5, 0.75, to 1.0, respectively. The\nα0was achieved by using spin-orbital torque-correlation\nmodel based on spin dependent electron band structures\nobtained above9,13.\nIt is significant to compare variations of the PMA\nandα0. Figures 4(a) & 4(b) show the KUandα0both\ndecrease with decreasing x. Similar variation trends of\nKUandα0have been observed for perpendicularly mag-\nnetized Pt/Co/Pt multilayers30. When the ξis smaller\nthan the exchange splitting, the magnetic anisotropy is\nthought to come from the second order energy correction\nof the SOC in the perturbation treatment and is roughly\nproportional to both the ξand the orbital angular\nmomentum. The orbital momentum in 3 dmagnetic\nmetallic films restored by the SOC is also proportional\nto theξand the PMA therefore is proportional to ξ2/W\nwith the bandwidth of 3 delectrons W36. Since the W\ndoes not change much with x, the enhanced PMA at\nhighxis attributed to a larger ξof Pt atoms compared\nwith that of Pd atoms27,37. Our calculations show ξ\nchange from 0 .20, 0.26, 0.41 to 0.58 (eV) when xvarying\nfrom 0, 0 .5, 0.75, to 1.0, as shown in Fig. 4(c). This is\nbecause the ξis 0.6, 0.20, and 0.06 (eV) for Pt, Pd, and\nFe atoms, respectively27,38and the effect of Fe atoms is\nnegligible compared with those of Pd and Pt atoms. The\npresent results of ξare in good agreement with previous\nab initio calculations38. Apparently, the PMA behavior\narises from the increase of ξat highx. As shown in\nFig. 4(b), measured and calculated results of α0are in\na good agreement. Since the lattice constant, D(EF),\nthe Curie temperature, the gyromagnetic ratio, and\nthe averaged spin are experimentally and theoretically\nshown to almost not change with x, the enhanced α0is\nmainly attributed to the ξincrease with increasing x.\nFigure 5 showsthat the α0is approximatelyproportional\ntoξ2, where the ξvalues at other xare exploited from\nthe fitted curve in Fig. 4(c). Since for the present\nL10ordered FePd 1−xPtxternary alloy films only ξis\ntuned with x, the present work has rigorously proven\nthe theoretical prediction about the ξ2scaling of α09. It\nis indicated that the α0at 200 K is mainly contributed\nby the interband contribution10,12,14. The electronic-\nscattering-based model of ferromagnetic relaxation is\ntherefore proved to be applicable for the α0in L10\nFePdPt ternary alloys9. In order to further verify the\nξ3dependence of α014, measurements of magnetization\nprecession at low temperatures need to be accomplished.\nIn summary, we have investigated the magneti-\nzation dynamics in L1 0FePdPt ternary alloy films\nusing TRMOKE. The intrinsic α0can be continuously\ntuned, showing a decrease with increasing Pd content\ndue to smaller ξcompared with that of Pt atoms. In4\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52/s48/s46/s48/s48/s46/s49/s48/s46/s50\n/s32/s32/s48/s32\n/s32/s40/s101/s86/s50\n/s41\nFIG. 5: The measured (solid square) and calculated (solid\ncircles)α0versusξ2as a function of x. The dashed curve\nrefers to the linear fit results.particular, the ξ2dependence of α0has been rigorously\ndemonstrated in experiments. The experimental results\ndeepen the understanding the mechanism of α0in\nmagnetic metallic materials and provide a new clue to\nexplore ideal ferromagnets with reasonably low α0and\nhigh PMA as storage media for the next generation\nmicrowave-assisted magnetic recording.\nAcknowledgements This work was supported by the\nMSTC under grant No. 2009CB929201, (US) DOE\ngrant No. DE-FG02-04ER46127, NSFC under Grant\nNos. 60908005, 51171129 and 10974032, and Shanghai\nPuJiang Program (10PJ004).\n1Th. Gerrits, H. A. M. van den Berg, J. Hohlfeld, L. B¨ ar,\nand Th. Rasing, Nature (London) 418, 509 (2002).\n2H.W. Schumacher, C. Chappert, R. C. Sousa, P. P. Freitas,\nand J. Miltat, Phys. Rev. Lett. 90, 017204 (2003).\n3S. I. Kiselev, J. C. Sankey, I.N. Krivorotov, N.C. Emley, R.\nJ. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature\n(London) 425, 380 (2003).\n4S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S.\nE. Russek, and J. A. Katine, Nature (London) 437, 389\n(2005)\n5E. 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Aziz, and J. Miguel et\nal, Phys. Rev. Lett. 104, 076402(2010).\n28S. Jeong et al., J. Appl. Phys. 91, 8813(2002)\n29G. J. Chen et al., Surf. Coat. Technol. 202, 937(2007)\n30S. Mizukami, E. P. Sajitha, F. Wu, and D. Watanabe et\nal, Appl. Phys. Lett. 96, 152502(2010)\n31J. W. Kim, H. S. Song, J. W. Jeong, K. D. Lee et al, Appl.\nPhys. Lett. 98, 092509(2011)\n32I. V. Solovyev, P. H. Dederichs, and I. Mertig, Phys. Rev.\nB52, 13419(1995)\n33J. Walowski, M. Djordjevic-Kaufmann, B. Lenk, and C.\nHamann et al, J. Phys. D: Appl. Phys. 41, 164016(2008)\n34R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev.\nLett.87, 217204(2001)\n35Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601(2002)\n36P. Bruno, Phys. Rev. B 39, 865(1989)\n37J. Friedel, in The Physics of Metals , edited by J. M. Ziman\n(Cambridge Univ. Press, Cambridge, 1969)\n38P. He, L. Ma, Z. Shi, G. Y. Guo, and S. M. Zhou, Chem-\nical Composition Tuning of the Anomalous Hall Ef-\nfect in Isoelectronic L1(0) FePd 1−xPtxAlloy Films ,\narXiv:1112.0834v1"    },    {        "title": "1506.05622v2.The_absence_of_intraband_scattering_in_a_consistent_theory_of_Gilbert_damping_in_metallic_ferromagnets.pdf",        "content": "arXiv:1506.05622v2  [cond-mat.str-el]  23 Oct 2015The absence of intraband scattering in a consistent theory o f Gilbert damping in\nmetallic ferromagnets\nD M Edwards\nDepartment of Mathematics, Imperial College London, Londo n SW7 2BZ, United Kingdom\nDamping of magnetization dynamics in a ferromagnetic metal , arising from spin-orbit coupling,\nis usually characterised by the Gilbert parameter α. Recent calculations of this quantity, using\na formula due to Kambersky, find that it is infinite for a perfec t crystal owing to an intraband\nscattering term which is of third order in the spin-orbit par ameterξ. This surprising result conflicts\nwith recent work by Costa and Muniz who study damping numeric ally by direct calculation of\nthe dynamical transverse susceptibility in the presence of spin-orbit coupling. We resolve this\ninconsistency by following the approach of Costa and Muniz f or a slightly simplified model where\nit is possible to calculate αanalytically. We show that to second order in ξone retrieves the\nKambersky result for α, but to higher order one does not obtain any divergent intrab and terms.\nThe present work goes beyond that of Costa and Muniz by pointi ng out the necessity of including\nthe effect of long-range Coulomb interaction in calculating damping for large ξ. A direct derivation\nof the Kambersky formula is given which shows clearly the res triction of its validity to second order\ninξso that no intraband scattering terms appear. This restrict ion has an important effect on the\ndamping over a substantial range of impurity content and tem perature. The experimental situation\nis discussed.\nI. INTRODUCTION\nMagnetization dynamics in a ferromagnetic metal is central to the fi eld of spintronics with its many applications.\nDamping is an essential feature of magnetization dynamics and is usu ally treated phenomenologically by means\nof a Gilbert term in the Landau-Lifshitz-Gilbert equation [1, 2]. For a system with spin-rotational invariance the\nuniform precession mode of the magnetization in a uniform external magnetic field is undamped and the fundamental\norigin of damping in ferromagnetic resonance is spin-orbit coupling (S OC). Early investigations of the effect include\nthose of Kambersky [3–5] and Korenman and Prange [6]. Kambersky ’s [4] torque-correlation formula for the Gilbert\ndamping parameter αhas been used by several groups [7–14], some of whom have given alt ernative derivations.\nHowever the restricted validity of this formula, as discussed below, has not been stressed. In this torque-correlation\nmodel contributions to αof both intraband and interband electronic transitions are usually c onsidered. The theory\nis basically developed for a pure metal with the effect of defects and /or phonons introduced as phenomenological\nbroadening of the one-electron states. Assuming that the electr on scattering-rate increases with temperature T due\nto electron-phonon scattering the intraband and interband tran sitions are found to play a dominant role in low and\nhigh T regimes, respectively. The intraband(interband) term is pre dicted to decrease(increase) with increasing T and\nto be proportional to ξ3(ξ2) whereξis the SOC parameter. Accordingly αis expected to achieve a minimum at an\nintermediate T. This is seen experimentally in Ni and hcp Co [15] but not in Fe [15] and FePt [16]. The ξ2dependence\nofαis well-established at high T [17, 18] but there seems to be no experim ental observation of the predicted ξ3\nbehaviourat lowT. The interband ξ2term in Kambersky’stheory canbe givenaverysimple interpretation in termsof\nsecond-orderperturbation theory [5]. A quite different phenomen ologicalapproach, not applicable in some unspecified\nlow scattering-rate regime, has been adopted to try and find a phy sical interpretation of the intraband term [5, 8].\nNo acceptable theoretical treatment of damping in this low scatter ing regime is available because the intraband term\nof Kambersky’s theory diverges to infinity in the zero-scattering lim it of a pure metal with translational symmetry at\nT=0 [9, 11, 13]. We consider it essential to understand the pure met al limit before introducing impurity and phonon\nscattering in a proper way.\nCosta and Muniz [19] recently studied damping numerically in this limit by direct calculation of the dynamical\nspin susceptibility in the presence of SOC within the random phase app roximation (RPA). They determine αfrom\nthe linewidth of the uniform (wave-vector q= 0) spin-wave mode which appears as a resonance in the transvers e\nsusceptibility. One of the main objects of this paper is to establish so me degree of consistency between the work of\nKambersky and that of Costa and Muniz. We follow the approach of t he latter authors for a slightly simplified model\nwhere it is possible to calculate αanalytically. We show that to second order in ξone retrieves the Kambersky result,\nbut to higher order no intraband terms occur, which removes the p roblem of divergent α. To confirm this point, in\nAppendix A we derive the Kambersky formula directly in a way that mak es clear its restriction to second order in\nξto which order the divergent terms in αarising from intraband transitions do not appear. This throws open the\ninterpretation of the minimum observed in the temperature depend ence ofαfor Ni and Co.\nAt this point we may mention an alternative theoretical approach to the calculation of Gilbert damping using2\nscattering theory [20, 21]. Starikov et al [21] find that, for Ni 1−xFexalloys at T=0, αbecomes large near the pure\nmetal limits x=0,1. They attribute this to the Kambersky intraband c ontribution although no formal correspondence\nis made between the two approaches.\nThe work of Costa and Muniz [19] follows an earlier paper [22] where it is shown that SOC has the effect of\ncoupling the transverse spin susceptibility to the longitudinal spin su sceptibility and the charge response. It is known\nthat a proper calculation of these last two quantities in a ferromagn et must take account of long-range Coulomb\ninteractions [23–27]. The essential role of these interactions is to e nsure conservation of particle number. Costa et\nal [19, 22] do not consider such interactions but we show here that this neglect is not serious for calculating αwith\nsufficiently small SOC. Howeverin the wider frameworkof this paper, where mixed charge-spinresponse is also readily\nstudied, long-rangeinteractions are expected to sometimes play a role. They also come into play, even to second order\ninξ, when inversion symmetry is broken.\nIn section II we establish the structure of spin-density response theory in the presence of SOC by means of exact\nspin-density functional theory in the static limit [28]. In section III w e introduce a spatial Fourier transform and an\napproximation to the dynamical response is obtained by introducing the frequency dependence of the non-interacting\nsusceptibilities. The theory then has the same structure as in the R PA. Section IV is devoted to obtaining an explicit\nexpression for the transverse susceptibility in terms of the non-in teracting susceptibilities. Expressions for these, in\nthe presence of SOC, are obtained within the tight-binding approxim ation in section V. In section VI we consider\nthe damping of the resonance in the q=0 transverse susceptibility a nd show how the present approach leads to the\nKambersky formula for the Gilbert damping parameter αwhere this is valid, namely to second order in the SOC\nparameter ξ. We do not give an explicit formula for αbeyond this order but it is clear that no intraband terms appear.\nIn section VII some experimental aspects are discussed with sugg estions for future work. The main conclusions are\nsummarized in section VIII.\nII. SPIN-DENSITY FUNCTIONAL THEORY WITH SPIN-ORBIT COUPLI NG\nThe Kohn-Sham equation takes the form\n/summationdisplay\nσ′[−δσσ′(/planckover2pi12/2m)∇2+Veff\nσσ′(r)+Hso\nσσ′]φnσ′(r) =ǫnφnσ(r) (1)\nwith the spin index σ=↑,↓corresponding to quantization along the direction of the ground-s tate magnetization in a\nferromagnet. This may be written in 2 ×2 matrix form with eigenvectors ( φn↑,φn↓)T. The density matrix is defined\nin terms of the spin components φnσ(r) of the one-electron orbitals by\nnσσ′=/summationdisplay\nnφnσ(r)φnσ′(r)∗θ(µ0−ǫn) (2)\nwhereθ(x) is the unit step function and µ0is the chemical potential. The electron density is given by\nρ(r) =/summationdisplay\nσnσσ(r) =/summationdisplay\nnσ|φnσ(r)|2θ(µ0−ǫn) (3)\nand the effective potential in (1) is\nVeff\nσσ′(r) =wσσ′(r)+δσσ′/integraldisplay\nd3r′ρ(r′)v(r−r′)+vxc\nσσ′(r) (4)\nwherewσσ′(r) is the external potential due to the crystal lattice and any magn etic fields and v(r) =e2/|r|is the\nCoulomb potential. The exchange-correlation potential vxc\nσσ′(r) is defined as δExc/δnσσ′(r), a functional derivative of\nthe exchange-correlation energy Exc. The term Hso\nσσ′in (1) is the SOC energy. A small external perturbation δwσσ′,\nfor example due to a magnetic field, changes the effective potential toVeff+δVeff, giving rise to new orbitals and\nhence to a change in density matrix δnσσ′. The equation\nδnσσ′(r) =−Ω−1/summationdisplay\nσ1σ′\n1/integraldisplay\nd3r1χ0\nσσ′σ1σ′\n1(r,r1)δVeff\nσ1σ′\n1(r1), (5)\nwhere Ω is the volume of the sample, defines a non-interacting respo nse function χ0and the full response function χ\nis defined by\nδnσσ′(r) =−Ω−1/summationdisplay\nττ′/integraldisplay\nd3r′χσσ′ττ′(r,r′)δwττ′(r′). (6)3\nFrom (4)\nδVeff\nσ1σ′\n1(r1) =δwσ1σ′\n1(r1)+/summationdisplay\nσ2σ′\n2/integraldisplay\nd3r2[v(r1−r2)δσ1σ′\n1δσ2σ′\n2+δvxc\nσ1σ′\n1(r1)\nδnσ2σ′\n2(r2)]δnσ2σ′\n2(r2) (7)\nand we may write\nδvxc\nσ1σ′\n1(r1)\nδnσ2σ′\n2(r2)=δ2Exc\nδnσ2σ′\n2(r2)δnσ1σ′\n1(r1)=Kσ1σ′\n1σ2σ′\n2(r1,r2). (8)\nCombining (5) - (8) we find the following integral equation for the spin -density response function χσσ′ττ′(r,r′):\nχσσ′ττ′(r,r′) =χ0\nσσ′ττ′(r,r′)−(Ω)−1/summationdisplay\nσ1σ′\n1/summationdisplay\nσ2σ′\n2/integraldisplay\nd3r1/integraldisplay\nd3r2χ0\nσσ′σ1σ′\n1(r,r1)[v(r1−r2)δσ1σ′\n1δσ2σ′\n2+Kσ1σ′\n1σ2σ′\n2(r1,r2)]\nχσ2σ′\n2ττ′(r2,r′).\n(9)\nThis equation is a slight generalisation of that given by Williams and von Ba rth [28]. In the static limit it is formally\nexact although the exchange-correlation energy Excis of course not known exactly. In the next section we generalise\nthe equation to the dynamical case approximately by introducing th e frequency dependence of the non-interacting\nresponse functions χ0, and also take a spatial Fourier transform. In the case where SOC is absent the result is directly\ncompared with results obtained using the RPA.\nIII. DYNAMICAL SUSCEPTIBILITIES IN THE PRESENCE OF SPIN-OR BIT COUPLING AND\nLONG-RANGE COULOMB INTERACTION\nIn general the response functions χ(r,r′) are not functions of r−r′and a Fourier representation of (9) for a\nspatially periodic system involves an infinite number of reciprocal latt ice vectors. There are two cases where this\ncomplication is avoided. The first is a homogeneous electron gas and t he second is in a tight-binding approximation\nwith a restricted atomic basis. We may then introduce Fourier trans forms of the form χ(r) =/summationtext\nqχ(q)eiq·ror\nχ(q) = (Ω)−1/integraltext\nd3rχ(r)e−iq·rand write (9) as\nχσσ′ττ′(q,ω) =χ0\nσσ′ττ′(q,ω)−/summationdisplay\nσ1σ′\n1/summationdisplay\nσ2σ′\n2χ0\nσσ′σ1σ′\n1(q,ω)Vσ1σ′\n1σ2σ′\n2(q)χσ2σ′\n2ττ′(q,ω), (10)\nwhere we have also introduced the ωdependence of χas indicated at the end of the last section. Here V(q) is an\nordinary Fourier transform, without a factor (Ω)−1, so that\nVσ1σ′\n1σ2σ′\n2(q) =v(q)δσ1σ′\n1δσ2σ′\n2+Kσ1σ′\n1σ2σ′\n2, (11)\nwherev(q) = 4πe2/q2is the usual Fourier transform of the Coulomb interaction and the s econd term is independent\nofqsinceKis a short-range spatial interaction. In the gas case it is proportio nal to a delta-function δ(r−r′) in the\nlocal-density approximation (LDA) [28] and in tight-binding it can be t aken as an on-site interaction. In both cases\nKmay be expressed in terms of a parameter Uas\nKσ1σ′\n1σ2σ′\n2=−U[δσ1σ′\n1δσ2σ′\n2δσ1σ2+δσ1σ′\n1δσ2σ′\n2δσ′\n1σ2] (12)\nwhereσ=↓,↑forσ=↑,↓. in the tight-binding case this form of Kcorresponds to a simple form of interaction\nwhich leads to a rigid exchange splitting of the bands ( [29], [22]). This is only appropriate for transition metals in a\nmodel with d bands only, hybridization with s and p bands being neglect ed. We adopt this model in order to obtain\ntransparent analytic results as far as possible. Although not as re alistic as ”first-principles” models of the electronic\nstructure it has been used, even with some quantitative success, in treating the related problem of magnetocrystalline\nanisotropy in Co/Pd structures as well as pure metals [30]. In (10) t he response functions χare per unit volume\nin the gas case but, more conveniently, may be taken as per atom in t he tight-binding case with v(q) modified to\nv(q) = 4πe2/(q2Ωa) where Ω ais the volume per atom.4\nTo show how equations (10) - (12) are related to RPA we examine two examples in the absence of SOC. First\nconsider the transverse susceptibility χ↓↑↑↓(q,ω) which is more usually denoted by χ−+(q,ω). Equation (10) becomes\nχ↓↑↑↓=χ0\n↓↑↑↓−χ0\n↓↑↑↓V↑↓↓↑χ↓↑↑↓ (13)\nand, from (11) and (12), V↑↓↓↑=K↑↓↓↑=−U. Hence\nχ↓↑↑↓=χ0\n↓↑↑↓(1−Uχ0\n↓↑↑↓)−1(14)\nwhich is just the RPA result of Izuyama et al [31] for a single-orbital Hubbard model and of Lowde and Windsor [32]\nfor a five-orbital d-band model. Clearly in the absence of SOC the Co ulomb interaction v(q) plays no part in the\ntransverse susceptibility, as is well-known. A more interesting case is the longitudinal susceptibility denoted by χmm\nin the work of Kim et al ( [26], [27]) and in [28]. This involves only the respon se functions χσσττwhich we abbreviate\ntoχστ. In fact [28]\nχmm=χ↑↑+χ↓↓−χ↑↓−χ↓↑. (15)\nWithout SOC χ0\nστtakes the form χ0\nσδστand (10) becomes\nχστ=χ0\nσδστ−/summationdisplay\nσ2χ0\nσVσσ2χσ2τ (16)\nwithVσσ2=v(q)−Uδσσ2. On solving the 2 ×2 matrix equation (16) for χστ, and using (15), we find the longitudinal\nsusceptibility in the form\nχmm=χ0\n↑+χ0\n↓−2[U−2v(q)]χ0\n↑χ0\n↓\n1+(χ0\n↑+χ0\n↓)[v(q)−U]+U[U−2v(q)]χ0\n↑χ0\n↓(17)\nwhich agrees with the RPA result that Kim et al ( [26], [27])found for a s ingle-orbitalmodel. The Coulomb interaction\nv(q) is clearly important, particularly for the uniform susceptibility with q =0, where v→ ∞. It plays an essential\nrole in enforcing particle conservation and hence in obtaining the cor rect result of Stoner theory. In view of the\ncorrespondence between our approach and the RPA method it see ms likely that when SOC is included our procedure\nusing equations (10) - (12) should be almost equivalent to that of Co sta and Muniz [19] in the case of a model with\nd-bands only. However our inclusion of the long-range Coulomb inter action will modify the results.\nIV. AN EXPLICIT EXPRESSION FOR THE TRANSVERSE SUSCEPTIBILI TY\nIn this section we obtain an explicit expression for the transverses usceptibility χ↓↑↑↓in terms of the non-interacting\nresponse functions χ0. We consider equation (10) as an equation between 4 ×4 matrices where σσ′=↓↑,↑↓,↑↑,↓↓\nlabels the rows in that order and ττ′labels columns similarly. The formal solution of (10) is then\nχ= (1+χ0V)−1χ0. (18)\nThis expression could be used directly as the basis of a numerical inve stigation similar to that of Costa and Muniz.\nHowever we wish to show that the present approach leads to a Gilber t damping parameter αin agreement with the\nKambersky formula, to second order in the SOC parameter ξwhere Kambersky’s result is valid. This requires some\nquite considerable analytic development of (18).\nFirst we partition each matrix in (18) into four 2 ×2 matrices. Thus from (11) and (12)\nV=/parenleftbigg\nV10\n0V2/parenrightbigg\n(19)\nwith\nV1=/parenleftbigg\n0−U\n−U0/parenrightbigg\n, V2=/parenleftbigg\nv−U v\nv v−U/parenrightbigg\n. (20)\nAlso\nχ=/parenleftbigg\nχ11χ12\nχ21χ22/parenrightbigg\n(21)5\nand similarly for χ0. If we write\n1+χ0V=/parenleftbigg\n1+χ0\n11V1χ0\n12V2\nχ0\n21V11+χ0\n22V2/parenrightbigg\n=/parenleftbigg\nA B\nC D/parenrightbigg\n(22)\n(18) becomes\nχ=/parenleftbigg\nS−1−S−1BD−1\n−D−1CS−1D−1+D−1CS−1BD−1/parenrightbigg/parenleftbigg\nχ0\n11χ0\n12\nχ0\n21χ0\n22/parenrightbigg\n(23)\nwhere\nS=A+BD−1C. (24)\nThe transverse susceptibility χ↓↑↑↓in which we are interested is the top right-hand element of χ11so this is the\nquantity we wish to calculate. From (23)\nχ11=S−1(χ0\n11−BD−1χ0\n21) (25)\nand, from (24) and (22),\nS= 1+χ0\n11V1−χ0\n12(V−1\n2+χ0\n22)−1χ0\n21V1. (26)\nThe elements of the 2 ×2 matrix S are calculated by straight-forward algebra and\nS11= 1−Uχ0\n↓↑↑↓+(U/Λ)[(X+χ0\n↓↓↓↓)χ0\n↑↑↑↓χ0\n↓↑↑↑−(Y+χ0\n↑↑↓↓)χ0\n↓↓↑↓χ0\n↓↑↑↑\n−(Y+χ0\n↓↓↑↑)χ0\n↑↑↑↓χ0\n↓↑↓↓+(X+χ0\n↑↑↑↑)χ0\n↓↓↑↑χ0\n↓↑↓↓](27)\nwhere\nX= [v−U]/[U(U−2v)], Y=−v/[U(U−2v)] (28)\nand\nΛ = (X+χ0\n↑↑↑↑)(X+χ0\n↓↓↓↓)−(Y+χ0\n↑↑↓↓)(Y+χ0\n↓↓↑↑) (29)\nThe other three elements of Sare given in Appendix B. The transverse susceptibility is obtained fro m (25) as\nχ↓↑↑↓= [S22(χ0\n11−BD−1χ0\n21)12−S12(χ0\n11−BD−1χ0\n21)22]/(S11S22−S12S21) (30)\nand\nBD−1=χ0\n12(V−1\n2+χ0\n22)−1. (31)\nA comparison of the fairly complex equation above for the transver se susceptibility with the simple well-known result\n(14) showsthe extent ofthe new physicsintroducedby SOC.This is due tothe coupling ofthe transversesusceptibility\nto the longitudinal susceptibility and the charge response, both of which involve the long-range Coulomb interaction.\nTo proceed further it is necessary to specify the non-interacting response functions χ0\nσσ′σ1σ′\n1which occur throughout\nthe equations above.\nV. THE NON-INTERACTING RESPONSE FUNCTIONS\nIn the tight-binding approximation the one-electron basis function s are the Bloch functions\n|kµσ∝angb∇acket∇ight=N−1/2/summationdisplay\njeik·Rj|jµσ∝angb∇acket∇ight (32)\nwherejandµare the site and orbital indices, respectively, and Nis the number of atoms in the crystal. The\nHamiltonian in the Kohn-Sham equation now takes the form\nHeff=/summationdisplay\nkµνσ(Tµν(k)+Veff\nσδµν)c†\nkµσckνσ+Hso(33)6\nwhereTµνcorresponds to electron hopping and\nVeff\nσ=−(σ/2)(∆+bex) (34)\nwhereσ= 1,−1 for spin ↑,↓respectively. Here ∆ = 2 U∝angb∇acketleftSz∝angb∇acket∇ight/NwhereSzis the total spin angular momentum, in\nunits of /planckover2pi1, and the Zeeman splitting bex= 2µBBex, whereBexis the external magnetic field and µBis the Bohr\nmagneton. The spin-orbit term Hso=ξ/summationtext\njLj·Sjtakes the second-quantized form\nHso= (ξ/2)/summationdisplay\nkµν[Lz\nµν(c†\nkµ↑ckν↑−c†\nkµ↓ckν↓)+L+\nµνc†\nkµ↓ckν↑+L−\nµνc†\nkµ↑ckν↓] (35)\nwhereLz\nµν,L±\nµνare matrix elements of the atomic orbital angular momentum operat orsLz,L±=Lx±iLyin units\nof/planckover2pi1. Within the basis of states (32) eigenstates of Hefftake the form\n|kn∝angb∇acket∇ight=/summationdisplay\nµσaσ\nnµ(k)|kµσ∝angb∇acket∇ight, (36)\nand satisfy the equation\nHeff|kn∝angb∇acket∇ight=Ekn|kn∝angb∇acket∇ight. (37)\nThus\nc†\nkµσ=/summationdisplay\nnaσ\nnµ(k)∗c†\nkn(38)\nwherec†\nkncreates the eigenstate |kn∝angb∇acket∇ight.\nThenon-interactingresponsefunction χ0\nσσ′σ1σ′\n1(q,ω) isconvenientlyexpressedastheFouriertransformofaretarde d\nGreen function by the Kubo formula\nχ0\nσσ′σ1σ′\n1(q,ω) =/summationdisplay\nk∝angb∇acketleft∝angb∇acketleft/summationdisplay\nµc†\nk+qµσckµσ′;/summationdisplay\nνc†\nkνσ1ck+qνσ′\n1∝angb∇acket∇ight∝angb∇acket∇ight0\nω (39)\nwhere the right-hand side is to be evaluated using the one-electron Hamiltonian Heff. Consequently, using (38), we\nhave\nχ0\nσσ′σ1σ′\n1(q,ω) =/summationdisplay\nkµν/summationdisplay\nmnaσ\nmµ(k+q)∗aσ′\nnµ(k)aσ1nν(k)∗aσ′\n1mν(k+q)∝angb∇acketleft∝angb∇acketleftc†\nk+qmckn;c†\nknck+qm∝angb∇acket∇ight∝angb∇acket∇ight0\nω\n=N−1/summationdisplay\nkµν/summationdisplay\nmnaσ\nmµ(k+q)∗aσ′\nnµ(k)aσ1\nnν(k)∗aσ′\n1mν(k+q)fkn−fk+qm\nEk+qm−Ekn−/planckover2pi1ω+iη.(40)\nThe last step uses the well-known form of the response function pe r atom for a non-interacting Fermi system (e.g. [33])\nandηis a small positive constant which ultimately tends to zero. The occup ation number fkn=F(Ekn−µ0) where\nFis the Fermi function with chemical potential µ0. Clearly for q= 0 the concept of intraband transitions ( m=n),\nfrequently introduced in discussions of the Kambersky formula, ne ver arises for finite ωsince the difference of the\nFermi functions in the numerator of (40) is zero. Equation (40) ma y be written in the form\nχ0\nσσ′σ1σ′\n1(q,ω) =N−1/summationdisplay\nkmnBσσ′\nmn(k,q)Bσ1′σ1\nmn(k,q)∗fkn−fk+qm\nEk+qm−Ekn−/planckover2pi1ω+iη(41)\nwhere\nBσσ′\nmn(k,q) =/summationdisplay\nµaσ\nmµ(k+q)∗aσ′\nnµ(k). (42)7\nVI. FERROMAGNETIC RESONANCE LINEWIDTH; THE KAMBERSKY FORM ULA\nWe now consider the damping of the ferromagnetic resonance in the q= 0 transverse susceptibility. The present\napproach, like the closely-related one of Costa and Muniz [19], is valid f or arbitrary strength of the SOC and can\nbe used as the basis of numerical calculations, as performed by the latter authors. However it is important to show\nanalytically that the present method leads to the Kambersky [4] for mula for the Gilbert damping parameter where\nthis is valid, namely to second order in the SOC parameter ξ. This is the subject of this section.\nIt is useful to consider first the case without SOC ( ξ= 0). The eigenstates nofHeffthen have a definite spin and\nmay be labelled nσ. It follows from (40) that χ0\nσσ′σ1σ′\n1∝δσσ′\n1δσ′σ1. Henceχ0\n12= 0 and, from (31), BD−1= 0. Also,\nfrom Appendix B, S12=S21= 0. Thus, (30) reduces to (14) as it should. Considering χ0\n↓↑↑↓(0,ω), given by (40)\nand (41), we note that state mis pure↓spin, labelled by m↓, andnis pure↑, labelled by n↑. Hence for ξ= 0\nB↓↑\nmn(k,0) =/summationdisplay\nµ∝angb∇acketleftkm|kµ∝angb∇acket∇ight∝angb∇acketleftkµ|kn∝angb∇acket∇ight=δmn (43)\nfrom closure. Thus\nχ0\n↓↑↑↓(0,ω) =N−1/summationdisplay\nknfkn↑−fkn↓\nEkn↓−Ekn↑−/planckover2pi1ω+iη(44)\nand it follows from (34) that Eknσmay be written as\nEknσ=Ekn−(σ/2)(∆+bex). (45)\nHence we find from (14) that for ξ= 0\nχ↓↑↑↓(0,ω) = (2∝angb∇acketleftSz∝angb∇acket∇ight/N)(bex−/planckover2pi1ω+iη)−1. (46)\nThus, as η→0,ℑχ↓↑↑↓(0,ω) has a sharp delta-function resonance at /planckover2pi1ω=bexas expected.\nWhen SOC is included /planckover2pi1ωacquires an imaginary part that corresponds to damping. We now pr oceed to calculate\nthis imaginary part to O(ξ2). To do this we can take ξ= 0 in the numerator of (30) so that\nχ↓↑↑↓(0,ω) =χ0\n↓↑↑↓(0,ω)/(S11−S12S21/S22) (47)\nIn factS12andS21are both O(ξ2) whileS22isO(1). Thus to obtain /planckover2pi1ωtoO(ξ2) we need only solve S11= 0.\nFurthermore all response functions such as χ0\n↑↑↑↓, with all but one spins in the same direction, are zero for ξ= 0 and\nneed only be calculated to O(ξ) in (27). We show below that to this order they vanish, so that to O(ξ2) the last term\ninS11is zero and we only have to solve the equation 1 −Uχ0\n↓↑↑↓= 0 for/planckover2pi1ω. This means that to second order in ξ\nthe shift in resonance frequency and the damping do not depend on the long-range Coulomb interaction.\nTo determine χ0\n↑↑↑↓(0,ω) to first order in ξfrom (40) we notice that states nmust be pure ↑spin, that is |kn∝angb∇acket∇ight=\n|kn↑∝angb∇acket∇ight, while states mmust be calculated using perturbation theory. The latter states m ay be written\n|km1∝angb∇acket∇ight=|km↑∝angb∇acket∇ight−ξ/summationdisplay\npσ∝angb∇acketleftkpσ|hso|km↑∝angb∇acket∇ight\nEkm↑−Ekpσ|kpσ∝angb∇acket∇ight (48)\n|km2∝angb∇acket∇ight=|km↓∝angb∇acket∇ight−ξ/summationdisplay\npσ∝angb∇acketleftkpσ|hso|km↓∝angb∇acket∇ight\nEkm↓−Ekpσ|kpσ∝angb∇acket∇ight, (49)\nwhere we have put Hso=ξhso, and to first order in ξ,\nχ0\n↑↑↑↓=1\nN/summationdisplay\nkµν/summationdisplay\nmn(a↑\nm1µ∗anµa∗\nnνa↓\nm1νfkn↑−fkm↑\nEkm↑−Ekn↑−/planckover2pi1ω+iη+a↑\nm2µ∗anµa∗\nnνa↓\nm2νfkn↑−fkm↓\nEkm↓−Ekn↑−/planckover2pi1ω+iη) (50)\nwithaσ\nmsµ=∝angb∇acketleftkµσ|kms∝angb∇acket∇ight,s= 1,2,andanµ=∝angb∇acketleftkµ|kn∝angb∇acket∇ightis independent of spin. Since a↓\nm1ν∼ξwe takea↑\nm1µ=amµin\nthe first term of (50). Also/summationtext\nµa∗\nmµanµ=δmnby closure so that the first term of (50) vanishes since the differen ce\nof Fermi functions is zero. Only the second term of χ0\n↑↑↑↓remains and this becomes, by use of (49),\nχ0\n↑↑↑↓=−ξ/summationdisplay\nkµν/summationdisplay\nmnp∝angb∇acketleftkp↑ |hso|km↓∝angb∇acket∇ight∗\nEkm↓−Ekp↑a∗\npµanµa∗\nnνamνfkn↑−fkm↓\nEkm↓−Ekn↑−/planckover2pi1ω+iη. (51)8\nAgain using closure only terms with p=m=nsurvive and the matrix element of hsobecomes\n∝angb∇acketleftkn↑ |/summationdisplay\njLj·Sj|kn↓∝angb∇acket∇ight=1\n2∝angb∇acketleftkn|L−|kn∝angb∇acket∇ight= 0 (52)\ndue to the quenching of total orbital angular momentum L=/summationtext\njLj[30]. Thus, to first order in ξ,χ0\n↑↑↑↓(0,ω), and\nsimilar response functions with one reversed spin, are zero. Hence we have only to solve 1 −Uχ0\n↓↑↑↓= 0 to obtain\nℑ(/planckover2pi1ω) toO(ξ2). Here we assume the system has spatial inversion symmetry witho ut which the quenching of orbital\nangular momentum, as expressed by (52), no longer pertains [30]. We briefly discuss the consequences of a breakdown\nof inversion symmetry at the end of this section.\nOn introducing the perturbed states (48) and (49) we write (41) in the form\nχ0\n↓↑↑↓(0,ω) =1\nN/summationdisplay\nkmn(|B↓↑\nm1n1|2fkn1−fkm1\nEkm1−Ekn1−/planckover2pi1ω+iη+|B↓↑\nm1n2|2fkn2−fkm1\nEkm1−Ekn2−/planckover2pi1ω+iη\n+|B↓↑\nm2n1|2fkn1−fkm2\nEkm2−Ekn1−/planckover2pi1ω+iη+|B↓↑\nm2n2|2fkn2−fkm2\nEkm2−Ekn2−/planckover2pi1ω+iη).(53)\nClearlyB↓↑\nm1n1andB↓↑\nm2n2are of order ξ,B↓↑\nm1n2isO(ξ2) andB↓↑\nm2n1isO(1). We therefore neglect the term |B↓↑\nm1n2|2\nand, using (48) and (49), we find\nB↓↑\nm1n1=−B↓↑\nm2n2=ξ\n2∝angb∇acketleftkm|L−|kn∝angb∇acket∇ight\nEkn↓−Ekm↑. (54)\nThe evaluation of |B↓↑\nm2n1|2requires more care. It appears at first sight that to obtain this to O(ξ2) we need to include\nsecond order terms in the perturbed eigenstates given by (48) an d (49). However it turns out that these terms do not\nin fact contribute to |B↓↑\nm2n1|2toO(ξ2) so we shall not consider them further. Then we find\nB↓↑\nm2n1=δmn−ξ∝angb∇acketleftkm|Lz|kn∝angb∇acket∇ight\nEkn−Ekm−ξ2\n4/summationdisplay\np∝angb∇acketleftkm|Lz|kp∝angb∇acket∇ight∝angb∇acketleftkp|Lz|kn∝angb∇acket∇ight\n(Ekm−Ekp)(Ekn−Ekp)(55)\nand hence to O(ξ2)\n|B↓↑\nm2n1|2=δmn(1−ξ2\n2/summationdisplay\np|∝angb∇acketleftkm|Lz|kp∝angb∇acket∇ight|2\n(Ekm−Ekp)2)+ξ2|∝angb∇acketleftkm|Lz|kn∝angb∇acket∇ight|2\n(Ekn−Ekm)2(56)\nThe contribution of this quantity to χ0\n↓↑↑↓(0,ω) in (53) may be written to O(ξ2) as\n1\nN/summationdisplay\nkmn|B↓↑\nm2n1|2fkn1−fkm2\nEkm2−Ekn1−bex+iη(1−bex−/planckover2pi1ω\nEkm2−Ekn1−bex+iη). (57)\nThisisobtainedbyintroducingtheidentity −/planckover2pi1ω=−bex+(bex−/planckover2pi1ω)intherelevantdenominatorin(53),andexpanding\nto first order in bex−/planckover2pi1ωwhich turns out to be O(ξ2). The remaining factors of this second term in (57) may then\nbe evaluated with ξ= 0, as at the beginning of this section, so that this term becomes ( /planckover2pi1ω−bex)/(2U2∝angb∇acketleftSz∝angb∇acket∇ight). By\ncombining equations (53), (54) and (57), and ignoring some real te rms, we find that the equation 1 −Uχ0\n↓↑↑↓(0,ω) = 0\nleads to the relation\nℑ(/planckover2pi1ω) =πξ2/(2∝angb∇acketleftSz∝angb∇acket∇ight)/summationdisplay\nknm[(fkn↑−fkm↓)|∝angb∇acketleftkm|Lz|kn∝angb∇acket∇ight|2δ(Ekm↓−Ekn↑−bex)\n+(1/4)(fkn↑+fkn↓−fkm↑−fkm↓)|∝angb∇acketleftkm|L−|kn∝angb∇acket∇ight|2δ(Ekm−Ekn−bex)](58)\nThe Gilbert damping parameter αis given by ℑ(/planckover2pi1ω)/bex(e.g. [39]) and in (58) we note that\n(fkn↑−fkm↓)δ(Ekm↓−Ekn↑−bex) = [F(Ekn↑−µ0)−F(Ekn↑+bex−µ0)]δ(Ekm↓−Ekn↑−bex)\n=bexδ(Ekn↑−µ0)δ(Ekm↑−µ0)(59)\nto first order in bexat temperature T= 0. Similarly\n(fknσ−fkmσ)δ(Ekm−Ekn−bex) =bexδ(Eknσ−µ0)δ(Ekmσ−µ0). (60)9\nThus from (58)\nα=πξ2/(2∝angb∇acketleftSz∝angb∇acket∇ight)/summationdisplay\nknm|∝angb∇acketleftkm|Lz|kn∝angb∇acket∇ight|2δ(Ekn↑−µ0)δ(Ekm↓−µ0)\n+πξ2/(8∝angb∇acketleftSz∝angb∇acket∇ight)/summationdisplay\nknmσ|∝angb∇acketleftkm|L−|kn∝angb∇acket∇ight|2δ(Eknσ−µ0)δ(Ekmσ−µ0)(61)\ncorrect to O(ξ2). We note that there is no contribution from intraband terms since ∝angb∇acketleftkn|L|kn∝angb∇acket∇ight= 0. It is straight-\nforward to show that to O(ξ2) this is equivalent to the expression\nα=π/(2∝angb∇acketleftSz∝angb∇acket∇ight)/summationdisplay\nknm/summationdisplay\nσσ′|Amσ,nσ′(k)|2δ(Ekmσ−µ0)δ(Eknσ′−µ0) (62)\nwhere\nAmσ,nσ′(k) =ξ∝angb∇acketleftkmσ|[S−,hso]|knσ′∝angb∇acket∇ight (63)\nandS−is the total spin operator/summationtext\njS−\njwithS−\nj=Sx\nj−iSy\nj. This may be written more concisely as\nα=π/(2∝angb∇acketleftSz∝angb∇acket∇ight)/summationdisplay\nknm|Amn(k)|2δ(Ekm−µ0)δ(Ekn−µ0) (64)\nwith\nAmn(k) =ξ∝angb∇acketleftkm|[S−,hso]|kn∝angb∇acket∇ight (65)\nand the understanding that the one-electron states km,knare calculated in the absence of SOC. Equation (64) is the\nstandard form of the Kambersky formula ( [4], [9]) but in the literatur e SOC is invariably included in the calculation\nof the one-electronstates. This means that the intraband terms withm=nno longer vanish. They involve the square\nof a delta-function and this problem is always addressed by invoking t he effect of impurity and/or phonon scattering\nto replace the delta-functions by Lorentzians of width proportion al to an inverse relaxation time parameter τ−1. Then\nas one approaches a perfect crystal ( τ→ ∞) the intraband contribution to αtends to infinity. This behaviour is\nillustrated in many papers ( [7], [10], [13], [14]). In fig. 1 of [14] it is shown c learly that αremains finite if one does\nnot include SOC in calculating the one-electron states. However the effect of not including SOC is not confined to\ntotal removal of the intraband contribution. The remaining interb and contribution is increased considerably in the\nlow scattering rate regime, by almost an order of magnitude in the ca se of Fe. This makes αalmost independent of\nscattering rate in Fe which may relate to its observed temperature independence [15]. The corresponding effect in\nCo is insufficient to produce the increase of αat low scattering rate inferred from its temperature dependence . The\nnon-inclusion of SOC in calculating the one-electron states used in th e Kambersky formula clearly makes a major\nqualitative and quantitative change in the results. This occurs as so on as intraband terms become dominant in\ncalculations where they are included. For Fe, Co and Ni this corresp onds to impurity content and temperature such\nthat the scattering rate 1 /τdue to defects and/or phonons is less that about 1014sec−1( [7, 14]). Typically these\nmetals at room temperature find themselves well into the high scatt ering-rate regime where the damping rate can\nbe reliably estimated from the Kambersky interband term, with or wit hout SOC included in the band structure [35].\nThe physics at room temperature is not particularly interesting. On e needs to lower the temperature into the low\nscattering-rate regime where intraband terms, if they exist, will d ominate and lead to an anomalous ξ3dependence of\nthe damping on spin-orbit parameter ξ( [4], [8], [14]). The origin of this behaviour is explained in [4], [14]. It arises\nin theksum of (64) from a striplike region on the Fermi surface around a line where two different energy bands cross\neach other in the absence of SOC. The strip width is proportional to ξ, or more precisely |ξ|. SinceAnn(k) is of\norderξthe contribution of intraband terms in (64) is proportional to |ξ|3. Thus the intraband terms lead to terms\ninαwhich diverge in the limit τ−1→0 and are non-analytic functions of ξ. The calculation of αin this section can\nbe extended to higher powers of ξthan the second. No intraband terms appear and the result is an an alytic power\nseries containing only even powers of ξ.\nThe interband term in Kambersky’s formula can be given a very simple in terpretation in terms of Fermi’s ”golden\nrule” for transition probability [5]. This corresponds to second orde r perturbation theory in the spin-orbit interaction.\nThe decay of a uniform mode ( q= 0) magnon into an electron-hole pair involves the transition of an ele ctron from\nan occupied state to an unoccupied state of the same wave-vecto r. This is necessarily an interband transition and\nthe states involved in the matrix element are unperturbed, that is c alculated in the absence of SOC. A quite different\napproach has been adopted to try and find a physical interpretat ion of Kambersky’s intraband term ( [5, 8]). This\nemploys Kambersky’s earlier ”breathing Fermi surface” model ( [3, 34]) whose range of validity is uncertain.10\nWe now briefly discuss the consequences of a breakdown of spatial inversion symmetry so that total orbital angular\nmomentum is not quenched. In general response functions such a sχ0\n↑↑↑↓(0,ω) with one reversed spin are no longer\nzero to first order in ξ. HenceS11is not given to order ξ2just by the first two terms of (27) but involves further terms\nwhich depend explicitly on the long-range Coulomb interaction. Conse quentlyαhas a similar dependence which\ndoes not emerge from the torque-correlation approach. In Appe ndix A it is pointed out how the direct proof of the\nKambersky formula breaks down in the absence of spatial inversion symmetry.\nVII. EXPERIMENTAL ASPECTS\nThe inclusion of intraband terms in the Kambersky formula, despite t heir singular nature, has gained acceptance\nbecause they appear to explain a rise in intrinsic damping parameter αat low temperature which is observed in some\nsystems [15]. The calculated intraband contribution to αis proportional to the relaxation time τand it is expected\nthat, due to electron-phonon scattering, τwill increase as the temperature is reduced. This is in qualitative agre ement\nwith data [15] for Ni and hcp Co. Also a small 10% increase in αis observed in Co 2FeAl films as the temperature is\ndecreased from 300 K to 80 K [36]. However in Fe the damping αis found to be independent of temperature down to\n4 K [15]. Very recent measurements [16] on FePt films, with varying an tisite disorder xintroduced into the otherwise\nwell-ordered structure, show that αincreases steadily as xincreases from 3 to 16%. Hence αincreases monotonically\nwith scattering rate 1 /τas expected from the Kambersky formula in the absence of intraba nd terms. Furthermore\nforx= 3% it is found that αremains almost unchanged when the temperature is decreased fro m 200 to 20 K. Ma et\nal [16] therefore conclude that there is no indication of an intraban d term in α. From the present point of view the\norigin of the observed low temperature increase of αin Co and Ni is unclear. Further experimental work to confirm\nthe results of Bhagat and Lubitz [15] is desirable.\nThe second unusual feature of the intraband term in Kambersky’s formula for αis its|ξ|3dependence on the SOC\nparameter ξ. This contrasts with the ξ2dependence of the interband contribution which has been observe d in a\nnumber of alloys at room temperature [17]. Recently this behaviour has been seen very precisely in FePd 1−xPtxalloys\nwhereξcan be varied over a wide range by varying x[18]. Unfortunately this work has not been extended to the\nlow temperature regime where the |ξ|3dependence, if it exists, should be seen. It would be particularly inte resting to\nsee low temperature data for NiPd 1−xPtxand CoPd 1−xPtxsince it is in Ni and Co where the intraband contribution\nhas been invoked to explain the low temperature behaviour of α. From the present point of view, with the intraband\nterm absent, one would expect ξ2behaviour over the whole temperature range.\nVIII. CONCLUSIONS AND OUTLOOK\nIn this paper we analyse two methods which are used in the literature to calculate the damping in magnetization\ndynamics due to spin-orbit coupling. The first common approach is to employ Kambersky’s[4] formula for the Gilbert\ndamping parameter αwhich delivers an infinite value for a pure metal if used beyond second order in the spin-orbit\nparameter ξ. The second approach [19] is to calculate numerically the line-width of the ferromagnetic resonance seen\nin the uniform transverse spin susceptibility. This is always found to b e finite, corresponding to finite α. We resolve\nthis apparent inconsistency between the two methods by an analyt ic treatment of the Costa-Muniz approach for the\nsimplified model of a ferromagnetic metal with d-bands only. It is sho wn that this method leads to the Kambersky\nresult correct to second order in ξbut Kambersky’s intraband scattering term, taking the non-analy tic form |ξ|3, is\nabsent. Higher order terms in the present work are analytic even p owers of ξ. The absence of Kambersky’s intraband\nterm is the main result of this paper and it is in agreement with the conc lusion that Ma et al [16] draw from their\nexperiments on FePt films. Further experimental work on the depe ndence of damping on electron scattering-rate and\nspin-orbit parameter in other systems is highly desirable.\nA secondaryconclusionis that beyond second orderin ξsome additional physics ariseswhich has not been remarked\non previously. This is the role of long-range Coulomb interaction which is essential for a proper treatment of the\nlongitudinal susceptibility and charge response to which the transv erse susceptibility is coupled by spin-orbit interac-\ntion. Costa and Muniz [19] stress this coupling but fail to introduce t he long-range Coulomb interaction. Generally,\nhowever, it seems unnecessary to go beyond second order in ξ[17, 18] and for most bulk systems Kambersky’s for-\nmula, with electron states calculated in the absence of SOC, should b e adequate. However in systems without spatial\ninversion symmetry, which include layered structures of practical importance, the Kambersky formulation may be\ninadequate even to second order in ξ. The long-range Coulomb interaction can now play a role.\nAn important property of ferromagnetic systems without inversio n symmetry is the Dzyaloshinskii-Moriya inter-\naction (DMI) which leads to an instability of the uniform ferromagnet ic state with the appearance of a spiral spin\nstructure or a skyrmion structure. This has been studied extens ively in bulk crystals like MnSi [37] and in layered11\nstructures [38]. The spiral instability appears as a singularity in the t ransverse susceptibility χ(q,0) at a value of q\nrelated to the DMI parameter. The method of this paper has been u sed to obtain a novel closed form expression for\nthis parameter which will be reported elsewhere.\nIn this paper we have analysed in some detail the transverse spin su sceptibility χ↓↑↑↓but combinations of some of\nthe 15 other response functions merit further study. Mixed char ge-spin response arising from spin-orbit coupling is\nof particular interest for its relation to phenomena like the spin-Hall effect.\nAppendix A: A direct derivation of the Kambersky formula\nIn this appendix we give a rather general derivation of the Kambers ky formula for the Gilbert damping parameter\nαwith an emphasis on its restriction to second order in the spin-orbit in teraction parameter ξ.\nWe consider a general ferromagnetic material described by the ma ny-body Hamiltonian\nH=H1+Hint+Hext (A1)\nwhereH1is a one-electron Hamiltonian of the form\nH1=Hk+Hso+V. (A2)\nHereHkis the total kinetic energy, Hso=ξhsois the spin-orbit interaction, Vis a potential term, Hintis the\nCoulomb interaction between electrons and Hextis due to an external magnetic field Bexin thezdirection. Thus\nHext=−Szbexwherebex= 2µBBex, as in (34), and Szis thezcomponent of total spin. Both HsoandVcan contain\ndisorderalthough in this paper we consider a perfect crystal. Follow ingthe general method of Edwardsand Fisher [40]\nwe use equations of motion to find that the dynamical transverse s usceptibility χ(ω) =χ−+(0,ω) satisfies [39]\nχ(ω) =−2∝angb∇acketleftSz∝angb∇acket∇ight\n/planckover2pi1ω−bex+ξ2\n(/planckover2pi1ω−bex)2(χF(ω)−ξ−1∝angb∇acketleft[F−,S+]∝angb∇acket∇ight) (A3)\nwhere\nχF(ω) =/integraldisplay\n∝angb∇acketleft∝angb∇acketleftF−(t),F+∝angb∇acket∇ight∝angb∇acket∇ighte−iωtdt (A4)\nwithF−= [S−,hso]. This follows since S−commutes with other terms in H1and with Hint. For small ω,χis\ndominated by the spin wave pole at /planckover2pi1ω=bext+/planckover2pi1δωwhereδω∼ξ2, so that\nχ(ω) =−2∝angb∇acketleftSz∝angb∇acket∇ight\n/planckover2pi1(ω−δω)−bex. (A5)\nFollowing [39] we compare (A3) and (A5) in the limit /planckover2pi1δω≪/planckover2pi1ω−bexto obtain\n−2∝angb∇acketleftSz∝angb∇acket∇ight/planckover2pi1δω=ξ2(χF(ω)−ξ−1∝angb∇acketleft[F−,S+]∝angb∇acket∇ight) =ξ2[ lim\n/planckover2pi1ω→bexχξ=0\nF(ω)−lim\nξ→0(1\nξ∝angb∇acketleft[F−,S+]∝angb∇acket∇ight)] (A6)\ncorrect to order ξ2. It is important to note that the limit ξ→0 within the bracket must be taken before putting\n/planckover2pi1ω=bex. If we put /planckover2pi1ω=bexfirst it is clear from (A3) that the quantity in brackets would vanish, giving the incorrect\nresultδω= 0. Furthermore it may be shown [M. Cinal, private communication] th at the second term in the bracket\nis real. Hence\nℑ(/planckover2pi1ω) =−ξ2\n2∝angb∇acketleftSz∝angb∇acket∇ightlim\n/planckover2pi1ω→bexℑ[χξ=0\nF(ω)]. (A7)\nKambersky [4] derived this result, using the approach of Mori and K awasaki ( [41] [42]), without noting its restricted\nvalidity to second order in ξ. This restriction is crucial since, as discussed in the main paper, it av oids the appearance\nof singular intraband terms. Oshikawa and Affleck emphasise strong ly a similar restriction in their related work on\nelectron spin resonance (Appendix of [43]).\nEquation (A7) is an exact result even in the presence of disorder in t he potential and spin-orbit terms of the\nHamiltonian. In the following we assume translational symmetry.12\nTo obtain the expression (61) for α=ℑ(/planckover2pi1ω)/bex, which is equivalent to Kambersky’s result (62), it is necessary to\nevaluate the response function χξ=0\nF(ω) in tight-binding-RPA. Using (35)we find\nF−=/summationdisplay\nkµν[Lz\nµνc†\nkµ↓ckν↑+(1/2)L−\nµν(c†\nkµ↓ckν↓−c†\nkµ↑ckν↑)]. (A8)\nHence\nχξ=0\nF=/summationdisplay\nµν/summationdisplay\nαβ[Lz\nµνLz\nβαGµ↓ν↑,β↑α↓+(1/4)L−\nµνL+\nβα(Gµ↓ν↓,β↓α↓+Gµ↑ν↑,β↑α↑−Gµ↓ν↓,β↑α↑−Gµ↑ν↑,β↓α↓)] (A9)\nwhere\nGµσνσ′,βτατ′=∝angb∇acketleft∝angb∇acketleft/summationdisplay\nkc†\nkµσckνσ′;/summationdisplay\nuc†\nuβτcuατ′∝angb∇acket∇ight∝angb∇acket∇ightω. (A10)\nThe Green function Gis to be calculated in the absence of SOC ( ξ= 0). Within RPA it satisfies an equation of the\nform\nGµσνσ′,βτατ′=G0\nµσνσ′,βτατ′−/summationdisplay\nµ1σ1ν1σ′\n1/summationdisplay\nµ2σ2ν2σ′\n2G0\nµσνσ′,µ1σ1ν1σ′\n1Vµ1σ1ν1σ′\n1,µ2σ2ν2σ′\n2Gµ2σ2ν2σ′\n2,βτατ′ (A11)\nwhereG0is the non-interacting (Hartree-Fock) Green function and\nVµ1σ1ν1σ′\n1,µ2σ2ν2σ′\n2=Vσ1σ′\n1σ2σ′\n2(q)δµ1ν1δµ2ν2 (A12)\nwithV(q) given by (11) and (12). Hence\nGµσνσ′,βτατ′=G0\nµσνσ′,βτατ′−/summationdisplay\nµ1σ1σ′\n1/summationdisplay\nµ2σ2σ′\n2G0\nµσνσ′,µ1σ1µ1σ′\n1Vσ1σ′\n1σ2σ′\n2Gµ2σ2µ2σ′\n2,βτατ′. (A13)\nThe form of the interaction Vgiven in (A12) is justified by the connection between (A13) and (10) , withq= 0. To\nsee this connection we note that χσσ′ττ′=/summationtext\nµνGµσµσ′,ντντ′and that (A13) then leads to (10) which is equivalent to\nRPA. On substituting (A13) into (A9) we see that the contributions from the second term of (A13) contain factors\nof the form\n/summationdisplay\nµνµ1Lz\nµνG0\nµ↓ν↑,µ1↑µ1↓,/summationdisplay\nµνµ1L−\nµνG0\nµσνσ,µ 1σ1µ1σ1. (A14)\nWe now show that such factors vanish owing to quenching of orbital angular momentum in the system without SOC\n(ξ= 0). Hence the Green functions Gin (A9) can be replaced by the non-interacting ones G0. The non-interacting\nGreen functions G0are of a similar form to χ0in (40) and for ξ= 0 may be expressed in terms of the quantities\nanµ=∝angb∇acketleftkµ|kn∝angb∇acket∇ightwhere|kn∝angb∇acket∇ightis a one-electron eigenstate as introduced in section VI. Hence we fi nd, in the same way\nthat (44) emerged,\n/summationdisplay\nµνµ1Lz\nµνG0\nµ↓ν↑,µ1↑µ1↓=/summationdisplay\nµν/summationdisplay\nknLz\nµνa∗\nnµanνfkn↑−fkn↓\n∆+bex−/planckover2pi1ω+iη. (A15)\nAlso by closure\n/summationdisplay\nµνLz\nµνa∗\nnµanν=∝angb∇acketleftkn|Lz|kn∝angb∇acket∇ight= 0, (A16)\nthe last step following from quenching of total orbital angular mome ntum. The proof that the second expression\nin (A14) vanishes is very similar.\nHence we can insert the non-interacting Green functions G0in (A9) and straight-forwardalgebra, with use of (A7),\nleads to (58). At the end of section VI this is shown to be equivalent t o the Kambersky formula for α. We emphasize\nagain that the present proof is valid only to order ξ2so that the one-electron states used to evaluate the formula\nshould be calculated in the absence of SOC.\nThis proof relies on the quenching of orbital angular momentum which does not occur in the absence of spatial\ninversion symmetry. When this symmetry is broken it is not difficult to s ee that the second term of (A13) gives a\ncontribution to the first term on the right of (A9) which contains th eq= 0 spin-wave pole and diverges as /planckover2pi1ω→bex.\nHence the proof of the torque-correlationformula (A7) collapses . The method of section VI must be used as discussed\nat the end of that section.13\nAppendix B: Elements of S\nThe element S11of the matrix Sis given in (27). The remaining elements are given below.\nS12=−Uχ0\n↓↑↓↑+(U/Λ)[(X+χ0\n↓↓↓↑)χ0\n↑↑↓↑χ0\n↓↑↑↑−(Y+χ0\n↑↑↓↓)χ0\n↓↓↓↑χ0\n↓↑↑↑\n−(Y+χ0\n↓↓↑↑)χ0\n↑↑↓↑χ0\n↓↑↓↓+(X+χ0\n↑↑↑↑)χ0\n↓↓↓↑χ0\n↓↑↓↓](B1)\nS21=−Uχ0\n↑↓↑↓+(U/Λ)[(X+χ0\n↓↓↓↓)χ0\n↑↑↑↓χ0\n↑↓↑↑−(Y+χ0\n↑↑↓↓)χ0\n↓↓↑↓χ0\n↑↓↑↑\n−(Y+χ0\n↓↓↑↑)χ0\n↑↑↑↓χ0\n↑↓↓↓+(X+χ0\n↑↑↑↑)χ0\n↓↓↑↓χ0\n↑↓↓↓](B2)\nS22= 1−Uχ0\n↑↓↓↑+(U/Λ)[(X+χ0\n↓↓↓↓)χ0\n↑↑↓↑χ0\n↑↓↑↑−(Y+χ0\n↑↑↓↓)χ0\n↓↓↓↑χ0\n↑↓↑↑\n−(Y+χ0\n↓↓↑↑)χ0\n↑↑↓↑χ0\n↑↓↓↓+(X+χ0\n↑↑↑↑)χ0\n↓↓↓↑χ0\n↑↓↓↓](B3)\nAcknowledgement\nMy recent interest in Gilbert damping arose through collaboration wit h O. Wessely, E. Barati, M. Cinal and A.\nUmerski. I am grateful to them for stimulating discussion and corre spondence. The specific work reported here arose\ndirectly from discussion with R.B.Muniz and I am particularly grateful t o him and his colleague A. T. Costa for this\nstimulation.\nReferences\n[1] Landau L D, Lifshitz E M and Pitaevski L P 1980 Statistical Physics part2 (Oxford: Pergamon)\n[2] Gilbert T L 1955 Phys. 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Lett. 98117601\n[18] He P, Ma X, Zhang JW, Zhao HB, L¨ upke G, Shi Z and Zhou SM 201 3Phys. Rev. Lett. 110077203\n[19] Costa A T and Muniz R B 2015 Phys. Rev. B92014419\n[20] Brataas A, Tserkovnyak Y and Bauer GEW 2008 Phys. Rev. Lett. 101037207\n[21] Starikov A, Kelly PJ, Brataas A, Tserkovnyak Y and Bauer GEW 2010 Phys. Rev. Lett. 105236601\n[22] Costa A T, Muniz R B, Lounis S, Klautau A B and Mills D L 2010 Phys. Rev. B82014428\n[23] Rajagopal A K, Brooks H and Ranganathan N R 1967 Nuovo Cimento Suppl. 5807\n[24] Rajagopal A K 1978 Phys. Rev. B172980\n[25] Low G G 1969 Adv. Phys. 18371\n[26] Kim D J, Praddaude H C and Schwartz B B 1969 Phys. Rev. Lett. 23419\n[27] Kim D J, Schwartz B B and Praddaude H C 1973 Phys. Rev. B7205\n[28] Williams A R and von Barth U 1983, in Theory of the Inhomogeneous Electron Gas eds. Lundqvist S and March N H\n(Plenum)14\n[29] Edwards D M 1984, in Moment Formation in Solids NATO Advanced Study Institute, Series B: Physics Vol. 117, e d.\nBuyers W J L (Plenum)\n[30] Cinal M and Edwards D M 1997 Phys. Rev B553636\n[31] Izuyama T, Kim D J and Kubo R 1963 J. Phys. Soc. Japan 181025\n[32] Lowde R D and Windsor C G 1970 Adv. Phys. 19813\n[33] Doniach S and Sondheimer E H 1998 Green’s Functions for Solid State Physicists Imperial College Press\n[34] Kunes J and Kambersky V 2002 Phys. Rev B65212411\n[35] Gilmore K, Garate I, MacDonald AH and Stiles MD Phys. Rev B84224412\n[36] Yuan HC, Nie SH, Ma TP, Zhang Z, Zheng Z, Chen ZH, Wu YZ, Zha o JH, Zhao HB and Chen LY 2014 Appl. Phys. Lett.\n105072413\n[37] Grigoriev SV, Maleyev SV, Okorokov AI, Chetverikov Yu. O, B¨ oni P, Georgii R, Lamago D, Eckerlebe H and Pranzas K\n2006Phys. Rev. B74214414\n[38] von Bergmann K, Kubetzka A, Pietzsch O and Wiesendanger R 2014J. Phys. Condens. Matter 26394002\n[39] Edwards D M and Wessely O 2009 J. Phys. Condens. Matter 21146002\n[40] Edwards D M and Fisher B 1971 J. Physique 32C1 697\n[41] Mori H 1965 Prog. Theor. Phys. 33423\n[42] Mori H and Kawasaki K 1962 Prog. Theor. Phys. 27529\n[43] Oshikawa M and Affleck I 2002 Phys. Rev. B65134410"    },    {        "title": "1507.06505v2.Nanomagnet_coupled_to_quantum_spin_Hall_edge__An_adiabatic_quantum_motor.pdf",        "content": "Nanomagnet coupled to quantum spin Hall edge: An adiabatic quantum\nmotor\nLiliana Arrachea\nDepartamento de F\u0013 \u0010sica, FCEyN, Universidad de Buenos Aires and IFIBA, Pabell\u0013 on I, Ciudad Universitaria, 1428 CABA\nand International Center for Advanced Studies, UNSAM, Campus Miguelete, 25 de Mayo y Francia, 1650 Buenos Aires,\nArgentina\nFelix von Oppen\nDahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universit at Berlin, 14195 Berlin, Germany\nAbstract\nThe precessing magnetization of a magnetic islands coupled to a quantum spin Hall edge pumps charge\nalong the edge. Conversely, a bias voltage applied to the edge makes the magnetization precess. We point\nout that this device realizes an adiabatic quantum motor and discuss the e\u000eciency of its operation based on\na scattering matrix approach akin to Landauer-B uttiker theory. Scattering theory provides a microscopic\nderivation of the Landau-Lifshitz-Gilbert equation for the magnetization dynamics of the device, including\nspin-transfer torque, Gilbert damping, and Langevin torque. We \fnd that the device can be viewed as\na Thouless motor, attaining unit e\u000eciency when the chemical potential of the edge states falls into the\nmagnetization-induced gap. For more general parameters, we characterize the device by means of a \fgure\nof merit analogous to the ZT value in thermoelectrics.\n1. Introduction\nFollowing Ref. [1], Meng et al. [2] recently showed\nthat a transport current \rowing along a quantum\nspin Hall edge causes a precession of the magneti-\nzation of a magnetic island which locally gaps out\nthe edge modes (see Fig. 1 for a sketch of the de-\nvice). The magnetization dynamics is driven by the\nspin transfer torque exerted on the magnetic island\nby electrons backscattering from the gapped region.\nIndeed, the helical nature of the edge state implies\nthat the backscattering electrons reverse their spin\npolarization, with the change in angular momen-\ntum transfered to the magnetic island. This e\u000bect\nis not only interesting in its own right, but may also\nhave applications in spintronics.\nCurrent-driven directed motion at the nanoscale\nhas also been studied for mechanical degrees of free-\ndom, as motivated by progress on nanoelectrome-\nchanical systems. Qi and Zhang [3] proposed that\na conducting helical molecule placed in a homo-\ngeneous electrical \feld could be made to rotate\naround its axis by a transport current and pointedout the intimate relations with the concept of a\nThouless pump [4]. Bustos-Marun et al. [5] devel-\noped a general theory of such adiabatic quantum\nmotors, used it to discuss their e\u000eciency, and em-\nphasized that the Thouless motor discussed by Qi\nand Zhang is optimally e\u000ecient.\nIt is the purpose of the present paper to em-\nphasize that the current-driven magnetization dy-\nnamics is another { perhaps more experimentally\nfeasible { variant of a Thouless motor and that\nthe theory previously developed for adiabatic quan-\ntum motors [5] is readily extended to this device.\nThis theory not only provides a microscopic deriva-\ntion of the Landau-Lifshitz-Gilbert equation for the\ncurrent-driven magnetization dynamics, but also al-\nlows one to discuss the e\u000eciency of the device and\nto make the relation with the magnetization-driven\nquantum pumping of charge more explicit.\nSpeci\fcally, we will employ an extension of the\nLandauer-B uttiker theory of quantum transport\nwhich includes the forces exerted by the electrons\non a slow classical degree of freedom [6, 7, 8, 9].\nPreprint submitted to Elsevier October 11, 2018arXiv:1507.06505v2  [cond-mat.mes-hall]  8 Sep 2015\u0001\u0001\u0001\u0001exeyezeθFigure 1: (Color online) Schematic setup. A nanomagnet\nwith magnetic moment Mcouples to a Kramers pair of edge\nstates of a quantum spin Hall insulator. The e\u000bective spin\ncurrent produces a spin-transfer toque and the magnetic mo-\nment precesses.\nMarkus B uttiker developed Landauer's vision of\nquantum coherent transport as a scattering prob-\nlem into a theoretical framework [10, 11] and ap-\nplied this scattering theory of quantum transport\nto an impressive variety of phenomena. These ap-\nplications include Aharonov-Bohm oscillations [12],\nshot noise and current correlations [11, 13, 14], as\nwell as edge-state transport in the integer Hall ef-\nfect [15] and topological insulators [16]. Frequently,\nB uttiker's predictions based on scattering theory\nprovided reference points with which other theo-\nries { such as the Keldysh Green-function formal-\nism [17, 18, 19, 20] or master equations [21] { sought\nto make contact.\nIn the present context, it is essential that scat-\ntering theory also provides a natural framework to\nstudy quantum coherent transport in systems un-\nder time-dependent driving. For adiabatic driving,\nB uttiker's work with Thomas and Pr^ etre [22] was\ninstrumental in developing a description of adia-\nbatic quantum pumping [4] in terms of scattering\ntheory [23, 24, 25, 26] which provided a useful back-\ndrop for later experiments [27, 28, 29, 30, 31]. Be-\nyond the adiabatic regime, Moskalets and B uttiker\ncombined the scattering approach with Floquet the-\nory to account for periodic driving [32]. These\nworks describe adiabatic quantum transport as a\nlimit of the more general problem of periodic driv-\ning and ultimately triggered numerous studies on\nsingle-particle emitters and quantum capacitors (as\nreviewed by Moskalets and Haack in this volume[33]).\nThe basic idea of the adiabatic quantum mo-\ntor [5] is easily introduced by analogy with the\nArchimedes screw, a device consisting of a screw\ninside a pipe. By turning the screw, water can be\npumped against gravity. This is a classical analog of\na quantum pump in which electrons are pumped be-\ntween reservoirs by applying periodic potentials to\na central scattering region. Just as the Archimedes\npump can pump water against gravity, charge can\nbe quantum pumped against a voltage. In addi-\ntion, the Archimedes screw has an inverse mode\nof operation as a motor : Water pushed through the\ndevice will cause the screw to rotate. The adiabatic\nquantum motor is a quantum analog of this mode\nof operation in which a transport current pushed\nthrough a quantum coherent conductor induces uni-\ndirectional motion of a classical degree of freedom\nsuch as the rotations of a helical molecule.\nThe theory of adiabatic quantum motors [5, 34]\nexploits the assumption that the motor degrees of\nfreedom { be they mechanical or magnetic { are\nslow compared to the electronic degrees of freedom.\nIn this adiabatic regime, the typical time scale of\nthe mechanical dynamics is large compared to the\ndwell time of the electrons in the interaction region\nbetween motor and electrical degrees of freedom. In\nthis limit, the dynamics of the two degrees of free-\ndom can be discussed in a mixed quantum-classical\ndescription. The motor dynamics is described in\nterms of a classical equation of motion, while a\nfully quantum-coherent description is required for\nthe fast electronic degrees of freedom.\nFrom the point of view of the electrons, the mo-\ntor degrees of freedom act as acpotentials which\npump charge through the conductor. Conversely,\nthe backaction of the electronic degrees of freedom\nenters through adiabatic reaction forces on the mo-\ntor degrees of freedom [6, 7, 8, 9]. When there is\njust a single (Cartesian) classical degree of freedom,\nthese reaction forces are necessarily conservative,\nakin to the Born-Oppenheimer force in molecular\nphysics [35]. Motor action driven by transport cur-\nrents can occur when there is more than one mo-\ntor degree of freedom (or a single angle degree of\nfreedom). In this case, the adiabatic reaction force\nneed no longer be conservative when the electronic\nconductor is subject to a bias voltage [6, 7, 8, 9].\nIn next order in the adiabatic approximation,\nthe electronic system also induces frictional and\nLorentz-like forces, both of which are linear in the\nslow velocity of the motor degree of freedom. In-\n2cluding the \ructuating Langevin force which ac-\ncompanies friction yields a classical Langevin equa-\ntion for the motor degree of freedom. This equation\ncan be derived systematically within the Keldysh\nformalism [35] and the adiabatic reaction forces ex-\npressed through the scattering matrix of the coher-\nent conductor [6, 7, 8].\nWhile these developments focused on mechani-\ncal degrees of freedom, it was also pointed out that\nthe scattering theory of adiabatic reaction forces\nextends to magnetic degrees of freedom [9]. In\nthis case, adiabaticity requires that the precessional\ntime scale of the magnetic moment is larger than\nthe electronic dwell time. The e\u000bective classical de-\nscription for the magnetic moment takes the form of\na Landau-Lifshitz-Gilbert (LLG) equation. Similar\nto nanoelectromechanical systems, the LLG equa-\ntion can be derived systematically in the adiabatic\nlimit for a given microscopic model and the coe\u000e-\ncients entering the LLG equation can be expressed\nalternatively in terms of electronic Green functions\nor scattering matrices [36, 37, 38, 39, 9]. In the fol-\nlowing, we will apply this general theory to a mag-\nnetic island coupled to a Kramers pair of helical\nedge states.\nThis work is organized as follows. Section 2\nreviews the scattering-matrix expressions for the\ntorques entering the LLG equation. Section 3 ap-\nplies this theory to helical edge states coupled to\na magnetic island and makes the relation to adia-\nbatic quantum motors explicit. Section 4 de\fnes\nand discusses the e\u000eciency of this device and de-\nrives a direct relation between charge pumping and\nspin transfer torque. Section 5 is devoted to con-\nclusions.\n2. S-matrix theory of spin transfer torques\nand Gilbert damping\n2.1. Landau-Lifshitz-Gilbert equation\nConsider a coherent (Landauer-B uttiker) con-\nductor coupled to a magnetic moment. The latter is\nassumed to be su\u000eciently large to justify a classical\ndescription of its dynamics but su\u000eciently small so\nthat we can treat it as a single macrospin. Then,\nits dynamics is ruled by a Landau-Lifshitz-Gilbert\nequation\n_M=M\u0002[\u0000@MU+Bel+\u000eB]: (1)\nNote that we use units in which Mis an angu-\nlar momentum and for simplicity of notation, Belas well as\u000eBdi\u000ber from a conventional magnetic\n\feld by a factor of gd, the gyromagnetic ratio of\nthe macrospin. The \frst term on the right-hand\nside describes the dynamics of the macrospin in the\nabsence of coupling to the electrons. It is derived\nfrom the quantum Hamiltonian\n^U=\u0000gd^M\u0001B+D\n2^M2\nz; (2)\nwhere M=h^Miis the uncoupled macrospin, Bthe\nmagnetic \feld, and D> 0 the easy-plane anisotropy\nof the macrospin. The coupling to the electrons\nleads to the additional e\u000bective magnetic \feld Bel.\nThis term can be derived microscopically from the\nHeisenberg equation of motion of the macrospin by\nevaluating the commutator of ^Mwith the interac-\ntion Hamiltonian between macrospin and electrons\nin the adiabatic approximation (see, e.g., Ref. [9]).\nKeeping terms up to linear order in the small mag-\nnetization \\velocity\" _M, we can write\nBel=B0(M)\u0000\r(M)_M: (3)\nHere, the \frst contribution B0can be viewed as the\nspin-transfer torque. The second term is a contribu-\ntion to Gilbert damping arising from the coupling\nbetween macrospin and electrons. In general, \rso\nderived is a tensor with symmetric and antisymmet-\nric components. However, it can be seen that only\nthe symmetric part plays a relevant role [9]. Finally,\nby \ructuation-dissipation arguments, the Gilbert\ndamping term is accompanied by a Langevin torque\n\u000eBwith correlator\nh\u000eBl(t)\u000eBk(t0)i=Dlk\u000e(t\u0000t0): (4)\nIts correlations are local in time as a consequence\nof the assumption of adiabaticity. As a result, we\n\fnd the LLG equation\n_M=M\u0002h\n\u0000@MU+B0\u0000\r_M+\u000eBi\n; (5)\nfor the macrospin M.\nThe spin-transfer torque, the Gilbert damping,\nand the correlator Dcan be expressed in terms\nof the scattering matrix of the coherent conductor,\nboth in and out of equilibrium [36, 37, 38, 39, 9].\nBefore presenting the S-matrix expressions, a few\ncomments are in order. First, the expression for the\nGilbert damping only contains the intrinsic damp-\ning originating from the coupling to the electronic\ndegrees of freedom. Coupling to other degrees of\nfreedom might give further contributions to Gilbert\n3damping which could be included phenomenologi-\ncally. Second, in the study of the nanomagnet cou-\npled to the helical modes we will consider the ex-\npressions to lowest order in the adiabatic approxi-\nmation presented in Sec. 2.2. The theory can actu-\nally be extended to include higher order corrections\n[9]. In Sec. 2.3 section, we brie\ry summarize the\nmain steps of the general procedure for complete-\nness.\n2.2. Coe\u000ecients of the LLG equation in the lowest\norder adiabatic approximation\nThis section summarizes the expressions for the\ncoe\u000ecients of the LLG equation that we will use\nto study the problem of the nanomagnet coupled\nto the helical edge states. These correspond to\nthe lowest order in the adiabatic approximation, in\nwhich we retain only terms linear in _MandeV.\nTo this order, we can write the coe\u000ecients of the\nLLG equation in terms of the electronic S-matrix\nfor a static macrospin M. The coupling between\nmacrospin and electronic degrees of freedom enters\nthrough the dependence of the electronic S-matrix\nS0=S0(M) on the (\fxed) macrospin. At this or-\nder, the spin-transfer torque and the Gilbert damp-\ning can be expressed as [36, 37, 38, 39, 9]\nB0(M) =X\n\u000bZd\"\n2\u0019if\u000bTr\u0014\n\u0005\u000b^Sy\n0@S0\n@M\u0015\n(6)\nand\n\rkl(M) =\u0000~X\n\u000bZd\"\n4\u0019f0\n\u000bTr\"\n\u0005\u000b@^Sy\n0\n@Mk@^S0\n@Ml#\ns;(7)\nrespectively. Finally, the \ructuation correlator Dis\nexpressed as [9]\nDkl(M) =~X\n\u000b;\u000b0Zd\"\n2\u0019f\u000b(1\u0000f\u000b0)\n\u0002Tr2\n4\u0005\u000b \n^Sy\n0@^S0\n@Mk!y\n\u0005\u000b0 \n^Sy\n0@^S0\n@Ml!3\n5\ns:(8)\nIn these expressions, \u000b=L;R denotes the reser-\nvoirs with electron distribution function f\u000b, \u0005\u000bis a\nprojector onto the channels of lead \u000b, and Tr traces\nover the lead channels.\n2.3. Corrections to the adiabatic approximation of\nthe S-matrix\nIn order to go beyond linear response in eVand\n_M, we must consider the electronic S-matrix in thepresence of the time-dependent magnetization M(t)\nand expand it to linear order in the magnetization\n\\velocity\" _M(t). This can be done, e.g., by starting\nfrom the full Floquet scattering matrix SF\n\u000b;\f(\"n;\")\nfor a periodic driving with period ![32]. The in-\ndices\u000band\flabel the scattering channels of the\ncoherent conductor and the arguments denote the\nenergies\"of the incoming electron in channel \u000band\n\"n=\"+n~!of the outgoing electron in channel \f.\nFor small driving frequency !, the Floquet scatter-\ning matrix can be expanded in powers of ~!,\n^SF(\"n;\") = ^S0\nn(\") +n~!\n2@^S0\nn(\")\n@\"\n+~!^An(\") +O(\"2): (9)\nHere ^S0\nn(\") is the Fourier transform of the frozen\nscattering matrix S0(M(t)) introduced above,\n^S0(M(t)) =1X\nn=\u00001e\u0000in!t^S0\nn(\"): (10)\nThe matrix ^An(\"), \frst introduced by Moskalets\nand B uttiker, is the \frst adiabatic correction to the\nadiabatic S-matrix and can be transformed in a sim-\nilar way to\n^A(t;\") =1X\nn=\u00001e\u0000in!t^An(\") = _M(t)\u0001^A(t;\"):(11)\nThe matrix ^An(\") can be straightforwardly calcu-\nlated from the retarded Green function of the device\n(see Refs. [20, 9]).\nWe are now in a position to give expressions for\nthe Gilbert damping to next order in the adiabatic\napproximation. (The spin-transfer torque and the\n\ructuation correlator remain unchanged.) To do\nso, we split the Gilbert matrix \rinto its symmetric\nand antisymmetric parts,\n\r=\rs+\ra: (12)\nStrictly speaking, it is only the symmetric part\nwhich corresponds to Gilbert damping. The anti-\nsymmetric part simply renormalizes the precession\nfrequency. One \fnds [9]\n\rkl\ns(M) =\u0000~X\n\u000bZd\"\n4\u0019f0\n\u000bTr\"\n\u0005\u000b@^Sy\n0\n@Mk@^S0\n@Ml#\ns\n+X\n\u000bZd\"\n2\u0019if\u000bTr\"\n\u0005\u000b \n@^Sy\n0\n@Mk^Al\u0000^Ay\nl@^S0\n@Mk!#\ns(13)\n4for the symmetric contribution. It can be seen, the\nsecond line is a pure nonequilibrium contribution\n(/eV~!). Similarly, the antisymmetric part of the\nGilbert damping can be written as [9]\n\rkl\na(M) =\u0000~X\n\u000bZd\"\n2\u0019if\u000b(\")\n\u0002Tr\"\n\u0005\u000b \n^Sy\n0@^Ak\n@Ml\u0000@^Ay\nk\n@Ml^S0!#\na;(14)\nwhich is really a renormalization of the precession\nfrequency as mentioned above.\n3. S-matrix theory of a nanomagnet coupled\nto a quantum spin Hall edge\nWe now apply the above theory to a magnetic\nisland coupled to a quantum spin Hall edge as\nsketched in Fig. 1. The quantum spin Hall edge sup-\nports a Kramers doublet of edge states. The mag-\nnetization M=M?cos\u0012ex+M?sin\u0012ey+Mzez\nof the magnetic island induces a Zeeman \feld JM\nacting on the electrons along the section of length\nLof the edge state which is covered by the magnet.\nThis Zeeman \feld causes backscattering between\nthe edge modes and induces a gap \u0001 = JM?~=2\n[1]. Linearizing the dispersion of the edge modes,\nthe electronic Hamiltonian takes the form [2]\n^H= (vp\u0000JMz)^\u001bz+\u0001(x) (cos\u0012^\u001bx+ sin\u0012^\u001by):(15)\nHere, the\u001bjdenote Pauli matrices in spin space\nand \u0001(x) is nonzero only over the region of length\nLcovered by the magnetic island. We have assumed\nfor simplicity that the spin Hall edge conserves \u001bz.\nThen, a static island magnetization induces a gap\nwhenever it has a component perpendicular to the\nz-direction. Indeed, ^His easily diagonalized for a\nspatially uniform coupling between edge modes and\nmagnet, and the spectrum\nEp=p\n(vp\u0000JMz)2+ \u00012 (16)\nhas a gap \u0001.\nIn the following, we assume that the easy-plane\nanisotropy D > 0 is su\u000eciently large so that the\nmagnetization entering the electronic Hamiltonian\ncan be taken in the xy-plane, i.e., Mz'0. (How-\never, we will have to keep Mzin the LLG equation\nwhen it is multiplied by the large anisotropy D.)\nThe electronic Hamiltonian (15) is equivalent to\nthe electronic Hamiltonian of the Thouless motor\nconsidered in Ref. [5]. Following this reference, wecan readily derive the frozen scattering matrix an-\nalytically [5],\n^S0=1\n\u0003\u0012\u0000iei\u0012\u0015 1\n1\u0000ie\u0000i\u0012\u0015\u0013\n; (17)\nwhere we have de\fned the shorthands\n\u0003 = cos \u001eL\u0000i\"p\n\"2\u0000\u00012sin\u001eL;\n\u0015=\u0001p\n\"2\u0000\u00012sin\u001eL (18)\nwith\n\u001eL(\") =L\n~vp\n\"2\u0000\u00012: (19)\nNote that these expressions are exact for any Land\nvalid for energies \"both inside and outside the gap.\nWe can now use this scattering matrix to eval-\nuate the various coe\u000ecients in the LLG equation,\nemploying the expressions given in Sec. 2.2. As-\nsuming zero temperature, we \fnd\nB0=eV\n2\u0019M\u0018(\u0016)e\u0012; (20)\nfor the spin transfer torque at arbitrary chemical\npotential\u0016. Here, we have de\fned the function\n\u0018(\u0016) =\u00012sin2\u001eL\nj\u00162\u0000\u00012jcos2\u001eL+\u00162sin2\u001eL(21)\nwith\u001eL=\u001eL(\u0016) (see Fig. 2). Below, we will iden-\ntify\u0018with the charge pumped between the reser-\nvoirs during one precessional period of the mag-\nnetization M. The vector B0points in the az-\nimuthal direction in the magnetization plane and\nindeed corresponds to a spin-transfer torque. Sim-\nilarly, we can substitute Eq. (17) into Eq. (13) for\nthe Gilbert damping and \fnd that the only nonzero\ncomponent of the tensor \ris\n\r\u0012\u0012=~\n2\u0019M2\u0018(\u0016): (22)\nSimilarly,\nD\u0012\u0012=~eV\n\u0019M2\u0018(\u0016) (23)\nis the only nonzero component of the \ructuation\ncorrelator. It is interesting to note that this yields\nan e\u000bective \ructuation-dissipation relation D\u0012\u0012=\n2Te\u000b\r\u0012\u0012with e\u000bective temperature Te\u000b=eV.\n5With these results, we can now write the LLG\nequation for the nanomagnet coupled to the helical\nedge state,\n_M =DM\u0002Mzez+\u0018(eV\u0000~_\u0012)\n2\u0019MM\u0002e\u0012\n+M\u0002\u000eB; (24)\nwhere\u0018=\u0018(\u0016), we have expressed _M'M_\u0012e\u0012, and\nassumed zero external magnetic \feld B. This com-\npletes our scattering-theory derivation of the LLG\nequation and generalizes the result obtained in Ref.\n[2] on phenomenological grounds in several respects.\nEquation (24) applies also for \fnite-length magnets\nand chemical potentials both inside and outside the\nmagnetization-induced gap of the edge-state spec-\ntrum. Moreover, the identi\fcation of the _\u0012-term\nas a damping term necessitates the inclusion of the\nLangevin torque \u000eB. Indeed, Ref. [2] refers to the\nentire term involving eV\u0000~_\u0012as the spin-transfer\ntorque. In contrast, our derivation produces the\nterm involving eValready in zeroth order in mag-\nnetization \\velocity\" _M, while the _\u0012term appears\nonly to linear order. Thus, the latter term is re-\nally a conrtribution to damping and related to the\nenergy dissipated in the electron system due to the\ntime dependence of the magnetization.\n4. E\u000eciency of the nanomagnet as a motor\nWhile the electronic Hamiltonian for the edge\nmodes is equivalent to that of the Thouless mo-\ntor discussed in Ref. [5], the LLG equation for the\nmacrospin di\u000ber from the equation of motion of the\nmechanical degrees of freedom discussed in Ref. [5].\nIn this section, we discuss the energetics and the\ne\u000eciency of the magnetic Thouless motor against\nthe backdrop of its mechanical cousin.\nThe dynamics of the macrospin is easily ob-\ntained from the LLG equation (24) [2]. For a large\nanisotropy and thus small Mz, we need to retain\nthez-component of Monly in combination with the\nlarge anisotropy D. Then, the steady-state value of\nMzis \fxed by the \u0012-component of the LLG equa-\ntion,\nMz=\u0000_\u0012\nD: (25)\nThe precessional motion of Mabout thez-axis is\ngoverned by the z-component of the LLG equation,\nwhich yields\n_\u0012=eV\n~(26)and hence Mz=\u0000eV=(~D). It is interesting to\nnote that the angular frequency _\u0012of the preces-\nsion is just given by the applied bias voltage, in-\ndependent of the damping strength. This should\nbe contrasted with the mechanical Thouless motor.\nHere, the motor degree of freedom satis\fes a New-\nton equation of motion which is second order in\ntime. Thus, the frequency of revolution is inversely\nproportional to the damping coe\u000ecient.\nIn steady state, the magnetic Thouless motor bal-\nances the energy provided by the voltage source\nthrough the spin-transfer torque B0against the dis-\nsipation through Gilbert damping due to the intrin-\nsic coupling between magnetic moment and elec-\ntronic degrees of freedom. It is instructive to look at\nthese contributions independently. The work per-\nformed by the spin-transfer torque per precessional\nperiod is given by\n\u0001Wspin\u0000transfer =Z2\u0019=_\u0012\n0dtB0\u0001_M: (27)\nWriting this as an integral over a closed loop of\nthe magnetization Mand inserting the S-matrix\nexpression (6), we \fnd\n\u0001Wspin\u0000transfer =X\n\u000bZd\"\n2\u0019if\u000b\n\u0002I\ndM\u0001Tr\"\n\u0005\u000b^Sy\n0@^S0\n@M#\n: (28)\nWithout applied bias, the integrand is just the gra-\ndient of a scalar function and the integral vanishes.\nThus, we expand to linear order in the applied bias\nand obtain\n\u0001Wspin\u0000transfer =ieV\n4\u0019\n\u0002X\n\u000bI\ndM\u0001Tr\"\n(\u0005L\u0000\u0005R)^Sy\n0@^S0\n@M#\n:(29)\nComparing Eq. (29) with the familiar S-matrix ex-\npression for the pumped charge [23], the right-hand\nside can now be identi\fed as the bias voltage multi-\nplied by the charge pumped between the reservoirs\nduring one revolution of the magnetization,\n\u0001Wspin\u0000transfer =QpV: (30)\nWith every revolution of the magnetization, a\nchargeQpis pumped between the reservoirs. The\ncorresponding gain QpVin electrical energy is driv-\ning the magnetic Thouless motor. This result can\nalso be written as\n_Wspin\u0000transfer =QpV\n2\u0019_\u0012 (31)\n6for the power provided per unit time by the voltage\nsource.\nThe relation between spin-transfer torque and\npumped charge also allows us to identify the func-\ntion\u0018(\u0016) appearing in the LLG equation as the\ncharge in units of epumped between the reservoirs\nduring one precessional period of the macrospin,\nQp=e\u0018: (32)\nThis can be obtained either by deriving the pumped\ncharge explicitly from the S-matrix expression or by\nevaluating Eq. (27) using the explicit expression Eq.\n(20).\nThe electrical energy gain is compensated by the\nenergy dissipated through Gilbert damping. The\ndissipated energy per period is given by\n\u0001WGilbert =Z2\u0019=_\u0012\n0dt_MT\r_MT\n= 2\u0019M2\r\u0012\u0012_\u0012: (33)\nUsing Eq. (22), this yields the dissipated energy\n\u0001WGilbert =\u0018~_\u0012 (34)\nper precessional period or\n_WGilbert =\u0018~\n2\u0019_\u00122(35)\nper unit time. These expressions have a simple\ninterpretation. Due to the \fnite frequency of the\nmagnetization precession, each pumped charge ab-\nsorbs on average an energy ~_\u0012which is then dissi-\npated in the reservoirs.\nArmed with these results, we can \fnally discuss\nthe e\u000eciency of a magnetic Thouless motor and fol-\nlow the framework introduced in Ref. [40] to de\fne\nan appropriate \fgure of merit (analogous to the ZT\nvalue of thermoelectrics). Imagine the same setup\nas in Fig. 1, but with an additional load coupled\nto the magnetization. We can now de\fne the e\u000e-\nciency of the magnetic Thouless motor as the ratio\nof the power delivered to the load and the electri-\ncal powerIVprovided by the voltage source. In\nsteady state, the power delivered to the load has\nto balance against the power provided by the elec-\ntrons, i.e., Bel\u0001_M. Thus, we can write the e\u000eciency\nas\n\u0011=_W\nIV; (36)\n00.20.40.60.81ξ,ηmax0\n0.5 1 1.5 2 2.5 3 µ/Δ\n00.20.40.60.81ξ,ηmaxFigure 2: (Color online) The parameter \u0018(dashed lines) en-\ntering the coe\u000ecients of the LLG equation and the maximal\ne\u000eciency\u0011max(solid lines) of the motor for a \fxed voltage\nV. Upper and lower panels correspond to nanomagnets of\nlengthL=~v=\u0001 andL= 10 ~v=\u0001, respectively.\nwhere\n_W= _Wspin\u0000torque\u0000_WGilbert\n=\u0018\n2\u0019eV_\u0012\u0000\u0018~\n2\u0019_\u00122: (37)\nThe total charge current \rowing along the topolog-\nical insulator edge averaged over the cycle is the\nsum of the dccurrentGVdriven by the voltage,\nwhereGis the dcconductance of the device, and\nthe pumping current Qp_\u0012=(2\u0019),\nI=GV+e\u0018\n2\u0019_\u0012: (38)\nWe can now optimize the e\u000eciency of the motor\nat a given bias Vas function of the frequency _\u0012of\nthe motor revolution. Note that due to the load,\nthe latter is no longer tied to the bias voltage eV.\nThis problem is analogous to the problem of the\noptimal e\u000eciency of a thermoelectric device which\nleads to the de\fnition of the important ZT value.\nThis analogy was discussed explicitly in Ref. [40].\nApplying the results of this paper to the present\ndevice yields the maximal e\u000eciency\n\u0011max=p1 +\u0010\u00001p1 +\u0010+ 1; (39)\nwith a \fgure of merit \u0010analogous to the ZT value\nde\fned by\n\u0010=e2\u0018(\u0016)\nhG(\u0016); (40)\n7where\u0018(\u0016) is de\fned in Eq. (21) and the conduc-\ntance reads\nG(\u0016) =e2\nhj\u00162\u0000\u00012j2\nj\u00162\u0000\u00012jcos2\u001eL+\u00162sin2\u001eL(41)\nas obtained from the Landauer-B uttiker equation.\nAs in thermoelectrics, the maximum e\u000eciency is\nrealized for \u0010!1 which requires a \fnite pumped\ncharge at zero conductance. Unlike thermoelectrics,\nthe motor e\u000eciency is bounded by \u0011= 1 instead of\nthe Carnot e\u000eciency. This re\rects the fact that\nelectrical energy can be fully converted into mag-\nnetic energy. Speci\fcally, unit e\u000eciency is reached\nin the limit of a true Thouless motor with zero\ntransmission when the Fermi energy falls into the\ngap and nonzero and quantized pumped charge per\nperiod. This can be realized to a good approxi-\nmation for a su\u000eciently long magnet, as seen from\nthe lower panel in Fig. 2. For chemical potentials\noutside the gap, the conductance and the pumped\ncharge exhibit Fabry-Perot resonances. This yields\na distinct sequence of maxima and minima in the\ne\u000eciency. For shorter magnets, the conductance\nremains nonzero within the gap, leading to lower\ne\u000eciencies. This is shown in the upper panel of\nFig. 2. Moreover, the Fabry-Perot resonances are\nwashed out, so that there is only a feature at the gap\nedge where the conductance vanishes while \u0018!1=2\nfor arbitrary L.\n5. Conclusions\nImplementing directional motion of a mechani-\ncal or magnetic degree of freedom is a fundamental\nproblem of nanoscale systems. An attractive gen-\neral mechanism relies on running quantum pumps\nin reverse. This is the underlying principle of adia-\nbatic quantum motors which drive periodic motion\nof a classical motor degree of freedom by applying a\ntransport current. In this paper, we emphasize that\na magnetic island coupled to a quantum spin Hall\nedge, recently discussed by Meng et al. [2], is just\nsuch an adiabatic quantum motor. We derive the\nLandau-Lifshitz-Gilbert equation for the magneti-\nzation dynamics from a general scattering-theory\napproach to adiabatic quantum motors, providing\na microscopic derivation of spin-transfer torque,\nGilbert damping, and Langevin torque. This ap-\nproach does not only provide a detailed microscopic\nunderstanding of the operation of the device but\nalso allows one to discuss its e\u000eciency. We \fndthat the device naturally approaches optimal e\u000e-\nciency when the chemical potential falls into the\nmagnetization-induced gap and the conductance is\nexponentially suppressed. This makes this system\na Thouless motor and possibly its most experimen-\ntally feasible variant to date.\nSeveral issues are left for future work. While we\nderived microscopic expressions for the Langevin\ntorque, we have not explored its consequences for\nthe motor dynamics. It should also be interesting\nto consider thermal analogs driven by a tempera-\nture gradient instead of a bias voltage. Inducing the\nmagnetization precession by a temperature gradient\nwould realize a quantum heat engine. Conversely,\nforcing a magnetic precession can be used to pump\nheat against a temperature gradient. Setups with\nseveral magnetic islands could be engineered to ef-\nfect exchange of charge and energy without employ-\ning a dcbattery. These devices have been explored\nin the literature on quantum pumps [41, 42, 43] and\ntheir e\u000eciencies could be analyzed in the thermo-\nelectric framework of Ref. [40].\nAcknowledgement\nWe thank Gil Refael and Ari Turner for dis-\ncussions. This work was supported by CON-\nICET, MINCyT and UBACyT (L.A.) as well\nas the Deutsche Forschungsgemeinschaft and the\nHelmholtz Virtual Institute New States of Matter\nand Their Excitations (F.v.O.). L.A. thanks the\nICTP Trieste for hospitality and the Simons Foun-\ndation for support. F.v.O. thanks the KITP Santa\nBarbara for hospitality during the \fnal preparation\nof this manuscript. 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In the\ndi\u000busive regime, the breathing Fermi-surface e\u000bect is balanced by the e\u000bects of interband scattering,\nwhich cause the Gilbert damping constant to saturate at around 0.005. In small samples, the system\nshape induces a large anisotropy in the Gilbert damping.\nI. INTRODUCTION\nThe magnetization dynamics of a ferromagnet can be\ndescribed phenomenologically by the Landau-Lifshitz-\nGilbert (LLG) equation:1,2\n1\n\rdM\ndt=\u0000M\u0002He\u000b+M\u0002\"~G(M)\n\r2M2sdM\ndt#\n:(1)\nHere,\ris the gyromagnetic ratio, He\u000bis the e\u000bective\nmagnetic \feld (which is the functional derivative of the\nfree energy He\u000b=\u0000\u000eF[M]=\u000eM),Mis the magnetiza-\ntion andMsis its magnitude. The Gilbert damping con-\nstant ~G(M) parameterizes the dissipative friction process\nthat drives the magnetization towards an equilibrium\nstate.3In the most general case, ~G(M) is a symmetric\npositive de\fnite matrix that depends on the magnetiza-\ntion direction; however, it is often assumed to be inde-\npendent of Mand proportional to the unit matrix, as-\nsumptions which are valid for isotropic systems. Gilbert\ndamping is important in magnetization dynamics. It de-\ntermines the magnitudes of the external magnetic \felds4\nand the current densities1that are required to reorient\nthe magnetization direction of a ferromagnet. Therefore,\na thorough understanding of its properties is essential for\nmodeling ferromagnetic systems.\nThe main contribution to the Gilbert damping process\nin metallic ferromagnets is the generation of electron-\nhole pairs.1,2,5,6A model that captures this process was\ndeveloped by Kambersky.5In this model, the electrons\nare excited by a time-varying magnetization via electron-\nmagnon coupling. If the ferromagnet is in metallic con-\ntact with other materials, the spin-pumping into the ad-\njacent leads provides an additional contribution to the\nmagnetization relaxation.7A general theory that cap-\ntures both of these e\u000bects was recently developed.8The\nmodel expresses the ~G(M) tensor in terms of the scatter-\ning matrix Sof the ferromagnetic system ( m\u0011M=Ms):\n~Gij(m) =\r2\u0016h\n4\u0019Re\u001a\nTr\u0014@S\n@mi@Sy\n@mj\u0015\u001b\n: (2)\nThe expression is evaluated at the Fermi energy. Instead\nof~G(M), one often parameterizes the damping by the\ndimensionless Gilbert damping parameter ~ \u000b\u0011~G=\rM s.\nEq. (2) allows studying both the e\u000bects of the systemshape and the disorder dependency of the magnetization\ndamping beyond the relaxation time approximation.9\nIn anisotropic systems, the Gilbert damping is ex-\npected to be a symmetric tensor with non-vanishing o\u000b-\ndiagonal terms. We are interested in how this tensor\nstructure in\ruences the dynamics of the precessing mag-\nnetization in (Ga,Mn)As. Therefore, to brie\ry discuss\nthis issue, let us consider a homogenous ferromagnet in\nwhich the magnetization direction m=m0+\u000empre-\ncesses with a small angle around the equilibrium direc-\ntionm0that points along the external magnetic \feld\nHext.10For clarity, we neglect the anisotropy in the\nfree energy and choose the coordinate system such that\nm0= (0 0 1) and \u000em= (mxmy0). For the lowest order\nof Gilbert damping, the LLG equation can be rewritten\nas:_m=\u0000\rm\u0002Hext+\rm\u0002(~\u000b[Hext\u0002m]), where ~\u000b[:::]\nis the dimensionless Gilbert damping tensor that acts on\nthe vector Hext\u0002m. Linearizing the LLG equation re-\nsults in the following set of equations for mxandmy:\n\u0012\n_mx\n_my\u0013\n=\u0000\rHext \n\u000b(0)\nyy (1\u0000\u000b(0)\nxy)\n\u0000(1 +\u000b(0)\nxy)\u000b(0)\nxx!\u0012\nmx\nmy\u0013\n:\n(3)\nHere,\u000b(0)\nijare the matrix elements of ~ \u000bwhen the tensor\nis evaluated along the equilibrium magnetization direc-\ntionm0. For the lowest order of Gilbert damping, the\neigenvalues of (3) are \u0015\u0006=\u0006i\rH ext\u0000\rHext\u000b, and the\neigenvectors describe a precessing magnetization with a\ncharacteristic life time \u001c= (\u000b\rH ext)\u00001. The e\u000bective\ndamping coe\u000ecient \u000bis:10\n\u000b\u00111\n2\u0010\n\u000b(0)\nxx+\u000b(0)\nyy\u0011\n: (4)\nThe value of \u000bis generally anisotropic and depends on\nthe static magnetization direction m0. The magnetiza-\ntion damping is accessible via ferromagnetic resonance\n(FMR) experiments by measuring the linear relation-\nship between the FMR line width and the precession fre-\nquency. This linear relationship is proportional to \u001c\u00001\nand thus depends linearly on \u000b. Therefore, an FMR ex-\nperiment can be used to determine the e\u000bective damping\ncoe\u000ecient\u000b. In contrast, the o\u000b-diagonal terms, \u000b(0)\nxy\nand\u000b(0)\nyx, do not contribute to the lowest order in the\ndamping and are di\u000ecult to probe experimentally.\nIn this paper, we use Eq. (2) to study the anisotropy\nand disorder dependency of the Gilbert damping in thearXiv:1105.4148v2  [cond-mat.mtrl-sci]  2 Nov 20112\nferromagnetic semiconductor (Ga,Mn)As. Damping co-\ne\u000ecients of this material in the range of \u000b\u00180:004\u00000:04\nfor annealed samples have been reported.11{14The damp-\ning is anisotropically dependent on the magnetization\ndirection.11,12,14The few previous calculations of the\nGilbert damping constant in this material have indicated\nthat\u000b\u00180:003\u00000:04.11,15{17These theoretical works\nhave included the e\u000bects of disorder phenomenologically,\nfor instance, by applying the relaxation time approxima-\ntion. In contrast, Eq. (2) allows for studying the disor-\nder e\u000bects fully and non-perturbatively for the \frst time.\nIn agreement with Ref. 15, we show that spin-pumping\nfrom the localized d-electrons to the itinerant holes dom-\ninates the damping process in the clean limit. In the\ndi\u000busive regime, the breathing Fermi-surface e\u000bect is bal-\nanced by e\u000bects of the interband transitions, which cause\nthe damping to saturate. In determining the anisotropy\nof the Gilbert damping tensor, we \fnd that the shape of\nthe sample is typically more important than the e\u000bects\nof the strain and the cubic symmetry in the GaAs crys-\ntal.18This shape anisotropy of the Gilbert damping in\n(Ga,Mn)As has not been reported before and provides\na new direction for engineering the magnetization relax-\nation.\nII. MODEL\nThe kinetic-exchange e\u000bective Hamiltonian approach\ngives a reasonably good description of the electronic\nproperties of (Ga,Mn)As.19The model assumes that the\nelectronic states near the Fermi energy have the character\nof the host material GaAs and that the spins of the itin-\nerant quasiparticles interact with the localized magnetic\nMn impurities (with spin 5/2) via the isotropic Heisen-\nberg exchange interaction. If the s-d exchange interac-\ntion is modeled by a mean \feld, the e\u000bective Hamiltonian\ntakes the form:19,20\nH=HHoles +h(r)\u0001s; (5)\nwhereHHoles is the k\u0001pKohn-Luttinger Hamiltonian de-\nscribing the valence band structure of GaAs and h(r)\u0001s\nis a mean \feld description of the s-d exchange interac-\ntion between the itinerant holes and the local magnetic\nimpurities ( sis the spin operator). The exchange \feld\nhis antiparallel to the magnetization direction m. The\nexplicit form of HHoles that is needed for realistic model-\ning of the band structure of GaAs depends on the doping\nlevel of the system. Higher doping levels often require\nan eight-band model, but a six- or four-band model may\nbe su\u000ecient for lower doping levels. In the four-band\nmodel, the Hamiltonian is projected onto the subspace\nspanned by the four 3/2 spin states at the top of the\nGaAs valence band. The six-band model also includes\nthe spin-orbit split-o\u000b bands with spin 1/2. The spin-\norbit splitting of the spin 3/2 and 1/2 states in GaAs\nis 341 meV.21We consider a system with a Fermi level\nof 77 meV when measured from the lowest subband. Inthis limit, the following four-band model gives a su\u000ecient\ndescription:\nH=1\n2m\u0014\n(\r1+5\n2\r2)p2\u00002\r3(p\u0001J)2+h\u0001J\u0015\n+\n\r3\u0000\r2\nm(p2\nxJ2\nx+c:p:) +Hstrain +V(r): (6)\nHere, pis the momentum operator, Jiare the spin 3/2\nmatrices22and\r1,\r2and\r3are the Kohn-Luttinger pa-\nrameters.V(r) =P\niVi\u000e(r\u0000Ri) is the impurity scat-\ntering potential, where Riis the position of the impurity\niandViare the scattering strengths of the impurities23\nthat are randomly and uniformly distributed in the in-\nterval [\u0000V0=2;V0=2].Hstrain is a strain Hamiltonian and\narises because the (Ga,Mn)As system is grown on top\nof a substrate (such as GaAs).24The two \frst terms in\nEq. (6) have spherical symmetry, and the term propor-\ntional to\r3\u0000\r2represents the e\u000bects of the cubic sym-\nmetry of the GaAs crystal. Both this cubic symmetry\nterm25and the strain Hamiltonian24are small compared\nto the spherical portion of the Hamiltonian. A numeri-\ncal calculation shows that they give a correction to the\nGilbert damping on the order of 10%. However, the un-\ncertainty of the numerical results, due to issues such as\nthe sample-to-sample disorder \ructuations, is also about\n10%; therefore, we cannot conclude how these terms in-\n\ruence the anisotropy of the Gilbert damping. Instead,\nwe demonstrate that the shape of the system is the dom-\ninant factor in\ruencing the anisotropy of the damping.\nTherefore, we disregard the strain Hamiltonian Hstrain\nand the term proportional to \r3\u0000\r2in our investigation\nof the Gilbert damping.\nGaAs GaAs (Ga,Mn)As \nyx\nFIG. 1: We consider a (Ga,Mn)As system attached along the\n[010] direction to in\fnite ballistic GaAs leads. The scattering\nmatrix is calculated for the (Ga,Mn)As layer and one lattice\npoint into each of the leads. The magnetization is assumed to\nbe homogenous. In this paper, we denote the [100] direction\nas the x-axis, the [010] direction as the y-axis and the [001]\ndirection as the z-axis\nWe consider a discrete (Ga,Mn)As system with\ntransverse dimensions Lx2 f17;19;21gnm,Lz2\nf11;15;17gnm andLy= 50 nm and connected to in\fnite\nballistic GaAs leads, as illustrated in Fig. 1. The leads\nare modeled as being identical to the (Ga,Mn)As system,\nexcept for the magnetization and disorder. The lattice\nconstant is 1 nm, which is much less than the Fermi wave-\nlength\u0015F\u001810 nm. The Fermi energy is 0.077 eV when3\nmeasured from the lowest subband edge. The Kohn-\nLuttinger parameters are \r1= 7:0 and\r2=\r3= 2:5,\nimplying that we apply the spherical approximation for\nthe Luttinger Hamiltonian, as mentioned above.25We\nusejhj= 0:032 eV for the exchange-\feld strength. To\nestimate a typical saturation value of the magnetization,\nwe useMs= 10j\rj\u0016hx=a3\nGaAs withx= 0:05 as the doping\nlevel andaGaAs as the lattice constant for GaAs.26\nThe mean free path lfor the impurity strength V0is\ncalculated by \ftting the average transmission probability\nT=hGi=GshtoT(Ly) =l=(l+Ly),27whereGshis the\nSharvin conductance and hGiis the conductance for a\nsystem of length Ly.\nThe scattering matrix is calculated numerically us-\ning a stable transfer matrix method.28The disorder ef-\nfects are fully and non-perturbatively included by the\nensemble average h\u000bi=PNI\nn=1\u000bn=NI, whereNIis the\nnumber of di\u000berent impurity con\fgurations. All the\ncoe\u000ecients are averaged until an uncertainty \u000eh\u000bi=r\u0010\nh\u000b2i\u0000h\u000bi2\u0011\n=NIof less than 10% is achieved. The\nvertex corrections are exactly included in the scattering\nformalism.\nIII. RESULTS AND DISCUSSION\nWithout disorder, the Hamiltonian describing our sys-\ntem is rotationally symmetric around the axis parallel\ntoh. Let us brie\ry discuss how this in\ruences the\nparticular form of the Gilbert damping tensor ~ \u000b.29For\nclarity, we choose the coordinate axis such that the ex-\nchange \feld points along the z-axis. In this case, the\nHamiltonian is invariant under all rotations Rzaround\nthe z-axis. This symmetry requires the energy dissipa-\ntion _E/_mT~\u000b_m(_mTis the transposed of _m) of the\nmagnetic system to be invariant under the coordinate\ntransformations r0=Rzr(i.e., ( _m0)T~\u000b0_m0=_mT~\u000b_m\nwhere m0=Rzmand ~\u000b0is the Gilbert damping ten-\nsor in the rotated coordinate system). Because ~ \u000bonly\ndepends on the direction of m, which is unchanged\nunder the coordinate transformation Rz, ~\u000b= ~\u000b0and\nRT\nz~\u000bRz= ~\u000b. Thus, ~\u000bandRzhave common eigenvectors\b\nj\u0006i\u0011 (jxi\u0006ijyi)=p\n2;jzi\t\n, and the spectral decompo-\nsition of ~\u000bis ~\u000b=\u000b+j+ih+j+\u000b\u0000j\u0000ih\u0000j +\u000bzjzihzj.\nRepresenting the damping tensor in the fjxi;jyi;jzig\ncoordinate basis yields \u000b\u0011~\u000bxx= ~\u000byy= (\u000b++\u000b\u0000)=2,\n~\u000bzz=\u000bz, ~\u000byx= ~\u000bxy= 0, and\u000b+=\u000b\u0000. The last\nequality results from real tensor coe\u000ecients. However,\n\u000bzzcannot be determined uniquely from the energy dis-\nsipation formula _E/_mT~\u000b_mbecause _mis perpendicu-\nlar to the z-axis. Therefore, \u000bzzhas no physical signi\f-\ncance and the energy dissipation is governed by the single\nparameter\u000b. For an in\fnite system, this damping pa-\nrameter does not depend on the speci\fc direction of the\nmagnetization, i.e., it is isotropic because the symmetry\nof the Hamiltonian is not directly linked to the crystallo-\ngraphic axes of the underlying crystal lattice (when the\n0 0.5 1 1.5 200.2 0.4 0.6 0.8 1\nφ / πθ / π\n44.5 55.5 6x 10 −3 \n0 0.5 1 1.5 200.2 0.4 0.6 0.8 1\nφ / πθ / π\n4.5 55.5 6x 10 −3 a\nbFIG. 2: ( a) The dimensionless Gilbert damping parameter\n\u000bas a function of the magnetization direction for a system\nwhereLx= 17 nm,Ly= 50 nm and Lz= 17 nm. ( b) The\ndimensionless Gilbert damping parameter \u000bas a function of\nthe magnetization direction for a system where Lx= 21 nm,\nLy= 50 nm and Lz= 11 nm. Here, \u0012and\u001eare the polar and\nazimuth angles, respectively, that describe the local magne-\ntization direction m= (sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012). In both\nplots, the mean free path is l\u001822 nm\ncubic symmetry term in Eq. (6) is disregarded). For a\n\fnite system, the shape of the system induces anisotropy\nin the magnetization damping. This e\u000bect is illustrated\nin Fig. 2, which plots the e\u000bective damping in Eq. (4) as\na function of the magnetization directions for di\u000berent\nsystem shapes. When the cross-section of the conductor\nis deformed from a regular shape to the shape of a thin-\nner system, the anisotropy of the damping changes. The\nmagnetization damping varies from a minimum value of\naround 0.004 to a maximum value of 0.006, e.g., the\nanisotropy is around 50%. The relaxation process is\nlargest along the axis where the ballistic leads are con-\nnected, i.e., the y-axis. This shape anisotropy is about\nfour- to \fve-times stronger than the anisotropy induced\nby the strain and the cubic symmetry terms in the Hamil-\ntonian (6), which give corrections of about 10 percent.\nFor larger systems, we expect this shape e\u000bect to be-4\ncome less dominant. In these systems, the anisotropy\nof the bulk damping parameter, which is induced by the\nanisotropic terms in the Hamiltonian, should play a more\nsigni\fcant role. The determination of the system size\nwhen the strain and cubic anisotropy become comparable\nto the shape anisotropy e\u000bects is beyond the scope of this\npaper because the system size is restricted by the com-\nputing time. However, this question should be possible to\ninvestigate experimentally by measuring the anisotropy\nof the Gilbert damping as a function of the \flm thickness.\n0 1 2 3 4 52345678910 x 10 −3 \nLy/l ααmin\nαmax \nαmean \nFIG. 3: The e\u000bective dimensionless Gilbert damping (4) as a\nfunction of the disorder. Here, lis the mean free path and\nLyis the length of the ferromagnetic system in the transport\ndirection.\u000bminand\u000bmaxare the minimum and maximum val-\nues of the anisotropic Gilbert damping parameter and \u000bmean is\nthe e\u000bective damping parameter averaged over all the magne-\ntization directions. The system dimensions are Lx= 19 nm,\nLy= 50 nm and Lz= 15 nm.\nWe next investigate how the magnetization relaxation\nprocess depends on the disorder. Ref. 15 derives an ex-\npression that relates the Gilbert damping parameter to\nthe spin-\rip rate T2of the system: \u000b/T2(1 + (T2)2)\u00001.\nIn the low spin-\rip rate regime, this expression scales\nwithT2as\u000b/T\u00001\n2, while the damping parameter is\nproportional to \u000b/T2in the opposite limit . As ex-\nplained in Ref. 15, the low spin-\rip regime is dominated\nby the spin-pumping process in which angular momen-\ntum is transferred to the itinerant particles; the trans-\nferred spin is then relaxed with a rate proportional to\nT\u00001\n2. This process appears inside the ferromagnet itself,\ni.e., the spin is transferred from the magnetic system to\nthe itinerant particles in the ferromagnet, which are then\nrelaxed within the ferromagnet. Therefore, this relax-\nation mechanism is a bulk process and should not be\nconfused with the spin-pumping interface e\u000bect across\nthe normal metal jferromagnet interfaces reported in\nRef. 7. In (Ga, Mn)As, this bulk process corresponds to\nspin-pumping from the d-electrons of the magnetic Mn\nimpurities to the itinerant spin 3/2 holes in the valence\nband of the host compound GaAs. The transfer of spin\nto the holes is then relaxed by the impurity scatteringwithin the ferromagnet. By contrast, the opposite limit\nis dominated by the breathing Fermi-surface mechanism.\nIn this mechanism, the spins of the itinerant particles are\nnot able to follow the local magnetization direction adia-\nbatically and lag behind with a delay time of T2. In our\nsystem, which has a large spin-orbit coupling in the band\nstructure, we expect the spin-\rip rate to be proportional\nto the mean free path ( l/T2).30The e\u000bective dimen-\nsionless Gilbert damping (4) is plotted as a function of\ndisorder in Fig. 3. The damping ( \u000bmean) partly shows\nthe same behavior as that reported in Ref. 15. For clean\nsystems (i.e., those with a low spin-\rip rate regime), the\ndamping increases with disorder. In such a regime, the\ntransfer of angular momentum to the spin 3/2 holes is the\ndominant damping process, i.e., the bulk spin-pumping\nprocess dominates. \u000bmean starts to decrease for smaller\nmean free paths, implying that the main contribution\nto the damping comes from the breathing Fermi-surface\nprocess. Refs. 11,16,17 have reported that the Gilbert\ndamping may start to increase as a function of disorder in\ndirtier samples. The interband transitions become more\nimportant with decreasing quasi-particle life times and\nstart to dominate the intraband transitions (The intra-\nband transitions give rise to the breathing Fermi-surface\ne\u000bect). We do not observe an increasing behavior in the\nmore di\u000busive regime, but we \fnd that the damping sat-\nurates at a value of around 0.0046 (See Fig. 3). In this\nregime, we believe that the breathing Fermi-surface e\u000bect\nis balanced by the interband transitions. The damping\ndoes not vanish in the limit 1 =l= 0 due to scattering\nat the interface between the GaAs and (Ga,Mn)As lay-\ners in addition to spin-pumping into the adjacent leads\n(an interface spin-pumping e\u000bect, as explained above).\nFig. 3 shows that the shape anisotropy of the damp-\ning is reduced by disorder because the di\u000berence between\nthe maximum ( \u000bmax) and minimum ( \u000bmin) values of the\ndamping parameter decrease with disorder. We antici-\npate this result because disorder increases the bulk damp-\ning e\u000bect, which is expected to be isotropic for an in\fnite\nsystem.\nIV. SUMMARY\nIn this paper, we studied the magnetization damping\nin the ferromagnetic semiconductor (Ga,Mn)As. The\nGilbert damping was calculated numerically using a\nrecently developed scattering matrix theory of mag-\nnetization dissipation.8We conducted a detailed non-\nperturbative study of the e\u000bects of disorder and an inves-\ntigation of the damping anisotropy induced by the shape\nof the sample.\nOur analysis showed that the damping process is\nmainly governed by three relaxation mechanisms. In the\nclean limit with little disorder, we found that the magne-\ntization dissipation is dominated by spin-pumping from\nthe d-electrons to the itinerant holes. For shorter mean\nfree paths, the breathing Fermi-surface e\u000bect starts to5\ndominate, which causes the damping to decrease. In\nthe di\u000busive regime, the breathing Fermi-surface e\u000bect\nis balanced by the interband transitions and the e\u000bective\ndamping parameter saturates at a value on the order of\n0.005.\nFor the small samples considered in this study, we\nfound that the shape of the system was typically more\nimportant than the anisotropic terms in the Hamiltonian\nfor the directional dependency of the damping parame-\nter. This shape anisotropy has not been reported beforeand o\u000bers a new way of manipulating the magnetization\ndamping.\nV. ACKNOWLEDGMENTS\nThis work was partially supported by the European\nUnion FP7 Grant No. 251759 \\MACALO\".\n1For a review, see D. C. Ralph and M. Stiles, J. Magn.\nMagn. Mater. 320, 1190 (2008), and reference therein.\n2B. Heinrich, D. Fraitov\u0013 a, and V. Kambersky, Phys. Status\nSolidi 23, 501 (1967); V. Kambersky, Can. J. Phys. 48,\n2906 (1970); V. Korenman and R.E. Prange, Phys. Rev.\nB6, 2769 (1972); V.S. Lutovinov and M.Y. Reizer, Zh.\nEksp. Teor. Fiz. 77, 707 (1979) [Sov. Phys. JETP 50, 355\n(1979)]; V.L. Safonov and H.N. Bertram, Phys. Rev. B 61,\nR14893 (2000); J. Kunes and V. Kambersky, Phys. Rev. B\n65, 212411 (2002); V. Kambersky Phys. Rev. B 76, 134416\n(2007).\n3T.L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n4J.A.C. Bland and B. Heinrich, Ultrathin Magnetic Struc-\ntures III Fundamentals of Nanomagnetism (Springer Ver-\nlag, Heidelberg, 2004).\n5V. Kambersky, Czech. J. Phys. B 26, 1366 (1976).\n6K. Gilmore, Y.U. Idzerda, and M.D. Stiles, Phys. Rev.\nLett. 99, 027204 (2007).\n7Y. Tserkovnyak, A. Brataas and G.E.W. Bauer, Phys. Rev.\nLett. 88, 117601 (2002).\n8A. Brataas, Y. Tserkovnyak, and G.E.W. Bauer, Phys.\nRev. Lett. 101, 037207 (2008); Phys. Rev. B 84, 054416\n(2011).\n9A. A. Starikov, P. J. Kelly, A. Brataas, Y. Tserkovnyak,\nand G. E. W. Bauer, Phys. Rev. Lett. 105, 236601\n(2010); Y. Liu, Z. Yuan, A. A. Starikov, and P. J. Kelly,\narXiv:1102.5305.\n10A similar analysis is presented in J. Seib, D. Steiauf, and\nM. F ahnle, Physical Review B 79, 092418 (2009).\n11J. Sinova, T. Jungwirth, X. Liu, Y. Sasaki, J.K. Furdyna,\nW.A. Atkinson, and A.H. MacDonald, Phys. Rev. B 69,\n085209 (2004).\n12Y.H. Matsuda, A. Oiwa, K. Tanaka, and H. Munekata,\nPhysica B 376-377 , 668 (2006).\n13A. Wirthmann et al. , Appl. Phys. Lett. 92, 232106 (2008).\n14Kh. Khazen et al. , Phys. Rev. B 78, 195210 (2008).\n15Y. Tserkovnyak, G.A. Fiete, and B.I. Halperin, Appl.\nPhys. Lett. 84, 5234 (2004).\n16I. Garate and A.H. MacDonald, Phys. Rev. B 79, 064403(2009).\n17I. Garate and A.H. MacDonald, Phys. Rev. B 79, 064404\n(2009).\n18This shape anisotropy in the Gilbert damping should not\nbe confused with the shape anisotropy (in the anisotropy\n\feld) caused by surface dipoles in non-spherical systems.\n19T. Jungwirth, J. Sinova, J. Ma\u0014 sek, J. Ku\u0014 cera, and A. H.\nMacDonald, Rev. Mod. Phys. 78, 809864 (2006).\n20M. Abolfath, T. Jungwirth, J. Brum, and A. H. MacDon-\nald, Phys. Rev. B 63, 054418 (2001).\n21P.Y. Yu and M. Cardona, Fundamentals of Semicon-\nductors: Physics and Materials Properties , 3rd Edition\n(Springer Verlag, Berlin, 2005).\n22Note that in Eq. (6) the spin operator s(in the p-d ex-\nchange term) is represented in the basis consisting of the\nfour spin 3/2 states ( s=J=3). The factor 1 =3 is absorbed\nin the exchange \feld h.\n23In the discrete version of Eq. (6), as used in the numerical\ncalculation, we have one impurity at each lattice site.\n24A. Chernyshov, M. Overby, X. Liu, J.K. Furdyna, Y.\nLyanda-Geller, and L.P. Rokhinson, Nature Physics 5, 656\n(2009).\n25A. Baldereschi and N.O. Lipari, Phys. Rev. B 8, 2697\n(1973).\n26The prefactor of 10 comes from 4 Ga atoms per unit cell\ntimes spin 5/2 per substitutional Mn, which are assumed to\nbe fully polarized. The reduction of the net magnetization\ndue to the interstitial Mn ions and p holes are disregarded\nin our estimate.\n27S. Datta, Electronic Transport in Mesoscopic Systems\n(Cambridge University Press, Cambridge, England, 1995).\n28T. Usuki, M. Saito, M. Takatsu, R. A. Kiehl, and N.\nYokoyama, Phys. Rev. B 52, 8244 (1995).\n29D. Steiauf and M. F ahnle, Physical Review B 72, 064450\n(2005).\n30The spin relaxation time of holes in GaAs is on the scale of\nthe momentum relaxation time. See D.J. Hilton and C.L.\nTang, Phys. Rev. Lett. 89, 146601 (2002), and references\ntherein."    },    {        "title": "2401.09938v2.Real_space_nonlocal_Gilbert_damping_from_exchange_torque_correlation_applied_to_bulk_ferromagnets_and_their_surfaces.pdf",        "content": "Real-space nonlocal Gilbert damping from exchange torque correlation applied to\nbulk ferromagnets and their surfaces\nBalázs Nagyfalusi,1,2,∗László Szunyogh,2,3,†and Krisztián Palotás1,2,‡\n1Institute for Solid State Physics and Optics, HUN-REN Wigner Research Center for Physics,\nKonkoly-Thege M. út 29-33, H-1121 Budapest, Hungary\n2Department of Theoretical Physics, Institute of Physics,\nBudapest University of Technology and Economics, Budafoki út 8, H-1111 Budapest, Hungary\n3HUN-REN-BME Condensed Matter Research Group,\nBudapest University of Technology and Economics, Budafoki út 8, H-1111 Budapest, Hungary\n(Dated: February 29, 2024)\nIn this work we present an ab initio scheme based on linear response theory of exchange torque\ncorrelation, implemented into the real-space Korringa-Kohn-Rostoker (RS-KKR) framework to cal-\nculate diagonal elements of the atomic-site-dependent intrinsic Gilbert damping tensor. The method\nis first applied to bcc iron and fcc cobalt bulk systems. Beside reproducing earlier results from the\nliterature for those bulk magnets, the effect of the lattice compression is also studied for Fe bulk,\nand significant changes for the Gilbert damping are found. Furthermore, (001)-oriented surfaces\nof Fe and Co are also investigated. It is found that the on-site Gilbert damping increases in the\nsurface atomic layer and decreases in the subsurface layer, and approaches the bulk value moving\nfurther inside the magnets. Realistic atomic relaxation of the surface layers enhances the identified\neffects. Thefirst-neighbordampingparametersareextremelysensitivetothesurfacerelaxation. De-\nspite their inhomogeneity caused by the surface, the transverse Gilbert damping tensor components\nremain largely insensitive to the magnetization direction.\nI. INTRODUCTION\nIt is highly demanded to understand and control\nthe dynamical processes governing the manipulation\nof various magnetic textures, such as atomic chains1,2,\nmagnetic skyrmions3,4or domain walls5, which can\nbe potentially used in future magnetic recording and\nlogic devices. These processes are often described by\nthe phenomenological Landau-Lifshitz-Gilbert (LLG)\nequation6,7,\n∂ ⃗ mi\n∂t=−γ ⃗ mi×⃗Beff\ni+α\nmi⃗ mi×∂ ⃗ mi\n∂t,(1)\nwhere ⃗ miis the magnetic moment at site i,mi=|⃗ mi|\nis its length, and γis the gyromagnetic ratio. The\nfirsttermonthe rhsofEq.(1)describestheprecession\nof⃗ miaround the effective magnetic field ⃗Beff\ni, while\nthe second term is the Gilbert damping due to the\nenergy dissipation to the lattice. Clearly, this latter\nterm causes the relaxation of the magnetization to its\nequilibrium value, which is controlled by the damping\nconstant αand plays a crucial role in the realization\nof high-speed spintronic devices.\nThe Gilbert damping constant αcan be deter-\nmined experimentally from the ferromagnetic reso-\nnance (FMR) spectroscopy where the damping pa-\nrameter is related to the line-width in the measured\nspectra8. FMR spectroscopy is a well-established\nmethod for bulk materials9,10, but especially in the\nlow temperature measurement it is controversial be-\ncause the intrinsic Gilbert damping needs to be sepa-\nrated from various extrinsic sources of the line-width,\ne.g., two-magnon scattering, eddy-current damping,\nradiative damping, spin-pumping, or the slow relaxer\nmechanism11–16. The comparison of experimental\nmeasurement to theoretical calculations is also made\ndifficult bythe sampleproperties likethe exactatomic\nstructure.From a theoretical perspective the ultimate goal is\nto develop a method to calculate the Gilbert damp-\ning parameters from the electronic structure of the\nmaterial. In the last decades there have been sev-\neral efforts to understand the damping process. The\nfirst successful method was developed by Kamberský\nwhorelatedthedampingprocesstothespin-orbitcou-\npling (SOC) in terms of the breathing Fermi surface\nmodel17, while he also proposed the spin-orbit torque\ncorrelation model18,19. Later on several other meth-\nods were introduced such as the spin-pumping20and\nlinear-response approaches11,21,22. A recent summary\nof these methods was published by Guimarães et al.23\nDue to the increased interest in noncollinear mag-\nnetism Fähnle et al.24suggested an inhomogeneous\ntensorial damping. The replacement of a scalar αby\na damping matrix αmeans that the damping field in\nEq.(1)isnolongerproportionaltothetimederivative\nof⃗ mi, it becomes a linear function of ∂ ⃗ mi/∂t. More-\nover, nonlocality of the damping process implies that\nthe damping field at site iexperiences ∂ ⃗ mj/∂tfor any\nsitej. The LLG equation (1) is then replaced by the\nset of equations25,\n∂ ⃗ mi\n∂t=⃗ mi×\n−γ⃗Beff\ni+X\njαij1\nmj∂ ⃗ mj\n∂t\n,(2)\nwhere the damping term is unfolded to pairwise con-\ntributions of strength αij. The appearance of non-\nlocal damping terms was evidenced for magnetic do-\nmain walls26,27by linking the Gilbert damping to the\ngradients of the magnetization. In NiFe, Co, and\nCoFeB thin films Li et al.28measured wave-number-\ndependent dissipation using perpendicular spin wave\nresonance, validating thus the idea of nonlocal damp-\ning terms. Different analytical expressions for αijare\nalready proposed22,25,29,30, and the nonlocal damp-\ning is found for bulk materials25,31as well as its ef-arXiv:2401.09938v2  [cond-mat.mtrl-sci]  28 Feb 20242\nfect on magnon properties of ferromagnets have been\ndiscussed32. Recent studies went further and, anal-\nogously to the higher order spin-spin interactions in\nspin models, introduced multi-body contributions to\nthe Gilbert damping33.\nThe calculation of the Gilbert damping prop-\nerties of materials has so far been mostly fo-\ncused on 3D bulk magnets, either in chemically\nhomogeneous11,19,23,25,34–36or heterogeneous (e.g.\nalloyed)11,22,31forms. There are a few studies avail-\nable reporting on the calculation of the Gilbert damp-\ning in 2D magnetic thin films12,23,37,38, or at surfaces\nand interfaces of 3D magnets31,35,37. The calculation\nof the Gilbert damping in 1D or 0D magnets is, due\nto our knowledge, not reported in the literature. Fol-\nlowing the trend of approaching the atomic scale for\nfunctional magnetic elements in future spintronic de-\nvices, the microscopic understanding of energy dissi-\npation through spin dynamics in magnets of reduced\ndimensions is inevitable and proper theoretical meth-\nods have to be developed.\nOur present work proposes a calculation tool for\nthediagonalelementsofthenon-localintrinsicGilbert\ndamping tensor covering the 3D to 0D range of mag-\nnetic materials on an equal footing, employing a real-\nspace embedding Green’s function technique39. For\nthis purpose, the linear response theory of the Gilbert\ndamping obtained by the exchange torque correlation\nis implemented in the real-space KKR method. As a\ndemonstration of the new method, elemental Fe and\nCo magnets in their 3D bulk form and their (001)-\noriented surfaces are studied in the present work. Go-\ning beyond comparisons with the available literature,\nnew aspects of the Gilbert damping in these materials\nare also reported.\nThe paper is organized as follows. In Sec. II the\ncalculation of the Gilbert damping parameters within\nthe linear response theory of exchange torque corre-\nlation using the real-space KKR formalism is given.\nSec. III reports our results on bulk bcc Fe and fcc Co\nmaterials and their (001)-oriented surfaces. We draw\nour conclusions in Sec. IV.\nII. METHOD\nA. Linear response theory within real-space\nKKR\nThe multiple-scattering of electrons in a finite clus-\nter consisting of NCatoms embedded into a 3D or 2D\ntranslation-invariant host medium is fully accounted\nfor by the equation39\nτC=τH\u0002\nI−(t−1\nH−t−1\nC)τH\u0003−1,(3)\nwhere τCand τHare the scattering path operator\nmatrices of the embedded atomic cluster and the host,\nrespectively, tCandtHare the corresponding single-\nsite scattering matrices, all in a combined atomic site\n(j, k∈ {1, ..., N C}) and angular momentum ( Λ,Λ′∈\n{1, ...,2(ℓmax+ 1)2}) representation: τ={τjk}=\n{τjk\nΛΛ′}andt={tj\nΛΛ′δjk}, where ℓmaxis the angularmomentum cutoff in describing the scattering events,\nand for simplicity we dropped the energy-dependence\nof the above matrices.\nFor calculating the diagonal Cartesian elements\nof the nonlocal Gilbert damping tensor connecting\natomic sites jandkwithin the finite magnetic atomic\ncluster, we use the formula derived by Ebert et al.22,\nαµµ\njk=2\nπmj\nsTr\u0010\nTj\nµ˜τjk\nCTk\nµ˜τkj\nC\u0011\n, (4)\nwhere µ∈ {x, y, z}, the trace is taken in the\nangular-momentum space and the formula has to\nbe evaluated at the Fermi energy ( EF). Here,\nmj\nsis the spin moment at the atomic site j,\n˜τjk\nC,ΛΛ′= (τjk\nC,ΛΛ′−(τkj\nC,Λ′Λ)∗)/2i, and Tj\nµis the\ntorque operator matrix which has to be calculated\nwithin the volume of atomic cell j,Ωj:Tj\nµ;ΛΛ′=R\nΩjd3rZj\nΛ(⃗ r)×βσµBxc(⃗ r)Zj\nΛ′(⃗ r),wherethenotationof\nthe energy-dependence is omitted again for simplicity.\nHere, βis a standard Dirac matrix entering the Dirac\nHamiltonian, σµare Pauli matrices, and Bxc(⃗ r)is the\nexchange-correlation field in the local spin density ap-\nproximation (LSDA), while Zj\nΛ(⃗ r)are right-hand side\nregular solutions of the single-site Dirac equation and\nthe superscript ×denotes complex conjugation re-\nstricted to the spinor spherical harmonics only22. We\nshould emphasize that Eq. (4) applies to the diagonal\n(µµ) elements of the Gilbert tensor only. To calcu-\nlate the off-diagonal tensor elements one needs to use,\ne.g., the more demanding Kubo-Bastin formula40,41.\nNote also that in noncollinear magnets the exchange\nfield Bxc(⃗ r)is sensitive to the spin noncollinearity42\nwhich influences the calculated torque operator ma-\ntrix elements, however, this aspect does not concern\nour present study including collinear magnetic states\nonly.\nNote that the nonlocal Gilbert damping is, in gen-\neral, not symmetric in the atomic site indices, αµµ\njk̸=\nαµµ\nkj, instead\nαµµ\nkj=mj\ns\nmksαµµ\njk(5)\nholds true. This is relevant in the present work for\nthe ferromagnetic surfaces. On the other hand, in\nferromagnetic bulk systems αµµ\njk=αµµ\nkjsince mj\ns=\nmk\ns=msfor any pair of atomic sites.\nIn practice, the Gilbert damping formula in Eq. (4)\nis not directly evaluated at the Fermi energy, but a\nsmall imaginary part ( η) of the complex energy is ap-\nplied, which is called broadening in the following, and\nits physical effect is related to the scattering rate in\nother damping theories19,25,37,43. Taking into account\nthe broadening η, the Gilbert damping reads\nαµµ\njk(η) =−1\n4h\n˜αµµ\njk(E+, E+) + ˜αµµ\njk(E−, E−)\n−˜αµµ\njk(E+, E−)−˜αµµ\njk(E−, E+)i\n,(6)\nwhere E+=EF+iηandE−=EF−iη, and the3\nindividual terms are\n˜αµµ\njk(E1, E2) =\n2\nπmj\nsTr\u0010\nTj\nµ(E1, E2)τjk\nC(E2)Tk\nµ(E2, E1)τkj\nC(E1)\u0011\n(7)\nwith E1,2∈ { E+, E−}, and the ex-\nplicitly energy-dependent torque opera-\ntor matrix elements are: Tj\nµ;ΛΛ′(E1, E2) =R\nΩjd3rZj×\nΛ(⃗ r, E1)βσµBxc(⃗ r)Zj\nΛ′(⃗ r, E2).\nB. Effective damping and computational\nparameters\nEq.(6)givesthebroadening-dependentspatiallydi-\nagonal elements of the site-nonlocal Gilbert damping\ntensor: αxx\njk(η),αyy\njk(η), and αzz\njk(η). Since no longitu-\ndinal variation of the spin moments is considered, the\ntwo transversal components perpendicular to the as-\nsumed uniform magnetization direction are physically\nmeaningful. Given the bulk bcc Fe and fcc Co sys-\ntems and their (001)-oriented surfaces with C4vsym-\nmetry under study in the present work, in the follow-\ning the scalar αrefers to the average of the xxand\nyyGilbert damping tensor components assuming a\nparallel magnetization with the surface normal z[001]-\ndirection: αjk= (αxx\njk+αyy\njk)/2 =αxx\njk=αyy\njk. From\nthe site-nonlocal spatial point of view in this work we\npresent results on the on-site (\" 00\"), first neighbor\n(denoted by \" 01\") and second neighbor (denoted by\n\"02\") Gilbert damping parameters, and an effective,\nso-called total Gilbert damping ( αtot), which can be\ndefined as the Fourier transform of αjkat⃗ q= 0. The\nFourier transform of the Gilbert damping reads\nα⃗ q=∞X\nj=0α0jexp(−i⃗ q(⃗ r0−⃗ rj))\n≈X\nr0j≤rmaxα0jexp(−i⃗ q(⃗ r0−⃗ rj)),(8)\nwhere r0j=|⃗ r0−⃗ rj|and the effective damping is\ndefined as\nαtot=α⃗ q=⃗0=∞X\nj=0α0j≈X\nr0j≤rmaxα0j.(9)\nSince we have a real-space implementation of the\nGilbert damping, the infinite summation for both\nquantities is replaced by an approximative summation\nfor neighboring atoms upto an rmaxcutoff distance\nmeasured from site \"0\". Moreover, note that for bulk\nsystems the effective damping αtotis directly related\nto the ⃗ q= 0mode of FMR experiments.\nThe accuracy of the calculations depends on many\nnumerical parameters such as the number of ⃗kpoints\nused in the Brillouin zone integration, the choice of\nthe angular momentum cutoff ℓmax, and the spatial\ncutoff rmaxused for calculating α⃗ qandαtot. Previ-\nous research25showed that the Gilbert damping heav-\nily depends on the broadening η, so we extended ourstudies to a wider range of η= 1meV to 1 eV. The\nsufficient k-point sampling was tested at the distance\nofrmax= 7a0(where a0is the corresponding 2D lat-\ntice constant) from the reference site with the broad-\nening set to 1mRy, and the number of ⃗kpoints was\nincreased up to the point, where the 5th digit of the\ndamping became stable. Maximally, 320400 ⃗kpoints\nwere used for the 2D layered calculation but the re-\nquested accuracy was reached with 45150 and 80600\n⃗kpoints for bulk bcc Fe and fcc Co systems, respec-\ntively.\nThe choice of ℓmaxwas tested through the whole η\nrange for bcc Fe, and it was based on the comparison\nof damping calculations with ℓmax= 2andℓmax= 3.\nThe maximal deviation for the on-site Gilbert damp-\ning was found at around η= 5mRy, but it was still\nless than 10%. The first and second neighbor Gilbert\ndamping parameters changed in a more significant\nway (by ≈50%) in the whole ηrange upon changing\nℓmax, yet the effective total damping was practically\nunchanged, suggesting that farther nonlocal damping\ncontributions compensate this effect. Since αtotis the\nmeasurable physical quantity we concluded that the\nlower angular momentum cutoff of ℓmax= 2is suffi-\ncient to be used further on.\nThe above choice of ℓmax= 2for the angular mo-\nmentumcutoff, themathematicalcriterionofpositive-\ndefinite αjk(which implies α⃗ q>0for all ⃗ qvectors),\nand the prescribed accuracy for the effective Gilbert\ndamping in the full considered η= 1meV to 1 eV\nrange set rmaxto 20 a0for both bcc Fe and fcc Co. It\nis worth mentioning that the consideration of lattice\nsymmetries made possible to decrease the number of\natomic sites in the summations for calculating α⃗ qand\nαtotby an order of magnitude.\nIII. RESULTS AND DISCUSSION\nOur newly implemented method was employed to\nstudy the Gilbert damping properties of Fe and Co\nferromagnetsintheirbulkand(001)-orientedsurfaces.\nIn these cases only unperturbed host atoms form\nthe atomic cluster, and the so-called self-embedding\nprocedure44is employed, where Eq. (3) reduces to\nτC=τHforthe3Dbulkmetalsand2Dlayeredmetal-\nvacuum interfaces.\nA. Bulk Fe and Co ferromagnets\nFirst we calculate and analyze the nonlocal and ef-\nfective dampings for bulk bcc Fe by choosing a 2D\nlattice constant of a0= 2.863Å. The magnitude of\nthe magnetic moments are obtained from the self-\nconsistent calculation. The spin and orbital moments\narems= 2.168µBandmo= 0.046µB, respectively.\nThe broadening is set to η= 68meV. The inset of Fig.\n1a) shows the typical function of the nonlocal Gilbert\ndamping α0jdepending on the normalized distance\nr0j/a0between atomic sites \"0\" and \" j\". In accor-\ndance with Ref. 25 the nonlocal Gilbert damping\nquickly decays to zero with the distance, and can be4\na)\n5 10 15 2005\n5 10 15 20−505\nr0j/a0α0j[×10−4]\nr0j/a0α0j·(r0j/a0)2[×10−4]\nb)\n5 10 15 20−202468\nrmax/a0αtot[×10−3]\nFIG. 1. a) Nonlocal Gilbert damping in bulk bcc Fe as a\nfunction of distance r0jbetween atomic sites \"0\" and \" j\"\nshown upto a distance of 20 a0(the 2D lattice constant is\na0= 2.863Å): the black squares are calculated α0jval-\nues times the normalized squared-distance along the [110]\ncrystallographic direction, and the red line is the corre-\nsponding fitted curve based on Eq. (10). The inset shows\nthe nonlocal Gilbert damping α0jvalues in the given dis-\ntance range. b) Convergence of the effective damping pa-\nrameter αtot, partial sums of α0jupto rmaxbased on Eq.\n(9), where rmaxis varied. The broadening is chosen to be\nη=68 meV.\nwell approximated with the following function:\nα(r)≈Asin (kr+ϕ0)\nr2exp(−βr).(10)\nTo test this assumption we assorted the atomic sites\nlying in the [110] crystallographic direction and fit-\nted Eq. (10) to the calculated data. In practice, the\nfit is made on the data set of α0j(r0j/a0)2, and is\nplotted in Fig. 1a). Although there are obvious out-\nliers in the beginning, the magnitude of the Gilbert\ndamping asymptotically follows the ∝exp(−βr)/r2\ndistance dependence. The physical reason for this de-\ncay is the appearance of two scattering path operators\n(Green’s functions) in the exchange torque correlation\nformula in Eq. (4) being broadened due to the finite\nimaginary part of the energy argument.\nIn our real-space implementation of the Gilbert\ndamping, an important parameter for the effective\ndamping calculation is the real-space cutoff rmaxin\nEq. (9). Fig. 1b) shows the evolution of the ef-\nfective (total) damping depending on the rmaxdis-\ntance, within which all nonlocal damping terms α0jare summed up according to Eq. (9). An oscillation\ncan similarly be detected as for the nonlocal damping\nitself in Fig. 1a), and this behavior was fitted with\na similar exponentially decaying oscillating function\nas reported in Eq. (10) in order to determine the ex-\npected total Gilbert damping αtotvalue in the asymp-\ntotic r→ ∞limit. In the total damping case it is\nfound that the spatial decay of the oscillation is much\nslower compared to the nonlocal damping case, which\nmakes the evaluation of αtotmore cumbersome. Our\ndetailed studies evidence that for different broaden-\ningηvalues the wavelength of the oscillation stays\nthe same but the spatial decay becomes slower as\nthe broadening is decreased (not shown). This slower\ndecay together with the fact that the effective (to-\ntal) damping value itself is also decreasing with the\ndecreasing broadening results that below the 10meV\nrange of ηthe amplitude of the oscillation at the dis-\ntance of 20 a0is much larger than its asymptotic limit.\nIn practice, since the total damping is calculated as\nther→ ∞limit of such a curve as shown in Fig. 1b),\nthis procedure brings an increased error for αtotbelow\nη= 10meV, and this error could only be reduced by\nincreasing the required number of atomic sites in the\nreal-space summation in Eq. (10).\nFig. 2 shows the dependence of the calculated on-\nsite, first- and second-neighbor and effective total\nGilbert damping parameters on the broadening η.\nThe left column shows on-site ( α00) and total ( αtot)\nwhile the right one the first ( α01) and second ( α02)\nneighbor Gilbert dampings. We find very good agree-\nment with the earlier reported results of Thonig et\nal.25, particularly that the on-site damping has the\nlargest contribution to the total damping being in the\nsame order of magnitude, while the first and second\nneighbors are smaller by an order of magnitude. The\nobtained dependence on ηis also similar to the one\npublished by Thonig et al.25:α00andαtotare in-\ncreasing with η, and α01andα02do not follow a\ncommon trend, and they are material-dependent, see,\ne.g., the opposite trend of α02with respect to ηfor\nFe and Co. The observed negative values of some of\nthesite-nonlocaldampingsarestillconsistentwiththe\npositive-definiteness of the full (infinite) αjkmatrix,\nwhich has also been discussed in Ref. 25.\nThe robustness of the results was tested against a\nsmall change of the lattice constant simulating the ef-\nfect of an external pressure for the Fe bulk. These re-\nsults are presented in the second row of Fig. 2, where\nthe lattice constant of Fe is set to a0= 2.789Å. In this\ncase the magnetic moments decrease to ms= 2.066µB\nandmo= 0.041µB. It can clearly be seen that the on-\nsite, first and second neighbor Gilbert dampings be-\ncome smaller upon the assumed 2.5% decrease of the\nlattice constant, but the total damping remains prac-\ntically unchanged in the studied ηrange. This sug-\ngests that the magnitudes of more distant non-local\ndamping contributions are increased.\nThe third row of Fig. 2 shows the selected damp-\ning results for fcc Co with a 2D lattice constant of\na0= 2.507Å. The spin and orbital moments are\nms= 1.654µBandmo= 0.078µB, respectively. The\nincreaseofthetotal, theon-site, andthefirst-neighbor5\n10−310−210−11000246810Fe -a0= 2.863˚A\nη(eV)α[×10−3]α00\nαtot\n10−310−210−1100−202468Fe -a0= 2.863˚A\nη(eV)α[×10−4]α01\nα02\n10−310−210−11000246810Fe -a0= 2.789˚A\nη(eV)α[×10−3]α00\nαtot\n10−310−210−1100−202468Fe -a0= 2.789˚A\nη(eV)α[×10−4]α01\nα02\n10−310−210−11000246810Co -a0= 2.507˚A\nη(eV)α[×10−3]α00\nαtot\n10−310−210−1100−202468Co -a0= 2.507˚A\nη(eV)α[×10−4]α01\nα02\nFIG. 2. Left column: Local on-site ( α00, black square)\nand total ( αtot, red triangle) Gilbert damping as a func-\ntionofthebroadening ηforbccFe(001)with a0= 2.863Å,\nbcc Fe(001) with a0= 2.789Å, and fcc Co(001) with\na0= 2.507Å. Right column: Nonlocal first nearest neigh-\nbor (α01, black square) and second nearest neighbor ( α02,\nred triangle) Gilbert damping for the same systems.\ndampings with increasing ηis similar to the Fe case,\nand the on-site term dominates αtot. An obvious\ndifference is found for the second-neighbor damping,\nwhich behaves as an increasing function of ηfor Co\nunlike it is found for Fe.\nConcerning the calculated damping values, there is\na large variety of theoretical methods and calculation\nparameters, as well as experimental setups used in\nthe literature, which makes ambiguous to compare\nour results with others. Recently, Miranda et al.31\nreported a comparison of total and on-site damping\nvalues with the available theoretical and experimen-\ntal literature in their Table S1. For bcc Fe bulk they\nreported total damping values in the range of 1.3–\n4.2×10−3and for fcc Co bulk within the range of 3.2–\n11×10−3, and our results fit very well within theseranges around η≈100meV for Fe and for η >100\nmeV for Co. Moreover, we find that our calculated on-\nsite damping values for bcc Fe are larger ( >5×10−3)\nthan the reported values of Miranda et al.(1.6×10−3\nand 3.6 ×10−3), but for fcc Co the agreement with\ntheir reported total (3.2 ×10−3) and on-site damping\n(5.3×10−3) values is very good at our η= 136meV\nbroadening value.\n10−310−210−110010−510−410−310−2Fe\nη(eV)αtot\nαSOC=1\nαSOC=0\n10−310−210−110010−510−410−310−2Co\nη(eV)αtot\nαSOC=1\nαSOC=0\nFIG. 3. Effective (total) Gilbert damping for bcc Fe\n(left) and fcc Co (right) as a function of broadening ηon\na log-log scale. The error bars are estimated from the\nfitting procedure of Eq. (10). The red triangles show the\ncase with normal SOC ( αSOC=1), and the blue diamonds\nwhere SOC is switched off ( αSOC=0).\nNext, weinvestigatethespin-orbit-coupling-(SOC)-\noriginated contribution to the Gilbert damping. Our\nmethodmakesitinherentlypossibletoincludeaSOC-\nscaling factor in the calculations45. Fig. 3 shows the\nobtained total Gilbert damping as a function of the\nbroadening ηwith SOC switched on/off for bcc Fe\nand fcc Co. It can be seen that the effect of SOC\nis not dominant at larger ηvalues, but the SOC\nhas an important contribution at small broadening\nvalues ( η < 10−2eV), where the calculated total\nGilbert damping values begin to deviate from each\nother with/without SOC. As discussed in Ref. 23,\nwithout SOC the damping should go toward zero for\nzero broadening, which is supported by our results\nshown in Fig. 3.\nB. (001)-oriented surfaces of Fe and Co\nferromagnets\nIn the following, we turn to the investigation of the\nGilbertdampingparametersatthe(001)-orientedsur-\nfaces of bcc Fe and fcc Co. Both systems are treated\nas a semi-infinite ferromagnet interfaced with a semi-\ninfinite vacuum within the layered SKKR method46.\nIn the interface region 9 atomic layers of the ferromag-\nnet and 3 atomic layers of vacuum are taken, which is\nsandwiched between the two semi-infinite (ferromag-\nnet and vacuum) regions. Two types of surface atomic\ngeometries were calculated: (i) all atomic layers hav-\ning the bulk interlayer distance, and (ii) the surface\nand subsurface atomic layers of the ferromagnets have6\nTABLE I. Geometry relaxation at the surfaces of the fer-\nromagnets: change of interlayer distances relative to the\nbulk interlayer distance at the surfaces of bcc Fe(001) and\nfcc Co(001), obtained from VASP calculations. \"L1\" de-\nnotes the surface atomic layer, \"L2\" the subsurface atomic\nlayer, and \"L3\" the sub-subsurface atomic layer. All other\ninterlayer distances are unchanged in the geometry opti-\nmizations.\nL1-L2 L2-L3\nbcc Fe(001) -13.7% -7.7%\nfcc Co(001) -12.4% -6.4%\nbeenrelaxedintheout-of-planedirectionusingtheVi-\nenna Ab-initio Simulation Package (VASP)47within\nLSDA48. For the latter case the obtained relaxed\natomic geometries are given in Table I.\nFigure 4 shows the calculated layer-resolved on-\nsite and first-neighbor Gilbert damping values (with\nη= 0.68eV broadening) for the bcc Fe(001) and fcc\nCo(001) surfaces. It can generally be stated that the\nsurface effects are significant in the first 4 atomic lay-\ners of Fe and in the first 3 atomic layers of Co. We\nfind that the on-site damping ( α00) increases above\nthe bulk value in the surface atomic layer (layer 1:\nL1), and decreases below the bulk value in the sub-\nsurface atomic layer (L2) for both Fe and Co. This\nfinding is interesting since the spin magnetic moments\n(ms, shown in the insets of Fig. 4) are also consider-\nably increased compared to their bulk values in the\nsurface atomic layer (L1), and the spin moment enters\nthe denominator when calculating the damping in Eq.\n(4).α00increases again in L3 compared to its value in\nL2, thus it exhibits a nonmonotonic layer-dependence\nin the vicinity of the surface. The damping results ob-\ntained with the ideal bulk interlayer distances and the\nrelaxed surface geometry (\"R\") are also compared in\nFig. 4. It can be seen that the on-site damping is in-\ncreasedinthesurfaceatomiclayer(L1), anddecreased\nin the subsurface (L2) and sub-subsurface (L3) atomic\nlayers upon atomic relaxation (\"R\") for both Fe and\nCo. The first-neighbor dampings ( α01) are of two\ntypes for the bcc Fe(001) and three types for the fcc\nCo(001), see caption of Fig. 4. All damping values\nare approaching their corresponding bulk value mov-\ning closer to the semi-infinite bulk (toward L9). In\nabsolute terms, for both Fe and Co the maximal sur-\nface effect is about 10−3for the on-site damping, and\n2×10−4for the first-neighbor dampings. Given the\ndamping values, the maximal relative change is about\n15% for the on-site damping, and the first-neighbor\ndampings can vary by more than 100% (and can even\nchangesign)inthevicinityofthesurfaceatomiclayer.\nNote that Thonig and Henk35studied layer-resolved\n(effective) damping at the surface of fcc Co within the\nbreathing Fermi surface model combined with a tight-\nbinding electronic structure approach. Although they\nstudied a different quantity compared to us, they also\nreported an increased damping value in the surface\natomic layer, followed by an oscillatory decay toward\nbulk Co.\nSo far the presented Gilbert damping results cor-\nrespond to spin moments pointing to the crystallo-\n1 3 5 7 90.81Fe\n1 92.43\nlayermsms\nmR\ns\nlayerα00[×10−2]\nα00\nαR\n00\n1 3 5 7 90.81\n1 91.71.8\nlayermsms\nmR\nsCo\nlayerα00[×10−2]α00\nαR\n00\n1 3 5 7 9−2−101Fe\nlayerα01[×10−4]\nα01+αR\n01+\nα01−αR\n01−\n1 3 5 7 91234\nCo\nlayerα01[×10−4]α01+αR\n01+\nα01−αR\n01−\nα01 αR\n01FIG. 4. Evolution of the layer-resolved Gilbert damping\nfrom the surface atomic layer (L1) of bcc Fe(001) and fcc\nCo(001) toward the bulk (L9), depending also on the out-\nof-plane atomic relaxation \"R\". On-site ( α00) and first\nneighbor ( α01) Gilbert damping values are shown in the\ntop two and bottom two panels, respectively. The broad-\nening is η= 0.68eV. The empty symbols belong to the\ncalculations with the ideal bulk interlayer distances, and\nthe full symbols to the relaxed surface geometry, denoted\nwith index \"R\". Note that α01is calculated for nearest\nneighbors of atomic sites in the neighboring upper, lower,\nand the same atomic layer (for fcc Co only), and they are\nrespectively denoted by \" +\" (L-(L+1)), \" −\" (L-(L −1)),\nand no extra index (L-L). The insets in the top two panels\nshow the evolution of the magnitudes of the layer-resolved\nspin magnetic moments ms. The horizontal dashed line in\nall cases denotes the corresponding bulk value.\ngraphic [001] ( z) direction, and the transverse compo-\nnents of the damping αxxandαyyare equivalent due\ntothe C4vsymmetryofthe(001)-orientedsurfaces. In\norder to study the effect of a different orientation of\nall spin moments on the transverse components of the\ndamping, we also performed calculations with an ef-\nfective field pointing along the in-plane ( x) direction:\n[100] for bcc Fe and [110] for fcc Co. In this case, due\nto symmetry breaking of the surface one expects an\nanisotropy in the damping, i.e., that the transverse\ncomponents of the damping tensor, αyyandαzz, are\nnot equivalent any more. According to our calcula-7\ntions, however, the two transverse components of the\non-site ( αyy\n00andαzz\n00) and nearest-neighbor ( αyy\n01and\nαzz\n01) damping tensor, at the Fe surface differed by less\nthan 0.1 % and at the Co surface by less than 0.2 %,\ni.e., despite the presence of the surface the damping\ntensor remained highly isotropic. The change of the\ndamping with respect to the orientation of the spin\nmoments in zorxdirection (damping anisotropy)\nturned out to be very small as well: the relative dif-\nference in αyy\n00was 0.1 % and 0.3 %, while 0.5 % and\n0.1 % in αyy\n01for the Fe and the Co surfaces, respec-\ntively. For the farther neighbors, this difference was\nless by at least two orders of magnitude.\nIV. CONCLUSIONS\nWe implemented an ab initio scheme of calculat-\ning diagonal elements of the atomic-site-dependent\nGilbert damping tensor based on linear response the-\nory of exchange torque correlation into the real-space\nKorringa-Kohn-Rostoker (KKR) framework. To val-\nidate the method, damping properties of bcc Fe and\nfcc Co bulk ferromagnets are reproduced in good com-\nparison with the available literature. The lattice com-\npression is also studied for Fe bulk, and important\nchanges for the Gilbert damping are found, most pro-\nnounced for the site-nonlocal dampings. By investi-\ngating (001)-oriented surfaces of ferromagnetic Fe andCo, we point out substantial variations of the layer-\nresolved Gilbert damping in the vicinity of the sur-\nfaces depending on various investigated parameters.\nThe effect of such inhomogeneous dampings should be\nincluded into future spin dynamics simulations aim-\ning at an improved accuracy, e.g., for 2D surfaces and\ninterfaces. We anticipate that site-nonlocal damping\neffects become increasingly important when moving\ntoward physical systems with even more reduced di-\nmensions (1D).\nACKNOWLEDGMENTS\nThe authors acknowledge discussions with Danny\nThonig. Financial support of the National Research,\nDevelopment, and Innovation (NRDI) Office of Hun-\ngary under Project Nos. 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China  \n5Beijing Academy of Quantum Information Sciences, Beijing 100193, P. R. China  \n6IBM Research - Almaden, San Jose, California 95120, USA   \n†These authors contributed equally to the work  \n*Correspondence to: weihan@pku.edu.cn (W.H.); seeyang@us.ibm.com (S.H.Y.) . \n \n \nAbstract   \nInterfaces between materials with differently ordered phases present unique opportunities for \nexotic physical properties, especially the interplay between ferromagnetism and superconductivity \nin the ferromagnet/superconductor heterostructures. The investig ation of zero- and π-junctions has \nbeen of particular interest for both fundamental physical science and emerging technologies.  Here, \nwe report the experimental observation of giant oscillatory Gilbert damping in the superconducting \nNb/NiFe/Nb junctions wi th respect to the NiFe thickness. This observation suggests an \nunconventional spin pumping and relaxation via zero-energy Andreev bound states that exist only \nin the Nb/NiFe/Nb π-junctions, but not in the Nb/NiFe/Nb zero-junctions. Our findings could be \nimportant for further exploring the exotic physical properties of ferromagnet/superconductor \nheterostructures, and potential applications of ferromagnet π-junctions in quantum computing, \nsuch as half -quantum flux qubits.  2 \n  \nOne sentence summary: Giant oscillat ory Gilbert damping is observed in \nsuperconductor/ferromagnet/superconductor junctions with varying the ferromagnet thickness.  \n \n \nIntroduction  \nThe interplay between ferromagnetism and superconductivity  has induced many exotic and \nexciting physical properties in ferromagnet (FM)/superconductor (SC) heterostructures (1-3). Of \nparticular interest is the unconventional π-phase ground state SC/FM/SC junction  that might be \nrealized for cert ain FM thicknesses arising from the quantum intermixing of the wave functions \nbetween spin -singlet Cooper pairs in SC and spin -polarized electrons in FM (1, 3, 4). At the FM/SC \ninterface, a Cooper pair moving into the FM will ha ve a finite center -of-mass momentum, resulting \nin the oscillation of the  real part of  superconducting order parameter  (Re {Ψ})  with respect to the \nFM thickness  (Fig. 1A ) (1, 5, 6). Depending on the FM thicknesses, the Cooper pair wavefunctions \nin the two superconductors on either side of the FM can have a phase difference from zero or π, \nforming so -called zero-junctions with positive Josephson coupling (Fig. 1B ) or π-junctions with  \nthe negative Josephson coupling  (Fig. 1C) . The FM π-junctions can be used for quantum \ncomputing applications (7, 8), as half quantum flux qubits (9). Due to the scientific and technical \nimportance, the research on the FM π-junctions has been active for the last tw o decades (6, 10-13). \nPrevious experimental studies have demonstrated the switching between zero- and π-junctions in \nSC/FM/SC structures by varying the temperature and the FM thickness (11, 14-17). These reports \nmainly focus on the electrical properties of the FM zero- and π-junctions. Recently, dynamic spin \ninjection into SCs has attracted considerable interest  both the exper imentally  (18-21) and \ntheoretically (22-26). However, the spin -dependent properties in FM zero- and π-junctions have \nnot been explored yet. The investigation of the spin -dependent properties requires the spin current \nprobes, such as the dynamical spin pumping (27). Furthemore, for the application of the FM π-\njunctio ns in quantum computing technologies (9), the magnetization/spin dynamic properties are \nextremely important to be studi ed. \nHere, we report the experimental observation of giant oscillatory Gilbert damping in the \nsuperconducting Nb/NiFe/Nb junctions with respect to the NiFe thickness, which can be 3 \n qualitatively explained by the different spin pumping efficiency via the Andr eev bound states \n(ABS) of Nb/NiFe/Nb zero- and π-junctions. Using a minimal model based on the ABS, we show \nthat an unconventional spin pumping into the zero -energy ABS  penetrat ed into SCs  could occur s \nonly for the π-junctions, which can lead to the oscillatory Gilbert damping as a function of the \nNiFe thick ness. \nResults  \nFigures 1D and 1E  show the schematic s of the spin pumping, magnetization dynamics, and \nenhanced Gilbert damping in the  SC/FM/SC  zero- and π-junction s. Spin pumping refers to the \nspin-polarized current injection to non -magnetic materials from a FM with precessing \nmagnetization around its ferromagnetic resonance (FMR) conditions (28, 29). In FM and its \nheterostructures, the Gilbert damping (\n) characterizes the magnetization dynamics , as described \nby the Landau -Lifshitz -Gilbert formula with an additiona l Slonczewski -torque term (30-32): \n 𝑑𝒎\n𝑑𝑡=−𝛾𝒎×𝑯𝒆𝒇𝒇+𝛼𝒎×𝑑𝒎\n𝑑𝑡+𝛾\n𝑀𝑠𝑉(ℏ\n4𝜋𝑔↑↓𝒎×𝑑𝒎\n𝑑𝑡) (1) \nwhere 𝒎=𝑴/|𝑴| is the magnetization unit vector,  𝛾 is the gyromagnetic ratio, 𝑯𝒆𝒇𝒇 is the total \neffective magnetic field, 𝑀𝑠=|𝑴| is the saturation magnetization, and 𝑔↑↓ is the interface spin  \nmixing conductance. The pumped spin current from FM into SCs can be expressed by Js =\nℏ\n4𝜋𝑔↑↓𝒎×𝑑𝒎\n𝑑𝑡 (29). The spin pumping into the SCs  give rise  to an enhanced Gilbert damping \nconstant  that is proportional to the spin pumping current  (αsp~ J s) (29). Fig. 1E illustrates the \npumped spin current mediated by the zero -energy ABS inside the superconducting gap  in π-\njunctions , which will be discussed later in details.  While for a zero-junction, the pumped spin \ncurrent is mediated by the ABS near the superconducting gap (Fig. 1D). The ABS can be formed \nwithin the FM layer and then extended into the interface of SCs with the superconducting coherent \nlength scale (33, 34).   \nThe SC/FM/SC junctions consist of a NiFe (Ni 80Fe20) layer (thickness: ~ 5 - 20 nm) \nsandwiched by two Nb layers (thickness: 100 nm) grown by magnetron sputt ering (see Methods \nand f ig. S1). To maximize the integrity of samples for a systematic study, more than tens of \nsamples are grown in each run via rotation mask technique in a sputtering system, which is the \nsame as in the previous study of the oscillatory exchange coupling in mag netic multilayer 4 \n structures (35). The Gilbert damping  and spin pumping  are measured by the ferromagnetic \nresonance  (FMR)  technique (see Methods for details) .  \nAbove the TC of Nb, spin pumping in the Nb/NiFe/Nb junctions leads to the spin accumulation \nin Nb near the interface, which can be described by the spin -dependent chemical potentials, as \nillustrated in Fig. 2 A. The Gilbert damping of NiFe in the Nb/NiFe/Nb junctions is determined \nfrom the microwave frequen cy-dependent FMR spectra ( fig. S2). A typical FMR curve with the \nLorentzian fitting is shown in Fig. 2 B, from which the  half linewidth (ΔH) can be obtained. The \nGilbert damping can be extracted from the best linear -fitting curve of ΔH vs. f (Fig. 2 C). Figure \n2D shows the NiFe thickness dependence of the Gilbert damping in the Nb/NiFe/Nb junctions \nmeasured at T = 10, 15, and 20 K, respectively. Interestingly,  an oscillating feature of the Gilbert \ndamping is observed as a function of 𝑑NiFe in the region of 𝑑NiFe < ~15 nm. This oscillating \nbehavior can be attributed to the quantum -interference effect of angular momentum transfer \nbetween the local precessing magnetic moment and conduction electrons in thin NiFe that was \ntheoretically predicted by Mills (36), but has not been experimentally reported yet. Above TC, the \ncontinuous energy bands of Nb, similar to the normal met al in the Mills theory, overlap with both \nspin-up and spin -down bands of NiFe at the interface, thus allowing the conducting electrons in \nNiFe to flip between the spin -down and spin -up states. As illustrated in the inset of Fig. 2 D, one \nspin-down electron scatters with the local magnetic moment and then flips to the spin -up \npolarization, giving rise to the angular momentum transfer between the spin -polarized electrons \nand the magnetic moment. Besides the change of angular momentum, the momentum of the \nelect ron also changes ( ∆𝑘), due to different Fermi vectors for spin -up (𝑘𝐹↑) and spin -down ( 𝑘𝐹↓) \nelectrons with exchange splitting (Fig. 2 A). When the NiFe layer is thin enough to become \ncomparable with 1\n∆𝑘, quantum -interference effect of the spin -polariz ed electrons shows up, which \ngives rise to the oscillating spin -transfer torque to the NiFe. When the NiFe thickness is \n2𝑛𝜋/[𝑘F↑−𝑘F↓] (n is an integer), the matching of the quantum levels between the spin -up and \nspin-down electrons in NiFe induces smaller Gilbert damping. On the other hand, when the NiFe \nthickness is (2n+1)𝜋/[𝑘F↑−𝑘F↓], a larger Gilbert damping is induced. Consequently, th e \nGilbert damping in the Nb/NiFe/Nb structures oscillates with a period of 2𝜋/[𝑘F↑−𝑘F↓] \n(Supplementary Materials S1) . Experimentally, an oscillating period ( λ) of ~ 1.8 nm is identified 5 \n (see the red dashed arrow in Fig. 2 D). At T = 50 K, the oscillating f eature disappears since the \nquantum -interference effect is smeared by thermal excitations ( fig. S3) . \nNext, we investigate the spin pumping and spin transfer torque of the Nb/NiFe/Nb junctions \nin the superconducting states below TC with a superconducting gap (Fig. 3 A). TC in the \nNb/NiFe/Nb junctions is obtained from typical four -probe resistance measurement as a function \nof the temperature. A typical temperature -dependent resistance curve measured on the Nb/NiFe \n(12 nm)/Nb junction is shown in Fig. 3b, indicating the TC of ~ 8.6 K. As dNiFe changes, TC of the \nNb/NiFe/Nb junctions exhibits little variat ion between ~ 8.4 and ~ 8.9 K ( fig. S4). Similar to the \nnormal states of Nb, the Gilbert damping below TC is also obtained from the be st linear -fitting \nresult of the  half linewidth vs. frequency ( fig. S5). During the FMR measurement, TC varies a little \n(< 1 K) ( Fig. S 6). As the temperature decreases, 𝛼 decreases abruptly from ~ 0.012 to ~ 0.0036 \nacross the TC (fig. 3C), which indicates the decrease of spin current injected into Nb due to the \nformation of superconducting gap below TC. This observation is consistent with previous reports \non spin pumping into SCs where the spin current is mediated by Bogoliubov quasiparticles (18, \n19, 37). As the tem perature decreases far below the TC, the quasiparticle population dramatically \ndecreases, leading to reduced spin pumping and Gilbert damping.  \nRemarkably, the oscillating amplitude of the Gilbert damping of the Nb/NiFe/Nb junctions as \na function of the NiF e thickness is dramatically enhanced as the temperature decreases into the \nsuperconducting states of Nb (Fig. 3 D). At T = 4 K, the oscillating magnitude of the Gilbert \ndamping constant is ~ 0.005 for the first three oscillations, which is comparable to the  background \nvalue of ~ 0.006. The obtained Gilbert damping values are not affected by thermal cycles, and the \nlarge oscillating feature has been confirmed on a different set of samples. Such a giant oscillation \nof the Gilbert damping cannot be explained by  spin pumping of Bogoliubov  quasiparticle -\nmediated spin current in SCs. Since as the temperature decreases, the population of the Bogoliubov  \nquasiparticles monotonically and rapidly decreases with an increase of the SC gap, which would \nlead to lower Gilber t damping and also smaller oscillation compared to the normal states. Note \nthat the oscillating period of the Gilbert damping at T = 4 K is the same as that at T = 10 K that is \nsupposed to be 2𝜋/[𝑘F↑−𝑘F↓] due to the quantum interference effect . Such oscillating period of \n2𝜋/[𝑘F↑−𝑘F↓] is the also same as that of the zero- and π-phase ground states transitions in FM \nJosephson devices, which is equal to the coherence length in NiFe film of 2𝜋/[𝑘F↑−𝑘F↓] in the 6 \n ballistic regime (1, 11, 17), and √ℏ𝐷𝑑𝑖𝑓𝑓 ∕𝐸𝑒𝑥  in the diffusive regime ( 𝐷𝑑𝑖𝑓𝑓 is the diffusion \ncoefficient, and 𝐸𝑒𝑥 is the exchange energy ). The observed oscillating period of ~ 1.8 nm in our \nstudy is similar to the zero - 𝜋 oscillating period measured in the NiFe Josephson junctions in the \ndiffusive regime reported previously (11, 17).  \nThe Gilbert damping difference  (∆𝛼) between the zero- and 𝜋-junctions is extracted as a \nfunction of NiFe thickness, as shown in Fig. 4 A. We assume the larger Gi lbert damping for the 𝜋-\njunctions and smaller value s for the zero-junctions, which will be discuss ed later in details . The \nthickness -dependent  Gilbert damping of  the zero- and 𝜋-junctions  are expected to both behave as \nα ~ 1 ⁄ dNiFe (29). Hence, we can treat them separately, as illustrated by the guide lines in the inse t \nof Fig. 4A, and ∆𝛼 is obtained by subtracting the fitted 1/d curve for the expected  zero-junctions  \n(black  dashed line) . Clearly, there is a pronounced oscillating feature of ∆𝛼 for the Nb/NiFe/Nb \njunctions with NiFe thickness from ~ 5 nm to ~ 11 nm. When the NiFe thickness is above ~ 11 \nnm, the oscillating feature of the Gilbert damping is largel y suppressed compared to thinner NiFe \njunctions. This feature might be associated wi th the strong Josephson coupling for thin NiFe \njunctions and the exponential decaying of the Josephson coupling as the NiFe thickness increases \n(11, 17). To confirm this, the Jose phson junctions are fabricated using the shadow mask technique, \nand a Josephson coupling is observed from the Nb/NiFe (5 nm and 10 nm)/Nb junctions \n(Supplementary Materials and fig. S7). \nDiscussion  \nLet us discuss the physical mechanism that induces the giant oscillating Gilbert damping in the \nfollowing. Apart from the spin pumping via ABS  discussed above (Fig. 1 D and 1E ), the spin \ncurrent in SCs can also be mediated by Bogoliubov quasiparticles  (fig. S8A) (18, 19, 22, 23, 38), \nspin-triplet pairs  (fig. S 8B) (3). Regarding Bogoliubov  quasiparticles, they populate around the \nedge of superconducting gap at elevated temperatures close to TC (39). As shown both theoretical \nand experimental studies, the enhanced Gilbert damping in the SC/FM/SC heterostructures \nhappens around TC (18, 19, 22, 23, 38). As the temperature decreases down to 0.5 TC, the \nBogoliubov quasiparticles are mostly frozen out, for which the spin pumping is forbidden that will \nno longer contribute to the enhanced Gilbert damping. Hence, the Bogoliubov quasi particles are \nvery unlikely to account for our experimental results. Regarding  the spin-triplet pairs, it has been \nshown in previous studies that the spin -triplet current under FMR conditions and spin triplet 7 \n correlations would be different for zero- and 𝜋-junctions (4, 38, 40, 41), which might result in \ndifferent Gilbert damping theoretically. However, in our study, there are not spin sinks adjacent to \nthe Nb layers , thus not allowing the spin -triplet Cooper pairs to be relaxed in the Nb. This is \ndiffe rent from previous report on the Pt/SC/FM/SC/Pt heterostructures (20), where the Pt is used \nas the spin sink. Experimentally, as the temperature below TC, the Gilbert damping exhibits a \nmonotonic decrease for the Nb/NiFe/Nb heterostructures (Fig. 3 C), which is different from the \nenhanced Gilbert damping due to spin -triplet pairs  (20). Furthermore, no Josephson current in the \nNb/NiFe/Nb heterostructures is observed i n Nb/NiFe (30 nm)/Nb junction ( fig. S7), which  \nindicates the absence of long -range spin -triplet Josephson coupling. Both these experimental \nresults indicate that the contribution from the spin -triplet pairs is not significant to the enhanced \nGilbert damping in the superconducting Nb/NiFe/Nb junctions . \nTo our best understanding, the most reasonable mechanism is the spin pumping via the ABS, \nwhich can qualitatively  describe our experimental observation. Previous studies have \ndemonstrated that the energy of ABS inside the superconducting gap depends on the  \nsuperconducting -phase (42, 43). For the FMR measurement under  open -circuit ed conditions , the \ninversion symmetry of the current -phase  (𝜑) relationships is preserved (43-45). For 𝜋-junctions, \nthere is a 𝜋-phase shift in the current -phase relationship curves compared to zero-junctions , i.e., \nthe properties of 𝜑 = 0 of a 𝜋-junction is the same as those of 𝜑 = 𝜋 of a zero-junction. Since this \n𝜋-phase shift is already taken into account by the FM exchange field , the ABS energy of the 𝜋-\njunctions  can be obtained at 𝜑 = 0 in the ground states , which is similar to that of 𝜑 = 𝜋 of zero-\njunctions .  \nFor 𝜋-junctions, ABS is located around the  zero-energy inside of the superconducting gap (Fig. \n1D). The ABS could penetrate into the superconducting Nb films with scale of superconducting \ncoherent length (~ 30 nm), which is evanescent to dissipate the spin angular moment um (25, 26, \n44). As shown in Fig. 4 B, the transfer efficiency of spin angular momentum via the zero -energy \nABS can lead to an enhanced Gilbert damping . Whileas, for  zero-junctions, the distribution of the \nABS is near the edge of the superconducting gap (Fig. 1C) , thus, the spin pumping effic iency is \nsuppressed due to the reduced population of the ABS at low temperatures  (Fig. 4 C). Furthermore , \nthe oscil latory energy levels of the ABS  between the zero- and 𝜋-junctions is also consistent with \nthe density of states (DOS)  oscillating in supercon ductors  between the zero- and 𝜋-junctions (1, 6, 8 \n 38, 44, 46). In consequence, as the NiFe thickness increas es, the oscillatory spin pumping \nefficiency via ABS  at the FM/SC interface  (or DOS in SC s) gives rise to the oscillatory Gilbert \ndamping.  We have proposed a simplified model for the case of ideal transparency of electrons \n(Supplementary Materials S2 and f ig. S9 ). For the less transparency cases, i.e., in diffusive regime , \n(42, 43), the energy level s of the ABS in  𝜋-junctions  locates away from zero -energy, but they are \nstill much smaller than those  of the ABS in zero-junctions. Actually, the similar oscillating \nbehaviors of ABS (or DOS) can be preserved in the diffusive regime  (6, 46). Hence, an oscillating \nspin pumping efficiency would also be expected in the diffusive regime, which could lead to the \noscillating Gilbert damping  observed in our experiment . To fully understand the experimental \nobservation of the oscillatory Gilbert damping and the detailed spin relaxation process in the \ndiffusive regime, further theoretical studies are needed.  \nFurthe rmore, the control samples of bilayer Nb/NiFe heterostructur es do not exhibit the large \noscillatory feature for the Gilbert damping as the NiFe thickness varies  at T = 4 K  (fig. S10), which \nfurther presents the important role of phase difference across NiFe  in the large the oscillatory \nGilbert damping  observed in t he trilayer  Nb/NiFe/Nb heterostructures .  \nIn conclusion, giant oscillatory Gilbert damping is observed in the superconducting \nNb/NiFe/Nb junctions with respect to the NiFe thickness. To our best knowledge, neither the \nBogoliubov quasiparticles, nor the spin -triplet pairs are relevant to this observation.  The most \npossible explanation for such giant oscillatory Gilbert damping could be related to the different \nABS energy levels and the DOS at the NiFe/SC interface in zero- and π- junctions. To full y \nunderstand these results, further theoretical studies are needed. Looking forward, our experimental \nresults  might pave the way for controlling the magnetization dynamics  by the superconducting \nphase  in a FM Josephson junction  in the SQUID setup, and could be important potential \napplications of ferromagnet π-junctions in quantum computing, such as half -quantum flux qubits.  \n \nMaterials and Methods  \nMaterials growth  \nThe SC/FM/SC heterostructures consisting of Nb (100 nm) and Ni 80Fe20 (NiFe ; ~ 5 - 20 nm) were \ngrown on thermally oxidized Si substrates in a d.c. magnetron sputtering system with a base \npressure of ∼1× 10−8 torr. To systematically vary the NiFe thickness that is crucial for the quantum -9 \n size effect, we adopted the rotating multi -platter technique that allows us to grow dozens of \nNb/NiFe/Nb samples in each run  (35). The thickness of the Nb layer is fixed to be ~100 nm that is \nmuch larger than the spin diffusion length of Nb  (20, 47). After the growth, a thin Al 2O3 layer (~ \n10 nm) was deposited in situ  as a capping layer to avoid sample degradation against air/water \nexposure. The crystalline properties of Nb/NiFe/Nb heterostructures were chara cterized by X -ray \ndiffraction (fig. S1A ) and high -resolution cross -section al tra nsmission electron microscopy (f ig. \nS1B) using a 200 -kV JEOL 2010F field -emission microscope. The NiFe thickness is determined \nby the growth rate that is calibrated by TEM measurement, where the uncertainty of the NiFe \nthickness is obtained to be  smaller than ~ 0.8  nm (f ig. S1B).  The resistivity of the NiFe layers \n(thickness: 5 - 20 nm) is ranging from 60 to 35 μΩ ·cm, which corresponds to the  mean free path \nbetween 2.3 and  3.9 nm.  \n \nFerromagnetic resonance measurement.   \nThe spin pumping of Nb/NiFe/Nb heterostructures was characterized via FMR  using the coplanar \nwave guide technique connected with a vector network analyzer (VNA; Agilent E5071C) in the \nvariable temperature insert of a Physical Properties Measurement System (PPMS; Quantum \nDesign)  (19). The FMR spectra were characterized by measuring the amplitudes of forward \ncomplex transmission coefficients (S 21) as the in -plane magnetic field decreases from 4000 to 0 \nOe under the microwave power of 1 mW. The typical FMR results measured on the Nb/NiFe (12 \nnm)/Nb het erostructures are shown in the fig. S2A  (T = 10 K) and fig. S 5A (T = 4 K). Weaker \nFMR signals are observed in the superconducting states compared to the normal states.  \nThe half linewidth ( ∆𝐻) can be obtained by the Lorentz fitting of the magnetic field -dependent \nFMR sign al following the relationship ( figs. S2B  and S4 B): \n 𝑆21∝𝑆0(∆𝐻)2\n(∆𝐻)2+(𝑯−𝑯𝒓𝒆𝒔)2 (3) \nwhere 𝑆0 is the coefficient for the transmitted microwave power,  𝑯 is the external in -plane \nmagnetic field, and 𝑯𝒓𝒆𝒔 is the resonance magnetic field.  The Gilbert damping constant (α) can be \nobtained from the slope of the best linear -fitting results of the ∆𝐻 vs. the microwave frequency ( f) \n(48-51): 10 \n  ∆𝐻=∆𝐻0+(2𝜋𝛼\n𝛾)𝑓 (4) \nwhere ∆𝐻0 is the zero -frequency line broadening that is related to the inhomogeneous properties, \nand  𝛾 is the gyromagnetic ratio. From the best linearly fits of the ∆𝐻 vs. f results measured on the \ntypical Nb/Py ( 12 nm)/Nb sample (red lines in f igs. S2 C and S 5C), 𝛼 is determined to be 0.012 \nand 0.0054 at T = 10 and 4 K, respectively. A larger zero -frequency line broadening ∆𝐻0 is \nobserved for the superconducting state compared to the normal state of Nb/Py/Nb heterostructures, \nwhich could be attributed to Meissn er screening effect and the formation of trapped magnetic \nfluxes in Nb  (51). The thickness dependent ∆𝐻0 is shown in f ig. S1 1C, and no obviously \noscillatory behaviors  are observed . \nThe effective magnetization and the gyromagn etic ratio can be fitted via the in -plane Kittel \nformula (51): \n 𝑓𝑟𝑒𝑠=𝛾\n2𝜋√(𝐻𝑟𝑒𝑠+ℎ)(𝐻𝑟𝑒𝑠+ℎ+4𝜋𝑀𝑒𝑓𝑓),                                      (5)  \nwhere 𝑓𝑟𝑒𝑠 and 𝐻𝑟𝑒𝑠 are the resonant microwave frequency and magnetic field respectively , \n4𝜋𝑀𝑒𝑓𝑓 is the effective saturated magnetization , and ℎ is the shifted magnetic field induced by \nsuperconducting proximity effect. The thickness -dependent gyromagnetic ratio and eff ective \nmagnetization can be found in fig. S1 1A and fig. S1 1B. Both parameters do not exhibit  any \noscillatory features  as the Gilbert damping does (Fig . 4A), which demonstrates that the oscillatory \nGilbert damping is not caused by any unintentional experimental error . \n \nSuperconducting transition temperature measurement.  \nThe superconducting transition temperature ( TC) of the Nb/NiFe/Nb heterostructures was \ndetermined via the zero -resistance temperature measured by four -probe method in a P PMS using \nstandard a.c. lock -in technique at a low frequency of 7 Hz. The TC of Nb (100 nm)/NiFe/Nb (100 \nnm) heterostructures exhibits little variation as a f unction of the NiFe thickness ( fig. S 4). It is \nnoticed that the FMR measurement can affect the TC a little (< 1 K), as shown in fig. S6.  \n 11 \n Supplementary Materials  \nSupplementary Materials and Methods  \nfig. S1 . The crystalline properties of the Nb/NiFe/Nb heterostructures.  \nfig. S2. Gilbert dampin g measurement of Nb/NiFe/Nb heterostructures at T = 10 K.  \nfig. S3. NiFe thickness dependence of Gilbert damping at T = 50 K.  \nfig. S4. NiFe thickness dependence of TC for the Nb/NiFe/Nb heterostructures.  \nfig. S5. Measurement of the Gilbert damping of Nb/ NiFe/Nb heterostructures at T = 4 K.  \nfig. S6. The effect of FMR measurement on the TC of Nb/NiFe/Nb heterostructures.  \nfig. S7. The measurement of Josephson coupling in Nb/NiFe/Nb junctions.  \nfig. S 8. Illustration of magnetization dynamics and spin pumping i n the SC/FM/SC \nheterostructures due to Bogoliubov quasiparticles and equal spin -triplet Cooper pairs.  \nfig. S9. Calculation of the enhanced Gilbert damping due to spin pumping via the ABS at T = 4 K.  \nfig. S10. Gilbert damping of control sample of bilayer Nb /NiFe junctions.  \nfig. S1 1. Thickness dependen ce of  gyromagnetic ratio, effective magnetization and \ninhomogeneous half -linewidth.  \n \n \nReferences and Notes:  \n1. A. I. Buzdin, Proximity effects in superconductor -ferromagnet heterostructures. Rev. 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Pratt, Spin -diffusion lengths in metals and alloys, and spin -flipping at \nmetal/metal interfaces: an experimentalist’s critical review. J. Phys. Condens. Matter.  19, \n183201 (2007).  \n62. A. A. Bannykh, J. Pfeiffer, V. S. St olyarov, I. E. Batov, V. V. Ryazanov, M. Weides, \nJosephson tunnel junctions with a strong ferromagnetic interlayer. Phys. Rev. B  79, 054501 \n(2009).  \n \n \n  15 \n Acknowledgments  \n \nGeneral : We acknowledge the fruitful discussion with Sadamichi Maekawa, Ziqiang Qiu, \nZhe Yuan, Ke Xia, Young Sun, and Kei Yamamoto. Y.Y., R.C., Y.M., W.X., Y.J., X.C.X., \nand W.H. acknowledge the financial support from National Basic Research Programs of \nChina (No. 2019YFA0308401), National Natural Science Foundation of China (No. \n11974025 ), Beijing Natural Science Foundation (No. 1192009), and the Strategic Priority \nResearch Program of the Chinese Academy of Sciences (No. XDB28000000). T.Y. is \nfinancially supported by DFG Emmy Noether program (SE 2558/2 -1).  \n \nAuthor contributions:  W.H. conceived and supervised the project. Y.Y. and R.C. \nperformed the ferromagnetic resonance measurements. Y.Y. and Y.M. performed X -ray \ndiffraction measurements. T.Y. performed the theor etical calculations. S.H.Y. synthesized \nthe Nb/NiFe/Nb heterostructures. Y.Y. and W.H. wrote the manuscript with the \ncontribution from all authors. All the authors discussed the results.  \n \nCompeting interests:  The authors declare no competing interests.  \n \nData Availability: All data needed to evaluate the conclusions in the paper are present in \nthe paper and/or the Supplementary Materials.  \n  16 \n  \nFig. 1. Illustration of magnetization dynamics and spin pumping in the SC/FM/SC \nheterostructures.  (A) The oscillatory real part of the superconducting order parameter (Re {Ψ}, \ngreen curve ) penetrated into FM leads to the zero-state and  𝜋-state. (B) The symmetric order \nparameter in the zero-junction s. (C) The anti -symmetric order parameter in the 𝜋-junction s. (D-E) \nSpin pumping via t he ABS  in SCs  in the zero- and 𝜋-junction s. M and 𝛼FM are the magnetization \nand Gilbert damping of the FM layer itself, and 𝛼sp is the enhanced Gilbert damping, which arises \nfrom the spin dissipation in SC layers during the spin pumping process.  \n \n17 \n  \nFig. 2. Oscillatory Gilbert damping of the Nb/NiFe/Nb heterostructures above TC. (A) The \nillustration of spin pumping into the normal states of Nb layers and the electronic band structure \nof NiFe with different spin-up and spin -down Fermi vectors (𝑘F↑ and 𝑘F↓) due to the exchange \nsplitting (2 𝐸𝑒𝑥). The spin pumping gives rise to the spin accumulation in the Nb layers, indicated \nby the spin -split chemical potential ( 𝜇↑ and 𝜇↓). (B) A typical FMR  curve measured with f = 12 \nGHz (black circles) and the Lorentzian fitting curve (red line) measured on Nb/Py (12 nm)/Nb . \nΔH is the half line width at the half maximum of FMR signal. (C) The determination of the Gilbert \n18 \n damping from ΔH vs. f. The red line represents the best linear -fitting curve. (D) The oscillatory \nGilbert damping as a function of NiFe thickness ( dNiFe) measured at T = 10, 15, and 20 K, \nrespectively. The experimental oscillating period ( 𝜆) is marked by the red dashed arrow. The inset: \nIllustration of the quantum -interference effect o f the angular momentum transfer between the local \nmagnetic moment and the spin -polarized electrons . When the NiFe thickness decreases to scale of \n1\n∆𝑘, the quantum -interference effect starts to be significant in the angular momentum transfer  and \nspin pumping into the Nb layers .  \n  \n \n \n \n \n \n \n \n \n \n \n 19 \n            \nFig. 3. Giant oscillatory Gilbert damping in Nb/NiFe/Nb heterostructures below TC. (A) The \nillustration of electronic band structures of Nb in the normal and superconducting states. (B) The \ndetermination of TC via zero -resistance temperature measured on the typical Nb/NiFe (12 nm)/Nb \nheterostructures. (C) Temperature dependence of Gilbert d amping of the typical Nb/NiFe (12 \nnm)/Nb heterostructures. (D) The oscillatory Gilbert damping as a function of the NiFe thickness \nin the Nb/NiFe/Nb heterostructures measured at T = 10, 7, 5, and 4 K, respectively. The oscillating \nfeature below TC (T = 4 and 5 K) is dramatically enhanced compared to that above TC (T = 10 K).  \n  \n20 \n  \nFig. 4. Physical mechanism of the giant oscillatory Gilbert dam ping in Nb/NiFe/Nb junctions. \n(A) The NiFe thickness dependence of the Gilbert damping difference ( ∆α) between the  \nNb/NiFe/Nb π- and zero-junctions  at T = 4 K . Inset: The Gilbert damping of zero- and π-junctions. \nThe solid balls represent the experimental data, the blue and black dash lines are the guide lines \nfor π- and zero-junctions, respectively. For bo th guide lines, the damping is expected to behave as \nα ~ 1 ⁄ dNiFe. (B-C) Illustration of the spin pumping via the ABS and the enhanced Gilbert damping \nfor Nb/NiFe/Nb π- and zero-junctions, respectively.  The red thick -arrows indicate the pumping \nand relaxation  of the spin current in SCs . \n21 \n Supplementary Materials for  \n \n \nGiant oscillatory Gilbert damping in \nsuperconductor/ferromagnet/superconductor junctions  \n \nAuthors  \nYunyan Yao1,2†, Ranran Cai1,2†, Tao Yu3, Yang Ma1,2, Wenyu Xing1,2, Yuan Ji1,2, Xin -Cheng \nXie1,2,4,5, See -Hun Yang6*, and Wei Han1,2* \n \n \nThis SM file includes : \n⚫ Supplementary Materials and Methods  \n⚫ fig. S1. The crystalline properties of the Nb/NiFe/Nb heterostructures.  \n⚫ fig. S2. Gilbert damping measurement of Nb/NiFe/Nb heterostructures at T = 10 K.  \n⚫ fig. S3. NiFe thickness dependence of  Gilbert damping at T = 50 K . \n⚫ fig. S4. NiFe thickness dependence of TC for the Nb/NiFe/Nb heterostructures.  \n⚫ fig. S5. Measurement of the Gilbert damping of Nb/NiFe/Nb heterostructures at T = 4 K.  \n⚫ fig. S6. The effect of FMR measurement on the TC of Nb/NiFe/N b heterostructures.  \n⚫ fig. S7. The measurement of Josephson coupling in Nb/NiFe/Nb junctions.  \n⚫ fig. S8. Illustration of magnetization dynamics and spin pumping in the SC/FM/SC \nheterostructures due to Bogoliubov quasiparticles and equal spin -triplet Cooper pai rs.  \n⚫ fig. S9. Calculation of the enhanced Gilbert damping due to spin pumping via the ABS at \nT = 4 K.  \n⚫ fig. S10. Gilbert damping of control sample of bilayer Nb/NiFe junctions.  \n⚫ fig. S11. Thickness dependence of gyromagnetic ratio, effective magnetization an d \ninhomogeneous half -linewidth . \n \n  22 \n Supplementary Materials and Methods  \nSection 1: Model of oscillating Gilbert damping above TC.  \nThe oscillatory Gilbert damping in normal metal (NM)/ferromagnet (FM)/NM \nheterostructures arising from quantum interference effect is analyzed based on previous theory by \nMills (36). Within the linear response theory, the enhanced Gilbert damping is related to the \ndynamical spin susceptibility ( 𝜒−+(𝛺)) of conduction electrons in a FM,  \n 𝛼sp=𝐽2𝑀𝑠𝑉\n2𝑁2ℏ3𝛾Λ2 (S1) \nwhere Λ2=Im(𝑑𝜒−+(𝛺)\n𝑑𝛺|\nΩ=0). Using one dimensional model,  we obtain   \n Λ2=1\n𝜋2∫𝑑𝑥𝑑 𝑥′Im[𝐺↑(𝑥,𝑥′,𝜖F)]Im[𝐺↓(𝑥,𝑥′,𝜖F)]\nFM (S2) \nwhere 𝐺𝜎(𝑥,𝑥′,𝜖F) is the Green’s function for conduction electrons with 𝜎-spin at the Fermi \nenergy ( 𝜖F). In a FM, 𝐺𝜎(𝑥,𝑥′,𝜖F) is related to the exchange energy.  \n [−ℏ2\n2𝑚𝑑\n𝑑𝑥2−𝜖±𝐸ex]𝐺𝜎(𝑥,𝑥′,𝜖F)=𝛿(𝑥−𝑥′) (S3) \nFor the FM film with a thickness dFM in − dFM/2 < x < dFM/2, the Green’s function satisfies the \nrelation  \n 𝐺𝜎(𝑥,𝑥′,𝜖F)=𝐺𝜎(𝑥′,𝑥,𝜖F)=𝐺𝜎(−𝑥,−𝑥′,𝜖F) (S4) \nHence, the imaginary part of the Green’s function could be expressed by  \n Im[𝐺𝜎(𝑥,𝑥′,𝜖F)]=−π{𝑁F𝜎cos [𝑘F𝜎(𝑥−𝑥′)]+𝑁F𝜎′cos [𝑘F𝜎(𝑥+𝑥′)]}     (S5) \nwhere 𝑘F𝜎=√2𝑚\nℏ2(𝜖F∓𝐸ex) is the Fermi wave -vector in the FM, 𝑁F𝜎 and 𝑁F𝜎′ are equivalent to \nthe density of states and the modulation amplitude of the local density of states, respectively. For \nthe same position of x, the local density of states is equal to  \n 𝑁𝜎(𝑥,𝜖F)=𝑁F𝜎+𝑁F𝜎′cos [2𝑘F𝜎𝑥] (S6) \nSince 𝐸ex is much smaller compared to 𝜖F, the spatial modulation of the local density of states is \nnegligible. The combination of  equations (S2) and (S5) leads to  \nΛ2=∫ 𝑑𝑥𝑑 𝑥′𝑑FM 2⁄\n−𝑑FM 2⁄{𝑁F↑cos[𝑘F↑(𝑥−𝑥′)]}∗{𝑁F↓cos[𝑘F↓(𝑥−𝑥′)]}        (S7) 23 \n                =2𝑁F↑𝑁F↓{1\n(𝑘F↑−𝑘F↓)2sin2[𝑘F↑−𝑘F↓\n2𝑑FM]+1\n(𝑘F↑+𝑘F↓)2sin2[𝑘F↑+𝑘F↓\n2𝑑FM]} \nClearly,  the enhanced  Gilbert  damping  is expected  to oscillate  as a function  of the FM \nthickness  with two periods  of  2𝜋/[𝑘F↑−𝑘F↓] and 2𝜋/[𝑘F↑+𝑘F↓]. For real FM materials,  such as \nNiFe  with (𝑘F↑+𝑘F↓)≫(𝑘F↑−𝑘F↓), the second  term in the equation  (S7) could  be negligible,  \nleaving  only one oscillating  period  of 2𝜋/[𝑘F↑−𝑘F↓]. When  the FM thickness  is equal  to \n2𝑛𝜋/[𝑘F↑−𝑘F↓], a lower  Gilbert  damping  is obtained.  On the other  hands  with FM thickness  of \n(2𝑛+1)𝜋/[𝑘F↑−𝑘F↓], a larger  Gilbert  damping  is obtained.   \nSection  2: Calculation  of the enhanced  Gilbert  damping  in Nb/NiFe/Nb  by spin pumping  via \nAndreev  bound  states  (ABS).   \nAs the reciprocal process of the spin transfer torque, conventional spin pumping is achieved \nby the magnetization torques provided by the driven quasiparticle carriers (29, 52-54), which are \nthe electrons in the normal metals. In SC/FM heterostructures, however, the quasiparticle carriers \ncan be either Bogoliubov quasiparticles or ABS (42), which lie above and within the \nsuperconducting gaps, respectively. Therefore, it is desirable to formulate and estim ate the \ncontribution to the spin pumping via the ABS (55), when the temperature is much smaller than the \nsuperconducting critical temperature.   \nWithout loss of generality, we start the analysis from a left -propagating electron of energy 𝜀 \nand spin 𝜎 = {↑, ↓} = {+, −,}. When the Zeeman splitting J is much smaller than the Fermi energy \nEF, it has momentum  \n 𝑘𝜎=𝑘𝐹+(𝜀+𝜎𝐽)/(ℏ𝜈𝐹),                                                             (S8) \nwhere 𝜈𝐹 is the Fermi velocity of the electron. When one electron goes from the FM to the SCs, it \nis reflected as a hole by the Andreev reflection at the right FM/SC interface; this hole has a phase \nshift χ=−arccos (𝜀/Δ) with respect to the electron (56), where Δ is the superconducting gap . \nSimilarly, when a hole goes from the metal to the superconductor at the left FM/SC interface, an \nelectron can be reflected. With a proper energy, the Andreev reflections can form a closed path, as \na result of which the ABS forms. This requires that the phase accumulated in the reflections \nsatisfies the Sommerfeld quantization condition, i.e. in the ballistic regime,  24 \n     𝜀𝐿\nℏ𝜈𝐹+𝜎𝐽𝑑NiFe\nℏ𝜈𝐹−arccos (𝜀\n∆)=𝑛𝜋+𝜑\n2,                                 (S9) \nwhere 𝜑 is the phase difference between the two superconductors , dNiFe is the thickness of the FM \nlayer and n is an integer. Since ℏ𝜈𝐹/∆  ≥ 100 nm with 𝜈𝐹 = 2.2 ×  105 m/s and ∆ = 1 meV at T = 4 \nK in our experiment (17, 57, 58), dNiFe < 19 nm << ℏ𝜈𝐹/∆ such that the first term in Eq. (S9) can \nbe safely disregarded. For the FMR measurements with open -circuited configuration, the junctions \nalways stay in the ground states (43-45). For 𝜋-junctions, there is a 𝜋-phase shift in the current -\nphase relationship curves compared to zero-junctions , i.e., the properties of 𝜑 = 0 of a 𝜋-junction \nis the same as those of 𝜑 = 𝜋 of a zero-junction. Since this 𝜋-phase shift is already taken into \naccount by the FM exchange field , the ABS energy of the 𝜋-junctions  can be obtained at 𝜑 = 0 in \nthe ground states , which is similar to that of 𝜑 = 𝜋 of zero-junctions . Hence, the energy of the ABS \ncan be described by   ε0=±∆cos (𝐽𝑑NiFe\nℏ𝜈𝐹) for ideal case with perfect transparency of \nelectrons/holes.  \nIn reality, the interfacial scattering and transport conditions (ballistic or diffusive regimes) of \nFM could affect th e energy of the ABS. Following previous studies (42, 59), a transmission \ncoefficient ( D) could be introduced to describe this issue , which is close to unity in the ballistic \nregime but can also be large in the  diffusive regime  with an ideal transparency at the interface (43, \n60, 61). In this work, we focus on the ideal cases with perfect transparency of electrons/holes . The \nenergy of the ABS oscillates from the ed ge of the superconducting gap to the zero -energy with \nrespect to the FM thickness (fig. S9A).   \nThe pumped spin current reads (29, 52-54), \n𝐉𝑠(𝑡)=ℏ\n4𝜋𝑔eff↑↓𝒎×𝑑𝒎\n𝑑𝑡,                                                          (S10)  \nwhere m is the magnetization unit vector, and we define the effective mixing spin conductivity \n𝑔eff↑↓ at the finite temperature via the zero -temperature one 𝑔↑↓ by (53, 55)  \n  𝑔eff↑↓=𝑛0∫𝑑𝜀𝑑𝑓(𝜀)\n𝑑𝜀Re[𝑔↑↓(𝜀)].                                              (S11)  \nHere, n0 is the number of the conduction channel that roughly corresponds the conduction electron \ndensity at the interface and 𝑓(𝜀)=1/{exp [𝜀/(𝑘𝐵𝑇)]+1} is the Fermi -Dirac distribution of \nelectron at the temperature T. Importantly, in the ballistic limit Re[𝑔↑↓(𝜀)]=1 when  𝜀=𝜀0; it 25 \n has width ∆𝜀 depending on the FM thickness dNiFe in the ballistic regime or the mean free path 𝑙𝑚 \nin the diffusive regime . By the uncertainty principle, ∆𝜀∆𝑡=2𝜋ℏ, where ∆𝑡=𝑙𝑚/𝜈𝐹 is the \npropagation time of the electron in the junction, leading to  ∆𝜀 ~ 2𝜋ℏ𝜈𝐹/𝑙𝑚. By further \nconsidering the degeneracy due to spin (× 2) and the existence of two interfaces (× 2), we thus can \nestimate  \n    𝑔eff↑↓ ~ 8𝜋𝑛0ℏ𝜈𝐹\n𝑙𝑚𝑑𝑓(𝜀0)\n𝑑𝜀.                                                         (S12)  \nThe pumped spin current carries the angular momentum away from the precessing magnetization \nand hence cause an enhanced Gilbert damping, which is described by  \n𝛿𝛼=2𝛾ℏ2𝜈𝐹\n𝑀𝑠𝑙𝑚𝑑NiFe𝑑𝑓(𝜀0)\n𝑑𝜀 ,                                                      (S13)  \nwhere 𝛾  is electron gyromagnetic ratio and 𝑀𝑠 is the saturated magnetization of the ferromagnet.  \n      We are now ready to estimate the contribution of ABS to  the Gilbert damping at T = 4 K with \nvarying transmission coefficient. We take 𝑛0 =0.5 ×  1016 m−2 following Ref. 44 , 𝑙𝑚~ 3 nm,  𝜈𝐹= \n2.2 ×  105 m/s, J = 400 meV and 𝜇0𝑀𝑠≈1 𝑇 from previous experimental results (17). With \nsuperconducting gaps ∆ ≈ 1 meV at T = 4 K for Nb (57, 58), Fig.S8A plots the normalized energy \nof ABS by the superconducting gap at T = 4 K as a function of dNiFe for the ideal transparency case . \nThe oscillation of the Gilbert damping can be resolved by using the FM exchange field -induced \nphase shift of 𝐽𝑑NiFe\nℏ𝜈𝐹 (fig. S9B) . For simplicity, we have disregarded the possible thickness \ndependence of the superconducting gaps and magnetizations. To  be noted, our theoretical \nestimation is based on a simplified model that assumes D = 1. For the diffusive regime or the case \nof non -perfect transparency of electrons at the interface  (42, 43), similar oscillating behaviors of \nABS (or DOS) in the SCs can also be preserved. For example, the oscillating ABS (or DOS) in the \nSCs have been shown to exist in the diffusive regime theoretically (6, 46), and indeed, the zero to \n𝜋 transitions have been experimentally observed in both the ballistic and diffusiv e regimes from \nthe supercurrent measurements (11, 17). To fully understand the experimental observation of the \noscillatory Gilbert damping in the diffusive regime, further theoreti cal studies are needed.  \n \n \n \n 26 \n Section 3: Measurement of the Josephson coupling in Nb/NiFe/Nb.  \nThe Nb/NiFe/Nb  Josephson devices are fabricated using the shadow mask techniques during \nthe films growth. As shown in figs. S7A and S7B, the Josephson devices have a junction area ( A) \nof ~ 80 μm  × 80 μm, and the other areas are electrically isolated by a 100 nm AlO x layer. The \nJosephson current is measured by standard a.c. lock -in technique. The normalized differential \nresistances (dV/dI) measured on the Nb/NiFe (5 nm)/Nb junction at various temperatures are \nshown in fig. S7C. The critical current ( Ic) is defined as poi nt where the differential resistance \nincreases above the value for the zero -bias current. The normal resistance (R n) is determined to be \nthe saturated value of the normal states of the Josephson coupling measurement. The measured \narea-resistance product (R nA) of ~  5×10−10 Ω𝑚2 is higher than that reported in metallic \nJosephson junction (17, 61), and comparable to that of FM Josephson junction with a thin tunnel \nbarrier (62). This behavior indicates that there is more likely a thin NiFeO x layer (indicated by Fig. \nS8B) in the junction formed during the AlO x growth step in the presence of oxygen gas. As the \ntemperature increases, I c and the characteristic voltage (I cRn) decrease (figs. S7C and S7D). Clear \nJosephson currents are observed on the Nb/NiFe (5 nm)/Nb junction and Nb/NiFe (10 nm)/Nb \njunction (figs. S7E and S78F). And the estimated SC gap energy is ~ 0.9 meV at T = 2 K (1, 43), \nwhich is comparable to the value of ~1.36 meV at T = 0 K estimated from TC of ~ 8.5 K (fig. S4). \nOn the other hand, no Josephson current could be observed in the Nb/NiFe (30 nm)/Nb junction \n(figs. S7E and S7F). The absence of Jo sephson current in Nb/NiFe (30 nm)/Nb junction indicates \nthat there is no long -range spin -triplet Josephson coupling in the Nb/NiFe/Nb heterostructures in \nour experiment.  \n  27 \n  \n \n \nfig. S1. The crystalline properties of the Nb/NiFe/Nb heterostructures. (A) The θ -2θ X -ray \ndiffraction results measured on the typical Nb/NiFe (12 nm)/Nb sample, where Nb (110) and NiFe \n(111) peaks are observed. ( B) High -resolution transmission electron micrographs mea sured on the \ntypical Nb/NiFe (12 nm)/Nb sample. The dashed lines show the interfaces between Nb and NiFe \nlayers. The red bars indicate the deviation of NiFe at the interface.  \n \n \n \n \n \n \n  \n28 \n  \n \n \nfig. S2. Gilbert damping measurement of Nb/NiFe /Nb heterostructures at T = 10 K. (A) The \ntypical FMR  spectra as a function of magnetic field with microwave frequency ( f) of 10, 12, 14, \n16, and 18 GHz, respectively. (B) The typical FMR  spectrum measured with f = 12 GHz (black \ncircles) and the Lorentz fi tting curve (red line). ΔH is the half linewidth of the FMR signal. (C) \nThe determination of the Gilbert damping from ΔH vs. f. The red line indicates the best linear -\nfitting curve. These results are obtained on the typical  Nb/Py (12 nm)/Nb sample.  \n  \n29 \n  \n \n \n \nfig. S3. NiFe thickness dependence of  Gilbert damping at T = 50 K .  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n30 \n  \nfig. S4. NiFe thickness dependence of TC for the Nb/NiFe/Nb heterostructures. The TC is \ndetermined from the zero -resistance temperature via four -probe resistance measurement.  \n \n31 \n  \nfig. S5. Measurement of the Gilbert damping of Nb/NiFe/Nb heterostructures at T = 4 K. (A) \nThe typical FMR  spectra as a function of magnetic field with microwave  frequency ( f) of 10, 12, \n14, 16, and 18 GHz, respectively. (B) The typical FMR  spectrum measured with f = 12 GHz (black \ncircles) and the Lorentz fitting curve (red line). ΔH is the half linewidth of the FMR signal. (C) \nThe determination of the Gilbert damping from ΔH vs. f. The red line indicates the best linear -\nfitting curve. These results are obtained on the typical  Nb/Py (12 nm)/Nb sample.  \n  \n32 \n  \n \nfig. S6. The effect of FMR measurement on the TC of Nb/NiFe/Nb heterostructures. The four -\nprobe resistances vs. temperature are probed from the typical Nb/NiFe  (12 nm)/Nb sample \nwith/without the presence of the in -plane magnetic field and microwave power.   \n33 \n  \n \nfig. S7. The measurement of Josephson coupling in Nb/NiFe/Nb junctions.  (A) The optical \nimage of a typical Nb/NiFe/Nb Josephson device and schematic of the electrical measurement \ngeometry.  (B) The cross -section of the Josephson devices with a junction area of ~ 80 μm  × 80 \nμm. At the junction, a thin oxide layer of NiFeO x is mostly liked formed on the top surface of NiFe \nduring the growth of A lOx in the presence of oxygen. ( C) The normalized differential resistance \n(dV/dI) as a function of the bias  current  measured on the Nb/NiFe (5 nm)/Nb junction from T = 2 \nto 6 K.  (D) The temperature dependence of the characteristic voltage (I cRn) of the Nb/NiFe (5 \nnm)/Nb Josephson  junction . (E) The normalized differential resistance as a function of the bias  \ncurrent  of the Nb/NiFe/Nb junctions ( dNiFe = 5, 10 and 30 nm) at T = 2 K. ( F) The NiFe thickness \ndependence of the characteristic voltages a t T = 2 K.  \n \n34 \n  \n \nfig. S8. Illustration of magnetization dynamics and spin pumping in the SC/FM/SC \nheterostructures due to Bogoliubov quasiparticles (A) and equal spin -triplet Cooper pairs \n(B). The dark and light balls represent the electron -like and hole-like quasiparticles respectively. \nThe red and blue arrows indicate the spin up and spin down respectively.  \n  \n35 \n   \n \nfig. S9.  Calculation of the enhanced Gilbert damping due to spin pumping via the ABS at T \n= 4 K. (A) The normalized energy of ABS by the superconducting gap at T = 4 K as a function of \ndNiFe for the ideal transparency case. ( B) The enhanced Gilbert damping via ABS as a function of \ndNiFe.  \n  \n36 \n  \n \nfig. S10. Gilbert damping of control sample of bilayer Nb/Ni Fe junctions. (A)  Gilbert damping \nof bilayer Nb/NiFe junctions at T = 4, 5, and 10  K. (B) Comparison of the Gilbert damping of \nbilayer Nb/NiFe and trilayer Nb/NiFe/Nb junctions at T = 4 K.  \n \n \n \n \n \n \n \n \n37 \n  \nfig. S11.  Thickness dependence of gyromagnetic ratio (and g factor) (A), effective \nmagnetization (B) and inhomogeneous half -linewidth (C) . The blue, black, green and red \ndotted -lines represent to temperature of T = 10, 7, 5 and 4 K, respectively.  \n \n \n"    },    {        "title": "1909.02738v2.The_interplay_of_large_two_magnon_ferromagnetic_resonance_linewidths_and_low_Gilbert_damping_in_Heusler_thin_films.pdf",        "content": "The interplay of large two-magnon ferromagnetic resonance linewidths and low\nGilbert damping in Heusler thin \flms\nW. K. Peria,1T. A. Peterson,1A. P. McFadden,2T. Qu,3C. Liu,1C. J. Palmstr\u001cm,2;4and P. A. Crowell1\n1School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455\n2Department of Electrical & Computer Engineering,\nUniversity of California, Santa Barbara, California 93106\n3Department of Electrical and Computer Engineering,\nUniversity of Minnesota, Minneapolis, Minnesota 55455\n4Department of Materials, University of California, Santa Barbara, California 93106\nWe report on broadband ferromagnetic resonance linewidth measurements performed on epitaxial\nHeusler thin \flms. A large and anisotropic two-magnon scattering linewidth broadening is observed\nfor measurements with the magnetization lying in the \flm plane, while linewidth measurements with\nthe magnetization saturated perpendicular to the sample plane reveal low Gilbert damping constants\nof (1:5\u00060:1)\u000210\u00003, (1:8\u00060:2)\u000210\u00003, and<8\u000210\u00004for Co 2MnSi/MgO, Co 2MnAl/MgO, and\nCo2FeAl/MgO, respectively. The in-plane measurements are \ft to a model combining Gilbert and\ntwo-magnon scattering contributions to the linewidth, revealing a characteristic disorder lengthscale\nof 10-100 nm.\nI. INTRODUCTION\nThe theoretical understanding of the damping mech-\nanism believed to govern longitudinal magnetization re-\nlaxation in metallic ferromagnets, originally due to Kam-\nbersk\u0013 y [1, 2], has in recent years resulted in quantita-\ntive damping estimates for realistic transition metal band\nstructures [3{5]. Although of great interest where engi-\nneering of damping is desired [6], these calculations re-\nmain largely uncompared to experimental data. Kam-\nbersk\u0013 y damping may be characterized by the so-called\nGilbert damping constant \u000bin the Landau-Lifshitz-\nGilbert macrospin torque equation of motion, and for-\nmally describes how the spin-orbit interaction in itinerant\nelectron systems results in damping of magnetization dy-\nnamics [2]. Schoen et al. [7] have reported that \u000bis mini-\nmized for Co-Fe alloy compositions at which the density-\nof-states at the Fermi level is minimized, in reasonable\nagreement with Kambersk\u0013 y model predictions [8]. Fur-\nthermore, half-metallic, or near half-metallic ferromag-\nnets such as full-Heusler compounds have been predicted\nto demonstrate an ultralow Kambersk\u0013 y \u000b(\u001410\u00003) due\nto their spin-resolved band structure near the Fermi level\n[9]. Finally, anisotropy of the Kambersk\u0013 y damping in sin-\ngle crystals has been predicted, which is more robust for\nFermi surfaces with single-band character [5, 10].\nThe Gilbert damping constant is often reported\nthrough measurements of the ferromagnetic resonance\n(FMR) linewidth \u0001 H, which may be expressed as a sum\nof individual contributions\n\u0001H=2\u000bf\n\r+ \u0001H0+ \u0001HTMS; (1)\nwhere the \frst term is the Gilbert damping linewidth\n(fis the FMR frequency, \ris the gyromagnetic ratio),\n\u0001H0is a frequency-independent inhomogeneous broad-\nening, and \u0001 HTMS represents an extrinsic two-magnon\nscattering (TMS) linewidth contribution [11, 12] that is,\nin general, a nonlinear function of frequency. In recentyears it has been realized that TMS linewidths are per-\nvasive for the conventional in-plane geometry of thin \flm\nFMR measurements, requiring either the perpendicular-\nto-plane FMR geometry [13] (for which TMS processes\nare suppressed) or su\u000eciently broadband measurements\n[14] to extract the bare Gilbert \u000b. For instance, recent\nFMR linewidth studies on Heusler compounds have re-\nported distinct TMS linewidths [15, 16], which challenged\nsimple inference of the Gilbert \u000b.\nIn this article, we present FMR linewidth measure-\nments for epitaxial Heusler thin \flms for all principal ori-\nentations of the magnetization with respect to the sym-\nmetry axes. For the in-plane con\fguration, large and\nanisotropic TMS-dominated linewidths are observed. In\nthe perpendicular-to-plane con\fguration, for which the\nTMS process is inactive [11], the Gilbert \u000band inhomo-\ngeneous broadening are measured. We \fnd evidence of a\nlow (\u001810\u00003) Gilbert\u000bin these Heusler thin \flms, accom-\npanied by a large and anisotropic TMS contribution to\nthe linewdith for in-plane magnetization. We conclude\nby discussing the interplay of low Gilbert \u000band large\nTMS, and we emphasize the nature by which the TMS\nmay conceal the presence of anisotropic Kambersk\u0013 y \u000b.\nII. SAMPLES\nThe Heusler alloy \flms used for these measurements\nwere grown by molecular beam epitaxy (MBE) by co-\nevaporation of elemental sources in ultrahigh vacuum\n(UHV). The MgO(001) substrates were annealed at\n700\u000eC in UHV followed by growth of a 20 nm thick MgO\nbu\u000ber layer by e-beam evaporation at a substrate temper-\nature of 630\u000eC. The 10 nm thick Co 2MnAl and Co 2MnSi\n\flms were grown on the MgO bu\u000ber layers at room tem-\nperature and then annealed at 600\u000eC for 15 minutes\nin situ in order to improve crystalline order and surface\nmorphology. The 24 nm thick Co 2FeAl sample was grown\nusing the same MgO substrate and bu\u000ber layer prepa-arXiv:1909.02738v2  [cond-mat.mtrl-sci]  9 Apr 20202\nration, but at a substrate temperature of 250\u000eC with\nno post-growth anneal. Re\rection high energy electron\ndi\u000braction (RHEED) was monitored during and after\ngrowth of all samples and con\frmed the expected epitax-\nial relationship of MgO(001) h110ijjHeusler(001)h100i.\nX-ray di\u000braction (XRD) demonstrated the existence of\na single phase of (001)-oriented Heusler, along with the\npresence of the (002) re\rection, con\frming at least B2\nordering in all cases. In addition, for the Co 2MnSi \flm\nonly, the (111) re\rection was observed, indicating L2 1\nordering [see Fig. 1(a)]. All of the \flms were capped\nwith several nm of e-beam evaporated AlOx for pas-\nsivation prior to atmospheric exposure. The e\u000bective\nmagnetization for the 24 nm thick Co 2FeAl \flm was\ndetermined from anomalous Hall e\u000bect saturation \feld\nto be 1200 emu/cm3, which is consistent with measure-\nments of Ref. [17] for L2 1or B2-ordered \flms, along\nwith 990 emu/cm3and 930 emu/cm3for the Co 2MnSi\nand Co 2MnAl \flms, respectively. Hereafter, we will refer\nto the Co 2MnSi(10 nm)/MgO as the \\CMS\" \flm, the\nCo2MnAl(10 nm)/MgO \flm as the \\CMA\" \flm, and the\nCo2FeAl(24 nm)/MgO \flm as the \\CFA\" \flm.\nIII. EXPERIMENT\nBroadband FMR linewidth measurements were per-\nformed at room temperature with a coplanar waveguide\n(CPW) transmission setup, similar to that discussed in\ndetail in Refs. [18, 19], placed between the pole faces\nof an electromagnet. A cleaved piece of the sample\n(\u00182 mm\u00021 mm) was placed face-down over the center-\nline of the CPW. A rectifying diode was used to detect\nthe transmitted microwave power, and a \u0018100 Hz mag-\nnetic \feld modulation was used for lock-in detection of\nthe transmitted power, resulting in a signal /d\u001f=dH\n(where\u001fis the \flm dynamic magnetic susceptibility).\nThe excitation frequency could be varied from 0-50 GHz,\nand a microwave power near 0 dBm was typically used. It\nwas veri\fed that all measurements discussed in this arti-\ncle were in the small precession cone angle, linear regime.\nThe orientation of the applied magnetic \feld could be\nrotated to arbitrary angle in the \flm plane (IP), or ap-\nplied perpendicular to the \flm plane (PP). We empha-\nsize again that TMS contributions are suppressed in the\nPP con\fguration [12]. The resonance \felds were \ft as a\nfunction of applied frequency in order to extract various\nmagnetic properties of the \flms.\nThe magnetic free energy per unit volume used to gen-\nerate the resonance conditions for these samples is given\nby\nFM=\u0000M\u0001H+K1sin2\u001ecos2\u001e+ 2\u0019M2\neffcos2\u0012;(2)\nwhere His the applied \feld, \u001eand\u0012are the azimuthal\nand polar angles of the magnetization, respectively, K1\nis a \frst order in-plane cubic anisotropy constant, and\n4\u0019Meffis the PP saturation \feld, which includes the\nusual demagnetization energy and a \frst order uniaxial\n-150-75075150-6-30361\n0 GHz20 GHz30 GHzdχ/dH (arb. u.)H\n - HFMR (Oe)40 GHzCMS\n090180270360101102103104 <111> \n<202>Intensity (arb. u.)φ\n (°)CMS\n-5000 5 00-101F\nield (Oe)M/MSH\n || 〈110〉H\n || 〈100〉H\n || 〈110〉CFA\n(d)(b)(\nc)(a)C\nFAFIG. 1. (a) Wide-angle x-ray di\u000braction \u001e-scans ofh202i\n(blue) andh111i(red) peaks for the CMS \flm. (b) Typical\nderivative susceptibility lineshapes for these samples at dif-\nferent microwave excitation frequencies. The \fts are shown\nas solid lines. (c) In-plane hysteresis loops for CFA obtained\nwith a vibrating-sample magnetometer (VSM). (d) Atomic\nforce microscopy (AFM) image of surface topography for\nCFA. RMS roughness is 0.2 nm.\nanisotropy due to interfacial e\u000bects. The parameters ob-\ntained by \ftting to Eq. 2 are shown in Table I. The uncer-\ntainty in these parameters was estimated by measuring a\nrange of di\u000berent sample pieces, and using the standard\ndeviation of the values as the error bar. The long-range\ninhomogeneity characteristic of epitaxial samples makes\nthis a more accurate estimate of the uncertainty than\nthe \ftting error. The magnetic-\feld-swept FMR line-\nshapes were \ft to the derivative of Lorentzian functions\n[19] in order to extract the full-width at half-maximum\nlinewidths \u0001 H[magnetic \feld units, Fig. 1(b)], which\nare the focus of this article. The maximum resonant fre-\nquency was determined by the maximum magnetic \feld\nthat could be applied for both IP and PP electromagnet\ncon\fgurations, which was 10.6 kOe and 29 kOe, respec-\ntively. For the IP measurement, the angle of the applied\n\feld in the plane of the \flm was varied to determine the\nin-plane magnetocrystalline anisotropy of our samples,\nwhich was fourfold-symmetric for the three \flms char-\nacterized in this article. The anisotropy was con\frmed\nusing vibrating-sample magnetometry (VSM) measure-\nments, an example of which is shown in Fig. 1(c), which\nshows IP easy and hard axis hysteresis loops for the\nCFA \flm. For the PP measurement, alignment was ver-\ni\fed to within\u00180.1\u000eto ensure magnetization saturation\njust above the PP anisotropy \feld, thus minimizing \feld-3\nTABLE I. Summary of the magnetic properties extracted from the dependence of the resonance \feld on applied frequency\nfor both \feld in-plane ( jj) and \feld perpendicular-to-plane ( ?) con\fgurations, along with the Gilbert \u000band inhomogeneous\nbroadening from the perpendicular-to-plane con\fguration. 2 K1=Msand 4\u0019Meffare the in-plane and perpendicular-to-plane\nanisotropy \felds, respectively (see Eq. 2), and gis the Land\u0013 e g-factor.\nSample 2 K1=Ms(Oe) 4\u0019Mjj\neff(kOe) 4 \u0019M?\neff(kOe) gjjg?\u000b001(\u000210\u00003) \u0001H0(Oe)\nCMS 280 12.3 13.3 2.04 2.04 1 :5\u00060:1 9\u00061\nCMA 35 11.3 11.7 2.06 2.08 1 :8\u00060:2 12\u00063\nCFA 230 15.1 15.5 2.06 2.07 <0.8 100\u00066\nCFA 500\u000eC anneal N/A N/A 15.1 N/A 2.07 1 :1\u00060:1 45\u00061\n01 02 03 04 05 00306090120α\n001 = 1.1×10-3CFA 500 °C annealCFAC\nMAα001 < 8×10-4α\n001 = 1.8×10-3ΔH (Oe)F\nrequency (GHz)α001 = 1.5×10-3CMS\nFIG. 2. Linewidths as a function of frequency with the \feld\napplied perpendicular to plane, for which two-magnon scat-\ntering is inactive. The black squares are data for the CMS\n\flm, the red circles are for the CMA \flm, and the blue trian-\ngles are for the CFA \flm. In addition, linewidths are shown\nfor a CFA \flm that was annealed at 500\u000eCex situ (magenta\ndiamonds). Corresponding linear \fts are shown along with\nthe extracted Gilbert damping factor \u000b. The blue dashed\nlines indicate an upper bound of \u000b001= 8\u000210\u00004and a lower\nbound of\u000b001= 0 for CFA.\ndragging contributions to the linewidth.\nIV. RESULTS AND ANALYSIS\nA. Perpendicular-to-plane linewidths\nFirst we discuss the results of the PP measurement. As\nstated in Sec. III, the TMS extrinsic broadening mecha-\nnism is suppressed when the magnetization is normal to\nthe plane of the \flm. We can thus \ft our data to Eq. 1\nwith \u0001HTMS = 0, greatly simplifying the extraction of\nthe Gilbert damping constant \u000band the inhomogeneous\nbroadening \u0001 H0. Prior knowledge of \u0001 H0is particu-\nlarly important for constraining the analysis of the IP\nmeasurements, as we shall discuss.\n3035404550C\nMS〈100〉C\nMS〈110〉ΔH (Oe)CMS2\n0 GHz2\n800300032003400H\nFMR(Oe)-\n450 4 5200400600C\nFA〈100〉ΔH (Oe)A\nngle (°)CFA2\n0 GHzCFA〈110〉2\n600280030003200H\nFMR(Oe)50100150200(a)(\nc)C\nMA〈100〉C\nMA〈110〉ΔH (Oe)CMA1\n5 GHz(b)2\n00020502100H\nFMR(Oe)FIG. 3. Azimuthal angular dependence of the linewidths (left\nordinate, blue circles) and resonance \felds (right ordinate,\nblack squares) for (a) CMS, (b) CMA, and (c) CFA. The\nexcitation frequency was 20 GHz for CMS, 15 GHz for CMA,\nand 20 GHz for CFA. The solid lines are sinusoidal \fts.\nThe dependence of \u0001 Hon frequency for the CMS,\nCMA, and CFA \flms in the PP con\fguration is\nsummarized in Fig. 2, in which \fts to Eq. 1 are\nshown with \u0001 HTMS set to zero. For the CMS\n\flm,\u000b001= (1:5\u00060:1)\u000210\u00003and \u0001H0= 9 Oe,\nwhile for the CMA \flm \u000b001= (1:8\u00060:2)\u000210\u000034\nand \u0001H0= 12 Oe. Co 2MnSi 2=3Al1=3/MgO and\nCo2MnSi 1=3Al2=3/MgO \flms (both 10 nm thick) were\nalso measured, with Gilbert damping values of \u000b001=\n(1:8\u00060:2)\u000210\u00003and\u000b001= (1:5\u00060:1)\u000210\u00003, re-\nspectively (not shown). For CFA, we obtained a damp-\ning value of \u000b001= 3\u000210\u00004with an upper bound of\n\u000b001<8\u000210\u00004and \u0001H0= 100 Oe. These \ft param-\neters are also contained in Table I. The source of the\nlarge inhomogeneous broadening for the CFA \flm is un-\nclear: AFM measurements [Fig. 1(d)] along with XRD in-\ndicate that the \flm is both crystalline and smooth. Note\nthat the range of frequencies shown in Fig. 2 are largely\ngoverned by considerations involving the Kittel equation\n[20]: measurements below 10 GHz were not used due to\nthe increasing in\ruence of slight misalignment on \u0001 H\n(through \feld-dragging) for resonant \felds just above\nthe saturation value. A piece of the CFA sample was\nannealed at 500\u000eCex situ , which reduced the inhomoge-\nnoeus broadening to \u001845 Oe (still a relatively large value)\nand increased the Gilbert damping to \u000b001= 1:1\u000210\u00003\n(similar behavior in CFA was seen in Ref. [21]). The\nconstraint of \u000b001<8\u000210\u00004is among the lowest of re-\nported Gilbert damping constants for metallic ferromag-\nnets, but the \u000b\u001810\u00004range is not unexpected based\non Kambersk\u0013 y model calculations performed for similar\nfull-Heusler compounds [9] or other recent experimental\nreports [22, 23]. It should be noted that Schoen et al. [7]\nhave recently reported \u000b= 5\u000210\u00004for Co 25Fe75thin\n\flms, where spin pumping and radiative damping con-\ntributions were subtracted from the raw measurement.\nSpin pumping contributions to the intrinsic damping are\nnot signi\fcant in our \flms, as no heavy-metal seed layers\nhave been used and the \flms have thicknesses of 10 nm\nor greater. For the radiative damping contribution [13]\nin the geometry of our CPW and sample, we calculate\ncontributions \u000brad<\u00181\u000210\u00004, which is below the uncer-\ntainty in our damping \ft parameter.\nB. In-plane linewidths\nWith the intrinsic damping and inhomogeneous broad-\nening characterized by the PP measurement, we turn our\nattention to the IP linewidth measurements, for which\nTMS contributions are present. For hard-axis measure-\nments, frequencies <\u00185 GHz were not used due to the\nin\ruence of slight magnetic \feld misalignment on the\nlinewidths. For easy-axis measurements, the lower limit\nis determined by the zero-\feld FMR frequency. Fig-\nure 3 shows the dependences of the resonance \felds and\nlinewidths on the angle of the in-plane \feld. An im-\nportant observation seen in Fig. 3 is that the linewidth\nextrema are commensurate with those of the resonance\n\felds and therefore the magnetocrystalline anisotropy en-\nergy. This rules out \feld-dragging and mosaicity contri-\nbutions to the linewidth, which can occur when the reso-\nnance \feld depends strongly on angle [24]. We note that\nsimilar IP angular dependence of the FMR linewidth,\n01 02 03 04 05 002004006008000204060801000\n60120180240300C\nFA(c)ΔH (Oe)F\nrequency (GHz)〈100〉ΔH (Oe)(a)C\nMSC\nMA[\n001]〈110〉(b)ΔH (Oe)FIG. 4. Linewidths along all three principal directions for\nCMS (a), CMA (b), and CFA (c). Heusler crystalline axes\nare labeled byh100i(black),h110i(red), and [001] (blue). In\nall three cases,h110iis the in-plane easy axis and h100iis the\nin-plane hard axis. The corresponding \fts are shown as the\nsolid curves, where the in-plane linewidths are \ft using Eq. 3\nand the out-of-plane linewidths are \ft to the Gilbert damping\nmodel. The \ft parameters are given in Table II.\nwhich was attributed to an anisotropic TMS mechanism\ncaused by a rectangular array of mis\ft dislocations, has\nbeen reported by Kurebayashi et al. [25] and Woltersdorf\nand Heinrich [14] for epitaxial Fe/GaAs(001) ultrathin\n\flms.\nTo further study the anisotropy of the IP \u0001 Hin our\n\flms, we have measured \u0001 Hat the angles correspond-\ning to the extrema of HFMR (and \u0001H) in Fig. 3 over\na range of frequencies. These data are shown in Fig. 4,\nalong with the PP ([001]) measurements for each sample.\nA distinguishing feature of the data shown in Fig. 4 is the\nsigni\fcant deviation between IP and PP linewidths in all\nbut one case (CMS h100i). Large and nonlinear frequency\ndependence of the IP linewidths is strongly suggestive of\nan active TMS linewidth broadening mechanism. In the\npresence of TMS, careful analysis is required to separate5\n0°90°1\n80°2\n70°01x10501 02 03 04 00.00.51.01.52\nx1045x1041\n1.411.82\n4 GHz32 GHz1\n6 GHz |q2M| (cm-1)ξ\n-1ξ = 100 nm10-41\n0-35\n×10-3ΔHTMS/H'2 (Oe-1) (×10-4)f\nFMR (GHz)α = 10-2(\nq || M)(q ^ M)(b) |q| (cm-1)q\n ^ Mωm (GHz)q\n || MD\negeneracyH = 1 kOe(a)\nFIG. 5. (a) Two-magnon scattering linewidth contribution\nfor values of Gilbert damping \u000b= 10\u00002;5\u000210\u00003;10\u00003;and\n10\u00004. The inset shows magnon dispersions for an applied\n\feld ofH= 1 kOe. (b) Contours of the degenerate mode\nwavenumber q2Min the \flm plane as a function of wavevector\nangle relative to the magnetization for fFMR = 16, 24, and\n32 GHz. The dashed circle indicates the wavenumber of a\ndefect with size \u0018= 100 nm.\nthe Gilbert damping from the TMS linewidth contribu-\ntions. We therefore describe the TMS mechanism in more\ndetail in the following section in order to analyze the IP\nlinewidths in Fig. 4 and extract the Gilbert damping.\nC. Two-magnon scattering model\nThe TMS mechanism leads to a characteristic nonlin-\near frequency dependence of \u0001 H[11, 12]. In Fig. 4,\nthe IP \u0001His not a linear function of frequency, but\npossesses the \\knee\" behavior characteristic of the fre-\nquency dependence of linewidths dominated by the TMS\nmechanism. We have \ft our data to the TMS model\ndescribed by McMichael and Krivosik [12], in which theTMS linewidth \u0001 HTMS is given by [26, 27]\n\u0001HTMS =\r2\u00182H02\ndf=dHjfFMRZ\n\u00000qCq(\u0018)\u000e\u000b(!\u0000!q)d2q;(3)\nwhere \u0000 0qis the defect-mediated interaction term be-\ntween magnons at wavevector 0 and q,Cq(\u0018) = (1 +\n(q\u0018)2)\u00003=2is the correlation function of the magnetic sys-\ntem with correlation length \u0018, andH0is the magnitude\nof the characteristic inhomogeneity (units of magnetic\n\feld). The \u000e\u000b-function in Eq. 3 selects only the magnon\nscattering channels that conserve energy. In the limit of\nzero intrinsic damping, it is identical to the Dirac delta\nfunction, but for \fnite \u000bit is replaced by a Lorentzian\nfunction of width \u000e!= (2\u000b!=\r )d!=dH . The magnon\ndispersion relation determining !qis the usual Damon-\nEshbach thin \flm result [26, 28] with the addition of mag-\nnetocrystalline anisotropy sti\u000bness \feld terms extracted\nfrom the dependence of the resonance \feld on the applied\nfrequency for the IP con\fguration. The \flm thickness\nda\u000bects the states available for two-magnon scattering\nthrough the dispersion relation, namely, the linear term\nwhich gives rise to negative group velocity for small q\n(/\u0000qd). The IP FMR linewidth data shown in Fig. 4\nwere \ft to Eq. 1 (with Eq. 3 used to evaluate \u0001 HTMS)\nwith\u0018,\u000b, andH0as \ftting parameters (shown in Table\nII). The correlation length \u0018remains approximately con-\nstant for di\u000berent in-plane directions, while the strength\nH0is larger for theh100idirections in the CMA and CFA\nsamples and the h110idirections in the CMS sample.\nSome degree of uncertainty results from this \ftting proce-\ndure, because for linewidth data collected over a limited\nfrequency range, \u0018and\u000bare not completely decoupled\nas \ftting parameters. In absolute terms, however, the\nlargest systematic errors come from the exchange sti\u000b-\nness, which is not well-known. The error bars given in\nTable II were calculated by varying the exchange sti\u000b-\nness over the range 400 meV \u0017A2to 800 meV \u0017A2, and\nrecording the change in the \ft parameters. This range of\nvalues was chosen based on previous Brillouin light scat-\ntering measurements of the exchange sti\u000bness in similar\nHeusler compounds [29, 30]. In addition, we note that\nin Eq. 1 \u0001 H0is taken to be isotropic, with the value\ngiven by the PP linewidth measurements shown in Fig.\n2. Although certain realizations of inhomogeneity may\nresult in an anisotropic \u0001 H0(see Ref. [14] for a good\ndiscussion), doing so here would only serve to create an\nadditional \ftting parameter.\nD. E\u000bect of low intrinsic damping\nThe e\u000bect of low intrinsic damping on the two-magnon\nlinewidth can be seen in Fig. 5(a). As \u000bdecreases, with\nall other parameters \fxed, \u0001 HTMS steadily increases\nand becomes increasingly nonlinear (and eventually non-\nmonotonic) with frequency. In particular, a \\knee\" in\nthe frequency dependence becomes more pronounced for6\nTABLE II. Summary of the \ftting parameters used to \ft the\nin-plane data of Fig. 4 (black squares and red circles) to Eqs.\n1 and 3. CFA refers to the unannealed Co 2FeAl sample.\nSample (Field Direction) \u000b(\u000210\u00003)\u0018(nm)H0(Oe)\nCMSh110i 1:6\u00060:2 40\u000625 55\u000630\nCMSh100i 1:5\u00060:1 40\u000625 30\u000615\nCMAh110i 3:1\u00060:2 70\u000620 30\u00065\nCMAh100i 4:7\u00060:4 55\u000610 90\u00065\nCFAh110i 2:0\u00060:3 20\u000610 175\u000660\nCFAh100i N/A N/A N/A\nlow damping (see e.g. Fig. 5(a) curve for \u000b= 10\u00004). The\nphysics giving rise to the knee behavior is illustrated in\nFig. 5(b). The TMS process scatters magnons from zero\nto non-zero wavevector at small q. There is assuemd\nto be su\u000ecient disorder to allow for the momentum q\nto be transferred to the magnon system. There will al-\nways be, however, a length scale \u0018below which the disor-\nder decreases, so that the \flm becomes e\u000bectively more\nuniform at large wavevectors. The corresponding FMR\nfrequencies are those for which the contours of constant\nfrequency (the \fgure eights in Fig. 5) in q-space have ex-\ntrema atq\u0018\u0018\u00001. The TMS rate is also determined by\nthe interplay of the magnon density of states, the e\u000bec-\ntive area in q-space occupied by the modes that conserve\nenergy, and the Gilbert damping. The knee behavior is\nmore pronounced for low \u000bdue to the increased weight\nof the van Hove singularity coming from the tips of the\n\fgure eights, in the integrand of Eq. 3. Although a larger\nwindow of energies, set by the width of \u000e\u000b, is available for\nlarger\u000b, this smears out the singularity in the magnon\ndensity of states, removing the sharp knee in the TMS\nlinewidth as a function of frequency. The PP measure-\nment con\frms that all of these epitaxial Heusler \flms lie\nwithin the range \u000b<2\u000210\u00003. Ferromagnetic \flms with\nultralow\u000bare therefore increasingly prone to large TMS\nlinewidths (particularly for metals with large Ms). The\nTMS linewidths will also constitute a larger fraction of\nthe total linewidth due to a smaller contribution from the\nGilbert damping. In practice, this is why experimental\nreports [7, 22, 23] of ultralow \u000bhave almost all utilized\nthe PP geometry.\nE. Discussion\nThe results of the IP linewidth \fts to Eqs. 1 and 3 are\nsummarized in Table II. In the case of CMS, the high-\nfrequency slopes in Fig. 4(a) approach the same value\nalong each direction, as would be expected when the fre-\nquency is large enough for the TMS wavevector to exceed\nthe inverse of any defect correlation length. In this limit,\n\u000bis isotropic (within error limits).\nNext, we discuss the CMA IP data shown in Fig. 4(b)\nand Table II. It is clear from this \fgure that a good \ftcan be obtained along both h100iandh110idirections.\nIn Table II it can be seen that the value of the defect cor-\nrelation length \u0018is approximately the same along both\ndirections. However, the values of \u000bwe obtain from \ft-\nting to Eqs. 1 and 3 do not agree well with the PP value\nof\u000b001= 1:8\u000210\u00003(Fig. 2). Anisotropic values of \u000b\nhave been both predicted [5, 10] and observed [31], and\nan anisotropic \u000bis possibly the explanation of our best-\n\ft results. The in-plane h100iand [001] directions are\nequivalent in the bulk, so the anisotropy would neces-\nsarily be due to an interface anisotropy energy [31] or\nperhaps a tetragonal distortion due to strain [32].\nFinally, we discuss the CFA linewidths shown in Fig.\n4(c) and Table II. This sample has by far the largest two-\nmagnon scattering contribution, which is likely related\nto the anomalously large inhomogeneous broadening and\nlow intrinsic damping [see Fig. 5(a)] observed in the PP\nmeasurement. A good \ft of the data was obtained when\nthe \feld was applied along the h110idirection. Notably,\nthe IPh110ibest \ft value of 2 :1\u000210\u00003is nearly a factor\nof 3 larger than the \u000b001upper bound on the same sample\n(Table I), strongly suggesting an anisotropic Gilbert \u000b. A\nstriking anisotropy in the IP linewidth was revealed upon\nrotating the magnetization to the h100iorientation. For\ntheh100icase, which yielded the largest TMS linewidths\nmeasured in this family of \flms, we were not able to \ft\nthe data to Eq. 3 using a set of physically reasonable in-\nput parameters. We believe that this is related to the\nconsideration that higher order terms in the inhomoge-\nneous magnetic energy (see Ref. [26]) need to be taken\ninto account. Another reason why this may be the case is\nthat the model of McMichael and Krivosik [12] assumes\nthe inhomogeneities to be grain-like, whereas the samples\nare epitaxial [see Fig. 1(a)]. Atomic force microscopy im-\nages of these samples [Fig. 1(d)] imply that grains, if they\nexist, are much larger than the defect correlation lengths\nlisted in Table II, which are of order 10's of nm. We also\nnote that there does not appear to be a correlation be-\ntween the strength of two-magnon scattering H0and the\ncubic anisotropy \feld 2 K1=Ms, which would be expected\nfor grain-induced two-magnon scattering.\nV. SUMMARY AND CONCLUSION\nWe conclude by discussing the successes and limita-\ntions of the McMichael and Krivosik [12] model in an-\nalyzing our epitaxial Heusler \flm FMR linewidth data.\nWe have shown that two-magnon scattering is the ex-\ntrinsic linewidth-broadening mechanism in our samples.\nAny model which takes this as its starting point will\npredict much of the qualitative behavior we observe,\nsuch as the knee in the frequency dependence and the\nlarge linewidths IP for low \u000b\flms. The TMS model\nused in this article (for the purpose of separating TMS\nand Gilbert linewidth contributions) is, however, only\nas accurate as its representation of the inhomogeneous\nmagnetic \feld and the underlying assumption for the7\nfunctional form of Cq(\u0018). Grain-like defects are as-\nsumed, which essentially give a random magnetocrys-\ntalline anisotropy \feld. We did not, however, explicitly\nobserve grains in our samples with AFM, at least below\nlengthscales of\u001810\u0016m [Fig. 1(d)]. Mis\ft dislocations, a\nmuch more likely candidate in our opinion, would cause\nan e\u000bective inhomogeneous magnetic \feld which could\nhave a more complicated spatial pro\fle and therefore\nlead to anisotropic two-magnon scattering (see Ref. [14]).\nThe perturbative nature of the model also brings its own\nlimitations, and we believe that the CFA h100idata, for\nwhich we cannot obtain a satisfactory \ft, are exemplary\nof a breakdown in the model for strong TMS. Future\nwork should go into methods of treating the two-magnon\nscattering di\u000berently based on the type of crystalline de-\nfects present, which will in turn allow for a more reli-\nable extraction of the Gilbert damping \u000band facilitate\nthe observation of anisotropic Gilbert damping, enabling\nquantitative comparison to \frst-principles calculations.\nRegardless of the limitations of the model, we empha-\nsize three critical observations drawn from the linewidth\nmeasurements presented in this article. First, in all cases\nwe observe large and anisotropic TMS linewidth contri-\nbutions, which imply inhomogeneity correlation length-\nscales of order tens-to-hundreds of nanometers. The mi-\ncroscopic origin of these inhomogeneities is the subjectof ongoing work, but are likely caused by arrays of mis\ft\ndislocations [14]. The relatively large lengthscale of these\ndefects may cause them to be easily overlooked in epi-\ntaxial \flm characterization techniques such as XRD and\ncross-sectional HAADF-STEM, but they still strongly in-\n\ruence magnetization dynamics. These defects and their\nin\ruence on the FMR linewidth through TMS complicate\ndirect observation of Kambersk\u0013 y's model for anisotropic\nand (in the case of Heusler compounds) ultralow intrinsic\ndamping in metallic ferromagnets. Second, we observed\nlow intrinsic damping through our PP measurement,\nwhich was<2\u000210\u00003for all of our samples. Finally, we\nhave presented the mechanism by which FMR linewidths\nin ultralow damping \flms are particularly likely to be en-\nhanced by TMS, the anisotropy of which may dominate\nany underlying anisotropic Kambersk\u0013 y damping.\nThis work was supported by NSF under DMR-1708287\nand by SMART, a center funded by nCORE, a Semi-\nconductor Research Corporation program sponsored by\nNIST. 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Rovisco Pais 1, P-1049-001 Lisboa, Portugal andGrupo de F´ ısica Matem´ atica,\nFaculdade de Ciˆ encias, Universidade de Lisboa, Campo Gran de, Edif´ ıcio C6,\nP-1749-016 Lisboa, Portugal\nJIˇR´I LIPOVSK ´Y\nDepartment of Physics, Faculty of Science, University of Hr adec Kr´ alov´ e,\nRokitansk´ eho 62, 50003 Hradec Kr´ alov´ e, Czechia\nAbstract. We evaluate the spectral determinant for the damped wave equat ion on\nan interval of length Twith Dirichlet boundary conditions, proving that it does not\ndepend on the damping. This is achieved by analysing the square of th e damped\nwave operator using the general result by Burghelea, Friedlander , and Kappeler on\nthe determinant for a differential operator with matrix coefficients .\nPACS: 46.40.Ff, 03.65.Ge\n1.Introduction\nWe consider the simple mathematical model of wave propagationon a damped string\nfixed at both ends given by\n∂2v(t,x)\n∂t2+2a(x)∂v(t,x)\n∂t=∂2v(t,x)\n∂x2, (1.1)\nwith the space variable xon an interval [0 ,T],v(0) =v(T) = 0 and a(x)∈C([0,T]).\nDespite its apparent simplicity, the problem is nontrivial and interest ing and has re-\nceived much attention over the last two decades – see, for instanc e, [GH11, FL17,\nCFN+91, CZ94, BF09].\nThe operator associated with (1.1) is non-selfadjoint and the asym ptotical location\nof its eigenvalues was determined to first order in [CFN+91, CZ94], wh ere it was shown\nthat eigenvalues λconverge to the vertical line Re λ=−/a\\}bracketle{ta/a\\}bracketri}htas their imaginary part\ngoes to±∞, where/a\\}bracketle{ta/a\\}bracketri}htdenotes the average of the damping function. The general\nasymptotic behaviour was analysed in [BF09], where further spectr al invariants were\ndetermined.\nComingfromothersourcesintheliterature, thenotionofdetermin antofamatrixhas\nbeen generalised to operators. In this analogy, we would like to obta in a regularisation\ncorresponding to the product of eigenvalues of the given operato r. If the considered\nE-mail addresses :psfreitas@fc.ul.pt, jiri.lipovsky@uhk.cz .\n12 DETERMINANT FOR THE DAMPED WAVE EQUATION\noperator Shas eigenvalues {λj}∞\nj=1, in agreement with [RS71] (see also [GY60, MP49])\nwe define the generalised zeta function associated with the operat orSby\nζS(s) =∞/summationdisplay\nj=1λ−s\nj,\nfor complex sin a half-plane such that the above Dirichlet series converges. The\nspectral determinant may then be defined by the formula\nDetS= e−ζ′\nS(0), (1.2)\nwhere the prime denotes the derivative with respect to the variable s. Note that the\nseries defining the zeta function will not, in general, be convergent fors= 0. We use\nthe definition of ζSfor the real part of slarge enough and understand the formula in\nthe sense of the analytic continuation of the generalized zeta func tion to the complex\nplane.\nThe spectral determinant was computed for the Sturm-Liouville op erator in [LS77],\nwhere an elegant expression using the solution of a corresponding C auchy problem was\npresented. This was extended to the case of quantum graphs in [AC D+00, Fri06]. We\npoint out that spectral determinants have several applications e .g. in quantum field\ntheory [Dun08].\nTo the best of our knowledge the spectral determinant for the da mped wave equation\nhad not been studied previously, so in this note we bring together th ese two topics and\nevaluate this object. From a mathematical perspecive there is also what we believe to\nbe the interesting feature of applying the concept of the determin ant of an operator to\na non-selfadjoint operator. Furthermore, and as we will see, the determinant does not,\nin fact, depend on the damping. This may be expected form a formal analysis, and\nour purpose is to give a rigorous justification of this fact.\nThis note is structured as follows. In the next section, we ellaborat e on the math-\nematical description of the model and state the main result. In Sec tion 3 we then\naddress the problem for the case without damping, as this already d isplays some of the\nimportant features which we will need to consider later, namely, the fact that the asso-\nciated zeta function will depend on the branch cut which is chosen fo r the logarithm.\nIn Section 4 we recall the general result of Burghelea, Friedlander , and Kappeler. In\nSection 5 we apply this result to the square of our operator, since a direct application\nis not possible. Finally, we find the sought determinant for the dampe d wave equation\nin Section 6.\n2.Basic setting and formulation of the main result\nEquation (1.1) may be written in a different form, namely,\n∂\n∂t/parenleftbigg\nv0(t,x)\nv1(t,x)/parenrightbigg\n=/parenleftbigg0 1\n∂2\n∂x2−2a(x)/parenrightbigg/parenleftbigg\nv0(t,x)\nv1(t,x)/parenrightbigg\nwhich will prove to be more convenient for our purposes. Using the a nsatzv0(t,x) =\neλtu0(x),v1(t,x) = eλtu1(x), we can translate the initial value problem into the follow-\ning spectral problem\nH/parenleftbigg\nu0(x)\nu1(x)/parenrightbigg\n=λ/parenleftbigg\nu0(x)\nu1(x)/parenrightbiggDETERMINANT FOR THE DAMPED WAVE EQUATION 3\nwhereHdenotes the matrix operator\nH=/parenleftbigg0 1\n∂2\n∂x2−2a(x)/parenrightbigg\n.\nThe domain of this operator consists of functions u(x) =/parenleftbigg\nu0(x)\nu1(x)/parenrightbigg\nwith components\nin the Sobolev spaces uj(x)∈W2,2([0,T]),j= 0,1 satisfying the Dirichlet boundary\nconditions\nuj(0) =uj(T) = 0, j= 0,1.\nOur main result is the following.\nTheorem 2.1. Assumea(x)∈C([0,T]), and let εbe a positive number such that there\nare no eigenvalues with phase on the interval [π−ε,π), Then the spectral determinant\nof the operator Hdoes not depend on the damping and equals ±2T, where the plus and\nminus signs correspond to whether we define λ−s\nj= e−slogλjin such a way that the\nbranch cut of the logarithm is λ=tei(π−ε), orλ=tei(2π−ε),t∈[0,∞), respectively.\nRemark 2.2. Note that a value of εas above always exists, since on any compact set\nthere are only a finite number of eigenvalues.\nRemark 2.3. The case of the damped wave equation where a potential is adde d to\nthe right-hand side of (1.1)may be treated in a similar fashion and the corresponding\ndeterminant also turns out to be independent of the damping t erm. We discuss this\nsituation in Remark 6.2.\n3.The case of a(x) = 0\nWe begin by considering the case without damping, and denote the co rresponding\noperator by H0. It is a simple exercise that its eigenvalues are of the form λj=ijπ\nT,\nj∈Z\\{0}. To obtain the spectral determinant in this instance, we start fro m the zeta\nfunction resulting from the definition (1.2). However, one must pro ceed carefully here,\nas the result depends on the definition of λ−s\njand, in particular, on which branch of\nthe logarithm we use when defining i−sand (−i)−s.\nFirst, we consider that the logarithm has the cut in the negative rea l axis, i.e. the\neigenvalues of H0in the upper half-plane are λj=jπ\nTeiπ\n2,j∈Nand the eigenvalues in\nthe lower half-plane are λj=jπ\nTe−iπ\n2,j∈N.\nThe generalized zeta function for this operator is\nζH0(s) =∞/summationdisplay\nj=1/bracketleftBigg/parenleftbiggjπ\nTeiπ\n2/parenrightbigg−s\n+/parenleftbiggjπ\nTe−iπ\n2/parenrightbigg−s/bracketrightBigg\n=∞/summationdisplay\nj=1/parenleftBig\ne−iπ\n2s+eiπ\n2s/parenrightBig/parenleftbiggjπ\nT/parenrightbigg−s\n= 2eslogT\nπcos/parenleftBigπs\n2/parenrightBig\nζR(s),\nwhereζR(s) =∞/summationdisplay\nj=1j−sis the Riemann zeta function. We obtain\n−ζ′\nH0(0) =−2logT\nπζR(0)−2ζ′\nR(0) = logT\nπ+log(2π) = log(2 T),4 DETERMINANT FOR THE DAMPED WAVE EQUATION\nwhere we have used ζR(0) =−1\n2andζ′\nR(0) =−1\n2log(2π). Hence the spectral determi-\nnant for the operator H0is given by\nDetH0= e−ζ′\nH0(0)= 2T .\nNow we are going to compute the determinant in the case where we ch oose the cut\nto be the positive real axis. The eigenvalues are λj=jπ\nTeiπ\n2,j∈N, for the upper half-\nplane and λj=jπ\nTe3iπ\n2,j∈Nfor the lower half-plane. The generalized zeta function is\nnow\nζH0(s) =∞/summationdisplay\nj=1/bracketleftBigg/parenleftbiggjπ\nTeiπ\n2/parenrightbigg−s\n+/parenleftbiggjπ\nTe3iπ\n2/parenrightbigg−s/bracketrightBigg\n= e−iπs∞/summationdisplay\nj=1/parenleftBig\neiπ\n2s+e−iπ\n2s/parenrightBig/parenleftbiggjπ\nT/parenrightbigg−s\n= 2e−iπseslogT\nπcos/parenleftBigπs\n2/parenrightBig\nζR(s).\nHence we have\n−ζ′\nH0(0) =−2logT\nπζR(0)+2iπζR(0)−2ζ′\nR(0) = logT\nπ−iπ+log(2π) =−iπ+log(2T).\nThe spectral determinant for the operator H0is\nDetH0= e−ζ′\nH0(0)=−2T .\n4.A general result\nThe starting point for finding the determinant for the damped wave equation is a\ngeneral result by Burghelea, Friedlander and Kappeler [BFK95]. Th is result gives a\nformula for the determinant of a more general matrix-valued oper ator on an interval.\nFor convenience, we state this result here, together with the nec essary definitions.\nDefinition 4.1. Let us for n∈Ndefine the operator A=2n/summationdisplay\nk=0ak(x)(−i)kdk\ndxk, where\nakarer×rmatrices, in general smoothly dependent on x∈[0,T]. We assume that\nthe leading term a2nis nonsingular and that there exist an angle θso thatspeca2n∩\n{ρeiθ,0≤ρ <∞}=∅. We assume the following boundary conditions at the end poin ts\nof the interval.\nαj/summationdisplay\nk=0bjku(k)(T) = 0,βj/summationdisplay\nk=0cjku(k)(0) = 0,1≤j≤n.\nHere,bjkandcjkare for each j,kconstant r×rmatrices and bjαj=cjβj=I(Idenotes\nther×ridentity matrix). The integer numbers αjandβjsatistfy\n0≤α1< α2<···< αn<2n−1,\n0≤β1< β2<···< βn<2n−1.\nMoreover, we define |α|=/summationtextn\nj=1αj,|β|=/summationtextn\nj=1βj. We define the 2n×2nmatrices\nB= (Bjk)andC= (Cjk), whose entries are r×rmatrices. Here 1≤j≤2n,DETERMINANT FOR THE DAMPED WAVE EQUATION 5\n0≤k≤2n−1and\nBjk:=/braceleftbigg\nbjkfor 1≤j≤nand 0≤k≤αj\n0 otherwise,\nCjk:=/braceleftbigg\nbj−n,kforn+1≤j≤2nand 0≤k≤αj−n\n0 otherwise.\nWe define a 2n×2nmatrixY(x) = (ykℓ(x)),0≤k,ℓ≤2n−1whose entries are\nr×rmatrices ykℓ(x)defined by\nykℓ(x) :=dkyℓ(x)\ndxk,\nwhereyℓ(x)is the solution of the Cauchy problem Ayℓ(x) = 0with the initial conditions\nykℓ(0) =δkℓI. We are interested in the value of the matrix Yat the point T.\nFinally, we introduce\ngα:=1\n2/parenleftbigg|α|\nn−n+1\n2/parenrightbigg\n,\nhα:= det\nwα1\n1... wα1n.........\nwαn\n1... wαnn\n,\nwherewk= exp/parenleftbig2k−n−1\n2nπi/parenrightbig\n. Similarly, we define gβandhβ. We denote by γj,j=\n1,...,rthe eigenvalues of the matrix a2nand define\n(deta2n)gα\nθ:=r/productdisplay\nj=1|γj|gαexp(igαarg(γj))\nwithθ−2π <argγj< θ.\nTheorem 4.2. (Burghelea, Friedlander and Kappeler)\nThe spectral determinant for the operator Ais\nDetA=Kθexp/parenleftbiggi\n2/integraldisplayT\n0Tr(a2n(x)−1a2n−1(x))dx/parenrightbigg\ndet(BY(T)−C),\nwhere\nKθ= [(−1)|β|(2n)nh−1\nαh−1\nβ]r(deta2n(0))gβ\nθ(deta2n(T))gα\nθ.\n5.The square of the operator H\nOurpurposeistouseTheorem4.2toobtainthespectraldetermina ntfortheoperator\nH. However, a direct application of the theorem is not possible since th e highest order\nderivative is only present in one of the entries of the matrix which give s the operator\nH. This would contradict the assumption that the matrix a2nis nonsingular. In order\nto overcome this difficuly, we shall consider the operator A=H◦H=H2for which it\nis then possible to apply Theorem 4.2.\nHowever, we have to be careful when computing the spectral det erminant of Hfrom\nthe spectral determinant of A, as the determinant of the composition of two operators\ndoes not necessarily have to equal the product of their determina nts. This is known in\nthe literature as the multiplicative anomaly and, as has been shown in [BS03, Woj01],\neven the determinant of the square of an operator is not always th e square of the\ndeterminant.6 DETERMINANT FOR THE DAMPED WAVE EQUATION\nA direct calculation yields\nA=/parenleftbigg0 1\n∂2\n∂x2−2a(x)/parenrightbigg2\n=/parenleftbigg∂2\n∂x2 −2a(x)\n−2a(x)∂2\n∂x2∂2\n∂x2+4a2(x)/parenrightbigg\n=/parenleftbigg\n−1 0\n2a(x)−1/parenrightbigg/parenleftbigg\n−i∂\n∂x/parenrightbigg2\n+/parenleftbigg\n0−2a(x)\n0 4a2(x)/parenrightbigg/parenleftbigg\n−i∂\n∂x/parenrightbigg0\n.\nHence we have n= 1,\na2(x) =/parenleftbigg\n−1 0\n2a(x)−1/parenrightbigg\n, a1=/parenleftbigg\n0 0\n0 0/parenrightbigg\n, a0=/parenleftbigg\n0−2a(x)\n0 4a2(x)/parenrightbigg\n.\nWe deal with matrices 2 ×2, sor= 2, and have α1=β1= 0 and hence |α|=|β|= 0.\nThe boundary conditions according to Definition 4.1 are\nb10u(T)+b11u′(T) = 0, c10u(0)+c11u′(0) = 0.\nWe have the Dirichlet boundary conditions u(T) =u(0) =0, and thus the matrices\nbjkandcjkmust be chosen as\nb10=c10=I , b 11=c11= 0,\nwhereIis the 2×2 identity matrix.\nThe matrices BandCare 2×2 matrices with the entries being 2 ×2 matrices. Their\nrows are indexed by 1 and 2, their columns by 0 and 1. According to De finition 4.1 we\nhave\nB10=b10=I, B 11=B20=B21= 0,\nC20=c10=I , C 11=C10=C21= 0.\nHence we have\nB=/parenleftbigg\nI0\n0 0/parenrightbigg\n, C=/parenleftbigg\n0 0\nI0/parenrightbigg\n.\nThe matrix a2n=a2clearly has eigenvalues γ1=γ2=−1. Moreover, we have\ngα=gβ=1\n2/parenleftbigg0\n2−1+1\n2/parenrightbigg\n=−1\n4, hα=hβ=w0\n1= 1.\nand\n(deta2n)gα\nθ=2/productdisplay\nj=11−1/4ei(−1/4)(−π)=i\nTo sum up,\nKθ= [(−1)0(2·1)11−2]2i2=−4.\nNow, we are going to compute the matrix Y(x). By its definition, we have\nY(x) =/parenleftbigg\ny00(x)y01(x)\ny10(x)y11(x)/parenrightbigg\n=/parenleftbigg\ny0(x)y1(x)\ny′\n0(x)y′\n1(x)/parenrightbigg\n,\nwhere the entries of this matrix are 2 ×2 matrices with the following boundary condi-\ntions atx= 0\ny0(0) =I, y′\n0(0) = 0, y1(0) = 0, y′\n1(0) =I . (5.1)DETERMINANT FOR THE DAMPED WAVE EQUATION 7\nFor the matrix in the formula in Theorem 4.2 we have\ndet(BY(T)−C) = det/bracketleftbigg/parenleftbigg\nI0\n0 0/parenrightbigg/parenleftbigg\ny0(T)y1(T)\ny′\n0(T)y′\n1(T)/parenrightbigg\n−/parenleftbigg\n0 0\nI0/parenrightbigg/bracketrightbigg\n= det/parenleftbigg\ny0(T)y1(T)\n−I0/parenrightbigg\n= dety1(T).\nNow, we will find the solutions of the Cauchy problem, i.e. the equation\nH2/parenleftbigg\nu0(x)\nu1(x)/parenrightbigg\n= 0\nwith the boundary conditions (5.1). We obtain\nu′′\n0(x)−2a(x)u1(x) = 0, (5.2)\n−2a(x)u′′\n0(x)+u′′\n1(x)+4a2(x)u1(x) = 0. (5.3)\nMultiplying (5.2) by 2 a(x) and adding the result to (5.3) we obtain\nu′′\n1(x) = 0. (5.4)\nThe boundary conditions are y(k)\nl(0) =δklI. Hence to obtain the matrix y1(x), we have\nto find two vectors/parenleftbigg\nu01(x)\nu11(x)/parenrightbigg\nand/parenleftbigg\nu02(x)\nu12(x)/parenrightbigg\n, for which\ny1(0) =/parenleftbigg\nu01(0)u02(0)\nu11(0)u12(0)/parenrightbigg\n= 0, y′\n1(0) =/parenleftbigg\nu′\n01(0)u′\n02(0)\nu′\n11(0)u′\n12(0)/parenrightbigg\n=I.(5.5)\nThe general form of the solution of (5.4) is u1(x) =ax+b, and from the first condition\nin (5.5) we have b= 0. The second condition in (5.5) yields\nu11(x) = 0, u12(x) =x.\nHence in the first case we have from (5.2) u′′\n01(x) = 0, hence we have u01(x) =cx+d.\nBy the first condition in (5.5) we have d= 0, by the second one u01(x) =x.\nIn the second case ( u12(x) =x) we obtain from (5.2) u′′\n02(x) = 2a(x)x. Hence with\nthe use of the second condition in (5.5) we have u′\n02(x) =/integraltextx\n02a(s)sdsand with the\nuse of the first condition in (5.5) we have u02(x) =/integraltextx\n0/integraltextq\n02a(s)sdsdq.\nTo sum up,\ny1(x) =/parenleftbigg\nx/integraltextx\n0/integraltextq\n02a(s)sdsdq\n0 x/parenrightbigg\nand hence det y1(T) =T2. We obtained\nDetA=−4T2.\n6.The determinant of H\nLet us consider the operator Hin the general case. In each horizontal strip |Imλ|<\nKthere are finitely many eigenvalues. Hence there exists a positive εso that there\nare no eigenvalues whose arguments are in the intervals (0 ,ε], [π−ε,π), (π,π+ε],\nand [2π−ε,2π). We will choose the cut of the logarithm as the half-line λ=tei(π−ε),\nt∈[0,∞). We denote the eigenvalues of Hin the upper half-plane by ˜ µj, and their\ncomplex conjugates by ¯˜µj, the positive real eigenvalues by ˜ νj,j∈I1(hereI1is a finite8 DETERMINANT FOR THE DAMPED WAVE EQUATION\nindex set) and the negative real eigenvalues as −˜ωj,j∈I2(hereI2is a finite index\nset). It can be easily proven that there are no zero eigenvalues of H.\nFor the operator A, we denote the eigenvalues as µ2\nj, ¯µ2\nj,ν2\njandω2\nj. We consider the\nphases of all these eigenvalues in the interval [0 ,2π−2ε), the cut of the logarithm for\nthe operator Ais the half-line λ=te−2iε. Hence one can easily obtain\nlog ˜µj=1\n2logµ2\nj,log¯˜µj=1\n2log ¯µ2\nj−πi,\nlog˜νj=1\n2logν2\nj,log(−˜ωj) =1\n2logω2\nj−πi.\nThe zeta functions for these operators are\nζA(s) =∞/summationdisplay\nj=1[(µ2\nj)−s+(¯µ2\nj)−s]+/summationdisplay\nj∈I1(ν2\nj)−s+/summationdisplay\nj∈I2((−ωj)2)−s\n=∞/summationdisplay\nj=1(e−slogµ2\nj+e−slog ¯µ2\nj)+/summationdisplay\nj∈I1e−slogν2\nj+/summationdisplay\nj∈I2e−slogω2\nj.\nζH(s) =∞/summationdisplay\nj=1[˜µ−s\nj+¯˜µj−s]+/summationdisplay\nj∈I1˜ν−s\nj+/summationdisplay\nj∈I2(−˜ωj)−s\n=∞/summationdisplay\nj=1(e−slog ˜µj+e−slog¯˜µj)+/summationdisplay\nj∈I1e−slog ˜νj+/summationdisplay\nj∈I2e−slog(−˜ωj)\n=∞/summationdisplay\nj=1(e−1\n2slogµ2\nj+e−1\n2slog ¯µ2\njeπis)+/summationdisplay\nj∈I1e−1\n2slogν2\nj+/summationdisplay\nj∈I2e−1\n2slogω2\njeπis.\nHence we have\nζH(s)−ζA/parenleftBigs\n2/parenrightBig\n= (eπis−1)/parenleftBigg∞/summationdisplay\nj=1e−1\n2slog ¯µ2\nj+/summationdisplay\nj∈I2e−1\n2slogω2\nj/parenrightBigg\n= 2ieπis\n2sinπs\n2/parenleftBigg∞/summationdisplay\nj=1e−1\n2slog ¯µ2\nj+/summationdisplay\nj∈I2e−1\n2slogω2\nj/parenrightBigg\n.\nFor the derivatives at zero we obtain, using the fact that the sine w ill vanish in that\ncase,\nζ′\nH(0)−1\n2ζ′\nA(0) =iπlim\ns→0∞/summationdisplay\nj=1e−1\n2slog ¯µ2\nj+iπcardI2\n=iπcardI2+iπlim\ns→0∞/summationdisplay\nj=1e−iπs\n2eslogT\nπe−slogj\n+iπlim\ns→0∞/summationdisplay\nj=1/parenleftBig\ne−1\n2slog ¯µ2\nj−e−iπs\n2eslogT\nπe−slogj/parenrightBig\n.\nWewillprove thatthelasttermiszero. Wewillusetheasymptotics (s eee.g. [BF09])\n¯µ2\nj=−j2π2\nT2/parenleftbigg\n1+O/parenleftbigg1\nj/parenrightbigg/parenrightbiggDETERMINANT FOR THE DAMPED WAVE EQUATION 9\nand by our assumption ¯ µ2\njis not close to zero. We have\nlim\ns→0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay\nj=1(e−1\n2slog ¯µ2\nj−e−iπs\n2eslogT\nπe−slogj)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= lim\ns→0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay\nj=1e−iπs\n2eslogT\nπe−slogj\n×(e−1\n2slog(1+O(1 /j))−1)/vextendsingle/vextendsingle/vextendsingle\n≤lim\ns→0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay\nj=1e−slogj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle|eslogT\nπ||e−sC−1|\n= lim\ns→0|ζR(s)||eslogT\nπ||e−sC−1|\n= 0.\nwhereCis a real constant and ζR(0) =−1\n2. Hence we have\nζ′\nH(0)−1\n2ζ′\nA(0) =iπcardI2+iπζR(0) =iπcardI2−iπ\n2. (6.1)\nThen the determinant is equal to\nDetH= e−ζ′\nH(0)= e−1\n2ζ′\nA(0)eiπ\n2e−iπcardI2=∓i√\nDetA=∓i√\n−4T2=±2T .\nThe number of negative real eigenvalues card I2is always even, so this term does\nnot influence the result. The aim of the rest of our analysis is to find t he sign of the\ndeterminant for the case of the cut we chose. First, we will follow an approach different\nfrom that which was used in Section 3 to find the determinant for the operator without\ndamping. The zeta function for the operator A0is\nζA0(s) = 2∞/summationdisplay\nj=1/parenleftbiggjπ\nT/parenrightbigg−2s\n(−1)−s= 2/parenleftbiggT\nπ/parenrightbigg2s\ne−slog(−1)∞/summationdisplay\nj=1j−2s= 2e−iπs/parenleftbiggT\nπ/parenrightbigg2s\nζR(2s).\nThen we have\n−ζ′\nA0(0) = 2πiζR(0)−4logT\nπζR(0)−4ζ′\nR(0) =−πi+2log(2 T).\nUsing equation (6.1) we obtain\nDetH0= e−ζH′\n0(0)= e−1\n2ζA′\n0(0)eiπ\n2= e−iπ\n2eiπ\n2elog(2T)= 2T .\nNow, we are going to generalize this approach to the operator with d amping. Our\naim is to find the imaginary part of ζ′\nH0(0). We denote the absolute value of µ2\njbyrj\nand its phase by π−ϕj. We have\nµ2\nj=rjei(π−ϕj),¯µ2\nj=rjei(π+ϕj).\nWe obtain\n(µ2\nj)−s+(¯µ2\nj)−s=r−s\nje−iπs(eiϕjs+e−iϕjs) = 2e−iπsr−s\njcos(ϕjs)\nand since r−s\njcos(ϕjs),ν2\njandω2\njare real\nImd\nds/braceleftBigg∞/summationdisplay\nj=1/bracketleftbig\n(µ2\nj)−s+(¯µ2\nj)−s/bracketrightbig\n+/summationdisplay\nj∈I1(ν2\nj)−s+/summationdisplay\nj∈I1(ω2\nj)−s/bracerightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ns=0=−2πlim\ns→0∞/summationdisplay\nj=0r−s\njcos(ϕjs)\n=−πlim\ns→0∞/summationdisplay\nj=1/bracketleftbig\n(µ2\nj)−s+(¯µ2\nj)−s/bracketrightbig10 DETERMINANT FOR THE DAMPED WAVE EQUATION\nWe will use the asymptotic behaviour of the eigenvalues of the opera torH(see\n[BF09])\nµj=jπ\nTi−/a\\}bracketle{ta/a\\}bracketri}ht−i/a\\}bracketle{ta2/a\\}bracketri}htT\n2πj+O/parenleftbigg1\nj2/parenrightbigg\n(here/a\\}bracketle{t·/a\\}bracketri}htdenotestheaverageofthefunctionontheinterval)whichleadsto thefollowing\nbehaviour of the eigenvalues of the operator A\nµ2\nj=−j2π2\nT2−2jπ/a\\}bracketle{ta/a\\}bracketri}ht\nTi+O(1).\nHence we obtain\n(µ2\nj)−s= (−1)−sj−2sT2s\nπ2s/parenleftbigg\n1−2T/a\\}bracketle{ta/a\\}bracketri}hts\nπji+O(j−2)/parenrightbigg\n.\nSince ¯µ2\njis the complex conjugate of µ2\nj, we have\n∞/summationdisplay\nj=1/bracketleftbig\n(µ2\nj)−s+(¯µ2\nj)−s/bracketrightbig\n= 2T2s\nπ2s(−1)−s∞/summationdisplay\nj=0j−2s+∞/summationdisplay\nj=1O(j−2s−2).\nThesecondseriesisabsolutelyconvergent downtoRe s= 0,andhencewecanexchange\nthe sum and the limit. Moreover, the term under the sum goes to zer o ass→0, as it\nis multiplied by s. The first term can be written as 2e−iπsT2s\nπ2sζR(2s), hence its limit is\nequal to −1. We conclude that\n−Imζ′\nA(0) =−π.\nUsing equation (6.1) similarly to the case of no damping leads to the pos itive sign for\nthe determinant with the cut of the logarithm taken just above the negative real axis.\nThis proves Theorem 2.1.\nRemark 6.1. It is possible prove in the similar manner that if one moves th e cut so\nthat it passes finitely many eigenvalues of H, the determinant does not change.\nRemark 6.2. If one considers the dampedwave equation with a potential te rm, namely,\n∂2v(t,x)\n∂t2+2a(x)∂v(t,x)\n∂t=∂2v(t,x)\n∂x2+b(x)v(t,x),\nthen it is possible to use a similar approach to the above. If t he operator on the right-\nhand side has only negative eigenvalues, the negative eigen values of the damped wave\nequation always appear in pairs, as in the case without the po tential. The only difference\nis in the solution of the Cauchy problem. We define y(x)satisfying y′′(x)+b(x)y(x) = 0,\ny(0) = 0,y′(0) = 1. The result for the determinant is DetH= 2y(T)for the cut of\nthe logarithm just above the negative real axis, and −2y(T)for the opposite case. If the\noperator on the right-hand side also has positive eigenvalu es (but no zero eigenvalue),\nthe phase (−1)cardI2appears multiplying the determinant, changing the sign acc ording\nto whether the number of negative eigenvalues of the damped w ave equation is odd or\neven.DETERMINANT FOR THE DAMPED WAVE EQUATION 11\nAcknowledgements\nP.F. was partially supported by the Funda¸ c˜ ao para a Ciˆ encia e a T ecnologia, Portu-\ngal, through project PTDC/MAT-CAL/4334/2014. J.L. was suppo rted by the project\n“International mobilities for research activities of the University o f Hradec Kr´ alov´ e”\nCZ.02.2.69/0.0/0.0/16 027/0008487. 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DOI: 10.1142/9789812810571 0004."    },    {        "title": "2304.11522v1.Decay_rates_for_a_variable_coefficient_wave_equation_with_nonlinear_time_dependent_damping.pdf",        "content": "arXiv:2304.11522v1  [math.AP]  23 Apr 2023Decay rates for a variable-coefficient wave equation\nwith nonlinear time-dependent damping\nMenglan Liao∗\nCollege of Science, Hohai University, Nanjing, 210098, China\nAbstract: In this paper, a class of variable-coefficient wave equations equipped with time-\ndependent damping and the nonlinear source is considered. W e show that the total energy\nof the system decays to zero with an explicit and precise deca y rate estimate under different\nassumptions on the feedback with the help of the method of wei ghted energy integral.\nKeywords: Variable coefficient; Time-dependent damping; Energy estim ates; Nonlinear\nsource.\nMSC(2020): 35B35; 35L70; 35A01.\n1 Introduction\nIn this paper, we are concerned with the following initial-b oundary value problem for a class\nof variable-coefficient wave equations with time-dependent damping\n\n\nutt−Au+γ(t)g(ut) =f(u)x∈Ω, t>0,\nu(x,t) = 0 x∈∂Ω, t>0,\nu(x,0) =u0(x), ut(x,0) =u1(x)x∈Ω,(1.1)\nwhere Ω ⊂R3, for simplicity, is a bounded domain with C∞boundary. In particular, present\nresults can be extended to bounded domain in Rnby adopting the Sobolev embeddings, and\nby adjusting some parameters imposed. Here γ(t)g(ut) is a time-dependent damping term, and\nf∈C1(R) such that |f′(s)| ≤C(|s|p−1+ 1) with 1 ≤p <6. To simplify computations, we\nchoosef(u) :=|u|p−1uwith 1≤p<6 in this paper. The second-order differential operator A\nis defined by\nAu= div(A(x)∇u) =∞/summationdisplay\ni,j=1∂\n∂xi/parenleftBig\naij(x)∂u\n∂xj/parenrightBig\n,\nwhereA(x) = (ai,j(x)) is symmetric and positive definite matrices with ai,j(x)∈C∞(¯Ω) and\nsatisfies the uniform ellipticity conditions\nn/summationdisplay\ni,j=1ai,j(x)ξiξj≥ωn/summationdisplay\ni=1ξ2\ni, x∈¯Ω, ω>0. (1.2)\nIt is easily verified that the bilinear form a(·,·) :H1\n0(Ω)×H1\n0(Ω)→Rdefined by\na(u,v) =n/summationdisplay\ni,j=1/integraldisplay\nΩai,j(x)∂u\n∂xj∂u\n∂xidx=/integraldisplay\nΩA∇u·∇vdx\n∗Corresponding author: Menglan Liao\nEmail addresses: liaoml@hhu.edu.cn2\nis symmetric and continuous. Further, it follows from ( 1.2) that\na(u,v)≥ω/ba∇dbl∇u/ba∇dbl2\n2. (1.3)\nThe problem of asymptotic stability of solutions of dissipa tive wave systems equipped with\ntime-dependent nonlinear damping forces has been given a lo t of attention. P. Pucci and J.\nSerrin [11] investigated the following nonlinear damped wave system w ith Dirichlet data\nutt−∆u+Q(t,x,u,ut)+f(x,u) = 0, (1.4)\nwhere the function Qrepresents a nonlinear damping satisfying ( Q(t,x,u,v),v)≥0,fis a\nrestoring force (that is (f(x,u),u)≥0). Based on the a priori existence of a suitable auxiliary\nfunction, they proved the natural energy Eu(t) associated with solutions of the system satisfies\nlimt→+∞Eu(t) = 0. Problem ( 1.4) is well-known in a variety of models in mathematical physic s,\nforinstance, elasticvibrationsinadissipativemedium,t hetelegraphicequation, andthedamped\nKlein-Gordon equation. P. Pucci and J. Serrin [ 13] studied again problem ( 1.4) not only for\npotential energies which arise from restoring forces, but a lso for the effect of amplifying forces .\nThey pointed out that global asymptotic stability can no lon ger be expected, and should be\nreplaced by local stability. However, whether an explicit a nd precise decay rate estimate of\nthe total energy of the system can be obtained was unknown. P. Martinez [ 10] considered the\nfollowing time-dependent dissipative systems\nutt−∆u+ρ(t,ut) = 0\nwith Dirichlet boundary, where ρ:R+×R→Ris a continuous function differentiable on\nR+×(−∞,0). By generalizing the method introduced to study autonomo us wave equation\ndamped with a boundary nonlinear velocity feedback ρ(ut) in [9], P. Martinez obtained that\nthe total energy of the system decays to zero with an explicit and precise decay rate estimate\nunder sharp assumptions on the feedback. M. Daoulatli [ 1] studied problem ( 1.1) withoutf(u),\nand they illustrated that if given suitable conditions on th e nonlinear terms, and the damping\nis modeled by a continuous monotone function without any gro wth restrictions imposed at the\norigin and infinity, the decay rate of the energy functional w as obtained by solving a nonlinear\nnon-autonomous ODE. The Cauchy problem for second order hyp erbolic evolution equations\nwith restoring force in a Hilbert space, under the effect of non linear time-dependent damping\nhas also been studied, the interested readers can refer to pa pers [5–8].\nIt is certainly beyond the scope of the present paper to give a comprehensive review for\ndissipative wave systems equipped with time-dependent non linear damping forces. However,\nthe literature on energy decay estimates is rare for the weak solution of the variable coefficient\nwave equations when it is concerned with the interaction bet ween time-dependent damping and\nsource term. Inspired by the paper [ 10], in this paper, the primary goal is to establish the energy\ndecay rates when the time-dependent damping satisfies differe nt assumptions. Because of the\nthe interaction between time-dependent damping and source term, some new difficulties need to\nbe solved. The outline of the paper is as follows. In Section 2, we shall give some assumptions,\nmain results and several remarks. In Section 3, we prove that the local (in time) existence of\nthe weak solution can be extended globally. Sections 4and5are used to prove the energy decay\nrates.\n2 Preliminaries and main results\nThroughout the paper, denote by Mthe optimal embedding constant for the embedding\ntheoremH1\n0(Ω)֒→Lp+1(Ω).The symbol Cis a generic positive constant, which may bedifferent3\nin various positions. Ci(i= 0,1,2,···,33) represent some positive constants. We impose the\nfollowing assumptions:\n(H1)γ∈W1,∞\nloc(R+) is bounded and nonnegative on R+= [0,+∞).\n(H2)gis continuous and monotone increasing feedback with g(0) = 0. In addition, there\nexist positive constants b1andb2such that\nb1|s|m+1≤g(s)s≤b2|s|m+1,wherem≥1 and|s|>1.\n(H3)pm+1\nm<6.\n(H4)u0(x)∈H1\n0(Ω), u1(x)∈L2(Ω).\nDefinition 2.1 (Weak solution) .A functionu(t) :=u(t,x) is said to a weak solution of problem\n(1.1) on [0,T] ifu∈C([0,T];H1\n0(Ω)) withut∈C([0,T];L2(Ω))∩Lm+1(Ω×(0,T)). In addition,\nfor allt∈[0,T],\n(ut(t),φ(t))−(ut(0),φ(0)) +/integraldisplayt\n0a(u(s),φ(s))ds−/integraldisplayt\n0/integraldisplay\nΩut(s)φt(s)dxds\n+/integraldisplayt\n0/integraldisplay\nΩγ(t)g(ut(s))φt(s)dxds=/integraldisplayt\n0/integraldisplay\nΩf(u(s))φ(s)dxds(2.1)\nforφ∈ {φ:φ∈C([0,T];H1\n0(Ω))∩Lm+1(Ω×(0,T)) withφt∈C([0,T];L2(Ω))}.\nTheorem 2.2 (Local existence) .Let(H1)−(H4)hold, then there exists a local (in time)\nweak solutions u(t)to problem (1.1)forT >0depending on the initial quadratic energy E(0).\nMoreover, the following identity holds\nE(t)+/integraldisplayt\n0/integraldisplay\nΩγ(t)g(ut)utdxds=E(0)+/integraldisplayt\n0/integraldisplay\nΩf(u(s))ut(s)dxds (2.2)\nwith the quadratic energy is defined by\nE(t) =1\n2/ba∇dblut(t)/ba∇dbl2\n2+1\n2a(u(t),u(t)). (2.3)\nThe following result illustrates that when the damping domi nates the source, then the solu-\ntion is global in time.\nTheorem 2.3 (Global existence I) .In addition to (H1)−(H4), ifu0(x)∈Lp+1(Ω)andm≥p,\nthen the weak solution of problem (1.1)is global in time.\nTheorem 2.2can be proved directly by employing the theory of monotone op erators and\nnonlinear semigroups combined with energy methods. Theore m2.3also follows from a standard\ncontinuation argument from ODE theory. The interested read ers can follow from line to line\nas shown in [ 2](see also a recent paper [ 3]) with slight differences to achieve it. The proof of\nTheorem 2.2and Theorem 2.3is not the point in question, so we omit it.\nForu∈Lp+1(Ω), define the total energy E(t) by\nE(t) :=E(t)−1\np+1/ba∇dblu/ba∇dblp+1=1\n2/ba∇dblut(t)/ba∇dbl2\n2+1\n2a(u(t),u(t))−1\np+1/ba∇dblu/ba∇dblp+1\np+1.(2.4)\nThen (2.2) can be transferred as\nE(t)+/integraldisplayt\n0/integraldisplay\nΩγ(t)g(ut)utdxds=E(0). (2.5)\nFurther, we obtain the total energy E(t) is monotone decreasing in time, and\nE′(t) =−/integraldisplay\nΩγ(t)g(ut)utdx. (2.6)4\nTheorem 2.4 (Global existence II) .Let1<p≤5and(H1)−(H4)hold. Assume\n0<E(0)</parenleftBig1\n2−1\np+1/parenrightBig/parenleftBigωp+1\n2\nMp+1/parenrightBig2\np−1, a(u0,u0)</parenleftBigωp+1\n2\nMp+1/parenrightBig2\np−1,\nthen the weak solution of problem (1.1)is global.\nWe need to impose additional conditions to discuss the energ y decay rates.\n(H5)γ:R+→R+a nonincreasing function of class C1and/integraltext∞\n0γ(t)dt=∞.\n(H6) There exists a strictly increasing and odd function h∈C1[1,1] such that\n|h(s)s| ≤g(s)s≤ |h−1(s)s|,where|s| ≤1,\nwhereh−1is the inverse function of h.\nTheorem 2.5 (Energy decay rates I) .In addition to all conditions of Theorem 2.4,(H5)and\n(H6), we assume 1≤m≤5. Then for all t≥0\n(1)ifh(s)is linear,\nE(t)≤E(0)e1−C/integraltextt\n0γ(s)ds. (2.7)\n(2)ifh(s)has polynomial growth,\nE(t)≤CE(0)/parenleftBigg\n1\n1+/integraltextt\n0γ(s)ds/parenrightBigg2\nm−1\nwithm>1. (2.8)\nWhenh(s) does not necessarily have polynomial growth, the energy de cay rates still can be\nobtained if we replace ( H5) by the following\n(H′\n5)γ(t)≥γ0>0, whereγ0is constant.\nTheorem 2.6 (Energy decay rates II) .In addition to all conditions of Theorem 2.4,(H′\n5)and\n(H6), we assume 1≤m≤5. Then for all t≥1\nE(t)≤CE(0)/parenleftbigg\nH−1/parenleftBig1\nt/parenrightBig/parenrightbigg2\n. (2.9)\nHereH(s) :=h(s)s.\nRemark 2.7.In the proof of Theorem 2.5, we only discuss the energy decay rates for m≤5, that\nis, the nonlinear damping is subcritical andcritical. The restraint results from the embedding\ntheoremH1\n0(Ω)֒→Lm+1(Ω). A recent paper [ 4] investigated the energy decay estimates for\nthe automous wave equation with supercritical nonlinear damping in the absence of the driving\nsource. Inspired by the paper [ 4], we reasonably conjecture that if there exists κ(t) satisfying\nsome suitable conditions, then\nE(t)≤CE(0)/parenleftBigg\n1\n1+/integraltextt\n0κ(t)γ(s)ds/parenrightBigg2\nm−1\n,\nform>5, that is the supercritical nonlinear damping.5\nRemark2.8.Inthis paper, wediscussthe variable-coefficient wave equat ion with nonlineartime-\ndependent damping and nonlinear source for standard growth conditions. However, the energy\ndecay rates for wave systems with nonstandard growth condit ion can be of equal importance.\nBy following this paper, it is possible to discuss the energy decay rates to the initial-boundary\nvalue problem\n\n\nutt−∆u+γ(t)|ut|m(x)−2ut=|u|p(x)−2uin Ω×(0,T),\nu(x,t) = 0 on ∂Ω×(0,T),\nu(x,0) =u0(x), ut(x,0) =u1(x) in Ω .\nRemark 2.9.The rest field u(t,x) = 0 will be called asymptotically stable in the mean , or simply\nasymptotically stable , if and only if\nlim\nt→∞E(t) = 0 for all solutions u(t) :=u(t,x) of problem ( 1.1).\nThis concept was proposed first by P. Pucci and J. Serrin [ 12]. Obviously, based on Theorem\n2.3or Theorem 2.4, by Lemma 3.2, we obtain lim t→∞E(t) = 0.Hence, the rest field u(x,t) = 0\nis asymptotically stable.\n3 Proof of Theorem 2.4\nLet us introduce a function Fas follows\nF(s) =1\n2s−Mp+1\n(p+1)ωp+1\n2sp+1\n2. (3.1)\nBy a direct computation, the function Fsatisfies that\n1.F(0) = 0;\n2. lims→+∞F(s) =−∞;\n3.Fis strictly increasing in (0 ,s1), and is strictly decreasing in ( s1,+∞);\n4.Fhas a maximum at s1with the maximum value F1. Here\ns1=/parenleftBigωp+1\n2\nMp+1/parenrightBig2\np−1,F1=/parenleftBig1\n2−1\np+1/parenrightBig\ns1.\nLemma 3.1. Ifu(t)is a solution for problem (1.1)andE(0)<F1, a(u0,u0)<s1,then there\nexists a positive constant s2satisfying 0<s2<s1such that\na(u(t),u(t))≤s2for allt≥0. (3.2)\nProof.Using (1.3) and (2.1), the embedding theorem H1\n0(Ω)֒→Lp+1(Ω), we obtain\nE(t)≥1\n2a(u(t),u(t))−1\np+1/ba∇dblu/ba∇dblp+1\np+1≥1\n2a(u(t),u(t))−Mp+1\np+1/ba∇dbl∇u/ba∇dblp+1\n2\n≥1\n2a(u(t),u(t))−Mp+1\n(p+1)ωp+1\n2[a(u(t),u(t))]p+1\n2:=F(a(u(t),u(t))).(3.3)6\nSinceE(0)<F1, there exists a s2< s1such that F(s2) =E(0). It follows from ( 3.3)\nthatF(a(u0,u0))≤E(0) =F(s2), which implies a(u0,u0)≤s2due to the given condition\na(u0,u0)< s1. To complete the proof of ( 3.2), we suppose by contradiction that for some\nt0>0,a(u(t0),u(t0))>s2.The continuity of a(u(t),u(t)) illustrates that we may choose t1such\nthats1>a(u(t1),u(t1))>s2, then we have E(0) =F(s2)<F(a(u(t1),u(t1)))≤E(t1).This is\na contradiction for E(t) is nonincreasing.\nLemma 3.2. Under all the conditions of Lemma 3.1,for allt≥0,the following holds\n0≤E(t)≤C0E(t)≤C0E(0). (3.4)\nProof.Similar to ( 3.3), and then using ( 3.2) and (2.4), then\n1\np+1/ba∇dblu/ba∇dblp+1\np+1≤Mp+1\np+1/ba∇dbl∇u/ba∇dblp+1\n2≤Mp+1\n(p+1)ωp+1\n2[a(u(t),u(t))]p+1\n2\n=Mp+1\n(p+1)ωp+1\n2[a(u(t),u(t))]p−1\n2a(u(t),u(t))\n≤Mp+1\n(p+1)ωp+1\n2sp−1\n2\n2/parenleftBig\n2E(t)+2\np+1/ba∇dblu/ba∇dblp+1\np+1/parenrightBig\n,\nwhich indicates\n/ba∇dblu/ba∇dblp+1\np+1≤ ME(t)≤ ME(0), (3.5)\nby recalling E(t) is monotone decreasing. Here\nM:=2Mp+1\nωp+1\n2sp−1\n2\n2\n1−2Mp+1\n(p+1)ωp+1\n2sp−1\n2\n2<2Mp+1\nωp+1\n2sp−1\n2\n1\n1−2Mp+1\n(p+1)ωp+1\n2sp−1\n2\n1=2(p+1)\np−1.\nCombining ( 3.5) with (2.4), we directly obtain\nE(t) =E(t)+1\np+1/ba∇dblu/ba∇dblp+1≤C0E(t)≤C0E(0),\nwhich yields ( 3.4).\nIt follows from Lemma 3.2that the weak solution u(t) of problem ( 1.1) exists globally, that\nisT=∞.\nRemark 3.3.By the well-known potential well theory, we can also prove th e global existence.\nIn fact,F1is equal to the potential well depth(the mountain pass level )ddefined by\nd:= inf\nu∈H1\n0(Ω)\\{0}sup\nλ≥0J(λu),whereJ(u) :=1\n2a(u,u)−1\np+1/ba∇dblu/ba∇dblp+1\np+1.\n4 Proof of Theorem 2.5\nThe proof of Theorem 2.5relies on the following crucial lemma. We refer to [ 9] for the\ndetailed proof.7\nLemma 4.1 ([9]).LetE:R+→R+be a non-increasing function and ψ:R+→R+be a\nstrictly increasing function of class C1such that\nψ(0) = 0andψ(t)→+∞ast→+∞.\nAssume that there exist σ≥0,σ′≥0,c≥0andω>0such that :\n/integraldisplay+∞\ntE1+σ(s)ψ′(s)ds≤1\nωEσ(0)E(t)+c\n(1+ψ(s))σ′Eq(0)E(s)\nthenEhas the following decay property :\n(1)ifσ=c= 0,thenE(t)≤E(0)e1−ωψ(t)for allt≥0;\n(2)ifσ>0,then there exists C >0such thatE(t)≤CE(0)(1+ψ(t))−1+σ′\nσfor allt≥0.\nInwhatfollows, letusproveTheorem 2.5. Itdosemakesensetomultiply( 1.1)byEβ(t)φ′(t)u(t)\nand to integrate over Ω ×[S,T]. Hereφ:R+→R+is a concave nondecreasing function of class\nC2, such that φ′is bounded, and β≥0 is constant. Then we obtain\n/integraldisplayT\nSEβ(t)φ′(t)/integraldisplay\nΩu(t)[utt(t)−Au(t)+γ(t)g(ut(t))]dxdt=/integraldisplayT\nSEβ(t)φ′(t)/ba∇dblu(t)/ba∇dblp+1\np+1dt.(4.1)\nIntegrating by parts yields\n/integraldisplayT\nSEβ(t)φ′(t)/integraldisplay\nΩu(t)utt(t)dxdt=Eβ(t)φ′(t)/integraldisplay\nΩu(t)ut(t)dx/vextendsingle/vextendsingle/vextendsingleT\nS−/integraldisplayT\nSEβ(t)φ′(t)/ba∇dblut(t)/ba∇dbl2\n2\n−/integraldisplayT\nS[βEβ−1(t)E′(t)φ′(t)+Eβ(t)φ′′(t)]/integraldisplay\nΩu(t)ut(t)dxdt.(4.2)\nSubstituting ( 4.2) into (4.1), one obtains\nEβ(t)φ′(t)/integraldisplay\nΩu(t)ut(t)dx/vextendsingle/vextendsingle/vextendsingleT\nS−/integraldisplayT\nSEβ(t)φ′(t)/ba∇dblut(t)/ba∇dbl2\n2dt\n−/integraldisplayT\nS[βEβ−1(t)E′(t)φ′(t)+Eβ(t)φ′′(t)]/integraldisplay\nΩu(t)ut(t)dxdt\n+/integraldisplayT\nSEβ(t)φ′(t)a(u(t),u(t))dt\n+/integraldisplayT\nSEβ(t)φ′(t)γ(t)/integraldisplay\nΩu(t)g(ut(t))dxdt=/integraldisplayT\nSEβ(t)φ′(t)/ba∇dblu(t)/ba∇dblp+1\np+1dt.(4.3)\nIt follows from ( 2.4) that (4.3) can be written as\n2/integraldisplayT\nSEβ+1(t)φ′(t)dt=−Eβ(t)φ′(t)/integraldisplay\nΩu(t)ut(t)dx/vextendsingle/vextendsingle/vextendsingleT\nS+2/integraldisplayT\nSEβ(t)φ′(t)/ba∇dblut(t)/ba∇dbl2\n2dt\n+/integraldisplayT\nS[βEβ−1(t)E′(t)φ′(t)+Eβ(t)φ′′(t)]/integraldisplay\nΩu(t)ut(t)dxdt\n+p−1\np+1/integraldisplayT\nSEβ(t)φ′(t)/ba∇dblu(t)/ba∇dblp+1\np+1dt−/integraldisplayT\nSEβ(t)φ′(t)γ(t)/integraldisplay\nΩu(t)g(ut(t))dxdt\n=J1+J2+J3+J4+J5.(4.4)8\nIn what follows, let us estimate every term on the right hand s ide above.\nUsing the embedding theorem H1\n0(Ω)֒→L2(Ω), (1.3) and (3.4) yields\n/ba∇dblu(t)/ba∇dbl2≤M/ba∇dbl∇u(t)/ba∇dbl2≤M/bracketleftBiga(u(t),u(t))\nω/bracketrightBig1\n2≤C1E1\n2(t). (4.5)\nApplying Cauchy’s inequality, the boundedness of φ′(t)(we can denote by θ), (4.5) and (3.4),\nwe arrive at\n|J1|=/vextendsingle/vextendsingle/vextendsingle−Eβ(t)φ′(t)/integraldisplay\nΩu(t)ut(t)dx/vextendsingle/vextendsingle/vextendsingleT\nS/vextendsingle/vextendsingle/vextendsingle\n≤θEβ(S)/bracketleftbig\n/ba∇dblut(T)/ba∇dbl2/ba∇dblu(T)/ba∇dbl2+/ba∇dblut(S)/ba∇dbl2/ba∇dblu(S)/ba∇dbl2/bracketrightbig\n≤C2Eβ+1(S),(4.6)\nmoreover,\n|J3|=/vextendsingle/vextendsingle/vextendsingle/integraldisplayT\nS[βEβ−1(t)E′(t)φ′(t)+Eβ(t)φ′′(t)]/integraldisplay\nΩu(t)ut(t)dxdt/vextendsingle/vextendsingle/vextendsingle\n≤ −θβ/integraldisplayT\nSEβ(t)E′(t)dt+C3/integraldisplayT\nSEβ+1(t)(−φ′′(t))dt\n≤θβ\nβ+1Eβ+1(S)+C4Eβ+1(S) =C5Eβ+1(S).(4.7)\nIt follows from ( 3.5) that\n|J4|=p−1\np+1/integraldisplayT\nSEβ(t)φ′(t)/ba∇dblu(t)/ba∇dblp+1\np+1dt≤p−1\np+1M/integraldisplayT\nSEβ+1(t)φ′(t)dt. (4.8)\nUsing Young’s inequality with ε1>0, (4.5), one has\n/integraldisplay\n|ut(t)|≤1u(t)g(ut(t))dx≤ε1\n2/ba∇dblu(t)/ba∇dbl2\n2+1\n2ε1/integraldisplay\n|ut(t)|≤1|g(ut(t))|2dx\n≤C6ε1E(t)+1\n2ε1/integraldisplay\n|ut(t)|≤1|g(ut(t))|2dx.\nIt follows from Young’s inequality with ε1>0, the embedding theorem H1\n0(Ω)֒→Lm+1(Ω),\n(1.3) and (3.4) that\n/integraldisplay\n|ut(t)|>1u(t)g(ut(t))dx≤ε2\nm+1/ba∇dblu(t)/ba∇dblm+1\nm+1+mε−1\nm\n2\nm+1/integraldisplay\n|ut(t)|>1|g(ut(t))|m+1\nmdx\n≤Mm+1ε2\nm+1/ba∇dbl∇u(t)/ba∇dblm+1\n2+mε−1\nm\n2\nm+1/integraldisplay\n|ut(t)|>1|g(ut(t))|m+1\nmdx\n≤Mm+1ε2\nm+1/bracketleftBiga(u(t),u(t))\nω/bracketrightBigm+1\n2+mε−1\nm\n2\nm+1/integraldisplay\n|ut(t)|>1|g(ut(t))|m+1\nmdx\n≤C7ε2E(t)+mε−1\nm\n2\nm+1/integraldisplay\n|ut(t)|>1|g(ut(t))|m+1\nmdx.9\nTherefore, by recalling the boundedness of γ(t)(we can denote by c1), then\n|J5|=/vextendsingle/vextendsingle/vextendsingle−/integraldisplayT\nSEβ(t)φ′(t)γ(t)/integraldisplay\nΩu(t)g(ut(t))dxdt/vextendsingle/vextendsingle/vextendsingle\n≤c1/integraldisplayT\nSEβ(t)φ′(t)/vextendsingle/vextendsingle/vextendsingle/integraldisplay\n|ut(t)|≤1u(t)g(ut(t))dx+/integraldisplay\n|ut(t)|>1u(t)g(ut(t))dx/vextendsingle/vextendsingle/vextendsingledt\n≤c1/integraldisplayT\nSEβ(t)φ′(t)/bracketleftBig\nC6ε1E(t)+1\n2ε1/integraldisplay\n|ut(t)|≤1|g(ut(t))|2dx\n+C7ε2E(t)+mε−1\nm\n2\nm+1/integraldisplay\n|ut(t)|>1|g(ut(t))|m+1\nmdx/bracketrightBig\ndt\n≤C8(ε1+ε2)/integraldisplayT\nSEβ+1φ′(t)dt+c1\n2ε1/integraldisplayT\nSEβ(t)φ′(t)/integraldisplay\n|ut(t)|≤1|g(ut(t))|2dx\n+c1mε−1\nm\n2\nm+1/integraldisplayT\nSEβ(t)φ′(t)/integraldisplay\n|ut(t)|>1|g(ut(t))|m+1\nmdxdt.(4.9)\nInserting ( 4.6)-(4.9) into (4.4) indicates\n/bracketleftBig\n2−p−1\np+1M−C8(ε1+ε2)/bracketrightBig/integraldisplayT\nSEβ+1(t)φ′(t)dt\n≤C9Eβ+1(S)+2/integraldisplayT\nSEβ(t)φ′(t)/ba∇dblut(t)/ba∇dbl2\n2dt\n+c1\n2ε1/integraldisplayT\nSEβ(t)φ′(t)/integraldisplay\n|ut(t)|≤1|g(ut(t))|2dxdt\n+c1mε−1\nm\n2\nm+1/integraldisplayT\nSEβ(t)φ′(t)/integraldisplay\n|ut(t)|>1|g(ut(t))|m+1\nmdxdt.(4.10)\nNote that 2 −p−1\np+1M>0, one gets\n2−p−1\np+1M−C8(ε1+ε2)>0\nfor sufficiently small positive constants ε1andε2.\n4.1 Case I: h(s)is linear\nLet usφ(t) =/integraltextt\n0γ(s)dsin this section. From ( H6), there exists two positive constant b3, b4\nsuch that\nb3s2≤g(s)s≤b4s2,where|s| ≤1. (4.11)10\nUsing (4.11), (H2) and (2.6) yields\n2/integraldisplayT\nSEβ(t)γ(t)/ba∇dblut(t)/ba∇dbl2\n2dt\n≤2/integraldisplayT\nSEβ(t)γ(t)/bracketleftBig/integraldisplay\n|ut(t)|≤1|ut(t)|2dx+/integraldisplay\n|ut(t)|>1|ut(t)|2dx/bracketrightBig\ndt\n≤2\nb3/integraldisplayT\nSEβ(t)/integraldisplay\nΩγ(t)g(ut(t))ut(t)dxdt+2/integraldisplayT\nSEβ(t)γ(t)/integraldisplay\n|ut(t)|>1|ut(t)|m+1dxdt\n≤2\nb3/integraldisplayT\nSEβ(t)/integraldisplay\nΩγ(t)g(ut(t))ut(t)dxdt+2\nb1/integraldisplayT\nSEβ(t)γ(t)/integraldisplay\nΩg(ut(t))ut(t)dxdt\n≤ −/bracketleftBig2\nb3+2\nb1/bracketrightBig/integraldisplayT\nSEβ(t)E′(t)dt≤C10Eβ+1(S).(4.12)\nBy using ( 4.11) and (2.6), we gets\nc1\n2ε1/integraldisplayT\nSEβ(t)γ(t)/integraldisplay\n|ut(t)|≤1|g(ut(t))|2dx\n≤c1b4\n2ε1/integraldisplayT\nSEβ(t)γ(t)/integraldisplay\n|ut(t)|≤1g(ut(t))ut(t)dx\n≤ −c1b4\n2ε1/integraldisplayT\nSEβ(t)E′(t)dt≤C11Eβ+1(S).(4.13)\nIt follows from ( H2) and (2.6) that\nc1mε−1\nm\n2\nm+1/integraldisplayT\nSEβ(t)γ(t)/integraldisplay\n|ut(t)|>1|g(ut(t))|m+1\nmdxdt\n=c1mε−1\nm\n2\nm+1/integraldisplayT\nSEβ(t)γ(t)/integraldisplay\n|ut(t)|>1|g(ut(t))|1\nm|g(ut(t))|dxdt\n≤c1mε−1\nm\n2b1\nm\n2\n(m+1)/integraldisplayT\nSEβ(t)γ(t)/integraldisplay\n|ut(t)|>1g(ut(t))ut(t)dxdt≤C12Eβ+1(S).(4.14)\nInserting ( 4.12)-(4.14) into (4.10), we easily deduce\n/integraldisplayT\nSEβ+1(t)γ(t)dt≤1\nC13Eβ(0)E(S). (4.15)\nWhenTgoes to + ∞, chooseβ= 0,ψ(t) =/integraltextt\n0γ(s)dsin Lemma 4.1, then we derive ( 2.7).\n4.2 Case II: h(s)has polynomial growth\nIn this subsection, we still choose φ(t) =/integraltextt\n0γ(s)ds. To simplicity, denote h(s) =b5|s|m−1s\nwithm>1,|s| ≤1, b5>0. From ( H6), there exists two positive constant b6, b7such that\nb6sm+1≤g(s)s≤b7|s|m+1\nm,where|s| ≤1. (4.16)11\nIt follows from H¨ older’s inequality, ( 4.16), Young’s inequality with ε3>0, (H2) and (2.6) yields\n2/integraldisplayT\nSEβ(t)γ(t)/ba∇dblut(t)/ba∇dbl2\n2dt\n≤2/integraldisplayT\nSEβ(t)γ(t)/bracketleftBig/integraldisplay\n|ut(t)|≤1|ut(t)|2dx+/integraldisplay\n|ut(t)|>1|ut(t)|2dx/bracketrightBig\ndt\n≤2|Ω|m−1\nm+1/integraldisplayT\nSEβ(t)γ(t)/parenleftBig/integraldisplay\n|ut(t)|≤1|ut(t)|m+1dx/parenrightBig2\nm+1dt+2\nb1/integraldisplayT\nSEβ(t)γ(t)/integraldisplay\nΩg(ut(t))ut(t)dxdt\n≤2|Ω|m−1\nm+11\nb2\nm+1\n6/integraldisplayT\nSEβ(t)γm−1\nm+1(t)/parenleftBig/integraldisplay\nΩγ(t)ut(t)g(ut(t))dx/parenrightBig2\nm+1dt−2\nb1/integraldisplayT\nSEβ(t)E′(t)dxdt\n≤C14ε3/integraldisplayT\nSEβ(m+1)\nm−1(t)γ(t)dt+C15ε−m−1\n2\n3/integraldisplayT\nS(−E′(t))dt+C16Eβ+1(S)\n≤C14ε3/integraldisplayT\nSEβ(m+1)\nm−1(t)γ(t)dt+C17E(S)+C16Eβ+1(S).\n(4.17)\nSimilar to ( 4.17), using the second inequality in ( 4.16), we obtain\nc1\n2ε1/integraldisplayT\nSEβ(t)γ(t)/integraldisplay\n|ut(t)|≤1|g(ut(t))|2dx\n≤c1\n2ε1|Ω|m−1\nm+1/integraldisplayT\nSEβ(t)γ(t)/parenleftBig/integraldisplay\n|ut(t)|≤1|g(ut(t))|m+1dx/parenrightBig2\nm+1dt\n≤/integraldisplayT\nSEβ(t)γm−1\nm+1(t)·c1\n2ε1|Ω|m−1\nm+1b2m\nm+1\n7/parenleftBig/integraldisplay\nΩγ(t)ut(t)g(ut(t))dx/parenrightBig2\nm+1dt\n≤C18ε4/integraldisplayT\nSEβ(m+1)\nm−1(t)γ(t)dt+C19ε−m−1\n2\n4/integraldisplayT\nS(−E′(t))dt\n≤C18ε4/integraldisplayT\nSEβ(m+1)\nm−1(t)γ(t)dt+C20E(S).(4.18)\nMoreover, ( 4.14) still holds in this case.\nInserting ( 4.14), (4.17) and (4.18) into (4.10), and choosing β=m−1\n2, we easily deduce\n/bracketleftBig\n2−p−1\np+1M−C8(ε1+ε2)−C21(ε3+ε4)/bracketrightBig/integraldisplayT\nSEβ+1(t)γ(t)dt≤C22Eβ+1(0)E(S).(4.19)\nWe also have\n2−p−1\np+1M−C8(ε1+ε2)−C21(ε3+ε4)>0\nfor sufficiently small positive constants ε1,ε2,ε3andε4.\nTherefore, we get/integraldisplayT\nSEβ+1(t)γ(t)dt≤1\nC23Eβ(0)E(S). (4.20)\nWhenTgoes to + ∞, note that β=m−1\n2,ψ(t) =/integraltextt\n0γ(s)dsin Lemma 4.1then we derive ( 2.8).125 Proof of Theorem 2.6\nTo prove Theorem 2.6, we need to the following lemma included in [ 9].\nLemma 5.1. The function φ: [1,+∞)→[1,∞)defined by\nφ(t) :=˜ψ−1(t) (5.1)\nis a strictly increasing concave function of class C2and satisfies\nlim\nt→+∞φ(t) = +∞,lim\nt→+∞φ′(t) = 0,\nand/integraltext∞\n1φ′(t)|h−1(φ′(t))|2dtconverges. Here\n˜ψ(t) := 1+/integraldisplayt\n11\nh(1/s)ds, (5.2)\nand the function hsatisfies (H6).\nIn this section, define\nχ(t) :=1\nφ(t)=1\n˜ψ−1(t). (5.3)\nNow we are in a position to estimate ( 4.10).\nFort≥1, set\nΩ1={x∈Ω :|ut(t)| ≤χ(t)};\nΩ2={x∈Ω :χ(t)<|ut(t)| ≤1};\nΩ3={x∈Ω :|ut(t)|>1}.\nUsing Lemma 5.1, (H6), (H′\n5) and (2.6), we gets\n/integraldisplayT\nSEβ(t)/integraldisplay\nΩ2φ′(t)|ut(t)|2dxdt\n=/integraldisplayT\nSEβ(t)/integraldisplay\nΩ2h(χ(t))|ut(t)|2dxdt≤/integraldisplayT\nSEβ(t)/integraldisplay\nΩ2h(ut(t))|ut(t)|2dxdt\n≤/integraldisplayT\nSEβ(t)1\nγ(t)/integraldisplay\nΩ2γ(t)g(ut(t))ut(t)dxdt≤1\nγ0Eβ+1(S).(5.4)\nIt follows from ( 2.6), (5.3) and Lemma 5.1that\n/integraldisplayT\nSEβ(t)φ′(t)/integraldisplay\nΩ1|ut(t)|2dxdt≤ |Ω|/integraldisplayT\nSEβ(t)φ′(t)χ2(t)dt\n≤ |Ω|Eβ(S)/integraldisplayT\nSφ′(t)χ2(t)dt≤ |Ω|Eβ(S)\nφ(S).(5.5)13\nRecalling ( 4.17), and using ( 5.4) and (5.5), one gets\n2/integraldisplayT\nSEβ(t)φ′(t)/ba∇dblut(t)/ba∇dbl2\n2dt\n≤2/integraldisplayT\nSEβ(t)φ′(t)/integraldisplay\n|ut(t)|≤1|ut(t)|2dxdt+2\nb1/integraldisplayT\nSEβ(t)φ′(t)\nγ(t)/integraldisplay\nΩ3γ(t)g(ut(t))ut(t)dxdt\n≤2/integraldisplayT\nSEβ(t)φ′(t)/integraldisplay\nΩ1∪Ω2|ut(t)|2dxdt+C24Eβ+1(S)\n≤C25Eβ+1(S)+|Ω|Eβ(S)\nφ(S).(5.6)\nNext we continue to estimate the remaining terms in ( 4.10).\nFort≥1 andφ′(t)≤1, set\nΩ1={x∈Ω :|ut(t)| ≤φ′(t)};\nΩ2={x∈Ω :φ′(t)<|ut(t)| ≤1};\nΩ3={x∈Ω :|ut(t)|>1}.\nFort≥1 andφ′(t)>1, set\nΩ4={x∈Ω :|ut(t)| ≤1<φ′(t)};\nΩ5={x∈Ω : 1<|ut(t)| ≤φ′(t)};\nΩ6={x∈Ω :|ut(t)|>φ′(t)>1}.\nHence\n{x∈Ω :|ut(t)| ≤1}= Ω1∪Ω2(or Ω4),{x∈Ω :|ut(t)|>1}= Ω3(or Ω5∪Ω6).\nSimilar to ( 4.14),\nc1mε−1\nm\n2\nm+1/integraldisplayT\nSEβ(t)φ′(t)/integraldisplay\nΩi|g(ut(t))|m+1\nmdxdt≤C26Eβ+1(S),i=3,5,6.\nThus\nc1mε−1\nm\n2\nm+1/integraldisplayT\nSEβ(t)φ′(t)/integraldisplay\n|ut(t)|>1|g(ut(t))|m+1\nmdxdt≤C27Eβ+1(S). (5.7)\nUsing (H6) and (H′\n5), then\nc1\n2ε1/integraldisplayT\nSEβ(t)φ′(t)/integraldisplay\nΩ2|g(ut(t))|2dxdt\n≤c1\n2ε1/integraldisplayT\nSEβ(t)/integraldisplay\nΩ2|ut(t)||g(ut(t))|2dxdt\n≤c1\n2ε1/integraldisplayT\nSEβ(t)/integraldisplay\nΩ2ut(t)g(ut(t))|h−1(ut(t))|dxdt\n≤c1\n2ε1h−1(1)/integraldisplayT\nSEβ(t)1\nγ(t)/integraldisplay\nΩ2γ(t)ut(t)g(ut(t))dxdt≤C28Eβ+1(S),(5.8)14\nand\nc1\n2ε1/integraldisplayT\nSEβ(t)φ′(t)/integraldisplay\nΩi|g(ut(t))|2dxdt\n≤c1\n2ε1/integraldisplayT\nSEβ(t)φ′(t)/integraldisplay\nΩi|h−1(ut(t))|2dxdt\n≤c1\n2ε1/integraldisplayT\nSEβ(t)φ′(t)/integraldisplay\nΩi|h−1(φ′(t))|2dxdt\n≤C29Eβ(S)/integraldisplayT\nSφ′(t)|h−1(φ′(t))|2dt,i=1,4.(5.9)\nSubstubiting ( 5.6)-(5.9) into (4.10), we get\n/bracketleftBig\n2−p−1\np+1M−C8(ε1+ε2)/bracketrightBig/integraldisplayT\nSEβ+1(t)φ′(t)dt\n≤C30Eβ+1(S)+|Ω|Eβ(S)\nφ(S)+C31Eβ(S)/integraldisplayT\nSφ′(t)|h−1(φ′(t))|2dt.(5.10)\nSince/integraltext∞\n1φ′(t)|h−1(φ′(t))|2dtconverges in Lemma 5.1, then\n/integraldisplayT\nSEβ+1(t)φ′(t)dt≤1\nC32Eβ(0)E(S) +C33\nφ(S)Eβ(0)E(S). (5.11)\nWhenTgoes to + ∞, chooseβ= 1, and choose ψ(t) =φ(t)−1 in Lemma 4.1, then we derive\nE(t)≤CE(0)\nφ2(t). (5.12)\nLet us choose s0such thath(1\ns0)≤1, then by ( 5.2) andH(s) :=h(s)s, fors≥s0\n˜ψ(s)≤1+(s−1)1\nh/parenleftBig\n1\ns/parenrightBig≤s1\nh/parenleftBig\n1\ns/parenrightBig=1\nH/parenleftBig\n1\ns/parenrightBig.\nHence, by ( 5.1), fors≥s0\ns≤φ\n1\nH/parenleftBig\n1\ns/parenrightBig\n=φ(t) witht=1\nH/parenleftBig\n1\ns/parenrightBig.\nFurther,\n1\nφ(t)≤1\ns=H−1/parenleftBig1\nt/parenrightBig\n.\nwhich together with ( 5.12) implies ( 2.9).\nAcknowledgements\nThisworkissupportedbytheFundamentalResearchFundsfor CentralUniversities(B230201033).15Competing Interests\nThe authors declare that they have no competing interests.\nData Availability\nData sharing is not applicable to this article as no new data w ere created or analyzed in this\nstudy.\nReferences\n[1] M. Daoulatli, Rates of decay for the wave systems with time-dependent damp ing, Discrete\nContin. Dyn. 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Serrin, Local asymptotic stability for dissipative wave systems , Israel J.\nMath.,104(1998), 29–50."    },    {        "title": "1810.04973v1.Propagating_spin_waves_in_nanometer_thick_yttrium_iron_garnet_films__Dependence_on_wave_vector__magnetic_field_strength_and_angle.pdf",        "content": "Propagating spin waves in nanometer-thick yttrium iron garnet \flms: Dependence on\nwave vector, magnetic \feld strength and angle\nHuajun Qin,1,\u0003Sampo J. H am al ainen,1Kristian Arjas,1Jorn Witteveen,1and Sebastiaan van Dijken1,y\n1NanoSpin, Department of Applied Physics, Aalto University School of Science, P.O. Box 15100, FI-00076 Aalto, Finland\n(Dated: October 12, 2018)\nWe present a comprehensive investigation of propagating spin waves in nanometer-thick yttrium\niron garnet (YIG) \flms. We use broadband spin-wave spectroscopy with integrated coplanar waveg-\nuides (CPWs) and microstrip antennas on top of continuous and patterned YIG \flms to characterize\nspin waves with wave vectors up to 10 rad/ \u0016m. All \flms are grown by pulsed laser deposition. From\nspin-wave transmission spectra, parameters such as the Gilbert damping constant, spin-wave dis-\npersion relation, group velocity, relaxation time, and decay length are derived and their dependence\non magnetic bias \feld strength and angle is systematically gauged. For a 40-nm-thick YIG \flm, we\nobtain a damping constant of 3 :5\u000210\u00004and a maximum decay length of 1.2 mm. Our experiments\nreveal a strong variation of spin-wave parameters with magnetic bias \feld and wave vector. Spin-\nwave properties change considerably up to a magnetic bias \feld of about 30 mT and above a \feld\nangle of\u0012H= 20\u000e, where\u0012H= 0\u000ecorresponds to the Damon-Eshbach con\fguration.\nPACS numbers:\nI. INTRODUCTION\nMagnonics aims at the exploitation of spin waves for\ninformation transport, storage, and processing1{7. For\npractical devices, it is essential that spin waves propa-\ngate over long distances in thin \flms. Because of its ul-\ntralow damping constant, ferrimagnetic YIG is a promis-\ning material. Bulk crystals and \u0016m-thick YIG \flms ex-\nhibit a Gilbert damping constant \u000b\u00193\u000210\u00005at GHz\nfrequencies. In recent years, nm-thick YIG \flms with\nultralow damping parameters have also been prepared\nsuccessfully. High-quality YIG \flms have been grown on\nGd3Ga5O12(GGG) single-crystal substrates using liquid\nphase epitaxy8{11, magnetron sputtering12{15, and pulsed\nlaser deposition (PLD)16{25. For thin YIG \flms, damp-\ning constants approaching the value of bulk crystals have\nbeen reported21,22. Meanwhile, YIG-based magnonic\ndevices such as logic gates, transistors, and multiplex-\ners have been demonstrated26{30. Spin-wave transmis-\nsion in nm-thick YIG \flms24,31{36and the excitation of\nshort-wavelength spin waves have been investigated as\nwell37{40. To advance YIG-magnonics further, knowledge\non the transport of spin waves in nm-thick YIG \flms and\nits dependence on wave vector and external magnetic bias\n\feld is essential.\nIn this paper, we present a broadband spin-wave spec-\ntroscopy study of PLD-grown YIG \flms with a thickness\nof 35 nm and 40 nm. Spin-wave transmission spectra are\nrecorded by patterning CPWs and microstrip antennas\non top of continuous and patterned YIG \flms. CPWs are\nused because they generate spin waves with well-de\fned\nwave vectors. This enables extraction of key parame-\nters such as the Gilbert damping constant ( \u000b), the spin-\nwave dispersion relation, group velocity ( \u001dg), relaxation\ntime (\u001c), and decay length ( ld). For a 40 nm YIG \flm,\nwe \fnd\u000b\u00193:5\u000210\u00004and a maximum group velocity\nand decay length of 3.0 km/s and 1.2 mm, respectively.\nWe show that spin-wave properties vary strongly withwave vector up to an in-plane external magnetic bias \feld\n\u00160Hext= 30 mT and below a \feld angle \u0012H= 20\u000e(\u0012H\n= 0 corresponds to the Damon-Eshbach geometry). Be-\nyond these \feld parameters, the dependence of spin-wave\nproperties on wave vector weakens. We demonstrate also\nthat broadband spectroscopy with integrated CPWs and\nmicrostrip antennas provide similar spin-wave parame-\nters.\nThe paper is organized as follows. In Sec. II,\nwe describe the PLD process, broadband spin-wave\nspectroscopy setup, and simulations of the CPW- and\nmicrostrip-antenna excitation spectra. In Sec. III, we\npresent vector network analyzer ferromagnetic resonance\n(VNA-FMR) results and broadband spin-wave transmis-\nsion spectra for CPWs. In Sec. IV, we \ft the ex-\nperimental data and extract parameters of propagating\nspin waves. Spin-wave transmission measurements using\nCPWs and microstrip antennas are compared in Sec. V.\nSection VI summarizes the paper.\nII. EXPERIMENT\nA. PLD of YIG thin \flms\nYIG \flms with a thickness of 35 nm and 40 nm were\ngrown on single-crystal GGG(111) substrates using PLD.\nPrior to loading into the PLD vacuum chamber, the\nsubstrates were ultrasonically cleaned in acetone, iso-\npropanol, and distilled water. The substrates were \frst\ndegassed at 550\u000eC for 15 minutes and then heated to\n800\u000eC at a rate of 5\u000eC per minute in an O 2pressure of\n0.13 mbar. YIG \flms were deposited under these condi-\ntions by ablation from a stoichiometric target using an\nexcimer laser with a pulse repetition rate of 2 Hz and\na \ruence of 1.8 J/cm2. After deposition, the YIG \flms\nwere \frst annealed at 730\u000eC for 10 minutes in 13 mbar\nO2before cooling down to room temperature at a rate ofarXiv:1810.04973v1  [cond-mat.mes-hall]  11 Oct 20182\n-1.0 -0.5 0.0 0.5 1.0-100-50050100\n49 50 51 52 53300 400 500 6000.00.51.0Ms (kA/m)\nMagnetic field (mT)GGG (444)Intensity (a.u.)\n2oYIG (444)\nM/Ms\nTemperature (K)(a) (b)\nFIG. 1: (a) XRD \u0012\u00002\u0012scan of the (444) re\rections from a\nPLD-grown YIG \flm on a GGG(111) substrate. The period\nof Laue oscillations surrounding the (444) peaks corresponds\nto a \flm thickness of 40 nm. (b) Room temperature VSM\nhysteresis loop of the same \flm. The inset shows how the\nYIG saturation magnetization varies with temperature.\n\u00003\u000eC per minute.\nB. Structural and magnetic characterization\nThe crystal structure of our YIG \flms was inspected\nby high-resolution X-ray di\u000braction (XRD) on a Rigaku\nSmartLab system. Figure 1(a) shows a XRD \u0012\u00002\u0012scan\nof a 40-nm-thick YIG \flm on GGG(111). Clear (444) \flm\nand substrate peaks are surrounded by Laue oscillations,\nsignifying epitaxial and smooth \flm growth. We used a\nvibrating sample magnetometer (VSM) in a PPMS Dy-\nnacool system from Quantum Design to characterize the\nmagnetic properties. Figure 1(b) depicts a VSM hystere-\nsis loop of a 40-nm-thick YIG \flm. At room temperature,\nthe coercive \feld of the YIG \flm is only 0.1 mT and the\nsaturation magnetization ( Ms) is 115 kA/m. The evo-\nlution ofMswith temperature is shown in the inset of\nFig. 1(b). From these data, we derive a Curie temper-\nature (TC) of around 500 K. The values of MsandTC\nare similar to those obtained in previous studies on nm-\nthick YIG \flms14,22,23and about 10% smaller compared\nto values of YIG bulk crystals ( Ms= 139 kA/m, TC=\n559 K). Minor o\u000b-stoichiometries in the YIG \flm might\nbe the reason for the small discrepancy41.\nC. Broadband spin-wave spectroscopy\nVNA-FMR and spin-wave transmission measurements\nwere performed using a two-port VNA and a microwave\nprobing station with a quadrupole electromagnet. In\nVNA-FMR experiments, the YIG \flm was placed face-\ndown onto a prepatterned CPW on a GaAs substrate.\nThe signal line and ground lines of this CPW had a\nwidth of 50 \u0016m and 800 \u0016m, respectively, and were sep-\narated by 30 \u0016m. Broadband spin-wave spectroscopyin transmission geometry was conducted by contacting\ntwo integrated CPWs or microstrip antennas on top of\na continuous YIG \flm or YIG waveguide. Most of the\nexperiments were performed with CPWs consisting of 2\n\u0016m-wide signal and ground lines with a separation of 1.6\n\u0016m. For comparison measurements, we used CPWs and\nmicrostrip antennas with 4- \u0016m-wide signal lines. All an-\ntenna structures were fabricated by electron-beam lithog-\nraphy and were composed of 3-nm Ta and 120-nm Au.\nA microwave current provided by the VNA was used to\ngenerate a rf magnetic \feld around one of the CPWs\nor microstrip antennas. We used CST microwave studio\nsoftware to simulate the excitation spectra of the antenna\nstructures (see next section).\nSpin waves that are excited by a rf magnetic \feld pro-\nduce an inductive voltage across a nearby antenna. At\nthe exciting CPW or microstrip antenna, this voltage is\ngiven by42:\nVind/Z\n\u001f(!;k)j\u001a(k)j2dk; (1)\nwhere\u001f(!;k) is the magnetic susceptibility and j\u001a(k)j2\nis the spin-wave excitation spectrum. Propagating spin\nwaves arriving at the receiving CPW or microstrip an-\ntenna produce an inductive voltage:\nVind/Z\n\u001f(!;k)j\u001a(k)j2exp(\u0000i(ks+ \b 0))dk; (2)\nwheresis the propagation distance and \b 0is the initial\nphase of the spin waves. In the experiments, we used the\n\frst and second port of the VNA to measure these induc-\ntive voltages by recording the S12scattering parameter.\nD. Simulations of CPW and microstrip antenna\nexcitation spectra\nWe used CST microwave studio software to simulate\nthe spin-wave excitation spectra of the di\u000berent antenna\nstructures43. This commercial solver of Maxwell's equa-\ntions uses a \fnite integration method to calculate the rf\nmagnetic \feld \u00160hrfand its in-plane ( \u00160hrf\nx,\u00160hrf\ny) and\nout-of-plane ( \u00160hrf\nz) components. Since the excitation\n\feld along the CPW or antenna ( \u00160hrf\nx) is nearly uni-\nform and\u00160hrf\nzis much smaller than \u00160hrf\ny, we Fourier-\ntransformed only the latter component. Figure 2 depicts\nseveral CPW and antenna con\fgurations used in the ex-\nperiments together with their simulated spin-wave exci-\ntation spectra. The large prepatterned CPW on a GaAs\nsubstrate (Fig. 2(a)), which is used for VNA-FMR mea-\nsurements, mainly excites spin waves with k\u00190 rad/\u0016m\n(Fig. 2(d)). The excitation spectrum of the smaller in-\ntegrated CPW with a 2- \u0016m-wide signal line (Fig. 2(b))\nincludes one main spin-wave mode with wave vector k1\n= 0.76 rad/ \u0016m and several high-order modes k2\u0000k7\n(Fig. 2(e)). The 4- \u0016m-wide microstrip antenna (Fig.\n2(c)) mainly excites spin waves with k1ranging from 03\n(a)\nGGS\n02468 1 0 1 20.00.51.0\n-20 -10 0 10 20Amplitude (Normalized)\nWave vector (rad/ m)0Hy (a. u.)\ny (m)S\n02468 1 0 1 20.00.51.0-100 -50 0 50 100Amplitude (Normalized)\nWave vector (rad/ m)GG0Hy (a. u.)\ny (m)S\n02468 1 0 1 20.00.51.0 -10 -5 0 5 10Amplitude (Normalized)\nWave vector (rad/ m)G G0Hy (a. u.)\ny (m)S\n(b) (c)\n(d) (e) (f)\nk1\nk2k3k4k5k6k7k1\nk2k3xyz\nxyz\nx yz\nFIG. 2: (a-c) Schematic illustrations of several measurement con\fgurations used in this study. (a) VNA-FMR measurements\nare performed by placing the YIG/GGG sample face-down onto a CPW. The CPW consists of a 50 \u0016m-wide signal line and\ntwo 800\u0016m-wide ground lines. The gap between the signal and ground lines is 30 \u0016m. (b-c) Spin-wave transmission through\nthe YIG \flm is characterized by patterning two CPWs (b) or two microstrip antennas (c) on top of a YIG \flm. The signal and\nground lines of the CPWs in (b) are 2 \u0016m wide and separated by 1.6 \u0016m gaps. The microstrip antennas, which are marked\nby red arrows in (c), are 4 \u0016m wide. (d-f) Simulated spin-wave excitation spectra of the di\u000berent antenna structures. The\nin-plane rf magnetic \felds ( \u00160hrf\ny) that are produced by passing a microwave current through the CPWs in (a) and (b) or the\nmicrostrip antenna in (c) are shown in the insets.\nto 1.5 rad/\u0016m and some higher order modes at k2\u00192:0\nrad/\u0016m andk3\u00193:8 rad/\u0016m (Fig. 2(f)). The insets of\nFigs. 2(d-f) show the simulated rf magnetic \felds \u00160hrf\ny\nalong they-axis for each antenna structure.\nIII. RESULTS\nA. VNA-FMR\nWe recorded FMR spectra for various in-plane exter-\nnal magnetic bias \felds by measuring the S12scatter-\ning parameter on a 40-nm-thick YIG \flm. As an ex-\nample, the imaginary part of S12recorded with a mag-\nnetic bias \feld \u00160Hext= 80 mT is shown in Fig. 3(a).\nThe spectrum was subtracted a reference measured at\na bias \feld of 200 mT for enhancing signal-to-noise ra-\ntio. The resonance at f= 4:432 GHz is \ftted by a\nLorentzian function. From similar data taken at other\nbias \felds, we extracted the \feld-dependence of FMR\nfrequency and the evolution of resonance linewidth (\u0001 f)\nwith frequency. Figures 3(b) and 3(c) summarize our\nresults. Fitting the data of Fig. 3(b) to the Kittel for-\nmulafres=\r\u00160\n2\u0019p\nHext(Hext+Meff) using\r=2\u0019= 28\nGHz/T, we \fnd Meff= 184 kA/m. The measured\nvalue ofMeffis comparable to those of other PLD-grown\nYIG thin \flms23,24, but it is large compared to Ms(115\nkA/m). Since Meff=Ms-Hani, this means that the\nanisotropy \feld Hani=\u000069 kA/m in our \flm. The\nnegative anisotropy \feld is caused by a lattice mismatch\nbetween the YIG \flm and GGG substrate23. Fitting the\n4.42 4.44 3456681012\n05 0 1 0 0 1 5 00246Im S12\nFrequency (GHz) Frequency (GHz)\nFrequency (GHz)\n0Hext (mT)(a) (b)\n= 3.5 × 10-4f(c)FIG. 3: (a) Imaginary part of the S12scattering parameter\nshowing FMR for an in-plane external magnetic bias \feld of\n80 mT along the CPW. The orange line is a Lorentzian func-\ntion \ft. (b) FMR frequency as a function of external magnetic\nbias \feld. The orange line represents a \ft to the experimen-\ntal data using the Kittel formula. (c) Dependence of FMR\nlinewidth (\u0001 f) on resonance frequency. From a linear \ft to\nthe data, we derive \u000b= 3:5\u000210\u00004.\ndata of Fig. 3(c) using \u0001 f= 2\u000bf+\u001dg\u0001kgives a Gilbert\ndamping constant \u000b= 3:5\u000210\u00004, which is comparable to\nother experiments on PLD-grown \flms17,18,20. In the \ft-\nting formula, \u001dgand \u0001kare the spin-wave group velocity\nand excitation-spectrum width, respectively44.4\n1.8 2.1 2.4 2.7 3.0 10 20 30 40 501234\n0Hext (mT)Frequency (GHz)Im S12k7k5k4 k6k3k2\nFrequency (GHz)k1\n-60 -30 0 30 601.82.12.42.73.03.3\nH (O)Frequency (GHz)(a) (b) (c)\nk1k7\nk1k7\nCPW 1\nCPW 2\n0Hext45 m\nFIG. 4: (a) Spin-wave transmission spectrum (imaginary part of S12) recorded on a 40-nm-thick YIG waveguide with an\nexternal magnetic bias \feld \u00160Hext= 15:5 mT along the CPWs. The inset shows a top-view schematic of the measurement\ngeometry. (b) 2D map of spin-wave transmission spectra measured as a function of magnetic bias \feld strength. (c) Angular\ndependence of spin-wave transmission spectra for a constant bias \feld of 15.5 mT. The \feld angle \u0012H= 0\u000ecorresponds to the\nDamon-Eshbach con\fguration.\nB. Propagating spin waves\nWe measured spin-wave transmission spectra on a 40-\nnm-thick YIG \flm. The measurement geometry con-\nsisted of two CPWs on top of YIG waveguides with 45\u000e\nedges (see the inset of Fig. 4(a)). The CPW parame-\nters were identical to those in Fig. 2(b) and their sig-\nnal lines were separated by 45 \u0016m. During broadband\nspin-wave spectroscopy, spin waves with characteristic\nwave vectors ki(i= 1, 2...) were excited by passing\na rf current through one of the CPWs. After propaga-\ntion through the YIG \flm, the other CPW inductively\ndetected the spin waves. Figure 4(a) shows the imagi-\nnary part of the S12scattering parameter for an external\nmagnetic bias \feld \u00160Hext= 15:5 mT parallel to the\nCPWs (Damon-Eshbach con\fguration). The graph con-\ntains seven envelope-type peaks ( k1\u0000k7) with clear pe-\nriodic oscillations. The peak intensities decrease with\nfrequency because of reductions in the excitation e\u000e-\nciency and spin-wave decay length. The oscillations sig-\nnify spin-wave propagation between the CPWs44. Fig-\nure 4(b) shows a 2D representation of spin-wave trans-\nmission spectra recorded at di\u000berent bias \felds. As the\n\feld strengthens, the frequency gaps between spin-wave\nmodes become smaller. Figure 4(c) depicts the angu-\nlar dependence of S12spectra at a constant magnetic\nbias \feld of 15.5 mT. In this measurement, the in-plane\nmagnetic bias \feld was rotated from -72\u000eto 72\u000e, where\n\u0012H= 0\u000ecorresponds to the Damon-Eshbach con\fgura-\ntion. As the magnetization rotates towards the wave vec-\ntor of propagating spin waves, the frequency and inten-\nsity of thek1\u0000k7modes drop. The frequency evolutions\nof the spin-wave modes in Figs. 4(b) and 4(c) are ex-\nplained by a \rattening of the dispersion relation with\nincreasing magnetic bias \feld strength and angle.\n1 . 71 . 81 . 92 . 02 . 12 . 22 . 3-101Im S12 (Normalized)  Fit\n  Exp.\nFrequency (GHz)k1\nk2FIG. 5: A \ft to the spectrum for \u00160Hext= 15:5 mT and\n\u0012H= 0\u000e(blue squares) using Eq. 3 (orange line).\nIV. DISCUSSION\nA. Fitting of spin-wave transmission spectra\nWe used Eq. 2 to \ft spin-wave transmission spectra.\nIn this equation, \u001f(!;k) is described by a Lorentzian\nfunction, while the excitation spectrum j\u001a(k)j2is ap-\nproximated by a Gaussian function (see Fig. 2(e)). For\nDamon-Eshbach spin waves with kd\u001c1, the wave vec-\ntor is given by k=2\nd(2\u0019f)2\u0000(2\u0019fres)2\n(\r\u00160Ms)2, wheredis the \flm\nthickness. Based on these approximations, we rewrite\nEq. 2 as:\nImS 12/\u0001f\n(f\u0000fres)2+ (\u0001f)2\u0002e\u00004ln2(k\u0000k0)2=\u0001k2\n\u0002sin(ks+ \b);(3)5\n02468 1 0 1 201234\n0369 1 2 1 51.52.02.53.040 mT 15.5 mTFrequency (GHz)\nWave vector (rad/ m)1 mT\n9070604836H = 0Frequency (GHz)\nWave vector (rad/ m)18(a) (b)\nFIG. 6: Spin-wave dispersion relations for di\u000berent external\nmagnetic bias \felds (a) and \feld angles (b). In (a) \u0012H= 0\u000e\nand in (b)\u00160Hext= 15.5 mT. The colored lines represent \fts\nto the disperion relations using Eq. 4.\nwhere \u0001fis theS12envelope width, \u0001 kis the width of\nthe spin-wave excitation spectrum, \b is the initial phase,\nandsis the propagation distance. Figure 5 shows a \ft-\nting result for a spin-wave transmission spectrum with\n\u00160Hext= 15.5 mT and \u0012H= 0\u000e. As input parameters,\nwe usedfres= 1.75 GHz, d= 40 nm,s= 45\u0016m, and\nMeff= 184 kA/m, which are either determined by ge-\nometry or extracted from measurements. \u0001 f, \u0001k,k0\nare \ftting parameters. For the k1peak, we obtained\nthe best \ft for \u0001 f= 0.25 GHz, \u0001 k= 0.6 rad/\u0016m, and\nk1= 0:72 rad/\u0016m. Thek2peak was \ftted with k2= 1:87\nrad/\u0016m. The values of \u0001 k,k1, andk2are in good agree-\nment with the simulated excitation spectrum of the CPW\n(Fig. 2(e)) and \u0001 fmatches the width of the envelope\npeak in the experimental S12spectrum.\nB. Spin-wave dispersion relations\nWe extracted spin-wave dispersion relations for di\u000ber-\nent magnetic bias \felds and \feld angles by \ftting the S12\ntransmission spectra shown in Figs. 4(b) and 4(c). The\nsymbols in Fig. 6 summarize the results. We also cal-\nculated the dispersion relations using the Kalinikos and\nSlavin formula45:\nf=\r\u00160\n2\u0019\u0014\nHext\u0000\nHext+Meff\u0002\n1\u0000Fsin2\u0012H\n+Meff\nHextF(1\u0000F) cos2\u0012H\u0003\u0001\u00151=2\n;(4)\nwithF= 1\u00001\u0000exp(\u0000kd)\nkd. The calculated dispersion re-\nlations for\r=2\u0019= 28 GHz/T, Meff= 184 kA/m, and d\n= 40 nm are shown as lines in Fig. 6.\nThe dispersion curves \ratten with increasing magnetic\nbias \feld. For instance, at \u00160Hext= 1 mT, the frequency\nof propagating spin waves changes from 0.5 GHz to 2.4\nGHz for wave vectors ranging from 0 to 10 rad/ \u0016m. At\n\u00160Hext= 40 mT, the frequency evolution with wave vec-\n0 1 02 03 04 05 00.51.01.52.02.53.0\n02 0 4 0 6 00.30.60.91.21.5g (k1)\ng (k2)\ng (k3)Group velocity g (km/s)\next (mT)\nGroup velocity g (km/s)\nH (o)g (k1)\ng (k2)\ng (k3)(a) (b)FIG. 7: Spin-wave group velocity \u001dgofk1\u0000k3modes as a\nfunction of external magnetic bias \feld (a) and \feld angle (b).\nIn (a)\u0012H= 0\u000eand in (b)\u00160Hext= 15.5 mT.\ntor is reduced to 3 \u00003:7 GHz. This magnetic-\feld de-\npendence of the dispersion relation narrows the spin-wave\ntransmission bands in Fig. 4(b) at large \u00160Hext.\nThe angular dependence of the spin-wave dispersion\ncurves in Fig. 6(b) is explained by in-plane magneti-\nzation rotation from M?k(\u0012H= 0\u000e) towardsMkk\n(\u0012H= 90\u000e). At\u0012H= 0\u000e, dispersive Damon-Eshbach spin\nwaves with positive group velocity propagate between the\nCPWs. The character of excited spin waves changes\ngradually with increasing \u0012Huntil it has fully trans-\nformed into a backward-volume magnetostatic mode at\n\u0012H= 90\u000e. This mode is only weakly dispersive and ex-\nhibits a negative group velocity.\nC. Group velocity\nThe phase relation between signals from the two CPWs\nis given by \b = k\u0002s31,44. Since the phase shift between\ntwo neighboring maxima ( \u000ef) in broadband spin-wave\ntransmission spectra corresponds to 2 \u0019, the group veloc-\nity can be written as:\n\u001dg=@!\n@k=2\u0019\u000ef\n2\u0019=s=\u000ef\u0002s; (5)\nwheres= 45\u0016m in our experiments. Using this equa-\ntion, we extracted the spin-wave group velocity for wave\nvectorsk1\u0000k3from the transmission spectra shown in\nFigs. 4(b) and 4(c). Figure 7 summarizes the variation\nof\u001dgwith external magnetic bias \feld and \feld angle.\nFor weak bias \felds ( \u00160Hext<30 mT), the group ve-\nlocity decreases swiftly, especially if kis small. For in-\nstance,\u001dg(k1) reduces from 3.0 km/s to 1.0 km/s in the\n0\u000030 mT \feld range, while \u001dg(k3) only changes from\n1.2 km/s to 0.8 km/s. At larger external magnetic bias\n\felds,\u001dgdecreases more slowly for all wave vectors. Fig-\nure 7(b) shows how \u001dgvaries as a function of \feld angle\nat\u00160Hext= 15.5 mT. For all wave vectors, the group\nvelocity is largest in the Damon-Eshbach con\fguration\n(\u0012H= 0\u000e). At larger \feld angles, \u001dgdecreases and its6\n0 1 02 03 04 05 0200400600\n02 0 4 0 6 0200250300Relaxation time  (ns)\next (mT) (k1)\n (k2)\nk3)\nRelaxation time (ns)\nH (o)  (k1)\n  (k2)\n  (k3)(a) (b)\nFIG. 8: Spin-wave relaxation time \u001cofk1\u0000k3modes as a\nfunction of external magnetic bias \feld (a) and \feld angle (b).\nIn (a)\u0012H= 0\u000eand in (b)\u00160Hext= 15.5 mT.\n0 1 02 03 04 05 0030060090012001500\n0 1 02 03 04 05 06 00100200300400Decay length ld (m)\n0Hext (mT) ld (k1)\n ld (k2)\n ld (k3)\nDecay length ld (m)\no ld (k1)\n ld (k2)\n ld (k3)(a) (b)\nFIG. 9: Spin-wave decay length ldofk1\u0000k3modes as a\nfunction of external magnetic bias \feld (a) and \feld angle\n(b). In (a) \u0012H= 0\u000eand in (b)\u00160Hext= 15.5 mT.\ndependence on wave vector diminishes. Variations of the\nspin-wave group velocity with wave vector and magnetic-\n\feld strength or angle are explained by a \rattening of the\ndispersion relations, as illustrated by the data in Fig. 6.\nD. Spin-wave relaxation time and decay length\nWe now discuss the relaxation time ( \u001c) and decay\nlength (ld) of spin waves in our YIG \flms. Following\nRef. 46, the relaxation time is estimated by \u001c= 1=2\u0019\u000bf.\nUsing\u000b= 3:5\u000210\u00004and spin-wave transmission data\nfrom Fig. 4, we determined \u001cfor wave vectors k1\u0000k3.\nThe dependence of \u001con external magnetic bias \feld and\n\feld angle is shown in Fig. 8. The maximum spin-wave\nrelaxation time in our 40-nm-thick YIG \flms is approx-\nimately 500 ns. Resembling the spin-wave group veloc-\nity,\u001cis largest for small wave vectors and it decreases\nwith increasing bias \feld (Fig. 8(a)). In contrast to \u001dg,\nthe spin-wave relaxation time is smallest in the Damon-\nEshbach con\fguration ( \u0012H= 0\u000e) and it evolves more\nstrongly with increasing \u0012Hifkis large (Fig. 8(b)). This\nresult is explained by \u001c/1=fand a lowering of thespin-wave frequency if the in-plane bias \feld rotates the\nmagnetization towards k(see Fig. 4(c)).\nThe spin-wave decay length is calculated using ld=\n\u001dg\u0002\u001cand data from Figs. 7 and 8. Figure 9(a) shows\nthe dependence of ldon\u00160Hextfor wave vectors k1\u0000k3.\nThe largest spin-wave decay length in our 40-nm-thick\nYIG \flms is 1.2 mm, which we measured for k1= 0:72\nrad/\u0016m and\u00160Hext= 2 mT. The decay length decreases\nwith magnetic bias \feld to about 100 \u0016m at\u00160Hext=\n50 mT. Figure 9(b) depicts the dependence of ldon the\ndirection of a 15.5 mT bias \feld. The spin-wave decay\nlength decreases substantially with \u0012Hfor smallk, but\nits angular dependence weakens for larger wave vectors.\nThe decay of propagating spin waves between the ex-\nciting and detecting CPW in the broadband spectroscopy\nmeasurement geometry is given by exp( \u0000s=ld)46. Based\non the results of Fig. 9, one would thus expect the in-\ntensity of spin waves to drop with increasing wave vector\nand in-plane bias \feld strength or angle. The spin-wave\ntransmission spectra of Fig. 4 con\frm this behavior.\nE. CPWs versus microstrip antennas\nFinally, we compare broadband spin-wave spec-\ntroscopy measurements on YIG thin \flms using CPWs\nand microstrip antennas. In these experiments, the\nCPWs and antenna structures have 4- \u0016m-wide signal\nlines and they were patterned onto the same 35-nm-thick\nYIG \flm. For comparison, we also recorded transmission\nspectra on 50- \u0016m wide YIG waveguides. The separation\ndistance (s) between the CPWs or microstrip antennas\nwas set to 110 \u0016m or 220\u0016m. Schematics of the di\u000berent\nmeasurement geometries are depicted on the sides of Fig.\n10. Transmission spectra that were obtained for Damon-\nEshbach spin waves in each con\fguration are also shown.\nIn all measurements, we used an in-plane external mag-\nnetic bias \feld of 10 mT. The plots focus on phase os-\ncillations in the \frst-order excitation at k1(higher-order\nexcitations were measured also, but are not shown). The\ndi\u000berently shaped outline of the S12peak for two CPWs\n(left) or two microstrip antennas (right) mimics the pro-\n\fle of their excitation spectra (Fig. 2). As expected from\n\u000ef=\u001dg=s, the period of frequency oscillations ( \u000ef) be-\ncomes smaller if the separation between antennas ( s) is\nenhanced (Figs. 10(c) and 10(f)).\nWe \ftted the spin-wave transmission spectra obtained\nwith CPWs (Figs. 10(a)-(c)) using the same procedure as\ndescribed earlier. Good agreements between experimen-\ntal data (blue squares) and calculations (orange lines)\nwere obtained by inserting Meff= 190\u00064 kA/m, \u0001f=\n0.18 GHz,k= 0.34 rad/ \u0016m, and \u0001k= 0.33 rad/ \u0016m into\nEq. 3. To \ft S12spectra measured by microstrip anten-\nnas, we approximated the wave vector of the excitation\nask=2\nd(2\u0019f)2\u0000(2\u0019fres)2\n(\r\u00160Meff)2H(f\u0000fres), whereHis a Heav-\niside step function47. The best results were achieved for\nMeff= 178\u00062 kA/m, \u0001 f= 0.25 GHz, k= 0 rad/\u0016m\nand \u0001k= 0.65 rad/ \u0016m. From the data comparison in7\n-101\n-101\n1.3 1.4 1.5-101-101\n-101\n1.3 1.4 1.5-101\nFrequency (GHz) Frequency (GHz)110 m\n220 mCPW1\nCPW2\n0Hext\n110 m110 mantenna2\n0Hextantenna1\n220 m(a) (d)\n(b) (e)\n(c) (f)Im S12 (Normalized)\nIm S12(Normalized)110 m\n220 m\nFIG. 10: (a)-(c) Spin-wave transmission spectra measured using CPWs on a continuous YIG \flm (a) and 50- \u0016m-wide YIG\nwaveguides ((b) and (c)). The YIG \flm and waveguides are 35 nm thick and the CPWs are separated by 110 \u0016m ((a) and\n(b)) and 220 \u0016m (c). (d)-(f) Spin-wave transmission spectra measured using microstrip antennas on the same YIG \flm and\nwaveguides. The signal lines of the CPWs and microstrip antennas are 4 \u0016m wide. The orange lines represent \fts to the\nexperimental data using Eq. 3. The measurement geometry for each spectrum is illustrated next to the graphs. In the\nschematics, the green areas depict a continuous YIG \flm or waveguide.\nFig. 10, we conclude that broadband spin-wave spec-\ntroscopy measurements with CPWs and microstrip an-\ntennas yield similar results for Meff. We also note that\ntheS12peak width (\u0001 f) obtained from measurements\non continuous YIG \flms and YIG waveguides are nearly\nidentical (\u0001 f= 0:18 GHz for CPWs, \u0001 f= 0:22 GHz for\nantennas). Thus, patterning of the YIG \flm into waveg-\nuides does not deteriorate the Gilbert damping constant.\nFrom the oscillation periods ( \u000ef) in the transmission\nspectra of Fig. 10, we extracted the properties of prop-\nagating spin waves. By averaging \u000efover the same fre-\nquency range in spectra measured by CPWs and mi-\ncrostrip antennas, we obtained \u001dg= 1:67 km/s and\n\u001dg= 1:53 km/s, respectively. The spin-wave relax-\nation time was determined as \u001c= 225 ns (CPW) and\n\u001c= 237 ns (antenna) and the decay length was extracted\nasld= 375\u0016m (CPW) and ld= 363\u0016m (antenna).\nThese results clearly demonstrate that broadband spin-\nwave spectroscopy measurements on YIG thin \flms us-\ning CPWs or microstrip antennas provide comparable\nresults.\nV. SUMMARY\nIn conclusion, we prepared 35 \u000040 nm thick epitaxial\nYIG \flms with a Gilbert damping constant \u000b= 3:5\u000210\u00004on GGG(111) substrates using PLD. The dependence of\nspin-wave transmission on the strength and angle of an\nin-plane magnetic bias \feld was systematically gauged.\nWe used the measurements to demonstrate strong tun-\ning of the spin-wave group velocity ( \u001dg), relaxation time\n(\u001c), and decay length ( ld) up to a \feld strength of about\n30 mT and above a \feld angle of 20\u000e. Maximum val-\nues of\u001dg= 3:0 km/s,\u001c= 500 ns, and ld= 1:2 mm\nwere extracted for Damon-Eshbach spin waves with k1\n= 0.72 rad/ \u0016m. Moreover, we demonstrated that broad-\nband spin-wave spectroscopy performed with integrated\nCPWs and microstrip antennas yield similar results.\nVI. ACKNOWLEDGEMENTS\nThis work was supported by the European Re-\nsearch Council (Grant Nos. ERC-2012-StG 307502-E-\nCONTROL and ERC-PoC-2018 812841-POWERSPIN).\nS.J.H. acknowledges \fnancial support from the V ais al a\nFoundation. Lithography was performed at the Mi-\ncronova Nanofabrication Centre, supported by Aalto\nUniversity. We also acknowledge the computational re-\nsources provided by the Aalto Science-IT project.\n\u0003Electronic address: huajun.qin@aalto.\f\nyElectronic address: sebastiaan.van.dijken@aalto.\f1A. A. Serga, A. V. Chumak, and B. Hillebrands, J. Phys.\nD: Appl. Phys. 43, 264002 (2010).8\n2V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J.\nPhys. D: Appl. 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Lett. 109, 012403 (2016)."    },    {        "title": "1209.3669v2.Nonlinear_emission_of_spin_wave_caustics_from_an_edge_mode_of_a_micro_structured_Co2Mn0_6Fe0_4Si_waveguide.pdf",        "content": "arXiv:1209.3669v2  [cond-mat.mes-hall]  12 Dec 2012Nonlinear emission of spin-wave caustics from an edge mode o f a\nmicro-structured Co 2Mn0.6Fe0.4Si waveguide\nT.Sebastian∗and T.Br¨ acher\nFachbereich Physik and Forschungszentrum OPTIMAS,\nTechnische Universit¨ at Kaiserslautern,\n67663 Kaiserslautern, Germany and\nGraduate School Materials Science in Mainz,\nGottlieb-Daimler-Straße 47, 67663 Kaiserslautern, Germa ny\nP.Pirro, A.A.Serga, and B.Hillebrands\nFachbereich Physik and Forschungszentrum OPTIMAS,\nTechnische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany\nT.Kubota\nWPI Advanced Institute for Materials Research, Tohoku Univ ersity,\nKatahira 2-1-1, Aoba-ku, Sendai 980-8577, Japan\nH.Naganuma, M.Oogane, and Y.Ando\nDepartment of Applied Physics, Graduate School of Engineer ing,\nTohoku University, Aoba-yama 6-6-05, Sendai 980-8579, Jap an\n(Dated: August 30, 2018)\nAbstract\nMagnetic Heusler materials with very low Gilbert damping ar e expected to show novel magnonic\ntransport phenomena. We report nonlinear generation of hig her harmonics leading to the emission\nofcausticspin-wave beamsinalow-damping, micro-structu redCo 2Mn0.6Fe0.4SiHeuslerwaveguide.\nThe source for the higher harmonic generation is a localized edge mode formed by the strongly\ninhomogeneous field distribution at the edges of the spin-wa ve waveguide. The radiation character-\nistics of the propagating caustic waves observed at twice an d three times the excitation frequency\nare described by an analytical calculation based on the anis otropic dispersion of spin waves in a\nmagnetic thin film.\n1In the last years nonlinear spin dynamics in magnetic microstructure s made of metallic\nferromagnetic thin films or layer stacks have gained large interest.[1 –5] The intrinsically\nnonlinear Landau-Lifshitz and Gilbert equation (LLG), that govern s the spin dynamics,\ngives rise to a variety of nonlinear effects.[6, 7]\nAmong the metallic ferromagnets, the class of Cobalt-based Heusle r materials is very\npromising for future magnon spintronics devices and the observation of new phenomena of\nmagnonic transport. The reasons for the interest in these mater ials are the small magnetic\nGilbert damping, the high spin-polarization, and the high Curie temper ature.[8, 9]\nAs shown recently, the full Heusler compound Co 2Mn0.6Fe0.4Si (CMFS) is a very suitable\nmaterial to be used as a micro-structured spin-wave waveguide du e to the increased decay\nlength which was observed for wave propagation in the linear regime.[ 10] The reason for\nthis observation is the low Gilbert damping of α= 3×10−3of CMFS compared to Ni 81Fe19\nwithα= 8×10−3, which is the material commonly used in related studies.[9] The decre ased\nmagnetic losses not only lead to an increase of the decay length but a lso to large precession\nangles of the magnetic moments and, thus, to the occurrence of n onlinear effects. Regarding\nfuture applications, the investigation and the thorough understa nding of phenomena related\nto the spin-wave propagation in the nonlinear regime in Heusler compo unds is crucial.\nIn this Letter, we report nonlinear higher harmonic generation fro m a localized edge\nmode [11, 12] causing the emission of caustic spin-wave beams [13, 14 ] in a micro-structured\nHeusler waveguide. Spin-wave caustics are characterized by the s mall transversal aperture\nof a beam, which practically does not increase during propagation, a nd the well-defined\ndirection of propagation.\nThe investigated sample is a 5 µm wide spin-wave waveguide structured from a 30nm\nthick film of the Heusler compound CMFS. Details about the fabricatio n and the material\nproperties can be found in Refs.15 and 16. The microfabrication of the waveguide was per-\nformedusing electron-beam lithography andion-milling. Forthe excit ation ofspin dynamics\nin the waveguide the shortened end of a coplanar waveguide made of copper was placed on\ntop of it. The Oersted field created by a microwave current in this an tenna structure can be\nused to excite spin dynamics in the Gigahertz range. The antenna ha s a thickness of 400nm\nand a width of ∆ x= 1µm.\nAll observations have been carried out using Brillouin light scattering microscopy\n(µBLS).[17] µBLS is a powerful tool to investigate spin dynamics in microstructur es with a\n2FIG. 1. (Color online) Sketch of the sample design. The short ened end of a coplanar waveguide\nis used as an antenna structure to excite spin dynamics in a 5 µm wide CMFS waveguide with a\nthickness of 30nm. The waveguide is positioned in the x-y-pl ane with its long axis pointing along\nthe x-direction. The external magnetic field is applied tran sversely to the waveguide in y-direction.\nThe figure includes a µBLS spectrum taken at a distance of 4.5 µm from the antenna in the center\nof the waveguide (see laser beam in the sketch) for an excitat ion frequency of fe= 3.5GHz, a\nmicrowave power of 20mW, and an external field µ0Hext= 48mT.\nspatial resolution of about 250nm and a frequency resolution of up to 50MHz.\nIn the following description, the waveguide is positioned in the x-y-pla ne with the long\naxis pointing in x-direction. The origin of the coordinate system is give n by the position\nof the antenna between x=−1µm andx= 0µm. An external magnetic field of µ0Hext=\n48mT was applied transversely to the waveguide in y-direction result ing in Damon-Eshbach\ngeometry [18] forspin waves propagatingalong thewaveguide. Ask etch of thesample layout\nis shown in Fig.1.\nIn addition, Fig.1 includes a spectrum taken by µBLS for an excitation frequency of\nfe= 3.5GHz and a microwave power of 20mW at a distance of 4.5 µm from the antenna in\nthe center of the waveguide. The spectrum shows not only a peak a tfe= 3.5GHz but also\nat 2fe= 7.0GHzand 3 fe= 10.5GHz. Furthermore, the intensity of the directly excited spin\nwaveat3.5GHzislowerthanforthehigherharmonicsatthepointofo bservation. Aswewill\nsee, the higher harmonics are excited resonantly by nonlinear magn on-magnon interactions\nand the intensity distribution is a consequence of the different prop agation characteristics\nof the observed spin-wave modes.\nA two-dimensional intensity distribution for the detection frequen cyfd=fe= 3.5GHz\nas well as the calculated dispersion relation [19] for the center of th e CMFS waveguide are\n3012345\ny position ( µm)123456789x position ( µm)fd=fe\n0123456\nwavevector ( µm−1)3456789101112frequency (GHz)\nfe2fe3fe\n(a) (b)\nFIG. 2. (Color online) (a) µBLS intensity distribution for the detection frequency fdbeing equal\nto the excitation frequency fe= 3.5GHz. The observed intensity of the edge mode is maximal near\nthe edges of the CMFS waveguide. Please note that the excitin g antenna is positioned between\nx=−1µm andx= 0µm. The blue dots in the graph indicates the position of the mea surement\npresented in Fig.1. (b) Calculated dispersion relation for the center of the CMFS waveguide\naccording to Ref.19 as well as excitation frequency and high er harmonics (dashed lines).\nshown in Fig.2. Figure2(a) reveals a strong localization of the intens ity at the edges of the\nwaveguide. The non-vanishing intensity close to the antenna and be tween the edges of the\nwaveguide ( y= 1−4µm) can be attributed to nonresonant, forced excitation by the Oe rsted\nfield created by the microwave current. Figure2(b) shows the spin -wave dispersion for the\nCMFS waveguide calculated according to Ref.19 assuming a homogen eous magnetization\noriented in y-direction by the external field. The material paramet ers used in all our calcula-\ntions were determined experimentally on the unstructured film via fe rromagnetic resonance\n(MS= 1003kA/m and Hani= 1kA/m) and via µBLS in the micro-structured waveguide\n(Aex= 13pJ/m) following a method described in [9]. The effective field of µ0Heff= 46mT\nused in the calculations was obtained by micromagnetic simulations. As can be seen, the\nlower cut-off frequency f= 6.9GHz is well above the excitation frequency of fe= 3.5GHz\n(see dashed line in Fig.2(b)). In the center of the waveguide, the a ssumption of a homoge-\nneous magnetizationis a very goodapproximation. Close to theedge s, demagnetizing effects\nare responsible for a strongly decreased effective field and an inhom ogeneous magnetization\nconfiguration. The inhomogeneity of the magnetization does not allo w for a quantitative\n45 10 15 20\nmicrowave power (mW)BLS intensity (arb. u.)fd=fe\nfd=2fe\nfd=3fe\nFIG. 3. (Color online) Power dependence of the BLS intensity for the detection frequencies fd=\n3,5GHz, 7GHz, and 10.5GHz for the fixed excitation frequency fe= 3.5GHz. Please note the\nlog-log presentation of the data. The lines in the graph corr espond to fits according to Eq.1. A\nleast square fit of the data yields s1f= 0.9±0.1,s2f= 2.1±0.1, ands3f= 2.8±0.3.\nmodeling of spin dynamics close to edges.[20] However, from previous work it is known that\nthis field and magnetization configuration allow for the existence of lo calized spin waves -\ncommonly referred to as edge modes - energetically far below the sp in-wave dispersion for\npropagating modes in the center.[11, 12] Therefore, we conclude t hat the spin-wave mode\natfd=fe= 3.5GHz is excited resonantly by the microwave field.\nAs can be seen from the spectrum in Fig.1, the applied microwave pow er of 20mW is\nsufficiently high to observe the nonlinear generation of higher harmo nics of the excitation\nfrequency fein theµBLS spectra. These resonant frequency multiplications to f= 2fe=\n7.0GHz and f= 3fe= 10.5GHz result in the excitation of propagating spin-wave modes in\nthe waveguide energetically above the cut-off frequency of the dis persion shown in Fig.2(b).\nFigure3 shows the dependence of the directly excited mode and the higher harmonics\non the applied microwave power. This data has been acquired close to the position of the\nedge mode and near the antenna at x= 0.7µm andy= 0.8µm. The data is presented on\na log-log scale with fits according to\nIn(p) =Anpsn+b, (1)\nwhereInis theµBLS intensity, Ana coupling parameter, pthe applied microwave power,\nandbthe noise-level in our measurement. As expected, these process es do not show a\n5threshold power level, but reliable detection on the background of t he noise is not possible\nfor powers below 5mW for fd= 3fe= 10.5GHz. The different slopes of the curves for the\ndifferent spin-wave modes nare caused by the different power-laws specified by the exponent\nsn. A least square fit of the data yields s1f= 0.9±0.1,s2f= 2.1±0.1, ands3f= 2.8±0.3.\nThese experimental findings close to the integer values 1, 2, and 3 a re in accordance with\nboth reported experimental data and theoretical predictions fo r the nonlinear generation of\nhigher harmonics.[3, 7]\nThe observation of the second harmonic can be understood qualita tively by considering\nthe strong demagnetizing fields caused by the out-of-plane compo nentmz(t) during the\nmagnetization precession. Due to the demagnetizing fields, the mag netization precession\nM(t) around the y-direction (defined by the effective field) follows an ellip tical trajectory\nrather than a circular one. In contrast to the case of a circular pr ecession, the resulting\nprojection of Mon the y-axis is time dependent and oscillating with the frequency 2 fe. The\nresulting dynamic dipolar field |hy(t)| ∝m2\nx−m2\nzcan be regarded as the source for the\nfrequency doubling. Similar considerations lead to the observation o f higher harmonics. A\nfull quantitative derivation of higher harmonic generation and othe r nonlinear effects based\non the expansion of the LLG in terms of the dynamic magnetization ca n be found in Ref.7.\nFigures4(a)and(b)showintensity mapsforthedetectionfreque nciesfd= 2fe= 7.0GHz\nandfd= 3fe= 10.5GHz, respectively. In both cases the intensity radiated from the\nposition of the edge mode is strongly directed, has a small transver sal aperture, and shows\nnondiffractive behavior. These radiation characteristics recorde d for the three spin-wave\nmodes at f= 3.5, 7.0 and 10.5GHz presented in Figs.2(a) and 4 are responsible for t he\nintensity distribution shown in the spectrum in Fig.1. The position of t his measurement\nis indicated in the corresponding intensity maps with a circle. Since the spin wave at\nfe= 3.5GHz is localized at the edges of the waveguide, its intensity is compar ably weak in\nthe center. In contrast, the higher harmonics have frequencies above the cut-off frequency of\nthe spin-wave dispersion and can propagate in the center of the wa veguide. The propagation\nanglesofthese modessupport theintensity distributionrecorded inourmeasurement. While\nan increased intensity can be found already for fd= 2fe, forfd= 3fethe two beams starting\nfrom both edges of the spin-wave waveguide even intersect at the measurement position\nresultinginthehighestintensityatthispoint. Becauseofthewell-de finedpropagationangles\nof the higher harmonics and the localization of the edge mode, the int ensity distribution is\n6012345\ny position ( µm)123456789x position ( µm)θ=78.0◦\nHextvG\nθfd=2fe\n012345\ny position ( µm)123456789x position ( µm)θ=67.0◦\nHextvG\nθfd=3fe\n(a) (b)\nFIG. 4. (Color online) µBLS intensity distribution for (a) fd= 2fe= 7.0GHz and (b) fd=\n3fe= 10.5GHz. Both intensity maps show strongly directed spin-wave beams along the angle\nθ=∠(Hext,vG). The lines in the maps are guides to the eye to identify the pr opagation angle θ.\nThe blue dots in the graphs indicate the position of the measu rement presented in Fig.1.\nstrongly depending on the measurement position.\nThis observation of spin-wave beams with small transversal apert ure is reminiscent of the\nresults in Refs.21 and 22, where nonlinear three-magnon scatter ing in yttrium iron garnet\nis reported. However, in that case, the propagation direction of t he nonlinearly generated\nspin wave is given by momentum conservation in the scattering proce ss. In contrary, in\nour case, due to the strong localization of the edge mode, the assu mption of a well-defined\ninitial wavevector and, thus, a strict momentum conservation is no t justified. In particular,\nit is not possible to find an initial wavevector that allows for the nonline ar generation of the\nsecond and third harmonic at the same time still respecting momentu m conservation.\nIn the following, we will describe the observed propagationcharact eristics using the prop-\nerties of the anisotropic spin-wave dispersion in a magnetic thin film.[13 , 14, 19] Because\nof this anisotropy, the direction of the flow of energy, which is given by the direction of\nthe group velocity vG= 2π∂f(k)/∂kof the investigated spin waves, can differ significantly\nfrom the direction of its wavevector k. To estimate the relevant range in k-space in our\nexperiment, we have to consider the lateral dimensions of the sour ce for the nonlinear pro-\ncesses. Since the excitation by the oscillating Oersted field is most effi cient directly below\nthe antenna, the edge mode has the highest intensity in this region g iven by the width of\nthe antenna of ∆ x= 1µm. The spread of the edge mode in y-direction can be estimated\n7from the intensity map in Fig.2(a) to be smaller than 1 µm. Because of this localization, the\nedge mode in our measurement at f= 3.5GHz can be regarded as a source for the nonlinear\nemission of the higher harmonics with lateral dimensions of approxima tely 1×1µm2. As a\nfirst approximation, the Fourier transformation of this geometry lets us estimate the maxi-\nmum wavevector that can be excited by the edge mode to be kmax≈6.3µm−1. As we will\nsee, the direction of the group velocity can be assumed to be const ant for most wavevectors\nthat can be excited. This finally leads to the formation of the caustic s in our experiment.\nFor given frequency and external field, the iso-frequency curve f(kx,ky) =constcan be\ncalculated analytically from the dispersion relation. Calculations for f= 2fe= 7.0GHz and\nf= 3fe= 10.5GHz are illustrated in Fig.5(a), where kyis shown as a function of kx. Using\nthis data, we calculate the direction θof the flow of energy of the spin waves relative to the\nexternally applied field by:\nθ=∠(Hext,vG) = arctan( vx/vy) = arctan( dky/dkx). (2)\nFigure5(b)showsthecalculatedpropagationangle θintheCMFSwaveguideasafunction\nofky. The most important feature in the trend of θis the small variation of ∆ θ≤2◦in the\nrange of ky= 2−7µm−1for both frequencies f= 2feandf= 3fe. While the wavevector\nkchanges, the direction of vG- and, thus, the flow of energy - keeps almost constant as\na function of k. In this range, which includes the maximum wavevector kmax≈6.3µm−1\nthat can be emitted from the edge mode (see considerations above ), the calculations yield\nθcalc(2f) = 79◦andθcalc(3f) = 66◦as mean values, respectively. The dash-dotted lines in\nFig.5(b) represent the propagation angles θof the spin-wave beams observed experimentally\nas shown in Fig.4 ( θexp(2f) = 78◦andθexp(3f) = 67◦). The comparison of experimental\nfindings and analytical calculations shows an agreement within the ex pected accuracy of\nour measurement setup and is, therefore, supporting our conclu sion. Higher harmonics with\nky≤2µm−1are emitted with strongly varying directions from the edge mode and can be\nregarded as a negligible background in our measurement.\nIn summary, we reported nonlinear higher harmonic generation fro m a localized source\nin a micro-structured CMFS waveguide leading to the emission of stro ngly directed spin-\nwave beams or caustics. This observation results from the complex interplay of different\nphenomena in magnonic transport in magnetic microstructures. Th e localization of an edge\nmode due to demagnetizing fields in the waveguide leads to the format ion of a source for\n80 2 4 6 8\nkx (µm−1)02468ky (µm−1)\nk\nvG\nHextθf=2fe\nf=3fe\n0 2 4 6 8\nky (µm−1)657075808590θ (degree)f=2fe\nf=3fe\n(a) (b)\nFIG. 5. (Color online) Analytical calculations according t o Ref.19. (a) Iso-frequency curves\nfconst=f(kx,ky) forf= 2fe= 7.0GHz and f= 3fe= 10.5GHz. Based on these calculations ex-\nemplary directions for k,vG,Hextand the propagation angle θ=∠(Hext,vG) = arctan( dky/dkx)\nare shown in the graph. (b) Radiation direction θcalculated from the iso-frequency curves shown\nin (a). Dash-dotted lines correspond to the angles θexpobserved in the experiment.\nthe following nonlinear processes. The nonlinear higher harmonic gen eration results in the\nresonant excitation and emission of propagating spin waves at 2 feand 3fein a wavevector\nrange corresponding to the localization of the edge mode. The expe rimentally observed\npower dependencies of the different spin-wave modes show the exp ected behavior for direct\nresonant excitation and nonlinear higher harmonic generation. As s hown by our calculation,\nthe anisotropic spin-wave dispersion yields a well-defined direction of the flow of energy\nof the emitted spin waves in the relevant range in k-space. The calcu lation is not only\nqualitatively in accordance with our experimental findings but does a lso show quantitative\nagreement.\nWe gratefully acknowledge financial support by the DFG Research U nit 1464 and the\nStrategic Japanese-German Joint Research from JST: ASPIMATT . Thomas Br¨ acher is sup-\nported by a fellowship of the Graduate School Materials Science in Ma inz (MAINZ) through\nDFG-fundingoftheExcellence Initiative(GSC266). Wethankourco lleaguesfromthe Nano\nStructuring Center of the TU Kaiserslautern for their assistance in sample preparation .\n∗tomseb@physik.uni-kl.de\n[1] H.Schultheiss, X.Janssens, M. van Kampen, F.Ciubotaru , S.J.Hermsdoerfer, B.Obry,\n9A.Laraoui, A.A.Serga, L.Lagae, A.N.Slavin, B.Leven, B.Hi llebrands, Phys. Rev. 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B79, 144428 (2009).\n11"    },    {        "title": "1407.0227v2.Transport_properties_of_Lévy_walks__an_analysis_in_terms_of_multistate_processes.pdf",        "content": "epl draft\nTransport properties of L\u0013 evy walks: an analysis in terms of mul-\ntistate processes\nGiampaolo Cristadoro1, Thomas Gilbert(a)2, Marco Lenci1;3andDavid P. Sanders4\n1Dipartimento di Matematica, Universit\u0012 a di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy.\n2Center for Nonlinear Phenomena and Complex Systems, Universit\u0013 e Libre de Bruxelles, C. P. 231, Campus Plaine,\nB-1050 Brussels, Belgium.\n3Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Via Irnerio 46, 40126 Bologna, Italy.\n4Departamento de F\u0013 \u0010sica, Facultad de Ciencias, Universidad Nacional Aut\u0013 onoma de M\u0013 exico, Ciudad Universitaria,\n04510 M\u0013 exico D.F., Mexico.\nPACS 05.40.Fb { Random walks and L\u0013 evy \rights\nPACS 05.60.-k { Transport processes\nPACS 02.50.-r { Probability theory, stochastic processes, and statistics\nPACS 02.30.Ks { Delay and functional equations\nAbstract { Continuous time random walks combining di\u000busive and ballistic regimes are intro-\nduced to describe a class of L\u0013 evy walks on lattices. By including exponentially-distributed waiting\ntimes separating the successive jump events of a walker, we are led to a description of such L\u0013 evy\nwalks in terms of multistate processes whose time-evolution is shown to obey a set of coupled delay\ndi\u000berential equations. Using simple arguments, we obtain asymptotic solutions to these equations\nand rederive the scaling laws for the mean squared displacement of such processes. Our calculation\nincludes the computation of all relevant transport coe\u000ecients in terms of the parameters of the\nmodels.\nRandom walks described by L\u0013 evy \rights give rise to\ncomplex di\u000busive processes [1{4] and have found many\napplications in physics and beyond [5{8]. Whereas the\nrandom walks associated with Brownian motion are char-\nacterized by Gaussian propagators whose variance grows\nlinearly in time, the propagators of L\u0013 evy \rights have in\f-\nnite variance [9{11]; they occur in models of random walks\nsuch that the probability of a long jump decays slowly with\nits length [12].\nBy considering the propagation time between the two\nends of a jump, one obtains a class of models known as\nL\u0013 evy walks [13{19]. A L\u0013 evy walker thus follows a contin-\nuous path between the two end points of every jump, per-\nforming each in a \fnite time; instead of having an in\fnite\nmean squared displacement, as happens in a L\u0013 evy \right\nwhose jumps take place instantaneously, a L\u0013 evy walker\nmoves with \fnite velocity and, ipso facto, has a \fnite mean\nsquared displacement, although it may increase faster than\nlinearly in time.\nA L\u0013 evy \right is characterized by its probability density\n(a)E-mail: thomas.gilbert@ulb.ac.beof jump lengths x, orfree paths , which we denote \u001e(x). It\nis assumed to have the asymptotic scaling, \u001e(x)\u0018x\u0000\u000b\u00001,\nwhose exponent, \u000b>0, determines whether the moments\nof the displacement are \fnite. In particular, for \u000b\u00142,\nthe variance diverges.\nIn the framework of continuous time random walks [5,\nChs. 10 & 13], a probability distribution \b( r;t) of mak-\ning a displacement rin a timetis introduced, such that,\nfor instance, in the so-called velocity picture, \b( r;t) =\n\u001e(jrj)\u000eD(t\u0000jrj=v), wherevdenotes the constant speed of\nthe particle and \u000eD(:) is the Dirac delta function. Consid-\nering the Fourier-Laplace transform of the propagator of\nthis process, one obtains, in terms of the parameter \u000b, the\nfollowing scaling laws for the mean squared displacement\nafter timet[13,20],\nhr2it\u00188\n>>>>>><\n>>>>>>:t2; 0<\u000b< 1;\nt2=logt; \u000b = 1;\nt3\u0000\u000b; 1<\u000b< 2;\ntlogt; \u000b = 2;\nt; \u000b> 2:(1)\np-1arXiv:1407.0227v2  [cond-mat.stat-mech]  10 Nov 2014G. Cristadoro et al.\nIn this Letter, we consider L\u0013 evy walks on lattices and\ngeneralize the above description, according to which a new\njump event takes place as soon as the previous one is\ncompleted, to include an exponentially-distributed wait-\ning time which separates successive jumps. This induces a\ndistinction between the states of particles which are in the\nprocess of completing a jump and those that are waiting to\nstart a new one. As shown below, such considerations lead\nto a theoretical formulation of the model as a multistate\ngeneralized master equation [21{23], which translates into\na set of coupled delay di\u000berential equations for the corre-\nsponding distributions.\nThe physical motivation for the inclusion of an\nexponentially-distributed waiting time between successive\njump events stems, for instance, in the framework of\nchaotic scattering, from the time required to escape a frac-\ntal repeller [24, 25], or, more generally, the time spent in\na chaotic transient [26]. In the framework of active trans-\nport, such as dealing with the motion of particles embed-\nded within living cells [27], such waiting times may help\nmodel the complex process related to changes in the di-\nrection of propagation of such particles. This is also rele-\nvant to laser cooling experiments [28], where a competition\nin the damping and increase of atomic momenta induces\na form of random walk in momentum space. The times\nspent by atoms in small momenta states typically follow\nexponential distributions.\nThe stop-and-go patterns of random walkers thus gen-\nerated have been studied in the context of animal foraging\n[29]. Such search strategies have been termed saltatory. In\ncontrast to classical strategies, according to which animals\neither move while foraging or stop to ambush their prey,\na saltatory searcher alternates between scanning phases,\nwhich are performed di\u000busively on a local scale, and re-\nlocation phases, during which motion takes place with-\nout search. Examples of such intermittent behaviour have\nbeen identi\fed in a variety of animal species [30, 31], as\nwell as in intracellular processes such as proteins binding\nto DNA strands [32]. Visual searching patterns whereby\ninformation is extracted through a cycle of brief \fxations\ninterspersed with gaze shifts [33] provide another illustra-\ntion in the context of neuroscience. One can also think of\napplications to sociological processes, for instance when\ninteractions between individuals is sampled at random\ntimes, independent of the underlying process [34].\nFrom a mathematical perspective, an important ques-\ntion that arises in the framework of foraging is that of\noptimal strategies [35]. As reported in [36], L\u0013 evy \right\nmotion can, under some conditions on the nature and dis-\ntribution of targets, emerge as an optimal strategy for\nnon-destructive search, i.e., when targets can be visited\nin\fnitely often. For destructive searches on the other\nhand, intermittent search strategies with exponentially-\ndistributed waiting times provide an alternative to L\u0013 evy\nsearch strategies, which turns out to minimize the search\ntime [37]. The processes we analyze in this Letter, al-\nthough they are restricted to motion on lattices, can bethought of as extensions of intermittent search processes to\npower-law distributed relocation phases which are typical\nof L\u0013 evy search strategies, thus opening a new perspective.\nWe show below that, inasmuch as the dispersive proper-\nties are concerned, a complete characterisation of the pro-\ncess can be obtained, which reproduces the scaling laws\n(1), as well as yields the corresponding transport coe\u000e-\ncients, whether normal or anomalous. These results also\nelucidate the incidence of exponential waiting times on\nthese coe\u000ecients.\nL\u0013 evy walks as multistate processes. { We call\npropagating the state of a particle which is in the process\nof completing a jump. In contrast, the state of a par-\nticle waiting to start a new jump is called scattering1.\nWhereas particles switch from propagating to scattering\nstates as they complete a jump, particles in a scattering\nstate can make transitions to both scattering and propa-\ngating states; as soon as their waiting time has elapsed,\nthey move on to a neighbouring site and, doing so, may\nswitch to a propagating state and carry their motion on\nto the next site, or start anew in a scattering state.\nWe consider a d-dimensional cubic lattice of individual\ncells n2Zd. The state of a walker at position nand\ntimetcan take on a countable number of di\u000berent values,\nspeci\fed by two integers, k\u00150 andj2f1;:::;zg, where\nz\u00112dis the coordination number of the lattice. Scatter-\ning states are labeled by the state k= 0 and propagating\nstates by the pair ( k;j), such that k\u00151 counts the re-\nmaining number of lattice sites the particle has to travel\nin direction jto complete its jump.\nTime-evolution proceeds as follows. After a random\nwaiting time t, exponentially-distributed with mean \u001cR,\na particle in the scattering state k= 0 changes its state to\n(k;j) with probability \u001ak=z, moving its location from site\nnto site n+ej. Conversely, particles which are at site n\nin a propagating state ( k;j),k\u00151, jump to site n+ej,\nin time\u001cB, changing their state to ( k\u00001;j).\nThe waiting time density of the process is the function\n k(t) =(\n\u001c\u00001\nRe\u0000t=\u001cR; k = 0;\n\u000eD(t\u0000\u001cB); k6= 0:(2)\nWhen a step takes place, the transition probability to go\nfrom state ( k;j) to state (k0;j0) is\np(k;j);(k0;j0)=(\n\u001ak0=z; k = 0;\n\u000ek\u00001;k0\u000ej;j0; k6= 0;(3)\nwhere\u000e:;:is the Kronecker symbol.\nFor de\fniteness, we consider below the following simple\nparameterisation of the transition probabilities,\n\u001ak=(\n1\u0000\u000f; k = 0;\n\u000f[k\u0000\u000b\u0000(k+ 1)\u0000\u000b]; k\u00151;(4)\n1B\u0013 enichou et al.[37] refer to these two states as respectively bal-\nlistic and di\u000busive.\np-2Transport properties of L\u0013 evy walks in terms of multistate processes\nin terms of the parameters 0 <\u000f< 1, which weights scat-\ntering states relative to propagating ones, and \u000b>0, the\nasymptotic scaling parameter of free path lengths.\nMaster equation. { The probability distribution of\nparticles at site nand timet,P(n;t), is a sum of the dis-\ntributions over the scattering states, P0(n;t), and propa-\ngating states, Pk;j(n;t),k\u00151 and 1\u0014j\u0014z. According\nto eqs. (2) and (3), changes in the distribution of ( k;j)-\nstates,k\u00151, in cell narise from particles located at cell\nn\u0000ejwhich make a transition from either state 0 or state\n(k+ 1;j). Since the latter transitions can be traced back\nto changes in the distribution of ( k+ 1;j)-states in cell\nn\u0000ejat time\u001cBearlier, we can write2\n@tPk;j(n;t)\u0000@tPk+1;j(n\u0000ej;t\u0000\u001cB)\n=\u001ak\nz\u001cR[P0(n\u0000ej;t)\u0000P0(n\u0000ej;t\u0000\u001cB)];(5)\nwhich accounts for the fact that a positive 0-state contri-\nbution at time tbecomes a negative one at time t+\u001cB.\nApplying this relation recursively, we have\nSee eq. (6)next page.\nTerms lost by (1 ;j)-states in cells n\u0000ej,j= 1;:::;z , are\ngained by the 0-state in cell n, which also gains contribu-\ntions from 0-state transitions. Since the scattering state\nalso loses particles at exponential rate 1 =\u001cR, we have\n@tP0(n;t) =1\nz\u001cRzX\nj=11X\nk=0\u001akP0(n\u0000(k+ 1)ej;t\u0000k\u001cB)\n\u00001\n\u001cRP0(n;t): (7)\nIt is straightforward to check that eqs. (6) and (7) are con-\nsistent with conservation of probability3,P\nnP(n;t) = 1.\nFraction of scattering particles. { As discussed\nbelow, an important role is played by the overall fraction of\nparticles in the scattering state, S0(t)\u0011P\nn2ZdP0(n;t).\nFrom eq. (7), this quantity is found to obey the following\nlinear delay di\u000berential equation,\n\u001cR_S0(t) =1X\nk=1\u001akS0(t\u0000k\u001cB)\u0000\u000fS0(t): (8)\nGiven initial conditions, e.g. S0(t) = 0,t<0, andS0(0) =\n1 (all particles start in a scattering state), this equation\ncan be solved by the method of steps [38]. Because the\nsum of the coe\u000ecients on the right-hand side of eq. (8) is\nzero, the solutions are asymptotically constant and can be\nclassi\fed in terms of the parameter \u000b.\n2The possible addition of source terms into this expression will\nnot be considered here.\n3A simpli\fcation occurs if one considers the distribution of prop-\nagating states in direction j,Pj(n;t) =P1\nk=1Pk;j(n;t). Us-\ning eq. (4), the time-evolution of this quantity is @tPj(n;t) =\n\u000f=(z\u001cR)P1\nk=1k\u0000\u000b[P0(n\u0000kej;t\u0000(k\u00001)\u001cB)\u0000P0(n\u0000kej;t\u0000k\u001cB)].For\u000b > 1, the average return time to the 0-state,P1\nk=0\u001ak(\u001cR+k\u001cB), is \fnite and given in terms of the Rie-\nmann zeta function, sinceP1\nk=0k\u001ak=\u000f\u0010(\u000b). The process\nis thus positive-recurrent and we have\nlim\nt!1S0(t) =\u001cR\n\u001cR+\u000f\u001cB\u0010(\u000b)(\u000b>1): (9a)\nIn the remaining range of parameter values, 0 < \u000b\u00141,\nthe process is null-recurrent: the average return time to\nthe 0-state diverges and lim t!1S0(t) = 0. If\u000b6= 1, the\ndecay is algebraic,\nlim\nt!1(t=\u001cB)1\u0000\u000bS0(t) =sin(\u0019\u000b)\n\u0019\u000f\u001cR\n\u001cB(0<\u000b< 1);(9b)\nwhich can be obtained from a result due to Dynkin [39];\nsee also Refs. [40, Vol. 2, xXIV.3] and [28,x4.4]. The\ncase\u000b= 1 is a singular limit with logarithmic decay,\nlim\nt!1log(t=\u001cB)S0(t) =1\n\u000f\u001cR\n\u001cB(\u000b= 1): (9c)\nMean squared displacement. { Assuming an ini-\ntial position at the origin, the second moment of the dis-\nplacement ishn2it=P\nn2Zdn2P(n;t). Its time-evolution\nis obtained by di\u000berentiating this expression with respect\nto time and substituting eqs. (6) and (7),\n\u001cRd\ndthn2it=S0(t) +\u000f1X\nk=12k+ 1\nk\u000bS0(t\u0000k\u001cB);(10)\nwhere, using eq. (4), we made use of the identityP1\nj=k\u001aj= 1 fork= 0 and\u000fk\u0000\u000botherwise. The time-\nevolution of the second moment is thus obtained by inte-\ngrating the fraction of 0-state particles,\n\u001cRhn2it=Zt\n0dsS0(s) +\u000fbt=\u001cBcX\nk=12k+ 1\nk\u000bZt\u0000k\u001cB\n0dsS0(s);\n(11)\nwhere, assuming the process starts at t= 0, we set S0(t) =\n0 fort<0.\nAs emphasized earlier, equation (8) can be solved an-\nalytically given initial conditions on the state of walkers.\nBy extension, so can equation (11), thus providing an ex-\nact time-dependent expression of the mean squared dis-\nplacement. This is particularly useful when one wishes to\nstudy transient regimes and the possibility of a crossover\nbetween di\u000berent scaling behaviours, or indeed when the\nasymptotic regime remains experimentally or numerically\nunaccessible. The analytic expression of the mean squared\ndisplacement and the issue of the transients will be stud-\nied elsewhere. Here, we focus on the asymptotic regime,\ni.e.,t\u001d\u001cB.\nSubstituting the asymptotic expressions (9), into\neq. (11), we retrieve the regimes described by eq. (1) and\nobtain the corresponding coe\u000ecients.\np-3G. Cristadoro et al.\n@tPk;j(n;t) =1\nz\u001cR1X\nk0=1\u001ak+k0\u00001h\nP0(n\u0000k0ej;t\u0000(k0\u00001)\u001cB)\u0000P0(n\u0000k0ej;t\u0000k0\u001cB)i\n: (6)\na=2.5\ne=0.5\n1011021031041.92.02.12.22.32.42.52.6\ntXn2\\tt\n1011021031040.59850.59900.59950.60000.60050.60100.6015S0HtL\n(a) Normal di\u000busion, \u000b= 5=2\na=2.\ne=0.5\n1011021031040.60.70.80.91.0\ntXn2\\t@tlogHtLD\n1011021031040.5500.5520.5540.5560.5580.560S0HtL\n(b) Weak super-di\u000busion, \u000b= 2\na=1.5\ne=0.5\n1011021031040.600.650.700.750.80\ntXn2\\tt1.5\n1011021031040.440.450.460.470.480.49S0HtL\n(c) Super di\u000busion, \u000b= 3=2\nFig. 1: Examples of numerical computations of hn2itfor pa-\nrameters values \u000b>1, rescaled by their respective asymptotic\nscalings with respect to time ( \u000f= 1=2 in all cases). The dotted\nlines correspond to eq. (12a). The insets show the evolution\nof the fraction of scattering states towards their asymptotic\nvalues, given by eq. (9a).\nStarting with the positive-recurrent regime, \u000b > 1,\neq. (9a), we have the three asymptotic regimes, t\u001d\u001cB,\nhn2it't\n\u001cR+\u000f\u001cB\u0010(\u000b)\na=0.5\ne=0.5\n1011021031040.460.470.480.490.50\ntXn2\\tt2\n1011021031040.6360.6380.6400.6420.6440.6460.648t0.5S0HtL(a) Ballistic di\u000busion, \u000b= 1=2\na=1.\ne=0.5\n1011021031040.800.850.900.951.00\ntXn2\\tlogHtLt2\n1011021031041.01.21.41.61.82.0logHtLS0HtL\n(b) Sub-ballistic di\u000busion, \u000b= 1\nFig. 2: Same as \fg. 1 for the range of parameters 0 < \u000b\u00141,\nwith comparisons to eq. (12b) and, in the insets, eqs. (9b) and\n(9c).\n\u00028\n><\n>:1 +\u000f[\u0010(\u000b) + 2\u0010(\u000b\u00001)]; \u000b> 2;\n2\u000flog(t=\u001cB); \u000b = 2;\n2\u000f\n(2\u0000\u000b)(3\u0000\u000b)(t=\u001cB)2\u0000\u000b; 1<\u000b< 2:(12a)\nWhereas the \frst regime, \u000b > 2, yields normal di\u000busion,\nthe other two correspond, for \u000b= 2, to a weak form of\nsuper-di\u000busion, and, for 1 < \u000b < 2, to super-di\u000busion,\nsuch that the mean squared displacement grows with a\npower of time 3\u0000\u000b>1, faster than linear4.\nBallistic di\u000busion occurs in the null-recurrent regime of\nthe parameter, 0 < \u000b\u00141. Using eqs. (9b) and (9c), we\n\fnd\nhn2it't2\n\u001c2\nB(\n1=log(t=\u001cB); \u000b = 1;\n1\u0000\u000b; 0<\u000b< 1:(12b)\nThe asymptotic regimes described by eqs. (12) gener-\nalize to continuous-time processes similar results found in\n4Equation (12a) assumes \u000f >0. If one takes the limit \u000f!0,\nsub-leading terms may become relevant. In particular, when \u000f= 0,\nnormal di\u000busion is recovered and the right-hand side of (12a) is t=\u001cR\nfor all\u000b.\np-4Transport properties of L\u0013 evy walks in terms of multistate processes\nthe context of countable Markov chains applied to discrete\ntime processes [41]. They can also be compared to results\nobtained in Ref. [42]. Although the L\u0013 evy walks considered\nby these authors do not include exponentially-distributed\nwaiting times separating successive propagating phases,\nour results are rather similar to theirs; the only actual\ndi\u000berences arise in the regime of normal di\u000busion, \u000b>2.\nIn \fgs. 1 and 2, the asymptotic results (12) are com-\npared to numerical measurements of the mean squared\ndisplacement of the process de\fned by eqs. (2), (3) and\nthe transition probabilities (4). Timescales were set to\n\u001cR\u0011\u001cB\u00111 and the lattice dimension to d= 1. The\nalgorithm is based on a classic kinetic Monte Carlo algo-\nrithm [43], which incorporates the possibility of a ballistic\npropagation of particles after they undergo a transition\nfrom a scattering to a propagating state. For each realisa-\ntion, the initial state is taken to be scattering. Positions\nare measured at regular intervals on a logarithmic time\nscale for times up to t= 104\u001cR. Averages are performed\nover sets of 108trajectories.\nConcluding remarks. { The speci\fcity of our ap-\nproach to L\u0013 evy walks lies in the inclusion of exponentially-\ndistributed waiting times that separate successive jumps.\nThis additional feature induces a natural description of\nthe process in terms of multiple propagating and scatter-\ning states whose distributions evolve according to a set of\ncoupled delay di\u000berential equations.\nThe mean squared displacement of the process depends\non the distribution of free paths and boils down to a simple\nexpression involving time-integrals of the fraction of scat-\ntering states. Using straightforward arguments, precise\nasymptotic expressions were obtained for this quantity,\nwhich reproduce the expected scaling regimes [13,20], and\nprovide values of the di\u000busion coe\u000ecients, whether normal\nor anomalous.\nOur results con\frm that, in the null-recurrent regime\nof ballistic transport, scattering events, are unimportant.\nFurthermore, these events do not modify the exponent\nof the mean squared displacement in the positive recur-\nrent regime; in other words, the addition of a scattering\nphase has no incidence on the scaling exponents. In this\nregime, however, the transport coe\u000ecients, whether nor-\nmal or anomalous, depend on the details of the model,\nunderlying the relevance of pausing times that separate\nlong \right events, for example, in the context of animal\nforaging [36].\nAlthough the results we reported are limited to walks\nwith exponentially distributed waiting times, our formal-\nism can be easily extended to include the possibility of\nwaiting times with power law distributions such as ob-\nserved in Ref. [44]. Such processes are known to allow\nfor sub-di\u000busive transport regimes [11]. The combination\nof two power law scaling parameters, one for the waiting\ntime and the other for the duration of \rights, indeed yields\na richer set of scaling regimes [45], which can be studied\nwithin our framework.Our results can on the other hand be readily applied to\nthe regime \u001cR=\u001cB\u001c1, i.e., such that the waiting times\nin the scattering state are typically negligible compared\nto the ballistic timescale. This is the regime commonly\nstudied in reference to L\u0013 evy walks.\nOur investigation simultaneously opens up new avenues\nfor future work. Among results to be discussed elsewhere,\nour formalism can be used to obtain exact solutions of the\nmean squared displacement as a function of time. As dis-\ncussed already, this is particularly useful to study transient\nregimes, such as can be observed when the distribution of\nfree paths has a cut-o\u000b or, more generally, when it crosses\nover from one regime to another, e.g. from a power law for\nsmall lengths to exponential decay for large ones, or when\nthe anomalous regime is masked by normal sub-leading\ncontributions which may nonetheless dominate over time\nscales accessible to numerical computations [46]. One can\nalso apply these ideas to the anomalous photon statistics\nof blinking quantum dots [47,48]. The on/o\u000b switchings of\na quantum dot typically exhibit power law distributions.\nIn the limit of strong \felds, however, the on-times display\nexponential cuto\u000bs.\nAnother interesting regime occurs when, in the positive-\nrecurrent range of the scaling parameter, \u000b>1, the like-\nlihood of a transition from a scattering to a propagating\nstate is small, \u000f\u001c1. A similar perturbative regime arises\nin the in\fnite horizon Lorentz gas in the limit of narrow\ncorridors [49]. As is well-known [50], the scaling parame-\nter of the distribution of free paths has the marginal value\n\u000b= 2, such that the mean squared displacement asymp-\ntotically grows with tlogt. Although it has long been ac-\nknowledged that the in\fnite horizon Lorentz gas exhibits\nfeatures similar to a L\u0013 evy walk [51, 52], we argue that\na consistent treatment of this model in such terms is not\npossible unless exponentially-distributed waiting times are\ntaken into account that separate successive jumps. In-\ndeed, the parameter \u000f, which weights the likelihood of\na transition from scattering to propagating states, is the\nsame parameter that separates the average relaxation time\nof the scattering state from the ballistic timescale, i.e.,\n\u001cB=\u001cR/\u000f\u001c1. This is the subject of a separate publica-\ntion [53].\n\u0003\u0003\u0003\nWe wish to thank Eli Barkai for useful comments and\nsuggestions. This work was partially supported by FIRB-\nProject No. RBFR08UH60 (MIUR, Italy), by SEP-\nCONACYT Grant No. CB-101246 and DGAPA-UNAM\nPAPIIT Grant No. IN117214 (Mexico), and by FRFC\nconvention 2,4592.11 (Belgium). T.G. is \fnancially sup-\nported by the (Belgian) FRS-FNRS.\nREFERENCES\n[1]Haus J. W. andKehr K. W. ,Physics Reports ,150\n(1987) 263.\np-5G. Cristadoro et al.\n[2]Weiss G. H. ,Aspects and Applications of the Random\nWalk (North-Holland, Amsterdam) 1994.\n[3]Krapivsky P. L., Redner S. andBen-Naim E. ,A Ki-\nnetic View of Statistical Physics (Cambridge University\nPress, Cambridge, UK) 2010.\n[4]Klafter J. andSokolov I. M. ,First steps in random\nwalks: from tools to applications (Oxford University Press,\nOxford, UK) 2011.\n[5]Shlesinger M. F., Zaslavsky G. 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Paris -Sud, Université Paris Saclay, 91767 Palaiseau, France  \n2LABSTICC, UMR 6285 CNRS, Université de Bretagne Occidentale, 29238 Brest, France  \n3 Thales Research and Technology, Thales 91767 Palaiseau , France  \n* Email : madjid.anane@u -psud.fr  \n \nA magnetic material combining low losses and  large Perpendicular Magnetic Anisotropy  (PMA) is still a \nmissing brick in the magnonic  and spintronic field s. We report  here  on the growth of ultrathin Bismuth \ndoped Y 3Fe5O12 (BiYIG ) films on Gd 3Ga 5O12 (GGG) and substituted GGG (sGGG) (111) oriented \nsubstrates. A fine tuning of the PMA is obtained  using both epitaxial strain and growth induced \nanisotrop ies. Both spontaneously in -plane and out -of-plane magnetized thin films can be elaborated . \nFerromagnetic Resonance  (FMR)  measurement s demonstrate the  high dynamic quality of these BiYIG \nultrathin  films , PMA films with Gilbert damping values as low as 3 10-4 and FMR linewidth  of 0.3 mT at \n8 GHz are achieved  even for films that do not exceed 30 nm in thickness . Moreover, w e measure \nInverse Spin Hall Effect (ISHE) on Pt/BiYIG stack s showing  that the magnetic insulator ’s surface  is \ntransparent to  spin current  making it appealing  for spintronic applications . \n \n  2 \n Introduction.  \nSpintronic s exploit s the electron’s spin in ferromagnetic transition metal s for data storage and data \nprocessing. Interestingly, as spintronics  codes information in the angular momentum degre es of \nfreedom , charge transport and therefore the use of conducting materials is not a requirement, opening \nthus electronics to  insulators . In magnetic insulators  (MI),  pure spin currents  are described using \nexcitation  states of the ferromagnetic background named magnons (or spin waves). Excitation, \npropagation  and  detection  of magnons are   at  the  confluent of the emerging concepts of magnonics  \n1,2, caloritronics3 and  spin -orbitronics4. Magnons, and their classical counterpart , the  spin waves (SWs) \ncan carry information over distances as large as millimeters in high quality thick YIG films, with \nfrequencies extending from the GHz to the THz regime5–7. The main figure of merit for magnonic \nmaterials is the Gilbert damping  1,5,8 which  has to be as small as possible. This makes  the number of \nrelevant materials for SW propagation quite limited and none of them has yet been found to possess a \nlarge enough perpendicular magnetic anisotropy (PMA ) to induce  spontaneous out -of-plane \nmagnetization . We report  here  on the  Pulsed  Laser  Deposition  (PLD)  growth  of  ultra -low loss   MI  \nnanometers -thick films  with large PMA : Bi substituted Yttrium Iron Garnet ( BixY3-xFe5O12 or BiYIG ) where  \ntunability of the PMA  is achieved through epitaxial strain and Bi doping level.  The peak -to-peak FMR  \nlinewidth  (that characterize the losses)  can be as low as  𝜇0𝛥𝐻pp=0.3 mT at 8 GHz for 30 nm thick \nfilms. This material thus opens new perspectives for both spintronic s and magnonic s fields as the SW \ndispersion relation can now be easily tuned through magnetic anisotropy without the need of  a large \nbias magnetic field. Moreover, energy efficient data storage devices based on magn etic textures existing \nin PMA materials like magnetic bubble s, chiral domain walls  and magnetic skyrmions  would benefit from \nsuch a low loss material for efficient operation9. \nThe study of micron -thick YIG films grown by liquid phase epitaxy (LPE) was among the hottest topics in \nmagnetism few decades ago . At this time,  it has been already noticed that unlike rare earths (Thulium, \nTerbium, Dysprosium …) substitutions, Bi substitution does not overwhelmingly increase the magnetic \nlosses10,11 even though it induces high uniaxial magnetic anisotropy12–14 . Very recently, ultra -thin MI \nfilms showing PMA have been the subject of an increasing interes t 15,16: Tm 3Fe5O12 or BaFe 12O19 \n(respectively a garnet and an hexaferrite) have been used to demonstrate spin -orbit -torque \nmagnetization reversal using a Pt over -layer as a source of spin current 4,17,18. However, their large \nmagnetic losses prohibit their use as a spin -wave medium  (reported va lue o f 𝜇0𝛥𝐻pp of TIG is  16.7  mT at \n9.5 GHz)19. Hence, whether  it is possible to fabricate ultra -low loss thin films  with a large PMA  that can \nbe used for both magnonics and spintronics applications remains to be demonstrated . Not only l ow 3 \n losses are important for long range spin wave propagation but they are also necessary for spin transfer \ntorque oscillators (STNOs) as the threshold current scales with the Gilbert damping20.  \nIn the quest for the  optimal  material platform , we explore here the growth of Bi doped YIG ultra -thin \nfilms using PLD  with different substitution;  BixY3-xIG (x= 0.7, 1 and 1.5)  and having a thickness ranging \nbetween 8 and 50 nm. We demonstrate fine tuning of the magnetic anisotropy using epitaxial strain  and  \nmeasure  ultra low Gilbert damping values ( 𝛼=3∗10−4) on ultrathin films with PMA . \nResults  \nStructural and magnetic characterization s \nThe two substrates that are used are Gallium Gadolinium Garnet  (GGG)  which is best lattice matched to \npristine YIG and substituted GGG  (sGGG) which is  traditionally used to accommodate  substituted YIG \nfilms for photonics  applications . The  difference between  Bi and Y ionic radii ( rBi = 113 pm and rY = 102 \npm)21 leads to a linear increase of  the  BixY3-xIG bulk lattice  parameter  with Bi content  (Fig. 1 -(a) and Fig. \n1-(b)). In Fig. 1,  we present the  (2−) X-ray diffraction patterns (Fig.1 -(c) and 1 -(d)) and reciprocal \nspace maps (RSM) (Fig.1 -(e) and 1 -(f)) of BiYIG on sGGG(111) and GGG(111) substrates  respectively . The \npresence of ( 222) family peaks in the diffraction spectra shown in Fig. 1 -(b) and 1 -(c) is a signature of  the \nfilms’ epitaxial quality  and the presence of Laue fringes attest s the coherent crystal structure existing \nover the whole thickness. As expected, all films on GGG are under compressive strain, whereas films \ngrown on sGGG exhibit a transition from a tensile (for x= 0.7 and 1) towards a compressive ( x= 1.5) \nstrain . Reciprocal Space Mapping  of these BiYIG samples  shown in Fig.1 -(e) and 1 -(f) evidences the \npseudomorphic nature of the growth for all films , which confirms the good epitaxy.  \nThe static magnetic properties of the films have been characterized using SQUID ma gnetometry, Faraday \nrotation measurements and Kerr microscopy. As the Bi doping has the effect of enhancing the magneto -\noptical response 22–24, we measure on average a large  Faraday rotation coefficients reaching up to 𝜃F =\n−3 °.𝑚−1 @ 632 nm  for  x= 1 Bi doping level  and 15 nm film thickness . Chern et al .25 performed PLD \ngrowth of BixY3-xIG on GGG  and reported an increase of 𝜃F\n𝑥= −1.9 °.𝜇𝑚−1 per Bi substitution  x @ 632 \nnm. The Faraday rotation coefficients we find  are slightly larger  and m ay be due to the much lower \nthickness of our films as 𝜃F is also dependent on  the film  thickness26. The saturation magnetization ( Ms) \nremains constant for all Bi content  (see Table 1)  within the  10%  experimental errors . We observe a clear  \ncorrelation between the strain and the shape of the in -plane and out -of-plane hysteresis loop s reflecting \nchanges in the magnetic anisotropy. Wh ile films under compressive strain exh ibit in -plane anisotropy, \nthose under  tensile strain show a large out -of-plane anisotrop y that can eventually lead to an out -of-\nplane easy axis  for x= 0.7 and x= 1 grown on sGGG.  The transition can be either induced by ch anging the 4 \n substrate  (Fig.2 -(a)) or the Bi content ( Fig. 2-(b)) since both act on the misfit strain. We ascribe the \nanisotropy change  in our films to  a combination of magneto -elastic anisotropy and growth induced \nanisotropy, this later term being the domin ant one (see Supplementary Note 1).  \nIn Fig. 2 -(c), we  show the magnetic domains structures at remanance  observed using polar Kerr \nmicroscopy  for Bi1Y2IG films  after demagnetization : µm-wide maze -like magnetic domains demonstrate s \nunambiguously that the magnetic easy axis is perpendicular to the film surface . We observe  a decrease \nof the domain width (Dwidth) when the film thickness ( tfilm) increases as expected from magnetostatic  \nenergy considerations. In fact, a s Dwidth is severa l order s of magnitude larger than tfilm, a domain wall \nenergy of σDW  0.7 and 0.65  mJ.m-2 (for x= 0.7 and 1 Bi doping) can inferred using the Kaplan and \nGerhing model27  (the fitting procedure is detailed  in the Supplementary Note 2).  \n \nDynamical characterization and spin transparency . \nThe most striking feature of these large PMA films is their extremely low magne tic losses that we \ncharacterize  using Ferromagnetic Resonance (FMR) measurements. First of all, we quantify  by in-plane  \nFMR the anisotropy field HKU deduced from the effective magnetization ( Meff): HKU = M S – Meff (the \nprocedure to derive Meff from in plan e FMR is presented in Supplementary Note 3 ). HKU values for BiYIG \nfilms with different doping levels grown on various substrates  are summarized  in Table 1.  As expected \nfrom out-of-plane  hysteresis curves , we observe different signs for HKU. For spontaneously out -of-plane \nmagnetized samples , HKU is positive and large enough to fully compensate the demagnetizing field while \nit is negative for in -plane magnetized films. From these results , one can expect that fine tuning of the Bi \ncontent allow s fine tuning of the effective magnetization and consequently of the FMR resonan ce \nconditions. We measure magnetic losses on a 30nm thick Bi1Y2IG//sGGG film  under tensile strain with \nPMA (Fig. 3 -(a)). We use the FMR absorption line shape  by extracting  the peak -to-peak linewidt h (𝛥𝐻pp) \nat different out-of-plane  angle  for a 30nm thick perpendicularly magnetized Bi 1Y2IG//sGGG film at 8 GHz \n(Fig.  3-(b)). This  yields an optimal value of  𝜇0𝛥𝐻pp  as low as 0.3 mT  (Fig. 3-(c)) for 27° out-of-plane  polar \nangle. We stress here that state of the art PLD grown YIG//GGG films exhibit similar values for 𝛥𝐻pp at \nsuch resonant conditions28. This angular dependence of 𝛥𝐻pp that shows pronounced variations at \nspecific angle is characteristic of a two magnons scattering relaxation process with few \ninhomogenei ties29. The value of this angle is sample dependent  as it is related to the distribution of the \nmagnetic inhomogeneities .  The dominance in our films of those two i ntrinsic relaxation processes \n(Gilbert damping and two magnons  scattering) confirms  the high films quality . We also derived the \ndamping value of th is film (Fig.  3-(d)) by selecting the lowest linewidth (corresponding to a specific out of 5 \n plane angle) at each frequency, the spread of the out of plane angle is ±3.5 ° around 30.5 °. The obtained \nGilbert damping value is α = 3.10-4 and the peak -to-peak extrinsic linew idth 𝜇0𝛥𝐻0 =0.23 mT a re \ncomparable to the one obtained for  the best PLD grown YIG//GGG nanometer thick films28 (α =2.10-4). \nFor x= 0.7  Bi doping, the smallest observed FMR linewidth is 0.5 mT  at 8 G Hz. \nThe low magnetic losses of BiYIG films could open  new perspectives for magnetization dynamics control \nusing spin-orbit torques20,30,31. For such phenomenon  interface transparency to spin curr ent is then the \ncritical parameter which is defined using the effective spin -mixing conductance ( 𝐺↑↓). We use spin \npumping experiments to estimate the increase of the Gilbert damping due to Pt deposition on Bi1Y2IG \nfilms. The spin mixing conductance can thereafter be calculated using 𝐺↑↓=4𝜋𝑀s𝑡film\n𝑔eff𝜇B(𝛥𝛼) where 𝑀s and \n𝑡film are the BiYIG magnetization saturation and thickness, 𝑔eff  is the effective Landé factor ( 𝑔eff=2), \n𝜇B is the Bohr magneton and 𝛥𝛼 is the increase in the Gilbert damping constant induced by the Pt top \nlayer. We obtain  𝐺↑↓=3.9 1018m−2  which is comparable to what is obtained on PLD grown  YIG//GGG \nsystems  28,32,33. Consequently , the doping in Bi should  not alter the spin orbit -torque efficiency  and spin \ntorque devices made out of  BiYIG will be as energy efficient as their  YIG counterpart . To further confirm \nthat spin current cross es the Pt/BiYIG interface , we measure Inverse Spin Hall Effect (ISHE) in Pt for a Pt/  \nBi1.5Y1.5IG(20nm)/ /sGGG in -plane magnetized film (to fulfill the ISHE geometry requirements  the \nmagnetization needs to be in -plane and perp endicular to the measured voltage ). We measure a \ncharacteristic voltage peak due to ISHE  that reverses its sign when the static in-plane  magnetic field is \nreversed (Fig. 4). We emphasize here that the amplitude of the s ignal is similar to that of Pt/ YIG//GGG in \nthe same experimental conditions.  \nConclusion  \nIn summary, this new material platform will be highly beneficial for magnon -spintronics and  related \nresearch fields like  caloritronics. In many aspects , ultra -thin BiYIG films offer new leverages for  fine \ntuning of the magnetic properties with no  drawbacks compared to the reference materials of th ese \nfields: YIG.  BiYIG with its higher Faraday rotation coefficient (almost two orders of magnitude more than \nthat of  YIG) will increase the sensitivity of light based detection technics  that can be used  (Brillouin light \nspectroscopy (BLS) or time resolved Kerr microscopy34). Innovative scheme s for on -chip magnon -light \ncoupler could be now  developed bridging the field of magnonics to th e one  of photonics. From a \npractical point of view , the design of future active devices will be much more flexible as it is possible to \neasily engineer the spin waves dispersion relation through magnetic anis otropy tuning without the need \nof large bias magnetic fields. For instance, working in the forward volume waves configuration comes 6 \n now cost free, whereas in sta ndard in -plane magnetized media  one has to overcome the demagnetizing  \nfield. As the development  of PMA tunnel junctions  was key in developing today  scalable MRAM \ntechnology , likewise, we believe that P MA in nanometer -thick low loss  insulator s paves the path to new \napproaches where the magnonic medium material could also be used to store information locally \ncombining therefore the memory and computational functions, a most desirable feature for the brain -\ninspired neuromorphic paradigm . \n  7 \n Methods  \nPulsed Laser Deposition (PLD) growth  \nThe PLD growth of BiYIG films is realized using stoichiometric BiYIG  target. The laser used is a frequency \ntripled Nd:YAG laser ( λ =355nm), of a 2.5Hz repetition rate and a fluency E varying from 0.95 to 1.43 \nJ.cm-2 depending upon the Bi doping in the target. The distance between target and substrate is fixed at \n44mm. Pri or to the deposition the substrate is annealed at 700°C under 0.4 mbar of O 2. For the growth, \nthe pressure is set at 0.25 mbar O 2 pressure. The optimum growth temperature varies with the Bi \ncontent from 400 to 550°C. At the end of the growth, the sample is  cooled down under 300 mbar of O 2. \n \nStructural characterization  \nAn Empyrean diffractometer with Kα 1 monochromator  is used  for measurement in Bragg -Brentano  \nreflection mode  to derive the (111) interatomic plan distance. Reciprocal Space Mapping is performed on \nthe same diffractometer and we used the diffraction along the (642) plane direction which allow to gain \ninformation on the in-plane  epitaxy relation along [20 -2] direc tion.  \n \nMagnetic characterization  \nA quantum design SQUID magnetometer was used to measure the films’ magnetic moment ( Ms) by \nperforming hysteresis curves along the easy magnetic direction at room temperature. The linear \ncontribution of the paramagnetic (sGG G or GGG) substrate is linearly subtracted.  \nKerr microscope (Evico Magnetics) is used in the polar mode to measure out-of-plane  hysteresis curves \nat room temperature.  The same microscope is also used to image the magnetic domains structure after \na demagn etization procedure. The spatial resolution of the system is 300 nm.  \nA broadband FMR setup with a motorized rotation stage was used. Frequencies from 1 to 20GHz have \nbeen explored. The FMR is measured as the derivative of microwave power absorption via a low \nfrequency modulation of the DC magnetic field. Resonance spectra were recorded with the applied static \nmagnetic field oriented in different geometries (in plane or tilted of an angle 𝜃 out of the strip line \nplane).  For o ut of plane magnetized samples the Gilbert damping parameter has been obtained by \nstudying the  angular  linewidth dependence.  The procedure assumes that close to the minimum \nlinewidth (Fig 3a) most of the linewidth angular dependence is dominated by the inhomogeneous \nbroadening, thus opt imizing the angle for each frequency within few degrees allows to estimate  better  8 \n the intrinsic contribution. To do so we varied the out of plane angle of the static field from 2 7° to 34 ° for \neach frequency  and w e select the lowest value of   𝛥𝐻pp.  \nFor Inverse spin Hall effect measurements, the same FMR setup was used, however here the modulation \nis no longer applied to the magnetic field but to the RF power at a frequency of 5kHz. A Stanford  \nResearch SR860 lock -in was used a signal demodulator.  \nData  availability : \nThe data that support the findings of this study are available within the article or from the corresponding \nauthor upon reasonable request . \nAcknowledgements:  \n We acknowledge J. Sampaio for preliminary Faraday rotation measurements and N. Rey ren and A. \nBarthélémy for fruitful discussions. This research was supported by the ANR Grant ISOLYIG (ref 15 -CE08 -\n0030 -01). LS is partially supported by G.I.E III -V Lab. France.   \n \nAuthor Contributions : \nLS performed the growth, all the measurements, the da ta analysis and wrote the manuscript  with AA . NB \nand JBY conducted the quantitative Faraday Rotation measurements and participated in the FMR data \nanalysis. LQ and fabricated the PLD targets . RL supervised the target fabrication and participated in the \ndesign of the study . EJ participated in the optimization of the film growth conditions. CC supervised the \nstructural characterization experiments. AA conceived the study and w as in charge of overall direction . \nPB and VC contributed to the design and implem entation of the research . 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Lett.  117,  1–5 (2016).  \n 11 \n Figures Captions :  \n \nFigure 1 -Structural properties  of ultra -thin BiYIG films . \n(a) and (b) : Evolution of the target  cubic lattice parameter of BixY3-xIG, the dashed line represents the \nsubstrate (sGGG  and GGG respectively) lattice parameter and allow s to infer the expected tensile or \ncompressive strain arising for each substrate/target combination.  \n(c) and ( d): 2𝜃−𝜔 X-Ray diffraction scan along the (111) out-of-plane  direction for BixY3-xIG films gr own \non sGGG (111) and GGG (111) respectively. From the film and substrate diffraction peak position , we can \nconclude about the nature of the strain. Compressive strain is observed for 1.5 doped films grown on \nsGGG substrate and for all films grown on GGG w hereas tensile strain occurs for films with x=  0.7 and x= \n1 Bi content grown on sGGG.  \n(e) and (f) : RSM along the evidence the (642) oblique plan showing pseudomorphic growth in films: both \nsubstrate and film the diffraction peak are aligned along the qx\\\\[20-2] direction. The relative position of \nthe diffraction peak of the film (up or down) along qx is related to the out-of-plane  misfit between the \nsubstrate and the film (tensile or compressive).  \nFigure 2 -Static magnetic  properties . \n(a) Out-of plane Kerr hysteresis loop performed in the polar mode for Bi0.7Y2.3IG films grown on the two \nsubstrates: GGG and sGGG  \n(b) Same measurement for BixY3-xIG grown on sGGG with  the three different Bi doping ( x= 0.7, 1 and 1.5) . \nBi0.7Y2.3IG//GGG is in -plane magnetized whereas perpendicular magnetic anisotropy (PMA) occurs for x= \n0.7 and x= 1 films grown on sGGG: square shaped loops with low saturation field ( µ0Hsat about 2.5 mT) are \nobserved. Those two films are experiencing tensile strain . Whereas the inset shows that the Bi1.5Y1.5IG \nfilm saturates at a much higher field wi th a curve characteristic of in -plane easy magnetization direction. \nNote that for Bi1.5Y1.5IG//sGGG µ0Hsat ≈290mT >µ0Ms≈162mT which points toward a negative uniaxial \nanisotropy term ( µ0HKU) of 128mT which is coherent with the values obtained from in plane FMR \nmeasurement .  \n(c) Magnetic domains structure imaged on Bi 1Y2IG//sGGG films of three different thicknesses at \nreman ant state after demagnetization . The scale bar, displayed in blue , equal s 20 µm . Periods of the \nmagnetic domains structure ( Dwidth) are derived using 2D Fast Fourier Transform . We obtained Dwidth =3.1,  \n1.6 and 0.4  µm for tBi1Y2IG= 32, 47 and 52  nm respectively. We note a decrease of Dwidth with increasing \ntBi1Y2IG  that is coherent with the Kaplan and Gehring model valide in the case  Dwidth>>tBiYIG.  12 \n Figure 3-Dynamical properties  of BiYIG films with PMA.  \n(a) Sketch of the epitaxial configuration for Bi1Y2IG films , films are grown under tensi le strain giving rise \nto tetragonal distortion of the unit cell.  \n(b) Out-of-plane  angular depend ence of the peak -to-peak FMR linewidth ( 𝛥𝐻pp) at 8 GHz on a 30 nm \nthick Bi1Y2IG//sGGG with PMA  (the continuous  line is a guide for the eye) . The geometry of the \nmeasurement is shown in top right of the graph. The wide disparity of the value for the peak to peak \nlinewidth 𝛥𝐻pp is attributed to the two magnons scattering process  and inhomogeneties in the sample . \n(c) FMR absorption linewidth o f 0.3 mT for the same film at measured at  𝜃=27°. (d) Frequency \ndependence of the FMR linewidth . The calculated  Gilbert damping parameter and the extrinsic linewidth \nare displayed  on the graph .  \nFigure 4- Inverse Spin Hall Effect  of BiYIG  films with in plane magnetic anisotropy.  \nInverse Spin Hall Effect (ISHE) voltage vs magnetic  field measured on the Pt / Bi1.5Y1.5IG//sGGG  sample in \nthe FMR resonant condition at 6 GHz proving  the interface transparency to spin current. The rf excitation \nfield is about 10-3 mT which corresponds to a linear regime of excitation. Bi1.5Y1.5IG//sGGG present s an in-\nplane  easy magnetization axis due to a growth under compressive strain.  \n  \n  13 \n Table 1 - Summary of the magnetic properties of BixY3-xIG films on GGG and sGGG \nsubstrates.  \nThe saturation magnetization is roughly unchanged. The effective magnetization Meff obtained through \nbroad -Band FMR measurements allow to deduce the out -of-plane anisotropy fields HKU (HKU =Ms-Meff) \nconfirming the dramatic changes of the out-of-plane magnetic anisotropy variations observed in the \nhysteresis curves . \n   \nBi doping  Substrate  µ0MS(mT)  µ0Meff(mT)  µ0HKU(mT)  \n0 GGG  157 200 -43 \n0.7 sGGG  180 -151 331 \n0.7 GGG  172 214 -42 \n1 sGGG  172 -29 201 \n1 GGG  160 189 -29 \n1.5 sGGG  162 278 -116 14 \n Figure 1  \n (f) \n0.5 1.0 1.51.2351.2401.2451.2501.255  Cubic Lattice parameter in nmBi content \n  sGGG\n45 50 55103106109\n Intensity (cps)\n2 angle(°)0.711.5\n0.711.5\n1.70 1.724.254.304.35qz //[444] (rlu)*10\nqx//[2-20] (rlu)*101E+01 1E+05\n1.74 1.754.234.274.30qz //[444] (rlu)*10\nqx//[2-20] (rlu)*101E+02 1E+05\n1.73 1.764.214.254.29qz //[444] (rlu)*10\nqx//[2-20] (rlu)*101E+02 1E+05\n0.7 1 1.5\nsGGGBi0.7Y2.3IG\nsGGGBi1Y2IG\nBi1.5Y1.5IGsGGG\n45 50 55102105108\n Intensity (cps)\n2 angle (°)\n0.5 1.0 1.51.2351.2401.2451.2501.255  Cubic Lattice parameter in nmBi content \n  \nGGG0.711.5\n0.7511.5\n1.75 1.774.204.304.40qz//[444] (rlu)*10\nqx//[2-20] (rlu)*101E+01 1E+05\n1.750 1.7754.284.324.35qz//[444] (rlu)*10\nqx//[2-20] (rlu)*101E+02 1E+05\n0.7\nGGG\nBi0.7Y2.3IGGGG\nBi1Y2IG1sGGG\nGGG(a)\n(b)(c)\n(d)(e)\n(f)15 \n Figure 2  \n \n \n \n16 \n Figure 3  \n \n \n  \n17 \n Figure 4  \n-130 -120 -110 110 120 130-4000400800V (nV)\nµ0H (mT)2 \n Supplementary Notes 1- Derivation of the magneto -elastic anisotropy  \n \nThe out -of-plan e anisotropy constant KU is ascribed to be a result  of, at least , two contributions : a \nmagneto -elastic anisotropy  term  induced by strain (𝐾MO) and a term that is due to preferential \noccupation of Bi atoms of non equivalent dodecahedral sites of the  cubic u nit cell. This last term is \nknown as the growth induced anisotropy term 𝐾GROWTH . From X -ray characterizations and f rom the \nknown properties of the thick BiYIG LPE grown films, it is possible to calculate the expected values of  \n𝐾MO in each doping/substra te combination. We thereafter deduce 𝐾GROWTH  from the relation  \n𝐾U=𝐾MO+ 𝐾GROWTH . KMO is directly proportional to the misfit between the film and the substrate:  \n𝐾MO=3\n2∙𝐸\n1−𝜇∙𝑎film −𝑎substrate\n𝑎film∙𝜆111 \nWhere E, µ and λ111 are respectively the Young modulus, the Poisson coefficient, and the \nmagnetostrictive constant along the (111) direction.  Those constant are well established for the bulk1: E= \n2.055.1011 J.m-3, µ= 0.29 . The magnetostriction  coefficient  λ111 for the thin film case is slightly higher \nthan that of the bulk and depends  upon the Bi rate x: 𝜆111(𝑥)=−2.819 ∙10−6(1+0.75𝑥) 2. The two \nlattice par ameter entering in the equation: afilm and asubstrate  correspond to the lattice parameter of the \nrelaxed film structure and of the subs trate. Under an elastic deformation afilm can be derived with the \nPoisson coefficient:  \n𝑎film =𝑎substrate −[1−𝜇111\n1+𝜇111]𝛥𝑎⊥  with  ∆𝑎⊥=4√3𝑑444film−𝑎substrate  \nAll values for the different target/substrate combinations are displayed in the Table S-1. We note here \nthat a negative (positive) misfit corresponding to a tensile (compressive) strain will favor an out-of-plane  \n(in-plane ) magnetic anisotropy which is coherent with what is observed in our samples. To estimate the \ncontribution to the magnetic  energy of the magneto elastic anisotropy term we compare it to the \ndemagnetizing field 𝜇0𝑀s that favors in-plane  magnetic anisotropy in thin films. Interestingly the \nmagneto elastic field ( 𝜇0𝐻MO) arising from 𝐾MO (𝜇0𝐻MO=2𝐾MO\n𝑀s) never exceed 30% o f the \ndemagnetizing fields and therefore cannot alon e be responsible of the observed PMA.  \n \nStudies on µm-thick BiYIG films  grown  by LPE showed that PMA in BiYIG arises due to the growth \ninduced anisotropy term 𝐾GROWTH  , this term is positive 3 for the case of Bi substitution . We have \ninferred 𝜇0𝐻GROWTH  values for all films using 𝐾U constants  measured by FMR. The results are \nsummarized in Supplementary Figure 1. One can clearly see that  𝐾GROWTH  is strongly substrate \ndependent and therefore does not depend sol ely on the Bi content. We conclude that strain play s a role \nin Bi3+ ion ordering within the unit cell . \n \n 3 \n  \n \nSupplementary Figure 1- Summary of the inferred values of the effective magnetic \nanisotropy out -of-plan fields.  \n \nHorizontal dash lines represent the magnitude of the demagnetization field µ0Ms. When µ0HKU is larger \nthan µ0Ms (dot line) films have a PMA, they are in -plane magnetized otherwise .   \n0200400µ0HKuvs Bi content \n HKu = Hstrain+ Hgrowth\nµ0Ms \n1.50.7µ0HKu (mT) µ0 Hstrain\n µ0 Hgrowth 1\nµ0Ms\n-1000100200\n0.7 µ0HKu (mT) µ0Hstrain\n µ0Hgrowth\n14 \n Supplementary Table 1 - Summary of the films’ calculated magneto -elastic \nanisotropy constant ( KMO) and the corresponding anisotropy field HMO \n \n \n  Bi content  substrate  afilm(Å) Δ a┴/afilm KMO (J.m-3) µ0 HMO(mT)  µ0 Hdemag (mT)  \n0.7 sGGG  12.45  0.6 5818  81 179 \n0.7 GGG  12.41  -0.4 -4 223  -61 172 \n1.0 sGGG  12.47  0.3 3 958  57 172 \n1.0 GGG  12.42  -0.6 -6 500  -102 160 \n1.5 sGGG  12.53  -0.6 -8 041  -124 157 5 \n Supplementary Notes 2- Derivation of the domain wall energy  \n \nTo derive the characteristic domain wall energy σDW for the maze shape like magnetic domains , we use \nthe Kaplan et al. model4. This model applies in our case as the ratio of the film thickness ( tfilm) to the  \nmagnetic domain width ( Dwidth) is small (𝑡film\n𝐷width~ 0.01). The domain wall width and the film thickness are \nthen expected to be linked by:  \n𝐷width =𝑡filme−π1.33𝑒𝜋𝐷0\n2𝑡film  where 𝐷0=2𝜎w\n𝜇0𝑀s2 is the dipolar length.  \nHence we expect a linear dependence of ln(𝐷width\n𝑡film) vs  1\n𝑡film : \nln(𝐷width\n𝑡film)=π𝐷0\n2∙ 1\n𝑡film+Cst.  (1) \nThe magnetic domain width of Bi xY3-xIG//sGGG ( x = 0.7 and 1) for several thicknesses are extracted from \n2D Fourier Transform of the Kerr microscopy images at remanence. In Supplementary Fig ure 2, we plot \nln(𝐷width\n𝑡film) vs  1\n𝑡film which follow s the expected linear dependence  of Equation (1) . We infer from the zero \nintercept an estimat ion of the dipolar length D0 of BiYIG films doped at 0.7 and 1 in Bi : D0 x=0.7= 16.5  µm \nand D0 x=1= 18.9µm. The corresponding domain wall energy are respectively 0.7 mJ .m-2 and 0.65 mJ .m-2. \nEven if the small  difference in domain wall energy between the two Bi content  may not be significant \nregarding the statistical fitting errors, it correlates to the decrease of the out-of-plane  anisotropy ( KU) \nwith increasing the Bi content.  6 \n Supplementary Figure 2- Evolution of the domain width vs film thickness  \n \nln(𝐷width\n𝑡film) vs  1\n𝑡film for Bi xY3-xIG//sGGG films doped at 0.7 (a) and 1 (b) in Bi. Dots correspond to the \nexperimental values. The dashed line is the linear fit that allows to extract the D0 parameter.  \n  \n25 50 752468\n  ln(Dwidth/tfilm)\ntfilm(µm-1)50 1002468\n  ln(Dwidth/tfilm)\n/tfilm(µm-1)Bi0.7Y2.3IG//sGGG\nD0=16.5µm\nBi1Y2IG//sGGG\nD0=18.9µm\n(a) \n(b) 7 \n  \nSupplementary Notes 3 - Damping and effective magnetic field derivation  \n \nFrom In Plane frequency dependent of FMR we can derive the effective magnetization  (𝑀eff) using the \nKittel law:  \n𝑓res=𝜇0𝛾√𝐻res(𝐻res+𝑀-eff) \nWhere  γ is the gyromagnetic ratio of the BiYIG  (assumed to be same as the one of the YIG) : γ=28 GHz.T-1. \n𝐻res  and 𝑓res are respectively the FMR resonant field and frequency . The uniaxial magnetic anisotropy \ncan thereafter  be derived using the saturation magnetizatio n form squid magnetometry using :   \nMeff=Ms-HKU. The Gilbert damping ( α) and the inhomogeneous linewidth ( ΔH 0) which are the two \nparamaters defining the magnetic relaxation are obtained from the evolution of the peak to peak \nlinewidth ( ΔHpp) vs the resonant frequency ( fres): \n𝛥𝐻pp=𝛥𝐻0+2\n√3𝛼𝑓res\n𝜇0𝛾  (2) \nThe first term is frequency independent and often attributed magnetic inhomogeneity’s (anisotropy, \nmagnetization).  \n  8 \n Supplementary Figure 3 - µ0ΔHpp vs fres on 18 nm thick Bi15.Y1.5IG//sGGG  \n \nThe l inewidth  frequency dependence from 5 to 19 GHz for in plane magnetized Bi 1.5Y1.5IG//sGGG  sample \nallow to ex tract the damping and the inhomogeneous linewidth parameter  using the Equation (2) . \n \n  \n0 5 10 150.00.51.01.5 \n Hpp(mT)\nfres (GHz)H0=0.5 mT\n=1.9*10-39 \n Supplementary References:  \n \n1. Hansen, P., Klages, C. ‐P. & Witter, K. Magnetic and magneto ‐optic properties of praseodymium ‐ \nand bismuth ‐substituted yttrium iron garnet films. J. Appl. Phys.  60, 721–727 (1986).  \n2. Ben Youssef, J., , Legall, H. & . U. P. et M. C. Characterisation and physical study of bismuth \nsubstituted thin garnet films grown by liquid phase epitaxy (LPE). (1989).  \n3. Fratello, V. J., Slusky, S. E. G., Brandle, C. D. & Norelli, M. P. Growth -induced anisotropy in \nbismuth: Rare -earth iron garnets. J. Appl. Phys.  60, 2488 –2497 (1986).  \n4. Kaplan, B. & Gehring, G. A. The domain structure in ultrathin magnetic films. J. Magn. Magn. \nMater.  128,  111–116 (1993).  \n "    },    {        "title": "1602.06673v3.Effects_of_Landau_Lifshitz_Gilbert_damping_on_domain_growth.pdf",        "content": "arXiv:1602.06673v3  [cond-mat.stat-mech]  1 Dec 2016Effects of Landau-Lifshitz-Gilbert damping on domain growt h\nKazue Kudo\nDepartment of Computer Science, Ochanomizu University,\n2-1-1 Ohtsuka, Bunkyo-ku, Tokyo 112-8610, Japan\n(Dated: May 25, 2021)\nDomain patterns are simulated by the Landau-Lifshitz-Gilb ert (LLG) equation with an easy-axis\nanisotropy. If the Gilbert damping is removed from the LLG eq uation, it merely describes the\nprecession of magnetization with a ferromagnetic interact ion. However, even without the damping,\ndomains that look similar to those of scalar fields are formed , and they grow with time. It is demon-\nstrated that the damping has no significant effects on domain g rowth laws and large-scale domain\nstructure. In contrast, small-scale domain structure is aff ected by the damping. The difference in\nsmall-scale structure arises from energy dissipation due t o the damping.\nPACS numbers: 89.75.Kd,89.75.Da,75.10.Hk\nI. INTRODUCTION\nCoarseningorphase-orderingdynamicsisobservedina\nwidevarietyofsystems. Whenasystemisquenchedfrom\na disordered phase to an ordered phase, many small do-\nmainsareformed, andtheygrowwithtime. Forexample,\nin the case of an Ising ferromagnet, up-spin and down-\nspin domains are formed, and the characteristic length\nscale increases with time. The Ising spins can be inter-\npreted as two different kinds of atoms in the case of a\nbinary alloy. At the late stage of domain growth in these\nsystems, characteristic length L(t) follows a power-law\ngrowth law,\nL(t)∼tn, (1)\nwherenis the growth exponent. The growth laws in\nscalarfieldshavebeenderivedbyseveralgroups: n= 1/2\nfornon-conservedscalarfields, and n= 1/3forconserved\nscalar fields [1–8].\nSimilar coarsening dynamics and domain growth have\nbeen observed alsoin Bose-Einstein condensates (BECs).\nThe characteristic length grows as L(t)∼t2/3in two-\ndimensional (2D) binary BECs and ferromagnetic BECs\nwith an easy-axis anisotropy [9–11]. The same growth\nexponent n= 2/3 is found in classical binary fluids in\nthe inertial hydrodynamic regime [1, 12]. It is remark-\nable that the same growth law is found in both quan-\ntum and classical systems. It should be also noted that\ndomain formation and coarsening in BECs occur even\nwithout energy dissipation. The dynamics in a ferro-\nmagnetic BEC can be described not only by the so-\ncalled Gross-Pitaevskii equation, which is a nonlinear\nSchr¨ odinger equation, but also approximately by a mod-\nified Landau-Lifshitz equation in which the interaction\nbetween superfluid flow and local magnetization is incor-\nporated[13–15]. Ifenergydissipationexists, theequation\nchanges to an extended Landau-Lifshitz-Gilbert (LLG)\nequation [9, 15, 16]. The normal LLG equation is usu-\nally used to describe spin dynamics in a ferromagnet.\nThe LLG equation includes a damping term which is\ncalled the Gilbert damping. When the system has an\neasy-axis anisotropy, the damping has the effect to directa spin to the easy-axis direction. The Gilbert damping\nin the LLG equation corresponds to energy dissipation\nin a BEC. In other words, domain formation without en-\nergy dissipation in a BEC implies that domains can be\nformed without the damping in a ferromagnet. However,\nthe LLG equation without the damping describes merely\nthe precession of magnetization with a ferromagnetic in-\nteraction.\nInthispaper, wefocusonwhateffectsthedampinghas\nondomainformationanddomaingrowth. UsingtheLLG\nequation (without flow terms), we investigate the mag-\nnetic domain growth in a 2D system with an easy-axis\nanisotropy. Since our system is simpler than a BEC, we\ncan also give simpler discussions on what causes domain\nformation. When the easy axis is perpendicular to the\nx-yplane, the system is an Ising-like ferromagnetic film,\nand domains in which the zcomponent of each spin has\nalmostthesamevalueareformed. Inordertoobservedo-\nmain formation both in damping and no-damping cases,\nwe limit the initial condition to almost uniform in-plane\nspins. Actually, without the damping, domain formation\ndoes not occur from an initial configuration of spins with\ntotally random directions. Without the damping, the z\ncomponent is conserved. The damping breaks the con-\nservation of the zcomponent as well as energy. Here,\nwe should note that the growth laws for conserved and\nnonconserved scalar fields cannot simply be applied to\nthe no-damping and damping cases, respectively, in our\nsystem. Although the zcomponent corresponds to the\norderparameterofascalarfield, oursystemhastheother\ntwo components. It is uncertain whether the difference\nin the number of degrees of freedom can be neglected in\ndomain formation.\nThe restofthe paperis organizedas follows. In Sec. II,\nwe describe the model and numerical procedures. Ener-\ngies and the characteristic length scale are also intro-\nduced in this section. Results of numerical simulations\nare shown in Sec. III. Domain patterns at different times\nand the time evolution of energies and the average do-\nmain size are demonstrated. Scaling behavior is con-\nfirmed in correlation functions and structure factors at\nlate times. In Sec. IV, we discuss why domain formation2\ncan occur even in the no-damping case, focusing on an\nalmost uniform initial condition. Finally, conclusions are\ngiven in Sec. V.\nII. MODEL AND METHOD\nThe model we use in numerical simulations is the LLG\nequation, which is widely used to describe the spin dy-\nnamics in ferromagnets. The dimensionless normalized\nform of the LLG equation is written as\n∂m\n∂t=−m×heff+αm×∂m\n∂t, (2)\nwheremis the unit vector of spin, αis the dimensionless\nGilbert damping parameter. We here consider the 2D\nsystemlyinginthe x-yplane,andassumethatthesystem\nhas a uniaxial anisotropy in the zdirection and that no\nlong-range interaction exists. Then, the dimensionless\neffective field is given by\nheff=∇2m+Canimzˆz, (3)\nwhereCaniis the anisotropy parameter, and ˆzis the unit\nvector in the zdirection.\nEquation (2) is mathematically equivalent to\n∂m\n∂t=−1\n1+α2m×heff+α\n1+α2m×(m×heff).\n(4)\nIn numerical simulations, we use a Crank-Nicolson\nmethod to solve Eq. (4). The initial condition is given as\nspins that are aligned in the xdirection with a little ran-\ndom noises: mx≃1 andmy≃mz≃0. Simulations are\nperformed in the 512 ×512 lattice with periodic bound-\nary conditions. Averages are taken over 20 independent\nruns.\nThe energy in this system is written as\nE=Eint+Eani\n=1\n2/integraldisplay\ndr(∇m(r))2−1\n2Cani/integraldisplay\ndrmz(r)2,(5)\nwhich gives the effective field as heff=−δE/δm. The\nfirst and second terms are the interfacial and anisotropy\nenergies, respectively. When Cani>0, thezcomponent\nbecomes dominant since a large m2\nzlowers the energy.\nWe take Cani= 0.2 in the simulations. The damping\nparameter αexpresses the rate of energy dissipation. If\nα= 0, the spatial average of mzas well as the energy E\nis conserved.\nConsidering mzas the order parameter of this system,\nwe here define the characteristic length scale Lof a do-\nmain pattern from the correlation function\nG(r) =1\nA/integraldisplay\nd2x/angb∇acketleftmz(x+r)mz(x)/angb∇acket∇ight,(6)\nwhereAis the area of the system and /angb∇acketleft···/angb∇acket∇ightdenotes an\nensemble average. The average domain size Lis defined\nby the distance where G(r), i.e., the azimuth average of\nG(r), first drops to zero, and thus, G(L) = 0.\nFIG. 1. (Color online) Snapshots of z-component mzat time\nt= 102((a) and (b)), 103((c) and (d)), and 104((e) and\n(f)). Snapshots (g) and (h) are enlarged parts of (e) and (f),\nrespectively. Profiles (i) and (j) of mzare taken along the\nbottom lines of snapshots (g) and (h), respectively. Left an d\nright columns are for the no-damping ( α= 0) and damping\n(α= 0.03) cases, respectively.\nIII. SIMULATIONS\nDomain patterns appear, regardless of the damping\nparameter α. The snapshots of the no-damping ( α= 0)\nand damping ( α= 0.03) cases are demonstrated in the\nleft and right columns of Fig. 1, respectively. Domain\npatterns at early times have no remarkable difference be-\ntweenthe twocases. Thecharacteristiclengthscalelooks\nalmostthesamealsoatlatertimes. However,asshownin\nthe enlarged snapshots at late times, difference appears\nespecially around domain walls. Domain walls, where\nmz≃0, are smooth in the damping case. However, in\nthe no-damping case, they look fuzzy. The difference ap-\npears more clearly in profiles of mz(Figs. 1(i) and 1(j)).\nWhile the profile in the damping case is smooth, that\nin the no-damping case is not smooth. Such an uneven\nprofile makes domain walls look fuzzy.\nThe difference in domain structure is closely connected\nwith energydissipation, which is shownin Fig. 2. The in-\nterfacial energy, which is the first term of Eq. (5), decays\nforα= 0.03 but increases for α= 0 in Fig. 2 (a). In con-3\n0 2000 4000 6000 8000 10000t00.020.040.060.080.1Eint α = 0\nα = 0.03(a)\n0 2000 4000 6000 8000 10000t-0.1-0.08-0.06-0.04-0.020Eaniα = 0\nα = 0.03(b)\nFIG. 2. (Color online) Time dependence of (a) the inter-\nfacial energy Eintand (b) the anisotropy energy Eani. The\ninterfacial energy increases with time in the no-damping ca se\n(α= 0) and decreases in the damping case ( α= 0.03). The\nanisotropy energy decreases with time in both cases.\ntrast, the anisotropy energy, which comes from the total\nofm2\nz, decreases with time for both α= 0 and α= 0.03.\nIn other words, the energy dissipation relating to the in-\nterfacial energy mainly causes the difference between the\ndamping and no-damping cases. In the damping case,\nthe interfacial energy decreases with time after a shot-\ntime increase as domain-wall structure becomes smooth.\nHowever, in the no-damping case, the interfacial energy\nincreases with time to conserve the total energy that is\ngiven by Eq. (5). This corresponds to the result that\nthe domain structure does not become smooth in the no-\ndamping case.\nBeforediscussinggrowthlaws, we shouldexaminescal-\ning laws. Scaled correlation functions of mzat different\ntimes are shown in Fig. 3. The functions look pretty\nsimilar in both damping and no-damping cases, which\nreflects the fact that the characteristic length scales in\nboth cases looks almost the same in snapshots. At late\ntimes, the correlation functions that are rescaled by the\naverage domain size L(t) collapse to a single function.\nHowever, the scaled correlation functions at early times\n(t= 100 and 1000) do not agree with the scaling func-\ntion especially in the short range. The disagreement at\nearly times is related with the unsaturation of mz. How\nmzsaturates is reflected in the time dependence of the0 0.5 1 1.5 2\nr/L(t)-0.200.20.40.60.8G(r)t =   100\nt =  1000\nt =  6000\nt =  8000\nt = 10000(a)\n0 0.5 1 1.5 2\nr/L(t)-0.200.20.40.60.8G(r)t =   100\nt =  1000\nt =  6000\nt =  8000\nt = 10000(b)\nFIG.3. (Color online) Scaledcorrelation functions atdiffe rent\ntimes in (a) no-damping ( α= 0) and (b) damping ( α= 0.03)\ncases. The correlation functions at late times collapse to a\nsingle function, however, the ones at early times do not.\nanisotropy energy which is shown in Fig. 2(b). At early\ntimes (t/lessorsimilar1000),Eanidecays rapidly. This implies that\nmzis not saturated enough in this time regime. The de-\ncreasein theanisotropyenergyslowsatlatetimes. In the\nlate-time regime, mzis sufficiently saturated except for\ndomain walls, and the decrease in the anisotropy energy\nis purely caused by domain growth. This corresponds to\nthe scaling behavior at late times.\nIn Fig. 4, the average domain size Lis plotted for\nthe damping and no-damping cases. In both cases,\nthe average domain size grows as L(t)∼t1/2at late\ntimes, although growth exponents at early times look\nliken= 1/3. Since scaling behavior is confirmed only\nat late times, the domain growth law is considered to be\nL(t)∼t1/2rather than t1/3in this system. In our pre-\nvious work, we saw domain growth as L(t)∼t1/3in a\nBEC without superfluid flow [9], which was essentially\nthe same system as the present one. However, the time\nregion shown in Ref. [9] corresponds to the early stage\n(t/lessorsimilar1830) in the present system.\nAlthough the growth exponent is supposed to be n=\n1/3 for conserved scalar fields, the average domain size\ngrows as L(t)∼t1/2, in our system, at late times even\nin the no-damping case. This implies that our system\nwithout damping cannot be categorized as a model of a4\n100 1000 10000t10100 Lα = 0\nα = 0.03\nt1/2\nt1/3\nFIG. 4. (Color online) Time dependence of the average do-\nmain size Lforα= 0 and 0 .03. In both damping and no-\ndamping cases, domain size grows as L(t)∼t1/2at late\ntimes. Before the scaling regime, early-time behavior look s\nas ifL(t)∼t1/3.\n1 10 100\nkL(t)10-1010-910-810-710-610-510-410-3S(k)/L(t)2\nt =  6000\nt =  8000\nt = 10000\n1 10 100\nkL(t)10-1010-910-810-710-610-510-410-3\nt =  6000\nt =  8000\nt = 10000(a) (b)\nk-3k-3\nFIG. 5. (Color online) Scaling plots of the structure factor\nscaled with L(t) at different times in (a) no-damping ( α= 0)\nand (b) damping ( α= 0.03) cases. In both cases, S(k)∼k−3\nin the high- kregime. However, they gave different tails in the\nultrahigh- kregime.\nconserved scalar field. Although we consider mzas the\norder parameter to define the characteristic length scale,\nthe LLG equation is described in terms of a vector field\nm.\nScaling behavior also appears in the structure factor\nS(k,t), which is given by the Fourier transformation of\nthe correlation function G(r). According to the Porod\nlaw, the structure factor has a power-law tail,\nS(k,t)∼1\nL(t)kd+1, (7)\nin the high- kregime [1]. Here, dis the dimension of\nthe system. Since d= 2 in our system, Eq. (7) leads\ntoS(k,t)/L(t)2∼[kL(t)]−3. In Fig. 5, S(k,t)/L(t)2is\nplotted as a function of kL(t). The data at different late\ntimes collapse to one curve, and they show S(k)∼k−3in the high- kregime (kL∼10) in both the damping and\nno-damping cases. In the ultrahigh- kregime (kL∼100),\ntails are different between the two cases, which reflects\nthe difference in domain structure. Since domain walls\nare fuzzy in the no-damping case, S(k) remains finite.\nHowever, in the damping case, S(k) decays faster in the\nultrahigh- kregime, which is related with smooth domain\nwalls.\nIV. DISCUSSION\nWe here have a naive question: Why does domain\npattern formation occur even in the no-damping case?\nWhenα= 0, Eq. (2) is just the equation of the pre-\ncession of spin, and the energy Eas well as mzis con-\nserved. We here discuss why similar domain patterns are\nformed from our initial condition in both damping and\nno-damping cases.\nUsing the stereographic projection of the unit sphere\nof spin onto a complex plane [17], we rewrite Eq. (4) as\n∂ω\n∂t=−i+α\n1+α2/bracketleftbigg\n∇2ω−2ω∗(∇ω)2\n1+ωω∗−Caniω(1−ωω∗)\n1+ωω∗/bracketrightbigg\n,\n(8)\nwhereωis a complex variable defined by\nω=mx+imy\n1+mz. (9)\nEquation (8) implies that the effect of the Gilbert damp-\ning is just a rescaling of time by a complex constant [17].\nThe fixed points of Eq. (8) are |ω|2= 1 and ω= 0.\nThelinearstabilityanalysisaboutthesefixedpointsgives\nsome clues about domain formation.\nAt the fixed point ω= 1,mx= 1 and my=mz= 0,\nwhich corresponds to the initial condition of the numer-\nical simulation. Substituting ω= 1 +δωinto Eq. (8),\nwe obtain linearized equations of δωandδω∗. Perform-\ning Fourier expansions δω=/summationtext\nkδ˜ωkeik·randδω∗=/summationtext\nkδ˜ω∗\n−keik·r, we have\nd\ndt/parenleftbiggδ˜ωk\nδ˜ω∗\n−k/parenrightbigg\n=/parenleftbigg\n˜α1(Cani−k2) ˜α1Cani\n˜α2Cani˜α2(Cani−k2)/parenrightbigg/parenleftbiggδ˜ωk\nδ˜ω∗\n−k/parenrightbigg\n,\n(10)\nwhere ˜α1=1\n2(−i+α)/(1+α2), ˜α2=1\n2(i+α)/(1+α2),\nk= (kx,ky), andk=|k|. The eigenvalues of the 2 ×2\nmatrix of Eq. (10) are\nλ(k) =α\n2(1+α2)(Cani−2k2)±/radicalbig\n4k2(Cani−k2)+α2C2\nani\n2(1+α2).\n(11)\nEven when α= 0,λ(k) has a positive real part for\nk <√Cani. Thus, the uniform pattern with mx= 1\nis unstable, and inhomogeneous patterns can appear.\nThe positive real parts of Eq. (11) for α= 0 and\nα= 0.03 have close values, as shown in Fig. 6. This cor-\nresponds to the result that domain formation in the early5\n0 0.1 0.2 0.3 0.4 0.5\nk00.020.040.060.080.1λ(k)α = 0\nα = 0.03\nFIG. 6. (Color online) Positive real parts of λ(k) that is given\nby Eq. (11), which has a positive real value for k <√Cani.\nThe difference between α= 0 and α= 0.03 is small.\nstage has no remarkable difference between the damping\n(α= 0.03) and no-damping ( α= 0) cases (See Fig. 1).\nFrom the view point of energy, the anisotropy energy\ndoes not necessarily keep decaying when α= 0. For con-\nservation of energy, it should be also possible that both\nanisotropy and interfacial energies change only a little.\nBecause of the instability of the initial state, mzgrows,\nand thus, the anisotropy energy decreases.\nThe initial condition, which is given as spins aligned in\nonedirection with somenoisesin the x-yplane, is the key\nto observe domain pattern formation in the no-damping\ncase. Actually, if spins have totally random directions,\nno large domains are formed in the no-damping case,\nalthough domains are formed in damping cases ( α >0)\nfrom such an initial state.\nWhenω= 0,mx=my= 0 and mz= 1, which is also\none of the fixed points. Substituting ω= 0 +δωinto\nEq. (8) and performing Fourier expansions, we have the\nlinearized equation of δ˜ωk,\nd\ndtδ˜ωk=i−α\n1+α2(k2+Cani)δ˜ωk. (12)\nThis implies that the fixed point is stable for α >0 andneutrally stable for α= 0. Although mz=−1 corre-\nsponds to ω→ ∞, the same stability is expected for\nmz=−1 by symmetry.\nSincetheinitialconditionisunstable, the z-component\nof spin grows. Moreover, linear instability is similar for\nα= 0 and α= 0.03. Since mz=±1 are not unstable,\nmzcan keep its value at around mz=±1. This is why\nsimilar domain patters are formed in both damping and\nno-damping cases. The main difference between the two\ncases is that mz=±1 are attracting for α >0 and neu-\ntrally stable for α= 0. Since mz=±1 are stable and at-\ntractingin the dampingcase, homogeneousdomainswith\nmz=±1 are preferable, which leads to a smooth profile\nofmzsuch as Fig. 1(j). In the damping case, mz=±1\nare neutrally stable (not attracting) fixed points, which\ndoes not necessarily make domains smooth.\nV. CONCLUSIONS\nWe have investigated the domain formation in 2D vec-\ntor fields with an easy-axis anisotropy, using the LLG\nequation. When the initial configuration is given as al-\nmost uniform spins aligned in an in-plane direction, sim-\nilar domain patterns appear in the damping ( α/negationslash= 0) and\nno-damping ( α= 0) cases. The average domain size\ngrows as L(t)∼t1/2in late times which are in a scal-\ning regime. The damping gives no remarkable effects\non domain growth and large-scale properties of domain\npattern. In contrast, small-scale structures are different\nbetween the two cases, which is shown quantitatively in\nthe structure factor. This difference is induced by the re-\nduction of the interfacial energy due to the damping. It\nshould be noted that the result and analysis especially\nin the no-damping case are valid for a limited initial\ncondition. Although domains grow in a damping case\neven from spins with totally random directions, domain\ngrowth cannot occur from such a random configuration\nin the no-damping case.\nACKNOWLEDGMENTS\nThis work was supported by MEXT KAKENHI\n(No. 26103514, “Fluctuation & Structure”).\n[1] A. Bray, Adv. Phys. 43, 357 (1994)\n[2] I. M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids\n19, 35 (1961)\n[3] C. Wagner, Z. Elektrochem 65, 581 (1961)\n[4] T. Ohta, D. Jasnow, and K. Kawasaki, Phys. Rev. Lett.\n49, 1223 (1982)\n[5] D. A. Huse, Phys. Rev. B 34, 7845 (1986)\n[6] A. J. Bray, Phys. Rev. Lett. 62, 2841 (1989)\n[7] A. J. Bray, Phys. Rev. B 41, 6724 (1990)\n[8] A. J. Bray and A. D. Rutenberg, Phys. Rev. E 49, R27\n(1994)[9] K. Kudo and Y. Kawaguchi, Phys. Rev. A 88, 013630\n(2013)\n[10] J. Hofmann, S. S. Natu, and S. Das Sarma, Phys. Rev.\nLett.113, 095702 (2014)\n[11] L. A. Williamson and P. B. Blakie, Phys. Rev. Lett. 116,\n025301 (2016)\n[12] H. Furukawa, Phys. Rev. A 31, 1103 (1985)\n[13] A. Lamacraft, Phys. Rev. A 77, 063622 (2008)\n[14] D. M. Stamper-Kurn and M. Ueda, Rev. Mod. Phys. 85,\n1191 (2013)\n[15] Y. Kawaguchi and M. Ueda, Phys. Rep. 520, 253 (2012)6\n[16] K. Kudo and Y. Kawaguchi, Phys. Rev. A 84, 043607\n(2011)\n[17] M. Lakshmanan and K. Nakamura, Phys. Rev. Lett. 53,\n2497 (1984)"    },    {        "title": "1412.1988v1.Calculating_linear_response_functions_for_finite_temperatures_on_the_basis_of_the_alloy_analogy_model.pdf",        "content": "arXiv:1412.1988v1  [cond-mat.mtrl-sci]  5 Dec 2014Calculating linear response functions for finite temperatu res on the basis of the alloy\nanalogy model\nH. Ebert, S. Mankovsky, K. Chadova, S. Polesya, J. Min´ ar, and D . K¨ odderitzsch\nDepartment Chemie/Phys. Chemie, Ludwig-Maximilians-Uni versit¨ at M¨ unchen,\nButenandtstrasse 5-13, D-81377 M¨ unchen, Germany\n(Dated: 8th December 2014)\nA scheme is presented that is based on the alloy analogy model and allows to account for thermal\nlattice vibrations as well as spin fluctuations when calcula ting response quantities in solids. Various\nmodels to deal with spin fluctuations are discussed concerni ng their impact on the resulting tem-\nperature dependent magnetic moment, longitudinal conduct ivity and Gilbert damping parameter.\nIt is demonstrated that using the Monte Carlo (MC) spin config uration as an input, the alloy ana-\nlogy model is capable to reproduce results of MC simulations on the average magnetic moment\nwithin all spin fluctuation models under discussion. On the o ther hand, response quantities are\nmuch more sensitive to the spin fluctuation model. Separate c alculations accounting for either the\nthermal effect due to lattice vibrations or spin fluctuations show their comparable contributions\nto the electrical conductivity and Gilbert damping. Howeve r, comparison to results accounting for\nboth thermal effects demonstrate violation of Matthiessen’ s rule, showing the non-additive effect of\nlattice vibrations and spin fluctuations. The results obtai ned for bcc Fe and fcc Ni are compared\nwith theexperimental data, showing rather good agreement f or thetemperature dependentelectrical\nconductivity and Gilbert damping parameter.\nI. INTRODUCTION\nFinite temperature has often a very crucial influence\non the response properties of a solid. A prominent ex-\nample for this is the electrical resistivity of perfect non-\nmagnetic metals and ordered compounds that only take\na non-zero value with a characteristic temperature ( T)\ndependence due to thermal lattice vibrations. While the\nHolstein transport equation1,2provides a sound basis for\ncorresponding calculations numerical work in this field\nhas been done so far either on a model level or for sim-\nplified situations.3–6In practice often the Boltzmann-\nformalism is adopted using the constant relaxation time\n(τ) approximation. This is a very popular approach in\nparticular when dealing with the Seebeck effect, as in\nthis case τdrops out.7,8The constant relaxation time\napproximation has also been used extensively when deal-\ning with the Gilbert damping parameter α.9–11Within\nthe description of Kambersky10,12the conductivity- and\nresistivity-like intra- and inter-band contributions to α\nshow a different dependency on τleading typically to\na minimum for α(τ) or equivalently for α(T).10,11A\nscheme to deal with the temperature dependent resistiv-\nity that is formally much more satisfying than the con-\nstant relaxation time approximation is achieved by com-\nbining the Boltzmann-formalism with a detailed calcula-\ntion of the phonon properties. As was shown by various\nauthors,13–16this parameter-free approach leads for non-\nmagneticmetalsingeneraltoaverygoodagreementwith\nexperimental data.\nAs an alternative to this approach, thermal lattice\nvibrations have also been accounted for within various\nstudies by quasi-static lattice displacements leading to\nthermallyinducedstructuraldisorderinthesystem. This\npoint of view provides the basis for the use of the al-\nloy analogy, i.e. for the use of techniques to deal withsubstitutional chemical disorder also when dealing with\ntemperature dependent quasi-static random lattice dis-\nplacements. An example for this are investigations on\nthe temperature dependence of the resistivity and the\nGilbert parameter αbased on the scattering matrix ap-\nproach applied to layered systems.17The necessary aver-\nageovermanyconfigurationsoflatticedisplacementswas\ntakenbymeansofthe supercelltechnique. Incontrastto\nthistheconfigurationalaveragewasdeterminedusingthe\nCoherent Potential Approximation (CPA) within invest-\nigations using a Kubo-Greenwood-like linear expression\nforα.18The same approach to deal with the lattice dis-\nplacements was also used recently within calculations of\nangle-resolved photo emission spectra (ARPES) on the\nbasis of the one-step model of photo emission.19\nAnother important contribution to the resistivity in\nthe case of magnetically ordered solids are thermally in-\nduced spin fluctuations.20Again, the alloy analogy has\nbeen exploited extensively in the past when dealing with\ntheimpactofspinfluctuationsonvariousresponsequant-\nities. Representing a frozen spin configuration by means\nof super cell calculations has been applied for calcula-\ntions of the Gilbert parameter for α17as well as the\nresistivity or conductivity, respectively.17,21,22Also, the\nCPA has been used for calculations of α23as well as the\nresistivity.20,24A crucial point in this context is obvi-\nously the modeling of the temperature dependent spin\nconfigurations. Concerning this, rather simple models\nhave been used,23but also quite sophisticated schemes.\nHere one should mention the transfer of data from Monte\nCarlo simulations based on exchange parameters calcu-\nlated in an ab-initio way25as well as work based on the\ndisordered local moment (DLM) method.24,26Although,\nthe standard DLM does not account for transversal spin\ncomponents it nevertheless allows to represent the para-\nmagnetic regime with no net magnetization in a rigor-2\nous way.Also, for the magnetically ordered regime below\nthe Curie-temperature it could be demonstrated that the\nuncompensated DLM (uDLM) leads for many situations\nstill to goodagreementwith experimentaldata on the so-\ncalled spin disorder contribution to the resistivity.20,24\nIn the following we present technical details and exten-\nsionsofaschemethatwasalreadyused beforewhendeal-\ning with the temperature dependence of response quant-\nities on the basis of Kubo’s response formalism. Various\napplications will be presented for the conductivity and\nGilbert damping parameter accounting simultaneously\nfor various types of disorder.\nII. THEORETICAL FRAMEWORK\nA. Configurational average for linear response\nfunctions\nMany important quantities in spintronics can be\nformulated by making use of linear response formal-\nism. Important examples for this are the electrical\nconductivity,27,28the spin conductivity29or the Gilbert\ndamping parameter.18,30Restricting here for the sake of\nbrevity to the symmetric part of the corresponding re-\nsponse tensor χµνthis can be expressed by a correlation\nfunction of the form:\nχµν∝Tr/angbracketleftbigˆAµℑG+ˆAνℑG+/angbracketrightbig\nc. (1)\nIt should be stressed that this not a real restriction as\nthe scheme described below has been used successfully\nwhen dealing with the impact of finite temperatures on\nthe anomalous Hall conductivity of Ni.31In this case the\nmore complex Kubo-Stˇ reda- or Kubo-Bastin formulation\nfor the full response tensor has to be used.32\nThe vector operator ˆAµin Eq. (1) stands for example\nin case of the electrical conductivity σµνfor the cur-\nrent density operator ˆjµ28while in case of the Gilbert\ndamping parameter αµνit stands for the torque oper-\natorˆTµ.9,18Within the Kubo-Greenwood-like equation\n(1) the electronic structure of the investigated system\nis represented in terms of its retarded Green function\nG+(r,r′,E). Within multiple scattering theory or the\nKKR (Korringa-Kohn-Rostoker)formalism, G+(r,r′,E)\ncan be written as:33–35\nG+(r,r′,E) =/summationdisplay\nΛΛ′Zm\nΛ(r,E)τmn\nΛΛ′(E)Zn×\nΛ′(r′,E)(2)\n−δmn/summationdisplay\nΛZn\nΛ(r,E)Jn×\nΛ′(r′,E)Θ(r′\nn−rn)\n+Jn\nΛ(r,E)Zn×\nΛ′(r′,E)Θ(rn−r′\nn).\nHerer,r′refer to points within atomic volumes around\nsitesRm,Rn, respectively, with Zn\nΛ(r,E) =ZΛ(rn,E) =\nZΛ(r−Rn,E) being a function centered at site Rn. Ad-\nopting a fully relativistic formulation34,35for Eq. (2) one\ngets in a natural way access to all spin-orbit inducedproperties as for example the anomalous and spin Hall\nconductivity29,32,36or Gilbert damping parameter.18In\nthis case, the functions Zn\nΛandJn\nΛstand for the reg-\nular and irregular, respectively, solutions to the single-\nsite Dirac equation for site nwith the associated single-\nsite scattering t-matrix tn\nΛΛ′. The corresponding scat-\ntering path operator τnn′\nΛΛ′accounts for all scattering\nevents connecting the sites nandn′. Using a suitable\nspinor representation for the basis functions the com-\nbined quantum number Λ = ( κ,µ) stands for the relativ-\nistic spin-orbit and magnetic quantum numbers κandµ,\nrespectively.34,35,37\nAs was demonstrated by various authors27,28,38rep-\nresenting the electronic structure in terms of the Green\nfunction G+(r,r′,E) allows to account for chemical dis-\norder in a random alloy by making use of a suitable al-\nloy theory. In this case ∝an}bracketle{t...∝an}bracketri}htcstands for the configura-\ntional average for a substitutional alloy concerning the\nsite occupation. Corresponding expressions for the con-\nductivity tensor have been worked out by Velick´ y27and\nButler28usingthe single-siteCoherentPotentialApprox-\nimation (CPA) that include in particular the so-called\nvertex corrections.\nThe CPA can be used to deal with chemical but also\nwith any other type of disorder. In fact, making use of\nthe different time scales connected with the electronic\npropagation and spin fluctuations the alloy analogy is\nexploited when dealing with finite temperature magnet-\nism on the basis of the disordered local moment (DLM)\nmodel.26,39Obviously, the same approach can be used\nwhen dealing with response tensors at finite temperat-\nures. In connection with the conductivity this is often\ncalled adiabatic approximation.40Following this philo-\nsophy, the CPA has been used recently also when calcu-\nlating response tensors using Eq. ( 1) with disorder in the\nsystem caused by thermal lattice vibrations18,31as well\nas spin fluctuations.20,41\nB. Treatment of thermal lattice displacement\nA way to account for the impact of the thermal dis-\nplacement of atoms from their equilibrium positions, i.e.\nfor thermal lattice vibrations, on the electronic struc-\nture is to set up a representative displacement configura-\ntion for the atoms within an enlargedunit cell (super-cell\ntechnique). In this case one has to use either a very large\nsuper-cell or to take the average over a set of super-cells.\nAlternatively, one may make use of the alloy analogy for\nthe averaging problem. This allows in particular to re-\nstrict to the standard unit cell. Neglecting the correla-\ntion between the thermal displacements of neighboring\natoms from their equilibrium positions the properties of\nthe thermal averaged system can be deduced by making\nuse of the single-site CPA. This basic idea is illustrated\nby Fig.1. To make use of this scheme a discrete set\nofNvdisplacement vectors ∆ Rq\nv(T) with probability xq\nv\n(v= 1,..,Nv) is constructed for each basis atom qwithin3\nFigure 1. Configurational averaging for thermal lattice dis -\nplacements: the continuous distribution P(∆Rn(T)) for the\natomic displacement vectors is replaced by a discrete set of\nvectors ∆ Rv(T) occurring with the probability xv. The con-\nfigurational average for this discrete set of displacements is\nmade using the CPA leading to a periodic effective medium.\nthe standard unit cell that is conform with the local sym-\nmetry and the temperature dependent root mean square\ndisplacement ( ∝an}bracketle{tu2∝an}bracketri}htT)1/2according to:\n1\nNvNv/summationdisplay\nv=1|∆Rq\nv(T)|2=∝an}bracketle{tu2\nq∝an}bracketri}htT. (3)\nIn the general case, the mean square displacement along\nthe direction µ(µ=x,y,z) of the atom ican be either\ntaken from experimental data or represented by the ex-\npression based on the phonon calculations42\n∝an}bracketle{tu2\ni,µ∝an}bracketri}htT=3/planckover2pi1\n2Mi/integraldisplay∞\n0dωgi,µ(ω)1\nωcoth/planckover2pi1ω\n2kBT,(4)\nwhereh= 2π/planckover2pi1the Planck constant, kBthe Boltzmann\nconstant, gi,µ(ω) is a partial phonon density of states.42\nOn the other hand, a rather good estimate for the root\nmean square displacement can be obtained using Debye’s\ntheory. In this case, for systems with one atom per unit\ncell, Eq. ( 4) can be reduced to the expression:\n∝an}bracketle{tu2∝an}bracketri}htT=1\n43h2\nπ2MkBΘD/bracketleftbiggΦ(ΘD/T)\nΘD/T+1\n4/bracketrightbigg\n(5)\nwith Φ(Θ D/T) the Debye function and Θ Dthe Debye\ntemperature43. Ignoring the zero temperature term 1 /4\nand assuming a frozen potential for the atoms, the situ-\nationcanbe dealt with in full analogytothe treatmentof\ndisorderedalloysonthebasisoftheCPA.Theprobability\nxvfor a specific displacement vmay normally be chosen\nas 1/Nv. The Debye temperature Θ Dused in Eq. ( 5) can\nbe either taken fromexperimental data orcalculated rep-\nresenting it in terms of the elastic constants44. In general\nthe latter approach should give more reliable results in\nthe case of multicomponent systems.\nTo simplify notation we restrict in the following to sys-\ntems with one atom per unit cell. The index qnumbering\nsites in the unit cell can therefore be dropped, while the\nindexnnumbers the lattice sites.\nAssuming a rigid displacement of the atomic potential\nin the spirit of the rigid muffin-tin approximation45,46\nthe correspondingsingle-site t-matrix tlocwith respect to\nthe local frame of reference connected with the displaced\natomic position is unchanged. With respect to the globalframe of reference connected with the equilibrium atomic\npositions Rn, however, the corresponding t-matrix tis\ngiven by the transformation:\nt=U(∆R)tlocU(∆R)−1. (6)\nThe so-called U-transformation matrix U(s) is given in\nits non-relativistic form by:45,46\nULL′(s) = 4π/summationdisplay\nL′′il+l′′−l′CLL′L′′jl′′(|s|k)YL′′(ˆs).(7)\nHereL= (l,m) represents the non-relativistic angu-\nlar momentum quantum numbers, jl(x) is a spherical\nBesselfunction, YL(ˆr) a realsphericalharmonics, CLL′L′′\na corresponding Gaunt number and k=√\nEis the\nelectronic wave vector. The relativistic version of the\nU-matrix is obtained by a standard Clebsch-Gordan\ntransformation.37\nThe various displacement vectors ∆ Rv(T) can be used\nto determine the properties of a pseudo-component of a\npseudo alloy. Each of the Nvpseudo-components with\n|∆Rv(T)|=∝an}bracketle{tu2∝an}bracketri}ht1/2\nTis characterized by a corresponding\nU-matrix Uvand t-matrix tv. As for a substitutional\nalloy the configurational average can be determined by\nsolving the multi-component CPA equations within the\nglobal frame of reference:\nτnn\nCPA=Nv/summationdisplay\nv=1xvτnn\nv (8)\nτnn\nv=/bracketleftbig\n(tv)−1−(tCPA)−1+(τnn\nCPA)−1/bracketrightbig−1(9)\nτnn\nCPA=1\nΩBZ/integraldisplay\nΩBZd3k/bracketleftbig\n(tCPA)−1−G(k,E)/bracketrightbig−1,(10)\nwhere the underline indicates matrices with respect to\nthe combined index Λ. As it was pointed out in the pre-\nvious work41, the cutoff for the angular momentum ex-\npansionin these calculations should be taken l≥lmax+1\nwith the lmaxvalue used in the calculations for the non-\ndistorted lattice.\nThe first of these CPA equations represents the re-\nquirement for the mean-field CPA medium that embed-\nding of a component vshould lead in the average to no\nadditional scattering. Eq. ( 9) gives the scattering path\noperator for the embedding of the component vinto the\nCPA medium while Eq. ( 10) gives the CPA scattering\npath operator in terms of a Brillouin zone integral with\nG(k,E) the so-called KKR structure constants.\nHaving solved the CPA equations the linear response\nquantity of interest may be calculated using Eq. ( 1)\nas for an ordinary substitutional alloy.27,28This im-\nplies that one also have to deal with the so-called ver-\ntex corrections27,28that take into account that one\nhas to deal with a configuration average of the type\n∝an}bracketle{tˆAµℑG+ˆAνℑG+∝an}bracketri}htcthat in general will differ from the\nsimpler product ∝an}bracketle{tˆAµℑG+∝an}bracketri}htc∝an}bracketle{tˆAνℑG+∝an}bracketri}htc.4\nC. Treatment of thermal spin fluctuations\nAs for the disorder connected with thermal displace-\nments the impact of disorder due to thermal spin fluc-\ntuations may be accounted for by use of the super-cell\ntechnique. Alternatively one may again use the alloy\nanalogy and determine the necessary configurational av-\nerage by means of the CPA as indicated in Fig. 2. As\nFigure 2. Configurational averaging for thermal spin fluc-\ntuations: the continuous distribution P(ˆen) for the orienta-\ntion of the magnetic moments is replaced by a discrete set of\norientation vectors ˆ efoccurring with a probability xf. The\nconfigurational average for this discrete set of orientatio ns is\nmade using the CPA leading to a periodic effective medium.\nfor the thermal displacements in a first step a set of rep-\nresentative orientation vectors ˆ ef(withf= 1,...,Nf) for\nthelocalmagneticmomentisintroduced(seebelow). Us-\ning the rigid spin approximation the spin-dependent part\nBxcoftheexchange-correlationpotentialdoesnotchange\nfor the local frame of reference fixed to the magnetic mo-\nment when the moment is oriented along an orientation\nvector ˆef. This implies that the single-site t-matrix tloc\nf\nin the local frame is the same for all orientation vectors.\nWith respect to the common global frame that is used\nto deal with the multiple scattering (see Eq. ( 10)) the\nt-matrix for a given orientation vector is determined by:\nt=R(ˆe)tlocR(ˆe)−1. (11)\nHere the transformation from the local to the global\nframe of reference is expressed by the rotation matrices\nR(ˆe) that are determined by the vectors ˆ eor correspond-\ning Euler angles.37\nAgain the configurational average for the pseudo-alloy\ncan be obtained by setting up and solvingCPAequations\nin analogy to Eqs. ( 8) to (10).\nD. Models of spin disorder\nThe central problem with the scheme described above\nis obviously to construct a realistic and representative\nset of orientation vectors ˆ efand probabilities xffor each\ntemperature T. A rather appealing approach is to cal-\nculate the exchange-coupling parameters Jijof a sys-\ntem in an ab-initio way25,47,48and to use them in sub-\nsequent Monte Carlo simulations. Fig. 3(top) shows\nresults for the temperature dependent average reduced\nmagnetic moment of corresponding simulations for bcc-\nFe obtained for a periodic cell with 4096 atom sites. The0 0.2 0.4 0.6 0.8 1 1.2\nT/TC00.20.40.60.81M(T)MC*\nKKR (MC*)\n0 0.2 0.4 0.6 0.8 1 1.2\nT/TC00.20.40.60.81M(T)\nMC\nMF-fit to MC (wMC(T))\nMF-fit to MC (w=const)\nExpt\nMF-fit to Expt (wExpt(T))\n0 0.2 0.4 0.6 0.8 1 1.2\nT/TC00.20.40.60.81M(T)MC\nKKR (MC)\nKKR (DLM)\nFigure 3. Averaged reduced magnetic moment M(T) =\n/angbracketleftmz/angbracketrightT/|/angbracketleftm/angbracketrightT=0|along the z-axis as a function of the tem-\nperature T. Top: results of Monte Carlo simulations using\nscheme MC* (full squares) compared with results of sub-\nsequent KKR-calculations (open squares). Middle: results\nof Monte Carlo simulations using scheme MC (full squares)\ncompared with results using a mean-field fit with a constant\nWeiss field wMC(TC) (open diamonds) and a temperature de-\npendent Weiss field wMC(T) (open squares). In addition ex-\nperimental data (full circles) together with a correspondi ng\nmean-field fit obtained for a temperature dependent Weiss\nfieldwexp(T). Bottom: results of Monte Carlo simulations\nusing scheme MC (full squares) compared with results sub-\nsequent KKR-calculations using the MC (triangles up) and\na corresponding DLM (triangle down) spin configuration, re-\nspectively.\nfull line gives the value for the reduced magnetic mo-\nmentMMC∗(T) =∝an}bracketle{tmz∝an}bracketri}htT/m0projected on the z-axis for\nthe lastMonteCarlostep (ˆ zis the orientationofthetotal\nmoment, i.e. ∝an}bracketle{tm∝an}bracketri}htT∝bardblˆz; the saturated magnetic moment at\nT= 0 K is m0=|∝an}bracketle{tm∝an}bracketri}htT=0|). This scheme is called MC∗\nin the following. In spite of the rather large number of\nsites (4096) the curve is rather noisy in particular when\napproaching the Curie temperature. Nevertheless, the5\nspin configuration of the last MC step was used as an\ninput for subsequent SPR-KKR-CPA calculations using\ntheorientationvectors ˆ efwiththeprobability xf= 1/Nf\nwithNf= 4096. As Fig. 3(top) shows, the temperature\ndependent reducedmagnetic moment MKKR(MC∗)(T) de-\nduced from the electronic structure calculations follows\none-to-one the Monte Carlo data MMC∗(T). This is a\nvery encouraging result for further applications (see be-\nlow) as it demonstrates that the CPA although being a\nmean-field method and used here in its single-site formu-\nlation is nevertheless capable to reproduce results of MC\nsimulations that go well beyond the mean-field level.\nHowever, using the set of vectors ˆ efof scheme MC*\nalso for calculations of the Gilbert damping parameters\nαas a function of temperature led to extremely noisy\nand unreliable curves for α(T). For that reason an av-\nerage has been taken over many MC steps (scheme MC)\nleading to a much smoother curve for MMC(T) as can\nbe seen from Fig. 3(middle) with a Curie temperature\nTMC\nC= 1082 K. As this enlarged set of vectors ˆ efgot\ntoo large to be used directly in subsequent SPR-KKR-\nCPA calculations, a scheme was worked out to get a set\nof vectors ˆ efand probabilities xfthat is not too large\nbut nevertheless leads to smooth curves for M(T).\nThe first attempt was to use the Curie temperature\nTMC\nCtodeduceacorrespondingtemperatureindependent\nWeiss-field w(TC) on the basis of the standard mean-field\nrelation:\nw(TC) =3kBTC\nm2\n0. (12)\nThis leads to a reduced magnetic moment curve MMF(T)\nthat shows by construction the same Curie temperature\nas the MC simulations. For temperatures between T=\n0 K and TC, however, the mean-field reduced magnetic\nmoment MMF(T) is well below the MC curve (see Fig. 3\n(middle) ).\nAs an alternative to this simple approach we intro-\nduced a temperature dependent Weiss field w(T). This\nallows to describe the temperature dependent magnetic\nproperties using the results obtained beyond the mean-\nfield approximation. At the same time the calculation\nof the statistical average can be performed treating the\nmodel Hamiltonian in termsofthe mean field theory. For\nthis reason the reduced magnetic moment M(T), being\na solution of equation (see e.g.49)\nM(T) =L/parenleftbiggwm2\n0M(T)\nkBT/parenrightbigg\n, (13)\nwas fitted to that obtained from MC simulations\nMMC(T)withtheWeissfield w(T)asafittingparameter,\nsuch that\nlim\nw→w(T)M(T) =MMC(T), (14)\nwithL(x) the Langevin function.\nThe corresponding temperature dependent probability\nx(ˆe) for an atomic magnetic moment to be oriented alongˆeis proportionalto exp( −w(T)ˆz·ˆe/kBT) (see, e.g.49). To\ncalculate this value we used NθandNφpoints for a reg-\nular grid for the spherical angles θandφcorresponding\nto the vector ˆ ef:\nxf=exp(−w(T)ˆz·ˆef/kBT)/summationtext\nf′exp(−w(T)ˆz·ˆef′/kBT).(15)\nFig.4shows for three different temperatures the θ-\ndependent behavior of x(ˆe). As one notes, the MF-fit\n0 30 60 90 120 150 180\nθ00.050.10.150.20.250.3P(θ)MC\nMF-fit to MC (wMC(T))T = 200 K\n0 30 60 90 120 150 180\nθ00.050.10.150.2P(θ)MC\nMF-fit to MC (wMC(T))T = 400 K\n0 30 60 90 120 150 180\nθ00.050.1P(θ)MC\nMF-fit to MC (wMC(T))T = 800 K\nFigure 4. Angular distribution P(θ) of the atomic magnetic\nmoment mobtained from Monte Carlo simulations (MC) for\nthe temperature T= 200, 400, and 800 K compared with field\nmean-field (MF) data, xf, (full line) obtained by fitting using\na temperature dependent Weiss field w(T) (Eq.13).\nto the MC-results perfectly reproduces these data for all\ntemperatures. This applies of course not only for the\nangular resolved distribution of the magnetic moments\nshown in Fig. 4but also for the average reduced mag-\nnetic moment recalculated using Eq.( 13), shown in Fig.\n3. Obviously, the MF-curve MMF(MC)(T) obtained using\nthe temperature dependent Weiss field parameter w(T)\nperfectly reproduces the original MMC(T) curve. The\ngreat advantage of this fitting procedure is that it al-\nlows to replace the MC data set with a large number6\nNMC\nfof orientation vectors ˆ ef(pointing in principle into\nany direction) with equal probability xf= 1/NMC\nfby a\nmuch smaller data set with Nf=NθNφwithxfgiven\nby Eq. (15).\nAccordingly, the reduced data set can straight for-\nwardly be used for subsequent electronic structure cal-\nculations. Fig. 3(bottom) shows that the calcu-\nlated temperature dependent reduced magnetic moment\nMKKR−MF(MC)(T) agrees perfectly with the reduced\nmagnetic moment MMC(T) given by the underlying MC\nsimulations.\nThe DLM method has the appealing feature that it\ncombines ab-initio calculations and thermodynamics in\na coherent way. Using a non-relativistic formulation, it\nwas shown that the corresponding averaging over all ori-\nentations of the individual atomic reduced magnetic mo-\nments can be mapped onto a binary pseudo-alloy with\none pseudo-component having up- and downward orient-\nation of the spin moment with concentrations x↑and\nx↓, respectively.24,50For a fully relativistic formulation,\nwith spin-orbitcoupling included, this simplificationcan-\nnot be justified anymore and a proper average has to be\ntaken over all orientations.51As we do not perform DLM\ncalculationsbut use hereonly the DLM picture to repres-\nent MC data, this complication is ignored in the follow-\ning. Having the set of orientation vectors ˆ efdetermined\nby MC simulations the corresponding concentrations x↑\nandx↓can straight forwardly be fixed for each temper-\nature by the requirement:\n1\nNfNf/summationdisplay\nf=1ˆef=x↑ˆz+x↓(−ˆz), (16)\nwithx↑+x↓= 1. Using this simple scheme electronic\nstructure calculations have been performed for a binary\nalloy having collinear magnetization. The resulting re-\nduced magnetic moment MKKR−DLM(MC) (T) is shown in\nFig.3(bottom). As one notes, again the original MC\nresults are perfectly reproduced. This implies that when\ncalculating the projected reduced magnetic moment Mz\nthat is determined by the averaged Green function ∝an}bracketle{tG∝an}bracketri}ht\nthe transversal magnetization has hardly any impact.\nFig.3(middle) gives also experimental data for\ntheM(T).52While the experimental Curie-temperature\nTexp\nC= 1044 K52is rather well reproduced by the MC\nsimulations TMC\nC= 1082 K one notes that the MC-curve\nMMC(T) is well below the experimental curve. In partic-\nular,MMC(T) drops too fast with increasing Tin the\nlow temperature regime and does not show the T3/2-\nbehavior. The reason for this is that the MC simulations\ndo not properly account for the low-energy long-ranged\nspinwaveexcitationsresponsibleforthelow-temperature\nmagnetization variation. Performing ab-initio calcula-\ntions for the spin wave energies and using these data for\nthe calculation of M(T) much better agreement with ex-\nperiment can indeed be obtained in the low-temperature\nregime than with MC simulations.53\nAs the fitting scheme sketched above needs only thetemperature reduced magnetic moment M(T) as input\nit can be applied not only to MC data but also to ex-\nperimental data. Fig. 3shows that the mean field fit\nMMF(exp)(T) again perfectly fits the experimental re-\nduced magnetic moment curve Mexp(T). Based on this\ngood agreement this corresponding data set {ˆef,xf}has\nalso been used for the calculation of responsetensors (see\nbelow).\nAn additional much simpler scheme to simulate the\nexperimental Mexp(T) curve is to assume the individual\natomic moments to be distributed on a cone, i.e. with\nNθ= 1 and Nφ>>1.23In this case the opening angle\nθ(T) of the cone is chosen such to reproduce M(T). In\ncontrasttothestandardDLMpicture,thissimplescheme\nallows already to account for transversal components of\nthe magnetization. Corresponding results for response\ntensor calculations will be shown below.\nFinally, it should be stressed here that the various spin\nconfiguration models discussed above assume a rigid spin\nmoment, i.e. its magnitude does not change with temper-\nature nor with orientation. In contrast to this Ruban et\nal.54usealongitudinalspinfluctuation Hamiltonianwith\nthe corresponding parameters derived from ab-initio cal-\nculations. As a consequence, subsequent Monte Carlo\nsimulations based on this Hamiltonian account in par-\nticular for longitudinal fluctuations of the spin moments.\nA similar approach has been used by Drchal et al.55,56\nleading to good agreement with the results of Ruban et\nal. However, the scheme used in these calculations does\nnot supply in a straightforward manner the necessary\ninput for temperature dependent transport calculations.\nThis is different from the work of Staunton et al.57who\nperformed self-consistent relativistic DLM calculations\nwithout the restriction to a collinear spin configuration.\nThis approach in particular accounts in a self-consistent\nway for longitudinal spin fluctuations.\nE. Combined chemical and thermally induced\ndisorder\nThe various types of disorder discussed above may be\ncombined with each other as well as with chemical i.e.\nsubstitution disorder. In the most general case a pseudo-\ncomponent ( vft) is characterized by its chemical atomic\ntypet, the spin fluctuation fand lattice displacement\nv. Using the rigid muffin-tin and rigid spin approxim-\nations, the single-site t-matrix tloc\ntin the local frame is\nindependent from the orientation vector ˆ efand displace-\nment vector ∆ Rv, and coincides with ttfor the atomic\ntypet. With respect to the common global frame one\nhas accordingly the t-matrix:\ntvft=U(∆Rv)R(ˆef)ttR(ˆef)−1U(∆Rv)−1.(17)\nWith this the corresponding CPA equations are identical\nto Eqs. ( 8) to (10) with the index vreplaced by\nthe combined index ( vft). The corresponding pseudo-\nconcentration xvftcombines the concentration xtof the7\natomic type twith the probability for the orientation\nvector ˆefand displacement vector ∆ Rv.\nIII. COMPUTATIONAL DETAILS\nThe electronic structure of the investigated ferro-\nmagnets bcc-Fe and fcc-Ni, has been calculated self-\nconsistently using the spin-polarized relativistic KKR\n(SPR-KKR) band structure method.58,59For the ex-\nchangecorrelationpotential the parametrizationas given\nby Vosko et al.60has been used. The angular-momentum\ncutoff of lmax= 3 was used in the KKR multiple scatter-\ning expansion. The lattice parameters have been set to\nthe experimental values.\nIn a second step the exchange-coupling parameters\nJijhave been calculated using the so-called Lichten-\nstein formula.25Although the SCF-calculations have\nbeen done on a fully-relativistic level the anisotropy of\nthe exchange coupling due to the spin-orbit coupling has\nbeen neglected here. Also, the small influence of the\nmagneto-crystallineanisotropyfor the subsequent Monte\nCarlo (MC) simulations has been ignored, i.e. these have\nbeen based on a classical Heisenberg Hamiltonian. The\nMC simulations were done in a standard way using the\nMetropolis algorithm and periodic boundary conditions.\nThe theoretical Curie temperature TMC\nChas been de-\nduced from the maximum of the magnetic susceptibility.\nThe temperature dependent spin configuration ob-\ntained during a MC simulation has been used to con-\nstruct a set of orientations ˆ efand probabilities xfac-\ncording to the schemes MC* and MC described in sec-\ntionIIDto be used within subsequent SPR-KKR-CPA\ncalculations (see above). For the corresponding calcu-\nlation of the reduced magnetic moment the potential\nobtained from the SCF-calculation for the perfect fer-\nromagnetic state ( T= 0K) has been used. The calcu-\nlation for the electrical conductivity as well as the Gil-\nbertdampingparameterhasbeenperformedasdescribed\nelsewhere.41,61\nIV. RESULTS AND DISCUSSION\nA. Temperature dependent conductivity\nEq. (1) has been used together with the various\nschemes described above to calculate the temperature\ndependent longitudinal resistivity ρ(T) of the pure fer-\nromagnets Fe, Co and Ni. In this case obviously disorder\ndue to thermal displacements of the atoms as well as spin\nfluctuations contribute to the resistivity.\nTo give an impression on the impact of the thermal\ndisplacementsaloneFig. 5givesthe temperaturedepend-\nent resistivity ρ(T) of pure Cu (Θ Debye= 315 K) that\nis found in very good agreement with corresponding ex-\nperimental data.62This implies that the alloy analogy\nmodel that ignores any inelastic scattering events should0 100 200 300 400 500\nTemperature (K)01234ρxx (10-6Ω⋅cm)Expt\nTheory - alloy analogy\nTheory - LOVA\nCu\nFigure 5. Temperature dependent longitudinal resistivity of\nfcc-Cuρ(T) obtained by accounted for thermal vibrations as\ndescribed in section IIBcompared with corresponding ex-\nperimental data.62In addition results are shown based on\nthe LOVA (lowest order variational approximation) to the\nBoltzmann formalism.14\nin general lead to rather reliable results for the resistivity\ninduced by thermal displacements. Accordingly, com-\nparison with experiment should allow for magnetically\nordered systems to find out the most appropriate model\nfor spin fluctuations.\nFig.6(top) shows theoretical results for ρ(T) of bcc-\nFe due to thermal displacements ρv(T), spin fluctuations\ndescribed by the scheme MC ρMC(T) as well as the com-\nbination of the two influences ( ρv,MC(T)). First of all\none notes that ρv(T) is not influenced within the adop-\ntedmodelbytheCurietemperature TCbutisdetermined\nonly by the Debye temperature. ρMC(T), on the other\nhand, reaches saturation for TCas the spin disorder does\nnot increase anymore with increasing temperature in the\nparamagnetic regime. Fig. 6also shows that ρv(T) and\nρMC(T)arecomparableforlowtemperaturesbut ρMC(T)\nexceedsρv(T) more and more for higher temperatures.\nMost interestingly, however, the resistivity for the com-\nbined influence of thermal displacements and spin fluctu-\nationsρv,MC(T) does not coincide with the sum of ρv(T)\nandρMC(T) but exceeds the sum for low temperatures\nand lies below the sum when approaching TC.\nFig.6(bottom) shows the results of three differ-\nent calculations including the effect of spin fluctuations\nas a function of the temperature. The curve ρMC(T)\nis identical with that given in Fig. 6(top) based on\nMonte Carlo simulations. The curves ρDLM(MC) (T) and\nρcone(MC)(T) are based on a DLM- and cone-like repres-\nentation of the MC-results, respectively. For all three\ncases results are given including as well as ignoring the\nvertex corrections. As one notes the vertex corrections\nplay a negligible role for all three spin disorder models.\nThis is fully in line with the experience for the longitud-\ninal resistivity of disordered transition metal alloys: as\nlong as the the states at the Fermi level have domin-\nantly d-character the vertex corrections can be neglected\nin general. On the other hand, if the sp-character dom-8\n0 0.2 0.4 0.6 0.8 1 1.2\nT/TC020406080100120ρxx  (10-6Ω⋅cm)vib\nfluct (MC)\nvib + fluct (MC)\n0 0.2 0.4 0.6 0.8 1 1.2\nT/TC020406080100120ρxx  (10-6Ω⋅cm)MC (VC)\nMC (NVC)\nDLM (VC)\nDLM (NVC)\ncone (VC)\ncone (NVC)\nFigure 6. Temperature dependent longitudinal resistivity of\nbcc-Feρ(T) obtained by accounted for thermal vibrations\nand spin fluctuations as described in section IIB. Top: ac-\ncounting for vibrations (vib, diamonds), spin fluctuations us-\ning scheme MC (fluct, squares) and both (vib+fluct, circles).\nBottom: accounting for spin fluctuations ˆ ef= ˆe(θf,φf) us-\ning the schemes: MC (squares) with 0 ≤θf≤π;0≤φf≤\n2π, DLM(MC) (triangles up) with θf1= 0,θf2=π, and\ncone(MC) (triangles down) θf=/angbracketleftθf/angbracketrightT;0≤φf≤2π. The\nfull and open symbols represent the results obtained with th e\nvertex corrections included (VC) and excluded (NV), respec t-\nively.\ninates inclusion of vertex corrections may alter the result\nin the order of 10 %.63,64\nComparing the DLM-result ρDLM(MC) (T) with\nρMC(T) one notes in contrast to the results for M(T)\nshown above (see Fig. 3(bottom)) quite an appreciable\ndeviation. This implies that the restricted collinear\nrepresentation of the spin configuration implied by the\nDLM-model introduces errors for the configurational\naverage that seem in general to be unacceptable, For\nthe Curie temperature and beyond in the paramagnetic\nregimeρDLM(MC) (T) andρMC(T) coincide, as it was\nshown formally before.20\nComparing finally ρcone(MC)(T) based on the conical\nrepresentationofthe MCspin configurationwith ρMC(T)\none notes that also this simplification leads to quite\nstrong deviations from the more reliable result. Never-\ntheless, one notes that ρDLM(MC) (T) agrees with ρMC(T)\nfor the Curie temperature and also accounts to some ex-\ntent for the impact of the transversal components of themagnetization.\nThe theoretical results for bcc-Fe (Θ Debye= 420 K)\nbased on the combined inclusion of the effects of thermal\ndisplacementsandspinfluctuationsusingtheMCscheme\n(ρv,MC(T)) are compared in Fig. 7(top) with experi-\nmental data ( ρexp(T)). For the Curie temperature ob-\n0 0.2 0.4 0.6 0.8 1 1.2 1.4\nT/TC020406080100120ρxx  (10-6Ω⋅cm)Expt: J. Bass and K.H. Fischer \nvib + fluct (MC)\nvib + fluct (exp)\n0 0.2 0.4 0.60.8 1 1.2 1.4 1.61.8\nT/TC01020304050ρxx  (10-6Ω⋅cm)Expt.: C.Y. Ho et al. (1983)\nvib\nvib (PM)\nfluct\nvib + fluct\nFigure 7. Top: Temperature dependent longitudinal res-\nistivity of bcc-Fe ρ(T) obtained by accounted for thermal\nvibrations and spin fluctuations using the scheme MC\n(vib+fluct(MC), squares) and a mean-field fit to the experi-\nmental temperature magnetic moment Mexp(vib+fluct(exp),\ndiamonds) compared with experimental data (circles).62Bot-\ntom: corresponding results for fcc-Ni. In addition results are\nshown accounting for thermal displacements (vib) only for\nthe ferromagnetic (FM) as well paramagnetic (PM) regime.\nExperimental data have been taken from Ref. 65.\nviously a very good agreement with experiment is found\nwhile for lower temperatures ρv,MC(T) exceeds ρexp(T).\nThis behavior correlates well with that of the temperat-\nure dependent reduced magnetic moment M(T) shown\nin Fig.3(middle). The too rapid decrease of MMC(T)\ncompared with experiment implies an essentially overes-\ntimated spin disorder at any temperature leading in turn\nto a too large resistivity ρv,MC(T). On the other hand,\nusing the temperature dependence of the experimental\nreducedmagneticmoment Mexp(T)tosetup thetemper-\nature dependent spin configuration as described above a\nvery satisfying agreement is found with the experimental\nresistivity data ρexp(T). Note also that above TCthe\ncalculated resistivity riches the saturation in contrast to\nthe experimental data where the continuing increase of9\nρexp(T) can be attributed to the longitudinal spin fluctu-\nations leading to a temperature dependent distribution\nof local magnetic moments on Fe atoms.54However, this\ncontribution was not taken into account because of re-\nstriction in present calculations using fixed value for the\nlocal reduced magnetic moments.\nFig.7(bottom) shows corresponding results for the\ntemperature dependent resistivity of fcc-Ni (Θ Debye=\n375 K). For the ferromagnetic (FM) regime that the\ntheoretical results are comparable in magnitude when\nonly thermal displacements ( ρv(T)) or spin fluctuations\n(ρMF(T)) are accounted for. In the later case the mean\nfieldw(T) has been fitted to the experimental M(T)-\ncurve. Taking both into account leads to a resistivity\n(ρv,MF(T)) that are well above the sum of the individual\ntermsρv(T) andρMF(T). Comparing ρv,MF(T) with ex-\nperimentaldata ρexp(T)ourfindingshowsthatthetheor-\netical results overshoots the experimental one the closer\none comes to the critical temperature. This is a clear\nindication that the assumption of a rigid spin moment\nis quite questionable as the resulting contribution to the\nresistivity due to spin fluctuations as much too small.\nIn fact the simulations of Ruban et al.54on the basis of\na longitudinal spin fluctuation Hamiltonian led on the\ncase of fcc-Ni to a strong diminishing of the averagelocal\nmagnetic moment when the critical temperature is ap-\nproachedfrom below (about 20% comparedto T= 0K).\nFor bcc-Fe, the change is much smaller (about 3 %) justi-\nfying on the case the assumption of a rigid spin moment.\nTaking the extreme point of view that the spin moment\nvanishescompletely abovethe criticaltemperature orthe\nparamagnetic (PM) regime only thermal displacements\nhave to be considered as a source for a finite resistivity.\nCorresponding results are shown in Fig. 7(bottom) to-\ngether with corresponding experimental data. The very\ngood agreementbetween both obviouslysuggeststhat re-\nmaining spin fluctuations above the critical temperature\nare of minor importance for the resistivity of fcc-Ni.\nB. Temperature dependent Gilbert damping\nparameter\nFig.8shows results for Gilbert damping parameter α\nof bcc-Fe obtained using different models for the spin\nfluctuations. All curves show the typical conductivity-\nlike behaviorfor low temperatures and the resistivity-like\nbehavior at high temperatures reflecting the change from\ndominating intra- to inter-band transitions.66The curve\ndenoted expt isbasedon aspin configurationtoted tothe\nexperimental Mexpt(T) data. Using the conical model to\nfitMexpt(T) as basis for the calculation of α(T) leads\nobviously to a rather good agreement with αM(expt)(T).\nHaving instead a DLM-like representation of Mexpt(T),\non the other hand, transverse spin components are sup-\npressed and noteworthy deviations from αM(expt)(T) are\nfound for the low temperature regime. Nevertheless, the\ndeviations are less pronounced than in the case of the0 200 400 600 800\nTemperature (K)02468α × 103fluct (MC)\nfluct (Expt)\nfluct (DLM)\nfluct (cone)\nFigure 8. Temperature dependent Gilbert damping α(T) for\nbcc-Fe, obtainedbyaccountedfor thermal vibrations andsp in\nfluctuations accounting for spin fluctuations using scheme\nMC (squares), DLM(MC) (triangles up), cone(MC) (triangles\ndown) and a MF fit to the experimental temperature reduced\nmagnetic moment (circles).\nlongitudinal resistivity (see Fig. 6(bottom)), where cor-\nresponding results are shown based on MMC(T) as a ref-\nerence. Obviously, the damping parameter αseems to\nbe less sensitive to the specific spin fluctuation model\nused than the resistivity. Finally, using the spin con-\nfiguration deduced from Monte Carlo simulations, i.e.\nbased on MMC(T) quite strong deviations for the result-\ningαM(MC)(T) fromαM(expt)(T) are found. As for the\nresistivity (see Fig. 6(bottom)) this seems to reflect the\ntoo fast drop of the reduced magnetic moment MMC(T)\nwith temperature in the low temperature regime com-\npared with temperature (see Fig. 3). As found before18\naccountingonly for thermal vibrations α(T) (Fig.6(bot-\ntom)) is found comparableto the casewhen only thermal\nspan fluctuations are allowed. Combing both thermal ef-\nfects does not lead to a curve that is just the sum of the\ntwoα(T) curves. As found for the conductivity (Fig. 6\n(top)) obviously the two thermal effects are not simply\nadditive. As Fig. 9(top) shows, the resulting damping\nparameter α(T) for bcc-Fe that accounts for thermal vi-\nbrationsaswellasspinfluctuationsisfoundinreasonable\ngood agreement with experimental data.18\nFig.9shows also corresponding results for the Gilbert\ndampingoffcc-Niasafunctionoftemperature. Account-\ning only for thermal spin fluctuations on the basis of the\nexperimental M(T)-curveleadsinthis casetocompletely\nunrealistic results while accounting only for thermal dis-\nplacements leads to results already in rather good agree-\nment with experiment. Taking finally both sources of\ndisorder into account again no simple additive behavior\nis found but the results are nearly unchanged compared\nto those based on the thermal displacements alone. This\nimplies that results for the Gilbert damping parameter\nof fcc-Ni hardly depend on the specific spin configura-\ntion model used but are much more governed by thermal\ndisplacements.10\n0 200 400 600 800\nTemperature (K)0246810α × 103vib\nvib + fluct (Expt)\nExpt 1\nExpt 2\n0 100 200 300 400 500\nTemperature (K)00.050.10.150.2αvib\nfluct (Expt)\nvib + fluct (Expt)\nExpt\nFigure 9. Top: Temperature dependent Gilbert damping\nα(T) for bcc-Fe, obtained byaccounted for thermal vibrations\nand spin fluctuations accounting for lattice vibrations onl y\n(circles) and lattice vibrations and spin fluctuations base d on\nmean-field fit to the experimental temperature reduced mag-\nnetic moment Mexpt(diamonds) compared with experimental\ndata (dashed and full lines).67,68Bottom: corresponding res-\nults for fcc-Ni. Experimental data have been taken from Ref.\n67.\nV. SUMMARY\nVarious schemes based on the alloy analogy that al-\nlow to include thermal effects when calculating responseproperties relevant in spintronics have been presented\nand discussed. Technical details of an implementation\nwithin the framework of the spin-polarized relativistic\nKKR-CPA band structure method have been outlined\nthat allow to deal with thermal vibrations as well as spin\nfluctuations. Various models to represent spin fluctu-\nations have been compared with each other concerning\ncorresponding results for the temperature dependence\nof the reduced magnetic moment M(T) as well as re-\nsponse quantities. It was found that response quantities\nare much more sensitive to the spin fluctuation model as\nthe reduced magnetic moment M(T). 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We use the classical Hei senberg model with Landau-Lifshitz-\nGilbert (LLG) dynamics including a stochastic field represe nting finite temperatures. The friction\nforce depends linearly on the velocity (provided it is small enough) for all temperatures. For\nlow temperatures, the corresponding friction coefficient is proportional to the phenomenological\ndamping constant of the LLG equation. This dependence is los t at high temperatures, where the\nfriction coefficient decreases exponentially. These finding s can be explained by properties of the\nspin polarisation cloud dragged along with the tip.\nIntroduction. – While on the macroscopic scale the\nphenomenology of friction is well known, several new as-\npects are currently being investigated on the micron and\nnanometer scale [1,2]. During the last two decades, the\nresearch on microscopic friction phenomena has advanced\nenormously, thanks to the development of Atomic Force\nMicroscopy (AFM, [3]), which allows to measure energy\ndissipation caused by relative motion of a tip with respect\nto a substrate.\nRecently the contribution of magnetic degrees of free-\ndom to energy dissipation processes has attracted increas-\ning interest [4–8]. Today, magnetic materials can be con-\ntrolleddowntothenanometerscale. Newdevelopmentsin\nthe data storage industry, spintronics and quantum com-\nputing require a better understanding of tribological phe-\nnomena in magnetic systems. For example, the reduction\nof heat generation in reading heads of hard disks which\nworkat nanometerdistancesis animportant issue, as heat\ncan cause data loss.\nMagnetic Force Microscopy(MFM), where both tip and\nsurface are magnetic, is used to investigate surface mag-\nnetism and to visualise domain walls. Although recent\nstudies have attempted to measure energy dissipation be-\ntween an oscillating tip and a magnetic sample [9,10], the\ndependenceofthefrictionforceonthetip’sslidingvelocity\nhasnot been consideredyet apartfroma workby C. Fusco\net al.[8] which is extended by the present work to temper-\naturesT∝negationslash=0. The relative motion of the tip with respect\nto the surface can lead to the creation of spin waves whichpropagate inside the sample and dissipate energy, giving\nrise to magnetic friction.\nWe will firstpresent asimulationmodel anddefine mag-\nnetic friction. The model contains classical Heisenberg\nspins located on a rigid lattice which interact by exchange\ninteraction with each other. Analogous to the reading\nhead of a hard disc or a MFM tip, an external fixed\nmagnetic moment is moved across the substrate. Using\nLangevin dynamics and damping, it is possible to simu-\nlate systems at finite temperatures. The main new results\nconcern the temperature dependence of magnetic friction.\nSimulation model and friction definition. – To\nsimulate a solid magnetic monolayer (on a nonmagnetic\nsubstrate), we consider a two-dimensional rigid Lx×Ly\nlattice of classical normalised dipole moments (“spins”)\nSi=µi/µs, where µsdenotes the material-dependent\nmagnetic saturationmoment(typically afew Bohrmagne-\ntons). These spins, located at z= 0 and with lattice spac-\ninga, represent the magnetic moments of single atoms.\nThey can change their orientation but not their absolute\nvalue, so that there are two degrees of freedom per spin.\nWeuseopenboundaryconditions. Aconstantpointdipole\nStippointing in the z-direction and located at z= 2arep-\nresentsthemagnetictip. Itismovedparalleltothesurface\nwith constant velocity v.\nThis model has only magnetic degrees of freedom and\nthus focusses on their contributions to friction. For a\nreal tip one could expect that magnetic, just like nonmag-\np-1M.P. Magiera 1 L. Brendel 1 D.E. Wolf 1 U. Nowak 2\nnetic [11,12] interactions might also lead to atomic stick-\nslip behaviour, and hence to phononic dissipation with\na velocity-independent friction contribution as described\nby the Prandtl-Tomlinson model [13,14]. However, this\nrequires a periodic potential between tip and substrate,\nthat is strong enough compared to the elastic deformation\nenergytoallowformultiple localpotentialenergyminima.\nThe magnetic tip-substrate interactions are unlikely to be\nstrong enough.\nThe Hamiltionan consists of two parts:\nH=Hsub+Hsub−tip. (1)\nThe first one represents the internal ferromagnetic short-\nrange interaction within the substrate. The second one\ndescribes the long-range coupling between the substrate\nand the tip.\nThe interaction between the substrate moments is mod-\neled by the anisotropic classical Heisenberg model,\nHsub=−J/summationdisplay\n/angbracketlefti,j/angbracketrightSi·Sj−dzN/summationdisplay\ni=1S2\ni,z. (2)\nJ >0 describes the ferromagnetic exchange interaction\nbetween two nearest neighbours, expressed by the an-\ngular brackets ∝angbracketlefti,j∝angbracketright.dz<0 quantifies the anisotropy,\nwhich prefers in-plane orientations of the spins. The\ndipole-dipole-interaction between the substrate spins is\nneglected, because it is much weaker than the exchange\ninteraction. A quantitative comparison of our simulation\nresults with the ones obtained in [8], where the dipole-\ndipole-interaction inside the substrate was taken into ac-\ncount, justifies this approximation, which reduces simula-\ntion time enormously.\nThe long-range interaction between substrate and tip is\ndescribed by a dipole-dipole interaction term\nHsub−tip=−wN/summationdisplay\ni=13 (Si·ei)(Stip·ei)−Si·Stip\nR3\ni,(3)\nwhereRi=|Ri|denotes the norm of the distance vector\nRi=ri−rtip, andeiits unit vector ei=Ri/Ri.riand\nrtipdenote the position vectors of the substrate spins and\nthe tip respectively. wquantifies the dipole-dipole cou-\npling of the substrate and the tip. Note, however, that\nthe results of the present study only depend on the com-\nbination w|Stip|, which is the true control parameter for\nthe tip-substrate coupling.\nTheproperequationofmotionofthemagneticmoments\nis theLandau-Lifshitz-Gilbert (LLG, [15]) equation,\n∂\n∂tSi=−γ\n(1+α2)µs[Si×hi+αSi×(Si×hi)],(4)\nwhich is equivalent to the Gilbert equation of motion [16]:\n∂\n∂tSi=−γ\nµsSi×/bracketleftbigg\nhi−αµs\nγ∂Si\n∂t/bracketrightbigg\n. (5)The first term on the right-hand side of eq. (4) describes\nthe dissipationless precession of each spin in the effective\nfieldhi(to be specified below). The second term de-\nscribes the relaxation of the spin towards the direction\nofhi.γdenotes the gyromagnetic ratio (for free electrons\nγ= 1.76086×1011s−1T−1), andαis a phenomenological,\ndimensionless damping parameter.\nThe effective field contains contributions from the tip\nand from the exchange interaction, as well as a thermally\nfluctuating term ζi[17,18],\nhi=−∂H\n∂Si+ζi(t). (6)\nThe stochastic, local and time-dependent vector ζi(t) ex-\npresses a “Brownian rotation”, which is caused by the\nheat-bath connected to each magnetic moment. In our\nsimulations this vector is realised by uncorrelated random\nnumbers with a Gaussian distribution, which satisfy the\nrelations\n∝angbracketleftζi(t)∝angbracketright= 0 and (7)\n∝angbracketleftζκ\ni(t)ζλ\nj(t′)∝angbracketright= 2αµs\nγkBTδi,jδκ,λδ(t−t′),(8)\nwhereTis the temperature, δi,jexpresses that the noise\nat different lattice sites is uncorrelated, and δκ,λrefers to\nthe absence of correlations among different coordinates.\nTofind aquantitywhichexpressesthe friction occurring\nin the system, it is helpful to discuss energy transfers be-\ntween tip, substrate and heat-bath first. It is straightfor-\nward to separate the time derivative of the system energy,\neq. (1), into an explicit and an implicit one,\ndH\ndt=∂H\n∂t+N/summationdisplay\ni=1∂H\n∂Si·∂Si\n∂t. (9)\nThe explicit time dependence is exclusively due to the tip\nmotion. The energy transfer between the tip and the sub-\nstrate is expressed by the first term of eq. (9), which jus-\ntifies to call it the “pumping power” Ppump:\nPpump=∂H\n∂t=∂Hsub−tip\n∂rtip·v\n=w/summationdisplay\nαvαN/summationdisplay\ni=13\nR4\ni/braceleftbigg\n(Si·eiei,α−Si,α)(Stip·ei) (10)\n+(Stip·eiei,α−Stip,α)(Si·ei)\n−Si·Stipei,α+3ei,α(Si·ei)(Stip·ei)/bracerightbigg\nAt any instance, the substrate exerts a force −∂Hsub−tip\n∂rtip\non the tip. Due to Newton’s third law, Ppumpis the work\nper unit time done by the tip on the substrate. Its time\nand thermal average ∝angbracketleftPpump∝angbracketrightis the average rate at which\nenergy is pumped into the spin system. In a steady state\nit must be equal to the average dissipation rate, i.e.to\np-2Spin excitations in a monolayer scanned by a magnetic tip\nh\nhSω\nδθδϕ\nFigure1: Asingle spininamagnetic fieldrotatingwithangul ar\nvelocity ω, is dragged along with a phase shift δϕand aquires\nan out of plane component δθ.\nthe net energy transferred to the heat bath per unit time\ndue to spin relaxation. The magnetic friction force can\ntherefore be calculated by\nF=∝angbracketleftPpump∝angbracketright\nv. (11)\nThe second term of eq. (9) describes the energy transfer\nbetween the spin system and the heat bath. Inserting eq.\n(5) intoPdiss=−/summationtextN\ni=1∂H\n∂Si·∂Si\n∂tleads to\nPdiss=N/summationdisplay\ni=1∂H\n∂Si·/bracketleftbiggγ\nµsSi×ζi−αSi×∂Si\n∂t/bracketrightbigg\n=−Ptherm+Prelax. (12)\nThe first term, Pthermcontaining ζi, describes, how much\nenergy is transferred into the spin system due to the ther-\nmal perturbation by the heat bath. The second term,\nPrelaxproportional to the damping constant α, describes\nthe rate of energy transfer into the heat bath due to the\nrelaxation of the spins.\nAtT= 0,Pthermis zero. The spins are only perturbed\nby the external pumping at v∝negationslash= 0. Then\nPrelax=Pdiss=γα\nµs(1+α2)N/summationdisplay\ni=1(Si×hi)2,(T= 0),\n(13)\nwhere for the last transformation we used eq. (4) in order\ntoshowexplicitlytherelationshipbetweendissipationrate\nand misalignment between spins and local fields.\nIt will be instructive to compare the magnetic substrate\nscanned by a dipolar tip with a much simpler system, inwhich the substrate is replaced by a single spin Ssub-\njected to an external field h(t) that rotates in the plane\nperpendicular to a constant angular velocity ω(replac-\ning the tip velocity). It is straight forward to obtain the\nsteady state solution for T= 0, where in the co-rotating\nframeSis at rest. Slags behind h/hby an angle δϕand\ngets a component δθinω-direction (cf. fig. 1), which are\nin first order given by\nδϕ\nα=δθ=ωµs\nhγ+O/parenleftBigg/parenleftbiggωµs\nhγ/parenrightbigg3/parenrightBigg\n.(14)\nInserting this into the ( N=1)-case of eq. (13) yields a dis-\nsipation rate of Pdiss=αω2µs/γ, which corresponds to a\n“viscous” friction F=Pdiss/ω∝αω.\nIt is instructive to give a simple physical explanation\nfor eq. (14), instead of presenting the general solution,\nwhich can be found in [19]. Two timescales exist in the\nsystem, which can be readily obtained from eq. (4); first,\nthe inverse Lamor frequency τLamor= (1+α2)µs/γh, and\nsecond, the relaxation time τrelax=τLamor/α. They govern\nthe time evolution of δϕandδθ. In leading order,\nδ˙θ=δϕ\nτLamor−δθ\nτrelax, (15)\nδ˙ϕ=ω−δθ\nτLamor−δϕ\nτrelax. (16)\nThefirstequationdescribes, how δθwouldincreasebypre-\ncession of the spin around the direction of the field, which\nis counteracted by relaxation back towards the equator.\nThe second equation describes that without relaxation\ninto the field direction, δϕwould increase with velocity\nωminus the azimuthal component of the precession veloc-\nity, which is in leading order proportional to δθ. Setting\nthe left hand sides to zeroin the steady state, immediately\ngives the solution (14).\nInthe (T= 0)-study[8], the timeaverage ∝angbracketleftPrelax∝angbracketright/vwas\nused to calculate the friction force. As pointed out above,\nthis quantity agrees with (11) in the steady state. For\nfinite temperatures, however, (11) is numerically better\nbehaved than ∝angbracketleftPrelax∝angbracketright/v. The reason is the following:\nForT∝negationslash= 0, the spins are also thermally agitated, even\nwithout external pumping, when the dissipation rate Pdiss\nvanishes. This shows that the two terms PthermandPrelax\nlargely compensate each other, and only their difference is\nthe dissipation rate we are interested in. This fact makes\nit difficult to evaluate (12) and is the reasonwhy we prefer\nto work with (11) as the definition of the friction force.\nWe have also analyzed the fluctuations of the friction\nforce (11). The power spectrum has a distinct peak at\nfrequency v/a, which means that the dominant temporal\nfluctuations are due to the lattice periodicity with lattice\nconstant a. A more complete investigation of the fluctu-\nations, which should also take into account, how thermal\npositional fluctuations influence the friction force, remains\nto be done.\np-3M.P. Magiera 1 L. Brendel 1 D.E. Wolf 1 U. Nowak 2\nTechnical remarks. – Because of the vector pro-\nduct in eq. (4), the noise ζienters in a multiplicative\nway, calling for special attention to the interpretation of\nthis stochastic differential equation (Stratonovich vs. Itˆ o\nsense). The physical origin of the noise renders it generi-\ncallycoloured and thus selects the Stratonovich interpre-\ntation as the appropriate one (Wong-Zakai theorem [20]),\nin which its appearance as white noise is an idealisation.\nAccordingly we employ the Heun integration scheme [21].\nAfter each time step the spins are rescaled so that their\nlength remains unchanged.\nTo get meaningful results, it is of prime importance to\nreach a steady state. The initial configuration turned out\nto be a crucial factor for achieving this within acceptable\ncomputing time. Therefore a long initialisation run is per-\nformed, before the tip motion starts.\nMoreover, the system size is another limiting factor. In\norder to avoid that the tip reaches the system boundary\nbefore the steady state is reached, we use a “conveyorbelt\nmethod” allowing to do the simulation in the comoving\nframe of the tip. The tip is placed in a central point above\nthe substrate plane, e.g.at ((Lx+1)/2,(Ly+1)/2). After\nan equilibration time, the tip starts to move with fixed\nvelocity in x-direction. When it passed exactly one lattice\nconstant a, the front line (at x=Lx) is duplicated and\nadded to the lattice at x=Lx+ 1. Simultaneously, the\nline at the opposite boundary of the system (at x= 1) is\ndeleted, so that the simulation cell is of fixed size and con-\nsists of the Lx×Lyspins centered around the tip position,\nwith open boundaries. Note that this is different from pe-\nriodicboundaryconditions, becausethespinconfiguration\ndeleted at one side is different from the one added at the\nopposite side. We compared the results obtained for a\nsmall system in the co-moving frame of the tip with some\nruns for a system that was long enough in x-direction that\nthe steady state could be reached in the rest frame of the\nsample, and confirmed that the same friction results and\nsteady state properties could be obtained with drastically\nreduced computation time.\nIt is convenient to rewrite the equations of motion in\nnatural units. An energy unit is prescribed by the ex-\nchange energy Jof two magnetic moments. It rescales the\nenergy related parameters dzandwas well as the simu-\nlated temperature,\nkBT′=kBT\nJ. (17)\nThe rescaled time further depends on the material con-\nstantsµsandγ,\nt′=Jγ\nµst. (18)\nA length scale is given by the lattice constant a, so a nat-\nural velocity for the system can be defined,\nv′=µs\nγJav. (19)-1-0.5 0 0.5 1\n-4-3-2-1 0 1 2 3 4mx(x)\nx-x0α = 0.3, v = 0.05\nα = 0.3, v = 0.10\nα = 0.5, v = 0.10\ntanh(0.75 x)-0.2 0\n 0  0.2 vx0/αα = 1.0\nα = 0.7\nα = 0.5\nFigure 2: Local magnetisation ( x-component) at T=0 along\nthe lattice axes in x-direction which are closest to the tip ( i.e.\naty=±0.5). Analogous to a domain wall one finds a tanh x-\nprofile. Depending on the damping constant αand the velocity\nv, its zero is shifted backwards from the tip position by a valu e\nx0≈−0.88αv, as shown in the inset.\nFrom the natural length and energy, the natural force re-\nsults to:\nF′=a\nJF (20)\nFrom now on, all variables are given in natural units, and\nwe will drop the primes for simplicity. The typical exten-\nsion of the simulated lattices is 50 ×30,which was checked\nto be big enough to exclude finite size-effects. For the tip\ncoupling, we chose large values ( e.g.wStip= (0,0,−10)),\nto get a large effective field on the substrate. Usually it\nis assumed that the dipole-dipole coupling constant has a\nvalue of about w= 0.01, which means that the magnetic\nmoment of the tip is chosen a factor of 1000 times larger\nthan the individual substrate moments. The anisotropy\nconstant is set to dz=−0.1 in all simulations. The damp-\ning constant αis varied from 0 .1 to the quite large value 1.\nAt finite temperatures typically 50 simulation runs with\ndifferent random number seeds are performed to get reli-\nable ensemble averages.\nSimulation results. – In [8] it was found that the\nmagnetic friction force depends linearly on the scanning\nvelocityvand the damping constant αfor small velocities\n(v≤0.3). For higher velocities the friction force reaches a\nmaximum and then decreases. In this work we focus on\nthe low-velocity regime with the intention to shed more\nlight on the friction mechanism and its temperature de-\npendence.1\nAdiabatic approximation at T=0.If we assume the\nfieldhfor each spin to vary slowly enough to allow the so-\nlution (14) to be attained as adiabatic approximation, the\nlinear dependence F∝αvfrom [8] follows immediately:\nAt every point, the temporal change of the direction of\nh, defining a local ωfor (14), is proportional to v. We\n1It should be noted that the smallest tip velocities we simula ted,\nare of the order of 10−2(aJγ/µ s), which is still fast compared to\ntypical velocities in friction force microscopy experimen ts.\np-4Spin excitations in a monolayer scanned by a magnetic tip\n-1-0.5 0 0.5 1\n-20-15-10-5 0 5 10 15 20mx\nx0.11.0\n2345678910|m|\nr0.11.0\n2345678910|m|\nrkBT=0.1\nkBT=0.5\nkBT=0.9\nkBT=1.5\nFigure 3: Left: Magnetisation profiles as in fig. 2 for several temperatures with wStip= 10,α= 0.5 andv= 0.01. Middle and\nright: Absolute value of the average magnetisation as a func tion of the distance rfrom (x,y) = (0,0), directly underneath the\ntip. For small temperatures (upper two curves) it decreases with a power law (cf. double-logarithmic plot, middle), for high\ntemperatures (lower two curves) it decreases exponentiall y (cf. semi-logarithmic plot, right).\nconfirmed the validity of the adiabatic approximation nu-\nmerically by decomposing S−h/hwith respect to the\nlocal basis vectors ∂t(h/h),h, and their cross-product, all\nof them appropriately normalized. In other words, we ex-\ntractedδϕandδθdirectly and found them in excellent\nagreement with (14).\nThe lag of Swith respect to hmanifests itself also\nmacroscopically in the magnetisation field as we will show\nnow. Thetip-dipoleisstrongenoughtoalignthesubstrate\nspins to nearly cylindrical symmetry. Since the anisotropy\nis chosen to generate an easy plane ( dz<0), spins far away\nfrom the tip try to lie in the xy-plane, while close to the\ntip they tilt into the z-direction. This is displayed in\nfig. 2 where the x-component of the local magnetisation is\nshown along a line in x-direction for a fixed y-coordinate.\nRemarkably, these magnetisation profiles for different val-\nues ofvandαcollapse onto a unique curve, if they are\nshifted by corresponding offsets x0with respect to the tip\nposition. As expected from (14), the magnetisation profile\nstays behind the tip by a ( y-dependent) shift x0∝αv(cf.\ninset of fig. 2).\nFriction at T>0.With increasing temperature the\nmagnetisation induced by the tip becomes smaller, as\nshown in fig. 3. One can distinguish a low tempera-\nture regime, where the local magnetisation decreases alge-\nbraically with the distance from the tip, and a high tem-\nperature regime, where it decreases exponentially. The\ntransitionbetween these regimeshappens around T≈0.7.\nFor all temperatures the friction force Fturns out to be\nproportional to the velocity (up to v≈0.3), as for T=0,\nwith a temperature dependent friction coefficient F/v.\nThe two temperature regimes manifest themselves also\nhere, as shown in fig. 4: For low temperatures the fric-\ntion coefficients depend nearly linearly on α, reflecting the\nT=0 behaviour. Towards the high temperature regime,\nhowever, the α-dependence vanishes, and all friction coef-\nficients merge into a single exponentially decreasing func- 0 2 4 6 8 10\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6F/v\nkBTwStip = 10, α=0.5\nwStip = 10, α=0.7\nwStip =  5, α=0.5\nwStip =  5, α=0.7\nFigure 4: Friction coefficients for different α,wStipandkBT.\nOne can distinguish between a low temperature regime, where\nthe friction coefficient depends on αbut not on wStip, and a\nhigh temperature regime, where it depends on wStipbut not\nonα.\ntion.\nThe low temperature behaviour can be understood es-\nsentially along the lines worked out for T=0, as result of\na delayed, deterministic response (precession and relax-\nation) to the time dependent tip field. At high tempera-\ntures, however,frictionresultsfromtheabilityofthetipto\npropagate partial order through the thermally disorderd\nmedium. The magnetisationpattern in the wakeofthe tip\nno longer adapts adiabatically to the dwindling influence\nofthe tip, but decaysdue to thermal disorder. Then, aris-\ningtemperatureletstheorderedareaaroundthetipshrink\nwhichleadstotheexponentialdecreaseofthefrictioncoef-\nficient. However, it increases with the tip strength, wStip,\nas stronger order can be temporarily forced upon the re-\ngion around the tip. By contrast, the tip strength looses\nits influence on friction in the limit T→0, because the\nsubstrate spins are maximally polarised in the tip field .\nThis picture of the two temperature regimes is sup-\np-5M.P. Magiera 1 L. Brendel 1 D.E. Wolf 1 U. Nowak 2\n-3-2.5-2-1.5-1-0.5 0\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6x0/v\nkBTwStip = 10, α=0.5\nwStip = 10, α=0.7\nwStip =  5, α=0.5\nwStip =  5, α=0.7\nFigure 5: Distance x0by which the magnetisation pattern lags\nbehind the tip is proportional to vfor all temperatures. The\nproportionality constant depends on αonly in the low temper-\nature regime.\nported by the distance x0, by which the magnetisation\npattern lags behind the tip. It is proportional to vfor all\ntemperatures, but α-dependent only in the low tempera-\nture regime, cf. fig. 5.\nConclusion and outlook. – In this work, we could\nexplain the low-velocity, zero-temperature findings from\n[8], namely that the magnetic friction force in the Heisen-\nberg model has a linear velocity dependence with a coeffi-\ncient proportional to the damping constant α. In the spin\npolarisation cloud dragged along with the tip, each sub-\nstratespin followsthe localfield with a lagproportionalto\nthe frequency of the field change and to α. Moreover, the\nmagnetisationpattern aroundthe tip getsdistorted due to\nprecession. These effects directly give rise to the observed\nmagneticfriction andcould be evaluated quantitativelyby\nmeans of a single spin model.\nSecond, for the first time the temperature dependence\nof magnetic friction in the Heisenberg model was investi-\ngated in the framework of Landau-Lifshitz-Gilbert (LLG)\ndynamics with a stochastic contribution to the magnetic\nfield. Two regimes were found, which can be charac-\nterised by their different relaxation behaviour. While in\nthe low-temperature regime the response of the system on\nthe perturbation due to the moving tip is dominated by\nthe deterministic precession and relaxation terms in the\nLLG equation, thermal perturbations competing with the\none caused by the moving tip are essential in the high-\ntemperature regime. This explains, why magnetic friction\ndepends on αbut noton wStipforlowtemperatures, while\nit depends on wStipbut not on αfor high temperatures\nwhere it decreases exponentially with T.\nImportant extensions of the present investigation in-\nclude the effects ofa tip magnetisationpointing in a differ-\nent than the z-direction, of the strength and sign of spin\nanisotropy, dz, or of the thickness of the magnetic layer.\nBoth, spin anisotropyand lattice dimension will be crucial\nfor the critical behaviour, as well as for the critical tem-perature itself. Studies dealing with these quantities are\nalready in progress and will be reported in a future work.\n∗∗∗\nThis work was supported by the German Research\nFoundation (DFG) via SFB 616 “Energy dissipation at\nsurfaces”. Computation time was granted in J¨ ulich by the\nJohn-von-Neumann Institute of Computing (NIC).\nReferences\n[1]Persson B. ,Sliding Friction (Springer, Berlin, Heidel-\nberg, New York) 1998.\n[2]Urbakh M., Klafter J., Gourdon D. andIs-\nraelachvili J. ,Nature,430(2004) 525.\n[3]Gnecco E., Bennewitz R., Gyalog T. andMeyer E. ,\nJ. Phys.: Condens. Matter ,13(2001) R619.\n[4]Acharyya M. andChakrabarti B. K. ,Phys. Rev. B ,\n52(1995) 6550.\n[5]Ort´ın J.andGoicoechea J. ,Phys. Rev. B ,58(1998)\n5628.\n[6]Corberi F., Gonnella G. andLamura A. ,Phys. Rev.\nLett.,81(1998) 3852.\n[7]Kadau D., Hucht A. andWolf D. E. ,Phys. Rev. Lett. ,\n101(2008) 137205.\n[8]Fusco C., Wolf D. E. andNowak U. ,Phys. Rev. B ,\n77(2008) 174426.\n[9]Grutter P., Liu Y., LeBlanc P. andDurig U. ,Appl.\nPhys. Lett. ,71(1997) 279.\n[10]Schmidt R., Lazo C., Holscher H., Pi U. H., Caciuc\nV., Schwarz A., Wiesendanger R. andHeinze S. ,\nNano Letters ,9(2009) 200.\n[11]Zw¨orner O., H ¨olscher H., Schwarz U. D. and\nWiesendanger R. ,Appl. Phys. A ,66(1998) S263.\n[12]Gnecco E., Bennewitz R., Gyalog T., Loppacher\nC., Bammerlin M., Meyer E. andG¨untherodt H.-\nJ.,Phys. Rev. Lett. ,84(2000) 1172.\n[13]Prandtl L. ,Zs. f. angew. Math. u. Mech. ,8(1928) 85.\n[14]Tomlinson G. A. ,Philos. Mag. ,7(1929) 905.\n[15]Landau L. D. andLifshitz E. M. ,Phys. Z. Sowjetunion ,\n8(1935) 153.\n[16]Gilbert T. L. ,IEEE Trans. Magn. ,40(2004) 3443.\n[17]N´eel L.,C. R. Acad. Sc. Paris ,228(1949) 664.\n[18]Brown W. F. ,Phys. Rev. ,130(1963) 1677.\n[19]Magiera M. P. ,Computer simulation of magnetic fric-\ntionDiploma Thesis, Univ. of Duisburg-Essen (2008).\n[20]Horsthemke W. andLefever R. ,Noise-Induced Tran-\nsitions(Springer) 1983.\n[21]Garc´ıa-Palacios J. L. andL´azaro F. J. ,Phys. Rev.\nB,58(1998) 14937.\np-6"    },    {        "title": "1310.7657v1.Observational_Study_of_Large_Amplitude_Longitudinal_Oscillations_in_a_Solar_Filament.pdf",        "content": "arXiv:1310.7657v1  [astro-ph.SR]  29 Oct 2013Nature of Prominences and their role in Space Weather\nProceedings IAU Symposium No. 300, 2014\nB. Schmieder, JM. Malherbe & S. Wu, eds.c/circlecopyrt2014 International Astronomical Union\nDOI: 00.0000/X000000000000000X\nObservational Study of Large Amplitude\nLongitudinal Oscillations in a Solar Filament\nKalman Knizhnik1,2,Manuel Luna3, Karin Muglach2,4\nHolly Gilbert2, Therese Kucera2, Judith Karpen2\n1Department of Physics and Astronomy\nThe Johns Hopkins University, Baltimore, MD 21218\nemail: kalman.knizhnik@nasa.gov\n2NASA/GSFC, Greenbelt, MD 20771, USA\n3Instituto de Astrof´ ısica de Canarias, E-38200 La Laguna, T enerife, Spain\n4ARTEP, Inc., Maryland, USA\nAbstract. On 20 August 2010 an energetic disturbance triggered damped large-amplitude lon-\ngitudinal (LAL) oscillations in almost an entire filament. I n the present work we analyze this\nperiodic motion in the filament to characterize the damping a nd restoring mechanism of the\noscillation. Our method involves placing slits along the ax is of the filament at different angles\nwith respect to the spine of the filament, finding the angle at w hich the oscillation is clearest,\nand fitting the resulting oscillation pattern to decaying si nusoidal and Bessel functions. These\nfunctions represent the equations of motion of a pendulum da mped by mass accretion. With\nthis method we determine the period and the decaying time of t he oscillation. Our preliminary\nresults support the theory presented by Luna and Karpen (201 2) that the restoring force of LAL\noscillations is solar gravity in the tubes where the threads oscillate, and the damping mechanism\nis the ongoing accumulation of mass onto the oscillating thr eads. Following an earlier paper, we\nhave determined the magnitude and radius of curvature of the dipped magnetic flux tubes host-\ning a thread along the filament, as well as the mass accretion r ate of the filament threads, via\nthe fitted parameters.\nKeywords. solar prominences, oscillations, magnetic structures\n1. Procedure\nLAL oscillations consist of periodic motions of the prominence thread s along the mag-\nnetic field that are disturbed by a small energetic event close to the filament (see Luna\net al. paper in this volume). Luna and Karpen (2012) argue that pro minence oscillations\ncan be modeled as a damped oscillating pendulum, whose equation of mo tion satis-\nfies a zeroth-order Bessel function. In their model, a nearby trig ger event causes quasi-\nstationary preexisting prominence threads sitting in the dips of the magnetic structure\nto oscillate back and forth, with the restoring force being the proj ected gravity in the\ntubes where the threads oscillate (e.g. Luna et al. (2012)). In this paper, we report pre-\nliminary results of comparisonsof observations of prominence oscilla tions with the model\npresented by Luna and Karpen (2012). More details will be available in the forthcoming\npaper by Luna et al. (2013).\nIn this analysis, we place slits along the filament spine and measure the intensity along\neach slit as a function of time. Fig. 1 (left) shows the filament in the AI A 171˚A filter\nwith the slits overlaid. Each slit is then rotated in increments of 0.5◦from 0◦to 60◦\nwith respect to the filament spine. We select the best slit according t o the following\ncriteria: (a) continuity of oscillations, (b) amplitude of the oscillation is maximized, (c)\nclear transition from dark to bright regions, (d) maximum number of cycles.\nThe oscillation for a representative slit is shown in Figure 1 (right), wh ich corresponds\nto the grey slit in Figure 1 (left). We identify the position of the cente r of mass of the\n12 K. Knizhnik, M. Luna, K. Muglach, H. Gilbert, T. Kucera, J. Karpen\nFigure 1. Left: Filament seen in AIA 171 with best slits overlaid. Right: An intensity distance–\ntime slit, showing an oscillation with the Bessel fit (white c urve) to equation (2.1) in Luna et\nal. (this volume). The sinusoidal fit was not as good as the Bes sel fit and is not shown.\nthread by finding the intensity minimum along the slit, indicated by black crosses in\nFigure 1 (right). These points are then fit to equation (2.1) of Luna et al. (this volume),\nand the resulting fit is shown in white.\n2. Results\nFitting our data to equation (2.1) of Luna et al. (this volume) yields va lues ofχ2\nranging between 1-13. Using equation (2.2) of Luna et al. (this volum e), we find the\naverageradiusofcurvatureofthe magneticfield dips that suppor t the oscillatingthreads.\nWe find it to be approximately 60 Mm. We also calculate a threshold value for the field\nitself that would allow it to support the observedthreads. Using equ ation (3.1) of Luna et\nal.(this volume), wefind anaveragemagneticfield of ∼20G, assumingatypicalfilament\nnumber density of 1011cm−3, in good agreement with measurements (e.g. Mackay et al.\n2010). On average, the oscillations form an angle of ∼25owith respect to the filament\nspine, and have a period of ∼0.8 hours. To explain the very strong damping mass must\naccrete onto the threads at a rate of about 60 ×106kg/hr.\n3. Conclusions\nWe conclude that the observedoscillationsarealongthe magneticfie ld, which formsan\nangle of∼25owith respect to the filament spine (Tandberg-Hanssen & Anzer, 19 70). We\nfind that both the curvature and the magnitude of the magnetic fie ld are approximately\nuniform on different threads. Both the Bessel and sinusoidal func tions are well fitted,\nindicating that mass accretion is a likely damping mechanism of LAL oscilla tions, and\nthat the restoring force is the projected gravity in the dips where the threads oscillate.\nThe mass accretion rate agrees with the theoretical value (Karpe n et al., 2006, Luna,\nKarpen, & DeVore, 2012).\nReferences\nKarpen, J. T., Antiochos, S. K., Klimchuk, J. A. 2006, ApJ, 63 7, 531\nLuna, M., Karpen, J. T., & Devore, C. R. 2012a, ApJ, 746, 30\nLuna, M., & Karpen, J. 2012, ApJ, 750, L1\nLuna, M., Knizhnik, K., Muglach, K., Gilbert, H, Kucera, T. & Karpen, J., this volume , 2014\nLuna, M., Knizhnik, K., Muglach, K., Gilbert, H, Kucera, T. & Karpen, J., ApJ, 2013,in prep.\nMackay, D., Karpen, J., Ballester, J., Schmieder, B., Aulan ier, G. 2010, Sp. Sci. Rev. , 151, 333\nTandberg-Hanssen, E. and Anzer, U. 1970, Solar Physics 15, 158T"    },    {        "title": "2006.16595v2.Polynomial_stabilization_of_non_smooth_direct_indirect_elastic_viscoelastic_damping_problem_involving_Bresse_system.pdf",        "content": "POLYNOMIAL STABILIZATION OF NON-SMOOTH DIRECT/INDIRECT\nELASTIC/VISCOELASTIC DAMPING PROBLEM INVOLVING BRESSE SYSTEM\nST\u0013EPHANE GERBI, CHIRAZ KASSEM, AND ALI WEHBE\nAbstract. We consider an elastic/viscoelastic problem for the Bresse system with fully Dirichlet or Dirichlet-\nNeumann-Neumann boundary conditions. The physical model consists of three wave equations coupled in\ncertain pattern. The system is damped directly or indirectly by global or local Kelvin-Voigt damping. Ac-\ntually, the number of the dampings, their nature of distribution (locally or globally) and the smoothness\nof the damping coe\u000ecient at the interface play a crucial role in the type of the stabilization of the corre-\nsponding semigroup. Indeed, using frequency domain approach combined with multiplier techniques and the\nconstruction of a new multiplier function, we establish di\u000berent types of energy decay rate (see the table of\nstability results at the end). Our results generalize and improve many earlier ones in the literature (see [7])\nand in particular some studies done on the Timoshenko system with Kelvin-Voigt damping (see for instance\n[9], [23] and [25]).\nContents\n1. Introduction 1\n2. Well-posedness of the problem 4\n3. Strong stability of the system 6\n4. Polynomial stability for non smooth damping coe\u000ecients at the interface 10\n5. The case of only one local viscoelastic damping with non smooth coe\u000ecient at the interface 17\n6. Lack of exponential stability 24\n7. Additional results and summary 25\nAcknowledgments 26\nReferences 26\n1.Introduction\n1.1.The Bresse system with Kelvin-Voigt damping. Viscoelasticity is the property of materials that\nexhibit both viscous and elastic characteristics when undergoing deformation. There are several mathemati-\ncal models representing physical damping. The most often encountered type of damping in vibration studies\nare linear viscous damping and Kelvin-Voigt damping which are special cases of proportional damping. Vis-\ncous damping usually models external friction forces such as air resistance acting on the vibrating structures\nand is thus called \\external damping\", while Kelvin-Voigt damping originates from the internal friction of\nthe material of the vibrating structures and thus called \\internal damping\". The stabilization of conser-\nvative evolution systems (wave equation, coupled wave equations, Timoshenko system ...) by viscoelastic\nKelvin-Voigt type damping has attracted the attention of many authors. In particular, it was proved that\nthe stabilization of wave equation with local Kelvin-Voigt damping is greatly in\ruenced by the smoothness of\nthe damping coe\u000ecient and the region where the damping is localized (near or faraway from the boundary)\neven in the one-dimensional case, see [6, 16]. This surprising result initiated the study of an elastic system\nwith local Kelvin-Voigt damping There are a few number of publications concerning the stabilization of\nBresse or Timoshenko systems with viscoelastic Kelvin-Voigt damping. (see Subsection 1.2 below).\n2010 Mathematics Subject Classi\fcation. 35B37, 35D05, 93C20, 73K50.\nKey words and phrases. Bresse system, Kelvin-Voigt damping, polynomial stability, non uniform stability, frequency domain\napproach.\n1arXiv:2006.16595v2  [math.AP]  22 Feb 2022Figure 1. After deformation the particle M0of the beam is at the position M.\nIn this paper, we study the stability of Bresse system with localized non-smooth Kelvin-Voigt damping\ncoe\u000ecient at the interface and we brie\ry state results when the Kelvin-Voigt damping coe\u000ecients are either\nglobal or localized but smooth at the interface since the tools used for the study of non-smooth coe\u000ecient\nare used in the same, but much simpler, way when the coe\u000ecients act on the totality of the domain or are\nsmooth enough at the interface. These results generalize and improve many earlier ones in the literature.\nThe Bresse system is usually considered in studying elastic structures of the arcs type (see [14]). It can\nbe expressed by the equations of motion:\n\u001a1'tt=Qx+`N\n\u001a2 tt=Mx\u0000Q\n\u001a1wtt=Nx\u0000`Q\nwhere\nN=k3(wx\u0000`') +F3; Q =k1('x+ +`w) +F1; M =k2 x+F2\nF1=D1('xt+ t+`wt); F 2=D2 xt; F 3=D3(wxt\u0000`'t)\nand whereF1,F2andF3are the Kelvin-Voigt dampings. When F1=F2=F3= 0,N,QandMdenote the\naxial force, the shear force and the bending moment. The functions ';  ; andwmodel the vertical, shear\nangle, and longitudinal displacements of the \flament. Here \u001a1=\u001aA; \u001a 2=\u001aI; k 1=k0GA; k 3=EA; k 2=\nEI; ` =R\u00001where\u001ais the density of the material, Eis the modulus of elasticity, Gis the shear modulus,\nk0is the shear factor, Ais the cross-sectional area, Iis the second moment of area of the cross-section, and\nRis the radius of curvature. see \fgure 1 reproduces from [7]. The damping coe\u000ecients D1,D2andD3are\nbounded non negative functions over (0 ;L).\nSo we will consider the system of partial di\u000berential equations given on (0 ;L)\u0002(0;+1) by the following form:\n(1.1)8\n>>>><\n>>>>:\u001a1'tt\u0000[k1('x+ +`w) +D1('xt+ t+`wt)]x\u0000`k3(wx\u0000`')\u0000`D3(wxt\u0000`'t) = 0;\n\u001a2 tt\u0000[k2 x+D2 xt]x+k1('x+ +`w) +D1('xt+ t+`wt) = 0;\n\u001a1wtt\u0000[k3(wx\u0000`') +D3(wxt\u0000`'t)]x+`k1('x+ +`w) +`D1('xt+ t+`wt) = 0;\nwith fully Dirichlet boundary conditions:\n(1.2) '(0;\u0001) ='(L;\u0001) = (0;\u0001) = (L;\u0001) =w(0;\u0001) =w(L;\u0001) = 0 in R+;\nor with Dirichlet-Neumann-Neumann boundary conditions:\n(1.3) '(0;\u0001) ='(L;\u0001) = x(0;\u0001) = x(L;\u0001) =wx(0;\u0001) =wx(L;\u0001) = 0 in R+;\n2in addition to the following initial conditions:\n(1.4)'(\u0001;0) ='0(\u0001);  (\u0001;0) = 0(\u0001); w(\u0001;0) =w0(\u0001);\n't(\u0001;0) ='1(\u0001);  t(\u0001;0) = 1(\u0001); wt(\u0001;0) =w1(\u0001);in (0;L):\nWe de\fne the three wave speeds as:\nc1=s\nk1\n\u001a1; c 2=s\nk2\n\u001a2; c 3=s\nk3\n\u001a1:\nIn the absence of the three Kelvin-Voigt damping terms, the system (1.1) is a system of three coupled wave\nequations. This system is conservative whereas when at least one of the three Kelvin-Voigt damping is\npresent, the system is dissipative. The combination of direct damping, that is damping that acts in the\nequation involving the unknown itself and indirect damping that acts on another unknown than the one\nconcerns by the equation, makes this study much delicate.\nWe note that when R!1 , then`!0 and the Bresse model reduces, by neglecting w, to the well-known\nTimoshenko beam equations:\n(1.5)8\n<\n:\u001a1'tt\u0000[k1('x+ ) +D1('xt+ t)]x= 0;\n\u001a2 tt\u0000[k2 x+D2 xt]x+k1('x+ ) +D1('xt+ t) = 0\nwith di\u000berent types of boundary conditions and with initial data.\n1.2.Motivation, aims and main results. The stability of elastic Bresse system with di\u000berent types of\ndamping (frictional, thermoelastic, Cattaneo, ...) has been intensively studied (see Subsection 1.3), but there\nare a few number of papers concerning the stability of Bresse or Timoshenko systems with local viscoelastic\nKelvin-Voigt damping. In fact, in [7], El Arwadi and Youssef studied the theoretical and numerical stability\non a Bresse system with Kelvin-Voigt damping under fully Dirichlet boundary conditions. Using multiplier\ntechniques, they established an exponential energy decay rate provided that the system is subject to three\nglobal Kelvin-Voigt damping. Later, a numerical scheme based on the \fnite element method was introduced\nto approximate the solution. Zhao et al. in [25], considered a Timoshenko system with Dirichlet-Neumann\nboundary conditions. They obtained the exponential stability under certain hypotheses of the smoothness\nand structural condition of the coe\u000ecients of the system, and obtain the strong asymptotic stability under\nweaker hypotheses of the coe\u000ecients. Tian and Zhang in [23] considered a Timoshenko system under\nfully Dirichlet boundary conditions and with two locally or globally Kelvin-Voigt dampings. First, in the\ncase when the two Kelvin-Voigt dampings are globally distributed, they showed that the corresponding\nsemigroup is analytic. On the contrary, they proved that the energy of the system decays exponentially or\npolynomially and the decay rate depends on properties of material coe\u000ecient function. In [9], Ghader and\nWehbe generalized the results of [25] and [23]. Indeed, they considered the Timoshenko system with only one\nlocally or globally distributed Kelvin-Voigt damping and subject to fully Dirichlet or to Dirichlet-Neumann\nboundary conditions. They established a polynomial energy decay rate of type t\u00001for smooth initial data.\nMoreover, they proved that the obtained energy decay rate is in some sense optimal. In [19], Maryati et al.\nconsidered the transmission problem of a Timoshenko beam composed by Ncomponents, each of them being\neither purely elastic, or a Kelvin-Voigt viscoelastic material, or an elastic material inserted with a frictional\ndamping mechanism. They proved that the energy decay rate depends on the position of each component.\nIn particular, they proved that the model is exponentially stable if and only if all the elastic components are\nconnected with one component with frictional damping. Otherwise, only a polynomial energy decay rate is\nestablished. So, the stability of the Bresse system with local viscoelastic Kelvin-Voigt damping is still an\nopen problem.\nThe purpose of this paper is to study the Bresse system in the presence of local non-smooth dampings\ncoe\u000ecient at interface and under fully Dirichlet or Dirichlet-Neumann-Neumann boundary conditions. The\nsystem is given by (1.1)-(1.2) or (1.1)-(1.3) with initial data (1.4).\nWhenD1,D2,D32L1(0;L), using frequency domain approach combined with multiplier techniques\nand the construction of new multiplier functions, we establish a polynomial stability of type1\nt(see Theorem\n34.2). Moreover, in the presence of only one local damping D2acting on the shear angle displacement\n(D1=D3= 0), we establish a polynomial energy decay estimate of type1p\nt(see Theorem 5.1).\nFinally, in the absence of at least one damping, we prove the lack of uniform stability for the system\n(1.1)-(1.3) even with smoothness of damping coe\u000ecients. In these cases, we conjecture the optimality of the\nobtained decay rate. For clarity, let\n;6=!= (\u000b;\f)\u001a(0;L):\nHere and thereafter, \u000band\fwill be considered as interfaces.\n1.3.Literature concerning the Bresse system. In [17], Liu and Rao considered the Bresse system with\ntwo thermal dissipation laws. They proved an exponential decay rate when the wave speed of the vertical\ndisplacement coincides with the wave speed of longitudinal displacement or of the shear angle displacement.\nOtherwise, they showed polynomial decays depending on the boundary conditions. These results are im-\nproved by Fatori and Rivera in [8] where they considered the case of one thermal dissipation law globally\ndistributed on the displacement equation. Wehbe and Najdi in [20] extended and improved the results of [8],\nwhen the thermal dissipation is locally distributed. Wehbe and Youssef in [24] considered an elastic Bresse\nsystem subject to two locally internal dissipation laws. They proved that the system is exponentially stable if\nand only if the waves propagate at the same speed. Otherwise, a polynomial decay holds. Alabau et al. in [2]\nconsidered the same system with one globally distributed dissipation law. The authors proved the existence\nof polynomial decays with rates that depend on some particular relation between the coe\u000ecients. In [10],\nGuesmia et al. considered Bresse system with in\fnite memories acting in the three equations of the system.\nThey established asymptotic stability results under some conditions on the relaxation functions regardless\nthe speeds of propagation. These results are improved by Abdallah et al. in [1] where they considered the\nBresse system with in\fnite memory type control and/or with heat conduction given by Cattaneo's law acting\nin the shear angle displacement. The authors established an exponential energy decay rate when the waves\npropagate at same speed. Otherwise, they showed polynomial decays. In [4], Benaissa and Kasmi, considered\nthe Bresse system with three control of fractional derivative type acting on the boundary conditions. They\nestablished a polynomial decay estimate.\n1.4.Organization of the paper. This paper is organized as follows: In Section 2, we prove the well-\nposedness of system (1.1) with either the boundary conditions (1.2) or (1.3). Next, in Section 3, we prove\nthe strong stability of the system in the lack of the compactness of the resolvent of the generator.\nIn Section 4 when the coe\u000ecient functions D1,D2, andD3are not smooth, we prove the polynomial\nstability of type1\nt. In section 5, we prove the polynomial energy decay rate of type1p\ntfor the system in\nthe case of only one local non-smooth damping D2acting on the shear angle displacement. In Section 6,\nunder boundary conditions (1.3), we prove the lack of uniform (exponential) stability of the system in the\nabsence of at least one damping. Finally in Section 7, we will brie\ry state the analytic stabilization of the\nsystem (1.1) when the three damping coe\u000ecient act on the whole spatial domain (0 ;L) and the exponential\nstability when the three damping coe\u000ecient are localized on ( \u000b;\f) and are smooth at the interfaces.\n2.Well-posedness of the problem\nIn this part, using a semigroup approach, we establish the well-posedness result for the systems (1.1)-(1.2)\nand (1.1)-(1.3). Let ( '; ;w ) be a regular solution of system (1.1)-(1.2), its associated energy is given by:\n(2.1)E(t) =1\n2\u001aZL\n0\u0010\n\u001a1j'tj2+\u001a2j tj2+\u001a1jwtj2+k1j'x+ +`wj2\u0011\ndx\n+ZL\n0\u0010\nk2j xj2+k3jwx\u0000`'j2\u0011\ndx\u001b\n;\nand it is dissipated according to the following law:\n(2.2) E0(t) =\u0000ZL\n0\u0000\nD1j'xt+ t+`wtj2+D2j xtj2+D3jwxt\u0000`'tj2\u0001\ndx\u00140:\n4Now, we de\fne the following energy spaces:\nH1=\u0000\nH1\n0(0;L)\u0002L2(0;L)\u00013andH2=H1\n0(0;L)\u0002L2(0;L)\u0002\u0000\nH1\n\u0003(0;L)\u0002L2\n\u0003(0;L)\u00012;\nwhere\nL2\n\u0003(0;L) =ff2L2(0;L) :ZL\n0f(x)dx= 0gandH1\n\u0003(0;L) =ff2H1(0;L) :ZL\n0f(x)dx= 0g:\nBoth spacesH1andH2are equipped with the inner product which induces the energy norm:\n(2.3)kUk2\nHj=k(v1;v2;v3;v4;v5;v6)k2\nHj\n=\u001a1\r\rv2\r\r2+\u001a2\r\rv4\r\r2+\u001a1\r\rv6\r\r2+k1\r\rv1\nx+v3+`v5\r\r2\n+k2\r\rv3\nx\r\r2+k3\r\rv5\nx\u0000`v1\r\r2; j= 1;2\nhere and afterk\u0001kdenotes the norm of L2(0;L) .\nRemark 2.1. In the case of boundary condition (1.2), it is easy to see that expression (2.3) de\fnes a norm\non the energy space H1. But in the case of boundary condition (1.3) the expression (2.3) de\fne a norm on\nthe energy space H2ifL6=n\u0019\n`for all positive integer n. Then, here and after, we assume that there does\nnot exist any n2Nsuch thatL=n\u0019\n`whenj= 2.\nNext, we de\fne the linear operator AjinHjby:\nD(A1) =\u001a\nU2H 1jv2;v4;v62H1\n0(0;L);\u0002\nk1\u0000\nv1\nx+v3+`v5\u0001\n+D1\u0000\nv2\nx+v4+`v6\u0001\u0003\nx2L2(0;L);\n\u0002\nk2v3\nx+D2v4\nx\u0003\nx2L2(0;L);\u0002\nk3(v5\nx\u0000`v1) +D3(v6\nx\u0000`v2)\u0003\nx2L2(0;L)\u001b\n;\nD(A2) =\u001a\nU2H 2jv22H1\n0(0;L);v4;v62H1\n\u0003(0;L);v3\nxj0;L=v5\nxj0;L= 0;\n\u0002\nk1\u0000\nv1\nx+v3+`v5\u0001\n+D1\u0000\nv2\nx+v4+`v6\u0001\u0003\nx2L2(0;L);\n\u0002\nk2v3\nx+D2v4\nx\u0003\nx2L2\n\u0003(0;L);\u0002\nk3(v5\nx\u0000`v1) +D3(v6\nx\u0000`v2)\u0003\nx2L2\n\u0003(0;L)\u001b\nand\n(2.4)\nAj0\nBBBBBBBB@v1\nv2\nv3\nv4\nv5\nv61\nCCCCCCCCA=0\nBBBBBBBBB@v2\n\u001a\u00001\n1\u0000\u0002\nk1(v1\nx+v3+`v5) +D1(v2\nx+v4+`v6)\u0003\nx+`k3(v5\nx\u0000`v1) +`D3(v6\nx\u0000`v2)\u0001\nv4\n\u001a\u00001\n2\u0000\n(k2v3\nx+D2v4\nx)x\u0000k1\u0000\nv1\nx+v3+`v5\u0001\n\u0000D1(v2\nx+v4+`v6)\u0001\nv6\n\u001a\u00001\n1\u0000\u0002\nk3(v5\nx\u0000`v1) +D3(v6\nx\u0000`v2)\u0003\nx\u0000`k1\u0000\nv1\nx+v3+`v5\u0001\n\u0000`D1(v2\nx+v4+`v6)\u00011\nCCCCCCCCCA\nfor allU=\u0000\nv1;v2;v3;v4;v5;v6\u0001T2D(Aj). So, ifU= (';'t; ; t;w;wt)Tis the state of (1.1)-(1.2) or\n(1.1)-(1.3), then the Bresse beam system is transformed into a \frst order evolution equation on the Hilbert\nspaceHj:\n(2.5)(\nUt=AjU; j = 1;2\nU(0) =U0(x);\nwhere\nU0(x) = ('0(x);'1(x); 0(x); 1(x);w0(x);w1(x))T:\nRemark 2.2. It is easy to see that there exists a positive constant c0such that for any ('; ;w )2\u0000\nH1\n0(0;L)\u00013forj= 1and for any ('; ;w )2H1\n0(0;L)\u0002\u0000\nH1\n\u0003(0;L)\u00012forj= 2,\n(2.6) k1k'x+ +`wk2+k2k xk2+k3kwx\u0000`'k2\u0014c0\u0010\nk'xk2+k xk2+kwxk2\u0011\n:\n5On the other hand, we can show by a contradiction argument the existence of a positive constant c1such\nthat, for any ('; ;w )2\u0000\nH1\n0(0;L)\u00013forj= 1and for any ('; ;w )2H1\n0(0;L)\u0002\u0000\nH1\n\u0003(0;L)\u00012forj= 2,\n(2.7) c1\u0010\nk'xk2+k xk2+kwxk2\u0011\n\u0014k1k'x+ +`wk2+k2k xk2+k3kwx\u0000`'k2:\nTherefore the norm on the energy space Hjgiven in (2.3) is equivalent to the usual norm on Hj.\nProposition 2.3. Assume that coe\u000ecients functions D1,D2andD3are non negative. Then, the operator\nAjis m-dissipative in the energy space Hj, forj= 1;2.\nProof. LetU=\u0000\nv1;v2;v3;v4;v5;v6\u0001T2D(Aj). By a straightforward calculation, we have:\n(2.8) Re ( AjU;U)Hj=\u0000ZL\n0\u0010\nD1\f\fv2\nx+v4+`v6\f\f2+D2\f\fv4\nx\f\f2+D3\f\fv6\nx\u0000`v2\f\f2\u0011\ndx:\nAsD1\u00150; D 2\u00150 andD3\u00150 , we get thatAjis dissipative.\nNow, we will check the maximality of Aj. For this purpose, let F=\u0000\nf1;f2;f3;f4;f5;f6\u0001T2Hj;we have\nto prove the existence of U=\u0000\nv1;v2;v3;v4;v5;v6\u0001T2D(Aj) unique solution of the equation \u0000AjU=F.\nLet\u0000\n'1;'3;'5\u0001\n2\u0000\nH1\n0(0;L)\u00013forj= 1 and\u0000\n'1;'3;'5\u0001\n2\u0000\nH1\n0(0;L)\u0002(H1\n\u0003(0;L))2\u0001\nforj= 2 be a test\nfunction. Writing \u0000AjUand replacing the \frst, third and \fth component of Uby\u0000f1;\u0000f3;\u0000f5and now\nmultiplying the second, the fourth and the sixth equation by respectively '1;'3;'5, after integrating by\nparts, we obtain the following form:\n(2.9)8\n><\n>:k1\u0000\nv1\nx+v3+`v5\u0001\n'1\nx\u0000`k3\u0000\nv5\nx\u0000`v1\u0001\n'1=h1;\nk2v3\nx'3\nx+k1\u0000\nv1\nx+v3+`v5\u0001\n'3=h3;\nk3\u0000\nv5\nx\u0000`v1\u0001\n'5\nx+`k1\u0000\nv1\nx+v3+`v5\u0001\n'5=h5;\nwhere\nh1=\u001a1f2'1+D1\u0000\nf1\nx+f3+`f5\u0001\n'1\nx\u0000`D3(f5\nx\u0000`f1)'1;\nh3=\u001a2f4'3+D2f3\nx'3\nx+D1\u0000\nf1\nx+f3+`f5\u0001\n'3;and\nh5=\u001a1f6'5+D3\u0000\nf5\u0000`f1\u0001\n'5\nx+`D1\u0000\nf1\nx+f3+`f5\u0001\n'5:\nUsing Lax-Milgram Theorem (see [21]), we deduce that (2.9) admits a unique solution in\u0000\nH1\n0(0;L)\u00013for\nj= 1 and in\u0000\nH1\n0(0;L)\u0002(H1\n\u0003(0;L))2\u0001\nforj= 2. Thus,\u0000AjU=Fadmits an unique solution U2D(Aj)\nand consequently 0 2\u001a(Aj). Then,Ajis closed and consequently \u001a(Aj) is open set of C(see Theorem\n6.7 in [13]). Hence, we easily get R(\u0015I\u0000Aj) =Hjfor su\u000eciently small \u0015 > 0. This, together with the\ndissipativeness of Aj, imply that D(Aj) is dense inHjand thatAjis m-dissipative in Hj(see Theorems\n4.5, 4.6 in [21]). The proof is thus complete.\n\u0003\nThanks to Lumer-Phillips Theorem (see [18, 21]), we deduce that Ajgenerates a C0-semigroup of contraction\netAjinHjand therefore problem (2.5) is well-posed. We have thus the following result.\nTheorem 2.4. For anyU02Hj, problem (2.5) admits a unique weak solution\nU2C(R+;Hj):\nMoreover, if U02D(Aj);then\nU2C(R+;D(Aj))\\C1(R+;Hj):\n3.Strong stability of the system\nIn this part, we use a general criteria of Arendt-Batty in [3] to show the strong stability of the C0-\nsemigroupetAjassociated to the Bresse system (1.1) in the absence of the compactness of the resolvent of\nAj. Before, we state our main result, we need the following stability condition:\n(SSC) There exist i2f1;2;3g; d0>0 and\u000b<\f2[0;L] such thatDi\u0015d0>0 on (\u000b;\f):\n6Theorem 3.1. Assume that condition (SSC) holds. Then the C0\u0000semigroupetAjis strongly stable in Hj,\nj= 1;2, i.e., for all U02Hj, the solution of (2.5) satis\fes\nlim\nt!+1\r\retAjU0\r\r\nHj= 0:\nFor the proof of Theorem 3.1, we need the following two lemmas.\nLemma 3.2. Under the same condition of Theorem 3.1, we have\n(3.1) ker ( i\u0015\u0000Aj) =f0g; j= 1;2;for all\u00152R:\nProof. We will prove Lemma 3.2 in the case D1=D3= 0 on (0;L) andD2\u0015d0>0 on (\u000b;\f)\u001a(0;L).\nThe other cases are similar to prove.\nFirst, from Proposition 2.3, we claim that 0 2\u001a(Aj):We still have to show the result for \u00152R\u0003. Suppose\nthat there exist a real number \u00156= 0 and 06=U=\u0000\nv1;v2;v3;v4;v5;v6\u0001T2D(Aj) such that:\n(3.2) AjU=i\u0015U:\nOur goal is to \fnd a contradiction by proving that U= 0. Taking the real part of the inner product in Hj\nofAjUandU, we get:\n(3.3) Re ( AjU;U)Hj=\u0000ZL\n0D2\f\fv4\nx\f\f2dx= 0:\nSince by assumption D2\u0015d0>0 on (\u000b;\f), it follows from equality (3.3) that:\n(3.4) D2v4\nx= 0 in (0 ;L) andv4\nx= 0 in (\u000b;\f):\nDetailing (3.2) we get:\nv2=i\u0015v1; (3.5)\nk1\u0000\nv1\nx+v3+`v5\u0001\nx+`k3\u0000\nv5\nx\u0000`v1\u0001\n=i\u001a1\u0015v2; (3.6)\nv4=i\u0015v3; (3.7)\u0000\nk2v3\nx+D2v4\nx\u0001\nx\u0000k1\u0000\nv1\nx+v3+`v5\u0001\n=i\u001a2\u0015v4; (3.8)\nv6=i\u0015v5; (3.9)\nk3\u0000\nv5\nx\u0000`v1\u0001\nx\u0000`k1\u0000\nv1\nx+v3+`v5\u0001\n=i\u001a1\u0015v6: (3.10)\nNext, inserting (3.4) in (3.7) and using the fact that \u00156= 0, we get:\n(3.11) v3\nx= 0 in (\u000b;\f):\nMoreover, substituting equations (3.5), (3.7) and (3.9) into equations (3.6), (3.8) and (3.10), we get:\n(3.12)8\n><\n>:\u001a1\u00152v1+k1\u0000\nv1\nx+v3+`v5\u0001\nx+`k3\u0000\nv5\nx\u0000`v1\u0001\n= 0;\n\u001a2\u00152v3+\u0000\nk2v3\nx+iD2\u0015v3\nx\u0001\nx\u0000k1\u0000\nv1\nx+v3+`v5\u0001\n= 0;\n\u001a1\u00152v5+k3\u0000\nv5\nx\u0000`v1\u0001\nx\u0000`k1\u0000\nv1\nx+v3+`v5\u0001\n= 0:\nNow, we introduce the functions bvi, fori= 1;::;6 bybvi=vi\nx:It is easy to see that bvi2H1(0;L).\nIt follows from equations (3.4) and (3.11) that:\n(3.13) bv3=bv4= 0 in (\u000b;\f)\nand consequently system (3.12) will be, after di\u000berentiating it with respect to x, given by:\n\u001a1\u00152bv1+k1\u0000\nbv1\nx+`bv5\u0001\nx+`k3\u0000\nbv5\nx\u0000`bv1\u0001\n= 0 in ( \u000b;\f); (3.14)\nbv1\nx+`bv5= 0 in ( \u000b;\f); (3.15)\n\u001a1\u00152bv5+k3\u0000\nbv5\nx\u0000`bv1\u0001\nx\u0000`k1\u0000\nbv1\nx+`bv5\u0001\n= 0 in ( \u000b;\f): (3.16)\n7Furthermore, substituting equation (3.15) into (3.14) and (3.16), we get:\n\u001a1\u00152bv1+`k3\u0000\nbv5\nx\u0000`bv1\u0001\n= 0 in ( \u000b;\f); (3.17)\nbv1\nx+`bv5= 0 in ( \u000b;\f); (3.18)\n\u001a1\u00152bv5+k3\u0000\nbv5\nx\u0000`bv1\u0001\nx= 0 in ( \u000b;\f): (3.19)\nDi\u000berentiating equation (3.17) with respect to x, a straightforward computation with equation (3.19) yields:\n\u001a1\u00152\u0000\nbv1\nx\u0000`bv5\u0001\n= 0 in (\u000b;\f):\nEquivalently\n(3.20) bv1\nx\u0000`bv5= 0 in (\u000b;\f):\nHence, from equations (3.18) and (3.20), we get:\n(3.21) bv5= 0 andbv1\nx= 0 in (\u000b;\f):\nPluggingbv5= 0 in (3.17), we get:\n(3.22)\u0000\n\u001a1\u00152\u0000`2k3\u0001\nbv1= 0:\nIn order to \fnish our proof, we have to distinguish two cases:\nCase 1:\u00156=`rk3\n\u001a1.\nUsing equation (3.22) , we deduce that:\nbv1= 0 in (\u000b;\f):\nSettingV=\u0000\nbv1;bv1\nx;bv3;bv3\nx;bv5;bv5\nx\u0001T. By continuity of bvion (0;L), we deduce that V(\u000b) = 0. Then system\n(3.12) could be given as:\n(3.23)\u001aVx=BV; in (0;\u000b)\nV(\u000b) = 0 ;\nwhere\n(3.24) B=0\nBBBBBBBBBBB@0 1 0 0 0 0\n\u0000\u00152\u001a1+`2k3\nk10 0 \u00001\u0000`(k1+k3)\nk10\n0 0 0 1 0 0\n0k1\nk2+i\u0015D 2k1\u0000\u00152\u001a2\nk2+i\u0015D 20`k1\nk2+i\u0015D 20\n0 0 0 0 0 1\n0`(k3+k1)\nk3`k1\nk30`2k1\u0000\u00152\u001a1\nk301\nCCCCCCCCCCCA:\nUsing ordinary di\u000berential equation theory, we deduce that system (3.23) has the unique trivial solution\nV= 0 in (0;\u000b). The same argument as above leads us to prove that V= 0 on (\f;L). Consequently, we\nobtainbv1=bv3=bv5= 0 on (0;L). It follows that bv2=bv4=bv6= 0 on (0;L), thusbU= 0. This gives that\nU=C, whereCis a constant. Finally, from the boundary condition (1.2) or (1.3), we deduce that U= 0.\nCase 2:\u0015=`rk3\n\u001a1.\nThe fact that bv1\nx= 0 on (\u000b;\f), we getbv1=con (\u000b;\f), wherecis a constant. By continuity of bv1on (0;L),\nwe deduce that bv1(\u000b) =c. We know also that bv3=bv5= 0 on (\u000b;\f) from (3.13) and (3.21). Hence, setting\nV(\u000b) = (c;0;0;0;0;0)T=V0, we can rewrite system (3.12) on (0 ;\u000b) under the form:\n\u001a\nVx=bBV;\nV(\u000b) =V0;\n8where\nbB=0\nBBBBBBBBBBBBB@0 1 0 0 0 0\n0 0 0 \u00001\u0000`(k1+k3)\nk10\n0 0 0 1 0 0\n0k1\nk2+i`q\nk3\n\u001a1D2k1\u0000\u00152\u001a2\nk2+i`q\nk3\n\u001a1\u0015D 20`k1\nk2+i`q\nk3\n\u001a1D20\n0 0 0 0 0 1\n0`(k3+k1)\nk3`k1\nk30`2(k1\u0000k3)\nk301\nCCCCCCCCCCCCCA:\nIntroducingeV=\u0000\nbv1\nx;bv3;bv3\nx;bv5;bv5\nx\u0001Tand\neB=0\nBBBBBBBBBBB@0 0 \u00001\u0000`(k1+k3)\nk10\n0 0 1 0 0\nk1\nk2+i`q\nk3\n\u001a1D2k1\u0000\u00152\u001a2\nk2+i`q\nk3\n\u001a1\u0015D 20`k1\nk2+i`q\nk3\n\u001a1D20\n0 0 0 0 1\n`(k3+k1)\nk3`k1\nk30`2(k1\u0000k3)\nk301\nCCCCCCCCCCCA:\nThen system (3.12) could be given as:\n(3.25)(\neVx=eBeV;in (0;\u000b);\neV(\u000b) = 0 :\nUsing ordinary di\u000berential equation theory, we deduce that system (3.25) has the unique trivial solution\neV= 0 in (0;\u000b). This implies that on (0 ;\u000b), we havebv3=bv5= 0. Consequently, v3=c3andv5=c5where\nc3andc5are constants. But using the fact that v3(0) =v5(0) = 0, we deduce that v3=v5= 0 on (0;\u000b).\nSubstituting v3andv5by their values in the second equation of system (3.12), we get that v1\nx= 0. This\nyieldsv1=c1, wherec1is a constant. But as v1(0) = 0, we get: v1= 0 on (0;\u000b). ThusU= 0 on (0;\u000b).\nThe same argument as above leads us to prove that U= 0 on (\f;L) and therefore U= 0 on (0;L). Thus\nthe proof is complete. \u0003\nLemma 3.3. Under the same condition of Theorem 3.1, (i\u0015I\u0000Aj);j= 1;2is surjective for all \u00152R.\nProof. We will prove Lemma 3.3 in the case D1=D3= 0 on (0;L) andD2\u0015d0>0 on (\u000b;\f)\u001a(0;L) and\nthe other cases are similar to prove.\nSince 02\u001a(Aj), we still need to show the result for \u00152R\u0003. For any\nF=\u0000\nf1;f2;f3;f4;f5;f6\u0001T2Hj; \u00152R\u0003;\nwe prove the existence of\nU=\u0000\nv1;v2;v3;v4;v5;v6\u0001T2D(Aj)\nsolution of the following equation:\n(3.26) ( i\u0015I\u0000Aj)U=F:\nEquivalently, we have the following system:\ni\u0015v1\u0000v2=f1; (3.27)\n\u001a1i\u0015v2\u0000k1\u0000\nv1\nx+v3+`v5\u0001\nx\u0000`k3\u0000\nv5\nx\u0000`v1\u0001\n=\u001a1f2; (3.28)\ni\u0015v3\u0000v4=f3; (3.29)\n\u001a2i\u0015v4\u0000(k2v3\nx+D2v4\nx)x+k1\u0000\nv1\nx+v3+`v5\u0001\n=\u001a2f4; (3.30)\ni\u0015v5\u0000v6=f5; (3.31)\n\u001a1i\u0015v6\u0000k3\u0000\nv5\nx\u0000`v1\u0001\nx+`k1\u0000\nv1\nx+v3+`v5\u0001\n=\u001a1f6: (3.32)\n9From (3.27),(3.29) and (3.31), we have:\n(3.33) v2=i\u0015v1\u0000f1; v3=i\u0015v3\u0000f3; v6=i\u0015v5\u0000f5:\nInserting (3.33) in (3.28), (3.30) and (3.32), we get:\n(3.34)8\n><\n>:\u0000\u00152v1\u0000k1\u001a\u00001\n1\u0000\nv1\nx+v3+`v5\u0001\nx\u0000`k3\u001a\u00001\n1\u0000\nv5\nx\u0000`v1\u0001\n=h1;\n\u0000\u00152v3\u0000\u001a\u00001\n2(k2+i\u0015D 2)v3\nxx+k1\u001a\u00001\n2\u0000\nv1\nx+v3+`v5\u0001\n=h3;\n\u0000\u00152v5\u0000k3\u001a\u00001\n1\u0000\nv5\nx\u0000`v1\u0001\nx+`k1\u001a\u00001\n1\u0000\nv1\nx+v3+`v5\u0001\n=h5;\nwhere\nh1=f2+i\u0015f1; h3=f4+i\u0015f3\u0000\u001a\u00001\n2D2f3\nxx; h5=f6+i\u0015f5:\nFor allv=\u0000\nv1;v3;v5\u0001T2\u0000\nH1\n0(0;L)\u00013forj= 1 andv=\u0000\nv1;v3;v5\u0001T2H1\n0(0;L)\u0002H1\n\u0003(0;L)2forj= 2, we\nde\fne the linear operator Lby:\nLv=0\nB@\u0000k1\u001a\u00001\n1\u0000\nv1\nx+v3+`v5\u0001\nx\u0000`k3\u001a\u00001\n1\u0000\nv5\nx\u0000`v1\u0001\n\u0000\u001a\u00001\n2(k2+i\u0015D 2)v3\nxx+k1\u001a\u00001\n2\u0000\nv1\nx+v3+`v5\u0001\n\u0000k3\u001a\u00001\n1\u0000\nv5\nx\u0000`v1\u0001\nx+`k1\u001a\u00001\n1\u0000\nv1\nx+v3+`v5\u00011\nCA:\nFor clarity, we consider the case j= 1. The proof in the case j= 2 is very similar. Using Lax-Milgram theo-\nrem, it is easy to show that Lis an isomorphism from ( H1\n0(0;L))3onto (H\u00001(0;L))3. Letv=\u0000\nv1;v3;v5\u0001T\nandh=\u0000\n\u0000h1;\u0000h3;\u0000h5\u0001T, then we transform system (3.34) into the following form:\n(3.35) ( \u00152I\u0000L )v=h:\nSince the operator Lis an isomorphism from ( H1\n0(0;L))3onto (H\u00001(0;L))3andIis a compact operator\nfrom (H1\n0(0;L))3onto (H\u00001(0;L))3, then using Fredholm's Alternative theorem, problem (3.35) admits a\nunique solution in ( H1\n0(0;L))3if and only if \u00152I\u0000L is injective. For that purpose, let ~ v=\u0000\n~v1;~v3;~v5\u0001Tin\nker(\u00152I\u0000L ). Then, if we set ~ v2=i\u0015~v1, ~v4=i\u0015~v3and ~v6=i\u0015~v5, we deduce that ~V= (~v1;~v2;~v3;~v5;~v6)\nbelongs toD(A1) and it is solution of:\n(i\u0015I\u0000A 1)~V= 0:\nUsing Lemma 3.2, we deduce that ~ v1= ~v3= ~v5= 0. This implies that equation (3.35) admits a unique\nsolution in v= (v1;v3;v5)2(H1\n0(0;L))3and\n\u0000k1\u001a\u00001\n1\u0000\nv1\nx+v3+`v5\u0001\nx\u0000`k3\u001a\u00001\n1\u0000\nv5\nx\u0000`v1\u0001\n2L2(0;L);\n\u0000\u001a\u00001\n2(k2v3\nx+i\u0015D 2v3\nx\u0000D2f3\nx)x+k1\u001a\u00001\n2\u0000\nv1\nx+v3+`v5\u0001\n2L2(0;L);\n\u0000k3\u001a\u00001\n1\u0000\nv5\nx\u0000`v1\u0001\nx+`k1\u001a\u00001\n1\u0000\nv1\nx+v3+`v5\u0001\n2L2(0;L):\nBy settingv2=i\u0015v1\u0000f1,v3=i\u0015v3\u0000f3andv6=i\u0015v5\u0000f5, we deduce that V= (v1;v2;v3;v4;v5;v6)\nbelongs toD(A1) and it is the unique solution of equation (3.26) and the proof is thus complete. \u0003\nProof of Theorem 3.1. Following a general criteria of Arendt-Batty in [3], the C0\u0000semigroupetAjof con-\ntractions is strongly stable if Ajhas no pure imaginary eigenvalues and \u001b(Aj)\\iRis countable. By Lemma\n3.2, the operator Ajhas no pure imaginary eigenvalues and by Lemma 3.3, R( i\u0015\u0000Aj) =Hjfor all\u00152R.\nTherefore the closed graph theorem of Banach implies that \u001b(Aj)\\iR=;. Thus, the proof is complete. \u0003\n4.Polynomial stability for non smooth damping coefficients at the interface\nBefore we state our main result, we recall the following results (see [11], [22] for part i), [5] for ii)and\n[21] for iii).\nTheorem 4.1. LetA:D(A)\u001aH!H be an unbounded operator generating a C0-semigroup of contractions\netAonH. Assume that i\u00152\u001a(A), for all\u00152R. Then, the C0-semigroup etAis:\ni) Exponentially stable if and only if\nlim\nj\u0015j!+1\u001a\nsup\n\u00152Rk(i\u0015I\u0000A)\u00001kL(H)\u001b\n<+1:\n10ii) Polynomially stable of order1\nl(l>0)if and only if\nlim\nj\u0015j!+1\u001a\nsup\n\u00152Rj\u0015j\u0000lk(i\u0015I\u0000A)\u00001kL(H)\u001b\n<+1:\niii) Analytically stable if and only if\nlim\nj\u0015j!+1\u001a\nsup\n\u00152Rj\u0015jk(i\u0015I\u0000A)\u00001kL(H)\u001b\n<+1:\nIt was proved that, see [6, 16], the stabilization of wave equation with local Kelvin-Voigt damping is greatly\nin\ruenced by the smoothness of the damping coe\u000ecient and the region where the damping is localized (near\nor faraway from the boundary) even in the one-dimensional case. So, in this section, we consider the Bresse\nsystems (1.1)-(1.2) and (1.1)-(1.3) subject to three local viscoelastic Kelvin-Voigt dampings with non smooth\ncoe\u000ecients at the interface. Using frequency domain approach combined with multiplier techniques and the\nconstruction of a new multiplier function, we establish the polynomial stability of the C0-semigroup etAj,\nj= 1;2. For this purpose, let ;6= (\u000bi;\fi)\u001a(0;L),i= 1;2;3;be an arbitrary nonempty open subsets of\n(0;L). We consider the following stability condition:\n(4.1)9di\n0>0 such that Di\u0015di\n0>0 in (\u000bi;\fi); i= 1;2;3;and3\\\ni=1(\u000bi;\fi) = (\u000b;\f)6=;:\nOur main result in this section can be given by the following theorem:\nTheorem 4.2. Assume that condition (4.1) holds. Then, there exists a positive constant c >0such that\nfor allU02D(Aj),j= 1;2;the energy of the system satis\fes the following decay rate:\n(4.2) E(t)\u0014c\ntkU0k2\nD(Aj):\nReferring to [5], (4.2) is veri\fed if the following conditions\n(H1) iR\u0012\u001a(Aj)\nand\n(H3) lim\n\u0015!+1sup\n\u00152R\u001a1\n\u00152\r\r\r(i\u0015Id\u0000Aj)\u00001\r\r\r\nL(Hj)\u001b\n=O(1)\nhold.\nCondition iR\u0012\u001a(Aj) is already proved in Lemma 3.2 and Lemma 3.3.\nWe will establish (H3) by contradiction. Suppose that there exist a sequence of real numbers ( \u0015n)n, with\nj\u0015nj!+1and a sequence of vectors\n(4.3) Un=\u0000\nv1\nn;v2\nn;v3\nn;v4\nn;v5\nn;v6\nn\u0001T2D(Aj) withkUnkHj= 1\nsuch that\n(4.4) \u00152\nn(i\u0015nUn\u0000AjUn) =\u0000\nf1\nn;f2\nn;f3\nn;f4\nn;f5\nn;f6\nn\u0001T!0 inHj; j = 1;2:\nWe will check the condition (H3) by \fnding a contradiction with (4.3)-(4.4) such as kUnkHj=o(1).\n11Equation (4.4) is detailed as:\ni\u0015nv1\nn\u0000v2\nn=f1\nn\n\u00152n; (4.5)\ni\u001a1\u0015nv2\nn\u0000\u0002\nk1\u0000\u0000\nv1\nn\u0001\nx+v3\nn+`v5\nn\u0001\n+D1\u0000\u0000\nv2\nn\u0001\nx+v4\nn+`v6\nn\u0001\u0003\nx\n\u0000`k3\u0002\u0000\nv5\nn\u0001\nx\u0000`v1\nn\u0003\n\u0000`D3\u0002\u0000\nv6\nn\u0001\nx\u0000`v2\nn\u0003\n=\u001a1f2\nn\n\u00152n; (4.6)\ni\u0015nv3\nn\u0000v4\nn=f3\nn\n\u00152n; (4.7)\ni\u001a2\u0015nv4\nn\u0000\u0002\nk2\u0000\nv3\nn\u0001\nx+D2\u0000\nv4\nn\u0001\nx\u0003\nx+k1\u0002\u0000\nv1\nn\u0001\nx+v3\nn+`v5\nn\u0003\n+D1\u0002\u0000\nv2\nn\u0001\nx+v4\nn+`v6\nn\u0003\n=\u001a2f4\nn\n\u00152n; (4.8)\ni\u0015nv5\nn\u0000v6\nn=f5\nn\n\u00152n; (4.9)\ni\u001a1\u0015nv6\nn\u0000\u0002\nk3\u0000\u0000\nv5\nn\u0001\nx\u0000`v1\nn\u0001\n+D3\u0000\u0000\nv6\nn\u0001\nx\u0000`v2\nn\u0001\u0003\nx+`k1\u0002\u0000\nv1\nn\u0001\nx+v3\nn+`v5\nn\u0003\n+`D1\u0002\u0000\nv2\nn\u0001\nx+v4\nn+`v6\nn\u0003\n=\u001a1f6\nn\n\u00152n: (4.10)\nFrom (4.3), (4.5), (4.7) and (4.9), we deduce that:\n(4.11) kv1\nnk=O(1\n\u0015n);kv3\nnk=O(1\n\u0015n);kv5\nnk=O(1\n\u0015n):\nFor clarity, we divide the proof into several lemmas. From now on, for simplicity, we drop the index n.\nLemma 4.3. Under all the above assumptions, we have:\n(4.12)kD1=2\n1\u0000\nv2\nx+v4+`v6\u0001\nk=o(1)\n\u0015;kD1=2\n2v4\nxk=o(1)\n\u0015;kD1=2\n3\u0000\nv6\nx\u0000`v2\u0001\nk=o(1)\n\u0015\nand\n(4.13)kv2\nx+v4+`v6k=o(1)\n\u0015;kv4\nxk=o(1)\n\u0015;kv6\nx\u0000`v2k=o(1)\n\u0015in (\u000b;\f):\nProof. Taking the inner product of (4.4) with UinHj, we get:\nRe\u0000\ni\u00153kUk2\u0000\u00152(AjU;U)\u0001\nHj=\u0000\u00152Re (AjU;U)Hj\n=\u00152ZL\n0\u0000\nD1jv2\nx+v4+`v6j2+D2jv4\nxj2+D3jv6\nx\u0000`v2j2\u0001\ndx=o(1): (4.14)\nThanks to (4.1), we obtain the desired asymptotic equations (4.12) and (4.13). Thus the proof is complete.\n\u0003\nRemark 4.4. These estimates are crucial for the rest of the prooof and they will be used to prove each point\nof the global proof divided in several lemmas.\nLemma 4.5. Under all the above assumptions, we have:\n(4.15)kv1\nx+v3+`v5k=o(1)\n\u00152;kv3\nxk=o(1)\n\u00152;kv5\nx\u0000`v1k=o(1)\n\u00152in (\u000b;\f):\nProof. First, using equations (4.5), (4.7) and (4.9), we obtain:\n(4.16) \u0015\u0000\nv1\nx+v3+`v5\u0001\n=\u0000i(v2\nx+f1\nx\n\u00152+v4+f3\n\u00152+`v6+`f5\n\u00152):\nConsequently,\n(4.17)Z\f\n\u000b\u00152jv1\nx+v3+`v5j2dx\u00142Z\f\n\u000bjv2\nx+v4+`v6j2dx+ 2Z\f\n\u000bjf1\nx+f3+`f5j2\n\u00154dx:\n12Using the \frst estimate of (4.13) and the fact that f1,f3,f5converge to zero in H1\n0(0;L) (or inH1\n?(0;L))\nin (4.17), we deduce:\n(4.18)Z\f\n\u000b\u00152jv1\nx+v3+`v5j2dx=o(1)\n\u00152:\nIn a similar way, one can prove:\n(4.19)Z\f\n\u000b\u00152jv3\nxj2dx=o(1)\n\u00152andZ\f\n\u000b\u00152jv5\nx\u0000`v1j2dx=o(1)\n\u00152:\nThe proof is thus complete. \u0003\nHere and after \u000fdesignates a \fxed positive real number such that 0 <\u000b+\u000f<\f\u0000\u000f<L . Then, we de\fne\nthe cut-o\u000b function \u00112C1\nc(R) by:\n\u0011= 1 on [\u000b+\u000f;\f\u0000\u000f];0\u0014\u0011\u00141; \u0011= 0 on (0 ;L)n(\u000b;\f):\nLemma 4.6. Under all the above assumptions, we have:\n(4.20)Z\f\u0000\u000f\n\u000b+\u000f\f\f\u0015v1\f\f2dx=o(1)\n\u0015;Z\f\u0000\u000f\n\u000b+\u000f\f\f\u0015v3\f\f2dx=o(1)\n\u0015;Z\f\u0000\u000f\n\u000b+\u000f\f\f\u0015v5\f\f2dx=o(1)\n\u0015:\nProof. First, multiplying equation (4.5) by i\u0015\u0011v1inL2(0;L) and integrating by parts, we get:\n(4.21) \u0000ZL\n0\u0011\f\f\u0015v1\f\f2dx\u0000iZL\n0\u0015\u0011v2v1dx=iZL\n0f1\n\u00152\u0011\u0015v1dx:\nAs\u0015v1is uniformly bounded in L2(0;L) andf1converges to zero in H1\n0(0;L), we get that the term on the\nright hand side of (4.21) converges to zero and consequently\n(4.22) \u0000ZL\n0\u0011\f\f\u0015v1\f\f2dx\u0000iZL\n0\u0015\u0011v2v1dx=o(1)\n\u00152:\nMoreover, multiplying (4.6) by \u001a\u00001\n1\u0011v1inL2(0;L), then integrating by parts we obtain:\niZL\n0\u0015\u0011v2v1dx+\u001a\u00001\n1ZL\n0\u0000\nk1\u0000\nv1\nx+v3+`v5\u0001\n+D1\u0000\nv2\nx+v4+`v6\u0001\u0001\u0010\n\u0011v1\u0011\nxdx\n\u0000`k3\u001a\u00001\n1ZL\n0\u0000\nv5\nx\u0000`v1\u0001\n\u0011v1dx\u0000`\u001a\u00001\n1ZL\n0D3\u0000\nv6\nx\u0000`v2\u0001\n\u0011v1dx=ZL\n0f2\n\u00152\u0011v1dx: (4.23)\nUsing (4.12), (4.15), the fact that f2converges to zero in L2(0;L) and\u0015v1,v1\nxare uniformly bounded in\nL2(0;L) in (4.23), we get:\n(4.24) iZL\n0\u0015\u0011v2v1dx=o(1)\n\u0015:\nFinally, using (4.24) in (4.22) and the de\fniton of \u0011, we get:\nZL\n0\u0011\f\f\u0015v1\f\f2dx=o(1)\n\u0015;Z\f\u0000\u000f\n\u000b+\u000f\f\f\u0015v1\f\f2dx=o(1)\n\u0015:\nIn a same way, we show:\nZ\f\u0000\u000f\n\u000b+\u000f\f\f\u0015v3\f\f2dx=o(1)\n\u0015;Z\f\u0000\u000f\n\u000b+\u000f\f\f\u0015v5\f\f2dx=o(1)\n\u0015:\nThe proof is thus complete. \u0003\nNow, we introduce new multiplier functions. For this purpose, let ;6=!\u000f= (\u000b+\u000f;\f\u0000\u000f):\n13Lemma 4.7. The solution (u;y;z )of the following system\n(4.25)8\n<\n:\u001a1\u00152u+k1(ux+y+`z)x+`k3(zx\u0000`u)\u0000i\u00151!\u000fu=v1;\n\u001a2\u00152y+k2yxx\u0000k1(ux+y+`z)\u0000i\u00151!\u000fy =v3;\n\u001a1\u00152z+k3(zx\u0000`u)x\u0000`k1(ux+y+`z)\u0000i\u00151!\u000fz=v5\nwith fully Dirichlet boundary conditions:\n(4.26) u(0) =u(L) =y(0) =y(L) =z(0) =z(L) = 0\nor with Dirichlet-Neumann-Neumann boundary conditions:\n(4.27) u(0) =u(L) =yx(0) =yx(L) =zx(0) =zx(L) = 0\nveri\fes the following inequality:\nZL\n0\u0012\n\u001a1j\u0015uj2+\u001a2j\u0015yj2+\u001a1j\u0015zj2+k2jyxj2\n+k1jux+y+`zj2+k3jzx\u0000`uj2\u0013\ndx\u0014CZL\n0\u0000\njv1j2+jv3j2+jv5j2\u0001\ndx; (4.28)\nwhereCis a constant independent of n.\nProof. We consider the following Bresse system subject to three local viscous dampings:\n(4.29)8\n>>>><\n>>>>:\u001a1utt\u0000k1(ux+y+`z)x\u0000lk3(zx\u0000`u) +1!\u000fut= 0;\n\u001a2ytt\u0000k2yxx+k1(ux+y+`z) +1!\u000fyt = 0;\n\u001a1ztt\u0000k3(zx\u0000`u)x+`k1(ux+y+`z) +1!\u000fzt= 0\nwith fully Dirichlet or Dirichlet-Neumann-Neumann boundary conditions. Systems (4.29)-(4.26) and (4.29)-\n(4.27) are well posed in the space H1=\u0000\nH1\n0(0;L)\u0002L2(0;L)\u00013and in the space H2=\u0000\nH1\n0(0;L)\u0002L2(0;L)\u0001\n\u0002\u0000\nH1\n\u0003(0;L)\u0002L2\n\u0003(0;L)\u00012respectively. In addition, both are exponentially stable (see [24]). Therefore, following\nHuang [11] and Pruss [22], we deduce that the resolvent of the associated operator:\nAauxj:D(Aauxj)\u001aHj!Hj\nde\fned by\nD(Aaux 1) =\u0000\nH1\n0(\n)\\H2(\n)\u00013\u0002\u0000\nH1\n0(\n)\u00013;\nD(Aaux 2) =\b\nU2H2:u2H1\n0\\H2;y;z2H1\n?\\H2;~u;yx;zx2H1\n0;~y;~z2H1\n?\t\nand\nAauxj0\nBBBBBBBB@u\n~u\ny\n~y\nz\n~z1\nCCCCCCCCA=0\nBBBBBBBBB@~u\n\u001a\u00001\n1[k1(ux+y+`z)x+`k3(zx\u0000`u)\u00001!\u000f~u]\n~y\n\u001a\u00001\n2[k2yxx\u0000k1(ux+y+`z)\u00001!\u000f~y]\n~z\n\u001a\u00001\n1[k3(zx\u0000`u)x\u0000`k1(ux+z+`z)\u00001!\u000f~z]1\nCCCCCCCCCA\nis uniformly bounded on the imaginary axis. So, by setting ~ u=i\u0015u, ~y=i\u0015yand ~z=i\u0015z, we deduce that:\n0\nBBBBBBBB@u\n~u\ny\n~y\nz\n~z1\nCCCCCCCCA=\u0000\ni\u0015\u0000Aauxj\u0001\u000010\nBBBBBBBBB@0\n\u00001\n\u001a1v1\n0\n\u00001\n\u001a2v3\n0\n\u00001\n\u001a1v11\nCCCCCCCCCA:\n14This yields:\nk(u;~u;y;~y;z;~z)k2\nHj\u0014k\u0000\ni\u0015\u0000Aauxj\u0001\u00001kL(Hj)k(0;\u00001\n\u001a1v1;0;\u00001\n\u001a2v3;0;\u00001\n\u001a1v5)kHj\n\u0014CZL\n0\u0000\njv1j2+jv3j2+jv5j2\u0001\ndx; (4.30)\nwhereCis a constant independent of n. Consequently, (4.28) holds. The proof is thus complete. \u0003\nLemma 4.8. Under all the above assumptions, we have:\n(4.31)ZL\n0j\u0015v1j2dx=o(1);ZL\n0j\u0015v3j2dx=o(1);ZL\n0j\u0015v5j2dx=o(1):\nProof. For clarity of the proof, we divide the proof into several steps.\nStep 1. First, multiplying (4.5) by i\u001a1\u0015u, whereuis a solution of system (4.25), we get:\n(4.32) \u0000ZL\n0\u001a1\u00152uv1dx\u0000iZL\n0\u001a1\u0015uv2dx=\u001a1ZL\n0if1\n\u0015udx:\nMoreover, multiplying (4.6) by uand integrating by parts, we obtain:\niZL\n0\u001a1\u0015uv2dx\u0000ZL\n0k1uxxv1dx\u0000ZL\n0`k3(\u0000`u)v1dx+ZL\n0k1uxv3dx+ZL\n0`k1uxv5dx\n+ZL\n0`k3uxv5dx+ZL\n0D1(v2\nx+v4+`v6)uxdx\u0000ZL\n0`D3(v6\nx\u0000`v2)udx=\u001a1ZL\n0f2\n\u00152udx: (4.33)\nNow, combining (4.32) and (4.33), we get:\nZL\n0\u0002\n\u001a1\u00152u+k1uxx+`k3(\u0000`u)\u0003\nv1dx\u0000ZL\n0k1uxv3dx\u0000ZL\n0`k1uxv5dx\u0000ZL\n0`k3uxv5dx\n\u0000ZL\n0D1(v2\nx+v4+`v6)uxdx+ZL\n0`D3(v6\nx\u0000`v2)udx=\u0000\u001a1ZL\n0\u0012if1\n\u0015+f2\n\u00152\u0013\nudx: (4.34)\nStep 2. Similarly to Step 1, multiplying (4.7) by i\u001a2\u0015yand (4.8) by y, whereyis a solution of system\n(4.25), we get:\nZL\n0\u0002\n\u001a2\u00152y+k2yxx\u0000k1y\u0003\nv3dx+ZL\n0k1yxv1dx\u0000ZL\n0`k1yv5dx\n\u0000ZL\n0D2v4\nxyxdx\u0000ZL\n0D1(v2\nx+v4+`v6)ydx=\u0000\u001a2ZL\n0\u0012if3\n\u0015+f4\n\u00152\u0013\nydx: (4.35)\nStep 3. As in Step 1 and Step 2, by multiplying (4.9) by i\u001a1\u0015zand (4.10) by z, wherezis a solution of\nsystem (4.25), we get:\nZL\n0\u0002\n\u001a1\u00152z+k3zxx\u0000`k1(`z)\u0003\nv5dx+ZL\n0`k3zxv1dx+ZL\n0`k1zxv1dx\n\u0000ZL\n0`k1zv3dx\u0000ZL\n0D3\u0000\nv6\nx\u0000`v2\u0001\nzxdx\u0000`ZL\n0D1(v2\nx+v4+`v6)zdx=\u0000\u001a1ZL\n0\u0012if5\n\u0015+f6\n\u00152\u0013\nzdx:(4.36)\n15Step 4. First, combining (4.34), (4.35) and (4.36), we obtain:\nZL\n0\u0002\n\u001a1\u00152u+k1(ux+y+`z)x+`k3(zx\u0000`u)\u0003\nv1dx+ZL\n0\u0002\n\u001a2\u00152y+k2yxx\u0000k1(ux+y+`z)\u0003\nv3dx\n+ZL\n0\u0002\n\u001a1\u00152z+k3(zx\u0000`u)x\u0000`k1(ux+y+`z)\u0003\nv5dx\u0000ZL\n0D1(v2\nx+v4+`v6)uxdx\n+ZL\n0`D3(v6\nx\u0000`v2)udx\u0000ZL\n0D2v4\nxyxdx\u0000ZL\n0D1(v2\nx+v4+`v6)ydx\u0000ZL\n0D3\u0000\nv6\nx\u0000`v2\u0001\nzxdx (4.37)\n\u0000`ZL\n0D1(v2\nx+v4+`v6)zdx=\u0000\u001a1ZL\n0\u0012if1\n\u0015+f2\n\u00152\u0013\nudx\u0000\u001a2ZL\n0\u0012if3\n\u0015+f4\n\u00152\u0013\nydx\n\u0000\u001a1ZL\n0\u0012if5\n\u0015+f6\n\u00152\u0013\nzdx:\nCombining equation (4.25) and (4.37), multiplying by \u00152, we get:\nZL\n0j\u0015v1j2dx+ZL\n0j\u0015v3j2dx+ZL\n0j\u0015v5j2dx=iZ\f\u0000\u000f\n\u000b+\u000f(\u00152u\u0015v1dx+\u00152y\u0015v3+\u00152z\u0015v5)dx\n+ZL\n0\u0015D 1(v2\nx+v4+`v6)\u0015uxdx\u0000ZL\n0`D3(v6\nx\u0000`v2)\u00152udx+ZL\n0\u0015D 2v4\nx\u0015yxdx\n+ZL\n0D1(v2\nx+v4+`v6)\u00152ydx+ZL\n0\u0015D 3\u0000\nv6\nx\u0000`v2\u0001\n\u0015zxdx+`ZL\n0D1(v2\nx+v4+`v6)\u00152zdx (4.38)\n\u0000\u001a1ZL\n0\u0000\nif1\u0015+f2\u0001\nudx\u0000\u001a2ZL\n0\u0000\nif3\u0015+f4\u0001\nydx\u0000\u001a1ZL\n0\u0000\nif5\u0015+f6\u0001\nzdx:\nUsing estimates (4.20) and the fact that \u00152u,\u00152yand\u00152zare uniformly bounded in L2(0;L) due to (4.28),\nwe get:\n(4.39) iZ\f\u0000\u000f\n\u000b+\u000f(\u00152u\u0015v1dx+\u00152y\u0015v3+\u00152z\u0015v5)dx=o(1)\n\u00151=2:\nIn addition, using (4.12) and the fact that \u0015ux,\u0015yxand\u0015zxare uniformly bounded in L2(0;L) due to (4.28).\nwe get:\n(4.40)ZL\n0\u0015D 1(v2\nx+v4+`v6)\u0015uxdx+ZL\n0\u0015D 2v4\nx\u0015yxdxZL\n0\u0015D 3\u0000\nv6\nx\u0000`v2\u0001\n\u0015zxdx=o(1):\nAlso, by using (4.12) and the fact that \u00152u,\u00152yand\u00152zare uniformly bounded in L2(0;L) due to (4.28),\nwe obtain:\n(4.41)ZL\n0`D3(v6\nx\u0000`v2)\u00152udx+ZL\n0D1(v2\nx+v4+`v6)\u00152ydx+`ZL\n0D1(v2\nx+v4+`v6)\u00152zdx=o(1)\n\u0015:\nMoreover, we have:\n(4.42)\u0000\u001a1ZL\n0\u0000\nif1\u0015+f2\u0001\nudx\u0000\u001a2ZL\n0\u0000\nif3\u0015+f4\u0001\nydx\u0000\u001a1ZL\n0\u0000\nif5\u0015+f6\u0001\nzdx=o(1);\nsincef1,f3,f5converge to zero in H1\n0(0;L) (or inH1\n?(0;L)),f2,f4,f6converge to zero in L2(0;L), and\n\u00152u,\u00152y,\u00152zare uniformly bounded in L2(0;L).\nFinally, inserting (4.39) - (4.42) into (4.38), we get the desired estimates in (4.31). Thus the proof is\ncomplete. \u0003\nLemma 4.9. Under all the above assumptions, we have:\n(4.43)ZL\n0jv1\nxj2dx=o(1);ZL\n0jv3\nxj2dx=o(1);ZL\n0jv5\nxj2dx=o(1):\n16Proof. First, multiplying (4.6) by v1and then integrating by parts, we get:\niZL\n0\u001a1\u0015v2v1dx+k1ZL\n0jv1\nxj2dx+k1ZL\n0\u0000\nv3+`v5\u0001\nv1xdx+ZL\n0D1\u0000\nv2\nx+v4+`v6\u0001\nv1xdx\n\u0000`k3ZL\n0\u0000\nv5\nx\u0000`v1\u0001\nv1dx\u0000`ZL\n0D3\u0000\nv6\nx\u0000`v2\u0001\nv1dx=\u001a1ZL\n0f2\n\u00152v1dx: (4.44)\nThen, using (4.11), (4.12) and the fact that v1\nx,\u0000\nv5\nx\u0000`v1\u0001\nare uniformly bounded in L2(0;L) due to (4.3),\nwe obtain:\nk1ZL\n0\u0000\nv3+`v5\u0001\nv1xdx+ZL\n0D1\u0000\nv2\nx+v4+`v6\u0001\nv1xdx\n\u0000`k3ZL\n0\u0000\nv5\nx\u0000`v1\u0001\nv1dx\u0000`ZL\n0D3\u0000\nv6\nx\u0000`v2\u0001\nv1dx=o(1): (4.45)\nAsf2converges to zero in L2(0;L) and\u0015v1is uniformly bounded in L2(0;L), we have:\n(4.46) \u001a1ZL\n0f2\n\u00152v1dx=o(1):\nNext, inserting (4.45) and (4.46) into (4.44), we get:\niZL\n0\u001a1\u0015v2v1dx+k1ZL\n0jv1\nxj2dx=o(1): (4.47)\nUsing Lemma 4.8 and the fact that v2is uniformly bounded in L2(0;L) due to (4.47), we deduce:\nZL\n0jv1\nxj2dx=o(1):\nSimilarly, one can prove that:\nZL\n0jv3\nxj2dx=o(1);ZL\n0jv5\nxj2dx=o(1):\nThus, the proof is complete. \u0003\nProof of Theorem 4.2. Using Lemma 4.8 and Lemma 4.9, we get that kUkHj=o(1). Therefore, we get a\ncontradiction with (4.3) and consequently (H3) holds. Thus the proof is complete \u0003\nRemark 4.10. It is known that for a single one-dimensional wave equation with damping coe\u000ecient D1=\nd0>0on!, the optimal solution decay rate is 1=t2. The new multipliers (one for each equation) we have\nused here, de\fned by system (4.25) , do not permit to obtain a decay rate of 1=t2but only 1=t. This may be\ndue to the coupling e\u000bects and we do not know if this decay rate of 1=tis optimal.\n5.The case of only one local viscoelastic damping with non smooth coefficient at the\ninterface\nIn control theory, it is important to reduce the number of control such as damping terms. So, this\nsection is devoted to show the polynomial stability of systems (1.1)-(1.2) and (1.1)-(1.3) subject to only one\nviscoelastic Kelvin-Voigt damping with non smooth coe\u000ecient at the interface. For this purpose, we consider\nthe following condition:\n(5.1)D1=D3= 0 in (0;L) and9d0>0 such that D2\u0015d0>0 in;6= (\u000b;\f)\u001a(0;L):\nThe main result of this section is given by the following theorem:\nTheorem 5.1. Assume that condition (5.1) is satis\fed. Then, there exists a positive constant c >0such\nthat for all U02D(Aj),j= 1;2;the energy of system (1.1) satis\fes the following decay rate:\n(5.2) E(t)\u0014cp\ntkU0k2\nD(Aj):\n17Referring to [5], (5.2) is veri\fed if the following conditions\n(H1) iR\u0012\u001a(Aj)\nand\n(H4) lim\nj\u0015j!+1sup\n\u00152R\u001a1\n\u00154\r\r\r(i\u0015I\u0000Aj)\u00001\r\r\r\nL(Hj)\u001b\n=O(1)\nhold.\nCondition iR\u0012\u001a(Aj) is already proved in Lemma 3.2 and Lemma 3.3.\nWe will establish (H4) by contradiction. Suppose that there exist a sequence of real numbers ( \u0015n)n, with\nj\u0015nj!+1and a sequence of vectors\n(5.3) Un=\u0000\nv1\nn;v2\nn;v3\nn;v4\nn;v5\nn;v6\nn\u0001T2D(Aj) withkUnkHj= 1\nsuch that\n(5.4) \u00154\nn(i\u0015nUn\u0000AjUn) =\u0000\nf1\nn;f2\nn;f3\nn;f4\nn;f5\nn;f6\nn\u0001T!0 inHj; j = 1;2:\nWe will check the condition (H4) by \fnding a contradiction with (5.3)-(5.4) such as kUnkHj=o(1).\nEquation (5.4) is detailed as:\ni\u0015nv1\nn\u0000v2\nn=f1\nn\n\u00154n; (5.5)\ni\u001a1\u0015nv2\nn\u0000k1\u0002\u0000\nv1\nn\u0001\nx+v3\nn+`v5\nn\u0003\nx\u0000`k3\u0002\u0000\nv5\nn\u0001\nx\u0000`v1\nn\u0003\n=\u001a1f2\nn\n\u00154n; (5.6)\ni\u0015nv3\nn\u0000v4\nn=f3\nn\n\u00154n; (5.7)\ni\u001a2\u0015nv4\nn\u0000\u0002\nk2\u0000\nv3\nn\u0001\nx+D2\u0000\nv4\nn\u0001\nx\u0003\nx+k1\u0002\u0000\nv1\nn\u0001\nx+v3\nn+`v5\nn\u0003\n=\u001a2f4\nn\n\u00154n; (5.8)\ni\u0015nv5\nn\u0000v6\nn=f5\nn\n\u00154n; (5.9)\ni\u001a1\u0015nv6\nn\u0000\u0002\nk3\u0000\u0000\nv5\nn\u0001\nx\u0000`v1\nn\u0001\u0003\nx+`k1\u0002\u0000\nv1\nn\u0001\nx+v3\nn+`v5\nn\u0003\n=\u001a1f6\nn\n\u00154n: (5.10)\nInserting (5.5), (5.7), and (5.9) into (5.6),(5.8) and (5.10) respectively, we get\n\u001a1\u00152\nnv1\nn+k1\u0002\u0000\nv1\nn\u0001\nx+v3\nn+`v5\nn\u0003\nx+`k3\u0002\u0000\nv5\nn\u0001\nx\u0000`v1\nn\u0003\n=\u0000i\u001a1f1\nn\n\u00153n\u0000\u001a1f2\nn\n\u00154n; (5.11)\n\u001a2\u00152\nnv3\nn+\u0002\nk2\u0000\nv3\nn\u0001\nx+D2\u0000\nv4\nn\u0001\nx\u0003\nx\u0000k1\u0002\u0000\nv1\nn\u0001\nx+v3\nn+`v5\nn\u0003\n=\u0000i\u001a2f3\nn\n\u00153n\u0000\u001a2f4\nn\n\u00154n; (5.12)\n\u001a1\u00152\nnv5\nn+\u0002\nk3\u0000\u0000\nv5\nn\u0001\nx\u0000`v1\nn\u0001\u0003\nx\u0000`k1\u0002\u0000\nv1\nn\u0001\nx+v3\nn+`v5\nn\u0003\n=\u0000i\u001a1f5\nn\n\u00153n\u0000\u001a1f6\nn\n\u00154n: (5.13)\nFrom (5.5), (5.7), (5.9) and (5.3), we deduce that:\n(5.14) kv1\nnk=O(1\n\u0015n);kv3\nnk=O(1\n\u0015n);kv5\nnk=O(1\n\u0015n):\nFor clarity, we divide the proof into several lemmas. From now on, for simplicity, we drop the index n.\nLemma 5.2. Under all the above assumptions, we have:\n(5.15)ZL\n0D2jv4\nxj2dx=o(1)\n\u00154;Z\f\n\u000bjv4\nxj2dx=o(1)\n\u00154\nand\n(5.16)ZL\n0\u0011jv3\nxj2dx=o(1)\n\u00156;Z\f\n\u000bjv3\nxj2dx=o(1)\n\u00156:\n18Proof. Taking the inner product of (5.4) with UinHj, we get:\nRe\u0000\ni\u00155kUk2\u0000\u00154(AjU;U)\u0001\nHj=\u0000\u00154Re (AjU;U)Hj=\u00154ZL\n0D2jv4\nxj2dx=o(1): (5.17)\nThanks to (5.1), we obtain the desired asymptotic equation (5.15).\nNext, di\u000berentiating equation (5.7), we get:\ni\u0015v3\nx=v4\nx+f3\nx\n\u00154;\nand consequentlyZ\f\n\u000bj\u0015v3\nxj2dx\u00142Z\f\n\u000bjv4\nxj2dx+ 2Z\f\n\u000bjf3\nxj2\n\u00158dx:\nUsing (5.15) and the fact that f3converges to zero in H1\n0(0;L) (or inH1\n\u0003(0;L) ) in the above equation, we\nget the desired estimate (5.16). Thus the proof is complete. \u0003\nRemark 5.3. Again, these estimates are crucial for the rest of the proof and they will be used to prove each\npoint of the global proof divided in several lemmas.\nLemma 5.4. Under all the above assumptions, we have:\n(5.18)ZL\n0\u0011j\u0015v3j2dx=O(1)\n\u00152andZ\f\u0000\u000f\n\u000b+\u000fj\u0015v3j2dx=O(1)\n\u00152:\nProof. First, multiplying (5.12) by \u001a\u00001\n2\u0011v3and integrating by parts, we get:\nZL\n0\u0011j\u0015v3j2dx=\u001a\u00001\n2ZL\n0\u0000\nk2v3\nx+D2v4\nx\u0001\u0010\n\u00110v3+\u0011v3x\u0011\ndx+\u001a\u00001\n2ZL\n0k1\u0000\nv1\nx+v3+`v5\u0001\n\u0011v3dx\n\u0000ZL\n0i\u0011f3\n\u00153v3dx\u0000ZL\n0f4\n\u00154\u0011v3dx: (5.19)\nThen, using (5.15), (5.16), kv3k=O(1\n\u0015) and the fact that f3,f4converge to zero in H1\n0(0;L) (or inH1\n?(0;L)),\nL2(0;L) respectively, we deduce that:\n\u001a\u00001\n2ZL\n0\u0000\nk2v3\nx+D2v4\nx\u0001\u0010\n\u00110v3+\u0011v3x\u0011\ndx\u0000ZL\n0i\u0011f3\n\u00153v3dx\u0000ZL\n0f4\n\u00154\u0011v3dx=o(1)\n\u00153: (5.20)\nNext, inserting (5.20) into (5.19), we obtain:\nZL\n0\u0011j\u0015v3j2dx=\u001a\u00001\n2ZL\n0k1\u0000\nv1\nx+v3+`v5\u0001\n\u0015\u0011\u0015v3dx+o(1)\n\u00153:\nUsing Cauchy-Shwartz and Young's inequalities in the above equation, we get:\nZL\n0\u0011j\u0015v3j2dx\u00142j\u001a\u00001\n2j2ZL\n0k2\n1\u0011\f\fv1\nx+v3+`v5\f\f2\n\u00152dx+1\n2ZL\n0\u0011j\u0015v3j2dx+o(1)\n\u00153;\nConsequently,\n1\n2ZL\n0\u0011j\u0015v3j2dx\u00142j\u001a\u00001\n2j2ZL\n0k2\n1\u0011\f\fv1\nx+v3+`v5\f\f2\n\u00152dx+o(1)\n\u00153:\nFinally, using the fact that\u0000\nv1\nx+v3+`v5\u0001\nis uniformly bounded in L2(0;L) and the de\fnition of \u0011, we get\nthe desired estimates in (5.18) and the proof is thus complete. \u0003\nLemma 5.5. Under all the above assumptions, we have:\n(5.21)ZL\n0\u0011jv1\nxj2dx=o(1);Z\f\u0000\u000f\n\u000b+\u000fjv1\nxj2dx=o(1)\nand\n(5.22)ZL\n0\u0011j\u0015v1j2dx=o(1);Z\f\u0000\u000f\n\u000b+\u000fj\u0015v1j2dx=o(1):\n19Proof. Our \frst aim here is to proveZL\n0\u0011jv1\nxj2dx=o(1):\nFor this sake, multiplying (5.8) by \u0011v1xand integrating by parts, we get:\n\u0000iZL\n0\u0015\u001a2v4\u00110v1dx\u0000iZL\n0\u0015\u001a2v4\nx\u0011v1dx+ZL\n0(k2v3\nx+D2v4\nx)(\u0011v1xx)dx+ZL\n0(k2v3\nx+D2v4\nx)(\u00110v1x)dx\n+ZL\n0\u0011k1jv1\nxj2dx+ZL\n0\u0011k1v3v1xdx+ZL\n0`k1\u0011v5v1xdx=ZL\n0\u001a2f4\n\u00154\u0011v1xdx: (5.23)\nNow, we need to estimate each term of (5.23):\n\u000fUsing (5.14), (5.18) and the fact that f3converges to zero in H1\n0(0;L) (orH1\n\u0003(0;L)), we get:\n(5.24)\u0000iZL\n0\u0015\u001a2v4\u00110v1dx=\u0000iZL\n0\u0015\u001a2(i\u0015v3\u0000f3\n\u00154)\u00110v1=ZL\n0\u001a2\u00152v3\u00110v1+iZL\n0f3\n\u00153\u00110v1=o(1):\n\u000fUsing (5.15) and the fact that \u0015v1is uniformly bounded in L2(0;L), we obtain:\n(5.25) \u0000iZL\n0\u0015\u001a2v4\nx\u0011v1dx=o(1)\n\u00152:\n\u000fFrom (5.6), we remark that1\n\u0015v1\nxxis uniformly bounded in L2(0;L). This fact combined with (5.15) and\n(5.16) yields\n(5.26)ZL\n0(k2\u0015v3\nx+D2\u0015v4\nx)(\u0011v1xx\n\u0015)dx=o(1)\n\u0015:\n\u000fUsing (5.15), (5.16) and the fact that v1\nxis uniformly bounded in L2(0;L), we get:\n(5.27)ZL\n0(k2v3\nx+D2v4\nx)(\u00110v1x)dx=o(1)\n\u00152:\n\u000fUsing (5.14) and the fact that v1\nxis uniformly bounded in L2(0;L), we obtain:\n(5.28)ZL\n0\u0011k1v3v1xdx+ZL\n0`k1\u0011v5v1xdx=o(1):\n\u000fUsing the fact that f4converges to zero in L2(0;L) andv1\nxis uniformly bounded in L2(0;L), we get:\n(5.29)ZL\n0\u001a2f4\n\u00154\u0011v1xdx=o(1)\n\u00154:\nFinally, inserting equations (5.24)-(5.29) into (5.23) and using the de\fnition of \u0011, we get the desired estimates\nin (5.21).\nNext, our second aim is to proveZL\n0\u0011j\u0015v1j2dx=o(1):\nFor this, multiplying (5.11) by \u001a\u00001\n1\u0011v1and integrating by parts, we get:\nZL\n0\u0011j\u0015v1j2dx=\u001a\u00001\n1ZL\n0k1(v1\nx+v3+`v5)(\u00110v1+\u0011v1x)dx\u0000\u001a\u00001\n1ZL\n0`k3(v5\nx\u0000`v1)\u0011v1 (5.30)\n\u0000ZL\n0\u0012f2\n\u00154+if1\n\u00153\u0013\n\u0011v1dx:\nSo, using (5.14), (5.21), the fact that ( v1\nx+v3+`v5), (v5\nx\u0000`v1) are uniformly bounded in L2(0;L) andf1,\nf2converge respectively to zero in H1\n0(0;L),L2(0;L) in the right hand side of the above equation and using\nthe de\fnition of \u0011, we get the desird estimates in (5.22). \u0003\n20Lemma 5.6. Under all the above assumptions, we have:\n(5.31)ZL\n0\u0011jv1\nxj2dx=o(1)\n\u00152;Z\f\u0000\u000f\n\u000b+\u000fjv1\nxj2dx=o(1)\n\u00152\nand\n(5.32)ZL\n0\u0011j\u0015v1j2dx=o(1)\n\u00152Z\f\u0000\u000f\n\u000b+\u000fj\u0015v1j2dx=o(1)\n\u00152:\nProof. For the clarity of the proof, we divide the proof into several steps:\nStep 1. In this step, we will prove\n(5.33) \u001a1ZL\n0\u0011j\u0015v1j2dx\u0000k1ZL\n0\u0011jv1\nxj2dx+`(k1+k3)Re(ZL\n0\u0011v5\nxv1dx)\n=o(1)\n\u00152:\nFor this sake, multiplying (5.11) by \u0011v1and integrating by parts, we get:\n\u001a1ZL\n0\u0011j\u0015v1j2dx\u0000k1ZL\n0\u0011jv1\nxj2dx\u0000k1Re\u001aZL\n0\u00110v1\nxv1dx\u001b\n+k1Re(ZL\n0\u0011v3\nxv1dx)\n+`(k1+k3)Re(ZL\n0\u0011v5\nxv1dx)\n\u0000`2k3ZL\n0\u0011jv1j2dx=o(1)\n\u00154: (5.34)\nNow, we need to estimate some terms of (5.34) as follows:\n\u000fWe get after integrating by parts\n\u0000k1Re\u001aZL\n0\u00110v1\nxv1dx\u001b\n=\u0000k1\n2ZL\n0\u00110(jv1j2)xdx=k1\n2ZL\n0\u001100jv1j2dx:\nUsing (5.22) in the previous equation, we obtain:\n(5.35) \u0000k1Re\u001aZL\n0\u00110v1\nxv1dx\u001b\n=o(1)\n\u00152:\n\u000fUsing (5.16) and (5.22), we deduce that\n(5.36) k1Re(ZL\n0\u0011v3\nxv1dx)\n\u0000`2k3ZL\n0\u0011jv1j2dx=o(1)\n\u00152:\nFinally, inserting (5.35) and (5.36) in (5.34), we get the desired estimate (5.33).\nStep 2. In this step, we will prove\n\u0012k1+k3\nk1\u0013\nRe\u001aZL\n0(k2v3\nx+D2v4\nx)\u0011v1xx)dx\u001b\n+ (k1+k3)ZL\n0\u0011jv1\nxj2dx\n\u0000`(k1+k3)Re\u001aZL\n0\u0011v5\nxv1dx\u001b\n=o(1)\n\u00152: (5.37)\nIn order to prove (5.37), multiplying (5.12) by the multiplier \u0000\u0012k1+k3\nk1\u0013\n\u0011v1xand integrating by parts, we\nget:\n\u0000\u001a2\u0012k1+k3\nk1\u0013\nRe\u001aZL\n0\u0011\u00152v3v1xdx\u001b\n| {z }\nI1\u0000\u0012k1+k3\nk1\u0013\nRe\u001aZL\n0(k2v3\nx+D2v4\nx)x\u0011v1xdx\u001b\n| {z }\nI2\n+ (k1+k3)ZL\n0\u0011jv1\nxj2dx+ (k1+k3)Re\u001aZL\n0\u0011v3v1xdx\u001b\n|{z }\nI3\u0000`(k1+k3)Re\u001aZL\n0\u0011v5\nxv1dx\u001b\n=o(1)\n\u00152: (5.38)\n21Next, we need to estimate I1;I2andI3.\n\u000fIntegrating by parts I1and then using (5.16), (5.18) and (5.22), we deduce that:\n(5.39) I1=\u001a2\u0012k1+k3\nk1\u0013\nRe\u001aZL\n0\u0010\n\u00110\u0015v3\u0015v1+\u0011\u0015v3\nx\u0015v1\u0011\ndx\u001b\n=o(1)\n\u00152:\n\u000fIntegrating by parts I2and then using (5.15), (5.16) and (5.21), we get:\nI2=\u0012k1+k3\nk1\u0013\nRe\u001aZL\n0\u0000\nk2v3\nx+D2v4\nx\u0001\n\u00110v1xdx\u001b\ndx+\u0012k1+k3\nk1\u0013\nRe\u001aZL\n0(k2v3\nx+D2v4\nx)\u0011v1xxdx\u001b\ndx\n=\u0012k1+k3\nk1\u0013\nRe\u001aZL\n0(k2v3\nx+D2v4\nx)\u0011v1xxdx\u001b\ndx+o(1)\n\u00152: (5.40)\n\u000fBy using (5.18) and (5.21), we deduce that\n(5.41) I3= (k1+k3)Re\u001aZL\n0\u0011v3v1xdx\u001b\n=o(1)\n\u00152:\nFinally, inserting (5.39), (5.40), and (5.41) into (5.38), we get the desired estimate (5.37).\nStep3. Combining (5.33) and (5.37), we get\n(5.42) \u001a1ZL\n0\u0011j\u0015v1j2dx+k3ZL\n0\u0011jv1\nxj2dx+\u0012k1+k3\nk1\u0013\nRe\u001aZL\n0\u0000\nk2v3\nx+D2v4\nx\u0001\n\u0011v1xxdx\u001b\n=o(1)\n\u00152:\nStep 4. In this step, we conclude the proof of the main estimates (5.31) and (5.32). For this aim, multiplying\n(5.11) by\u0011\u0010\nk2v3x+D2v4x\u0011\n, we get:\nRe(ZL\n0\u001a1\u0011\u0015v1\u0015\u0010\nk2v3x+D2v4x\u0011\ndx)\n+k1Re(ZL\n0\u0011\u0010\nk2v3x+D2v4x\u0011\nv1\nxxdx)\n+k1Re(ZL\n0\u0011\u0000\nv3\nx+`v5\nx\u0001\u0010\nk2v3x+D2v4x\u0011\ndx)\n+`k3Re(ZL\n0\u0000\nv5\nx\u0000`v1\u0001\n\u0011\u0010\nk2v3x+D2v4x\u0011\ndx)\n| {z }\nI=o(1)\n\u00152:(5.43)\nUsing the fact that v3\nxand (v5\nx\u0000`v1) are uniformly bounded in L2(0;L), (5.15) and (5.16), we get:\nI=k1Re(ZL\n0\u0011\u0000\nv3\nx+`v5\nx\u0001\u0010\nk2v3x+D2v4x\u0011\ndx)\n+`k3Re(ZL\n0\u0000\nv5\nx\u0000`v1\u0001\n\u0011\u0010\nk2v3x+D2v4x\u0011\ndx)\n=o(1)\n\u00152: (5.44)\nSubstitute (5.44) in (5.43), we get:\n(5.45) k1Re(ZL\n0\u0011\u0010\nk2v3x+D2v4x\u0011\nv1\nxxdx)\n=\u0000Re(ZL\n0\u001a1\u0011\u0015v1\u0015\u0010\nk2v3x+D2v4x\u0011\ndx)\n+o(1)\n\u00152:\nNow, substitute (5.45) in (5.42), we obtain:\n(5.46)\u001a1ZL\n0\u0011j\u0015v1j2dx+k3ZL\n0\u0011jv1\nxj2dx=\u0000\u0012k1+k3\nk2\n1\u0013\nRe(ZL\n0\u001a1\u0011\u0015v1\u0015\u0010\nk2v3x+D2v4x\u0011\ndx)\n+o(1)\n\u00152:\n22We will now apply Young's inequality in (5.46). For this sake , let \u000f>0 be given. We get:\n\u001a1ZL\n0\u0011j\u0015v1j2dx+k3ZL\n0\u0011jv1\nxj2dx\u00141\n\u000f\u0012k1+k3\nk2\n1\u00132ZL\n0\u001a1\u0011\u00152\f\f\fk2v3x+D2v4x\f\f\f2\ndx\n| {z }\n=o(1)\n\u00152+\u000fZL\n0\u001a1\u0011j\u0015v1j2dx+o(1)\n\u00152\n\u0014\u000fZL\n0\u001a1\u0011j\u0015v1j2dx+o(1)\n\u00152:\nConsequently, we have:\n(1\u0000\u000f)\u001a1ZL\n0\u0011j\u0015v1j2dx+k3ZL\n0\u0011jv1\nxj2dx=o(1)\n\u00152:\nFinally, it is su\u000ecient to take \u000f=1\n2in the previous equation to get the desired estimates in (5.31) and\n(5.32). The proof is thus complete. \u0003\nLemma 5.7. Under all the above assumptions, we have:\n(5.47)Z\f\u0000\u000f\n\u000b+\u000fjv5\nxj2dx=o(1)\nProof. Multiplying (5.11) by \u0011v5xand integrating over (0 ;L), we get:\n(`k1+`k3)ZL\n0\u0011jv5\nxj2=\u0000\u001a1ZL\n0\u0011\u00152v1v5xdx+k1ZL\n0v1\nx\u00110v5xdx+k1ZL\n0\u0011\u0015v1\nxv5xx\n\u0015dx (5.48)\n\u0000k1ZL\n0\u0011v3\nxv5xdx+`2k3ZL\n0\u0011v1v5xdx+o(1)\n\u00153:\nFinally, using (5.16), (5.31), (5.32), the fact that v5\nxis uniformly bounded in L2(0;L) and1\n\u0015v5\nxxis uniformly\nbounded in L2(0;L) due to (5.10) in the right hand side of the previous equation, we get the desired estimate\n(5.47). The proof is thus complete. \u0003\nLemma 5.8. Under all the above assumptions, we have:\n(5.49)Z\f\u0000\u000f\n\u000b+\u000fj\u0015v5j2dx=o(1):\nProof. Multiplying (5.13) by \u0011\u001a\u00001\n1v5, we get:\nZL\n0\u0011j\u0015v5j2dx=\u001a\u00001\n1ZL\n0k3(v5\nx\u0000`v1)(\u00110v5+\u0011v5x)dx+\u001a\u00001\n1ZL\n0`k1(v1\nx+v3+`v5)\u0011v1dx (5.50)\n\u0000ZL\n0\u0012f6\n\u00154+if5\n\u00153\u0013\n\u0011v5dx:\nUsing (5.14), (5.47), the fact that ( v1\nx+v3+`v5), (v5\nx\u0000`v1) are uniformly bounded in L2(0;L),f5,f6\nconverge to zero respectively in H1\n0(0;L) (or inH1\n\u0003(0;L)),L2(0;L) in the right hand side of the above\nequation, we deduce:ZL\n0\u0011j\u0015v5j2dx=o(1):\nFinally, using the de\fnition of \u0011, we get the desired estimate (5.49). The proof is thus complete. \u0003\nProof of Theorem 5.1 It follows from Lemmas 5.2, 5.4, 5.5, 5.7 and 5.8 that kUnkHj=o(1) on\n(\u000b+\u000f;\f\u0000\u000f):So one can use estimate (4.38) with D1=D3= 0 and Lemma 4.9 to conclude that kUnkHj=o(1)\non (0;L) which is a contradiction with (5.3). Consequently, condition (H4) holds and the energy of smooth\nsolutions of system (1.1) decays polynomially as tgoes to in\fnity. \u0003\n236.Lack of exponential stability\nIt was proved that the Bresse system subject to one or two viscous dampings is exponentially stable if\nand only if the wave propagate at the same speed (see [24] and [1]). In the case of viscoelastic damping, the\nsituation is more delicate. In this section, we prove that the Bresse system (1.1)-(1.3) subject to two global\nviscoelastic dampings is not exponentially stable even if the waves propagate at same speed. So, we assume\nthat:\n(6.1) D1= 0 and D2=D3= 1 in (0 ;L):\nTheorem 6.1. Under hypothesis (6.1) , the Bresse system (1.1) -(1.3) , is not exponentially stable in the\nenergy spaceH2.\nProof. For the proof of Theorem 6.1, it su\u000eces to show that there exists\n\u000fa sequence ( \u0015n)\u001aRwith lim\nn!+1j\u0015nj= +1, and\n\u000fa sequence ( Vn)\u001aD(A2),\nsuch that (i\u0015nI\u0000A 2)Vnis bounded inH2and lim\nn!+1kVnk= +1. For the sake of clarity, we skip the index\nn. LetF= (0;0;0;f4;0;0)2H 2with\nf4(x) = cos\u0010n\u0019x\nL\u0011\n; \u0015=n\u0019p\u001a2k2\nL\u001a2; n2N:\nWe solve the following equations:\n(6.2) i\u0015v1\u0000v2= 0;\n(6.3) i\u0015\u001a 1v2\u0000k1\u0000\nv1\nxx+v3\nx+`v5\nx\u0001\n\u0000`k3\u0000\nv5\nx\u0000`v1\u0001\n\u0000`\u0000\nv6\nx\u0000`v2\u0001\n= 0;\n(6.4) i\u0015v3\u0000v4= 0;\n(6.5) i\u0015\u001a 2v4\u0000k2v3\nxx+k1\u0000\nv1\nx+v3+`v5\u0001\n=\u001a2f4;\n(6.6) i\u0015v5\u0000v6= 0;\n(6.7) i\u0015\u001a 1v6\u0000k3\u0000\nv5\nxx\u0000`v1\nx\u0001\n+`v2\nx+`k1\u0000\nv1\nx+v3+`v5\u0001\n= 0:\nEliminating v2,v4andv6in (6.3), (6.5) and (6.7) by (6.2), (6.4) and (6.6), we get:\n(6.8) \u00152\u001a1v1+k1\u0000\nv1\nxx+v3\nx+`v5\nx\u0001\n+`(k3+i\u0015)\u0000\nv5\nx\u0000`v1\u0001\n= 0;\n(6.9) \u00152\u001a2v3+k2v3\nxx\u0000k1\u0000\nv1\nx+v3+`v5\u0001\n=\u0000\u001a2f4;\n(6.10) \u00152\u001a1v5+k3\u0000\nv5\nxx\u0000`v1\nx\u0001\n\u0000i\u0015`v1\nx\u0000`k1\u0000\nv1\nx+v3+`v5\u0001\n= 0:\nThis can be solved by the ansatz:\n(6.11) v1=Asin\u0010n\u0019x\nL\u0011\n; v3=Bcos\u0010n\u0019x\nL\u0011\n; v5=Ccos\u0010n\u0019x\nL\u0011\nwhereA,BandCdepend on \u0015are constants to be determined. Notice that k2\u0000n\u0019\nL\u00012\u0000\u001a2\u00152= 0, and\ninserting (6.11) in (6.8)-(6.10) we obtain that:\n(6.12)\u0012\u0010n\u0019\nL\u00112\nk1\u0000\u00152\u001a1+ (k3+i\u0015)`2\u0013\nA+k1\u0010n\u0019\nL\u0011\nB+ (k1+k3+i\u0015)`\u0010n\u0019\nL\u0011\nC= 0;\n(6.13) k1\u0010n\u0019\nL\u0011\nA+k1B+`k1C=\u001a2;\n(6.14) ( k1+k3+i\u0015)`\u0010n\u0019\nL\u0011\nA+`k1B+\u0014\nk3\u0010n\u0019\nL\u00112\n\u0000\u00152\u001a1+`2k1\u0015\nC= 0:\n24Equivalently,\n(6.15)0\nB@\u0000n\u0019\nL\u00012k1\u0000\u00152\u001a1+ (k3+i\u0015)`2k1\u0000n\u0019\nL\u0001\n(k1+k3+i\u0015)`\u0000n\u0019\nL\u0001\nk1\u0000n\u0019\nL\u0001\nk1 `k1\n(k1+k3+i\u0015)`\u0000n\u0019\nL\u0001\n`k1k3\u0000n\u0019\nL\u00012\u0000\u00152\u001a1+`2k11\nCA0\n@A\nB\nC1\nA=0\n@0\n\u001a2\n01\nA:\nThis implies that:\n(6.16) A=(k2\u001a1\u0000\u001a2k3)\u001a2\n2L\n\u0019(k2\u001a2\n1\u0000k3\u001a1\u001a2+\u001a2`2)k2n+O(n\u00002);\n(6.17) B=\u001a2\u0000\nk1k3\u001a2\n2+\u0000\n(\u0000k1\u0000k3)\u001a1+`2\u0001\nk2\u001a2+k2\n2\u001a2\n1\u0001\nk1((\u0000k3\u001a1+`2)\u001a2+k2\u001a2\n1)k2+O(n\u00001);\n(6.18) C=i`\u001a2\n2Lp\u001a2k2\n\u0019((\u0000k3\u001a1+`2)\u001a2+k2\u001a2\n1)k2n+O(n\u00002):\nNow, letVn=\u0000\nv1; i\u0015v1;v3; i\u0015v3;v5; i\u0015v5\u0001\n, wherev1;v3andv5are given by (6.11) and (6.16)-(6.18). It is\neasy to check that\nkVnkH2\u0015p\u001a2k\u0015v3k\u0018jB\u0015j\u0018jnj!+1asn!+1:\nOn the other hand, using (6.2)-(6.7), we deduce that\nk(i\u0015I\u0000A 2)Vnk2\nH2=k(0;0;0;\u001a2f4\u0000i\u0015D 2v3\nxx;0;i\u0015D 3v5\nxx)k2\nH2\u0014c:\nConsequently,k(i\u0015I\u0000A 2)Vnk2\nH2is bounded as ntends to +1. Thus the proof is complete. \u0003\nRemark 6.2. By a similar way, we can prove that the Bresse system (1.1) -(1.3) subject to only one vis-\ncoelastic damping is also not exponentially stable even if the waves propagate at same speed. \u0003\n7.Additional results and summary\nGlobal Kelvin{Voigt damping : analytic stability . In [12], Huang considered a one-dimensional wave\nequation with global Kelvin-Voigt damping and he proved that the semigroup associated to the equation is\nnot only exponentially stable, but also is analytic. So, it is logic that in the case of three waves equations\nwith three global dampings, the decay will be also analytic.\nIn this part, we state the analytic stability of the Bresse systems (1.1)-(1.2) and (1.1)-(1.3) provided that\nthere exists a positive constant d0such that:\n(7.1) D1; D 2D3\u0015d0>0 for every x2(0;L):\nTheorem 7.1. Assume that condition (7.1) holds. Then, the C0-semigroup etAj, forj= 1;2;is analytically\nstable.\nThe proof relies on the characterization of the analytic stability stated in theorem 4.1 and on the same\nkind of proof used for the preceding results : we use a contradiction argument and much simpler estimation\nto obtain the result. This much simpler proof is left to the reader.\nLocalized smooth damping : exponential stability . In [15], K. Liu and Z. Liu considered a one-\ndimensional wave equation with Kelvin-Voigt damping distributed locally on any subinterval of the region\noccupied by the beam. They proved that the semigroup associated with the equation for the transversal\nmotion of the beam is exponentially stable, although the semigroup associated with the equation for the\nlongitudinal motion of the beam is not exponentially stable.\nAnd in [16], K. Liu and Z. Liu reconsidered the one-dimensional linear wave equation with the Kelvin-\nVoigt damping presented on a subinterval but with smooth transition at the end of the interval. They proved\nthat the smoothness of the damping coe\u000ecient at the interface leads to an exponential stability. They were\nthe \frst researchers to suggest that discontinuity of material properties at the interface and the \\type\" of\nthe damping can a\u000bect the qualitative behavior of the energy decay. The smoothness of the coe\u000ecient at\nthe interface plays a crucial role in the stabilization of the wave equation. In this part, we generalize these\nresults on Bresse system.\n25So we consider the Bresse systems (1.1)-(1.2) and (1.1)-(1.3) subject to three local viscoelastic Kelvin-\nVoigt dampings with smooth coe\u000ecients at the interface. We establish uniform (exponential) stability of the\nC0-semigroup etAj,j= 1;2. For this purpose, let ;6=!= (\u000b;\f)\u001a(0;L) be the biggest nonempty open\nsubset of (0 ;L) satis\fying:\n(7.2) 9d0>0 such that Di\u0015d0;for almost every x2!; i= 1;2;3:\nTheorem 7.2. Assume that condition (7.2) holds. Assume also that D1,D2,D32W1;1(0;L). Then, the\nC0-semigroup etAjis exponentially stable in Hj,j= 1;2, i.e., for all U02Hj, there exist constants M\u00151\nand\u000e>0independent of U0such that:\n\r\retAjU0\r\r\nHj\u0014Me\u0000\u000etkU0kHj; t\u00150; j= 1;2:\nAgain, the proof relies on the characterization of the exponential stability stated in theorem 4.1 and on\nthe same kind of arguments used for the proof of the preceding results : we use a contradiction argument\nand simpler estimation to obtain the result. This proof is left to the reader.\nThe following table summarizes the results of this study:\nRegularity of D1Regularity of D2Regularity of D3 Localization Energy decay rate\nL1(0;L)L1(0;L)L1(0;L)Di\u0015d0>0 in (0;L)\ni= 1;2;3Analytic stability\nW1;1(0;L)W1;1(0;L)W1;1(0;L)Di\u0015d0>0 in!\ni= 1;2;3Exponential stability\nL1(0;L)L1(0;L)L1(0;L)3\\\ni=1suppDi=! Polynomial of type1\nt\n0 L1(0;L) 0 D2\u0015d0>0 in! Polynomial of type1p\nt\nAcknowledgments\nThe authors thanks professor Kais Ammari for his valuable discussions and comments.\nChiraz Kassem would like to thank the AUF agency for its support in the framework of the PCSI project\nuntitled Theoretical and Numerical Study of Some Mathematical Problems and Applications .\nAli Wehbe would like to thank the CNRS and the LAMA laboratory of Mathematics of the Universit\u0013 e Savoie\nMont Blanc for their supports.\nThe authors thank also the referees for very useful comments.\nReferences\n[1]F. Abdallah, M. Ghader, and A. Wehbe ,Stability results of a distributed problem involving Bresse system with history\nand/or Cattaneo law under fully Dirichlet or mixed boundary conditions , Math. Methods Appl. Sci., 41 (2018), pp. 1876{\n1907.\n[2]F. Alabau Boussouira, J. E. Mu ~noz Rivera, and D. d. S. Almeida J \u0013unior ,Stability to weak dissipative Bresse system ,\nJ. Math. Anal. Appl., 374 (2011), pp. 481{498.\n[3]W. Arendt and C. J. K. Batty ,Tauberian theorems and stability of one-parameter semigroups , Trans. Amer. Math.\nSoc., 306 (1988), pp. 837{852.\n[4]A. Benaissa and A. Kasmi ,Well-posedness and energy decay of solutions to a Bresse system with a boundary dissipation\nof fractional derivative type , Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), pp. 4361{4395.\n[5]A. Borichev and Y. Tomilov ,Optimal polynomial decay of functions and operator semigroups , Math. Ann., 347 (2010),\npp. 455{478.\n[6]S. Chen, K. Liu, and Z. Liu ,Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping , SIAM\nJ. Appl. Math., 59 (1999), pp. 651{668.\n26[7]T. El Arwadi and W. Youssef ,On the stabilization of the Bresse beam with Kelvin{Voigt damping , Applied Mathematics\n& Optimization, (2019), pp. 1{27.\n[8]L. H. Fatori and J. E. M. n. Rivera ,Rates of decay to weak thermoelastic Bresse system , IMA J. Appl. Math., 75\n(2010), pp. 881{904.\n[9]M. Ghader and A. Wehbe. ,A transmission problem for the Timoshenko system with one local Kelvin{Voigt damping\nand non-smooth coe\u000ecient at the interface . arXiv: 2005.12756, 2020.\n[10]A. Guesmia and M. Kafini ,Bresse system with in\fnite memories , Math. Methods Appl. Sci., 38 (2015), pp. 2389{2402.\n[11]F. L. Huang ,Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces , Ann.\nDi\u000berential Equations, 1 (1985), pp. 43{56.\n[12] ,On the mathematical model for linear elastic systems with analytic damping , SIAM J. Control Optim., 26 (1988),\npp. 714{724.\n[13]T. Kato ,Perturbation theory for linear operators , Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the\n1980 edition.\n[14]J. E. Lagnese, G. Leugering, and E. J. P. G. Schmidt ,Modeling, analysis and control of dynamic elastic multi-link\nstructures , Systems & Control: Foundations & Applications, Birkh auser Boston, Inc., Boston, MA, 1994.\n[15]K. Liu and Z. Liu ,Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping ,\nSIAM J. Control Optim., 36 (1998), pp. 1086{1098.\n[16] ,Exponential decay of energy of vibrating strings with local viscoelasticity , Z. Angew. Math. Phys., 53 (2002),\npp. 265{280.\n[17]Z. Liu and B. Rao ,Energy decay rate of the thermoelastic Bresse system , Z. Angew. Math. Phys., 60 (2009), pp. 54{69.\n[18]Z. Liu and S. Zheng ,Semigroups associated with dissipative systems , vol. 398 of Chapman & Hall/CRC Research Notes\nin Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 1999.\n[19]T. K. Maryati, J. E. Mu ~noz Rivera, A. Rambaud, and O. Vera ,Stability of an N-component Timoshenko beam with\nlocalized Kelvin-Voigt and frictional dissipation , Electron. J. Di\u000berential Equations, (2018), pp. Paper No. 136, 18.\n[20]N. Najdi and A. Wehbe ,Weakly locally thermal stabilization of Bresse systems , Electron. J. Di\u000berential Equations,\n(2014), pp. No. 182, 19.\n[21]A. Pazy ,Semigroups of linear operators and applications to partial di\u000berential equations , vol. 44 of Applied Mathematical\nSciences, Springer-Verlag, New York, 1983.\n[22]J. Pr uss,On the spectrum of C0-semigroups , Trans. Amer. Math. Soc., 284 (1984), pp. 847{857.\n[23]X. Tian and Q. Zhang ,Stability of a Timoshenko system with local Kelvin-Voigt damping , Z. Angew. Math. Phys., 68\n(2017), pp. Paper No. 20, 15.\n[24]A. Wehbe and W. Youssef ,Exponential and polynomial stability of an elastic Bresse system with two locally distributed\nfeedbacks , J. Math. Phys., 51 (2010), pp. 103523, 17.\n[25]H. L. Zhao, K. S. Liu, and C. G. Zhang ,Stability for the Timoshenko beam system with local Kelvin-Voigt damping ,\nActa Math. Sin. (Engl. Ser.), 21 (2005), pp. 655{666.\nLaboratoire de Math \u0013ematiques UMR 5127 CNRS, Universit \u0013e de Savoie Mont Blanc, Campus scientifique, 73376\nLe Bourget du Lac Cedex, France\nEmail address :stephane.gerbi@univ-smb.fr\nUniversit \u0013e Libanaise, Facult \u0013e des Sciences 1, EDST, Equipe EDP-AN, Hadath, Beyrouth, Liban\nEmail address :shiraz.kassem@hotmail.com\nUniversit \u0013e Libanaise, Facult \u0013e des Sciences 1, EDST, Equipe EDP-AN, Hadath, Beyrouth, Liban\nEmail address :ali.wehbe@ul.edu.lb\n27"    },    {        "title": "2205.13364v1.Ergodic_results_for_the_stochastic_nonlinear_Schrödinger_equation_with_large_damping.pdf",        "content": "arXiv:2205.13364v1  [math.PR]  26 May 2022Ergodic results for the stochastic nonlinear\nSchr¨ odinger equation with large damping\nZdzis/suppress law Brze´ zniak∗, Benedetta Ferrario†and Margherita Zanella‡\nMay 27, 2022\nAbstract\nWe study the nonlinear Schr¨ odinger equation with linear da mping, i.e.\na zero order dissipation, and additive noise. Working in Rdwithd= 2\nord= 3, we prove the uniqueness of the invariant measure when the\ndamping coefficient is sufficiently large.\n1 Introduction\nThe nonlinear Schr¨ odinger equation occurs as a basic model in many areas\nof physics: hydrodynamics, plasma physics, optics, molecular biolog y, chemical\nreaction,etc. Itdescribesthepropagationofwavesinmediawithb othnonlinear\nand dispersive responses.\nInthisarticle,weinvestigatethelongtimebehaviorofthenonlinearS chr¨ odinger\nequation with a linear damping term and an additive noise. This is our mod el\n(1.1)/braceleftBigg\ndu(t)+/bracketleftbig\ni∆u(t)+iα|u(t)|2σu(t)+λu(t)/bracketrightbig\ndt= ΦdW(t)\nu(0) =u0,\nThe unknown is u:Rd→C. We consider σ >0,λ >0 andα∈ {−1,1}; for\nα= 1 this is called the focusing equation and for α=−1 this is the defocusing\none.\nMany results are known about existence and uniqueness of solution s, in\ndifferent spatial domains and with different noises; see [ BRZ14,BRZ16,BRZ17,\nBHW19,CHS19,DBD99,DBD03]. Basically these results are obtained without\ndamping but can be easily extended to the case with λ>0.\nWhen there is no damping and no forcing term ( λ= 0 and Φ = 0), the\nequation is conservative. However, with a noise and a damping term, we expect\nthat the energy injected by the noise is dissipated by the damping te rm; because\nof this balance it is meaningful to look for stationary solutions or inva riant\nmeasures. Ekren, Kukavica and Ziane [ EKZ17] and Kim [ K06] provide the\n∗Department of Mathematics, University of York, Heslington , York, YO105DD, U E-mail\naddress: zdzislaw.brzezniak@york.ac.uk\n†Dipartimento di Scienze Economiche e Aziendali, Universit ` a di Pavia, 27100 Pavia, Italy\nE-mail address : benedetta.ferrario@unipv.it\n‡Dipartimento di Matematica ”Francesco Brioschi”, Politec nico di Milano, Via Bonardi\n13, 20133 Milano, Italy E-mail address : margherita.zanella@polimi.it\n1existence of invariant measures of the equation ( 1.1) for any damping coefficient\nλ >0; see also the more general setting of [ BFZ21] for the two dimensional\ncase in a different spatial domain and with multiplicative noise and the bo ok\n[HW19] for the numerical analysis approach. Notice that the damping λuis\nweaker than the dissipation given by a Laplacian −λ∆u; for this reason we say\nthatλuis a zero-order dissipation. This implies that the results of existence\nor uniqueness of invariant measures for the damped Schr¨ odinger equation are\nless easy than for the stochastic parabolic equations (see, e.g., [ DPZ96]). A\nsimilar issue appears in the stochastic damped 2D Euler equations, fo r which\nthe existence of invariant measures has been recently proven in [ BF20]; there\nagain the difficulty comes from the absence of the strong dissipation , given by\nthe Laplacian in the Navier-Stokes equations.\nThe question of the uniqueness of invariant measures is quite challen ging for\nthe SPDE’s with azero-orderdissipation. Debussche and Odasso[ DO05] proved\nthe uniqueness ofthe invariant measurefor the cubic focusing Sch r¨ odingerequa-\ntion (1.1), i.e.σ=α= 1, when the spatial domain is a bounded interval;\nhowever no uniqueness results are known for larger dimension. For the one-\ndimensional stochastic damped KdV equation there is a recent resu lt by Glatt-\nHoltz, Martinez and Richards [ GHMR21 ]. However for nonlinear SPDE’s of\nparabolic type, i.e. with the stronger dissipation term, the uniquene ss issue\nhas been solved in many cases; see, e.g., the book [ DPZ96] by Da Prato and\nZabczyk, and the many examples in the paper [ GHMR17 ] by Glatt-Holtz, Mat-\ntingly and Richards, dealing with the coupling technique. Let us point o ut that\nthe coupling technique allows for the uniqueness result without rest riction on\nthe damping parameter λbut all the examples solved so far areset in a bounded\nspatial domain and not in Rd.\nThe aim of our paper is to investigate the uniqueness of the invariant mea-\nsures for equation ( 1.1) in dimension d= 2 andd= 3, with some restrictions on\nthe nonlinearity when d= 3. However our technique fails for larger dimension.\nNotice that also the results for the attractor in the deterministic s etting are\nknown for d≤3 (see [L95]). Our main result is Theorem 5.1; it provides a\nsufficient condition to get the uniqueness of the invariant measure, involvingλ\nand the intensity of the noise. To optimize this condition ( 5.1) we perform a\ndetailed analysis on how the solution depends on λ.\nAs far as the contents of this paper are concerned, in Section 2we introduce\nthe mathematical setting; in Section 3by means of the Strichartz estimates we\nprove a regularity result on the solutions; this will allow to prove in Sec tion4\nthat the support of any invariant measure is contained in V∩L∞(Rd) and some\nestimates of the moments are given. Finally Section 5presents the uniqueness\nresult. The four appendices contain auxiliary results.\n2 Assumptions and basic results\nForp≥1,Lp(Rd) is the classical Lebesgue space of complex valued functions,\nand the inner product in the real Hilbert space L2(Rd) is denoted by\n/an}b∇acketle{tu,v/an}b∇acket∇i}ht=/integraldisplay\nRdu(y)v(y)dy.\n2We consider the Laplace operator ∆ as a linear operator in L2(Rd); so\nA0=−∆, A 1= 1−∆\nare non-negative linear operators and {eitA0}t∈Ris a unitary group in L2(Rd).\nMoreover for s≥0 we consider the power operator As/2\n1inL2(Rd) with domain\nHs={u∈L2(Rd) :/ba∇dblAs/2\n1u/ba∇dblL2(Rd)<∞}. Our two main spaces are H:=\nL2(Rd) andV:=H1(Rd). We setH−s(Rd) for the dual space of Hs(Rd) and\ndenote again by /an}b∇acketle{t·,·/an}b∇acket∇i}htthe duality bracket.\nWe define the generalized Sobolev spaces Hs,p(Rd) with norm given by\n/ba∇dblu/ba∇dblHs,p(Rd) =/ba∇dblAs/2\n1u/ba∇dblLp(Rd). We recall the Sobolev embedding theorem, see\ne.g. [BL76][Theorem 6.5.1]: if 1 <q<p< ∞with\n1\np=1\nq−r−s\nd,\nthen the following inclusion holds\nHr,q(Rd)⊂Hs,p(Rd)\nand there exists a constant Csuch that /ba∇dblu/ba∇dblHs,p(Rd)≤C/ba∇dblu/ba∇dblHr,q(Rd)for all\nu∈Hr,q(Rd).\nRemark 2.1. Ford= 1the spaceVis a subset of L∞(R)and is a multiplicative\nalgebra. This simplifies the analysis of the Schr¨ odinger eq uation(1.1). However\nford≥2the analysis is more involved.\nWe write the nonlinearity as\n(2.1) Fα(u) :=α|u|2σu.\nLemmaC.1provides a priori estimates on it.\nAs far as the stochastic term is concerned, we considera realHilbe rt spaceU\nwith an orthonormal basis {ej}j∈Nand a complete probability space (Ω ,F,P).\nLetWbe aU-canonical cylindrical Wiener process adapted to a filtration F\nsatisfying the usual conditions. We can write it as a series\nW(t) =∞/summationdisplay\nj=1Wj(t)ej,\nwith{Wj}ja sequence of i.i.d. real Wiener processes. Hence\n(2.2) Φ W(t) =∞/summationdisplay\nj=1Wj(t)Φej\nfor a given linear operator Φ : U→V.\nNow we rewrite the Schr¨ odinger equation ( 1.1) in the abstract form as\n(2.3)/braceleftBigg\ndu(t)+[−iA0u(t)+iFα(u(t))+λu(t)] dt= ΦdW(t)\nu(0) =u0\nWe work under the following assumptions on the noise and the nonlinea rity.\nThe initial data u0is assumed to be V.\n3Assumption 2.2 (on the noise) .We assume that Φ :U→Vis a Hilbert-\nSchmidt operator, i.e.\n(2.4) /ba∇dblΦ/ba∇dblLHS(U,V):=/parenleftbig∞/summationdisplay\nj=1/ba∇dblΦej/ba∇dbl2\nV/parenrightbig1/2<∞.\nThis means that\n/ba∇dblΦ/ba∇dbl2\nLHS(U;V)=∞/summationdisplay\nj=1/ba∇dblA1/2\n1Φej/ba∇dbl2\nH=∞/summationdisplay\nj=1/ba∇dblΦej/ba∇dbl2\nH+∞/summationdisplay\nj=1/ba∇dbl∇Φej/ba∇dbl2\nH.\nIn order to compare our setting with the more general one of our p revious\npaper [BFZ21] in the two-dimensional setting, we point out that Φ is also\na Hilbert-Schmidt operator from UtoH(and we denote /ba∇dblΦ/ba∇dblLHS(U;H):=/parenleftBig/summationtext\nj∈N/ba∇dblΦej/ba∇dbl2\nH/parenrightBig1/2\n) and, ford= 2, aγ-radonifying operator from UtoLp(R2)\nfor any finite p.\nAssumption 2.3 (on the nonlinearity ( 2.1)).\n•Ifα= 1(focusing), let 0≤σ<2\nd.\n•Ifα=−1(defocusing), let/braceleftBigg\n0≤σ<2\nd−2,ford≥3\nσ≥0, ford≤2\nWe recall the continuous embeddings\nH1(R2)⊂Lp(R2)∀p∈[2,∞)\nH1(Rd)⊂Lp(Rd)∀p∈[2,2d\nd−2] ford≥3\nHence forσchosen as in Assumption 2.3there is the continuous embedding\n(2.5) H1(Rd)⊂L2+2σ(Rd).\nMoreover if σd<2(σ+1) the following Gagliardo-Niremberg inequality holds\n(2.6) /ba∇dblu/ba∇dblL2+2σ(Rd)≤C/ba∇dblu/ba∇dbl1−σd\n2(1+σ)\nL2(Rd)/ba∇dbl∇u/ba∇dblσd\n2(1+σ)\nL2(Rd).\nIn particular this holds for the values of σspecified in the Assumption 2.3. In\nthe focusing case, thanks to the Young inequality for any ǫ >0 there exists\nCǫ>0 such that\n(2.7) /ba∇dblu/ba∇dbl2+2σ\nL2+2σ(Rd)≤ǫ/ba∇dbl∇u/ba∇dbl2\nL2(Rd)+Cǫ/ba∇dblu/ba∇dbl2+4σ\n2−σd\nL2(Rd).\nWe recall the classical invariant quantities for the deterministic unf orced\nSchr¨ odinger equation ( λ= 0, Φ = 0), the mass and the energy (see [ C03]):\nM(u) =/ba∇dblu/ba∇dbl2\nH, (2.8)\nH(u) =1\n2/ba∇dbl∇u/ba∇dbl2\nH−α\n2(1+σ)/ba∇dblu/ba∇dbl2+2σ\nL2+2σ(Rd). (2.9)\nThey are both well defined on V, thanks to ( 2.5).\n4Remark 2.4. In the defocusing case α=−1, we have\n(2.10) H(u)≥1\n2/ba∇dbl∇u/ba∇dbl2\nH≥0∀u∈V.\nIn the focusing case α= 1, the energy has no positive sign but we can modify\nit by adding a term and recover the sign property. We introduc e the modified\nenergy\n(2.11) ˜H(u) =1\n2/ba∇dbl∇u/ba∇dbl2\nH−1\n2(1+σ)/ba∇dblu/ba∇dbl2+2σ\nL2(1+σ)(Rd)+G/ba∇dblu/ba∇dbl2+4σ\n2−σd\nH\nwhereGis the constant appearing in the following particular form o f(2.7)\n(2.12)1\n2(1+σ)/ba∇dblu/ba∇dbl2+2σ\nL2(1+σ)(Rd)≤1\n4/ba∇dbl∇u/ba∇dbl2\nH+G/ba∇dblu/ba∇dbl2+4σ\n2−σd\nH.\nTherefore\n(2.13) ˜H(u)≥1\n4/ba∇dbl∇u/ba∇dbl2\nH≥0∀u∈V.\nAbove we denoted by Ca generic positive constant which might vary from\none line to the other, except Gwhich is the particular constant in ( 2.12) coming\nfrom the Gagliardo-Niremberg inequality. Moreover we shall use this notation:\nifa,b≥0 satisfy the inequality a≤CAbwith a constant CA>0 depending on\nthe expression A, we writea/lessorsimilarAb; for a generic constant we put no subscript.\nIf we havea/lessorsimilarAbandb/lessorsimilarAa, we writea≃Ab.\nNow we recall the known results on solutions and invariant measures ; then\nwe provide the improved estimates for the mass and the energy.\n2.1 Basic results\nWe recall from [ DBD03] the basic results on the solutions; for any u0∈V\nthere exists a unique global solution u={u(t;u0)}t≥0, which is a continuous\nV-valued process. Here uniqueness is meant as pathwise uniqueness . Actually\ntheir result is given without damping but one can easily pass from λ= 0 to\nanyλ>0. The difference consists in getting uniform in time estimates for the\ndamped equationoverthe time interval [0 ,+∞), whereasthey hold on any finite\ntime interval [0 ,T] whenλ= 0. Let us state the result from De Bouard and\nDebussche [ DBD03].\nTheorem 2.5. Under Assumptions 2.2and2.3, for every u0∈Vthere exists\na uniqueV-valued and continuous solution of (2.3). This is a Markov process\ninV. Moreover for any finite T >0and integer m≥1there exist positive\nconstantsC1andC2(depending on T,mand/ba∇dblu0/ba∇dblV) such that\nEsup\n0≤t≤T[M(u(t))m]≤C1\nand\nE/bracketleftbigg\nsup\n0≤t≤TH(u(t))/bracketrightbigg\n≤C2.\n5We notice that the last estimate can be obtained for any power m>1 for the\nenergy in the defocusing case as well as for the modified energy in th e focusing\ncase. Similar computations can be found in [ EKZ17] for the defocusing case;\nhowever their Lemma 5.1 has to be modified in the focusing case. The s trategy\nis the same as that given in the next Section 2.2.\nAs soon as a unique solution in Vis defined, we can introduce the Markov\nsemigroup. Let us denote by u(t;x) the solution evaluated at time t>0, with\ninitial value x. We define\n(2.14) Ptf(x) =E[f(u(t;x))]\nfor any Borelian and bounded function f:V→R.\nA probability measure µon the Borelian subsets of Vis said to be an in-\nvariant measure for ( 2.3) when\n(2.15)/integraldisplay\nVPtf dµ=/integraldisplay\nVf dµ ∀t≥0,f∈Bb(V).\nWe recall Theorem 3.4 from [ EKZ17].\nTheorem 2.6. Under Assumptions 2.2and2.3, there exists an invariant mea-\nsure supported in V.\n2.2 Mean estimates\nIn this section we revise some bounds on the moments of the mass, t he en-\nergy and the modified energy, in order to see how these quantities d epend\non the damping coefficient λ. This improves the results by [ DBD03][§3] and\n[EKZ17][Lemma 5.1].\nThis is the result for the mass M(u) =/ba∇dblu/ba∇dbl2\nH.\nProposition 2.7. Letu0∈V. Then under assumptions 2.2and2.3, for every\nm≥1there exists a positive constant C(depending on m) such that\n(2.16) E[M(u(t))m]≤e−λmtM(u0)m+C/ba∇dblΦ/ba∇dbl2m\nLHS(U;H)λ−m\nfor anyt≥0.\nProof.Let us start by proving the estimate ( 2.16) form= 1. We apply the Itˆ o\nformula to M(u(t))\ndM(u(t))+2λM(u(t))dt=/ba∇dblΦ/ba∇dbl2\nLHS(U;H)dt+2Re/an}b∇acketle{tu(t),ΦdW(t)/an}b∇acket∇i}ht.\nTaking the expected value and using the fact that the stochastic in tegral is a\nmartingale by Theorem 2.5, we obtain, for any t≥0,\nd\ndtE[M(u(t))] =−2λE[M(u(t))]+/ba∇dblΦ/ba∇dbl2\nLHS(U,H).\nSolving this ODE we obtain\nE[M(u(t))] =e−2λtM(u0)+/ba∇dblΦ/ba∇dbl2\nLHS(U,H)/integraldisplayt\n0e−2λ(t−s)ds\n≤e−2λtM(u0)+1\n2λ/ba∇dblΦ/ba∇dbl2\nLHS(U,H),\n6which proves ( 2.16) form= 1.\nFor larger values of m, let us apply the Itˆ o formula to M(u(t))mto obtain\nM(u(t))m=M(u0)m−2λm/integraldisplayt\n0M(u(s))mds\n+2m/integraldisplayt\n0M(u(s))m−1Re/an}b∇acketle{tu(s),ΦdW(s)/an}b∇acket∇i}ht\n+m/ba∇dblΦ/ba∇dbl2\nLHS(U,H)/integraldisplayt\n0M(u(s))m−1ds\n+2(m−1)m/integraldisplayt\n0M(u(s))m−2∞/summationdisplay\nj=1[Re/an}b∇acketle{tu(s),Φej/an}b∇acket∇i}ht]2ds.(2.17)\nWith the Young inequality we get\nm/ba∇dblΦ/ba∇dbl2\nLHS(U,H)M(u)m−1+2(m−1)mM(u)m−2∞/summationdisplay\nj=1[Re/an}b∇acketle{tu,Φej/an}b∇acket∇i}ht]2\n≤m(2m−1)/ba∇dblΦ/ba∇dbl2\nLHS(U,H)M(u)m−1\n≤λmM(u)m+/ba∇dblΦ/ba∇dbl2m\nHS(U;H)/parenleftbiggm−1\nm/parenrightbiggm−1\n(2m−1)mλ1−m.(2.18)\nBy Theorem 2.5we know that the stochastic integral in ( 2.17) is a martingale,\nso taking the expected value on both sides of ( 2.17) we obtain\nEM(u(t))m≤ M(u0)m−λm/integraldisplayt\n0EM(u(s))mds+/ba∇dblΦ/ba∇dbl2m\nHS(U;H)Cmλ1−mt.\nBy means of Gronwall inequality we get\nEM(u(t))m≤e−λmtM(u0)m+/ba∇dblΦ/ba∇dbl2m\nHS(U;H)Cmλ1−m/integraldisplayt\n0e−λm(t−s)ds\n≤e−λmtM(u0)m+/ba∇dblΦ/ba∇dbl2m\nHS(U;H)Cm\nmλ−m\nfor anyt≥0.\nNotice that the estimates on the mass do not depend on α, whereas this\nhappens in the following result. Indeed, consider the energy H(u) given in ( 2.9)\nand the modified energy ˜H(u) given in ( 2.11). We deal with d≥2, since the\ncased= 1 is easier and already analysed in [ DO05]. Therefore the condition\nσ<2\ndimplies that σ<1 whend≥2. We introduce the function\n(2.19) φ1(σ,λ,Φ) =/ba∇dblΦ/ba∇dbl2\nLHS(U;V)+/ba∇dblΦ/ba∇dbl2+2σ\nLHS(U;V)λ−σ.\nNotice that the mapping λ/ma√sto→φ1(σ,λ,Φ) is strictly decreasing. The same prop-\nerty holds for the other function φ2(d,σ,λ,m, Φ) appearing the next statement.\nThe expression of φ2(d,σ,λ,1,Φ) is given in ( 2.34); however for general mthe\nexpression is similar but longer and we avoid to specify it.\n7Proposition 2.8. Letd≥2andu0∈V. Under Assumptions 2.2and2.3, we\nhave the following estimates:\ni)Whenα=−1, for everym≥1there exists a positive constant C=C(d,σ,m)\nsuch that\n(2.20) EH(u(t))m≤e−λmtH(u0)m+Cφm\n1λ−m\nfor anyt≥0.\nii)Whenα= 1, for every m≥1there exist a smooth positive function φ2=\nφ2(d,σ,λ,m, Φ)and positive constants a=a(d,σ),C1=C(d,σ,m)andC2=\nC(d,σ,m)such that\n(2.21)E˜H(u(t))m\n≤e−maλt/parenleftBig\n˜H(u0)m+C1(λ−m+λ−m−1\n2)M(u0)m(1+2σ\n2−σd)/parenrightBig\n+C2φm\n2λ−m\nfor anyt≥0. Moreover the mapping λ/ma√sto→φ2(d,σ,λ,m, Φ)is strictly decreasing.\nProof.The Itˆ o formula for H(u(t)) is\n(2.22)dH(u(t))+2λH(u(t))dt=αλσ\nσ+1/ba∇dblu(t)/ba∇dbl2σ+2\nL2σ+2(Rd)dt\n−∞/summationdisplay\nj=1Re/an}b∇acketle{t∆u(t)+α|u(t)|2σu(t),Φej/an}b∇acket∇i}htdWj(t)+1\n2/ba∇dbl∇Φ/ba∇dbl2\nLHS(U;H)dt\n−α\n2/ba∇dbl|u(t)|σΦ/ba∇dbl2\nLHS(U;H)dt−ασ∞/summationdisplay\nj=1/an}b∇acketle{t|u(t)|2σ−2,[Re(u(t)Φej)]2/an}b∇acket∇i}htdt.\nBelow we repeatedly use the H¨ older and Young inequalities. In partic ular\n(2.23) Am−1B≤ǫλAm+Cǫλ1−mBm\nand\n(2.24) Am−2B≤ǫλAm+Cǫλ1−m\n2Bm\n2\nfor positive A,B,λ,ǫ .\n•In the defocusing case α=−1, we neglect the first term in the r.h.s. in ( 2.22),\ni.e.\n(2.25)dH(u(t))+2λH(u(t))dt≤ −∞/summationdisplay\nj=1Re/an}b∇acketle{t∆u(t)−|u(t)|2σu(t),Φej/an}b∇acket∇i}htdWj(t)\n+/bracketleftBig1\n2/ba∇dbl∇Φ/ba∇dbl2\nLHS(U;H)+1\n2/ba∇dbl|u(t)|σΦ/ba∇dbl2\nLHS(U;H)+σ∞/summationdisplay\nj=1/an}b∇acketle{t|u(t)|2σ−2,[Re(u(t)Φej)]2/an}b∇acket∇i}ht/bracketrightBig\ndt.\nMoreover thanks to the Assumption 2.3we use the H¨ older and Young inequali-\n8ties to get\n1\n2/ba∇dbl|u|σΦ/ba∇dbl2\nLHS(U;H)+σ∞/summationdisplay\nj=1/an}b∇acketle{t|u|2σ−2,[Re(u(t)Φej)]2/an}b∇acket∇i}ht\n≤1\n2/ba∇dbl|u|σ/ba∇dbl2\nL2σ+2\nσ(Rd)∞/summationdisplay\nj=1/ba∇dblΦej/ba∇dbl2\nL2σ+2(Rd)+σ/ba∇dbl|u|2σ/ba∇dbl\nL2σ+2\n2σ(Rd)∞/summationdisplay\nj=1/ba∇dbl|Φej|2/ba∇dblLσ+1(Rd)\n≤2σ+1\n2/ba∇dblu/ba∇dbl2σ\nL2σ+2(Rd)∞/summationdisplay\nj=1/ba∇dblΦej/ba∇dbl2\nL2σ+2(Rd)\n≤2σ+1\n2/ba∇dblu/ba∇dbl2σ\nL2σ+2(Rd)/ba∇dblΦ/ba∇dbl2\nLHS(U;V)by (2.5)\n≤λ\n2+2σ/ba∇dblu/ba∇dbl2+2σ\nL2+2σ(Rd)+Cσ/ba∇dblΦ/ba∇dbl2+2σ\nLHS(U;V)λ−σ\n≤λH(u)+C/ba∇dblΦ/ba∇dbl2+2σ\nLHS(U;V)λ−σ(2.26)\nNow we insert this estimate in ( 2.25) and take the mathematical expectation to\nget rid of the stochastic integral\nd\ndtEH(u(t))+2λEH(u(t))≤1\n2/ba∇dblΦ/ba∇dbl2\nLHS(U;V)+λEH(u(t))+C/ba∇dblΦ/ba∇dbl2+2σ\nLHS(U;V)λ−σ,\ni.e.\nd\ndtEH(u(t))+λEH(u(t))≤1\n2/ba∇dblΦ/ba∇dbl2\nLHS(U;V)+C/ba∇dblΦ/ba∇dbl2+2σ\nLHS(U;V)λ−σ.\nBy Gronwall lemma we get\nEH(u(t))≤e−λtH(u0)+1\n2/ba∇dblΦ/ba∇dbl2\nLHS(U;V)λ−1+C/ba∇dblΦ/ba∇dbl2+2σ\nLHS(U;V)λ−σ−1\nfor anyt≥0. This proves ( 2.20) form= 1.\nFor higher powers m>1, by means of Itˆ o formula we get\n(2.27)dH(u(t))m=mH(u(t))m−1dH(u(t))\n+m(m−1)\n2H(u(t))m−2∞/summationdisplay\nj=1[Re/an}b∇acketle{t∆u(t)−|u(t)|2σu(t),Φej/an}b∇acket∇i}ht]2dt.\nWe estimate the latter term using H¨ older and Young inequality:\n1\n2/summationdisplay\nj[Re/an}b∇acketle{t∆u−|u|2σu,Φej/an}b∇acket∇i}ht]2\n≤/summationdisplay\nj[Re/an}b∇acketle{t∆u,Φej/an}b∇acket∇i}ht]2+/summationdisplay\nj[Re/an}b∇acketle{t|u|2σu,Φej/an}b∇acket∇i}ht]2\n≤ /ba∇dbl∇u/ba∇dbl2\nH/summationdisplay\nj/ba∇dbl∇Φej/ba∇dbl2\nH+/ba∇dbl|u|2σu/ba∇dbl2\nL2σ+2\n2σ+1(Rd)/summationdisplay\nj/ba∇dblΦej/ba∇dbl2\nL2+2σ(Rd)\n≤ /ba∇dbl∇u/ba∇dbl2\nH/ba∇dblΦ/ba∇dbl2\nLHS(U;V)+/ba∇dblu/ba∇dbl2(2σ+1)\nL2+2σ(Rd)/ba∇dblΦ/ba∇dbl2\nLHS(U;V)\n≤ǫλH(u)2+Cǫ,σ/parenleftBig\n/ba∇dblΦ/ba∇dbl4\nLHS(U;V)+/ba∇dblΦ/ba∇dbl4(1+σ)\nLHS(U;V)λ−2σ/parenrightBig\nλ−1\n9for anyǫ>0. Inserting in ( 2.27) and using the Young inequality ( 2.24) we get\n(2.28)dH(u(t))m≤mH(u(t))m−1dH(u(t))\n+1\n2mλH(u(t))mdt+C/parenleftBig\n/ba∇dblΦ/ba∇dbl4\nLHS(U;V)+/ba∇dblΦ/ba∇dbl4(σ+1)\nLHS(U;V)λ−2σ/parenrightBigm/2\nλ−m+1dt\nWe estimate H(u(t))m−1dH(u(t)) using ( 2.25), (2.26), and the Young in-\nequality ( 2.23). Then we take the mathematical expectation in ( 2.28) and ob-\ntain\n(2.29)d\ndtEH(u(t))m+mλEH(u(t))m\n≤Cσ,m/parenleftBig\n/ba∇dblΦ/ba∇dbl2\nLHS(U;V)+/ba∇dblΦ/ba∇dbl2(1+σ)\nLHS(U;V)λ−σ/parenrightBigm\nλ−m+1.\nBy Gronwall lemma we get ( 2.20).\n•In the focusing case α= 1, we neglect the last two terms in the r.h.s. in ( 2.22)\nand get\n(2.30)dH(u(t))+2λH(u(t))dt≤λσ\nσ+1/ba∇dblu(t)/ba∇dbl2+2σ\nL2+2σ(Rd)dt\n−∞/summationdisplay\nj=1Re/an}b∇acketle{t∆u(t)+|u(t)|2σu(t),Φej/an}b∇acket∇i}htdWj(t)+1\n2/ba∇dbl∇Φ/ba∇dbl2\nLHS(U;H)dt.\nWe write the Itˆ o formula for the modified energy ˜H(u) =H(u)+GM(u)1+2σ\n2−σd.\nKeeping in mind ( 2.17) and (2.18) for the power m= 1+2σ\n2−σdof the mass, we\nhave\n(2.31)d˜H(u(t))+2λ˜H(u(t))dt\n≤λσ\nσ+1/ba∇dblu(t)/ba∇dbl2σ+2\nL2σ+2(Rd)dt−2λ2σ\n2−σdGM(u(t))1+2σ\n2−σddt\n+CM(u(t))1+2σ\n2−σddt+C/ba∇dblΦ/ba∇dbl2+4σ\n2−σd\nLHS(U;H)dt+1\n2/ba∇dbl∇Φ/ba∇dbl2\nLHS(U;H)dt\n−/summationdisplay\njRe/an}b∇acketle{t∆u(t)+|u(t)|2σu(t),Φej/an}b∇acket∇i}htdWj(t)\n+2(1+2σ\n2−σd)GM(u(s))2σ\n2−σdRe/an}b∇acketle{tu(t),ΦdW(t)/an}b∇acket∇i}ht.\nSince (1−2\n2−σd)≤0 by Assumption 2.3, we get\nσ\nσ+1/ba∇dblu/ba∇dbl2σ+2\nL2σ+2(Rd)−4σ\n2−σdGM(u)1+2σ\n2−σd\n≤\n(2.12)σ\n2/ba∇dbl∇u/ba∇dbl2\nH+2σ(1−2\n2−σd)GM(u)1+2σ\n2−σd\n≤σ\n2/ba∇dbl∇u/ba∇dbl2\nH≤\n(2.13)2σ˜H(u).\n10Then\n(2.32)d˜H(u(t))+2(1−σ)λ˜H(u(t))dt\n≤/parenleftBig\nCM(u(t))1+2σ\n2−σd+C/ba∇dblΦ/ba∇dbl2+4σ\n2−σd\nLHS(U;H)+1\n2/ba∇dbl∇Φ/ba∇dbl2\nLHS(U;H)/parenrightBig\ndt\n−/summationdisplay\njRe/an}b∇acketle{t∆u(t)+|u(t)|2σu(t),Φej/an}b∇acket∇i}htdWj(t)\n+2(1+2σ\n2−σd)GM(u(s))2σ\n2−σdRe/an}b∇acketle{tu(t),ΦdW(t)/an}b∇acket∇i}ht.\nSo considering the mathematical expectation we obtain\nd\ndtE˜H(u(t))+2(1−σ)λE˜H(u(t))\n≤CE[M(u(t))1+2σ\n2−σd]+C/ba∇dblΦ/ba∇dbl2+4σ\n2−σd\nLHS(U;H)+1\n2/ba∇dbl∇Φ/ba∇dbl2\nLHS(U;H)\n≤Ce−λ(1+2σ\n2−σd)tM(u0)1+2σ\n2−σd+C/ba∇dblΦ/ba∇dbl2(1+2σ\n2−σd)\nLHS(U;H)λ−(1+2σ\n2−σd)\n+C/ba∇dblΦ/ba∇dbl2+4σ\n2−σd\nLHS(U;H)+1\n2/ba∇dbl∇Φ/ba∇dbl2\nLHS(U;H)(2.33)\nthanks to ( 2.16). By means of the Gronwall lemma, setting a= min(2 −2σ,1+\n2σ\n2−σd)>0 we obtain\nE˜H(u(t))≤e−2(1−σ)λt˜H(u0)+e−aλtλ−1CM(u0)1+2σ\n2−σd+Cφ2λ−1\nwhereφ2=φ2(d,σ,λ,1,Φ) is equal to\n(2.34) /ba∇dblΦ/ba∇dbl2(1+2σ\n2−σd)\nLHS(U;H)/parenleftBig\nλ−1−2σ\n2−σd+1/parenrightBig\n+/ba∇dbl∇Φ/ba∇dbl2\nLHS(U;H).\nThis proves ( 2.21) form= 1.\nForm>1, we have by Itˆ o formula\n(2.35)d˜H(u(t))m≤m˜H(u(t))m−1d˜H(u(t))+m(m−1)\n2˜H(u(t))m−22r(t)dt,\nwhere\nr(t) =∞/summationdisplay\nj=1[Re/an}b∇acketle{t∆u(t)+|u(t)|2σu(t),Φej/an}b∇acket∇i}ht]2+4G2(1+2σ\n2−σd)2M(u(t))4σ\n2−σd∞/summationdisplay\nj=1[Re/an}b∇acketle{tu(t),Φej/an}b∇acket∇i}ht]2.\nKeeping in mind the previous estimates we get\nr(t)/lessorsimilar/ba∇dbl∇u(t)/ba∇dbl2\nH/ba∇dblΦ/ba∇dbl2\nLHS(U;V)+/ba∇dblu(t)/ba∇dbl2(2σ+1)\nL2σ+2(Rd)/ba∇dblΦ/ba∇dbl2\nLHS(U;V)\n+4G2(1+2σ\n2−σd)2M(u(t))1+4σ\n2−σd/ba∇dblΦ/ba∇dbl2\nLHS(U;H).\nNow we use ( 2.13), i.e./ba∇dbl∇u(t)/ba∇dbl2\nH≤4˜H(u), and by means of ( 2.7) we get\n/ba∇dblu/ba∇dbl2(2σ+1)\nL2σ+2(Rd)≤ǫ\n4/ba∇dbl∇u/ba∇dbl22σ+1\nσ+1\nH+Cǫ,σM(u)2σ+1\nσ+1(1+2σ\n2−σd)\n≤ǫ˜H(u)2σ+1\nσ+1+Cǫ,σM(u)2σ+1\nσ+1(1+2σ\n2−σd)\n11for anyǫ>0. Thus\n˜H(u(t))m−2r(t)/lessorsimilar˜H(u(t))m−1/ba∇dblΦ/ba∇dbl2\nLHS(U;V)+˜H(u(t))m−1\nσ+1/ba∇dblΦ/ba∇dbl2\nLHS(U;V)\n+˜H(u(t))m−2M(u)2σ+1\nσ+1(1+2σ\n2−σd)/ba∇dblΦ/ba∇dbl2\nLHS(U;V)\n+˜H(u(t))m−2M(u(t))1+4σ\n2−σd/ba∇dblΦ/ba∇dbl2\nLHS(U;H).\nIn (2.35) we insert this estimate and the previous estimates for d˜H(u(t)),\nintegrate in time, take the mathematical expectation to get rid of t he stochastic\nintegrals and use Young inequality; hence we obtain\nd\ndtE[˜H(u(t))m]+2m(1−σ)λE[˜H(u(t))m]≤m(1−σ)λE[˜H(u(t))m]\n+Cλ1−mE[M(u(t))m(1+2σ\n2−σd)]\n+Cλ1−m\n2/parenleftBig\n/ba∇dblΦ/ba∇dblm\nLHS(U;V)E[M(u(t))m\n22σ+1\nσ+1(1+2σ\n2−σd)]+/ba∇dblΦ/ba∇dblm\nLHS(U;H)E[M(u(t))m\n2(1+4σ\n2−σd)]/parenrightBig\n+C/parenleftbigg\n/ba∇dblΦ/ba∇dbl2\nLHS(U;V)+/ba∇dblΦ/ba∇dbl2+4σ\n2−σd\nLHS(U;H)/parenrightbiggm\nλ1−m+/ba∇dblΦ/ba∇dbl2m(σ+1)\nLHS(U;V)λ1−m(σ+1).\nBearing in mind the estimates ( 2.16) for the mass, we get an inequality for\nE[˜H(u(t))m] thanks to Gronwall lemma. Then a repeated use of the Young\ninequality with long but elementary calculations provides that there e xist a\nconstanta=a(d,σ)>0 and a smooth function φ2=φ2(d,σ,λ,m, Φ), strictly\ndecreasing w.r.t. λ, such that\nE˜H(u(t))m≤e−maλt/parenleftBig\n˜H(u0)m+C(λ−m+λ−m−1\n2)M(u0)m(1+2σ\n2−σd)/parenrightBig\n+Cφm\n2λ−m\nfor anyt≥0.\nMerging the results for the mass and the energy, we obtain the res ult for the\nV-norm. Indeed /ba∇dblu/ba∇dbl2\nV=/ba∇dbl∇u/ba∇dbl2\nH+/ba∇dblu/ba∇dbl2\nHand\n/ba∇dbl∇u/ba∇dbl2\nH= 2H(u)+α\nσ+1/ba∇dblu/ba∇dbl2σ+2\nL2σ+2(Rd).\nForα=−1 we trivially get\n/ba∇dblu/ba∇dbl2\nV≤2H(u)+M(u).\nForα= 1, we have from ( 2.13)\n/ba∇dblu/ba∇dbl2\nV≤4˜H(u)+M(u).\nHereφ1andφ2are the functions appearing in Proposition 2.8.\nCorollary 2.9. Letd≥2andu0∈V. Under Assumptions 2.2and2.3, for\neverym≥1we have the following estimates:\ni)whenα=−1\n(2.36)E[/ba∇dblu(t)/ba∇dbl2m\nV]/lessorsimilare−mλt[H(u0)m+M(u0)m]+[φ1+/ba∇dblΦ/ba∇dbl2\nLHS(U;H)]mλ−m\n12for anyt≥0;\nii)whenα= 1, there is a positive constant a=a(d,σ)such that\n(2.37)\nE[/ba∇dblu(t)/ba∇dbl2m\nV]/lessorsimilare−maλt[˜H(u0)m+(λ−m+λ−m−1\n2)M(u0)m(1+2σ\n2−σd)+M(u0)m]\n+[φ2+/ba∇dblΦ/ba∇dbl2\nLHS(U;H)]mλ−m\nfor anyt≥0.\nThe constant providing the above estimates /lessorsimilardepends on m,σanddbut\nnot onλ.\n3 Regularity results for the solution\nForthesolutionofequation( 2.3)weknowthat u∈C([0,+∞);V)a.s. ifu0∈V.\nNow we look for the L∞(Rd)-space regularity of the paths. When d= 1, this\nfollows directly from the Sobolev embedding H1(R)⊂L∞(R). But such an\nembedding does not hold for d>1. However for d= 2 ord= 3 one can obtain\ntheL∞(Rd)-regularity by means of the deterministic and stochastic Strichar tz\nestimates.\nLetφ1andφ2be the functions appearing in Proposition 2.8.\nProposition 3.1. Letd= 2ord= 3. In addition to the Assumptions 2.2and\n2.3we suppose that σ<1+√\n17\n4whend= 3.\nGiven any finite T >0andu0∈Vthe solution of equation (2.3)is in\nL2σ(Ω;L2σ(0,T;L∞(Rd))). Moreover there exist positive constants b1=b1(σ),\nb2=b2(σ)andC=C(σ,d,T)such that\n(3.1)E/ba∇dblu/ba∇dbl2σ\nL2σ(0,T;L∞(Rd))\n≤C/parenleftBig\n/ba∇dblu0/ba∇dbl2σ\nV+λ−b1ψ(u0)σ(2σ+1)+φσ(2σ+1)\n3λ−σ(2σ+1)+/ba∇dblΦ/ba∇dbl2σ\nLHS(U;V)/parenrightBig\n.\nwhere\n(3.2)ψ(u0) =/braceleftBigg\nH(u0)+M(u0), α =−1\n˜H(u0)+(λ−1+λ−b2)M(u0)1+2σ\n2−σd+M(u0), α= 1\nand\n(3.3)φ3(d,σ,λ,Φ) =/braceleftBigg\nφ1(σ,λ,Φ)+/ba∇dblΦ/ba∇dbl2\nLHS(U;H), α =−1\nφ2(d,σ,λ,σ(2σ+1),Φ)+/ba∇dblΦ/ba∇dbl2\nLHS(U;H), α= 1\nsoλ/ma√sto→φ3(d,σ,λ,Φ)is a strictly decreasing function.\nProof.Firstletusconsider d= 2. Werepeatedlyusetheembedding H1,q(R2)⊂\nL∞(R2) valid for any q >2. So our target is to prove the estimate for the\nL2σ(Ω;L2σ(0,T;H1,q(R2)))-norm of ufor someq>2.\nWe introduce the operator Λ := −iA0+λ. It generates the semigroup\ne−iΛt=e−λteiA0t,t≥0.\n13Let us fixT >0. We write equation ( 2.3) in the mild form (see [ DBD03])\niu(t) = ie−Λtu0+/integraldisplayt\n0e−Λ(t−s)Fα(u(s))ds+i/integraldisplayt\n0e−Λ(t−s)ΦdW(s)\n=:I1(t)+I2(t)+I3(t) (3.4)\nand estimate\nE/ba∇dblIi/ba∇dbl2σ\nL2σ(0,T;H1,q(R2)), i= 1,2,3\nfor someq>2.\nFor the estimate of I1we set\n(3.5) q=/braceleftBigg\n2σ\nσ−1ifσ>1\n6\n3−σif 0<σ≤1\nNotice that q >2. Now, before using the homogeneous Strichartz inequality\n(A.1) we neglect the term e−λt, sincee−λt≤1. First, assuming σ>1 we work\nwith the admissible Strichartz pair (2 σ,2σ\nσ−1) and get\n/ba∇dblI1/ba∇dbl\nL2σ(0,T;H1,2σ\nσ−1(R2))=/vextenddouble/vextenddouble/vextenddoublee−λ·eiA0·A1/2\n1u0/vextenddouble/vextenddouble/vextenddouble\nL2σ(0,T;L2σ\nσ−1(R2))\n≤/vextenddouble/vextenddouble/vextenddoubleeiA0·A1/2\n1u0/vextenddouble/vextenddouble/vextenddouble\nL2σ(0,T;L2σ\nσ−1(R2))\n/lessorsimilar/ba∇dblA1/2\n1u0/ba∇dblL2(R2)=/ba∇dblu0/ba∇dblV\nFor smaller values, i.e. 0 <σ≤1, we choose ˜ σ=3\nσ>2>σso2˜σ\n˜σ−1=6\n3−σand\n/ba∇dblI1/ba∇dbl\nL2σ(0,T;H1,2˜σ\n˜σ−1(R2))/lessorsimilar/ba∇dblI1/ba∇dbl\nL2˜σ(0,T;H1,2˜σ\n˜σ−1(R2))/lessorsimilar/ba∇dblu0/ba∇dblV\nby the previous computations.\nFor the estimate of I2, we use the Strichartz inequality ( A.2) and then the\nestimate from Lemma C.1on the nonlinearity. We bear in mind the notation γ′\nfor the conjugate exponent of γ∈(1,∞), i.e.1\nγ+1\nγ′= 1. First, consider σ>1;\nthe pair (2σ,2σ\nσ−1) is admissible. Then\n/ba∇dblI2/ba∇dbl\nL2σ(0,T;H1,2σ\nσ−1(R2))=/ba∇dblA1/2\n1I2/ba∇dbl\nL2σ(0,T;L2σ\nσ−1(R2))\n/lessorsimilar/ba∇dblA1/2\n1Fα(u)/ba∇dblL4\n3(0,T;L4\n3(R2))by (A.2)\n=/ba∇dblFα(u)/ba∇dblL4\n3(0,T;H1,4\n3(R2))\n/lessorsimilar/ba∇dblu/ba∇dbl2σ+1\nL4\n3(2σ+1)(0,T;V)by (C.1) and (C.2)\nFor 0<σ≤1 we proceed in a similar way; consideringthe admissible Strichartz\npair (2+σ,2+4\nσ) we have\n/ba∇dblI2/ba∇dblL2σ(0,T;H1,2+4\nσ(R2))/lessorsimilar/ba∇dblI2/ba∇dblL2+σ(0,T;H1,2+4\nσ(R2))\n=/ba∇dblA1/2\n1I2/ba∇dblL2+σ(0,T;L2+4\nσ(R2))\n/lessorsimilar/ba∇dblA1/2\n1Fα(u)/ba∇dblLγ′(0,T;Lr′(R2))by (A.2)\n=/ba∇dblFα(u)/ba∇dblLγ′(0,T;H1,r′(R2))\n14where (r,γ) is an admissible Strichartz pair. According to ( C.1) we choose\n(3.6) (1 ,2)∋r′=/braceleftBigg\n2\n1+2σ,0<σ<1\n2\n4\n3,1\n2≤σ≤1\nHence\n(3.7) γ′=2r′\n3r′−2=/braceleftBigg\n1\n1−σ,0<σ<1\n2\n4\n3,1\n2≤σ≤1\nIn this way by means of the estimate ( C.2) of the polynomial nonlinearity\n/ba∇dblFα(u))/ba∇dblH1,r′(R2)/lessorsimilar/ba∇dblu/ba∇dbl1+2σ\nVwe obtain\n/ba∇dblI2/ba∇dblL2σ(0,T;H1,2+4\nσ(R2))/lessorsimilar/ba∇dblu/ba∇dbl2σ+1\nLγ′(2σ+1)(0,T;V).\nSumming up, we have shown that for any σ >0 there exists q >2 andγ′\nsuch that\n(3.8) E/ba∇dblI2/ba∇dbl2σ\nL2σ(0,T;H1,q(R2))/lessorsimilarE/parenleftBigg/integraldisplayT\n0/ba∇dblu(t)/ba∇dblγ′(2σ+1)\nVdt/parenrightBigg2σ\nγ′\n.\nBearing in mind Corollary 2.9, we get the second and third terms in the r.h.s.\nof (3.1). The details are given in the Appendix D.1.\nIt remains to estimate the term I3. We choose qas in (3.5). Using the\nstochastic Strichartz estimate ( A.3), we get for σ>1\nE/ba∇dblI3/ba∇dbl2σ\nL2σ(0,T;H1,2σ\nσ−1(R2))=E/ba∇dblA1/2\n1I3/ba∇dbl2σ\nL2σ(0,T;L2σ\nσ−1(R2))\n/lessorsimilar/ba∇dblA1/2\n1Φ/ba∇dbl2σ\nLHS(U;H)=/ba∇dblΦ/ba∇dbl2σ\nLHS(H;V).\nFor smaller values of σ, we proceed as before for I1.\nNow consider d= 3. The additional assumption on σappears because of\nthe stronger conditions on the parameters given later on.\nForq≥1 we haveHθ,q(R3)⊂L∞(R3) whenθq >3. So for each Iiin (3.4)\nwe look for an estimate in the norm L2σ(0,T;Hθ,q(R3)) for some parameters\nθq>3.\nWe estimate I1for any 0<σ<2. When 0 <σ≤1 we consider the admis-\nsible Strichartz pair (2 ,6). By means of the homogeneous Strichartz estimate\n(A.1) we proceed as before\n/ba∇dblI1/ba∇dblL2σ(0,T;H1,6(R3))/lessorsimilar/ba∇dblI1/ba∇dblL2(0,T;H1,6(R3))\n=/vextenddouble/vextenddouble/vextenddoublee−λ·eiA0·A1/2\n1u0/vextenddouble/vextenddouble/vextenddouble\nL2(0,T;L6(R3))\n≤/vextenddouble/vextenddouble/vextenddoubleeiA0·A1/2\n1u0/vextenddouble/vextenddouble/vextenddouble\nL2(0,T;L6(R3))\n/lessorsimilar/ba∇dblA1/2\n1u0/ba∇dblL2(R3)=/ba∇dblu0/ba∇dblV.\nWhenσ>1 we work with the admissible Strichartz pair (2 σ,6σ\n3σ−2) and get\n/ba∇dblI1/ba∇dbl\nL2σ(0,T;H1,6σ\n3σ−2(R3))/lessorsimilar/ba∇dblA1/2\n1u0/ba∇dblL2(R3)=/ba∇dblu0/ba∇dblV;\n15since6σ\n3σ−2>3 for 1<σ<2, we obtain the L∞(R3)-norm estimate.\nThe estimate for I2is more involved and therefore we postpone it to the\nAppendix D.2.\nIt remains to estimate the term I3. For any σ >0 we use the H¨ older\ninequality and the stochastic Strichartz estimate ( A.3) for the admissible pair\n(2+σ2\n2,64+σ2\n4+3σ2); therefore\nE/ba∇dblI3/ba∇dbl2σ\nL2σ(0,T;H1,64+σ2\n4+3σ2(R3))/lessorsimilarTE/ba∇dblI3/ba∇dbl2σ\nL2+σ2\n2(0,T;H1,64+σ2\n4+3σ2(R3))\n/lessorsimilar/ba∇dblΦ/ba∇dbl2σ\nLHS(U;V).\nNoticethattherestriction σ<1+√\n17\n4onthepowerofthe nonlinearityaffects\nonly the defocusing case.\nWe conclude this section by remarking that there is no similar result fo r\nd≥4.\nRemark 3.2. For larger dimension, there is no result similar to those in t his\nsection. Indeed, if one looks for u∈L2σ(0,T;H1,q(Rd))⊂L2σ(0,T;L∞(Rd))it\nis necessary that\nq>d\nin order to have H1,q(Rd)⊂L∞(Rd). Already the estimate for I1does not\nhold under this assumption. Indeed the homogeneous Stricha rtz estimate (A.1)\nprovides\nI1∈C([0,T];H1(Rd))∩L2σ(0,T;H1,q(Rd))\nif\n1\nσ=d/parenleftbigg1\n2−1\nq/parenrightbigg\nand 2≤q≤2d\nd−2.\nSince2d\nd−2≤4ford≥4, the latter condition q≤2d\nd−2and the condition q >d\nare incompatible for d≥4.\nLet us notice that also in the deterministic setting the resu lts on the attractors\nare known for d≤3, see [L95].\n4 The support of the invariant measures\nNowweshowsomemorepropertiesonthe invariantmeasuresandth eirsupport.\nIn dimension d= 2 andd= 3, thanks to the regularity results of Section 3we\nprovide an estimate for the moments in the VandL∞(Rd)-norm.\nProposition 4.1. Letd= 2ord= 3and the Assumptions 2.2and2.3hold.\nLetµbe an invariant measure for equation (2.3). Then for any finite m≥1we\nhave\n(4.1)/integraldisplay\n/ba∇dblx/ba∇dbl2m\nVdµ(x)≤φm\n4λ−m\nfor some smooth function φ4=φ4(d,σ,λ,m, Φ)which is strictly decreasing\nw.r.t.λ.\n16Moreover, supposing in addition that σ<1+√\n17\n4whend= 3, we have\n(4.2)/integraldisplay\n/ba∇dblx/ba∇dbl2σ\nL∞dµ(x)≤φ5(λ),\nwhereφ5(λ)is a smooth decreasing function, depending also on the param eters\nd,σand on/ba∇dblΦ/ba∇dblLHS(U;V),\nProof.As far as ( 4.1) is concerned, we define the bounded mapping Ψ konVas\nΨk(x) =/braceleftBigg\n/ba∇dblx/ba∇dbl2m\nV,if/ba∇dblx/ba∇dblV≤k\nk2m,otherwise\nBy the invariance of µand the boundedness of Ψ k, we have\n(4.3)/integraldisplay\nVΨkdµ=/integraldisplay\nVPsΨkdµ∀s>0.\nSo\nPsΨk(x) =E[Ψk(u(s;x))]≤E/ba∇dblu(s;x)/ba∇dbl2m\nV.\nMoreover, from Corollary 2.9we get an estimate for E/ba∇dblu(s;x)/ba∇dbl2m\nV, and letting\ns→+∞the first terms in ( 2.36) and (2.37) vanish so we get\nlim\ns→+∞PsΨk(x)≤φm\n4λ−m∀x∈V\nwhere\nφ4=/braceleftBigg\nφ1+/ba∇dblΦ/ba∇dbl2\nLHS(U;H),forα=−1\nφ2+/ba∇dblΦ/ba∇dbl2\nLHS(U;H),forα= 1\nThe same holds for the integral, that is\nlim\ns→+∞/integraldisplay\nVPsΨk(x)dµ(x)≤φm\n4λ−m\nby the Bounded Convergence Theorem. From ( 4.3) we get\n/integraldisplay\nVΨkdµ≤φm\n4λ−m\nas well. Since Ψ kconverges pointwise and monotonically from below to /ba∇dbl·/ba∇dbl2m\nV,\nthe Monotone Convergence Theorem yields ( 4.1).\nAs far as ( 4.2) is concerned, we consider the estimate ( 3.1) forT= 1 and\nset˜Ψ(u) =/ba∇dblu/ba∇dbl2σ\nL∞(Rd); this defines a mapping ˜Ψ :V→R+∪ {+∞}. Its\napproximation ˜Ψk:V→R+, given by\n˜Ψk(u) =/braceleftBigg\n/ba∇dblu/ba∇dbl2σ\nL∞(Rd),if/ba∇dblu/ba∇dblL∞(Rd)≤k\nk2σ, otherwise\ndefines a bounded mapping ˜Ψk:V→R+.\nIt obviously holds\n/integraldisplay\nV˜Ψkdµ=/integraldisplay1\n0/parenleftbigg/integraldisplay\nV˜Ψkdµ/parenrightbigg\nds.\n17By the invariance of µand the boundedness of ˜Ψk, it also holds\n/integraldisplay\nV˜Ψkdµ=/integraldisplay\nVPs˜Ψkdµ∀s>0.\nThus, by Fubini-Tonelli Theorem, since ˜Ψk(u) =/ba∇dblu/ba∇dbl2σ\nL∞(Rd)∧k2σ≤ /ba∇dblu/ba∇dbl2σ\nL∞(Rd),\nwe get by Proposition 3.1\n/integraldisplay\nV˜Ψkdµ=/integraldisplay1\n0/integraldisplay\nVPs˜Ψkdµds=/integraldisplay\nV/integraldisplay1\n0E/bracketleftBig\n˜Ψk(u(s;x))/bracketrightBig\ndsdµ(x)\n≤/integraldisplay\nVE/integraldisplay1\n0/ba∇dblu(s;x)/ba∇dbl2σ\nL∞(Rd)dsdµ(x)\n≤C/integraldisplay\nV/parenleftBig\n/ba∇dblx/ba∇dbl2σ\nV+λ−bψ(x)σ(2σ+1)+φσ(2σ+1)\n3λ−σ(2σ+1)+/ba∇dblΦ/ba∇dbl2σ\nLHS(U;V)/parenrightBig\ndµ(x)\nfrom (3.1).\nKeeping in mind ( 3.2) and (4.1) we obtain the existence of a smooth decreasing\nfunctionφ5(λ), depending alsoon the parameters d,σand on/ba∇dblΦ/ba∇dblLHS(U;V), such\nthat /integraldisplay\nV˜Ψkdµ≤φ5(λ)\nfor anyk.\nSince˜Ψkconvergespointwise and monotonicallyfrom belowto ˜Ψ, the Mono-\ntone ConvergenceTheorem yields the same bound for/integraltext\nV/ba∇dblx/ba∇dbl2σ\nL∞(Rd)dµ(x). This\nproves (4.2).\n5 Uniqueness of the invariant measure for suffi-\nciently large damping\nWe will prove that, if the damping coefficient λis sufficiently large, then the\ninvariant measure is unique.\nTheorem 5.1. Letd= 2ord= 3. In addition to the Assumptions 2.2and2.3\nwe suppose that σ<1+√\n17\n4whend= 3.\nIf\n(5.1) λ>2φ5(λ)\nwhereφ5is the function appearing in Proposition 4.1, then there exists a unique\ninvariant measure for equation (2.3).\nProof.The existence of an invariant measure comes from Theorem 2.6. Now\nwe prove uniqueness by means of a reductio ad absurdum. Let us su ppose that\nthere exists more than one invariant measure. In particular there exist two\ndifferent ergodic invariant measures µ1andµ2. For both of them Proposition\n4.1holds. Fix either i= 1 ori= 2 and consider any f∈L1(µi). Then by the\nBirkhoff Ergodic Theorem (see, e.g., [ CFS82]) forµi-a.e.xi∈Vwe have\n(5.2) lim\nt→+∞1\nt/integraldisplayt\n0f(u(s;xi))ds=/integraldisplay\nVf dµiP−a.s.\n18Hereu(t;x) is the solution at time t, with initial value u(0) =x∈V.\nNow fix two initial data x1andx2belonging, respectively, to the support of\nthe measure µ1andµ2. We have\n/integraldisplay\nVf dµ1−/integraldisplay\nVf dµ2= lim\nt→+∞1\nt/integraldisplayt\n0[f(u(s;x1))−f(u(s;x2))]ds\nP-a.s.. Taking any arbitrary fin the set G0defined in ( B.2), we get\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nVf dµ1−/integraldisplay\nVf dµ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Llim\nt→+∞1\nt/integraldisplayt\n0/ba∇dblu(s;x1)−u(s;x2)/ba∇dblHds.\nIf we prove that\n(5.3) lim\nt→+∞/ba∇dblu(t;x1)−u(t;x2)/ba∇dblH= 0P−a.s.,\nthen we conclude that /integraldisplay\nVf dµ1−/integraldisplay\nVf dµ2= 0\nsoµ1=µ2thanks to Lemma B.1. So let us focus on the limit ( 5.3).\nWith a short notation we write ui(t) =u(t;xi). Then consider the difference\nw=u1−u2fulfilling\n/braceleftBigg\nd\ndtw(t)−iA0w(t)+iFα(u1(t))−iFα(u2(t))+γw(t) = 0\nw(0) =x1−x2\nso\n1\n2d\ndt/ba∇dblw(t)/ba∇dbl2\nH+γ/ba∇dblw(t)/ba∇dbl2\nH≤/integraldisplay\nRd/vextendsingle/vextendsingle/vextendsingle[|u1(t)|2σu1(t)−|u2(t)|2σu2(t)]w(t)/vextendsingle/vextendsingle/vextendsingledy.\nUsing the elementary estimate\n||u1|2σu1−|u2|2σu2| ≤Cσ[|u1|2σ+|u2|2σ]|u1−u2|,\nwe bound the nonlinear term in the r.h.s. as\n/integraldisplay\nRd|[|u1|2σu1−|u2|2σu2]w|dy≤[/ba∇dblu1/ba∇dbl2σ\nL∞(Rd)+/ba∇dblu2/ba∇dbl2σ\nL∞(Rd)]/ba∇dblw/ba∇dbl2\nL2(Rd).\nTherefore\nd\ndt/ba∇dblw(t)/ba∇dbl2\nH+2λ/ba∇dblw(t)/ba∇dbl2\nH≤2/parenleftBig\n/ba∇dblu1(t)/ba∇dbl2σ\nL∞(Rd)+/ba∇dblu2(t)/ba∇dbl2σ\nL∞(Rd)/parenrightBig\n/ba∇dblw(t)/ba∇dbl2\nH.\nGronwall inequality gives\n/ba∇dblw(t)/ba∇dbl2\nH≤ /ba∇dblw(0)/ba∇dbl2\nHe−2λt+2/integraltextt\n0/parenleftBig\n/bardblu1(s)/bardbl2σ\nL∞(Rd)+/bardblu2(s)/bardbl2σ\nL∞(Rd)/parenrightBig\nds\nthat is\n(5.4) /ba∇dblw(t)/ba∇dbl2\nH≤ /ba∇dblx1−x2/ba∇dbl2\nHe−2t/bracketleftBig\nλ−1\nt/integraltextt\n0/parenleftBig\n/bardblu1(s)/bardbl2σ\nL∞(Rd)+/bardblu2(s)/bardbl2σ\nL∞(Rd)/parenrightBig\nds/bracketrightBig\n.\nThis is a pathwise estimate.\n19We know from Proposition 4.1thatf(x) =/ba∇dblx/ba∇dbl2σ\nL∞(Rd)∈L1(µi); therefore\n(5.2) becomes\nlim\nt→+∞1\nt/integraldisplayt\n0/ba∇dblu(s;xi)/ba∇dbl2σ\nL∞(Rd)ds=/integraldisplay\nV/ba∇dblx/ba∇dbl2σ\nL∞(Rd)dµi(x)≤φ5(λ)\nP-a.s., for either i= 1 ori= 2. Therefore, if\nλ>2φ5(λ),\nthe exponential term in the r.h.s. of ( 5.4) vanishes as t→+∞. This proves\n(5.3) and concludes the proof.\nA Strichartz estimates\nIn this section we recall the deterministic and stochastic Strichart z estimates on\nRd.\nDefinition A.1. We say that a pair (p,r)is admissible if\n2\np+d\nr=d\n2and(p,r)/ne}ationslash= (2,∞)\nand \n\n2≤r≤2d\nd−2ford≥3\n2≤r<∞ford= 2\n2≤r≤ ∞ ford= 1\nIf (p,r) is an admissible pair, then 2 ≤p≤ ∞.\nGiven 1≤γ≤ ∞, we denote by γ′its conjugate exponent, i.e.1\nγ+1\nγ′= 1.\nLemma A.2. Let(p,r)be an admissible pair of exponents. Then the following\nproperties hold\ni)for everyϕ∈L2(Rd)the function t/ma√sto→eitA0ϕbelongs toLp(R;Lr(Rd))∩\nC(R;L2(Rd)). Furthermore, there exists a constant Csuch that\n(A.1) /ba∇dblei·A0ϕ/ba∇dblLp(R;Lr(Rd))≤C/ba∇dblϕ/ba∇dblL2(Rd),∀ϕ∈L2(Rd).\nii)LetIbe an interval of Rand0∈J=I. If(γ,ρ)is an admissible pair\nandf∈Lγ′(I;Lρ′(Rd)), then the function t/ma√sto→Gf(t) =/integraltextt\n0ei(t−s)A0f(s)ds\nbelongs toLq(I;Lr(Rd))∩C(J;L2(Rd)). Furthermore, there exists a constant\nC, independent of I, such that\n(A.2) /ba∇dblGf/ba∇dblLp(I;Lr(Rd))≤C/ba∇dblf/ba∇dblLγ′(I;Lρ′(Rd)),∀f∈Lγ′(I;Lρ′(Rd)).\nProof.See [C03, Proposition 2.3.3].\nLemma A.3 (stochastic Strichartz estimate) .Let(p,r)be an admissible pair.\nThen for any a∈(1,∞)andT <∞there exists a constant Csuch that\n(A.3)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplay·\n0ei(·−s)A0Ψ(s)dW(s)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nLa(Ω,Lp(0,T;Lr(Rd)))≤C/ba∇dblΨ/ba∇dblL2(0,T;LHS(U,L2(Rd)))\nfor anyΨ∈L2(0,T;LHS(U,L2(Rd))).\nProof.See [H18, Proposition 2].\n20B Determining sets\nThe set\n(B.1) G1=/braceleftBigg\nf∈Cb(V) : sup\nu/negationslash=vf(u)−f(v)\n/ba∇dblu−v/ba∇dblV<∞/bracerightBigg\nis a determining set for measures on V(see, e.g., [ B13] Theorem 1.2). This\nmeans that given two probability measures µ1andµ2onVwe have\n/integraldisplay\nVf dµ1=/integraldisplay\nVf dµ2∀f∈ G1=⇒µ1=µ2.\nFollowing Remark 2.2 in [ GHMR17 ] we can consider as a determining set for\nmeasures on Vthe set\n(B.2) G0=/braceleftBigg\nf∈Cb(V) : sup\nu/negationslash=vf(u)−f(v)\n/ba∇dblu−v/ba∇dblH<∞/bracerightBigg\ninvolving the weaker H-norm instead of the V-norm. Indeed we have\nLemma B.1. Letµ1andµ2be two invariant measures. If\n/integraldisplay\nVf dµ1=/integraldisplay\nVf dµ2∀f∈ G0,\nthenµ1=µ2.\nProof.We show the proof since we work in Rdwhereas [ GHMR17 ] deals with\na bounded domain.\nSetPNxto be the element whose Fourier transform is 1 |ξ|≤NF(x); hence\n/ba∇dblPNx/ba∇dblV≤√\n1+N2/ba∇dblx/ba∇dblH. Now we show that any function f∈ G1can be\napproximated by a function fN∈ G0by settingfN(x) =f(PNx). Indeed\n|fN(x)−fN(y)| ≤L/ba∇dblPNx−PNy/ba∇dblV≤L/radicalbig\n1+N2/ba∇dblx−y/ba∇dblH\nBy assumption we know that\n/integraldisplay\nVfNdµ1=/integraldisplay\nVfNdµ2\nTaking the limit as N→+∞, by the bounded convergence theorem we get the\nsame identity for f∈ G1. Henceµ1=µ2.\nC Estimate of the nonlinearity\nWe consider F(u) =|u|2σu.\nLemma C.1. Letd= 2. For anyσ>0, ifp∈(1,2)is defined as\n(C.1) p=/braceleftBigg\n2\n2σ+1,0<σ<1\n2\n4\n3, σ ≥1\n2\n21then\n(C.2) /ba∇dblF(u)/ba∇dblH1,p(R2)/lessorsimilar/ba∇dblu/ba∇dbl2σ+1\nH1(R2)∀u∈H1(R2).\nLetd= 3. For anyσ∈(0,3\n2]we have\n(C.3) /ba∇dblF(u)/ba∇dbl\nH1,6\n2σ+3(R3)/lessorsimilar/ba∇dblu/ba∇dbl2σ+1\nH1(R3)∀u∈H1(R3)\nand for any σ∈[1,3\n2]we have\n(C.4) /ba∇dblF(u)/ba∇dblH2−σ,6\n5(R3)/lessorsimilar/ba∇dblu/ba∇dbl2σ+1\nH1(R3)∀u∈H1(R3).\nProof.We start with the case d= 2. To estimate the H1,p-norm ofFit is\nenough to deal with /ba∇dblF/ba∇dblLp(Rd)and/ba∇dbl∂F/ba∇dblLp(Rd). We compute\n(C.5)∂F(u) =σ|u|2σ−2(¯u∂u+u∂¯u)u+|u|2σ∂u,for an arbitrary u∈V,\nand thus |∂F(u)|/lessorsimilarσ|u|2σ|∂u|.\nWe have\n/ba∇dblF(u)/ba∇dblLp(Rd)=/ba∇dblu/ba∇dbl2σ+1\nL(2σ+1)p(Rd)(C.6)\nand the H¨ older inequality, for 1 ≤p<2, gives\n/ba∇dbl∂F(u)/ba∇dblLp(Rd)≤ /ba∇dbl|u|2σ/ba∇dbl\nL2p\n2−p(Rd)/ba∇dbl∂u/ba∇dblL2(Rd) (C.7)\n≤ /ba∇dblu/ba∇dbl2σ\nL4σp\n2−p(Rd)/ba∇dblu/ba∇dblV\nWe recall the Sobolev embedding\nH1(R2)⊂Lr(R2) for any 2 ≤r<∞.\nTherefore, if\n\n2≤(2σ+1)p\n2≤4σp\n2−p\nthen both the r.h.s. of ( C.6) and (C.7) can be estimated by a quantity involving\ntheH1(R2)-norm. The two latter inequalities are the same as\np≥2\n2σ+1\nsooneeasilyseesthat the choice( C.1) allowsto fulfil the tworequiredestimates,\ni.e./ba∇dblF(u)/ba∇dblLp(Rd)/lessorsimilar/ba∇dblu/ba∇dbl2σ+1\nVand/ba∇dbl∂F(u)/ba∇dblLp(Rd)/lessorsimilar/ba∇dblu/ba∇dbl2σ+1\nV. This proves ( C.2).\nFord= 3, first we show that for any σ∈(0,3\n2]\n(C.8) /ba∇dblF(u)/ba∇dblH1,p(R3)/lessorsimilar/ba∇dblu/ba∇dbl2σ+1\nH1(R3)∀u∈H1(R3)\nwithp=6\n2σ+3∈[1,2).\n22To this end we notice that the r.h.s. of ( C.6) and (C.7) are estimated by a\nquantity involving the H1(R3)-norm ifH1(R3)⊂L(2σ+1)p(R3) andH1(R3)⊂\nL4σp\n2−p(R3). Recalling the Sobolev embedding\nH1(R3)⊂Lr(R3) for any 2 ≤r≤6\nwe get the conditions\n2≤(2σ+1)p≤6 (equivalent to2\n2σ+1≤p≤6\n2σ+1) (C.9)\n2≤4σp\n2−p≤6 (equivalent to2\n2σ+1≤p≤6\n2σ+3) (C.10)\nNotice that ( C.10) is stronger than ( C.9); moreover ( C.10) has a solution p∈\n[1,2) only ifσ∈(0,3\n2]. Choosing\n(C.11) p=6\n2σ+3∈[1,2)\nwe fulfil all the requirements and so we have proved ( C.3).\nNow for 1 ≤σ≤3\n2there is the continuous embedding H1,6\n2σ+3(R3)⊆\nH2−σ,6\n5(R3). Hence from ( C.3) we get ( C.4).\nD Computations in the proof of Proposition 3.1\nD.1 From (3.8)to(3.1)\nFrom (3.8) we proceed as follows. We distinguish different values of the param-\neterσ.\n•σ∈(0,1\n4): wehaveγ′=1\n1−σ, so2σ\nγ′= 2σ(1−σ)<3\n8andγ′(2σ+1) =2σ+1\n1−σ<2.\nWith the H¨ older inequality twice\nE/parenleftBigg/integraldisplayT\n0/ba∇dblu(t)/ba∇dblγ′(2σ+1)\nVdt/parenrightBigg2σ\nγ′\n≤/parenleftBigg\nE/integraldisplayT\n0/ba∇dblu(t)/ba∇dblγ′(2σ+1)\nVdt/parenrightBigg2σ\nγ′\n/lessorsimilarT/parenleftBigg\nE/integraldisplayT\n0/ba∇dblu(t)/ba∇dbl2\nVdt/parenrightBigg2σ\nγ′γ′2σ+1\n2\nWe conclude by means of the estimate of Corollary 2.9form= 1; for instance\nin casei)\n/parenleftBigg\nE/integraldisplayT\n0/ba∇dblu(t)/ba∇dbl2\nVdt/parenrightBiggσ(2σ+1)\n/lessorsimilard,σ/parenleftBigg/integraldisplayT\n0/bracketleftBig\ne−λt[H(u0)+M(u0)]+[φ1+/ba∇dblΦ/ba∇dbl2\nLHS(U;H)]λ−1/bracketrightBig\ndt/parenrightBiggσ(2σ+1)\n/lessorsimilard,σ,T/parenleftBig\n[H(u0)+M(u0)]λ−1+[φ1+/ba∇dblΦ/ba∇dbl2\nLHS(U;H)]λ−1/parenrightBigσ(2σ+1)\n/lessorsimilard,σ,T[H(u0)+M(u0)]σ(2σ+1)λ−σ(2σ+1)+[φ1+/ba∇dblΦ/ba∇dbl2\nLHS(U;H)]σ(2σ+1)λ−σ(2σ+1)\n23•σ∈[1\n4,1\n2): wehaveγ′=1\n1−σ, so2σ\nγ′= 2σ(1−σ)≤1\n2andγ′(2σ+1) =2σ+1\n1−σ≥2.\nWith the H¨ older inequality\nE/parenleftBigg/integraldisplayT\n0/ba∇dblu(t)/ba∇dblγ′(2σ+1)\nVdt/parenrightBigg2σ\nγ′\n≤/parenleftBigg\nE/integraldisplayT\n0/ba∇dblu(t)/ba∇dblγ′(2σ+1)\nVdt/parenrightBigg2σ\nγ′\nand then we conclude by means of the estimate of Corollary 2.9for 2m=\nγ′(2σ+1); for instance in case ii)\n/parenleftBigg\nE/integraldisplayT\n0/ba∇dblu(t)/ba∇dblγ′(2σ+1)\nVdt/parenrightBigg2σ\nγ′\n/lessorsimilar/parenleftBig/integraldisplayT\n0e−aγ′\n2(2σ+1)λtdt[˜H(u0)γ′\n2(2σ+1)+(λ−γ′\n2(2σ+1)+λ−γ′\n2(2σ+1)−1\n2)M(u0)γ′\n2(2σ+1)(1+2σ\n2−σd)\n+M(u0)γ′\n2(2σ+1)]+T[φ2+/ba∇dblΦ/ba∇dbl2\nLHS(U;H)]γ′\n2(2σ+1)λ−γ′\n2(2σ+1)/parenrightBig2σ\nγ′\n/lessorsimilara,σ,Tλ−2σ(1−σ)/bracketleftbig˜H(u0)+(λ−1+λ−4σ−1\n4σ(1−σ)(2σ+1))M(u0)1+2σ\n2−σd+M(u0)/bracketrightbigσ(2σ+1)\n+[φ2+/ba∇dblΦ/ba∇dbl2\nLHS(U;H)]σ(2σ+1)λ−σ(2σ+1)\n•σ∈[1\n2,2\n3): we have γ′=4\n3, so2σ\nγ′=3\n2σ <1 andγ′(2σ+1) =4\n3(2σ+1)≥8\n3.\nSo we proceed as in the previous case.\n•σ≥2\n3: we haveγ′=4\n3, so2σ\nγ′≥1 andγ′(2σ+ 1)≥28\n9. With the H¨ older\ninequality\nE/parenleftBigg/integraldisplayT\n0/ba∇dblu(t)/ba∇dblγ′(2σ+1)\nVdt/parenrightBigg2σ\nγ′\n/lessorsimilarTE/integraldisplayT\n0/ba∇dblu(t)/ba∇dblγ′(2σ+1)2σ\nγ′\nVdt\nand then we conclude by means ofthe estimate ofCorollary 2.9form= 2σ(2σ+\n1).\nD.2 Estimate of I2whend= 3\nWe distinguish two ranges of values for σ.\n•For 0<σ≤1 we haveL2(0,T;L6(R3))⊆L2σ(0,T;L6(R3)). So we con-\nsider the admissible Strichartz pair (2 ,6) and get for any admissible Strichartz\npair (γ,r)\n/ba∇dblI2/ba∇dblL2σ(0,T;H1,6(R3))/lessorsimilar/ba∇dblI2/ba∇dblL2(0,T;H1,6(R3))\n=/ba∇dblA1/2\n1I2/ba∇dblL2(0,T;L6(R3))\n/lessorsimilar/ba∇dblA1/2\n1Fα(u)/ba∇dblLγ′(0,T;Lr′(R3))by (A.2)\n/lessorsimilar/ba∇dblFα(u)/ba∇dblLγ′(0,T;H1,r′(R3))\nThe parameters are such that γ′=4r′\n7r′−6. From the Definition A.1we have the\ncondition 2 ≤r≤6, equivalent to6\n5≤r′≤2. Choosing\nr′=6\n3+2σ,\n24we haver′∈[6\n5,2) when 0<σ≤1 andγ′=2\n2−σ∈(1,2]; thus we can use ( C.3)\nto estimate the nonlinearity Fα(u). Summing up, we have\n/ba∇dblI2/ba∇dblL2σ(0,T;L∞(R3))/lessorsimilar/ba∇dblI2/ba∇dblL2σ(0,T;H1,6(R3))/lessorsimilar/ba∇dblu/ba∇dbl2σ+1\nL22σ+1\n2−σ(0,T;V).\nHence\nE/ba∇dblI2/ba∇dbl2σ\nL2σ(0,T;L∞(R3))/lessorsimilarE/parenleftBigg/integraldisplayT\n0/ba∇dblu(t)/ba∇dbl2\n2−σ(2σ+1)\nVdt/parenrightBiggσ(2−σ)\n/lessorsimilar/parenleftBigg\nE/integraldisplayT\n0/ba∇dblu(t)/ba∇dbl22σ+1\n2−σ\nVdt/parenrightBiggσ(2−σ)\nby H¨ older inequality since σ(2−σ)≤1.\nFrom here, bearing in mind Corollary 2.9we conclude as in the previous sub-\nsection and we obtain the second and third terms in the r.h.s. of ( 3.1).\n•Forσ>1 we use the admissible Strichartz pair (2 σ,6σ\n3σ−2) so\n/ba∇dblI2/ba∇dbl\nL2σ(0,T;Hθ,6σ\n3σ−2(R3))=/ba∇dblAθ/2\n1I2/ba∇dbl\nL2σ(0,T;L6σ\n3σ−2(R3))\n/lessorsimilar/ba∇dblAθ/2\n1Fα(u)/ba∇dblL2(0,T;L6\n5(R3))\n/lessorsimilar/ba∇dblFα(u)/ba∇dblL2(0,T;Hθ,6\n5(R3))\nwhere we used ( A.2) withγ′= 2 andρ′=6\n5, corresponding to the admissible\nStrichartzpair( γ,ρ) withγ= 2andρ= 6. Notice that6\n5isthe minimal allowed\nvalue forρ′whend= 3.\nNow, assuming 1 <σ≤3\n2we use the estimate ( C.4) withθ= 2−σ. Summing\nup, we obtain\n/ba∇dblI2/ba∇dbl\nL2σ(0,T;H2−σ,6σ\n3σ−2(R3))/lessorsimilar/ba∇dblu/ba∇dbl2σ+1\nL2(2σ+1)(0,T;H1(R3)).\nWhen\n(2−σ)6σ\n3σ−2>3\nwe haveH2−σ,6σ\n3σ−2(R3)⊂L∞(R3). This gives the condition σ<1+√\n17\n4. Hence\n/ba∇dblI2/ba∇dblL2σ(0,T;L∞(R3))/lessorsimilar/ba∇dblu/ba∇dbl2σ+1\nL2(2σ+1)(0,T;V).\nSinceσ>1, we conclude with the H¨ older inequality that\nE/ba∇dblI2/ba∇dbl2σ\nL2σ(0,T;L∞(R3))/lessorsimilarE/parenleftBigg/integraldisplayT\n0/ba∇dblu(t)/ba∇dbl2(2σ+1)\nVdt/parenrightBiggσ\n/lessorsimilarTE/integraldisplayT\n0/ba∇dblu(t)/ba∇dbl2σ(2σ+1)\nVdt\nFinally we obtain the second and third term of ( 3.1) by means of Corollary 2.9\nas before.\n25Acknowledgements\nB.FerrarioandM.Zanellagratefullyacknowledgesfinancialsuppor tfromGNAMPA-\nINdAM.\nReferences\n[BRZ14] V. Barbu, M. R¨ ockner and D. Zhang. 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A stochastic nonlinear\nSchr¨ odinger equation with multiplicative noise. Commun. Math.\nPhys., 205(1):161–181, 1999.\n[DBD03] A. De Bouard and A. Debussche, The Stochastic Nonlinear\nSchr¨ odinger Equation in H1.Stochastic Analysis and Applications\n21:1, 97-126, 2003.\n[DO05] A. Debussche and C. Odasso. Ergodicity for a weakly damped\nstochastic non-linear Schr¨ odinger equation. J. Evol. Equ. 5(3):317–\n356, 2005.\n[EKZ17] I. Ekren, I. Kukavicaand M. Ziane. Existence of invariant m easures\nfor the stochastic damped Schr¨ odingerequation. Stoch. Partial Dif-\nfer. Equ. Anal. Comput. 5(3): 343–367, 2017.\n[GHMR21] N.E. Glatt-Holtz, V.R. Martinez and G.H. Richards. On the lon g-\ntime statistical behavior of smooth solutions of the weakly damped,\nstochastically-driven KdV equation. arXiv:2103.12942v1\n[GHMR17] N.E. Glatt-Holtz, J. Mattingly and G.H. Richards. On Unique E r-\ngodicity in Nonlinear Stochastic Partial Differential Equations. J.\nStat. Phys. 1–32, 2016.\n[HW19] J. Hong and X. Wang. Invariant measures for stochastic nonlinear\nSchr¨ odinger equations. Numerical Approximations and Symplectic\nStructures, Springer, 2019.\n[H18] F. Hornung. The nonlinear stochastic Schr¨ odinger equation via\nstochastic Strichartz estimates. J. Evol. Equ. 18, 1085–1114, 2018.\n[K06] J.U. Kim. Invariant Measures for a Stochastic Nonlinear\nSchr¨ odinger Equation Indiana University Mathematics Journal ,\n55(2): 687–717, 2006\n[L95] P. Lauren¸ cot. Long-time behaviour for weakly damped drive n non-\nlinear Schr¨ odinger equations in RN,N≤3.NoDEA 2, 357–369,\n1995.\n27"    },    {        "title": "1402.6899v1.On_the_longitudinal_spin_current_induced_by_a_temperature_gradient_in_a_ferromagnetic_insulator.pdf",        "content": "arXiv:1402.6899v1  [cond-mat.mtrl-sci]  27 Feb 2014On the longitudinal spin current induced by a temperature gr adient in a\nferromagnetic insulator\nS. R. Etesami,1,2L. Chotorlishvili,2A. Sukhov,2and J. Berakdar2\n1Max-Planck-Institut f¨ ur Mikrostrukturphysik, 06120 Hal le, Germany\n2Institut f¨ ur Physik, Martin-Luther-Universit¨ at Halle- Wittenberg, 06120 Halle, Germany\n(Dated: March 3, 2022)\nBased on the solution of the stochastic Landau-Lifshitz-Gi lbert equation discretized for a ferro-\nmagnetic chain subject to a uniform temperature gradient, w e present a detailed numerical study\nof the spin dynamics with a focus particularly on finite-size effects. We calculate and analyze the\nnet longitudinal spin current for various temperature grad ients, chain lengths, and external static\nmagnetic fields. In addition, we model an interface formed by a nonuniformly magnetized finite-size\nferromagnetic insulator and a normal metal and inspect the e ffects of enhanced Gilbert damping on\nthe formation of the space-dependent spin current within th e chain. A particular aim of this study\nis the inspection of the spin Seebeck effect beyond the linear response regime. We find that within\nour model the microscopic mechanism of the spin Seebeck curr ent is the magnon accumulation effect\nquantified in terms of the exchange spin torque. According to our results, this effect drives the spin\nSeebeck current even in the absence of a deviation between th e magnon and phonon temperature\nprofiles. Our theoretical findings are in line with the recent ly observed experimental results by M.\nAgrawal et al., Phys. Rev. Lett. 111, 107204 (2013).\nPACS numbers: 85.75.-d, 73.50.Lw, 72.25.Pn, 71.36.+c\nI. INTRODUCTION\nThermal magneto- and electric effects have a long his-\ntory and are the basis for a wide range of contemporary\ndevices. Research activities revived substantially upon\nthe experimental demonstration of the correlation be-\ntween an applied temperature gradient and the observed\nspindynamics, includingaspincurrentalongthetemper-\naturegradientinanopen-circuitmagneticsample,theso-\ncalled spin Seebeck effect (SSE)1. Meanwhile an impres-\nsive body work has accumulated on thermally induced\nspin- and spin-dependent currents1–11(for a dedicated\ndiscussion we referto the topical review12). The SSE was\nobserved not only in metallic ferromagnets (FMs) like\nCo2MnSi or semiconducting FMs, e.g. GaMnAs, (Ref.4),\nbut also in magnetic insulators LaY 2Fe5O12(Ref.5) and\n(Mn, Ze)Fe 2O4(Ref.7). The Seebeck effect is usually\nquantified by the Seebeck coefficient Swhich is defined,\nin a linear response manner, as the ratio of the gener-\nated electric voltage ∆ Vto the temperature difference\n∆T:S=−∆V\n∆T. The magnitude of the Seebeck coeffi-\ncientSdepends on the scattering rate and the density\nof electron states at the Fermi level, and thus it is ma-\nterial dependent variable. In the case of SSE, the spin\nvoltage is formally determined by µ↑−µ↓, whereµ↓(↑)\nare the electrochemical potentials for spin-up and spin-\ndown electrons, respectively. The density of states and\nthe scattering rate for spin-up and spin-down electrons\nare commonly different, which results in various Seebeck\nconstants for the two spin channels. In a metallic magnet\nsubjected to a temperature gradient, one may think of\nthe electrons in different spin channels to generate differ-\nent driving forces, leading to a spin voltage that induces\na nonzero spin current. When a magnetic insulator is\nin contact with a normal metal (NM) and the system issubjected to a thermal gradient, the total spin current\nflowing through the interface is a sum of two oppositely\ndirected currents. The current emitted from the FM into\nthe NM, is commonly identified as a spin pumping cur-\nrentIspand originatesfromthe thermally activatedmag-\nnetization dynamics in the FM, while the other current\nIflis associated with the thermal fluctuations in the NM\nand is known as spin torque13. The competition between\nthe spin pump and the spin torque currents defines the\ndirectionofthetotalspincurrentwhichisproportionalto\nthe thermal gradient applied to the system. The theory\nofthe magnon-drivenSSE5presupposesthat the magnon\ntemperature follows the phonon temperature profile and\ninalinearresponseapproximationprovidesagoodagree-\nment with experiments.\nIn a recent study14the theory of the magnon-driven\nSSE was extended beyond the linear response approxi-\nmation. In particular, it was shown that the nonlinearity\nleads to a saturation of the total spin current and nonlin-\near effects become dominant when the following inequal-\nity holds H0/Tm\nF< kB/(MsV), where H0is the constant\nmagnetic field applied to the system, Tm\nFis the magnon\ntemperature, Msis the saturation magnetization and V\nis the volume of the sample. The macrospin formulation\nofthestochasticLandau-Lifshitz-Gilbert(LLG)equation\nand the Fokker-Planck approach utilized in Ref.14is in-\nappropriate for non-uniformly magnetized samples with\ncharacteristic lengths exceeding several 10 nm. Beyond\nthe macrospin formulation the SSE effect for nonuni-\nformly magnetized samples can be described by intro-\nducing a local magnetization vector15/vector m(/vector r,t). In this\ncase, however, the correspondingFokker-Planckequation\nturns into an integro-differential equation and can only\nbe solved after a linearization16. Recently17, the lon-\ngitudinal SSE was studied in a NM-FM-NM sandwich2\nstructure in the case of a nonuniform magnetization pro-\nfile. The linear regime, however, can not totaly embrace\nnontrivial and affluent physics of the SSE.\nIn the present study we inspect the SSE for a nonuni-\nformlymagnetized finite-size FM-NM interfacesubjected\nto an arbitrary temperature gradient. Our purpose is to\ngo beyond linear response regime which is relevant for\nthe nonlinear magnetization dynamics. It is shown that\nin analogy with the macrospin case14the spin current in\nthe nonlinear regime depends not only on the tempera-\nture gradient, but on the absolute values of the magnon\ntemperature as well. In finite-size non-uniformly magne-\ntized samples, however, the site-dependent temperature\nprofile may lead to new physical important phenomena.\nFor instance, we show that the key issue for the spin cur-\nrent flowing through a nonuniformly magnetized mag-\nnetic insulator is the local exchange spin torque and the\nlocal site-dependent magnon temperature profile, result-\ning in a generic spatial distribution of the steady state\nspin current in a finite chain subject to a uniform tem-\nperaturegradient. The maximalspincurrentispredicted\nto be located at the middle of the chain.\nII. THEORETICAL FRAMEWORK\nFor the description of the transversal magnetiza-\ntion dynamics we consider propagation of the normal-\nized magnetization direction /vector m(/vector r,t) as governed by the\nLaundau-Lifshitz-Gilbert (LLG) equation18,36\n∂/vector m\n∂t=−γ/bracketleftBig\n/vector m×/vectorHeff/bracketrightBig\n+α/bracketleftbigg\n/vector m×∂/vector m\n∂t/bracketrightbigg\n−γ/bracketleftBig\n/vector m×/vectorh(/vector r,t)/bracketrightBig\n,(1)\nwhere the deterministic effective field /vectorHeff=−1\nMSδF\nδ/vector mde-\nrives from the free energy density Fand is augmented by\na Gaussian white-noise random field h(/vector r,t) with a space-\ndependent local intensity and autocorrelation function.\nαis the Gilbert damping, γ= 1.76·1011[1/(Ts)] is the\ngyromagnetic ratio and MSis the saturation magnetiza-\ntion.Freads\nF=1\nV/integraldisplay/bracketleftbiggA\n2|/vector∇m|2+Ea(/vector m)−µ0MS/vectorH0·/vector m/bracketrightbigg\ndV,(2)\nwhere/vectorH0is the external constant magnetic field, Ea(/vector m)\nis the anisotropy energy density and Ais the exchange\nstiffness. Vis the system volume. We employ a dis-\ncretized version of the integro-differential equation (1)\nby defining Ncells with a characteristic length a= /radicalbig\n2A/µ0M2sof the exchange interaction between the\nmagnetic moments. a3= Ω0is the volume of the re-\nspective cell. Assuming negligible variations of /vector m(/vector r,t)\nover a small a, one introduces a magnetization vector\n/vectorMnaveraged over the nth cell/vectorMn=MS\nV/integraltext\nΩ0/vector m(/vector r,t)dVand the total energy density becomes\nε=−/vectorH0·/summationdisplay\nn/vectorMn+K1\nM2\nS/summationdisplay\nn/parenleftbig\nM2\nS−(Mz\nn)2/parenrightbig\n−2A\na2M2\nS/summationdisplay\nn/vectorMn·/vectorMn+1.(3)\n/vectorH0is the external magnetic field and K1is the uniaxial\nanisotropy density (with the easy axis: /vector ez). The effective\nmagnetic field actingon the n-th magnetic moment reads\n/vectorHeff\nn=−∂ε\n∂/vectorMn=/vectorH0+2K1\nM2\nSMz\nn/vector ez\n+2A\na2M2\nS/parenleftBig\n/vectorMn+1+/vectorMn−1/parenrightBig\n.(4)\nThermal activation is introduced by adding to the total\neffectivefieldastochasticfluctuatingmagneticfield /vectorhn(t)\nso that\n/vectorHeff\nn(t) =/vectorH0+/vectorHanis\nn+/vectorHexch\nn+/vectorhn(t).(5)\nHere/vectorHanis\nnis the magnetic anisotropy field, /vectorHexch\nnis the\nexchange field. The random field /vectorhn(t) has a thermal\norigin and simulates the interaction of the magnetization\nwith a thermal heat bath (cf. the review Ref. [19] and\nreferences therein). The site dependence of /vectorhn(t) reflects\ntheexistenceofthelocalnonuniformtemperatureprofile.\nOn the scale ofthe volumeΩ 0the heat bath is considered\nuniform at a constant temperature. The random field is\ncharacterized via the standard statistical properties of\nthe correlation function\n/angb∇acketlefthik(t)/angb∇acket∇ight= 0,\n/angb∇acketlefthik(t)hjl(t+∆t)/angb∇acket∇ight=2kBTiαi\nγMSa3δijδklδ(∆t).(6)\niandjdefine the corresponding sites of the FM-chain\nandk,lcorrespond to the cartesian components of the\nrandom magnetic field, Tiandαiare the site-dependent\nlocal temperature and the dimensionless Gilbert damp-\ning constant, respectively, kB= 1.38·10−23[J/K] is the\nBoltzmann constant.\nIn what follows we employ for the numerical calcu-\nlations the material parameters related to YIG, e.g. as\ntabulated in Ref.6(Table I). Explicitlythe exchangestiff-\nness isA≈10 [pJ/m], the saturation magnetization has\na value of 4 πMS≈106[A/m]. The anisotropy strength\nK1can be derived from the estimate for the frequency\nω0=γ2K1/MS≈10·109[1/s]6. The size of the FM\ncell is estimated from a=/radicalbig\n2A/µ0M2syielding about 20\n[nm]. Fordampingparameterwetakethevalue α= 0.01,\nwhich exceeds the actual YIG value5,6. This is done to\noptimize the numerical procedure in order to obtain rea-\nsonable calculation times. We note that although the\nquasi-equilibrium is assured when tracking the magne-\ntization trajectories on the time scale longer than the\nrelaxation time, the increased αquantitatively alters the3\nFIG. 1: a) Schematics of the FM chain considered in the\ncalculations. b) Suggested alignment for measurements.\nstrength in the correlation function (eq. (6)) and there-\nfore indirectly has an impact on the values of the spin\ncurrent.\nWe focus on a system representing a junction of a FM\ninsulatorand a NM which is schematically shownin FIG.\n1. This illustration mimics the experimental setup for\nmeasuring the longitudinal SSE20, even though the anal-\nysis performed here does not include all the aspects of\nthe experimental setting. The direction of the magnetic\nmoments in the equilibrium is parallel to the FM-NM in-\nterface. Experimentally it was suggested to pick up the\nlongitudinal spin current by means of the inverse spin\nHall effect20. If it is so possible then, the electric field\ngenerated via the inverse spin Hall effect (ISHE) reads− →E=D− →Is×− →σ. Here− →Edenotes electric field related\nto the inverse spin Hall effect,− →Isdefines the spatial di-\nrection of the spin current, and− →σis spin polarization of\nthe electrons in the NM, and Dis the constant. We note,\nhowever, that our study is focused on the spin dynamics\nonly and makes no statements on ISHE.\nIII. DEFINITION OF THE SPIN CURRENT\nFor convenience we rewrite the Gilbert equation with\nthe total energy density (3) in the form suggested in\nRef.17\n∂/vectorSn\n∂t+γ/bracketleftBig\n/vectorSn×/parenleftBig\n/vectorHeff\nn(t)−/vectorHex/parenrightBig/bracketrightBig\n+αγ\nMS/bracketleftBigg\n/vectorSn×∂/vectorSn\n∂t/bracketrightBigg\n+∇·/vectorJ/vector s\nn= 0,\n(7)where/vectorSn=−/vectorMn/γand the expression for the spin cur-\nrent density tensor reads\n∇·/vectorJ/vector s\nn=γ/bracketleftBig\n/vectorSn×/vectorHex\nn/bracketrightBig\n. (8)\nHere\n/vectorQn=−γ[/vectorSn×/vectorHex\nn] (9)\nis the local exchange spin torque.\nFor the particular geometry (FIG. 1) the only nonzero\ncomponents of the spin current tensor are Isxn,Isy\nn,Iszn.\nTaking into account eqs. (4) and (7), we consider a dis-\ncrete version of the gradient operator and for the com-\nponents of the spin current tensor Is\nn=a2Js\nnwe deduce:\nIα\nn=Iα\n0−2Aa\nM2\nSn/summationdisplay\nm=1Mβ\nm(Mγ\nm−1+Mγ\nm+1)εαβγ,(10)\nwhereεαβγis the Levi-Civita antisymmetric tensor,\nGreek indexes define the current components and the\nLatin ones denote sites of the FM-chain. In what fol-\nlows we will utilize eq. (10) for quantifying the spin cur-\nrent in the spin chain. We consider different temperature\ngradients applied to the system taking into account the\ndependence of the magnon temperature on the phonon\ntemperature profile5. Since the temperature in the chain\nis not uniform, we expect a rich dynamics of different\nmagnetic moments /vectorMn. In this case only nonuniform\nsite-dependent spin current Incan fulfil the equation (7).\nIn order to prove this we will consider different configu-\nrations of magnetic fields for systems of different lengths.\nModeling the interface effects between the FM insulator\nand the NM proceeds by invoking the concept of the en-\nhanced Gilbert damping proposed in a recent study21.\nThe increased damping constant in the LLG equation of\nthe last magnetic moment describes losses of the spin\ncurrent due to the interface effect. In order to evaluate\nthe spin current flowing from the NM to the FM insula-\ntor we assume that the dynamics of the last spin in the\ninsulator chain is influenced by the spin torque flowing\nfrom the NM to the magnetic insulator. The magnetic\nanisotropy is considered to have an easy axis5.\nIV. NUMERICAL RESULTS ON ISOLATED\nFERROMAGNETIC INSULATOR CHAIN\nFor the study of thermally activated magnetization\ndynamics we generate from 1000 to 10000 random tra-\njectories for each magnetic moment of the FM-chain.\nAll obtained observables are averaged over the statis-\ntical ensemble of stochastic trajectories. The number\nof realizations depends on the thermal gradient applied\nto the system. For long spin chains (up to 500 mag-\nnetic moments) the calculations are computationally in-\ntensive even for the optimized advanced numerical Heun-\nmethod22, which converges in quadratic mean to the so-\nlution of the LLG equation when interpreted in the sense4\nof Stratonovich23. For the unit cell of the size 20 [nm],\nthe FM-chain of 500 spins is equivalent to the magnetic\ninsulator sample of the width around 10 [ µm]. We make\nsure in our calculations that the magnetization dynam-\nics is calculated on the large time scale exceeding the\nsystem’s relaxation time which can be approximated via\nτrel≈MS/(γ2K1α)≈10 [ns]24.\nA. Role of the local temperature and local spin\nexchange torque\nPrior to studying a realistic finite-size system we con-\nsider a toy model of three coupled magnetic moments.\nOur aim is to better understand the role of local tem-\nperature and local exchange spin torque Qn(eq. (9)) in\nthe formation of the spin current In. Considering eqs.\n(8, 9), we can utilize a recursive relation for the site-\ndependent spin current Inand the local exchange spin\ntorqueIn=In−1+a3\nγQnfor different temperatures of\nthe site in the middle of the chain above T2> Tavand\nbelowT2< Tav. The mean temperature in the system is\nTav=/parenleftbig\nT1+T2+T3/parenrightbig\n/3. The calculations are performed\nfor different values of the site temperatures. We find that\nthe exchange spin torques Qnrelated to magnetic mo-\nmentsMnwith a temperature above the mean temper-\natureTn> Tavhave a positive contribution to the spin\ncurrent in contrast to the exchange spin torques Qmof\nthe on-average-”cold”magnetic momentswith Tm< Tav.\nThis finding hints on the existence of a maximum spin\ncurrent in a finite chain of magnetic moments and/or\nstrong temperature gradient. This means that the site-\ndependent spin current Inincreases if Qn>0 until the\nlocal site temperature drops below the mean tempera-\ntureTn< Tav, in which case the exchange spin torque\nbecomes negative Qn<0 and the spin current decreases.\nIn order to prove that the negative contribution in the\nspin current of the on-average-cold magnetic moments\nis not an artefact of the three magnetic moments only,\nwe studied long spin chains which mimic non-uniformly\nmagnetized magnetic insulators. In the thermodynamic\nlimit for a large number of magnetic moments N≫1\nwe expect to observe a formation of the equilibrium pat-\nterns in the spin current profile correspondingto the zero\nexchange spin torque Qn= 0 between nearest adjacent\nmoments.\nB. Longitudinal spin current\nIn FIG. 2 a dependence of distinct components of the\nspin current on the site is plotted. As inferred from the\nfigure the current is not uniformly distributed along the\nchain. Evidently, the spin current has a maximum in\nthe middle of the chain. The site-dependent spin cur-\nrent is an aftermath of the nonuniform magnon temper-/SolidSquare/SolidSquare\n/SolidSquare\n/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare/Square/Square/Square/Square/Square/Square\n/Square/Square/Square\n/Square/Square\n/Square/Square/Square/Square\n/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square\n/Square/Square/Square/Square/Square/Square\n/Square\n/Square/Square/Square\n/Square/Square/Square/Square\n/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle/SolidSquareInSx\n/SquareInSy\n/SolidCircleInSz\n0 10 20 30 40 5001234\nSitenumber, nSpincurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1\nFIG. 2: Different cartesian components of the statistically\naveraged longitudinal spin current as a function of the site\nnumber. Numerical parameters are ∆ T= 50 [K], α= 0.01\nandH0= 0 [T]. The temperature gradient is defined ∆ T=\nT1−T50, whereT1= 50 [K]. The only nonzero component of\nthe spin current is ISzn. Other two components ISxn, ISy\nnare\nzero because of the uniaxial magnetic anisotropy field which\npreserves XOYsymmetry of the magnetization dynamics.\nature profile applied to the system. This effect was not\nobserved in the single macro spin approximation and is\nonlyrelevantforthe non-uniformlymagnetizedfinite-size\nmagnetic insulator sample. In addition one observesthat\nthe amplitude of the spin current increases with increas-\ningthe thermalgradient. Thisispredictablynatural; less\nsohowever, is the presenceof amaximum ofthe spin cur-\nrent observed in the middle of the chain. We interpret\nthis observation in terms of a collective cumulative av-\neraged influence of the surrounding magnetic moments\non particular magnetic moment. For a linear tempera-\nture gradient, as in FIG. 2, we have ∆ T=T1−TN\naNwhich\nmeans that half of the spins with i < N/2 possess tem-\nperatures abovethe mean temperature of the chain T1/2,\nwhile the other half have temperatures below the mean\ntemperature. Further, the main contributors in the total\nspin current are the hot magnetic moments with temper-\natures above the mean temperature Tn> Tavand with\na positive exchange spin torque Qn>0. While magnetic\nmoments with a temperature below the mean tempera-\ntureTn< Tav,Qn<0absorbthe spincurrentandhavea\nnegativecontribution in the total spin current. This non-\nequivalence of magnetic moments results in a maximum\nof the total spin current in the center of the chain. In\nwhat follows the magnetic moments with temperatures\nhigher than the mean temperature in the chain are re-\nferred to as hot magnetic moments, while the magnetic\nmoments with temperatures lower than Tavwe refer to\nas cold magnetic moments (i.e., our reference tempera-\nture isTav). The idea we are following is that the hot\nmagnetic moments form the total spin current which is\npartly utilized for the activation of the cold magnetic\nmoments. FIG. 3 illustrates the motivation of this state-\nment. The maximum of the spin current (solid circles) is5\n/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidCircleInSz\n/SolidUpTriangle/ScriptA3\nΓQnz\n0 10 20 30 40 5001234\nSitenumber, n1011/LBracket1/HBars/Minus1/RBracket1\nFIG. 3: Z-componentofthestatistically averaged spincurr ent\nISzn(blue solid circles) and the distribution of the exchange\nspin torquea3\nγQz\nn(red solid triangles), both site-dependent.\nDirect correlation between the behavior of the spin current\nand the exchange spin torque can be observed: the change\nof the sign of the exchange spin torque exactly matches the\nmaximum of the spin current.\nobserved in the vicinity of the sites where the exchange\nspin torque term Qnchanges its sign from positive to\nnegative (solid triangles), highlighting the role of the hot\nand cold magnetic moments in finite-size systems. To\nfurther affirm we consider two different temperature pro-\nfiles - linear and exponential - with slightly shifted values\nof the mean temperature (FIG. 4). The dependence of\nthe maximum spin current on the mean temperature is\na quite robust effect and a slight shifts of the mean tem-\nperature to the left lead to a certain shifting of the spin\ncurrent’s maximum. The effect of the nonuniform spin\ncurrent passing through the finite-size magnetic insula-\ntormightbetestedexperimentallyusingtheSSEsetupin\nwhich the spin current’s direction is parallel to the tem-\nperature gradient. One may employ the inverse spin Hall\neffect using FM insulator covered by a stripe of param-\nagnetic metal, e.g. Ptat different sites (cf Ref.20), albeit\nthe chain must be small ( <∼1µm).\nFurthermore, from FIG. 2 we infer that the only\nnonzero component of the spin current is Iz\nn. Due to\nthe uniaxial magnetic anisotropy all orientations of the\nmagnetic moments in the XOYplane are equivalent and\nIx\nn,Iy\nncomponents of the spin current vanish.\nC. Role of boundary conditions\nTo elaborate on the origin of the observed maximum\nof the spin current we inspect the role of boundary con-\nditions. In fact, in spite of employing different bound-\nary conditions for the chain we observe the same effect\n(FIG. 5), from which we can conclude that the effect of/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle/SolidSquareTn/EquΑlLinear\n/CircleTn/EquΑlExponential\n0 10 20 30 40 500.00.51.01.52.02.5\nSitenumber, nSpincurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1\n1 25 502031.234.450\nnTn/LBracket1K/RBracket1Temperature\nprofile\nFIG. 4: Z-component of the statistically averaged spin cur-\nrent for the linear ∆ T=T1−T50and exponential ∆ T(n) =\n50[K]e−(n−1)/50temperature gradients. The slight shift of the\nmean temperature to the left leads to a certain shifting of th e\nmaximum spin current to the left.\nthe cold and hot magnetic moments is inherent to the\nspin dynamics within the chain, which is independent\nfrom the particular choice of the boundary conditions.\nFurthermore, we model the situation with the extended\nregion at the ends of the FM chain (FIG. 6), in which\nthe end temperatures are constant (i.e., one might imag-\nine the heat reservoirs to have finite spatial extensions).\nModeling the ends of the FM-chain with zero temper-\nature gradient by means of the LLG equations is cer-\ntainly an approximation, which can be improved by em-\nploying the Landau-Lifshitz-Bloch equations reported in\nRef.12. It captures, however, the main effects at rela-\ntively low temperatures: the flow of the spin current for\nthe decaying spin density away from the T= const-∆ T-\ninterfaceand anon-zerointegralspin currentfor the sites\n0< n <50 and 150 < n <200. As we see even in the\nfragments of the chain with a zero temperature gradient\nthespincurrentisnotzero. Thereasonisthattheforma-\ntion of the spin current profile is a collective many body\neffect of the interacting magnetic moments. Therefore,\nthe fragment of the chain with nonzero temperature gra-\ndient (sites 50 < n <150) has a significant influence on\nthe formation of the spin current profiles in the left and\nrightregionsofthe chainwhere the temperaturegradient\nvanishes.\nD. Temperature dependence of the longitudinal\nspin current\nInFIG.7thedependenceofthe z-componentoftheav-\neraged longitudinal spin current on the temperature gra-\ndient is shown. The dependence In(∆T) (inset ofFIG. 7)\nis linear and the amplitude of the spin current increases6\n/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle\n/DownTriangle\n/DownTriangle\n/DownTriangle\n/DownTriangle\n/DownTriangle\n/DownTriangle\n/DownTriangle\n/DownTriangle\n/DownTriangle/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square\n/Square\n/Square\n/Square\n/Square\n/Square\n/Square\n/Square\n/Square\n/Square/SolidCircleM0/EquΑlMN/Plus1/EquΑl/LParen10,0,0/RParen1\n/UpTriangleM0/EquΑl/LParen10,0,0/RParen1,MN/Plus1/EquΑl/LParen10,0,Ms/RParen1\n/DownTriangleM0/EquΑl/LParen10,0,Ms/RParen1,MN/Plus1/EquΑl/LParen10,0,0/RParen1\n/SquareM0/EquΑlMN/Plus1/EquΑl/LParen10,0,Ms/RParen1\n0 10 20 30 40 5001234\nSitenumber, nSpinnurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1\nFIG. 5: Effect of different boundary conditions on the aver-\naged spin current. Numerical parameters are ∆ T= 50 [K],\nα= 0.01 andH0= 0 [T]. The temperature gradient is defined\n∆T=T1−T50, whereT1= 50 [K]. Inspite of different bound-\nary conditions we observe the same maximal spin current for\nthe site number corresponding to the mean temperature of\nthe system. Thus, the effect of the cold and hot magnetic\nmoments is independent of the particular choice of boundary\nconditions.\n/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet\n/Bullet\n0 50 100 150 200/Minus1012345\nSitenumber, nSpinnurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1\n1501001502002060100\nnTn/LBracket1K/RBracket1Temperature\nprofile\nFIG. 6: Effect of boundary conditions in the case of different\ntemperature profiles at the boundaries: linear temperature\ngradient (thick curve), constant temperature for 0 < n <50\nand 150 < n <200 (thin curve). Even in the fragments of\nthe chain with zero temperature gradient the spin current is\nnot zero, which results from the formation of the spin cur-\nrent profile as a collective many body effect of the interactin g\nmagnetic moments. Therefore, the fragment of the chain with\nnonzero temperature gradient (sites 50 < n <150) has a sig-\nnificant influence on the formation of the spin current profile s\nin the left and right zero temperature gradient parts of the\nchain.\nwith the temperature gradient. This result is consistent\nwith the experimental facts (Refs.4,5) and our previous\nanalytical estimations obtained via the single macrospin\nmodel14./Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n0 10 20 30 40 5001234\nSitenumber, nSpincurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1\n/Circle /Circle/SolidCircle /SolidCircle/Square /Square/SolidSquare /SolidSquare/UpTriangle /UpTriangle/SolidUpTriangle /SolidUpTriangle\n0 5003\n/DifferenceDeltaT/LBracket1K/RBracket1I26/Multiply1011/LBracket1/HBars/Minus1/RBracket1\n/DifferenceDeltaT=50 [K]\n/DifferenceDeltaT=40 [K]\n/DifferenceDeltaT=0 [K]/DifferenceDeltaT=10 [K]/DifferenceDeltaT=20 [K]/DifferenceDeltaT=30 [K]\nFIG. 7: Dependence of the averaged spin current on the\nstrength of the temperature gradient. Numerical parameter s\nareα= 0.01 andH0= 0 [T]. The temperature gradient is\ndefined as ∆ T=T1−T50, whereT1= 50 [K]. The inset shows\nthe averaged spin current for the 26-th site. The maximum\ncurrent increases with elevating the temperature gradient .\nE. Finite-size effects\nFinite-size effects are considered relevant for the ex-\nperimental observations (e.g. Ref.4). In the thermody-\nnamic limit N≫1 we expect the formation of equilib-\nrium patterns in the spin currentprofile correspondingto\nthe zero exchange spin torque Qn= 0 between nearest\nadjacent moments. To address this issue, the spin cur-\nrent for chains of different lengths is shown in FIG. 8. As\nwe see in the case of N= 500 magnetic moments large\npattern of the uniform spin current corresponding to the\nsites 50< n <450 is observed. In order to understand\nsuch a behavior of the spin current for a large system\nsize, we plotted the dependence on the site number of\nthe exchange spin torque Qn(FIG. 9). As we see, the\nexchange spin torque corresponding to the spin current\nplateau is characterizedby largefluctuations aroundzero\nvalue, while nonzero positive (negative) values of the ex-\nchange spin torque Qnobserved at the left (right) edges\ncorrespond to the nonmonotonic left and right wings of\nthe spin torque profile. One may try to interpret the ob-\nservedresults in terms ofthe so called magnon relaxation\nlength (MRL) λm≈2/radicalBig\n(DkBT/¯h2)τmmτmp(Refs.5,6),\nwhereDis the spin-wave stiffness constant and τmm,mp\nare the magnon-magnon, and the magnon-phonon relax-\nation times, respectively. The MRL is a characteristic\nlength which results from the solution of the heat-rate\nequation for the coupled magnon-phonon system5. The\nphysical meaning of λmis an exponential drop of the\nspace distribution of the local magnon temperature for\nthe given external temperature gradient ∆ T. In other\nwords, although the externally applied temperature bias\nis kept constant, the thermal distribution for magnons\nis not necessarily linear. In general, one may suggest a\nsinh(x)-like spatial dependence5and a temperature de-7\npendence λm(T). Estimates of the MRL for the material\nparameters related to YIG (suppl. mater. of Ref.5) and\nTN= 0.2 [K] yield the following λm≈10 [µm]26. As\nseen from FIG. 8 the length starting from which the sat-\nuration of the spin current comes into play as long as\nthe FM-chain exceeds the length 20 [nm] ×100≈2 [µm].\nHowever,werecallthat MRLisawitnessofthe deviation\nbetween the magnon and phonon temperature profiles.\nTherefore, for interpretingthe nonmonotonicpartsofthe\nspin current profile (FIG. 8) in terms of the MRL one has\nto prove the pronounced deviation between magnon and\nphonontemperaturesattheboundaries. Forfurtherclar-\nification we calculate the magnon temperature profile.\nThis can be done self-consistently via the Langevin func-\ntion< Mz\nn>=L/parenleftbig\n< Mz\nn> Hn/kBTm\nn/parenrightbig\n. HereHz\nnis the\nzcomponent of the local magnetic field which depends\non the external magnetic field and the mean values of the\nadjacent magnetic moments < Mz\nn−1>, < Mz\nn+1>(\nsee eq. (4)). As inferred from the FIG. 10 the magnon\ntemperature profile follows the phonon temperature pro-\nfile. Prominent deviation between the phonon and the\nmagnon temperatures is observed only at the beginning\nof the chain and gradually decreases and becomes small\non the MRL scale. Close to the end of the chain the\ntemperature difference becomes almost zero. This means\nthat left nonmonotonic parts of the spin current profile\nFIG. 8 can be interpreted in terms of none-equilibrium\nprocesses. Comparing this result with the exchange spin\ntorque profile (FIG. 9) we see that in this part of the\nspin chain the exchange spin torque is positive. This\nis the reason why the spin current Inis increasing with\nthe site number n. The saturated plateau of the spin\ncurrent shown in FIG. 8 corresponds to the zero ex-\nchange spin torque Qn= 0 (cf. FIG. 9) and the decay\nof the spin Seebeck current Inat the right edge corre-\nsponds to the negative spin exchange torque Qn<0.\nThus, for the formation of the convex spin current profile\nthe key issue is not the difference between magnon and\nphonontemperatures,whichasweseeisprettysmall,but\nthe magnon temperature profile itself. The existence of\nthe hot(cold) magnetic moments with the local magnon\ntemperature up (below) the mean magnon temperature\ngenerates the spin current. This difference in the local\nmagnon temperature of the different magnetic moments\ndrives the spin current in the chain. On the other hand\nany measurement of the spin current done in the vicin-\nity of the right edge of the current profile will demon-\nstrate a non-vanishing spin current in the absence of the\ndeviation between the magnon and phonon temperature\nprofiles. This may serve as an explanation of the re-\ncent experiment25, where a non-vanishing spin current\nwas observed in the absence of the deviation between the\nmagnon and the phonon temperature profiles. We note\nthat zero values of the spin current shown in FIG. 8 is\nthe artefact of isolated magnetic insulator chain. Real\nmeasurement of the spin currents usually involve FM-\ninsulator/NM-interfaces. As will be shown below the in-\nterface effect described by an enhanced Gilbert dampingN=50N=500\nN=200\nN=150\nN=100\n1 100 200 300 400 500012\nSitenumber, nSpincurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1\nFIG. 8: The dependence of the averaged spin current on the\nlength of the FM-chain. Numerical parameters are α= 0.01\nandH0= 0 [K]. The temperature gradient is linear and the\nmaximum temperature is on the left-hand-side of the chain\n(T1= 100 [K]). In all cases the per-site temperature gradient\nis ∆T/N= 0.2 [K].\n0100200300400500/Minus0.10/Minus0.050.000.050.100.150.20\nSitenumber, n/ScriptA3\nΓQnz/Multiply1011/LBracket1/HBars/Minus1/RBracket1\nFIG. 9: The dependence of the exchange spin torque on\nthe site number. Numerical parameters are α= 0.01 and\nH0= 0 [K]. The temperature gradient is linear and the\nmaximum temperature is on the left-hand-side of the chain\n(T1= 100 [K]). The per-site temperature gradient is ∆ T/N=\n0.2 [K]. The exchange spin torque profile consists of three\nparts, the positive part corresponds to the high temperatur e\ndomain and low temperature domain corresponds to the neg-\native exchange spin torque. In the middle of the chain where\nthe spin current is constant, the exchange spin torque fluctu -\nates in the vicinity of the zero value.\nand the spin torque lead to a nonzero spin current at the\ninterfaces which is actually measured in the experiment.\nF. Role of the external magnetic filed ( H0/negationslash= 0)\nIt follows from our calculations that the dependence of\nthe longitudinal spin current on the magnetic field is not8\n/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bu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200 300 400 500020406080100\nSitenumber, nTemperature/LBracket1K/RBracket1\nFIG. 10: The magnon temperature profile (line) formed in the\nsystem. Numerical parameters are α= 0.01 andH0= 0 [K].\nBlue line corresponds to the applied linear phonon tempera-\nture profile. The maximum temperature on the left-hand-side\nof the chain is( T1= 100 [K]). The per-site temperature gradi-\nent is ∆T/N= 0.2 [K]. The maximal deviation between the\nphonon and magnon temperatures is observed only at the left\nedge of chain. The difference between temperatures graduall y\ndecreases and becomes almost zero for the sites with n >400.\ntrivial. Once the external static magnetic field is applied\nperpendicularly to the FM-chain and along the easy axis\nat the same time, we can suppress the spin current at ele-\nvated magnetic fields (FIG. 11). The threshold magnetic\nfield is - asexpected - the strength ofthe anisotropyfield,\ni.e. 2K1/MS∼0.056 [T]. By applying magnetic fields\nmuch higher than 0 .056 [T], the magnetic moments are\nfully aligned along the field direction and hence the X-,\nY-components of the magnetization required to form the\nZ-component of the longitudinal averaged spin current\nvanish.\nIn the case of the magnetic field being applied perpen-\ndicularly to the easy axis, the behavior becomes more\nrich (FIG. 12). In analogy with the situation observed\nin FIG. 11 there are no sizeable changes for the In(∆T)-\ndependence at low static fields. This is the regime where\nthe anisotropy field is dominant. In contrast to the Hz\n0\napplied field, the spin current does not linearly depend\non the strength of the field (inset of FIG. 11), which is\nexplained by the presence of different competing contri-\nbutions in the total energy density and not a simple cor-\nrection of the Z-component of the anisotropy field illus-\ntrated in the previous figure. Surprisingly, the magnetic\nfield oriented along the FM-chain can also suppress the\nappearance of the spin current’s profile. Also in this case\nthe strong magnetic field destroys the formation of the\nmagnetization gradient resulting from the applied tem-\nperature bias./SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle\n/Circle/Circle/Circle/Circle/Circle/Circle\n/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/Square/Square/Square/Square/Square/Square/SolidCircleH0z/EquΑl0/LBracket1T/RBracket1\n/CircleH0z/EquΑl10/Minus2/LBracket1T/RBracket1\n/SolidSquareH0z/EquΑl10/Minus1/LBracket1T/RBracket1\n/SquareH0z/EquΑl1/LBracket1T/RBracket1\n0 10 20 30 40 500123\n/DifferenceDeltaT/LBracket1K/RBracket1Spincurrent I26/Multiply1011/LBracket1/HBars/Minus1/RBracket1\nFIG. 11: Effect of the external magnetic field applied paralle l\nto the easy axis on the averaged spin current. Numerical\nparameters are ∆ T= 50 [K], α= 0.01 andN= 50. The\ntemperature gradient is linear and the maximum temperature\nis on the left-hand-side of the chain.\n/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle\n/Circle/Circle/Circle/Circle/Circle/Circle\n/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare /Square/Square/Square/Square/Square/Square/SolidCircleH0x/EquΑl0/LBracket1T/RBracket1\n/CircleH0x/EquΑl10/Minus2/LBracket1T/RBracket1\n/SolidSquareH0x/EquΑl10/Minus1/LBracket1T/RBracket1\n/SquareH0x/EquΑl1/LBracket1T/RBracket1\n0 10 20 30 40 5001234\n/DifferenceDeltaT/LBracket1K/RBracket1Spincurrent I26/Multiply1011/LBracket1/HBars/Minus1/RBracket1\n/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/DifferenceDeltaT/EquΑl50/LBracket1K/RBracket1\n0.020.040.060.080.1001234\nH0x/LBracket1T/RBracket1I26/Multiply1011/LBracket1/HBars/Minus1/RBracket1\nFIG. 12: Effect of the external magnetic field applied perpen-\ndicularly to the easy axis on the averaged spin current. Nu-\nmerical parameters are ∆ T= 50 [K], α= 0.01 andN= 50.\nThe temperature gradient is linear and the maximum tem-\nperature is on the left-hand-side of the chain.\nV. INTERFACE EFFECTS\nThe experimental setup to detect the spin current\nmight involve a NM adjacent to the spin-current gener-\nating substance, e.g. a FM insulator. This NM converts\nthe injected spin current from the FM to an electric cur-\nrent via ISHE1,5,27. So it is of interest to see the effect\nof the adjacent NM on the generated spin current in the\nconsidered chain. Obviously, the main effects appear in\nthe FM-NM interface. The interface effect can be di-\nvided into two parts which is described in the following\nsubsections.9\nA. Spin pumping and enhanced Gilbert damping\nInmagnetic insulators , chargedynamicsislessrelevant\n(in our model, anyway), and in some cases the dissipa-\ntive losses associated with the magnetization dynamics\nare exceptionally low (e.g. in YIG28α= 6.7×10−5).\nWhen a magnetic insulator is brought in contact with\nanormal metal , magnetization dynamics results in spin\npumping, which in turn causes angular momentum being\npumped to the NM. Because of this nonlocal interaction,\nthe magnetization losses become enhanced21.\nIf we consider the normal metal as a perfect spin\nsinkwhich remains in equilibrium even though spins are\npumped into it (which means there is a rapid spin relax-\nation and no back flow of spin currents to the magnetic\ninsulator), the magnetization dynamics is described by\nthe LLG equation with an additional torque originating\nfrom the FM-insulator/NM interfacial spin pumping21\n∂/vectorM\n∂t=−γ/bracketleftBig\n/vectorM×/vectorHeff/bracketrightBig\n+α\nMS/bracketleftBigg\n/vectorM×∂/vectorM\n∂t/bracketrightBigg\n+/vector τsp,(11)\nwhere\n/vector τsp=γ¯h\n4πM2\nSgeffδ(x−L)/bracketleftBigg\n/vectorM×∂/vectorM\n∂t/bracketrightBigg\n,(12)\nwhereLis the position of the interface, eis the elec-\ntron charge and geffis the real part of the effective\nspin-mixing conductance. In the YIG-Pt bilayer the\nmaximum measured effective spin-mixing conductance is\ngeff= 4.8×1020[m−2] Ref.21. In fact if the spin pumping\ntorque should be completely described, one should add\nanother torque containing the imaginary part of geff29.\nHowever, we omit this imaginary part here because it\nhas been found to be too small at FM-NM interfaces30.\nThe aforementioned spin pumping torque concerns the\ncases that we characterized with /vectorM. In our discrete\nmodel which includes a chain of Nferromagnetic cells,\nwe describe the above phenomena as follows\n∂/vectorMn\n∂t=−γ/bracketleftBig\n/vectorMn×/vectorHeff\nn/bracketrightBig\n+α\nMS/bracketleftBigg\n/vectorMn×∂/vectorMn\n∂t/bracketrightBigg\n+/vector τsp\nn,\n(13)\nwhere\n/vector τsp\nn=γ¯h2\n2ae2M2\nSg⊥δnN/bracketleftbigg\n/vectorMn×∂Mn\n∂t/bracketrightbigg\n,(14)\nwhich means the spin pumping leads to an enhanced\nGilbert damping in the last site\n∆α=γ¯h\n4πaMsgeff. (15)\nAs mentioned, the above enhanced Gilbert damping\ncould solely describe the interfacial effects as long as\nwe treat the adjacent normal metal as a perfect spinsink without any back flow of the spin current from the\nNM17,21. The latter is driven by the accumulated spins\nin the normal metal. If we model the normal metal as\na perfect spin sink for the spin current, spin accumu-\nlation does not build up. This approximation is valid\nwhen the spin-flip relaxation time is very small and so\nit prevents any spin-accumulation build-up. So the spins\ninjected by pumping decayand/orleavethe interfacesuf-\nficiently fast and there won’t be any backscattering into\nthe ferromagnet13,31. We note by passing that in a re-\ncentstudy concerningthis phenomena, it hasbeen shown\nthat spin pumping (and so enhanced Gilbert damping)\ndepends on the transverse mode number and in-plane\nwave vector21.\nB. Spin transfer torque\nIt was independently proposed by Slonczewski32and\nBerger33that the damping torque in the LLG equation\ncould have a negative sign as well, corresponding to a\nnegative sign of α. This means that the magnetization\nvector could move into a final position antiparallel to the\neffective field. In order to achieve this, energy has to be\nsupplied to the FM system to make the angle between\nthe magnetization and the effective field larger. This en-\nergy is thought to be provided by the injection of a spin\ncurrent/vectorIincidentto the FM13,29,34\n/vector τs=−γ\nM2\nSV/bracketleftBig\n/vectorM×/bracketleftBig\n/vectorM×/vectorIinjected/bracketrightBig/bracketrightBig\n,(16)\nwhich describes the dynamics of a monodomain ferro-\nmagnet of volume Vthat is subject to the spin current\n/vectorIincidentand modifies the right-hand side of the LLG\nequation as a source term. In general, a torque-term\nadditional to the Slonczewskis torque (eq. (16)) is also\nallowed29,35\n/vector τsβ=−γ\nMSVβ/bracketleftBig\n/vectorM×/vectorIincident/bracketrightBig\n, (17)\nwhereβgives the relative strength with respect to the\nSlonczewski’s torque (eq. (16)).\nFor the case of a FM-chain, again we assume that the\nabovespin-transfertorques act solelyon the last FM cell.\nC. Numerical results for interface effects\nInordertosimulatetheenhancedGilbertdampingand\nthe spin–transfer torque we assume that they act only on\nthechainend(motivatedbytheiraforementionedorigin).\nSo the dynamics of our FM-chain is described by the\nfollowing LLG18,36equations\n∂/vectorMn\n∂t=−γ\n1+α2/bracketleftBig\n/vectorMn×/vectorHeff\nn/bracketrightBig\n−γα\n(1+α2)MS/bracketleftBig\n/vectorMn×/bracketleftBig\n/vectorMn×/vectorHeff\nn/bracketrightBig/bracketrightBig\n,\nn= 1,...,(N−1),(18)10\nand\n∂/vectorMN\n∂t=−γ\n1+α2\nN/bracketleftBig\n/vectorMN×/vectorHeff\nN/bracketrightBig\n−γαN\n(1+α2\nN)MS/bracketleftBig\n/vectorMN×/bracketleftBig\n/vectorMN×/vectorHeff\nN/bracketrightBig/bracketrightBig\n−γ\nM2\nSa3/bracketleftBig\n/vectorMN×/bracketleftBig\n/vectorMN×/vectorIinjected/bracketrightBig/bracketrightBig\n−γ\nMSa3β/bracketleftBig\n/vectorMN×/vectorIincident/bracketrightBig\n,(19)\nwhereαN=α+γ¯hgeff/(4πaMs).\nEq.(18) and (19) describe the magnetizationdynamics\nin the presence of the interface effects and include both\nspinpump and spintorqueeffects. Results inthe absence\nof the spin torque are presented at the FIG. 13. The en-\nhanced Gilbert damping captures losses of the spin cur-\nrent associated with the interface effect. A nonzero spin\ncurrent corresponding to the last n= 500 spin quantifies\nthe amount of the spin current pumped into the normal\nmetal from the magnetic insulator. However, the con-\nvex profile of the spin current is observed as well in the\npresence ofthe interface effects. The influence ofthe spin\ntorqueonthespincurrentprofileisshowninFIG.14. We\nsee from these results, the large spin torque reduces the\ntotalspincurrentfollowingthroughtheFM-insulato/NM\ninterfaces. The spin torque current is directed opposite\nto the spin pump current and therefore compensates it.\n/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSqu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ircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle\n/SmallCircle\n0 100 200 300 400 500012\nSitenumber, nSpincurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1\n/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle\n/SmallCircle\n450 475 500012\nnWith enhanced \ndamping, /DifferenceDeltaΑ=0.5\nWithout enhanced \ndamping, /DifferenceDeltaΑ=0.5\nFIG. 13: Statistically averaged spin current in the chain of\nN= 500-sites. Numerical parameters are ∆ T= 100 [K],\nα= 0.01 andH0= 0 [T]. The temperature gradient is lin-\near and the maximum temperature is on the left-hand-side\nof the chain ( T1). The blue curve shows the averaged spin\ncurrent when no enhanced Gilbert damping and no spin–\ntransfer torque is present. The red curve shows the aver-\naged spin current when the enhanced Gilbert damping with\ngeff= 1.14×1022[m−2] is present. The inset shows the aver-\naged spin current of the last fifty sites only./SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare\n/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidS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SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle\n/SmallCircle\n0 100 200 300 400 500012\nSitenumber, nSpincurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1\n/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare\n/SmallSolidSquare/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle\n/SmallCircle\n450 475 500012\nnIn both cases /DifferenceDeltaΑ=0.5\nincidentIincident/EquΑl\n1.03/Multiply1015/LParen1/Minus1, 0, 0/RParen1/LBracket2/HBars/Minus1/RBracket2\nIincident/EquΑl\n5.15/Multiply1015/LParen1/Minus1, 0, 0/RParen1/LBracket2/HBars/Minus1/RBracket2\nFIG. 14: Statistically averaged spin current in the chain of\nN= 500 when there are both the enhanced Gilbert damp-\ning and the spin–transfer torque. Numerical parameters are\n∆T= 100 [K], α= 0.01,H0= 0 [T], geff= 1.14×1022[m−2]\nandβ= 0.01. The temperature gradient is linear and the\nmaximum temperature is on the left-hand-side of the chain\n(T1). The blue curve has /vectorIincident= 1×1015(−1,0,0) [¯hs−1]\nand the red curve is with /vectorIincident= 5×1015(−1,0,0) [¯hs−1].\nVI. MECHANISMS OF THE FORMATION OF\nSPIN EXCHANGE TORQUE AND SPIN\nSEEBECK CURRENT\nIn the previous sections we demonstrated the direct\nconnection between the spin Seebeck current profile and\nthe exchange spin torque. Here we consider the mecha-\nnisms of the formation of the exchange spin torque. For\nthis purpose we investigate changes in the magnetization\nprofile associatedwith the changeof the magnontemper-\nature<∆Mz\nn>=< Mz\nn>−< Mz\n0n>, where< Mz\nn>\nis the mean component of the magnetization moment\nfor the case of the applied linear thermal gradient, while\n< Mz\n0n>correspondstothemeanmagnetizationcompo-\nnent in the absence of thermal gradient ∆ T= 0. Quan-\ntity<∆Mz\nn>defines the magnon accumulation as the\ndifferencebetweentherelativeequilibriummagnetization\nprofile and excited one Ref.37and is depicted in FIG. 15.\nWe observe a direct connection between the magnon ac-\ncumulation effect and the exchange spin torque. A pos-\nitive magnon accumulation, meaning an excess of the\nmagnonscomparedtotheequilibriumstateisobservedin\nthe high temperature part of the chain. While in the low\ntemperature part the magnon accumulation is negative\nindicatingalackofmagnonscomparedtotheequilibrium\nstate. The exchange spin torque is positive in the case of\nthe positive magnon accumulation and is negative in the\ncase of the negative magnon accumulation (the exchange\nspin torque vanishes in the equilibrium state). From the\nphysical point of view, the result is comprehensible: the\nspin Seebeck current is generated by the magnon accu-\nmulation, transmitted through the equilibrium part of\nthe chain and partially absorbed in the part of the chain\nwith a negative magnon accumulation.11\n/SolidCircle/SolidCircle/SolidCircle\n/SolidCircle/SolidCircle\n/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle\n/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle\n/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle\n/SolidCircle/SolidCircle/SolidCircle\n/SolidCircle\n/SolidCircle\n50 100 150 200/Minus4/Minus202\nSitenumber, nMagnon accumulation10/Minus5/Multiply/LParen1m/Minusm0/RParen1\n15010015020020\nnTn/LBracket1K/RBracket1\n050100150200/Minus0.10/Minus0.050.000.050.100.15\nn/ScriptA3\nΓQnz/Multiply1011/LBracket1/HBars/Minus1/RBracket1\nFIG. 15: Site dependence of the exchange spin torque and\nthe magnon accumulation effect. We observe a direct connec-\ntion between magnon accumulation effect and the exchange\nspin torque. A positive magnon accumulation, i.e. excess of\nthe magnons, is observed in the high temperature part of the\nchain. While in the low temperature part magnon accumu-\nlation is negative (lack of magnons compare to the equilib-\nrium state). The exchange spin torque is positive for posi-\ntive magnon accumulation, and negative for negative magnon\naccumulation. The spin Seebeck current is generated by ex-\ncess magnons, transmitted through the equilibrium part of\nthe chain and partially absorbed in the region with magnon\ndrain.\nVII. CONCLUSIONS\nBased on the solution of the stochastic Landau-\nLifshitz-Gilbert equation discretized for a ferromagnetic\nchain in the presence of a temperature gradient formed\nalong the chain, we studied the longitudinal spin See-\nbeck effect with a focus on the space-dependent effects.\nIn particular, we calculated a longitudinal averaged spin\ncurrent as a function of different temperature gradients,\ntemperature gradient strengths, distinct chain lengths\nand differently oriented external static magnetic fields.\nOur particular interest was to explain the mechanisms\nof the formation of the spin Seebeck current beyond the\nlinear response regime. The merit was in pointing out a\nmicroscopicmechanismfortheemergenceofthespinSee-beck current in a finite-size system. We have shown that,\nwithin our model, the microscopic mechanism of the spin\nSeebeck current is the magnon accumulation effect quan-\ntified in terms of the exchange spin torque. We proved\nthat the magnon accumulation effect drives the spin See-\nbeck current even in the absence of significant deviation\nbetween magnon and phonon temperature profiles. Our\ntheoretical findings are in line with recently observed ex-\nperimental results25where non-vanishing spin Seebeck\ncurrent was observed in the absence of a temperature\ndifference between phonon and magnon baths.\nConcerningthe influence ofthe external constant mag-\nnetic fields on the spin Seebeck current we found that\ntheir role is nontrivial: An external static magnetic field\napplied perpendicularly to the FM-chain and along the\neasy axis may suppress the spin current at elevated mag-\nnetic fields (FIG. 11). The threshold magnetic field has a\nstrengthofthe anisotropyfield, i.e. 2 K1/MS∼0.056[T].\nIn the case of the magnetic field applied perpendicu-\nlarly to the easy axis, we observe a more complex be-\nhavior (FIG. 12). In analogy with the situation seen in\nFIG. 11 there are no sizeable changes for the In(∆T)-\ndependence at low static fields. This is the regime where\nthe anisotropy field is dominant. In contrast to the Hz\n0\napplied field, it does not linearly depend on the strength\nof the field (inset of FIG. 11), which is explained by the\npresence of different competing contributions in the total\nenergyand not a simple correctionof the Z-componentof\nthe anisotropy field. Notably, the magnetic field oriented\nalong the FM-chain can also suppress the emergence of\nthe spin current’s profile. Also in this case a strong mag-\nnetic field destroys the formation of the magnetization\ngradient resulting from the applied temperature bias.\nIn addition, we modeled an interface formed by a\nnonuniformly magnetized finite size ferromagnetic insu-\nlator and a normal metal (e.g., YIG-Platinum junction)\nto inspect the effects of the enhanced Gilbert damping\non the formation of space-dependent spin current within\nthe chain.\nVIII. ACKNOWLEDGEMENTS\nThe financial support by the Deutsche Forschungsge-\nmeinschaft (DFG) is gratefully acknowledged.\n1K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae,\nK. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778\n(2008).\n2M. Hatami, G. E. W. Bauer, Q. Zhang, and P. J. Kelly,\nPhys. Rev.B 79, 174426 (2009); A. D.Avery, M. R. Pufall,\nand B. L. Zink, Phys. Rev. Lett. 109, 196602 (2012); C.\nH. Wong, H. T. C. Stoof, and R. A. Duine, Phys. Rev. A\n85, 063613 (2012).\n3S. Bosu, Y. Sakuraba, K. Uchida, K. Saito, T. 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B.89, 024409 (2014)."    },    {        "title": "1702.08408v2.Current_Induced_Damping_of_Nanosized_Quantum_Moments_in_the_Presence_of_Spin_Orbit_Interaction.pdf",        "content": "Current Induced Damping of Nanosized Quantum Moments in the Presence of\nSpin-Orbit Interaction\nFarzad Mahfouzi\u0003and Nicholas Kioussisy\nDepartment of Physics and Astronomy, California State University, Northridge, CA, USA\n(Dated: November 10, 2021)\nMotivated by the need to understand current-induced magnetization dynamics at the nanoscale,\nwe have developed a formalism, within the framework of Keldysh Green function approach, to study\nthe current-induced dynamics of a ferromagnetic (FM) nanoisland overlayer on a spin-orbit-coupling\n(SOC) Rashba plane. In contrast to the commonly employed classical micromagnetic LLG simula-\ntions the magnetic moments of the FM are treated quantum mechanically . We obtain the density\nmatrix of the whole system consisting of conduction electrons entangled with the local magnetic\nmoments and calculate the e\u000bective damping rate of the FM. We investigate two opposite limiting\nregimes of FM dynamics: (1) The precessional regime where the magnetic anisotropy energy (MAE)\nand precessional frequency are smaller than the exchange interactions, and (2) The local spin-\rip\nregime where the MAE and precessional frequency are comparable to the exchange interactions. In\nthe former case, we show that due to the \fnite size of the FM domain, the \\Gilbert damping\"does\nnot diverge in the ballistic electron transport regime, in sharp contrast to Kambersky's breathing\nFermi surface theory for damping in metallic FMs. In the latter case, we show that above a critical\nbias the excited conduction electrons can switch the local spin moments resulting in demagnetization\nand reversal of the magnetization. Furthermore, our calculations show that the bias-induced anti-\ndamping e\u000eciency in the local spin-\rip regime is much higher than that in the rotational excitation\nregime.\nPACS numbers: 72.25.Mk, 75.70.Tj, 85.75.-d, 72.10.Bg\nI. INTRODUCTION\nUnderstanding the current-induced magnetization\nswitching (CIMS) at the nanoscale is mandatory for the\nscalability of non-volatile magnetic random access mem-\nory (MRAM) of the next-generation miniaturized spin-\ntronic devices. However, the local magnetic moments of a\nnanoisland require quantum mechanical treatment rather\nthan the classical treatment of magnetization commonly\nemployed in micromagnetic simulations, which is the cen-\ntral theme of this work.\nThe \frst approach of CIMS employs the spin transfer\ntorque (STT)1,2in magnetic tunnel junctions (MTJ) con-\nsisting of two ferromagnetic (FM) layers (i.e., a switch-\nable free layer and a \fxed layer) separated by an insulat-\ning layer, which involves spin-angular-momentum trans-\nfer from conduction electrons to local magnetization3,4.\nAlthough STT has proven very successful and brings the\nprecious bene\ft of improved scalability, it requires high\ncurrent densities ( \u00151010A/cm2) that are uncomfort-\nably high for the MTJ's involved and hence high power\nconsumption. The second approach involves an in-plane\ncurrent in a ferromagnet-heavy-metal bilayer where the\nmagnetization switching is through the so-called spin-\norbit torque (SOT) for both out-of-plane and in-plane\nmagnetized layers.5{8The most attractive feature of the\nSO-STT method is that the current does not \row through\nthe tunnel barrier, thus o\u000bering potentially faster and\nmore e\u000ecient magnetization switching compared to the\nMTJs counterparts.\nAs in the case of STT, the SO-STT has two compo-\nnents: a \feld-like and an antidamping component. Whilethe \feld-like component reorients the equilibrium direc-\ntion of the FM, the antidamping component provides the\nenergy necessary for the FM dynamics by either enhanc-\ning or decreasing the damping rate of the FM depending\non the direction of the current relative to the magneti-\nzation orientation as well as the structural asymmetry\nof the material. For su\u000eciently large bias the SOT can\novercome the intrinsic damping of the FM leading to ex-\ncitation of the magnetization precession.8The underlying\nmechanism of the SOT for both out-of-plane and in-plane\nmagnetized layers remains elusive and is still under de-\nbate. It results from either the bulk Spin Hall E\u000bect\n(SHE)9{12, or the interfacial Rashba-type spin-orbit cou-\npling,13{16or both17{19.\nMotivated by the necessity of scaling down the size\nof magnetic bits and increasing the switching speed, the\nobjective of this work is to develop a fully quantum me-\nchanical formalism, based on the Keldysh Green function\n(GF) approach, to study the current-induced local mo-\nment dynamics of a bilayer consisting of a FM overlayer\non a SOC Rashba plane, shown in Fig. 1.\nUnlike the commonly used approaches to investigate\nthe magnetization dynamics of quantum FMs, such as the\nmaster equation20, the scattering21or quasi-classical22\nmethods, our formalism allows the study of magnetiza-\ntion dynamics in the presence of nonequilibrium \row of\nelectrons.\nWe consider two di\u000berent regimes of FM dynamics: In\nthe \frst case, which we refer to as the single domain\ndynamics, the MAE and the precession frequency are\nsmaller than the exchange interactions, and the FM can\nbe described by a single quantum magnetic moment, of\na typically large spin, S, whose dynamics are governedarXiv:1702.08408v2  [cond-mat.mes-hall]  27 Apr 20172\nFIG. 1: (Color online) Schematic view of the FM/Rashba\nplane bilayer where the FM overlayer has length Lxand is\nin\fnite (\fnite) along the y-direction for the case of a single\ndomain (nano-island) discussed in Sec. III (IV). The magne-\ntization,~ m, of the FM precesses around the direction denoted\nby the unit vector, ~ nM, with frequency !and cone angle, \u0012.\nThe Rashba layer is attached to two normal (N) leads which\nare semi-in\fnite along the x-direction, across which an exter-\nnal bias voltage, V, is applied.\nmainly by the quantized rotational modes of the magne-\ntization. We show that the magnetic degrees of freedom\nentering the density matrix of the conduction electron-\nlocal moment entagled system simply shift the chemical\npotential of the Fermi-Dirac distribution function by the\nrotational excitations energies of the FM from its ground\nstate. We also demonstrate that the e\u000bective damping\nrate is simply the netcurrent along the the auxiliarym-\ndirection, where m= -S, -S+1, :::, +S, are the eigenval-\nues of the total Szof the FM. Our results for the change\nof the damping rate due to the presence of a bias volt-\nage are consistent with the anti-damping SOT of clas-\nsical magnetic moments,16,23, where due to the Rashba\nspin momentum locking, the anti-damping SOT, to low-\nest order in magnetic exchange coupling, is of the form,\n~ m\u0002(~ m\u0002^y), where ^yis an in-plane unit vector normal\nto the transport direction.\nIn the adiabatic and ballistic transport regimes due to\nthe \fnite S value of the nanosize ferromagnet our formal-\nism yields a \fnite \\Gilbert damping\", in sharp contrast to\nKambersky's breathing Fermi surface theory for damp-\ning in metallic FMs.24On the other hand, Costa and\nMuniz25and Edwards26demonstrated that the prob-\nlem of divergent Gilbert damping is removed by takinginto account the collective excitations. Furthermore, Ed-\nwards points out26the necessity of including the e\u000bect of\nlong-range Coulomb interaction in calculating damping\nfor large SOC.\nIn the second case, which corresponds to an indepen-\ndent local moment dynamics, the FM has a large MAE\nand hence the rotational excitation energy is compara-\nble to the local spin-\rip excitation (exchange energy).\nWe investigate the e\u000bect of bias on the damping rate of\nthe local spin moments. We show that above a criti-\ncal bias voltage the \rowing conduction electrons can ex-\ncite (switch) the local spin moments resulting in demag-\nnetization and reversal of the magnetization. Further-\nmore, we \fnd that, in sharp contrast to the single do-\nmain precessional dynamic, the current-induced damping\nis nonzero for in-plane and out-of-plane directions of the\nequilibrium magnetization. The bias-induced antidamp-\ning e\u000eciency in the local moment switching regime is\nmuch higher than that in the single domain precessional\ndynamics.\nThe paper is organized as follows. In Sec. II we present\nthe Keldysh formalism for the density matrix of the en-\ntagled quantum moment-conduction electron system and\nthe e\u000bective dampin/antdamping torque. In Sec. III we\npresent results for the current-induced damping rate in\nthe single domain regime. In Sec. IV we present results\nfor the current-induced damping rate in the independent\nlocal regime. We conclude in Sec. V.\nII. THEORETICAL FORMALISM\nFig. 1 shows a schematic view of the ferromagnetic\nheterostructure under investigation consisting of a 2D\nferromagnet-Rashba plane bilayer attached to two semi-\nin\fnite normal (N) leads whose chemical potentials are\nshifted by the external bias, Vbias. The magnetization\nof the FM precesses around the axis speci\fed by the\nunit vector, ~ nM, with frequency !and cone angle \u0012.\nThe FM has length, LFM\nx, along the transport direction.\nThe total Hamiltonian describing the coupled conduc-\ntion electron-localized spin moment system in the het-\nerostructure in Fig. 1 can be written as,\nHtot=X\nrr0;\u001b\u001b0Trfsdgh\u0010\n1s^H\u001b\u001b0\nrr0+\u000err0\u000e\u001b\u001b01s\u0016r+\u000err0Jsd~ \u001b\u001b\u001b0\u0001~sd(r) +\u000e\u001b\u001b0\u000err0HM\u0011\n \u0003\nfs0\ndgr0\u001b0 fsdgr\u001bi\n: (1)\nHere,~sd(r) is the local spin moment at atomic position\nr, the trace is over the di\u000berent con\fgurations of the lo-\ncal spin moments, fsdg, fsdgr\u001b=jfsdgi\n e\nr\u001bis the\nquasi-particle wave-function associated with the conduc-tion electron (  e) entangled to the FM states ( jfsdgi),\nJsdis thes\u0000dexchange interaction, 1sis the identity ma-\ntrix in spin con\fguration space, and ^ \u001bx;y;z are the Pauli\nmatrices. We use the convention that, except for r, bold3\nsymbols represent operators in the magnetic con\fgura-\ntion space and symbols with hat represent operators in\nthe single particle Hilbert space of the conduction elec-\ntrons. The magnetic Hamiltonian HMis given by\nHM=\u0000g\u0016BX\nr~Bext(r)\u0001~sd(r) (2)\n\u0000X\nhr;r0iJdd\nrr0\ns2\nd~sd(r0)\u0001~sd(r)\u0000X\nrJsd\nsd~sc(r)\u0001~sd(r);\nwhere, the \frst term is the Zeeman energy due to the\nexternal magnetic \feld, the second term is the magnetic\ncoupling between the local moments and the third term\nis the energy associated with the intrinsic magnetic \feld\nacting on the local moment, ~sd(r), induced by the local\nspin of the conduction electrons, ~sc(r).\nThe Rashba model of a two-dimensional electron gas\nwith spin orbit coupling interacting with a system of\nlocalized magnetic moments has been extensively em-\nployed14,27,28to describe the e\u000bect of enhanced spin-orbit\ncoupling solely at the interface on the current-induced\ntorques in ultrathin ferromagnetic (FM)/heavy metal\n(HM) bilayers. The e\u000bects of (i) the ferromagnet induc-\ning a moment in the HM and (ii) the HM with strong\nspin-orbit coupling inducing a large spin-orbit e\u000bect in\nthe ferromagnet (Rashba spin-orbit coupling) lead to a\nthin layer where the magnetism and the spin-orbit cou-\npling coexist.27\nThe single-electron tight-binding Hamiltonian29for\nthe conduction electrons of the 2D Rashba plane, H\u001b\u001b0\nrr0\nwhich is \fnite along the transport direction xand in\fnite\nalong theydirection is of the form,\n^H\u001b\u001b0\nxx0(kya) = [tcos(kya)\u000e\u001b\u001b0\u0000tsosin(kya)\u001bx\n\u001b\u001b0]\u000exx0(3)\n+t(\u000ex;x0+1+\u000ex+1;x0)\u000e\u001b\u001b0+itso(\u000ex;x0+1\u0000\u000ex+1;x0)\u001by\n\u001b\u001b0:\nHere,x;x0denote atomic coordinates along the trans-\nport direction, ais the in-plane lattice constant, and tso\nis the Rashba SOI strength. The values of the local e\u000bec-\ntive exchange interaction, Jsd= 1eV, and of the nearest-\nneighbor hopping matrix element, t=1 eV, represent a\nrealistic choice for simulating the exchange interaction of\n3dferromagnetic transition metals and their alloys (Fe,\nCo).30{32The Fermi energy, EF=3.1 eV, is about 1 eV\nbelow the upper band edge at 4 eV consistent with the\nab initio calculations of the (111) Pt surface33. Further-\nmore, we have used tso=0.5 eV which yields a Rashba\nparameter, \u000bR=tsoa\u00191.4 eV \u0017A (a=2.77 \u0017A is the in-\nplane lattice constant of the (111) Pt surface) consis-\ntent with the experimental value of about 1-1.5 eV \u0017A34\nand the ab initio value of 1 eV \u0017A28. However, because\nother experimental measurements for Pt/Co/Pt stacks\nreport35a Rashba parameter which is an order of mag-\nnitude smaller, in Fig.3 we show the damping rate for\ndi\u000berent values of the Rashba SOI.. For the results in\nSec. IV, we assume a real space tight binding for propa-\ngation along y-axis.The single particle propagator of the coupled electron-\nspin system is determined from the equation of motion\nof the retarded Green function,\n\u0012\nE\u0000i\u0011\u0000^\u0016\u0000^H\u0000HM\u0000Jsd\n2^~ \u001b\u0001^~sd\u0013\n^Gr(E) =^1;(4)\nwhere,\u0011is the broadening of the conduction electron\nstates due to inelastic scattering from defects and/or\nphonons, and for simplicity we ignore writing the identity\nmatrices ^1 and 1in the expression. The density matrix\nof the entire system consisting of the noninteracting elec-\ntrons (fermionic quasi-particles) and the local magnetic\nspins is determined (see Appendix A for details of the\nderivation for a single FM domain) from the expression,\n^\u001a=ZdE\n\u0019^Gr(E)\u0011f(E\u0000^\u0016\u0000HM)^Ga(E): (5)\nIt is important to emphasize that Eq. (5) is the central\nresult of this formalism which demonstrates that the ef-\nfect of the local magnetic degrees of freedom is to shift the\nchemical potential of the Fermi-Dirac distribution func-\ntion by the eigenvalues, \"m, ofHMjmi=\"mjmi, i.e.,\nthe excitation energies of the FM from its ground state.\nHere,jmiare the eigenstates of the Heisenberg model\ndescribing the FM. The density matrix can then be used\nto calculate the local spin density operator of the con-\nduction electrons, [ ~sc(r)]mm0=P\nss0\u001amm0\nss0;rr~ \u001bss0=2, which\nalong with Eqs. (2), (4), and (5) form a closed set\nof equations that can be solved self consistently. Since,\nthe objective of this work is the damping/anti-damping\n(transitional) behavior of the FM in the presence of bias\nvoltage, we only present results for the \frst iteration.\nEq. (5) shows that the underlying mechanism of the\ndamping phenomenon is the \row of conduction electrons\nfrom states of higher chemical potential to those of lower\none where the FM state relaxes to its ground state by\ntransferring energy to the conduction electrons. There-\nfore, the FM dynamical properties in this formalism is\ncompletely governed by its coupling to the conduction\nelectrons, where conservation of energy and angular mo-\nmentum dictates the excitations as well as the \ructua-\ntions of the FM sate through the Fermi distribution func-\ntion of the electrons coupled to the reservoirs. This is\ndi\u000berent from the conventional Boltzmann distribution\nfunction which is commonly used to investigate the ther-\nmal and quantum \ructuations of the magnetization.\nDue to the fact that the number of magnetic con\fgura-\ntions (i.e. size of the FM Hilbert space) grows exponen-\ntially with the dimension of the system it becomes pro-\nhibitively expensive to consider all possible eigenstates\nof theHMoperator. Thus, in the following sections we\nconsider two opposite limiting cases of magnetic con\fg-\nurations. In the \frst case we assume a single magnetic\nmoment for the whole FM which is valid for small FMs\nwith strong exchange coupling between local moments\nand small MAE. In this case the dynamics is mainly gov-\nerned by the FM rotational modes and local spin \rips can4\nFIG. 2: (Color online) Schematic representation of the quasi-\nparticles of the FM and conduction electron entangled states.\nThe horizontal planes denote the eigenstates, jS;miof the\ntotalSzof the FM with eigenvalues m=\u0000S;\u0000S+ 1;:::; +S\nalong the auxiliarym-direction. Excitation of magnetic state\ninduces a shift of the chemical potential of the Fermi-Dirac\ndistribution function leading to \row of quisiparticles along the\nm-direction which corresponds to the damping rate of the FM.\nThe FM damping involves two processes: (1) An intra-plane\nprocess involving spin reversal of the conduction electron via\nthe SOC; and (2) An inter-plane process involving quasiparti-\ncle \row of majority (minority) spin along the ascending (de-\nscending)m-direction due to conservation of total angular\nmomentum, where the interlayer hopping is accompanied by\na spin \rip of conduction electrons.\nbe ignored. In the second case we ignore the correlation\nbetween di\u000berent local moments and employ a mean \feld\napproximation such that at each step we focus on an indi-\nvidual atom by considering the local moment under con-\nsideration as a quantum mechanical object while the rest\nof the moments are treated classically. We should men-\ntion that a more accurate modeling of the system should\ncontain both single domain rotation of the FM as well\nas the local spin \ripping but also the e\u000bect of nonlocal\ncorrelations between the local moments and conduction\nelectrons, which are ignored in this work.\nIII. SINGLE DOMAIN ROTATIONAL\nSWITCHING\nIn the regime where the energy required for the excita-\ntion of a single local spin moment ( \u0019meV ) is much larger\nthan the MAE (\u0019\u0016eV) the low-energy excited states cor-\nrespond to rotation of the total angular momentum of the\nFM acting as a single domain and the e\u000bects of local spin\n\rips described as the second term in Eq 2, can be ignored.\nIn this regime all of the local moments behave collectively\nand the local moment operators can be replaced by the\naverage spin operator, ~sd(r) =P\nr0~sd(r0)=Nd=sd~S=S,\nwhereNdis the number of local moments and ~Sis the\ntotal angular momentum with amplitude S. The mag-\nnetic energy operator is given by HM=\u0000~B\u0001S, where,\n~B=g\u0016B~Bext+Jsd~ sc. Here, for simplicity we assume\n~ scto be scalar and independent of the FM state. The\neigenstates of HMoperator are then simply the eigen-states,jS;mi, of the total angular momentum Sz, with\neigenvalues m!=\u0000S!;:::; +S!, where!=Bzis the\nLarmor frequency. Thus, the wave function of the cou-\npled electron-spin con\fguration system, shown schemat-\nically in Fig. 2 is of the form,  ms0r(t) =jS;mi\n s0r(t).\nOne can see that the magnetic degrees of freedom corre-\nsponding to the di\u000berent eigenstates of the Szoperator,\nenters as an additional auxiliary dimension for the elec-\ntronic system where the variation of the magnetic energy,\nhS;mjHMjS;mi=m!, shifts the chemical potentials of\nthe electrons along this dimension. The gradient of the\nchemical potential along the auxiliary direction, is the\nLarmor frequency ( \u0016eV\u0019GHz ) which appears as an\ne\u000bective \\electric \feld\"in that direction.\nSubstituting Eq (5) in Eq (A1)(b) and averaging over\none precession period we \fnd that the average rate of\nangular momentum loss/gain, which we refer to as the\ne\u000bective \\ damping rate \"per magnetic moment, can be\nwritten as\nTm=1\n2=(T\u0000\nm\u0000T+\nm); (6)\nwhere,\nT\u0006\nm=Jsd\n2SNdTrel[^\u001b\u0007S\u0006\nm^\u001am;m\u00061]: (7)\nis the current along the auxiliarym-direction in Fig. 2\nfrom them$m+ 1 (\u0006sign) state of the total Szof the\nFM. Here,Trel, is the trace over the conduction electron\ndegrees of freedom, and S\u0006\nm=p\nS(S+ 1)\u0000m(m\u00061)\nare the ladder operators. It is important to note that\nwithin this formalism the damping rate is simply the net\ncurrent across the mth-layer along the auxiliary direction\nassociated with the transition rate of the FM from state\nmto its nearest-neighbor states ( m\u00061).\nFig. 3 shows the damping rate as a function of the pre-\ncession cone angle, \u0012= cos\u00001(m\nS), for di\u000berent values of\nbias and for an in-plane e\u000bective magnetic \feld (a) along\nand (b) normal to the transport direction, and (c) an out-\nof-plane magnetic \feld. For cases (a) and (c) the damp-\ning rate is negative and relatively independent of bias for\nlow bias values. A negative damping rate implies that the\nFM relaxes towards the magnetic \feld by losing its angu-\nlar momentum, similar to the Gilbert damping rate term\nin the classical LLG equation, where its average value\nover the azimuthal precession angle, '=!t, is of the\nform,T=\u0000\u000bsdRd'\n2\u0019~ m\u0002(~ m\u0002~B)\u0001~ nM, which is nonzero\n(zero) when the unit vector ~ nMis along (perpendicular\nto) the e\u000bective magnetic \feld. The dependence of the\ndamping rate on the bias voltage when the e\u000bective mag-\nnetic \feld~Bis inplane and normal to the transport direc-\ntion can be understood by the spin-\rip re\rection mech-\nanism accompanied by Rashba spin-momentum locking\ndescribed in Ref.16. One can see that a large enough bias\ncan result in a sign reversal of the damping rate and hence\na magnetization reversal of the FM. It's worth mention-\ning that due to the zero-point quantum \ructuations of5\nthe magnetization, at \u0012= 0;\u0019(i.e.m=\u0006S) we have\nT 6=0 which is inversely proportional to the size of the\nmagnetic moment, S.\nIn Fig. 4(a) we present the e\u000bective damping rate ver-\nsus bias for di\u000berent values of the Rashba SOC. The re-\nsults show a linear response regime with respect to the\nbias voltage where both the zero-bias damping rate and\nthe slope,dT=dV increases with the Rashba SOC. This\nis consistent with Kambersky's mechanism of Gilbert\ndamping due to the SOC of itinerant electrons,24and the\nSOT mechanism16. Fig. 4(b) shows that in the absence\nof bias voltage the damping rate is proportional to t2\nsoand\nthe e\u000bect of the spin current pumped into the left and\nright reservoirs is negligible. This result of the t2\nsodepen-\ndence of the zero-bias damping rate is in agreement with\nrecent calculations of Costa and Muniz25and Edwards26\nwhich took into account the collective excitations. In the\npresence of an external bias, Tvaries linearly with the\nSOC, suggesting that to the lowest order it can be \ftted\nto\nT= sin2(\u0012)tso(c1tso~!+c2eVbias); (8)\nwherec1andc2are \ftting parameters.\nThe bias-induced e\u000eciency of the anti-damping SOT,\n\u0002\u0011~!(T(Vbias)\u0000T(0))=eVbiasT(0), describes how e\u000e-\ncient is the energy conversion between the magnetization\ndynamics and the conduction electrons. Accordingly, for\na given bias-induced e\u000eciency, \u0002, one needs to apply an\nexternal bias equal to ~!=e\u0002 to overcome the zero-bias\ndamping of the FM. Fig. 5 displays the anti-damping ef-\n\fciency versus the position of the Fermi energy of the FM\nfrom the bottom (-4 t=-4 eV) to the top (4 t=4eV) of the\nconduction electron band for the two-dimensional square\nlattice. The result is independent of the bias voltage and\nthe Larmor frequency in the linear response regime ( i.e.\nVbias;!\u001ct). We \fnd that the e\u000eciency peaks when the\nFermi level is in the vicinity of the bottom or top of the\nenergy band where the transport is driven by electron- or\nhole-like carriers and the Gilbert damping is minimum.\nThe sign reversal of the antidamping SOT is due to the\nelectron- or hole-like driven transport similar to the Hall\ne\u000bect.36\nClassical Regime of the Zero Bias Damping rate |\nIn the following we show that in the case of classical\nmagnetic moments ( S!1 ) and the adiabatic regime\n(!!0), the formalism developed in this paper leads to\nthe conventional expressions for the damping rate. In this\nlimit the system becomes locally periodic and one can\ncarry out a Fourier transformation from m\u0011Szspace\nto azimuthal angle of the magnetization orientation, ',\nspace. Conservation of the angular momentum suggests\nthat the majority- (minority-) spin electrons can propa-\ngate only along the ascending (descending) m-direction,\nwhere the hopping between two nearest-neighbor m-\nlayers is accompanied by a spin-\rip. As shown in Fig.\n2 the existence of spin-\rip hopping requires the presence\nof intralayer SOC-induced noncollinear spin terms which\nrotate the spin direction of the conduction electrons as\n04590135180−10−505B=[|B|,0,0]Damping Rate ( µeV)\n  \n4590135180B=[0,|B|,0]\nCone Angle, θ (deg)4590135180B=[0,0,|B|]\nVbias=−5 mV\nVbias=0\nVbias=5 mV(a) (b) (c)\nStudent Version of MATLABFIG. 3: (Color online) E\u000bective damping rate for a single\nFM domain as a function of the precession cone angle, \u0012, for\nvarious bias values under an e\u000bective magnetic \feld which is\nin-plane (a) along and (b) normal to the transport direction\nand (c) out-of-plane. The length of the FM along the xdi-\nrection isLx= 25awhile it is assumed to be in\fnite in the\ny-direction, ~!= 10\u0016eV, the broadening parameter \u0011= 0,\nkBT= 10meV and the domain magnetic moment S= 200.\nThe results are robust with larger values of Sin either the\nballistic,\u0011\u001c~!, or dirty,\u0011\u001d~!, regimes.\n−2−1012−3−2−101\nBias Voltage (mV)Damping Rate ( µeV)\n  \n−0.5 0 0.5−15−10−505\nSpin Orbit Coupling, tso (eV)  \ntso=0\ntso=0.1 eV\ntso=0.2 eVVbias=−5 mV\nVbias=0\nVbias=5 mV(a) (b)\nStudent Version of MATLAB\nFIG. 4: (Color online) Damping torque versus (a) bias voltage\nand (b) spin-orbit coupling strength, for m= 0 corresponding\nto the precession cone angle of 90o. The precession axis of the\nFM is along the y-direction and the rest of the parameters are\nthe same as in Fig. 3. The zero-bias damping rate versus SOC\nshows at2\nsodependence while the damping rate under non-\nzero bias exhibits nearly linear SOC dependence.\nthey propagate in each m-layer. This is necessary for\nthe persistent \row of electrons along the 'auxiliary di-\nrection and therefore damping of the magnetization dy-\nnamics. Using the Drude expression of the longitudinal\nconductivity along the '-direction for the damping rate,\nwe \fnd that, within the relaxation time approximation,\n\u0011=!!1 , where the relaxation time of the excited con-\nduction electrons is much shorter than the time scale of6\n−4−3−2−101234−3\n−2\n−1\n0\n1\n2\n3\nFermi Energy (eV)Antidamping Efficiency (%)\n−4−2024−40−200Damping Rate ( µeV)\n  \nVbias=5 mV\nVbias=0\nVbias=−5 mV\nStudent Version of MATLAB\nFIG. 5: (Color online) Bias-induced precessional anti-\ndamping e\u000eciency, \u0002 = ~!(T(Vbias)\u0000T(0))=eVbiasT(0), ver-\nsus the Fermi energy of the 2D Rashba plane in Fig.1, where\nthe energy band ranges from -4 eV to +4 eV. The magnetiza-\ntion precesses around the in-plane direction ( y\u0000axis) normal\nto the transport direction and the rest of the parameters of\nthe system are the same as in Fig. 3. Note, for magnetization\nprecession around the xandzaxis,T(Vbias) =T(0) for all\nprecession cone angles and hence \u0002=0. Inset shows the damp-\ning rate versus the Fermi energy for di\u000berent bias values used\nto calculate precessional anti-damping e\u000eciency.\nthe FM dynamics, Tis given by\nT=\u0000!\n\u0011X\nnZdkxdkyd'\n(2\u0019)3(v'\nn~k)2f0(\"n~k(')):(9)\nHere,v'\nn~k=@\"n~k(')=@' is the group velocity along the\n'-direction in Fig. 2, and \"n;~k=\"0(j~kj)\u0006j~h(~k)jfor the\n2D-Rashba plane, where \"0(j~kj) is the spin independent\ndispersion of the conduction electrons and ~h=atso^ez\u0002\n~k+1\n2Jsd~ m, is the spin texture of the electrons due to\nthe SOC and the s\u0000dexchange interaction. For small\nprecession cone angle, \u0012, the Gilbert damping constant\ncan be determined from \u000b=\u0000T=sd!sin2(\u0012), where the\nzero-temperature Tis evaluated by Eq. (9). We \fnd\nthat\n\u000b\u00191\n\u0011t2\nso\u0002\n(k+\nFa)2D+(EF) + (k\u0000\nFa)2D\u0000(EF)\u0003\n(1+cos2(\r));\n(10)\nwhereD+(\u0000)(E) is the density of states of the majority\n(minority) band, \ris the angle between the precession\naxis and the normal to the Rashba plane, and the Fermi\nwave-vectors ( k\u0006\nF) are obtained from, \"0(k\u0006\nF) =EF\u0007\nJsd=2. Eq. (10) shows that the Gilbert damping increases\nas the precession axis changes from in-plane ( \r=\u0019=2) to\nout of plane ( \r= 0),37which can also be seen in Fig. 3.\nIt is important to emphasize that in contrast to Eq. (9)\nthe results shown in Fig. 4 correspond to the ballistic\nregime with \u0011= 0 in the central region and the relaxation\nof the excited electrons occurs solely inside the metallic\nreservoirs. To clarify how the damping rate changes from\n10−810−610−410−2100−80−60−40−20020\nBroadening (eV)Damping Rate ( µeV)\n  \nS=200, Vbias=0\nS=200, Vbias=3 mV\nS=300, Vbias=0\nS=300, Vbias=3 mV\nStudent Version of MATLABFIG. 6: (Color online) Precessional damping rate versus\nbroadening of the states in the presence (solid lines) and ab-\nsence (dashed lines) of bias voltage for two values of the do-\nmain sizeS= 200 and S= 300. In both ballistic, \u0011=!\u00190,\nand di\u000busive, \u0011=!\u001d1, regimes the precessional damping rate\nis independent of the domain size, while in the intermediate\ncase, the amplitude of the minimum of damping rate shows a\nlinear dependence versus S. Note that the value of the broad-\nening at which the damping rate is minimum varies inversely\nproportional to the domain size, S.\nthe ballistic to the di\u000busive regime we present in Fig.\n6 the damping rate versus the broadening, \u0011, of states\nin the presence (solid line) and absence (dashed line) of\nbias voltage. We \fnd that in both ballistic ( \u0011=!\u00190)\nand di\u000busive ( \u0011=!\u001d1) regimes the damping rate is\nindependent of the size of the FM domain, S. On the\nother hand, in the intermediate regime the FM dynamics\nbecome strongly dependent on the e\u000bective domain size\nwhere the minimum of the damping rate varies linearly\nwithS. This can be understood by the fact that the\ne\u000bective chemical potential di\u000berence between the \frst,\nm=\u0000Sand last,m=Slayers in Fig.3 is proportional\ntoSand for a coherent electron transport the conduc-\ntance is independent of the length of the system along\nthe transport direction. Therefore, in this case the FM\nmotion is driven by a coherent dynamics.\nIV. DEMAGNETIZATION MECHANISM OF\nSWITCHING\nIn Sec. III we considered the case of a single FM do-\nmain where its low-energy excitations, involving the pre-\ncession of the total angular momentum, can be described\nby the eigenstates jmiofSzand local spin \rip processes\nwere neglected. However, for ultrathin FM \flms or FM\nnanoclusters, where the MAE per atom ( \u0019meV ) is com-\nparable to the exchange energy between the local mo-\nments (Curie temperature), the low-energy excitations\ninvolve both magnetization rotation and local moments\nspin-\rips due to conduction electron scattering which can\nin turn change also S. In this case the switching is ac-7\nFIG. 7: (Color online) Spatial dependence of the local damp-\ning rate for the spin-1 =2 local moments of a FM island under\ndi\u000berent bias voltages ( \u00060:4V) and magnetization directions.\nFor the parameters we chose the size of the FM island to be\n25\u000225a2, the e\u000bective magnetic \feld, jBj= 20meV , the\nbroadening, \u0011= 0, andkBT= 10 meV.\ncompanied by the excitation of local collective modes that\ne\u000bectively lowers the amplitude of the magnetic ordering\nparameter. For simplicity we employ the mean \feld ap-\nproximation for the 2D FM nanocluster where the spin\nunder consideration at position ris treated quantum me-\nchanically interacting with all remaining spins through\nan e\u000bective magnetic \feld, ~B. The spatial matrix ele-\nments of the local spin operator are\n[^~sd;r]r1r2=~ sd(r1)\u000er1r2(1\u0000\u000er1r)1s+1\n2\u000er1r2\u000er1r~ \u001c;(11)\nwhere,~\u001cs are the Pauli matrices. The magnetic energy\ncan be expressed as, HM(r) =\u0000~B(r)\u0001~\u001c=2, where, the\ne\u000bective local magnetic \feld is given by,\n~B(r) =g\u0016B~Bext+ 4X\nr0Jdd\nrr0~ sd(r0) + 2Jsd~ sc(r):(12)\nThe equation of motion for the single particle propa-\ngator of the electronic wavefunction entangled with the\nlocal spin moment under consideration can then be ob-\ntained from,\n\u0012\nE\u0000^\u0016\u0000HM(r)\u0000^H\u0000Jsd\n2^~ \u001b\u0001^~sd;r\u0013\n^Gr\nr(E) =^1:(13)\nThe density matrix is determined from Eq. (5) which\ncan in turn be used to calculate the spin density of the\nconduction electrons, ~ sc(r) =Tr(^~ \u001b^\u001arr)=2, and the di-\nrection and amplitude of the local magnetic moments,\n~ sd(r) =Tr(~\u001c^\u001arr)=2.\nFig. 7 shows the spatial dependence of the spin-1\n2local\nmoment switching rate for a FM/Rashba bilayer (Fig.\n−101−30−20−100102030\n  Damping Torque (meV)B=[|B|,0,0]\n−101\nBias Voltage (V)B=[0,|B|,0]\n−101B=[0,0,|B|]\n|B|=1 meV\n|B|=20 meV(c) (a) (b)\nStudent Version of MATLABFIG. 8: (Color online) Bias dependence of the average (over\nall sites) damping rate of the FM island for in-plane e\u000bective\nmagnetic \feld (or equilibrium magnetization) (a) along and\n(b) normal to the transport direction and (c) out-of-plane\nmagnetic \feld for two values of jBj.\n1) for two bias values ( Vbias=\u00060:4V) and for an in-\nplane e\u000bective magnetic \feld (a) along and (b) normal\nto the transport direction, and (c) an out-of-plane mag-\nnetic \feld. The size of the FM island is 25 a\u000225a, where\nais the lattice constant. Negative local moment switch-\ning rate (blue) denotes that, once excited, the local mo-\nment relaxes to its ground state pointing along the di-\nrection of the e\u000bective magnetic \feld; however positive\nlocal damping rate (red) denotes that the local moments\nremain in the excited state during the bias pulse dura-\ntion. Therefore, the damping rate of the local moments\nunder bias voltage can be either enhanced or reduced\nand even change sign depending on the sign of the bias\nvoltage and the direction of the magnetization. We \fnd\nthat the bias-induced change of the damping rate is high-\nest when the FM magnetization is in-plane and normal to\nthe transport directions similar to the single domain case.\nFurthermore, the voltage-induced damping rate is peaked\nclose to either the left or right edge of the FM (where the\nreservoirs are attached) depending on the sign of the bias.\nNote that there is also a \fnite voltage-induced damping\nrate when the magnetization is in-plane and and along\nthe transport direction ( x) or out-of-the-plane ( z).\nFig. 8 shows the bias dependence of the average (over\nall sites) damping rate for in- (a and b) and out-of-plane\n(c) directions of the e\u000bective magnetic \feld (direction\nof the equilibrium magnetization) and for two values of\njBj. This quantity describes the damping rate of the\namplitude of the magnetic order parameter. For an in-\nplane magnetization and normal to the transport direc-\ntion (Fig. 8) the bias behavior of the damping rate is lin-\near and \fnite in contrast to the single domain [Fig. 3(a)]\nwhere the damping rate was found to have a negligible\nresponse under bias. On the other hand, the bias behav-\nior of the current induced damping rate shows similar\nbehavior to the single domain case when the equilibrium8\n−4 −2 0 2 4−30−20−100102030\nFermi Energy (eV)Antidamping Efficiency (%)\n  \nB=[20,0,0] meV\nB=[0,20,0] meV\nB=[0,0,20] meV\nStudent Version of MATLAB\nFIG. 9: (Color online) Bias-induced local anti-damping\ne\u000eciency due to local spin-\rip, \u0002 = jBj(T(Vbias)\u0000\nT(0))=eVbiasT(0), versus Fermi energy for di\u000berent equilib-\nrium magnetization orientations. For the calculation we chose\nVbias= 0:2 V and the rest of the Hamiltonian parameters are\nthe same as in Fig. 7.\nmagnetization direction is in-plane and normal to the\ntransport direction (Fig. 8(b)). For an out-of-plane ef-\nfective magnetic \feld [Fig. 8(c)] the damping torque has\nan even dependence on the voltage bias.\nIn order to quantify the e\u000eciency of the voltage in-\nduced excitations of the local moments, we calculate the\nrelative change of the average of the damping rate in the\npresence of a bias voltage and present the result versus\nthe Fermi energy for di\u000berent orientations of the magne-\ntization in Fig 9. We \fnd that the e\u000eciency is maximum\nfor an in-plane equilibrium magnetization normal to the\ntransport direction and it exhibits an electron-hole asym-\nmetry. The bias-induced antidamping e\u000eciency due to\nspin-\rip can reach a peak around 20% which is much\nhigher than the peak e\u000eciency of about 2% in the sin-\ngle domain precession mechanism in Fig. 5 for the same\nsystem parameters.\nFuture work will be aimed in determining the switch-\ning phase diagram16by calculating the local antidamping\nand \feld-like torques self consistently for di\u000berent FM\ncon\fgurations.\nV. CONCLUDING REMARKS\nIn conclusion, we have developed a formalism to in-\nvestigate the current-induced damping rate of nanoscale\nFM/SOC 2D Rashba plane bilayer in the quantum\nregime within the framework of the Kyldysh Green func-\ntion method. We considered two di\u000berent regimes of FM\ndynamics, namely, the single domain FM and indepen-\ndent local moments regimes. In the \frst regime we as-\nsume the rotation of the FM as the only degree of free-\ndom, while the second regime takes into account only\nthe local spin-\rip mechanism and ignores the rotation ofthe FM. When the magnetization (precession axis) is in-\nplane and normal to the transport direction, similar to\nthe conventional SOT for classical FMs, we show that the\nbias voltage can change the damping rate of the FM and\nfor large enough voltage it can lead to a sign reversal. In\nthe case of independent spin-1 =2 local moments we show\nthat the bias-induced damping rate of the local quantum\nmoments can lead to demagnetization of the FM and has\nstrong spatial dependence. Finally, in both regimes we\nhave calculated the bias-induced damping e\u000eciency as a\nfunction of the position of the Fermi energy of the 2D\nRashba plane.\nAppendix A: Derivation of Electronic Density\nMatrix\nUsing the Heisenberg equation of motion for the an-\ngular momentum operator, ~S(t), and the commutation\nrelations for the angular momentum, we obtain the fol-\nlowing Landau-Lifshitz equations of motion,\n\u0007i@\n@tS\u0006(t) =hzS\u0006(t)\u0000h\u0006(t)Sz(t) (A1a)\n\u0000i@\n@tSz(t) =1\n2\u0000\nh+(t)S\u0000(t)\u0000h\u0000(t)S+(t)\u0001\n(A1b)\n~hmm0(t) =1\n~X\nrJsd~smm0\nc(r) +g\u0016B\u000emm0~B(t);(A1c)\nwhere,S\u0006=Sx\u0006Sy(\u001b\u0006=\u001bx\u0006\u001by), is the angu-\nlar momentum (spin) ladder operators, ~smm0\nc(r) =\n1\n2P\n\u001b\u001b0~ \u001b\u001b\u001b0\u001amm0\n\u001b\u001b0;rris the local spin density of the con-\nduction electrons which is an operator in magnetic con-\n\fguration space. Here, \u001ais the density matrix of the\nsystem, and the subscripts, r;m;\u001b refer to the atomic\ncite index, magnetic state and spin of the conduction\nelectrons, respectively. In the following we assume a pre-\ncessing solution for Eq (A1)(a) with a \fxed cone angle\nand Larmor frequency !=hz. Extending the Hilbert\nspace of the electrons to include the angular momentum\ndegree of freedom we de\fne  ms0i(t) =jS;mi\n s0i(t).\nThe equation of motion for the Green function (GF) is\nthen given\n\u0010\nE\u0000i\u0011\u0000^H(k) +n!\u0000n\n2SJsd(k)\u001bz\u0011\n^Gr\nnm(E;k) (A2)\n\u0000p\nS(S+ 1)\u0000n(n+ 1)\n2SJsd(k)\u001b\u0000^Gr\nn+1m(E;k)\n\u0000p\nS(S+ 1)\u0000n(n\u00001)\n2SJsd(k)\u001b+^Gr\nn\u00001m(E;k) =^1\u000enm\nwhere,n= (\u0000S;\u0000S+1;:::;S ) and the gauge transforma-\ntion n\u001bi(t)!ein!t n\u001bi(t) has been employed to remove\nthe time dependence. The density matrix of the system\nis of the form\n^\u001anm=e\u0000i(n\u0000m)!tSX\np=\u0000SZdE\n2\u0019^Gr\nnp2\u0011fp^\u0016^Ga\npm (A3)9\nwhere,fp^\u0016(E) =f(E\u0000p!\u0000^\u0016) is the equilibrium Fermi\ndistribution function of the electrons. Due to the fact\nthatp!are the eigenvalues of HM=\u0000g\u0016B~B\u0001S, one\ncan generalize this expression by transforming into a ba-\nsis where the magnetic energy is not diagonal which in\nturn leads to Eq (5) for the density matrix of the con-\nduction electron-local moment entagled system.\nAppendix B: Recursive Relation for GFs\nSince in this work we are interested in diagonal blocks\nof the GFs and in general for FMs at low temperaturewe haveS\u001d1, we need a recursive algorithm to be able\nto solve the system numerically. The surface Keldysh\nGFs corresponding to ascending ^ gu;r=<, and descending\n^gd;r=<, recursion scheme read,\n^gu;r\nn(E;k) =1\nE\u0000!n\u0000i\u0011n\u0000^H(k)\u0000^\u0006rn(E;k)\u0000n\n2SJsd(k)\u001bz\u0000(S\u0000\nn)2\n4S2Jsd(k)\u001b+^gu;r\nn\u00001(E;k)\u001b\u0000Jsd(k)(B1)\n^\u0006u;<\nn(E;k) =\u0000X\n\u000b\u0010\n2i\u0011n+^\u0006r\nn;\u000b(E;k)\u0000^\u0006a\nn;\u000b(E;k)\u0011\nfn\u000b+(S\u0000\nn)2\n4S2Jsd\u001b+^gu;r\nn\u00001^\u0006u;<\nn\u00001^gu;a\nn\u00001\u001b\u0000Jsd (B2)\n^gd;r\nn(E;k) =1\nE\u0000!n\u0000i\u0011n\u0000^\u0006rn(E;k)\u0000^H(k)\u0000n\n2SJsd(k)\u001bz\u0000(S+\nn)2\n4S2Jsd(k)\u001b\u0000^gu;r\nn+1(E;k)\u001b+Jsd(k)(B3)\n^\u0006d;<\nn(E;k) =\u0000X\n\u000b\u0010\n2i\u0011n+^\u0006r\nn;\u000b(E;k)\u0000^\u0006a\nn;\u000b(E;k)\u0011\nfn\u000b+(S+\nn)2\n4S2Jsd\u001b\u0000^gd;r\nn+1^\u0006d;<\nn+1^gd;a\nn+1\u001b+Jsd (B4)\nwhere, ^\u0006r\nn(E;k) =P\n\u000b^\u0006r\n\u000b(E\u0000!n;k) corresponds to the\nself energy of the leads, \u000b=L;R refers to the left and\nright leads in the two terminal device in Fig. 3 and S\u0006\nm=p\nS(S+ 1)\u0000m(m\u00061). Using the surface GFs we can\ncalculate the GFs as follows,\n^Gr\nn;m(E;k) =1\nE\u0000!n\u0000i\u0011n\u0000^H(k)\u0000^\u0006rn\u0000n\n2SJsd(k)\u001bz\u0000^\u0006r;u\nn\u0000^\u0006r;d\nn; n =m (B5)\n=S+\nn\n2S^gu;r\nn(E;k)Jsd(k)\u001b\u0000^Gr\nn+1;m(E;k); n6=m (B6)\n=S\u0000\nn\n2S^gd;r\nn(E;k)Jsd(k)\u001b+^Gr\nn\u00001;m(E;k); n6=m (B7)\nwhere the ascending and descending self energies are given by,\n^\u0006r;u\nn=(S\u0000\nn)2\n4S2Jsd(k)\u001b+^gu;r\nn\u00001(E;k)\u001b\u0000Jsd(k) (B8)\n^\u0006r;d\nn=(S+\nn)2\n4S2Jsd(k)\u001b\u0000^gd;r\nn+1(E;k)\u001b+Jsd(k) (B9)\nThe average rate of angular momentum loss/gain can be obtained from the real part of the loss of angular momentum\nin one period of precession,\nT0\nn=1\n2(T0\u0000\nn\u0000T0+\nn) =1\n2= X\nkTr[S\u0000\nn\n2S\u001b+Jsd(k)^\u001ann+1(k)\u0000S+\nn\n2S\u001b\u0000Jsd(k)^\u001ann\u00001(k)]!\n(B10)10\nwhich can be interpreted as the current \rowing across the layer n.\nT0\u0000=+\nn =X\nkZdE\n2\u0019iTrnh\n^\u0006d=u;r\nn(E;k)\u0000^\u0006d=u;a\nn(E;k)i\n^G<\nnn(E;k) +^\u0006d=u;<\nn (E)h\n^Gr\nnn(E;k)\u0000^Ga\nnn(E;k)io\n;(B11)\nAcknowledgments\nThe work at CSUN is supported by NSF-Partnership\nin Research and Education in Materials (PREM) GrantDMR-1205734, NSF Grant No. ERC-Translational Ap-\nplications of Nanoscale Multiferroic Systems (TANMS)-\n1160504, and US Army of Defense Grant No. W911NF-\n16-1-0487.\n\u0003Electronic address: Farzad.Mahfouzi@gmail.com\nyElectronic address: nick.kioussis@csun.edu\n1J. 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The mechanical damp-\ning in the damped part is present in the shear angle equation only,\nand the damped part may be of arbitrary positive length. We prove\nwell-posedness of the corresponding PDE system in energy space and\nestablish existence of a regular global attractor under certain conditions\non nonlinearities and coefficients of the damped part only. Moreover, we\nstudy singular limits of the problem when l→0 orl→0 simultaneously\nwith ki→+∞and perform numerical modelling for these processes.\n2 Keywords:\nBresse beam, transmission problem, global attractor, singular limit\n1arXiv:2309.15171v2  [math.AP]  22 Jan 20243 Introduction\nIn this paper we consider a contact problem for the Bresse beam. Originally\nthe mathematical model for homogeneous Bresse beams was derived in [ 4].\nWe use the variant of the model described in [ 20, Ch. 3]. Let the whole beam\noccupies a part of a circle of length Land has the curvature l=R−1. We\nconsider the beam as a one-dimensional object and measure the coordinate\nxalong the beam. Thus, we say that the coordinate xchanges within the\ninterval (0 , L). The parts of the beam occupying the intervals (0 , L0) and\n(L0, L) consist of different materials. The part lying in the interval (0 , L0) is\npartially subjected to a structural damping (see Figure 1). The Bresse system\nFigure 1: Composite Bresse beam.\ndescribes evolution of three quantities: transversal displacement, longitudinal\ndisplacement and shear angle variation. We denote by φ,ψ, and ωthe\ntransversal displacement, the shear angle variation, and the longitudinal\ndisplacement of the left part of the beam lying in (0 , L0). Analogously, we\ndenote by u,v, and wthe transversal displacement, the shear angle variation,\nand the longitudinal displacement of the right part of the beam occupying\nthe interval ( L0, L). We assume the presence of the mechanical dissipation in\nthe equation for the shear angle variation for the left part of the beam. We\nalso assume that both ends of the beam are clamped. Nonlinear oscillations\n2of the composite beam can be described by the following system of equations\nρ1φtt−k1(φx+ψ+lω)x−lσ1(ωx−lφ) +f1(φ, ψ, ω ) =p1(x, t), (1)\nβ1ψtt−λ1ψxx+k1(φx+ψ+lω) +γ(ψt) +h1(φ, ψ, ω ) =r1(x, t), x∈(0, L0), t >0,\n(2)\nρ1ωtt−σ1(ωx−lφ)x+lk1(φx+ψ+lω) +g1(φ, ψ, ω ) =q1(x, t), (3)\nand\nρ2utt−k2(ux+v+lw)x−lσ2(wx−lu) +f2(u, v, w ) =p2(x, t), (4)\nβ2vtt−λ2vxx+k2(ux+v+lw) +h2(u, v, w ) =r2(x, t), x ∈(L0, L), t >0,\n(5)\nρ2wtt−σ2(wx−lu)x+lk2(ux+v+lw) +g2(u, v, w ) =q2(x, t), (6)\nwhere ρj, βj, kj, σj, λjare positive parameters, fj, gj, hj:R3→Rare\nnonlinear feedbacks, pj, qj, rj: (0, L)×R3→Rare known external loads,\nγ:R→Ris a nonlinear damping. The system is subjected to the Dirichlet\nboundary conditions\nφ(0, t) =u(L, t) = 0 , ψ(0, t) =v(L, t) = 0 , ω(0, t) =w(L, t) = 0 ,(7)\nthe transmission conditions\nφ(L0, t) =u(L0, t), ψ(L0, t) =v(L0, t), ω(L0, t) =w(L0, t),(8)\nk1(φx+ψ+lω)(L0, t) =k2(ux+v+lw)(L0, t), (9)\nλ1ψx(L0, t) =λ2vx(L0, t), (10)\nσ1(ωx−lφ)(L0, t) =σ2(wx−lu)(L0, t), (11)\nand supplemented with the initial conditions\nφ(x,0) = φ0(x), ψ(x,0) = ψ0(x), ω(x,0) = ω0(x), (12)\nφt(x,0) = φ1(x), ψ t(x,0) = ψ1(x), ω t(x,0) = ω1(x), (13)\n3u(x,0) = u0(x), v(x,0) = v0(x), w (x,0) = w0(x), (14)\nut(x,0) = u1(x), v t(x,0) = v1(x), w t(x,0) = w1(x). (15)\nOne can observe patterns in the problem which appear to have physical\nmeaning:\nQi(ξ, ζ, η ) =ki(ξx+ζ+lη) are shear forces ,\nNi(ξ, ζ, η ) =σi(ηx−lξ) are axial forces ,\nMi(ξ, ζ, η ) =λiζxare bending moments\nfor damped ( i= 1) and undamped ( i= 2) parts respectively. Later we will\nuse them to rewrite the problem in a compact and physically natural form.\nThe paper is devoted to the well-posedness and long-time behaviour of\nthe system (1)-(15). Our main goal is to establish conditions under which\nthe assumed amount of dissipation is sufficient to guarantee the existence of\na global attractor.\nThe paper is organized as follows. In Section 2 we represent functional\nspaces and pose the problem in an abstract form. In Section 3 we prove\nthat the problem is well-posed and possesses strong solutions provided\nnonlinearities and initial data are smooth enough. Section 4 is devoted\nto the main result on the existence of a compact attractor. The nature\nof dissipation prevents us from proving dissipativity explicitly, thus we\nshow that the corresponding dynamical system is of gradient structure and\nasymptotically smooth. We establish the unique continuation property by\nmeans of the observability inequality obtained in [ 27] to prove the gradient\nproperty. The compensated compactness approach is used to prove the\nasymptotic smoothness. In Section 5 we show that solutions to (1)-(15)tend\nto solutions to a transmission problem for the Timoshenko beam when l→0\nand to solutions to a transmission problem for the Euler-Bernoulli beam\nwhen l→0 and ki→ ∞ as well as perform numerical modelling of these\nsingular limits.\n44 Preliminaries and Abstract formulation\n4.1 Spaces and notations\nLet us denote\nΦ1= (φ, ψ, ω ),Φ2= (u, v, w ),Φ = (Φ1,Φ2).\nThus, Φ is a six-dimensional vector of functions. Analogously,\nFj= (fj, gj, hj) :R3→R3, F = (F1, F2),\nPj= (pj, qj, rj) : [(0 , L)×R+]3→R3, P = (P1, P2),\nRj=diag{ρj, βj, ρj}, R =diag{ρ1, β1, ρ1, ρ2, β2, ρ2},\nΓ(Φ t) = (0 , γ(ψt),0,0,0,0),\nwhere j= 1,2. The static linear part of the equation system can be formally\nrewritten as\nAΦ =\n−∂xQ1(Φ1)−lN1(Φ1)\n−∂xM1(Φ1) +Q1(Φ1)\n−∂xN1(Φ1) +lQ1(Φ1)\n−∂xQ2(Φ2)−lN2(Φ2)\n−∂xM2(Φ2) +Q2(Φ2)\n−∂xN2(Φ2) +lQ2(Φ2)\n. (16)\nThen transmission conditions (8)-(11) can be written as follows\nΦ1(L0, t) = Φ2(L0, t),\nQ1(Φ1(L0, t)) =Q2(Φ2(L0, t)),\nM1(Φ1(L0, t)) =M2(Φ2(L0, t)),\nN1(Φ1(L0, t)) =N2(Φ2(L0, t)).\nThroughout the paper we use the notation ||·||for the L2-norm of a function\nand (·,·) for the L2-inner product. In these notations we skip the domain,\non which functions are defined. We adopt the notation || · || L2(Ω)only when\n5domain is not evident. We also use the same notations || · || and (·,·) for\n[L2(Ω)]3.\nTo write our problem in an abstract form we introduce the following spaces.\nFor the velocities of the displacements we use the space\nHv={Φ = (Φ1,Φ2) : Φ1∈[L2(0, L0)]3,Φ2∈[L2(L0, L)]3}\nwith the norm\n||Φ||2\nHv=||Φ||2\nv=2X\nj=1||p\nRjΦj||2,\nwhich is equivalent to the standard L2-norm.\nFor the beam displacements we use the space\nHd={Φ∈Hv: Φ1∈[H1(0, L0)]3,Φ2∈[H1(L0, L)]3,\nΦ1(0, t) = Φ2(L, t) = 0 ,Φ1(L0, t) = Φ2(L0, t)\t\nwith the norm\n||Φ||2\nHd=||Φ||2\nd=2X\nj=1\u0000\n||Qj(Φj)||2+||Nj(Φj)||2+||Mj(Φj)||2\u0001\n.\nThis norm is equivalent to the standard H1-norm. Moreover, the equivalence\nconstants can be chosen independent of lforlsmall enough (see [ 24], Remark\n2.1). If we set\nΨ(x) =(\nΦ1(x), x ∈(0, L0),\nΦ2(x), x ∈[L0, L)\nwe see that there is isomorphism between Hdand [ H1\n0(0, L)]3.\n4.2 Abstract formulation\nThe operator A:D(A)⊂Hv→Hvis defined by formula (16), where\n6D(A) ={Φ∈Hd: Φ1∈H2(0, L0),Φ2∈H2(L0, L), Q1(Φ1(L0, t)) =Q2(Φ2(L0, t)),\nN1(Φ1(L0, t)) =N2(Φ2(L0, t)), M1(Φ1(L0, t)) =M2(Φ2(L0, t))}\nArguing analogously to Lemmas 1.1-1.3 from [ 23] one can prove the following\nlemma.\nLemma 4.1. The operator Ais positive and self-adjoint. Moreover,\n(A1/2Φ, A1/2B) =1\nk1(Q1(Φ1), Q1(B1)) +1\nσ1(N1(Φ1), N1(B1)) +1\nλ1(M1(Φ1), M1(B1))+\n1\nk2(Q2(Φ2), Q2(B2)) +1\nσ2(N2(Φ2), N2(B2)) +1\nλ2(M2(Φ2), M2(B2))\n(17)\nandD(A1/2) =Hd⊂Hv.\nThus, we can rewrite equations (1)-(6) in the form\nRΦtt+AΦ + Γ(Φ t) +F(Φ) = P(x, t), (18)\nboundary conditions (7) in the form\nΦ1(0, t) = Φ2(L, t) = 0 , (19)\nand transmission conditions (8)-(11) can be written as\nΦ1(L0, t) = Φ2(L0, t), (20)\nQ1(Φ1(L0, t)) =Q2(Φ2(L0, t)), (21)\nM1(Φ1(L0, t)) =M2(Φ2(L0, t)), (22)\nN1(Φ1(L0, t)) =N2(Φ2(L0, t)). (23)\nInitial conditions have the form\nΦ(x,0) = Φ 0(x), Φt(x,0) = Φ 1(x). (24)\nWe use H=Hd×Hvas a phase space.\n75 Well-posedness\nIn this section we study strong, generalized and variational (weak) solutions\nto (18)-(24).\nDefinition 5.1. Φ∈C(0, T;Hd)TC1(0, T;Hv)such that Φ(x,0) = Φ 0(x),\nΦt(x,0) = Φ 1(x)is said to be a strong solution to (18)-(24) if\n•Φ(t)lies in D(A)for almost all t;\n•Φ(t)is an absolutely continuous function with values in HdandΦt∈\nL1(a, b;Hd)for0< a < b < T ;\n•Φt(t)is an absolutely continuous function with values in HvandΦtt∈\nL1(a, b;Hv)for0< a < b < T ;\n•equation (18) is satisfied for almost all t.\nDefinition 5.2. Φ∈C(0, T;Hd)TC1(0, T;Hv)such that Φ(x,0) = Φ 0(x),\nΦt(x,0) = Φ 1(x)is said to be a generalized solution to (18)-(24) if there\nexists a sequence of strong solutions Φ(n)to(18)-(24) with the initial data\n(Φ(n)\n0,Φ(n)\n1)and right hand side P(n)(x, t)such that\nlim\nn→∞max\nt∈[0,T]\u0010\n||Φ(n)(·, t)−Φ(·, t)||d+||Φ(n)\nt(·, t)−Φt(·, t)||v\u0011\n= 0.\nWe also need a definition of a variational solution. We use six-dimensional\nvector-functions B= (B1, B2),Bj= (βj, γj, δj) from the space\nFT={B∈L2(0, T;Hd), Bt∈L2(0, T;Hv), B(T) = 0}\nas test functions.\nDefinition 5.3. Φis said to be a variational (weak) solution to (18)-(24) if\n•Φ∈L∞(0, T;Hd),Φt∈L∞(0, T;Hv);\n•Φsatisfies the following variational equality for all B∈FT\n8−TZ\n0(RΦt, Bt)(t)dt−(RΦ1, B(0)) +ZT\n0(A1/2Φ, A1/2B)(t)dt+\nZT\n0(Γ(Φ t), B)(t)dt+ZT\n0(F(Φ), B)(t)dt−ZT\n0(P, B)(t)dt= 0; (25)\n•Φ(x,0) = Φ 0(x).\nNow we state a well-posedness result for problem (18)-(24).\nTheorem 5.4 (Well-posedness) .Let\nfi, gi, hi:R3→Rare locally Lipschitz i.e.\n|fi(a)−fi(b)| ≤L(K)|a−b|,provided |a|,|b| ≤K; (N1)\nthere exist Fi:R3→Rsuch that (fi, hi, gi) =∇Fi;\nthere exists δ >0such that Fj(a)≥ −δfor all a∈R3; (N2)\nP∈L2(0, T;Hv); (R1)\nand the nonlinear dissipation satisfies\nγ∈C(R)and non-decreasing , γ(0) = 0 . (D1)\nThen for every initial data Φ0∈Hd,Φ1∈Hvand time interval [0, T]there\nexists a unique generalized solution to (18)-(24) with the following properties:\n•every generalized solution is variational;\n•energy inequality\nE(T) +ZT\n0(γ(ψt), ψt)dt≤ E(0) +ZT\n0(P(t),Φt(t))dt (26)\n9holds, where\nE(t) =1\n2h\n||R1/2Φt(t)||2+||A1/2Φ(t)||2i\n+LZ\n0F(Φ(x, t))dx\nand\nF(Φ(x, t)) =(\nF1(φ(x, t), ψ(x, t), ω(x, t)), x ∈(0, L0),\nF2(u(x, t), v(x, t), w(x, t)), x ∈(L0, L).\n•If, additionally, Φ0∈D(A),Φ1∈Hdand\n∂tP(x, t)∈L2(0, T;Hv) (R2)\nthen the generalized solution is also strong and satisfies the energy\nequality.\nProof. The proof essentially uses monotone operators theory. It is rather\nstandard by now (see, e.g., [ 9]), so in some parts we give only references to\ncorresponding arguments. However, we give some details which demonstrate\nthe peculiarity of 1D problems.\nStep 1. Abstract formulation. We need to reformulate problem (18)-(24)as\na first order problem. Let us denote\nU= (Φ,Φt), U 0= (Φ 0,Φ1)∈H=Hd×Hv,\nTU= \nI0\n0R−1! \n0−I\nA0!\nU+ \n0\nΓ(Φ t)!\n.\nConsequently, D(T) =D(A)×Hd⊂H. In what follows we use the notations\nB(U) = \nI0\n0R−1! \n0\nF(Φ)!\n,P(x, t) = \n0\nP(x, t)!\n.\n10Thus, we can rewrite problem (18)-(24) in the form\nUt+TU+B(U) =P, U (0) = U0∈H. (27)\nStep 2. Existence and uniqueness of a local solution. Here we use Theorem 7.2\nfrom [9]. For the reader’s convenience we formulate it below.\nTheorem 5.5 ([9]).Consider the initial value problem\nUt+TU+B(U) =f, U (0) = U0∈H. (28)\nSuppose that T:D(T)⊂H→His a maximal monotone mapping, 0∈ T0\nandB:H→His locally Lipschitz, i.e. there exits L(K)>0such that\n||B(U)−B(V)||H≤L(K)||U−V||H,||U||H,||V||H≤K.\nIfU0∈D(T),f∈W1\n1(0, t;H)for all t >0, then there exists tmax≤ ∞ such\nthat(28) has a unique strong solution Uon(0, tmax).\nIfU0∈D(T),f∈L1(0, t;H)for all t >0, then there exists tmax≤ ∞ such\nthat(28) has a unique generaized solution Uon(0, tmax).\nIn both cases\nlim\nt→tmax||U(t)||H=∞provided tmax<∞.\nFirst, we need to check that Tis a maximal monotone operator. Mono-\ntonicity is a direct consequence of Lemma 4.1 and (D1).\nTo prove Tis maximal as an operator from HtoH, we use Theorem 1.2\nfrom [ 3, Ch. 2]. Thus, we need to prove that Range (I+T) =H. Let\nz= (Φ z,Ψz)∈Hd×Hv. We need to find y= (Φ y,Ψy)∈D(A)×Hd=D(T)\nsuch that\n−Ψy+ Φ y= Φ z,\nAΦy+ Ψ y+ Γ(Ψ y) = Ψ z,\n11or, equivalently, find Ψ y∈Hdsuch that\nM(Ψy) =1\n2AΨy+1\n2AΨy+ Ψ y+ Γ(Ψ y) = Ψ z−AΦz= Θ z\nfor an arbitrary Θ z∈H′\nd=D(A1/2)′. Naturally, due to Lemma 4.1 Ais a\nduality map between HdandH′\nd, thus the operator Mis onto if and only\nif1\n2AΨy+ Ψ y+ Γ(Ψ y) is maximal monotone as an operator from Hdto\nH′\nd. According to Corollary 1.1 from [ 3, Ch. 2], this operator is maximal\nmonotone if1\n2Ais maximal monotone (it follows from Lemma 4.1) and\nI+ Γ(·) is monotone, bounded and hemicontinuous from HdtoH′\nd. The last\nstatement is evident for the identity map, now let’s prove it for Γ.\nMonotonicity is evident. Due to the continuity of the embedding H1(0, L0)⊂\nC(0, L0) in 1D every bounded set XinH1(0, L0) is bounded in C(0, L0) and\nthus due to (D1) Γ(X) is bounded in C(0, L0) and, consequently, in L2(0, L0).\nTo prove hemicontinuity we take an arbitrary Φ = ( φ, ψ, ω, u, v, w )∈Hd, an\narbitrary Θ = ( θ1, θ2, θ3, θ4, θ5, θ6)∈Hdand consider\n(Γ(Ψ y+tΦ),Θ) =ZL0\n0γ(ψy(x) +tψ(x))θ2(x)dx,\nwhere Ψ y= (φy, ψy, ωy, uy, vy, wy). Since ψy+tψ→ψy,ast→0 inH1(0, L0)\nand in C(0, L0), we obtain that γ(ψy(x) +tϕ(x))→γ(ψy(x)) as t→0 for\nevery x∈[0, L0], and has an integrable bound from above due to (D1).\nThis implies γ(ψy(x) +tϕ(x))→γ(ψy(x)) in L1(0, L0) ast→0 . Since\nθ2∈H1(0, L0)⊂L∞(0, L0), then\n(Γ(Ψ y+tΦ),Θ)→(Γ(Ψ y),Θ), t→0.\nHemicontinuity is proved.\nFurther, we need to prove that Bis locally Lipschitz on H, that is, Fis\nlocally Lipschitz from HdtoHv. The embedding H1/2+ε(0, L)⊂C(0, L)\nand (N1) imply\n|Fj(eΦj(x))−Fj(bΦj(x))| ≤C(max(||eΦ||d,||bΦ||d))||eΦj−bΦj||1 (29)\n12for all x∈[0, L0], ifj= 1 and for all x∈[L0, L], ifj= 2. This, in turn,\ngives us the estimate\n||F(eΦ)−F(bΦ)||v≤C(max(||eΦ||d,||bΦ||d))||eΦ−bΦ||d.\nThus, all the assumptions of Theorem 5.5 are satisfied and existence of a\nlocal strong/generalized solution is proved.\nStep 3. Energy inequality and global solutions. It can be verified by direct\ncalculations, that strong solutions satisfy energy equality. Using the same\narguments, as in proof of Proposition 1.3 [11], and (D1) we can pass to the\nlimit and prove (26) for generalized solutions.\nLet us assume that a local generalised solution exists on a maximal interval\n(0, tmax),tmax<∞. Then (26) implies E(tmax)≤ E(0). Since due to (N2)\nc1||U(t)||H≤ E(t)≤c2||U(t)||H,\nwe have ||U(tmax)||H≤C||U0||H. Thus, we arrive to a contradiction which\nimplies tmax=∞.\nStep 4. Generalized solution is variational (weak). We formulate the\nfollowing obvious estimate as a lemma for future use.\nLemma 5.6. Let(N1) holds and eΦ,bΦare two weak solutions to (18)-\n(24) with the initial conditions (eΦ0,eΦ1)and(bΦ0,bΦ1)respectively. Then the\nfollowing estimate is valid for all x∈[0, L], t > 0andϵ∈[0,1/2):\n|Fj(eΦj(x, t))−Fj(bΦj(x, t))| ≤C(max(||(eΦ0,eΦ1)||H,||(bΦ0,bΦ1)||H))||eΦj(·, t)−bΦj(·, t)||1−ϵ, j = 1,2.\nProof. The energy inequality and the embedding H1/2+ε(0, L)⊂C(0, L)\nimply that for every weak solution Φ\nmax\nt∈[0,T],x∈[0,L]|Φ(x, t)| ≤C(||Φ0||d,||Φ1||v).\nThus, using (N1) and (29), we prove the lemma.\nEvidently, (25)is valid for strong solutions. We can find a sequence of\n13strong solutions Φ(n), which converges to a generalized solution Φ strongly\ninC(0, T;Hd), and Φ(n)\ntconverges to Φ tstrongly in C(0, T;Hv). Using\nLemma 5.6, we can easily pass to the limit in nonlinear feedback term in\n(25). Since the test function B∈L∞(0, T;Hd)⊂L∞((0, T)×(0, L)), we\ncan use the same arguments as in the proof of Proposition 1.6 [ 11] to pass\nto the limit in the nonlinear dissipation term. Namely, we can extract from\nΦ(n)\nta subsequence that converges to Φ talmost everywhere and prove that\nit converges to Φ tstrongly in L1((0, T)×(0, L)).\nRemark 1. In space dimension greater then one we do not have the embed-\nding H1(Ω)⊂C(Ω), therefore, we need to assume polynomial growth of the\nderivative of the nonlinearity to obtain estimates similar to Lemma 5.6.\n6 Existence of attractors.\nIn this section we study long-time behaviour of solutions to problem (18)-(24)\nin the framework of dynamical systems theory. From Theorem 5.4 we have\nCorollary 1. Let, additionally to conditions of Theorem 5.4, P(x, t) =P(x).\nThen (18)-(24) generates a dynamical system (H, S t)by the formula\nSt(Φ0,Φ1) = (Φ( t),Φt(t)),\nwhere Φ(t)is the weak solution to (18)-(24) with initial data (Φ0,Φ1).\nTo establish the existence of the attractor for this dynamical system we\nuse Theorem 6.8 below, thus we need to prove the gradientness and the\nasymptotic smoothness as well as the boundedness of the set of stationary\npoints.\n6.1 Gradient structure\nIn this subsection we prove that the dynamical system generated by (18)-(24)\npossesses a specific structure, namely, is gradient under some additional\nconditions on the nonlinearities.\n14Definition 6.1 ([8,10,12]).LetY⊆Xbe a positively invariant set of\n(X, S t).\n•a continuous functional L(y)defined on Yis said to be a Lyapunov\nfunction of the dynamical system (X, S t)on the set Y, if a function\nt7→L(Sty)is non-increasing for any y∈Y.\n•the Lyapunov function L(y)is said to be strict onY, if the equality\nL(Sty) =L(y) for all t >0implies Sty=yfor all t >0;\n•A dynamical system (X, S t)is said to be gradient , if it possesses a\nstrict Lyapunov function on the whole phase space X.\nThe following result holds true.\nTheorem 6.2. Let, additionally to the assumptions of Corollary 1, the\nfollowing conditions hold\nf1=g1= 0, h 1(φ, ψ, ω ) =h1(ψ), (N3)\nf2, g2, h2∈C1(R3), (N4)\nγ(s)s >0for all s̸= 0. (D2)\nThen the dynamical system (H, S t)is gradient.\nProof. We use as a Lyapunov function\nL(Φ(t)) =L(t) =1\n2\u0010\n||R1/2Φt(t)||2+||A1/2Φ(t)||2\u0011\n+LZ\n0F(Φ(x, t))dx+(P,Φ(t)).\n(30)\nEnergy inequality (26) implies that L(t) is non-increasing. The equality\nL(t) =L(0) together with (D2) imply that ψt(t)≡0 on [0 , T]. We need\nto prove that Φ( t)≡const , which is equivalent to Φ( t+h)−Φ(t) = 0 for\nevery h >0. In what follows we use the notation Φ( t+h)−Φ(t) =Φ(t) =\n(φ,ψ,ω,u,v,w)(t) .\nStep 1. Let us prove that Φ1≡0. In this step we use the distribution theory\n(see, e.g., [ 5]) because some functions involved in computations are of too low\n15smoothness. Let us set the test function B= (B1,0) = ( β1, γ1, δ1,0,0,0).\nThen Φ(t) satisfies\n−TZ\n0(R1Φ1\nt, Bt)(t)dt−(R1(Φ1\nt(h)−Φ1\n1), B1(0))+\nTZ\n0\u00141\nk1(Q1(Φ1), Q1(B1))(t)dt+1\nσ1(N1(Φ1), N1(B1))(t)\u0015\n+\nTZ\n0(h1(ψ(t+h))−h1(ψ(t)), γ1(t))dt= 0.\nThe last term equals to zero due to (N3) and ψ(t)≡const .\nSetting in turn B= (0, γ1,0,0,0,0),B= (0,0, δ1,0,0,0),B= (β1,0,0,0,0,0)\nwe obtain\nφx+lω= 0 almost everywhere on (0 , L0)×(0, T),\n(31)\nρ1ωtt−lσ1(ωx−lφ)x= 0 almost everywhere on (0 , L0)×(0, T),\n(32)\nρ1φtt−σ1(ωx−lφ) = 0 in the sense of distributions on (0 , L0)×(0, T).\n(33)\nThese equalities imply\nφttx= 0,ωtt= 0 in the sense of distributions . (34)\nSimilar to regular functions, if partial derivative of a distribution equals to\nzero, then the distribution ”does not depends” on the corresponding variable\n(see [5, Ch. 7], Example 2.) That is,\nωt=c1(x)×1(t) in the sense of distributions\n16However, Theorem 5.4 implies that ωtis a regular distribution, thus, we can\ntreat the equality above as the equality almost everywhere. Furthermore,\nω(x, t) =ω(x,0) +Zt\n0c1(x)dτ=ω(x,0) +tc1(x).\nSince||ω(·, t)|| ≤Cfor all t∈R+,c1(x) must be zero. Thus,\nω(x, t) =c2(x), (35)\nwhich together with (31) implies\nφx=−lc2(x),\nφ(x, t) =φ(0, t)−lxZ\n0c2(y)dy=c3(x),\nφtt= 0.\nThe last equality together with (31),(33)and boundary conditions (19)give\nus that φ,ωare solutions to the following Cauchy problem (with respect to\nx):\nωx=lφ,\nφx=−lω,\nω(0, t) =φ(0, t) = 0 .\nConsequently, ω≡φ≡0.\nStep 2. Let us prove, that u≡v≡w≡0. Due to (N4), we can use the\nTaylor expansion of the difference F2(Φ2(t+h))−F2(Φ2(t)) and thus ( u,v,w)\n17satisfies on (0 , T)×(L0, L)\nρ2utt−k2uxx+gu(∂xΦ2,Φ2) +∇f2(ζ1,h(x, t))·Φ2= 0, (36)\nβ2vtt−λ2vxx+gv(∂xΦ2,Φ2) +∇h2(ζ2,h(x, t))·Φ2= 0, (37)\nρ2wtt−σ2wxx+gw(∂xΦ2,Φ2) +∇g2(ζ3,h(x, t))·Φ2= 0 (38)\nu(L0, t) =v(L0, t) =w(L0, t) = 0 , (39)\nu(L, t) =v(L, t) =w(L, t) = 0 , (40)\nk2(ux+v+lw)(L0, t) = 0 , (41)\nvx(L0, t) = 0 , σ 2(wx−lu)(L0, t) = 0 , (42)\nΦ2(x,0) = Φ2(x, h)−Φ2\n0,Φ2\nt(x,0) = Φ2\nt(x, h)−Φ2\n1, (43)\nwhere gu, gv, gware linear combinations of ux, vx, wx, u, v, w with the constant\ncoefficients, ζj,h(x, t) are 3D vector functions which components lie between\nu(x, t+h) and u(x, t),v(x, t+h) and v(x, t),w(x, t+h) and w(x, t) respectively.\nThus, we have a system of linear equations on ( L0, L) with overdetermined\nboundary conditions. L2-regularity of ux, vx, wxon the boundary for solutions\nto a linear wave equation was established in [ 21], thus, boundary conditions\n(41)-(42) make sense.\nIt is easy to generalize the observability inequality [ 27, Th. 8.1] for the case\nof the system of the wave equations.\nTheorem 6.3 ([27] ).For the solution to problem (36)-(43) the following\nestimate holds:\nZT\n0[|ux|2+|vx|2+|wx|2](L0, t)dt≥C(E(0) + E(T)),\nwhere\nE(t) =1\n2\u0000\n||ut(t)||2+||vt(t)||2+||wt(t)||2+||ux(t)||2+||vx(t)||2+||wx(t)||2\u0001\n.\nTherefore, if conditions (41),(42)hold true, then u=v=w= 0. The\ntheorem is proved.\n186.2 Asymptotic smoothness.\nDefinition 6.4 ([8,10,12]).A dynamical system (X, S t)is said to be\nasymptotically smooth if for any closed bounded set B⊂Xthat is positively\ninvariant ( StB⊆B) one can find a compact set K=K(B)which uniformly\nattracts B, i. e. sup{distX(Sty,K) :y∈B} →0ast→ ∞ .\nIn order to prove the asymptotical smoothness of the system considered\nwe rely on the compactness criterion due to [ 17], which is recalled below in\nan abstract version formulated in [12].\nTheorem 6.5. [12] Let (St, H)be a dynamical system on a complete metric\nspace Hendowed with a metric d. Assume that for any bounded positively\ninvariant set BinHand for any ε >0there exists T=T(ε, B)such that\nd(STy1, STy2)≤ε+ Ψ ε,B,T(y1, y2), yi∈B, (44)\nwhere Ψε,B,T(y1, y2)is a function defined on B×Bsuch that\nlim inf\nm→∞lim inf\nn→∞Ψε,B,T(yn, ym) = 0\nfor every sequence yn∈B. Then (St, H)is an asymptotically smooth\ndynamical system.\nTo formulate the result on the asymptotic smoothness of the system\nconsidered we need the following lemma.\nLemma 6.6. Let assumptions (D1) hold. Let moreover, there exists a\npositive constant Msuch that\nγ(s1)−γ(s2)\ns1−s2≤M, s 1, s2∈R, s1̸=s2. (D3)\nThen for any ε >0there exists Cε>0such that\n\f\f\f\f\f\fL0Z\n0(γ(ξ1)−γ(ξ2))ζdx\f\f\f\f\f\f≤ε∥ζ∥2+CεL0Z\n0(γ(ξ1)−γ(ξ2))(ξ1−ξ2)dx (45)\n19for any ξ1, ξ2, ζ∈L2(0, L0).\nThe proof is similar to that given in [12, Th.5.5]).\nTheorem 6.7. Let assumptions of Theorem 5.4, (D3), and\nm≤γ(s1)−γ(s2)\ns1−s2, s 1, s2∈R, s1̸=s2. (D4)\nwithm > 0hold. Let, moreover,\nk1=σ1 (46)\nρ1\nk1=β1\nλ1. (47)\nThen the dynamical system (H, S t)generated by problem (1)–(11) is asymp-\ntotically smooth.\nProof. In this proof we perform all the calculations for strong solutions and\nthen pass to the limit in the final estimate to justify it for weak solutions.\nLet us consider strong solutions ˆU(t) = ( ˆΦ(t),ˆΦt(t)) and ˜U(t) = ( ˜Φ(t),˜Φt(t))\nto problem (1)–(11)with initial conditions ˆU0= (ˆΦ0,ˆΦ1) and ˜U0= (˜Φ0,˜Φ1)\nlying in a ball, i.e. there exists R >0 such that\n∥˜U0∥H+∥ˆU0∥H≤R. (48)\nDenote U(t) =˜U(t)−ˆU(t) and U0=˜U0−ˆU0. Obviously, U(t) is a weak\n20solution to the problem\nρ1φtt−k1(φx+ψ+lω)x−lσ1(ωx−lφ) +f1( ˜φ,˜ψ,˜ω)−f1( ˆφ,ˆψ,ˆω) = 0\n(49)\nβ1ψtt−λ1ψxx+k1(φx+ψ+lω) +γ(˜ψt)−γ(ˆψt) +h1( ˜φ,˜ψ,˜ω)−h1( ˆφ,ˆψ,ˆω) = 0\n(50)\nρ1ωtt−σ1(ωx−lφ)x+lk1(φx+ψ+lω) +g1( ˜φ,˜ψ,˜ω)−g1( ˆφ,ˆψ,ˆω) = 0\n(51)\nρ2utt−k2(ux+v+lw)x−lσ2(wx−lu) +f2(˜u,˜v,˜w)−f2(ˆu,ˆv,ˆw) = 0\n(52)\nβ2vtt−λ2vxx+k2(ux+v+lw) +h2(˜u,˜v,˜w)−h2(ˆu,ˆv,ˆw) = 0 , (53)\nρ2wtt−σ2(wx−lu)x+lk2(ux+v+lw) +g2(˜u,˜v,˜w)−g2(ˆu,ˆv,ˆw) = 0\n(54)\nwith boundary conditions (7),(8)–(11)and the initial conditions U(0) =\n˜U0−ˆU0. It is easy to see by the energy argument that\nE(U(T)) +TZ\ntL0Z\n0(γ(˜ψs)−γ(ˆψs))ψsdxds =E(U(t)) +TZ\ntH(ˆU(s),˜U(s))ds,\n(55)\nwhere\nH(ˆU(t),˜U(t)) =L0Z\n0(f1( ˆφ,ˆψ,ˆω)−f1( ˜φ,˜ψ,˜ω))φtdx+L0Z\n0(h1( ˆφ,ˆψ,ˆω)−h1( ˜φ,˜ψ,˜ω))ψtdx\n+L0Z\n0(g1( ˆφ,ˆψ,ˆω)−g1( ˜φ,˜ψ,˜ω))ωtdx+LZ\nL0(f2(ˆu,ˆv,ˆw)−f2(˜u,˜v,˜w))utdx\n+LZ\nL0(h2(ˆu,ˆv,ˆw)−h2(˜u,˜v,˜w))vtdx+LZ\nL0(g2(ˆu,ˆv,ˆw)−g2(˜u,˜v,˜w))wtdx,\n(56)\n21and\nE(t) =E1(t) +E2(t), (57)\nhere\nE1(t) =ρ1L0Z\n0ω2\ntdxdt +ρ1L0Z\n0φ2\ntdxdt +β1L0Z\n0ψ2\ntdx+σ1L0Z\n0(ωx−lφ)2dx+\n+k1L0Z\n0(φx+ψ+lω)2dx+λ1L0Z\n0ψ2\nxdx(58)\nand\nE2(t) =ρ2L0Z\n0w2\ntdxdt +ρ2L0Z\n0u2\ntdxdt +β2L0Z\n0v2\ntdx+σ2L0Z\n0(wx−lu)2dx+\n+k2L0Z\n0(ux+v+lw)2dx+λ2L0Z\n0v2\nxdx. (59)\nIntegrating in (55) over the interval (0 , T) we come to\nTE(U(T))+TZ\n0TZ\ntL0Z\n0(γ(˜ψs)−γ(ˆψs))ψsdxdsdt =TZ\n0E(U(t))dt+TZ\n0TZ\ntH(ˆU(s),˜U(s))dsdt.\n(60)\nNow we estimate the first term in the right-hand side of (60). In what follows\nwe present formal estimates which can be performed on strong solutions.\nStep 1. We multiply equation (51)byωandx·ωxand sum up the results.\nAfter integration by parts with respect to twe obtain\n22ρ1TZ\n0L0Z\n0ωtxωtxdxdt +ρ1TZ\n0L0Z\n0ω2\ntdxdt +σ1TZ\n0L0Z\n0(ωx−lφ)xxωxdxdt\n+σ1TZ\n0L0Z\n0(ωx−lφ)xωdxdt −k1lTZ\n0L0Z\n0(φx+ψ+lω)xωxdxdt\n−k1lTZ\n0L0Z\n0(φx+ψ+lω)ωdxdt −TZ\n0L0Z\n0(g1( ˜φ,˜ψ,˜ω)−g1( ˆφ,ˆψ,ˆω))(xωx+ω)dxdt\n=ρ1L0Z\n0ωt(x, T)xωx(x, T)dx+ρ1L0Z\n0ωt(x, T)ω(x, T)dx\n−ρ1L0Z\n0ωt(x,0)xωx(x,0)dx−ρ1L0Z\n0ωt(x,0)ω(x,0)dx. (61)\nIntegrating by parts with respect to xwe get\nρ1TZ\n0L0Z\n0ωtxωtxdxdt =−ρ1\n2TZ\n0L0Z\n0ω2\ntdxdt +ρ1L0\n2TZ\n0ω2\nt(L0, t)dt (62)\nand\n23σ1TZ\n0L0Z\n0(ωx−lφ)xxωxdxdt−k1lTZ\n0L0Z\n0(φx+ψ+lω)xωxdxdt\n=σ1TZ\n0L0Z\n0(ωx−lφ)xx(ωx−lφ)dxdt +σ1lTZ\n0L0Z\n0(ωx−lφ)xxφdxdt\n−k1lTZ\n0L0Z\n0(φx+ψ+lω)xωxdxdt =−σ1\n2TZ\n0L0Z\n0(ωx−lφ)2dxdt\n+σ1L0\n2TZ\n0(ωx−lφ)2(L0, t)dt−σ1lTZ\n0L0Z\n0(ωx−lφ)φdxdt\n−2σ1lTZ\n0L0Z\n0(ωx−lφ)x(φx+ψ+lω)dxdt +σ1lTZ\n0L0Z\n0(ωx−lφ)x(ψ+lω)dxdt\n−σ1lL0TZ\n0(ωx−lφ)(L0, t)φ(L0, t)dt−k1l2TZ\n0L0Z\n0(φx+ψ+lω)xφdxdt.\n(63)\nAnalogously,\nσ1TZ\n0L0Z\n0(ωx−lφ)xωdxdt =−σ1TZ\n0L0Z\n0(ωx−lφ)2dxdt\n+σ1TZ\n0(ωx−lφ)(L0, t)ω(L0, t)dt−lσ1TZ\n0L0Z\n0(ωx−lφ)φdxdt. (64)\nIt follows from Lemma 5.6, energy relation (26), and property (N2) that\nTZ\n0L0Z\n0|g1( ˜φ,˜ψ,˜ω)−g1( ˆφ,ˆψ,ˆω)|2dxdt≤C(R, T) max\nt∈[0,T]∥Φ(·, t)∥2\nH1−ϵ,0< ϵ < 1/2.\n(65)\n24Therefore, for every ε >0\n\f\f\f\f\f\fTZ\n0L0Z\n0(g1( ˜φ,˜ψ,˜ω)−g1( ˆφ,ˆψ,ˆω))(xωx+ω)dxdt\f\f\f\f\f\f≤εTZ\n0∥ωx−lφ∥2dt+C(ε, R, T )lot,\n(66)\nwhere we use the notation\nlot= max\nt∈[0,T](∥φ(·, t)∥2\nH1−ϵ+∥ψ(·, t)∥2\nH1−ϵ+∥ω(·, t)∥2\nH1−ϵ\n+∥u(·, t)∥2\nH1−ϵ+∥v(·, t)∥2\nH1−ϵ+∥w(·, t)∥2\nH1−ϵ),0< ϵ < 1/2.(67)\nSimilar estimates hold for nonlinearities g2,fi,hi,i= 1,2.\nWe note that for any η∈H1(0, L0) (or analogously η∈H1(L0, L))\nη(L0)≤sup\n(0,L0)|η| ≤C∥η∥H1−ϵ,0< ϵ < 1/2. (68)\nSince due to (46)\n2σ1l\f\f\f\f\f\fTZ\n0L0Z\n0(ωx−lφ)x(φx+ψ+lω)dxdt\f\f\f\f\f\f\n≤σ1\n16TZ\n0L0Z\n0(ωx−lφ)2dxdt + 16k1l2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)2dxdt,\nthe following estimate can be obtained from (61)– (66)\nρ1\n2TZ\n0L0Z\n0ω2\ntdxdt +ρ1L0\n2TZ\n0ω2\nt(L0, t)dt+13σ1L0\n8TZ\n0(ωx−lφ)2(L0, t)dt\n≤13σ1\n8TZ\n0L0Z\n0(ωx−lφ)2dxdt+17k1l2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)2dxdt+C(R, T)lot\n+C(E(0) + E(T)),(69)\n25where C >0.\nStep 2. Multiplying equation (51)byωand ( x−L0)·ωxand arguing as\nabove we come to the estimate\nρ1\n2TZ\n0L0Z\n0ω2\ntdxdt+13σ1L0\n8TZ\n0(ωx−lφ)2(0, t)dt≤13σ1\n8TZ\n0L0Z\n0(ωx−lφ)2dxdt\n+ 17k1l2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)2dxdt +C(R, T)lot+C(E(0) + E(T)).(70)\nSumming up estimates (69)and(70)and multiplying the result by1\n2we get\nρ1\n2TZ\n0L0Z\n0ω2\ntdxdt +ρ1L0\n4TZ\n0ω2\nt(L0, t)dt+3σ1L0\n16TZ\n0(ωx−lφ)2(L0, t)dt\n+3σ1L0\n16TZ\n0(ωx−lφ)2(0, t)dt≤13σ1\n8TZ\n0L0Z\n0(ωx−lφ)2dxdt\n+ 17k1l2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)2dxdt +C(R, T)lot+C(E(0) + E(T)).(71)\nStep 3. Next we multiply equation (49)by−1\nl(ωx−lφ), equation (51)by\n1\nlφx, summing up the results and integrating by parts with respect to twe\narrive at\n26ρ1\nlTZ\n0L0Z\n0φt(ωtx−lφt)dxdt +k1\nlTZ\n0L0Z\n0(φx+ψ+lω)x(ωx−lφ)dxdt\n+σ1TZ\n0L0Z\n0(ωx−lφ)2dxdt−1\nlTZ\n0L0Z\n0(f1( ˜φ,˜ψ,˜ω)−f1( ˆφ,ˆψ,ˆω))(ωx−lφ)dxdt\n+ρ1\nlTZ\n0L0Z\n0ωtφtxdxdt +σ1\nlTZ\n0L0Z\n0(ωx−lφ)xφxdxdt\n−k1TZ\n0L0Z\n0(φx+ψ+lω)φxdxdt−TZ\n0L0Z\n0(g1( ˜φ,˜ψ,˜ω)−g1( ˆφ,ˆψ,ˆω))φxdxdt\n=ρ1\nlL0Z\n0φt(x, T)(ωx−lφ)(x, T)dx−ρ1\nlL0Z\n0φt(x,0)(ωx−lφ)(x,0)dx\n+ρ1\nlL0Z\n0ωt(x, T)φx(x, T)dx−ρ1\nlL0Z\n0ωt(x,0)φx(x,0)dx.\n(72)\nIntegrating by parts with respect to xwe obtain\n\f\f\f\f\f\fρ1\nlTZ\n0L0Z\n0φtωtxdxdt +ρ1\nlTZ\n0L0Z\n0ωtφtxdxdt\f\f\f\f\f\f=\f\f\f\f\f\fρ1\nlTZ\n0φt(L0, t)ωt(L0, t)dt\f\f\f\f\f\f\n≤ρ1L0\n8TZ\n0ω2\nt(L0, t)dt+2ρ1\nl2L0TZ\n0φ2\nt(L0, t)dt.\n(73)\nTaking into account (46) we get\n27k1\nlTZ\n0L0Z\n0(φx+ψ+lω)x(ωx−lφ)dxdt +σ1\nlTZ\n0L0Z\n0(ωx−lφ)xφxdxdt\n=k1\nlTZ\n0(φx+ψ+lω)(L0, t)(ωx−lφ)(L0, t)dt−k1\nlTZ\n0(φx+ψ+lω)(0, t)(ωx−lφ)(0, t)dt\n+k1\nlTZ\n0L0Z\n0ψx(ωx−lφ)dxdt+σ1TZ\n0L0Z\n0(ωx−lφ)2dxdt+σ1lTZ\n0L0Z\n0(ωx−lφ)φdxdt.\n(74)\nUsing the estimates\n\f\f\f\f\f\fk1\nlTZ\n0(φx+ψ+lω)(L0, t)(ωx−lφ)(L0, t)dt\f\f\f\f\f\f\n≤4k1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt+σ1L0\n16TZ\n0(ωx−lφ)2(L0, t)dt,\n\f\f\f\f\f\fk1\nlTZ\n0L0Z\n0ψx(ωx−lφ)dxdt\f\f\f\f\f\f≤4k1\nl2TZ\n0L0Z\n0ψ2\nxdxdt +σ1\n16TZ\n0L0Z\n0(ωx−lφ)2dxdt\nand (72)–(74) we infer\n2815σ1\n8TZ\n0L0Z\n0(ωx−lφ)2dxdt≤ρ1TZ\n0L0Z\n0φ2\ntdxdt+ 2k1TZ\n0L0Z\n0(φx+ψ+lω)2dxdt\n+4k1\nl2TZ\n0L0Z\n0ψ2\nxdxdt+4k1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt+σ1L0\n8TZ\n0(ωx−lφ)2(L0, t)dt\n+4k1\nl2L0TZ\n0(φx+ψ+lω)2(0, t)dt+σ1L0\n8TZ\n0(ωx−lφ)2(0, t)dt\nρ1L0\n8TZ\n0ω2\nt(L0, t)dt+2ρ1\nl2L0TZ\n0φ2\nt(L0, t)dt+C(R, T)lot+C(E(0) + E(T)).\n(75)\nAdding (75) to (71) we obtain\nσ1\n4TZ\n0L0Z\n0(ωx−lφ)2dxdt+ρ1\n2TZ\n0L0Z\n0ω2\ntdxdt+ρ1L0\n8TZ\n0ω2\nt(0, t)dt+σ1L0\n16TZ\n0(ωx−lφ)2(L0, t)dt\n+σ1L0\n16TZ\n0(ωx−lφ)2(L0, t)dt≤ρ1TZ\n0L0Z\n0φ2\ntdxdt+k1(2+17 l2L2\n0)TZ\n0L0Z\n0(φx+ψ+lω)2dxdt\n+4k1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt+4k1\nl2L0TZ\n0(φx+ψ+lω)2(0, t)dt\n+4k1\nl2TZ\n0L0Z\n0ψ2\nxdxdt +2ρ1\nl2L0TZ\n0φ2\nt(L0, t)dt+C(R, T)lot+C(E(0) + E(T)).\n(76)\nStep 4. Now we multiply equation (49)by−16\nl2L2\n0xφxand−16\nl2L2\n0(x−L0)φx\nand sum up the results. After integration by parts with respect to twe get\n2916ρ1\nl2L2\n0TZ\n0L0Z\n0φtxφtxdxdt +16ρ1\nl2L2\n0TZ\n0L0Z\n0φt(x−L0)φtxdxdt\n+16k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)xxφxdxdt+16k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)x(x−L0)φxdxdt\n+16σ1\nlL2\n0TZ\n0L0Z\n0(ωx−lφ)xφxdxdt +16σ1\nlL2\n0TZ\n0L0Z\n0(ωx−lφ)(x−L0)φxdxdt\n−16\nl2L2\n0TZ\n0L0Z\n0(f1( ˜φ,˜ψ,˜ω)−f1( ˆφ,ˆψ,ˆω))(2x−L0)φxdxdt\n=16ρ1\nl2L2\n0L0Z\n0φt(x, T)(2x−L0)φx(x, T)dx−16ρ1\nl2L2\n0L0Z\n0φt(x, T)(2x−L0)φx(x, T)dx.\n(77)\nIt is easy to see that\n16ρ1\nl2L2\n0TZ\n0L0Z\n0φtxφtxdxdt +16ρ1\nl2L2\n0TZ\n0L0Z\n0φt(x−L0)φtxdxdt\n=−16ρ1\nl2L2\n0TZ\n0L0Z\n0φ2\ntdxdt+8ρ1\nl2L0TZ\n0φ2\nt(L0, t)dt\n(78)\nand\n3016k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)xxφxdxdt+16k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)x(x−L0)φxdxdt\n=−16k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)2dxdt +8k1\nl2L0TZ\n0(φx+ψ+lω)2(0, t)dt\n+8k1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt−16k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)xx(ψ+lω)dxdt\n−16k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)x(x−L0)(ψ+lω)dxdt\n=−16k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)2dxdt +8k1\nl2L0TZ\n0(φx+ψ+lω)2(0, t)dt\n+8k1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt−16k1\nl2L0TZ\n0(φx+ψ+lω)(L0, t)(ψ+lω)(L0, t)dt\n+32k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)(ψ+lω)dxdt++16k1\nlL2\n0TZ\n0L0Z\n0(φx+ψ+lω)(2x−L0)(ωx−lφ)dxdt\n+16k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)(2x−L0)ψxdxdt+16k1\nL2\n0TZ\n0L0Z\n0(φx+ψ+lω)(2x−L0)φdxdt.\n(79)\nMoreover,\n16σ1\nlL2\n0TZ\n0L0Z\n0(ωx−lφ)xφxdxdt +16σ1\nlL2\n0TZ\n0L0Z\n0(ωx−lφ)(x−L0)φxdxdt\n=16σ1\nlL2\n0TZ\n0L0Z\n0(ωx−lφ)(2x−L0)(φx+ψ+lω)dxdt\n31−16σ1\nlL2\n0TZ\n0L0Z\n0(ωx−lφ)(2x−L0)(ψ+lω)dxdt. (80)\nCollecting (77)–(80) and using the estimates\n\f\f\f\f\f\f32k1\nlL2\n0TZ\n0L0Z\n0(φx+ψ+lω)(2x−L0)(ωx−lφ)dxdt\f\f\f\f\f\f\n≤σ1\n8TZ\n0L0Z\n0(ωx−lφ)2dxdt +2046k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)2dxdt\nand\n\f\f\f\f\f\f16k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)(2x−L0)ψxdxdt\f\f\f\f\f\f\n≤k1\nl2TZ\n0L0Z\n0ψ2\nxdxdt +64k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)2dxdt\nwe come to\n7k1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt+7k1\nl2L0TZ\n0(φx+ψ+lω)2(0, t)dt\n(81)\n+8ρ1\nl2L0TZ\n0φ2\nt(L0, t)dt≤16ρ1\nl2L2\n0TZ\n0L0Z\n0φ2\ntdxdt+2150k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)2dxdt\n+k1\nl2TZ\n0L0Z\n0ψ2\nxdxdt+3σ1\n16TZ\n0L0Z\n0(ωx−lφ)2dxdt+C(R, T)lot+C(E(0)+E(T)).\n(82)\nAdding (82) to (76) we arrive at\n32σ1\n16TZ\n0L0Z\n0(ωx−lφ)2dxdt +ρ1\n2TZ\n0L0Z\n0ω2\ntdxdt +ρ1L0\n8TZ\n0ω2\nt(L0, t)dt\n+σ1L0\n16TZ\n0(ωx−lφ)2(L0, t)dt+σ1L0\n16TZ\n0(ωx−lφ)2(0, t)dt\n+3k1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt+3k1\nl2L0TZ\n0(φx+ψ+lω)2(0, t)dt\n+6ρ1\nl2L0TZ\n0φ2\nt(L0, t)dt≤ρ1\u0012\n1 +16\nl2L2\n0\u0013TZ\n0L0Z\n0φ2\ntdxdt\n+k1\u0012\n2 + 17 l2L2\n0+2150\nl2L2\n0\u0013TZ\n0L0Z\n0(φx+ψ+lω)2dxdt\n+5k1\nl2TZ\n0L0Z\n0ψ2\nxdxdt +C(R, T)lot+C(E(0) + E(T)).(83)\nStep 5. Next we multiply equation (49)by−\u0010\n1 +18\nl2L2\n0\u0011\nφand integrate by\nparts with respect to t\nρ1\u0012\n1 +18\nl2L2\n0\u0013TZ\n0L0Z\n0φ2\ntdxdt +k1\u0012\n1 +18\nl2L2\n0\u0013TZ\n0L0Z\n0(φx+ψ+lω)xφdxdt\n+lσ1\u0012\n1 +18\nl2L2\n0\u0013TZ\n0L0Z\n0(ωx−lφ)φdxdt−\u0012\n1 +18\nl2L2\n0\u0013TZ\n0L0Z\n0(f1( ˜φ,˜ψ,˜ω)−f1( ˆφ,ˆψ,ˆω))φdxdt =\nρ1\u0012\n1 +18\nl2L2\n0\u0013L0Z\n0(φt(x, T)φ(x, T)−φt(x,0)φ(x,0))dx. (84)\nSince\n33k1\u0012\n1 +18\nl2L2\n0\u0013TZ\n0L0Z\n0(φx+ψ+lω)xφdxdt =−k1\u0012\n1 +18\nl2L2\n0\u0013TZ\n0L0Z\n0(φx+ψ+lω)2dxdt\n+k1\u0012\n1 +18\nl2L2\n0\u0013TZ\n0(φx+ψ+lω)(L0, t)φ(L0, t)dt\n+k1\u0012\n1 +18\nl2L2\n0\u0013TZ\n0(φx+ψ+lω)(ψ+lω)dxdt (85)\nwe obtain the estimate\nρ1\u0012\n1 +17\nl2L2\n0\u0013TZ\n0L0Z\n0φ2\ntdxdt≤k1\u0012\n2 +18\nl2L2\n0\u0013TZ\n0L0Z\n0(φx+ψ+lω)2dxdt\n+k1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt+σ1\n32TZ\n0L0Z\n0(ωx−lφ)2dxdt\n+C(R, T)lot+C(E(0) + E(T)).(86)\nSumming up (83) and (86) we get\n34σ1\n32TZ\n0L0Z\n0(ωx−lφ)2dxdt +ρ1\n2TZ\n0L0Z\n0ω2\ntdxdt +ρ1L0\n8TZ\n0ω2\nt(L0, t)dt\n+σ1L0\n16TZ\n0(ωx−lφ)2(L0, t)dt+σ1L0\n16TZ\n0(ωx−lφ)2(0, t)dt\n+2k1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt+2k1\nl2L0TZ\n0(φx+ψ+lω)2(0, t)dt\n+6ρ1\nl2L0TZ\n0φ2\nt(L0, t)dt+1\nl2L2\n0TZ\n0L0Z\n0φ2\ntdxdt\n≤k1\u0012\n4 + 17 l2L2\n0+2200\nl2L2\n0\u0013TZ\n0(φx+ψ+lω)2dxdt\n+6k1\nl2TZ\n0L0Z\n0ψ2\nxdxdt+C(R, T)lot+C(E(0)+E(T)).\n(87)\nStep 6. Next we multiply equation (50)byC1(φx+ψ+lω) and equation (49)\nbyC1β1\nρ1ψx, where C1= 2(6 + 17 l2L2\n0+2200\nl2L2\n0). Then we sum up the results\nand integrate by parts with respect to t. Taking into account (46),(47)we\ncome to\n35−β1C1TZ\n0L0Z\n0φtψtxdxdt−λ1C1TZ\n0L0Z\n0(φx+ψ+lω)xψxdxdt\n−lC1λ1TZ\n0L0Z\n0(ωx−lφ)ψxdxdt+C1β1\nρ1TZ\n0L0Z\n0(f1( ˜φ,˜ψ,˜ω)−f1( ˆφ,ˆψ,ˆω))ψxdxdt\n−β1C1TZ\n0L0Z\n0ψt(φxt+ψt+lωt)dxdt−λ1C1TZ\n0L0Z\n0ψxx(φx+ψ+lω)dxdt\n+k1C1TZ\n0L0Z\n0(φx+ψ+lω)2dxdt +C1TZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))(φx+ψ+lω)dxdt\n+C1TZ\n0L0Z\n0(h1( ˜φ,˜ψ,˜ω)−h1( ˆφ,ˆψ,ˆω))(φx+ψ+lω)dxdt =β1C1L0Z\n0φt(x,0)ψx(x,0)dx\n−β1C1L0Z\n0φt(x, T)ψx(x, T)dx+β1C1L0Z\n0ψt(x,0)(φx+ψ+lω)(x,0)dx\n−β1C1L0Z\n0ψt(x, T)(φx+ψ+lω)(x, T)dx. (88)\nIntegrating by parts with respect to xwe get\n\f\f\f\f\f\fβ1C1TZ\n0L0Z\n0φtψtxdxdt +β1C1TZ\n0L0Z\n0ψt(φxt+lωt)dxdt\f\f\f\f\f\f\n≤\f\f\f\f\f\fβ1C1TZ\n0φt(L0, t)ψt(L0, t)dt+β1C1lTZ\n0L0Z\n0ψtωtdxdt\f\f\f\f\f\f≤ρ1\nl2L0TZ\n0φ2\nt(L0, t)dt\n+β2\n1C2\n1l2L0\n4ρ1TZ\n0ψ2\nt(L0, t)dt+ρ1\n4TZ\n0L0Z\n0ω2\ntdxdt +β2\n1C2\n1l2\nρ1TZ\n0L0Z\n0ψ2\ntdxdt (89)\nand\n36\f\f\f\f\f\fλ1C1TZ\n0L0Z\n0(φx+ψ+lω)xψxdxdt +λ1C1TZ\n0L0Z\n0ψxx(φx+ψ+lω)dxdt\f\f\f\f\f\f\n=\f\f\f\f\f\fλ1C1TZ\n0(φx+ψ+lω)(L0, t)ψx(L0, t)dt−λ1C1TZ\n0(φx+ψ+lω)(0, t)ψx(0, t)dt\f\f\f\f\f\f\n≤k1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt+k1\nl2L0TZ\n0(φx+ψ+lω)2(0, t)dt\n+l2L0λ2\n1C2\n1\n4k1TZ\n0ψ2\nx(L0, t)dt+l2L0λ2\n1C2\n1\n4k1TZ\n0ψ2\nx(0, t)dt.(90)\nMoreover,\n\f\f\f\f\f\flC1λ1TZ\n0L0Z\n0(ωx−lφ)ψxdxdt\f\f\f\f\f\f≤σ1\n64TZ\n0L0Z\n0(ωx−lφ)2dxdt+16l2C2\n1λ2\n1\nσ1TZ\n0L0Z\n0ψ2\nxdxdt.\n(91)\nIt follows from Lemma (6.6) with ε=k1C1\n4\n\f\f\f\f\f\fC1TZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))(φx+ψ+lω)dxdt\f\f\f\f\f\f\n≤k1C1\n4TZ\n0L0Z\n0(φx+ψ+lω)2dxdt +CTZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))ψtdxdt (92)\nConsequently, collecting (88)–(92) we obtain\n37C1k1\n2TZ\n0L0Z\n0(φx+ψ+lω)2dxdt≤σ1\n64TZ\n0L0Z\n0(ωx−lφ)2dxdt\n+20l2C2\n1λ2\n1\nσ1TZ\n0L0Z\n0ψ2\nxdxdt +C1\u0012\nβ1+β2\n1l2\nρ1\u0013TZ\n0L0Z\n0ψ2\ntdxdt+\nk1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt+k1\nl2L0TZ\n0(φx+ψ+lω)2(0, t)dt\n+l2L0λ2\n1C2\n1\n4k1TZ\n0ψ2\nx(L0, t)dt+l2L0λ2\n1C2\n1\n4k1TZ\n0ψ2\nx(0, t)dt\n+ρ1\nl2L0TZ\n0φ2\nt(L0, t)dt+β2\n1C2\n1l2L0\n4ρ1TZ\n0ψ2\nt(L0, t)dt+ρ1\n4TZ\n0L0Z\n0ω2\ntdxdt\n+CTZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))ψtdxdt +C(R, T)lot+C(E(0) + E(T)).(93)\nCombining (93) with (87) we get\n38σ1\n64TZ\n0L0Z\n0(ωx−lφ)2dxdt +ρ1\n4TZ\n0L0Z\n0ω2\ntdxdt +ρ1L0\n8TZ\n0ω2\nt(L0, t)dt\n+σ1L0\n16TZ\n0(ωx−lφ)2(L0, t)dt+σ1L0\n16TZ\n0(ωx−lφ)2(0, t)dt\n+k1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt+k1\nl2L0TZ\n0(φx+ψ+lω)2(0, t)dt\n+5ρ1\nl2L0TZ\n0φ2\nt(L0, t)dt+1\nl2L2\n0TZ\n0L0Z\n0φ2\ntdxdt + 2k1TZ\n0L0Z\n0(φx+ψ+lω)2dxdt\n≤\u00126k1\nl2+20l2C2\n1λ2\n1\nσ1\u0013TZ\n0L0Z\n0ψ2\nxdxdt +C1\u0012\nβ1+β2\n1l2\nρ1\u0013TZ\n0L0Z\n0ψ2\ntdxdt\n+l2L0λ2\n1C2\n1\n4k1TZ\n0ψ2\nx(L0, t)dt+l2L0λ2\n1C2\n1\n4k1TZ\n0ψ2\nx(0, t)dt+β2\n1C2\n1l2L0\n4TZ\n0ψ2\nt(L0, t)dt\n+CTZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))ψtdxdt +C(R, T)lot+C(E(0) + E(T)).(94)\nStep 7. Our next step is to multiply equation (50)by−C2xψx−C2(x−L0)ψx,\nwhere C2=l2λ1C2\n1\nk1. After integration by parts with respect to twe obtain\nβ1C2TZ\n0L0Z\n0ψtxψxtdxdt+β1C2TZ\n0L0Z\n0ψt(x−L0)ψxtdxdt\n39+λ1C2TZ\n0L0Z\n0ψxxxψxdxdt +λ1C2TZ\n0L0Z\n0ψxx(x−L0)ψxdxdt\n−k1C2TZ\n0L0Z\n0(φx+ψ+lω)(2x−L0)ψxdxdt−C2TZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))(2x−L0)ψxdxdt\n+TZ\n0L0Z\n0(h1( ˜φ,˜ψ,˜ω)−h1( ˆφ,ˆψ,ˆω))(2x−L0)ψxdxdt\n=β1C2L0Z\n0ψt(x, T)(2x−L0)ψx(x, T)dx−β1C2L0Z\n0ψt(x,0)(2x−L0)ψx(x,0)dx.\n(95)\nAfter integration by parts with respect to xwe get\nβ1C2TZ\n0L0Z\n0ψtxψxtdxdt +β1C2TZ\n0L0Z\n0ψt(x−L0)ψxtdxdt\n=−β1C2TZ\n0L0Z\n0ψ2\ntdxdt +β1C2L0\n2TZ\n0ψ2\nt(L0, t)dt(96)\nand\nλ1C2TZ\n0L0Z\n0ψxxxψxdxdt +λ1C2TZ\n0L0Z\n0ψxx(x−L0)ψxdxdt\n=λ1C2L0\n2TZ\n0ψ2\nx(L0, t)dt+λ1C2L0\n2TZ\n0ψ2\nx(0, t)dt−λ1C2TZ\n0L0Z\n0ψ2\nxdxdt. (97)\nFurthermore,\n40\f\f\f\f\f\fk1C2TZ\n0L0Z\n0(φx+ψ+lω)(2x−L0)ψxdxdt\f\f\f\f\f\f\n≤k1TZ\n0L0Z\n0(φx+ψ+lω)2dxdt +k1C2\n2L2\n0\n4TZ\n0L0Z\n0ψ2\nxdxdt. (98)\nBy Lemma (6.6) with ε=k1C22L2\n0\n4we have\n\f\f\f\f\f\fC2TZ\n0L0Z\n0ψt(2x−L0)ψxdxdt\f\f\f\f\f\f≤k1C2\n2L2\n0\n4TZ\n0L0Z\n0ψ2\nxdxdt+CTZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))ψtdxdt.\n(99)\nAs a result of (95)– (99) we obtain the estimate\nβ1C2L0\n2TZ\n0ψ2\nt(L0, t)dt+λ1C2L0\n2TZ\n0ψ2\nx(L0, t)dt+λ1C2L0\n2TZ\n0ψ2\nx(0, t)dt\n≤k1TZ\n0L0Z\n0(φx+ψ+lω)2dxdt+\u0000\nk1C2\n2L2\n0+λ1C2\u0001TZ\n0L0Z\n0ψ2\nxdxdt+β1C2TZ\n0L0Z\n0ψ2\ntdxdt\n+CTZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))ψtdxdt +C(R, T)lot+C(E(0) + E(T)).(100)\nSumming up (94) and (100) and using (47) we infer\n41σ1\n64TZ\n0L0Z\n0(ωx−lφ)2dxdt +ρ1\n4TZ\n0L0Z\n0ω2\ntdxdt +ρ1L0\n8TZ\n0ω2\nt(L0, t)dt\n+σ1L0\n16TZ\n0(ωx−lφ)2(L0, t)dt+σ1L0\n16TZ\n0(ωx−lφ)2(0, t)dt\n+k1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt+k1\nl2L0TZ\n0(φx+ψ+lω)2(0, t)dt\n+5ρ1\nl2L0TZ\n0φ2\nt(L0, t)dt+1\nl2L2\n0TZ\n0L0Z\n0φ2\ntdxdt +k1TZ\n0L0Z\n0(φx+ψ+lω)2dxdt\nl2L0λ2\n1C2\n1\n4k1TZ\n0ψ2\nx(L0, t)dt+l2L0λ2\n1C2\n1\n4k1TZ\n0ψ2\nx(0, t)dt\n+β2\n1C2\n1l2L0\n4ρ1TZ\n0ψ2\nt(L0, t)dt≤\u00126k1\nl2+20l2C2\n1λ2\n1\nσ1+λ1C2+k1C2\n2L2\n0\u0013TZ\n0L0Z\n0ψ2\nxdxdt\n+\u0012\n(C1+C2)β1+C1β2\n1l2\nρ1\u0013TZ\n0L0Z\n0ψ2\ntdxdt\n+CTZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))ψtdxdt +C(R, T)lot+C(E(0) + E(T)).(101)\nStep 8. Now we multiply equation (50)byC3ψ, where C3=2\nλ1\u0010\n6k1\nl2+20l2C2\n1λ2\n1\nσ1+λ1C2+k1C2\n2L2\n0\u0011\nand integrate by parts with respect to t\n42−C3β1TZ\n0L0Z\n0ψ2\ntdxdt−λ1C3TZ\n0L0Z\n0ψxxψdxdt +k1C3TZ\n0L0Z\n0(φx+ψ+lω)ψdxdt\n+C3TZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))ψdxdt +C3TZ\n0L0Z\n0(h1( ˜φ,˜ψ,˜ω)−h1( ˆφ,ˆψ,ˆω))ψdxdt\n(102)\n=C3β1L0Z\n0ψt(x,0)ψ(x,0)dx−C3β1L0Z\n0ψt(x, T)ψ(x, T)dx\n(103)\nAfter integration by parts we infer the estimate\nλ1C3TZ\n0L0Z\n0ψ2\nxdxdt≤k1\n2TZ\n0(φx+ψ+lω)2dxdt+C3β1TZ\n0L0Z\n0ψ2\ntdxdt+l2L0λ2\n1C2\n1\n8k1TZ\n0ψ2\nx(L0, t)dt\n+CTZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))ψtdxdt +C(R, T)lot+C(E(0) + E(T)).(104)\nCombining (104) with (101) we obtain\nσ1\n64TZ\n0L0Z\n0(ωx−lφ)2dxdt +ρ1\n4TZ\n0L0Z\n0ω2\ntdxdt +ρ1L0\n8TZ\n0ω2\nt(L0, t)dt\n+σ1L0\n16TZ\n0(ωx−lφ)2(L0, t)dt+σ1L0\n16TZ\n0(ωx−lφ)2(0, t)dt\n+k1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt+k1\nl2L0TZ\n0(φx+ψ+lω)2(0, t)dt\n43+5ρ1\nl2L0TZ\n0φ2\nt(L0, t)dt+1\nl2L2\n0TZ\n0L0Z\n0φ2\ntdxdt +k1\n2TZ\n0L0Z\n0(φx+ψ+lω)2dxdt\nl2L0λ2\n1C2\n1\n8k1TZ\n0ψ2\nx(L0, t)dt+l2L0λ2\n1C2\n1\n4k1TZ\n0ψ2\nx(0, t)dt+β2\n1C2\n1l2L0\n4ρ1TZ\n0ψ2\nt(L0, t)dt\n+\u00126k1\nl2+20l2C2\n1λ2\n1\nσ1+λ1C2+k1C2\n2L2\n0\u0013TZ\n0L0Z\n0ψ2\nxdxdt\n≤\u0012\n(C1+C2)β1+C1β2\n1l2\nρ1+C3β1\u0013TZ\n0L0Z\n0ψ2\ntdxdt\n+CTZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))ψtdxdt +C(R, T)lot+C(E(0) + E(T)).(105)\nStep 9. Consequently, it follows from (105) and assumption (D4) for any\nl >0 where exist constants Mi,i={1,3}(depending on l) such that\nTZ\n0E1(t)dt+TZ\n0B1(t)dt≤M1TZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))ψtdxdt\n+M2(R, T)lot+M3(E(T) +E(0)),(106)\nwhere\nB1(t) =TZ\n0(ωx−lφ)2(L0, t)dt+TZ\n0(φx+ψ+lω)2(L0, t)dt+TZ\n0ψ2\nx(L0, t)dt\n+TZ\n0ω2\nt(L0, t)dt+TZ\n0ψ2\nt(L0, t)dt+TZ\n0φ2\nt(L0, t)dt.(107)\nStep 10. Finally, we multiply equation (52)by (x−L)ux, equation (53)by\n(x−L)vx, and (54)by (x−L)wx. Summing up the results and integrating\n44by parts with respect to twe arrive at\n−ρ2TZ\n0LZ\nL0ut(x−L)utxdxdt−k2TZ\n0LZ\nL0(ux+v+lw)x(x−L)uxdxdt\n−lσ2TZ\n0LZ\nL0(wx−lu)(x−L)uxdxdt+TZ\n0LZ\nL0(f2(˜u,˜v,˜w)−f2(ˆu,ˆv,ˆw))(x−L)uxdxdt\n−β2TZ\n0LZ\nL0vt(x−L)vxtdxdt−λ2TZ\n0LZ\nL0vxx(x−L)vxdxdt\n+k2TZ\n0LZ\nL0(ux+v+lw)(x−L)vxdxdt+TZ\n0LZ\nL0(h2(˜u,˜v,˜w)−h2(ˆu,ˆv,ˆw))(x−L)vxdxdt\n−ρ2TZ\n0LZ\nL0wt(x−L)wxtdxdt−σ2TZ\n0LZ\nL0(wx−lu)x(x−L)wxdxdt\n+lk2TZ\n0LZ\nL0(ux+v+lw)(x−L)wxdxdt+TZ\n0LZ\nL0(g2(˜u,˜v,˜w)−g2(ˆu,ˆv,ˆw))(x−L)wxdxdt =\n−ρ2LZ\nL0(x−L)((utux)(x, T)−(utux)(x,0))dx−β2LZ\nL0(x−L)((vtvx)(x, T)−(vtvx)(x,0))dx\n−ρ2LZ\nL0(x−L)((wtwx)(x, T)−(wtwx)(x,0))dx. (108)\nAfter integration by parts with respect to xwe infer\n45−ρ2TZ\n0LZ\nL0ut(x−L)utxdx−β2TZ\n0LZ\nL0vt(x−L)vxtdxdt−ρ2TZ\n0LZ\nL0wt(x−L)wxtdxdt\n=ρ2\n2TZ\n0LZ\nL0u2\ntdx+β2\n2TZ\n0LZ\nL0v2\ntdxdt +ρ2\n2TZ\n0LZ\nL0w2\ntdxdt\n−ρ2(L−L0)\n2TZ\n0u2\nt(L0)dt−β2(L−L0)\n2TZ\n0v2\nt(L0)dt−ρ2(L−L0)\n2TZ\n0w2\nt(L0)dt\n(109)\nand\n−k2TZ\n0LZ\nL0(ux+v+lw)x(x−L)uxdxdt−lσ2TZ\n0LZ\nL0(wx−lu)(x−L)uxdxdt\n−λ2TZ\n0LZ\nL0vxx(x−L)vxdxdt +k2TZ\n0LZ\nL0(ux+v+lw)(x−L)vxdxdt\n46−σ2TZ\n0LZ\nL0(wx−lu)x(x−L)wxdxdt+lk2TZ\n0LZ\nL0(ux+v+lw)(x−L)wxdxdt =\n−k2TZ\n0LZ\nL0(ux+v+lw)x(x−L)(ux+v+lw)dxdt\n−σ2TZ\n0LZ\nL0(wx−lu)x(x−L)(wx−lu)dxdt−λ2TZ\n0LZ\nL0vxx(x−L)vxdxdt\n−lσ2(L−L0)TZ\n0(wx−lu)(L0)u(L0)dt+k2(L−L0)TZ\n0(ux+v+lw)(L0)v(L0)dt\n+lk2(L−L0)TZ\n0(ux+v+lw)(L0)w(L0)dt=\n−k2(L−L0)\n2TZ\n0(ux+v+lw)2(L0)dt+k2\n2TZ\n0LZ\nL0(ux+v+lw)2dxdt\n+σ2\n2TZ\n0LZ\nL0(wx−lu)2dxdt−σ2(L−L0)\n2TZ\n0(wx−lu)2(L0)dt\n+λ2\n2TZ\n0LZ\nL0v2\nxdxdt−λ2(L−L0)\n2TZ\n0v2\nx(L0)dt−lσ2(L−L0)TZ\n0(wx−lu)(L0)u(L0)dt\n+k2(L−L0)TZ\n0(ux+v+lw)(L0)v(L0)dt\n+lk2(L−L0)TZ\n0(ux+v+lw)(L0)w(L0)dt.(110)\nConsequently, it follows from (108) –(110) that for any l >0 where exist\n47constants M4, M5, M6>0 such that\nTZ\n0E2(t)dt≤M4TZ\n0B2(t)dt+M5(R, T)lot+M6(E(T) +E(0)),(111)\nwhere\nB2(t) =TZ\n0(wx−lu)2(L0, t)dt+TZ\n0(ux+v+lw)2(L0, t)dt+TZ\n0v2\nx(L0, t)dt\n+TZ\n0w2\nt(L0, t)dt+TZ\n0v2\nt(L0, t)dt+TZ\n0u2\nt(L0, t)dt.(112)\nThen, due to transmission conditions (8)–(11) there exist δ, M 7, M8>0\n(depending on l), such that\nTZ\n0E(t)dt≤δTZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))ψtdxdt+M7(R, T)lot+M8(E(T)+E(0)).\n(113)\nIt follows from (55) that there exists C >0 such that\nTZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))ψtdxdt≤C\nE(0) +TZ\n0|H(ˆU(t),˜U(t))|dt\n.(114)\nBy Lemma 5.6 we have that for any ε >0 there exists C(ε, R)>0 such that\nTZ\n0|H(ˆU(t),˜U(t))|dt≤εTZ\n0L0Z\n0E(t)dxdt +C(ε, R, T )lot. (115)\nCombining (115) with (114) we arrive at\nTZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))ψtdxdt≤CE(0) + C(R, T)lot. (116)\n48Substituting (116) into (113) we obtain\nTZ\n0E(t)dt≤C(R, T)lot+C(E(T) +E(0)) (117)\nfor some C, C(R, T)>0.\nOur remaining task is to estimate the last term in (60).\n\f\f\f\f\f\fTZ\n0TZ\ntH(ˆU(s),˜U(s))dsdt\f\f\f\f\f\f≤TZ\n0E(t)dt+T3C(R)lot. (118)\nThen, it follows from (60) and (118) that\nTE(T)≤CTZ\n0E(t)dt+C(T, R)lot. (119)\nThen the combination of (119) with (117) leads to\nTE(T)≤C(R, T)lot+C(E(T) +E(0)). (120)\nChoosing Tlarge enough one can obtain estimate (44)which together with\nTheorem 6.5 immediately leads to the asymptotic smoothness of the system.\n6.3 Existence of attractors.\nThe following statement collects criteria on existence and properties of\nattractors to gradient systems.\nTheorem 6.8 ([10,12]).Assume that (H, S t)is a gradient asymptotically\nsmooth dynamical system. Assume its Lyapunov function L(y)is bounded\nfrom above on any bounded subset of Hand the set WR={y:L(y)≤R}\nis bounded for every R. If the set Nof stationary points of (H, S t)is\nbounded, then (St, H)possesses a compact global attractor. Moreover, the\n49global attractor consists of full trajectories γ={U(t) :t∈R}such that\nlim\nt→−∞distH(U(t),N) = 0 and lim\nt→+∞distH(U(t),N) = 0 (121)\nand\nlim\nt→+∞distH(Stx,N) = 0 for any x∈H; (122)\nthat is, any trajectory stabilizes to the set Nof stationary points.\nNow we state the result on the existence of an attractor.\nTheorem 6.9. Let assumptions of Theorems 6.2, 6.7, hold true, moreover,\nlim inf\n|s|→∞h1(s)\ns>0, (N5)\n∇F2(u, v, w )(u, v, w )−a1F2(u, v, w )≥ −a2, a i≥0.\nThen, the dynamical system (H, S t)generated by (1)-(11)possesses a compact\nglobal attractor Apossessing properties (121) ,(122) .\nProof. In view of Theorems 6.2, 6.7, 6.8 our remaining task is to show the\nboundedness of the set of stationary points and the set WR={Z:L(Z)≤R},\nwhere Lis given by (30).\nThe second statement follows immediately from the structure of function\nLand property (N5).\nThe first statement can be easily shown by energy-like estimates for\nstationary solutions taking into account (N5).\n7 Singular Limits on finite time intervals\n7.1 Singular limit l→0\nLet the nonlinearities fj, hj, gjare such that\nf1(φ, ψ, ω ) =f1(φ, ψ), h 1(φ, ψ, ω ) =h1(φ, ψ), g1(φ, ψ, ω ) =g1(ω),\nf2(u, v, w ) =f2(u, v), h 2(u, v, w ) =h2(u, v), g 2(u, v, w ) =g2(w).(N6)\n50If we formally set l= 0 in (18)-(24), we obtain the contact problem for a\nstraight Timoshenko beam\nρ1φtt−k1(φx+ψ)x+f1(φ, ψ) =p1(x, t), (x, t)∈(0, L0)×(0, T),\n(123)\nβ1ψtt−λ1ψxx+k1(φx+ψ) +γ(ψt) +h1(φ, ψ) =r1(x, t), (x, t)∈(0, L0)×(0, T),\n(124)\nρ2utt−k2(ux+v)x+f2(u, v) =p2(x, t), (x, t)∈(L0, L)×(0, T),\n(125)\nβ2vtt−λ2vxx+k2(ux+v) +h2(u, v) =r2(x, t), (x, t)∈(L0, L)×(0, T),\n(126)\nφ(0, t) =ψ(0, t) = 0 , u(L, t) =v(L, t) = 0 , (127)\nφ(L0, t) =u(L0, t), ψ(L0, t) =v(L0, t), (128)\nk1(φx+ψ)(L0, t) =k2(ux+v)(L0, t), λ1ψx(L0, t) =λ2vx(L0, t), (129)\nand an independent contact problem for wave equations\nρ1ωtt−σ1ωxx+g1(ω) =q1(x, t), (x, t)∈(0, L0)×(0, T),\n(130)\nρ2wtt−σ2wxx+g2(w) =q2(x, t), (x, t)∈(L0, L)×(0, T),\n(131)\nσ1ωx(L0, t) =σ2wx(L0, t), ω(L0, t) =w(L0, t), (132)\nw(L, t) = 0 , ω(0, t) = 0 . (133)\nThe following theorem gives an answer, how close are solutions to (18)-(24)\nto the solution of decoupled system (123)-(133) when l→0.\nTheorem 7.1. Assume that the conditions of Theorem 5.4, (D3) and(N6)\nhold. Let Φ(l)be the solution to (18)-(24) with the fixed land the initial data\nΦ(x,0) = ( φ0, ψ0, ω0, u0, v0, w0)(x),Φt(x,0) = ( φ1, ψ1, ω1, u1, v1, w1)(x).\n51Then for every T >0\nΦ(l)∗⇀(φ, ψ, ω, u, v, w ) inL∞(0, T;Hd)asl→0,\nΦ(l)\nt∗⇀(φt, ψt, ωt, ut, vt, wt) inL∞(0, T;Hv)asl→0,\nwhere (φ, ψ, u, v )is the solution to (123) -(129) with the initial conditions\n(φ, ψ, u, v )(x,0) = ( φ0, ψ0, u0, v0)(x),(φt, ψt, ut, vt)(x,0) = ( φ1, ψ1, u1, v1)(x),\nand(ω, w)is the solution to (130) -(133) with the initial conditions\n(ω, w)(x,0) = ( ω0, w0)(x),(ωt, wt)(x,0) = ( ω1, w1)(x).\nThe proof is similar to that of Theorem 3.1 [ 24] for the homogeneous\nBresse beam with obvious changes, except for the limit transition in the\nnonlinear dissipation term. For the future use we formulate it as a lemma.\nLemma 7.2. Let(D3) holds. Then\nZT\n0ZL0\n0γ(ψ(l)(x, t))γ1(x, t)dxdt→ZT\n0ZL0\n0γ(ψ(x, t))γ1(x, t)dxdt asl→0\nfor every γ1∈L2(0, T;H1(0, L0)).\nProof. Since (D1) and (D3) hold |γ(s)| ≤Ms, therefore\n||γ(ψ(l))||L∞(0,T;L2(0,L0))≤C(||ψ(l)||L∞(0,T;L2(0,L0))).\nThus, due to Lemmas 4.1, 5.6 the sequence\nRΦ(l)\ntt=AΦ(l)+ Γ(Φ(l)\nt) +F(Φ(l)) +P\nis bounded in L∞(0, T;H−1(0, L)) and we can extract from Φ(l)\ntta subse-\nquence, that converges ∗-weakly in L∞(0, T;H−1(0, L)). Thus,\nΦ(l)\nt→Φtstrongly in L2(0, T;H−ε(0, L)), ε > 0.\n52Consequently,\n\f\f\f\fZT\n0ZL0\n0(γ(ψ(l)(x, t))−γ(ψ(x, t)))γ1(x, t)dxdt\f\f\f\f≤\nC(L)ZT\n0ZL0\n0|ψ(l)(x, t)−ψ(x, t)||γ1(x, t)|dxdt→0.\nWe perform numerical modelling for the original problem with l=\n1,1/3,1/10,1/30,1/100,1/300,1/1000 and the limiting problem ( l= 0) with\nthe following values of constants ρ1=ρ2= 1, β1=β2= 2, σ1= 4,\nσ2= 2,λ1= 8,λ2= 4,L= 10, L0= 4 and the right-hand sides\np1(x) = sin x, r 1(x) =x, q 1(x) = sin x, (134)\np2(x) = cos x, r 2(x) =x+ 1, q 2(x) = cos x. (135)\nIn this subsection we consider the nonlinearities with the potentials\nF1(φ, ψ, ω ) =|φ+ψ|4− |φ+ψ|2+|φψ|2+|ω|3,\nF2(u, v, w ) =|u+v|4− |u+v|2+|uv|2+|w|3.\nConsequently, the nonlinearities have the form\nf1(φ, ψ, ω ) = 4( φ+ψ)3−2(φ+ψ) + 2φψ2, f 2(u, v, w ) = 4( u+v)3−2(u+v) + 2uv2,\nh1(φ, ψ, ω ) = 4( φ+ψ)3−2(φ+ψ) + 2φ2ψ, h 2(u, v, w ) = 4( u+v)3−2(u+v) + 2u2v,\ng1(φ, ψ, ω ) = 3|ω|ω, g 2(u, v, w ) = 3|w|w.\nFor modelling we choose the following (globally Lipschitz) dissipation\nγ(s) =\n\n1\n100s3,|s| ≤10,\n10s, |s|>10\n53and the following initial data:\nφ(x,0) =−3\n16x2+3\n4x, u (x,0) = 0 ,\nψ(x,0) =−1\n12x2+7\n12x, v (x,0) =−1\n6x+5\n3,\nω(x,0) =1\n16x2−1\n4x, w (x,0) =−1\n12x2+7\n6x−10\n3,\nφt(x,0) =x\n4, u t(x,0) =−1\n6(x−10),\nψt(x,0) =x\n4, v t(x,0) =−1\n6(x−10),\nωt(x,0) =x\n4, w t(x,0) =−1\n6(x−10).\nFigures 2-7 show the behavior of solutions when l→0 for the chosen\ncross-sections of the beam.\n0 1 2 3 4 5 6 7 8-2-1.5-1-0.500.51\nFigure 2: Transversal displacement of the beam, cross-section x= 2.\n7.2 Singular limit ki→ ∞, l→0\nThe singular limit for the straight Timoshenko beam ( l= 0) as ki→+∞is\nthe Euler-Bernoulli beam equation [ 19, Ch. 4]. We have a similar result for\nthe Bresse composite beam when ki→ ∞ , l→0.\nTheorem 7.3. Let the assumptions of Theorem 5.4, (N6) and(D3) hold.\n540 1 2 3 4 5 6 7 8-2-1.5-1-0.500.511.5Figure 3: Transversal displacement of the beam, cross-section x= 6.\n0 1 2 3 4 5 6 7 8-2-1.5-1-0.500.51\nFigure 4: Shear angle variation of the beam, cross-section x= 2.\nMoreover,\n(φ0, u0)∈\b\nφ0∈H2(0, L0), u0∈H2(L0, L), φ0(0) = u0(L) = 0 ,\n∂xϕ0(0) = ∂xu0(L) = 0 , ∂xφ0(L0, t) =∂xu0(L0, t)};(I1)\nψ0=−∂xφ0, v0=−∂xu0; (I2)\n(φ1, u1)∈ {φ1∈H1(0, L0), u1∈H1(L0, L), φ1(0) = u1(L) = 0 , φ1(L0, t) =u1(L0, t)};\n(I3)\nω0=w0= 0; (I4)\nh1, h2∈C1(R2); (N6)\nr1∈L∞(0, T;H1(0, L0)), r2∈L∞(0, T;H1(L0, L)),\nr1(L0, t) =r2(L0, t)for almost all t >0.(R3)\n550 1 2 3 4 5 6 7 8-2-1.5-1-0.500.511.5Figure 5: Shear angle variation of the beam, cross-section x= 6.\n0 1 2 3 4 5 6 7 800.20.40.60.81\nFigure 6: Longitudinal displacement of the beam, cross-section x= 2.\nLetk(n)\nj→ ∞ ,l(n)→0asn→ ∞ , and Φ(n)be weak solutions to (18)-(24)\nwith fixed k(n)\nj, l(n)and the same initial data\nΦ(x,0) = ( φ0, ψ0, ω0, u0, v0, w0)(x),Φt(x,0) = ( φ1, ψ1, ω1, u1, v1, w1).\nThen for every T >0\nΦ(n)∗⇀(φ, ψ, ω, u, v, w ) inL∞(0, T;Hd)asn→ ∞ ,\nΦ(n)\nt∗⇀(φt, ψt, ωt, ut, vt, wt) inL∞(0, T;Hv)asn→ ∞ ,\nwhere\n560 1 2 3 4 5 6 7 800.511.522.53Figure 7: Longitudinal displacement of the beam, cross-section x= 6.\n•(φ, u)is a weak solution to\nρ1φtt−β1φttxx+λ1φxxxx−γ′(−φtx)φtxx+∂xh1(φ,−φx) +f1(φ,−φx) =\np1(x, t) +∂xr1(x, t),(x, t)∈(0, L0)×(0, T),\n(136)\nρ2utt−β2uttxx+λ2uxxxx+∂xh2(u,−ux) +f2(u,−ux) =\np2(x, t) +∂xr2(x, t),(x, t)∈(L0, L)×(0, T),\n(137)\nφ(0, t) =φx(0, t) = 0 , u(L, t) =ux(L, t) = 0 , (138)\nφ(L0, t) =u(L0, t), φx(L0, t) =ux(L0, t), λ1φxx(L0, t) =λ2uxx(L0, t),\n(139)\nλ1φxxx(L0, t)−β1φttx(L0, t) +h1(φ(L0, t),−φx(L0, t)) +γ(−φtx(L0, t)) =\nλ2uxxx(L0, t)−β2uttx(L0, t) +h2(u(L0, t),−ux(L0, t)),\n(140)\nwith the initial conditions\n(φ, u)(x,0) = ( φ0, u0)(x),(φt, ut)(x,0) = ( φ1, u1)(x).\n•ψ=−φx, v=−ux;\n57•(ω, w)is the solution to\nρ1ωtt−σ1ωxx+g1(ω) =q1(x, t),(x, t)∈(0, L0)×(0, T),(141)\nρ2wtt−σ2wxx+g2(w) =q2(x, t),(x, t)∈(L0, L)×(0, T),(142)\nω(0, t) = 0 , w(L, t) = 0 , (143)\nσ1ωx(L0, t) =σ2wx(L0, t), ω(L0, t) =w(L0, t) (144)\nwith the initial conditions\n(ω, w)(x,0) = (0 ,0),(ωt, wt)(x,0) = ( ω1, w1)(x).\nProof. The proof uses the idea from [ 19, Ch. 4.3] and differs from it mainly\nin transmission conditions. We skip the details of the proof, which coincide\nwith [19].\nEnergy inequality (26) implies\n∂t(φ(n), ψ(n), ω(n), u(n), v(n), w(n)) bounded in L∞(0, T;Hv),(145)\nψ(n)bounded in L∞(0, T;H1(0, L0)),(146)\nv(n)bounded in L∞(0, T;H1(L0, L)) (147)\nω(n)\nx−l(n)φ(n)bounded in L∞(0, T;L2(0, L0)),(148)\nw(n)\nx−l(n)u(n)bounded in L∞(0, T;L2(L0, L)),(149)\nk(n)\n1(φ(n)\nx+ψ(n)+l(n)ω(n)) bounded in L∞(0, T;L2(0, L0)),(150)\nk(n)\n2(u(n)\nx+v(n)+l(n)w(n)) bounded in L∞(0, T;L2(L0, L)),(151)\nThus, we can extract subsequences which converge in corresponding spaces\nweak-∗. Similarly to [19] we have\nφ(n)\nx+ψ(n)+l(n)ω(n)∗⇀0 in L∞(0, T;L2(0, L0)),\ntherefore\nφx=−ψ.\n58Analogously,\nux=−v.\n(146)-(151) imply\nω(n)∗⇀ ω inL∞(0, T;H1(0, L0)), w(n)∗⇀ w inL∞(0, T;H1(L0, L)),\n(152)\nφ(n)∗⇀ φ inL∞(0, T;H1(0, L0)), u(n)∗⇀ u inL∞(0, T;H1(L0, L)).\n(153)\nThus, the Aubin’s lemma gives that\nΦ(n)→Φ strongly in C(0, T; [H1−ε(0, L0)]3×[H1−ε(L0, L)]3) (154)\nfor every ε >0 and then\n∂xφ0+ψ0+l(n)ω0→0 strongly in H−ε(0, L0),\nThis implies that\n∂xφ0=−ψ0, ω 0= 0.\nAnalogously,\n∂xu0=−v0, w 0= 0.\nLet us choose a test function of the form B= (β1,−β1\nx,0, β2,−β2\nx,0)∈FT\nsuch that β1\nx(L0, t) =β2\nx(L0, t) for almost all t. Due to (152) -(154) and\nLemma 7.2 we can pass to the limit in variational equality (25)asn→ ∞ .\nThe same way as in [ 19, Ch. 4.3] we obtain that the limiting functions φ, u\nare of higher regularity and satisfy the following variational equality\n59ZT\n0ZL0\n0\u0000\nρ1φtβ1\nt−β1φtxβ1\ntx\u0001\ndxdt +ZT\n0ZL\nL0\u0000\nρ2utβ2\nt−β1utxβ2\ntx\u0001\ndxdt−\nZL0\n0\u0000\nρ1(φtβ1\nt)(x,0)−β1(φtxβ1\ntx)(x,0)\u0001\ndx+ZL\nL0\u0000\nρ2(utβ2\nt)(x,0)−β1(utxβ2\ntx)(x,0)\u0001\ndx+\nZT\n0ZL0\n0λ1φxxβ1\nxxdxdt+ZT\n0ZL\nL0λ2uxxβ2\nxxdxdt−ZT\n0ZL0\n0γ′(−φxt)φtxxβ1dxdt+\nZT\n0ZL0\n0\u0000\nf1(φ,−φx)β1−h1(φ,−φx)β1\nx\u0001\ndxdt+ZT\n0ZL\nL0\u0000\nf2(u,−ux)β2−h2(u,−ux)β2\nx\u0001\ndxdt =\nZT\n0ZL0\n0\u0000\np1β1−r1β1\nx\u0001\ndxdt +ZT\n0ZL\nL0\u0000\np2β2−r2β2\nx\u0001\ndxdt. (155)\nProvided φ, uare smooth enough, we can integrate (155) by parts with\nrespect to x, tand obtain\nZT\n0ZL0\n0(ρ1−β1∂xx)φttβ1dxdt +ZT\n0ZL\nL0(ρ2−β2∂xx)uttβ2dxdt+\nZT\n0[β1φttx(t, L0)−β2uttx(t, L0)]β1(t, L0)dt+\nZT\n0ZL0\n0λ1φxxxxβ1dxdt +ZT\n0ZL\nL0λ2uxxxxβ2dxdt+\nZT\n0[λ1φxx−λ2uxx] (t, L0)β1\nx(t, L0)dt−ZT\n0[λ1φxxx−λ2uxxx] (t, L0)β1(t, L0)dt−\nZT\n0ZL0\n0γ′(−φxt)φxxtβ1dxdt−ZT\n0γ(−φxt(L0, t))β1(L0, t)+\nZT\n0ZL0\n0(f1(φ,−φx) +∂xh1(φ,−φx))β1dxdt+ZT\n0ZL\nL0(f2(u,−ux) +∂xh2(u,−ux))β2dxdt+\nZT\n0(h2(u(L0, t),−ux(L0, T))−h1(φ(L0, t),−φx(L0, T)))β1(L0, t)dt=\nZT\n0ZL0\n0(p1+∂xr1)β1dxdt+ZT\n0ZL\nL0(p2+∂xr2)β2dxdt+ZT\n0[r2(t, L0)−r1(t, L0)]β1(t, L0)dt.\n(156)\nRequiring all the terms containing β1(L0, t),β1\nx(L0, t) to be zero, we get\ntransmission conditions (139) -(137) . Equations (136) -(137) are recovered\n60from the variational equality (156).\nProblem (141)-(144) can be obtained in the same way.\nWe perform numerical modelling for the original problem with the initial\nparameters\nl(1)= 1, k(1)\n1= 4, k(1)\n2= 1;\nWe model the simultaneous convergence l→0 and k1, k2→ ∞ in the\nfollowing way: we divide lby the factor χand multiply k1, k2by the factor\nχ. Calculations performed for the original problem with\nχ= 1, χ= 3, χ= 10, χ= 30, χ= 100 , χ= 300\nand the limiting problem (136) -(140) . Other constants in the original problem\nare the same as in the previous subsection and we choose the functions in\nthe right-hand side of (134)-(135) as follows:\nr1(x) =x+ 4, r2(x) = 2 x.\nThe nonlinear feedbacks are\nf1(φ, ψ, ω ) = 4 φ3−2φ, f 2(u, v, w ) = 4 u3−8u,\nh1(φ, ψ, ω ) = 0 , h 2(u, v, w ) = 0 ,\ng1(φ, ψ, ω ) = 3|ω|ω, g 2(u, v, w ) = 6|w|w.\nWe use linear dissipation γ(s) =sand choose the following initial displace-\nment and shear angle variation\nφ0(x) =−13\n640x4+6\n40x2−23\n40x2,\nu0(x) =41\n2160x4−68\n135x3+823\n180x2−439\n27x+520\n27.\nψ0(x) =−\u0012\n−13\n160x3+27\n40x2−23\n20x\u0013\n,\n61v0(x) =−\u001241\n540x3−68\n45x2+823\n90x−439\n27\u0013\n.\nand set\nω0(x) =w0(x) = 0 .\nWe choose the following initial velocities\nφ1(x) =−1\n32x3+3\n16x2, u 1(x) =1\n108x3−7\n36x2+10\n9x−25\n27,\nω1(x) =ψ1(x) =3\n5x,\nw1(x) =v1(x) =−2\n5x+ 4.\nThe double limit case appeared to be more challenging from the point\nof view of numerics, then the case l→0. The numerical simulations of\nthe coupled system in equations (1)-(7)including the interface conditions\nin(8)-(11)were done by a semidiscretization of the functions ϕ, ψ, ω, u, v, w\nwith respect to the position xand by using an explicit scheme for the time\nintegration. That allows to choose the discretized values at grid points near\nthe interface in a separate step so that they obey the transmission conditions.\nIt was necessary to solve a nonlinear system of equations for the six functions\nat three grid points (at the interface, and left and right of the interface) in\neach time step. Any attempt to use a full implicit numerical scheme led to\nextremely time-expensive computations due to the large nonlinear system\nover all discretized values which was to solve in each time step. On the\nother hand, increasing k1, k2increase the stiffness of the system of ordinary\ndifferential equations which results from the semidiscretization, and the CFL-\nconditions requires small time steps — otherwise numerical oscillations occur.\nFigures 8-13 present smoothed numerical solutions, particularly necessary\nfor large factors χ, e. g. χ= 300. When the parameters k1, k2are large,\nthe material of the beam gets stiff, and so does the discretized system of\ndifferential equations. Nevertheless the oscillations are still noticeable in the\ngraph. The observation that the factor χcannot be arbitrarily enlarged,\nunderlines the importance of having the limit problem for χ→ ∞ in(1)-(15).\n620 1 2 3 4 5 6 7 8-1-0.500.511.52Figure 8: Transversal displacement of the beam, cross-section x= 2.\n0 1 2 3 4 5 6 7 8-4-2024\nFigure 9: Transversal displacement of the beam, cross-section x= 6.\n8 Discussion\nThere is a number of papers devoted to long-time behaviour of linear ho-\nmogeneous Bresse beams (with various boundary conditions and dissipation\nnature). If damping is present in all three equations, it appears to be sufficient\nfor the exponential stability of the system without additional assumptions\non the parameters of the problems (see, e.g., [2]).\nThe situation is different if we have a dissipation of any kind in one or\ntwo equations only. First of all, it matters in which equations the dissipation\nis present. There are results on the Timoshenko beams [ 25] and the Bresse\n630 1 2 3 4 5 6 7 8-20246Figure 10: Shear angle variation of the beam, cross-section x= 2.\n0 1 2 3 4 5 6 7 80510\nFigure 11: Shear angle variation of the beam, cross-section x= 6.\nbeams [ 13] that damping in only one of the equations does not guarantee the\nexponential stability of the whole system. It seems that for the Bresse system\nthe presence of the dissipation in the shear angle equation is necessary for the\nstability of any kind. To get the exponential stability, one needs additional\nassumptions on the coefficients of the problem, usually, the equality of the\npropagation speeds:\nk1=σ1,ρ1\nk1=β1\nλ1.\nOtherwise, only polynomial (non-uniform) stability holds (see, e.g., [ 1] for\nmechanical dissipation and [ 13] for thermal dissipation). In [ 6] analogous\n640 1 2 3 4 5 6 7 8-3-2-10123Figure 12: Longitudinal displacement of the beam, cross-section x= 2.\n0 1 2 3 4 5 6 7 8-3-2-1012\nFigure 13: Longitudinal displacement of the beam, cross-section x= 6.\nresults are established in case of nonlinear damping.\nIf dissipation is present in all three equations of the Bresse system,\ncorresponding problems with nonlinear source forces of local nature possesses\nglobal attractors under the standard assumptions for nonlinear terms (see,\ne.g., [ 24]). Otherwise, nonlinear source forces create technical difficulties and\nmay cause instability of the system. To the best of our knowledge, there is\nno literature on such cases.\nIn the present paper we study a transmission problem for the Bresse\nsystem.\nTransmission problems for various equation types have already had\n65some history of investigations. One can find a number of papers concern-\ning their well-posedness, long-time behaviour and other aspects (see, e.g.,\n[26] for a nonlinear thermoelastic/isothermal plate, or [ 14] for the Euler-\nBernoulli/Timoshenko beam and [ 15] for the full von Karman beam). Prob-\nlems with localized damping are close to transmission problems. In the recent\nyears a number of such problems for the Bresse beams were studied in, e.g.,\n[24,6]. To prove the existence of attractors in this case a unique continuation\nproperty is an important tool, as well as the frequency method.\nThe only paper we know on a transmission problem for the Bresse system\nis [28]. The beam in this work consists of a thermoelastic (damped) and elastic\n(undamped) parts, both purely linear. Despite the presence of dissipation\nin all three equations for the damped part, the corresponding semigroup is\nnot exponentially stable for any set of parameters, but only polynomially\n(non-uniformly) stable. In contrast to [ 28], we consider mechanical damping\nonly in the equation for the shear angle for the damped part. However, we\ncan establish exponential stability for the linear problem and existence of an\nattractor for the nonlinear one under restrictions on the coefficients in the\ndamped part only. The assumption on the nonlinearities can be simplified in\n1D case (cf. e.g. [16]).\nConflict of Interest Statement\nThe research was conducted in the absence of any commercial or financial\nrelationships that could be construed as a potential conflict of interest.\nAcknowledgements\nThe research is supported by the Volkswagen Foundation project ”From\nModeling and Analysis to Approximation”. The first and the third authors\nwere also successively supported by the Volkswagen Foundation project\n”Dynamic Phenomena in Elasticity Problems” at Humboldt-Universit¨ at zu\nBerlin, Funding for Refugee Scholars and Scientists from Ukraine.\n66References\n[1]Fatiha Alabau-Boussouira and Jaime E. Munoz-Rivera and Dilberto\nda S. Almeida Junior, Stability to weak dissipative Bresse system,\nJournal of Mathematical Analysis and Applications, 374 (2011), 481-\n498, https://doi.org/10.1016/j.jmaa.2010.07.046.\n[2]Dilberto da S. Almeida Junior and M L Santos, Numerical exponential\ndecay to dissipative Bresse system, Journal of Applied Mathematics,\n2010, Art. ID 848620, 17 pp.\n[3]Viorel Barbu, Nonlinear Semigroups and Differential Equations in\nBanach Spaces, Nordhof, 1976.\n[4]J. A. C. Bresse, Cours de Mechanique Appliquee, Mallet Bachelier,\nParis, 1859.\n[5] R.P. Kanwal, Generalized functions. Theory and applications,\nBirkh¨ auser, 2004.\n[6]Wenden Charles and J.A. 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Yao, Carleman Estimates with No Lower-\nOrder Terms for General Riemann Wave Equations. Global Uniqueness\nand Observability in One Shot, Appl. Math. Optim., 46 (2002), 331-375.\n[28]W. Youssef, Asymptotic behavior of the transmission problem of the\nBresse beam in thermoelasticity, Z. Angew. Math. Phys., 73 (2022),\n10.1007/s00033-022-01797-7.\n69"    },    {        "title": "1610.06661v1.Spin_transport_and_dynamics_in_all_oxide_perovskite_La___2_3__Sr___1_3__MnO__3__SrRuO__3__bilayers_probed_by_ferromagnetic_resonance.pdf",        "content": "Spin transport and dynamics in all-oxide perovskite La 2=3Sr1=3MnO 3/SrRuO 3bilayers\nprobed by ferromagnetic resonance\nSatoru Emori,1,\u0003Urusa S. Alaan,1, 2Matthew T. Gray,1, 2Volker\nSluka,3Yizhang Chen,3Andrew D. Kent,3and Yuri Suzuki1, 4\n1Geballe Laboratory for Advanced Materials, Stanford University, Stanford, CA 94305 USA\n2Department of Materials Science and Engineering, Stanford University, Stanford, CA 94305 USA\n3Department of Physics, New York University, New York, NY 10003, USA\n4Department of Applied Physics, Stanford University, Stanford, CA 94305 USA\n(Dated: November 11, 2021)\nThin \flms of perovskite oxides o\u000ber the possibility of combining emerging concepts of strongly\ncorrelated electron phenomena and spin current in magnetic devices. However, spin transport and\nmagnetization dynamics in these complex oxide materials are not well understood. Here, we ex-\nperimentally quantify spin transport parameters and magnetization damping in epitaxial perovskite\nferromagnet/paramagnet bilayers of La 2=3Sr1=3MnO 3/SrRuO 3(LSMO/SRO) by broadband ferro-\nmagnetic resonance spectroscopy. From the SRO thickness dependence of Gilbert damping, we\nestimate a short spin di\u000busion length of <\u00181 nm in SRO and an interfacial spin-mixing conductance\ncomparable to other ferromagnet/paramagnetic-metal bilayers. Moreover, we \fnd that anisotropic\nnon-Gilbert damping due to two-magnon scattering also increases with the addition of SRO. Our\nresults demonstrate LSMO/SRO as a spin-source/spin-sink system that may be a foundation for\nexamining spin-current transport in various perovskite heterostructures.\nI. INTRODUCTION\nManipulation and transmission of information by spin\ncurrent is a promising route toward energy-e\u000ecient mem-\nory and computation devices1. Such spintronic devices\nmay consist of ferromagnets interfaced with nonmagnetic\nconductors that exhibit spin-Hall and related spin-orbit\ne\u000bects2{4. The direct spin-Hall e\u000bect in the conductor\ncan convert a charge current to a spin current, which ex-\nerts torques on the adjacent magnetization and modi\fes\nthe state of the device5,6. Conversely, the inverse spin-\nHall e\u000bect in the conductor can convert a propagating\nspin current in the magnetic medium to an electric signal\nto read spin-based information packets7. For these device\nschemes, it is essential to understand the transmission of\nspin current between the ferromagnet and the conductor,\nwhich is parameterized by the spin-mixing conductance\nand spin di\u000busion length. These spin transport parame-\nters can be estimated by spin pumping at ferromagnetic\nresonance (FMR), in which a spin current is resonantly\ngenerated in the ferromagnet and absorbed in the adja-\ncent conductor8,9. Spin pumping has been demonstrated\nin various combinations of materials, where the magnetic\nlayer may be an alloy (e.g., permalloy) or insulator (e.g.,\nyttrium iron garnet) and the nonmagnetic conductor may\nbe a transition metal, semiconductor, conductive poly-\nmer, or topological insulator10{16.\nTransition metal oxides, particularly those with the\nperovskite structure, o\u000ber the intriguing prospect of\nintegrating a wide variety of strongly correlated elec-\ntron phenomena17,18with spintronic functionalities19,20.\nAmong these complex oxides, La 2=3Sr1=3MnO 3(LSMO)\nand SrRuO 3(SRO) are attractive materials for epitaxial,\nlattice-matched spin-source/spin-sink heterostructures.\nLSMO, a metallic ferromagnet known for its colossalmagnetoresistance and Curie temperature of >300 K,\ncan be an excellent resonantly-excited spin source be-\ncause of its low magnetization damping21{26. SRO, a\nroom-temperature metallic paramagnet with relatively\nhigh conductivity27, exhibits strong spin-orbit coupling28\nthat may be useful for emerging spintronic applications\nthat leverage spin-orbit e\u000bects2{4.\nA few recent studies have reported dc voltages at FMR\nin LSMO/SRO bilayers that are attributed to the in-\nverse spin-Hall e\u000bect in SRO generated by spin pump-\ning24{26. However, it is generally a challenge to separate\nthe inverse spin-Hall signal from the spin recti\fcation sig-\nnal, which is caused by an oscillating magnetoresistance\nmixing with a microwave current in the conductive mag-\nnetic layer29{31. Moreover, while the spin-mixing con-\nductance is typically estimated from the enhancement in\nthe Gilbert damping parameter \u000b, the quanti\fcation of \u000b\nis not necessarily straightforward in epitaxial thin \flms\nthat exhibit pronounced anisotropic non-Gilbert damp-\ning23,32{37. It has also been unclear how the Gilbert and\nnon-Gilbert components of damping in LSMO are each\nmodi\fed by an adjacent SRO layer. These points above\nhighlight the need for an alternative experimental ap-\nproach for characterizing spin transport and magnetiza-\ntion dynamics in LSMO/SRO.\nIn this work, we quantify spin transport parameters\nand magnetization damping in epitaxial LSMO/SRO bi-\nlayers by broadband FMR spectroscopy with out-of-plane\nandin-plane external magnetic \felds. Out-of-plane FMR\nenables straightforward extraction of Gilbert damping as\na function of SRO overlayer thickness, which is repro-\nduced by a simple \\spin circuit\" model based on di\u000busive\nspin transport38,39. We \fnd that the spin-mixing conduc-\ntance at the LSMO/SRO interface is comparable to other\nferromagnet/conductor interfaces and that spin current\nis absorbed within a short length scale of <\u00181 nm in thearXiv:1610.06661v1  [cond-mat.mtrl-sci]  21 Oct 20162\n42 44 46 48 50LSMO(10)\n  /SRO(18)LSMO(10)\n  log(intensity) (a.u.)\n2 (deg.)LSAT(002)  \nLSMO(002)  \nSRO(002)  \nFigure 1. 2 \u0012-!x-ray di\u000braction scans of a single-layer\nLSMO(10 nm) \flm and LSMO(10 nm)/SRO(18 nm) bilayer.\nconductive SRO layer. From in-plane FMR, we observe\npronounced non-Gilbert damping that is anisotropic and\nscales nonlinearly with excitation frequency, which is ac-\ncounted for by an existing model of two-magnon scat-\ntering40. This two-magnon scattering is also enhanced\nwith the addition of the SRO overlayer possibly due to\nspin pumping. Our \fndings reveal key features of spin\ndynamics and transport in the prototypical perovskite\nferromagnet/conductor bilayer of LSMO/SRO and pro-\nvide a foundation for future all-oxide spintronic devices.\nII. SAMPLE AND EXPERIMENTAL DETAILS\nEpitaxial \flms of LSMO(/SRO) were grown on\nas-received (001)-oriented single-crystal (LaAlO 3)0:3\n(Sr2AlTaO 6)0:7(LSAT) substrates using pulsed laser de-\nposition. LSAT exhibits a lower dielectric constant than\nthe commonly used SrTiO 3substrate and is therefore\nbetter suited for high-frequency FMR measurements.\nThe lattice parameter of LSAT (3.87 \u0017A) is also closely\nmatched to the pseudocubic lattice parameter of LSMO\n(\u00193.88 \u0017A). By using deposition parameters similar to\nthose in previous studies from our group41,42, all \flms\nwere deposited at a substrate temperature of 750\u000eC with\na target-to-substrate separation of 75 mm, laser \ruence\nof\u00191 J/cm2, and repetition rate of 1 Hz. LSMO was de-\nposited in 320 mTorr O 2, followed by SRO in 100 mTorr\nO2. After deposition, the samples were held at 600\u000eC\nfor 15 minutes in \u0019150 Torr O 2and then the substrate\nheater was switched o\u000b to cool to room temperature. The\ndeposition rates were calibrated by x-ray re\rectivity mea-\nsurements. The thickness of LSMO, tLSMO , in this study\nis \fxed at 10 nm, which is close to the minimum thickness\nat which the near-bulk saturation magnetization can be\nattained.\nX-ray di\u000braction results indicate that both the LSMO\n\flms and LSMO/SRO bilayers are highly crystalline\nand epitaxial with the LSAT(001) substrate, with high-\nresolution 2 \u0012-!scans showing distinct Laue fringes\naround the (002) Bragg re\rection (Fig. 1). In this study,\nthe maximum thickness of the LSMO and SRO layers\n770 780 790 800 810 main mode\n sec. mode\n  dIFMR/dH (a.u.)\n0H (mT) data\n fit(a) (b) \n770 780 790 800 810 data\n fit\n  dIFMR/dH (a.u.)\n0H (mT)Figure 2. Exemplary FMR spectra and \ftting curves: (a)\none mode of Lorentzian derivative; (b) superposition of a main\nmode and a small secondary mode due to slight sample inho-\nmogeneity.\ncombined is less than 30 nm and below the threshold\nthickness for the onset of structural relaxation by mis\ft\ndislocation formation41,42. The typical surface roughness\nof LSMO and SRO measured by atomic force microscopy\nis<\u00184\u0017A, comparable to the roughness of the LSAT sub-\nstrate surface.\nSQUID magnetometry con\frms that the Curie tem-\nperature of the LSMO layer is \u0019350 K and the room-\ntemperature saturation magnetization is Ms\u0019300 kA/m\nfor 10-nm thick LSMO \flms. The small LSMO thickness\nis desirable for maximizing the spin-pumping-induced en-\nhancement in damping, since spin pumping scales in-\nversely with the ferromagnetic layer thickness8,9. More-\nover, the thickness of 10 nm is within a factor of \u00192\nof the characteristic exchange lengthp\n2Aex=\u00160M2s\u00195\nnm, assuming an exchange constant of Aex\u00192 pJ/m in\nLSMO (Ref. 43), so standing spin-wave modes are not\nexpected.\nBroadband FMR measurements were performed at\nroom temperature. The \flm sample was placed face-\ndown on a coplanar waveguide with a center conductor\nwidth of 250 \u0016m. Each FMR spectrum was acquired at a\nconstant excitation frequency while sweeping the exter-\nnal magnetic \feld H. The \feld derivative of the FMR\nabsorption intensity (e.g., Fig. 2) was acquired using an\nrf diode combined with an ac (700 Hz) modulation \feld.\nEach FMR spectrum was \ftted with the derivative of the\nsum of the symmetric and antisymmetric Lorentzians, as\nshown in Fig. 2, from which the resonance \feld HFMR\nand half-width-at-half-maximum linewidth \u0001 Hwere ex-\ntracted. In some spectra (e.g., Fig. 2(b)), a small sec-\nondary mode in addition to the main FMR mode was\nobserved. We \ft such a spectrum to a superposition of\ntwo modes, each represented by a generalized Lorentzian\nderivative, and analyze only the HFMR and \u0001Hof the\nlarger-amplitude main FMR mode. The secondary mode\nis not a standing spin-wave mode because it appears\nabove or below the resonance \feld of the main mode\nHFMR with no systematic trend in \feld spacing. We\nattribute the secondary mode to regions in the \flm with3\n0.360.400.440.48\n  0Meff (T)\n0 5 10 15 201.952.002.05\ntSRO (nm)\n  gop\n0 5 10 15 200.00.20.40.60.81.01.2\nLSMO/SRO\nLSMO\n  0HFMR (T)\nf (GHz)(a) (b) \n(c) \nFigure 3. (a) Out-of-plane resonance \feld\nHFMR versus excitation frequency ffor\na single-layer LSMO(10 nm) \flm and a\nLSMO(10 nm)/SRO(3 nm) bilayer. The\nsolid lines indicate \fts to the data using\nEq. 1. (b,c) SRO-thickness dependence of\nthe out-of-plane Land\u0013 e g-factor (b) and ef-\nfective saturation magnetization Me\u000b(c).\nThe dashed lines indicate the values aver-\naged over all the data shown.\nslightly di\u000berent Msor magnetic anisotropy. More pro-\nnounced inhomogeneity-induced secondary FMR modes\nhave been observed in epitaxial magnetic \flms in prior\nreports22,44.\nIII. OUT-OF-PLANE FMR AND ESTIMATION\nOF SPIN TRANSPORT PARAMETERS\nOut-of-plane FMR allows for conceptually simpler ex-\ntraction of the static and dynamic magnetic properties\nof a thin-\flm sample. For \ftting the frequency depen-\ndence ofHFMR, the Land\u0013 e g-factor gopand e\u000bective sat-\nuration magnetization Me\u000bare the only adjustable pa-\nrameters in the out-of-plane Kittel equation. The fre-\nquency dependence of \u0001 Hfor out-of-plane FMR arises\nsolely from Gilbert damping, so that the conventional\nmodel of spin pumping8,9,38,39can be used to analyze the\ndata without complications from non-Gilbert damping.\nThis consideration is particularly important because the\nlinewidths of our LSMO(/SRO) \flms in in-plane FMR\nmeasurements are dominated by highly anisotropic non-\nGilbert damping (as shown in Sec. IV). Furthermore, a\nsimple one-dimensional, time-independent model of spin\npumping outlined by Boone et al.38is applicable in the\nout-of-plane con\fguration, since the precessional orbit of\nthe magnetization is circular to a good approximation.\nThis is in contrast with the in-plane con\fguration with\na highly elliptical orbit from a large shape anisotropy\n\feld. By taking advantage of the simplicity in out-of-\nplane FMR, we \fnd that the Gilbert damping parame-\nter in LSMO is approximately doubled with the addition\nof a su\u000eciently thick SRO overlayer due to spin pump-\ning. Our results indicate that spin-current transmission\nat the LSMO/SRO interface is comparable to previously\nreported ferromagnet/conductor bilayers and that spin\ndi\u000busion length in SRO is <\u00181 nm.\nWe \frst quantify the static magnetic properties of\nLSMO(/SRO) from the frequency dependence of HFMR.\nThe Kittel equation for FMR in the out-of-plane con\fg-\nuration takes a simple linear form,\nf=gop\u0016B\nh\u00160(HFMR\u0000Me\u000b); (1)\nwhere\u00160is the permeability of free space, \u0016Bis the Bohrmagneton, and his the Planck constant. As shown in\nFig. 3(a), we only \ft data points where \u00160HFMR is at\nleast 0.2 T above \u00160Me\u000bto ensure that the \flm is sat-\nurated out-of-plane. Figures 3(b) and (c) plot the ex-\ntractedMe\u000bandgop, respectively, each exhibiting no\nsigni\fcant dependence on SRO thickness tSROto within\nexperimental uncertainty. The SRO overlayer therefore\nevidently does not modify the bulk magnetic proper-\nties of LSMO, and signi\fcant interdi\u000busion across the\nSRO/LSMO interface can be ruled out. The averaged\nMe\u000bof 330\u000610 kA/m (\u00160Me\u000b= 0:42\u00060:01 T) is close\ntoMsobtained from static magnetometery and implies\nnegligible out-of-plane magnetic anisotropy; we thus as-\nsumeMs=Me\u000bin all subsequent analyses. The SRO-\nthickness independence of gop, averaging to 2 :01\u00060:01,\nimplies that the SRO overlayer does not generate a signif-\nicant orbital contribution to magnetism in LSMO. More-\nover, the absence of detectable change in gopwith in-\ncreasingtSRO may indicate that the imaginary compo-\nnent of the spin-mixing conductance8,9is negligible at\nthe LSMO/SRO interface.\nThe Gilbert damping parameter \u000bis extracted from\nthe frequency dependence of \u0001 H(e.g., Figure 4(a)) by\n\ftting the data with the standard linear relation,\n\u0001H= \u0001H0+h\ngop\u0016B\u000bf: (2)\nThe zero-frequency linewidth \u0001 H0is typically attributed\nto sample inhomogeneity. We observe sample-to-sample\nvariation of \u00160\u0001H0in the range\u00191\u00004 mT with no\nsystematic correlation with tSRO or the slope in Eq. 2.\nMoreover, similar to the analysis of HFMR, we only \ft\ndata obtained at \u00150.2 T above \u00160Me\u000bto minimize spu-\nrious broadening of \u0001 Hat low \felds. The linear slope of\n\u0001Hplotted against frequency up to 20 GHz is therefore\na reliable measure of \u000bdecoupled from \u0001 H0in Eq. 2.\nFigure 4(a) shows an LSMO single-layer \flm and an\nLSMO/SRO bilayer with similar \u0001 H0. The slope, which\nis proportional to \u000b, is approximately a factor of 2 greater\nfor LSMO/SRO. Figure 4(b) summarizes the dependence\nof\u000bon SRO-thickness, tSRO. For LSMO single-layer\n\flms we \fnd \u000b= (0:9\u00060:2)\u000210\u00003, which is on the same\norder as previous reports of LSMO thin \flms21{23,26.\nThis low damping is also comparable to the values re-4\n0 5 10 15 200123\n   (10-3)\ntSRO (nm)\n1/Gext 1/G↑↓m LSMO SRO\n0 5 10 15 201.01.52.02.53.0\nLSMO\n  0H (mT)\nf (GHz)LSMO/SRO(a) (b) (c) \nFigure 4. (a) Out-of-plane FMR linewidth \u0001 Hversus excitation frequency for LSMO(10 nm) and LSMO(10 nm)/SRO(3 nm).\nThe solid lines indicate \fts to the data using Eq. 2. (b) Gilbert damping parameter \u000bversus SRO thickness tSRO. The solid\ncurve shows a \ft to the di\u000busive spin pumping model (Eq. 5). (c) Schematic of out-of-plane spin pumping and the equivalent\n\\spin circuit.\"\nported in Heusler alloy thin \flms45,46and may arise from\nthe half-metal-like band structure of LSMO (Ref. 47).\nLSMO can thus be an e\u000ecient source of spin current\ngenerated resonantly by microwave excitation.\nWith a few-nanometer thick overlayer of SRO, \u000bin-\ncreases to\u00192\u000210\u00003(Fig. 4(b)). This enhanced damping\nwith the addition of SRO overlayer may arise from (1)\nspin scattering48,49at the LSMO/SRO interface or (2)\nspin pumping8,9where nonequilibrium spins from LSMO\nare absorbed in the bulk of the SRO layer. Here, we\nassume that interfacial spin scattering is negligible, since\n<\u00181 nm of SRO overlayer does not enhance \u000bsigni\fcantly\n(Fig. 4(b)). This is in contrast with the pronounced in-\nterfacial e\u000bect in ferromagnet/Pt bilayers48,49, in which\neven<1 nm of Pt can increase \u000bby as much as a fac-\ntor of\u00192 (Refs. 50{52). In the following analysis and\ndiscussion, we show that spin pumping alone is su\u000ecient\nfor explaining the enhanced damping in LSMO with an\nSRO overlayer.\nWe now analyze the data in Fig. 4(b) using a one-\ndimensional model of spin pumping based on di\u000busive\nspin transport38,39. The resonantly-excited magnetiza-\ntion precession in LSMO generates non-equilibrium spins\npolarized along ^ m\u0002d ^m=dt, which is transverse to the\nmagnetization unit vector ^ m. This non-equilibrium spin\naccumulation di\u000buses out to the adjacent SRO layer\nand depolarizes exponentially on the characteristic length\nscale\u0015s. The spin current density ~jsat the LSMO/SRO\ninterface can be written as38,53\n~jsjinterface =~2\n2e2^m\u0002d^m\ndt\u0010\n1\nG\"#+1\nGext\u0011; (3)\nwhere ~is the reduced Planck constant, G\"#is the inter-\nfacial spin-mixing conductance per unit area, and Gextis\nthe spin conductance per unit area in the bulk of SRO.\nIn Eq. 3, 1/ G\"#and 1/Gextconstitute spin resistors in\nseries such that the spin transport from LSMO to SRO\ncan be regarded analogously as a \\spin circuit,\" as il-\nlustrated in Fig. 4(c). In literature, these interfacialand bulk spin conductances are sometimes lumped to-\ngether as an \\e\u000bective spin-mixing conductance\" Ge\u000b\n\"#=\n(1=G\"#+ 1=Gext)\u00001(Refs. 10{13, 16, 20, 23, 26, 44). We\nalso note that the alternative form of the (e\u000bective) spin-\nmixing conductance g(e\u000b)\ne\u000b, with units of m\u00002, is related to\nG(e\u000b)\n\"#, with units of \n\u00001m\u00002, byg(e\u000b)\ne\u000b= (h=e2)G(e\u000b)\n\"#\u0019\n26 k\n\u0002G(e\u000b)\n\"#.\nThe functional form of Gextis obtained by solving the\nspin di\u000busion equation with appropriate boundary condi-\ntions38,39,53. In the case of a ferromagnet/nonmagnetic-\nmetal bilayer, we obtain\nGext=1\n2\u001aSRO\u0015stanh\u0012tSRO\n\u0015s\u0013\n; (4)\nwhere\u001aSROis the resistivity of SRO, tSROis the thick-\nness of the SRO layer, and \u0015sis the di\u000busion length of\npumped spins in SRO. Finally, the out\row of spin cur-\nrent (Eq. 3) is equivalent to an enhancement of Gilbert\ndamping9with respect to \u000b0of LSMO with tSRO = 0\nsuch that\n\u000b=\u000b0+gop\u0016B~\n2e2MstLSMO\u00141\nG\"#+ 2\u001aSRO\u0015scoth\u0012tSRO\n\u0015s\u0013\u0015\u00001\n:\n(5)\nThus, two essential parameters governing spin transport\nG\"#and\u0015scan be estimated by \ftting the SRO-thickness\ndependence of \u000b(Fig. 4(b)) with Eq. 5.\nIn carrying out the \ft, we \fx \u000b0= 0:9\u000210\u00003. We note\nthat\u001aSROincreases by an order of magnitude compared\nto the bulk value of \u00192\u000210\u00006\nm astSROis reduced to\na few nm; also, at thicknesses of 3 monolayers ( \u00191.2 nm)\nor below, SRO is known to be insulating54. We there-\nfore use the tSRO-dependent \u001aSRO shown in Appendix\nA while assuming \u0015sis constant. An alternative \ftting\nmodel that assumes a constant \u001aSRO, which is a common\napproach in literature, is discussed in Appendix A.\nThe curve in Fig. 4(b) is generated by Eq. 5 with G\"#=\n1:6\u00021014\n\u00001m\u00002and\u0015s= 0:5 nm. Given the scatter of5\n170175180185\n170\n175\n180\n185[010]\n[110]\n[100] \n  \n0HFMR (mT)\nLSMO\nLSMO/SRO\n0 5 10 15 200100200300400500\n  0HFMR (mT)\nf (GHz)H||[100]\nH||[110](a) (b) (c) \n0 5 10 15 201.952.002.05\ntSRO (nm)\n  gip\n-6-4-20\n  0H||,4 (mT)\n(d) \n14 15330360 \n \n  \nFigure 5. (a) Angular dependence of HFMR at 9 GHz for LSMO(10 nm) and LSMO(10 nm)/SRO(7 nm). The solid curves\nindicate \fts to the data using Eq. 6. (b) Frequency dependence of HFMR for LSMO(10 nm)/SRO(7 nm) with \feld applied in the\n\flm plane along the [100] and [110] directions. Inset: close-up of HFMR versus frequency around 14-15 GHz. In (a) and (b), the\nsolid curves show \fts to the Kittel equation (Eq. 6). (c,d) SRO-thickness dependence of the in-plane cubic magnetocrystalline\nanisotropy \feld (c) and in-plane Land\u0013 e g-factor (d). The dashed lines indicate the values averaged over all the data shown.\nthe experimental data, acceptable \fts are obtained with\nG\"#\u0019(1:2\u00002:5)\u00021014\n\u00001m\u00002and\u0015s\u00190:3\u00000:9\nnm. The estimated ranges of G\"#and\u0015salso depend\nstrongly on the assumptions behind the \ftting model.\nFor example, as shown in Appendix A, the constant- \u001aSRO\nmodel yields G\"#>\u00183\u00021014\n\u00001m\u00002and\u0015s\u00192:5 nm.\nNevertheless, we \fnd that the estimated G\"#\nis on the same order of magnitude as those\nof various ferromagnet/transition-metal heterostruc-\ntures39,55,56, signifying that the LSMO/SRO interface\nis reasonably transparent to spin current. More impor-\ntantly, the short \u0015simplies the presence of strong spin-\norbit coupling that causes rapid spin scattering within\nSRO. This \fnding is consistent with a previous study on\nSRO at low temperature in the ferromagnetic state show-\ning extremely fast spin relaxation with Gilbert damping\n\u000b\u00181 (Ref. 28). The short \u0015sindicates that SRO may be\nsuitable as a spin sink or detector in all-oxide spintronic\ndevices.\nIV. IN-PLANE FMR AND ANISOTROPIC\nTWO-MAGNON SCATTERING\nIn epitaxial thin \flms, the analysis of in-plane FMR\nis generally more complicated than that of out-of-plane\nFMR. High crystallinity of the \flm gives rise to a non-\nnegligible in-plane magnetocrystalline anisotropy \feld,\nwhich manifests in an in-plane angular dependence of\nHFMR and introduces another adjustable parameter in\nthe nonlinear Kittel equation for in-plane FMR. More-\nover, \u0001Hin in-plane FMR of epitaxial thin \flms often\ndepends strongly on the magnetization orientation and\nexhibits nonlinear scaling with respect to frequency due\nto two-magnon scattering, a non-Gilbert mechanism for\ndamping23,32{37. We indeed \fnd that damping of LSMO\nin the in-plane con\fguration is anisotropic and domi-\nnated by two-magnon scattering. We also observe ev-idence of enhanced two-magnon scattering with added\nSRO layers, which may be due to spin pumping from\nnonuniform magnetization precession.\nFigure 5(a) plots HFMR of a single-layer LSMO \flm\nand an LSMO/SRO bilayer as a function of applied \feld\nangle within the \flm plane. For both samples, we observe\nclear four-fold symmetry, which is as expected based on\nthe epitaxial growth of LSMO on the cubic LSAT(001)\nsubstrate. Similar to previous FMR studies of LSMO on\nSrTiO 3(001)57,58, the magnetic hard axes (corresponding\nto the axes of higher HFMR) are alongh100i. The in-\nplane Kittel equation for thin \flms with in-plane cubic\nmagnetic anisotropy is59,\nf=gip\u0016B\nh\u00160\u0002\nHFMR +Hjj;4cos(4\u001e)\u00031\n2\u0002\n\u0014\nHFMR +Me\u000b+1\n4Hjj;4(3 + cos(4\u001e))\u00151\n2\n;\n(6)\nwheregipis the Land\u0013 e g-factor that is obtained from in-\nplane FMR data, Hjj;4is the e\u000bective cubic anisotropy\n\feld, and\u001eis the in-plane \feld angle with respect to\nthe [100] direction. Given that LSMO is magnetically\nvery soft (coercivity on the order of 0.1 mT) at room\ntemperature, we assume that the magnetization is par-\nallel to the \feld direction, particularly with \u00160H\u001d10\nmT. In \ftting the angular dependence (e.g., Fig. 5(a))\nand frequency dependence (e.g., Fig. 5(b)) of HFMR to\nEq. 6, we \fx Me\u000bat the values obtained from out-of-\nplane FMR (Fig. 3(b)) so that Hjj;4andgipare the\nonly \ftting parameters. For the two samples shown in\nFig. 5(a), the \fts to the angular dependence and fre-\nquency dependence data yield consistent values of Hjj;4\nandgip. For the rest of the LSMO(/SRO) samples, we\nuse the frequency dependence data with Hjj[100] and\nHjj[110] to extract these parameters. Figures 5(c) and\n(d) show that Hjj;4andgip, respectively, exhibit no sys-\ntematic dependence on tSRO, similar to the \fndings from6\nout-of-plane FMR (Figs. 3(b),(c)). The in-plane cubic\nmagnetocrystalline anisotropy in LSMO(/SRO) is rela-\ntively small, with \u00160Hjj;4averaging to\u00192.5 mT.gipav-\nerages out to 1 :99\u00060:02, which is consistent with gop\nfound from out-of-plane FMR.\nWhile the magnetocrytalline anisotropy in\nLSMO(/SRO) is found to be modest and indepen-\ndent oftSRO, we observe much more pronounced\nin-plane anisotropy and tSRO dependence in linewidth\n\u0001H, as shown in Figs. 6(a) and (b). Figure 6(a)\nindicates that the in-plane dependence of \u0001 His four-\nfold symmetric for both LSMO(10 nm) and LSMO(10\nnm)/SRO(7 nm). \u0001 His approximately a factor of 2\nlarger when the sample is magnetized along h100icom-\npared to when it is magnetized along h110i. One might\nattribute this pronounced anisotropy to anisotropic\nGilbert damping60, such that the sample magnetized\nalong the hard axes h100imay lead to stronger damp-\ning. However, we \fnd no general correlation between\nmagnetocrystalline anisotropy and anisotropic \u0001 H: As\nwe show in Appendix B, LSMO grown on NdGaO 3(110)\nwith pronounced uniaxial magnetocrystalline anisotropy\nexhibits identical \u0001 Hwhen magnetized along the easy\nand hard axes. Moreover, whereas Gilbert damping\nshould lead to a linear frequency dependence of \u0001 H,\nfor LSMO(/SRO) the observed frequency dependence\nof \u0001His clearly nonlinear as evidenced in Fig. 6(b).\nThe pronounced anisotropy and nonlinear frequency\ndependence of \u0001 Htogether suggest the presence of a\ndi\u000berent damping mechanism.\nA well-known non-Gilbert damping mechanism in\nhighly crystalline ultrathin magnetic \flms is two-magnon\nscattering23,32{37,40,61,62, in which uniformly precessing\nmagnetic moments (a spin wave, or magnon mode, with\nwavevector k= 0) dephase to a k6= 0 magnon mode with\nadjacent moments precessing with a \fnite phase di\u000ber-\nence. By considering both exchange coupling (which re-\nsults in magnon energy proportional to k2) and dipolar\ncoupling (magnon energy proportional to \u0000jkj) among\nprecessing magnetic moments, the k= 0 andk6= 0\nmodes become degenerate in the magnon dispersion re-\nlation61as illustrated in Fig. 6(c).\nThe transition from k= 0 tok6= 0 is activated by\ndefects that break the translational symmetry of the\nmagnetic system by localized dipolar \felds40,61,62. In\nLSMO(/SRO), the activating defects may be faceted such\nthat two-magnon scattering is more pronounced when\nthe magnetization is oriented along h100i. One possibil-\nity is that LSMO thin \flms naturally form pits or islands\nfaceted alongh100iduring growth. However, we are un-\nable to consistently observe signs of such faceted defects\nin LSMO(/SRO) samples with an atomic force micro-\nscope (AFM). It is possible that these crystalline defects\nare smaller than the lateral resolution of our AFM setup\n(<\u001810 nm) or that these defects are not manifested in sur-\nface topography. Such defects may be point defects or\nnanoscale clusters of distinct phases that are known to\nexist intrinsically even in high-quality crystals of LSMO(Ref. 63).\nAlthough the de\fnitive identi\fcation of defects that\ndrive two-magnon scattering would require further in-\nvestigation, we can rule out (1) atomic step terraces\nand (2) mis\ft dislocations as sources of anisotropic two-\nmagnon scattering. (1) AFM shows that the orienta-\ntion and density of atomic step terraces di\u000ber randomly\nfrom sample to sample, whereas the anisotropy in \u0001 H\nis consistently cubic with larger \u0001 HforHjjh100ithan\nHjjh110i. This is in agreement with the recent study\nby Lee et al. , which shows anisotropic two-magnon scat-\ntering in LSMO to be independent of regularly-spaced\nparallel step terraces on a bu\u000bered-oxide etched SrTiO 3\nsubstrate23. (2) Although Woltersdorf and Heinrich have\nfound that mis\ft dislocations in Fe/Pd grown on GaAs\nare responsible for two-magnon scattering33, such dis-\nlocations are expected to be virtually nonexistent in\nfully strained LSMO(/SRO) \flms on the closely-latticed\nmatched LSAT substrates41,42.\nWe assume that the in-plane four-fold anisotropy and\nnonlinear frequency dependence of \u0001 Hare entirely due\nto two-magnon scattering. For a sample magnetized\nalong a given in-plane crystallographic axis hhk0i=h100i\norh110i, the two-magnon scattering contribution to \u0001 H\nis given by40\n\u0001Hhhk0i\n2m = \u0000hhk0i\n2m sin\u00001sp\nf2+ (fM=2)2\u0000fM=2p\nf2+ (fM=2)2+fM=2;(7)\nwherefM= (gip\u0016B=h)\u00160Msand \u0000hhk0i\n2m is the two-\nmagnon scattering parameter. The angular dependence\nof \u0001His \ftted with33\n\u0001H= \u0001H0+h\ngip\u0016B\u000bf\n+ \u0001Hh100i\n2m cos2(2\u001e) + \u0001Hh110i\n2m cos2(2[\u001e\u0000\u0019\n4]):(8)\nSimilarly, the frequency dependence of \u0001 Hwith the sam-\nple magnetized along [100] or [110], i.e., \u001e= 0 or\u0019=4,\nis well described by Eqs. 7 and 8. In principle, it should\nbe possible to \ft the linewidth data with \u0001 H0,\u000b, and\n\u00002mas adjustable parameters. In practice, the \ft car-\nried out this way is overspeci\fed such that wide ranges\nof these parameters appear to \ft the data. We there-\nfore impose a constraint on \u000bby assuming that Gilbert\ndamping for LSMO(/SRO) is isotropic: For each SRO\nthicknesstSRO,\u000bis \fxed to the value estimated from\nthe \ft curve in Fig. 4(c) showing out-of-plane FMR data.\n(This assumption is likely justi\fed, since the damping\nfor LSMO(10 nm) on NdGdO 3(110) with strong uniaxial\nmagnetic anisotropy is identical for the easy and hard\ndirections, as shown in Appendix B.) To account for the\nuncertainty in the Gilbert damping in Fig. 4(c), we vary \u000b\nby\u000625% for \ftting the frequency dependence of in-plane\n\u0001H. Examples of \fts using Eqs. 7 and 8 are shown in\nFig. 6(a),(b).\nFigure 6(d) shows that the SRO overlayer enhances\nthe two-magnon scattering parameter \u0000 2mby up to a7\n0 5 10 15 2004812\nLSMOLSMO/SRO\n  0H (mT)\nf (GHz)(a) (b) \n(c) (d) \n  \nFMR \nfreq. \nk f \nk=0 k≠0 \n0 5 10 15 200102030\n  02m (mT)\ntSRO (nm)H||[100]\nH||[110]\n0510\n0\n5\n10\nLSMO\nLSMO/SRO[110][010]\n[100]\n   0H (mT)\nFigure 6. (a) In-plane angular dependence of\nlinewidth \u0001H at 9 GHz for LSMO(10 nm) and\nLSMO(10 nm)/SRO(7 nm). The solid curves\nindicate \fts to Eq. 8. (b) Frequency depen-\ndence of \u0001H for LSMO(10 nm) and LSMO(10\nnm)/SRO(7 nm) with Happlied along the [100]\ndirection. The solid curves indicate \fts to\nEq. 7. The dashed and dotted curves indicate\nestimated two-magnon and Gilbert damping\ncontributions, respectively. (c) Schematic of a\nspin wave dispersion curve (when the magne-\ntization is in-plane and has a \fnite component\nparallel to the spin wave wavevector k) and two-\nmagnon scattering. (d) Two-magnon scattering\ncoe\u000ecient \u0000 2m, estimated for the cases with H\napplied along the [100] and [110] axes, plotted\nagainst SRO thickness tSRO. The dashed curve\nis the same as that in Fig. 4(c) scaled to serve\nas a guide for the eye for \u0000 2mwith H along\n[100].\nfactor of\u00192 forHjj[100]. By contrast, for Hjj[110], al-\nthough LSMO/SRO exhibits enhanced \u0001 Hcompared to\nLSMO, the enhancement in \u0000 2mis obscured by the un-\ncertainty in Gilbert damping. In Table I, we summa-\nrize the Gilbert and two-magnon contributions to \u0001 H\nfor LSMO single layers and LSMO/SRO (averaged val-\nues for samples with tSRO>4 nm) with Hjj[100] and\nHjj[110]. Comparing the e\u000bective spin relaxation rates,\n(gip\u0016B=h)\u00160Ms\u000band (gip\u0016B=h)\u00160\u00002m, reveals that two-\nmagnon scattering dominates over Gilbert damping.\nWe now speculate on the mechanisms behind the\nenhancement in \u0000 2min LSMO/SRO, particularly for\nHjj[100]. One possibility is that SRO interfaced with\nLSMO directly increases the rate of two-magnon scat-\ntering, perhaps due to formation of additional defects at\nthe surface of LSMO. If this were the case we might ex-\npect a signi\fcant increase and saturation of \u0000 2mat small\ntSRO. However, in reality, \u0000 2mincreases for tSRO>1 nm\n(Fig. 6(d)), which suggests spin scattering in the bulk\nof SRO. We thus speculate another mechanism, where\nk6= 0 magnons in LSMO are scattered by spin pump-\nTable I. Spin relaxation rates extracted from in-plane FMR\n(106s\u00001)\nLSMO LSMO/SRO*\nGilbert:gip\u0016B\nh\u00160Ms\u000b 11\u00062 23\u00064\ntwo-magnon:gip\u0016B\nh\u00160\u00002m(Hjj[100]) 290\u000650 550\u0006100\ntwo-magnon:gip\u0016B\nh\u00160\u00002m(Hjj[110]) 140\u000660 250\u000660\n* Averaged over samples with tSRO>4 nm.ing into SRO. As shown by the guide-for-the-eye curve\nin Fig. 6(d), the tSROdependence of \u0000 2m(forHjj[100])\nmay be qualitatively similar to the tSROdependence of\n\u000bmeasured from out-of-plane FMR (Fig. 4(c)); this cor-\nrespondence would imply that the same spin pumping\nmechanism, which is conventionally modeled to act on\nthek= 0 mode, is also operative in the degenerate k6= 0\nmagnon mode in epitaxial LSMO. Indeed, previous stud-\nies have electrically detected the presence of spin pump-\ning fromk6= 0 magnons by the inverse spin-Hall e\u000bect in\nY3Fe5O12/Pt bilayers64{66. However, we cannot conclu-\nsively attribute the observed FMR linewidth broadening\nin LSMO/SRO to such k6= 0 spin pumping, since it\nis unclear whether faster relaxation of k6= 0 magnons\nshould necessarily cause faster relaxation of the k= 0\nFMR mode. Regardless of its origin, the pronounced\nanisotropic two-magnon scattering introduces additional\ncomplexity to the analysis of damping in LSMO/SRO\nand possibly in other similar ultrathin epitaxial magnetic\nheterostructures.\nV. SUMMARY\nWe have demonstrated all-oxide perovskite bilayers\nof LSMO/SRO that form spin-source/spin-sink systems.\nFrom out-of-plane FMR, we deduce a low Gilbert damp-\ning parameter of \u00191\u000210\u00003for LSMO. The two-fold en-\nhancement in Gilbert damping with an SRO overlayer\nis adequately described by the standard model of spin\npumping based on di\u000busive spin transport. We ar-\nrive at an estimated spin-mixing conductance G\"#\u0019\n(1\u00002)\u00021014\n\u00001m\u00002and spin di\u000busion length \u0015s<\u00181\nnm, which indicate reasonable spin-current transparency\nat the LSMO/SRO interface and strong spin scattering8\nwithin SRO. From in-plane FMR, we reveal pronounced\nnon-Gilbert damping, attributed to two-magnon scatter-\ning, which results in a nonlinear frequency dependence\nand anisotropy in linewidth. The magnitude of two-\nmagnon scattering increases with the addition of an SRO\noverlayer, pointing to the presence of spin pumping from\nnonuniform spin wave modes. Our \fndings lay the foun-\ndation for understanding spin transport and magneti-\nzation dynamics in epitaxial complex oxide heterostruc-\ntures.\nACKNOWLEDGEMENTS\nWe thank Di Yi, Sam Crossley, Adrian Schwartz, Han-\nkyu Lee, and Igor Barsukov for helpful discussions, and\nTianxiang Nan and Nian Sun for the design of the copla-\nnar waveguide. This work was funded by the National\nSecurity Science and Engineering Faculty Fellowship of\nthe Department of Defense under Contract No. N00014-\n15-1-0045.\nAPPENDIX A: SPIN PUMPING AND SRO\nRESISTIVITY\nWhen \ftting the dependence of the Gilbert damping\nparameter\u000bon spin-sink thickness, a constant bulk re-\nsistivity for the spin sink layer is often assumed in lit-\nerature. By setting the resistivity of SRO to the bulk\nvalue\u001aSRO= 2\u000210\u00006\nm and \ftting the \u000b-versus-tSRO\ndata (Fig. 4(c) and reproduced in Fig. 7(a)) to Eq. 5,\nwe arrive at G\"#>\u00183\u00021014\n\u00001m\u00002and\u0015s\u00192:5 nm.\nThe \ft curve is insensitive to larger values of G\"#because\nthe bulk spin resistance 1/ Gext, with the relatively large\nresistivity of SRO, dominates over the interfacial spin re-\nsistance 1/G\"#(see Eqs. 4 and 5). As shown by the dot-\nted curve in Fig. 7, this simple constant- \u001aSROmodel ap-\npears to mostly capture the tSRO-dependence of \u000b. This\nmodel of course indicates \fnite spin pumping at even\nvery small SRO thickness <\u00181 nm, which is likely non-\nphysical since SRO should be insulating in this thickness\nregime54. Indeed,\u0015sestimated with this model should\nprobably be considered a phenomenological parameter:\nAs pointed out by recent studies, strictly speaking, a\nphysically meaningful estimation of \u0015sshould take into\naccount the thickness dependence of the resistivity of the\nspin sink layer39,56,67, especially for SRO whose thickness\ndependence of resistivity is quite pronounced.\nFigure 7(b) plots the SRO-thickness dependence of the\nresistivity of SRO \flms deposited on LSAT(001) mea-\nsured in the four-point van der Pauw geometry. The\ntrend can be described empirically by\n\u001aSRO=\u001ab+\u001as\ntSRO\u0000tth; (9)\nwhere\u001ab= 2\u000210\u00006\nm is the resistivity of SRO in the\nbulk limit,\u001as= 1:4\u000210\u000014\nm2is the surface resistivity\n0 5 10 15 200123\n   (10-3)\ntSRO (nm)\n0 10 20 3010-61x10-51x10-4\n  SRO (m)\ntSRO (nm)(a) (b) Figure 7. (a) Gilbert damping parameter \u000bversus SRO\nthicknesstSRO. The solid curve is a \ft taking into account\nthetSROdependence of SRO resistivity, whereas the dotted\ncurve is a \ft assuming a constant bulk-like SRO resistivity.\n(b) Resistivity of SrRuO 3\flms on LSAT(001) as a function\nof thickness.\n0 5 10 15 20012345\nhard\neasy\n  0H (mT)\nf (GHz)\n0 5 10 15 200123450H (mT)\n  \nf (GHz)hard\neasy(b) (a) \nFigure 8. Frequency dependence of in-plane FMR\nlinewidth \u0001 Hof LSMO(10 nm) on (a) LSAT(001) and (b)\nNdGaO 3(110), with the magnetization along the magnetic\neasy and hard axes. The solid curves are \fts to Eq. 7 with\nthe Gilbert damping parameter \u000b\fxed to 0:9\u000210\u00003.\ncoe\u000ecient, and tth= 1 nm is the threshold thickness\nbelow which the SRO layer is essentially insulating. The\nvalue oftthagrees with literature reporting that SRO\nis insulating at thickness of 3 monolayers ( \u00191.2 nm) or\nbelow54. Given the large deviation of \u001aSROfrom the bulk\nvalue, especially at small tSRO, the trend in Fig. 7(b)\nsuggests that taking into account the tSRO dependence\nof\u001aSROis a sensible approach.\nAPPENDIX B: IN-PLANE DAMPING OF LSMO\nON DIFFERENT SUBSTRATES\nIn Fig. 8, we compare the frequency dependence of \u0001 H\nfor 10-nm thick LSMO \flms deposited on di\u000berent sub-\nstrates: LSAT(001) and NdGaO 3(110). (NdGaO 3is an\northorhombic crystal and has ap\n2-pseudocubic param-\neter of\u00193.86 \u0017A, such that (001)-oriented LSMO grows\non the (110)-oriented surface of NdGaO 3.) As shown\nin Sec. 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To this end,\nwe start by phenomenologically constructing the stochastic Landau-Lifshitz-Gilbert equation for an\nantiferromagnet with the aid of the \ructuation-dissipation theorem. We then derive the Langevin\nequation for the soliton's center of mass by the collective coordinate approach. An antiferromagentic\nsoliton behaves as a classical massive particle immersed in a viscous medium. By considering a\nthermodynamic ensemble of solitons, we obtain the Fokker-Planck equation, from which we extract\nthe average drift velocity of a soliton. The di\u000busion coe\u000ecient is inversely proportional to a small\ndamping constant \u000b, which can yield a drift velocity of tens of m/s under a temperature gradient\nof 1 K/mm for a domain wall in an easy-axis antiferromagnetic wire with \u000b\u001810\u00004.\nPACS numbers: 75.78.-n, 66.30.Lw, 75.10.Hk\nIntroduction. |Ordered magnetic materials exhibit\nsolitons and defects that are stable for topological rea-\nsons [1]. Well-known examples are a domain wall (DW)\nin an easy-axis magnet or a vortex in a thin \flm. Their\ndynamics have been extensively studied because of fun-\ndamental interest as well as practical considerations such\nas the racetrack memory [2]. A ferromagnetic (FM) soli-\nton can be driven by various means, e.g., an external\nmagnetic \feld [3] or a spin-polarized electric current [4].\nRecently, the motion of an FM soliton under a temper-\nature gradient has attracted a lot of attention owing to\nits applicability in an FM insulator [5{8]. A temperature\ngradient of 20 K/mm has been demonstrated to drive a\nDW at a velocity of 200 \u0016m/s in an yttrium iron garnet\n\flm [9].\nAn antiferromagnet (AFM) is of a great current inter-\nest in the \feld of spintronics [10{12] due to a few advan-\ntages over an FM. First, the characteristic frequency of\nan AFM is several orders higher than that of a typical\nFM, e.g., a timescale of optical magnetization switching\nis an order of ps for AFM NiO [13] and ns for FM CrO 2\n[14], which can be exploited to develop faster spintronic\ndevices. Second, absence of net magnetization renders\nthe interaction between AFM particles weak, and, thus,\nleads us to prospect for high-density AFM-based devices.\nDynamics of an AFM soliton can be induced by an elec-\ntric current or a spin wave [15{17].\nA particle immersed in a viscous medium exhibits a\nBrownian motion due to a random force that is required\nto exist to comply with the \ructuation-dissipation theo-\nrem (FDT) [18, 19]. An externally applied temperature\ngradient can also be a driving force, engendering a phe-\nnomenon known as thermophoresis [20]. Dynamics of an\nFM and an AFM includes spin damping, and, thus, in-\nvolves thermal \ructuations at a \fnite temperature [21].\nThe corresponding thermal stochastic \feld in\ruences dy-\nnamics of a magnetic soliton [8, 22, 23], e.g., by assisting\na current-induced motion of an FM DW [24].\nFIG. 1. (Color online) A thermal stochastic force caused by\na temperature gradient pushes an antiferromagnetic domain\nwall to a colder region. The di\u000busion coe\u000ecient of the domain\nwall is inversely proportional to a small damping constant,\nwhich may give rise to a sizable drift velocity.\nIn this Rapid Communication, we study the Brown-\nian motion of a soliton in an AFM under a tempera-\nture gradient. We derive the stochastic Landau-Lifshitz-\nGilbert (LLG) equation for an AFM with the aid of the\nFDT, which relates the \ructuation of the staggered and\nnet magnetization to spin damping. We then derive the\nLangevin equation for the soliton's center of mass by em-\nploying the collective coordinate approach [16, 25]. We\ndevelop the Hamiltonian mechanics for collective coordi-\nnates and conjugate momenta of a soliton, which sheds\nlight on stochastic dynamics of an AFM soliton; it can be\nconsidered as a classical massive particle moving in a vis-\ncous medium. By considering a thermodynamic ensemble\nof solitons, we obtain the Fokker-Planck equation, from\nwhich we extract the average drift velocity. As a case\nstudy, we compute the drift velocity of a DW in a quasi\none-dimensional easy-axis AFM.\nThermophoresis of a Brownian particle is a multi-\nfaceted phenomenon, which involves several competing\nmechanisms. As a result, a motion of a particle depends\non properties of its environment such as a medium or\na temperature T[26]. For example, particles in pro-\ntein (e.g., lysozyme) solutions move to a colder region\nforT > 294 K and otherwise to a hotter region [20, 27].arXiv:1503.07854v2  [cond-mat.mes-hall]  8 Jul 20152\nThermophoresis of an AFM soliton would be at least as\ncomplex as that of a Brownian particle. We focus on one\naspect of it in this Rapid Communication; the e\u000bect of\nthermal stochastic force on dynamics of the soliton. We\ndiscuss two other possible mechanisms, the e\u000bects of a\nthermal magnon current and an entropic force [5], later\nin the Rapid Communication.\nMain results .|Before pursuing details of derivations,\nwe \frst outline our three main results. Let us consider\na bipartite AFM with two sublattices that can be trans-\nformed into each other by a symmetry transformation\nof the crystal. Its low-energy dynamics can be devel-\noped in terms of two \felds: the unit staggered spin\n\feldn\u0011(m1\u0000m2)=2 and the small net spin \feld\nm\u0011(m1+m2)=2 perpendicular to n. Here, m1and\nm2are unit vectors along the directions of spin angular\nmomentum in the sublattices.\nStarting from the standard Lagrangian description of\nthe antiferromagnetic dynamics [28], we will show below\nthat the appropriate theory of dissipative dynamics of\nantiferromagnets at a \fnite temperature is captured by\nthe stochastic LLG equation\ns(_n+\fn\u0002_m) =n\u0002(h+hth); (1a)\ns(_m+\fm\u0002_m+\u000bn\u0002_n) =n\u0002(g+gth)\n+m\u0002(h+hth);(1b)\nin conjunction with the correlators of the thermal\nstochastic \felds gthandhth,\nhgth\ni(r;t)gth\nj(r0;t0)i= 2kBT\u000bs\u000e ij\u000e(r\u0000r0)\u000e(t\u0000t0);(2a)\nhhth\ni(r;t)hth\nj(r0;t0)i= 2kBT\fs\u000e ij\u000e(r\u0000r0)\u000e(t\u0000t0);(2b)\nwhich are independent of each other [29]. This is our \frst\nmain result. Here, \u000band\fare the damping constants\nassociated with _nand _m,g\u0011\u0000\u000eU=\u000enandh\u0011\u0000\u000eU=\u000em\nare the e\u000bective \felds conjugate to nandm,U[n;m]\u0011\nU[n]+R\ndVjmj2=2\u001fis the potential energy ( \u001frepresents\nthe magnetic susceptibility), and s\u0011~S=Vis the spin\nangular momentum density ( Vis the volume per spin)\nper each sublattice. The potential energy U[n(r;t)] is\na general functional of n, which includes the exchange\nenergyR\ndVA ij@in\u0001@jnat a minimum [28].\nSlow dynamics of stable magnetic solitons can often\nbe expressed in terms of a few collective coordinates\nparametrizing slow modes of the system. The center of\nmassRrepresents the proper slow modes of a rigid soli-\nton when the translational symmetry is weakly broken.\nTranslation of the stochastic LLG equation (1) into the\nlanguage of the collective coordinates results in our sec-\nond main result, a Langevin equation for the soliton's\ncenter of mass R:\nMR+ \u0000_R=\u0000@U=@R+Fth; (3)\nwhich adds the stochastic force Fthto Eq. (5) of Tveten\net al. [17]. The mass and dissipation tensors are symmet-\nric and proportional to each other: Mij\u0011\u001aR\ndV(@in\u0001@jn) and \u0000 ij\u0011Mij=\u001c;where\u001c\u0011\u001a=\u000bs is the relaxation\ntime,\u001a\u0011\u001fs2is the inertia of the staggered spin \feld\nn. The correlator of the stochastic \feld Fthobeys the\nEinstein relation\nhFth\ni(t)Fth\nj(t0)i= 2kBT\u0000ij\u000e(t\u0000t0): (4)\nA temperature gradient causes a Brownian motion of\nan AFM soliton toward a colder region. In the absence\nof a deterministic force, the average drift velocity is pro-\nportional to a temperature gradient V/kBrTin the\nlinear response regime. The form of the proportional-\nity constant can be obtained by a dimensional analy-\nsis. Let us suppose that the mass and dissipation ten-\nsors are isotropic. The Langevin equation (3) is, then,\ncharacterized by three scalar quantities: the mass M,\nthe viscous coe\u000ecient \u0000, and the temperature T, which\nde\fne the unique set of natural scales of time \u001c\u0011M=\u0000,\nlengthl\u0011pkBTM= \u0000, and energy \u000f\u0011kBT. Using\nthese scales to match the dimension of a velocity yields\nV=\u0000c\u0016(kBrT);where\u0016\u0011\u0000\u00001is the mobility of an\nAFM soliton and cis a numerical constant. The explicit\nsolution of the Fokker-Planck equation, indeed, shows\nc= 1. This simple case illustrates our last main result;\na drift velocity of an AFM soliton under a temperature\ngradient in the presence of a deterministic force Fis given\nby\nV=\u0016F\u0000\u0016(kBrT): (5)\nFor a DW in an easy-axis one-dimensional AFM, the\nmobility is \u0016=\u0015=2\u000bs\u001b, where\u0015is the width of the\nwall and\u001bis the cross-sectional area of the AFM. For\na numerical estimate, let us take an angular momen-\ntum density s= 2~nm\u00001, a width\u0015= 100 nm, and a\ndamping constant \u000b= 10\u00004following the previous stud-\nies [17, 30]. For these parameters, the AFM DW moves\nat a velocity V= 32 m/s for the temperature gradient of\nrT= 1 K/mm.\nStochastic LLG equation. |Long-wave dynamics of an\nAFM on a bipartite lattice at zero temperature can de-\nscribed by the Lagrangian [28]\nL=sZ\ndVm\u0001(n\u0002_n)\u0000U[n;m]: (6)\nWe use the potential energy U[n;m]\u0011R\ndVjmj2=2\u001f+\nU[n] throughout the Rapid Communication, which re-\nspects the sublattice exchange symmetry ( n!\u0000n;m!\nm). Minimization of the action subject to nonlinear con-\nstraintsjnj= 1 and n\u0001m= 0 yields the equations of mo-\ntion for the \felds nandm. Damping terms that break\nthe time reversal symmetry can be added to the equa-\ntions of motion to the lowest order, which are \frst order\nin time derivative and zeroth order in spatial derivative.\nThe resultant phenomenological LLG equations are given3\nby\ns(_n+\fn\u0002_m) =n\u0002h; (7a)\ns(_m+\fm\u0002_m+\u000bn\u0002_n) =n\u0002g+m\u0002h (7b)\n[16, 30, 31]. The damping terms can be derived from the\nRayleigh dissipation function\nR=Z\ndV(\u000bsj_nj2+\fsj_mj2)=2; (8)\nwhich is related to the energy dissipation rate by \u0000_U=\n2R. The microscopic origin of damping terms does not\nconcern us here but it could be, e.g., caused by thermal\nphonons that deform the exchange and anisotropy inter-\naction.\nAt a \fnite temperature, thermal agitation causes \ruc-\ntuations of the spin \felds nandm. These thermal \ruc-\ntuations can be considered to be caused by the stochas-\ntic \felds gthandhthwith zero mean, which are con-\njugate to nandm, respectively; their noise correlators\nare then related to the damping coe\u000ecients by the FDT.\nThe standard procedure to construct the noise sources\nyields the stochastic LLG equation (1). The correlators\nof the stochastic \felds are obtained in the following way\n[18, 32]. Casting the linearized LLG equation (7) into the\nformfh;gg= ^\r\nf_n;_mgprovides the kinetic coe\u000ecients\n^\r. Symmetrizing the kinetic coe\u000ecients ^ \rproduces the\ncorrelators (2) of the stochastic \felds consistent with the\nFDT.\nLangevin equation. |For slow dynamics of an AFM,\nthe energy is mostly dissipated through the temporal\nvariation of the staggered spin \feld ndue toj_mj2'\n(\u000b\u001c)2jnj2\u001cj_nj2(from Eq. (7)), which allows us to set\n\f= 0 to study long-term dynamics of the magnetic soli-\nton [17]. At this point, we switch to the Hamiltonian for-\nmalism of an AFM [33], which sheds light on the stochas-\ntic dynamics of a soliton. The canonical momentum \feld\n\u0019conjugate to the staggered spin \feld nis\n\u0019\u0011\u000eL=\u000e _n=sm\u0002n: (9)\nThe stochastic LLG equations (1) can be interpreted as\nHamilton's equations,\n_n=\u000eH=\u000e \u0019=\u0019=\u001a; _\u0019=\u0000\u000eH=\u000en\u0000\u000eR=\u000e _n+gth;(10)\nwith the Hamiltonian\nH\u0011Z\ndV\u0019\u0001_n\u0000L=Z\ndVj\u0019j2\n2\u001a+U[n]: (11)\nLong-time dynamics of magnetic texture can often be\ncaptured by focusing on a small subset of slow modes,\nwhich are parametrized by the collective coordinates\nq=fq1;q2;\u0001\u0001\u0001g. A classical example is a DW in a one-\ndimensional easy-axis magnet described by the position\nof the wall Xand the azimuthal angle \b [3, 33]. An-\nother example is a skyrmion in an easy-axis AFM \flm,which is described by the position R= (X;Y ) [34, 35].\nTranslation from the \feld language into that of collective\ncoordinates can be done as follows. If the staggered spin\n\feldnis encoded by coordinates qasn(r;t) =n[r;q(t)],\ntime dependence of nre\rects evolution of the coordi-\nnates: _n= _qi@n=@qi. With the canonical momenta p\nde\fned by\npi\u0011@L\n@_qi=Z\ndV@n\n@qi\u0001\u0019; (12)\nHamilton's equations (10) translate into\nM_q=p;_p+ \u0000_q=F+Fth; (13)\nwhere F\u0011\u0000@U=@qis the deterministic force and Fth\ni\u0011R\ndV@ qin\u0001gthis the stochastic force. Hamilton's equa-\ntions (13) can be derived from the Hamiltonian in the\ncollective coordinates and conjugate momenta,\nH\u0011pTM\u00001p=2 +U(q); (14)\nwith the Poisson brackets fqi;pjg=\u000eij;fqi;qjg=\nfpi;pjg= 0. An AFM soliton, thus, behaves as a classi-\ncal particle moving in a viscous medium.\nWe focus on a translational motion of a rigid AFM\nsoliton by choosing its center of mass as the collective\ncoordinates q=R;n(r;t) =n(r\u0000R(t)). Eliminat-\ning momenta from Hamilton's equations (13) yields the\nLangevin equation for the soliton's center of mass:\n\u001cR+_R=\u0016F+\u0011; (15)\nwhere \u0011\u0011\u0016Fthis the stochastic velocity. Here the mo-\nbility tensor of the soliton \u0016\u0011\u0000\u00001relates a deterministic\nforce to a drift velocity h_Ri=\u0016Fat a constant temper-\nature [36]. The mobility is inversely proportional to a\ndamping constant, which can be a small number for an\nAFM, e.g., \u000b\u001810\u00004for NiO [37]. The correlator (2) of\nthermal stochastic \felds is translated into the correlator\nof the stochastic velocity,\nh\u0011i(t)\u0011j(t0)i= 2kBT\u0016ij\u000e(t\u0000t0)\u00112Dij\u000e(t\u0000t0):(16)\nFrom Eq. (16), we see that di\u000busion coe\u000ecient and the\nmobility of the soliton respect the Einstein-Smoluchowski\nrelation:D=\u0016kBT, which is expected on general\ngrounds. It can also be explicitly veri\fed as follows.\nA system of an ensemble of magnetic solitons at ther-\nmal equilibrium is described by the partition function\nZ\u0011R\n\u0005i[dpidxi=2\u0019~] exp(\u0000H=k BT), which provides the\nautocorrelation of the velocity, h_xi_xji=2 =M\u00001\nijkBT=2\n(the equipartition theorem). In the absence of an ex-\nternal force, multiplying \u001cxi+ _xi=\u0011i(15) byxjand\nsymmetrizing it with respect to indices iandjgive the\nequation,\u001cd2hxixji=dt2+dhxixji=dt= 2\u001ch_xi_xji, where\nthe \frst term can be neglected for long-term dynamics\nt\u001d\u001c. This equation in conjunction with the autocorre-\nlation of the velocity allows us to obtain the di\u000busion co-\ne\u000ecientDijin Eq. (16),hxixji= 2kBT\u001cM\u00001\nijt= 2Dijt,4\nwithout prior knowledge about the correlator (2) of the\nstochastic \felds.\nAverage dynamics. |An AFM soliton exhibits Brown-\nian motion at a \fnite temperature. The following Fokker-\nPlanck equation for an ensemble of solitons in an inho-\nmogeneous medium describes the evolution of the density\n\u001a(R;t) at timet\u001d\u001c:\n@\u001a\n@t+r\u0001j= 0;withj\u0011\u0016F\u001a\u0000Dr\u001a\u0000DT(kBrT);(17)\nwhereDT\u0011\u0016\u001ais the thermophoretic mobility (also\nknown as the thermal di\u000busion coe\u000ecient) [20, 38]. A\nsteady-state current density j=\u0016F\u001a0\u0000DT(kBrT) with\na constant soliton density \u001a(r;t) =\u001a0solves the Fokker-\nPlanck equation (17), from which the average drift veloc-\nity of a soliton can be extracted [39]:\nV=\u0016F\u0000\u0016(kBrT): (18)\nLet us take an example of a DW in a quasi one-\ndimensional easy-axis AFM with the energy U[n] =R\ndV(Aj@xnj2\u0000Kn2\nz)=2. A DW in the equilibrium\nisn(0)= (sin\u0012cos \b;sin\u0012sin \b;cos\u0012) with cos \u0012=\ntanh[(x\u0000X)=\u0015], where\u0015\u0011p\nA=K is the width of\nthe wall. The position Xand the azimuthal angle \b\nparametrize zero-energy modes of the DW, which are en-\ngendered by the translational and spin-rotational symme-\ntry of the system. Their dynamics are decoupled, \u0000 X\b=\n0, which allows us to study the dynamics of Xseparately\nfrom \b. The mobility of the DW is \u0016=\u0015=2\u000bs\u001b, where\u001b\nis the cross-sectional area of the AFM. The average drift\nvelocity (18) is given by\nV=\u00001\n2\u000bkB\u0015rT\ns\u001b: (19)\nDiscussion |The deterministic force Fon an AFM\nsoliton can be extended to include the e\u000bect of an elec-\ntric current, an external \feld, and a spin wave [15{17]. It\ndepends on details of interaction between the soliton and\nthe external degrees of freedom, whose thorough under-\nstanding would be necessary for a quantitative theory for\nthe deterministic drift velocity \u0016F. The Brownian drift\nvelocity V(18) is, however, determined by local property\nof the soliton. We have focused on the thermal stochastic\nforce as a trigger of thermophoresis of an AFM soliton in\nthis Rapid Communication. There are two other possi-\nble ingredients of thermophoresis of a magnetic soliton.\nOne is a thermal magnon current, scattering with which\ncould exert a force on a soliton [40]. The other is an\nentropic force, which originates from thermal softening\nof the order-parameter sti\u000bness [5]. E\u000bects of these two\nmechanisms have not been studied for an AFM soliton;\nfull understanding of its thermophoresis is an open prob-\nlem.\nIn order to compare di\u000berent mechanisms of thermally-\ndriven magnetic soliton motion, let us address a closelyrelated problem of thermophoresis of a DW in a quasi\none-dimensional FM wire with an easy- xz-plane easy-\nz-axis [3], which has attracted a considerable scrutiny\nrecently. To that end, we have adapted the approach de-\nveloped in this Rapid Communication to the FM case,\nwhich leads to the conclusion that a DW drifts to a\ncolder region by a Brownian stochastic force at the ve-\nlocity given by the same expression for an AFM DW,\nVB=\u0000kB\u0015rT=2\u000bs\u001b [41]. A thermal magnon cur-\nrent pushes a DW to a hotter region at the velocity\nVM=kBrT=6\u00192s\u0015m, where\u0015m\u0011p\n~A=sT is the\nthermal-magnon wavelength [7]. According to Schlick-\neiser et al. [5], an entropic force drives a DW to a hotter\nregion at the velocity VE=kBrT=4sa, whereais the\nlattice constant. The Brownian stochastic force, there-\nfore, dominates the other forces for a thin wire, \u001b\u001c\u0015a=\u000b\n(supposing rigid motion) [42].\nWithin the framework of the LLG equations that are\n\frst order in time derivative, the thermal noise is white\nas long as slow dynamics of a soliton is concerned,\ni.e., the highest characteristic frequency of the natural\nmodes parametrized by the collective coordinates is much\nsmaller than the temperature scale, ~!\u001ckBT. The\nthermal noise could be colored in general [26], e.g., for\nfast excitations of magnetic systems, which may be ex-\namined in the future. In addition, local energy dissi-\npation (8) allowed us to invoke the standard FDT at\nthe equilibrium to derive the stochastic \felds. It would\nbe worth pursuing to understand dissipative dynamics\nof general magnetic systems, e.g., with nonlocal energy\ndissipation with the aid of generalized FDTs at the out-\nof-equilibrium [43].\nWe have studied dynamics of an AFM soliton in the\nHamiltonian formalism. Hamiltonian's equations (13) for\nthe collective coordinates and conjugate momenta can be\nderived from the Hamiltonian (14) with the conventional\nPoisson bracket structure. By replacing Poisson brackets\nwith commutators, the coordinates and conjugate mo-\nmenta can be promoted to quantum operators. This may\nprovide a one route to study the e\u000bect of quantum \ruc-\ntuations on dynamics of an AFM soliton [44].\nAfter the completion of this work, we became aware\nof two recent reports. One is on thermophoresis of an\nAFM skyrmion [35], whose numerical simulations sup-\nport our result on di\u000busion coe\u000ecient. 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Usp. 23,\n21 (1980).\n[29] Employing the quantum FDT would change the corre-\nlator of the stochastic \felds to hhth\ni(r;!)hth\nj(r0;!0)i=\n[2\u0019\u000eij\u000bs~!=tanh( ~!=2kBT)]\u000e(r\u0000r0)\u000e(!\u0000!0) in the fre-\nquency space [18]. We focus on slow dynamics of an AFM\nsoliton in the manuscript, ~!\u001ckBT, which allows us to\nreplace ~!=tanh( ~!=2kBT) with 2kBT, yielding Eq. (2).\n[30] S. Takei, B. I. Halperin, A. Yacoby, and Y. Tserkovnyak,\nPhys. Rev. B 90, 094408 (2014).\n[31] B. A. Ivanov and D. D. Sheka, Phys. Rev. Lett. 72, 404\n(1994); N. Papanicolaou, Phys. Rev. B 51, 15062 (1995);\nH. V. Gomonay and V. M. Loktev, Phys. Rev. B 81,\n144427 (2010); A. C. Swaving and R. A. Duine, Phys.\nRev. B 83, 054428 (2011).\n[32] Y. Tserkovnyak and C. H. Wong, Phys. Rev. B 79,\n014402 (2009).\n[33] H. J. Mikeska, J. Phys. C: Solid St. Phys. 13, 2913 (1980);\nF. D. M. Haldane, Phys. Rev. Lett. 50, 1153 (1983).\n[34] I. Rai\u0014 cevi\u0013 c, D. Popovi\u0013 c, C. Panagopoulos, L. Benfatto,\nM. B. Silva Neto, E. S. Choi, and T. Sasagawa, Phys.\nRev. Lett. 106, 227206 (2011); X. Zhang, Y. Zhou, and\nM. Ezawa, arXiv:1504.01198.\n[35] J. Barker and O. A. Tretiakov, arXiv:1505.06156.\n[36] The relation between the mobility and dissipation tensor\ncan also be understood by equating the (twice) Rayleigh\nfunction, 2R=VT\u0000V, to the dissipation governed by\nthe mobility,\u0000_E=V\u0001F=VT\u0016\u00001V.\n[37] T. Kampfrath, A. Sell, G. Klatt, A. Pashkin, S. Mahrlein,\nT. Dekorsy, M. Wolf, M. Fiebig, A. Leitenstorfer, and\nR. Huber, Nature Photon. 5, 31 (2011).\n[38] N. van Kampen, IBM J. Res. Dev. 32, 107 (1988);\nT. Kuroiwa and K. Miyazaki, J. Phys. A: Math. Theor.\n47, 012001 (2014).\n[39] M. Braibanti, D. Vigolo, and R. Piazza, Phys. Rev. Lett.\n100, 108303 (2008).\n[40] The e\u000bect is, however, negligible at a temperature lower\nthan the magnon gap ( \u001840 K for NiO [37]). Also for our\nexample|a DW in a 1D easy-axis AFM|the conserva-\ntive force and torque exerted by magnons vanish [17].\n[41] Unlike 1D domain walls, Brownian motions are drasti-\ncally distinct between 2D FM and AFM solitons due to\nthe gyrotropic force, which signi\fcantly slows down fer-\nromagnetic di\u000busion [23, 35].\n[42] A DW in a wire with a large crosssection \u001b\u001da2forms\na 2D membrane. Its \ructuations foment additional soft\nmodes of the dynamics, which needs to be taken into\naccount to understand the dynamics of such a DW [25].\n[43] M. Baiesi, C. Maes, and B. Wynants, Phys. Rev. Lett.\n103, 010602 (2009); U. Seifert and T. Speck, Europhys.\nLett. 89, 10007 (2010).\n[44] S.-Z. Lin and L. N. Bulaevskii, Phys. Rev. B 88, 060404\n(2013).\n[45] P. Yan, Y. Cao, and J. Sinova, arXiv:1504.00651."    },    {        "title": "1510.01894v1.Tunable_damping__saturation_magnetization__and_exchange_stiffness_of_half_Heusler_NiMnSb_thin_films.pdf",        "content": "Tunable damping, saturation magnetization, and exchange sti\u000bness of half-Heusler\nNiMnSb thin \flms\nP. D urrenfeld,1F. Gerhard,2J. Chico,3R. K. Dumas,1, 4M. Ranjbar,1A. Bergman,3\nL. Bergqvist,5, 6A. Delin,3, 5, 6C. Gould,2L. W. Molenkamp,2and J. \u0017Akerman1, 4, 5\n1Department of Physics, University of Gothenburg, 412 96 Gothenburg, Sweden\n2Physikalisches Institut (EP3), Universit at W urzburg, 97074 W urzburg, Germany\n3Department of Physics and Astronomy, Uppsala University, Box 520, 752 20 Uppsala, Sweden\n4NanOsc AB, 164 40 Kista, Sweden\n5Materials and Nano Physics, School of ICT, KTH Royal Institute of Technology, Electrum 229, 164 40 Kista, Sweden\n6Swedish e-Science Research Centre (SeRC), 100 44 Stockholm, Sweden\nThe half-metallic half-Heusler alloy NiMnSb is a promising candidate for applications in spin-\ntronic devices due to its low magnetic damping and its rich anisotropies. Here we use ferromagnetic\nresonance (FMR) measurements and calculations from \frst principles to investigate how the com-\nposition of the epitaxially grown NiMnSb in\ruences the magnetodynamic properties of saturation\nmagnetization MS, Gilbert damping \u000b, and exchange sti\u000bness A.MSandAare shown to have a\nmaximum for stoichiometric composition, while the Gilbert damping is minimum. We \fnd excellent\nquantitative agreement between theory and experiment for MSand\u000b. The calculated Ashows the\nsame trend as the experimental data, but has a larger magnitude. Additionally to the unique in-\nplane anisotropy of the material, these tunabilities of the magnetodynamic properties can be taken\nadvantage of when employing NiMnSb \flms in magnonic devices.\nI. INTRODUCTION\nInterest in the use of half-metallic Heusler and half-\nHeusler alloys in spintronic and magnonic devices is\nsteadily increasing,1{3as these materials typically exhibit\nboth a very high spin polarization4{8and very low spin-\nwave damping.9{12One such material is the epitaxially\ngrown half-Heusler alloy NiMnSb,13,14which not only has\none of the lowest known spin-wave damping values of\nany magnetic metal, but also exhibits an interesting and\ntunable combination of two-fold in-plane anisotropy15\nand moderate out-of-plane anisotropy,10all potentially\ninteresting properties for use in both nanocontact-\nbased spin-torque oscillators16{22and spin Hall nano-\noscillators23{27. To successfully employ NiMnSb in such\ndevices, it is crucial to understand, control, and tailor\nboth its magnetostatic and magnetodynamic properties,\nsuch as its Gilbert damping ( \u000b), saturation magnetiza-\ntion (MS), and exchange sti\u000bness ( A).\nHere we investigate these properties in Ni 1-xMn1+xSb\n\flms using ferromagnetic resonance (FMR) measure-\nments and calculations from \frst principles for compo-\nsitions of -0.1\u0014x\u00140.4.MSandAare shown experi-\nmentally to have a maximum for stoichiometric compo-\nsition, while the Gilbert damping is minimum; this is\nin excellent quantitative agreement with calculations of\nand experiment on MSand\u000b. The calculated Ashows\nthe same trend as the experimental data, but with an\noverall larger magnitude. We also demonstrate that the\nexchange sti\u000bness can be easily tuned over a wide range\nin NiMnSb through Mn doping, and that the ultra-low\ndamping persists over a wide range of exchange sti\u000b-\nnesses. This unique behavior makes NiMnSb ideal for\ntailored spintronic and magnonic devices. Finally, by\ncomparing the experimental results with \frst-principlescalculations, we also conclude that the excess Mn mainly\noccupies Ni sites and that interstitial doping plays only\na minor role.\nII. METHODS\nA. Thin Film Growth\nThe NiMnSb \flms were grown by molecular beam\nepitaxy onto InP(001) substrates after deposition of a\n200 nm thick (In,Ga)As bu\u000ber layer.15The \flms were\nsubsequently covered in situ by a 10 nm thick mag-\nnetron sputtered metal cap to avoid oxidation and sur-\nface relaxation.28The Mn content was controlled dur-\ning growth via the temperature, and hence the \rux, of\nthe Mn e\u000busion cell. Six di\u000berent samples (see table I)\nwere grown with increasing Mn concentration, sample 1\nhaving the lowest and sample 6 the highest concentra-\ntion of Mn. High-resolution x-ray di\u000braction (HRXRD)\nmeasurements give information on the structural proper-\nties of these samples, con\frming the extremely high crys-\ntalline quality of all samples with di\u000berent Mn concentra-\nSamplevertical\nlattice\nconstant ( \u0017A)thickness\n(nm)uniaxial\neasy axis2K1\nMS(Oe)\n1 5.94 38 [110] 170\n2 5.97 38 [110] 8.4\n3 5.99 40 [110] 0\n4 6.02 45 [1 \u001610] 9.0\n5 6.06 45 [1 \u001610] 14.2\n6 6.09 38 [1 \u001610] 25.5\nTable I. Overview of NiMnSb \flms investigated in this study.arXiv:1510.01894v1  [cond-mat.mtrl-sci]  7 Oct 20152\ntion, even in the far from stoichiometric cases (samples 1\nand 6).15The vertical lattice constant is found to increase\nwith increasing Mn concentration and, assuming a linear\nincrease,29we estimate the di\u000berence in Mn concentra-\ntion across the whole set of samples to be about 40 at. %.\nWe will thus represent the Mn concentration in the fol-\nlowing experimental results by the measured vertical lat-\ntice constant. Stoichiometric NiMnSb exhibits vertical\nlattice constants in the range of 5.96{6.00 \u0017A, leading to\nthe expectation of stoichiometric NiMnSb in samples 2\nand 3.15Finally, the layer thicknesses are also determined\nfrom the HRXRD measurements, giving an accuracy of\n\u00061 nm.\nB. Ferromagnetic Resonance\nBroadband \feld-swept FMR spectroscopy was per-\nformed using a NanOsc Instruments PhaseFMR system\nwith a coplanar waveguide for microwave \feld excitation.\nMicrowave \felds hrfwith frequencies of up to 16 GHz\nwere applied in the \flm plane, perpendicularly oriented\nto an in-plane dc magnetic \feld H. The derivative of\nthe FMR absorption signal was measured using a lock-\nin technique, in which an additional low-frequency mod-\nulation \feld Hmod<1 Oe was applied using a pair of\nHelmholtz coils parallel to the dc magnetic \feld. The\n\feld directions are shown schematically in Fig. 1(a) and\na typical spectrum measured at 13.6 GHz is given in the\ninset of Fig. 1(b). In addition to the zero wave vector\nuniform FMR mode seen at about H=2.1 kOe, an addi-\ntional weaker resonance is observed at a much lower \feld\nof about 500 Oe, and is identi\fed as the \frst exchange-\ndominated perpendicular standing spin wave (PSSW)\nmode. The PSSW mode has a nonzero wave vector point-\ning perpendicular to the thin \flm plane and a thickness-\ndependent spin-wave amplitude and phase.30,31This can\nbe e\u000eciently excited in the coplanar waveguide geome-\ntry due to the nonuniform strength of the microwave \feld\nacross the \flm thickness.32\nThe \feld dependence of the absorption spectra (inset\nof Fig. 1(b)) can be \ft well (red line) by the sum of a sym-\nmetric and an antisymmetric Lorentzian derivative:33,34\ndP\ndH(H) =\u00008C1\u0001H(H\u0000H0)\nh\n\u0001H2+ 4 (H\u0000H0)2i2\n+2C2\u0000\n\u0001H2\u00004(H\u0000H0)2\u0001\nh\n\u0001H2+ 4 (H\u0000H0)2i2; (1)\nwhereH0is the resonance \feld, \u0001 Hthe full width at\nhalf maximum (FWHM), and C1andC2\ftted param-\neters representing the amplitude of the symmetric and\nantisymmetric Lorentzian derivatives, respectively. Both\nthe FMR and the PSSW peaks can be \ftted indepen-\ndently, as they are well separated by the exchange \feld\n\u00160Hex/(\u0019=d)2, wheredis the thickness of the layer.\neasy axis\nH, Hmod\nhard axis\nhrf p = 0\n(FMR)p = 1\n(PSSW)z\nHexx\ny\nFMRPSSW(a)\n(b)\n0\n500\n1000\n1500\n2000\n2500\n0\n5\n10\n15\nf(GHz)\nField (Oe)\n400\n600\n2000\n2200\n-0.5\n0.0\n0.5\n1.0\n1.5\ndP/dH (a. u.)\nField (Oe)\nf= 13.6 GHzFigure 1. (a) Schematic diagram of the FMR measurement\nshowing \feld directions. In our setup, the FMR mode and the\n\frst PSSW mode are excited. (b) Frequency vs. resonance\n\felds of the PSSW (red) and uniform FMR (black) mode for\nsample 2. The solid lines are \fts to the Kittel equation, and\nboth modes are o\u000bset horizontally by Hex. Inset: Resonance\ncurves forf=13.6 GHz. The \frst PSSW mode on the left and\nthe FMR mode on the right were \ft with Eq. 1\nFor our chosen sample thicknesses, the di\u000berences in res-\nonance \felds are always much larger than the resonance\nlinewidths.\nThe \feld dependence of both resonances is shown in\nFig. 1(b) and can now be used to extract information\nabout the magnetodynamic properties and anisotropies\nof the \flms. The curves are \fts to the Kittel equation,\nincluding internal \felds from the anisotropy and the ex-\nchange \feld for the PSSW excitation:15,35\nf=\r\u00160\n2\u0019\u0014\u0012\nH0+2KU\nMS\u00002K1\nMS+Hex\u0013\n\u0002\u0012\nH0+2KU\nMS+K1\nMS+Hex+Me\u000b\u0013\u00151=2\n;(2)\nwhereH0is the resonance \feld, \r=2\u0019the gyromagnetic\nratio, and\u00160the permeability of free space. Me\u000bis the ef-\nfective magnetization, which has a value close to the satu-\nration magnetization MS. 2KU=MSand 2K1=MSstands\nfor the internal anisotropy \felds coming from the uniaxial\n(KU) and biaxial ( K1) anisotropy energy densities in the\nhalf-Heusler material. The e\u000bective magnetic \feld also\nincludes an exchange \feld \u00160Hex= (2A=M S)(p\u0019=d )2,3\nwhich is related to the exchange sti\u000bness A, the \flm\nthicknessd, and the integer order of the PSSW mode\np, wherep= 0 denotes the uniform FMR excitation and\np= 1 the \frst PSSW mode. This mode numbering re-\n\rects the boundary conditions with no surface pinning of\nthe spins, which is expected for the in-plane measurement\ngeometry.36\nWe stress that the expression for the anisotropy con-\ntribution in Eq. 2 is only valid for the case in which the\nmagnetization direction is parallel to the uniaxial easy\naxis and also parallel to the applied \feld. A full angular-\ndependent formulation of the FMR condition is described\nin Ref. 15. To ful\fll the condition of parallel alignment\nfor all resonances, we perform the FMR measurements\nwith the dc magnetic \feld being applied along the domi-\nnant uniaxial easy axis of each \flm, which changes from\nthe [110] crystallographic direction to the [1 \u001610]-direction\nwith increasing Mn concentration (see Table I).\nThe values of the biaxial anisotropy2K1\nMShave been de-\ntermined in a previous study by \fxed-frequency in-plane\nangular dependent FMR measurements,15and were thus\ntaken as constant values in the \ftting process for Eq. 2;\na simultaneous \ft of both contributions can yield arbi-\ntrary combinations of anisotropy \felds due to their great\ninterdependence. The values for the uniaxial anisotropy\n2KU\nMSobtained from the frequency-dependent \ftting are\nin very good agreement with the previously obtained val-\nues in Ref. 15. The gyromagnetic ratio was measured\nto be\r=2\u0019= (28.59\u00060.20) GHz/T for all investigated\nsamples, and was therefore \fxed for all samples to allow\nbetter comparison of the e\u000bective magnetization values.\nThe Gilbert damping \u000bof the \flms is obtained by\n\ftting the FMR linewidths \u0001 Hwith the linear depen-\ndence:37\n\u00160\u0001H=\u00160\u0001H0+4\u0019\u000b\n\rf; (3)\nwhere \u0001H0is the inhomogeneous linewidth broaden-\ning of the \flm. The parallel alignment between mag-\nnetization and external magnetic \feld ensures that the\nlinewidth is determined by the Gilbert damping process\nonly.38\nC. Calculations from First Principles\nThe electronic and magnetic properties of the NiMnSb\nhalf-Heusler system were studied via \frst-principles cal-\nculations. The material was assumed to be ordered in a\nface-centered tetragonal structure with an in-plane lat-\ntice parameter ak\nlat= 5.88 \u0017A, close to the lattice con-\nstant of the InP substrate, and an out-of-plane lattice\nconstant of a?\nlat= 5.99 \u0017A, matching the value for the\nstoichiometric composition. Fixed values for the lattice\nparameters were chosen since an exact relation between\nthe o\u000b-stoichiometric composition and the experimen-\ntally measured vertical lattice constants cannot be es-\ntablished. Moreover, calculations with a varying verticallattice parameter for a constant composition showed only\na negligible e\u000bect on M S,A, and\u000b. The calculations\nwere performed using the multiple scattering Korringa-\nKohn-Rostocker (KKR) Green's function formalism as\nimplemented in the SPRKKR package.39Relativistic ef-\nfects were fully taken into account by solving the Dirac\nequation for the electronic states, the shape of the poten-\ntial was considered via the Atomic Sphere Approximation\n(ASA), and the local spin density approximation (LSDA)\nwas used for the exchange correlation potential. The co-\nherent potential approximation (CPA) was used for the\nchemical disorder of the system.\nThe Gilbert damping \u000bof the material was calculated\nusing linear response theory40, including the temperature\ne\u000bects from interatomic displacements and spin \ructua-\ntions.41,42\nThe exchange interactions Jijbetween the atomic\nmagnetic moments were calculated using the magnetic\nforce theorem, as considered in the LKAG formalism.43,44\nThe interactions were calculated for up to 4.5 times the\nlattice constant in order to take into account any long-\nrange interactions. Given the interatomic exchange in-\nteractions, the spin-wave sti\u000bness Dcan be calculated.\nDue to possible oscillations in the exchange interactions\nas a function of the distance, it becomes necessary to in-\ntroduce a damping parameter, \u0011, to assure convergence\nof the summation. Dcan then be obtained by evaluating\nthe limit\u0011!0 of\nD=2\n3X\nijJijp\nMiMjr2\nijexp\u0012\n\u0000\u0011rij\nalat\u0013\n; (4)\nas described in [45]. Here, MiandMjare the local mag-\nnetic moments at sites iandj,Jijis the exchange cou-\npling between the magnetic moments at sites iandj,\nandrijis the distance between the atoms iandj. This\nformalism can be extended to a multisublattice system46.\nTo calculate the e\u000bect of chemical disorder on the ex-\nchange sti\u000bness of the system, the obtained exchange in-\nteractions were summed over a supercell with a random\ndistribution of atoms in the chemically disordered sub-\nlattice. The e\u000bect that distinct chemical con\fgurations\ncan have over the calculation of the exchange sti\u000bness\nwas treated by taking 200 di\u000berent supercells. The re-\nsults were then averaged and the standard deviation was\ncalculated. The cells were obtained using the atomistic\nspin dynamics package UppASD.47\nFinally, with the spin-wave sti\u000bness determined as de-\nscribed above, the exchange sti\u000bness Acan be calculated\nfrom:48\nA=DM S(T)\n2g\u0016B: (5)\nHere,gis the Land\u0013 e g-factor of the electron, \u0016Bthe Bohr\nmagneton, and MS(T) the magnetization density of the\nsystem for a given temperature T, which for T= 0 K\ncorresponds to the saturation magnetization.\nFrom the \frst-principles calculations, the magnetic\nproperties for ordered NiMnSb and chemically disordered4\n0.60.70.80.95\n.956 .006 .056 .100123(b) m0MS \nm0Meffm0M (T)(a)4  µB/u.f.KS (mJ/m2)v\nertical lattice constant (Å)\nFigure 2. (a) MSandMe\u000bas functions of vertical lattice\nconstant. The theoretical value of 4.0 \u0016B=u.f. is shown by\nthe blue dashed line. (b) The calculated surface anisotropy\ndensity follows from the di\u000berence between MSandMe\u000b.\nNi1-xMn1+xSb were studied. To obtain the values of the\nexchange sti\u000bness AforT= 300 K, the exchange interac-\ntions from the ab initio calculations were used in conjunc-\ntion with the value of the magnetization at T= 300 K\nobtained from Monte Carlo simulations.\nIII. RESULTS\nA. Magnetization\nThe values of \u00160Me\u000bare plotted in Fig. 2(a) as red\ndots. The e\u000bective magnetization is considerably lower\nthan the saturation magnetization \u00160MS, which was in-\ndependently assessed using SQUID measurements and al-\nternating gradient magnetometry (AGM). The values for\n\u00160MScorrespond to a saturation magnetization between\n3.5\u0016B=unit formula and 3.9 \u0016B=u.f., with the latter\nvalue being within the error bars of the theoretically ex-\npected value of 4.0 \u0016B=u.f. for stoichiometric NiMnSb.49\nA reduction of MSis expected in Mn-rich NiMnSb alloys,\ndue to the antiferromagnetic coupling of the Mn Nidefects\nto the Mn lattice in the C1 bstructure of the half-Heusler\nmaterial.29An even stronger reduction is observed for\nthe Ni-rich sample 1, which is in accordance with the\nformation of Ni Mnantisites.50\nWhile the measurement error for MSis comparatively\nlarge due to uncertainties in the volume determination,\nthe error bars for Me\u000b, as obtained from ferromagnetic\nresonance, are negligible. NiMnSb \flms have been shown\nto possess a small but substantial perpendicular mag-\nnetic anisotropy, which can arise from either interfacial\nanisotropy or lattice strain.10,12To quantify the di\u000ber-\nence observed between MSandMe\u000b, we assume a uniax-\nial perpendicular anisotropy due to a surface anisotropy\n2\n4\n6\n8\n10\n5.95\n6.00\n6.05\n6.10\n0\n2\n4\n(b)\nA (pJ/m)\n(a)\nα(10-3)\nvertical lattice constant (Å)\n2\n4\n6\n8\n10\n-0.1\n0.0\n0.1\n0.2\n0.3\n0.4\n0\n2\n4\n(d)\n(c)\nxantisitesFigure 3. (a) and (b) show respectively the exchange sti\u000bness\nand Gilbert damping constant obtained from FMR measure-\nments, plotted as a function of the vertical lattice constant.\n(c) and (d) show the corresponding values obtained from \frst-\nprinciple calculations for T= 300 K. Negative values for x\nimply the introduction of Ni Mnantisites and positive values\nare related to Mn Niantisite defects. The error bars in (c) are\nthe standard deviations from repeated \frst-principles calcu-\nlations with 200 randomized supercells.\nenergy density KS, which is known to follow the rela-\ntion:51\n\u00160Me\u000b=\u00160MS\u00002KS\nMSd: (6)\nTheKScalculated in this way has values between\n0.5mJ=m2and 1.5mJ=m2, as shown in Fig. 2(b); these\nare comparable to the surface anisotropies obtained in\nother crystalline thin \flm systems.52. Although the \flm\nthicknesses in our set vary unsystematically, we can ob-\nserve systematic behavior of KSwith the vertical lat-\ntice constant, with an apparent minimum under the con-\nditions where stoichiometric NiMnSb is expected|that\nis, for samples 2 and 3. The increasing values for o\u000b-\nstoichiometric NiMnSb can be thus attributed to the con-\ncomitant increase in lattice defects, and thus of surface\ndefects, in these \flms.\nB. Exchange Sti\u000bness and Gilbert Damping\nThe experimentally determined exchange sti\u000bness, as\na function of the vertical lattice constant, and the Gilbert\ndamping parameter are shown in Fig. 3(a) and (b), re-\nspectively. The minimum damping observed in our mea-\nsurements is 1 :0\u000210\u00003for sample 3, and so within sto-\nichiometric composition. Sample 1, with a de\fciency\nof Mn atoms, showed nonlinear linewidth behavior at\nlow frequencies, which vanished for out-of-plane measure-\nments (not shown). This is typical with the presence of\ntwo-magnon scattering processes.52However, the damp-\ning is considerably lower in all samples than in a permal-\nloy \flm of comparable thickness.\nThe exchange sti\u000bness and Gilbert damping ob-\ntained from the \frst-principles calculations are shown in5\nFig. 3(c) and (d), respectively. For both parameters, the\nexperimental trends are reproduced quantitatively, with\nAhaving a maximum and \u000ba minimum value at stoi-\nchiometry.\nAs the concentration of both Mn or Ni antisites in-\ncreases, the exchange sti\u000bness decreases. This behavior\ncan be explained by analyzing the terms in the expres-\nsion for the spin-wave sti\u000bness, Eq. 4. It turns out that\nthe new exchange couplings Jij, which appear when an-\ntisites are present, play a major role, whereas changes in\nthe atomic magnetic moments or the saturation magne-\ntization appear to be relatively unimportant. Mn anti-\nsites in the Ni sublattice (i.e., excess Mn) have a strong\n(2 mRy) antiferromagnetic coupling to the Mn atoms in\nthe adjacent Mn layers. This results in a negative contri-\nbution toDcompared to the stoichiometric case, where\nthis interaction is not present. On the other hand, Ni\nantisites in the Mn sublattice have a negative in-plane\nexchange coupling of 0.3 mRy to their nearest-neighbor\nMn atoms, with a frustrated antiferromagnetic coupling\nto the Ni atoms in the adjacent Ni plane. The net e\u000bect is\na decreasing spin-wave sti\u000bness as the composition moves\naway from stoichiometry. The calculated values of Aare\naround 30 % larger than the experimental results, which\nis the same degree of overestimation we recently observed\nin a study of doped permalloy \flms53. It thus seems to\nbe inherent in our calculations from \frst principles.\nThe calculated Gilbert damping also agrees well with\nthe experimental values. The damping has its minimum\nvalue of 1.0\u000210\u00003at stoichiometry and increases with\na surplus of Ni faster than with the same surplus of\nMn. Both Mn and Ni antisites will act as impurities and\nit is thus reasonable to attribute the observed increase\nin damping at o\u000b-stoichiometry to impurity scattering.\nWhile the damping at stoichiometry also agrees quanti-\ntatively, the increase in damping is underestimated in the\ncalculations compared to the experimental values.\nDespite the fact that the calculations here focus purely\non the formation of Mn Nior Ni Mnantisites, they are\nnonetheless capable of reproducing the experimental\ntrends well. However, interstitials|that is, Mn or Ni sur-\nplus atoms in the vacant sublattice|may also be a possi-\nble o\u000b-stoichiometric defect in our system.50We have cal-\nculated their e\u000bects and can therefore discuss about the\nexistence of interstitials in our samples. A large fraction\nof Mn interstitials seems unlikely, as an increase in the\nsaturation magnetization can be predicted through calcu-\nlations, contrary to the experimental trend; see Fig. 2(a).On the other hand, the existence of Ni interstitials may\nbe compatible with the observed experimental trend, as\nthey decrease the saturation magnetization|albeit at a\nslower rate than Ni antisites and slower than experimen-\ntally observed. Judging from the measured data, it is\ntherefore likely that excess Ni exists in the samples as\nboth antisites and interstitials.\nIV. CONCLUSIONS\nIn summary, we have found that o\u000b-stoichiometry in\nthe epitaxially grown half-Heusler alloy NiMnSb has a\nsigni\fcant impact on the material's magnetodynamic\nproperties. In particular, the exchange sti\u000bness can be\naltered by a factor of about 2 while keeping the Gilbert\ndamping very low ( \u00195 times lower than in permalloy\n\flms). This is a unique combination of properties and\nopens up for the use of NiMnSb in, e.g., magnonic cir-\ncuits, where a small spin wave damping is desired. At the\nstoichiometric composition, the saturation magnetization\nand exchange sti\u000bness take on their maximum values,\nwhereas the Gilbert damping parameter is at its mini-\nmum. These experimentally observed results are repro-\nduced by calculations from \frst principles. Using these\ncalculations, we can also explain the microscopic mecha-\nnisms behind the observed trends. We also conclude that\ninterstitial Mn is unlikely to be present in the samples.\nThe observed e\u000bects can be used to \fne-tune the mag-\nnetic properties of NiMnSb \flms towards their speci\fc\nrequirements in spintronic devices.\nACKNOWLEDGMENTS\nWe acknowledge \fnancial support from the G oran\nGustafsson Foundation, the Swedish Research Council\n(VR), Energimyndigheten (STEM), the Knut and Alice\nWallenberg Foundation (KAW), the Carl Tryggers Foun-\ndation (CTS), and the Swedish Foundation for Strate-\ngic Research (SSF). F.G. acknowledges \fnancial support\nfrom the University of W urzburg's \\Equal opportunities\nfor women in research and teaching\" program. This work\nwas also supported initially by the European Commission\nFP7 Contract ICT-257159 \\MACALO\". A.B acknowl-\nedges eSSENCE. 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B 92, 024427\n(2015)."    },    {        "title": "1602.06201v2.A_systematic_study_of_magnetodynamic_properties_at_finite_temperatures_in_doped_permalloy_from_first_principles_calculations.pdf",        "content": "arXiv:1602.06201v2  [cond-mat.mtrl-sci]  23 Jun 2016A systematic study of magnetodynamic properties at finite te mperatures in doped\npermalloy from first principles calculations\nFan Pan,1,2,∗Jonathan Chico,3Johan Hellsvik,1Anna Delin,1,2,3Anders Bergman,3and Lars Bergqvist1,2\n1Department of Materials and Nano Physics, School of Informa tion and Communication Technology,\nKTH Royal Institute of Technology, Electrum 229, SE-16440 K ista, Sweden\n2Swedish e-Science Research Center (SeRC), KTH Royal Instit ute of Technology, SE-10044 Stockholm, Sweden\n3Department of Physics and Astronomy, Materials Theory Divi sion,\nUppsala University, Box 516, SE-75120 Uppsala, Sweden\n(Dated: June 24, 2018)\nBy means of first principles calculations, we have systemati cally investigated how the magnetody-\nnamicproperties Gilbert damping, magnetization andexcha ngestiffness areaffected whenpermalloy\n(Py) (Fe 0.19Ni0.81) is doped with 4d or 5d transition metal impurities. We find th at the trends in\nthe Gilbert damping can be understood from relatively few ba sic parameters such as the density of\nstates at the Fermi level, the spin-orbit coupling and the im purity concentration. The temperature\ndependence of the Gilbert damping is found to be very weak whi ch we relate to the lack of intraband\ntransitions in alloys. Doping with 4 delements has no major impact on the studied Gilbert damping,\napart from diluting the host. However, the 5 delements have a profound effect on the damping and\nallows it to be tuned over a large interval while maintaining the magnetization and exchange stiff-\nness. As regards spin stiffness, doping with early transitio n metals results in considerable softening,\nwhereas late transition metals have a minor impact. Our resu lt agree well with earlier calculations\nwhere available. In comparison to experiments, the compute d Gilbert damping appears slightly\nunderestimated while the spin stiffness show good general ag reement.\nI. INTRODUCTION\nSpintronics and magnonic applications have attracted\na large degree of attention due to the potential of cre-\nating devices with reduced energy consumption and im-\nprovedperformancecomparedtotraditionalsemiconduc-\ntor devices1–3. An important ingredient for understand-\ning and improving the performance of these devices is\na good knowledge of the magnetic properties. In this\nstudy, we focus on the saturation magnetization Ms, the\nexchange stiffness Aand the Gilbert damping α4. The\nlatter is related to the energy dissipation rate of which a\nmagnetic system returns to its equilibrium state from an\nexcited state, e.g. after the system has been subjected to\nanexternalstimuliisuchasanelectricalcurrentwhichal-\ntersits magneticstate. The threeparameters, Ms,Aand\nαdescribe the magnetodynamical properties of the sys-\ntem of interest. Ultimately one would like to have com-\nplete independent control and tunability of these proper-\nties. In this study, the magnetodynamical properties of\nPermalloy (Py) doped with transition metal impurities\nare systematically investigated within the same compu-\ntational framework.\nThe capability of tuning the damping for a material\nwith such a technological importance as Py is important\nfor the development of possible new devices in spintron-\nics and magnonics. The understanding of how transi-\ntion metals or rare earth dopants can affect the prop-\nerties of Py has been the focus of a number of recent\nexperimental studies5–9. Typically in these studies, the\nferromagnetic resonance10(FMR) technique is employed\nandαandMsare extracted from the linewidth of the\nuniform precession mode while Ais extracted from the\nfirst perpendicularstandingspin-wavemode11,12. On thetheory side, calculations of Gilbert damping from first-\nprinciples density functional theory methods have only\nrecently become possible due to the complexity of such\ncalculations. Two main approaches have emerged, the\nbreathing Fermi surface model13,14and the torque corre-\nlation models15,16. Common to both approaches is that\nspin-orbit coupling along with the density of states at\nthe Fermi level are the main driving forces behind the\ndamping. The breathing Fermi surface model only takes\nonly into account intraband transitions while torque cor-\nrelation model also includes interband transitions. The\ntorque correlation model in its original form contains a\nfree parameter, namely the scattering relaxation time.\nBrataaset. al17later lifted this restriction by employing\nscattering theory and linear response theory. The result-\ning formalism provides a firm foundation of calculating\nαquantitative from first principles methods and allows\nfurther investigations of the source of damping. Gilbert\ndamping in pure Py as well as doping with selected el-\nements have been calculated in the past9,18–20, however\nno systematic study of the magnetodynamic properties\nwithin the same computational framework has been con-\nducted which the present paper aims to address.\nThe paper is outlined as follows: In Section II we\npresent the formalism and details of the calculations, in\nSection III we present the results of our study and in\nSection IV we summarize our findings and provide an\noutlook.2\nII. THEORY\nA. Crystal structure of permalloy and treatment of\ndisorder in the first-principles calculations\nPure Permalloy (Py), an alloy consisting of iron (Fe)\nandnickel(Ni) withcompositionFe 0.19Ni0.81, crystallizes\nin the face centered cubic (fcc) crystal structure, where\nFe and Ni atoms are randomly distributed. Additional\ndoping with 4 dand 5dimpurities (M) substitutes Fe (or\nNi) sothat it becomes a threecomponent alloywith com-\nposition Py 1−xMx, wherexis the concentration of the\ndopant.\nAll first principles calculations in this study were\nperformed using the spin polarized relativistic (SPR)\nKorringa-Kohn-Rostoker (KKR)21Green’s function\n(GF) approach as implemented in the SPR-KKR\nsoftware22. The generalized gradient approximation\n(GGA)23wasusedintheparametrizationoftheexchange\ncorrelation potential and both the core and valence elec-\ntrons were solved using the fully relativistic Dirac equa-\ntion. The broken symmetry associated with the chemical\nsubstitution in the system was treated using the coherent\npotential approximation (CPA)24,25.\nB. Calculation of magnetodynamical properties of\nalloys: Gilbert damping within linear response\ntheory and spin stiffness\nOne of the merits with the KKR-CPA method is that\nit has a natural way of incorporating calculations of re-\nsponse properties using linear response formalism17,19,20.\nThe formalism for calculating Gilbert damping in the\npresent first principles method has been derived in Refs.\n[17] and [20], here we only give a brief outline of the\nmost important points. The damping can be related as\nthe dissipation rate of the magnetic energy which in turn\ncan be associated to the Landau-Lifshitz-Gilbert (LLG)\nequation4, leading to the expression\n˙E=Heff·dM\ndτ=1\nγ2˙ˆm[˜G(m)˙ˆm], (1)\nwhereˆm=M/Msdenotesthe normalizedmagnetization\nvector,Msthe saturation magnetization, γthe gyromag-\nnetic ratio and ˜G(m) the Gilbert relaxation rate tensor.\nPerturbing a magnetic moment from its equilibrium\nstate by a small deviation, ˆm(τ) =ˆm0+u(τ), gives an\nalternative expression of the dissipation rate by employ-\ning linear response theory\n˙Edis=π/planckover2pi1/summationdisplay\nij/summationdisplay\nµν˙uµ˙uν∝angbracketleftψi|∂ˆH\n∂uµ|ψj∝angbracketright∝angbracketleftψj|∂ˆH\n∂uν|ψi∝angbracketright×\nδ(EF−Ei)δ(EF−Ej),(2)where the δ-functions restrict the summation over\neigenstates to the Fermi level which can be rewrit-\nten in terms of Green’s function as Im G+(EF) =\n−π/summationtext\ni|ψi∝angbracketright∝angbracketleftψi|δ(EF−Ei). By comparing Eqs. (1) and\n(2), the Gilbert damping parameter αis obtained, which\nis dimensionless and is related to the Gilbert relaxation\ntensorα=˜G/(γMs). This can be expressed as a trans-\nport Kubo-Greenwood-like equation26,27in terms of the\nretarded single-particle Green’s functions\nαµν=−/planckover2pi1γ\nπMsTrace/angbracketleftBig∂ˆH\n∂uµImG+(EF)∂ˆH\n∂uνImG+(EF)/angbracketrightBig\nc,\n(3)\nwhere∝angbracketleft...∝angbracketrightcdenotes a configurational average. For the\ncubic systems treated in this study, the tensorial form of\nthe damping can with no loss of generality be replaced\nwith a scalar damping parameter. Thermal effects from\natomicdisplacementsandspinfluctuationswereincluded\nusing the alloy-analogy model28within CPA.\nThe spin-wave stiffness Dis defined as the curvature\nof the spin wave dispersion spectrum at small wave vec-\ntors (ω(q)≈Dq2).Din turn is directly related to the\nexchange interactions in the Heisenberg model which are\nobtained using the LKAG formalism29,30such that\nD=2\n3/summationdisplay\nijJijR2\nij√mimj, (4)\nwhereJijis the interatomic exchangeparameterbetween\nthei-thandj-thmagneticmoment, Rijthe distancecon-\nnecting the atomic sites with index iandjandmi(mj)\nthe magneticmoment atsite i(j). It is worthnotingthat\nEq. (4) only holds for cubic systems as treated here, for\nlower symmetries the relation needs modifications. The\nexchange couplings in metallic systems are typically long\nranged and could have oscillations of ferromagnetic and\nantiferromagnetic character, such as present in RKKY\ntype interactions. Due to the oscillations in exchange in-\nteractions, care is needed to reach numerical convergence\nof the series in Eq. (4) and it is achieved following the\nmethodology as outlined in Refs. 31 and 32.\nC. Calculation of finite temperature magnetic\nproperties\nOnce the exchange interactions within the Heisen-\nberg model have been calculated, we obtained finite\ntemperature properties from Metropolis33Monte Carlo\nsimulations as implemented in the UppASD software\npackage34,35. In particular, the temperature dependent\nmagnetization was obtained, and enters the expression\nfor micromagnetic exchange stiffness A, defined as36–39\nA(T) =DM(T)\n2gµB, (5)3\nwhereµBis the Bohr magneton, gis the Land´ e g-factor\nandM(T) the magnetization at temperature T.\nD. Details of the calculations\nForeachconcentrationofthedifferentimpuritiesinPy,\nthe lattice parameter was optimized by varying the vol-\nume and finding the energy minimum. The k-point mesh\nfor the self consistent calculations and exchange interac-\ntions was set to 223giving around 800 k-points in the ir-\nreducible wedge of the Brillouin zone (IBZ). The Gilbert\ndamping calculation requires a very fine mesh to resolve\nallthe Fermisurfacefeaturesandthereforeasignificantly\ndenser k-point mesh of 2283(∼1.0×106k-points in\nIBZ) was employed in these calculations to ensure nu-\nmericalconvergence. Moreover,vertexcorrections40were\nincluded in the damping calculationssince it has been re-\nvealedtobeimportantinpreviousstudies20forobtaining\nquantitative results.\nIII. RESULTS\nA. Equilibrium volumes and induced magnetic\nmoments\n6.656.706.756.806.856.906.957.00\nNbMoTcRuRhPdAgTaWReOsIrPtAuLattice constant (Bohr)   \n10%M\n15%M\nPy\nFIG. 1. Calculated equilibrium volumes of Py-M, where M\nstands for a 4 d(left) or 5 d(right) transition metal. Values\nfor 10% and 15% doping concentrations are shown. Reference\nvalue of pure Py is diplayed with a dashed line.\nFigure 1 shows the calculated equilibrium volume of\ndoped Pyfor twodifferent concentrations(10%and 15%)\nofimpuritiesfrom the 4 dand5dseriesofthe PeriodicTa-\nble. Firstofall, itis notedthat thevolumeincreaseswith\nthe concentration, and the volume within a series (4 dor\n5d) has a parabolicshape with minimum in the middle of\nthe series. This is expected since bonding states are con-secutivelyfilledand maximizedinthe middle ofthe series\nandthusthebondingstrengthreachesamaximum. Mov-\ning further through the series, anti-bonding states start\nto fill, giving rise to weaker bonding and larger equilib-\nriumvolumes. Thisisconsistentwiththeatomicvolumes\nwithin the two series41.\n0.500.600.700.800.901.001.10Total moment ( µΒ)\nM 5%\n10%\n15%\n20%\nPy\n-0.4-0.20.00.20.40.6\nNbMoTcRuRhPdAgTaWReOsIrPtAuLocal moment ( µΒ)M\nFIG.2. (Upper)Totalmagneticmoment(spinandorbital)for\ndifferent impurities and concentrations. Reference value f or\npure Py marked with a dashed line. (Lower) Local impurity\nmagnetic moment for Py 0.95M0.05\nThe local moments of the host atoms are only weakly\ndependent on the type of impurity atom present. More-\nover, the magnetic moments are dominated by the spin\nmomentµSwhile the orbital moments µLare much\nsmaller. As an example, in pure Py without additional\ndoping, the spin (orbital) moments of Fe is calculated to\n≈2.64 (0.05) µBand for Ni ≈0.64 (0.05) µB, respec-\ntively. This adds up to an average spin (orbital) moment\nof≈1.04(0.05)µBby taking into account the concentra-\ntion of Fe and Ni in Py. The total moment is analyzed\nin more detail in Fig. 2 (upper panel). As mentioned\nabove, one would like to achieve tunable and indepen-\ndent control of the saturation magnetization. Reducing\nthe magnetization reduces the radiative extrinsic damp-\ning but could at the same time affect the other properties\nin an unwanted manner. In many situations, one strive\nfor keeping the value of the total moment (saturation\nmagnetization) at least similar to pure Py, even for the\ndoped systems. It is immediately clear from Fig. 2 that\ndoping elements late in the series are the most preferable\nin that respect, for instance Rh and Pd in the 4 dseries\nand Ir, Pt and Au in the 5 dseries.\nIn Fig. 2 (lower panel) we show the local impurity\nmagnetic moments for 5% impurities in Py. In the be-\nginning of the 4 d(5d) series, the impurity atoms have an\nantiferromagnetic coupling, reflected in the negative mo-\nments compared to the host (Fe and Ni) atoms while lat-4\nter in the series couples ferromagnetically (positive mo-\nments). The antiferromagnetic coupling may not be pre-\nferred since it will tend to soften the magnetic properties\nand maybe even cause more complicated non-collinear\nmagnetic configurations to occur.\nB. Band structure\nSince Py and doped-Py are random alloys, they lack\ntranslational symmetry and calculations using normal\nband structure methods are more challenging due to the\nneed for large supercells. However, employing CPA re-\nstores the translational symmetry and more importantly,\nthe band structure of disordered systems can be ana-\nlyzedthroughtheBlochspectralfunction (BSF) A(E,k),\nwhich can be seen as a wave vector k-dependent density\nof states (DOS) function. For ordered systems the BSF\nis aδ-like function at energy E( k) while for disordered\nsystems each peak has an associated broadening with a\nlinewidth proportional to the amount of disorder scatter-\ning. In the upper panel of Fig. 3 the calculated BSF for\npure Py is displayed. Despite being a disordered system,\nthe electron bands are rather sharp below the Fermi level\nwhile in the vicinity ofthe Fermi level the bands becomes\nmuch more diffuse indicating that most of the disorder\nscattering takes place around these energies.\nIf Py is doped with 20% Pt impurities, the positions\nof the electron bands do not change much as shown by\nthe BSF in the lower panel of Fig. 3. The most strik-\ning change is the large increase of the disorder scatter-\ning compared to than Py causing diffuse electron bands\nthroughout the Brillouin zone and energies. However,\nexactly at the Fermi level the differences between the\ndoped and undoped system is not very pronounced and\nthesestatesarethe mostimportantforthe determination\nof the Gilbert damping, as seen from Eq. 3.\nC. Gilbert damping: effect of doping\nThe calculated Gilbert damping of the doped Py sys-\ntems fordifferent concentrationsofimpurities isshownin\nFig. 4 (upper panel). The 4 dimpurities only marginally\ninfluence the damping while the 5 dimpurities dramati-\ncally change the damping. The first observation is that\nweobtainverygoodagreementasinthe previousstudy19\nfor the 5dseries with 10% impurities, howevernot so sur-\nprising since we use same methodology. Secondly, the\nmost dramatic effect on damping upon doping is for the\ncase of Py doped with 20% Os impurities in which the\ndamping increases with approximately 800% compared\nto pure Py, as previously reported in Ref. [20]. How-\never, in the present study we have systematically var-\nied the impurity elements and concentrations and tried\nto identify trends over a large interval. Compared to\nexperiments5, the calculated values of the Gilbert damp-\ning are consistently underestimated. However it is worth\nFIG. 3. The Bloch spectral function A(E,k) of Py (upper\npanel) and Py doped with 20% Pt impurities (lower panel).\nThe Fermi level is indicated with a horizontal black line at\nzero energy.\nremembering that calculations only shows the intrinsic\npart of the damping while experiments may still have\nsome additional portion of extrinsic damping left such\nas Eddy current damping and radiation damping, since\nit is difficult to fully separate the different contributions.\nMoreover,incalculationsacompleterandomdistribution\nof atoms is assumed while there may be sample inhomo-\ngeneities such as clustering in the real samples.\nFrom most theoretical models, the two main material\nproperties that determine the damping are the density\nof states (DOS) at the Fermi level and the strength of\nthe spin-orbit coupling. In the following, we first inves-\ntigate separately how these properties affect the damp-\ning and later the combination of the two. In the lower\npanel of Fig. 4, the total DOS and the impurity-DOS are\ndisplayed for 10% impurity concentration of 4 dand 5d\nseries transition metals. In the both 4 dand 5dseries the\nimpurity-DOS exhibits a maximum in the middle of the\nseries. However, the value of the DOS are similar for the\n4dand 5dseries and therefore cannot solely explain the\nlarge difference in damping found between the two series.5\nFor the 4dseries, the calculated damping is not directly\nproportional to the DOS while there is a significant cor-\nrelation of the DOS and damping in the 5 dseries.\n0.000.010.020.030.04Gilbert damping αM 5%\n10%\n15%\n20%\nPy\nExp.5%\n0.200.400.600.801.001.201.40\nNbMoTcRuRhPdAgTaWReOsIrPtAun(EF) (sts./eV)\nM\nPy-M\nFIG. 4. (Upper) Calculated Gilbert damping parameter for\nPy+M in different concentrations of 4 dand 5dtransition\nmetal M at low temperatures ( T= 10K). Experimental re-\nsults from Ref. [5] measured at room temperature are dis-\nplayed by solid squares and dashed line indicate reference\nvalue for pure Py. (Lower) Total (blue) and impurity (black)\ndensity of states at the Fermi level EFfor 10% impurities in\nPy.\nIn order to analyze the separate influence of spin-orbit\ncoupling on the damping, we show in upper panel of\nFig. 5 the spin-orbit parameter ξ∝1\nrdV(r)\ndr, where V(r)\nis the radial potential, of the impurity d-states. The\ncalculations include all relativistic effects by solving the\nDirac equation but here we have specifically extracted\nthe main contribution from the spin-orbit coupling. As\nexpected, the spin-orbit parameter increases with atomic\nnumberZ, and is therefore considerably larger in the\n5dseries compared to the 4 dseries. This is the most\nlikely explanation why the damping is found to be larger\nin the 5dseries than the 4 dseries. However, within a\nsingle element in either the 4d or 5d series, the damp-\ning is quadratically dependent on the relatve strength of\nthe spin-orbit strength20. The calculated values of the\nspin-orbit parameter are in good agreement with previ-\nous calculations42,43and reaches large values of 0.6-0.9\neV for the late 5 delements Ir,Pt and Au while all val-ues are below 0.3 eV for the 4 dseries. If the damping\nacross elements would only be proportional to the spin-\norbit coupling, then the damping would monotonously\nincrease with atomic number and since this is not what\nhappens, we conclude that there is a delicate balance be-\ntween spin-orbit coupling and DOS that determines the\ndamping which is further highlighted through a qualita-\ntive analysis of the involved scattering processes.\n0.000.300.600.90ξ (eV)Spin-orbit coupling\n0.000.010.02\nNbMoTcRuRhPdAgTaWReOsIrPtAuα (Norm.)TC model\nCalculation\nFIG. 5. Upper: the spin-orbit parameter of d-electrons of\nthe impurity atoms. Lower: qualitative comparison between\ncalculations and torque correlation (TC) model for damping\nwith 10% impurity concentration.\nIn the torque correlation model, the dominant con-\ntribution to damping is through the scattering44,45and\ntakes the following form\nα=1\nγMs(γ\n2)2n(EF)ξ2(g−2)2/τ, (6)\nwhereτis the relaxation time between scattering events,\nandgthe Lande g-factor, for small orbital contributions,\ncan be related as46g= 2(1 +µL\nµS). We assume that τ\nis the same for all impurities, which is clearly an ap-\nproximation but calculating τis beyond the scope of the\npresent study. By normalizing the damping from Eq. (6)\nsuch that the value for Os (10% concentration) coincides\nwith the first principles calculations, we obtain a quali-\ntatively comparison between the model and calculations,\nas illustrated in lower panel of Fig. 5. It confirms the6\ntrend in which 5 dseries lead to a larger damping than\nthe4dseriesandcapturesqualitativelythemainfeatures.\nHowever, the peak value of the damping within the 5 d\nseries in the TC model occurs for Ir while calculations\ngive Os as in experiment. Another model developed for\nlow dimensional magnetic systems such as adatoms and\nclusters suggests that the damping is proportional to the\nproduct of majority and minority density of states at the\nFermi level47. It produces a parabolic trend but with\nmaximum at incorrect position and fails to capture the\nincreased damping of the 5 delements.\nTo further analyse the role of impurity atoms on the\ndamping we also performed calculations where instead of\nimpurities we added vacancies in the system, i.e. void\natoms. The results are shown in Fig. 6 where damping\nas a function of concentrationofAg (4 d), Os (5d) and va-\ncancies are compared to each other along with Os results\nfrom experimental5and previous calculations. Surpris-\ningly, vacancies have more or less the same effect as Ag\nwith the damping practically constant when increasing\nconcentration. Since Ag has a zero moment, small spin-\norbit coupling and small density of states at the Fermi\nlevel, the net effect of Ag from a damping (or scatter-\ning) point of view is mainly diluting the host similar to\nadding vacancies. In contrast, in the Os case, being a\n5dmetal, there is a strong dependence on the concentra-\ntion that was previously analyzed in terms of density of\nstates and Os having a strong spin-orbit coupling. Our\nresults from Os is slightly lower than the previous re-\nported values19,20, despite using same software. How-\never, the most likely reason for the small discrepancy is\nthe use of different exchange-correlationpotentials in the\ntwo cases.\n0.000.010.020.030.040.050.060.070.08\n 0  5  10  15  20Gilbert damping α\n(%) of MPyOs\nPyAg\nPyVac\n10%Os ref(theo.1)\n15%Os ref(theo.2)\nPyOs ref(expt.)\nFIG. 6. Calculated Gilbert damping as a function of Os,\nAg and vacancy (Vac) concentration in Py. Open red circle:\ncalculation from Ref. [19], solid red circle: calculation f rom\nRef. [20] and red solid square: experimental data from Ref. [ 5]D. Gilbert damping: effect of temperature\nIntheprevioussectionwestudiedhowthedampingde-\npends on the electronic structure and spin-orbit coupling\nat low temperatures. However, with increasing tempera-\nture additional scattering mechanisms contribute to the\ndamping, most importantly phonon and magnon scatter-\ning. The phonon scattering is indirectly taken into ac-\ncount by including a number of independent atomic dis-\nplacementsbringingtheatomsoutfromtheirequilibrium\npositions and magnon scattering is indirectly included by\nreducing the magnetic moment for a few configurations\nand then average over all atomic and magnetic configu-\nrations within CPA. It should be noted that the present\nmethodology using the alloy-analogymodel28has limita-\ntions for pure systems at very low temperatures where\nthe damping diverges, but we are far from that situation\nin this study since all systems have intrinsic chemical\ndisorder. However, the limitations for pure systems can\nbe lifted using a more advanced treatment using explicit\ncalculation of the dynamical susceptibility48.\n0.000.010.02α(T)\n1.001.25\n 0 50 100 150 200 250 300 350 400α(T)pho+mag/α(T)pho\nTemperature (K) Py+20% Mo\nPy+20% Rh\nPy+20% W\nPy+20% Pt\nFIG. 7. Gilbert damping parameter including temperature\neffects from both atomic displacements and spin fluctuations\n(upper panel). The effect of spin fluctuations on the Gilbert\ndamping (lower panel), see text.\nThe temperature dependence of damping for a few se-\nlected systems is displayed in Fig. 7 where both atomic\ndisplacements and spin fluctuations are taken into ac-\ncount. From the 4 d(5d) series, we choose to show results7\nfor Mo and Rh (W and Pt), where Mo (W) has a small\nantiferromagnetic moment and Rh (Pt) a sizeable ferro-\nmagnetic moment, from Fig. 2. All systems display an\noverall weak temperature dependence on damping which\nonly marginally increases with temperature, as shown\nin upper panel of Fig. 7. However, in order to sepa-\nrate the temperature contributions from atomic displace-\nments and spin fluctuations, we show the ratio between\nthe total damping and damping where only atomic dis-\nplacements are taken into account in the lower panel of\nFig. 7. The two systems with sizable moments (Rh and\nPt), clearly have a dominant contribution from spin fluc-\ntuations when the moments are reduced upon increased\nscattering due to temperature. In contrast, the two sys-\ntems with (small) antiferromagnetic moments (Mo and\nW), the effect of the spin fluctuations on the damping is\nnegligible and atomic displacements are solely responsi-\nble. The weak temperature dependence found in these\ndoped Py systems is somewhat surprising since in pure\nmetals like Fe and Ni, a strong temperature dependence\nhas been both measured and calculated20, however data\nfor other random alloy systems is scarce.\nThe temperature dependence of damping from the\nband structure is often attributed to interband and\nintraband transitions which arises from the torque-\ncorrelation model. Intraband transitions has conduc-\ntivity like dependence on temperature while interband\nshows resistivty-like dependence. The weak overall de-\npendence found in the systems in Fig. 7 suggests lack of\nintraband transitions but a more detailed analysis of the\nband structure and thermal disorder are left for a future\nstudy.\nE. Spin-wave stiffness and exchange stiffness\nIn the previous sections, we investigated saturation\nmagnetization and damping and we are therefore left\nwith the exchange stiffness. The calculated spin-wave\nstiffnessDatT= 0 K, from Eq. 4, is displayed in the up-\nperpanelofFig.8. Dcanbedirectlymeasuredfromneu-\ntron scattering experiments but as far as we are aware,\nno such data exist. For the late elements in the 4 dand\n5dseries, the spin wave stiffness is maximized and have\nvalues rather similar to pure Py, however with a reduc-\ntion of approximately 20%. In micromagnetic modelling,\nit is common to use the exchange stiffness Ainstead of\nD.Ais proportional to D, from Eq. 5, and the sole\ntemperature dependence of Atherefore comes from the\nmagnetization. In the lower panel of Fig. 8, we show the\ncalculated room temperature ( T= 300 K) values of A,\ntogether with values for pure Py and available experi-\nmental data. In the beginning of the 4 d(5d) series, the\nexchangestiffnessbecomes smalluponincreasingconcen-\ntration of impurities and the systems are magnetically\nvery soft. It follows from the fact that magnetization\nis small because the systems are close to their ordering\ntemperature. Contrary, for the late elements in the 4 d100200300400500Spin−wave stiffness\nD (meV Å2)5% M\n10% M15% M\nPy\n  5 10 15\nNbMoTcRuRhPdAgTaWReOsIrPtAuExchange stiffness\nA (pJ/M)\nPy(expt.) 15%(expt.)\nFIG. 8. Spin-wave stiffness Dof Py-M in the ground state\n(top) and exchange stiffness constant Aat room temperature\nT= 300K (bottom) as a function of doping concentration.\nThe strict dashed lines show the reference value of pure Py\nfrom calculation and experiments. The scattered dots indi-\ncate the experimental data for Py+15%M (Ag/Pt/Au) from\nRef.9\n(5d)series,themagnetizationhasalargefinitevalueeven\nat room temperature and thereforethe exchangestiffness\nalso has a large value, howeverreduced by approximately\n15% compared to pure Py.\nIV. SUMMARY AND CONCLUSIONS\nAsystematic study ofthe intrinsicmagnetic properties\nof transition metal doped Py has been presented. It is\nfound that the Gilbert damping is strongly dependent on\nthe spin-orbit coupling of the impurity atoms and more\nweakly dependent on the density of states that deter-\nmines disorder scattering. The strong influence of the\nspin-orbit coupling makes the 5 delements much more\neffective to change the Gilbert damping and more sen-\nsible to the concentration. As a result, the damping\ncan be increased by an order of magnitude compared to\nundoped Py. Overall, the damping features are quali-\ntatively rather well explained by the torque correlation\nmodel, yet it misses some quantitative predictive power\nthatonlyfirstprinciplesresultscanprovide. Moreover,it8\nisfoundthatthedampingoverallhasaweaktemperature\ndependence, howeverit is slightly enhanced with temper-\nature due to increased scattering caused by atomic dis-\nplacements and spin fluctuations. Elements in the begin-\nning of the 4 dor 5dseries are found to strongly influence\nthe magnetization and exchange stiffness due to antifer-\nromagnetic coupling between impurity and host atoms.\nIncontrast,elementsinthe endofthe4 dor5dserieskeep\nthe magnetization and exchange stiffness rather similar\nto undoped Py. More specifically, doping of the 5 dele-\nments Os, Ir and Pt are found to be excellent candidates\nfor influencing the magnetodynamical properties of Py.\nRecently, there have been an increasing attention for\nfinding metallic materials with small intrinsic damping,\nfor instance half metallic Heusler materials and FeCoalloys32,49. Controlling and varying the magnetodynam-\nical properties in these systems through doping or by\nother means, like defects, are very relevant and left for a\nfuture study.\nACKNOWLEDGMENTS\nThe work was financed through VR (the Swedish Re-\nsearch Council) and GGS (G¨ oran Gustafssons Founda-\ntion). A.B. acknowledge support from eSSENCE. The\ncomputations were performed on resources provided by\nSNIC (Swedish National Infrastructure for Computing)\nat NSC (National Supercomputer Centre) in Link¨ oping.\n∗fanpan@kth.se\n1I.ˇZuti´ c, J. Fabian, and S. Das Sarma,\nRev. Mod. Phys. 76, 323 (2004).\n2A. Brataas, A. D. Kent, and H. 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Shaw,\narXiv:1512:3610 1512, 3610 (2015)."    },    {        "title": "1802.05548v1.Damping_s_effect_on_the_magnetodynamics_of_spin_Hall_nano_oscillators.pdf",        "content": "Damping's e\u000bect on the magnetodynamics of spin Hall nano-oscillators\nYuli Yin,1, 2,\u0003Philipp D urrenfeld,2Mykola Dvornik,2Martina\nAhlberg,2Afshin Houshang,2Ya Zhai,1and Johan \u0017Akerman2, 3\n1Department of Physics, Southeast University, 211189 Nanjing, China\n2Department of Physics, University of Gothenburg, 412 96 Gothenburg, Sweden\n3Department of Materials and Nano Physics, School of Information and Communication Technology,\nKTH Royal Institute of Technology, Electrum 229, 164 40 Kista, Sweden\n(Dated: July 27, 2021)\nWe study the impact of spin wave damping ( \u000b) on the auto-oscillation properties of nano-\nconstriction based spin Hall nano-oscillators (SHNOs). The SHNOs are based on a 5 nm Pt layer\ninterfaced to a 5 nm Py 100\u0000x\u0000yPtxAgymagnetic layer, where the Pt and Ag contents are co-varied\nto keep the saturation magnetization constant (within 10 %), while \u000bvaries close to a factor of\nthree. We systematically investigate the in\ruence of the Gilbert damping on the magnetodynamics\nof these SHNOs by means of electrical microwave measurements. Under the condition of a constant\n\feld, the threshold current scales with the damping in the magnetic layer. The threshold current as\na function of \feld shows a parabolic-like behavior, which we attribute to the evolution of the spatial\npro\fle of the auto-oscillation mode. The signal linewidth is smaller for the high-damping materials\nin low magnetic \felds, although the lowest observed linewidth was measured for the alloy with least\ndamping.\nPACS numbers: 75.70.-i, 76.50.+g, 75.78.-n\nINTRODUCTION\nSpin Hall nano-oscillators (SHNO) are spintronic de-\nvices in which magnetization oscillations are induced by\npure spin currents [1]. These pure spin currents can be\nexperimentally realized via the spin Hall e\u000bect (SHE)\nin an adjacent heavy metal layer [2{4] or by non-local\nspin injection [5, 6]. SHNOs, which use the SHE in\na heavy metal layer, have been fabricated in a vari-\nety of device layouts, which all utilize the focusing of\ncharge current into a region with a lateral size of tens\nto hundreds of nanometers. This focusing is commonly\ndone via a nano-gap between two highly conductive elec-\ntrodes [3, 7, 8], with a nanoconstriction [9{13], or with\na nanowire [14, 15]. Most recently, nanoconstriction-\nSHNOs have attracted large interest, due to their rel-\native ease of fabrication, their direct optical access to\nthe magnetization oscillation area, and their potential for\nlarge scale and large distance synchronization of multiple\nSHNOs [16, 17].\nNanoconstriction-SHNOs consist of a bilayer of a fer-\nromagnetic free layer and a SHE inducing heavy metal\nlayer. Since the SHE and the concomitant spin accumu-\nlation at the bilayer interface are only in\ruenced by the\ncurrent density in the heavy metal layer, magnetization\noscillations of the device under a constant current can\nbe directly linked to the magnetodynamic properties of\nthe magnetic free layer. Until now, the variety of materi-\nals from which SHNOs has been fabricated is limited to a\nfew standards like permalloy (Py, Ni 80Fe20), (Co,Fe)B, or\nyttrium iron garnet (YIG). However, these materials are\ndi\u000berent from each other in every one of the key magneto-\ndynamic parameters, such as magnetization ( M), Gilbertdamping (\u000b), or exchange constant ( A).\nIn a recent study, we have shown how the magneto-\ndynamic properties of Py can be engineered by alloying\nwith the noble metals Pt, Au, and Ag [18]. While alloy-\ning with Pt leads to a large increase in damping but only\na small decrease in magnetization, alloying with Ag has\nonly a weak e\u000bect on the damping but reduces the mag-\nnetization relatively strongly. Co-alloying with both ele-\nments Pt and Ag thus results in Py 100\u0000x\u0000yPtxAgy\flms,\nwhoseMand\u000bcan be tuned independently, e.g. the\nmagnetization can be kept constant, while the damping\nis strongly increased with increasing Pt concentration.\nHere, we employ a series of alloyed Py 100\u0000x\u0000yPtxAgy\nthin \flms in nanoconstriction-SHNOs, where we vary the\ne\u000bective damping of the free layer by a factor of three,\nwhile we keep the magnetization of the \flms constant.\nBased on these \flms, we fabricate geometrically identical\nnanoconstriction-SHNOs and compare their microwave\nauto-oscillation characteristics. This allows us to directly\nanalyze the in\ruence of one single magnetodynamic prop-\nerty, namely the Gilbert damping, on the spectral char-\nacteristics, i.e. the onset current ( Ith\nDC), the output power\n(P), and the linewidth (\u0001 f).\nSPIN HALL NANO-OSCILLATOR DEVICES\nBilayers of 5 nm Py 100\u0000x\u0000yPtxAgyand 5 nm Pt were\nsputter-deposited onto sapphire substrates in a high-\nvacuum chamber with a base pressure of less than\n3\u000210\u00008Torr. The deposition was carried out with\n3 mTorr argon gas at a \row rate of 30 sccm. The alloyed\nlayers were co-sputtered from up to 3 targets, and the\nPy target power was kept constant at 350 W, while thearXiv:1802.05548v1  [cond-mat.mes-hall]  15 Feb 20182\n150 nm\n200 nm\n(a) (c)\n(b)\nHIPCurrent\nFIG. 1. (a) Schematic representation of the sputtered bilayer\nstructure. (b) SEM micrograph of a nanoconstriction-SHNO\nshowing the relative orientations of current and \feld. (c) Op-\ntical micrograph showing the microwave wave guide used for\ncontacting the SHNOs.\nnoble metal sputtering powers and the sputtering time\nwas adjusted for composition and thickness, respectively.\nThe top Pt layer was magnetron sputtered with a dc\npower of 50 W. The alloy compositions are Py 84Ag16\n(S01), Py 77:5Pt10Ag12:5(S02), Py 75Pt15Ag10(S03), and\nPy73Pt19Ag8(S04), chosen to result in a constant satura-\ntion magnetization throughout the series of SHNOs [18].\nDevices for electrical measurements were fabricated\nfrom these bilayers by electron beam lithography and ar-\ngon ion beam etching, using the negative resist as an etch-\ning mask. Nanoconstrictions were formed by two sym-\nmetrical indentations with a 50 nm tip radius into 4 µm\nwide stripes, see Fig. 1(b). The width of the nanocon-\nstrictions is 150 nm. Finally, 1 µm thick copper waveg-\nuides with a 150 µm pitch were fabricated by optical\nlithography and lift-o\u000b, see Fig. 1(c).\nFILM CHARACTERIZATION\nCharacterization of the extended bilayer samples was\nperformed by ferromagnetic resonance (FMR), and two-\npoint anisotropic magneotresistance (AMR) measure-\nments. The FMR was carried out with in-plane applied\n\felds using a NanOsc Instruments PhaseFMR with a\n200µm wide coplanar waveguide (CPW). An asymmet-\nric Lorentzian was \ft to the absorption peaks. The fre-\nquency dependence of the determined resonance \felds\nand linewidths was subsequently used to extract the ef-\nfective magnetization ( \u00160Me\u000b) and the damping param-\n0.0\n0.2\n0.4\n0.6\n0.8\nμ0M eff\nα\n0.02\n0.03\n0.04\n0.05\n0.06\nα  \n0\n5\n10\n15\n20\n0.3\n0.4\n0.5\nAMR (%)\nxPt(%)φ (°)\n0\n90\n180\n270\n360\n0.0\n0.1\n0.2\n0.3\n0.4\nMR (%)\nS01μ0Meff(T) μ0Meff = 0.617 TFIG. 2. (a) Magnetization and damping of the alloyed \flms\nin the bilayer as measured by CPW-based FMR. (b) AMR of\nthe extended layer structure. The inset shows the angular-\ndependent relative resistance of the Py 84Ag16/Pt (S01) bi-\nlayer, together with a \ft to a cos2-function.\neter (\u000b), respectively [18]. Figure 2(a) shows the two\nparameters, \u00160Me\u000band\u000b, as a function of Pt concen-\ntration. The magnetization is constant throughout the\nsample series ( \u00160Me\u000b= 0.617(34) T), while the damp-\ning increases linearly from 0 :023(1) to 0 :058(3) as the\nPt concentration increases from 0 (Py 84Ag16) to 19 %\n(Py73Pt19Ag8). The small layer thickness compared to\nthe \flms in Ref. 18 results in a slightly lower magnetiza-\ntion, whereas the damping is enhanced as a consequence\nof spin pumping into the adjacent Pt layer [19{21].\nThe AMR was determined by probing the resistance\nof 4µm wide stripes in a rotating 90 mT in-plane mag-\nnetic \feld. A representative AMR measurement is pre-\nsented in the inset of Fig. 2(b), together with a \ft of a\ncos2-function to the data. The angle '= 0\u000edenotes a\nperpendicular orientation between current and \feld, and\nthe AMR (Fig. 2(b)) is calculated by the di\u000berence in\nresistance at perpendicular and parallel alignments via\nAMR =Rk\u0000R?\nR?. The AMR is below 1 %, which is a re-\nsult of the majority of the current \rowing through the\nnonmagnetic platinum layer, which has a higher conduc-\ntivity than the Py alloys. The AMR reduces by \u001930 %\nacross the samples series, but the absolute resistance of\nthe bilayers decreases by less than 5 %. The AMR magni-\ntude is therefore most likely governed by the alloy compo-\nsition, since the amount of current in the magnetic layer\ndoes not change signi\fcantly.3\nf(GHz)\nCurrent (mA)\n2.2 2.4 2.6 2.8 3.0 3.2 3.45.96.06.1\n2468S01, H = 500 mT\n5.85 5.95 6.05 6.150246\nf(GHz)PSD (nV2/Hz)\nFIG. 3. Power spectral density (PSD) of the Py 84Ag16/Pt\n(S01) SHNO as a function of current in an external \feld of\n\u00160Hext= 0:5 T, tilted 80\u000eOOP. The inset shows the PSD at\nIDC= 3:26 mA and the solid line is a Lorentzian \ft resulting\nin \u0001f= 5:98 MHz and P= 1:02 pW.\nMICROWAVE EMISSION MEASUREMENTS\nAND DISCUSSION\nThe microwave measurements were conducted with the\ndevices placed in a magnetic \feld oriented at an out-of-\nplane (OOP) angle of 80\u000efrom the \flm plane, and an\nin-plane angle of '= 0\u000e. The in-plane component of the\nmagnetic \feld ( HIP\next) was thus perpendicular to the cur-\nrent \row direction, as sketched in Fig. 1(b). The relative\norientation of the current and HIP\nextyields a spin-torque\ncaused by the spin current from the Pt layer, which re-\nduces the damping in the Py layer and leads to auto-\noscillations for su\u000eciently large positive applied dc cur-\nrents (IDC) [22]. The current was applied to the samples\nvia the dc port of a bias-tee and the resulting microwave\nsignals from the devices were extracted from the rf port of\nthe bias-tee. The microwave signals were then ampli\fed\nby a broadband (0 :1 to 40 GHz) low-noise ampli\fer with\na gain of +32 dB before being recorded by a spectrum an-\nalyzer (Rohde&Schwarz FSV-40) with a resolution band-\nwidth of 500 kHz. All measurements were carried out at\nroom temperature.\nA typical microwave measurement of a Py 84Ag16/Pt\n(S01) device in a constant \feld of \u00160Hext= 0:5 T and\na varying current is displayed in Fig. 3. The peak fre-\nquency \frst decreases slightly after the oscillation onset\natIth\nDC= 2:26 mA, then reaches a minimum at \u00182:6 mA,\nand \fnally increases up to the maximum applied current\nof 3:4 mA. A Lorentzian peak function \fts well to the\nauto-oscillation signal, see inset of Fig. 3, allowing for de-\ntermination of the full-width at half-maximum linewidth\n(\u0001f) and the integrated output power ( P). Besides the\n10\n100\nΔf(MHz)\n5.9\n6.0\n6.1\nS01\nS02\nS03\nS04\nf(GHz)\n2.2\n2.4\n2.6\n2.8\n3.0\n3.2\n3.4\n0.1\n1\nPower (pW)\nI (mA)(a)\n(b)\n(c)FIG. 4. (a) Frequency, (b) linewidth, and (c) integrated power\nof the microwave auto-oscillations as a function of current for\nfour di\u000berent SHNOs with increasing damping. The applied\n\feld is\u00160Hext= 0:5 T, tilted 80\u000eOOP.\nhighly coherent auto-oscillation mode, no other modes\nare excited under these \feld conditions.\nFigure 4 shows the determined auto-oscillation char-\nacteristics of SHNOs with di\u000berent alloy composition\nand damping. The measurements were again made in\na constant \feld of 0 :5 T. The oscillation frequencies in\nFig. 4(a) lie around 6.0 \u00060.1 GHz for all samples, and the\ncurrent-frequency dependence is virtually identical above\nthe individual threshold currents. However, the current\nrange where fdecreases with IDCis missing for the S04\nsample, which suggests that the threshold current is un-\nderestimated for this device. The comparable frequencies\nof all samples con\frm that the saturation magnetization\nis constant throughout the alloy series. Furthermore, the\nquantitatively similar current tunability implies that the\nincreased damping does not change the fundamental na-\nture of the excited auto-oscillation mode.\nThe linewidth of the SHNOs decreases rapidly after the\nauto-oscillation onset and then levels o\u000b for higher IDC\nvalues, as shown in Fig. 4(b). This behavior is consistent\nwith previous studies on nanoconstriction-SHNOs made4\n300\n400\n500\n600\n700\n800\n2.0\n2.4\n2.8\n3.2\nS02\nS01\nS03\nS04\nIth\nDC(mA)\nField (mT)\nFIG. 5. Threshold current ( Ith\nDC) as a function of external\nmagnetic \feld for the four devices of this study.\nof permalloy \flms [11, 17]. The low-damping device S01\nreaches its minimum level at \u0001 f\u001811 MHz, while the\nSHNOs with higher damping materials all have a simi-\nlar minimum linewidth of \u0001 f\u00185 MHz. The linewidth\nis inversely proportional to the mode volume [23], and\nthe decrease in \u0001 fcan therefore be attributed to a spa-\ntial growth of the auto-oscillation region as the damping\nincreases. Nevertheless, the active area of the device is\ncon\fned to the nanoconstriction, which limits the reduc-\ntion in linewidth.\nThe output power of the four nanoconstriction-SHNOs\nis shown as a function of IDCin Fig. 4(c). The power\ngrows almost exponentially with increasing current for\nall samples. However, Pdrops dramatically as the Pt\nconcentration increases. The AMR also decreases in the\nhigher damping samples, but the reduction is too small\nto fully account for the drop in power. Together with the\ntrend in linewidth, the evolution of the power contradicts\nthe general assumption \u0001 f/\u000b=P [23{25]. This equa-\ntion is only valid in the vicinity of the threshold current\nand a direct comparison to the data is problematic, due\nto the experimental di\u000eculties of determining Ith\nDC. Still,\nthe direct relation between the intrinsic oscillator power\nand the electrically measured power is put into question\ndue to the remarkable decrease in the measured P. A\nnumber of factors could in\ruence the signal strength, e.g.\nrecti\fcation, spin-pumping, and the inverse spin-Hall ef-\nfect.\nThe onset current for auto-oscillations was determined\nby current scans for external \felds ranging between 0 :3 T\nand 0:8 T, and the results are shown in Fig. 5. The\n\feld dependence of Ith\nDCis parabola-like for all samples.\nThis kind of behavior has been predicted in a numerical\nstudy by Dvornik et al. [13]. The non-monotonic behav-\nior of threshold current as a function of applied \feld is\na result of a re-localization of the auto-oscillation mode\nand a corresponding change in the spin-transfer-torque\n(STT) e\u000eciency. In weak oblique magnetic \felds, the\nmode is of edge type and samples a signi\fcant portion of\nthe pure spin current, which is largest at the nanocon-striction edges due to the inhomogeneous current den-\nsity. When the \feld strength increases, the mode shows\nan even stronger localization towards the region of the\nhigher current density. Thereby, the STT e\u000eciency in-\ncreases and the threshold current drops. When the \feld\nstrength increases further, the mode detaches from the\nedges and eventually transforms to the bulk type. As\nthis transformation gradually reduces the spatial corre-\nlation between the spin current density and the location\nof the mode, the STT e\u000eciency drops and the threshold\ncurrent increases. The lower \feld tunability of Ith\nDCof\nthe high damping samples imply an initially larger mode\nvolume, which also was suggested by the evolution of the\nlinewidth.\nThe \feld and current range with detectable auto-\noscillations is strongly dependent on \u000b. The threshold\ncurrent should increase linearly with damping [13] and\nthe minimum Ith\nDCindeed scales with \u000b. The enhance-\nment is smaller than predicted (a factor of three), which\nindicates that the increase in damping is accompanied\nwith a higher STT e\u000eciency. A possible reason for the\nimproved e\u000eciency is a larger SHE through a more trans-\nparent interface for alloyed \flms. The origin of the ob-\nserved damping dependence of the threshold \feld is un-\nclear at this stage, calling for a closer inspection of the\nimpact of the applied \feld on the spectral characteristics.\nThus, a further investigation of our devices is targeted\ntowards the microwave emission as a function of \feld\nwith a constant IDC= 3.2 mA, i.e. above or at the pre-\nviously measured auto-oscillation threshold for all \felds.\nWhile the peak frequencies are virtually identical for all\nthe samples, see Fig. 6(a), the varied damping manifests\nin a clear pattern in Pand \u0001f. The microwave power,\nshown in Fig. 6(c), \frstly increases for all samples with\nincreasing \feld, peaks for an intermediate \feld, and \f-\nnally drops relatively sharp until a point where no more\noscillations are detectable. An opposite behavior can be\nseen for \u0001f, which shows a minimum for intermediate\n\felds. The \feld at which the SHNOs emit their maxi-\nmum output power decreases monotonically from 0.64 T\nto 0.4 T with increasing damping. The same trend is\nvisible for the point of minimum linewidth, which de-\ncreases with increased damping from 0.71 T to 0.49 T,\nand is therefore at a typically \u00180.1 T larger \feld than the\nrespective maximum power. The lowest overall linewidth\ncan be achieved for the lowest damping SHNO (S01) at\nhigh \felds, where only this device still shows a detectable\nsignal, i.e., \u0001 f= 1:2 MHz at\u00160Hext= 0:71 T. How-\never, at low applied \felds \u00160Hext\u00140:48 T a clear trend\nis noticeable towards smaller linewidths for the alloyed\npermalloy \flms with larger damping.\nIn light of this inverse trend, we can argue that auto-\noscillations in nanoconstriction-SHNO should also be de-\nscribed in the framework of non-linear auto-oscillators,\nalthough the study in Ref. 13 has shown that oscilla-\ntions in nanoconstriction-SHNOs emerge from a local-5\nΔf(MHz)\n(b)\n(c)\n4\n6\n8\n10\nS01\nS02S03\nS04\n1\n10\n100\n300\n400\n500\n600\n700\n800\n0.1\n1\nPower (pW)\nField (mT)(a)f(GHz)\nFIG. 6. (a) Frequency, (b) linewidth, and (c) integrated power\nof the auto-oscillations as a function of the applied external\nmagnetic \feld at a constant drive current IDC= 3:2 mA.\nized linear mode. The generation linewidth of nanocon-\ntact spin torque oscillators, which are a prime example\nof non-linear auto-oscillators, has been studied analyti-\ncally [23, 26] and experimentally [27]. The linewidth as\na function of current and magnetic \feld angle was shown\nto follow the expression:\n\u0001f=\u00000\n2\u0019\u0012kBT\nE0\u0013\"\n1 +\u0012N\n\u0000e\u000b\u00132#\n; (1)\nwherekB,T, andE0(IDC=Ith\nDC) are the Boltzmann con-\nstant, temperature and the average oscillator energy, re-\nspectively. Nis the nonlinear frequency shift, a material\nproperty that is determined by the internal magnetic \feld\nand the magnetization [28]. \u0000 e\u000bis the e\u000bective nonlinear\ndamping rate and \u0000 0is the positive damping rate, and\nboth have an explicit linear dependence on the Gilbert\ndamping\u000b[23]. Assuming everything else equal amongst\nour devices, a decrease of the linewidth with \u000bcan be\nthus expected, when the second term in the brackets in\nEq. 1 dominates. This is likely for low to intermediate\felds, since Ncan be calculated to take up the largest\nvalues under these conditions [28], which are thus in ac-\ncordance with the range of \felds, where we observe the\ndiscussed linewidth vs. damping behavior in our devices.\nCONCLUSIONS\nIn conclusion, we have fabricated a series of sam-\nples where the magnetization is constant, while the\nspin wave damping is varied by a factor of three. We\nhave shown that the damping of the magnetic layer in\nnanoconstriction-SHNOs has an important in\ruence on\nall the spectral characteristics of the devices. The re-\nsults of our study will encourage the application of tai-\nlored materials in SHNOs and can be used for a further\nunderstanding of the magnetodynamics in nanodevices,\ne.g. the coupling mechanisms in mutually synchronized\nSHNOs.\nACKNOWLEDGMENTS\nWe acknowledge \fnancial support from the China\nScholarship Council (CSC), the G oran Gustafsson\nFoundation, the Swedish Research Council (VR), the\nKnut and Alice Wallenberg Foundation (KAW), and\nthe Swedish Foundation for Strategic Research (SSF).\nThis work was also supported by the European Re-\nsearch Council (ERC) under the European Communitys\nSeventh Framework Programme (FP/2007-2013)/ERC\nGrant 307144 \\MUSTANG\".\n\u0003yuri@seu.edu.cn\n[1] T. Chen, R. K. Dumas, A. Eklund, P. K. Muduli,\nA. Houshang, A. A. Awad, P. D urrenfeld, B. G. Malm,\nA. Rusu, and J. \u0017Akerman, \\Spin-Torque and Spin-Hall\nNano-Oscillators,\" Proc. IEEE 104, 1919 (2016).\n[2] L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and\nR. A. 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B 76, 024437 (2007)."    },    {        "title": "1308.0450v2.Spin_pumping_damping_and_magnetic_proximity_effect_in_Pd_and_Pt_spin_sink_layers.pdf",        "content": "arXiv:1308.0450v2  [cond-mat.mes-hall]  5 Apr 2016Spin pumping damping and magnetic proximity effect in Pd and P t spin-sink layers\nM. Caminale,1,2,∗A. Ghosh,2,†S. Auffret,2U. Ebels,2K. Ollefs,3F. Wilhelm,4A. Rogalev,4and W.E. Bailey1,5,‡\n1Fondation Nanosciences, F-38000 Grenoble, France\n2SPINTEC, Univ. Grenoble Alpes / CEA / CNRS, F-38000 Grenoble , France\n3Fakult¨ at f¨ ur Physik and Center for Nanointegration (CENI DE),\nUniversit¨ at Duisburg-Essen, 47057 Duisburg, Germany\n4European Synchrotron Radiation Facility (ESRF), 38054 Gre noble Cedex, France\n5Dept. of Applied Physics & Applied Mathematics,\nColumbia University, New York NY 10027, USA\n(Dated: April 6, 2016)\nWe investigated the spin pumping damping contributed by par amagnetic layers (Pd, Pt) in both\ndirect and indirect contact with ferromagnetic Ni 81Fe19films. We find a nearly linear dependence\nof the interface-related Gilbert damping enhancement ∆ αon the heavy-metal spin-sink layer thick-\nnesses t Nin direct-contact Ni 81Fe19/(Pd, Pt) junctions, whereas an exponential dependence is o b-\nservedwhenNi 81Fe19and(Pd, Pt)areseparated by3nmCu. Weattributethequasi-l inear thickness\ndependence to the presence of induced moments in Pt, Pd near t he interface with Ni 81Fe19, quan-\ntified using X-ray magnetic circular dichroism (XMCD) measu rements. Our results show that the\nscattering of pure spin current is configuration-dependent in these systems and cannot be described\nby a single characteristic length.\nI. INTRODUCTION\nAs a novel means of conversion between charge- and\nspin-currents,spinHallphenomenahaverecentlyopened\nup new possibilities in magneto-electronics, with poten-\ntial applications in mesocale spin torques and electrical\nmanipulation of domain walls1–9. However, several as-\npects of the scattering mechanisms involved in spin cur-\nrent flow across thin films and interfaces are not entirely\nunderstood. Fundamental studies of spin current flow in\nferromagnet/non-magnetic-meal (F/N) heterostructures\nin the form of continuous films have attempted to iso-\nlate the contributions of interface roughness, microstruc-\nture and impurities10–12. magnet/non-magnetic-meal\n(F/N) heterostructures in the form of continuous films\nhave attempted to isolate the contributions of interface\nroughness, microstructure and impurities10–12. Proto-\ntypical systems in this class of studies are Ni 81Fe19/Pt\n(Py/Pt)3,5–7,13–18andNi 81Fe19/Pd(Py/Pd)8,11,14,16,19,20\nbilayers. Inthesesystems, PtandPdareemployedeither\nas efficient spin-sinks or spin/charge current transform-\ners, in spin pumping and spin Hall experiments, respec-\ntively. Pd and Pt are metals with high paramagnetic\nsusceptibility and when placed in contact with a ferro-\nmagnetic layer (eg. Py, Ni, Co or Fe) a finite magnetic\nmoment is induced at the interface by direct exchange\ncoupling21–24.\nThe role of the magnetic proximity effect on interface\nspin transport properties is still under debate. Zhang et\nal.25havereportedthat induced magnetic momentsin Pt\nand Pd films in direct contact with Py correlate strongly\nreduced spin Hall conductivities. This is ascribed to a\nspin splitting of the chemical potential and on the energy\ndependence of the intrinsic spin Hall effect. In standard\nspin pumping theory26, possible induced moments in N\nare supposed to be a priori included in calculations of\nthe spin-mixing conductance g↑↓of a F|N interface27,28,which tends to be insensitive to their presence.\nRecent theoretical works, on the other hand, propose\nthe need of a generalized spin pumping formalisms in-\ncluding spin flip and spin orbit interaction at the F |N\ninterface, in order to justify discrepancies between exper-\nimental and calculated values of mixing conductance29,30.\nAt present, it is still an open issue whether and how\nproximity-induced magnetic moments in F/N junctions\narelinked to the varietyofthe spin-transportphenomena\nreported in literature10,17,31.\nHere, we present an experimental study of the pro-\ntotypical systems: Py/Pd, Pt and Py/Cu/Pd, Pt het-\nerostructures. The objective of our study is to address\nthe role of proximity induced magnetic moments in spin\npumpingdamping. Tothisend, weemployedtwocomple-\nmentary experimental techniques. X-ray magnetic circu-\nlar dichroism (XMCD) is an element sensitive technique\nwhich allows us to quantify any static proximity-induced\nmagnetic moments in Pt and Pd. Ferromagnetic reso-\nnance(FMR) measurementsprovideindirectinformation\non the spin currents pumped out the Py layer by the pre-\ncessing magnetization, through the characterization of\nthe Pd, Pt thickness dependence of the interface-related\nGilbert damping α. In Fig. 3(Sec.IIIB), comparative\nmeasurements in Py/Cu/N and Py/N structures show a\nchange of the N thickness dependence of ∆ α(tN) from\nan exponential to a linear-like behavior. A change in\n∆α(tN) indicates a transformation in the spin scattering\nmechanism occurring at the interface, ascribed here to\nthe presence of induced moments in directly exchange\ncoupled F/N systems. Theoretical works predicted a\ndeviation from a conventional N-thickness dependence\nwhen interface spin-flip scattering is considered in the\npumping model29,30, howeverno functional form waspro-\nvided. For Py/N systems, we find that the experimental\nthickness dependence cannot be described by standard\nmodels16,26,32, but rather a linear function reproduces2\nthe data to a better degree of accuracy, by introducing\na different characteristic length. We speculate that the\nspatial extent of spin current absorption in F/N systems\nshows an inverse proportionality to interfacial exchange\ncoupling energy, obtained from XMCD, as proposed be-\nfore for spin polarized, decoupled interfaces in F 1/Cu/F 2\nheterostructures14.\nII. EXPERIMENT\nThe heterostructures were fabricated by DC mag-\nnetron sputtering on ion-cleaned Si/SiO 2substrates in\nthe form of substrate/seed/multilayer/cap stacks, where\nTa(5nm)/Cu(5nm) bilayerwasemployedas seed. Ta/Cu\nis employed to promote <111>growth in Py and subse-\nquent fcc layers (Pd, Pt), and Ta is known to not affect\nthe dampingstrongly17,32,33. Different stacksweregrown\nasmultilayer for each measurement.\nFor FMR measurements, we have multilayer =\nPy(tF)/N(tN), Py(tF)/Cu(3nm)/N( tN) with N = Pd,\nPt; an Al(3nm) film, oxidized in air, was used as\ncap. The smallest N layer thickness tNdeposited is\n0.4nm, the maximum interdiffusion length observed\nfor similar multilayers34. Samples with multilayer =\nPy(tN)/Cu(3nm) and no sink layer were also fabricated\nas reference for evaluation of the Gilbert damping en-\nhancement due tothe Pd orPtlayer. The tN-dependence\nmeasurementsofFMRweretakenforPythicknesses tF=\n5and 10nm. Results from the tF= 10nm data set are\nshown in Appendix A. Measurements of the FMR were\ncarried out at fixed frequency ωin the 4-24 Ghz range,\nby means of an in-house apparatus featuring an external\nmagnetic field up to 0.9T parallel to a coplanar waveg-\nuide with a broad center conductor width of 350 µm.\nFor XMCD measurements, given the low X-ray ab-\nsorption cross-sectionpresented by Pt and Pd absorption\nedges, a special set of samples was prepared, consisting\nof 20 repeats per structure in order to obtain sufficiently\nhighsignal-to-noiseratio. Inthiscase,wehave multilayer\n= [Py(5nm)/N] 20, with N = Pd(2.5nm) and Pt(1nm);\nCu(5nm)/Py(5nm)/Al(3nm) was deposited as cap. The\nPt and Pd thicknesses were chosen to yield a damping\nenhancement equal about to half of the respective satu-\nration value (as it will be shown later), i.e. a thicknesses\nfor which the F/N interface is formed but the damp-\ning enhancement is still increasing. XMCD experiments\nwere carried out at the Circular Polarization Beamline\nID-12 of the European Synchrotron Radiation Facility\n(ESRF)35. Measurementsweretakenintotalfluorescence\nyield detection mode, at grazing incidence of 10◦, with\neither left or right circular helicity of the photon beam,\nswitching a 0.9T static magnetic field at each photon en-\nergy value (further details on the method are in Ref.22).\nNo correction for self-absorption effects is needed; how-\never XMCD spectra measured at the L 2,3edges of Pd\nhave to be corrected for incomplete circular polarization\nrate of monochromatic X-rays which is 12% and 22% at/s49/s49/s46/s53/s52 /s49/s49/s46/s53/s54 /s49/s49/s46/s53/s56 /s49/s49/s46/s54/s48 /s49/s49/s46/s54/s50 /s49/s51/s46/s50/s54 /s49/s51/s46/s50/s56 /s49/s51/s46/s51/s48 /s49/s51/s46/s51/s50 /s49/s51/s46/s51/s52/s48/s46/s48/s48/s46/s51/s48/s46/s54/s48/s46/s57/s49/s46/s50/s49/s46/s53\n/s32/s80/s116/s32/s88/s65/s83\n/s32/s65/s117/s32/s88/s65/s83\n/s80/s100/s88/s65/s83/s32/s40/s97/s46/s117/s46/s41\n/s80/s104/s111/s116/s111/s110/s32/s101/s110/s101/s114/s103/s121 /s32/s40/s107/s101/s86/s41/s80/s116\n/s45/s48/s46/s48/s54/s45/s48/s46/s48/s51/s48/s46/s48/s48/s48/s46/s48/s51/s48/s46/s48/s54\n/s32\n/s88/s77/s67/s68/s32/s40/s97/s46/s117/s46/s41\n/s51/s46/s49/s52 /s51/s46/s49/s54 /s51/s46/s49/s56 /s51/s46/s50/s48 /s51/s46/s50/s50 /s51/s46/s51/s48 /s51/s46/s51/s50 /s51/s46/s51/s52 /s51/s46/s51/s54 /s51/s46/s51/s56/s48/s46/s48/s48/s46/s51/s48/s46/s54/s48/s46/s57/s49/s46/s50/s49/s46/s53\n/s32/s80/s100/s32/s88/s65/s83\n/s32/s65/s103/s32/s88/s65/s83/s88/s65/s83/s32/s40/s97/s46/s117/s46/s41\n/s80/s104/s111/s116/s111/s110/s32/s101/s110/s101/s114/s103/s121 /s32/s40/s107/s101/s86/s41/s45/s48/s46/s48/s57/s45/s48/s46/s48/s54/s45/s48/s46/s48/s51/s48/s46/s48/s48/s48/s46/s48/s51/s48/s46/s48/s54/s48/s46/s48/s57\n/s88/s77/s67/s68/s32/s40/s97/s46/s117/s46/s41\nFIG. 1. (Color online) X-ray absorption (XAS, left axis) and\nmagnetic circular dichroism (XMCD, right axis) spectra at\nthe L-edges of Pt (top panel) and Pd (bottom panel) for\n[Py(5nm)/Pt(1nm)] 20and [Py(5nm)/Pd(2.5nm)] 20multilay-\ners. The dashed traces represent XAS spectra at L-edge of\nAg and Au used as background of Pd and Pt, respectively, to\nextract the values of induced magnetic moment reported in\nTab.I.\nL3and L2, respectively. The circular polarization rate is\nin excess of 95 % at the L 2,3edges of Pt.\nIII. RESULTS AND ANALYSIS\nIn order to study how the proximity-induced mag-\nnetic moments may affect the absorption of spin-currents\nthrough interfaces, the static moment induced in Pt, Pd\nlayers in direct contact with ferromagnetic Py in char-\nacterized first, by means of XMCD. The value of the\ninduced moment extracted for the two Py/N systems is\nusedtoestimatetheinterfacialexchangeenergyactingon\nthe two paramagnets. Afterwards, the dynamic response\nofthe magnetization is addressedby FMR measurements\nin Py/N(direct contact) and Py/Cu/N(indirect contact)\nheterostructures. From FMR measurements carried out\non both configurations as a function of N thickness, the\ndamping enhancement due to the presence of the spin-\nsink layers Pt and Pd is obtained from the frequency-\ndependence of the FMR linewidth. The relation between\nthe static induced moment and the spin pumping damp-\ning is discussed by comparing the results of the direct\nwith indirect contact systems.3\n-0.06-0.04-0.020.00XMCD\n11.5911.5811.5711.5611.55\nPhoton Energy (keV)1.0\n0.5\n0.0\n3210tCu (nm) Area\n 1 nm Cu\n 0.5 nm Cu\n 0 nm Cu\nFIG. 2. (Color online) XMCD spectra at the L 3edge of Pt\nfor [Py(5nm)/Cu(t Cu)/Pt(1nm)] 15, with t Cu= 0, 0.5 and\n1nm. As inset the ares of the peak is plot as a function of Cu\nthickness.\nA. XMCD: Probing the induced magnetic moment\nIn Fig.1we report X-ray absorption (XAS) and mag-\nnetic circular dichroism (XMCD) spectra at the L2,3\nedges of Pt (top panel) and Pd (bottom panel) taken on\nPy(5nm)/Pt(1nm) |20and Py(5nm)/Pd(2.5nm) |20, re-\nspectively. Rather intense XMCD signals have been de-\ntected at both Pt and Pd L 2,3edges, showing unambigu-\nously that a strong magnetic moment is induced by di-\nrect exchange coupling at the Py |N interface. The static\ninduced moment is expected to be ferromagnetically cou-\npled with the magnetization in Py21. From the integrals\nof XMCD spectra, the induced magnetic moment on the\nPt, Pd sitesis determined byapplying the sum rulesasin\nRef.22(and references therein). In Py/Pd(2.5nm) |20, Pd\natoms bear a moment of 0.12 µB/at, averaged over the\nwhole volume of the volume, with an orbital-to-spinratio\nmL/mS= 0.05. In Py/Pt(1nm) |20, a magnetic moment\n0.27µB/at is found on Pt, comparable to that reported\nfor Ni/Pt epitaxial multilayers23, with a relatively high\norbitalcharacter mL/mS= 0.18, ascomparedwithPdin-\nduced moment. The large difference in volume-averaged\ninduced moment per atom comes from the different film\nthickness, hence volume, for Pt and Pd. Assuming that\nthe induced magnetic moment is confined to the first\natomic layers at the interface with Py23,24, one could\nestimate 0.32 µB/at for Pd and 0.30 µB/at for Pt36.\nWhen a3nm thick Cu interlayeris introducedbetween\nPy and N, a two ordersofmagnitude smaller induced mo-\nment (0.0036 µB/at) was found for 2.5nm Pd22, while Pt\nshowed an XMCD signal of the order of the experimen-\ntal sensitivity, ∼0.5·10−3µB/at. In Fig. 2XMCD\nspectra at the L 3edge of Pt are shown for Cu interlayer\nthicknesses 0, 0.5 and 1nm. For 0.5nm Cu the integral\nof XMCD signal at the L 3edge shrinks to 30%, while\nfor 1nm it is reduced to zero within experimental error.\nThis result could be explained either by a 3d growth of\nthe Cu layer, allowing a fraction of the Pt layer to be in\ndirect contact with Py for Cu coverages of 0.5nm, or byNχmol37S37N0abulktN/angbracketleftM/angbracketrightMiJex\n[cm3/mol][1/eV·at][nm][nm][µB/at][µB/at][meV]\n10−4\nPd5.5±0.29.30.83±0.030.3892.50.1160.3242\nPt1.96±0.13.70.74±0.040.3921.00.270.30109\nTABLE I. Spin-sink layer N properties in Py/N heterostruc-\ntures: experimental molar susceptibility χmolat 20◦C; den-\nsity of states N0calculated from tabulated χmol; Stoner pa-\nrameter S from Ref.37; bulk lattice parameter a; layer thick-\nnessestN; volume averaged induced magnetic moment /angbracketleftM/angbracketright\nfrom XMCD measurement in Fig. 1; interface magnetic mo-\nmentMi36; Py|N interfacial exchange energy per interface\natomJex(Eq.1).\ndiffusion of magnetic Ni atoms in Cu on a scale shorter\nthan 1nm. The film then becomes continuous, and at\n1nm coverage, no direct exchange coupling takes place\nbetween Py and Pt layers. For FMR measurements pre-\nsented in the followingsection, a 3nm thick Cu interlayer\nis employed, reducing also any possible indirect exchange\ncoupling.\nFrom the values of induced moments in Pd and Pt,\nwe can make a step forward and estimate the interfacial\nexchange coupling energies for the two cases. Equating\ninteratomic exchange energy Jexand Zeeman energy for\nan interface paramagnetic atom, we have (see Appendix\nB1for the derivation)\nJex=1\n2/angbracketleftM/angbracketright\nµBN0StN\nti(1)\nwhere/angbracketleftM/angbracketrightis the thickness-averaged paramagnetic\nmoment, N0is the single-spin density of states (in\neV−1at−1), S is the Stoner factor and ti= 2∗a/√\n3\nis the polarized interface-layer thickness36. The 1/2fac-\ntor accounts for the fact that in XMCD measurements\nthe N layer has both interfaces in contact with F. Un-\nder the simplifying assumption that all the magnetic mo-\nment is confined to the interface N layer and assuming\nexperimental bulk susceptibility parameters for χv, we\nobtainJPd\nex= 42 meV for Pd and JPt\nex= 109 meV for\nPt (results and properties are summarized in Tab. I).\nHere the difference in estimated Jex, despite roughly\nequalMi, comes from the larger Stoner factor S for\nPd. A stronger interfacial exchange energy in Pt de-\nnotes a stronger orbital hybridization, yielding possibly\na higher orbital character of the interfacial magnetic mo-\nment in the ferromagnetic Py counterpart21. For com-\nparison, we consider the interatomic exchange param-\netersJexin ferromagnetic Py and Co, investigated in\nRef.14.Jexis estimated from the respective Curie tem-\nperatures TC, through Jex≃6kBTC/(m/µB)2, wherem\nis the atomic moment in µB/at (see Appendix B2). Ex-\nperimental Curie temperatures of 870K and 1388K give\nJCo\nex=293meV for Co and JPy\nex=393meV for Py, which\nare of the same order of the value calculated for Pt (de-\ntails about calculation in Appendix B2).4\n1.0\n0.5\n0.0\n14121086420\ntPt (nm) Py/Pt\n Py/Cu/Pt12\n10\n8\n6\n4\n2\n0\n468\n12468\n10\ntPt (nm) Py/Pt\n tc = 2.4 nm\n Py/Cu/Pt\n λα = 1.8 nm\n1.0\n0.5\n0.0∆α / ∆α0\n14121086420\ntPd (nm) Py/Pd\n Py/Cu/Pd6\n5\n4\n3\n2\n1\n0∆α (x103)\n468\n12468\n10\ntPd (nm) Py/Pd\n tc = 5.0 nm\n Py/Cu/Pd\n λα = 5.8 nma) b)\nc) d)\nFIG. 3. (Color online) Damping enhancement ∆ α, due to\npumped spin current absorption, as a function of thickness tN\nfor Py(5nm)/N and Py(5nm)/Cu(3nm)/N heterostructures,\nwith N = Pd( tN) (panels a,c), Pt( tN) (panels b,d). Solid\nlines result from a fit with exponential function (Eq. 2) with\ndecay length λα. Dashed lines represents instead a linear-\ncutoff behavior (Eq. 3) fortN< tc. Please notice in panels a,\nc the x-axis is in logarithmic scale. In panels b, c the dampin g\nenhancement is normalized to the respective saturation val ue\n∆α0.\nIn the following, the effect of these static induced mo-\nments on the spin pumping damping of the heterostruc-\ntures characterized will be discussed.\nB. FMR: damping enhancement\nThe main result of our work is now shown in Figure\n3. In Fig. 3the damping enhancement ∆ αis plotted as\na function of the spin-sink layer thickness tN, for Py/Pd,\nPy/Cu/Pd (panels a, c) and Py/Pt, Py/Cu/Pt (panels\nb, d). The enhancement ∆ αis compared with the damp-\ningαof a reference structure Py(5nm)/Cu, excluding\nthe sink layer N. Each value of αresults from established\nanalysis of the linewidth of 11 FMR traces13,14, employ-\ning ag-factor equal to 2.09 as a constant fit parameter\nfor all samples.\nIn Py/Cu/N systems (Fig. 3, green square markers),\n∆αrises with increasing tNthickness to similar satura-\ntion values ∆ α0= 0.0027, 0.0031 for Pd and Pt, but\nreachedondifferentlengthscales,giventhedifferentchar-Ng↑↓\neff(Py|Cu/N) λαg↑↓\neff(Py|N)tc\n[nm−2][nm] [nm−2][nm]\nPd 7.2 5.8±0.2145.0±0.3\nPt 8.3 2.4±0.1321.8±0.2\nTABLE II. Mixing conductance values extracted from the\ndamping enhancement ∆ αat saturation in Fig. 3, and re-\nspective length scales (see text for details).\nacteristic spin relaxation lengths of the two materials.\nFrom the saturation value, an effective mixing conduc-\ntanceg↑↓\neff(Py|Cu/N) = 7 .2−8.3nm−2is deduced in the\nframeworkof standardspin pumping picture13,17,19, with\nPy saturation magnetization µ0Ms= 1.04T. The fact\nthat the spin-mixing conductance is not material depen-\ndent indicates that similar Cu |N interfaces are formed.\nThe thickness dependence is well described by the expo-\nnentialfunction14,20\n∆α(tN) = ∆α0(1−exp(−2tN/λN\nα)) (2)\nas shown by the fit in Fig. 3a-b (continuous line). As\na result, exponential decay lengths λPt\nα= 1.8nm and\nλPd\nα= 5.8nm are obtained for Pt and Pd, respectively.\nWhen the Pt, Pd spin-sink layers come into direct con-\ntact with the ferromagnetic Py, the damping enhance-\nment ∆α(tN) changes dramatically. In Py/N systems\n(Fig.3a-b, trianglemarkers),the damping saturationval-\nues become ∆ αPt\n0= 0.0119 and ∆ αPd\n0= 0.0054 for Pt\nand Pd, respectively a factor ∼2 and∼4 larger as com-\npared to Py/Cu/N. Within the spin-pumping descrip-\ntion, a largerdamping enhancement implies a largerspin-\ncurrentdensitypumped outoftheferromagnetacrossthe\ninterface and depolarized in the sink.\nIn Py/N heterostructures, because of the magnetic\nproximity effect, few atomic layers in N are ferromagnet-\nically polarized, with a magnetic moment decaying with\ndistance from the Py |N interface. The higher value of\ndamping at saturation might therefore be interpreted as\nthe result of a magnetic bi-layer structure, with a thin\nferromagnetic N characterized by high damping αN\nhigh\ncoupled to a low damping αF\nlowferromagnetic Py38. To\ninvestigate whether damping is of bi-layer type, or truly\ninterfacial, in Fig 4we show the tFthickness dependence\nof the damping enhancement ∆ α, for a Py( tF)/Pt(4 nm)\nseries of samples. The power law thickness dependence\nadheres very closely to t−1\nF, as shown in the logarithmic\nplot. The assumption of composite damping for syn-\nchronous precession, as ∆ α(t1) = (α1t1+α2t2)/(t1+t2),\nshown here for t2= 0.25nm and 1.0nm, cannot follow\nan inverse thickness dependence over the decade of ∆ α\nobserved. Damping is therefore observed to have a pure\ninterfacial character.\nIn this case, the mixing conductances calculated from\nthe saturation values are g↑↓\neff(Py|Pd) = 14nm−2and\ng↑↓\neff(Py|Pt) = 32nm−2. From ab initio calculations\nwithin a standard spin-pumping formalism in diffusive\nfilms10,29, it is found g↑↓\neff(Py|Pd) = 23nm−2for Pd and5\n10 3 4 56789 20 30 40\ntF (nm)110\n2345678920∆α(10−3)Py(tF)/Pt(4)\ninterfacial, K=0.055\nbilayer, t2=0.25 nm\n             t2=1.0 nm\n26101520 30\ntF (nm)051015202530α(10−3)Py(t)/Pt(4nm)\nPy(t)\nFIG. 4. Logarithmic plot of the damping enhancement ∆ α\n(triangle markers) as a function of the Py layer thickness tF,\nin Py(t F)/Pt(4nm). Solid and dashed lines represents, re-\nspectively, fits according to the spin pumping ( interfacial )\nmodel ∆ α=Kt−1\nFand to a αlow(tF)/αhigh(t2)bilayermodel,\nwitht2= 0.25,1.0nm.Inset: Gilbert damping αfor Py(t F)\n(square markers) and Py(t F)/Pt(4nm) (round markers).\ng↑↓\neff(Py|Pt) = 22nm−2for Pt. Theoretical spin mixing\nconductance from a standard picture does reproduce the\nexperimental order of magnitude, but it misses the 2.3\nfactor between the Py |Pt and Py |Pd interfaces. Beyond\na standard pumping picture, Liu and coworkers29intro-\nduce spin-flipping scattering at the interface and calcu-\nlate from first principles, for ideal interfaces in finite dif-\nfusive films: g↑↓\neff(Py|Pd) = 15nm−2, in excellent agree-\nment with the experimental value here reported for Pd\n(Tab.II), andg↑↓\neff(Py|Pt) = 25nm−2. Zhang et al.10\nsuggest an increase up to 25% of the mixing conductance\ncan be obtained by introducing magnetic layers on the\nPt side. The results here reported support the emerging\nidea that a generalized model of spin pumping including\nspin-orbit coupling and induced magnetic moments at\nF|N interfaces may be required to describe the response\nof heterostructures involving heavy elements.\nThe saturation value of damping enhancement at ∆ α0\nas a function of the Cu interlayer thickness is shown in\nFig.5to follow the same trend of the XMCD signal\n(dashed line), reported from Fig. 2. Indeed, it is found\nthat the augmented ∆ α0in Py/N junctions is drasti-\ncally reduced by the insertion of 0.5nm Cu at the Py |N\ninterface17, and the saturation of the Py/Cu/N configu-\nration is already reached for 1nm of Cu interlayer. As\nsoonasacontinuousinterlayerisformedandnomagnetic\nmoment is induced in N, ∆ α0is substantially constant\nwith increasing Cu thickness.\nThe N-thickness dependence of ∆ α(tN) in Py/N sys-\ntems before saturation is addressed in the following. At\nvariance with the Py/Cu/N case, the thickness depen-\ndence of ∆ αis not anymore well described by an expo-1.0\n0.5\n0.0\n3210\ntCu (nm)1.0\n0.5\n0.0L3 XMCD area (norm.)Pt(3nm)\n XMCD Pt\n3210\ntCu (nm)1.0\n0.5\n0.0∆α0 (norm.)Pd(7nm)\n XMCD Pt\nFIG. 5. (Color online) Normalized damping enhancement ∆ α\n(left axis), due to spin pumping, as a function of interlayer\nthickness tCufor Py(5nm)/Cu( tCu)/N heterostructures, with\nN = Pd(7nm), N = Pt(3nm). The dashed line represents the\nXMCD signal (right axis) reported from inset in Fig. 2.\nnential behavior, as an exponential fit (with exponential\ndecay length as only free fit parameter) fails to repro-\nduce the increase of ∆ αtowards saturation (solid lines\nin Fig.3a-b). More rigorous fitting functions employed\nin spin pumping experiments, within standard spin trans-\nport theory16,26,32, cannot as well reproduce the experi-\nmental data (see Appendix. A). It is worth mentioning\nthat the same change of trend between the two configu-\nrations was observed for the same stacks with a 10nm\nthick Py layer (data shown in Appendix A, Fig.7). A\nchange of the functional dependence of ∆ αontNre-\nflects a change in the spin-depolarization processes the\npumped spin current undergoes, as for instance shown in\nRef.30when interfacial spin-orbit coupling is introduced\nin the spin-pumping formalism. Experimentally, a linear\nthickness dependence with sharp cutoff has been shown\nto characterize spin-current absorption in spin-sink lay-\ners exhibiting ferromagnetic order at the interface, as re-\nported for F 1/Cu/F 2(tF2) junctions with F = Py, Co,\nCoFeB14. Given the presence of ferromagnetic order in\nN at the interface of F/N structures, the data are tenta-\ntively fit with a linear function\n∆α= ∆α0tN/tN\nc (3)\nThis linear function better reproduce the sharp rise of\n∆α(dashed lines in Fig. 3a-b) and gives cutoff thick-\nnessestPt\nc= 2.4±0.2nm and tPd\nc= 5.0±0.3nm for Pt\nand Pd, respectively. The linearization is ascribed to the\npresence of ferromagnetic order in the paramagnetic Pd,\nPt spin-sink layers at the interface with the ferromag-\nnetic Py. The linear trend extends beyond the thickness\nfor which a continuous layer is already formed (less than\n1nm), and, especially for Pd, far beyond the distance\nwithin the non-uniform, induced moment is confined (up\nto0.9nm). InRef.14, thecutoff tcinF/Cu/Fheterostruc-\nturesis proposedto be on the orderofthe transversespin\ncoherence length λJin ferromagnetically ordered layers.\nλJcan be expressed in terms of the exchange splitting6\n6\n5\n4\n3\n2\n1\n0tc(nm)\n2520151050\n1/Jex(eV-1)PtPd\nPy\nCo\nFIG. 6. Effect of direct exchange strength on length scale of\nspin current absorption. Cutoff thickness tcextracted from\nthe ∆α(tN) data in Fig. 3as a function of reciprocal interfa-\ncial exchange energy 1 /Jexextracted from XMCD in Fig. 1.\nLabels are given in terms of Jex. The Co and Py points are\nfrom Ref.14.\nenergyJex,\nλJ=hvg\n2Jex(4)\nwherevgis the electronic group velocity at the Fermi\nlevel. This form, found from hot-electron Mott\npolarimetry1, is expressed equivalently for free electrons\nasπ/|k↑−k↓|, which is a scaling length for geometrical\ndephasing in spin momentum transfer2. Electrons which\nenter the spin-sink at E Fdo so at a distribution of angles\nwith respect to the interface normal, traverse a distribu-\ntionofpathlengths, andprecessbydifferentangles(from\nminority to majority or vice versa ) before being reflected\nback into the pumping ferromagnet. For a constant vg,\nit is therefore predicted that tcis inversely proportional\nto the exchange energy Jex.\nIn Figure 6we plot the dependence of the cutoff thick-\nnesstN\ncupontheinverseoftheestimatedexchangeenergy\nJex(Tab.I),asextractedfromtheXMCDmeasurements.\nA proportionality is roughly verified, as proposed for the\ntransverse spin coherence length across spin polarized in-\nterfaces. Under the simplistic assumption that tc=λJ,\nfrom the slope of the line we extract a Fermi velocity\nof∼0.1·106m/s (Eq.4), of the order of magnitude ex-\npectedforthematerialsconsidered39,40. Thesedatashow\nthat, up to a certain extent, length scale for spin-current\nscattering shares common physical origin in ferromag-\nnetic layers and paramagnetic heavy-metals, such as Pd\nand Pt, under the influence of magnetic proximity effect.\nThis unexpected results is observed in spite of the fact\nthat F 1/Cu/F 2and F/N systems present fundamental\ndifferences. In F/N structure, the induced moment in N\nis expected to be directly exchange coupled with the fer-\nromagneticcounterpart. Whereasin F 1/Cu/F 2, themag-\nnetic moment in F 2(off-resonance) are only weakly cou-\npled with the precession occurring in F 1(in-resonance),\nthrough spin-orbit torque and possible RKKY interac-tion. Magnetization dynamics in N might therefore be\nexpected with its own pumped spin current, albeit, to\nthe best of our knowledge, no experimental evidence\nof a dynamic response of proximity induced moments\nwas reported so far. From these considerations and the\nexperimental findings, counter-intuitively the proximity-\ninduced magnetic moments appear not to be involved in\nthe production of spin current, but rather to contribute\nexclusively with an additional spin-depolarization mech-\nanism at the interface.\nIV. CONCLUSIONS\nWe have investigated the effect of induced magnetic\nmoments in heavy metals at Py/Pt and Py/Pd inter-\nfaces on the absorption of pumped spin currents, by\nanalyzing ferromagnetic resonance spectra with varying\nPt, Pd thicknesses. Static, proximity-induced magnetic\nmoments amount to 0.32 and 0.3 µB/atom in Pd and\nPt, respectively, at the interface with Py, as deduced\nfrom XMCD measurements taken at the L 2,3edges. We\nhave shown that when the proximity induced moment\nin Pt and Pd is present, an onset of a linear-like thick-\nness dependence of the damping is observed, in con-\ntrast with an exponential trend shown by Py/Cu/Pd\nand Py/Cu/Pt systems, for which no induced moment\nis measured. These results point to the presence of an\nadditional spin-flip process occurringat the interface and\nto a change of the character of spin current absorption\nin the ultrathin Pd and Pt paramagnets because of the\ninterfacial spin polarization. The range of linear increase\nis proposed to be inversely proportional to the interfa-\ncial exchange energy in Py/Pt and Py/Pd, inferred from\nXMCD data.\nWEB acknowledges the Universit´ e Joseph Fourier and\nFondation Nanosciences for his research stay at SPIN-\nTEC. This work was supported in part by the U.S. NSF-\nECCS-0925829 and the EU EP7 NMP3-SL-2012-280879\nCRONOS. MC is financed by Fondation Nanosciences.\nAppendix A: N-thickness dependence\nIn order to confirm the results presented in the\nmanuscript, additional sample series with thicker Py\nlayer were fabricated and measured. The experimental\nresults for 10nm thick Py layer are shown in Fig.s 7and\n8forPdandPt, respectively. Wehavepresentedthedata\nhere, rather than including them with the other plots in\nFigure3, to keep the figures from being overcrowded. As\nexpected when doubling the ferromagnet thickness, the\nsaturation values ∆ α0are about half of those measured\nfor 5nm Py (Fig. 3). Confirming the data presented in\nthe manuscript, it is observedagaina changeof thickness\ndependence of ∆ α(tN), fromexponential for Py/Cu/N\n(solid lines; Eq. 2,λα= 4.8nm and 1.4nm for Pd and Pt7\n3\n2\n1\n0∆α (x10-3)\n345678\n12345678\n10\ntPd (nm)Py/Pd\nPy/Cu/Pd\n Eq. 3\n,  Eq. 2\n [16, 32] - ρPd=1.4E-7\n [16, 32] - ρPd->ρ(tPd)\nFIG. 7. Damping enhancement ∆ α, due to pumped\nspin current absorption, as a function of thickness tPd\nfor Py(10nm)/Pd and Py(10nm)/Cu(3nm)/Pd heterostruc-\ntures. Solid lines result from a fit with exponential functio n\n(Eq.2) with decay λα. Dashed lines represents instead a\nlinear-cutoff behavior (Eq. 3) fortPd< tc. Short-dash and\npoint-dash traces are fit to the data, employing equations\nfrom standard spin transport theory (see text for details)16,32.\nIn bottom panel, ∆ αis normalized to the respective satura-\ntion value.\nrespectively) to linear-like for Py/N (dashed lines; Eq. 3,\ntc= 5.3nm and 2nm for Pd and Pt respectively).\nThe experimental data are also fitted with a set of\nequations derived from standard theory of diffusive spin\ntransport16,26,32, describing the the dependence of ∆ α\non the thickness of adjacent metallic layers (either N or\nCu/N in our case) as follow\n∆α=γ¯h\n4πMstFMg↑↓\n1+g↑↓/gx\next(A1)\nwith (Eq. 7 in Ref.16, and Eq. 6 in Ref.32)\ngN\next=gNtanhtN/λN\nsd\ngCu/N\next=gCugCucothtN/λN\nsd+gNcothtCu/λCu\nsd\ngCucothtN/λN\nsdcothtCu/λCu\nsd+gN(A2)\nwheregx=σx/λx\nsd,σxandλx\nsdare the electrical conduc-\ntivity and spin diffusion length of the non magnetic layer\nx. For the thin Cu layer, we used a resistivity ρCu=\n1×107Ωm and a spin diffusion length λCu\nsd= 170nm32.\nFor the Pt and Pd layers, two fitting models in which the\nconductivity of the films is either constant or thickness\ndependent areconsidered, as recently proposed by Boone\nand coworkers16. The values of conductivity, as taken di-\nrectly from Ref.16, will influence the spin diffusion length\nλN\nsdand spin mixing conductance g↑↓resulting from the\nfit, but will not affect the conclusions drawn about the\noverall trend. When a constant resistivity is used (short-\ndash, blue lines), the model basically corresponds to the6\n5\n4\n3\n2\n1\n0∆α (x10-3)\n345678\n12345678\n10\ntPt (nm) Py/Pt\n Py/Cu/Pt\n Eq. 3\n, Eq. 2\n [16, 32] - ρPt=1.7E-7\n [16, 32] - ρPt->ρ(tPt)\nFIG. 8. Damping enhancement ∆ α, due to pumped\nspin current absorption, as a function of thickness tPtfor\nPy(10nm)/Pt and Py(10nm)/Cu(3nm)/Pt heterostructures.\nSolid lines result from a fit with exponential function (Eq. 2)\nwith decay λα. Dashed lines represents instead a linear-cutoff\nbehavior (Eq. 3) fortPt< tc. Short-dash and point-dash\ntraces are fit to the data, employing equations from standard\nspin transport theory (see text for details)16,32. In bottom\npanel, ∆ αis normalized to the respective saturation value.\nsimple exponential function in Eq. 2. It nicely repro-\nduces the data in the indirect contact case (Py/Cu/N)\nfor both Pd (Fig. 7) and Pt (Fig. 8), but it fails to fit\nthe direct contact (Py/N) configuration. When a thick-\nness dependent resistivity of the form ρN=ρb\nN+ρs\nN/tNis\nused (dash-point, cyan lines)16, in Py/Cu/N systems, no\nsignificant difference with the other functions is observed\nfor Pt, while for Pd a deviation from experimental trend\nis observed below 1.5nm. In Py/Nsystems, the fit better\ndescribes the rise at thicknesses shorter than the charac-\nteristic relaxation length, while deviates from the data\naround the saturation range.\nModels from standard spin transport theory cannot\nsatisfactorily describe the experimental data for the di-\nrect contact Py/N systems. For this reason a different\nmechanismforthe spin depolarizationprocesseshasbeen\nproposed, considering the presence of induced magnetic\nmoments in N in contact with the ferromagnetic layer.\nAppendix B: Interfacial interatomic exchange\n1. Paramagnets\nWe will show estimates for exchange energy based\non XMCD-measured moments in [Py/(Pt, Pd)] repeatsu-\nperlattices. Calculations of susceptibility are validated\nagainst experimental data for Pd and Pt. Bulk suscepti-\nbilities will be used to infer interfacial exchange parame-8\ntersJi\nex.\na. Pauli susceptibility For an itinerant electron sys-\ntem characterized by a density of states at the Fermi\nenergyN0, if an energy ∆ Esplits the spin-up and spin-\ndown electrons, the magnetization resulting from the\n(single-spin) exchange energy ∆ Eis\nM=µB/parenleftbig\nN↑−N↓/parenrightbig\n= 2µBN0S∆E(B1)\nwhereN0is the density of states in # /eV/at,Sis the\nStoner parameter, and 2∆ Eis the exchange splitting in\neV. Moments are then given in µB/at. Solving for ∆ E,\n∆E=M\n2µBN0S(B2)\nIf the exchange splitting is generated through the ap-\nplication of a magnetic field, ∆ E=µBH,\nµBH=M\n2µBN0S(B3)\nand the dimensionless volume magnetic susceptibility\ncan be expressed\nχv≡M\nH= 2µ2\nBN0S (B4)\nIn this expression, the prefactor can be evaluated\nthrough\nµ2\nB= 59.218 eV ˚A3(B5)\nso with N0[=]/eV/at, χvtakes units of volume per\natom, and is then also called an atomic susceptibility, in\ncm3/at, as printed in Ref37.\nb. Molar susceptiblity Experimentalvaluesaretabu-\nlated as molar susceptibilities. The atomic susceptibility\nχvcanbe contrastedwith the masssusceptibility χmand\nmolar susceptibility χmol\nχmass=χv\nρχmol=ATWT\nρχv(B6)\nwhere ATWT is the atomic weight (g/mol) and ρis\nthe density (g/cm3). These have units of χmass[=]cm3/g\nandχmol[=]cm3/mol. The molar susceptibility χmolis\nthen\nχmol= 2µ2\nBN0NAS (B7)\nin cm3/mol, where µBis the Bohr magneton, and\n2N0S=χmol\nNAµ2\nB(B8)Eq.B8providesaconvenentmethodtoestimateexper-\nimental unknowns, the density of states N0and Stoner\nparameter S, from measurements of χmol.\nExample: for Pd, the low-temperature measurement\n(differentfromtheroom-temperaturemeasurementinTa-\nbleI) isχmol∼7.0×10−4cm3/mol. In the denomina-\ntor, (NAµ2\nB) = 2.622 ×10−6Ry·cm3/mol, The value\n2N0Sconsistent with the experiment is 266/(Ry-at) or\n19.6/(eV-at). For the tabulated measurement of S= 9.3,\nthe inferred density of states is then N0= 1.05/eV/at.\nc. Interfacial exchange We canassumethat the Zee-\nman energy per interface atom is equal to its exchange\nenergy, through the Heisenberg form\nM2\np\nχvVat= 2Ji\nexsfsp (B9)\nwhereMpis the magnetization of the paramagnet,\nwith the atomic moment of the paramagnet mpin terms\nof its per-atom spin sp,\nMp=mp\nVatmp= 2µBsp (B10)\nVatis the volume of the paramagnetic site, sf,pare the\nper-atom spin numbers for the ferromagnetic and para-\nmagnetic sites, and Ji\nexis the (interatomic) exchange en-\nergy acting on the paramagnetic site from the ferromag-\nneticlayersonthe othersideoftheinterface. Interatomic\nexchangeenergyhasbeen distinguished fromintraatomic\n(Stoner) exchange involved in flipping the spin of a single\nelectron. Rewriting Eq B9,\nM2\np\nχvVat= 2Ji\nexsfMp\n2µBVat (B11)\nifsf= 1/2, appropriate for 4 πMs∼10 kG,\nJi\nex= 2µBMp\nχv(B12)\nand substituting for χvthrough Eq B4,\nJi\nex=Mp\nµBN0S(B13)\nIn the XMCD experiment, we measure the thickness-\naveraged magnetization as < M > in a [F/N]nsuper-\nlattice. We make a simplifying assumption that the ex-\nchange acts only on nearest-neighbors and so only the\nnear-interface atomic layer has a substantial magnetiza-\ntion. We can then estimate Mpfrom< M >through\n< M > t p= 2Mpti (B14)9\nwheretiis the polarized interface-layer thickness of\nN36. Since the interface exists on both sides of the N\nlayer, 2tiis the thickness in contact with F. Finally,\nJex=1\n2< M >\nµBN0Stp\nti(B15)\nThe exchange energy acting on each interface atom,\nfrom all neighbors, is JPt\nex= 109 meV for Pt and\nJPd\nex= 42 meV for Pd. Per nearest neighbor for an\nideal F/N(111) interface, it is JPy|Pt= 36 meV and\nJPy|Pd= 14 meV. Per nearest neighbor for an inter-\nmixed interface (6 nn), the values drop to 18 meV and 7\nmeV, respectively.\nSince explicit calculations for these systems are not\nin the literature, we can compare indirectly with theo-\nretical values. Dennler41showed that at a (3 d)F/(4d)N\ninterface (e.g. Co/Rh), there is a geometrical enhance-\nment in the moment induced in Nper nearest-neighbor\nofF. The 4d Natoms near the Finterface have larger\ninduced magnetic moments per nn of Fby a factor of\nfour. Specific calculations exist of JF|N(per neighbor)\nfor dilute Co impurities in Pt and dilute Fe impurties in\nPd42.JFe−Pd∼3 meV is calculated, roughly indepen-\ndent of composition up to 20% Fe. If this value is scaled\nup by a factor of four, to be consistent with the inter-\nface geometry in the XMCD experiment, it is ∼12 meV,\ncomparable with the value for Pd, assuming intermixing.\nTherefore the values calculated have the correct order of\nmagnitude.\n2. Ferromagnets\nThe Weiss molecular field,\nHW=βMs (B16)\nwhereβis a constant of order 103, can be used to give\nan estimate of the Curie temperature, as\nTC=µBgJJ(J+1)\n3kBHW (B17)\nDensity functional theory calculations have been used\ntoestimatethemolecularfieldrecently42,43; forspintype,\ntheJ(J+1) term is substutited with < s >2, giving an\nestimate of\nTC=2< s >2J0\n3kB(B18)where< s >is the number of spins on the atom as in\nEqB10; see the text by St¨ ohr and Siegmann44.< s >\ncan be estimated from m=1.07µBfor Py and 1.7 µBfor\nCo, respectively. Then\nJ0≃6kBTC\n(m/µB)2(B19)\nwith experimental Curie temperatures of 870 and 1388\nK, respectively, gives estimates of J0= 293 meV for Co\nandJ0= 393 meV for Py.\nNote that there is also a much older, simpler method.\nKikuchi45has related the exchange energies to the Curie\ntemperature for FCC lattices through\nJ= 0.247kBTC (B20)\nTaking 12 NN, 12 Jgives a total energy of 222 meV for\nPy (870 K) and 358 meV for FCC Co (1400K), not too\nfar off from the DFT estimates.\nd. Other estimates TheJ0exchange parameter is\ninteratomic, describing the interaction between spin-\nclusters located on atoms. Reversing the spin of one of\nthese clusters would change the energy J0. The Stoner\nexchange ∆ is different, since it is the energy involved in\nreversing the spin of a single electron in the electron sea.\nGenerally ∆ is understood to be greater than J0because\nit involvesmore coloumbrepulsion; interatomic exchange\ncan be screened more easily by spelectrons.\nThis exchange energy is that which is measured by\nphotoemissionandinversephotoemission. Measurements\nare quite different for Py and Co. Himpsel40finds an\nexchange splitting of ∆ = 270 meV for Py, which is not\ntoo far away from the Weiss J0value. For Co, however,\nthe value is between 0.9 and 1.2 eV, different by a factor\nof four. For Co the splitting needs to be estimated by a\ncombination of photoemission and inverse photoemission\nbecause the splitting straddles EF.46.\nFor comparison with the paramagnetic values of Ji\nex,\nwe use the J0estimates, since they both involve a bal-\nance between Zeeman energy (here in the Weiss field)\nand Heisenberg interatomic exchange. 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Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and\nR. a. Buhrman, Science 336, 555 (2012).\n34J. Kohlhepp, G. Strijkers, H. Wieldraaijer, and W. J. M.\nde Jonge, Phys. status solidi 704, 701 (2002).\n35A. Rogalev and F. Wilhelm, Phys. Mat. Mat. 116, 1285\n(2015).\n36The values of interface moment M iin Pt and Pd are calcu-\nlated confining the volume-averaged moment to the first 2\natomic planes at both the Py |N|Py interfaces of the stack.\nGiven the bulk (111)-plane distance a/ sqrt(3) (with a as in\nTab.I), thepolarized interface-layer thickness ti= 2∗a/√\n3\nis therefore defined. Considering the decay of the moment\nwith distance from the interface, tiwould capture about\n70% of the total induced moment23,24.\n37K. K. S. Misawa, 3d, 4d and 5d Elements, Alloys and Com-\npounds(SpringerMaterials, 1986), vol.19a, chap.1.3.1: In-\ntroduction to the paramagnetism of 4d and 5d transition\nmetals.\n38H. Song, L. Cheng, and W. Bailey, J. Appl. Phys. 95, 6592\n(2004).\n39D. H. Dye, J. 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Wisser,1Alexander J. Grutter,2Dustin A. Gilbert,3Alpha T. N'Diaye,4\nChristoph Klewe,4Padraic Shafer,4Elke Arenholz,4, 5Yuri Suzuki,1and Satoru Emori6,\u0003\n1Department of Applied Physics, Stanford University, Stanford, CA, USA\n2NIST Center for Neutron Research, Gaithersburg, MD, USA\n3Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN, USA\n4Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, CA, USA\n5Cornell High Energy Synchrotron Source, Ithaca, NY, USA\n6Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA\n(Dated: October 29, 2019)\nHigh-quality epitaxial ferrites, such as low-damping MgAl-ferrite (MAFO), are promising\nnanoscale building blocks for all-oxide heterostructures driven by pure spin current. However, the\nimpact of oxide interfaces on spin dynamics in such heterostructures remains an open question. Here,\nwe investigate the spin dynamics and chemical and magnetic depth pro\fles of 15-nm-thick MAFO\ncoherently interfaced with an isostructural \u00191-8-nm-thick overlayer of paramagnetic CoCr 2O4\n(CCO) as an all-oxide model system. Compared to MAFO without an overlayer, e\u000bective Gilbert\ndamping in MAFO/CCO is enhanced by a factor of >3, irrespective of the CCO overlayer thickness.\nWe attribute this damping enhancement to spin scattering at the \u00181-nm-thick chemically disordered\nlayer at the MAFO/CCO interface, rather than spin pumping or proximity-induced magnetism. Our\nresults indicate that damping in ferrite-based heterostructures is strongly in\ruenced by interfacial\nchemical disorder, even if the thickness of the disordered layer is a small fraction of the ferrite\nthickness.\nI. INTRODUCTION\nEmerging spintronic device schemes leverage magnon\nspin currents in electrically insulating magnetic oxides\n(e.g., ferrites), unaccompanied by dissipative motion\nof electrons, for computing and communications\napplications1,2. Low-dissipation spintronic devices\nbecome particularly attractive if insulating ferrite thin\n\flms with low magnetic damping can serve as sources\nof magnon spin currents. Such low-damping ferrites\ninclude not only epitaxial garnet ferrites (e.g., YIG)3{11\nthat have been widely used in studies of insulating\nspintronics2{4,12{15, but also coherently strained epitaxial\nspinel ferrites16{18with crucial technical advantages over\ngarnets, such as lower thermal budget for crystallization,\nhigher magnon resonance frequencies, and potential to be\nintegrated coherently with other spinels and perovskites\nwith various functionalities19{22.\nIn general, low-damping ferrite thin \flms must be\ninterfaced with other materials to realize spintronic\ndevices. It is therefore essential to understand whether\nand how damping in the ferrite is impacted by the\nproximity to another material. For instance, to convert\nbetween electronic and magnonic signals through direct\nand inverse spin Hall or Rashba-Edelstein e\u000bects23,\nthe low-damping ferrite needs to be interfaced with\na nonmagnetic metal with strong spin-orbit coupling.\nSpin transport and enhanced damping through spin\npumping24in ferrite/spin-orbit-metal structures has\nalready been extensively studied3,4,12{15,25. Moreover,\nthe low-damping ferrite can be interfaced with an\ninsulating antiferromagnetic or paramagnetic oxide, in\nwhich signals can be transmitted as a pure magnon\nspin current26{40. While interfacing low-damping ferriteswith insulating anti/paramagnetic oxides has enabled\nprototypes of magnon spin valves37{39, the fundamental\nimpact of insulating oxide interfaces on spin dynamics\nhas remained mostly unexplored. In particular, it is an\nopen question whether or how damping of the ferrite is\nenhanced from spin dissipation within the bulk of the\nadjacent anti/paramagnetic oxide or from spin scattering\nat the oxide interface.\nHere, we investigate how room-temperature magnetic\ndamping in epitaxial ferrimagnetic spinel MgAl-ferrite\n(MgAl 1=2Fe3=2O4, MAFO) is impacted when interfaced\nwith an overlayer of insulating paramagnetic spinel\nCoCr 2O4(CCO)41,42. This epitaxial MAFO/CCO\nbilayer is an isostructural model system, possessing\na coherent interface with continuous crystal lattices\nbetween the spinel ferrite and paramagnet. We \fnd that\nthe presence of MAFO/CCO interface increases damping\nby more than a factor of >3 compared to MAFO without\nan overlayer. We attribute this damping enhancement {\nwhich is comparable to or greater than spin pumping\ne\u000bects reported for ferrite/spin-orbit-metal bilayers { to\nspin scattering by the ultrathin ( \u00181 nm) chemically\ndisordered layer at the MAFO/CCO interface. Our\n\fndings show that spin scattering at oxide interfaces\nhas a profound in\ruence on damping, even when the\nchemically disordered layer is a small fraction of the total\nmagnetic layer thickness.\nII. FILM GROWTH AND STRUCTURAL\nCHARACTERIZATION\nEpitaxial thin \flms of 15-nm-thick MAFO interfaced\nwith 1.3-8 nm of CCO overlayer were grown on as-arXiv:1908.08629v2  [cond-mat.mtrl-sci]  26 Oct 20192\n40 42 44 46MAO (004)\nMAFO/CCO\n      (004)\nMAFO (004)log10(Intensity) (arb. units)\n2q (deg)CCO (004)\n-0.180 -0.175 -0.1700.570.580.590.600.610.620.63-\n \n   (115)-(a) (c)\nqip(Å-1)qop(Å-1)\nCCO\n(25 nm)MAO(b)\n-0.2-0.1 0.00.10.2MAFOCCOIntensity (arb. units)\nDw004 (deg)MAFO/CCO\nFigure 1. (a) 2 \u0012-!scans of epitaxial MAFO(15 nm), CCO(25 nm), and MAFO(15 nm)/CCO(8 nm). The data are o\u000bset for\nclarity. (b) Rocking curve scans about the (004) \flm peak for the \flms shown in (a). (c) Reciprocal space map of epitaxial\nCCO(25 nm) coherently strained to the MAO substrate.\nreceived single-crystal MgAl 2O4(MAO) substrates via\npulsed laser deposition. A KrF 248 nm laser was\nincident on stoichiometric targets of MAFO and CCO\nwith \ruences of \u00191.5 J/cm2and\u00191.3 J/cm2,\nrespectively. Both \flms were grown in 10 mTorr (1.3\nPa) O 2and were cooled in 100 Torr (13 kPa) O 2.\nMAFO \flms were grown at 450\u000eC, whereas CCO \flms\nwere deposited at 300\u000eC in an attempt to minimize\nintermixing between the MAFO and CCO layers. These\ngrowth temperatures, much lower than >700\u000eC typically\nrequired for epitaxial garnets3{11, are su\u000ecient to fully\ncrystallize MAFO and CCO. The low crystallization\ntemperatures of the spinels o\u000ber an advantage over\nthe oft-studied garnets, with more opportunities for\nisostructural integration with coherent interfaces. The\nMAFO \flms exhibit a room-temperature saturation\nmagnetization of\u0019100 kA/m and a Curie temperature of\n\u0019400 K18. To obtain consistent ferromagnetic resonance\nresults, MAFO \flms were grown and subsequently\ncharacterized by ferromagnetic resonance (FMR) ex-situ;\nafter surface cleaning with ultrasonication in isopropanol,\nCCO overlayers were then deposited as described above.\nGrowth rates were calibrated via X-ray re\rectivity.\nOur structural characterization of MAFO and\nCCO shows high-quality, coherently strained \flms.\nIn symmetric 2 \u0012-!X-ray di\u000braction scans, only\npeaks corresponding to the (00 `) re\rections are\nobserved, indicating that the \flms are highly epitaxial.\nAdditionally, as seen in Fig. 1(a), Laue oscillations\naround the (004) Bragg re\rections in both single-layer\nMAFO and CCO layers as well as MAFO/CCO bilayers\ndenote smooth interfaces. Furthermore, MAFO, CCO,\nand MAFO/CCO samples all exhibit essentially the\nsame \flm-peak rocking curve widths (FWHM) of \u00190.06\u000e\n(Fig. 1(b)). Reciprocal space mapping of the ( \u00161\u001615)\nre\rection in 25-nm-thick single-layer CCO on MAO\n(Fig. 1(c)) reveals that the in-plane lattice parameter of\nthe \flm coincides with that of the substrate, indicating\nCCO is coherently strained to MAO. We note thatdespite the relatively large lattice mismatch between\nCCO and MAO of \u00193 %, coherently strained growth of\nCCO of up to 40 nm has been previously reported on\nMAO substrates41. For our CCO \flm, we calculate an\nout-of-plane lattice constant c\u00198:534\u0017A from the 2 \u0012-!\nscan; taking the in-plane lattice parameter a= 8:083\u0017A of\nthe MAO substrate, the resulting tetragonal distortion of\ncoherently strained CCO is c=a\u00191:055, similar to that\nfor coherently strained MAFO18.\nStructural characterization results underscore the\nquality of these epitaxial \flms grown as single layers and\nbilayers. Considering the comparable high crystalline\nquality for MAFO, CCO, and MAFO/CCO { as\nevidenced by the presence of Laue oscillations and narrow\n\flm-peak rocking curves { we conclude that MAFO/CCO\nbilayers (with the total thickness limited to \u001423 nm) are\ncoherently strained to the substrate. In these samples\nwhere the substrate and \flm layers are isostructural, we\nalso do not expect antiphase boundaries43{46. Indeed,\nwe \fnd no evidence for frustrated magnetism, i.e., high\nsaturation \feld and coercivity, that would arise from\nantiphase boundaries in spinel ferrites43{46; MAFO/CCO\nbilayers studied here instead exhibit soft magnetism, i.e.,\nsquare hysteresis loops with low coercivity <0.5 mT,\nsimilar to our previous report on epitaxial MAFO thin\n\flms18. Thus, MAFO/CCO is a high-quality all-oxide\nmodel system, which permits the evaluation of how spin\ndynamics are impacted by a structurally clean, coherent\ninterface.\nIII. FERROMAGNETIC RESONANCE\nCHARACTERIZATION OF DAMPING\nTo quantify e\u000bective damping in coherently strained\nMAFO(/CCO) thin \flms, we performed broadband\nFMR measurements at room temperature in a coplanar\nwaveguide setup using the same procedure as our prior\nwork16,18. We show FMR results with external bias3\nmagnetic \feld applied in the \flm plane along the [100]\ndirection of MAFO(/CCO); essentially identical damping\nresults were obtained with in-plane \feld applied along\n[110]47. Figure 2(a) shows the frequency fdependence of\nhalf-width-at-half-maximum (HWHM) linewidth \u0001 Hfor\na single-layer MAFO sample and a MAFO/CCO bilayer\nwith a CCO overlayer thickness of just 1.3 nm, i.e., less\nthan 2 unit cells. The linewidth is related to the e\u000bective\nGilbert damping parameter \u000beffvia the linear equation:\n\u0001H= \u0001H0+h\u000beff\ng\u00160\u0016Bf (1)\nwhere \u0001H0is the zero-frequency linewidth, his Planck's\nconstant,g\u00192:05 is the Land\u0013 e g-factor derived from the\nfrequency dependence of resonance \feld HFMR ,\u00160is the\npermeability of free space, and \u0016Bis the Bohr magneton.\nIt is easily seen from Fig. 2(a) that with the addition\nof ultrathin CCO, the damping parameter is drastically\nincreased, i.e., >3 times its value in bare MAFO.\nFigure 2(b) shows that the damping enhancement\nseen in MAFO/CCO is essentially independent of\nthe CCO thickness. This trend suggests that\nthe damping enhancement is purely due to the\nMAFO/CCO interface, rather than spin dissipation in\nthe bulk of CCO akin to the absorption of di\u000busive\nspin current reported in antiferromagnetic NiO26,35,48.\nWe note that other bulk magnetic properties of\nMAFO (e.g., e\u000bective magnetization, Land\u0013 e g-factor,\nmagnetocrystalline anisotropy) are not modi\fed by the\nCCO overlayer in a detectable way. We also rule\nout e\u000bects from solvent cleaning prior to CCO growth\nor thermal cycling in the deposition chamber up to\n300\u000eC, as subjecting bare MAFO to the same ex-\nsitu cleaning and in-situ heating/cooling processes as\ndescribed in Section II, but without CCO deposition,\nresults in no measurable change in damping. The\ndamping enhancement therefore evidently arises from the\nproximity of MAFO to the CCO overlayer.\nWe consider two possible mechanisms at the\nMAFO/CCO interface for the observed damping\nenhancement:\n(1) Spin current excited by FMR in MAFO\nmay be absorbed via spin transfer in an interfacial\nproximity-magnetized layer49of CCO, whose magnetic\nmoments may not be completely aligned with those of\nMAFO. While CCO by itself is paramagnetic at room\ntemperature, prior studies have shown that Co2+and\nCr3+cations in epitaxial CCO interfaced with a spinel\nferrite (e.g., Fe 3O4) can develop measurable magnetic\norder50. Such damping enhancement due to interfacial\nmagnetic layer is analogous to spin dephasing reported\nfor ferromagnets interfaced directly with proximity-\nmagnetized paramagnetic metal (e.g., Pt, Pd)49.\n(2) Even if CCO does not develop proximity-induced\nmagnetism, chemical disorder at the MAFO/CCO\ninterface may enhance spin scattering. For instance,\nchemical disorder may lead to an increase of Fe2+\n0 10 20 300246810HWHM Linewidth (mT)\nFrequency (GHz)(a)\n(b)MAFO/CCO\neff≈ 0.007\nMAFO\neff≈ 0.002\n0 2 4 6 80.0000.0020.0040.0060.0080.0100.012eff\nCCO thickness (nm)Figure 2. (a) HWHM FMR linewidth versus frequency\nfor MAFO(15 nm) and MAFO(15 nm)/CCO(1.3 nm). The\ne\u000bective Gilbert damping parameter \u000beffis derived from\nthe linear \ft. (b) \u000beffplotted against the CCO overlayer\nthickness. The dashed horizontal line indicates the average of\n\u000befffor MAFO without an overlayer.)\ncations at the MAFO surface, thereby increasing\nthe spin-orbit spin scattering contribution to Gilbert\ndamping in MAFO compared to its intrinsic composition\ndominated by Fe3+with weak spin-orbit coupling18,51.\nAnother possibility is that chemical disorder at the\nMAFO/CCO interface introduces magnetic roughness\nthat gives rise to additional spin scattering, perhaps\nsimilar to two-magnon scattering recently reported for\nferromagnet/spin-orbit-metal systems52.\nIn the following section, we directly examine interfacial\nproximity magnetism and chemical disorder to gain\ninsight into the physical origin of the observed damping\nenhancement in MAFO/CCO.\nIV. CHARACTERIZATION OF INTERFACE\nCHEMISTRY AND MAGNETISM\nTo evaluate the potential formation of a magnetized\nlayer in the interfacial CCO through the magnetic\nproximity e\u000bect, we performed depth-resolved\nand element-speci\fc magnetic characterization\nof MAFO/CCO bilayers using polarized neutron\nre\rectometry (PNR) and soft magnetic X-ray\nspectroscopy. PNR measurements were performed\nusing the PBR instrument at the NIST Center for\nNeutron Research on nominally 15-nm-thick MAFO\nlayers capped with either thick (5 nm) or thin (3 nm)4\nCCO overlayers. PNR measurements were performed in\nan in-plane applied \feld of 3 T at temperatures of 300\nK and 115 K, the latter case being slightly above the\nnominal 97 K Curie temperature of CCO41,42. Incident\nneutrons were spin-polarized parallel or anti-parallel to\nthe applied \feld both before and after scattering from\nthe sample, and the re\rected intensity was measured\nas a function of the perpendicular momentum transfer\nvector Q. The incident spin state of measured neutrons\nwere retained after scattering, corresponding to the\ntwo non-spin-\rip re\rectivity cross sections ( \"\"and##).\nSince all layers of the \flm are expected to saturate well\nbelow the applied \feld of 3 T, no spin-\rip re\rectivity is\nexpected and these cross sections were not measured.\nSince PNR is sensitive to the depth pro\fles of the\nnuclear and magnetic scattering length density (SLD),\nthe data can be \ftted to extract the chemical and\nmagnetic depth pro\fles of the heterostructure. In this\ncase, we used the Re\r1D software package for this\npurpose53. Figure 3(a,b) shows the 300 K re\rectivities\nand spin asymmetry curves of a nominal MAFO (15\nnm)/CCO (5 nm) sample alongside the depth pro\fle\n(Fig. 3(c)) used to generate the \fts shown. The\nbest \ft pro\fle (Fig. 3(c)) provides no evidence of a\nlayer with proximity-induced magnetization in the CCO.\nRather, we note that there appears to be a layer of\nmagnetization suppression near both the MAO/MAFO\nand MAFO/CCO interfaces. Further, the interfacial\nroughnesses of both the MAO/MAFO and MAFO/CCO,\n0.9(1) nm and 1.35(5) nm respectively, are signi\fcantly\nlarger than the CCO surface roughness of 0.27(3) nm\nand the bare MAFO surface roughness of <\u00180.5 nm54.\nThe interfacial roughnesses are signatures of chemical\nintermixing at the spinel-spinel interface leading to\ninterfacial suppression of the magnetization and/or Curie\ntemperature. Thus, we \fnd that the MAFO/CCO\ninterface, although structurally coherent, exhibits a\nchemically intermixed region on the order of one spinel\nunit cell thick on either side.\nTo obtain an upper limit of the proximity-induced\ninterfacial magnetization in CCO, we performed Markov-\nchain Monte-carlo simulations as implemented in the\nDREAM algorithm of the BUMPS python package.\nThese simulations suggest an upper limit (95% con\fdence\ninterval of) 7 emu/cc in the 1.5 nm of the CCO closest\nto the interface. In this case, the model evaluated the\nMAFO as a uniform structural slab but allowed for total\nor partial magnetization suppression at both interfaces,\nwhile the CCO layer was treated as a uniform slab with\nan allowed magnetization layer of variable thickness at\nthe interface.\nHowever, we note that equivalently good \fts are\nobtained using simpler models that \ft a single MAFO\nlayer with magnetically dead layers at the interfaces and\na completely nonmagnetic CCO layer. Equivalent results\nwere obtained for the thick CCO sample at 115 K and\nfor the thin CCO sample. We therefore conclude that the\nPNR results strongly favor a physical picture in which the\nFigure 3. (a) Spin-polarized neutron re\rectivity and (b)\nspin asymmetry of a MAFO (15 nm)/CCO (5 nm) bilayer\nalongside theoretical \fts. (c) Nuclear and magnetic scattering\n(scaled \u000210) length density pro\fle used to generate the \fts\nshown. Error bars represent \u00061 standard deviation.\nCCO is notmagnetized through the magnetic proximity\ne\u000bect.\nTo con\frm the PNR results and examine the e\u000bect\nof a CCO overlayer on the local environment of Fe\ncations in MAFO, we performed temperature-dependent\nX-ray absorption (XA) spectroscopy and X-ray magnetic\ncircular dichroism (XMCD) measurements at Beamline\n4.0.2 of the Advanced Light Source at Lawrence Berkeley\nNational Laboratory. We note that the detection\nmode (total electron yield) used here for XA/XMCD\nis sensitive to the top \u00195 nm of the sample, such that\nFe L edge signals from CCO-capped MAFO primarily\ncapture the cation chemistry near the MAFO/CCO\ninterface. Measurements were performed in an applied\n\feld of 400 mT along the circularly polarized X-ray beam,\nincident at 30\u000egrazing from the \flm plane. To minimize\ndrift e\u000bects during the measurement, multiple successive\nenergy scans were taken and averaged, switching both\napplied \feld direction and photon helicity so that all\nfour possible combinations of \feld direction and helicity\nwere captured at least once. XA and XMCD intensities\nwere normalized such that the pre-edge is zero and\nthe maximum value of the average of the (+) and\n(\u0000) intensities is unity. In the case of the Co L-\nedge, measurements were taken with energy sweeps\ncovering both Fe and Co edges, and for consistency\nboth edges were normalized to the highest XAS signal,\ncorresponding to the Fe L 3-edge.\nFigure 4(a) compares the XA of a bare MAFO \flm5\nFigure 4. (a) 300 K X-ray absorption spectra of MAFO and\nMAFO/CCO (3 nm) grown on MAO. (b) Photon helicity-\ndependent XA spectra and XMCD of the Fe L-edge for a\nMAFO/CCO (3 nm) bilayer at 300 K. (c) Co and (d) Cr\nL-edge XA and XMCD of the same bilayer.\nwith one capped by 3 nm of CCO. The two XA lineshapes\nare nearly identical, indicating the same average Fe\noxidation state and site-distribution in CCO-capped\nand uncapped MAFO \flms. It is therefore likely that\nthe reduced interfacial magnetization observed through\nPNR is a result of a defect-induced Curie temperature\nreduction, rather than preferential site-occupation of Co\nand Cr that might increase the Fe2+content in the\nintermixed interfacial region.\nWe further note that although a large XMCD signal\nis observed on the Fe-edge at 300 K (Fig. 4(b)), neither\nthe Co nor Cr L edges exhibit any signi\fcant magnetic\ndichroism, as shown in Figs. 4(c)-(d). Similar results\nare obtained on the Cr L edge at 120 K. Consistent\nwith the PNR results, we thus \fnd no evidence for\na net magnetization induced in the CCO through the\ninterfacial magnetic proximity e\u000bect.\nOur \fnding of suppressed interfacial magnetism\nin MAFO/CCO is reminiscent of earlier reports\nof magnetic dead layers in epitaxially-grown ferrite-\nbased heterostructures55{57. For example, prior\nPNR experiments have revealed magnetic dead layers\nat the interfaces of ferrimagnetic spinel Fe 3O4and\nantiferromagnetic rock-salt NiO or CoO, even when the\ninterfacial roughness is small (e.g., only 0.3 nm)55,56.\nA magnetic dead layer of 1 spinel unit cell has also\nbeen reported at the interface of Fe 3O4and diamagnetic\nrock-salt MgO grown by molecular beam epitaxy57.\nWe note that in these prior studies, the spinel ferrite\flms interfaced with the rock salts (NiO, CoO, MgO)\npossess antiphase boundaries. Suppressed magnetism\nis known to result from antiphase boundaries, as they\nfrustrate the long-range magnetic order and reduce\nthe net magnetization of the ferrite44. By contrast,\nthere is no evidence for antiphase boundaries in all-\nspinel MAFO/CCO grown on spinel MAO; therefore,\nthe suppressed magnetism at the MAFO/CCO interface\ncannot be attributed to antiphase-boundary-induced\nmagnetic frustration.\nAnother possible scenario is that magnetic dead layer\nformation is a fundamental consequence of the charge\nimbalance between di\u000berent lattice planes, as recently\nshown in a recent report of (polar) Fe 3O4undergoing\natomic reconstruction to avoid \\polar catastrophe\" when\ngrown on (nonpolar) MgO58. In our study on all-\nspinel heterostructures, there may also be some degree of\ncharge mismatch depending on the relative populations\nof cations on the tetrahedrally- and octahedrally-\ncoordinated sites at the MAFO/CCO interface, although\nthe charge mismatch is expected to be only \u0019\u00061, i.e.,\na factor of\u00195-6 smaller than that in MgO/Fe 3O458.\nThus, atomic reconstruction driven by charge imbalance\nappears unlikely as a dominant source of the magnetic\ndead layer in MAFO/CCO. We instead tentatively\nattribute the dead layer to atomic intermixing driven by\ndi\u000busion across the MAFO/CCO interface during CCO\noverlayer deposition.\nV. DISCUSSION\nOur PNR and XA/XMCD results (Section IV) indicate\nthat the damping enhancement observed in Section III\narises from chemical disorder, rather than proximity-\ninduced magnetism, at the MAFO/CCO interface.\nWe emphasize that this interfacial disordered layer\nis con\fned to within \u00192 spinel unit cells. We\nalso note that this interfacial disorder is due to\natomic intermixing, but not structural defects (e.g.,\ndislocations, antiphase boundaries), in this coherent\nbilayer system of MAFO/CCO. Nevertheless, this\nultrathin chemically disordered layer alone is evidently\nsu\u000ecient to signi\fcantly increase spin scattering.\nConsidering that the cation chemistry of Fe in MAFO\ndoes not change substantially (Fig. 4(a)), the interfacial\nspin scattering is likely driven by magnetic roughness,\nleading to a mechanism similar to two-magnon scattering\nthat accounts for a large fraction of e\u000bective damping in\nmetallic ferromagnet/Pt bilayers52.\nWe now put in context the magnitude of the damping\nenhancement \u0001 \u000beff, i.e., the di\u000berence in the e\u000bective\nGilbert damping parameter between CCO-capped and\nbare MAFO,\n\u0001\u000beff=\u000bbilayer\neff\u0000\u000bferrite\neff; (2)\nby comparing it with ferrite/spin-orbit-metal systems\nwhere spin pumping is often considered as the source6\n0.0000.0020.0040.0060.008\n MAFO/CCO\n   [this study] MAFO/W\n [Riddiford] MAFO/Pt\n [Riddiford]YIG/Pt\n[Wang]Daeff\nYIG/Pt\n [Sun]\nFigure 5. Comparison of the enhancement of the e\u000bective\nGilbert damping parameter \u0001 \u000befffor MAFO/CCO and\nferrite/spin-orbit-metal bilayers. YIG/Pt [Sun], YIG/Pt\n[Wang], and MAFO/Pt(W) [Riddiford] are adapted from\nRefs.59,60, and61respectively. The values of \u0001 \u000befffrom the\nliterature are normalized for the saturation magnetization\nof 100 kA/m and magnetic thickness of 15 nm for direct\ncomparison with our MAFO/CCO result.\nof damping enhancement. Since damping enhancement\nfrom spin pumping or interfacial scattering scales\ninversely with the product of the saturation of\nmagnetization Msand the magnetic layer thickness tm,\nthe values of \u0001 \u000befftaken from the literature59{61are\nnormalized for direct comparison with the MAFO \flms\nstudied here with Ms= 100 kA/m and tm= 15 nm.\nAs summarized in Fig. 5, \u0001 \u000befffor MAFO/CCO\nis comparable to { or even greater than { \u0001 \u000beff\nfor ferrite/metal bilayers. This \fnding highlights that\nthe strength of increased spin scattering in a ferrite\ndue to interfacial chemical disorder can be on par\nwith spin dissipation due to spin pumping in metallic\nspin sinks. More generally, this \fnding suggests that\nspecial care may be required in directly relating \u0001 \u000beff\nto spin pumping across bilayer interfaces (i.e., spin-\nmixing conductance52), particularly when the FMR-\ndriven magnetic layer is directly interfaced with a spin\nscatterer.\nFurthermore, the strong interfacial spin scattering {\neven when the oxide interface is structurally coherent\nand the chemically disordered layer is kept to just <\u00182\nunit cells { poses a signi\fcant challenge for maintaining\nlow damping in ferrite/insulator heterostructures. This\nchallenge is partially analogous to the problem of reduced\nspin polarization in tunnel junctions consisting of spinelFe3O4and oxide barriers (e.g., MgO)62{65, which is also\nlikely due to interfacial chemical disorder and magnetic\ndead layers. However, we emphasize that the problems of\nantiphase boundaries43{46and charge-imbalance-driven\natomic reconstruction58, which have posed intrinsic\nchallenges for devices with MgO/Fe 3O4interfaces, are\nlikely not applicable to all-spinel MAFO/CCO. It is\ntherefore possible that deposition schemes that yield\nsharper interfaces, e.g., molecular beam epitaxy, can be\nemployed to reduce interfacial imperfections and hence\nspin scattering at MAFO/CCO for low-loss all-oxide\ndevice structures.\nVI. CONCLUSIONS\nWe have shown that e\u000bective damping in epitaxial\nspinel MgAl-ferrite (MAFO) increases more than\nthreefold when interfaced coherently with an insulating\nparamagnetic spinel of CoCr 2O4(CCO). This damping\nenhancement is not due to spin pumping into the\nbulk of CCO. Our depth-resolved characterization of\nMAFO/CCO bilayers also reveals no proximity-induced\nmagnetization in CCO or signi\fcant change in the\ncation chemistry of MAFO. We attribute the giant\ndamping enhancement to spin scattering in an ultrathin\nchemically disordered layer, con\fned to within 2 spinel\nunit cells across the MAFO/CCO interface. Our results\ndemonstrate that spin dynamics in ferrite thin \flms are\nstrongly impacted by interfacial disorder.\nAcknowledgements - This work was supported in\npart by the Vannevar Bush Faculty Fellowship program\nsponsored by the Basic Research O\u000ece of the Assistant\nSecretary of Defense for Research and Engineering and\nfunded by the O\u000ece of Naval Research through grant\nno. N00014-15-1-0045. 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Phys.\nLett.105, 102410 (2014)."    },    {        "title": "1606.06610v1.Torsion_Effects_and_LLG_Equation.pdf",        "content": "arXiv:1606.06610v1  [hep-th]  21 Jun 2016Torsion Effects and LLG Equation.\nCristine N. Ferreira†a, Cresus F. L. Godinho‡b, J. A. Helay¨ el Neto∗c\n†N´ ucleo de Estudos em F´ ısica, Instituto Federal de Educa¸ c ˜ ao, Ciˆ encia e Tecnologia\nFluminense, Rua Dr. Siqueira 273, Campos dos Goytacazes, 28 030-130 RJ, Brazil\n‡Grupo de F´ ısica Te´ orica, Departamento de F´ ısica, Univer sidade Federal Rural do Rio de\nJaneiro, BR 465-07, 23890-971, Serop´ edica, RJ, Brazil\n∗Centro Brasileiro de Pesquisas F´ ısicas (CBPF), Rua Dr. Xav ier Sigaud 150, Urca,\n22290-180, Rio de Janeiro, Brazil\nAbstract\nBased on the non-relativistic regime of the Dirac equation coupled to a torsion\npseudo-vector, we study the dynamics of magnetization and how it is affected by the\npresence of torsion. We consider that torsion interacting terms in Dirac equation\nappear in two ways one of these is thhrough the covariant derivativ e considering the\nspin connection and gauge magnetic field and the other is through a n on-minimal\nspin torsion coupling. We show within this framework, that it is possible to obtain\nthe most general Landau, Lifshitz and Gilbert (LLG) equation includ ing the torsion\neffects, where we refer to torsion as a geometric field playing an impo rtant role in the\nspin coupling process. We show that the torsion terms can give us tw o important\nlandscapes in the magnetization dynamics: one of them related with d amping and the\nother related with the screw dislocation that give us a global effect lik e a helix damping\nsharped. These terms are responsible for changes in the magnetiz ation precession\ndynamics.\nacrisnfer@iff.edu.br ,bcrgodinho@ufrrj.br ,chelayel@cbpf.br\n11 Introduction\nThe discovery of the graphene-like systems and topological insulators systems introduced a\nnew dynamic in the applications of the framework of the high e nergy physics in low energy\nsystems in special condensed matter systems. The fact that t hese systems can be described\nby Dirac equations give us new possibilities for theoretica l and experimental applications.\nIn this direction there are some effects in condensed matter sy stems still without a full\ndescription as the magnetizable systems that we described i n this work. In this sense the\nconstructionofthetheoreticalframeworksthatcan,insom elimit, beobtainedinlowenergy\nsystems is the crucial importance to understand the invaria nces and interactions in certain\nlimits. One of the important effects that we can study is relate d with the spin systems.\nSo, in this work we deal with the new framework to study the spi n systems considering\nthe Dirac equation in non-relativistic limit with torsion i nteraction[1]. Spin systems are\ngenerally connected with magnetic systems. It is well known that spin angular momentum\nisanintrinsicpropertyofquantumsystems. Whenamagnetic fieldisapplied, eachmaterial\npresents some level of magnetization, and Quantum Mechanic s says that magnetization is\nrelated to the expectation value of the spin angular momentu m operator. In the case\nof ferromagnetic materials, they can have a large magnetiza tion even under the action\nof a small magnetic field and the magnetization process is alw ays followed by hysteresis,\nand the magnetization is uniform and lined up with the magnet ic field, usually these\nmaterials exhibit a strong ordering process that results in a parallel line up the spins[2]. In\nmaterials graphene type we also can generate magnetic momen t. In this form it is possible\nto study the transport phenomena [3]. Overlapping between e lectronic wave functions are\ninteractions well understood, again thanks to Quantum Mech anics, however thereare other\nkinds of interactions occurring such as magnetocrystallin e anisotropy, connected with the\ntemperature dependence[4] and demagnetization fields [5], acting in low range. In such\nsystems if we only consider the precession we will not reach t he right limit. Certainly, the\nprecession equation has to include a damping term providing the magnetization alignment\nwith the magnetic field after a finite time [6]. In order to simu late these phenomena,\nseveral physical models have been presented. However, the L andau-Lifshitz model is still\nthe one widely used in the description of the dynamics of ferr omagnetic media. In their\npioneering work [7] in 1935, Landau and Lifshitz proposed a n ew theory based on the\nfollowing dynamical equation:\n∂tM=/vectorHeff×/vectorM+α\nM2s/vectorM×(/vectorM×/vectorHeff), (1)\nwhere/vectorHeffdenote an effective magnetic field, with the gyromagnetic rati o absorbed, inter-\nacting with the magnetization M=|/vectorM|. The first term is the precession of the magnetiza-\ntion vector around the direction of the effective magnetic fiel d and the second one describes\na damping of the dynamics. With this theory we are able to comp ute the thickness of walls\nbetween magnetic domains, and also understand the domain fo rmation in ferromagnetic\n2materials. This theory, which now goes under the name of micr omagnetics, has been in-\nstrumental in the understanding and development of magneti c memories. Landau and\nLifshitz considered the Gibbs energy G of a magnetic materia l to be composed of three\nterms: exchange, anisotropy and Zeeman energies (due to the external magnetic field), and\npostulated that the observed magnetization per unit volume M field would correspond to\na local minimum of the Gibbs energy. Later researchers added other terms to G such as\nmagnetoelastic energy and demagnetization energy. They al so derived the Landau Lifshitz\n(LL) equation using only physical arguments and not using th e calculus of variations. In\nsubsequent work, Gilbert [8] realized a more convincing for m for the damping term, based\non a variational approach, and the new combined form was then called Landau Lifshitz\nGilbert (LLG) equation, today it is a fundamental dynamic sy stem in applied magnetism.\nNowadays, the scientific and technological advances provid e a wide spectrum of ma-\nnipulations to the spin degrees of freedom. The complete for mulation for magnetization\ndynamics also include the excitation of magnons and their in teraction with other degrees\nof freedom, that remains as a challenge for modern theory of m agnetism [9]. These amaz-\ning and reliable kinds of procedures are propelling spintro nics as a consolidated sub-area\nof Condensed Matter Physics [10]. Since the experimental ad vances are increasingly pro-\nviding high-precision data, many theoretical works are bei ng presented [11, 14, 12, 13]\nand including strange materials, as the topological insula tors, connected with the mag-\nnetocondutivity [15] and graphene like structures [16] for a deeper understanding of the\nphenomenon including the spin polarization super currents for spintronics [17], holographic\nunderstanding of spin transport phenomena [18] and non-rel ativistic background [19].\nThe torsion field appears as one of the most natural extension s of General Relativity\nalong with the metric tensor, which couples to the energy-mo mentum distribution, inspects\nthe details of the spin density tensor. Actually, in General Relativity, fermions naturally\ncouple to torsion by means of their spin.\nIn this work we consider that the torsion interacts with the m atter in two types one of\nthese is present in covariant derivative that contains the s pin connection related with the\nChristoffell symbol given by the metric of the curve space time and the contorsion given\nby the torsion that have two antisymmetric index. The other c ontribution is given by the\nnon minimal spin torsion coupling that is important to the co nsistence of the theory. It is\npossible to study the non relativistic approach to the torsi on in connection with the spin\nparticles [20], in this work we only consider the torsion con tribution considering the plane\nspace-time.\nOur work is organized as follows, in Section 2, we analyse the torsion coupling in\nrelativistic limit, in Section 3 the modified version of Paul i equation is presented. Our\napproach starts with a field theoretical action where a Dirac fermion is non-minimally\ncoupled in the presence of a torsion term, a low relativistic approximation is considered\nand the equivalent Pauli equation is then obtained. We deriv e a very similar expression to\nthe Landau-Lifshitz-Gilbert (LLG) equation from our Pauli equation with torsion. Under\nspecific conditions for the magnetic moment, we show that the LLG equation can be\n3established with damping and dislocations terms.\n2 TheRelativistic andNon- Relativistic Discussions for Sp in\nCoupling with Torsion\nIn this Section, let us understand the way to describe the spi n interaction by taking into\naccount the torsion coupling. In our framework there are two terms in Dirac action, one\nof these is connected with spin current effect from the spin con nection, the other is a non-\nminimal spin torsion coupling whose effects are the subject of this work. Dirac’s equation\nis relativistic and we should justify why we use it in our mode l. Dispersion relations in\nCondensed Matter Physics (CMP) linear in the velocity appea r in a wide class of models\nand one adopts the framework of Dirac’s equation to approach them. However, the speed\nof light, c, is suitably replaced by the Fermi velocity, vF. Here, this is not what we are\ndoing. We actually start off from the Dirac’s equation and we t ake, to match with effects\nof CMP, the non-relativistic regime, for the electron moves with velocities v≤c\n300. So,\ncontrary to an analogue model where we describe the phenomen a by a sort of relativity\nwith c replaced by vF, we here consider that the non-relativistic electrons of ou r system is\na remnant of a more fundamental relativistic world. The non- relativistic limit is also more\ncomplete because it brings effects that do not directly appear in Galilean Physics. This is\nwhy we have taken the viewpoint of associating our physics to the Dirac’s equation.\n2.1 The Dirac Model for Torsion and the Spin Current Interpre tation\nIn this sub-section, we consider the microscopic discussio n that gives the explicit form of\nthe spin current in function of the gauge potential and torsi on coupling. The scenario we\nare setting up is justified by the following chain of argument s: (i) We are interested in\nspin effects. We assume that there is a space-time structures ( torsion) whose coupling with\nthe matter spin becomes relevant. But, we are actually inter ested in the possible non-\nrelativistic effects stemming from this coupling, which is mi nimal and taken into account\nin the covariant derivative though the spin connection. (ii ) The other point we consider\nis that, amongst the three irreducible torsion components, its pseudo-vector piece is the\nonly one that couples to the charged leptons. Then, with this results in mind, we realize\nthat the electron spin density may non-minimally couple, in a Pauli- like interaction, to\nthe field-strength of the torsion pseudo-vector degree of fr eedom.\nSo, our scenario is based on the relevant role space-time tor sion, here modeled by a\npseudo-vector, may place in the non-relativistic electron s of spin systems in CMP. The spin\ncurrent that we talk about is the spin magnetic moment and in g eneral is not conserved\nalone. Thequantity that is conserved is the total magnetic m oment that is the composition\nbetween both /vectorJ=/vectorJS+/vectorJL. In form that ∂µJµ= 0 where the spin current can be defined\nin related to the three component spin current as Jµ\nS=ǫµab\nρJρ\naband/vectorJLis the spin orbit\n4coupling . In analogy with the charge current, defined by the d erivative of the action in\nrelation to the gauge field Aµwe used the definition where the spin current is the derivativ e\nof the action in relation with the spin connection. We consid er here the spin current as\nthe derivative of the action in relation to the spin connecti onωab\nµthen the spin current is\ngiven by\nJµ\nab=δS\nδωabµ, (2)\nwhereωab\nµis\nωab\nµ=ea\nν∇µeνb+ea\nλΓλ\nµνeνb+ea\nλKλ\nµνeνb, (3)\nandKλ\nµνis the contorsion given by\nKλ\nµν=−1\n2(Tλ\nµ ν+Tλ\nν µ−Tλ\nµν), (4)\nwhereTλ\nµνis the torsion. In this work we consider two terms for torsion , on of these is the\ntotally anti symmetric tensor, that respecting the duality relation given by Tµνλ=ǫµνλρSρ\nwhereSρis the pseudo-vector part of torsion.1The other term that we consider is the 2-\nform tensor TµνwhereTµν=∂µSν−∂νSµthis term is analog to the field strength of the\nelectromagnetic gauge potential Aµthat in our case is changed to pseudo-vector Sµ. The\ninvariant fermionic action that contained these contribut ions for torsion is given by\nS=/integraldisplay\nd4xi¯ψ(γµDµ+λTµνΣµν+m)ψ, (5)\nwhere the covariant derivative is Dµ=Dµ−iηωab\nµΣabthat contain the covariant gauge\nderivativeDµ=∂µ−ieAµand the spin connection covariant derivative.\nWe consider the flat space-time where the only contribution f or the spin connection is\nthe contortion. In this form we have a spin current given by\nJµ\nab=1\n2¯ψγµΣabψ. (6)\nWe consider the ansatz where ωab\nµcontain the total antisymmetric part of the contorsion\ngiven by\nωκλ\nµ=µǫκλρ\nµSρ. (7)\nWe used the splitting γµΣκλ=ǫµ\nκλργργ5+δµ\nκγλ−δµ\nλγκThat give us the current in the\nform\nJµ\nαλ=δS\nδΓabµ=1\n2ǫµ\nαλρ¯ψγργ5ψ=Ji\njk+J0\nij (8)\n1Considering space times with torsion Tα\nβγ, the afine connection is not symmetric, Tα\nβγ= Γα\nβγ−Γα\nγβ, and\nwe can split it into three irreducible components, where one of them is the pseudo-trace Sκ=1\n6ǫαβγκTαβγ.\n5where the currents are\nJi\njk=1\n2ǫi\njk¯ψγ0γ5ψ; (9)\nJ0\nij=1\n2ǫijk¯ψγkγ5ψ, (10)\nthen the current part of the action coming from the covariant derivative is given by\nScurr=−η/integraldisplay\nd4xγµγ5Sµ=η/integraldisplay\nd4x/vectorS·/vectorJ. (11)\nThe action (5) considers the temporal component of the torsi on pseudo-vector S0= 0,\nwe have can be written as\nS=/integraldisplay\nd4xi¯ψ/parenleftBig\nγµ∂µ+igγµAµ+iηγµγ5Sµ+λTµνΣµν+m/parenrightBig\nψ, (12)\nOur sort of gravity background does not exhibit metric fluctu ations. The space-time is\ntaken to be flat, and we propose a scenario such that the type of gravitational background\nis parametrized by the torsion pseudo-trace Sµ, whose origin may be traced back to one\ngeometrical defect.\n2.2 Dirac equation in presence of torsion\nNow, we discuss the Dirac equation given by eq. (12). From the action above, taking the\nvariation with respect to ( δS/δ¯ψ), a modified Dirac’s equation reads as below:\n[iγµ∂µ−ηγµγ5Sµ−eγµAµ+λΣµν∂µSν+m]ψ= 0. (13)\nFor a vanishing λ−parameter, the equation (13) has been carefully studied in [ 21, 22, 23,\n24, 26].\nThe generation, manipulation, and detection of a spin curre nt, as well as the flow of\nelectron spins, are the main challenges in the field of spintr onics, which involves the study\nof active control and manipulation of the spin degree of free dom in solid-state systems,\n[27, 10, 28]. A spin current interacts with magnetization by exchanging the spin-angular\nmomentum, enabling the direct manipulation of magnetizati on without using magnetic\nfields [29, 30]. The interaction between spin currents and ma gnetization provides also\na method for spin current generation from magnetization pre cession, which is the spin\npumping [31, 32]. We showed in last Sections that there are tw o type of deformations\ninduced by the torsion. Both of these can be generating a spin current.\nAfter a suitable separation of components ( µ= 0,1,2,3), the equation of motion can\nbe written as,\ni∂tψ=iαi∂iψ+ηγ5S0ψ−ηαiγ5Siψ+eA0ψ+\n−eαiAiψ−iλ\n4ǫijkβγ5αk∂iSjψ+βmψ. (14)\n6Definingagauge-invariant momentum, πj=i∂j−eAj, andusingthatΣ ij=−i\n4ǫijkγ5αk\nwithγ0=β, the effective Hamiltonian takes the form,\nH=αkπk+ηγ5S0−ηαkγ5Sk+eA0+\n−iλ\n4ǫijkβγ5αk∂iSj+βm. (15)\nwhere we have the matricial definitions:\nαi=/parenleftbigg0σi\nσi0/parenrightbigg\n,γ5=/parenleftbigg0 1\n1 0/parenrightbigg\n,β=/parenleftbigg1 0\n0−1/parenrightbigg\n, (16)\nIn the Heisenberg picture, the position, /vector x, and momentum, /vector π, operators obey two\ndifferent kinds of relations; we consider the torsion as a func tion of position only, S=S(/vector x),\nso that\n˙/vector x=/vector α\n˙/vector π=e(/vector α×/vectorB)+e/vectorE+ηγ5∂\n∂x(/vector α·/vectorS)ˆx. (17)\nOne reproduces the usual relation for ˙/vector x, while the equation for ˙/vector πpresents a new term\napparently giving some tiny correction to the Lorentz force .\nHowever, if we consider the torsion in a broader context, now as a momentum- and\nposition-dependent background field, S=S(/vector x,/vectork), we have to deal with the following\npicture,\n˙/vector x=/vector α−ηγ5∂\n∂k(/vector α·/vectorS)ˆk\n˙/vector π=e(/vector α×/vectorB)+e/vectorE+ηγ5∂\n∂x(/vector α·/vectorS)ˆx+\n+eηγ5∂A\n∂x∂\n∂k(/vector α·/vectorS)ˆk. (18)\nThe two sets of dynamical equations above are clearly showin g us the small corrections\ninducedbythetorsionterm. Nevertheless, wearestill here intherelativistic domain, andit\nis necessary change this framework for a better understandi ng of the SHE phenomenology.\nFor this reason, in next Section we are going to approach the s ystem by going over into its\nnon-relativistic regimen\n2.3 Non-Relativistic Approach with torsion\nIn this sub-section, we consider the Dirac equation in its no n-relativistic limit. One im-\nportant requirement for the Dirac equation is that it reprod uces what we know from non-\nrelativistic quantum mechanics. We can show that, in the non -relativistic limit, two com-\nponents of the Dirac spinor are large and two are quite small. To make contact with the\n7non-relativistic description , we go back to the equations w ritten in terms of ϕandχof\nthe four component spinor ψ=eimt√\n2m/parenleftbiggϕ\nχ/parenrightbigg\n, just prior to the introduction of the γma-\ntrices. we obtain two equation on of these for ϕand the other for χ. We can solved in\nχand substiuted in the Dirac equation given by (12) and take th e non-relativistic regi-\nmen (|/vector p|<< m). So, in this physical landscape, from now and hereafter, ou r goal is to\nconsider a low-relativistic approximation based on an exte nded Pauli equation version by\nincluding torsion as presented before. Employing the Hamil tonian (15), we could carry\nout our calculations in the framework of the Fouldy-Wouthuy sen transformations; however\nfor the sake of our approximation at lowest order in v/c, we take that SHE is adequately\nwell described by the low-relativistic Pauli equation. We a re considering that the electron\nvelocities are in the range of Fermi’s velocity. In this case , we arrive at the version given\nbelow for the Pauli’s equation:\ni∂ϕ\n∂t=/bracketleftBig(/vector p−e/vectorA)2\n2m−e\n2m(/vector σeff·/vectorB)+eA0−(σ·/vectorSeff)+\n−λ\n8m(/vector∇×/vectorS)·(/vector σ×/vector p)+iλ\n8m(/vector∇×/vectorS)·/vector p+\n−eλ\n8m(/vector∇×/vectorS)·(/vector σ×/vectorA)+ieλ\n8m(/vector∇×/vectorS)·/vectorA/bracketrightBig\nϕ. (19)\nThe equation above displays the usual Pauli terms, but corre cted by new terms due to\nthe torsion coupling. The second and the fourth contributio ns in the RHS of eq.(19) can\nbe thought of as effective terms for /vector σand/vectorS, respectively given by\nσeff=/vector σ+η\n2m/vectorS+iη\n2m(/vector σ×/vectorS) (20)\n/vectorSeff=η/vectorS+iλ\n4(/vector∇×/vectorS). (21)\nThe fifth contribution in the RHS is proportional to the Rashb a SO coupling term; this\nterm yields an important effect on the behavior of spin.\n3 From the Modified Pauli Equation to Unfold in LLG\nIn this Section, we consider the magnetization equation der ivation given by Dirac non-\nrelativistic limit take into account the presence of torsio n. Let us start by considering the\nmodified Pauli equation eq.(19) and find the magnetization eq uation. By using the Landau\ngauge/vectorA=H/vector xand taking that /vector x·/vector σ= 0 (the spins are aligned orthogonally to the plane\nof motion), give us the Hamiltonian of the full system in the n on-relativistic limit as:\n8H=(/vector p−e/vectorA)2\n2m−e\n2m(/vector σeff·/vectorB)+eA0−(σ·/vectorSeff)−λ\n8m(/vector∇×/vectorS)·(/vector σ×/vector p)+iλ\n8m(/vector∇×/vectorS)·/vector p,\n(22)\nwith/vectorB(t) =µ0/vectorH(t), whereµ0is the gyromagnetic ratio. The Pauli equation associated\nwith (22) reads as follows below\ni∂ϕ\n∂t=/bracketleftBig(/vector p−eA)2\n2m−e\n2m(/vector σeff·/vectorH)+eA0−(/vector σ·/vectorSeff)+\n−λ\n8m(/vector∇×/vectorS)·(/vector σ×/vector p))+iλ\n8m(/vector∇×/vectorS)·p/bracketrightBig\nϕ. (23)\nLet us consider the magnetization vector equation related w in the spin magnetic mo-\nment/vector µ=e\n2m/vector σ. In our approach we consider the magnetization is defined by /vectorM=\n(/vector µϕ)†ϕ−ϕ†(/vector µϕ) whereϕgiven by Pauli equation (23) and ϕ†ϕ= 1 and/vectorˆSϕ=/vectorSϕ,\nwith the notation/vectorˆSis a torsion operator and /vectorSis the torsion autovalue. We have, by\nthe manipulation of Pauli equation the magnetization equat ion associated with a fermionic\nstate when we applied a external magnetic field /vectorHconsidering the Pauli product algebra\nas1\n2(σiσj−σjσi) =iǫijkσkand1\n2(σiσj+σjσi) =δij. The magnetization equation that\narrive that is\n∂/vectorM\n∂t=/vectorM×/vectorH+η/vectorM×/vectorS+β(/vectorM×/vectorL) +\n+ηe\n2m2(/vectorS×/vectorH) +eλ\n2m(/vector∇×/vectorS), (24)\nwith the magnetic moment given by /vectorL=/vector r×/vector pand/vectorSas the torsion pseudo-vector. We can\nobserved that there are two terms that arrived by the covaria nt derivative Dµdefined by\nthe coupling constant ηand the other is the parameter that arrived by non-minimal sp in\ntorsion coupling with the coupling constant λ. Where the effect of the new terms given\nwhenη/ne}ationslash= 0 andλ/ne}ationslash= 0.\nWe consider the scalar product of the magnetization /vectorM, the magnetic field /vectorHand the\ntorsion pseudo-vector /vectorSwith the equation (28) and we obtain2\n∂t/bracketleftBig1\n2(/vectorM·/vectorM)/bracketrightBig\n=ηe\n2m2/bracketleftBig\n/vectorM·(/vectorS×/vectorH)+λm\nη/vectorM·(/vector∇×/vectorS)/bracketrightBig\n; (25)\n2We used the vectorial relating given by A·(B×C) =B·(C×A) =C·(A×B) the other is\nA×(B×C) = (A·C)B−(A·B)Cand∇·(A×B) =B·(∇×A)−A·(∇×B).\n9∂t/bracketleftBig1\n2(/vectorH·/vectorM)/bracketrightBig\n=/bracketleftBig\nη/vectorM·(/vectorS×/vectorH)+β/vectorM·(/vectorL×/vectorH) +\n+e\n2mλ/vectorH·(/vector∇×/vectorS)/bracketrightBig\n; (26)\n∂t/bracketleftBig1\n2(/vectorS·/vectorM)/bracketrightBig\n=/vectorM·(/vectorS×/vectorH)+β/vectorM·(/vectorL×/vectorH). (27)\n∂t/bracketleftBig1\n2(/vectorL·/vectorM)/bracketrightBig\n=/vectorM·(/vectorL×/vectorH)+η/vectorM·(/vectorL×/vectorS) +\n+ηe\n2m2/vectorS·(/vectorL×/vectorH)+eλ\n2m/vectorL·(/vector∇×/vectorS),. (28)\nWith the equations dysplayed in (25)-(27), it is possible to inspect the general behavior of\nthe magnitude of the magnetization, /vectorM, that precesses around the magnetic field, /vectorH. In\nour framework, the magnetization also precesses around the torsion vector /vectorM·/vectorS. Without\ntorsion, we have∂/vectorM\n∂t=/vectorM×/vectorH, so that/vectorM·/vectorM= constant and /vectorM·/vectorH= constant as in the\nusual case of the electron under the action of a time-depende nt external magnetic field,\nwith the Zeeman term given by the Hamiltonian HM=/vectorM·/vectorH.\n3.1 Planar torsion analysis with damping\nHere, we intend to analyze some possibilities of solutions t o the magnetization that respect\nthe conditions given by (25)-( 27). The magnitude of the magn etization is not constant in\ngeneral, as we can see in equation (25), but, if this quantity is constant, there comes out a\nconstraint given by\n/vectorM·(/vectorS×/vectorH) =−λm\nη/vectorM·(/vector∇×/vectorS). (29)\nIf we considerd(/vectorL·/vectorM)\ndt= 0, we have\n/vectorS·(/vectorL×/vectorH) =−mλ\nη/vectorL·(/vector∇×/vectorS). (30)\nThis expression describes us the case where /vectorM·/vectorH/ne}ationslash= 0 and/vectorS·/vectorM/ne}ationslash= 0; then, there is a the\ndamping angle in both directions given by the precession aro und the magnetic field /vectorHand\naround the torsion pseudo-vector /vectorS:\n∂t/bracketleftBig1\n2(/vectorH·/vectorM)/bracketrightBig\n=λm(e\n2m2/vectorH−/vectorM)·(/vector∇×/vectorS); (31)\n∂t/bracketleftBig1\n2(/vectorS·/vectorM)/bracketrightBig\n=−λm\nη/vectorM·(/vector∇×/vectorS). (32)\n10Let us consider the first proposal in a very particular and ver y simple case for a planar\ntorsion field, /vectorS=1\n2χ(xˆy−yˆx); this choice allows us to realize the curl of torsion as an\neffective magnetic field, /vector∇×/vectorS=/vectorBeff=χˆz.\nIf we pick up the configuration of Fig. 1, we find the relation of the angle between the\nmagnetic field /vectorHand the magnetization /vectorM.\nFigure 1: Magnetization vector rotating around the magneti c field/vectorHwith damping given\nby the dynamics of the angle ζ. The system {M,H}rotates around the vector /vectorSin the xy-\nplane also with damping given by the angle φ. We consider φ=ωφtwithωφ=ωθ=2λmχ\nηS;\nwith this configuration ζ=ωζt=λχm\nηHM/parenleftBig\nH+ηM/parenrightBig\nt.\nWe started off by discussing the case where the torsion is plan ar with the magnitude\nof the magnetization being constant, /vectorM·/vectorM= 0, and\n∂t/bracketleftBig1\n2(/vectorS·/vectorM)/bracketrightBig\n=−λχm\nη/vectorM·/vector z,. (33)\nthis gives us the magnetic momentum precession around the to rsion. For consistency, we\nshow that this result is compatible with the equation\n∂t/bracketleftBig1\n2(/vectorH·/vectorM)/bracketrightBig\n=λmχ(e\n2m2/vectorH−/vectorM)·/vector z. (34)\nIn the case of the Fig. 1, the magnetization precesses around the magnetic field and around\nthe planar torsion vector both with damping.\n113.2 Helix-Damping Sharped Effect in a Planar Torsion Configur ation\nNow, let us consider the most general case, where the magnitu de of the magnetization is\nnot constant, but with ( /vectorL·/vectorM) = 0. The configuration is considered in Fig. 2. With the\nFigure 2: In this picture we show the effect of the torsion in mag netization dynamics. The\ngreen vector is the magnetization vector and the blue vector is the external magnetic field.\nexpressions (25)-(27), we can readily write the magnitude o f the magnetization\n∂t/bracketleftBig1\n2(/vectorM·/vectorM)/bracketrightBig\n=eη\n2m2/bracketleftBig\n∂t/bracketleftBig1\n2(/vectorS·/vectorM)/bracketrightBig\n+λm\nη/vectorM·(/vector∇×/vectorS)/bracketrightBig\n. (35)\nBy using of the equation (35) and considering ∂t/bracketleftBig\n1\n2(/vectorS·/vectorM)/bracketrightBig\n/ne}ationslash= 0, we can see, the example\nof the Fig. 3, that ∂t/bracketleftBig\n1\n2(/vectorM·/vectorM)/bracketrightBig\n/ne}ationslash= 0. This possibility gives us that the magnitude of\nmagnetization is not constant, as in the usual LLG. This effect is the effect of torsion that\ngives us that the rotational lines do not return around thems elves.\nEquation (35 ) does not involve the explicit dependence of th e magnetic field. We\nchoose to work out the equation\n∂t/bracketleftBig1\n2(/vectorHeff·/vectorM)/bracketrightBig\n=e\n2mλχ/vectorH·/vector z, (36)\nwhere/vectorHeff=/vectorH−η/vectorSgives us the explicit form of the magnetic field interaction w ith the\n12Figure 3: In this draw we show the effect of the torsion in magnet ization dynamics. The\ngreen vector is the magnetization vector and the blue vector is the external magnetic field.\nIn this representation we used |/vectorM|=M(t),|/vectorS|= constant and |/vectorH|= constant with\nωθ=2mλχ\nηSthenM(t) =λeχ\nωθmsinωθt.\nmagnetization.\nWe notice that this quantity is different from zero, then the an gle between the magnetic\nfield and the magnetization is not constant; this yields us th e damping precessing effect of\nthe magnetization vector around the magnetic field. The comp osition between these two\neffects, dislocation and damping, is what we refer to as the hel ix-sharped with damping,\neffect where the damping effect can be see in Fig. 4. We can show tha t there are two\nmagnetization effects: the damping given by the longitudinal magnetization function m(t)l\nand dislocations given by the longitudinal magnetization f unctionm(t)tas we can see in\nFig. 2. The trajectory of the magnetization is the conical in creasing spiral, where the\nmodulus of magnetization increases with the time. In the tor sion plan the behavior is\ngiven by Fig3.\n13Figure 4: Damping behavior in torsion plane. Show the behavi or of theφ=ωφtdynamic.\n4 Concluding Remarks\nIn this work, we have considered that the magnetization equa tion is a non-relativistic\nremnant of the non-relativistic limit of the Dirac equation with torsion couplings. We\nhave considered two types of couplings: one of these related with the spin current in Dirac\nequation, defined by the spin connection. When we derived the action in relation with\nthe spin connection we obtain the spin current, this descrip tion is analog to the charged\ncurrent when we have the derivation of the action in relation with the gauge field.\nWerefertotheothertermasthenon-minimaltorsiontermand Itgives ustherotational\nof the torsion. We have analyzed this term in the general cont ext and observed that it is\npossible to recover the Landau Lifshitz in the case were the t orsion is zero. Then, we can\npoint out that the non-relativistic limit of the Dirac equat ion reproduces the usual case\nwhere the magnetization vector precesses around the magnet ic field. When we introduce\nthe torsion terms we analyze, in the general regime the magni tude of magnetization /vectorM·/vectorM,\nthe precession of the magnetization around the magnetic fiel d/vectorH·/vectorM, and the precession\nof the magnetization around the torsion pseudo-vector /vectorS·/vectorMis not constant.\nWhen the magnitude of the magnetization is constant, in the c ase where the torsion is\nplanar, there are two possible magnetization precessions o ne around the magnetic field and\nother around the planar torsion pseudo-vector. In both dyna mics, there occurs damping.\nAn interesting example has been analyzed in Fig. 1, where we s how that it is possible\nto realized an apparatus in some experimental device. In thi s sense, our framework can\nreproduce the LLG equation. The most general approach shoul d consider that the mag-\nnitude of the magnetization is not constant. In this case, as we can see from Fig. 2, the\nloop drawn by the magnetization damping but the is not remain in the same plane. This\n14effect is typically a torsion effect, were the lines are not close d. This effect seems to be\nlike a dislocation in the material that presents topologica l defects like solitons and vortices.\nBoth dislocation and damping give us what we refer to as the he lix-damping sharp effect,\nwich is a new feature of the models with torsion[33].\nWe can observethat this resultis thenewfeatureintroduced bytheplanartorsion, ifwe\nconsider the comparation with dampingand dislocations ter ms presented in LLG equation.\nThis may help in the task of setting up new apparatuses and may be experimental purposes\nto explore such characteristics in this phenomenon. We have found that /vectorM·/vectorM= constant,\nits consequence is the dislocation effect. The damping effect is the usual one, where the\nangle dynamic can crease and decrease with the time. In this w ork we does not study the\npolarization of the spins that is subject of next work when we will consider these systems in\nterms or the spinup and spin down dynamic. In the literature, this effect is named pumped\nspin current [34], and we shall study the possibility of this current when the system is in\na helix-sharped configuration[35].\nReferences\n[1] A. Dyrdal, J. Barnas, Phys. Rev. B 92, 165404 (2015)\n[2] Z. Wang, C. Tang, R. Sachs, Y. Barlas, and J. Shi, Phys. Rev . Lett.114, 016603\n(2015).\n[3] K. M. McCreary, A. G. Swartz, W. Han, J. Fabian, R.K. Kawak ami Phys. Rev. Lett.\n109, 186604 (2012)\n[4] I. A. Zhuravlev, V. P. Antropov andK. D. Belashchenko, Ph ys.Rev. Lett. 115, 217201\n(2015).\n[5] V. Flovik, F. Maci, J. M. Hernndez, R. Brucas, M. Hanson an d E. Wahlstrm, Phys.\nRev. B92, 104406 (2015).\n[6] I. Turek, J. Kudrnovsky and V. Drchal, Physical Review B 92, 214407 (2015).\n[7] L.D. Landau, E.M. 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Phys. 76, 323 (2004).\n[28] Maekawa S., Concepts in Spin Electronics (Oxford: Oxfo rd University Press), (2006).\n[29] Grollier J, Cros V, Hamzic A, George J M, Jaffr‘es H, Fert A, Faini G, Youssef J B\nand Legall H., Appl.Phys. Lett. 78, 3663 (2001).\n[30] Ando K, Takahashi S, Harii K, Sasage K, Ieda J, Maekawa S. and Saitoh E., Phys.\nRev. Lett. 101, 036601 (2008).\n16[31] Tserkovnyak Y, Brataas A. and Bauer G. E. W. , Phys. Rev. L ett.88, 117601 (2002).\n[32] Mizukami S, Ando Y and Miyazaki T., Phys. Rev. B 66104413 (2002).\n[33] S. Azevedo, J. Phys. A 34, 6081 (2001).\n[34] P. A. Andreev, Phys. Rev. E 91, 033111 (2015)\n[35] TaikiYoda, TakehitoYokoyama, ShuichiMurakami, Scie ntificReports5, 12024(2015).\n17"    },    {        "title": "1001.2845v1.Resonance_Damping_in_Ferromagnets_and_Ferroelectrics.pdf",        "content": "arXiv:1001.2845v1  [cond-mat.other]  16 Jan 2010Resonance Damping in Ferromagnets and Ferroelectrics\nA. Widom\nPhysics Department, Northeastern University, Boston, MA U SA\nS. Sivasubramanian\nNSF Nanoscale Science & Engineering Center for High-rate Na nomanufacturing,\nNortheastern University, Boston MA USA\nC. Vittoria and S. Yoon\nDepartment of Electrical and Computer Engineering, Northe astern University, Boston, MA USA\nY.N. Srivastava\nPhysics Department and & INFN, University of Perugia, Perug ia IT\nThe phenomenological equations of motion for the relaxatio n of ordered phases of magnetized\nand polarized crystal phases can be developed in close analo gy with one another. For the case of\nmagnetized systems, thedrivingmagnetic fieldintensityto ward relaxation was developedbyGilbert.\nFor the case of polarized systems, the driving electric field intensity toward relaxation was developed\nby Khalatnikov. The transport times for relaxation into the rmal equilibrium can be attributed to\nviscous sound wave damping via magnetostriction for the mag netic case and electrostriction for the\npolarization case.\nPACS numbers: 76.50.+g, 75.30.Sg\nI. INTRODUCTION\nIt has long been of interest to understand the close\nanalogiesbetween orderedelectric polarized systems, e.g.\nferroelectricity , and ordered magnetic systems, e.g. fer-\nromagnetism . Atthe microscopiclevel, the sourceofsuch\nordering must depend on the nature of the electronic en-\nergy spectra. The relaxation mechanism into thermal\nequilibrium state must be described by local electric field\nfluctuationsforthe electricpolarizationcaseandbymag-\nnetic intensity fluctuations for the magnetization case;\nSpecifically, the field fluctuations for each case\nGpol\nij(r,r′,t) =1\n¯h/integraldisplayβ\n0/an}bracketle{t∆Ej(r′,−iλ)∆Ei(r,t)/an}bracketri}htdλ,\nGmag\nij(r,r′,t) =1\n¯h/integraldisplayβ\n0/an}bracketle{t∆Hj(r′,−iλ)∆Hi(r,t)/an}bracketri}htdλ,\nwherein β=¯h\nkBT,(1)\ndetermine the relaxation time tensor for both cases via\nthe fluctuation-dissipation formula1–4\nτij=/integraldisplay∞\n0lim\nV→∞/bracketleftbigg1\nV/integraldisplay\nV/integraldisplay\nVGij(r,r′,t)d3rd3r′/bracketrightbigg\ndt.(2)\nWe have unified the theories of relaxation in ordered po-\nlarized systems and ordered magnetized systems via the\nKubo transport time tensor in Eqs.(1) and (2).\nThe transport describing the relaxation of or-\ndered magnetization is the Landau-Lifshitz-Gilbert\nequation5–7. This equation has been of considerable\nrecent interest8–10in describing ordered magnetic reso-\nnancephenomena11–14. The equationdescribingthe elec-tric relaxation of an ordered polarization is the Landau-\nKhalatinikov-Tani equation15–17. This equation can be\nsimplymodeled18–21witheffectiveelectricalcircuits22–25.\nInformation memory applications26–29of such polarized\nsystem are of considerable recent interest30–32.\nThe unification of the magnetic Gilbert-Landau-\nLifshitz equations and the electric Landau-Khalatnikov-\nTani equations via the relaxation time tensor depends\non the notion of a nonequilibrium driving field . For the\nmagnetic case, the driving magnetic intensity Hddeter-\nmines the relaxation of the magnetization via the torque\nequation\n˙M=γM×Hd, (3)\nwhereinγisthegyromagneticratio. Fortheelectriccase,\nthe driving electric field Eddetermines the relaxation of\nthe polarization via the equation of motion for an ion of\nchargeze\nm¨r=zeEd. (4)\nThe unification of both forms of relaxation lies in the\nclose analogy between the magnetic driving intensity Hd\nand the electric driving field Ed.\nIn Sec.II the thermodynamics of ordered magnetized\nand polarized systems is reviewed. The notions of mag-\nnetostrictionand electrostrictionaregiven aprecise ther-\nmodynamic definition. In Sec.III, the phenomenology of\nthe relaxation equations are presented. The magnetic\ndriving intensity Hdand the electric driving field Edare\ndefined in terms of the relaxation time tensor Eq.(2). In\nSec.IV, we introduce the crystal viscosity tensor. From a\nKubo formula viewpoint, the stress fluctuation correlax-2\nation\nFijkl(r,r′,t) =1\n¯h/integraldisplayβ\n0/an}bracketle{t∆σkl(r′,−iλ)∆σij(r,t)/an}bracketri}htdλ,(5)\ndetermines the crystal viscosity\nηijkl=/integraldisplay∞\n0lim\nV→∞/bracketleftbigg1\nV/integraldisplay\nV/integraldisplay\nVFijkl(r,r′,t)d3rd3r′/bracketrightbigg\ndt.(6)\nFor models of magnetic relaxation wherein acoustic heat-\ning dominates via magnetostriction33and for models of\nelectric relaxation wherein acoustic heating dominates\nvia electrostriction, the relaxation time tensor in Eq.(2)\ncan be related to the viscosity tensor Eq.(6). An inde-\npendent microscopic derivation of viscosity induced re-\nlaxationisgiveninAppendixA. IntheconcludingSec.V,\nthe sound wave absorption physics of the viscous damp-\ning mechanism will be noted.\nII. THERMODYNAMICS\nOur purpose is to review the thermodynamic proper-\nties of both magnetically ordered crystals and polariza-\ntion ordered crystals. The former is characterized by\na remnant magnetization Mfor vanishing applied mag-\nnetic intensity H→0 while the latter is characterized by\na remnant polarization Pfor vanishing applied electric\nfieldE→0.\nA. Magnetically Ordered Crystals\nLetwbe the enthalpy per unit volume. The funda-\nmental thermodynamic law determining the equations of\nstate for magnetically ordered crystals is given by\ndw=Tds+H·dM−e:dσ, (7)\nwhereinsistheentropyperunitvolume, Tisthetemper-\nature,eis the crystal strain and σis the crystal stress.\nThe magnetic adiabatic susceptibility is defined by\nχ=/parenleftbigg∂M\n∂H/parenrightbigg\ns,σ. (8)\nIf\nN=M\nM⇒N·N= 1 (9)\ndenotes a unit vector in the direction of the magnetiza-\ntion, then the tensor Λ ijkldescribing adiabatic magne-\ntostriction coefficients may be defined as34\n2ΛijklNl=M/parenleftbigg∂eij\n∂Mk/parenrightbigg\ns,σ=−M/parenleftbigg∂Hk\n∂σij/parenrightbigg\ns,M.(10)When the system is out of thermal equilibrium, the driv-\ning magnetic intensity is\nHd=H−/parenleftbigg∂w\n∂M/parenrightbigg\ns,σ−τ·/parenleftbigg∂M\n∂t/parenrightbigg\n,(11)\nwherein τare the relaxation time tensor transport co-\nefficients which determine the relaxation of the ordered\nmagnetic system into a state of thermal equilibrium.\nB. Ordered Polarized Crystals\nThe fundamental thermodynamic law determining the\nequations of state for ordered polarized crystals is given\nby\ndw=Tds+E·dP−e:dσ, (12)\nwherein wis the enthalpy per unit volume, sis the en-\ntropy per unit volume, Tis the temperature, eis the\ncrystal strain and σis the crystal stress. The electric\nadiabatic susceptibility is defined by\nχ=/parenleftbigg∂P\n∂E/parenrightbigg\ns,σ. (13)\nThe tensor βijkdescribing adiabatic electrostriction co-\nefficients may be defined as34\nβijk=/parenleftbigg∂eij\n∂Pk/parenrightbigg\ns,σ=−/parenleftbigg∂Ek\n∂σij/parenrightbigg\ns,P.(14)\nThe piezoelectric tensor is closely related to the elec-\ntrostriction tensor via\nγijk=/parenleftbigg∂eij\n∂Ek/parenrightbigg\ns,σ=/parenleftbigg∂Pk\n∂σij/parenrightbigg\ns,E=βijmχmk.(15)\nWhen the system is out of thermal equilibrium, the driv-\ning electric field is\nEd=E−/parenleftbigg∂w\n∂P/parenrightbigg\ns,σ−τ·/parenleftbigg∂P\n∂t/parenrightbigg\n,(16)\nwherein τis the relaxation time tensor transport coef-\nficients which determine the relaxation of the ordered\npolarized system into a state of thermal equilibrium.\nIII. RESONANCE DYNAMICS\nHere we shall show how the magnetic intensity Hd\ndrives the magnetic resonance equations of motion in\nmagnetically ordered systems. Similarly, we shall show\nhowtheelectricfield Eddrivesthe polarizationresonance\nequations of motion for polarized ordered systems.3\nA. Gilbert-Landau-Lifshitz Equations\nThe driving magnetic intensity determines the torque\non the magnetic moments according to\n∂M\n∂t=γM×Hd. (17)\nEmploying Eqs.(11) and (17), one finds the equations for\nmagnetic resonance in the Gilbert form\n∂M\n∂t=γM×/bracketleftBigg\nH−/parenleftbigg∂w\n∂M/parenrightbigg\ns,σ−/parenleftbiggα\nγM/parenrightbigg\n·∂M\n∂t/bracketrightBigg\n,(18)\nwherein the Gilbert dimensionless damping tensor αis\ndefined as\nα= (γM)τ. (19)\nOnemaydirectlysolvetheGilbert equationsforthe driv-\ning magnetic intensity according to\nHd+α·/parenleftbig\nN×Hd/parenrightbig\n=H−/parenleftbigg∂w\n∂M/parenrightbigg\ns,σ.(20)\nEqs.(17) and (20) expressthe magneticresonancemotion\nin the Landau-Lifshitz form.\nB. Landau-Khalatnikov-Tani Equations\nThe driving electric field gives rise to a polarization\nresponse according to\n∂2P\n∂t2=/parenleftBigg\nω2\np\n4π/parenrightBigg\nEd, (21)\nwhereinωpis the plasma frequency. A simple derivation\nof Eq.(21) may be formulated as follows. In a large vol-\numeV, the polarization due to charges {zje}is given\nby\nP=/parenleftbigg/summationtext\njzjerj\nV/parenrightbigg\n. (22)\nIf the drivingelectric field acceleratesthe chargesaccord-\ning to\nmj¨rj=zjeEd, (23)\nthen Eq.(21) holds true with the plasma frequency\nω2\np= 4πe2lim\nV→∞/bracketleftBigg/summationtext\nj(z2\nj/mj)\nV/bracketrightBigg\n= 4πe2/summationdisplay\nanaz2\na\nma,(24)\nwhereinnais the density of charged particles of type a.\nThe polarization resonance equation of motion follows\nfrom Eqs.(16) and (21) as17\n/parenleftbigg4π\nω2p/parenrightbigg∂2P\n∂t2+τ·∂P\n∂t+∂w(P,s,σ)\n∂P=E.(25)The electric field Einduces the polarization Pat reso-\nnant frequencies which are eigenvalues of the tensor Ω\nfor which\nΩ2=ω2\npχ−1\n4π≡ω2\np(ǫ−1)−1. (26)\nThedecayratesforthepolarizationoscillationsareeigen-\nvalues of the tensor Γfor which\nΓ=ω2\npτ\n4π. (27)\nIfthedecayratesarelargeonthescaleofthetheresonant\nfrequencies, then the equation of motion is over damped\nso that\nmin\njΓj≫max\niΩiimplies\nτ·∂P\n∂t+∂w(P,s,σ)\n∂P=E. (28)\nEq.(28) represents the Landau-Khalatnikov equation for\npolarized systems.\nIV. HEATING RATE PER UNIT VOLUME\nLet us here consider the heating rate implicit in relax-\nation processes. Independently of the details of the mi-\ncroscopic mechanism for generating such heat, the rates\nof energy dissipation are entirely determined byτ. Ex-\nplicitly, the heating rates per unit volume for magnetiza-\ntion and polarization are given, respectively, by\n˙qM=∂M\n∂t·τ·∂M\n∂t, (29)\nand\n˙qP=∂P\n∂t·τ·∂P\n∂t. (30)\nFinally, the notion of crystal viscosity ηijklis introduced\ninto elasticity theory35via the heating rate per unit vol-\nume from rates of change in the strain ∂e/∂t; It is\n˙qe=∂eij\n∂tηijkl∂ekl\n∂t. (31)\nCrystal viscosity is employed to describe, among other\nthings, sound wave attenuation. Our purpose is to de-\nscribe how heating rates in Eqs.(29) and (30) can be re-\nlated to the heating rate in Eq.((31)). This allows us to\nexpressthetransportcoefficients τintermsofthecrystal\nviscosity.\nA. Relaxation via Magnetostriction\nFrom the magnetostriction Eq.(10), it follows that\nmagnetic relaxation gives rise to a strain\n∂eij\n∂t=2\nMΛijklNk∂Ml\n∂t, (32)4\nand thereby to the heating rate,\n˙q=4\nM2∂Mi\n∂t(ΛmnqiNq)ηmnrs(ΛrskjNk)∂Mj\n∂t,(33)\nin virtue of Eq.(31). Employing Eqs.(29) and (33), we\nfind that the magnetic relaxation transport coefficient in\nthe magnetostriction model\nτij=4\nM2(ΛmnqiNq)ηmnrs(ΛrskjNk).(34)\nThe Gilbert damping tensor follows from Eqs.(19) and\n(34) as\nαij=4γ\nM(ΛmnqiNq)ηmnrs(ΛrskjNk).(35)\nThe central relaxation tensor Eq.(35) describes the mag-\nnetic relaxation in terms of the magnetostriction coeffi-\ncients and the crystal viscosity.\nB. Relaxation via Electrostriction\nFromtheelectrostrictionEq.(14), it followsthatatime\nvarying polarization gives rise to a time varying strain\n∂eij\n∂t=βijk∂Pk\n∂t, (36)\nand thereby to the heating rate,\n˙q=∂Pi\n∂tβkliηklmnβmnj∂Pj\n∂t, (37)\nin virtue of Eq.(31). Employing Eqs.(30) and (37), we\nfind that the electric relaxation transport coefficient in\nthe electroostriction model\nτij=βkliηklmnβmnj. (38)\nThecentralrelaxationtensorEq.(38) describesthe polar-\nization relaxation time tensor coefficients in terms of the\nelectrostriction coefficients and the crystal viscosity. The\nimplications ofthe electrostrictionmodel forthe Landau-\nKhalatnikov equation is to the authors knowledge a new\nresult.\nV. CONCLUSIONS\nFor ordered polarized and magnetized systems, we\nhave developed phenomenological equations of motion inclose analogywith one another. For the magnetized case,\nthe relaxation is driven by the magnetic intensity Hd\nyielding the Gilbert equation of motion7. For the polar-\nized case, the relaxation is driven by the electric field Ed\nyielding the Tani equation of motion17. In both cases,\nthe relaxation time tensor τis determined by the crystal\nviscosity as derived in the Appendix A; i.e. in Eqs.(A3)\nand (A6). The viscosity can be measured independently\nfrom the magnetic or electrical relaxation by employing\nsound absorption techniques36.\nAppendix A: Kubo formulae\nFrom the thermodynamic Eq.(10), the fluctuations in\nthe magnetic intensity are given by magnetostriction, i.e.\n∆Hk(r,t) =−/parenleftbigg2ΛijklNl\nM/parenrightbigg\n∆σij(r,t).(A1)\nEqs.(A1), (1) and (5) imply\nGmag\nij(r,r′,t) =\n4\nM2(ΛmnqiNq)Fmnrs(r,r′,t)(ΛrskjNk).(A2)\nEmploying Eqs.(A2), (2) and (6), one finds the central\nresult for the magnetic relaxation time tensor; It is\nτmag\nij=4\nM2(ΛmnqiNq)ηmnrs(ΛrskjNk) =αij\nγM.(A3)\nFrom the thermodynamic Eq.(14), the fluctuations in\nthe electric intensity are given by electrostriction, i.e.\n∆Ek(r,t) =−βijk∆σij(r,t). (A4)\nEqs.(A4), (1) and (5) imply\nGpol\nij(r,r′,t) =βkliFklmn(r,r′,t)βmnj.(A5)\nEmploying Eqs.(A5), (2) and (6), one finds the central\nresult for the electric relaxation time tensor; It is\nτpol\nij=βkliηklmnβmnj. (A6)\n1S. Machlup and L. Onsager, Phys. Rev. 91, 1505 (1953).\n2S. Machlup and L. Onsager, Phys. Rev. 91, 1512 (1953).\n3H.B. Callen, Fluctuation, Relaxation, and Resonance inMagnetic Systems , Editor D. ter Haar, p 176, Oliver &\nBoyd Ltd., London (1962).\n4R. Kubo, M. Toda and N. Hashitsume, Statistical Physics5\nII, Nonequilibrium Statistical Mechanics Springer, Berlin\n(1998).\n5L. Landau and L. Lifshitz, Phys. Zeit. Sowjetunion 8,153\n(1935).\n6T. L. Gilbert, Armor Research Foundation Rep. No. 11\nChicago, IL. (1955).\n7T. L. Gilbert, IEEE Trans. Mag. 40, 3443 (2004).\n8M. Fahnle, D. Steiauf, and J. Seib, J. Phys. D: Appl. Phys.\n41, 164014 (2008).\n9E. Rossi, O.G. Heinonen, andA.H. MacDonald, Phys. Rev.\nB 72, 174412 (2005).\n10V. Kambersky, Phys. Rev. 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Bhatia, Ultrasonic Absorption , Oxford University\nPress, Oxford (1967)."    },    {        "title": "1111.6272v1.A_two_stage_approach_to_relaxation_in_billiard_systems_of_locally_confined_hard_spheres.pdf",        "content": "A two-stage approach to relaxation in billiard systems of locally\ncon\fned hard spheres\nPierre Gaspard1,a)and Thomas Gilbert1,b)\nCenter for Nonlinear Phenomena and Complex Systems,\nUniversit\u0013 e Libre de Bruxelles, Code Postal 231, Campus Plaine, B-1050 Brussels,\nBelgium\nWe consider the three-dimensional dynamics of systems of many interacting hard spheres, each individually\ncon\fned to a dispersive environment, and show that the macroscopic limit of such systems is characterized\nby a coe\u000ecient of heat conduction whose value reduces to a dimensional formula in the limit of vanishingly\nsmall rate of interaction. It is argued that this limit arises from an e\u000bective loss of memory. Similarities with\nthe di\u000busion of a tagged particle in binary mixtures are emphasized.\nPACS numbers: 05.20.Dd,05.45.-a,05.60.-k,05.70.Ln\nThe derivation of the macroscopic transport\nequations of hydrodynamics and the computa-\ntion of the associated coe\u000ecients for systems de-\nscribed at the microscopic level by Hamilton's\nequations of classical mechanics is a central prob-\nlem of non-equilibrium statistical physics. The\nperiodic Lorentz gas provides an example where\nthis program can be achieved and Fick's law of\ndi\u000busion established1{3. Furthermore, by tweak-\ning the system's geometry so that tracer particles\nhop from cell to cell at nearly vanishing rates,\nmemory e\u000bects disappear and the dynamics of\ntracers is well approximated by a continuous time\nrandom walk4. In this regime, the di\u000busion coef-\n\fcient takes on a simple limiting value, given by\na dimensional formula5, by which we mean that\nits expression reduces to the square of the length\nscale of the cell separation multiplied by the hop-\nping rate. Similarly, it was found that heat trans-\nport in systems of con\fned hard disks with rare\ninteractions reduces to a stochastic process of\nenergy exchanges which obeys Fourier's law of\nheat conduction, with the coe\u000ecient of heat con-\nductivity also given by a dimensional formula6{8,\nwhere the timescale is that of energy exchanges\namong particles in neighboring cells. Here we ex-\ntend these \fndings to mechanical systems of con-\n\fned hard spheres, whose stochastic limit was al-\nready studied elsewhere9, and review in some de-\ntails the analogy with the problem of mass trans-\nport in the periodic Lorentz gas.\nI. INTRODUCTION\nA long-standing problem in non-equilibrium statisti-\ncal mechanics has been to provide a derivation from\na)Electronic mail: gaspard@ulb.ac.be\nb)Electronic mail: thomas.gilbert@ulb.ac.be\frst principles of Fourier's law of heat conduction in\ninsulating materials in the framework of Hamiltonian\nmechanics10. A centerpiece of the puzzle is to identify\nthe conditions under which scale separation occurs so the\ntime evolution at the microscopic level can somehow be\nreduced to a hydrodynamic equation at the macroscopic\nlevel, which is an especially challenging problem for in-\nteracting particle systems.\nAs emphasized in earlier papers6{8, the following two-\nstep program, which consists of (i) identifying an in-\ntermediate level of description|a mesoscopic scale|\nwhere the Newtonian dynamics can be consistently ap-\nproximated by a set of stochastic equations, and (ii)\nsubsequently analyzing the statistical properties of this\nstochastic system and computing its transport properties\nin the hydrodynamic scaling limit, can be successfully\nachieved in a class of chaotic billiard systems composed\nof many hard disks trapped in a semi-porous material\nwhich prevents mass transfer and yet allows energy trans-\nfer through elastic collisions among neighboring disks.\nUnder the assumption that collisions among moving\ndisks are rare compared to wall collision events, the global\nmulti-particle probability distribution of the system typ-\nically reaches local equilibrium at the kinetic energy of\neach individual particle before energy exchanges proceed.\nThis mechanism naturally yields a stochastic description\nfor the process of energy exchanges in the system. In a\nsubsequent paper9, it was shown that the same reduction\napplies to systems of con\fned hard spheres.\nOur purpose in this paper is twofold. Our main objec-\ntive is to establish a comparison between the process of\nenergy transfer in models of interacting hard spheres with\nlocal con\fnement rules, on the one hand, and the di\u000bu-\nsive motion of tracer particles in low-dimensional billiard\ntables such as the \fnite-horizon periodic Lorentz gas, on\nthe other. Speci\fcally, we show that the two systems, un-\nder equivalent assumptions of local equilibration, whose\naccuracy can be precisely controlled by tuning the sys-\ntems' parameters down to a critical geometry, are both\namenable to stochastic descriptions in the form of mas-\nter equations, whether for the distribution of energies in\nthe case of the former systems, or that of mass in thearXiv:1111.6272v1  [cond-mat.stat-mech]  27 Nov 20112\nlatter. Taking the hydrodynamic limit of these stochas-\ntic systems, we obtain explicit values of the transport\ncoe\u000ecients of heat conduction and di\u000busion respectively,\nwhich turn out to share the remarkable property that\nthey are given by simple dimensional formulae, i. e. by\nthe square of the mesoscopic length scale multiplied re-\nspectively by the rates of energy or mass transfer.\nTurning back to the billiard dynamics, we address our\nsecond objective, which is to show that, under the as-\nsumption of separation of two characteristic timescales,\none associated with the local dynamics, the other with\nenergy transfers, the process of heat transport in three-\ndimensional billiard systems of con\fned hard spheres is\nwell-approximated by the corresponding stochastic pro-\ncess of energy exchanges. We do so by considering the\nHelfand moment of thermal conductivity11and compute\nthe linear divergence in time of its mean squared change.\nPlotting our results as functions of the system's sizes for\ndi\u000berent parameter values, we compute the in\fnite-size\nextrapolations to obtain the heat conductivity and ob-\nserve a very good convergence to the value obtained for\nthe stochastic systems for parameter values where an ef-\nfective separation of timescales is observed.\nThe paper is organized as follows. The problem of mass\ntransport in a spatially periodic billiard table is consid-\nered in Sec. II. Starting from the pseudo-Liouville equa-\ntion for the billiard dynamics, we derive a continuous-\ntime random walk which describes the stochastic jumps\nof particles across a lattice, and analyze its transport\nproperties, comparing it to that of the billiard. The\nsame procedure is applied in Sec. III to two-dimensional\nbilliards of con\fned hard disks. In Sec. IV, we turn to\na three-dimensional billiard system and discuss our nu-\nmerical results. Conclusions are drawn in Sec. V.\nII. A CONTINUOUS-TIME RANDOM WALK\nAPPROACH TO MASS TRANSPORT\nWe consider for the sake of the example a periodic ar-\nray of two-dimensional semi-dispersing Sinai billiard ta-\nbles in the form of square billiard cells bounded by \rat\nwalls of sizes l, and connected to each other by small\nopenings of relative widths \u000e. All the cells are identical\nand contain circular obstacles arranged so that there are\nno periodic trajectory that avoid these obstacles. Inde-\npendent pointwise tracer particles move across this array,\nperforming elastic collisions on the obstacles and walls.\nAn example is shown in \fgure 1. The di\u000busive properties\nof this model were studied in some details in Ref. [12].\nThere the attention was on memory e\u000bects and their im-\npact on the value of the di\u000busion coe\u000ecient. Here we\nwill focus on the derivation of the kinetic prediction of\nthis coe\u000ecient, which yields the aforementioned dimen-\nsional formula, disregarding the memory e\u000bects. The\nconclusions are similar to those obtained by Machta and\nZwanzig5in the framework of the periodic Lorentz gas.\nThe continuous-time approach we present below is some-\nFIG. 1. Periodic billiard table on a square lattice with a\ntypical trajectory. Here the central disks in the initial and\n\fnal cells are color-\flled. One considers the process of mass\ntransport across the periodic cells. The model has several\nparameters in terms of which a dimensional formula of the\ndi\u000busion coe\u000ecient is computed.\nwhat similar to that of Zwanzig4.\nThe phase-space con\fguration of this system refers to\na single tracer particle and is speci\fed by the triplet\nfn;r;vg, where n= (nx;ny)2Z2denotes the lattice\nindex of the cell where the tracer is located, r, its posi-\ntion within the cell and vits velocity.\nLetp(n;r;v;t) denote the probability distribution of\nthe tracer. Its time evolution is determined by the\npseudo-Liouville operator13, which comprises four di\u000ber-\nent types of contributions, corresponding to as many dif-\nferent types of events:\n1. Free advection of the tracer inside the cell is ac-\ncounted for by the term \u0000v\u0001@r;\n2. Collisions of the tracer with any of the circular ob-\nstacles inside the cell are determined by the opera-\ntorsK(d), wheredrefers to a speci\fc disk of radius\n\u001adat position qd:\nK(d)p(n;r;v;t) =\u001adZ\n^e\u0001v>0d^e(^e\u0001v)\n\u0002h\n\u000e(r\u0000qd\u0000\u001a^e)p(n;r;v\u00002^e(^e\u0001v);t)\n\u0000\u000e(r\u0000qd+\u001a^e)p(n;r;v;t)i\n; (1)\n3. Collisions of the tracer with any of the \rat walls\nof cell n, with operator W(j), wherejtakes on\nthe valuesj= 1;:::; 4, corresponding to the right,\nbottom, left and top walls: letting rxandvxdenote\nthe position and velocity components of the tracer\nalong the horizontal axis, wall #1 is the right wall3\nat positionl=2 so that, if the wall is continuous with\nrespect to the vertical axis, i. e. \u000e\u00110, the collision\noperator acts according to\nW(1)p(n;r;v;t) =jvxj\u000e(rx\u0000l=2) (2)\n\u0002h\n\u0012(\u0000vx)p(n;r;\u0000vx;vy;t)\u0000\u0012(vx)p(n;r;v;t)i\n;\nwith similar expressions for W(2),W(3), andW(4);\n4. Jump events between neighboring cells take place\nwhen a wall collision event occurs at a position\nwhere the wall is actually open, which amounts\nto subtracting the action of the wall operator (2)\nabove, keeping track of the cell indices: using the\nsame set of four indices as above for every wall and\nassuming that wall #1 has a slit of width \u000ecentered\nabouty= 0, the jump operator J(1)\n\u000eis\nJ(1)\n\u000ep(n;r;v;t) =jvxj\u000e(rx\u0000l=2)\u0002\u000e(ry)\u0012(\u0000vx) (3)\n\u0002h\np(n+ (1;0);r;v;t)\u0000p(n;r;\u0000vx;vy;t)i\n;\nwhere \u0002\u000e(x) = 1 ifjxj\u0014\u000e=2 and 0 otherwise.\nCollecting these terms together, we obtain the time evo-\nlution ofp(n;r;v;t), given by the pseudo-Liouville equa-\ntion:\n@tp(n;r;v;t) = (4)n\n\u0000v\u0001@r+P\ndK(d)+P\njW(j)o\np(n;r;v;t)\n+P\njJ(j)\n\u000ep(n;r;v;t):\nThe terms on the right-hand side of this equation are\ngrouped so as to distinguish the local terms, which do\nnot act on the cell index in the distribution, from the\nnon-local ones, i. e. the jump terms, which act on the\ncell index. Also note that the velocity amplitude is in-\nvariant under every term of this equation and is therefore\na conserved quantity which plays no role once it has been\n\fxed by the initial condition.\nIt it an immediate consequence of the ergodicity of the\nlocal dynamics that the following distribution,\nP(leq)(n;t) =Z\ndrZ\ndv\u000e(v2\u0000v2\n0)p(n;r;v;t);(5)\nis invariant under the local terms of the pseudo-Liouville\nequation (4). We thus refer to P(leq)(n;t) as the lo-\ncal equilibrium distribution . Its time evolution occurs\nthrough jump events only :\n@tP(leq)(n;t) =\nP\njR\ndrR\ndv\u000e(v2\u0000v2\n0)J(j)\n\u000ep(n;r;v;t): (6)\nIn order to close this equation for P(leq)(n;t), we make\nthelocal equilibrium approximation ,\np(n;r;v;t) =1\n\u0019A\u000e(v2\u0000v2\n0)P(leq)(n;t); (7)whereAis the area of the billiard, e. g. A=l2\u0000\u0019(\u001a2\n1+\u001a2\n2)\nin the case of \fgure 1 (where \u001a1denotes the radius of the\ndisk at the center of the cell and \u001a2that of the disks at\nthe cell corners).\nPlugging Eq. (7) into (6), the pseudo-Liouville equa-\ntion (4) reduces to the following continuous-time random\nwalk\n@tP(leq)(n;t) =X\njv0\u000e\n\u0019A[P(leq)(n+ej;t)\u0000P(leq)(n;t)];\n(8)\nwhere ej2f(\u00061;0);(0;\u00061)g.\nIn the continuum scaling limit, ~ r=ln,l!0,v0\u0018l\u00002,\nthis reduces to a di\u000busion equation,\n@tP(~ r;t) =Dr2P(~ r;t); (9)\nwith di\u000busion coe\u000ecient\nD=l2\u0017(J); (10)\nhere written in terms of the frequency \u0017(J)of a jump\nevent in a speci\fc direction, identical for all four direc-\ntions,\n\u0017(J)=v0\u000e\n\u0019A: (11)\nEquation (10) is what we refer to as a dimensional\nformula : the di\u000busion coe\u000ecient associated with the\ncontinuous-time random walk process (8) is the product\nbetween the scale of displacements squared and hopping\nrate. The fact that it is an exact property of the stochas-\ntic process is remarkable.\nAs far as the transport properties of the billiard table\ngo, Eq. (10) is a mere approximation: its validity relies\non the local equilibrium approximation (7). In contrast,\nEq. (11) is an exact result for the billiard dynamics as\nwell|valid for all parameter values|as it relies solely on\nthe ergodicity of the billiard table14.\nThe closure approximation (7) is expected to be exact\nin the limit of vanishing window widths, \u000e!0, whereby\nthe hopping rate (11) diverges and the di\u000busion coe\u000e-\ncient (10) vanishes. In practice however, we may expect\np(n;r;v;t) to converge to the local equilibrium distribu-\ntionP(leq)(n;t) between successive jumps so long as the\ntypical number of disk collision events within a cell is\nlarge. This can be veri\fed numerically, as shown on the\ntop panel of \fgure 2. These numerical results were al-\nready reported in Ref. [12]. Similar results are equally\nobtainable for three dimensional billiard systems such as\nthe three-dimensional periodic Lorentz gas15.\nIII. HEAT TRANSPORT IN TWO-DIMENSIONAL\nCONFINING BILLIARDS\nSimilar considerations apply to models of heat trans-\nport with mass con\fnement6. Classes of such mod-\nels were initially introduced by Bunimovich et. al in4\nææææææææææææææææææææææææææææææææææ\n0.00.10.20.30.40.50.60.70.30.40.50.60.70.80.91.0\ndDl-2nHJL-1\n0.00.10.20.30.40.50.60.70.00.10.20.30.40.50.6\ndnHJL-1\nFIG. 2. (Top) Di\u000busion coe\u000ecient as measured from the\nmean-squared displacement divided by the number of jumps\nfor the Sinai billiard table shown in \fgure 1. (Bottom) Corre-\nsponding decay time. The dots show numerical measurements\nand the red line is the formula (11).\nRef. [16]. There, the authors proved the ergodicity of\ntwo-dimensional billiard tables consisting of an arbitrary\nlarge number of unit cells placed side by side, each con-\ntaining a single disk trapped within the cell's boundaries,\nbut in such a way that collisions may still take place be-\ntween disks belonging to neighboring cells.\nFigure 3 shows an example of such a system17. In\nRef. [7], we introduce the following parametrization of\nthe model in terms of two parameters: \u001a, which char-\nacterizes the timescale of wall collision events, and \u001am,\nwhich characterizes that of binary collisions.\nOn the one hand, the dynamics within an isolated cell\nof widthlboils down to the motion of a point particle\nin an area bounded by the exterior intersection of four\ndisks of radius \u001a,l=2\u0014\u001a < l=p\n2. In the absence of\ninteraction with neighboring particles, the mean free path\nof a particle is given by `=\u0019jB\u001aj=j@B\u001aj, whereB\u001aand\n@B\u001adenote respectively the area and perimeter of the\nbilliard cell. The corresponding wall collision timescale\nis obtained by multiplying the mean free path by the\nspeed of the particle.\nInteractions among neighboring particles on the other\nhand take place provided the radius of the moving par-\nticles\u001amis larger than a critical value, \u001am> \u001a c= p\n\u001a2\u0000l2=4. The corresponding timescale can be com-\nFIG. 3. Example of a two-dimensional billiard table consist-\ning of many moving hard disks (colored), each trapped within\ncon\fning walls (black disks), here arranged in a square tiling.\nThe centers of the moving disks are bound to the areas delim-\nited by the exterior intersections of the black circles around\nthe \fxed disks (the solid broken lines sample their trajecto-\nries). The parameters (radii of the black circles \u001aand moving\ndisks\u001am) are so chosen that (i) black circles overlap, so the\nmoving disks are con\fned, and (ii) collisions are allowed (such\nas in the upper right corner), whereby energy exchanges take\nplace. The mobile disks are color coded from blue to red with\ngrowing kinetic energies.\nputed and shown to diverge with ( \u001am\u0000\u001ac)\u00003. Letting\n\u0017(W)(T) and\u0017(B)(T) respectively denote the frequencies\nof wall collision and binary collision events measured at\nequilibrium temperature T, the following separation of\ntimescales is assumed,\n\u0017(B)(T)\u001c\u0017(W)(T); (12)\nwhich occurs when \u001am!\u001ac.\nUnder this assumption, we showed6,8that the heat\nconductivity of spatially extended billiard systems, which\nare in\fnite size limits of systems such as depicted in\nFig. 3, reduces, up to a dimensional factor l2, to the\nfrequency of binary collisions,\n\u0014(B)(T) =l2\u0017(B)(T): (13)\nFurthermore the heat conductivity scales with the ther-\nmal speed, \u0014(B)(T)\u0018p\nT.\nAlthough Eqs. (10) and (13) are very much alike, the\nderivation of the latter is much more involved than that\nof the former.\nProceeding as in Sec. II, the \frst part of the pro-\ngram, which is to reduce the pseudo-Liouville equation\ndescribing the time evolution of the billiard system to a5\ncontinuous-time stochastic process of energy exchanges,\nfollows closely along lines which led from Eqs. (4) to\n(8). Indeed, considering the time evolution acting on the\nN-particle phase-space distribution pN(fri;vig;t), one\nreadily notices that the local equilibrium distribution\nP(leq)\nN(\u000f1;:::;\u000fN;t)\u0011 (14)\nZNY\na=1dradva\u000e(\u000fa\u0000mv2\na=2)pN(fri;vig;t)\nis left unchanged by the advection and wall collisionterms. Therefore, and provided binary collision events\nare rare with respect to wall collision events, we can make\na closure approximation similar to Eq. (7) to obtain the\ntime evolution of the local equilibrium distribution in the\nform of a stochastic process.\nThe result is a master equation which describes the\ntime evolution of the Ncell system with energy variables\nf\u000f1;:::;\u000fNgin terms of energy exchanges between neigh-\nboring cells at respective energies \u000faand\u000fbof amount\u0011,\nspeci\fed by a stochastic kernel W:\n@tP(leq)\nN(\u000f1;:::;\u000fN;t) =1\n2NX\na;b=1Z\nd\u0011h\nW(\u000fa+\u0011;\u000fb\u0000\u0011j\u000fa;\u000fb)P(leq)\nN(:::;\u000fa+\u0011;:::;\u000fb\u0000\u0011;:::;t )\n\u0000W(\u000fa;\u000fbj\u000fa\u0000\u0011;\u000fb+\u0011)P(leq)\nN(:::;\u000fa;:::;\u000fb;:::;t )i\n; (15)\nwhere the expression of Wcan be obtained by direct com-\nputation of the collision integrals, which, after rescaling\nthe time variable to the units of the frequency of binary\ncollision events and thus absorbing all the parameters\ninto the time scale, yields the universal function8:\nW(\u000fa;\u000fbj\u000fa\u0000\u0011;\u000fb+\u0011) =r\n2\n\u00193\u00028\n>>>>>><\n>>>>>>:q\n1\n\u000faK\u0010\n\u000fb+\u0011\n\u000fa\u0011\nq\n1\n\u000fb+\u0011K\u0010\n\u000fa\n\u000fb+\u0011\u0011\nq\n1\n\u000fa\u0000\u0011K\u0010\n\u000fb\n\u000fa\u0000\u0011\u0011\nq\n1\n\u000fbK\u0010\n\u000fa\u0000\u0011\n\u000fb\u0011;\n(16)\nwhose de\ftion intervals correspond respectively to \u0000\u000fb<\n\u0011<\u0000max(\u000fb\u0000\u000fa;0),\u0000max(\u000fb\u0000\u000fa;0)<\u0011< 0, 0<\u0011<\nmax(\u000fa\u0000\u000fb;0), and max( \u000fa\u0000\u000fb;0)<\u0011<\u000fa. HereK(m)\ndenotes the complete elliptic intergal of the \frst kind18.\nA system of Nisolated cells whose time evolution is\nspeci\fed by the master equation (15) reaches a micro-\ncanonical equilibrium state whose total energy can be\nparametrized in terms of the temperature according to\n\u000f1+:::+\u000fN=NT. The corresponding energy exchange\nfrequency is\n\u0017N(T) =p\nTh\n1 +O(1=N)i\n; (17)\nwhose in\fnite-size limit is simply \u0017(T) =\nlimN!1\u0017N(T) =p\nT(in the chosen units of time).\nAn expression of the heat conductivity of the system\ndescribed by Eq. (15) is obtained by considering the\nEinstein-type relation satis\fed by the variance of the as-\nsociated Helfand moment, which measures the spread of\nenergy as a function of time.\nFor the system of Nenergy cells aligned along a one-\ndimensional ring, the Helfand moment is de\fned accord-ing to\nHN(t) =NX\ni=1i\u000fi(t); (18)\nwhere\u000fi(t) is the state of the energy at site iat timet.\nThis quantity evolves in time by discrete steps, when en-\nergy exchanges occur. Let f\u001cngn2Ndenote the sequence\nof times at which successive energy exchanges take place.\nAssuming cells iandi+ 1 exchange some amount of en-\nergy at time \u001cn, we can write the corresponding change\nin the Helfand moment as \u000fi(\u001cn\u00000)\u0000\u000fi(\u001cn+ 0).\nComputing the mean squared change in the Helfand\nmoment as a function of time, we obtain an expression\nof the thermal conductivity according to\n\u0014(T) = lim\nN!1\u0014N(T); (19)\nwhere we de\fned the \fnite Nconductivity to be\n\u0014N(T) =1\nN(kBT)2lim\nn!11\n2\u001cnD\n[HN(\u001cn)\u0000HN(\u001c0)]2E\n:\n(20)\nIn Ref. [8], it was argued that the in\fnite Nlimit of\nthis quantity is determined by the static correlations only,\nyielding the result\n\u0014N(T) =l2p\nTh\n1 +O(1=N)i\n: (21)\nComparing Eqs. (17) and (21), we obtain the an-\nnounced result\n\u0014(T) =l2\u0017(T); (22)\nwhich is an exact result for the stochastic system evolved\nby the master equation (15) and does not make explicit6\nuse of the form (16), except for some symmetries9. The\ncorresponding result (13) for the billiard dynamics is ob-\ntained by plugging back the proper timescale of binary\ncollisions and letting \u001am!\u001acso that the separation of\ntimescales (12) is e\u000bective.\nUnlike mass transport in Sinai billiard tables for which\nthe transport equation (9) follows directly from the\ncontinuous-time random walk (8), so that there only re-\nmains the problem of comparing the di\u000busion coe\u000ecient\nof the billiard to the dimension formula (10) in the ap-\npropriate parameter regime, the problem of computing\nthe heat conductivity associated with heat transport in\nbilliard systems of many con\fned particles proceeds in\ntwo separate steps.\nHaving carried out the reduction of the billiard's\npseudo-Liouville equation to a master equation for a\nstochastic energy exchange system, the \frst step is to es-\ntablish the cancellation of dynamical correlations in such\na system, which yields the identity lim N!1\u0014N=\u0017N=l2.\nThis is however a delicate result19and requires in-depth\nknowledge of the spectral properties of the master equa-\ntion (15). The approach we took in Ref. [8], though\nequivalent, uses di\u000berent techniques, based on analyz-\ning the \frst few orders of the gradient expansion of the\nkinetic equation to obtain the expression of the heat cur-\nrent in terms of the local temperature gradient in a non-\nequilibrium stationary state, i. e. Fourier's law.\nThe second step of the program is to go back to the\nbilliard dynamics and investigate the convergence of the\nheat conductivity to the binary collision frequency in the\nlimit\u001am!\u001ac.\nFor two-dimensional billiard systems such as the ones\nshown in Fig. 3, the agreement was found to be rather\nsatisfactory7. For the three-dimensional billiard systems\nwe turn to now, this agreement is even better.\nIV. THREE-DIMENSIONAL BILLIARD FOR HEAT\nTRANSPORT\nConsider a three-dimensional billiard of con\fned hard\nspheres such as shown in Fig. 4. The reduction of the\npseudo-Liouville equation governing the phase-space evo-\nlution of probability densities pN(fri;vig;t) to a master\nequation similar to Eq. (15) was already carried out in\nRef. [9].\nThe corresponding kernel, in the appropriate time\nunits, is found to have the universal form\nW(\u000fa;\u000fbj\u000fa\u0000\u0011;\u000fb+\u0011) =r\u0019\n8\u00028\n>>><\n>>>:q\n\u000fb+\u0011\n\u000fa\u000fb\n1p\nmax(\u000fa;\u000fb)q\n\u000fa\u0000\u0011\n\u000fa\u000fb;(23)\nwhose de\fnition intervals correspond respectively to\n\u0000\u000fb< \u0011 <\u0000max(\u000fb\u0000\u000fa;0),\u0000max(\u000fb\u0000\u000fa;0)< \u0011 <\nmax(\u000fa\u0000\u000fb;0), and max( \u000fa\u0000\u000fb;0)<\u0011<\u000fa.\nFIG. 4. (Top) Hard-sphere particle trapped in a cuboid cell\nwith cylindrical edges. (Bottom) A system made out of many\ncopies of such cells which form a spatially periodic structure.\nThe cells are semi-porous in the sense that particles are pre-\nvented from escaping, and can yet partially penetrate into\nthe neighboring cells, thus allowing energy transfer through\ncollisions among neighboring particles. As with the two-\ndimensional case, the likelihood of binary collision events can\nbe controlled by tuning the geometry of the cell.\nExtensive numerical investigations of the master equa-\ntion (15) with the stochastic kernel (23) were presented\nin Ref. [9], supporting with high precision the validity of\nEq. (22). Here we focus on the billiard dynamics and\nconsider the mean squared change in time of the Helfand\nmoment associated with the distribution of energy in bil-\nliard systems formed by one-dimensional lattices of bil-\nliard cells such as shown in Fig. 4.\nLetf\u001cngn2Zdenote the times at successive binary col-7\nlision events. As we now need to account for the motion\nof particles within their respective cells, there are two\ntypes of contributions to changes in the Helfand moment\nbetween two successive binary collision events. The lead-\ning contribution arises from the energy exchanges which\ntake place at binary collision events. Thus, when a binary\ncollision occurs between particles jandk, the Helfand\nmoment changes by the amount [ xj(\u001cn)\u0000xk(\u001cn)][\u000fj(\u001cn+\n0)\u0000\u000fj(\u001cn\u00000)], wherexj(\u001cn) andxk(\u001cn) denote the po-\nsitions of particles jandkalong the direction of the\nlattice spatial extension at time \u001cn. The other contri-\nbution to the Helfand moment arises from the advec-\ntion of particles within their respective cells according toP\na[xa(\u001cn)\u0000xa(\u001cn\u00001)]\u000fa(\u001cn\u00001).\nThe range of allowed parameter values is identical to\nthe two-dimensional case: l=2\u0014\u001a < l=p\n2, and\u001ac<\n\u001am<\u001a, where\u001ac\u0011p\n\u001a2\u0000l2=4.\nTakingl= 1, we \fx \u001a= 0:50 so\u001ac= 0 and use the six\nparameter values \u001am= 0:20;0:25;:::; 0:45. For each one\nof them, we \fx the total energy to be E= 3N=2 (T= 1)\nand vary the system size from N= 3 and up to N= 100\ncells. We typically run 103trajectories, each for a dura-\ntion of 103\u0002Nunits of the average time between collision\nevents. We compute: (i) the Helfand moment versus time\nand take the mean and standard deviation of the trans-\nposition of Eq. (20) to obtain the conductivities \u0014(B)\nN; (ii)\nthe binary and wall collision frequencies \u0017(B)\nNand\u0017(W)\nN\nby directly averaging their numbers with respect to time.\nWe then use linear \fts in 1 =NforN\u001510 of\u0014(B)\nNand\n\u0017(B)\nNto obtain the extrapolation \u0017(B)\n1= limN!1\u0017(B)\nNand\n\u0014(B)\n1= limN!1\u0014(B)\nN. Here and below, explicit tempera-\nture dependencies are dropped.\nN=25\n0.200.250.300.350.400.450.0010.010.11\nrm-rcnNHBLnNHWL\nFIG. 5. Ratio between the binary and wall collision frequen-\ncies as a function of the parameter \u001amforN= 25 (\u001a= 0:50,\n\u001ac= 0).\nMeasurements of the ratio between \u0017(B)\nNand\u0017(W)\nN, by\nwhich we can assess the e\u000bectiveness of the separation\nof timescales (12), are displayed in Fig. 5 for N= 25\n(other system sizes yield similar values). The results of\nthe computations of \u0014(B)\n1=\u0017(B)\n1for the di\u000berent parame-\nter values\u001amare displayed in Fig. 6 and are found to be\nin very good agreement (within two digits) of the dimen-\n0.200.250.300.350.400.450.00.51.01.52.0\nrm-rck¥HBLn¥HBL-1FIG. 6. Ratio between the heat conductivity of the in\fnite\nlength system and the corresponding collision frequency be-\ntween neighboring particles as a function of the parameter \u001am\n(\u001a= 0:50 and\u001ac= 0). The error bars show the widths of the\ncomputed 0.95 con\fdence intervals.\nsional formula (13) for values of \u001amas large as 0 :30. In\nFig. 7, the details of the \ftting procedure used to obtain\nthe values of \u0014(B)\n1and\u0017(B)\n1are shown for the di\u000berent\nparameter values as functions of N.\nV. CONCLUDING REMARKS\nBy controlling the rate at which a tracer hops from\ncell to cell in a Sinai billiard table or that of interaction\namong neighboring disks or spheres in a high-dimensional\nbilliard with local con\fnement rules, one identi\fes a limit\nof vanishing rate where the complicated phase-space dy-\nnamics are replaced by stochastic processes which ac-\ncount for the transport of mass in the \frst case, or (ki-\nnetic) energy in the second. Furthermore, the trans-\nport coe\u000ecients of these stochastic processes have the\nsame simple dimensional expression, given by the length\nscale of transfers squared multiplied by the corresponding\nrates.\nAs emphasized in this paper, the dimensional formulae\nwe obtained for the transport coe\u000ecients of the models\nof mass and heat transport we have considered are but\ntwo faces of the same coin. Indeed, in both cases the ac-\ncuracy of our approximation of the transport coe\u000ecients\nof the billiards in terms of hopping or collision rates relies\non the e\u000ecient separation of two timescales, namely the\ntimescale of local collision events must be much shorter\nthan that characterizing the transfer of mass or energy.\nIn other words, the relaxation to local equilibrium pre-\ncedes mass or energy transfers.\nWe can in fact view the notion of relaxation to local\nequilibrium as a low-dimensional transposition of that of\nlocal thermal equilibrium, which is at the heart of many\ntheories involving hydrodynamic scaling limits and typi-\ncally assumes a large number of degrees of freedom10. In\nour billiards, the relaxation to local equilibrium occurs8\nrm=0.45\nk¥HBLn¥HBL=1.957\n0204060801001.01.21.41.61.82.0\nNkNHBLn¥HBL,nNHBLn¥HBL\nrm=0.4\nk¥HBLn¥HBL=1.335\n0204060801000.91.01.11.21.3\nNkNHBLn¥HBL,nNHBLn¥HBL\nrm=0.35\nk¥HBLn¥HBL=1.066\n0204060801000.800.850.900.951.001.051.10\nNkNHBLn¥HBL,nNHBLn¥HBL\nrm=0.3\nk¥HBLn¥HBL=0.986\n0204060801000.750.800.850.900.951.00\nNkNHBLn¥HBL,nNHBLn¥HBL\nrm=0.25\nk¥HBLn¥HBL=0.987\n0204060801000.700.750.800.850.900.951.00\nNkNHBLn¥HBL,nNHBLn¥HBL\nrm=0.2\nk¥HBLn¥HBL=0.992\n0204060801000.750.800.850.900.951.00\nNkNHBLn¥HBL,nNHBLn¥HBL\nFIG. 7. Heat conductivities \u0014(B)\nNand binary collision frequencies \u0017(B)\nNmeasured as functions of the system size Nfor di\u000berent\nvalues of the parameter \u001am(\u001a= 0:50). The solid curves correspond to the linear \fts of \u0014(B)\nNand\u0017(B)\nNas functions of 1 =N. The\nintercepts yield the in\fnite size estimates \u0014(B)\n1and\u0017(B)\n1.\non the constant energy surface of a single particle. But\none could instead consider systems of trapped gases, in\nwhich case the relaxation to local equilibrium would in-\nvolve transfers of energy among the particles in the same\ntrap. In such a case, provided relaxation to local equi-\nlibrium takes place on timescales much shorter than that\nof energy transfer between neighboring traps, a similar\ndimensional expression of the heat conductivity of the\ncorresponding stochastic process in terms of the prod-\nuct of length scale squared and rate of energy transfer\nwould yield an accurate approximation of the transport\ncoe\u000ecient of the billiard system.\nWe end with a remark concerning the closure (7) which\nrelies on the ergodicity of the local dynamics on the sur-face of constant energy. As noted already by Zwanzig4,\nSinai billiards cannot be replaced by polygonal ones, for\nwhich the notion of ergodicity is weaker since the ve-\nlocity directions take on values in a discrete set. In\nenergy exchange processes, however, ergodicity of the\nmany-particle billiard may be restored through the sole\ninteraction among neighboring particles. As shown in\nRef. [20], an elastic string-type interaction between par-\nticles trapped in polygonal boxes provides a simple model\nof a system which, on the one hand, cannot be accurately\ndescribed by a master equation similar to Eq. (15), even\nwhen interactions are rare, but on the other hand, has\na well-de\fned heat conductivity which is well approxi-\nmated by a dimensional formula. An understanding of9\nthe transport properties of this system beyond the Boltz-\nmann hypothesis remains to be elucidated.\nACKNOWLEDGMENTS\nWe dedicate this paper to the memory of Sasha Losku-\ntov in appreciation for his co-organizing the conference\nBilliards'2011 in Ubatuba, SP, Brazil. TG also wishes\nto thank E. Leonel for his warm hospitality. The au-\nthors would further like to thank F. Barra, L. Buni-\nmovich, R. Lefevere, M. Lenci, C. Liverani, V. Rom-\nKedar, D. P. Sanders and D. Sz\u0013 asz for stimulating discus-\nsions which took place at di\u000berent stages of this project.\nThey acknowledge \fnancial support by the Belgian Fed-\neral Government under the Interuniversity Attraction\nPole project NOSY P06/02 and FRS-FNRS under con-\ntract C-Net NR/FVH 972. TG is \fnancially supported\nby the Fonds de la Recherche Scienti\fque FRS-FNRS\nand receives additional support through FRFC conven-\ntion 2,4592.11.\n1L. A. Bunimovich and Y. G. 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Gilbert, \\On the derivation of Fourier's law\nin stochastic energy exchange systems,\" J. Stat. Mech. P11021\n(2008).\n9P. Gaspard and T. Gilbert, \\Heat transport in stochastic energy\nexchange models of locally con\fned hard spheres,\" J. Stat. Mech.\n, P08020 (2009).\n10F. Bonetto, J. L. Lebowitz, and L. Rey-Bellet, \\Fourier's law:\na challenge for theorists,\" Mathematical Physics 2000, 2052, A.\nFokas, A. Grigoryan, T. Kibble, and B. Zegarlinski eds. (Imperial\nCollege, London, 2000)..\n11E. Helfand, \\Transport coe\u000ecients from dissipation in a canoni-\ncal ensemble,\" Phys. Rev. 119, 1{9 (1960).\n12T. Gilbert and D. P. Sanders, \\Persistence e\u000bects in deterministic\ndi\u000busion,\" Phys. Rev. E 80, 41121 (2009).\n13J. R. Dorfman and M. H. Ernst, \\Hard-sphere binary-collision\noperators,\" J. Stat. Phys. 57, 581{593 (1989).\n14N. Chernov and R. Markarian, \\Introduction to the ergodic the-\nory of chaotic billiards, 2nd edition\" Instituto de Matem\u0013 atica\nPura e Aplicada, Rio de Janeiro (2003).\n15T. Gilbert, H. C. Nguyen, and D. P. Sanders, \\Di\u000busive prop-\nerties of persistent walks on cubic lattices with application to\nperiodic lorentz gases,\" J. Phys. A 44, 065001 (2011).\n16L. A. Bunimovich, C. Liverani, A. Pellegrinotti, and Y. M.\nSuhov, \\Ergodic systems of n balls in a billiard table,\" Commun.\nMath. Phys. 146, 357 (1992).\n17For technical reasons pertaining to the probability of direct re-\ncollisions between two particles, this example actually falls out\nof the class of systems considered in Ref. [16]. This is however\nunimportant for our own considerations.\n18M. Abramowitz and I. A. Stegun, \\Handbook of mathematical\nfunctions : with formulas, graphs, and mathematical tables,\"\n(Dover, New York, 1972).\n19A. Grigo, K. Khanin, and D. Sz\u0013 asz, \\Mixing rates of par-\nticle systems with energy exchange,\" arXiv math-ph (2011),\n1109.2356v1.\n20T. Gilbert and R. Lefevere, \\Heat conductivity from molecular\nchaos hypothesis in locally con\fned billiard systems,\" Phys. Rev.\nLett. 101, 200601 (2008)."    },    {        "title": "0805.1320v2.Spin_dynamics_in__III_Mn_V_ferromagnetic_semiconductors__the_role_of_correlations.pdf",        "content": "arXiv:0805.1320v2  [cond-mat.str-el]  25 Aug 2008Spin dynamics in (III,Mn)V ferromagnetic semiconductors: the role of correlations\nM. D. Kapetanakis and I. E. Perakis\nDepartment of Physics, University of Crete, and Institute o f Electronic Structure & Laser,\nFoundation for Research and Technology-Hellas, Heraklion , Crete, Greece\n(Dated: November 6, 2018)\nWe address the role of correlations between spin and charge d egrees of freedom on the dynamical\nproperties of ferromagnetic systems governed by the magnet ic exchange interaction between itiner-\nant and localized spins. For this we introduce a general theo ry that treats quantum fluctuations\nbeyond the Random Phase Approximation based on a correlatio n expansion of the Green’s function\nequations of motion. We calculate the spin susceptibility, spin–wave excitation spectrum, and mag-\nnetization precession damping. We find that correlations st rongly affect the magnitude and carrier\nconcentration dependence of the spin stiffness and magnetiz ation Gilbert damping.\nPACS numbers: 75.30.Ds, 75.50.Pp, 78.47.J-\nIntroduction— Semiconductors displaying carrier–\ninduced ferromagnetic order, such as Mn–doped III-V\nsemiconductors, manganites, chalcogenides, etc, have re-\nceived a lot of attention due to their combined magnetic\nand semiconducting properties [1, 2]. A strong response\nof their magnetic properties to carrier density tuning via\nlight, electrical gates, or current[3, 4, 5] canlead to novel\nspintronics applications [6] and multifunctional magnetic\ndevices combining information processing and storage on\na single chip. One of the challenges facing such magnetic\ndevices concerns the speed of the basic processing unit,\ndetermined by the dynamics of the collective spin.\nTwo key parameters characterize the spin dynam-\nics in ferromagnets: the spin stiffness, D, and the\nGilbert damping coefficient, α.Ddetermines the long–\nwavelength spin–wave excitation energies, ωQ∼DQ2,\nwhereQis the momentum, and other magnetic prop-\nerties.Dalso sets an upper limit to the ferromagnetic\ntransition temperature: Tc∝D[1]. So far, the Tcof\n(Ga,Mn)As has increased from ∼110 K [2] to ∼173 K\n[1, 7]. It is important for potential room temperature\nferromagnetism to consider the theoretical limits of Tc.\nTheGilbertcoefficient, α, characterizesthedampingof\nthe magnetization precession described by the Landau–\nLifshitz–Gilbert (LLG) equation [1, 8]. A microscopic\nexpression can be obtained by relating the spin suscepti-\nbility of the LLG equation to the Green’s function [9]\n≪A≫=−iθ(t)<[A(t),S−\nQ(0)]> (1)\nwithA=S+\n−Q,S+=Sx+iSy.∝angbracketleft···∝angbracketrightdenotes the\naverage over a grand canonical ensemble and SQ=\n1/√\nN/summationtext\njSje−iQRj, whereSjare spins localized at N\nrandomly distributed positions Rj. The microscopic ori-\ngin ofαisstill notfully understood[9]. Amean–fieldcal-\nculation of the magnetization damping due to the inter-\nplay between spin–spin interactions and carrier spin de-\nphasingwasdevelopedin Refs.[9, 10]. Themagnetization\ndynamics can be probed with, e.g., ferromagnetic res-\nonance [11] and ultrafast magneto–optical pump–probe\nspectroscopy experiments [5, 12, 13, 14]. The interpre-tation of such experiments requires a better theoretical\nunderstanding of dynamical magnetic properties.\nIn this Letter we discuss the effects of spin–charge cor-\nrelations, due to the p–d exchange coupling of local and\nitinerant spins, on the spin stiffness and Gilbert damp-\ningcoefficient. Wedescribequantumfluctuationsbeyond\nthe Random Phase Approximation (RPA) [15, 16] with a\ncorrelationexpansion[17]ofhigherGreen’sfunctionsand\na 1/S expansion of the spin self–energy. To O(1/S2), we\nobtain a strong enhancement, as compared to the RPA,\nof the spin stiffness and the magnetization damping and\na different dependence on carrier concentration.\nEquations of motion— The magnetic propertiescan be\ndescribedby the Hamiltonian [1] H=HMF+Hcorr, where\nthe mean field Hamiltonian HMF=/summationtext\nknεkna†\nknaknde-\nscribes valence holes created by a†\nkn, wherekis the mo-\nmentum, nis the band index, and εknthe band disper-\nsion in the presenceof the mean field created by the mag-\nnetic exchangeinteraction[16]. The Mn impurities act as\nacceptors, creating a hole Fermi sea with concentration\nch, and also provide S= 5/2 local spins.\nHcorr=βc/summationdisplay\nq∆Sz\nq∆sz\n−q+βc\n2/summationdisplay\nq(∆S+\nq∆s−\n−q+h.c.),(2)\nwhereβ∼50–150meV nm3in (III,Mn)V semiconductors\n[1] is the magnetic exchane interaction. cis the Mn spin\nconcentration and sq= 1/√\nN/summationtext\nnn′kσnn′a†\nk+qnakn′the\nhole spin operator. ∆ A=A− ∝angbracketleftA∝angbracketrightdescribes the quan-\ntum fluctuations of A. The ground state and thermo-\ndynamic properties of (III,Mn)V semiconductors in the\nmetallic regime ( ch∼1020cm−3) are described to first\napproximation by the mean field virtual crystal approxi-\nmation,HMF, justified for S→ ∞[1]. Most sensitive to\nthe quantum fluctuations induced by Hcorrare the dy-\nnamical properties. Refs.[9, 15] treated quantum effects\ntoO(1/S) (RPA). Here we study correlations that first\narise atO(1/S2). By choosing the z–axis parallel to the\nground state local spin S, we have S±= 0 and Sz=S.\nThe mean hole spin, s, is antiparallel to S,s±= 0 [1].2\nThe spin Green’s function is given by the equation\n∂t≪S+\n−Q≫=−2iSδ(t)+βc≪(s×S−Q)+≫\n−i∆≪s+\n−Q≫+βc\nN×\n/summationdisplay\nkpnn′≪(σnn′×∆Sp−k−Q)+∆[a†\nknapn′]≫,(3)\nwhere ∆ = βcSis the mean field spin–flip energy gap\nands= 1/N/summationtext\nknσnnfknis the ground state hole spin.\nfkn=∝angbracketlefta†\nknakn∝angbracketrightis the hole population. The first line on\nthe right hand side (rhs) describes the mean field pre-\ncession of the Mn spin around the mean hole spin. The\nsecond line on the rhs describes the RPA coupling to the\nitinerant hole spin [10], while the last line is due to the\ncorrelations. The hole spin dynamics is described by\n(i∂t−εkn′+εk−Qn)≪a†\nk−Q↑ak↓≫\n=βc\n2√\nN/bracketleftbigg\n(fk−Qn−fkn′)≪S+\n−Q≫\n+/summationdisplay\nqm≪(σn′m·∆Sq)∆[a†\nk−Qnak+qm]≫\n−/summationdisplay\nqm≪(σmn·∆Sq)∆[a†\nk−Q−qmakn′]≫/bracketrightbigg\n.(4)\nThe firstterm on the rhsgivesthe RPAcontribution[10],\nwhile the last two terms describe correlations.\nThe correlation contributions to Eqs.(3) and (4) are\ndetermined by the dynamics of the interactions be-\ntween a carrier excitation and a local spin fluctuation.\nThis dynamics is described by the Green’s functions\n≪∆Sp−k−Q∆[a†\nknapn′]≫, whose equations of motion\ncouple to higher Green’s functions, ≪Sa†aa†a≫and\n≪SSa†a≫, describingdynamicsof threeelementaryex-\ncitations. To truncate the infinite hierarchy, we apply a\ncorrelation expansion [17] and decompose ≪Sa†aa†a≫\ninto all possible products of the form ∝angbracketlefta†aa†a∝angbracketright ≪S≫,\n∝angbracketleftS∝angbracketright∝angbracketlefta†a∝angbracketright ≪a†a≫,∝angbracketlefta†a∝angbracketright ≪∆S∆[a†a]≫, and∝angbracketleftS∝angbracketright ≪\na†aa†a≫c, where≪a†aa†a≫cis obtained after sub-\ntracting all uncorrelated contributions, ∝angbracketlefta†a∝angbracketright ≪a†a≫,\nfrom≪a†aa†a≫(we include all permutations of mo-\nmentum and band indices) [18]. Similarly, we decompose\n≪SSa†a≫into products of the form ∝angbracketleftSS∝angbracketright ≪a†a≫,\n∝angbracketleftS∝angbracketright∝angbracketlefta†a∝angbracketright ≪S≫,∝angbracketleftS∝angbracketright ≪∆S∆[a†a]≫, and∝angbracketlefta†a∝angbracketright ≪\n∆S∆S≫. This corresponds to decomposing all opera-\ntorsAinto average and quantum fluctuation parts and\nneglecting products of three fluctuations. We thus de-\nscribe all correlations between any twospin and charge\nexcitations and neglect correlations among threeor more\nelementary excitations (which contribute to O(1/S3))\n[18]. In the case of ferromagnetic β, as in the mangan-\nites, we recover the variational results of Ref.[19] and\nthus obtain very good agreement with exact diagonaliza-\ntionresultswhilereproducingexactlysolvablelimits (one\nelectron, half filling, and atomic limits, see Refs.[18, 19]).When treating correlations in the realistic (III,Mn)V\nsystem, the numerical solution of the above closed sys-\ntem of equations of motion is complicated by the cou-\npling of many momenta and bands and by unsettled is-\nsues regarding the role on the dynamical and magnetic\nanisotropy properties of impurity bands, strain, localized\nstates, and sp–d hybridization [1, 20, 21, 22, 23]. In the\nsimpler RPA case, which neglects inelastic effects, a six–\nband effective mass approximation [16] revealed an order\nof magnitude enhancement of D. The single–band RPA\nmodel [15] also predicts maximum Dat very small hole\nconcentrations, while in the six–band model Dincreases\nand then saturates with hole doping. Here we illustrate\nthe main qualitative features due to ubiquitous corre-\nlations important in different ferromagnets [19, 24] by\nadopting the single–band Hamiltonian [15]. We then dis-\ncuss the role of the multi–band structure of (III,Mn)V\nsemiconductors by using a heavy and light hole band\nmodel.\nIn the case of two bands of spin– ↑and spin– ↓states\n[15], we obtain by Fourier transformation\n≪S+\n−Q≫ω=−2S\nω+δ+ΣRPA(Q,ω)+Σcorr(Q,ω),(5)\nwhereδ=βcsgives the energy splitting of the local spin\nlevels. Σ RPAis the RPA self energy [15, 16].\nΣcorr=βc\n2N/summationdisplay\nkp/bracketleftBigg\n(Gpk↑+Fpk)ω+εk−εk+Q\nω+εk−εk+Q+∆+iΓ\n−(Gpk↓−Fpk)ω+εp−Q−εp\nω+εp−Q−εp+∆+iΓ/bracketrightBigg\n(6)\nis the correlated contribution, where\nGσ=≪S+∆[a†\nσaσ]≫\n≪S+≫, F=≪∆Sza†\n↑a↓≫\n≪S+≫.(7)\nΓ∼10-100meV is the hole spin dephasing rate [25]. Sim-\nilar to Ref.[10] and the Lindblad method calculation of\nRef.[14], we describe such elastic effects by substituting\nthe spin–flip excitation energy∆ by ∆+ iΓ. We obtained\nGandFbysolvingthecorrespondingequationstolowest\norder in 1/S, with βSkept constant, which gives Σcorrto\nO(1/S2). More details will be presented elsewhere [18].\nResults— Firstwestudythe spinstiffness D=DRPA+\nDcorr\n++Dcorr\n−. The RPA contribution DRPAreproduces\nRef.[15]. The correlated cotributions Dcorr\n+>0 and3\n0 0.1 0.2 0.3 0.4 0.5\np00.020.040.060.08D/D0 D\nDRPA\nDRPA+D(-)\n0 0.2 0.4\np00.020.040.06\n50 100 150\nβc  (meV)00.010.02D/D0\n50 100 150\nβc  (meV)00.020.040.06a) βc =70meV b) βc =150meV\nc) p =0.1 d) p =0.5\nFIG. 1: (Color online) Spin stiffness Das function of hole\ndoping and interaction strength for the single–band model.\nc= 1nm−3, Γ=0,D0=/planckover2pi12/2mhh,mhh= 0.5me.\nDcorr\n−<0 were obtained to O(1/S2) from Eq.(6) [18]:\nDcorr\n−=−/planckover2pi12\n2mhS2N2/summationdisplay\nkp/bracketleftBigg\nfk↓(1−fp↓)εp(ˆp·ˆQ)2\nεp−εk\n+fk↑(1−fp↑)εk(ˆk·ˆQ)2\nεp−εk/bracketrightBigg\n, (8)\nDcorr\n+=/planckover2pi12\n2mhS2N2/summationdisplay\nkpfk↓(1−fp↑)×\n/bracketleftBig\nεk(ˆk·ˆQ)2+εp(ˆp·ˆQ)2/bracketrightBig\n×\n/bracketleftbigg2\nεp−εk+1\nεp−εk+∆−∆\n(εp−εk)2/bracketrightbigg\n,(9)\nwhereˆQ,ˆk, andˆ pdenote the unit vectors.\nFor ferromagnetic interaction, as in the manganites\n[19, 24], the Mn and carrier spins align in parallel. The\nHartree–Fock is then the state of maximum spin and\nan exact eigenstate of the many–body Hamiltonian (Na-\ngaoka state). For anti–ferromagnetic β, as in (III,Mn)V\nsemiconductors, the ground state carrier spin is anti–\nparallel to the Mn spin and can increase via the scat-\ntering of a spin– ↓hole to an empty spin– ↑state (which\ndecreases Szby 1). Such quantum fluctuations give rise\ntoDcorr\n+, Eq.(9), which vanishes for fk↓= 0.Dcorr\n−comes\nfrom magnon scattering accompanied by the creation of\naFermi seapair. In the caseofaspin– ↑Fermi sea, Eq.(8)\nrecovers the results of Refs.[19, 24].\nWe evaluated Eqs.(8) and (9) for zero temperature\nafter introducing an upper energy cutoff corresponding\nto the Debye momentum, k3\nD= 6π2c, that ensures the\ncorrect number of magnetic ion degrees of freedom [15].0 0.1 0.2 0.3 0.4 0.5\np00.20.4D/D0\n0 0.1 0.2 0.3 0.4 0.5\np00.20.4\n0 0.1 0.2 0.3\nεF  (eV)00.010.02D/D0\n0 0.1 0.2 0.3 0.4 0.5\nεF (eV)00.020.04a) βc =70meV b) βc =150meV\nc) βc =70meV d) βc =150meV\nFIG. 2: (Color online) Spin stiffness Dfor the parameters of\nFig. 1. (a)and(b): two–bandmodel, (c)and(d): dependence\non the Fermi energy within the single–band model.\nFigs. 1(a) and (b) show the dependence of Don hole\ndoping, characterized by p=ch/c, for two couplings β,\nwhile Figs. 1(c) and (d) show its dependence on βfor\ntwo dopings p. Figure 1 also compares our full result, D,\nwithDRPAandDRPA+Dcorr\n−. It is clear that the cor-\nrelations beyond RPA have a pronounced effect on the\nspin stiffness, and therefore on Tc∝D[1, 7] and other\nmagnetic properties. Similar to the manganites [19, 24],\nDcorr\n−<0 destabilizes the ferromagneticphase. However,\nDcorr\n+stronglyenhances Das comparedto DRPA[15] and\nalso changes its p–dependence.\nThe ferromagnetic order and Tcvalues observed in\n(III,Mn)V semiconductors cannot be explained with the\nsingle–band RPA approximation [15], which predicts a\nsmallDthat decreases with increasing p. Figure 1\nshows that the correlations change these RPA results in\na profound way. Even within the single–band model,\nthe correlations strongly enhance Dand change its p–\ndependence: Dnow increases with p. Within the RPA,\nsuch behavior can be obtained only by including multiple\nvalence bands [16]. As discussed e.g. in Refs.[1, 7], the\nmain bandstructure effects can be understood by con-\nsidering two bands of heavy ( mhh=0.5me) and light (\nmlh=0.086me) holes. Dis dominated and enhanced by\nthe more dispersive light hole band. On the other hand,\nthe heavily populated heavy hole states dominate the\nstatic properties and EF. By adopting such a two–band\nmodel, we obtain the results of Figs. 2(a) and (b). The\nmain difference from Fig. 1 is the order of magnitude en-\nhancement of all contributions, due to mlh/mhh= 0.17.\nImportantly,thedifferencesbetween DandDRPAremain\nstrong. Regarding the upper limit of Tcdue to collective\neffects, we note from Ref.[7] that is is proportional to D\nand the mean field Mn spin. We thus expect an enhance-\nment, as compared to the RPA result, comparable to the4\n0 0.5 100.020.04αα\nαRPA\n0 0.5 100.020.04\n0 0.5 1\np00.020.04α\n0 0.5 1\np00.020.04a) βc =70meV b) βc =100meV\nc) βc =120meV d) βc =150meV\nFIG. 3: (Color online) Gilbert damping as function of hole\ndoping for different interactions β.c= 1nm−3,Γ = 20meV.\ndifference between DandDRPA.\nThe dopingdependence of Dmainlycomesfromits de-\npendence on EF, shown in Figs. 2(c) and (d), which dif-\nfers strongly from the RPA result. Even though the two\nband model captures these differences, it fails to describe\naccuratelythe dependence of EFonp, determined by the\nsuccessive population of multiple anisotropicbands. Fur-\nthermore, thespin–orbitinteractionreducesthe holespin\nmatrix elements [22]. For example, |σ+\nnn′|2is maximum\nwhen the bandstates arealsospin eigenstates. The spin–\norbit interaction mixes the spin– ↑and spin– ↓states and\nreduces|σ+\nnn′|2. From Eq.(3) we see that the deviations\nfromthe meanfield resultaredetermined bythe coupling\nto the Green’s functions ≪σ+\nnn′∆[a†\nnan′]≫(RPA),≪\n∆Szσ+\nnn′∆[a†\nnan′]≫(correctiontoRPAdueto Szfluctu-\nationsleadingto Dcorr\n+>0), and≪∆S+σz\nnn′∆[a†\nnan′]≫\n(magnon–Fermi sea pair scattering leading to Dcorr\n−<0).\nBoth the RPA and the correlation contribution arising\nfrom ∆Szare proportional to σ+\nnn′. Our main result, i.e.\ntherelativeimportance of the correlation as compared to\nthe RPA contribution, should thus also hold in the real-\nistic system. The full solution will be pursued elsewhere.\nWe now turn to the Gilbert damping coefficient, α=\n2S/ω×Im≪S+\n0≫−1atω→0 [9]. We obtain to\nO(1/S2) thatα=αRPA+αcorr, where αRPArecovers\nthe mean–field result of Refs [9, 10] and predicts a linear\ndependence on the hole doping p, while\nαcorr=∆2\n2N2S2/summationdisplay\nkpIm/bracketleftBigg\nfk↓(1−fp↑)\n∆+iΓ×\n/parenleftbigg1\nεp−εk−δ+1\nεp−εk+∆+iΓ/parenrightbigg/bracketrightBigg\n(10)\narises from the carrier spin–flip quantum fluctuations.Fig.(3) compares αwith the RPA result as function of\np. The correlations enhance αand lead to a nonlinear\ndependence on p, which suggests the possibility of con-\ntrolling the magnetization relaxation by tuning the hole\ndensity. A nonlinear dependence of αon photoexcitation\nintensity was reported in Ref.[13] (see also Refs.[12, 21]).\nWe conclude that spin–charge correlations play an im-\nportant role on the dynamical properties of ferromag-\nnetic semiconductors. For quantitative statements, they\nmust be addressed together with the bandstructure ef-\nfects particular to the individual systems. 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Lett. 81, 4029 (2002)."    },    {        "title": "1506.04851v1.Fast_energy_decay_for_wave_equations_with_variable_damping_coefficients_in_the_1_D_half_line.pdf",        "content": "arXiv:1506.04851v1  [math.AP]  16 Jun 2015Fast Energy Decay for Wave Equations with Variable\nDamping Coefficients in the 1-D Half Line\nRyo IKEHATA∗and Takeshi KOMATSU\nDepartment of Mathematics, Graduate School of Education, Hiro shima University\nHigashi-Hiroshima 739-8524, Japan\nApril 15, 2019\nAbstract\nWe derive fast decay estimates of the total energy for wave equa tions with localized\nvariable damping coefficients, which are dealt with in the one dimensiona l half line (0 ,∞).\nThe variable damping coefficient vanishes near the boundary x= 0, and is effective critically\nnear spatial infinity x=∞.\n1 Introduction\nWe consider the 1-dimensional initial-boundary value prob lem for linear dissipative wave\nequation in the half line (0 ,+∞):\nutt(t,x)−uxx(t,x)+V(x)ut(t,x) = 0,(t,x)∈(0,∞)×(0,∞), (1.1)\nu(0,x) =u0(x), ut(0,x) =u1(x), x∈(0,∞), (1.2)\nu(t,0) = 0, t∈(0,∞), (1.3)\nwhere the initial data ( u0,u1) are compactly supported with the size R >0:\nu0∈H1\n0(0,∞), u1∈L2(0,∞),suppui⊂[0,R],(i= 0,1).\nLet us first talk about several related results of the Cauchy p roblem in RNof the equation:\nutt(t,x)−∆u(t,x)+V(x)ut(t,x) = 0. (1.4)\nIn the case when V(x) = constant >0, Matsumura [9] derived a historical Lp-Lqdecay estimates\nof solutions of (1.4). Later on more detail investigations s uch as the asymptotic profiles of\nsolutions were studied by Nishihara [14] and the references therein.\nIn the case of mixed problems of (1.4) in the exterior domain Ω ⊂RNNakao [13] and\nIkehata [4] have derived precise decay estimates of the tota l energy for the initial data belonging\nto the energy class and weighted L2-class, respectively. In these cases they have dealt with th e\nlocalized damping coefficient V(x), which is effective only near infinity such as V(x)> V0for\n|x| ≫1, and the star-shaped boundary ∂Ω. In particular, fast decay estimates of the total\n∗Corresponding author: ikehatar@hiroshima-u.ac.jp\nKeywords and Phrases: Localized damping; Wave equation; Mu ltiplier method; Total energy decay; Weighted\ninitial data.\n2010 Mathematics Subject Classification. Primary 35L20; Se condary 35L05, 35B33, 35B40.\n1energy like E(t) =O(t−2) was obtained in [4] by a special multiplier method develope d in [6],\nwhich modified the Morawetz one [12]. Racke [15] also derived theLp-Lqestimates for solutions\nto (1.4) with constant V(x) in the exterior domain case by relying on so called the gener alized\nFourier transform (in fact, more general form of equations a nd systems are considered). For\nrelated results in the case when the damping is effective near i nfinity one can cite the papers\nwritten by Aloui-Ibrahim-Khenissi [1] and Daoulatli [2], w here they studied the fast decay of\nthe total and local energy under the so called GCC assumption due to Lebeau [8]. Exponential\ndecay of the total energy for wave equations with mass (i.e., Klein-Gordon type) and localized\ndamping terms was investigated by Zuazua [19] in the exterio r domain case. Furthermore, by\nWatanabe [18] an effective application of the method develope d in [4] is recently introduced,\nand there a global existence and decay estimates of solution s to a nonlinear Cauchy problem of\nthe quasilinear wave equation with a localized damping near infinity are studied.\nOn the other hand, recently in [7] the critically decaying da mping coefficient V(x) near\nspatial infinity was considered for (1.4) in the whole space RN, in fact, they have studied so\ncalled the critical case with δ= 1 below:\nV0\n(1+|x|)δ≤V(x)≤V1\n(1+|x|)δx∈RN, (1.5)\nand they have announced the fact that E(t) =O(t−N) ifV0> N, while if V0≤N, then (roughly\nspeaking) E(t) =O(t−V0). In this connection, we call it sub-critical damping in the case of (1.5)\nwithδ∈[0,1), while in the case when δ >1, (1.5) is called as super-critical damping. The\nprecise decay order of several norms of solutions for the Cau chy problem in RNof (1.4) with\n(1.5) were investigated by Todorova-Yordanov [16] in the su b-critical case. Furthermore, in the\nsuper critical case it is well-known by Mochizuki [10] that t he solution of the Cauchy problem\nof (1.4) is asymptotically free.\nWhile, in the constant damping case with V(x) = constant of (1.1)-(1.3) (i.e., problem in\nthe half line) was studied in [3], and he derived E(t) =O(t−2) (t→+∞). In the localized\ndamping case, it seems that there are no any previous researc h papers that are dealing with the\nhalf space problem and critical damping near infinity. In thi s connection, one notes that in [5]\nIkehata-Inoue have derived E(t) =O(t−1) ast→+∞forV0>1 (see also Mochizuki-Nakazawa\n[11] and Uesaka [17] for the topics on the energy decay proper ty). In fact, they treated a more\ngeneral nonlinear equation like\nutt(t,x)−∆u(t,x)+V(x)ut(t,x)+|u(t,x)|p−1u(t,x) = 0,\nin the critical damping case of (1.5).\nSo, a natural question arises whether one can derive fast dec ay result like E(t) =O(t−2) in\nthe case when V(x) vanishes near the boundary and is effective critically (i.e. ,δ= 1 in (1.5))\nonly near spatial infinity x=∞. The main purpose of this paper is to answer this natural\nquestion by modifying a technical method originally develo ped in [5].\nOur assumptions on V(x) are as follows:\n(A)V∈L∞(0,∞)∩C([0,∞)), and there exist three constants\nV0>2, V1∈[V0,+∞), L2>0\nsuch thatV0\n1+x≤V(x)≤V1\n1+x, x∈[L2,+∞),\n0≤V(x) (x∈[0,∞)).\nOur new results read as follows.\n2Theorem 1.1 LetV(x)satisfy the assumption (A). If the initial data (u0,u1)∈H1\n0(0,∞)×\nL2(0,∞)further satisfies\nsuppu 0∪suppu 1⊂[0,R]\nfor some R >0, then for the solution u∈C([0,∞);H1\n0(0,∞))∩C1([0,∞);L2(0,∞))to problem\n(1.1)−(1.3)one has\nE(t) =O(t−2),(t→ ∞).\nRemark 1.1 In the 1-D whole space case, if we consider the corresponding Cauchy problem,\nwe can have\nE(t) =O(t−1),(t→ ∞)\nprovided that V0>1 (cf., [7], [17]). So, the results seem to reflect the half spa ce property\nitself. This discovery of the new number V0>2in the half space case is one of our maximal\ncontribution. Even if one takes L2= 0 formally, the obtained result in Theorem 1.1 is new.\nRemark 1.2 In the case when V0≤2, we still do not know the exact rate of decay for E(t).\nHowever, from [7, Theorem 4.1] one can state a conjecture tha tE(t) =O(t−V0) (t→+∞) if\nV0≤2. Furthermore, the result will be generalized to the N-dimensional half space problems of\n(1.1)-(1.3). We can present a conjecture that if V0> N+1, then E(t) =O(t−(N+1)) (t→+∞)\nin the half space case.\nRemark 1.3 The assumption ( A) implies that the damping coefficient V(x) can vanish near\nthe boundary x= 0, so the damping is effective only near infinity. But, the effect iveness of the\ndamping V(x) near infinity corresponds to so called the critical case of ( 1.5) with δ= 1 (see\n[7]). In this sense, the result obtained in Theorem 1.1 is der ived under two types of difficult\nsituations, that is, one is vanishing damping, and the other is critical one.\nThis paper is organized as follows. In section 2 we shall prov e Theorem 1.1 by relying on a\nmultiplier method which was originally introduced by the fir st author’s collaborative work [5]\nand [6].\nNotation. Throughout this paper, /bardbl·/bardblmeans the usual L2(Ω)-norm. The total energy E(t) to the\nsolution u(t,x) of (1.1) is defined by\nE(t) :=1\n2(/bardblut(t,·)/bardbl2+/bardblux(t,·)/bardbl2).\nFurthermore, one sets as a L2(0,∞) inner product:\n(u,v) :=/integraldisplay+∞\n0u(x)v(x)dx.\nA weighted function space is defined as follows: u∈L1,1/2(0,∞) iffu∈L1(0,∞) and\n/bardblu/bardbl1,1/2:=/integraldisplay∞\n0√\n1+x|u(x)|dx <+∞.\nFurthermore, one sets\nX1(0,∞) :=C([0,∞);H1\n0(0,∞))∩C1([0,∞);L2(0,∞)).\n32 Proof of Results\nWe first prepare an important lemma, which is borrowed from [3 , Lemma 2.1].\nLemma 2.1 It holds that\nsup\n0≤x<+∞(|u(x)|√1+x)≤ /bardblux/bardbl\nfor allu∈H1\n0(0,∞).\nLemma 2.2 Letu∈X1(0,∞)be the solution to problem (1.1)−(1.3). Then it is true that\nd\ndtE(t)+F(t) = 0,(t≥0)\nwhere\nE(t) =1\n2d\ndt/integraldisplay∞\n0[f(u2\nt+u2\nx)+2guut+(gV−gt)u2+2huxut]dx,\nF(t) =1\n2/integraldisplay∞\n0(2fV−ft−2g+hx)u2\ntdx+1\n2/integraldisplay∞\n0(2g−ft+hx)u2\nxdx\n+1\n2/integraldisplay∞\n0(gtt−gtV)u2dx+/integraldisplay∞\n0(hV−ht)uxutdx−1\n2/integraldisplay∞\n0d\ndx(hux)dx.\nMoreover, f=f(t), g=g(t)and h=h(t,x)are all smooth functions specified later on.\nProof.Since one considers the weak solutions u(t,x) of the problem (1.1)-(1.3), we can assume\nto check the desired identity that the solution u(t,x) is sufficiently smooth, and vanishes for\nlargex≫1.\nStep 1. Multiplying the both sides of (1.1) by futand rearranging it one can get:\nfututt−futuxx+fVu2\nt\n=f1\n2d\ndtu2\nt−d\ndx(futux)+1\n2d\ndtfu2\nx−1\n2ftu2\nx+fVu2\nt\n=1\n2d\ndt[f(u2\nt+u2\nx)]+(fV−1\n2ft)u2\nt−1\n2ftu2\nx−d\ndx(futux) = 0.\nStep 2. Multiplying the both sides of (1.1) by guand rearranging it one has:\nguutt−guuxx+gVutu\n=gd\ndtuut−gu2\nt−d\ndx(guux)+gu2\nx+gV1\n2d\ndtu2\n=d\ndtguut−gtuut−gu2\nt−d\ndx(guux)+gu2\nx+1\n2d\ndt(gVu2)−1\n2gtVu2\n=d\ndt(guut)−1\n2d\ndt(gtu2)+1\n2gttu2−gu2\nt−d\ndx(guux)+gu2\nx+1\n2d\ndt(gVu2)−1\n2gtVu2= 0.\nStep 3. Multiplying the both sides of (1.1) by huxand rearranging it one obtains:\nhuxutt−huxuxx+huxVut\n=h(d\ndtuxut)−hd\ndt(ux)ut−1\n2hd\ndx(u2\nx)+hVutux\n=d\ndt(huxut)−htuxut−1\n2hd\ndx(u2\nt)−1\n2d\ndx(hu2\nx)+1\n2hxu2\nx+hVutux\n4=d\ndt(hutux)−htuxut−1\n2d\ndx(hu2\nt)+1\n2hxu2\nt−1\n2d\ndx(hu2\nx)+1\n2hxu2\nx+hVutux= 0.\nStep 4. Add all final identities from Step 1 to Step 3 and integrate ove r [0,∞). Then, it follows\nfrom the boundary condition (1.3) that one can get:\n1\n2d\ndt/integraldisplay∞\n0[f(u2\nt+u2\nx)+2guut+(gV−gt)u2+2hutux]dx\n+1\n2/integraldisplay∞\n0(2fV−ft−2g+hx)u2\ntdx+1\n2/integraldisplay∞\n0(2g−ft+hx)u2\nxdx\n+1\n2/integraldisplay∞\n0(gtt−gtV)u2dx+/integraldisplay∞\n0(hV−ht)uxutdx−1\n2/integraldisplay∞\n0d\ndx(hu2\nx)dx= 0,\nwhich implies the desired identity. ✷\nSince the finite speed of propagation property can be applied to the solution u(t,x) of the\ncorresponding problem (1 .1)−(1.3), in order to estimate the functions E(t) andF(t) it suffices\nto consider all the spatial integrand over the closed interv al [0,R+t] (t≥0).\nNow, because of the assumption ( A) such as V(L2)>0 and the continuity of V(x), there\nexist constants L1∈[0,L2) andVm>0 such that\nV(x)≥Vm(x∈[L1,L2]).\nBy using these constants we choose the functions f(t),g(t) andh(t,x) as follow:\nf(t) =ǫ1(1+t)2, g(t) =ǫ2(1+t), h(t,x) =ǫ3(1+t)φ(x),\nwhere the monotone increasing function φ∈C∞([0,∞)) can be defined to satisfy\nφ(x) =\n\n(1+x) (0≤x < L1),\n1+L2(L2≤x),(2.1)\nwhereǫi>0 (i= 1,2,3) are some constants determined later on.\nLemma 2.3 Letf,gandhbe defined by (2.1). Then one has the following estimates:for each\nt≥0,\n(i)/integraldisplay∞\n0(hV−ht)utuxdx≥ −k\n2/integraldisplay∞\n0hVu2\ntdx−1\n2k/integraldisplay∞\n0hVu2\nxdx−1\n2/integraldisplay∞\n0htu2\ntdx−1\n2/integraldisplay∞\n0htu2\nxdx,\n(ii)/integraldisplay∞\n0d\ndx(hu2\nx)dx≤0,\nwherek >0is a constant.\nProof.(i): This is easily derived from the following inequality wi th some positive number p:\n|utux| ≤1\n2pu2\nt+p\n2u2\nx.\n(ii): Because of the finite speed of propagation property, on e has\n/integraldisplay∞\n0d\ndx(hu2\nx)dx=h(t,∞)ux(t,∞)2−h(t,0)ux(t,0)2\n=−h(t,0)ux(t,0)2≤0.✷\nThus, it follows from Lemmas 2.2 and 2.3 that for t≥0\n1\n2d\ndt/integraldisplay∞\n0[f(u2\nt+u2\nx)+2guut+(gV−gt)u2+2hutux]dx\n5+1\n2/integraldisplay∞\n0(2fV−ft−2g+hx−khV−ht)u2\ntdx\n+1\n2/integraldisplay∞\n0(2g−ft+hx−1\nkhV−ht)u2\nxdx+1\n2/integraldisplay∞\n0(gtt−gtV)u2dx≤0. (2.2)\nFurthermore, one can get the estimates below:\nLemma 2.4 Let f, g and h be defined by (2.1), and assume (A). If all parameters ǫi(i= 1,2,3)\nare well-chosen, then there exists a large t0>0such that for all t≥t0≫0,one has\n(iii)2f(t)V(x)−ft(t)−2g(t)+hx(t,x)−kh(t,x)V(x)−ht(t,x)>0, x∈[0,∞),\n(iv)2g(t)−ft(t)+hx(t,x)−1\nkh(t,x)V(x)−ht(t,x)>0, x∈[0,∞)\nwith some constant k >0.\nProof.We first check (iii) by separating the integrand of xinto 3 parts [0 ,L1], [L1,L2], and\n[L2,+∞).\n(iii)The case 0 ≤x≤L1:\n2fV−ft−2g+hx−khV−ht\n= 2ǫ1(1+t)2V(x)−2ǫ1(1+t)−2ǫ2(1+t)+ǫ3(1+t)−kǫ3(1+t)(1+x)V(x)−ǫ3(1+x)\n≥(1+t)2{2ǫ1−kǫ31+L1\n1+t}V(x)+(1+t){ǫ3−2ǫ1−2ǫ2−ǫ31+L1\n1+t}.\nThe case L1≤x≤L2:\n2fV−ft−2g+hx−khV−ht\n= 2ǫ1(1+t)2V(x)−2ǫ1(1+t)−2ǫ2(1+t)+ǫ3(1+t)φ′(x)−kǫ3(1+t)φ(x)V(x)−ǫ3φ(x)\n≥(1+t)2{2ǫ1Vm−2ǫ11\n1+t−2ǫ21\n1+t−kǫ3VM1+L2\n1+t−ǫ31+L2\n(1+t)2},\nwhereVM:= sup{V(x)|0≤x≤L2}>0, and we have just used the fact that φ′(x)≥0.\nThe case L2≤x: Here we use the finite speed of propagation property of the so lution below:\n2fV−ft−2g+hx−khV−lht\n= 2ǫ1(1+t)2V(x)−2ǫ1(1+t)−2ǫ2(1+t)−kǫ3(1+t)(1+L2)V(x)−ǫ3(1+L2)\n≥2ǫ1(1+t)2V0\n1+x−2ǫ1(1+t)−2ǫ2(1+t)−kǫ3(1+t)(1+L2)V1\n1+x−ǫ3(1+L2)\n≥(1+t)2\n1+x{2ǫ1V0−2ǫ11+R+t\n1+t−2ǫ21+R+t\n1+t−kǫ3(1+L2)V1\n1+t−ǫ3(1+L2)1+R+t\n(1+t)2}.\nNote that in this region, φ′(x) = 0.\nNext one can check the condition (iv) similarly to the case (i ii).\n(iv)The case 0 ≤x≤L1:\n2g−ft+hx−1\nkhV−ht\n= 2ǫ2(1+t)−2ǫ1(1+t)+ǫ3(1+t)−ǫ3\nk(1+t)(1+x)V(x)−ǫ3(1+x)\n6≥(1+t){2ǫ2−2ǫ1+ǫ3−ǫ3\nk(1+L1)VM−ǫ31+L1\n1+t}.\nThe case L1≤x≤L2:\n2g−ft+hx−1\nkhV−ht\n= 2ǫ2(1+t)−2ǫ1(1+t)+ǫ3(1+t)φ′(x)−ǫ3\nk(1+t)φ(x)V(x)−ǫ3φ(x)\n≥(1+t){2ǫ2−2ǫ1−ǫ3\nk(1+L2)VM−ǫ31+L2\n1+t}.\nThe case L2≤x:\n2g−ft+hx−1\nkhV−ht\n= 2ǫ2(1+t)−2ǫ1(1+t)−ǫ3\nk(1+t)(1+L2)V(x)−ǫ3\nl(1+L2)\n≥(1+t){2ǫ2−2ǫ1−ǫ2\nk(1+L2)V1\n1+x−ǫ21+L2\n1+t}\n≥(1+t){2ǫ2−2ǫ1−ǫ3\nkV1−ǫ21+L2\n1+t}.\nSo, in order to obtain (iii) and (iv) for large t≫1 the following five conditions should be\nsatisfied to ǫi(i= 1,2,3) andk >0:\nǫ3−2ǫ1−2ǫ2>0, (2.3)\n2ǫ1V0−2ǫ1−2ǫ2>0, (2.4)\n2ǫ2−2ǫ1+ǫ3−ǫ3\nk(1+L1)VM>0, (2.5)\n2ǫ2−2ǫ1−ǫ3\nk(1+L2)VM>0, (2.6)\n2ǫ2−2ǫ1−ǫ3\nkV1>0. (2.7)\n(2.3) and (2.4) come from the check of (iii), while (2.5)-(2. 7) have its origin in the case when we\ncheck (iv) as t→+∞.\nWe need to look for the constants ǫ1,ǫ2,ǫ3,k >0 satisfying these five conditions (2.3)-(2.7)\nabove.\nFirst, the condition (2 .5) and (2 .6) and (2 .7) can be unified by the following one condition\n(2.8):\n2ǫ2−2ǫ1+ǫ3−ǫ3\nk(1+L1)VM≥2ǫ2−2ǫ1−ǫ3\nk(1+L2)VM\n≥2ǫ2−2ǫ1−ǫ3\nkV∗>0, (2.8)\nwhere\nV∗:= max{(1+L2)VM, V1}>0.\n7Thus it suffices to check only 3 conditions (2 .3), (2.4) and (2 .8) above. However, for its purpose\nit is enough to choose all constants ǫi>0 (i= 1,2,3) andk >0 as follows:\nǫ1:= 1, ǫ2:=V0\n2, ǫ3:= 2V0,\nand\nk:=4V0V∗\nV0−2,\nwhere the assumption V0>2 is essentially used. Therefore, one has the desired estima tes if one\ntakest≫1 sufficiently large. ✷\nNext lemma is a direct consequence of (2.2) and Lemma 2.4.\nLemma 2.5 Letu∈X1(0,∞)be the solution to problem (1.1)−(1.3), andf,g,hbe defined by\n(2.1). Under the condition (A), the following estimate holds true:\nd\ndt{f(t)E(t)+g(t)(ut,u)+2(hux,ut)} ≤1\n2d\ndt/integraldisplay∞\n0(gt−gV)u2dx+1\n2/integraldisplay∞\n0(gtV−gtt)u2dx(2.9)\nfor allt≥t0, wheret0≫1is a fixed time defined in Lemma 2.4.\nThen we integrate the both sides of (2 .9) over [t0,t]:\nf(t)E(t)+g(t)(ut,u)+2(hux,ut)\n≤f(t0)E(t0)+g(t0)(ut(t0),ut(t0))+2(h(t0)ux(t0),ut(t0))\n+1\n2/integraldisplay∞\n0(gt−gV)u2dx−1\n2/integraldisplay∞\n0(gt(t0)−g(t0)V)u2(t0)dx+1\n2/integraldisplayt\nt0/integraldisplay∞\n0(gtV−gtt)u2dxds\n=C+1\n2/integraldisplay∞\n0(gt−gV)u2dx+1\n2/integraldisplayt\nt0/integraldisplay∞\n0(gtV−gtt)u2dxds, (2.10)\nwhere the constant C >0, which is independent from t≥t0, is defined by\nC:=f(t0)E(t0)+g(t0)(ut(t0),ut(t0))+2(h(t0)ux(t0),ut(t0))\n−1\n2/integraldisplay∞\n0(gt(t0)−g(t0)V)u2(t0)dx.\nFurthermore, we have the following estimates.\nLemma 2.6 Letgbe defined by (2.1). The the smooth function gsatisfies the following two\nestimates:\n(v)gt−gV(x)≤C1, x∈[0,∞), t≥0,\n(vi)gtV−gtt≤C2V(x), x∈[0,∞), t≥0,\nwhereCi>0 (i= 1,2)are some constants.\nProof.The proof can be easily checked, so we omit it. ✷\nOntheotherhand,weshallpreparethefollowingcruciallem mabasedontheonedimensional\nHardy-Sobolev inequality in the half space case, which is st ated in Lemma 2.1.\n8Lemma 2.7 Letu∈X1(0,∞)be the solution to problem (1.1)−(1.3). Then it is true that\n/bardblu(t,·)/bardbl2+/integraldisplayt\n0/integraldisplay∞\n0V(x)|u(s,x)|2dxds≤C(/bardblu0/bardbl2+/bardbl(V(·)u0+u1)/bardbl2\n1,1/2), (2.11)\nprovided that /bardbl(V(·)u0+u1)/bardbl1,1/2<+∞.\nProof.The original idea comes from [6]. We introduce an auxiliary f unction\nw(t,x) :=/integraldisplayt\n0u(s,x)ds.\nThenw(t,x) satisfies\nwtt−wxx+V(x)wt=V(x)u0+u1,(t,x)∈(0,∞)×(0,∞), (2.12)\nw(0,x) = 0, wt(0,x) =u0(x), x∈(0,∞). (2.13)\nMultiplying (2 .12) bywtand integrating over [0 ,t]×[0,∞) we get\n1\n2(/bardblwt(t,·)/bardbl2+/bardblwx(t,·)/bardbl2)+/integraldisplayt\n0/bardbl/radicalBig\nV(·)ws(s,·)/bardbl2ds\n=1\n2/bardblu0/bardbl2+/integraldisplayt\n0(V(·)u0+u1,ws)ds. (2.14)\nNext step is to use Lemma 2.1 to obtain a series of inequalitie s below:\n/integraldisplayt\n0(V(·)u0+u1,ws)ds=/integraldisplayt\n0d\nds(V(·)u0+u1,w)ds\n≤/integraldisplay∞\n0√\n1+x|V(x)u0+u1||w(t,x)|√1+xdx\n≤( sup\nx∈[0,∞)|w(t,x)|√1+x)/bardblV(·)u0+u1/bardbl1,1/2\n≤1\n4/bardblwx/bardbl2+/bardblV(·)u0+u1/bardbl2\n1,1/2. (2.15)\nCombining (2 .14) with (2 .15) we can derive\n1\n2/bardblwt(t,·)/bardbl2+1\n4/bardblwx(t,·)/bardbl2+/integraldisplayt\n0/integraldisplay∞\n0V(x)wt(s,x)dxds\n≤1\n2/bardblu0/bardbl2+/bardblV(·)u0+u1/bardbl2\n1,1/2.\nThe desired estimate follows from the estimate above and the fact that wt=u.✷\nIt follows from (2.10) and Lemmas 2.6 and 2.7 that there exist s a constant C >0 such that\nf(t)E(t)+g(t)(u(t,·),ut(t,·))+2(hux,ut)≤C(t≥t0), (2.16)\nprovided that /bardbl(V(·)u0+u1)/bardbl1,1/2<+∞.\nFinally, we can derive the following lemma.\n9Lemma 2.8 Lethbe defined by (2.1). Then, for all t≥t0≫1it is true that\nf(t)E(t)+2(hux,ut)≥Cf(t)E(t),\nwhereC >0is a constant.\nProof.Indeed, one has\nf(t)E(t)+2(hux,ut)≥1\n2/integraldisplay∞\n0f(t)(u2\nt+u2\nx)dx−/integraldisplay∞\n0h(t,x)(u2\nx+u2\nt)dx\n≥1\n2/integraldisplay∞\n0(f(t)−2h(t,x))(u2\nx+u2\nt)dx.\nOn the other hand, if necessarily, by choosing t0≫1 further large enough, one can derive\nthe following estimates for t≥t0:\nf(t)−2h(t,x)≥ǫ1(1+t)2−2ǫ3(1+t)(1+L2)\n= (1+t)2{ǫ1−2ǫ3(1+L2)\n1+t} ≥C(1+t)2≥Cf(t),\nwith some constant C >0. Here we have just used the monotonicity of the function φ(x) closely\nrelated with the definition of the function h(t,x).✷\nNow we can finalize the proof of Theorem 1.1.\nProof of Theorem 1.1. We first note that one can use Lemma 2.7 because one can check\n/bardbl(V(·)u0+u1)/bardbl1,1/2<+∞under the assumption on the initial data stated in Theorem 1. 1.\nThus, by using the Schwarz inequality, (2.16) and Lemma 2.8 w e get\nC(1+t)2E(t)≤g(t)/bardblu(t,·)/bardbl/bardblut(t,·)/bardbl+C≤Cg(t)/radicalBig\nE(t)+C, t≥t0.\nFurthermore, if we set X(t) =/radicalbig\nE(t) fort∈[0,+∞), then one has\nf(t)X(t)2−Cg(t)X(t)−C≤0, t≥t0. (2.17)\nBy solving the quadratic inequality (2 .17) forX(t) we have\n/radicalBig\nE(t)≤Cg(t)+/radicalbig\nC2g(t)2+4Cf(t)\n2f(t)(t≥t0).\nThis inequality leads to\nE(t)≤C/parenleftbiggg(t)\nf(t)/parenrightbigg2\n+C/parenleftbigg1\nf(t)/parenrightbigg\n, t≥t0,\nwhich implies the desired decay estimates. ✷\nAcknowledgment. The work of the first author (R. IKEHATA) was supported in part by Grant-\nin-Aid for Scientific Research (C) 15K04958 of JSPS.\n10References\n[1] L. Aloui, S. Ibrahim and M. Khenissi, Energy decay for lin ear dissipative wave equations\nin exterior domains, J. Diff. Eqns 259 (2015), 2061-2079.\n[2] M. Daoulatli, Energy decay rates for solutions of the wav e equation with linear damping in\nexterior domain, arXiv:1203.6780v4.\n[3] R. Ikehata, A remark on a critical exponent for the semili near dissipative wave equation in\nthe one dimensional half space, Diff. Int. Eqns 16, No. 6 (2003) , 727-736.\n[4] R. Ikehata, Fast decay of solutions for linear wave equat ions with dissipation localized near\ninfinity in an exterior domain, J. Diff. Eqns 188 (2003), 390-40 5.\n[5] R. Ikehata and Y. Inoue, Total energy decay for semilinea r wave equations with a critical\npotential type of damping, Nonlinear Anal. 69 (2008), 1396- 1401.\n[6] R. Ikehata and T. Matsuyama, L2-behaviour of solutions to the linear heat and wave\nequations in exterior domains, Sci. Math. Japon. 55 (2002), 33-42.\n[7] R. Ikehata, G. Todorova and B. Yordanov, Optimal decay ra te of the energy for linear wave\nequations with a critical potential, J. Math. Soc. Japan 65, No. 1 (2013), 183-236.\n[8] G. Lebeau, ´Equations des ondes amorties, Algebraic and geometric meth ods in Math.\nPhysics, A. Boutet de Monvel and V. Marchenko (eds), Kluwer A cademic, The Nether-\nlands, (1996), 73-109.\n[9] A. Matsumura, On the asymptotic behavior of solutions of semilinear wave equations, Publ.\nRIMS Kyoto Univ. 12 (1976), 169-189.\n[10] K. Mochizuki, Scattering theory for wave equations wit h dissipative terms, Publ. Res. Inst.\nMath. Sci. 12 (1976), 383-390.\n[11] K. Mochizuki and H. Nakazawa, Energy decay and asymptot ic behavior of solutions to the\nwave equations with linear dissipation, Publ. Res. Inst. Ma th. Sci. 32 (1996), 401-414.\n[12] C. Morawetz, The decay of solutions of the exterior init ial-boundary value problem for the\nwave equation, Comm. Pure Appl. Math. 14 (1961), 561-568.\n[13] M. Nakao, Energy decay for the linear and semilinear wav e equations in exterior domains\nwith some localized dissipations, Math. Z. 238 (2001), 781- 797.\n[14] K. Nishihara, Lp-Lqestimates to the damped wave equation in 3-dimensional spac e and\ntheir application, Math. Z. 244 (2003), 631-649.\n[15] R. Racke, Decay rates for solutions of damped systems an d generalized Fourier transforms,\nJ. Reine Angew. Math. 412 (1990), 1-19.\n[16] G. Todorova and B. Yordanov, Weighted L2-estimates of dissipative wave equations with\nvariable coefficients, J. Diff. Eqns 246 (2009), 4497-4518.\n[17] H. Uesaka, The total energy decay of solutions for the wa ve equation with a dissipative\nterm, J. Math. Kyoto Univ. 20 (1980), 57-65.\n[18] T. Watanabe, Global existence and decay estimates for q uasilinear wave equations with\nnonuniform dissipative term, Funk. Ekvac. 58 (2015), 1-42.\n11[19] E. Zuazua, Exponential decay for the semilinear wave eq uation with localized damping in\nunbounded domains, J. Math. Pures Appl. 70 (1991), 513-529.\n12"    },    {        "title": "1104.3002v1.Lagrangian_approach_and_dissipative_magnetic_systems.pdf",        "content": "arXiv:1104.3002v1  [cond-mat.stat-mech]  15 Apr 2011Lagrangian approach and dissipative magnetic systems\nThomas Bose and Steffen Trimper\nInstitute of Physics, Martin-Luther-University, D-06099 Halle, Germany∗\n(Dated: September 18, 2018)\nAbstract\nA Lagrangian is introduced which includes the coupling betw een magnetic moments mand the\ndegrees of freedom σof a reservoir. In case the system-reservoir coupling break s the time reversal\nsymmetry the magnetic moments perform a damped precession a round an effective field which is\nself-organized by the mutual interaction of the moments. Th e resulting evolution equation has the\nform of the Landau-Lifshitz-Gilbert equation. In case the b ath variables are constant vector fields\nthe moments mfulfill the reversible Landau-Lifshitzequation. Applying Noether’s theorem we find\nconserved quantities under rotation in space and within the configuration space of the moments.\nPACS numbers: 75.78.-n, 11.10.Ef, 75.10.Hk\n∗thomas.bose@physik.uni-halle.de; steffen.trimper@physik.uni-halle.de\n1I. INTRODUCTION\nThe dynamics of magnetic systems is described in a wide range of time a nd length scales\nfrom a quantum approach up to a macroscopic thermodynamic acce ss. On a coarse-grained\nmesoscopic level the relevant electronic degrees of freedom are g rouped into effective mag-\nnetic moments. As the consequence the magnetization is characte rized by a spatiotemporal\nvector field m(r,t). Owing to the mutual interaction between the magnetic moments t hey\nperform a precession motion around a local effective field giving rise t o the propagation of\nspin-wave excitations. Due to a system-reservoir coupling the pre cession of the moments\nshould be a damped one. To analyze this situation one has to specify t he coupling between\nthe system and the bath. The most popular approach to incorpora te dissipation is the em-\nbedding of the relevant system into a quantum-statistical environ ment which is assumed to\nremain in thermal equilibrium. The reservoir is often represented by harmonic oscillators or\nspin moments which are analyzed by path integral techniques [1–3]. A specification of the\npath integral approach for spin systems can be found in [4–7]. A mor e generic description\nconcerning dissipative semiclassical dynamics is presented in [8]. Altho ugh the application\nof path integrals can be considered as an intuitive formalism analytica l calculations are often\nimpossible and numerical schemes are necessary. In the present p aper we propose an alter-\nnative way to include dissipative effects for mesoscopic magnetic sys tems. On this level the\nanalysis of magnetodynamics is performed properly by applying the L andau-Lifshitz-Gilbert\nequation designated as LLG [9, 10]. A comprehensive survey of magn etization dynamics is\ngiven in [11]. Our investigation can be grouped in the effort to underst and dissipative mech-\nanisms in magnets. So, a non-linear dissipative model for magnetic sy stems was discussed in\n[12]. On the relation between fluctuation-dissipation theorems and d amping terms like that\none occurring in the LLG was reported in [13]. The dynamical respons e of ferromagnetic\nshape memory alloy actuators can be modeled by means of a dissipativ e Euler-Lagrange\nequation as performed in [14]. Likewise, the pinning of magnetic domain walls in multifer-\nroics is discussed in terms of the EL equations in [15]. An alternative a nsatz is introduced in\n[16], where a Lagrangian density is obtained based on a projection on to the complex plane.\nThe procedure gives rise to a dynamical equation which is equivalent t o the Landau-Lifshitz\nequation. Different to the mentioned approaches the present pap er is aimed to derive an\nequation of motion for a magnetic system on a mesoscopic scale unde r the influence of a\n2bath which likewise consists of mesoscopic moments. Following this idea we propose a La-\ngrangian comprising both fields, m(r,t) as the system variables and σas the bath variables.\nThe bath becomes dynamically active by the coupling to the system. I n case the coupling\nbetween system and reservoir breaks the time reversal symmetr y the motion of the moments\nm(r,t) is damped. The Lagrangian is modified in such a manner that dissipatio n can occur.\nII. THE LAGRANGIAN\nAs indicated we are interested to construct a Lagrangian describin g the motion of a magne-\ntization vector field within a bath of spins. This reservoir should influe nce the measurable\nmagnetization due to the mutual interaction. Let us formulate the general assumptions for\nthe underlying model. The magnetic order is originated by single magne tic atoms which\noccupy equivalent crystal positions. Here we refer to a continuou s description in terms of\na field vector denoted as m(r,t). Because the ferromagnet is considered below the Curie\ntemperature a sufficient number of microscopic spins preferring a p arallel alignment are in-\ncluded in m, i.e. the effective magnetic moment is given by m(r,t) =/summationtext\niµiwhere the\nsum is extended over all microscopic moments within a small volume aro und the spatial\ncoordinate rat timet. As each axial vector the moment fulfills m(−t) =−m(t). The bath\nin which the moments are embedded consists likewise of mesoscopic sp ins. They are denoted\nasσand are also composed of microscopic moments ηi. This bath moments which play\nthe role of ’virtual’ moments are also axial vectors changing their sig n by time inversion.\nA further new aspect is that the coupling between the real and the virtual moments is not\nassumed to be weak. As the result the complete system consists of two subsystems. One of\nthem abbreviated as L1 is occupied exclusively by the real spins with t he moments mand\nthe other one denoted as L2 is occupied by the bath spins σ. The situation is illustrated in\nFIG. 1. Now let us introduce the action\nS[{qα}] =/integraldisplay\ndt/integraldisplay\nd3xL[{qα}], (1)\nwhere the set {qα}consists of the set of both moments σandm. The Lagrange density\ncomprises three terms\nL[m,˙m,∇m;σ,∇σ] =L(m)[m,˙m,∇m]+L(σ)[∇σ]+L(mσ)[σ,˙m], (2)\n3µi−1 µi µi+1 ηj−k ηj ηj+1 ηj+1+kJµηJηηJµµ\nFIG. 1. (Color online) Schematic illustration of the basic m odel. The red spins represent the\nmagnetic moments µiand refer to the lattice L1 introduced in the text. The green s pin vectors\nηibuild the bath lattice L2. Interactions are possible betwee n theµiandηj,µiandµjandηi\nandηj. The respective coupling strengths correspond to the coupl ing parameters in Eqs. (3)-(5)\nas follows: Jµη↔J(mσ),Jµµ↔J(m)andJηη↔J(σ).\nwhereL(m)indicates the Lagrangian of the magnetic system, L(σ)represents the reservoir\nand the interaction term is denoted as L(mσ). To be more specific the magnetic moments of\nthe system interact via exchange coupling defined by the Lagrangia n\nL(m)=1\n2J(m)\nαβ∂mν\n∂xα∂mν\n∂xβ+Aν(m) ˙mν, (3)\nwhereJ(m)\nαβisthecouplingparameter, diagonalintheisotropiccase. Thefirstt ermrepresents\nthe energy density of the magnetic system. Because we are not co nsidering the acceleration\nof magnetic moments a term of the order ˙m2is missing. Moreover, the magnetic moments\nperformaprecessionaroundaneffective magneticfield, whichisself -organizedbythemutual\ninteraction. Thereforethevectorpotential Adependsonthemoments, i. e. A=A(m(r,t)).\nThe coupling has the same form as the minimal coupling in electrodynam ics. The bath\nLagrangian is defined in a similar manner as\nL(σ)=1\n2J(σ)\nαβ∂σν\n∂xα∂σν\n∂xβ, (4)\nwith the coupling constant J(σ)\nαβ. Eventually, the interacting part between system and bath\nis written as\nL(mσ)=J(mσ)\nαβ∂mν\n∂xα∂σν\n∂xβ+Bν(σ) ˙mν, (5)\n4with the coupling strength J(mσ)\nαβ. The second term is constructed in the same manner as\nin Eq. (3), where the potential B(σ) will be specified below, see Eq. (8). The dynamics\nof the bath variable σremains unspecified for the present, i.e. the Lagrangian does not\ninclude a term of the form ∝˙σ. Owing to the constraint, introduced in the next section,\nthe dynamically passive bath is sensitive to a change of the system va riablesmin such a\nmanner that small variations of the system variables mare related to small variations of\nσ. This procedure leads to a coupling between bath and system so tha t the time reversal\nsymmetry is broken.\nIII. RELATION TO THE LANDAU-LIFSHITZ-GILBERT EQUATION\nIn this section we find the equation of motion for the magnetization m(r,t) from Eq. (2)\ncombined with Eqs. (3)-(5). Using the principle of least action it follow s\n/bracketleftbigg∂L\n∂σβ−∂\n∂xα∂L\n∂/parenleftBig\n∂σβ\n∂xα/parenrightBig/bracketrightbigg\nδσβ+/bracketleftbigg∂L\n∂mβ−∂\n∂t∂L\n∂˙mβ−∂\n∂xα∂L\n∂/parenleftBig\n∂mβ\n∂xα/parenrightBig/bracketrightbigg\nδmβ= 0,(6)\nwhereδmβandδσβare the small variations which drive the value for the action out of th e\nstationary state. In general, one derives a system of coupled par tial differential equations.\nHowever, to proceed further let us impose a constraint on the sys tem. A small variation of\nσβshould be related to a small variation of mβ. Thus, we make the ansatz\nδσβ=−κδmβ,withκ= const>0. (7)\nNotice that this condition should be valid only locally but not globally. Ins ofar Eq. (7) is\ncomparable to an anholonom condition in mechanics. Moreover relatio n (7) is in accordance\nwith thebehavior of themoments mandσunder timeinversion. Physically the last relation\nmeans that the bath reacts to a change of the system only tempor arily. Because the system-\nreservoir coupling should typically break the time reversal symmetr y the expansion of the\nfunctionBν(σ) in terms of σincludes only odd terms. In lowest order we get from Eq. (5)\nBν(σ) =−cσν,withc= const. (8)\nDue to Eqs. (7) and (8) the second term in Eq. (5) is of the form ∝σ·˙m. Such a term is not\ninvariant under time reversal symmetry t→ −t. As demonstrated below the broken time\n5inversion invariance gives rise to damping effects. Inserting Eqs. (7 ) and (8) into Eq. (2)\nand performing the variation according to Eq. (6) we get\n0 =/parenleftbigg∂Aν\n∂mβ−∂Aβ\n∂mν/parenrightbigg\n˙mν+c˙σβ+κc˙mβ\n−[J(m)−κJ(mσ)]∇2mβ−[J(mσ)−κJ(σ)]∇2σβ.(9)\nHere we have assumed for simplicity that all coupling tensors Jare diagonal: Jαβ=Jδαβ.\nThe first term on the right hand side in Eq. (9) reminds of the field str ength tensor in\nelectrodynamics [17]. Thus, we rewrite\n/parenleftbigg∂Aν\n∂mβ−∂Aβ\n∂mν/parenrightbigg\n˙mν≡Fβν˙mν=/bracketleftBig\n˙m×(∇m×A(m))/bracketrightBig\nβ. (10)\nAs mentioned above the vector function A(m) is regarded as vector potential which depends\non space-time coordinates via the magnetic moment m(r,t). In vector notation the last\nequation reads\n˙m×(∇m×A) = [J(m)−κJ(mσ)]∇2m−κc˙m−c˙σ+[Jmσ−κJ(σ)]∇2σ.(11)\nIf one is interested in weak excited states of a ferromagnet it is rea sonable to assume that\nthe direction of the magnetization in space changes slowly while its abs olute value is fixed,\nthat ism2= 1. Without loss of generality we have set the amplitude of mto unity. In order\nto proceed it is necessary to specify the condition which should be fu lfilled by the function\nA(m). Having in mind the LLG then we make the ansatz ∇m×A(m) =gm,g= const.\nBased on these assumptions we get from Eq. (11)\n∂m\n∂t=1\ng/parenleftBigg\nm×Heff/parenrightBigg\n−κc\ng/parenleftBigg\nm×∂m\n∂t/parenrightBigg\n. (12)\nHere the effective field is given by the expression\nHeff=/parenleftBig\nJ(m)−κJ(mσ)/parenrightBig\n∇2m−c∂σ\n∂t+/parenleftBig\nJ(mσ)−κJ(σ)+/parenrightBig\n∇2σ. (13)\nEq. (12) is nothing else than the Gilbert equation [10] by relating the prefactors as follows\nγ=−1\ng, α=−κc\ng=κcγ, (14)\nwhereγandαare the gyromagnetic ratio and the Gilbert damping parameter, res pectively.\nSince bothparameters arepositive quantities itfollows that g<0aswell as κc>0. Further,\n6Eq. (12) can be converted into the form of the equivalent and widely used Landau-Lifshitz-\nGilbert equation which reads\n∂m\n∂t=−γ\n(1+α)2(m×Heff)−αγ\n(1+α2)/bracketleftBig\nm×(m×Heff)/bracketrightBig\n, (15)\nBoth quantities γandαare still related to the model parameters by the expressions in\nEq. (14) whereas the the effective field Heffis given by Eq. (13). Now we want to analyze\nthis expression and in particular, to assign a physical meaning to the more or less ad hoc\nintroduced quantity σ. In doing so one can distinguish four different cases:\n(i) The bath is not included which corresponds formally to σis a constant vector depend-\ning neither on coordinates nor on time. Then obviously all derivatives with respect to the\ncoordinates and the time of σdisappear in Eq. (13) and consequently, the set of {qα}in\nEq. (1) does not include σ. From here we conclude that the variation fulfills δσ= 0 in\nEq. (6) which can be easily realized setting κ= 0, cf. Eq. (7). Thus, the effective field\nin Eq. (13) comprises the pure exchange interaction J(m)between the magnetic moments\nand the damping term in Eq. (12) is absent due to α= 0 in Eq. (14). A constant bath\nfieldσlead to the Landau-Lifshitz equation in the exchange interaction ap proach without\ndamping, compare [18]. It describes the precession of magnetic mom ents of an effective field\nwhich is self-organized by the mutual interaction of the moments.\n(ii)σ=σ(t) depends only on the time and not on the spatial coordinates. Rega rding\nEq. (13) the effective field is modified by two additional contributions , namely one propor-\ntional to ∇2m, originated in the exchange interaction of the magnetic moments, a nd the\nother one ∝˙σ. The latter one could be associated with an external time dependen t field\nor, ifσpoints into a fixed direction, gives rise to magnetic anisotropy. In th at case the\nanisotropy axis is spatially constant but the amount of the anisotro py is changing in time.\nSuch a situation could be realized for instance when the ferromagne tic sample is excited by\nthe irradiation with electromagnetic waves. As already mentioned th e exchange coupling\nJ(m)is supplemented by a term −κJ(mσ). In this manner the exchange interaction is influ-\nenced by the coupling between mandσalthough the spatial dependence of σis not taken\ninto account explicitly.\n(iii)σ=σ(r) depends only on the spatial coordinates and not on the time. In th is case we\nfirst recognize that the coupling strength J(m)in the term ∝ ∇2mis influenced in the same\nmanner as in case when σ=σ(t), see the previous point. Different to the former cases the\n7expression ∝ ∇2σbecomes important for the effective field in Eq. (13). The appearan ce of\nthis term suggests that spatial inhomogeneities of the surroundin gs of the magnetic system\nrepresented by mhave to be incorporated into the effective field. It seems to be reas onable\nthat the origin of this term is an inherent one and should not be led bac k to external fields.\nAs possible sources we have in mind local varying fields like inner and out er demagnetization\nfields as well as accessible fields created for instance by different loc al temperatures.\n(iv)σ=σ(r,t) isthemost general case. Then external aswell asinternal fields arecaptured\nin the model. Thus, the effective field in Eq. (13) can be rewritten as\nHeff(r,t) =Hexch(r)+h(r,t), (16)\nwhereHeffconsists of two parts. The term Hexch= (J(m)−κJ(mσ))∇2mis due to the\nexchange interaction between the magnetic moments whereas h(r,t) represents other possi-\nble influences as discussed under the points (ii) and (iii). The function his related to the\nquantity σby\nh(r,t) =−c∂σ(r,t)\n∂t+/bracketleftbig\nJ(mσ)−κJ(σ)/bracketrightbig\n∇2σ(r,t). (17)\nRemark that the formerly introduced quantity σis related to the physically relevant effec-\ntive field by the first derivation with respect to the time and the seco nd derivation with\nrespect to the spacial coordinates via Eq. (17). This equation is an inhomogeneous diffusion\nequation which can be generally solved by means of the expansion into Fourier series and the\nassumption of accurate initial and boundary conditions which depen d on the actual physical\nproblem.\nIV. SYMMETRY AND CONSERVATION\nAfter regarding the special example of the LLG we proceed with the investigation of more\ngeneral aspects. The Lagrangian density allows to discuss the beh avior under space-time\ndependent group transformation. For this purpose we apply Noet her’s theorem [19] to our\nmodel. To be more precise we consider the conservation equation [20 ]\n∂\n∂Xα/bracketleftbigg/parenleftBig\nLδαβ−∂L\n∂(∂αΨγ)∂βΨγ/parenrightBig\n∆Xβ+∂L\n∂(∂αΨγ)∆Ψγ/bracketrightbigg\n= 0. (18)\nHere, the expression in the square brackets are the components of the Noether current Iα.\nThe term∂/∂Xαin front ofIαshould be interpreted as an implicit derivative with respect\n8to time and three spatial coordinates. The symmetry operations ∆ Xαand ∆Ψ αwill be\nspecified below. With regard to the Lagrangian in Eq. (2) we introduc e the components\nΨα= (mx,my,mz,σx,σy,σz) and their partial derivatives with respect to the independent\nvariables∂βΨα=∂Ψα/∂Xβ. Since we examine an Euclidean field theory a distinction\nbetween upper and lower indices is not necessary. Eq. (18) can be r ewritten by using\nEq. (6). This yields\n∂\n∂tL∆t+∂\n∂xαL∆xα+∂L\n∂Ψα/parenleftbig\n∆Ψα−∂\n∂tΨα∆t−∂\n∂xβΨα∆xβ/parenrightbig\n= 0. (19)\nIn this equation we distinguish between the time and space variables tandxαexplicitly.\nEq. (19) is the basis for the application of the following symmetry ope rations. Now we study\nthe rotation around a certain axis as a relevant one. Here we select for instance the z-axis.\nPerforming a rotation in coordinate space with the infinitesimal angle ∆Θ the change of the\nxandy-coordinates obeys\n∆t= 0,∆xα= ∆Rαβxβ,∆R=\n0 ∆Θ\n−∆Θ 0\n. (20)\nIn the same manner one can perform the rotation in the configurat ion space of the moments\nmandσsymbolized by the before introduced vector Ψ α={mx,my,mz,σx,σy,σz}. The\ntransformation reads ∆Ψ α= ∆Sαβ(∆Φ)Ψ β, where the rotation matrix is a 6 ×6-matrix\ndetermined by the rotation angle ∆Φ. Because both rotations in coo rdinate space and\nconfiguration space, respectively, are in general independent fr om each other we find two\nconserved quantities. Using Eq. (19) it results\nˆDzL= 0,ˆΓzL= 0. (21)\nHere the two operators ˆDzandˆΓzare expressed by\nˆDz=ˆLz−/parenleftBig\nˆLzψα/parenrightBig∂\n∂Ψα,\nˆΓz=ˆS(m)\nz+ˆS(σ)\nz.(22)\nThe quantity ˆLzis the generator of an infinitesimal rotation around the z-axis in the coor-\ndinate space\nˆLz=y∂\n∂x−x∂\n∂y, (23)\n9and therefore, it is identical with the angular momentum operator. The other quantities\nˆS(m)\nzandˆS(σ)\nzare the corresponding generators in the configuration space of t he moments.\nThey are defined as\nˆS(m)\nz=my∂\n∂mx−mx∂\n∂my,\nˆS(σ)\nz=σy∂\n∂σx−σx∂\n∂σy.(24)\nThese operators reflect the invariance of the total magnetic mom entm+σunder rotation.\nMoreover the system is invariant under the combined transformat ion expressed by ˆDzand\nˆΓz, where ˆDzoffers due to the coupling between system and bath variables as well as the\nbreaking of time reversal invariance a coupling between magnetic mo ments and the angular\nmomentum.\nV. CONCLUSION\nIn this paper we have presented an approach for a mesoscopic mag netic system with dissi-\npation. The Lagrangian consists of two interacting subsystems ch aracterized by the active\nmagnetic moments of the system mand the dynamically inactive moments of the bath\ndenoted as σ. Both systems are in contact so that a small local alteration of the system\nvariables mis related as well to a small change of the bath variables σand vice versa.\nDue to this constraint we are able to describe the system by a commo n Lagrangian which\nincorporates both degrees of freedom and their coupling. In case the bath variables are\nconstant then the coupling between both systems is absent and th e whole system decays\ninto two independent subsystems. The magnetic moments mperform a precession around\nan effective field which is self-organized by the mutual interaction of the moments. If the\ncoupling between both subsystems breaks the time reversal symm etry the related evolution\nequation of the moments mis associated with the Landau-Lifshitz-Gilbert equation which\ndescribes both the precession of magnetic moments as well as their damping. It turned\nout that the bath variable σcan be linked to the effective magnetic field which drives the\nmotion of the magnetic moments. As consequence the motion of the moments is influenced\nby the additional bath degrees of freedom. This influence is formula ted mathematically and\nis described by an inhomogeneous diffusion equation. Finally, we have f ound conservation\nlaws by means of symmetry considerations based on Noether’s theo rem. Aside from the\n10expected symmetry transformation in the coordinate space and t he configuration space of\nthe moments, the analysis offers in a non-relativistic Euclidean field th eory an unexpected\ncoupling between both. This point deserves further consideration . Our approach could be\nalso considered as starting point for a further analysis in magnetic a nd multiferroic systems.\nEspecially, we are interested in more refined models which include for in stance higher order\ncouplings or anisotropy in the Lagrangian. In multiferroic systems o ne could study the case\nthat the magnetic and the polar subsystem have their own reservo irs.\nOne of us (T.B.) is grateful to the Research Network ’Nanostructu red Materials’, which\nis supported by the Saxony-Anhalt State, Germany.\n11[1] R. Feynman, A. Hibbs, and D. Styer, Quantum Mechanics and Path Integrals: Emended\nEdition(Dover Publications, 2010).\n[2] H. Kleinert, Path integrals in quantum mechanics, statistics, polymer p hysics, and financial\nmarkets (World Scientific, 2009).\n[3] U. Weiss, Quantum dissipative systems , Series in modern condensed matter physics (World\nScientific, 1999).\n[4] L. Schulman, Phys. Rev. 176, 1558 (1968).\n[5] D. C. Cabra, A. Dobry, A. Greco, and G. L. Rossini, J. Phys. A30, 2699 (1997).\n[6] V. V. Smirnov, J. Phys. A 32, 1285 (1999).\n[7] H. Grinberg, Phys. Lett. A 311, 133 (2003).\n[8] W. Koch, F. Großmann, J. T. Stockburger, and J. Ankerhold ,\nPhys. Rev. Lett. 100, 230402 (2008).\n[9] L. Landau and E. Lifshitz, Zeitschr. d. Sowj. 8, 153 (1935).\n[10] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n[11] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halp erin, Rev. Mod. Phys. 77, 1375\n(2005).\n[12] P. Durand and I. Paidarov ˜A¡, EPL 89, 67004 (2010).\n[13] V. L. Safonov and H. N. Bertram, Phys. Rev. B 71, 224402 (2005).\n[14] P. Weetman and G. Akhras, J. Appl. Phys. 105, 023917 (2009).\n[15] Z. V. Gareeva and A. K. Zvezdin, EPL 91, 47006 (2010).\n[16] I. V. Ovchinnikov and K. L. Wang, Phys. Rev. B 82, 024410 (2010).\n[17] J. Jackson, Classical electrodynamics (Wiley, 1999).\n[18] L. D. Landau, E. Lifshitz, and L. Pitaevskii, Statistical Physics Part 2: Theory of the Con-\ndensed State (Pergamon Press, Oxford, 1980).\n[19] E. Noether, Nachr. Ges. Wiss. G¨ ottingen , 235 (1918).\n[20] E. L. Hill, Rev. Mod. Phys. 23, 253 (1951).\n12"    },    {        "title": "2212.01815v2.Inverse_problem_of_recovering_the_time_dependent_damping_and_nonlinear_terms_for_wave_equations.pdf",        "content": "Inverse problem of recovering the time-dependent damping and\nnonlinear terms for wave equations\nSong-Ren Fu\nAbstract. In this paper, we consider the inverse boundary problems of recovering\nthe time-dependent nonlinearity and damping term for a semilinear wave equation on\na Riemannian manifold. The Carleman estimate and the construction of Gaussian\nbeams together with the higher order linearization are respectively used to derive the\nuniqueness results of recovering the coe\u000ecients.\nKey words: semilinear wave equation, damping term, Carleman estimate, higher\norder linearization, Gaussian beams\n1 Introduction\nLet (\n;g) be a Riemannian manifold of dimension n\u00152 with smooth boundary @\n =\n\u0000:LetM= \n\u0002(0;T) and \u0006 = \u0000\u0002(0;T):Assume that ( x;t) = (x1;\u0001\u0001\u0001;xn;x0=t)\nare local coordinates on M:The nonlinear wave equation considered in this paper is\ngiven by 8\n><\n>:utt\u0000\u0001gu+b(x;t)ut+f(x;t;u ) = 0;(x;t)2M;\nu(x;t) =h(x;t);(x;t)2\u0006;\nu(0;x) =u0(x); ut(0;x) =u1(x); x2\n;(1.1)\nwheref:\n\u0002R1\u0002C!Cis a smooth function. In the local coordinates,\n\u0001gu= divgDu= (detg)\u00001\n2nX\nij=1@xj[(detg)1\n2gij@xiu];\nwhere div gandDare the divergence operator and Levi-Civita connection in the metric\ng, respectively.\nThe main goal in this paper is to study the inverse problem of recovering the time-\ndependent coe\u000ecient b(x;t) (damping term), and the nonlinear term f(x;t;z ) by some\nAcademy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P. R. China;\nSchool of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, P. R. China;\ne-mail: songrenfu@amss.ac.cn.\n1arXiv:2212.01815v2  [math.AP]  7 Dec 2022suitable boundary measurements. There are lots of literature concerning the inverse\nproblem of recovering time-dependent or time-independent coe\u000ecients in PDEs. We\nwill \frstly consider the inverse problem of recovering the particular case, where the\nnonlinear term fis of time-independent. More precisely, we will recovery f(x;z) in\n(1.1) by means of the Carleman estimates. It is pointed out that, after the methodology\ncreated by [6], the Carleman estimates are well used in inverse problems of uniquely\ndetermining the time-independent coe\u000ecients.\nSecondly, for the time-dependent case, we will recovery b(x;t) andf(x;t;z ) simul-\ntaneously by the higher order linearization method together with constructing some\nGaussian beams. We mention that [18] devoted to an inverse boundary problem for a\nnonlinear parabolic equation, in which the \frst order linearization of the DN map was\nproposed. This approach has been developed and applied in di\u000berent other contexts.\nFor simplicity, we write the inverse problem of recovering f(x;z) as inverse problem\n(I), recovering b(x;t) andf(x;t;z ) as inverse problem (II).\nLetmbe a positive integer. We introduce the following energy space\nEm=m\\\nk=0Ck([0;T];Hm\u0000k(\n))\nwith the norm\njjujj2\nEm= sup\n0\u0014t\u0014TmX\nk=0jj@k\ntu(t)jj2\nHm\u0000kforu2Em:\nThe well-posedness of system (1.1) is discussed in the appendix of this present paper.\n1.1 Recovery of the nonlinear term f(x;z)\nWe consider the following equation.\n8\n><\n>:utt\u0000\u0001gu+b(x;t)ut+f(x;u) = 0;(x;t)2\n\u0002(0;T);\nu(x;t) =h(x;t);(x;t)2\u0000\u0002(0;T);\nu(x;0) =\u0016(x); ut(x;0) = 0; x2\n:(1.2)\nLetu=u(f;\u0016) be a solution to (1.2) with respect to fand\u0016(x):De\fne the input-to-\noutput map as\n\u0003f\nT(\u0016(x)) =@\u0017u(f;\u0016)j\u0000\u0002(0;T);\nwhere\u0017denotes the unit normal \feld pointing outside \n along \u0000 :For simplicity, we\nassume that \u0016(x)2C1(\n):Based on the initial data \u0016(x);the boundary data is given\nby\nu(x;t)j\u0006=h(x;t) =\u0016(x)+t2\n2[\u0001g\u0016(x)\u0000f(x;\u0016(x))]+mX\nk=3tk\nk!@k\ntu(x;0); x2\u0000; m\u00152;\n2where\n@k+2\ntu(x;0) = \u0001g(@k\ntu(x;0))\u0000@k\nt(but)jt=0\u0000@k\ntf(x;u)jt=0; k = 0;1;\u0001\u0001\u0001:\nIt is easy to see that the compatibility conditions hold for system (1.2) up to order m:\nMain assumptions.\nThe following are the main assumptions for the inverse problem (I).\n(A.1)f(x;z) is analytic on Cwith values in C1(M). Moreover, we assume that\nf(x;0) = 0;andf(k)\nz(x;0)j\n0=f0k(x)j\n0;wheref0k(x) is known for each k= 1;2;\u0001\u0001\u0001;\nand \n 0is an arbitrary small neighborhood of \u0000 inside \n :\n(A.2) There exists a non-negative strictly convex function  :\n!R;of classC3\nin the metric g:There exists a positive constant \u001aon\n such that  (x) satis\fes\n(i)D2 (x)(X;X )\u00152\u001ajXj2\ng; x2\n; X2\nx:\n(ii) (x) has no critical point for x2\n:Namely, inf x2\njD (x)j>0:\n(A.3)b(x;t)2C1(M) withb(x;0) = 0:Let\u000e >0 be a small constant. Assume\nthatj\u0016(x)j\u0015\u00160>0 forx2\n;such that\njjhjjHm+1(\u0006)+jj\u0016(x)jjHm+1(\n)\u0014\u000e\n2; m>n\n2;\nWe give the admissible sets of functions b(x;t) andf(x;z) as follows. Let M0be a\npositive constant. We de\fne\nU1=fb(x;t)2C1(M) :b(x;0) = 0;jjbjjC1(M)\u0014M0g; (1.3)\nU2=ff(x;z)2C1(\n\u0002C);fsatis\fes assumption (A :1);\njjf(k)\nz(x;0)jjL1(\n)\u0014M0; k= 1;2;\u0001\u0001\u0001g: (1.4)\nRemark 1.1. Assume that fhas the following Taylor expansion\nf(x;u) =1X\nk=0f(k)\nz(x;0)uk\nk!: (1.5)\nLetf1;f22U 2and letuj=uj(x;t;\u0016;fj) be the solution to (1.2) with respect to fj\nforj= 1;2:We see that assumption (A.1) implies that h(x;t;f1) =h(x;t;f2);where\nh(x;t;fj) is the given boundary data corresponding to the equation\nujtt\u0000\u0001guj+bujt+fj(x;uj) = 0;forj= 1;2:\nAssumption (A.2) are usually used in the Carleman estimates, see for example [5, 30].\nThe existence of convex functions depends on the curvature of (\n ;g):Particularly, for\nthe Euclidean case, we can take  (x) =jx\u0000x0j2withx02Rnn\n:For general Rie-\nmannian manifolds, such  exists locally. There are a number of non-trivial examples\nto give such  ;see [48, Chapter 2.3].\n3We are now in a position to state the main theorem of recovering f(x;z):\nTheorem 1.1. Let assumptions (A.1)-(A.3) hold. Assume that b(x;t)2U 1and\nf1;f22U 2:LetT >T\u0003;whereT\u0003is given by (2.2). Then\n\u0003f1\nT(\u0016(x)) = \u0003f2\nT(\u0016(x)) implies f1(x;z) =f2(x;z);(x;z)2\n\u0002C: (1.6)\nRemark 1.2. Assume that \u0000 0\u001a\u0000 such that\nfx2\u0000 :hD ;\u0017i\u00150g\u001a\u00000:\nThen the measurement can be replaced by @\u0017uj\u00000\u0002(0;T):\n1.2 Recovery of the time-dependent coe\u000ecient b(x;t)and the nonlinear\ntermf(x;t;z )\nWe here consider the nonlinear equation (1.1). Let u(x;t;h;b;f ) be the solution to (1.1)\nwith respect to h;b;f: With the boundary data, we de\fne the Dirichlet-to-Neumann\nmap as\n\u0003b;f\nT(h(x;t)) =@u(x;t;h)\n@\u0017\f\f\f\n\u0006:\nBefore giving the main assumptions for inverse problem (II), we give two de\fnitions\nas follows.\nDe\fnition 1.1. A Riemannian manifold (\n;g)is called a simple manifold if it is\nsimply connected, any geodesic in \nhas no conjugate points, and the boundary \u0000is\nstrictly convex with respect to the metric g(the second fundamental form is positive for\nevery point on the boundary).\nDe\fnition 1.2. A compact Riemannian manifold (\n;g)satis\fes the foliation con-\ndition if there is a smooth strictly convex function, and the boundary \u0000is strictly convex\nwith respect to the metric g.\nThe above conditions are crucial in the inverse problems. They are always used as\na su\u000ecient condition in the geodesic ray transform. The inversion of the geodesic ray\ntransform on \n with depends on the geometric properties of \n. See more details in\nsection 4. Similar to [9], let\nD=f(x;t)2M : dist(x;\u0000)<t<T\u0000dist(x;\u0000)g\nbe the domain of in\ruence. It is clear that no information can be obtained about\nthe coe\u000ecients on MnD;due to the \fnite speed of propagation of waves. Let T >\n2Diam(\n) be given, where\nDiam(\n) = supflengths of all geodesics in (\n;g)g<1:\n4We de\fne a subset of Dby\nE=f(x;t)2M :Dg(x)<t<T\u0000Dg(x)g;\nwhereDg(x) denotes the length of the longest geodesic passing through the point x2\n:\nThe following assumptions are the main assumptions for the inverse problem (II).\n(B.1)f(x;t;z ) is analytic on Cwith values in C1(M);andf(x;t;0) = 0:We de\fne\nan admissible set for fas\nU3=ff2C1(\n\u0002R\u0002C) :fsatis\fes assumption (B :1);\nsuppf(k)\nz(x;t;0)\u001aE;fork= 1;2g: (1.7)\n(B.2)b(x;t)2C1(M):Similar tof;letb0(x;t) be a known smooth function. We\nde\fne an admissible set for bas\nU4=fb(x;t)2C1(M) : suppb(x;t)\u001aEg: (1.8)\n(B.3) Assume that the initial data u(x;0) =u0(x);ut(x;0) =u1(x);and\n(u0(x);u1(x);h(x;t))2Hm+1(\n)\u0002Hm(\n)\u0002Hm+1(\u0006)\nsatisfying the compatibility conditions (3.30) up to order m>n\n2;such that\njju0jjHm+1(\n)+jju1jjHm(\n)+jjh(x;t)jjHm+1(\u0006)\u0014\u000e\n2;\nwhere\u000e>0 is a given small constant.\nSuppose that (\n ;g) is either simple or satis\fes the foliation condition, based on the\nGaussian beams, which can allow the existence of conjugate points, we have\nTheorem 1.2. Assume that uj=u(x;t;bj;fj)solves (1.1) with respect to bjandfj\nfor eachj= 1;2:Let assumptions (B.1), (B.2) and (B.3) hold, and let T >2Diam(\n):\nSuppose that b1;b22U 4; f1;f22U 3andf(2)\nzis known. Then\n\u0003b1;f1\nT(h(x;t)) = \u0003b2;f2\nT(h(x;t)) implies b1=b2inE;andf1=f2inD\u0002C:\n1.3 Literature review\nInverse problems of PDEs have attracted much attention with lots of literature on in-\nverse elliptic equations, parabolic equations, hyperbolic equations, and plate equations,\netc. For example, for the inverse elliptic problems, the well known Calder\u0013 on type in-\nverse problems were investigated in [13, 19, 31, 42] and many subsequent papers. The\nlinear inverse problems of determining either the time-independent coe\u000ecients or the\ntime-dependent coe\u000ecients were widely studied. It is known to us that the Carleman\nestimates method created by [6] have well used in inverse problems to derive the strong\n5Lipschit stability results for the time-independent coe\u000ecients. In this present paper,\nthe Carleman estimate will be used for the recovery of the time-independent nonlin-\nearityf(x;z). However such method deals with the time-independent case only. Also,\nthere are some other methods for the recovery of the time-independent case (or for the\nreal analytic time-dependent coe\u000ecients), such as the boundary control (BC) method\nsteaming from [2, 3] together with Tataru's sharp unique continuation theorem [43].\nWe are not going to list more literature concerning the linear inverse problems.\nThe inverse problems of nonlinear PDEs are much less. Among them, the early\nworks [18, 20, 21] used the linearization procedure to study the recovery of nonlinear\nterms appearing in elliptic or parabolic equations. It turns out that the nonlinear\ninteraction of waves can generate new waves, which are essential for the nonlinear\ninverse problems. For instance, [11, 17, 18, 22, 27, 41] devoted to the unique recovery\nof nonlinear terms or coe\u000ecients appearing in nonlinear elliptic equations. [8] dealt\nwith the stable recovery of a semilinear term appearing in a parabolic equation, and\n[26] studied the fractional semilinear Schr odinger equations.\nFor the inverse problems of hyperbolic equations, we refer to [34, 35], which re-\nspectively concerned the the recovery of a conductivity and quadratic coe\u000ecients. We\nmention that [29] devoted to the recovery of nonlinear terms from source-to-solution\nmap, where method of the nonlinear interactions of distorted plane waves, originated\nfrom [25], was used. Such method has been successfully used in inverse problems of\nnonlinear hyperbolic equations, see for example [15, 25, 45, 47]. Similarly, for some\nsemilinear wave equations, instead of the distorted plane waves, Gaussian beams to-\ngether with the higher order linearization and the stationary phase method are used to\nrecover the coe\u000ecients. For this method, for instance, we refer to [12, 14, 46]. Among\nthose, [14] studied an inverse boundary value problem for a semilinear wave equation\non a time-dependent Lorentzian manifolds. Both the distorted plane waves and the\nGaussian beams were used to derive the uniqueness. In their paper, due to the un-\nsolved inverse problems of recovering the zeroth order term on general manifolds, they\nassumed the nonlinearity has the following form\nH(x;z) =1X\nk=2hk(z)zk; hk2C1:\nIn their paper, the recovery of the quadric term fuuis much complex and interesting.\nThe distorted plane waves were used to construct four future light-like vectors to re-\ncovery the quadric term, but without recovering the zeroth term fu(x;t;0):However,\nthe quadric term is unable to recovery by using the Gaussian beams.\nIt is worth noting that [24] considered the inverse problem of determining a gen-\neral nonlinear term appearing in a semilinear hyperbolic equation on a Riemannian\nmanifold (M;g) with boundary of dimension n= 2;3:They determined the expres-\n6sionF(t;x;u ) both on the boundary x2Mand inside the manifold x2Mfrom some\npartial knowledge of the solutions uon the boundary of the time-space cylindrical man-\nifold (0;T)\u0002Mor on the lateral boundary (0 ;T)\u0002@M: Let us also point out that [32]\ninvestigated inverse boundary problems associated with a time-dependent semilinear\nhyperbolic equation with variable coe\u000ecients. They developed a new method, which\ncombined the observability inequality and a Runge approximation with higher order\nlinearization, to derive the uniqueness of both the initial velocity and the nonlinearity.\nThe measurements they used were either active or passive. For the stability of recover-\ning some coe\u000ecients, [28] devoted to the uniqueness and stability of an inverse problem\nfor a semi-linear wave equation: utt\u0000\u0001u+a(x;t)um= 0;where (x;t)2Rn\u0002Rand\nm\u00152. They used the higher order linearization together with the Radon transform\nto prove the stability results of recovering a(x;t) by the Dirichlet-to-Neumann map. It\nis worth mentioning that, instead of di\u000berentiating the nonlinear equation by@\u000fm\n@\u000f1\u0001\u0001\u0001@\u000fm;\nthey used the \fnite di\u000berences operator Dm\n\u000f:We note that, all the above literature\nconcerning the nonlinear wave equations do not consider the recovery of the \frst order\ncoe\u000ecients, such as the damping term. Motivated by these previous works, we study\nthe recovery of the time-dependent damping term and the nonlinearity simultaneously,\nwhich can be seen as an extended studying of the existing literature.\nThe rest of this present paper is organized as follows: In Section 2, we prove Theorem\n1.1 by the Carleman setimates. We also add some additional contents in this section.\nSection 4 focuses on the Gaussian beams and proofs of Theorem 1.4. Finally, in the\nappendix, the well-posedness of the semilinear wave equation (1.1) is discussed.\n2 Proofs of Theorem 1.1 and an additional result\nIn this section, we focus on proving Theorem 1.1 and give an additional result of\nrecovering a leading coe\u000ecient.\n2.1 Proofs of Theorem 1.1\nLetN\u00151 be a positive inter, and \u000f= (\u000f1;\u0001\u0001\u0001;\u000fN). Let\u0016k(x)6= 0 forx2\n;and\n\u0016(x) =nX\nk=1\u000fk\u0016k(x)2Hm+1(\n)\nwithj\u000fj=NP\nk=1j\u000fkjsu\u000eciently small, such that\njjh(x;t)jjHm+1(\u0006)+jj\u000f1\u00161+\u0001\u0001\u0001+\u000fN\u0016NjjHm+1(\n)\u0014\u000e\n2:\nBy the same output, we have\n\u0003f1(\u000f1\u00161+\u0001\u0001\u0001\u000fN\u0016N) = \u0003f2(\u000f1\u00161+\u0001\u0001\u0001\u000fN\u0016N);\n7which gives\n@j\u000fj\n@\u000f1\u0001\u0001\u0001@\u000fN\f\f\f\n\u000f=0\u0003f1(\u000f1\u00161+\u0001\u0001\u0001\u000fN\u0016N) =@j\u000fj\n@\u000f1\u0001\u0001\u0001@\u000fN\f\f\f\n\u000f=0\u0003f2(\u000f1\u00161+\u0001\u0001\u0001\u000fN\u0016N):(2.1)\nClearly, \u0003f(NP\nk=1\u000fk\u0016k) contains more information than f\u0003f(\u0016k)gk=1;\u0001\u0001\u0001;N:Indeed, useful\ninformation can be extracted from\n@N\n@\u000f1\u0001\u0001\u0001@\u000fN\f\f\f\n\u000f=0\u0003f(NX\nk=1\u000fk\u0016k):\nLet\neQ= \n\u0002(\u0000T;T);e\u0006 = \u0000\u0002(\u0000T;T):\nExtend the domains of ujandb(x;t) to the region eQevenly as usual. Notice that\nu1(x;t) =u1t=b(x;0) = 0;thenu1ttt(x;0) = 0;which implies that the extension of\nu1(x;t) (alsouj\n1(x;t)) is smooth.\nSimilar to [5, Chapter 5.2], we set\n\u001e(x;t) = (x)\u0000\ft2+\f0; ' (x;t) =e\u0015\u001e(x;t);(x;t)2M;\nwhere\u0015is a positive constant, \f2(0;\u001a) and\u001ais given by (i) in assumption (A.2).\nMoreover,\f0>0 is chose such that \u001e(x)>0. Let\nT\u0003=1p\u001a\u0012\nmax\nx2\n (x)\u00131\n2\n: (2.2)\nWe assume that T >T\u0003:Then we can choose \u000e>0 and\f >0 such that\n\u001aT2>max\nx2\n (x) + 4\u000e; \fT2>max\nx2\n (x) + 4\u000e:\nThus,\u001e(x;t) has the following properties:\n(1)\u001e(x;T)\u0014\f0\u00004\u000euniformly for x2\n:\n(2) There exists a small constant \">0 such that\n\u001e(x;t)\u0014\f0\u00002\u000efor (x;t)2\n\u0002[T\u00002\";T][[\u0000T;\u0000T+ 2\"]:\nTherefore\n'(x;t)\u0014e\u0015(\f0\u00002\u000e)=:d1<d 0=:e\u0015\f0uniformly in \n\u0002[T\u00002\";T]:\nWe next give a key Carleman estimate for the linear wave operator\nLv=vtt\u0000\u0001gv+b(x;t)vt+q(x;t)v;\nwhereq;b2L1(M):Let\nH1\n0(eQ) =fu2H1\n0(\n\u0002[\u0000T;T]) :uj\u0000= 0; @l\ntu(x;\u0006T) = 0; l= 0;1g:\nWe have the following from [4].\n8Theorem 2.1. Under assumptions (A.1) and (A.2), there exist constants C=\nC(M0)>0and\u0015\u0003>0, such that for any \u0015>\u0015\u0003, there exists s0=s(\u0015)such that\nZ\neQs[(jDvj+v2\nt) +s2v2]e2s'dgdt\u0014CZ\neQjPvj2e2s'dgdt +CZ\n\u0006j@\u0017vj2e2s'd\u0006 (2.3)\nholds for all v2H1\n0(eQ)ands>s 0>1:\nProof of Theorem 1.1. Based on the above linearization procedure and the\nCarleman estimate, we divide the proofs into two steps.\nStep 1. First order linearization. Let uj=uj(x;t;fj;\u000f)2Em+1be a solution\nto (1.2) with respect to fjand\u0016(x) =\u000f1\u00161(x) +\u0001\u0001\u0001+\u000fN\u0016N(x) forj= 1;2:Then\neuj=uj(x;t;fj;0) solves\n8\n><\n>:eujtt\u0000\u0001geuj+b(x;t)eujt+fj(x;euj) = 0;(x;t)2\n\u0002(0;T);\neuj(x;t) = 0;(x;t)2\u0000\u0002(0;T);\neuj=eujt(x;0) = 0; x2\n;(2.4)\nwhich admits a zero solution euj= 0 sincefj(x;0) = 0 for each j= 1;2:We next\nlinearize the system (1.2) around euj= 0.\nLetN= 1 and let uj\n1=@\n@\u000f1\f\f\f\n\u000f=0uj:Thenuj\n1satis\fes\n8\n>><\n>>:uj\n1tt\u0000\u0001guj\n1+b(x;t)uj\n1t+fju(x;0)uj\n1= 0;(x;t)2eQ;\nuj\n1(x;t) =@\n@\u000f1\f\f\f\n\u000f=0h:=h1;(x;t)2e\u0006;\nuj\n1(x;0) =\u00161(x); uj\n1t(x;0) = 0; x2\n:(2.5)\nSetu1=u1\n1\u0000u2\n1andq(x) =f2u(x;0)\u0000f1u(x;0);then\n8\n><\n>:u1tt\u0000\u0001gu1+b(x;t)u1t+f1u(x;0)u1=q(x)u2\n1;(x;t)2eQ;\nu1(x;t) = 0;(x;t)2e\u0006;\nu1(x;0) =u1t(x;0) = 0; x2\n:(2.6)\nLety1=u1t:Then\n8\n><\n>:y1tt\u0000\u0001gy1+by1t+ (bt+f1u(x;0))y1=q(x)u2\n1t;(x;t)2eQ;\ny1(x;t) = 0;(x;t)2e\u0006;\ny1(x;0) = 0; y1t(x;0) =q(x)\u00161(x); x2\n:(2.7)\nNext, we chose a cut-o\u000b function \u001f(t)2C1\n0([\u0000T;T]) satisfying\n0\u0014\u001f(t)\u00141; \u001f(t) =(\n0; t2[\u0000T;\u0000T+\")[(T\u0000\";T];\n1; t2[\u0000T+ 2\";T\u00002\"]:(2.8)\nLet ^y1=\u001f(t)y1:Then\n^y1tt\u0000\u0001g^y1+b^y1t+ (bt+f1u(x;0))^y1=\u001fq(x)u2\n1t+\u001ft(2y1t+by1) +\u001ftty1:\n9Since the assumption that j\u00161(x)j\u0015c1>0;then\nZ\n\njq(x)j2e2s'(x;0)dg\u0014CZ\n\njq(x)\u00161(x)j2e2s'(x;0)dg\n=\u0000ZT\n0@\n@tZ\n\njy1t(x;t)\u001f(t)j2e2s'dtdg\n=\u00002Z\nQ(\u001f2y1ty1tt+\u001f\u001fty2\n1t+s't\u001f2y2\n1t)e2s'dgdt\n\u0014CsZ\nQ[j(\u001fy1)tj2+\u001f2\nt(y2\n1+y2\n1t)]e2s'dgdt + 2Z\nQ\u001f2y1ty1tte2s'dgdt: (2.9)\nWe compute the term \u001f2y1ty1tte2s'as follows.\n\u001f2y1ty1tte2s'=\u001f2y1t[\u0001gy1\u0000by1t\u0000(bt+f1u(x;0))y1+q(x)u2\n1t]e2s'\n=\u001f2div (y1te2s'Dy1)\u00001\n2(\u001f2e2s'jDy1j2)t+ (\u001f\u001ft+s\u001f2't)jDy1j2e2s'\n\u00002s\u001f2y1thDy1;D'ie2s'\u0000\u001f2[(bt+f1u(x;0))y1\u0000q(x)u2\n1t]e2s'\n\u0014\u001f2div (y1te2s'Dy1)\u00001\n2(\u001f2e2s'jDy1j2)t\n+Cs[jD(\u001fy1)j2+j(\u001fy1)tj2+\u001f2\nt(y2\n1t+jDy1j2)]e2s'+\u001f2q(x)u2\n1te2s':\nMoreover, for system (3.24), the hyperbolic regularity (see, Lemma A.1 in the ap-\npendix) implies that u2\n1t2C([\u0000T;T];Hm(\n)) form>n\n2:Then the Sobolev embedding\ntheorem shows that u2\n1t2L1(eQ):Thus, there exists a positive constant C=C(M0)\nsuch that\n\u0000Z\nQf\u001f2y1t[by1t+ (bt+f1u(x;0))y1\u0000q(x)u2\n1t]ge2s'dgdt\n\u0014CZ\nQ[\u001f2\nty2\n1+j(\u001fy1)tj2+q(x)]e2s'dgdt:\nNotice that1\n2(\u001f2jDy1j2e2s')jT\n0= 0:Therefore, by (2.9) and (2.10), with v= ^y;it follows\nfrom the Carleman estimate (2.3) that\nZ\n\nq2e2s'(x;0)dg\u0014CsZ\nQ(^y2+jD^yj2+ ^y2\nt)e2s'dgdt\n+CsZ\nQ\u001f2\nt(y2\n1+y2\n1t+jDy1j2)e2s'dgdt +CZ\nQq2e2s'dgdt\n\u0014CZ\nQq2e2s'dgdt +Z\nQ(\u001f2\nt+\u001f2\ntt)(y2\n1+y2\n1t+jDy1j2)e2s'dgdt +CeCsjj@\u0017y1jj2\nL2(\u0006):\nBy the standard energy estimate of system (2.7), there exists a positive constant C=\nC(T;M 0) such that\nZ\n\n(y2\n1+y2\n1t+jDy1j2)dg\u0014CZ\n\nq2dg+CZ\n\u0006j@\u0017y1j2d\u0006: (2.10)\n10Since\u001ft;\u001ftt6= 0 in the case where '(x;t)\u0014d1;with (2.10), we have\nZ\n\nq2e2s'(x;0)dg\u0014CZ\nQq2e2s'dgdt +CeCsjj@\u0017y1jj2\nL2(\u0006): (2.11)\nMoreover, by the Lebesgue's theorem, we have\nZ\nQq2e2s'dgdt =Z\n\nq2e2s'(x;0)ZT\n0e\u00002s('(x;0\u0000'(x;t)))dt=o(1)Z\n\nq2e2s'(x;0)dg:\nThus,\njjqjj2\nL2(\n)\u0014CeCsjj@\u0017y1jj2\nL2(\u0006)=CeCsjj@\u0017(u1\n1\u0000u2\n1)tjj2\nL2(\u0006):\nHence@\u0017u1\n1=@\u0017u2\n1on \u0006 implies that f1u(x;0) =f2u(x;0) =fu(x;0) forx2\n:\nStep 2. Higher order linearization. Based on Step 1, we set f1u(x;0) =f2u(x;0) =\nfu(x;0) for simplicity. Let uj\n2=@2\n@\u000f1@\u000f2\f\f\f\n\u000f=0ujforj= 1;2:Then\n8\n><\n>:uj\n2tt\u0000\u0001guj\n2+b(x;t)uj\n2t+fu(x;0)uj\n2+fjuu(x;0)u1\n1u2\n1= 0;(x;t)2\n\u0002(0;T);\nuj\n2(x;t) = 0;(x;t)2\u0000\u0002(0;T);\nuj\n2(x;0) = 0; uj\n2t(x;0) = 0; x2\n:\n(2.12)\nHereu1\n1andu2\n1satisfy\n8\n>><\n>>:uj\n1tt\u0000\u0001guj\n1+b(x;t)uj\n1t+fu(x;0)uj\n1= 0;(x;t)2\n\u0002(0;T);\nuj\n1(x;t) =@\n@\u000fj\f\f\f\n\u000f=0:=hj(x;t);(x;t)2\u0000\u0002(0;T);\nuj\n1(x;0) =\u0016j(x); uj\n1t(x;0) = 0; x2\n:(2.13)\nSety2= (u1\n2\u0000u2\n2)t;then\n8\n><\n>:y2tt\u0000\u0001gy2+b(x;t)y2t+ (bt+fu(x;0))y2=F(x;t);(x;t)2\n\u0002(0;T);\ny2(x;t) = 0;(x;t)2\u0000\u0002(0;T);\ny2(x;0) = 0; y2t(x;0) =\u0000(f1uu\u0000f2uu)(x;0)\u00161(x)\u00162(x); x2\n;(2.14)\nwhereF(x;t) =\u0000(f1uu\u0000f2uu)(x;0)(u1\n1tu2\n1+u1\n1u2\n1t):By a similar argument as that in\nStep 1, we have\n@\u0017u1\n2=@\u0017u2\n2on \u0006)f1uu(x;0) =f2uu(x;0))u1\n2=u2\n2:=u2:\nSuppose that\nf(N\u00001)\n1u(x;0) =f(N\u00001)\n2u(x;0); x2\n:\nLetwN\nj=@N\n@\u000f1\u0001\u0001\u0001\u000fN\f\f\f\n\u000f=0uj:By the recursive assumption, for the expression\nfj(x;u) =1X\nk=0f(k)\nu(x;0)uk\nk!;\n11andvk=@\n@\u000fk\f\f\f\n\u000f=0u;we know that\n@N\n@\u000f1\u0001\u0001\u0001\u000fN\f\f\f\n\u000f=0fj(x;u)\u0000fu(x;0)wN\nj\u0000f(N)\nju(x;0)v1\u0001\u0001\u0001vN\nis already known.\nThe above procedure can be proceeded for N-th linearization to obtain\nf(N)\n1u(x;0) =f(N)\n2u(x;0); x2\n:\nUp to now, by the analyticity of f;we have\nf1(x;z) =1X\nk=1f(k)\n1u(x;0)zk\nk!=1X\nk=1f(k)\n2u(x;0)zk\nk!=f2(x;z):\nThus, the uniqueness result in Theorem 1.1 holds and the proof of Theorem 1.1 is\ncomplete.\n2.2 An additional result of recovering a leading coe\u000ecient\nBased on the Carleman estimate, we introduce here a stability result of recovering a\nleading coe\u000ecient. Suppose that there is a leading coe\u000ecient appearing in the wave\nequation\n%(x)utt\u0000\u0001gu+but+f(x;t;u ) = 0;(x;t)2M: (2.15)\nThe unique recovery of the mass density %(x) for a wave equation is essential in inverse\nproblems. There are literature concerning this topic, see for example [1, 33, 40]. We\ngive a brief discussion of recovering %(x):\nDue to the presence of %(x);we consider a new metric ^ g=%g;then\n\u0001^gu=1\n%\u0001gu+n\u00002\n2%2hD%;Dui:\nLet ^Dbe the Levi-Civita connection in the metric ^ g:It is well known that, for any\nvectorsX;Y\n^DXY=DXY+1\n2g(Dln%;Y)X+1\n2g(Dln%;X)Y+1\n2g(X;Y )Dln%:\nIf (x) is strictly convex in the metric ^ g, then one needs\n^D2 (X;X ) =%g(^DX^D ;X ) =%[g(DX(%\u00001D );X)]\n+1\n2g(D (ln%)X+X(ln%)D#+X( )D(ln%);X)\n=D2 (X;X ) +1\n2D#(ln%)jXj2\ng\u0015(2 +1\n2D (ln%))jXj2\ng\n=1\n%(2 +1\n2D (ln%))jXj2\n^g\u0015#0jXj2\n^g;\n12where#0>0 is a constant. Therefore, we need further assumption on %(x):\ng(D ;D% )\u00152%(#0\u00002):\nLetuj(x;t) =u(x;t;\u001aj) be a solution to (2.15) with respect to %jfor eachj= 1;2:Let\nw=u1\u0000u2;^%=\u001a1\u0000\u001a2:Then\n%2wtt\u0000\u0001gw+bwt+cw=\u0000^%u1tt; (2.16)\nwherec(x;t) =R1\n0fu(x;t;ru 1+ (1\u0000r)u2)dr2L1(M) for su\u000eciently smooth u1and\nu2:Suppose thatj\u0001g\u0016(x)\u0000f(x;0;\u0016(x))j>0 forx2\n;where (uj(x;0);ujt(x;0)) =\n(\u0016(x);0) forj= 1;2:Then a similar argument with the proof of Theorem 1.1 yields\nthe following stability of recovering %(x):\njj%1(x)\u0000%2(x)jjL2(\n)\u0014Cjj@\u0017(u1\u0000u2)tjjL2(\u0006):\nRemark 2.1. Clearly, as we have discussed above, the non-degeneracy of initial\ndatau(x;0);which is viewed as the input is needed. Moreover, the existence of some\nstrictly convex functions, which seems not sharp, should be assumed. Such functions\nguarantee that interior information of solutions to the system arrives at boundary in a\n\fnite time.\n3 Gaussian beams and proofs of Theorem 1.4\nIn this section, as usual, the boundary data his the input. A geodesic \f(t)\u001aM is\ncalled a null geodesic in the metric g=\u0000dt2+g, ifDg\n_\f_\f=g(_\f;_\f) = 0;whereDgis the\nconnection in the metric g:We intend to construct some Gaussian beams around a null\ngeodesic. For completeness, we give the details of constructing the Gaussian beams.\n3.1 Gaussian beams\nLet\f(t) be a null geodesic. We \frstly introduce the Fermi coordinates in a neighbor-\nhood of the null geodesic \f. We follow the constructions in [10], see also [12]. Recall that\n\n\u001a\u001a\n1and the functions are extended smoothly to \n 1:Let\f(t) = (t;\r(t))\u001aR\u0002\n1;\nwhere\r(t) is a unit-speed geodesic in the Riemannian manifold (\n 1;g):Assume that\n\f(t) passes through a point ( t0;x0);wheret02(0;T) and\r(t0) =x02\n:Let\fjoin\ntwo points ( t\u0000;\r(t\u0000)) and (t+;\r(t+)) witht\u00062(0;T) and\r(t\u0006)2\u0000:We extend\fto a\nlarger manifoldM1= (0;T)\u0002\n1such that\r(t) is well de\fned on [ t\u0000\u0000\u000f;t++\u000f]\u001a(0;T)\nwith\u000f>0 su\u000eciently small.\nSince the geodesic \ris parallel along itself, we can choose fe2;\u0001\u0001\u0001;engsuch that\nf_\r(t0);e2\u0001\u0001\u0001;engforms an orthonormal basis of \n x0:Letsdenote the arc length along\n\rfromx0:LetEk(s)2\n\r(s)be the parallel transport of ekalong\rto the point \r(s):\n13We now de\fne a map\nF1:Rn+1!M 1\nsuch that\nF1(y0=t;s=y1;y2\u0001\u0001\u0001;yn) = (t;exp\r(s)(y2E2(s) +\u0001\u0001\u0001+ynEn(s)));\nwhere expp(\u0001) denotes the exponential map on \n 1at the point p. In the new coordinates,\nwe have\ngj\r=nX\nk=1dy2\nkand@gij\n@yk\f\f\f\n\r= 0 for 1\u0014i;j;k\u0014n:\nOn the Lorentzian manifold ( M1;\u0000dt2+g);we introduce the well known Fermi coor-\ndinates near the null geodesic \f: (t\u0000\u0000\u000f;t++\u000f)!M 1as follows.\nLet\na=p\n2(t\u0000\u0000\u000f); b =p\n2(t++\u000f); a 0=p\n2(t\u0000\u0000\u000fp\n2); b 0=p\n2(t++\u000fp\n2);\nand\nz0=1p\n2(t+s) +a\n2; z 1=1p\n2(\u0000t+s) +a\n2; zj=yjfor 2\u0014j\u0014n:\nThen, we have\ngj\f= 2dz0dz1+nX\nk=2dz2\nkand@gij\n@zk\f\f\f\n\f= 0 for 0\u0014i;j;k\u0014n: (3.1)\nFor simplicity, we use the notation z= (z0;z0) = (z0;z1;z00) to denote the so called\nFermi coordinates. The following lemma from [10, Lemma 3.1] (see also [12, Lemma\n1]) is essential for the construction of Gaussian beams.\nLemma 3.1. Let\f: (t\u0000\u0000\u000f;t++\u000f)!M 1be a null geodesic as above. Then\nthere exists a coordinate neighborhood (U;\b)of\f(t\u0000\u0000\u000f\n2;t++\u000f\n2);with the coordinates\ndenoted by (z0;z0)such that\nV= \b(U) = (a;b)\u0002B(0;\u000e);\nwhereB(0;\u000e)denotes a ball in Rnwith a small radius \u000e:\nBased on the above coordinates, we will construct some approximate Gauss beams\nin a neighborhood of \fby de\fning\nV=f(z0;z0)2M 1:z02(a0;b0);jz0j\u0014\u000e0g\nwith 0<\u000e0<\u000esu\u000eciently small such that the set Vdoes not intersect the sets f0g\u0002\nandfTg\u0002\n:\n14We use the shorthand notation Lb;qu=utt\u0000\u0001gu+but+qu:We consider the WKB\nansatz\nu\u001b=ei\u001b'a+r\u001b;\nwhere\u001b >0 is a constant, r\u001bis the reminder term. 'andaare called the amplitude\nand phase respectively. In particular, we will construct '2C1(V) anda2C1\n0(V):\nDirectly calculation yields\nLb;q(ei\u001b'a) =ei\u001b'Lb;qa+ 2 i\u001b('tat\u0000hDa;D'ig)ei\u001b'\n+ i\u001ba('tt\u0000\u0001g'+b't)ei\u001b'+a\u001b2(jD'j2\ng\u0000'2\nt)ei\u001b': (3.2)\nBased on the above equation, we respectively solve\nThe eikonal equation\nS'=jD'j2\ng\u0000'2\nt=hd';d'ig=nX\nk;l=0gkl@k'@l'= 0;\nand the transport equation\nTb(a;') =\u00002hd';daig\u0000(\u0001g'\u0000b't)a\n= 2('tat\u0000hDa;D'ig) + ('tt\u0000\u0001g'+b't)a= 0: (3.3)\nTo achieve this, we make the following ansatz for ';a; namely\n'(z0;z0) =NX\nk=0'k(z0;z0); a =NX\nk=0\u001b\u0000k\u001f(jz0j\n\u000e0)ak(z0;z0); ak=NX\nj=0ak;j(z0;z0):\nHere'kis a homogeneous polynomial of degree kwith respect to the variables zifor\nk= 0;1;\u0001\u0001\u0001;N:In terms of the Fermi coordinates z= (z0;z1;\u0001\u0001\u0001;zn) forg=\u0000dt2+g;\nwe need\n@j\u0002j\n@z\u0002(S')(z0;0) =@j\u0002j\n@z\u0002hd';d'igj\f= 0 for \u0002 = (0 ;\u00121;\u0001\u0001\u0001;\u0012n); z 02(a0;b0);\nwhere\u0012j\u00150 are integers for 1 \u0014j\u0014n;andj\u0002j=nP\nj=1\u0012j\u0014N;Moreover, for\nk= 1;\u0001\u0001\u0001;N;we need\n@j\u0002j\n@z\u0002T(a0;')(z0;0) = 0;@j\u0002j\n@z\u0002( iT(ak;') +Lb;qak\u00001)(z0;0) = 0; z 02(a0;b0):\nConstruction of the phase. We \frstly solve equation (3.4) with j\u0002j= 0:That is,\nnX\nk;l=0gkl@k'@l'= 0 on\f:\n15By (3.1), this reduces to\n2@0'@1'+nX\nk=2(@k')2= 0: (3.4)\nSimilar, forj\u0002j= 1;we have\nnX\nkl=0@2\njk'@l'= 0 for 1\u0014j\u0014n: (3.5)\nClearly, equations (3.4) and (3.5) are satis\fed by respectively setting\n'0= 0; ' 1=z1=1p\n2(\u0000t+s) +a\n2:\nFor the case where j\u0002j= 2;we set\n'2(z0;z0) =nX\ni;j=1Hij(z0)zizj; (3.6)\nwhereHij=Hjiis a complex-valued matrix such that Im His positive de\fnite. It\nfollows from (3.4) and (3.6) that\nnX\nk;l=0(@2\nijgkl@k'@l'+ 2gkl@3\nkij'@l'+ 2gkl@2\nki'@2\nlj'+ 4@igkl@2\njk'@l')j\f= 0: (3.7)\nBy the choices of '0;'1and'2;(3.7) implies that\n(@2\nijg11+ 2g10@3\n0ij'+ 2nX\nk=2@2\nki'@2\nkj')j\f= 0:\nWe \fnally obtain the following Riccati equation for H(z0);namely,\nd\ndz0H+HAH +B= 0; H (s) =H0;with ImH0>0 andz02(a0;b0);(3.8)\nwhereB=1\n4@2\nijg11; s=p\n2t\u0000and the components of A= (Aij) satisfy\n8\n><\n>:A11= 0;\nAii= 2; i= 2;\u0001\u0001\u0001;n;\nAij= 0;otherwise:\nFor the above Riccati equation, we have\nLemma 3.2. [23, Section 8] The Riccati equation (3.8) admits a unique solution.\nThe solution His symmetric and Im (H(z0))>0for allz02(a0;b0):We haveH(z0) =\nZ(z0)Y\u00001(z0);where the matrix valued functions Z(z0);Y(z0)solve the \frst order linear\nsystem\ndZ\ndz0=\u0000BY;dY\ndz0=AZ; subject to Y(s) =I; Z (s) =H0:\n16Moreover, the matrix Y(z0)is non-degenerate on (a0;b0), and there holds\ndet(ImH(z0))\u0001jdet(Y(z0))j2= det(ImH0):\nFor the case where j\u0002j= 3;4\u0001\u0001\u0001;the polynomials 'jof higher degree are con-\nstructed analogously. We omit the details.\nConstruction of the amplitude. Let us consider the transport equation (3.3). Let\nj\u0002j= 0:It follows from (3.1) that b't=\u0000bp\n2and\n\u0001g'=nX\nk;l=0gklD2\ng'(@k;@l) =nX\nk;l=0gkl@2\nkl'= Tr(AH) on\f:\nTherefore, the transport equation (3.3) reduces to\n2@z0a0;0+ [Tr(AH)\u0000bp\n2]a0;0= 0; z 02(a0;b0): (3.9)\nNotice that\nTr(AH) = Tr(A(z0)Z(z0)Y\u00001(z0)) = Tr(dY\ndz0Y\u00001(z0)) =d\ndz0log(detY(z0));\nthen\na0;0(z0) = (det (Y(z0)))\u00001\n2e1\n2p\n2Rz0\nsb(\u001c;0)d\u001c; z 02(a0;b0) (3.10)\nis a solution to (3.9). The subsequent terms ak;0can be constructed by solving some\nlinear ODEs of \frst order. We refer to [12] for more details.\nConstruction of the remainder terms. By a similar proof with [12, Lemma 2],\nthe Gaussian beam has the following property.\nLemma 3.3. Letu\u001b=ei\u001b'abe an approximate Gaussian beam of order Nalong\nthe null geodesic \f:Then for all \u001b>0\njjLb;qu\u001bjjHk(M)\u0014C\u001b\u0000K;\nwhereK=N+1\u0000k\n2\u00001:\nBased on the above lemma and the Sobolev embedding theorem, for su\u000ecient large\nN;the remainder term r\u001bsatis\fes the estimate (cf. [14], [36, Proposition 2.2])\njjr\u001bjjHk+1(M)\u0014C\u001b\u0000K)jjr\u001bjjC(M)\u0014C\u001b\u0000n+1\n2\u00002: (3.11)\n3.2 Proofs of Theorem 1.4\nLet us introduce some basic notations on the geodesic ray transform, which are ex-\nplicitly discussed in, e.g., [9, 10, 38]. The following contents are mainly from [9], we\npresent here for completeness.\n17LetS\n2T\n be the unit sphere bundle of (\n ;g), and by\r(\u0001;x;v) the geodesic with\ninitial data ( x;v)2S\n:For all (x;v)2S\nint;we de\fne the exist times as\n\u001c\u0006(x;v) = inffr>0 :\r(\u0006r;x;v)2\u0000g:\nAssume that (\n ;g) is simple, then \u001c\u0006<Diam(\n):De\fne\n@\u0006S\n =f(x;v)2S\n :x2\u0000;\u0006hv;\u0017(x)ig>0g:\nAll geodesics in \nintcan be parametrized by \r(\u0001;x;v)\u001a\n for (x;v)2@\u0000S\n:The\ngeodesic ray transform on (\n ;g) is de\fned for f2C1(\n) by\nIf(x;v) =Z\u001c+(x;v)\n0f(\r(r;x;v))drfor (x;v)2@\u0000S\n:\nLet\fbe a null geodesic (also called light ray). By the product structure of the\nLorentzian manifold R\u0002\n, we can parametrize the null geodesic \fas\n\f(r;s;x;v ) = (r+s;\r(r;x;v));8(s;x;v )2R\u0002@\u0000S\n:\nThen, all the null geodesics \fthrough\f(\u0001;s;x;v ) with (s;x;v )2R\u0002@\u0000S\n over their\nmaximal intervals [0 ;\u001c+(x;v)] can be identi\fed. The so called light ray transform on\nR\u0002\n can be de\fned as\nGf(s;x;v ) =Z\u001c+(x;v)\n0f(r+s;\r(r;x;v))dr8(s;x;v )2R\u0002@\u0000S\n:\nBy [9, Proposition 1.3] and the discussions in [10, Section 2.2], together with [37,\nTheorem 1.6], we have\nProposition 3.1. Suppose that either (\n;g)is simple or (\n;g)satis\fes the foliation\ncondition. Let f2C1(M))vanish on the setMnE:ThenGf= 0for all maximal null\ngeodesic\f\u001aD impliesf= 0.\nRemark 3.1. For the non-simple case, such ray transform has been also discussed\ne.g., in [39] and references therein.\nWe begin with the \frst order linearized equation\n8\n><\n>:Lbj;qjuj=ujtt\u0000\u0001guj+bjujt+qjuj= 0;(t;x)2M;\nuj(t;x) =h1(t;x); (t;x)2\u0006;\nuj(0;x) =ujt(0;x) = 0; x2\n;(3.12)\nwhereqj=fju(x;t;0):By the de\fnition of Lb;q;we have\nL\u0003\nb;qu=utt\u0000\u0001gu\u0000but+ (q+bt)u;\n18whereL\u0003\nb;qdenotes the formal adjoint of Lb;qwith respect to the L2(M) inner product.\nThe formal adjoint system of (3.12) with j= 1 is given by\n(\nL\u0003\nb1;q1v=vtt\u0000\u0001gv\u0000b1vt+ (q1+b1t)v= 0;(t;x)2M;\nv(T;x) =vt(T;x) = 0; x2\n;(3.13)\nBased on the above constructions of Gaussian beams, we seek such solutions for\nsystems (3.12) and (3.13), respectively. More precisely, we let\nu2=ei\u001b'a1+r1\u001b=ei\u001b'\u001bn\n4\u001f(jz0j\n\u000e0)a10;0+r1\u001b;\nv=e\u0000i\u001b'a2+r2\u001b=ei\u001b'\u001bn\n4\u001f(jz0j\n\u000e0)a20;0+r2\u001b;\nwhere\u0001means the conjugate of \u0001:Thenr1\u001bandr2\u001brespectively solve\n8\n><\n>:Lb2;q2r1\u001b=\u0000Lb2;q2(ei\u001b'a1) inM;\nr1\u001b= 0 on \u0006 ;\nr1\u001b(0) =r1\u001bt(0) = 0 in \n ;(3.14)\nand 8\n><\n>:L\u0003\nb1;q1r2\u001b=\u0000L\u0003\nb1;q1(e\u0000i\u001b'a2) inM;\nr2\u001b= 0 on \u0006 ;\nr2\u001b(T) =r2\u001bt(T) = 0 in \n ;(3.15)\nAccording to (3.2), we have\nLb2;q2(ei\u001b'a1) =ei\u001b'[Lb2;q2a1+\u001b2(S')a1+ i\u001bTb2(a1;')];\nand\nL\u0003\nb1;q1(e\u0000i\u001b'a2) =e\u0000i\u001b'[L\u0003\nb1;q1a2+\u001b2(S')a2\u0000i\u001bT\u0000b1(a2;')]:\nAs we have discussed in Section 4.1, we choose\na10;0(z0) = (detY(z0))\u00001\n2e1\n2p\n2Rz0\nsb2(\u001c;0)d\u001c; z 02(a0;b0);\nand\na20;0(z0) = (detY(z0))\u00001\n2e\u00001\n2p\n2Rz0\nsb1(\u001c;0)d\u001c; z 02(a0;b0):\nClearly,L\u0003\nb1;q1(e\u0000i\u001b'a2) andL\u0003\nb1;q1(e\u0000i\u001b'a2) are compactly supported in a small tubu-\nlar region around the null geodesic where the Fermi coordinates are well de\fned.\nThe following lemma shows that the remainder terms r1\u001bandr2\u001bvanish as\u001b!+1:\nLemma 3.4. [10] Let the remainder terms r1\u001bandr2\u001brespectively solve (3.14) and\n(3.15). Then rj\u001b2C([0;T];H1\n0(\n))\\C1([0;T];L2(\n));and\nlim\n\u001b!1(jjrj\u001bjjL2(M)+\u001b\u00001jjrj\u001bjjH1(M)) = 0;forj= 1;2:\n19We are now in a position to prove Theorem 1.4.\nWe only prove Theorem 1.4 for the case where ( M;g) is simple. If (\n ;g) satis\fes\nthe foliation condition, as in [44] (see also [14]), for any point q2\u0000, there exists a\nwedge-shaped neighborhood Oq\u001a\n ofqsuch that any geodesic in ( Oq;g) has no\nconjugate points. Therefore, we can now recover fk\nz(x;t;0) fork\u00153:Then the folia-\ntion condition allows a layer stripping scheme to recover the coe\u000ecients in the whole\ndomain. However, the recovery of fz(x;t;0) andf(2)\nz(x;t;0) is quite di\u000berent, which\nneeds the inversion of some new ray transforms.\nProof of Theorem 1.4. We divide the proof into three steps.\nStep 1. Leth1=ei\u001b'a1j\u0006in (3.12). Let\nw=u1\u0000u2; b =b1\u0000b2; q =q1\u0000q2:\nThen 8\n><\n>:wtt\u0000\u0001gw+b1wt+q1w=\u0000(bu2t+qu2);(t;x)2M;\nw(t;x) = 0; (t;x)2\u0006;\nw(0;x) =wt(0;x) = 0; x2\n:(3.16)\nNotice that vsolves system (3.13) and \u0003 b1;q1(h1) = \u0003b2;q2(h1) on \u0006 implies @\u0017wj\u0006= 0;\nwe multiply the \frst equation of (3.16) by vand integrate over Mto obtain\n(Lb1;q1w;v)L2(M)= (w;L\u0003\nb1;q1v)L2(M)= 0)Z\nM(bu2t+qu2)vdVg= 0: (3.17)\nHeredVg=jgj1\n2dt^dxdenotes the volume form of the metric g=\u0000dt2+g:Recalling\nthatu2=ei\u001b'a1+r1\u001bandv=e\u0000i\u001b'a2+r2\u001bin a neighborhood of the null geodesic\n\f;we have\n0 = i\u001bZ\nMb'ta1a2e\u00002\u001bIm'dVg+Z\nMqa1a2e\u00002\u001bIm'dVg\n+Z\nMb[a1ta2e\u00002\u001bIm'+ ( i\u001b'ta1+a1t)r2\u001bei\u001b'+e\u0000i\u001b'a2r1\u001bt+r1htr2\u001b]dVg\n+Z\nMq(ei\u001b'a1r2\u001b+e\u0000i\u001b'a2r1\u001b+r1\u001br2\u001b)dVg (3.18)\nIt then follows from Lemma 3.4 that\nlim\n\u001b!1Z\nMb'ta1a2e\u00002\u001bIm'dVg= 0: (3.19)\nSince the functions a1;a2are supported in a small tubular neighborhood of the null\ngeodesic\f, the integrand in (3.19) is supported near \f:Therefore we can use the Fermi\ncoordinates z= (z0;z1;z00) to compute the limit. Recall that\na1a2=\u001bn\n2a10;0a20;0\u001f2(jz0j\n\u000e0) +O(\u001b\u00001) =\u001bn\n2jdetY(z0)j\u00001e\u00001\n2p\n2Rz0\nsb(\u001c;0)d\u001c+O(\u001b\u00001):\n20Moreover, by [10, Lemma 3.6], we know that 'tdoes not vanish in V:Notice that\nb;q= 0 on the setM1nM:Then (3.19) yields\nlim\n\u001b!1\u001bn\n2Zb0\na0Z\njz0j<\u000e0\u001f2(jz0j\n\u000e0)e\u00002\u001bIm'jdetY(z0)j\u00001b(z0;z0)e\u00001\n2p\n2Rz0\nsb(\u001c;0)d\u001cdz0^z0= 0:\nWe proceed to calculate the term\n\u001bn\n2Z\njz0j<\u000e0\u001f2(jz0j\n\u000e0)b(z0;z0)e\u00002\u001bIm'dz0=Z\nRne\u00002\u001bxTPx\u0011(x)dx;\nwhere\u0011(x) =\u001f2(jxj\n\u000e0)b(z0;x) is a smooth function with compact support B(0;\u000e0),P=\nImH(z0) is a positive-de\fnite matrix. By the following well-known formula\nF(e\u00001\n2xTPx)(\u0018) =(2\u0019)n\n2\n(detP)1\n2e\u00001\n2\u0018TP\u00001\u0018;\nwhereFdenotes the Fourier transform, we have\nF(e\u00002\u001bxTPx)(\u0018) =(2\u0019)n\n2\n(detP)1\n2(4\u001b)n\n2e\u00001\n8\u001b\u0018TP\u00001\u0018:\nEquivalently, we have\nF[e\u00001\n2xT(1\n4\u001bP\u00001)x] =(2\u0019)n\n2\ndet (1\n4\u001bP\u00001)1\n2e\u00002\u001bxTPx:\nSince\nZ\nRnF[f](x)g(x)dx=Z\nRnf(x)F[g](x)dx; forf;g2Lp(Rn); p2(1;+1);\nwe have\nZ\nRne\u00001\n8\u001b\u0018TP\u00001\u0018F\u0011(\u0018)d\u0018=Z\nRnF[e\u00001\n8\u001bxTP\u00001x]\u0011(x)dx\n= (2\u0019)n\n2(4\u001b)n\n2(detP)1\n2Z\nRne\u00002\u001bxTPx\u0011(x)dx:\nLetP(\u0018) :Rn!Cbe a measurable function. The well known theory of pseudo-\ndi\u000berential operators tells us\nF[P(D)u](\u0018) =P(\u0018)^u(\u0018) =P(\u0018)F[a](\u0018);P(\u0018)2Sm;\nwhereDis the di\u000berential operator, Smis the symbol class of order m;and\nP(D)u=1\n(2\u0019)nZ\nRneix\u0001\u0018P(\u0018)^u(\u0018)d\u0018:\n21ThereforeZ\nRne\u00002hxTPx\u0011(x)dx=1\n(2\u0019)n\n2(4h)n\n2(detP)1\n2Z\nRne\u00001\n8h\u0018TP\u00001\u0018F\u0011(\u0018)d\u0018\n=1\n(2\u0019)n\n2(4\u001b)n\n2(detP)1\n2+1X\nk=01\nk!(\u00001\n8\u001b)kZ\nRn(\u0018TP\u00001\u0018)kF\u0011(\u0018)d\u0018\n=1\n(4\u001b)n\n2(detP)1\n2+1X\nk=01\nk!(\u00001\n8\u001b)k(PP\u00001(D))k\u0011(0)\n=1\n(4\u001b)n\n2(detP)1\n2(\u0011(0) +O(\u001b\u00001));\nwherePP\u00001(D)\u0011(\u0018) is de\fned by\n(\u0018TP\u00001\u0018)kF\u0011(\u0018) =F[(PP\u00001(D))k\u0011(\u0018)]:\nBy Lemma 3.2, we have\njdetY(z0)j\u00001= (det ImH(z0))1\n2(det ImH0)\u00001\n2\nThen\nlim\n\u001b!1\u001bn\n2Z\njz0j<\u000e0\u001f2(jz0j\n\u000e0)e\u00002\u001bIm'jdetY(z0)j\u00001e\u00001\n2p\n2Rz0\nsb(\u001c;0)d\u001cdz0\n=1\n4n(det ImH0)\u00001\n2b(z0;0)e\u00001\n2p\n2Rz0\nsb(\u001c;0)d\u001c: (3.20)\nCombine (3.19) with (3.20), we \fnd that\n\u00002p\n2Zb0\na0d\ndz0e\u00001\n2p\n2Rz0\nsb(\u001c;0)d\u001cdz0=Zb0\na0b(z0;0)e\u00001\n2p\n2Rz0\nsb(\u001c;0)d\u001cdz0= 0: (3.21)\nTherefore Zb0\na0b(z0;0)dz0= 0;\nwhich implies that Z\n\fb(\f) = 0\nholds for any maximal null geodesic in R\u0002\n. Then Proposition 3.1 is applied to obtain\nb= 0;which implies that b1=b2inE:\nWe return to the equation (3.18) with b= 0 to have\nlim\n\u001b!1Z\nMqa1a2e\u00002\u001bIm'dVg= 0:\nBy a similar argument, this reduces\nlim\n\u001b!1\u001bn\n2Zb0\na0Z\njz0j<\u000e0q\u001f2(jz0j\n\u000e0)jdetY(z0)j\u00001e\u00002\u001bIm'dz0^dz0= 0: (3.22)\n22Then Z\n\fq(\f) = 0\nholds for all maximal null geodesics \finR\u0002\n:Together with Proposition 3.1, we\nconclude that q= 0:\nStep 3. Sincef(2)\nzis known, we proceed with the third and higher order lineariza-\ntion. Let\nW(123)=@3\n@\u000f1@\u000f2@\u000f3\f\f\f\n\u000f=0u(f); W(ij)=@2\n@\u000fi@\u000fj\f\f\f\n\u000f=0u(f);1\u0014i;j\u00143:\nSetm(x;t) =fuuu(x;t;0):Let \u0006(3) be the permutation group of f1;2;3g:Then\n8\n>><\n>>:Lb;qW(123)+fuu(x;t;0)\n2P\n\u00102\u0006(3)W(\u0010(1)\u0010(2))v\u0010(3)+mv1v2v3= 0 inM;\nW(123)= 0 on \u0006 ;\nW(123)(x;0) =W(123)\nt(x;0) = 0 in \n ;(3.23)\nwhere\nLb;qW(123)=W(123)\ntt\u0000\u0001gW(123)+bW(123)\nt+qW(123);\nand for each k= 1;2;3; vk=@\n@\u000fk\f\f\f\n\u000f=0u(f) solves\n8\n><\n>:vktt\u0000\u0001gvk+bvkt+qvk= 0 inM;\nvk=hk on \u0006;\nvk(x;0) =vkt(x;0) = 0 in \n :(3.24)\nLetv0solve the following adjoint system of (3.24)\n8\n><\n>:v0tt\u0000\u0001gv0\u0000bv0t+ (q+bt)v0= 0 inM;\nv0=h0 on \u0006;\nv0(x;T) =v0t(x;T) = 0 in \n :(3.25)\nIntegrating by parts over Myields\nZ\n\u0006v0@3\n@\u000f1@\u000f2@\u000f3\f\f\f\n\u000f=0\u0003(\u000fh1+\u000f2h2+\u000f3h2)d\u0006\n=Z\nMmv1v2v3v0+Z\nMfuu(x;t;0)\n2X\n\u00102\u0006(3)W(\u0010(1)\u0010(2))v\u0010(3)v0dgdt: (3.26)\nSincef2uu(x;t;0) is known. Then the Dirichlet to Neumann map \u0003 determines\nZ\nMmv1v2v3v0dVg: (3.27)\nWe will use special solutions v1;v2;v3;v0in the above identity. Concretely, we shall use\nthe following Gaussian beam solutions ei\u001b'a+r\u001bconstructed in section 4.1.\n23For givenp= (t0;x0)2E;we choose local coordinates such that gcoincides with\nthe standard Minkowski metric at p:De\fne the light cone at pas\nC(p) =f(t;X)2TpM:t2=jXj2\ngg:\nSimilar to [7, Lemma 1] (see also, [14, section 3.2], [12, Section 5]), we can assume\nwithout loss of generality that \u00100;\u001012C(p) satisfying\n\u00100= (1;\u0000p\n1\u0000\u00122;\u0012;0;\u0001\u0001\u0001;0); \u0010 1= (1;1;0;\u0001\u0001\u0001;0)\nfor some\u00122[0;1]:Takinge\u0012>0 small and introduce\n\u00102= (1;q\n1\u0000e\u00122;e\u0012;0;\u0001\u0001\u0001;0); \u0010 3= (1;q\n1\u0000e\u00122;\u0000e\u0012;0;\u0001\u0001\u0001;0):\nBy [7, Lemma 1], \u00100;\u00101are linear-independent, and there are constants k0;k1;k2;k3\nsuch that\nk0\u00100+k1\u00101+k2\u00102+k3\u00103= 0:\nDenote\fkto be the null geodesic with cotangent vector \u0010kandp:Taking\nvk=ei\u001bkk'kak+rk\u001bfork= 0;1;2;3\nas the Gaussian beams concentrating near the null geodesic \fk:Notice that the manifold\n(\n;g) is simple, the null geodesic \fk(k= 0;1;2;3) can intersect only at p:\nInsertingvk=ei\u001bkk'kak+rk\u001binto (3.27), with estimate (3.11), the Dirichlet-to-\nNeumann map determines\n\u001bn+1\n2Z\nMmv0v1v2v3dVg\n=\u001bn+1\n2Z\nMmei\u001b(k0'0+k1'1+k2'2+k3'3)a0a1a2a3dVg+O(\u001b\u00001): (3.28)\nClearly, the product a0a1a2a3is supported in a neighborhood of p:We introduce a\nlemma to deal with the above integral.\nLemma 3.5. [12, Lemma 5]. The function\nS:=k0'0+k1'1+k2'2+k3'3\nis well-de\fned in a neighborhood of pand\n(1)S(p) = 0;\n(2)DgS(p) = 0;\n(3)ImS(p1)\u0015cd(p1;p)forp1in a neighborhood of p;wherec>0is a constant.\nBased on the above lemma, applying the stationary phase (e.g., see [16, Theorem\n7.75]) to (3.28), we\nc\u001bn+1\n2Z\nMmv0v1v2v3dVg=m(p)(a0a1a2a3)(p) +O(\u001b\u00001);\n24wherecdenotes some explicit constant. Thus, the Dirichlet-to-Neumann map deter-\nminesm(p):\nFor the recovery the higher order coe\u000ecients f(k)\nu(x;t;0) fork\u00154;we can achieve\nthis by induction. We refer to [14, Section 4] for such an operation and omit the details.\nTherefore, the proof of Theorem 1.4 is complete.\nRemark 3.2. We notice that the recovery of qandbis much di\u000berent from that of\nhigher order terms f(k)\nu(k\u00153). Lemma 3.4 is a special case of Lemma 3.3, and the\nGaussian beam of order N= 0 is enough for the linear inverse problem of recovering b\nandq.\nIt seems that we can not recover fuuby the same method as that for terms f(k)\nufor\nk= 1 ork\u00153. One of the reason is that we can not choose three time-like vectors\n\u00100;\u00101;\u00102such that\u00100;\u00101are linear-dependent but \u00100;\u00101;\u00102are linear-dependent. We\nmention that, in [46], due to the presence of two di\u000berent matrics gPandgS;the\nauthors have chosen three di\u000berent vectors satisfying the above Lemma 3.5. Therefore,\nthey proved the unique recovery of coe\u000ecients by the Gaussian beams in place of the\ndistorted plane waves method used in e.g., [14].\nAppendix Well-posedness\nWe prove the well-posendess result to (1.1). We begin with the following linear wave\nequation 8\n><\n>:utt\u0000\u0001gu=FinM;\nu=h on \u0006;\nu(x;0) =u0;ut(x;0) =u1(x) in \n:(3.29)\nLet\n(u0;u1)2Hm+1(\n)\u0002Hm(\n); h2Hm+1(\u0006); F2L1([0;T];Hm(\n))\nwith@k\ntF(x;t)2L1([0;T];Hm\u0000k(\n)) fork= 0;1;\u0001\u0001\u0001m:Moreover, we assume that the\ncompatibility conditions hold up to order m;which are given by\n(\nh(x;0) =u0(x)j\u0000; ht(x;0) =u1(x)j\u0000; htt(x;0) = [\u0001gu0(x) +F(x;0)]j\u0000;\n@k\nth(x;0) =@k\ntu(x;0)j\u0000; k= 3;\u0001\u0001\u0001;m:(3.30)\nAccording to [23, Theorem 2.45], for system 3.29 with the above conditions, we have\nLemma 3.6. Letmbe a positive integer and T > 0:Then system (3.29) admits a\nunique solution u2Em+1and@\u0017u2Hm(\u0006):Moreover, the dependence of uand@\u0017u\nonu0;u1;h;F is continuous in the corresponding spaces, i.e.,\njjujjEm+1+jj@\u0017vjjHm(\u0006)\n\u0014C(T)(jju0jjHm+1(\n)+jju1jjHm(\n)+jjhjjHm+1(\u0006)+jjFjjXm); (3.31)\n25wherejjFjj2\nXm=mP\nk=0jj@k\ntFjj2\nL1([0;T];Hm\u0000k):\nAssume further that F2Emandb;q2Cm(M) withm >n\n2:Assume that the\ncompatibility conditions hold for (3.32) up to order m. Applying lemma 3.6, and by\nthe fact that Emis a Banach algebra when m>n\n2;we know that\n8\n><\n>:vtt\u0000\u0001gv+bvt+qv=FinM;\nv=h on \u0006;\nv(x;0) =v0;vt(x;0) =v1(x) in \n:(3.32)\nadmits a unique solution v2Em+1with@\u0017v2Hm(\u0006):Moreover, the following energy\nestimate holds\njjvjjEm+1+jj@\u0017vjjHm(\u0006)\n\u0014C(T)(jjv0jjHm+1(\n)+jjv1jjHm(\n)+jjhjjHm+1(\u0006)+jjFjjXm): (3.33)\nWe now in a position to prove the well-posedness of the nonlinear system (1.1) with\nsmall initial data ( u0;u1) and small boundary data h:\nLetvsolve the following non-homogeneous linear wave equation\n8\n><\n>:vtt\u0000\u0001gv+bvt+fz(x;t;0)v= 0 inM;\nv=h on \u0006;\nv(x;0) =u0;vt(x;0) =u1(x) in \n:(3.34)\nForL>0 given, let\nBm+1(L) =f(y1;y2)2Hm+1(\n)\u0002Hm(\n) :jjy1jjHm+1+jjy2jjHm\u0014Lg;\nNm+1(L) =fh2Hm+1(\u0006) :jjhjjHm+1(\u0006)\u0014Lg\u001aHm+1(\u0006):\nLetu0;u12Bm+1(\"0=3);andh2Nm+1(\"0=3) for\"0su\u000eciently small. Then\njjvjjEm+1+jj@\u0017vjjHm(\u0006)\u0014C(T)\"0: (3.35)\nFor given\u000e0>0 su\u000eciently small, let\nZm+1(\u000e0) =f^w2Em+1:jj^wjjEm+1\u0014\u000e0g\u001aEm+1:\nLetwsolve the following homogeneous equation\n8\n>><\n>>:wtt\u0000\u0001gw+bwt+fz(x;t;0)w+1P\nk=2f(k)(x;t;0)(v+ ^w)k\nk!= 0 inM;\nw= 0 on \u0006 ;\nw(x;0) =wt(x;0) = 0 in \n ;(3.36)\nwherev2Em+1is the solution of (3.34) with estimate (3.35). We de\fne a map\nA:Zm+1(\u000e0)!Em+1;\n26which sends the given ^ w2Zm+1(\u000e0) to the solution of (3.36). For any positive integers\nkandR;one has\njjf(k)(x;t;0)jjEm+1\u0014k!\nRksup\njzj=Rjjf(x;t;z )jjEm+1:\nBy the a priori estimate of (3.36) and the property of Banach algebra, we have\njjA^wjjEm+1\u0014C(T)1X\nk=21\nk!jjf(k)(x;t;0)jjEm+1jjv+ ^wjjk\nEm+1\n\u0014C(T)1X\nk=21\nRksup\njzj=Rjjf(x;t;z )jjEm+12k\u00001(jjvjjk\nEm+1+jj^wjjk\nEm+1)\n\u0014C(T)\nRsup\njzj=Rjjf(x;t;z )jjEm+11X\nk=12k\nRk(\u000e0\u000ek\n0+\"0\"k\n0)\n=2C(T)\nRsup\njzj=Rjjf(x;t;z )jjEm+1(\u000e0\u000e0\nR\u00002\u000e0+\"0\"0\nR\u00002\"0): (3.37)\nLetR= 2 and\"0=\u000e0\u00141\n2small enough such that\nC(T) sup\njzj=Rjjf(x;t;z )jjEm+1\u000e0\u00141\n2:\nThen we have\njjA^wjjEm+1\u0014\u000e0;\nwhich implies that A:Zm+1(\u000e0)!Zm+1(\u000e0) is well de\fned. Let\nF(x;t;^w) =1X\nk=2f(k)(x;t;0)(v+ ^w)k\nk!:\nThen\nF(x;t;^w1)\u0000F(x;t;^w2)\n=1X\nk=2f(k)(x;t;0)\n(k\u00001)!Z1\n0[\u001c(v+ ^w1) + (1\u0000\u001c)(v+ ^w2)]k\u00001d\u001c:\nTherefore, taking \u000e0small enough, we see that\njjA^w1\u0000A^w2jjEm+1=jjw1\u0000w2jjEm+1\n\u0014C(T)1X\nk=2k\nRksup\njzj=Rjjf(x;t;z )jjEm+1(3\u000e0)k\u00001jj^w1\u0000^w2jjEm+1\n\u00141\n2jj^w1\u0000^w2jjEm+1: (3.38)\nThus,A:Zm+1(\u000e0)!Zm+1(\u000e0) is a contraction. The Banach's \fxed point theorem\nimplies that u=v+w2Em+1is a solution to system (1.1). Moreover, we have\njjujjEm+1+jjujjC(M)+jj@\u0017ujjHm(\u0006)\u0014C(jju0jjHm+1(\n)+jju1jjHm(\n)+jjhjjHm+1(\u0006));\nwhereC > 0 is independent of u0;u1andh:\n27References\n[1] L. Beilina, M. Cristofol, S. Li, and M. Yamamoto, Lipschitz stability for an in-\nverse hyperbolic problem of determining two coe\u000ecients by a \fnite number of\nobservations, Inverse Problems 34 (2018) 015001.\n[2] M. 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Yao, Modeling and Control in Vibrational and Structural Dynamics. A Dif-\nferential Geometric Approach, Chapman and Hall/CRC Applied Mathematics and\nNonlinear Science Series, CRC Press, Boca Raton, FL, 2011.\n31"    },    {        "title": "2011.03311v1.A_generalized_finite_element_method_for_the_strongly_damped_wave_equation_with_rapidly_varying_data.pdf",        "content": "arXiv:2011.03311v1  [math.NA]  6 Nov 2020A GENERALIZED FINITE ELEMENT METHOD FOR THE\nSTRONGLY DAMPED WAVE EQUATION WITH RAPIDLY\nVARYING DATA\nPER LJUNG1, AXEL M ˚ALQVIST1, AND ANNA PERSSON2\nAbstract. We propose a generalized finite element method for the strong ly\ndamped wave equation with highly varying coefficients. The pr oposed method\nis based on the localized orthogonal decomposition introdu ced in [30], and\nis designed to handle independent variations in both the dam ping and the\nwave propagation speed respectively. The method does so by a utomatically\ncorrecting for the damping in the transient phase and for the propagation\nspeed in the steady state phase. Convergence of optimal orde r is proven in\nL2(H1)-norm, independent of the derivatives of the coefficients. W e present\nnumerical examples that confirm the theoretical findings.\n1.Introduction\nThis paper is devoted to the study of numerical solutions to the str onglydamped\nwaveequation with highly varying coefficients. The equation takes th e generalform\n(1.1) ¨ u−∇·(A∇˙u+B∇u) =f,\non a bounded domain Ω. Here, AandBrepresent the system’s damping and\nwave propagation respectively, fdenotes the source term, and the solution uis a\ndisplacement function. This equation commonly appears in the modellin g of vis-\ncoelastic materials, where the strong damping −∇ ·A∇˙uarises due to the stress\nbeing represented as the sum of an elastic part and a viscous part [6 , 13]. Viscoelas-\ntic materials have several applications in engineering, including noise d ampening,\nvibration isolation, and shock absorption (see [21] for more applica tions). In partic-\nular, in multiscale applications, such as modelling of porous medium or co mposite\nmaterials, AandBare both rapidly varying.\nThere have been much recent work regarding strongly damped wav e equations.\nFor instance, well-posedness of the problem is discussed in [7, 20, 2 2], asymptotic\nbehavior in [8, 3, 31, 35] solution blowup in [12, 2], and decay estimates in [18].\nIn particular, FEM for the strongly damped wave equation has been analyzed in\n[25] using the Ritz–Volterra projection, and [24] uses the classical Ritz-projection\nin the homogeneous case with Rayleigh damping. In these papers, co nvergence\nof optimal order is shown. However, in the case of piecewise linear po lynomials,\nthe convergence relies on at least H2-regularity in space. Consequently, since the\nKey words and phrases. Strongly damped wave equation, multiscale, localized orth ogonal de-\ncomposition, finite element method, reduced basis method.\n1Department of Mathematical Sciences, Chalmers University of Technology and University of\nGothenburg, SE-412 96 Gothenburg, Sweden.\n2Department of Mathematics, KTH Royal Institute of Technolo gy,, SE-100 44 Stockholm,\nSweden.\n12 PER LJUNG, AXEL M ˚ALQVIST, AND ANNA PERSSON\nH2-norm depends on the derivatives of the coefficients, the error is b ounded by\n/⌊a∇d⌊lu/⌊a∇d⌊lH2∼max(ε−1\nA,ε−1\nB) whereεAandεBdenote the scales at which AandBvary\nrespectively. The convergenceorderisthus onlyvalidwhen the mes hwidthhfulfills\nh <min(εA,εB). In other words, we require a mesh that is fine enough to resolve\nthe variations of AandB, which becomes computationally challenging. This type\nof difficulty is common for equations with rapidly varying data, an issue for which\nseveral numerical methods have been developed (see e.g. [5, 4, 23 , 29, 19]). None\nof these methods are however applicable to the strongly damped wa ve equation,\nwhere two different multiscale coefficients have to be dealt with. In th is paper, we\nproposea novelmultiscale method based on the localized orthogona ldecomposition\n(LOD) method.\nThe LOD method is based on the variational multiscale method presen ted in\n[17]. It was first introduced in [30], and has since then been further d eveloped and\nanalyzed for several types of problems (see e.g. [27, 28, 1, 16, 1 5]). In particular,\n[26] studies the LOD method for quadratic eigenvalue problems, whic h correspond\nto time-periodic wave equations with weak damping. The main idea of th e method\nis based on a decomposition of the solution space into a coarse and a fi ne part.\nThe decomposition is done by defining an interpolant that maps funct ions from\nan infinite dimensional space into a finite dimensional FE-space. In th is way, the\nkernel of the interpolant captures the finescale features that t he coarse FE-space\nmisses, and hence defines the finescale space. Subsequently, one may use the or-\nthogonal complement to this finescale space with respect to a prob lem-dependent\nRitz-projection as a modified FE-space. In the case of time-depen dent problems,\nthe LODmethod performs particularlywell in the sense that the mod ified FE-space\nonly needs to be computed once, and can then be re-used in each tim e step.\nMultiscale methods, as the localized orthogonal decomposition, are usually de-\nsigned to handle problems with a single multiscale coefficient. In this sen se, the\nstrongly damped wave equation is different, as an extra coefficient a ppears due to\nthe strong damping. Hence, one of the main challenges for the nove l method is\nhow to incorporate the finescale behavior of both coefficients in the computation.\nNevertheless, it should be noted that existing multiscale methods ar e applicable\nfor some special cases of this equation. An example is the case of Ra yleigh damp-\ning where the coefficients are proportional to each other. Other e xamples are the\nsteady state case, the transient phase in which the solution evolve s rapidly in time,\nas well as the case of weak damping where no spatial derivatives are present on the\ndamping term.\nIn this paper we present a generalized finite element method (GFEM) for solving\nthe strongly damped wave equation. The method uses both the dam ping and\ndiffusion coefficients to construct a generalized finite element space , similar to those\nin e.g. [30, 28]. The solution is then evaluated in this space, but to acco unt for the\ntime dependence, an additional correction is added to it. However, this correction\nis evaluated on the fine scale, and thus expensive to compute. To ov ercome this\nissue, we prove spatial exponential decay for the corrections so that we can restrict\nthe problems to patches in a similar manner as for the modified basis fu nctions in\n[30]. The effect of the proposed method is that the multiscale basis co mpensates\nfor the damping early on in the simulation when it is dominant and then gr adually\nstarts to compensate for the wave propagation which is dominant a t steady state.\nThis is done seamlessly and automatically by the method. Furthermor e, we proveA GENERALIZED FEM FOR THE STRONGLY DAMPED WAVE EQUATION 3\noptimal order convergence in L2(H1)-norm for this method. Following this, we\nshow that it is sufficient to compute the finescale corrections for on ly a few time\nsteps by applying reduced basis (RB) techniques. For related work on RB methods,\nsee e.g. [14, 10, 9], and for an introduction to the topic we refer to [ 33].\nThe outline of the paper is as follows: In Section 2 we present the wea k formula-\ntionandclassicalFEMforthestronglydampedwaveequation, along withnecessary\nassumptions. Section 3 is devoted to the generalized finite element m ethod and its\nlocalizationprocedure. In Section 4 errorestimates for the metho d areproven. Sec-\ntion 5 covers the details of the RB approach, and finally in Section 6 we illustrate\nnumerical examples that confirm the theory derived in this paper.\n2.Weak formulation and classical FEM\nWe consider the wave equation with strong damping of the following fo rm\n¨u−∇·(A∇˙u+B∇u) =f,in Ω×(0,T], (2.1)\nu= 0,on Γ×(0,T], (2.2)\nu(0) =u0,in Ω, (2.3)\n˙u(0) =v0in Ω, (2.4)\nwhereT >0 and Ω is a polygonal (or polyhedral) domain in Rd, d= 2,3,and\nΓ :=∂Ω. The coefficients AandBdescribe the damping and propagation speed\nrespectively,and fdenotesthesourcefunction ofthesystem. Weassume A=A(x),\nB=B(x) andf=f(x,t), i.e. the multiscale coefficients are independent of time.\nDenote by H1\n0(Ω) the classical Sobolev space with norm\n/⌊a∇d⌊lv/⌊a∇d⌊l2\nH1(Ω)=/⌊a∇d⌊lv/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊l∇v/⌊a∇d⌊l2\nL2(Ω)\nwhose functions vanish on Γ. Moreover, let Lp(0,T;B) be the Bochner space with\nnorm\n/⌊a∇d⌊lv/⌊a∇d⌊lLp(0,T;B)=/parenleftbigg/integraldisplayT\n0/⌊a∇d⌊lv/⌊a∇d⌊lp\nBdt/parenrightbigg1/p\n, p∈[1,∞),\n/⌊a∇d⌊lv/⌊a∇d⌊lL∞(0,T;B)= esssup\nt∈[0,T]/⌊a∇d⌊lv/⌊a∇d⌊lB,\nwhereBis a Banach space with norm /⌊a∇d⌊l·/⌊a∇d⌊lB. In this paper, following assumptions\nare made on the data.\nAssumptions. The damping and propagation coefficients A,B∈L∞(Ω,Rd×d)are\nsymmetric and satisfy\n0< α−:= essinf\nx∈Ωinf\nv∈Rd\\{0}A(x)v·v\nv·v<esssup\nx∈Ωsup\nv∈Rd\\{0}A(x)v·v\nv·v=:α+<∞,\n0< β−:= essinf\nx∈Ωinf\nv∈Rd\\{0}B(x)v·v\nv·v<esssup\nx∈Ωsup\nv∈Rd\\{0}B(x)v·v\nv·v=:β+<∞.\nIn addition, we assume that f∈L∞([0,T];L2(Ω))and˙f∈L2([0,T];L2(Ω)).\nFor the spatial discretization, let {Th}h>0denote a family of shape regular el-\nements that form a partition of the domain Ω. For an element K∈ Th, let the4 PER LJUNG, AXEL M ˚ALQVIST, AND ANNA PERSSON\ncorresponding mesh size be defined as hK:= diam( K), and denote the largest di-\nameter of the partition by h:= max K∈ThhK. We now define the classical FE-space\nusing continuous piecewise linear polynomials as\nSh:={v∈ C(¯Ω) :v/vextendsingle/vextendsingle\nΓ= 0,v/vextendsingle/vextendsingle\nKis a polynomial of partial degree ≤1,∀K∈ Th},\nand letVh=Sh∩H1\n0(Ω). The semi-discrete FEM becomes: find uh(t)∈Vhsuch\nthat\n(2.5) (¨ uh,v)+a(˙uh,v)+b(uh,v) = (f,v),∀v∈Vh, t >0,\nwith initial values uh(0) =uh,0and ˙uh(0) =vh,0whereuh,0,vh,0∈Vhare appropri-\nate approximations of u0andv0respectively. Here ( ·,·) denotes the usual L2-inner\nproduct, a(·,·) = (A∇·,∇·), andb(·,·) = (B∇·,∇·).\nFor the temporal discretization, let 0 = t0< t1< ... < t N=Tbe a uniform\npartition with time step tn−tn−1=τ. The time step here is chosen uniformly for\nsimplicity, but the choice of varying time step is still viable. We apply a ba ckward\nEuler scheme to get the fully discrete system: find un\nh∈Vhsuch that\n(2.6) ( ¯∂2\ntun\nh,v)+a(¯∂tun\nh,v)+b(un\nh,v) = (fn,v),∀v∈Vh\nforn≥2. Here, the discrete derivative is defined as ¯∂tun\nh= (un\nh−un−1\nh)/τ.\nFor results on regularityand errorestimates for the FEM solution o f the strongly\ndamped wave equation, we refer to [24]. Moreover, existence and u niqueness of a\nsolution to (2.6) is guaranteed by Lax–Milgram.\nIn the analysis, we use the notations /⌊a∇d⌊l · /⌊a∇d⌊l2\na:=a(·,·),/⌊a∇d⌊l · /⌊a∇d⌊l2\nb:=b(·,·), as well as\n|||·|||2= ˜a(·,·) :=a(·,·) +τb(·,·), and the fact that these are equivalent with the\nH1-norm. That is, there exist positive constants Ca,Cb,C˜a,ca,cb,c˜a∈R, such that\nca/⌊a∇d⌊lv/⌊a∇d⌊l2\nH1≤ /⌊a∇d⌊lv/⌊a∇d⌊l2\na≤Ca/⌊a∇d⌊lv/⌊a∇d⌊l2\nH1,∀v∈H1(Ω),\ncb/⌊a∇d⌊lv/⌊a∇d⌊l2\nH1≤ /⌊a∇d⌊lv/⌊a∇d⌊l2\nb≤Cb/⌊a∇d⌊lv/⌊a∇d⌊l2\nH1,∀v∈H1(Ω), (2.7)\nc˜a/⌊a∇d⌊lv/⌊a∇d⌊l2\nH1≤ |||v|||2≤C˜a/⌊a∇d⌊lv/⌊a∇d⌊l2\nH1,∀v∈H1(Ω).\nTheorem 2.1. The solution un\nhto(2.6)satisfies the following bounds\n/⌊a∇d⌊l¯∂tun\nh/⌊a∇d⌊l2\nL2+n/summationdisplay\nj=2τ/⌊a∇d⌊l¯∂tuj\nh/⌊a∇d⌊l2\nH1+/⌊a∇d⌊lun\nh/⌊a∇d⌊l2\nH1≤Cn/summationdisplay\nj=2τ/⌊a∇d⌊lfj/⌊a∇d⌊l2\nH−1+C(/⌊a∇d⌊l¯∂tu1\nh/⌊a∇d⌊l2\nL2+/⌊a∇d⌊lu1\nh/⌊a∇d⌊l2\nH1),(2.8)\nn/summationdisplay\nj=2τ/⌊a∇d⌊l¯∂2\ntuj\nh/⌊a∇d⌊l2\nL2+/⌊a∇d⌊l¯∂tun\nh/⌊a∇d⌊l2\nH1≤Cn/summationdisplay\nj=2τ/⌊a∇d⌊lfj/⌊a∇d⌊l2\nL2+C(/⌊a∇d⌊l¯∂tu1\nh/⌊a∇d⌊l2\nH1+/⌊a∇d⌊lu1\nh/⌊a∇d⌊l2\nH1),. (2.9)\nforn≥2.\nProof.To prove (2.8), choose v=τ¯∂tun\nhin (2.6) to get\nτ(¯∂2\ntun\nh,¯∂tun\nh)+τ/⌊a∇d⌊l¯∂tun\nh/⌊a∇d⌊l2\na+τb(un\nh,¯∂tun\nh) =τ(fn,¯∂tun\nh). (2.10)\nDue to Cauchy–Schwarz and Young’s inequality we have the following lo wer bound\nτ(¯∂2\ntun\nh,¯∂tun\nh) =/⌊a∇d⌊l¯∂tun\nh/⌊a∇d⌊l2\nL2−(¯∂tun−1\nh,¯∂tun\nh)≥1\n2/⌊a∇d⌊l¯∂tun\nh/⌊a∇d⌊l2\nL2−1\n2/⌊a∇d⌊l¯∂tun−1\nh/⌊a∇d⌊l2\nL2,\nand similarly\nτb(un\nh,¯∂tun\nh)≥1\n2/⌊a∇d⌊lun\nh/⌊a∇d⌊l2\nb−1\n2/⌊a∇d⌊lun−1\nh/⌊a∇d⌊l2\nb.A GENERALIZED FEM FOR THE STRONGLY DAMPED WAVE EQUATION 5\nSimilar bounds will be used repeatedly throughout the paper. Summin g (2.10) over\nngives\n1\n2/⌊a∇d⌊l¯∂tun\nh/⌊a∇d⌊l2\nL2−1\n2/⌊a∇d⌊l¯∂tu1\nh/⌊a∇d⌊l2\nL2+n/summationdisplay\nj=2τ/⌊a∇d⌊l¯∂tuj\nh/⌊a∇d⌊l2\na+1\n2/⌊a∇d⌊lun\nh/⌊a∇d⌊l2\nb−1\n2/⌊a∇d⌊lu1\nh/⌊a∇d⌊l2\nb≤n/summationdisplay\nj=2τ/⌊a∇d⌊lfj/⌊a∇d⌊lH−1/⌊a∇d⌊l¯∂tuj\nh/⌊a∇d⌊lH1.\nUsing the equivalence of the norms (2.7), Cauchy–Schwarz and You ng’s (weighted)\ninequality to subtract/summationtextn\nj=2τ/⌊a∇d⌊l¯∂tuj\nh/⌊a∇d⌊l2\nH1from both sides, we get exactly (2.8).\nThe proof of (2.9) is similar. We choose v=τ¯∂2\ntun\nhin (2.6) and sum over nto\nget\nn/summationdisplay\nj=2τ/⌊a∇d⌊l¯∂2\ntuj\nh/⌊a∇d⌊l2\nL2+1\n2/⌊a∇d⌊l¯∂tun\nh/⌊a∇d⌊l2\na−1\n2/⌊a∇d⌊l¯∂tu1\nh/⌊a∇d⌊l2\na+n/summationdisplay\nj=2τb(uj\nh,¯∂2\ntuj\nh)≤n/summationdisplay\nj=2τ/⌊a∇d⌊lfj/⌊a∇d⌊lL2/⌊a∇d⌊l¯∂2\ntuj\nh/⌊a∇d⌊lL2.\nFor the sum involving the bilinear form b(·,·) we use summation by parts to get\nn/summationdisplay\nj=2τb(uj\nh,¯∂2\ntuj\nh) =n/summationdisplay\nj=3−τb(¯∂tuj\nh,¯∂tuj−1\nh)−b(u2\nh,¯∂tu1\nh)+b(un\nh,¯∂tun\nh).\nUsing (2.8), the equivalence of the norms (2.7), and Young’s weighte d inequality\nwe have/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen/summationdisplay\nj=3τb(¯∂tuj\nh,¯∂tuj−1\nh)+b(u2\nh,¯∂tu1\nh)−b(un\nh,¯∂tun\nh)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤Cn/summationdisplay\nj=3τ/⌊a∇d⌊l¯∂tuj\nh/⌊a∇d⌊l2\nH1+C(/⌊a∇d⌊lu2\nh/⌊a∇d⌊l2\nH1+/⌊a∇d⌊l¯∂tu1\nh/⌊a∇d⌊l2\nH1)+C/⌊a∇d⌊lun\nh/⌊a∇d⌊l2\nH1+Cǫ/⌊a∇d⌊l¯∂tun\nh/⌊a∇d⌊l2\na\n≤Cn/summationdisplay\nj=2τ/⌊a∇d⌊lfj/⌊a∇d⌊l2\nH−1+C(/⌊a∇d⌊l¯∂tu1\nh/⌊a∇d⌊l2\nH1+/⌊a∇d⌊lu1\nh/⌊a∇d⌊l2\nH1)+Cǫ/⌊a∇d⌊l¯∂tun\nh/⌊a∇d⌊l2\na.\nSinceCǫcan be made arbitrarily small, it can be kicked to the left hand side. Usin g\nthat/⌊a∇d⌊lfj/⌊a∇d⌊l2\nH−1≤C/⌊a∇d⌊lfj/⌊a∇d⌊l2\nL2we deduce (2.9).\n/square\n3.Generalized finite element method\nThis section is dedicated to the development of a multiscale method ba sed on\nthe framework of the standard LOD. First of all, we introduce some notation for\nthe discretization. Let VHbe a FE-space defined analogously to Vhin previous\nsection, but with larger mesh size H > h. Moreover, we assume that corresponding\nfamily of partitions {TH}H>his, in addition to shape-regular, also quasi-uniform.\nDenote by Nthe set of interior nodes of VHand byλxthe standard hat function\nforx∈ N, such that VH= span({λx}x∈N). Finally, we make the assumption that\nThis a refinement of TH, such that VH⊆Vh.\n3.1.Ideal method. Todefineageneralizedfiniteelementmethod forourproblem,\nwe aim to construct a multiscale space Vmsof the same dimension as VH, but with\nbetter approximation properties. For the construction of such a multiscale space,\nletIH:Vh→VHbe an interpolation operator that has the projection property\nIH=IH◦IHand satisfies\n(3.1)H−1\nK/⌊a∇d⌊lv−IHv/⌊a∇d⌊lL2(K)+/⌊a∇d⌊l∇IHv/⌊a∇d⌊lL2(K)≤CI/⌊a∇d⌊l∇v/⌊a∇d⌊lL2(N(K)),∀K∈ TH, v∈Vh,6 PER LJUNG, AXEL M ˚ALQVIST, AND ANNA PERSSON\nwhereN(K) :={K′∈ TH:K′∩K/\\e}atio\\slash=∅}.Furthermore, for a shape-regular and\nquasi-uniform partition, the estimate (3.1) can be summed into the g lobal estimate\nH−1/⌊a∇d⌊lv−IH/⌊a∇d⌊lL2(Ω)+/⌊a∇d⌊l∇IHv/⌊a∇d⌊lL2(Ω)≤Cγ/⌊a∇d⌊l∇v/⌊a∇d⌊lL2(Ω),\nwhereCγdepends on the interpolation constant CIand the shape regularity pa-\nrameter defined as\nγ:= max\nK∈THγK,whereγK=diam(BK)\ndiam(K).\nHereBKdenotes the largest ball inside K. A commonly used example of such an\ninterpolant is IH=EH◦ΠH, where Π His the piecewise L2-projection onto P1(TH),\nthe space of functions that are affine on each triangle K∈ TH, andEH:P1(TH)→\nVHis an averaging operator that, to each free node x∈ N, assigns the arithmetic\nmean of corresponding function values on intersecting elements, i.e .\n(EH(v))(x) =1\ncard{K∈ TH:x∈K}/summationdisplay\nK∈TH:x∈Kv/vextendsingle/vextendsingle\nK(x).\nFor more discussion regarding possible choices of interpolants, see e.g. [11] or [32].\nLet the space Vfbe defined by the kernel of the interpolant, i.e.\nVf= ker(IH) ={v∈Vh:IHv= 0}.\nThat is, Vfis a finescale space in the sense that it captures the features that are\nexcluded from the coarse FE-space. This consequently leads to th e decomposition\nVh=VH⊕Vf,\nsuch that every function v∈Vhhas a unique decomposition v=vH+vf, where\nvH∈VHandvf∈Vf.\nIn the case of the LOD method for the standard wave equation (se e [1]), one\nconsiders a Ritz-projection based solely on the B-coefficient to construct a multi-\nscale space. Instead, the goal is to define a multiscale space based on the inner\nproduct a(·,·) +τb(·,·) (for a fixed τ) and add additional correction to account\nfor the time-dependency. This particular choice of scalar product comes from the\nbackward Euler time-stepping formulation and both simplifies the ana lysis and is\nmorenaturalin the implementation. Another possibility is to choose a(·,·) asscalar\nproduct. For v∈VH, we consider the Ritz-projection Rf:VH→Vfdefined by\na(Rfv,w)+τb(Rfv,w) =a(v,w)+τb(v,w),∀w∈Vf.\nUsing this projection, we may define the multiscale space Vms:=VH−RfVHsuch\nthat\n(3.2) Vh=Vms⊕Vf,anda(vms,vf)+τb(vms,vf) = 0.\nNote that dim( Vms) = dim( VH), and hence we can view Vmsas a modified coarse\nspace that contains finescale information of AandB. Next, we may use the Ritz-\nprojection to define the basis functions for the space Vms. Forx∈ N, denote by\nφx:=Rfλx∈Vfthe solution to the (global) corrector problem\n(3.3) a(φx,w)+τb(φx,w) =a(λx,w)+τb(λx,w),∀w∈Vf.\nWecannowconstructourbasisfor Vmsas{λx−φx}x∈Nwhichincludesthebehavior\nof the coefficients. For an illustration of the Ritz-projected hat fu nction, as well as\nthe modified basis function for Vms, see Figure 1.A GENERALIZED FEM FOR THE STRONGLY DAMPED WAVE EQUATION 7\n(a)λx−φx. (b)φx.\nFigure 1. The modified basis function λx−φxand the Ritz-\nprojected hat function φx.\nWe may now formulate our ideal (but impractical) method. Since the s olution\nspace can be decomposed as Vh=Vms⊕Vf, the idea is to solve a coarse scale\nproblem in Vms, and then add additional correction from a problem on the fine\nscale. The method reads: find un\nlod=vn+wn, wherevn∈Vmsandwn∈Vfsuch\nthat\nτ(¯∂2\ntvn,z)+a(vn,z)+τb(vn,z) =τ(fn,z)+a(un−1\nlod,z),∀z∈Vms, (3.4)\na(wn,z)+τb(wn,z) =a(un−1\nlod,z), ∀z∈Vf, (3.5)\nforn≥2 with initial data u0\nlod=u0\nh∈Vmsandu1\nlod=u1\nh∈Vms. The initial\ndata is chosen in Vmsto simplify the implementation of the finescale correctors. We\nfurther discuss this choice in Section 4.4.\nRemark 3.1.Note that in (3.5), we do not take neither the source function nor\nthe second derivative into account. This is because we can subtrac t an interpolant\nwithin the L2-product, so that corresponding error converges at the same o rder as\nthe method itself. Moreover, the vn-part and wn-part have been excluded from the\nbilinear form a(·,·)+τb(·,·) in (3.4) and (3.5) respectively, due to the orthogonality\nbetween VmsandVf.\nNote that the multiscale space Vmsis created using (3.3) with small τ. Thus,\ntheA-coefficient dominates the system for short times. Moreover, we n ote from\n(3.5) that for Nlarge enough, we reach a steady state so that wN≈wN−1and\nvN≈vN−1. We get for z∈Vf\na(wN,z)+τb(wN,z)≈a(uN\nlod,z) =a(vN,z)+a(wN,z) =−τb(vN,z)+a(wN,z),\ndue to the orthogonality. Hence, by rearranging terms we have th at\nb(vN,z)+b(wN,z) =b(uN\nlod,z)≈0,\nwhich shows that the solution converges to a state where it is ortho gonal with\nrespect to B.\n3.2.Localized method. The method we have considered so far is based on the\nglobal projection (3.3) onto the finescale space Vf, which results in a large linear\nsystem that is expensive to solve. Moreover, the basis corrector s yield a global\nsupport that makes the linear system (3.4) not sparse, but dense . Hence, we wish8 PER LJUNG, AXEL M ˚ALQVIST, AND ANNA PERSSON\nto localize the computations onto coarse grid patches in order to yie ld a sparse\nmatrix system.\nTo localize the corrector problem, we first introduce the patches t o which the\nsupport of each basis function is to be restricted. For ω⊂Ω, letN(ω) :={K∈\nTH:K∩ω/\\e}atio\\slash=∅}, and define a patch Nk(ω) of size kas\nN1(ω) :=N(ω),\nNk(ω) :=N(Nk−1(ω)),fork≥2.\nGiven these coarse grid patches, we may restrict the finescale spa ceVfto them by\ndefining\nVω\nf,k:={v∈Vf: supp(v)⊆Nk(ω)},\nfor a subdomain ω⊂Ω. In particular, we will commonly use ω=T∈ THand\nω=x∈ N.\nNext, define the element restricted Ritz-projection RT\nfsuch that RT\nfv∈Vfis the\nsolution to the system\na(RT\nfv,z)+τb(RT\nfv,z) =/integraldisplay\nT(A+τB)∇v·∇zdx,∀z∈Vf.\nNote that we may construct the global Ritz-projection as the sum\nRfv=/summationdisplay\nT∈THRT\nfv.\nFork∈N, we may restrict the projection to a patch by letting RT\nf,k:VH→VT\nf,kbe\nsuch that RT\nf,kv∈VT\nf,ksolves\na(RT\nf,kv,z)+τb(RT\nf,kv,z) =/integraldisplay\nT(A+τB)∇v·∇zdx,∀z∈VT\nf,k.\nBy summation we yield the corresponding global version as\nRf,kv=/summationdisplay\nT∈THRT\nf,kv.\nFinally, we may construct a localized multiscale space as Vms,k:=VH−Rf,kVH,\nspanned by {λx−Rf,kλx}x∈N.\nIn order to justify the act of localization, it is required that a corre ctorφx\nvanishes rapidly outside an area of its corresponding node x. Indeed, the following\ntheorem from [27] shows that the corrector φxsatisfy an exponential decay away\nfrom its node, making the localization procedure viable.\nTheorem 3.2. There exists a constant c≥(8CIγ(2+CI))−1, that only depends\non the mesh constant γ, such that for any T∈ THand any v∈H1\n0(Ω)the solution\nφ∈Vfof the variational problem\n˜a(φ,w) =/integraldisplay\nT(˜A∇v)·∇wdx,∀w∈Vf\nsatisfies\n/⌊a∇d⌊l˜A1/2∇φ/⌊a∇d⌊lL2(Ω\\Nk(T))≤√\n2exp/parenleftbig\n−cα−+τβ−\nα++τβ+k/parenrightbig\n/⌊a∇d⌊l˜A1/2∇v/⌊a∇d⌊lL2(T),∀k∈N,\nwhere˜A=A+τB.A GENERALIZED FEM FOR THE STRONGLY DAMPED WAVE EQUATION 9\nWith the space Vms,kdefined, we are able to localize the computations on the\ncoarse scale system in (3.4) by replacing the multiscale space by its loc alized coun-\nterpart. It remains to localize the computations of the finescale sy stem in (3.5),\nwhich equivalently can be written as\na(¯∂twn,z)+b(wn,z) =1\nτa(vn−1,z).\nWe replace the right hand side by its localized version vn−1\nk∈Vms,kand note that\nvn−1\nk=/summationtext\nx∈Nαn−1\nx(λx−Rf,kλx). Thus, we seek our localized finescale solution as\nwn\nk=/summationtext\nx∈Nwn\nk,x, wherewn\nk,x∈Vx\nf,ksolves\n(3.6) a(¯∂twn\nk,x,z)+b(wn\nk,x,z) =1\nτa(αn−1\nx(λx−Rf,kλx),z),∀z∈Vx\nf,k,\nso that the computation of this equation is localized to a patch surro unding the\nnodex∈ N. We introduce the functions ξl\nk,x∈Vx\nf,kas solution to the parabolic\nequation\n(3.7) a(¯∂tξl\nk,x,z)+b(ξl\nk,x,z) =a(1\nτχ(0,τ)(λx−Rf,kλx),z),∀z∈Vx\nf,k,\nwith initial value ξ0\nk,x= 0, and where χ(0,τ)is an indicator function on the interval\n(0,τ). We claim that wn\nk,x=/summationtextn\nl=1αn−l\nxξl\nk,xis the solution to (3.6). This follows as\nfor allz∈Vx\nf,k\na(¯∂twn\nk,x,z)+b(wn\nk,x,z) =a(¯∂tn/summationdisplay\nl=1αn−l\nxξl\nk,x,z)+b(n/summationdisplay\nl=1αn−l\nxξl\nk,x,z)\n=n/summationdisplay\nl=2αn−l\nx/parenleftbig\na(¯∂tξl\nk,x,z)+b(ξl\nk,x,z)/parenrightbig\n+αn−1\nx/parenleftbig\na(¯∂tξ1\nk,x,z)+b(ξ1\nk,x,z)/parenrightbig\n= 0+a(αn−1\nx(λx−Rf,kλx),z).\nWith the localized computations established, the GFEM reads: find un\nlod,k=vn\nk+\nwn\nk, wherevn\nk=/summationtext\nx∈Nαn\nx(λx−Rf,kλx)∈Vms,ksolves\n(3.8)τ(¯∂2\ntvn\nk,z)+a(vn\nk,z)+τb(vn\nk,z) =τ(fn,z)+a(un−1\nlod,k,z),∀z∈Vms,k,\nandwn\nk=/summationtext\nx∈N/summationtextn\nl=1αn−l\nxξl\nk,x, whereξl\nk,x∈Vx\nf,ksolves (3.7).\nTo justify the fact that we localize the finescale equation, we requir e a result\nsimilar to that of Theorem 3.2, but for the functions {ξl\nx}N\nl=1. We finish this section\nabout localization by proving that these functions satisfy the expo nential decay\nrequired for the localization procedure to be viable.\nTheorem 3.3. For any node x∈ N, letξn\nx∈Vfbe the solution to\na(¯∂tξn\nx,z)+b(ξn\nx,z) =a(1\nτχ(0,τ)(λx−Rfλx),z),∀z∈Vf,\nwith initial value ξ0\nx= 0. Then there exist constants c >0andC >0such that for\nanyk≥1\n/⌊a∇d⌊lξn\nx/⌊a∇d⌊lH1(Ω\\Nk(x))≤Ce−ck/⌊a∇d⌊lλx/⌊a∇d⌊lH1,\nfor sufficiently small time step τ.10 PER LJUNG, AXEL M ˚ALQVIST, AND ANNA PERSSON\nProof.First, we analyze the problem for the first time step, which when mult iplied\nbyτcan be written as\n(3.9) a(ξ1\nx,z)+τb(ξ1\nx,z) =a(λx−φx,z),∀z∈Vf,\nwhereφx=Rfλx. We denote ˜ a=a+τbsuch that ˜ a(φx,z) = ˜a(λx,z) for allz∈Vf.\nFurthermore we use the energy norm |||·|||:=/radicalbig\n˜a(·,·), and by |||·|||Dwe denote the\nrestriction of the norm onto a domain D. As seen in the proof of Theorem 4.1 in\n[27], the result in Theorem 3.2 can be written as\n|||φx|||Ω\\Nk(x)≤Cφµ⌊k/4⌋|||λx|||,\nfor some µ <1. Moreover we define the cut-off function ηk∈VHby\nηk:=/braceleftBigg\n1,in Ω\\Nk+1(x),\n0,inNk(x),\nforx∈ N. Now let ν=ηk−3. Then we have that\nsupp(ν) = Ω\\Nk−3(x),\nsupp(∇ν) =Nk−2(x)\\Nk−3(x).\nWith this setting, we note that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nΩ\\Nk≤/integraldisplay\nΩν˜A∇ξ1\nx·∇ξ1\nxdx=/integraldisplay\nΩ˜A∇ξ1\nx·∇(νξ1\nx)dx−/integraldisplay\nΩ˜A∇ξ1\nx·ξ1\nx∇νdx\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nΩ˜A∇ξ1\nx·∇(1−IH)(νξ1\nx)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n=:M1+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nΩ˜A∇ξ1\nx·∇IH(νξ1\nx)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=:M2\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nΩ˜A∇ξ1\nx·ξ1\nx∇νdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=:M3,\nwherewe havedenoted ˜A=A+τB. We now proceedto estimate the terms M1,M2\nandM3separately. For M1, we use the problem (3.9) with z= (1−IH)(νξ1\nx)∈Vf\nto get\nM1=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nΩA∇(λx−φx)·∇(1−IH)(νξ1\nx)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleτ/integraldisplay\nΩ\\Nk−3B∇(φx)·∇(1−IH)(νξ1\nx)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle,\nwhere we have used the ˜ a-orthogonality between VmsandVf, that the integral is\nzero on supp( λx), and that the support of the remaining integrand is Ω \\Nk−3.\nThus, we get that\nM1≤τβ+\nα−|||φx|||Ω\\Nk−3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nΩ\\Nk−3≤τβ+\nα−Cφµ⌊k−3\n4⌋|||λx|||/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nΩ\\Nk−4.\nMoreover, by similar calculations as in the proof of Theorem 4.1 in [27], f romM2\nandM3we get\nM2,M3≤˜C/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nNk\\Nk−4,A GENERALIZED FEM FOR THE STRONGLY DAMPED WAVE EQUATION 11\nfor a constant ˜C >0. In total, for ε∈(0,1), we find that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nΩ\\Nk≤τβ+\nα−Cφµ⌊k−3\n4⌋|||λx|||/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nΩ\\Nk−4+˜C/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nNk\\Nk−4\n≤(β+Cφ)2\nα2\n−ετ2µ2⌊k−3\n4⌋|||λx|||2+ε/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nΩ\\Nk−4\n+˜C(/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nΩ\\Nk−4−/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nΩ\\Nk).\nLetδ:= (ε+˜C)(1+˜C)−1<1, and set κ= max(δ,µ)<1. Then, by rearranging\nthe terms we get the inequality\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nΩ\\Nk≤(β+Cφ)2\nα2\n−ε(1+˜C)τ2κ2⌊k−3\n4⌋|||λx|||2+κ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nΩ\\Nk−4.\nRepeating the estimate, we end up with\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nΩ\\Nk≤κ⌊k/4⌋/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nΩ+(β+Cφ)2\nα2\n−ε(1+˜C)|||λx|||2⌊k/4⌋−1/summationdisplay\ni=0τ2κiκ2⌊k−3−4i\n4⌋.\nWe proceed by estimating/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nΩ. By choosing z=ξ1\nxin (3.9) we get\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2≤ |||λx−φx|||/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ |||λx|||/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle,\nsince\n|||λx−φx|||2= ˜a(λx−φx,λx−φx)≤ |||λx−φx||||||λx|||.\nMoreover, for i= 0,1,2,...,⌊k/4⌋−1, we note that\nκi+2⌊k−3−4i\n4⌋≤κ⌊k/4⌋−1+2⌊k−3−4(k/4−1)\n4⌋=κ⌊k/4⌋−1+2⌊1\n4⌋=κ⌊k/4⌋−1(3.10)\nso in total we have the estimate\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nΩ\\Nk≤/radicalbig\n1+C0τ2κ1\n2⌊k/4⌋|||λx|||,withC0=(β+Cφ)2κ−1\nα2\n−ε(1+˜C)(⌊k/4⌋−1).\nRecall that this is for the first time step. In next time step, we cons ider the problem\na(ξ2\nx,z)+τb(ξ2\nx,z) =a(ξ1\nx,z),∀z∈Vf.\nAs for the first time step, we split the estimate into the similar integra lsM1,M2,\nandM3, and get\nM1≤τβ+\nα−/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nΩ\\Nk−3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ2\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nΩ\\Nk−3≤τβ+\nα−/radicalbig\n1+C0τ2κ1\n2⌊k−3\n4⌋|||λx|||/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ2\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nΩ\\Nk−4,\nwhileM2andM3remain the same. In total, we get the estimate\n(1+˜C)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ2\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nΩ\\Nk≤β2\n+\nα2\n−ετ2(1+C0τ2)κ⌊k−3\n4⌋|||λx|||2+(ε+˜C)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ2\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nΩ\\Nk−4.\nOnce again, by letting δ= (ε+˜C)/(1+˜C) and since δ≤κ, we get\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ2\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nΩ\\Nk≤β2\n+\nα2\n−ε(1+˜C)τ2(1+C0τ2)κ⌊k−3\n4⌋|||λx|||2+κ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ2\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nΩ\\Nk−4\n≤κ⌊k/4⌋|||λx|||2+β2\n+\nα2\n−ε(1+˜C)(1+C0τ2)⌊k/4⌋−1/summationdisplay\ni=0τ2κiκ⌊k−3−4i\n4⌋.12 PER LJUNG, AXEL M ˚ALQVIST, AND ANNA PERSSON\nOnce again we use (3.10) to conclude that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ2\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nΩ\\Nk≤/radicalbig\n1+C1τ2(1+C0τ2)κ1\n2⌊k/4⌋|||λx|||\n=/radicalbig\n1+C1τ2+C1τ2C0τ2κ1\n2⌊k/4⌋|||λx|||,\nwhere\nC1=β2\n+κ−1\nα2\n−ε(1+˜C)(⌊k/4⌋−1).\nInductively, we get for arbitrary time step nthe estimate\n|||ξn\nx|||Ω\\Nk≤κ1\n2⌊k/4⌋|||λx|||/radicaltp/radicalvertex/radicalvertex/radicalbtn−1/summationdisplay\ni=0(C1τ2)i+(C1τ2)nC0τ2.\nSinceκ1\n2⌊k/4⌋≤Ce−ckfor some c >0 andC >0, and since the energy norm is\nequivalent to the H1-norm, the theorem holds.\n/square\nRemark 3.4.Note that the constant that appears in the final inequality conver ges\nto1\n1−C1τ2,\nwhichmeansthat theconstantbehavesnicelyforsufficientlysmallt ime steps. More\nspecifically, for time steps\nτ≤/radicalBigg\nα2\n−εκ(1+˜C)\nβ2\n+(⌊k/4⌋−1).\n4.Error estimates\nIn this section we derive error estimates of the ideal method (3.4)- (3.5). The\nadditional error due to localization can be controlled in terms of the lo calization\nparameter k. This is further discussed in Remark 4.9. We begin by considering an\nauxiliary problem.\n4.1.Auxiliary problem. The auxiliary problem is defined as the standard vari-\national formulation for the strongly damped wave equation, but we exclude the\nsecond order time derivative. Moreover, we let the starting time t=t0be general\nand set the time discretization to t=t0< t1< ... < t N=T. Thus, the auxiliary\nproblem is to find Zn\nh∈Vhforn= 1,...,N, such that\n(4.1) a(¯∂tZn\nh,v)+b(Zn\nh,v) = (fn,v),∀v∈Vh,\nwith initial value Z0\nh∈Vms. Equivalently, multiply (4.1) by τand we may consider\n(4.2) a(Zn\nh,v)+τb(Zn\nh,v) =τ(fn,v)+a(Zn−1\nh,v),∀v∈Vh.\nExistence of a solution to this problem is guaranteed by Lax–Milgram. For sim-\nplicity, we make the assumption that the initial data for the damped w ave equation\n(2.6) is already in the multiscale space Vms, such that\nu0\nh=u0\nlod∈Vms, u1\nh=u1\nlod∈Vms.\nFor general initial data we refer to Section 4.4 below. Furthermore , to limit the\ntechnical details in the proof we have chosen to analyze the error in theL2(H1)-\nnorm instead of the pointwise (in time) H1-norm.A GENERALIZED FEM FOR THE STRONGLY DAMPED WAVE EQUATION 13\nThe solution space can be decomposed as Vh=Vms⊕Vf, such that the solution\ncan be written as Zn\nh=vn+wnwherevn∈Vmsandwn∈Vf. If we insert this into\nthe system in (4.2) and consider test functions z∈Vms, the left hand side becomes\na(Zn\nh,z)+τb(Zn\nh,z) =a(vn,z)+τb(vn,z),\nwhere we have used the orthogonality between VmsandVfwith respect to a(·,·)+\nτb(·,·). Likewise, if test functions z∈Vfare considered, the left hand side becomes\na(Zn\nh,z)+τb(Zn\nh,z) =a(wn,z)+τb(wn,z).\nWith these findings, we define the approximation to the auxiliary prob lem as to\nfindZn\nlod=vn+wn, wherevn∈Vmsandwn∈Vfsuch that\na(vn,z)+τb(vn,z) =τ(fn,z)+a(Zn−1\nlod,z),∀z∈Vms, (4.3)\na(wn,z)+τb(wn,z) =a(Zn−1\nlod,z), ∀z∈Vf, (4.4)\nwith initial data Z0\nlod∈Vms. Note that if f= 0, then Zn\nh=Zn\nlodfor every n,\nmeaning that the method reproduces Zn\nhexactly. For the auxiliary problem, we\nprove the following error estimates.\nTheorem 4.1. LetZn\nhbe the solution to (4.1)andZn\nlodthe solution to (4.3)-(4.4).\nAssume that Z0\nlod−Z0\nh= 0, then the error is bounded by\n/⌊a∇d⌊lZn\nh−Zn\nlod/⌊a∇d⌊lH1≤CHn/summationdisplay\nj=1τ/⌊a∇d⌊lfj/⌊a∇d⌊lL2. (4.5)\nIffn∈L2(Ω), forn≥0, then we have\nn/summationdisplay\nj=1τ/⌊a∇d⌊lZj\nh−Zj\nlod/⌊a∇d⌊l2\nL2≤CH2n/summationdisplay\nj=1τ/⌊a∇d⌊lfj/⌊a∇d⌊l2\nL2, (4.6)\nand iffn=¯∂tgn, for some {gn}N\nn=0such that gn∈Vh, then\nn/summationdisplay\nj=1τ/⌊a∇d⌊lZj\nh−Zj\nlod/⌊a∇d⌊l2\nL2≤CH2/parenleftBiggn/summationdisplay\nj=1τ/⌊a∇d⌊lgj/⌊a∇d⌊l2\nL2+/⌊a∇d⌊lg0/⌊a∇d⌊l2\nL2/parenrightBigg\n, (4.7)\nwhere C does not depend on the variations in AorB.\nProof.SinceZn\nh∈Vhthere are ¯ vn∈Vmsand ¯wn∈Vfsuch that Zn\nh= ¯vn+ ¯wn.\nLeten=Zn\nh−Zn\nlod, and consider\n|||en|||2:=a(en,en)+τb(en,en)\n=τ(fn,en)+a(Zn−1\nh,en)−a(vn,en)−τb(vn,en)−a(wn,en)−τb(wn,en).\nForvn∈Vmswe have due to the orthogonality and (4.3)\na(vn,en)+τb(vn,en) =a(vn,¯vn−vn)+τb(vn,¯vn−vn)\n=τ(fn,¯vn−vn)+a(Zn−1\nlod,¯vn−vn).\nSimilarly, for wn∈Vfwe use the orthogonality and (4.4) to get\na(wn,en)+τb(wn,en) =a(Zn−1\nlod,¯wn−wn).14 PER LJUNG, AXEL M ˚ALQVIST, AND ANNA PERSSON\nHence,\n|||en|||2=τ(fn,en)+a(Zn−1\nh,en)−τ(fn,¯vn−vn)\n−a(Zn−1\nlod,¯vn−vn)−a(Zn−1\nlod,¯wn−wn)\n=τ(fn,¯wn−wn)+a(Zn−1\nh−Zn−1\nlod,en).\nThe first term can be bounded by using the interpolation operator IH\nτ|(fn,¯wn−wn)| ≤τ/⌊a∇d⌊lfn/⌊a∇d⌊lL2/⌊a∇d⌊l¯wn−wn−IH(¯wn−wn)/⌊a∇d⌊lL2\n≤CHτ/⌊a∇d⌊lfn/⌊a∇d⌊lL2/⌊a∇d⌊l¯wn−wn/⌊a∇d⌊lH1\n≤CHτ/⌊a∇d⌊lfn/⌊a∇d⌊lL2/⌊a∇d⌊len/⌊a∇d⌊lH1\n≤CHτ/⌊a∇d⌊lfn/⌊a∇d⌊lL2|||en|||.\nFor the second term we note that Zn−1\nh−Zn−1\nlod=en−1so that\n|||en||| ≤CHτ/⌊a∇d⌊lfn/⌊a∇d⌊lL2+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleen−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nUsing this bound repeatedly and e0= 0 we get\n|||en||| ≤CHn/summationdisplay\nj=1τ/⌊a∇d⌊lfj/⌊a∇d⌊lL2.\nThis concludes the proof since /⌊a∇d⌊len/⌊a∇d⌊lH1≤C|||en|||.\nTo prove the remaining bounds in L2-norm, we define the forward difference\noperator ˜∂txn= (xn+1−xn)/τand consider the dual problem: find xj\nh∈Vhfor\nj=n−1,...,0, such that xn\nh= 0 and\na(−˜∂txj\nh,z)+b(xj\nh,z) = (ej+1,z),∀z∈Vh. (4.8)\nNote that this problem moves backwards in time. By choosing z=xj\nhin (4.8) and\nperforming a classical energy argument, we deduce\n/⌊a∇d⌊lxj\nh/⌊a∇d⌊l2\nH1+n/summationdisplay\nk=jτ/⌊a∇d⌊lxk\nh/⌊a∇d⌊l2\nH1≤Cn/summationdisplay\nk=j+1τ/⌊a∇d⌊lek/⌊a∇d⌊l2\nL2. (4.9)\nSimilarly, by choosing z=−˜∂txj\nh, we achieve\n/⌊a∇d⌊lxj\nh/⌊a∇d⌊l2\nH1+n/summationdisplay\nk=jτ/⌊a∇d⌊l˜∂txk\nh/⌊a∇d⌊l2\nH1≤Cn/summationdisplay\nk=j+1τ/⌊a∇d⌊lek/⌊a∇d⌊l2\nL2. (4.10)\nNow, use (4.8) to get\nn/summationdisplay\nj=1τ/⌊a∇d⌊lej/⌊a∇d⌊l2\nL2=n/summationdisplay\nj=1τa(−˜∂txj−1\nh,ej)+τb(xj−1\nh,ej).\nSummation by parts gives\nn/summationdisplay\nj=1τ/⌊a∇d⌊lej/⌊a∇d⌊l2\nL2=n/summationdisplay\nj=1τa(−˜∂txj−1\nh,ej)+τb(xj−1\nh,ej) (4.11)\n=n/summationdisplay\nj=1τa(xj−1\nh,¯∂tej)+τb(xj−1\nh,ej),A GENERALIZED FEM FOR THE STRONGLY DAMPED WAVE EQUATION 15\nwherewehaveused xn=e0= 0. Furthermore, weusethe equations(4.1) and(4.3),\nand the orthogonality in (3.2), to show that the following Galerkin ort hogonality\nholds for zms∈Vms\na(¯∂tej,zms)+b(ej,zms) =a(¯∂tZj\nh,zms)+b(Zj\nh,zms)−1\nτa(vj,zms) (4.12)\n−b(vj,zms)+1\nτa(Zj−1\nlod,zms) (4.13)\n= (fj,zms)−(fj,zms) = 0.\nLetxj\nh=xms+xf, for some xms∈Vms, xf∈Vf. Using the orthogonality (4.12)\nand the equations (4.4) and (4.1) we deduce\nn/summationdisplay\nj=1τa(xj−1\nh,¯∂tej)+τb(xj−1\nh,ej) =n/summationdisplay\nj=1τa(xj−1\nf,¯∂tej)+τb(xj−1\nf,ej)\n=n/summationdisplay\nj=1τa(xj−1\nf,¯∂tZj\nh)+τb(xj−1\nf,Zj\nh)\n=n/summationdisplay\nj=1τ(xj−1\nf,fj).\nIffj∈L2(Ω), then we may subtract IHxf= 0 and use (3.1) to achieve\nn/summationdisplay\nj=1τ(xj−1\nf,fj)≤CHn/summationdisplay\nj=1τ/⌊a∇d⌊lxj−1\nf/⌊a∇d⌊lH1/⌊a∇d⌊lfj/⌊a∇d⌊lL2\n≤CH/parenleftBiggn/summationdisplay\nj=1τ/⌊a∇d⌊lxj−1\nf/⌊a∇d⌊l2\nH1/parenrightBigg1/2/parenleftBiggn/summationdisplay\nj=1τ/⌊a∇d⌊lfj/⌊a∇d⌊l2\nL2/parenrightBigg1/2\n.\nNote that/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglexj−1\nf/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglexj−1\nms/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglexj−1\nh/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n. Hence the energy estimate (4.9) can\nnow be used to achieve (4.6). If fj=¯∂tgjone may use summation by parts to\nachieve\nn/summationdisplay\nj=1τ/⌊a∇d⌊lej/⌊a∇d⌊l2\nL2=n/summationdisplay\nj=1τ(xj−1\nf,¯∂tgj)≤n/summationdisplay\nj=1τ(−˜∂txj−1\nf,gj)−(x0\nf,g0)\n≤CHn/summationdisplay\nj=1τ/⌊a∇d⌊l˜∂txj−1\nf/⌊a∇d⌊lH1/⌊a∇d⌊lgj/⌊a∇d⌊lL2+CH/⌊a∇d⌊lx0\nf/⌊a∇d⌊lH1/⌊a∇d⌊lg0/⌊a∇d⌊lL2,\nwhere we have used xn\nf=xn\nh= 0. Using (4.10) we conclude (4.7). /square\nRemark 4.2.The bound in (4.6) is not of optimal order, but it is useful in the error\nanalysis.\nThe next lemmagiveserrorestimatesforthe discrete time derivativ eofthe error.\nIn the analysis of the (full) damped wave equation we use g=¯∂tuh, see Lemma 4.5.\nIf the initial data is nonzero we expect ¯∂tg1below to be of order t−1inL2-norm. A\ndetailed explanation of this is given below. Hence, we have a blow up clos e to zero\ndue to low regularity of the initial data. Therefore, we need to multip ly the error\nbytj. This is similar to the parabolic case for nonsmooth initial data see, e.g ., [34].16 PER LJUNG, AXEL M ˚ALQVIST, AND ANNA PERSSON\nLemma 4.3. LetZn\nhbe the solution to (4.1)andZn\nlodthe solution to (4.3)-(4.4).\nAssumeZ0\nlod−Z0\nh= 0. If¯∂tfn∈L2(Ω), forn≥1, then\nn/summationdisplay\nj=2τ/⌊a∇d⌊l¯∂t(Zj\nh−Zj\nlod)/⌊a∇d⌊l2\nL2≤CH2/parenleftBiggn/summationdisplay\nj=2τ/⌊a∇d⌊l¯∂tfj/⌊a∇d⌊l2\nL2+/⌊a∇d⌊lf1/⌊a∇d⌊l2\nL2/parenrightBigg\n(4.14)\nand iffn=¯∂tgn, for some {gn}N\nn=0, such that gn∈Vh, then\nn/summationdisplay\nj=2τ/⌊a∇d⌊l¯∂t(Zj\nh−Zj\nlod)/⌊a∇d⌊l2\nL2≤CH2/parenleftBiggn/summationdisplay\nj=2τ/⌊a∇d⌊l¯∂tgj/⌊a∇d⌊l2\nL2+/⌊a∇d⌊l¯∂tg1/⌊a∇d⌊l2\nL2/parenrightBigg\n(4.15)\nand, in addition, the following bound holds\nn/summationdisplay\nj=2τt2\nj/⌊a∇d⌊l¯∂t(Zj\nh−Zj\nlod)/⌊a∇d⌊l2\nL2≤CH2/parenleftBiggn/summationdisplay\nj=2τ/⌊a∇d⌊l¯∂tgj/⌊a∇d⌊l2\nL2+t2\n1/⌊a∇d⌊l¯∂tg1/⌊a∇d⌊l2\nL2/parenrightBigg\n, (4.16)\nwhere C does not depend on the variations in AorB.\nProof.The proof of (4.14) is similar to (4.6). Let ej=Zj\nh−Zj\nlodand define the\ndual problem\na(−˜∂txj\nh,z)+b(xj\nh,z) = (¯∂tej+1,z),∀z∈Vh, j=n−1,...,0, (4.17)\nwithxn= 0. Choosing z=¯∂tej+1and performing summation by parts we deduce\nn/summationdisplay\nj=2τ/⌊a∇d⌊l¯∂tej/⌊a∇d⌊l2\nL2=n/summationdisplay\nj=2τa(−˜∂txj−1,¯∂tej)+τb(xj−1\nh,¯∂tej) (4.18)\n=n/summationdisplay\nj=2τa(xj−1\nh,¯∂2\ntej)+τb(xj−1\nh,¯∂tej)+a(x1\nh,¯∂te1),\nwhere we used that xn= 0. Following the same argument as for (4.7), but with a\ndifference quotient, we arrive at\nn/summationdisplay\nj=2/⌊a∇d⌊l¯∂tej/⌊a∇d⌊l2\nL2=n/summationdisplay\nj=2τa(xj−1\nh,¯∂2\ntej)+τb(xj−1\nh,¯∂tej)+a(x1\nh,¯∂te1)\n=n/summationdisplay\nj=2τ(xj−1\nf,¯∂tfj)+a(x1\nh,¯∂te1).\nUsinge0= 0, we deduce\na(x1\nh,¯∂te1) =1\nτa(x1\nh,e1)≤C\nτH/⌊a∇d⌊lx1\nh/⌊a∇d⌊lH1τ/⌊a∇d⌊lf1/⌊a∇d⌊lL2≤CH/⌊a∇d⌊lx1\nh/⌊a∇d⌊lH1/⌊a∇d⌊lf1/⌊a∇d⌊lL2,\nand with ¯∂tfj∈L2(Ω) we get\nn/summationdisplay\nj=2/⌊a∇d⌊l¯∂tej/⌊a∇d⌊l2\nL2≤CHn/summationdisplay\nj=2τ/⌊a∇d⌊lxj−1\nf/⌊a∇d⌊lH1/⌊a∇d⌊l¯∂tfj/⌊a∇d⌊lL2+CH/⌊a∇d⌊lx1\nh/⌊a∇d⌊lH1/⌊a∇d⌊lf1/⌊a∇d⌊lL2,\nand (4.14) follows by using an energy estimate of xj\nhsimilar to (4.9), but with ¯∂tej\non the right hand side.\nIffj=¯∂tgjwe proceed as for (4.7) to achieve\nn/summationdisplay\nj=2τ(xj−1\nf,¯∂tfj)≤CHn/summationdisplay\nj=2τ/⌊a∇d⌊l˜∂txj−1\nf/⌊a∇d⌊lH1/⌊a∇d⌊l¯∂tgj/⌊a∇d⌊lL2+CH/⌊a∇d⌊lx1\nf/⌊a∇d⌊lH1/⌊a∇d⌊l¯∂tg1/⌊a∇d⌊lL2A GENERALIZED FEM FOR THE STRONGLY DAMPED WAVE EQUATION 17\nand (4.15) follows by using energy estimates similar to (4.10).\nFor (4.16) we consider the dual problem\na(−˜∂txj\nh,z)+b(xj\nh,z) = (tj+1¯∂tej+1,z),∀z∈Vh, j=n−1,...,0. (4.19)\nA simple energy estimate shows\n/⌊a∇d⌊lxj\nh/⌊a∇d⌊l2\nH1+n−1/summationdisplay\nk=jτ/⌊a∇d⌊l˜∂txk\nh/⌊a∇d⌊l2\nH1≤Cn−1/summationdisplay\nk=jτt2\nk+1/⌊a∇d⌊l¯∂tek+1/⌊a∇d⌊l2\nL2, j= 0,...,n−1. (4.20)\nNow choosing z=tj+1¯∂tej+1in (4.19) and performing summation by parts gives\nn/summationdisplay\nj=2τt2\nj/⌊a∇d⌊l¯∂tej/⌊a∇d⌊l2\nL2=n/summationdisplay\nj=2τa(−˜∂txj−1\nh,tj¯∂tej)+τb(xj−1\nh,tj¯∂tej) (4.21)\n=n/summationdisplay\nj=2/parenleftbig\nτa(xj−1\nh,tj¯∂2\ntej)+τb(xj−1\nh,tj¯∂tej)\n+a(xj−1\nh,(tj−tj−1)¯∂tej−1)/parenrightbig\n+a(x1\nh,t1¯∂te1).\nThe first two terms of the sum can be handled similarly to (4.15),\nn/summationdisplay\nj=2τa(xj−1\nh,tj¯∂2\ntej)+τb(xj−1\nh,tj¯∂tej) =n/summationdisplay\nj=2τ(xj−1\nf,tj¯∂tfj).\nNow, using summation by parts we achieve\nn/summationdisplay\nj=2τ(xj−1\nf,tj¯∂tfj) =n/summationdisplay\nj=2τ/parenleftbig\n(−˜∂txj−1\nf,tjfj)+(xj−1\nf,fj)/parenrightbig\n+(x1\nf,t1f1)\n≤CH/parenleftBiggn/summationdisplay\nj=2τ/parenleftbig\ntj/⌊a∇d⌊l˜∂txj−1\nf/⌊a∇d⌊lH1/⌊a∇d⌊lfj/⌊a∇d⌊lL2+/⌊a∇d⌊lxj−1\nf/⌊a∇d⌊lH1/⌊a∇d⌊lfj/⌊a∇d⌊lL2/parenrightbig\n+t1/⌊a∇d⌊lx1\nf/⌊a∇d⌊lH1/⌊a∇d⌊lf1/⌊a∇d⌊lL2/parenrightBigg\nwhere we can use (4.20). Note that in the first term we can use the ( crude) bound\nt2\nj≤t2\nnand let the constant Cdepend on T. We get\nn/summationdisplay\nj=2τ(xj−1\nf,tj¯∂tfj)≤CH/parenleftBiggn/summationdisplay\nj=1τt2\nj/⌊a∇d⌊l¯∂tej/⌊a∇d⌊l2\nL2/parenrightBigg1/2/parenleftBigg/parenleftBiggn/summationdisplay\nj=2τ/⌊a∇d⌊lfj/⌊a∇d⌊l2\nL2/parenrightBigg1/2\n+t1/⌊a∇d⌊lf1/⌊a∇d⌊lL2/parenrightBigg\n.\nForthe thirdterm in(4.21), weuse tj−tj−1=τandonceagainperformsummation\nby parts to get\nn/summationdisplay\nj=2τa(xj−1\nh,¯∂tej−1) =n/summationdisplay\nj=2τa(−˜∂txj−1\nh,ej−1),\nwhere we have used xn\nh=e0= 0. Combining (4.20) and (4.5) we get\nn/summationdisplay\nj=2τa(−˜∂txj−1\nh,ej−1)≤Cmax\nj=1,...,n/⌊a∇d⌊lej/⌊a∇d⌊lH1/parenleftBiggn/summationdisplay\nj=2τ/parenrightBigg1/2/parenleftBiggn/summationdisplay\nj=2τ/⌊a∇d⌊l˜∂txj−1\nh/⌊a∇d⌊l2\nH1/parenrightBigg1/2\n≤CHn/summationdisplay\nj=1τ/⌊a∇d⌊lfj/⌊a∇d⌊lL2/parenleftBiggn/summationdisplay\nj=1τt2\nj/⌊a∇d⌊l¯∂tej/⌊a∇d⌊l2\nL2/parenrightBigg1/2\n.18 PER LJUNG, AXEL M ˚ALQVIST, AND ANNA PERSSON\nFor the last term in (4.21) we use (4.20) and (4.5) for n= 1 to achieve\na(x1\nh,t1¯∂te1) =a(x1\nh,e1)≤CH/parenleftBiggn/summationdisplay\nj=1τt2\nj/⌊a∇d⌊l¯∂tej/⌊a∇d⌊l2\nL2/parenrightBigg1/2\nt1/⌊a∇d⌊lf1/⌊a∇d⌊lL2,\nand (4.16) follows by letting fj=¯∂tgj.\n/square\n4.2.The damped wave equation. For the error analysis of the full damped\nwave equation we shall make use of the projection corresponding t o the auxiliary\nproblem. For un\nh∈Vh, letXn=Xn\nv+Xn\nw∈VhwithXn\nv∈VmsandXn\nw∈Vfsuch\nthat\na(Xn\nv−un\nh,z)+τb(Xn\nv−un\nh,z) =a(Xn−1−un−1\nh,z),∀z∈Vms, (4.22)\na(Xn\nw,z)+τb(Xn\nw,z) =a(Xn−1,z), ∀z∈Vf. (4.23)\nNote that since un\nhsolves (2.6), the system (4.22)-(4.23) is equivalent to\na(Xn\nv,z)+τb(Xn\nv,z) =τ(fn−¯∂2\ntun\nh,z)+a(Xn−1,z),∀z∈Vms, (4.24)\na(Xn\nw,z)+τb(Xn\nw,z) =a(Xn−1,z), ∀z∈Vf. (4.25)\nThat is, we may view un\nhandXnas the solution and approximationto the auxiliary\nproblem with source data fn−¯∂2\ntun\nh. We deduce following lemma.\nLemma 4.4. Letun\nhbe the solution to (2.6)andXnthe solution to (4.22)-(4.23).\nThe error satisfies the following bounds\n/⌊a∇d⌊lXn−un\nh/⌊a∇d⌊lH1≤CHn/summationdisplay\nj=2τ/⌊a∇d⌊lfj−¯∂2\ntuj\nh/⌊a∇d⌊lL2, n≥2, (4.26)\nn/summationdisplay\nj=2τ/⌊a∇d⌊lXn−un\nh/⌊a∇d⌊l2\nL2≤CH2/parenleftBiggn/summationdisplay\nj=2τ(/⌊a∇d⌊lfj/⌊a∇d⌊l2\nL2+/⌊a∇d⌊l¯∂tuj\nh/⌊a∇d⌊l2\nL2)+/⌊a∇d⌊l¯∂tu1\nh/⌊a∇d⌊l2\nL2/parenrightBigg\n, n≥2,(4.27)\nwhere C does not depend on the variations in AorB.\nProof.We let the auxiliaryproblem(4.1) startat t1. In this case e0=u1\nh−u1\nms= 0,\nsinceu1\nms=u1\nh∈Vms. The bound (4.26) now follows directly from (4.5) with\nfn−¯∂2\ntun\nhas right hand side. The second bound (4.27) follows from (4.6) and (4 .7)\nwithfn∈L2(Ω) and gn=¯∂tun+1\nh. /square\nIn a similar way me may deduce bounds for the (discrete) time derivat ive of the\nerror. As a direct consequence of Lemma 4.3, we get the following re sult.A GENERALIZED FEM FOR THE STRONGLY DAMPED WAVE EQUATION 19\nLemma 4.5. Letun\nhbe the solution to (2.6)andXnthe solution to (4.22)-(4.23).\nThe following bounds hold\nn/summationdisplay\nj=3τ/⌊a∇d⌊l¯∂t(Xj−uj\nh)/⌊a∇d⌊l2\nL2(4.28)\n≤CH2/parenleftBiggn/summationdisplay\nj=3τ(/⌊a∇d⌊l¯∂tfj/⌊a∇d⌊l2\nL2+/⌊a∇d⌊l¯∂2\ntuj\nh/⌊a∇d⌊l2\nL2)+/⌊a∇d⌊lf2/⌊a∇d⌊l2\nL2+/⌊a∇d⌊l¯∂2\ntu2\nh/⌊a∇d⌊l2\nL2/parenrightBigg\n,\nn/summationdisplay\nj=3τt2\nj/⌊a∇d⌊l¯∂t(Xj−uj\nh)/⌊a∇d⌊l2\nL2(4.29)\n≤CH2/parenleftBiggn/summationdisplay\nj=3τ(/⌊a∇d⌊l¯∂tfj/⌊a∇d⌊l2\nL2+/⌊a∇d⌊l¯∂2\ntuj\nh/⌊a∇d⌊l2\nL2)+t2\n2/⌊a∇d⌊lf2/⌊a∇d⌊l2\nL2+t2\n2/⌊a∇d⌊l¯∂2\ntu2\nh/⌊a∇d⌊l2\nL2/parenrightBigg\n,\nwhere C does not depend on the variations in AorB.\nLemma 4.6. Letun\nhandun\nlodbe the solutions to (2.6)and(3.4)-(3.5), respectively.\nAssume that u0=u1= 0. The error is bounded by\nn/summationdisplay\nj=2τ/⌊a∇d⌊luj\nlod−uj\nh/⌊a∇d⌊l2\nH1≤CH2/parenleftBiggn/summationdisplay\nj=1τ(/⌊a∇d⌊lfj/⌊a∇d⌊l2\nL2+/⌊a∇d⌊l¯∂tfj/⌊a∇d⌊l2\nL2)+ max\nj=1,...,n/⌊a∇d⌊lfj/⌊a∇d⌊l2\nL2/parenrightBigg\n,\nforn≥2, where C does not depend on the variations in AorB.\nProof.Begin by splitting the error into two contributions\nun\nlod−un\nh=un\nlod−Xn+Xn−un\nh=:θn+ρn,\nwhereXnis the solution to the simplified problem in(4.22)-(4.23). By Lemma 4.4\nρnis bounded by\n/⌊a∇d⌊lρn/⌊a∇d⌊lH1≤CHn/summationdisplay\nj=2τ/parenleftbig\n/⌊a∇d⌊lfj/⌊a∇d⌊lL2+/⌊a∇d⌊l¯∂2\ntuj\nh/⌊a∇d⌊lL2/parenrightbig\n,\nand we can now apply the energy bound (2.9). It remains to bound θn. Recall that\nfor anyz∈Vhwe have z=zms+zffor some zms∈Vmsandzf∈Vf. Using that\nun\nlod=vn+wnsatisfies (3.4) and the orthogonality (3.2) we get\n(¯∂2\ntun\nlod,zms)+a(¯∂tun\nlod,zms)+b(un\nlod,zms) = (fn,zms)+(¯∂2\ntwn,zms).\nSimilarly, due to (3.5) and the orthogonality,\n(¯∂2\ntun\nlod,zf)+a(¯∂tun\nlod,zf)+b(un\nlod,zf) = (¯∂2\ntun\nlod,zf).\nForXnwe use (4.22)-(4.23) and the orthogonality to deduce\n(¯∂2\ntXn,z)+a(¯∂tXn,z)+b(Xn,z) = (¯∂2\ntXn,z)+(fn−¯∂2\ntun\nh,zms), z∈Vh,\nHence,θnsatisfies\n(¯∂2\ntθn,z)+a(¯∂tθn,z)+b(θn,z) = (−¯∂2\ntρn,z)−(¯∂2\ntun\nh,zf)\n+(¯∂2\ntun\nlod,zf)+(¯∂2\ntwn,zms), z∈Vh,20 PER LJUNG, AXEL M ˚ALQVIST, AND ANNA PERSSON\nwithθ0=θ1= 0, since u0\nlod=u0\nh=X0andu1\nlod=u1\nh=X1. Let˜θn=/summationtextn\nj=2τθj.\nMultiplying by τand summing over ngives\n(¯∂tθn,z)+a(θn,z)+b(˜θn,z)≤(−¯∂tρn,z)−(¯∂tun\nh−¯∂tu1\nh,zf)(4.30)\n+(¯∂tun\nlod−¯∂tu1\nlod,zf)+(¯∂twn−¯∂tw1,zms),\nwhere we have used that θ1=θ0=ρ1=ρ0= 0. Using the interpolant IHwe\ndeduce\n(¯∂tun\nh,zf)+(¯∂tun\nlod,zf)+(¯∂twn,zms)≤CH(/⌊a∇d⌊l¯∂tun\nh/⌊a∇d⌊lL2+/⌊a∇d⌊l¯∂tun\nlod/⌊a∇d⌊lL2)/⌊a∇d⌊lz/⌊a∇d⌊lH1\n+CH/⌊a∇d⌊l¯∂tun\nlod/⌊a∇d⌊lH1/⌊a∇d⌊lzms/⌊a∇d⌊lL2,\nfor 1≤n≤N. Letα(n) =/⌊a∇d⌊l¯∂tun\nh/⌊a∇d⌊lL2+/⌊a∇d⌊l¯∂tun\nlod/⌊a∇d⌊lH1. Since/⌊a∇d⌊lzms/⌊a∇d⌊lL2≤C/⌊a∇d⌊lz/⌊a∇d⌊lH1and\nα(1) = 0 due to the vanishing initial data, we get\n(¯∂tθn,z)+a(θn,z)+b(˜θn,z)≤(−¯∂tρn,z)+CHα(n)/⌊a∇d⌊lz/⌊a∇d⌊lH1, z∈Vh.\nNow, choose z=θn=¯∂t˜θnin (4.30). We get\n1\n2/⌊a∇d⌊lθn/⌊a∇d⌊l2\nL2−1\n2/⌊a∇d⌊lθn−1/⌊a∇d⌊l2\nL2+τ/⌊a∇d⌊lθn/⌊a∇d⌊l2\na+1\n2/⌊a∇d⌊l˜θn/⌊a∇d⌊l2\nb−1\n2/⌊a∇d⌊l˜θn−1/⌊a∇d⌊l2\nb\n≤τ/⌊a∇d⌊l¯∂tρn/⌊a∇d⌊lL2/⌊a∇d⌊lθn/⌊a∇d⌊lL2+CHτα(n)/⌊a∇d⌊lθn/⌊a∇d⌊lH1.\nSumming over ngives\n/⌊a∇d⌊lθn/⌊a∇d⌊l2\nL2+n/summationdisplay\nj=2τ/⌊a∇d⌊lθj/⌊a∇d⌊l2\nH1+/⌊a∇d⌊l˜θn/⌊a∇d⌊l2\nH1≤n/summationdisplay\nj=2τ/⌊a∇d⌊l¯∂tρj/⌊a∇d⌊lL2/⌊a∇d⌊lθj/⌊a∇d⌊lL2+CHn/summationdisplay\nj=2τα(j)/⌊a∇d⌊lθj/⌊a∇d⌊lH1.\nNowusing that /⌊a∇d⌊lθn/⌊a∇d⌊lL2≤ /⌊a∇d⌊lθn/⌊a∇d⌊lH1and Young’sweighted inequality, θjcan be kicked\nback to the left hand side. We deduce\nn/summationdisplay\nj=2τ/⌊a∇d⌊lθj/⌊a∇d⌊l2\nH1≤Cn/summationdisplay\nj=2τ/⌊a∇d⌊l¯∂tρj/⌊a∇d⌊l2\nL2+CH2n/summationdisplay\nj=2τα(j)2.\nUsing Lemma 4.5 we have\nn/summationdisplay\nj=2τ/⌊a∇d⌊lθj/⌊a∇d⌊l2\nH1≤CH2/parenleftBiggn/summationdisplay\nj=2τ(/⌊a∇d⌊l¯∂tfj/⌊a∇d⌊l2\nL2+/⌊a∇d⌊l¯∂2\ntuj\nh/⌊a∇d⌊l2\nL2)+/⌊a∇d⌊l¯∂2\ntu2\nh/⌊a∇d⌊l2\nL2/parenrightBigg\n+CH2n/summationdisplay\nj=2τα(j)2.\nTo bound /⌊a∇d⌊l¯∂2\ntu2\nh/⌊a∇d⌊l2\nL2, we consider (2.6) for n= 2 and choose v=¯∂2\ntu2\nh, which gives\n(¯∂2\ntu2\nh,¯∂2\ntu2\nh)+a(¯∂tu2\nh,¯∂2\ntu2\nh)+b(u2\nh,¯∂2\ntu2\nh) = (¯∂tf2,¯∂2\ntu2\nh).\nDue to the vanishing initial data ¯∂tu2\nh=τ−1u2\nhand¯∂2\ntu2\nh=τ−2u2\nh. We get\n/⌊a∇d⌊l¯∂2\ntu2\nh/⌊a∇d⌊l2\nL2+1\nτ3/⌊a∇d⌊lu2\nh/⌊a∇d⌊l2\na+1\nτ2/⌊a∇d⌊lu2\nh/⌊a∇d⌊l2\nb= (f2,¯∂2\ntu2\nh), (4.31)\nand we deduce\n/⌊a∇d⌊l¯∂2\ntu2\nh/⌊a∇d⌊l2\nL2≤C/⌊a∇d⌊lf2/⌊a∇d⌊l2\nL2.\nAll terms, except/summationtextn\nj=2τ/⌊a∇d⌊l¯∂tuj\nlod/⌊a∇d⌊l2\nH1that appears in/summationtextn\nj=2α2(j), can now be\nbounded by using the regularity in Theorem 2.1. To bound/summationtextn\nj=2τ/⌊a∇d⌊l¯∂tuj\nlod/⌊a∇d⌊l2\nH1weA GENERALIZED FEM FOR THE STRONGLY DAMPED WAVE EQUATION 21\nchoosez=¯∂tvnandz=¯∂twnin (3.4) and (3.5) respectively. Adding the two\nequations and using the orthogonality between VmsandVfwe achieve\n(¯∂2\ntvn,¯∂tvn)+a(¯∂tun\nlod,¯∂tun\nlod)+b(un\nlod,¯∂tun\nlod) = (fn,¯∂tvn)\n≤Cǫ/⌊a∇d⌊lfn/⌊a∇d⌊l2\nL2+ǫ/⌊a∇d⌊l¯∂tvn/⌊a∇d⌊l2\nL2.\nNote that /⌊a∇d⌊l¯∂tvn/⌊a∇d⌊lL2≤C/⌊a∇d⌊l∇¯∂tvn/⌊a∇d⌊lL2≤C/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle¯∂tvn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=C/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle¯∂tun\nlod/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C/⌊a∇d⌊l¯∂tun\nlod/⌊a∇d⌊la, so\nwe may choose ǫsmall enough such that/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle¯∂tun\nlod/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglecan be kicked to the left hand\nside. As in the proof of Theorem 2.1 we may now deduce\n/⌊a∇d⌊l¯∂tvn/⌊a∇d⌊l2\nL2+n/summationdisplay\nj=2τ/⌊a∇d⌊l¯∂tuj\nlod/⌊a∇d⌊l2\nH1+/⌊a∇d⌊lun\nlod/⌊a∇d⌊l2\nH1≤C/parenleftBiggn/summationdisplay\nj=2/⌊a∇d⌊lfj/⌊a∇d⌊l2\nL2+/⌊a∇d⌊l¯∂tu1\nh/⌊a∇d⌊l2\nL2+/⌊a∇d⌊lu1\nh/⌊a∇d⌊l2\nH1/parenrightBigg\n,(4.32)\nwhere we have used that v1=u1\nlod=u1\nh∈Vms. However, we have assumed\nvanishing initial data so these terms disappear. The lemma follows. /square\nLemma 4.7. Letun\nhandun\nlodbe the solutions to (2.6)and(3.4)-(3.5), respectively.\nAssume that f= 0. The error is bounded by\nn/summationdisplay\nj=2τt2\nj/⌊a∇d⌊luj\nlod−uj\nh/⌊a∇d⌊l2\nH1≤CH2(/⌊a∇d⌊l¯∂tu1\nh/⌊a∇d⌊l2\nH1+/⌊a∇d⌊lu1\nh/⌊a∇d⌊l2\nH1+/⌊a∇d⌊lu0\nh/⌊a∇d⌊l2\nH1), n≥2,\nwhere C does not depend on the variations in AorB.\nProof.We follow the steps in the proof of Lemma 4.6 to equation (4.30). Note that\n/⌊a∇d⌊lρn/⌊a∇d⌊lH1can be bounded by Lemma 4.4 and the energy bound in (2.9) with f= 0.\nNow, let ˜θn=/summationtextn\nj=2τθj. Choose z=θn=¯∂t˜θnin (4.30) and multiply by τt2\nn.\nWe get\nt2\nn\n2/⌊a∇d⌊lθn/⌊a∇d⌊l2\nL2−t2\nn−1\n2/⌊a∇d⌊lθn−1/⌊a∇d⌊l2\nL2+τt2\nn/⌊a∇d⌊lθn/⌊a∇d⌊l2\na+t2\nn\n2/⌊a∇d⌊l˜θn/⌊a∇d⌊l2\nb−t2\nn−1\n2/⌊a∇d⌊l˜θn−1/⌊a∇d⌊l2\nb\n≤τt2\nn/⌊a∇d⌊l¯∂tρn/⌊a∇d⌊lL2/⌊a∇d⌊lθn/⌊a∇d⌊lL2+CHt2\nnτ(α(n)+α(1))/⌊a∇d⌊lθn/⌊a∇d⌊lH1\n+(t2\nn−t2\nn−1)\n2/⌊a∇d⌊lθn−1/⌊a∇d⌊l2\nL2+(t2\nn−t2\nn−1)\n2/⌊a∇d⌊l˜θn−1/⌊a∇d⌊l2\nb.\nSumming over nand using t2\nn−t2\nn−1≤2τtngives\nt2\nn/⌊a∇d⌊lθn/⌊a∇d⌊l2\nL2+n/summationdisplay\nj=2τt2\nj/⌊a∇d⌊lθj/⌊a∇d⌊l2\nH1+t2\nn/⌊a∇d⌊l˜θn/⌊a∇d⌊l2\nH1≤Cn/summationdisplay\nj=2τt2\nj/⌊a∇d⌊l¯∂tρj/⌊a∇d⌊lL2/⌊a∇d⌊lθj/⌊a∇d⌊lL2(4.33)\n+CHn/summationdisplay\nj=2τt2\nj(α(j)+α(1))/⌊a∇d⌊lθj/⌊a∇d⌊lH1+Cn/summationdisplay\nj=2τtj/⌊a∇d⌊lθj/⌊a∇d⌊l2\nL2+Cn/summationdisplay\nj=2τtj/⌊a∇d⌊l˜θj/⌊a∇d⌊l2\nb.\nFrom the first two sums on the right hand side we can kick tj/⌊a∇d⌊lθj/⌊a∇d⌊lL2≤tj/⌊a∇d⌊lθj/⌊a∇d⌊lH1\nandtj/⌊a∇d⌊lθj/⌊a∇d⌊lH1to the left hand side. The remaining two sums needs to be bounded\nby other energy estimates.22 PER LJUNG, AXEL M ˚ALQVIST, AND ANNA PERSSON\nMultiply (4.30) by τand sum over nto get\n(θn,z)+a(˜θn,z)+b/parenleftBiggn/summationdisplay\nj=2τ˜θj,z/parenrightBigg\n≤(ρn,z)−(un\nh−u1\nh,zf)+(un\nlod−u1\nlod,zf)(4.34)\n+(wn−w1,zms)+tn((¯∂tu1\nh,zf)−(¯∂tu1\nlod,zf)−(¯∂tw1,zms)).\nwhere we have used θ1=ρ1= 0. As in the proof of Lemma 4.6 we get\n(un\nh,zf)+(un\nlod,zf)+(wn,zms)≤CH(/⌊a∇d⌊lun\nh/⌊a∇d⌊lL2+/⌊a∇d⌊lun\nlod/⌊a∇d⌊lL2)/⌊a∇d⌊lz/⌊a∇d⌊lH1\n+CH/⌊a∇d⌊lun\nlod/⌊a∇d⌊lH1/⌊a∇d⌊lzms/⌊a∇d⌊lL2,\nfor 1≤n≤N. Letβ(n) =/⌊a∇d⌊lun\nh/⌊a∇d⌊lL2+/⌊a∇d⌊lun\nlod/⌊a∇d⌊lH1. Choose z=˜θn=¯∂t/summationtextn\nj=1τ˜θj.\nSimilar to above energy estimates, we get\n/⌊a∇d⌊l˜θn/⌊a∇d⌊l2\nL2+n/summationdisplay\nj=2τ/⌊a∇d⌊l˜θj/⌊a∇d⌊l2\na+/⌊a∇d⌊ln/summationdisplay\nj=2τ˜θj/⌊a∇d⌊l2\nb≤n/summationdisplay\nj=2τ/⌊a∇d⌊lρj/⌊a∇d⌊l2\nL2+CH2n/summationdisplay\nj=2τ(β(j) +β(1)+α(1))2.(4.35)\nSince/summationtextn\nj=2τtj/⌊a∇d⌊l˜θj|2\nb≤C(tn)/summationtextn\nj=2τ/⌊a∇d⌊l˜θj/⌊a∇d⌊l2\nawe may use (4.35) in (4.33). This gives\nt2\nn/⌊a∇d⌊lθn/⌊a∇d⌊l2\nL2+n/summationdisplay\nj=2τt2\nj/⌊a∇d⌊lθj/⌊a∇d⌊l2\nH1+t2\nn/⌊a∇d⌊l˜θn/⌊a∇d⌊l2\nH1≤Cn/summationdisplay\nj=2τ(t2\nj/⌊a∇d⌊l¯∂tρj/⌊a∇d⌊l2\nL2+/⌊a∇d⌊lρj/⌊a∇d⌊l2\nL2)(4.36)\n+Cn/summationdisplay\nj=2τtj/⌊a∇d⌊lθj/⌊a∇d⌊l2\nL2+CH2n/summationdisplay\nj=2τ(t2\nj(α(j)+α(1))2+(β(j)+β(1)+α(1))2).\nIt remains to bound C/summationtextn\nj=2τtj/⌊a∇d⌊lθj/⌊a∇d⌊l2\nL2. For this purpose, choose z=θn=¯∂t˜θnin\n(4.34). Multiply by tnτand sum over nto achieve\nn/summationdisplay\nj=2τtj/⌊a∇d⌊lθj/⌊a∇d⌊l2\nL2+tn/⌊a∇d⌊l˜θn/⌊a∇d⌊l2\na+n/summationdisplay\nj=2tjτb/parenleftBiggj/summationdisplay\nk=2τ˜θk,¯∂t˜θj/parenrightBigg\n≤n/summationdisplay\nj=1τtj/⌊a∇d⌊lρj/⌊a∇d⌊l2\nL2+n/summationdisplay\nj=2τ/⌊a∇d⌊l˜θj/⌊a∇d⌊l2\na(4.37)\n+CHn/summationdisplay\nj=2τtj(β(j)+β(1)+α(1))/⌊a∇d⌊lθj/⌊a∇d⌊lH1.\nNote that /⌊a∇d⌊lθj/⌊a∇d⌊lH1in the last sum in only present in the right hand side. The\nsecond term on the right hand side is bounded by (4.35). For the ter m involving\nthe bilinear form b(·,·) we use summation by parts to get\n−n/summationdisplay\nj=2τb/parenleftBigg\ntjj/summationdisplay\nk=2τ˜θk,¯∂tj/summationdisplay\nk=2τθk/parenrightBigg\n≤n/summationdisplay\nj=2τb/parenleftBigg\ntj˜θj+j/summationdisplay\nk=2τ˜θk,˜θj−1/parenrightBigg\n−b/parenleftBigg\ntnn/summationdisplay\nj=2τ˜θj,˜θn/parenrightBigg\n≤Cn/summationdisplay\nj=2τtj/⌊a∇d⌊l˜θj/⌊a∇d⌊l2\nb+C/⌊a∇d⌊ln/summationdisplay\nj=2τ˜θj/⌊a∇d⌊l2\nb+Cǫt2\nn/⌊a∇d⌊l˜θn/⌊a∇d⌊l2\nH1.A GENERALIZED FEM FOR THE STRONGLY DAMPED WAVE EQUATION 23\nHere the constant Cǫcan be made arbitrarily small due to Young’s weighted in-\nequality. The first two terms can be bounded by (4.35). Thus, (4.37 ) becomes\nn/summationdisplay\nj=2τtj/⌊a∇d⌊lθj/⌊a∇d⌊l2\nL2+tn/⌊a∇d⌊l˜θn/⌊a∇d⌊l2\na≤n/summationdisplay\nj=1τ(tj/⌊a∇d⌊lρj/⌊a∇d⌊l2\nL2+/⌊a∇d⌊lρj/⌊a∇d⌊l2\nL2)+CHn/summationdisplay\nj=2τtj(β(j)+β(1)\n+α(1))/⌊a∇d⌊lθj/⌊a∇d⌊lH1+CH2n/summationdisplay\nj=2τ(β(j) +β(1)+α(1))2.\nUsing this in (4.36) we arrive at\nt2\nn/⌊a∇d⌊lθn/⌊a∇d⌊l2\nL2+n/summationdisplay\nj=2τt2\nj/⌊a∇d⌊lθj/⌊a∇d⌊l2\nH1+t2\nn/⌊a∇d⌊l˜θn/⌊a∇d⌊l2\nH1≤Cn/summationdisplay\nj=2τ(t2\nj/⌊a∇d⌊l¯∂tρj/⌊a∇d⌊l2\nL2+tj/⌊a∇d⌊lρj/⌊a∇d⌊l2\nL2)\n+CH2n/summationdisplay\nj=2τ(t2\nj(α(j)+α(1))2+(β(j)+β(1)+α(1))2) (4.38)\n+CHn/summationdisplay\nj=2τtj(β(j)+β(1)+α(1))/⌊a∇d⌊lθj/⌊a∇d⌊lH1+Cǫt2\nn/⌊a∇d⌊l˜θn/⌊a∇d⌊l2\nH1.\nUsing Lemma 4.5 and Lemma 4.4 with f= 0 we deduce for the first two terms in\n(4.38)\nCn/summationdisplay\nj=2τ(t2\nj/⌊a∇d⌊l¯∂tρj/⌊a∇d⌊l2\nL2+tj/⌊a∇d⌊lρj/⌊a∇d⌊l2\nL2)≤CH2/parenleftBiggn/summationdisplay\nj=2τ/⌊a∇d⌊l¯∂2\ntuj\nh/⌊a∇d⌊l2\nL2+t2\n2/⌊a∇d⌊l¯∂2\ntu2\nh/⌊a∇d⌊l2\nL2/parenrightBigg\n,\nwhere we can use (2.9) for n= 2 and f= 0 to bound ¯∂2\ntu2\nh. We get\nt2\n2/⌊a∇d⌊l¯∂2\ntu2\nh/⌊a∇d⌊l2\nL2≤Cτ(/⌊a∇d⌊l¯∂tu1\nh/⌊a∇d⌊l2\nH1+/⌊a∇d⌊lu1\nh/⌊a∇d⌊l2\nH1).\nFor the remaining terms in (4.38) we note that tj/⌊a∇d⌊lθj/⌊a∇d⌊lH1now may be kicked to\nleft hand side using Cauchy–Schwarz and Young’s weighted inequality . The term\ninvolving Cǫcan also be moved to the left hand side. All terms involving α(j) and\nβ(j) can be bounded by (2.8) and (4.32). This finishes the proof after u sing the\nregularity in Theorem 2.1 with f= 0. /square\n4.3.Error bound for the ideal method. We get the final result by combining\nthe two previous lemmas.\nCorollary 4.8. Letun\nhandun\nlodbe the solutions to (2.6)and(3.4)-(3.5), respec-\ntively. The solutions can be split into un\nh=un\nh,1+un\nh,2andun\nlod=un\nlod,1+un\nlod,2,\nwhere the first part has vanishing initial data, and the secon d part a vanishing right\nhand side. The error is bounded by\nn/summationdisplay\nj=2τ/⌊a∇d⌊luj\nh,1−uj\nlod,1/⌊a∇d⌊l2\nH1≤CH2/parenleftBiggn/summationdisplay\nj=1τ(/⌊a∇d⌊lfj/⌊a∇d⌊l2\nL2+/⌊a∇d⌊l¯∂tfj/⌊a∇d⌊l2\nL2)+ max\nj=1,...,n/⌊a∇d⌊lfj/⌊a∇d⌊l2\nL2/parenrightBigg\n,\nand\nn/summationdisplay\nj=2τt2\nj/⌊a∇d⌊luj\nh,2−uj\nlod,2/⌊a∇d⌊l2\nH1≤CH2(/⌊a∇d⌊l¯∂tu1\nh/⌊a∇d⌊l2\nH1+/⌊a∇d⌊lu1\nh/⌊a∇d⌊l2\nH1+/⌊a∇d⌊lu0\nh/⌊a∇d⌊l2\nH1),\nforn≥2.24 PER LJUNG, AXEL M ˚ALQVIST, AND ANNA PERSSON\nProof.This is a direct consequence of Lemma 4.6 and Lemma 4.7 together with\nthe fact that the problem is linear so the error can be split into two co ntributions\nsatisfying the conditions of each lemma. /square\nRemark 4.9.The result from Corollary 4.8 is derived for the ideal method pre-\nsented in (3.4)-(3.5). The GFEM in (3.7)-(3.8) will yield yet another er ror from the\nlocalization procedure. However, due to the exponential decay in T heorem 3.2 and\nTheorem 3.3, it holds for the choice k≈ |log(H)|that the perturbation from the\nideal method is of higher order and the derived result in Corollary 4.8 is still valid.\nFor the details regarding the error from the localization procedure , we refer to [30].\n4.4.Initial data. For general initial data u0\nh,u1\nh∈Vhwe consider the projections\nRmsu0\nhandRmsu1\nh, whereRms=I−Rfis the Ritz-projection onto Vms. Letvbe\nthe differencebetween twosolutionsto the dampedwaveequationw ith the different\ninitial data. From (2.8) it follows that\n/⌊a∇d⌊lv/⌊a∇d⌊l2\nH1≤C(/⌊a∇d⌊l¯∂t(u1\nh−Rmsu1\nh)/⌊a∇d⌊l2\nL2+/⌊a∇d⌊lu1\nh−Rmsu1\nh/⌊a∇d⌊l2\nb),\nwhere we have chosen to keep the b-norm. For the first term we may use the\ninterpolant IHto achieve H. For the second term use\n/⌊a∇d⌊lu1\nh−Rmsu1\nh/⌊a∇d⌊l2\nb≤β+\nα−+τβ−(/⌊a∇d⌊lu1\nh−Rmsu1\nh/⌊a∇d⌊l2\na+τ/⌊a∇d⌊lu1\nh−Rmsu1\nh/⌊a∇d⌊l2\nb), (4.39)\nIf the initial data fulfills the following condition for some g∈L2(Ω)\na(u1\nh,v)+τb(u1\nh,v) = (g,v),∀v∈Vh, (4.40)\nthen we may deduce\n/⌊a∇d⌊lu1\nh−Rmsu1\nh/⌊a∇d⌊l2\na+τ/⌊a∇d⌊lu1\nh−Rmsu1\nh/⌊a∇d⌊l2\nb= (g,u1\nh−Rmsu1\nh)≤CH/⌊a∇d⌊lg/⌊a∇d⌊lL2/⌊a∇d⌊lu1\nh−Rmsu1\nh/⌊a∇d⌊lH1.\nHence, the error introduced by the projection of the initial data is of order H.\nThe condition (4.40) appears when applying the LOD method to classic al wave\nequations, see [1], where it is referred to as “well prepared data” . We note in our\ncase that if Bis small compared to A, that is if the damping is strong, then the\nconstant in (4.39) is small. In some sense, this means that the condit ion in (4.40) is\nof “less importance”, which is consistent with the fact that strong damping reduces\nthe impact of the initial data over time.\n5.Reduced basis method\nThe GFEM as it is currently stated requires us to solve the system in ( 3.7) for\neachcoarsenodeineachtimestep, i.e. Nnumberoftimes. Wewillalterthemethod\nby applying a reduced basis method, such that it will suffice to find the solutions\nforM < N time steps, and compute the remaining in a significantly cheaper and\nefficient way.\nFirst of all, we note how the system (3.7) that ξn\nk,xsolves resembles a parabolic\ntype equation with no source term. That is, the solution will decay ex ponentially\nuntil it is completely vanished. An example of how ξn\nk,xvanish with increasing n\ncan be seen in Figure 2, where the coefficients are given as\nA(x) =/parenleftBig\n2−sin/parenleftbig2πx\nεA/parenrightbig/parenrightBig−1\nandB(x) =/parenleftBig\n2−cos/parenleftbig2πx\nεB/parenrightbig/parenrightBig−1\n,\nwithεA= 2−4andεB= 2−6.A GENERALIZED FEM FOR THE STRONGLY DAMPED WAVE EQUATION 25\nIn Figure2it isalsoseenhowthe solutionsdecaywith asimilarshapethr oughall\ntime steps. This gives the idea that it is possible to only evaluate the so lutions for\na few time steps, and utilize these solutions to find the remaining ones . This idea\ncan be further investigated by storing the solutions {ξn\nk,x}N\nn=1and analyzing the\ncorresponding singular values. The singular values are plotted and s een in Figure\n3. It is seen how the values decrease rapidly, and that most of the v alues lie on\nmachine precision level. In practice, this means that the information in{ξn\nk,x}N\nn=1\ncan be extracted from only a few ξn\nk,x’s. We use this property to decrease the\ncomputational complexity by means of a reduced basis method. We r emark that\nsingular value decomposition is not used for the method itself, but is m erely used\nas a tool to analyze the possibility of applying reduced basis methods .\n0.3 0 .4 0 .5 0 .6 0 .7−0.0010−0.00050.00000.00050.0010n= 1\nn= 25\nn= 50\nn= 100\nn= 150\nn= 200\nFigure 2. The behavior of the correction functions ξn\nk,xwith in-\ncreasing n. The time step is τ= 0.01 andkis here chosen so that\nthe support covers the entire interval.\n0 20 40 60 80 100\nn10−1710−1510−1310−1110−910−710−510−310−1\nFigure 3. The singular values obtained when performing a sin-\ngular value decomposition of the matrix created by storing the\nfinescale corrections {ξn\nk,x}N\nn=1withN= 100.26 PER LJUNG, AXEL M ˚ALQVIST, AND ANNA PERSSON\nThe main idea behind reduced basis methods is to find an approximate s olu-\ntion in a low-dimensional space VRB\nM,k,x, which is created using a number of al-\nready computed solutions. More precisely, to construct a basis fo r this space, one\nfirst computes Msolutions {ξm\nk,x}M\nm=1, whereM < N. By orthonormalizing these\nsolutions using e.g. Gram–Schmidt orthonormalization, we yield a set o f vectors\n{ζm\nk,x}M\nm=1, called the reduced basis. Consequently, the reduced basis space be-\ncomesVRB\nM,k,x= span({ζm\nk,x}M\nm=1) for each node x∈ N. With this space created,\nthe procedure of finding {ξn\nk,x}N\nn=1is now reduced to finding {ξn\nk,x}M\nn=1, and then\napproximate the remaining solutions by {ξn,rb\nk,x}N\nn=M+1⊂VRB\nM,k,x. The matrix sys-\ntem to solve for a solution in VRB\nM,k,xis of dimension M×M, so when Mis chosen\nsmall, the last N−Msolutions are significantly cheaper to compute, which solves\nthe issue of computing Nproblems on the finescale space.\nWhen constructing the reduced basis {ζm\nk,x}M\nm=1, it is important to be aware of\nthe fact that the solution corrections {ξn\nk,x}N\nn=1all show very similar behavior. In\npractice, this implies that many of the ξn\nk,x’s are linearly dependent, hence causing\nfloating point errors to become of significant size in the RB-space VRB\nM,k,x. To work\naround this issue, one may include a relative tolerance level that rem oves a vector\nfrom the basis if it is too close to being linearly dependent to one of the previously\northonormalized vectors. One may moreover use this tolerance lev el as a criterion\nfor the amount of solutions, M, to pre-compute. That is, once the first vector is\nremovedfromthe orthonormalizationprocess, then the RB-spac econtainssufficient\ninformation and no more solutions need to be added.\nIn total, the novel method first requires that we solve NHnumber of systems on\nthe localized fine scale in order to construct the multiscale space Vms,k. Moreover,\nwe require to solve a localized fine system NHtimes for Mtime steps to create\nthe RB-space VRB\nM,k,xfor each coarse node x∈ N. By utilizing the RB-space, the\nremaining N−Mfinescale corrections are then solved for in an M×Mmatrix\nsystem, and we yield the sought solution uN,rb\nlod,kby computing a matrix system on\nthe coarse grid with the multiscale space Vms,k.\n6.Numerical examples\nIn this section we present numerical examples that illustrate the pe rformance of\nthe established theory. For all examples, we consider the domain to be the unit\nsquare Ω = [0 ,1]×[0,1]. The coefficients A(x,y) andB(x,y) used in all examples\naregeneratedrandomlywith valuesin the interval [10−1,103], and examples ofsuch\nareseen in Figure 4. Moreover,as initial value for eachexample we se tu0=u1= 0,\nand the source function is given by f= 1.\nThe first example is used to show how the performance is effected by the lo-\ncalization parameter k. Here, we evaluate the solution on the full grid, un\nlod, and\ncompare it with the localized solution, un\nlod,k, askvaries. For the example the\ntime step τ= 0.02 was used and final time was set to T= 1. The fine and coarse\nmeshes were set to h= 2−7andH= 2−4respectively, and we let k= 2,3,...,7.\nThe relative error between the functions can be seen in Figure 5. He re we can see\nhow the error decays exponentially as kincreases, verifying the theoretical findings\nregarding the localization procedure.\nFor the second example, the performance of the GFEM in (3.7)-(3.8 ) depending\non the coarse mesh width His shown. For this example, the fine mesh width isA GENERALIZED FEM FOR THE STRONGLY DAMPED WAVE EQUATION 27\n200400600800\n(a)A(x,y).\n200400600800\n(b)B(x,y).\nFigure 4. The two different coefficients used for the numerical\nexamples. The contrast is α+/α−=β+/β−= 104.\n2 3 4 5 6\nk10−510−410−310−210−1\nFigure 5. Relative H1-error/⌊a∇d⌊lun\nlod−un\nlod,k/⌊a∇d⌊lH1//⌊a∇d⌊lun\nlod/⌊a∇d⌊lH1between\nthe non-localized and localized method, plotted against the layer\nnumberk.\nset toh= 2−8, and for each coarse mesh width the localization parameter is set to\nk= log2(1/H). Moreover, the time step is set to τ= 0.02 (for the GFEM as well\nas the reference solution) and the solution is evaluated at T= 1. To compute the\nerror, we use a FEM solution on the fine mesh as a reference solution . The error\nas a function of 1 /Hcan be seen in Figure 6. Here it is seen how the error for the\nnovel method decays faster than linearly, confirming the error es timates derived\nin Section 4. For comparison, Figure 6 also shows the error of the st andard FEM\nsolution, as well as the solution using the standard LOD method with c orrection\nsolely on AandBrespectively, i.e. corrections based on the bilinear forms a(·,·)\nandb(·,·) respectively and without finescale correctors. As expected, th e error of\nthese methods stay at a constant level through all coarse grid siz es.\nAt last, we compute the solution where the system (3.7) is computed using the\nreduced basis approach. For this example, we let the number of pre -computed\nsolutions Mvary, and see how the error between the solutions un\nlod,kandun,rb\nlod,k28 PER LJUNG, AXEL M ˚ALQVIST, AND ANNA PERSSON\n2122232425\n1/H10−210−1\nNew method\nFEM\nLOD (A)\nLOD (B)\nO(H)\nFigure 6. Relative H1-error/⌊a∇d⌊lun\nref−un\nlod,k/⌊a∇d⌊lH1//⌊a∇d⌊lun\nref/⌊a∇d⌊lH1between\nthe referencesolutionandthe approximatesolutioncomputed with\nthe proposed method (without reduced basis computations).\n5 10 15 20\nM10−410−310−210−1100\nFigure 7. Relative H1-error/⌊a∇d⌊lun\nlod,k−un,rb\nlod,k/⌊a∇d⌊lH1//⌊a∇d⌊lun\nlod,k/⌊a∇d⌊lH1be-\ntween the solution with and without the reduced basis approach,\nplotted against the number of pre-computed solutions.\nbehaves. In the example we have the fine mesh h= 2−8, the coarse mesh H= 2−5,\nthe time step τ= 0.02, and the final time T= 1. The result can be seen in Figure\n7. Here it is seen how the error decreases rapidly with the amount of pre-computed\nsolutions. Note that it is sufficient to compute approximately 10 solut ions to yield\nan errorsmaller than the discretization error for the main method in Figure 6. This\nfor the case when the number of time steps are N= 50. We emphasize that a large\nincrement in time steps does not impact the number of pre-compute d solutions M\nsignificantly, making the RB-approach relatively more efficient the mo re time steps\nthat are considered.A GENERALIZED FEM FOR THE STRONGLY DAMPED WAVE EQUATION 29\nReferences\n[1] A. Abdulle and P. Henning. Localized orthogonal decompo sition method for the wave equa-\ntion with a continuum of scales. Math. Comput. , 86(304):549–587, 2017.\n[2] G. Avalos and I. Lasiecka. Optimal blowup rates for the mi nimal energy null control of the\nstrongly damped abstract wave equation. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) , 2(3):601–\n616, 2003.\n[3] J. Azevedo, C. Cuevas, and H. Soto. Qualitative theory fo r strongly damped wave equations.\nMathematical Methods in the Applied Sciences , 40, 08 2017.\n[4] I. Babuska and J. E. Osborn. Generalized finite element me thods: Their performance and\ntheir relation to mixed methods. 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Existence and asymptotic behavior for a stro ngly damped nonlinear wave equa-\ntion.Canadian Journal of Mathematics , 32(3):631–643, 1980."    },    {        "title": "1806.04881v1.Low_magnetic_damping_of_ferrimagnetic_GdFeCo_alloys.pdf",        "content": "1 \n Low magnetic damping of ferrimagnetic GdFeCo alloys \nDuck-Ho Kim1†*, Takaya Okuno1†, Se Kwon Kim2, Se-Hyeok Oh3, Tomoe Nishimura1, \nYuushou Hirata1, Yasuhiro Futakawa4, Hiroki Yoshikawa4, Arata Tsukamoto4, Yaroslav \nTserkovnyak2, Yoichi Shiota1, Takahiro Moriyama1, Kab-Jin Kim5, Kyung-Jin Lee3,6,7, and \nTeruo Ono1,8* \n1Institute for Chemical Research, Kyoto University, Uji, Kyoto 6 11-0011, Japan \n2Department of Physics and Astronomy, University of California, Los Angeles, California \n90095, USA \n3Department of Nano-Semiconductor and Engineering, Korea Univers ity, Seoul 02841, \nRepublic of Korea \n4College of Science and Technology, Nihon University, Funabashi,  Chiba 274-8501, Japan \n5Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon \n34141, Republic of Korea \n6Department of Materials Science & Engineering, Korea University , Seoul 02841, Republic \nof Korea \n7KU-KIST Graduate School of Converging Science and Technology, K orea University, Seoul \n02841, Republic of Korea \n8Center for Spintronics Research Network (CSRN), Graduate School  of Engineering Science, \nOsaka University, Osaka 560-8531, Japan \n \n† These authors contributed equally to this work. \n* E-mail: kim.duckho.23z@st.kyoto-u.ac.jp , ono@scl.kyoto-u.ac.jp  2 \n We investigate the Gilbert damping parameter  for rare earth (RE)–\ntransition metal (TM) ferrimagnets over a wide temperature rang e. Extracted from the \nfield-driven magnetic domain-wall mobility,  was as low as 7.2 × 10-3 and was almost \nconstant across the angular momentum compensation temperature 𝑻𝐀, starkly \ncontrasting previous predictions that  should diverge at 𝑻𝐀 due to vanishing total \nangular momentum. Thus, magnetic damping of RE-TM ferrimagnets is not related to \nthe total angular momentum but is dominated by electron scatter ing at the Fermi level \nwhere the TM has a dominant damping role.  \n  3 \n Magnetic damping, commonly described by the Gilbert damping par ameter, \nrepresents the magnetization relaxation phenomenon, describing how quickly magnetization \nspins reach equilibrium [1–3]. Understanding the fundamental or igin of the damping as well \nas searching for low damping materials has been a central theme  of magnetism research. \nSeveral theoretical models for magnetic damping have been propo sed [4–11] and compared \nwith experiments [12–20]. Ultra-low damping was predicted in fe rromagnetic alloys using a \nlinear response damping model [11] and was demonstrated experim entally for CoFe alloys \n[20]. However, the majority of these studies have focused only on ferromagnetic systems. \nAntiferromagnets, which have alt ernating orientations of their neighboring magnetic \nmoments, have recently received considerable attention because of their potential importance \nfor spintronic applications [21– 30]. Antiferromagnetic spin sys tems can have much faster \nspin dynamics than their ferromagnetic counterparts, which is a dvantageous in spintronic \napplications [21, 25, 31–39]. However, the manipulation and con trol of antiferromagnets is \nchallenging because the net magnetic moment is effectively zero . Recently, antiferromagnetic \nspin dynamics have been successfully demonstrated using the mag netic domain-wall (DW) \ndynamics in ferrimagnets with finite magnetization in the vicin ity of the angular momentum \ncompensation temperature, at which the net angular momentum van ishes [38]. This field-\ndriven antiferromagnetic spin dyn amics is possible because the time evolution of the \nmagnetization is governed by the commutation relation of the an gular momentum rather than \nthe commutation relation of the magnetic moment.  \nMotivated by the aforementioned result, in this letter, we inve stigate the magnetic \ndamping of ferrimagnets across th e angular momentum compensatio n temperature, which \nwill allow us to understand magnetic damping in antiferromagnet ically coupled system. We 4 \n selected rare earth (RE)–transition metal (TM) ferrimagnets for  the material platforms \nbecause they have an angular momentum compensation temperature 𝑇 w h e r e  \nantiferromagnetic spin dynamics are achieved [38, 40, 41]. The magnetic-field-driven DW \nmotion was explored over a wide range of temperatures including  𝑇, and the Gilbert \ndamping parameter was extracted from the measured DW mobility a t each temperature by \nemploying the collective coordina te model initially developed f or ferrimagnetic spin \ndynamics [38]. Contrary to the previous prediction that the Gil bert damping parameter would \ndiverge at 𝑇 due to the vanishing of the total angular momentum [42, 43], w e found that the \nGilbert damping parameter remained nearly constant over a wide range of temperatures \nacross 𝑇 with the estimated value as low as 7.2 × 10-3, which was similar to the reported \nvalues of TM-only ferromagnets [20]. These results suggested th at Gilbert damping was \nmainly governed by electron scattering at the Fermi level, and hence, the 4f electron of the \nR E  e l e m e n t ,  w h i c h  l i e s  f a r  b e l o w  t h e  F e r m i  l e v e l ,  d i d  n o t  p l a y  an important role in the \nmagnetic damping of RE–TM ferrimagnets. \nFor this study, we prepared perpendicularly magnetized ferrimag netic GdFeCo films \nin which the Gd and FeCo moments were coupled antiferromagnetic ally. Specifically, the \nfilms were 5-nm SiN/30-nm Gd 23.5Fe66.9Co9.6/100-nm SiN on an intrinsic Si substrate. The \nGdFeCo films were then patterned  into 5-µm-wide and 500-µm-long  microwires with a Hall \ncross structure using electron beam lithography and Ar ion mill ing. For current injection, \n100-nm Au/5-nm Ti electrodes were stacked on the wire. A Hall b ar was designed to detect \nthe DW velocity via the anomalous Hall effect (AHE). \nWe measured the magnetic DW motion using a real-time DW detecti on technique [38, \n40, 41, 44, 45] [see Fig. 1(a) for a schematic]. We first appli ed a magnetic field of –200 mT 5 \n to saturate the magnetization al ong the –z direction. Subsequen tly, a constant perpendicular \nmagnetic field 𝜇𝐻, which was lower than the coercive field, was applied along +z  direction. \nNext, a d.c. current was applied along the wire to measure the anomalous Hall voltage. Then, \na current pulse (12 V , 100 ns) was injected through the writing  line to nucleate the DW in the \nwire. The created DW was moved along the wire and passed throug h the Hall bar because of \nthe presence of 𝜇𝐻. The DW arrival time was detected by monitoring the change in the Hall \nvoltage using a real-time oscillo scope. The DW velocity could t hen be calculated from the \narrival time and the travel dis tance between the writing line a nd Hall bar (500 µm). \nFigure 1(b) shows the averaged DW velocity 〈𝑣〉 as a function of the perpendicular \nmagnetic field 𝜇𝐻 for several temperatures 𝑇∗. Here, we used the d.c. current density of \n|𝐽|ൌ1.3×1010 A / m2 to measure the AHE change due to DW motion. Note that 𝑇∗ i s  a n  \nelevated temperature that considers Joule heating by d.c. curre nt [46]. To eliminate the \nundesired current-induced spin-transfer-torque effect, we avera ged the DW velocity for 𝐽 \nand –𝐽, i.e., 〈𝑣〉ൌሾ𝑣ሺ𝐽ሻ𝑣ሺെ𝐽ሻሿ/2. Figure 1(b) shows that 〈𝑣〉 increases linearly with \n𝜇𝐻 for all 𝑇∗. Such linear behavior can be described by 〈𝑣〉ൌ𝜇ሾ𝜇𝐻െ𝜇 𝐻ሿ, where 𝜇 \nis the DW mobility and 𝜇𝐻 is the correction field, which generally arises from \nimperfections in the sample or complexities of the internal DW structure [47, 48]. We note \nthat 𝜇𝐻 can also depend on the temperature dependence of the magnetic properties of \nferrimagnets [45]. Figure 1(c) shows 𝜇 as a function of 𝑇∗ at several current densities \n(|𝐽|ൌ1.3, 1.7, and 2.0 ×1010 A / m2). A sharp peak clearly occurs for 𝜇 a t  𝑇∗ൌ241.5 K \nirrespective of |𝐽|. The drastic increase of 𝜇 is evidence of antiferromagnetic spin dynamics \nat 𝑇, as demonstrated in our pre vious report [38, 40, 41]. \nThe obtained DW mobility was theoretically analyzed as follows.  The DW velocity 6 \n of ferrimagnets in the precessional regime is given by [38, 39]  \n          𝑉 ൌ 𝜆𝛼ሺ𝑠ଵ𝑠 ଶሻሺ𝑀ଵെ𝑀 ଶሻ\nሾ𝛼ሺ𝑠ଵ𝑠 ଶሻሿଶሺ𝑠ଵെ𝑠 ଶሻଶ𝜇𝐻,                                                                                ሺ1ሻ  \nwhere 𝑉 is the DW velocity, 𝜆 is the DW width, 𝜇𝐻 is the perpendicular magnetic field, \n𝛼 is the Gilbert damping parameter, 𝑀 and 𝑠 are the magnetization and the spin angular \nmomentum of one sublattice, respectively. The spin angular mome ntum densities are given \nby 𝑠ൌ𝑀 /𝛾 [49], where 𝛾ൌ𝑔 𝜇/ℏ is the gyromagnetic ratio of lattice 𝑖, 𝑔 i s  t h e  \nLandé g factor of lattice 𝑖, 𝜇 is the Bohr magneton, and ℏ is the reduced Plank’s constant. \nThe Gilbert damping is in principle different for two sublattic e s ,  b u t  f o r  s i m p l i c i t y ,  w e  \nassume that it is the same, which can be considered as the aver age value of the damping \nparameters for the two sublattices weighted by the spin angular  momentum density. We note \nthat this assumption does not alter our main conclusion: low da mping and its insensitivity to \nthe temperature. Equation (1) gives the DW mobility 𝜇 a s  𝜆𝛼ሺ𝑠ଵ𝑠 ଶሻሺ𝑀ଵെ𝑀 ଶሻ/\nሼሾ𝛼ሺ𝑠ଵ𝑠 ଶሻሿଶሺ𝑠ଵെ𝑠 ଶሻଶሽ, which can be rearranged as \n          𝜇 ሺ𝑠ଵ𝑠 ଶሻଶ𝛼ଶെ𝜆ሺ𝑠ଵ𝑠 ଶሻሺ𝑀ଵെ𝑀 ଶሻ𝛼𝜇 ሺ𝑠ଵെ𝑠 ଶሻଶൌ 0                                           ሺ2ሻ  \nUsing Eq. (2) to find the solution of 𝛼, we find \n          𝛼 േൌ𝜆ሺ𝑀ଵെ𝑀 ଶሻേඥሾ𝜆ଶሺ𝑀ଵെ𝑀 ଶሻଶെ4 𝜇ଶሺ𝑠ଵെ𝑠 ଶሻଶሿ\n2𝜇ሺ𝑠ଵ𝑠 ଶሻ.                                              ሺ3ሻ  \nEquation (3) allows us to estimate 𝛼 for the given 𝜇. We note that for each value of 𝜇, 𝛼 \nca n h av e t w o v a lu e s,  𝛼ା and 𝛼ି because of the quadratic nature of Eq. (2). Only one of \nthese two solutions is physically sound, which can be obtained using the following energy \ndissipation analysis.  7 \n  The energy dissipation (per unit cross section) through the DW  dynamics is given by \n𝑃ൌ2 𝛼 ሺ 𝑠 ଵ𝑠 ଶሻ𝑉ଶ/𝜆  2𝛼ሺ𝑠 ଵ𝑠 ଶሻ 𝜆Ωଶ [38, 39], where Ω is the angular velocity of the \nDW. The first and the second terms represent the energy dissipa tion through the translational \nand angular motion of the DW, respectively. In the precessional  regime, the angular velocity \nis proportional to the translational velocity: Ωൌ ሺ𝑠ଵെ𝑠 ଶሻ𝑉/𝛼ሺ𝑠 ଵ𝑠 ଶሻ𝜆. Replacing Ω b y  \nthe previous expression yields 𝑃ൌ𝜂 𝑉ଶ w h e r e  𝜂ൌ2 ሺ 𝑀 ଵെ𝑀 ଶሻ/𝜇 is the viscous \ncoefficient for the DW motion:  \n          𝜂 ൌ2\n𝜆ቊ𝛼ሺ𝑠ଵ𝑠 ଶሻ ሺ𝑠ଵെ𝑠 ଶሻଶ\n𝛼ሺ𝑠ଵ𝑠 ଶሻቋ .                                                                                         ሺ4ሻ  \nThe first and the second terms in parenthesis capture the contr ibutions to the energy \ndissipation from the translational and angular dynamics of the DW, respectively. The two \nsolutions for the Gilbert damping parameter,  𝛼ା and 𝛼ି, can yield the same viscous \ncoefficient 𝜂. The case of the equal solutions,  𝛼ାൌ𝛼 ି, corresponds to the situation when \nthe two contributions are identical:  𝛼േൌሺ 𝑠 ଵെ𝑠 ଶሻ/ሺ𝑠ଵ𝑠 ଶሻ. For the larger solution 𝛼ൌ\n𝛼ା, the energy dissipation is dominated by the first term, i.e., through the translational DW \nmotion, which should be the case in the vicinity of 𝑇 where the net spin density ሺ𝑠ଵെ𝑠 ଶሻ \nis small and thus the angular velocity is negligible. For examp le, at exact 𝑇, the larger \nsolution 𝛼ା is the only possible solution because the smaller solution is zero, 𝛼ିൌ0, and \nthus unphysical. For the smaller solution 𝛼ൌ𝛼 ି, the dissipation is dominated by the second \nterm, i.e., through the precessional motion, which should descr ibe cases away from 𝑇. \nTherefore, in the subsequent analysis, we chose the larger solu tion  𝛼ା in the vicinity of 𝑇 \nand the smaller solution 𝛼ି far away from 𝑇 and connected the solution continuously in \nbetween. 8 \n The other material parameters such as 𝑀ଵ, 𝑀ଶ, 𝑠ଵ, and 𝑠ଶ a r e  e s t i m a t e d  b y  \nmeasuring the net magnetic moment of GdFeCo film, |𝑀୬ୣ୲|, for various temperatures. \nBecause 𝑀୬ୣ୲ includes contributions from both  the Gd and FeCo sub-moments, the sub-\nmagnetic moments, 𝑀ଵ a n d  𝑀ଶ, could be decoupled based on the power law criticality [see \ndetails in refs. 38, 40]. The spin angular momentums, 𝑠ଵ and 𝑠ଶ, were calculated using the \nknown Landé g factor of FeCo and Gd (the Landé g factor of FeCo  is 2.2 and that of Gd is \n2.0) [50–52]. \nFigures 2(a)–(c) show the temperature-dependent DW mobility 𝜇, sub-magnetic \nmoment 𝑀, and sub-angular momentum  𝑠, respectively. Here, we used the relative \ntemperature defined as ∆𝑇 ൌ 𝑇∗െ𝑇  to investigate the Gilbert damping near 𝑇. The \nGilbert damping parameter 𝛼 was obtained based on Eq. (3) and the information in Fig. \n2(a)–(c). Figure 2(d) shows the resulting values of 𝛼േ as a function of ∆𝑇. For ∆𝑇ଵ൏\n∆𝑇 ൏ ∆𝑇 ଶ, 𝛼ା is nearly constant, while 𝛼ି varies significantly. For ∆𝑇 ൏ ∆𝑇 ଵ and ∆𝑇 \n∆𝑇ଶ, on the other hand,  𝛼ି is almost constant, while 𝛼ା varies significantly. At ∆𝑇 ൌ ∆𝑇 ଵ \nand ∆𝑇 ൌ ∆𝑇 ଶ, the two solutions are equal, corresponding to the aforementio ned case when \nthe energy dissipation through the translational and angular mo tion of the DW are identical. \nThe proper damping solution can be selected by following the gu ideline obtained \nfrom the above analysis. For ∆𝑇ଵ൏∆ 𝑇൏∆ 𝑇 ଶ, which includes 𝑇, the energy dissipation \nshould be dominated by the translational motion, and thus 𝛼ା is a physical solution. Note \nalso that 𝛼ି becomes zero at 𝑇, which results in infinite DW mobility in contradiction with \nthe experimental observation. For ∆𝑇 ൏ ∆𝑇 ଵ and ∆𝑇  ∆𝑇 ଶ, where the energy dissipation is \ndominated by the angular motion of the DW, 𝛼ି is the physical solution. 9 \n Figure 3 shows the resultant Gil bert damping parameter in all t ested temperature \nranges. The Gilbert damping parameter was almost constant acros s 𝑇 with 𝛼ൌ7.2 × 10-3 \n(see the dotted line in Fig. 3). This result is in stark contra st to the previous prediction. In ref. \n[42], Stanciu et al. investigated the temperature dependence of the effective Gilb ert damping \nparameter based on a ferromagnet-based model and found that the  damping diverged at 𝑇. \nBecause they analyzed the magnetic resonance in ferrimagnetic m aterials based on a \nferromagnet-based model, which cannot describe the antiferromag netic dynamics at 𝑇 a t  \nwhich the angular momentum vanis hes, it exhibits unphysical res ults. However, our \ntheoretical analysis for field-driven ferromagnetic DW motion b ased on the collective \ncoordinate approach can properly describe both the antiferromag netic dynamics in the \nvicinity of 𝑇 and the ferromagnetic dynamics away from 𝑇 [38]. Therefore, the \nunphysical divergence of the Gilbert damping parameter at 𝑇 is absent in our analysis. \nOur results, namely the insensitivity of damping to the compens ation condition and \nits low value, have important implications not only for fundame ntal physics but also for \ntechnological applications. From the viewpoint of fundamental p hysics, nearly constant \ndamping across 𝑇 indicates that the damping is almost independent of the total angular \nmomentum and is mostly determined by electron spin scattering n ear the Fermi level. \nSpecifically, our results suggest that the 4f electrons of RE e lements, which lie in a band far \nbelow the Fermi level, do not play an important role in the mag netic damping of RE-TM \nferrimagnets, whereas the 3d and 4s bands of TM elements have a  governing role in magnetic \ndamping. This result is consistent with the recently reported t heoretical and experimental \nresults in FeCo alloys [20]. From the viewpoint of practical ap plication, we note that the \nestimated damping of 𝛼ൌ7.2 × 10-3 is the upper limit, as the damping estimated from DW 10 \n dynamics is usually overestimated due to disorders [53]. The ob tained value of the Gilbert \ndamping parameter is consistent with our preliminary ferromagne t i c  r e s o n a n c e  ( F M R )  \nmeasurements. The experimental results from FMR measurements an d the corresponding \ntheoretical analysis will be publ ished elsewhere. This low valu e of the Gilbert damping \nparameter suggests that ferrimagne ts can serve as versatile pla t f o r m s  f o r  l o w - d i s s i p a t i o n  \nhigh-speed magnetic devices such as spin-transfer-torque magnet ic random-access memory \nand terahertz magnetic oscillators. \nIn conclusion, we investigated the field-driven magnetic DW mot ion in ferrimagnetic \nG d F e C o  a l l o y s  o v e r  a  w i d e  r a n g e  o f  t e m p e r a t u r e s  a c r o s s  𝑇 and extracted the Gilbert \ndamping parameter from the DW mobility. The estimated Gilbert d amping parameter was as \nlow as 7.2 × 10-3 and almost constant over the temperature range including 𝑇, which is in \nstark contrast to the previous prediction in that the Gilbert d amping parameter would diverge \nat 𝑇 due to the vanishing total angular momentum. 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Shinjo, Science 284, 468 \n(1999). \n[49] In this Letter , the parameters such as the spin angular mo mentum density 𝑠 r e p r e s e n t  \nthe magnitudes of the quantities.  Their directions are separate ly handled through the signs in \nthe equations of motion. \n[50] C. Kittel, Phys. Rev. 76, 743 (1949). \n[51] G. G. Scott, Rev. Mod. Phys. 34, 102 (1962). \n[52] B. I. Min and Y.-R. Jang, J. Phys. Condens. Matter 3, 5131 (1991). \n[53] H. Min, R. D. McMichael, M. J. Donahue, J. Miltat, and M. D. Stiles, Phys. Rev. Lett. \n104, 217201 (2010). \n   16 \n Figure Captions \nFigure 1(a) Schematic illustration of the GdFeCo microwire devi ce. (b) The averaged DW \nvelocity 〈𝑣〉 as a function of the perpendicular magnetic field 𝜇𝐻 for several temperatures \n𝑇∗ (202, 222, 242, 262, and 282 K). The dots indicate the best li n e a r  f i t s .  ( c )  T h e  D W  \nmobility 𝜇 as a function of 𝑇∗ at several current densities ( |𝐽|ൌ1.3, 1.7, and 2.0 ×1010 \nA/m2). \nFigure 2 The temperature-dependent (a) DW mobility 𝜇, (b) sub-magnetic moment 𝑀, and \n(c) sub-angular momentum  𝑠. Here, we use the relative temperature defined as ∆𝑇 ൌ 𝑇∗െ\n𝑇. (d) The Gilbert damping parameter 𝛼േ as a function of ∆𝑇. Here, we use 𝜆ൌ15 nm for \nproper solutions of Eq. (3). \nFigure 3 The resultant Gil bert damping parameter 𝛼 in all tested temperature ranges. \n 17 \n Acknowledgements  \nThis work was supported by the JSPS KAKENHI (Grant Numbers 15H0 5702, 26103002, and \n26103004), Collaborative Research Program of the Institute for Chemical Research, Kyoto \nUniversity, and R & D project for ICT Key Technology of MEXT fr om the Japan Society for \nthe Promotion of Science (JSPS). This work was partly supported  by The Cooperative \nResearch Project Program of the Research Institute of Electrica l Communication, Tohoku \nUniversity. D.H.K. was supported as an Overseas Researcher unde r the Postdoctoral \nFellowship of JSPS (Grant Number P16314). S.H.O. and K.J.L. wer e supported by the \nNational Research Foundation of Korea (NRF-2015M3D1A1070465, 20 17R1A2B2006119) \nand the KIST Institutional Program (Project No. 2V05750). S.K.K . was supported by the \nArmy Research Office under Contract No. W911NF-14-1-0016. K.J.K . was supported by the \nNational Research Foundation of Korea (NRF) grant funded by the  Korea Government \n(MSIP) (No. 2017R1C1B2009686).  \nCompeting financial interests \nThe authors declare no competing financial interests. 200 225 250 275 3000.00.51.01.52.0\n 1.3\n1.7\n2.0\n [104 m/sT]\nT* [K]J [1010 A/m2]0 50 100 1500.00.51.01.5\n  202\n 222\n 242\n 262\n 282<v> [km/s]\n0H [mT]T* [K]\nFigure 1b\nca\nWriting line\n\tܫ\nܸ\nߤܪ\ny xz-60 -40 -20 0 20 40 60 801.52.02.53.0 s1\n s2s [10-6 Js/m3]\nT [K]-60 -40 -20 0 20 40 60 800.00.51.01.52.0\n [104 m/sT]\nT [K]\n-60 -40 -20 0 20 40 60 8010-610-510-410-310-210-1100\n \n +\n -\nT [K]T1T2-60 -40 -20 0 20 40 60 800.30.40.50.6  M1\n M2M [MA/m]\nT [K]a\nb\nc\nd\nFigure 2Figure 3-60 -40 -20 0 20 40 60 8010-610-510-410-310-210-1100\n \nT [K]"    },    {        "title": "1412.3783v1.Deviation_From_the_Landau_Lifshitz_Gilbert_equation_in_the_Inertial_regime_of_the_Magnetization.pdf",        "content": "arXiv:1412.3783v1  [cond-mat.mtrl-sci]  11 Dec 2014Deviation From the Landau-Lifshitz-Gilbert equation in th e Inertial regime of the\nMagnetization\nE. Olive and Y. Lansac\nGREMAN, UMR 7347, Universit´ e Fran¸ cois Rabelais-CNRS, Pa rc de Grandmont, 37200 Tours, France\nM. Meyer, M. Hayoun, and J.-E. Wegrowe\nLaboratoire des Solides Irradi´ es, ´Ecole Polytechnique, CEA-DSM, CNRS, F-91128 Palaiseau, Fr ance\n(Dated: February 7, 2018)\nWe investigate in details the inertial dynamics of a uniform magnetization in the ferromagnetic\nresonance (FMR) context. Analytical predictions and numer ical simulations of the complete equa-\ntions within the Inertial Landau-Lifshitz-Gilbert (ILLG) model are presented. In addition to the\nusual precession resonance, the inertial model gives a seco nd resonance peak associated to the nuta-\ntion dynamics provided that the damping is not too large. The analytical resolution of the equations\nof motion yields both the precession and nutation angular fr equencies. They are function of the in-\nertial dynamics characteristic time τ, the dimensionless damping αand the static magnetic field H.\nA scaling function with respect to ατγHis found for the nutation angular frequency, also valid for\nthe precession angular frequency when ατγH≫1. Beyond the direct measurement of the nutation\nresonance peak, we show that the inertial dynamics of the mag netization has measurable effects on\nboth the width and the angular frequency of the precession re sonance peak when varying the applied\nstatic field. These predictions could be used to experimenta lly identify the inertial dynamics of the\nmagnetization proposed in the ILLG model.\nPACS numbers:\nI. INTRODUCTION\nThe Landau-Lifshitz-Gilbert (LLG) equation is a ki-\nnetic equation that does not contain acceleration terms,\ni.e. that does not contain inertia. The corresponding\ntrajectory is reduced to a damped precession around\nthe axis defined by the effective field. The measurement\nof this precession is usually performed by the mean of\nferromagnetic resonance (FMR). The power absorbed by\nthe system is then measured at steady state while adding\nan oscillatory field to the effective field, and tuning the\nfrequency close to the resonance frequency. However,\nthe validity of the LLG equation is limited to large\ntime scales1, or low frequency regimes (similarly to the\nDebye model of electric dipoles2). Indeed, the precession\nwith damping described by the LLG equation is a\ndiffusion process in a field of force, for which the angular\nmomentum has reached equilibrium. Accordingly, if\nthe measurements are performed at fast enough time\nscales, or high enough frequencies, inertial terms should\nbe expected to play a role in the dynamics, which is no\nlonger reduced to a damped precession3–9. A nutation\ndynamics is therefore expected, giving a second resonant\npeak at the nutation frequency, and this new absorption\nshould be measurable with dedicated spectroscopy (e.g.\nusing infrared spectroscopy).\nDespite its fundamental importance, a systematic\nexperimental investigation of possible inertial effects of\nthe uniform magnetization has however been overlooked.\nIn order to evidence experimentally the consequences\nof inertia in the dynamics of a uniform magnetization,\nit is first necessary to establish the characteristics\nthat would allow to discriminate inertia from spuriouseffects in spectroscopy experiments. We propose in this\npaper some simple theoretical and numerical tools than\ncan be used by experimentalists in order to evidence\nunambiguouslytheeffectsofinertiaofthemagnetization.\nThe LLG equation reads :\ndM\ndt=γM×/bracketleftbigg\nHeff−ηdM\ndt/bracketrightbigg\n(1)\nwhereMisthemagnetization, Hefftheeffectivemagnetic\nfield,ηthe Gilbert damping, and γthe gyromagnetic ra-\ntio. If the description is extended to the fast degrees\nof freedom (i.e. the degrees of freedom that includes the\ntime derivative of the angularmomentum), a supplemen-\ntary inertial term should be added with the correspond-\ningrelaxationtime τ. FromthisInertialLandau-Lifshitz-\nGilbert (ILLG) model, the new equation reads3–7:\ndM\ndt=γM×/bracketleftbigg\nHeff−η/parenleftbiggdM\ndt+τd2M\ndt2/parenrightbigg/bracketrightbigg\n(2)\nOne of the main consequences of the new dynami-\ncal equation is the emergence of the second resonance\npeak associated to the nutation at high frequencies, as\nreported in our previous study7. In the literature the\nnutation dynamics of magnetic moments has been in-\nvestigated using various theoretical approaches though\nnot yet evidenced experimentally. B¨ ottcher and Henk\nstudied the significance of nutation in magnetization dy-\nnamics of nanostructures such as a chain of Fe atoms,\nand Co islands on Cu(111)8. They found that the nu-\ntation is significant on the femtosecond time scale with\na typical damping constant of 0.01 up to 0.1. Moreover,2\nthey concluded that nutation shows up preferably in low-\ndimensional systems but with a small amplitude with\nrespect to the precession. Zhu et al.predicted a nuta-\ntion dynamics for a single spin embedded in the tunnel-\ning barrierbetween twosuperconductors10. This unusual\nspin dynamics is caused by coupling to a Josephson cur-\nrent. They argue that this prediction might be directly\ntested for macroscopic spin clusters. The nutation is also\ninvolved in the dynamics of a single spin embedded in\nthe tunnel junction between ferromagnets in the pres-\nence of an alternating current11. In an atomistic frame-\nwork, Bhattacharjee et al.showed that first-principle\ntechniques used to calculate the Gilbert damping factor\nmay be extended to calculate the moment of inertia ten-\nsor associated to the nutation9.\nOur previous work7was focussed on the short time\nnutation dynamics generated by the ILLG equation, and\nwas limited to fixed values of the inertial characteristic\ntime scale τ, the dimensionless damping αand the static\nfieldH. In this paper we present a combined analytical\nand numerical simulation study of the ILLG equation\nwith new results. In particular we derive analytical re-\nsults in the small inclination limit that can be used in\nferromagnetic resonance (FMR) experiments, and which\nallow to predict both the precession and nutation reso-\nnance angular frequencies. We also investigate the ILLG\nequation while varying the three parameters α,τandH,\nand scaling functions are found. Finally, we present im-\nportant indications for experimental investigations of the\ninertial dynamics of the magnetization. Indeed, a conse-\nquence of the ILLG equation is the displacement of the\nwell-known FMR peak combined with a modified shape\nwith respect to that given by the LLG equation. This\ndisplacement could not be without consequences on the\ndetermination of the gyromagnetic factor γby ferromag-\nnetic resonance.\nThe paper is organized as follows. In section II we\nshow analytical solutions of the precession and nutation\ndynamics for the uniform magnetization in a static ap-\nplied field H. The small inclination limit is investigated\nin order to reproduce the usual experimental FMR con-\ntext. In section III we describe the numerical simula-\ntions of the magnetization inertial dynamics in both a\nstatic anda smallperpendicular sinusoidalmagneticfield\n(Heff=H+h⊥(ω)). The resonance curves are computed\nand, provided that the damping is not too large, a nu-\ntation resonance peak appears in addition to the usual\nferromagnetic resonance peak associated to the magne-\ntization precession. In section IV the behavior of the\nILLG equation is investigated in details while varying\nthe characteristic time τof the inertial dynamics, the di-\nmensionless damping αand the static field H. A very\ngood agreement is found between the analytical and nu-\nmerical simulation results, and a scaling function with\nrespect to ατγHis found. In section V we propose ex-\nperiments in the FMR context that should evidence the\ninertial dynamics of the magnetization described in the\nILLGmodel. Inparticular,whenthestaticfieldisvaried,the ILLG precession resonance peak has different behav-\niors compared to the usual LLG precession peak with\nshifted resonance angular frequency and modified shape.\nWe show that the differences between LLG and ILLG\nprecession peaks are more pronounced in large damping\nmaterials and increase with the static field. Finally, we\nderive the conclusions in section VI.\nII. ANALYTICAL SOLUTIONS FOR THE ILLG\nEQUATION\nThe magnetization position is described in spherical\ncoordinates ( Ms,θ,φ), where Msis the radius coordinate\nfixed at a constant value for the uniformly magnetized\nbody,θis the inclination and φis the azimuthal angle.\nIn a static magnetic field Hˆ zapplied in the zdirection,\ni.e.H=H(cosθer−sinθeθ) in the spherical basis\n(er,eθ,eφ), Eq. (2) gives the following system :\n¨θ=−1\nτ˙θ−1\nτ1˙φsinθ+˙φ2sinθcosθ\n−ω2\nτ1sinθ (3a)\n¨φsinθ=1\nτ1˙θ−1\nτ˙φsinθ−2˙φ˙θcosθ (3b)\nwhere the characteristic times are τandτ1=ατ,ω2=\nγHis the Larmor angular frequency, and α=γηMsis\nthe dimensionless damping.\nUsingthedimensionlesstime t′=t/τ, Eqs. (3)become\nθ′′=−θ′−/tildewideτ1φ′sinθ+φ′2sinθcosθ\n−/tildewideω2/tildewideτ1sinθ (4a)\nφ′′sinθ=/tildewideτ1θ′−φ′sinθ−2φ′θ′cosθ(4b)\nwhere\nθ′=dθ/dt′, θ′′=d2θ/dt′2, φ′=dφ/dt′, φ′′=d2φ/dt′2,\nand\n/tildewideτ1=τ\nτ1=1\nα\n/tildewideω2=ω2τ=τγH\nIn the following subsections we extract analytical re-\nsults that can be used to predict the positions in the\nangular frequency domain of the precession and nutation\nresonance peaks. We will consider the small inclination\nlimit which holds in the FMR context.3\nA. Precession : exact and approximate solutions\nTo determine the precession dynamics of the iner-\ntial model we search for the long time scale solution\nφ′(t′) =φ′\nprec, whereφ′\nprecis the constant precession ve-\nlocity. Since the damping progressivelyshifts the magne-\ntization to the zaxis, we investigate the small inclination\nlimit where φ′(t′) =φ′\nprecshould hold. With sin θ∼θ\nand cosθ∼1, Eqs. (4) therefore reads :\nθ′′+θ′+/tildewideω2\n0θ= 0 (5a)\nφ′\nprec=/tildewideτ1θ′\nθ+2θ′(5b)\nwhere the natural angular frequency of the overdamped\nharmonic oscillator θ(t′) defined by Eq. (5a) is given by\n/tildewideω0=/radicalBig\n/tildewideτ1(φ′prec+/tildewideω2)−φ′2prec (6)\nThe characteristic equation associated to the differential\nequation Eq. (5a) is β2+β+/tildewideω2\n0= 0 which gives in the\naperiodic regime the two solutions\nβ±=−1±/radicalbig\n1−4/tildewideω2\n0\n2(7)\nSince|β+|<|β−|, the inclination of the magnetization\nbehaves at long time scales as\nθ(t′)∼eβ+t′,\nwhich inserted in Eq. (5b) gives\nφ′\nprec=/tildewideτ1β+\n1+2β+(8)\nIn original time units, the precession velocity ˙φprecis\ntherefore the solution of\n˙φprec=β+(˙φprec)\nατ/parenleftBig\n1+2β+(˙φprec)/parenrightBig (9)\nwhere the function β+(˙φprec) is given by\nβ+(˙φprec) =−1+/radicalbigg\n1−4τ/parenleftBig˙φprec+γH\nα−τ˙φ2prec/parenrightBig\n2(10)\nEquation 9 may be numerically solved to extract the\nprecession velocity, and therefore the precession reso-\nnance peak when a sinusoidal magnetic field h⊥(ω) is\nsuperimposed perpendicular to the static field Hˆ z.\nForτ≪10−11sandα≤0.1, theprecessionvelocity ˙φprec\nfor small applied static fields may be accurately evalu-\nated from a quadratic equation : in this case /tildewideω2\n0≪1 and\nEq. (7) leads to β+≈ −/tildewideω2\n0. Eq. (8) therefore gives a\ncubic equation in φ′\nprecwhere the cubic term −2αφ′3\nprecis negligeable. In this case the solution of the resulting\nquadratic equation is in original time units\n˙φprec=−b−/radicalbig\nb2+12τγH/α\n6τ(11)\nwithb= 2τγH−α−1/α. We choose the negative so-\nlution of the quadratic equation in order to agree with\nthe negative velocity ˙φLLG=−γH/(1+α2) given by the\nLLG model.\nB. Nutation : angular frequency\nUnlike the precession, the nutation properties should\nbederivedconsideringintermediatetime scaleswherethe\nprecession has not yet reached a constant velocity. Eqs.\n(4) should therefore be reconsidered. To derive the nuta-\ntion properties, it is convenient to examine the angular\nvelocityθ′. For simplicity we note θ′=/tildewideωθandφ′=/tildewideωφ.\nEqs. (4) therefore rewrite\n/tildewideω′\nθ=−/tildewideωθ−/tildewideτ1/tildewideωφsinθ+/tildewideω2\nφsinθcosθ\n−/tildewideω2/tildewideτ1sinθ (12a)\n/tildewideω′\nφsinθ=/tildewideτ1/tildewideωθ−/tildewideωφsinθ−2/tildewideωφ/tildewideωθcosθ(12b)\nWe derive Eq. (12a) with respect to time t′which gives\n/tildewideω′′\nθ=−/tildewideω′\nθ+(2/tildewideωφcosθ−/tildewideτ1)/tildewideω′\nφsinθ−/tildewideτ1/tildewideωφ/tildewideωθcosθ\n+/tildewideω2\nφ/tildewideωθ(cos2θ−sin2θ)−/tildewideω2/tildewideτ1/tildewideωθcosθ\nwhere the term /tildewideω′\nφsinθmay be replaced with the expres-\nsion in Eq. (12b). We therefore obtain\n/tildewideω′′\nθ+/tildewideω′\nθ+/parenleftbig\n/tildewideτ2\n1+/tildewideω2/tildewideτ1cosθ/parenrightbig\n/tildewideωθ=\n/tildewideτ1/tildewideωφsinθ+3/tildewideτ1/tildewideωφ/tildewideωθcosθ−2/tildewideω2\nφcosθsinθ\n−(3cos2θ+sin2θ)/tildewideω2\nφ/tildewideωθ (13)\nEq. (13)shouldbecloselyrelatedtothenutationdynam-\nics since it describes the /tildewideωθoscillator. This assumption\nwill be confirmed in section IVA2 for a broad range of\nparameters. Eq. (13) defines the damped oscillator /tildewideωθ\nwhich is non-linearly coupled to the /tildewideωφoscillator. This\nexpression shows that, in the absence of coupling and in\nthe smallinclination limit θ≪1rad, the/tildewideωθoscillatoros-\ncillatesatthenaturalangularfrequency/radicalbig\n/tildewideτ2\n1+/tildewideω2/tildewideτ1. We\ntherefore deduce an approximate expression for the nu-\ntation angularfrequency in the weak coupling case which\nis given by the expression\n/tildewideωweak\nnu=/radicalBig\n/tildewideτ2\n1+/tildewideω2/tildewideτ1 (14)\nwhich in original time units gives\nωweak\nnu=√1+ατγH\nατ(15)4\nFrom Eq. (15) we deduce the following asymptotic\nbehaviors : when τ≪1/αγHthenωweak\nnu∼1/ατ, and\nwhenτ≫1/αγHthenωweak\nnu∼1/√ατ.\nBecause of the non-linear coupling terms in the right-\nhand side of Eq. (13), the true position of the nuta-\ntion resonancepeak in FMR experiments may differ from\nthe approximate angular frequency defined by Eq. (15).\nHoweverthe simulation of the resonancecurves with a si-\nnusoidal magnetic field h⊥(ω) superimposed perpendic-\nular to the static field Hˆ zwill show in section IVA2\nthat the non-linear coupling terms only slightly shift the\nnutation resonance peak from the approximate angular\nfrequency.\nIII. NUMERICAL SIMULATIONS OF THE\nRESONANCE CURVES IN THE ILLG MODEL\nWe apply a fixed magnetic field H=Hˆ zalong the\nzdirection, and a small sinusoidal magnetic field h⊥=\nh⊥cosωtˆ xin thexdirection. In the spherical basis the\ncomponents of the total magnetic field Heff=H+h⊥in\nEq. (2) are\nHeff\nr=Hcosθ+h⊥sinθcosφcosωt\nHeff\nθ=−Hsinθ+h⊥cosθcosφcosωt\nHeff\nφ=−h⊥sinφcosωt.\nwhich lead to the following dynamical equations for the\nspherical angles ( θ,φ) of the magnetization\n¨θ=−1\nτ˙θ−1\nτ1˙φsinθ+˙φ2sinθcosθ\n−ω2\nτ1sinθ+ω3\nτ1cosθcosφcosωt(16a)\n¨φsinθ=1\nτ1˙θ−1\nτ˙φsinθ−2˙φ˙θcosθ\n−ω3\nτ1sinφcosωt (16b)\nwhereω3=γh⊥is the angular frequency associated to\nthe sinusoidal field.\nUsing the dimensionless time t′=t/τ, Eqs. (16) be-\ncome\nθ′′=−θ′−/tildewideτ1φ′sinθ+φ′2sinθcosθ\n−/tildewideω2/tildewideτ1sinθ+/tildewideω3/tildewideτ1cosθcosφcos/tildewideωt′(17a)\nφ′′sinθ=/tildewideτ1θ′−φ′sinθ−2φ′θ′cosθ\n−/tildewideω3/tildewideτ1sinφcos/tildewideωt′(17b)\nwhere\nθ′=dθ/dt′, θ′′=d2θ/dt′2, φ′=dφ/dt′, φ′′=d2φ/dt′2,and\n/tildewideτ1=τ\nτ1=1\nα\n/tildewideω2=ω2τ=τγH\n/tildewideω3=ω3τ=τγh⊥\n/tildewideω=ωτ\nWe useγ= 1011rad.s−1.T−1, and we vary the charac-\nteristic time τfor three different values of the dimension-\nless damping α= 0.1, 0.01 and 0 .5. We investigate sev-\neralvalues of the static magnetic field from H= 0.2Tup\ntoH= 200T. We numerically integrate Eqs. (17) using\neither a double precision second order Runge-Kutta algo-\nrithm or a double precision five order Gear algorithm12.\nTypically, we use time steps 10−7< δt′<10−3depend-\ning on the values of τandω.\nThe resonance curves are obtained by investigating\nthe magnetization response to the small oscillating field\nh⊥(ω) =h⊥cosωtˆ xapplied perpendicular to the static\nfieldH=Hˆ z. We analyse the permanent dynami-\ncal regime where the magnetization components oscil-\nlate around well defined mean values. For fixed values\nof the oscillating field angular frequency ωand oscil-\nlating field amplitude h⊥, we compute the mean value\n< M⊥>(averaged over time) of the transverse magneti-\nzationM⊥(t) =/radicalBig\nM2x(t)+M2y(t), fromwhichweextract\nfor fixed values of ωthe transverse susceptibility defined\nbyχ⊥=d < M ⊥> /dh ⊥. We choose values of the\noscillating field amplitude h⊥= 10−1,10−2,10−3,10−4\nand 10−5T, and we plot < M⊥>with respect to h⊥\nfor each ω. As an example, we show the case α= 0.1,\nτ= 2×10−10s,H= 2Tandω= 1.2×1011rad.s−1.\nThe inset of Fig. 1 shows that the response is linear\n< M⊥>=χ⊥h⊥wherefrom we extract the transverse\nsusceptibility χ⊥using a linear fitting. We repeat the\nsame procedure for each oscillating field angular fre-\nquencyωwhich gives the resonance curve χ⊥(ω) of the\ntransverse susceptibility shown in Fig. 1. Two peaks\nclearly appear, the usual FMR peak associated to the\nprecession velocity, and the nutation peak associated to\nthe nutation dynamics originatingfrom the inertial term.\nIV. RESULTS\nA. Effects of τ\nWe now examine the ILLG model when varying the\ncharacteristic time τ. For different values of the parame-\nterτ, we show in Fig. 2 the typical profiles of the trans-\nverse susceptibility χ⊥versus the angular frequency ωof\nthe applied oscillating field. The four resonance curves\nplotted in figure 2 are obtained by numerical simulations\nwithH= 2Tandα= 0.1. They show how the nu-\ntation resonance peak position depend on the value of\nτ. Asτis increased, the nutation peak moves towards5\n01×10112×10113×1011\nω (rad.s-1)1234567χ⊥10-510-410-310-2\nh⊥10-510-410-310-210-1< Μ⊥>Precession peak\nNutation peak\nFigure 1: Resonance curves of the transverse susceptibil-\nityχ⊥(ω) with respect to the oscillating field angular fre-\nquencyω. The resonance curves are computed within the\nILLG model with τ= 2×10−10s, for dimensionless damping\nα= 0.1 and for an applied static field H= 2T. Two reso-\nnance peaks are observed : the precession resonance at lower\nangular frequency which is the usual FMR and the nutation\nresonance at higher angular frequency. Inset : Example of\nthe calculation of χ⊥such that < M⊥>=χ⊥h⊥obtained for\nω= 1.2×1011rad.s−1.\nthe precession peak with an increasing intensity which is\nan order of magnitude smaller than the precession one\nforτ= 10−11s. Note that the transverse susceptibil-\nity at the resonance follows a power law of the form\nχ⊥(ωILLG\nnu)∝1/ωILLG\nnu, whereωILLG\nnuis defined as the nu-\ntation resonance angular frequency. A similar power law\nis reported for the precession peak obtained for different\nstatic fields H(see section IVB).\nWe now compare the analytical and numerical sim-\nulation results concerning the positions in the angular\nfrequency domain of both the precession and nutation\nresonance peaks.\n1. Precession peak\nWe define ωprec=|˙φprec|as the angular frequency of\nthe precession. When computed from the exact expres-\nsions (9) and (10) we will refer to ωexact\nprec, and when com-\nputed from the approximate expression (11) we will refer\ntoωapprox\nprec. Finally, we will denote by ωILLG\nprecthe angular\nfrequency of the precession resonance peak obtained in\nthe numerical simulations of the ILLG model. Eq. (9)\nmaybeeasilynumericallysolvedtofindthesolution ˙φprec\nfor several values of αandτ. The behavior with respect\ntoτofωprecobtained either analytically or from the sim-\nulated FMR curves is shown in Fig. 3. There is an excel-10 -4 10 -3 10 -2 10 -1 10 0\n10 11 10 12 10 13 10 14 10 15 χ\n⊥\nω ( rad.s -1 )τ=1×10 -113UHFHVVLRQ \n1utation \nτ=1×10 -12\nτ=1×10 -13\nτ=1×10 -14\nFigure 2: Resonance curves of the transverse susceptibilit y\nshowing the displacement of the nutation peak caused by the\nvariation of τ:τ= 10−11s (open circles), 10−12s (filled\ncircles), 10−13s (crosses), and 10−14s (open squares). These\ncurves are simulated using the ILLG model with α= 0.1,\nandH= 2T. Note that the precession peak positions are\nonly slightly affected. The dotted line shows the power law\nfitted on χ⊥∝1/ωILLG\nnu, whereωILLG\nnuis the resonance angular\nfrequency of the nutation.\nlent agreement between the analytical prediction ωexact\nprec\nand the precession resonance peak ωILLG\nprecobtained in nu-\nmericalsimulations. WealsoshowinFig.3theprecession\nangular frequency ωapprox\nprec. Forτ <10−11sandα= 0.1,\nit nicely agrees with the exact analytical value and with\nthenumericalsimulationresults, but the approximateso-\nlution becomes no longer valid for τ >10−11s. To quan-\ntify the validity of the approximate solution we compute,\nforτ= 10−12sand for three different dampings, the\nrelative difference\nδana\nprec=ωapprox\nprec−ωexact\nprec\nωexactprec×100\nWe show in the inset of Fig. 3 the evolution of δana\nprecwith\nrespect to the applied static field H. ForH <20T\nthe relative difference remains less than 0 .1% for small\ndamping α= 0.01, and remains less than 3% for mod-\nerate damping α= 0.1. For large damping α= 0.5 the\napproximate solution remains valid for small fields, but\nfor 12T < H < 20Tthe error becomes larger than 10%.\n2. Nutation peak\nFigure 4 displays both the analytical prediction of the\nnutation angular frequency ωweak\nnugiven by Eq. (15) and\nthe angular frequency ωIILG\nnuof the nutation resonance6\n10-1610-1510-1410-1310-1210-1110-1010-9\nτ (s)01×10112×1011ωprec (rad.s-1)\n0 5 10 15 20 25 30\nH (T)10-410-310-210-1100101102δprecana (%)α=0.5\nα=0.1\nα=0.01\nFigure 3: (Color online) Comparaison of the analytical and\nnumerical simulation results for the precession angular fr e-\nquency obtained for α= 0.1 andH= 2T. Filled circles\n(black) are the precession angular frequency ωexact\nprec, open cir-\ncles (red) are the position of the precession resonance peak s\nωILLG\nprec, stars (orange) are the approximate precession angular\nfrequencies ωapprox\nprecvalid for small values of τ. Thedashed line\n(black) is the LLG precession angular frequency, i.e.without\ninertial term. Inset : relative difference δana\nprecfor three differ-\nent dampings.\nobtained in the numerical simulations. The agreement\nis excellent for τ <10−11s, and indicates that the non-\nlinearcouplingtermsofEq. (13)donotsignificantlyshift\nthe angular frequency of the nutation resonance from the\napproximate angular frequency ωweak\nnu. On the contrary,\nin the range 10−11s < τ < 10−8s, the simulated nuta-\ntion resonance angular frequency is slightly higher than\nωweak\nnu, as shown in the upper inset of Fig. 4. In the lower\ninset ofFig. 4 we show the relative difference δnubetween\nthe approximate nutation angular frequency ωweak\nnuand\nthe nutation resonance angular frequency ωILLG\nnuof the\nnumerical simulations, i. e.\nδnu=ωILLG\nnu−ωweak\nnu\nωILLGnu×100\nWe therefore see that in the range 10−11s < τ <10−8s,\nthe approximate nutation angular frequency remains\nless than 15% close to the simulated nutation resonance\nangular frequency.\nB. Scaling and overview of the ILLG equation\nIn the preceding section we investigated the behav-\nior of the ILLG model when varying the characteristic\ntime scale τwhich drives the inertial dynamics. We also10-1610-1510-1410-1310-1210-1110-1010-9\nτ (s)1011101210131014101510161017ωnu (rad.s-1)\n10-1110-1010-9\nτ (s)10111012ωnu (rad.s-1)\n10-1310-1210-1110-1010-910-8\nτ (s)010\nδnu (%)\nFigure 4: (Color online) Comparaison of the analytical and\nnumerical simulation results for the nutation angular fre-\nquency obtained for α= 0.1 andH= 2T. Filled cir-\ncles (black) are the approximate nutation angular frequenc ies\nωweak\nnuand open circles (red) are the positions ωILLG\nnuof the\nsimulated nutation resonance peaks. Upper inset : enlarge-\nment showing the effect of the non-linear coupling terms of\nEq. (13). Lower inset : relative difference δnubetween ωweak\nnu\nandωILLG\nnu.\nvary the static field Hand the dimensionless damping\nα. Increasing Hmoves both the precession and nuta-\ntion resonance peaks to higher angular frequencies, with\nsmaller and broadened peaks, while increasing the di-\nmensionless damping moves both peaks to lower angular\nfrequencies with still smaller and broadened peaks. Note\nthat the ILLG precession resonances obtained when the\nstatic field His varied show that the transverse suscep-\ntibility follows a power law χ⊥∝1/ωIILG\nprec(not shown).\nThis law is the same as the one resulting from the LLG\nmodel13.\nEq. (15) suggests a scaling function\nωnu\nγH=√1+x\nx\nwherex=ατγH. Scaling curves obtained for different\nvalues of τ,αandHare shown in the inset of Fig. 5\nwhere both the precession and nutation resonance an-\ngular frequencies are dispayed with respect to ατγH.\nFig. 5 is an enlargement of the intermediate region of\nthe inset where we added the points obtained by the nu-\nmerical simulations for H= 2Tandα= 0.1. The\ntwo asymptotic behaviors of the nutation are highlighted\nwith the dashed lines in agreement with Eq. (15) :\nwhenατγH≪1 thenωweak\nnu/γH= 1/ατγH, and when\nατγH≫1 thenωweak\nnu/γH= 1/√ατγH. Remarquably,\nwe see that the precession peak position divided by γH\nalso scales as ωprec/γH∼1/√ατγHwhenατγH≫1.7\n0.01 0.1 1 10 100\nατγH0.010.1110100ωnu/γH , ωprec/γH\n10-610-410-210010210-2100102104106\n1/ατγH\n1/(ατγH)1/2\nFigure 5: (Color online) Scaling curves : nutation ωnuand\nprecession ωprecpeak positions in the angular frequency do-\nmain divided by γHwith respect to ατγH. Open circles\n(red) are the nutation and precession resonance peak posi-\ntions obtained in the numerical simulations for α= 0.1 and\nH= 2T. Other points are ωweak\nnucomputed from Eq. (15),\nandωexact\npreccomputed from Eq. (9). Different values of the\nstatic field Hand the dimensionless damping αare reported :\nH= 0.2Tandα= 0.1 (red open diamonds), H= 2T(blue\nopen squares for α= 0.1 and blue filled squares for α= 0.01),\nH= 20Tandα= 0.1 (green open triangles), H= 200T\nandα= 0.1 (black crosses). The dashed lines are the two\nasymptotic behaviors of the nutation in agreement with Eq.\n(15). Inset : same scaling curves (without red open circles)\ndisplayed on larger scales.\nThe two asymptotic behaviors intersect at ατγH= 1\nandω/γH= 1. This point corresponds to the max-\nimum value of the LLG precession angular frequency\nωLLG/γH= 1/(1 +α2) which is obtained in the limit\ncase of no damping α= 0.\nThe inset of Figure 5 indicates that only one resonance\npeakisexpectedwhen ατγH→ ∞. Inthiscase,boththe\nnutation and the precession contribute to a unique peak.\nOn the contrary, for finite ατγHthey remain separated.\nThere are two different well-defined peaks in the investi-\ngated range ( ατγH≤100). For ατγH≪1 the preces-\nsion peak is close to the usual LLG precession peak, and\nthe nutation peak shifts rapidly ( ωweak\nnu/γH∼1/ατγH)\nto high angular frequencies. In other words, the nutation\noscillator defined by Eq. (13) is independent of the pre-\ncession for ατγH≪1, whereas both synchronize at the\nsame frequency for ατγH→ ∞.\nAccurate predictions about the precession and nutation\npeak positions in the angular frequency domain can be\nmade, as long as the non-linear coupling terms of Eq.\n(13) remain weak or compensate each other.V. TOWARDS EXPERIMENTAL EVIDENCE\nOF THE INERTIAL DYNAMICS OF THE\nMAGNETIZATION\nThroughout the preceding sections we studied the new\nproperties of the inertial dynamics of the magnetization\nwithin the ILLG model. We specifically considered the\nFMR framework where a small perpendicular sinusoidal\nfield is applied implying that the small inclination limit\nholds. We now focus on possible simple experiments in\nsuch FMR framework that should highlight the inertial\ndynamics of the magnetization.\nThe first direct evidence would of course be the measure\nof the nutation resonance peak at frequencies larger than\nthe precession resonance peak. Since the expected nuta-\ntion resonance peak is given by Eq. (15), the evolution\nwith the static field Hmay be used to discriminate the\nnutation resonnce peak from possible other higher fre-\nquency peaks.\nHowever the amplitude of the nutation resonance peak is\nsmallerthan forthe precessionpeak, and itmaybe tricky\nin unfavorable situations to measure such a peak, for ex-\nample in materials with small characteristic time τ. Fur-\nthermore, for large dimensionless damping αboth peaks\nhave smaller amplitude and are rounded. It may even\nappear that the nutation resonance peak of the magneti-\nzationinthe ILLGmodel disappearsforalargedamping,\nlikethe resonantpeakofthe classicaldrivendamped har-\nmonic oscillator. For exampleFig. 6showsthat for mate-\nrials with a large damping ( α= 0.5) the resonance peaks\nare smaller and rounded compared to smaller damping\n(α= 0.1), and the nutation resonance peak disappears\nforH≤5T.\nIt is therefore necessary to find measurable characteris-\nticsofthemagnetizationinertialdynamicsotherthanthe\ndirect measure of the nutation resonance peak. Actually,\nwe show in the following that beyond the nutation reso-\nnance peak, the inertial dynamics has measurable effects\non the precession resonance peak. Indeed, as shown in\nFig. 7, the shape of the precession peak and its position\nintheangularfrequencydomainaremodifiedbytheiner-\ntial dynamics. And the effects are shown to be more pro-\nnounced for large damping materials and for large static\nmagnetic fields H. To show these effects we compare the\nprecession resonance angular frequencies ωILLG\nprecandωLLG\nprec\nobtained in the numerical simulations of both the ILLG\nand non-inertial LLG models. We use two different di-\nmensionless damping α= 0.1 andα= 0.5, and vary the\namplitude Hof the static magnetic field. For the ILLG\nmodel, we choose, as in Ref. 4, a rough estimation of the\ncharacteristic time scale τ= 10−12s.\nA. Angular frequency of the precession resonance\npeak\nWefirstlookatthepositionoftheprecessionresonance\npeakin the angularfrequencydomain. Fig.8(a)and 8(b)8\n02×10124×10126×1012\nω (rad.s-1)1×10-21×10-11×100χ⊥α=0.5\nFigure 6: (Color online) Resonance curves of the transverse\nsusceptibility χ⊥(ω) with respect to the oscillating field angu-\nlar frequency ω. The resonance curves are computed within\nthe ILLG model with τ= 10−12s, for a large dimension-\nless damping α= 0.5 and for various applied static fields :\nH= 2T(black filled circles), H= 5T(green filled trian-\ngles),H= 20T(blue open circles) and H= 50T(red open\ntriangles). Both resonance peaks clearly appear for H= 20T\nandH= 50T. ForH= 5TandH= 2Tthe nutation reso-\nnance peak dissapears due to the large damping.\n00.10.20.3\n1×10 12 2×10 12 3×10 12 20 T\n00.20.40.6\n0 1×10 12 2×10 12 10 T2 T\n0123\n0 1×10 12 \n00.050.1\n3×10 12 4×10 12 5×10 12 6×10 12 50 T2 T 2 T \nχ\n⊥ILLG LLG ILLG LLG χ\n⊥\nILLG LLG \nILLG LLG \nω (rad.s -1 ) ω (rad.s -1 )\nFigure 7: Precession resonance curves of the transverse sus -\nceptibility simulated for differentvalues ofthestatic fiel dH=\n2T, 10T, 20T, and 50 T. ILLG model (filled circles) and\nnon-inertial LLGmodel (opencircles). γ= 1011rad.s−1.T−1,\nα= 0.1, andτ= 10−12s.\ndisplay the evolution of the resonance angular frequencyωprecwith respect to Hobtained for α= 0.1 andα=\n0.5 within the numerical simulations of both the ILLG\nand LLG models. As expected the resonance angular\n0 10 20 30 40 50 60\nH (T)0123456ωprec (1012 rad.s-1)\n0 10 20 30 40 50 60\nH (T)0123456ωprec (1012 rad.s-1)\n0 10 20 30 40 50 60\nH (T)01020304050δ prec (%)a) b) α=0.1\nc)α=0.5\nα=0.5\nα=0.1LLGLLG\nILLG\nILLG\nFigure 8: (Color online) (a) and (b) Precession resonance an -\ngular frequency with respect to the applied static field. Re-\nsults obtainedin thenumerical simulations of theILLGmode l\n(withτ= 10−12s) and non-inertial LLG model, for dimen-\nsionless damping (a) α= 0.1 (blue open circles for LLG and\nred filled circles for ILLG) and (b) α= 0.5 (blue open squares\nfor LLG and red filled squares for ILLG). (c) Relative differ-\nenceδprecbetween LLG and ILLG precession resonance an-\ngular frequencies for α= 0.1 (green filled circles) and α= 0.5\n(green filled squares).\nfrequency of the LLG precession is linear with Hsince\nωLLG\nprec=γH/(1+α2) whereas the behavior is not linear in\nHfor the ILLG model. In Fig. 8(c) we plot the relative\ndifference\nδprec=ωLLG\nprec−ωILLG\nprec\nωLLGprec×100\nbetweenbothresonanceangularfrequencies. Therelative\ndistance between both precession peaks increases with H\nand with the dimensionless damping α.\nB. Width of the precession resonance peak\nWe now examine the evolution with Hof the shape\nof the precession resonance peak obtained in the sim-\nulations of the ILLG and LLG models. For α= 0.1,\nthe full width at half maximum (FWHM) is shown in\nFig. 9(a) while Fig. 9(b) displays the FWHM divided\nby the resonance angular frequency. For large damping\nα= 0.5 we change the criterion since the reduced\namplitude of the resonant peak does not allow anymore\nto compute the FWHM. We therefore compute the\nbandwith defined by the width of the peak at Amax/√\n29\n0 10 20 30 40 50 60\nH (T)00.10.20.3FWHM / ωprec\n0 10 20 30 40 50 60\nH (T)0.00.51.01.52.0FWHM (1012 rad.s-1)\n0 10 20 30 40 50 60\nH (T)00.51Bandwidth / ωprec\n0 10 20 30 40 50 60\nH (T)012345Bandwidth (1012 rad.s-1)a)\nb)\nd)c)α=0.1\nα=0.1\nα=0.5\nα=0.5LLGLLG\nLLGLLGILLGILLG\nILLGILLG\nFigure 9: (Color online) (a) Full width at half maximum\n(FWHM) for the precession resonance peak for α= 0.1 within\nthe LLG (blue open circles) and the ILLG (red filled circles)\nmodels. (b)FWHMdividedeither by ωLLG\nprec(blueopencircles)\nor byωILLG\nprec(red filled circles). (c) Bandwidth of the preces-\nsion resonance peak for α= 0.5 within the LLG (blue open\nsquares)andILLG(redfilledsquares)models. (d)Bandwidth\ndivided either by ωLLG\nprec(blue open squares) or by ωILLG\nprec(red\nfilled squares).\nThe numerical simulations of the ILLG model are computed\nwithτ= 10−12s\nwhereAmaxis the maximum value of the peak. The\nbandwidth for α= 0.5 is shown in Fig. 9(c) and the\nbandwidth divided by the resonance angular frequency\nis plotted in Fig. 9(d). The numerical simulations of the\nILLG and LLG models lead to different behaviors for\nthe shape of the precession resonance peak. In the LLG\nmodel the FWHM and the bandwidth exhibit a linear\nevolution with the applied static field which results in\na constant evolution when divided by the resonance\nangular frequency. Very different behaviors are observed\nwithin the ILLG model where no linear evolution of the\nFWHM or the bandwidth is measured.\nFigs. 8 and 9 show that high applied static fields in\nlarge damping materials produce large differences be-\ntween the positions and shapes of the precession reso-\nnance peaks originating from the LLG and ILLG mod-\nels. Therefore, applying high static fields in large damp-\ning materials better allows to differentiate the precession\npeak originating from the ILLG and LLG models.\nAlthough the theory is clear and allows in principle to\ndifferentiate inertialfromnon-inertialdynamicswhen ex-\naminingboth precessionresonancepeaks, the experimen-\ntal investigations are rather more complex. Indeed, the\nexperimental demonstration of inertial effects first ne-\ncessitate to identify and control the different contribu-tions to the effective field (anisotropy, dipolar interac-\ntion, magnetostriction, ...) other than the applied static\nfield.\nVI. CONCLUSION\nThe magnetization dynamics in the ILLG model that\ntakes into account inertial effects has been studied from\nboth analytical and numerical points ofview. Within the\nFMR context, a nutation resonance peak is expected in\naddition to the usual precession resonance peak.\nAnalytical solutions of the inertial precession and nuta-\ntion angular frequencies are presented. The analytical\nsolutions nicely agree with the numerical simulations of\nthe resonance curves in a broad range of parameters.\nAt first, we investigated the effects of the time scale τ\nwhich drives the additional inertial term introduced in\nEq. (2)comparedtotheusualLLGequationEq. (1). We\nalso varied the dimensionless damping αand the static\nmagnetic field H, and a scaling function with respect to\nατγHis found for the nutation angular frequency. Re-\nmarquably, the same scaling holds for the precession an-\ngular frequency when ατγH≫1.\nIn the second part of the paper we focussed on the sig-\nnatures of the inertial dynamics which could be detected\nexperimentallywithintheFMRcontext. Weshowedthat\nbeyondthemeasureofthenutationresonancepeakwhich\nwould be a direct signature of the inertial dynamics, the\nprecession is modified by inertia and the ILLG preces-\nsion resonance peak is different from the usual LLG pre-\ncession peak. Indeed, whereas a linear evolution with\nrespect to His expected for the LLG precession reso-\nnance angular frequency, the ILLG precession resonance\nangular frequency is clearly non-linear. Furthermore, the\nshape of the precession resonance peak is different in the\nLLG and ILLG models. Again, the width variation of\nthe precession resonance peak is non-linear in the ILLG\ndynamics as opposed to the linear evolution with Hin\nthe LLG dynamics. We also showed that the difference\nbetween both LLG and ILLG precession peaks is more\npronounced when the damping is increased and when τ\nis increased. For example the discrepancy between the\nLLG and ILLG precession resonance angular frequencies\natH= 20Tforτ= 1psis expected to be of the order\nof 20% for α= 0.1 and 30% for α= 0.5. Therefore, large\ndamping materials are better candidates to experimen-\ntallyevidencetheinertialdynamicsofthemagnetization.\nFinally, a specific behavior of the amplitude of the mag-\nnetic susceptibility as a function of the nutation reso-\nnance angular frequency ωnuis predicted, of the form\nχ⊥(ωnu)∝ω−1\nnu(analogousto that ofthe usual FMR sus-\nceptibility). This law could be a useful criterion in order\nto discriminate the nutation peak among the other exci-\ntations that could also occur close to the infrared region\n(100 GHz up to 100 THz) in a ferromagnetic material.10\n1W.F. Brown Thermal Fluctuations of a Single-Domain\nParticle, Phys. Rev. 130, 1677 (1963).\n2R. Kubo, M. Toda, N. Hashitzume, Statistical physics II,\nNonequilibrium Statistical Mechanics , Springer Series in\nSolid-State Sciences 31, Berlin 1991 (second edition), Ed.\nP. Fulde, Chap 3, Paragraph 3.4.3, p. 131.\n3M.-C. Ciornei, Role of magnetic inertia in damped\nmacrospin dynamics , Ph. D. thesis, Ecole Polytechnique,\nPalaiseau France 2010.\n4M.-C. Ciornei, J. M. Rub´ ı, and J.-E. Wegrowe, Magnetiza-\ntion dynamics in the inertial regime : Nutation predicted\nat short time scales , Phys. Rev. B 83, 020410(R) (2011).\n5M. F¨ ahnle, D. Steiauf, and Ch. Illg, Generalized Gilbert\nequation including inertial damping : Derivation from an\nextended breathing Fermi surface model , Phys. Rev. B 84,\n172403 (2011).\n6J.-E. Wegrowe, C. Ciornei Magnetization dynamics, Gyro-\nmagnetic Relation, and Inertial Effects , Am J. Phys. 80,\n607 (2012).\n7E. Olive, Y. Lansac, and J.-E. wegrowe, Beyond ferromag-\nnetic resonance : the inertial regime of the magnetization ,Appl. Phys. Lett. 100, 192407 (2012).\n8D. B¨ ottcher, and J. Henk Significance of nutation in mag-\nnetization dynamics of nanostructures , Phys. Rev. B 86,\n020404(R) (2012).\n9S. Bhattacharjee, L. Nordstr¨ om, and J. Fransson Atomistic\nspin dynamic method with both damping and moment of\ninertia effects included from first principles , Phys. Rev.\nLett.108, 057204 (2012).\n10J.-X. Zhu, Z. Nussinov, A. Shnirman, and A. V. Balatsky\nNovel spin dynamics in a Josephson junction , Phys. Rev.\nLett.92, 107001 (2004).\n11J. Fransson, and J. Xi. Zhu Spin dynamics in a tunnel\njunction between ferromagnets , New J. Phys. 10, 013017\n(2008).\n12C. W. Gear, Numerical initial value problems in ordinary\ndifferential equations , Prentice Hall, Englewood Cliffs (N.\nJ.) 1971.\n13A. G. Gurevich and G. A. Melkov, Magnetization Oscilla-\ntion and Waves , CRC Press, 1996, p. 19."    },    {        "title": "1408.2160v1.Local_existence_results_for_the_Westervelt_equation_with_nonlinear_damping_and_Neumann_as_well_as_absorbing_boundary_conditions.pdf",        "content": "arXiv:1408.2160v1  [math.AP]  9 Aug 2014LOCAL EXISTENCE RESULTS FOR THE WESTERVELT\nEQUATION WITH NONLINEAR DAMPING AND NEUMANN AS\nWELL AS ABSORBING BOUNDARY CONDITIONS\nVANJA NIKOLI ´C\nAbstract. We investigate the Westervelt equation with several versio ns of\nnonlinear damping and lower order damping terms and Neumann as well as ab-\nsorbingboundary conditions. We prove localintimeexisten ce ofweaksolutions\nunder the assumption that the initial and boundary data are s ufficiently small.\nAdditionally, we prove local well-posedness in the case of s patially varying\nL∞coefficients, a model relevant in high intensity focused ultr asound (HIFU)\napplications.\n1.Introduction\nHigh intensity focused ultrasound (HIFU) is crucial in many medical a nd in-\ndustrial applications including lithotripsy, thermotherapy, ultraso und cleaning or\nwelding and sonochemistry. Widely used mathematical model for non linear wave\npropagation is the Westervelt equation, which can either be written in terms of the\nacoustic pressure p\n(1−2kp)ptt−c2∆p−b∆pt= 2k(pt)2, (1.1)\nor in terms of the acoustic velocity potential ψ\n(1−2˜kψt)ψtt−c2∆ψ−b∆ψt= 0, (1.2)\nwith̺ψt=p. Here,cdenotes the speed and bthe diffusivity of sound, k=βa/λ,\nβa= 1+B/(2A),B/Arepresents the parameter of nonlinearity, ̺is the mass den-\nsity,λ=̺c2is the bulk modulus and ˜k=̺k. For a detailed derivation of (1.1) and\n(1.2) we refer the reader to [4], [9], [13].\nWell-posedness and exponential decay of small and H2−spatially regular solu-\ntionsisestablishedforthe Westerveltequationwith homogeneous[ 6] andinhomoge-\nneous [7] Dirichlet and Neumann [8] boundary conditions as well as with boundary\ninstead of interior damping [5].\nA significant task in the analysis of the Westervelt equation is avoiding degener-\nacyofthe coefficient 1 −2kpforthe secondtime derivative pttin (1.1) and, similarly,\nof the term 1 −2˜kψtin the formulation (1.2). At the same time, in applications\nthe existence of spatially less regular solutions is important, e.g. in th e coupling of\nacoustic with acoustic or elastic regions with different material para meters. In [2],\n2010Mathematics Subject Classification. Primary: 35L05; Secondary: 35L20.\nKey words and phrases. nonlinear acoustics, Westervelt’s equation, local existe nce.\nResearch supported by the Austrian Science Fund (FWF): P249 70.\n12 V. NIKOLI ´C\nBrunnhuber, Kaltenbacher and Radu treated this issue by introdu cing nonlinear\ndamping terms to the Westervelt equation and considering the follow ing equations\n(1−2ku)utt−c2∆u−bdiv/parenleftBig\n((1−δ)+δ|∇ut|q−1)∇ut/parenrightBig\n= 2k(ut)2,(1.3)\n(1−2ku)utt−c2div(∇u+ε|∇u|q−1∇u)−b∆ut= 2k(ut)2, (1.4)\nutt−c2\n1−2˜kut∆u−bdiv/parenleftBig\n((1−δ)+δ|∇ut|q−1)∇ut/parenrightBig\n= 0, (1.5)\nwith homogeneous Dirichlet boundary data. First two equations are derived from\nthe Westervelt equation in the acoustic pressure formulation (1.1) , while the third\nequation comes from the acoustic potential formulation (1.2) (with the notation\nchanged to p→u,ψ→u). Added nonlinear damping terms make obtaining\nL∞(0,T;L∞(Ω)) estimate on u(ut) possible, without the need to estimate ∆ u\n(∆ut) and thus refraining from too high regularity.\nThe central aim of the present paper is to investigate this relaxatio n of regu-\nlarity by nonlinear damping, but equipped with practically relevant abs orbing and\nNeumann boundary data. This is motivated by many applications of hig h intensity\nfocused ultrasoundwherethe need formorerealisticboundaryco nditions is evident.\nE.g. in lithotripsy one faces the problem of a physically unbounded dom ain, as typ-\nical in acoustics, which should be truncated for numerical computa tions. Absorbing\nboundary conditions are then used to avoid reflections on the artifi cial boundary ˆΓ\nof the computational domain.\nUltrasound excitation, e.g. by piezoelectric transducers, can be m odeled by Neu-\nmann boundary conditions on the rest of the boundary Γ = ∂Ω\\ˆΓ.\nIn our case, the design of the nonlinear absorbing and inhomogeneo us Neumann\nboundary conditions is influenced by the presence of the nonlinear s trong damping\nin the equations. We will study initial boundary value problems of the f ollowing\ntype:\n\n\n(1−2ku)utt−c2∆u−bdiv/parenleftBig\n((1−δ)+δ|∇ut|q−1)∇ut/parenrightBig\n+βut\n= 2k(ut)2in Ω×(0,T],\nc2∂u\n∂n+b((1−δ)+δ|∇ut|q−1)∂ut\n∂n=gon Γ×(0,T],\nαut+c2∂u\n∂n+b((1−δ)+δ|∇ut|q−1)∂ut\n∂n= 0 onˆΓ×(0,T],\n(u,ut) = (u0,u1) onΩ×{t= 0},(1.6)\n\n\n(1−2ku)utt−c2∆u−bdiv/parenleftBig\n((1−δ)+δ|∇ut|q−1)∇ut/parenrightBig\n+γ|ut|q−1ut\n= 2k(ut)2in Ω×(0,T],\nc2∂u\n∂n+b((1−δ)+δ|∇ut|q−1)∂ut\n∂n=gon Γ×(0,T],\nαut+c2∂u\n∂n+b((1−δ)+δ|∇ut|q−1)∂ut\n∂n= 0 onˆΓ×(0,T],\n(u,ut) = (u0,u1) onΩ×{t= 0},(1.7)LOCAL EXISTENCE RESULTS 3\n\n\n(1−2ku)utt−c2div(∇u+ε|∇u|q−1∇u)−b∆ut+βut\n= 2k(ut)2in Ω×(0,T],\nc2∂u\n∂n+c2ε|∇u|q−1∂u\n∂n+b∂ut\n∂n=gon Γ×(0,T],\nαut+c2∂u\n∂n+c2ε|∇u|q−1∂u\n∂n+b∂ut\n∂n= 0 onˆΓ×(0,T],\n(u,ut) = (u0,u1) onΩ×{t= 0},(1.8)\n\n\nutt−c2\n1−2˜kut∆u−bdiv/parenleftBig\n((1−δ)+δ|∇ut|q−1)∇ut/parenrightBig\n+γ|ut|q−1ut\n= 0 in Ω ×(0,T],\nc2\n1−2˜kut∂u\n∂n+b((1−δ)+δ|∇ut|q−1)∂ut\n∂n=gon Γ×(0,T],\nαut+c2\n1−2˜kut∂u\n∂n+b((1−δ)+δ|∇ut|q−1)∂ut\n∂n= 0 onˆΓ×(0,T],\n(u,ut) = (u0,u1) onΩ×{t= 0}.(1.9)\nNote that in the case of b= 0,α=cand˜k= 0 the absorbing conditions\nprescribed in (1.6)-(1.9) would reduce to the standard linear absor bing boundary\nconditions of the form ut+c∂u\n∂n= 0.\nIn the equations, we assume that the parameters βandγare nonnegative; the\ncaseβ=γ= 0 reduces them to (1.3)-(1.4). Another task of the present pap er is to\ninvestigate possible introduction of these lower order linearandnonlinear damping\nterms to the equations (1.3)-(1.4), this becomes beneficial when d eriving energy\nestimates.\nAdditionally, in the context ofHIFU devices based on the acoustic len s immersed\nin a fluid medium, a problem of Westervelt’s equation coupled with other equations\nor with jumping coefficients arises. We will treat acoustic-acoustic c oupling which\ncan be modeled by Westervelt’s equation in the pressure formulation with spatially\nvarying coefficients (see [1] for the linear case and [2] for the nonlin ear case with\nhomogeneous Dirichlet boundary conditions):\n\n\n1\nλ(x)(1−2k(x)u)utt−div(1\n̺(x)∇u)−div/parenleftBig\nb(x)(((1−δ(x))+δ(x)|∇ut|q−1)∇ut/parenrightBig\n=2k(x)\nλ(x)(ut)2in Ω×(0,T],\n1\n̺(x)∂u\n∂n+b(x)((1−δ(x))+δ(x)|∇ut|q−1)∂ut\n∂n=gon Γ×(0,T],\nα(x)ut+1\n̺(x)∂u\n∂n+b(x)((1−δ(x))+δ(x)|∇ut|q−1)∂ut\n∂n= 0 onˆΓ×(0,T],\n(u,ut) = (u0,u1) onΩ×{t= 0}.\n(1.10)\n1.1.Notations and Preliminaries. We assume Ω ⊂Rd,d∈ {1,2,3}to be an\nopen, connected, bounded set with Lipschitz boundary; ∂Ω is assumed to be a\ndisjoint union of Γ and ˆΓ. We denote by nthe outward unit normal vector.\nWewillstudytheproblemswithstrongdamping b>0andwithc2>0,δ∈(0,1),\nε>0 andk,˜k∈R. Our results will hold for αassumed to be nonegative; the case\nα= 0 reduces (1.6)-(1.9) to problems with only Neuman boundary cond itions.\nNote that, in general, we will assume that q≥1, but this condition will have\nto be strenghtened at several instances to assure well-posedne ss of (1.6), (1.7) and4 V. NIKOLI ´C\nexistence results for (1.8) and (1.9). We will often make use of the c ontinuous\nembeddings\nH1(Ω)֒→L4(Ω),with the norm CΩ\nH1,L4,and\nW1,q+1(Ω)֒→L∞(Ω),with the norm CΩ\nW1,q+1,L∞,\nwith the latter being valid for q+1>d. In Section 2 and 5 we will need to employ\nthe embedding Lq+1(Ω)֒→L4(Ω), which holds true for q≥3.\nWe denote with Ctr\n1the norm of the trace mapping\nTr:W1,q+1(Ω)→W1−1\nq+1,q+1(Γ),\nand withCtr\n2the norm of the trace mapping tr:H1(Ω)→H−1/2(Γ) (withCtr\n1=\nCtr\n2forq= 1).\nThroughout the paper we assume t∈[0,T], whereTis a finite time horizon.\n1.2.Outline of the paper. The rest of the paper is organized as follows. Subsec-\ntion 1.3 contains the derivation of L∞-bounds on uandutas well as several useful\ninequalities that will be employed in the paper.\nIn Section 2, we start by looking at a linearized version of (1.6) and (1 .7) with\nβ=γ= 0, with nonlinearity appearing only through damping, and show local\nwell-posedness. Then we discuss linearized versions of (1.6) and (1.7 ) withβ,γ >0.\nBy employing the result for the linearized version we proceed to prov e local well-\nposedness for (1.6) and (1.7).\nSection 3 deals with the short time well-posedness of the acoustic-a coustic cou-\npling modeled by (1.10).\nIn Section 4 and 5 we consider (1.8) and (1.9), respectively. We again begin by\ninvestigating the linearized versions of the problems at hand for β= 0 andγ= 0 re-\nspectively, and continue with introducing lower order damping terms and the proof\nof local existence of solutions.\n1.3.Inequalities. In the case of problems with inhomogeneous Neumann bound-\nary data it is often necessary to employ Poincar´ e’s inequality valid fo r functions in\nW1,q+1(Ω). We recall such inequality (cf. Theorem 12.23, [10]), namely that there\nexists a constant CP>0 depending on qand Ω such that\n|ϕ−1\n|Ω|/integraldisplay\nΩϕdx|Lq+1(Ω)≤CP|∇ϕ|Lq+1(Ω), (1.11)\nfor allϕ∈W1,q+1(Ω).\nThe nonlinear damping term appearing in the equations (1.6)-(1.9) will enable\nus to avoid degeneracy of the coefficients 1 −2kuand 1−2kutby deriving L∞\nestimates on uandut. From (1.11) we can obtain\n|u(t)|W1,q+1(Ω)≤(1+CP)|∇u(t)|Lq+1(Ω)+CΩ\n1/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nΩu(t)dx/vextendsingle/vextendsingle/vextendsingle,(1.12)\nand by replacing uwithutalso\n|ut(t)|W1,q+1(Ω)≤(1+CP)|∇ut(t)|Lq+1(Ω)+CΩ\n2|ut(t)|L2(Ω), (1.13)\nwhereCΩ\n1=|Ω|−q\nq+1andCΩ\n2=|Ω|−q−1\n2(q+1).\nFrom (1.13), by making use of the embedding W1,q+1(Ω)֒→L∞(Ω),q>d−1, weobtain anL∞estimate on ut\n/ba∇dblut/ba∇dblL∞(0,T;L∞(Ω))≤CΩ\nW1,q+1,L∞/bracketleftBig\n(1+CP)/ba∇dbl∇ut/ba∇dblL∞(0,T;Lq+1(Ω))\n+CΩ\n2/ba∇dblut/ba∇dblL∞(0,T;L2(Ω))/bracketrightBig\n,(1.14)\nwhich will be used to avoid degeneracy of the factor 1 −2kutin the problem (1.9).\nEmploying (1.12) and the estimate\n/ba∇dblut/ba∇dbl2\nL2(0,T;L2(Ω))≤T/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω), (1.15)\nwe can get an L∞estimate on u\n/ba∇dblu/ba∇dblL∞(0,T;L∞(Ω))≤CΩ\nW1,q+1,L∞/bracketleftBig\n(1+CP)|∇u(t)|Lq+1(Ω)\n+CΩ\n1/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nΩ(u0+/integraldisplayt\n0ut(s)ds)dx/vextendsingle/vextendsingle/vextendsingle/bracketrightBig\n≤CΩ\nW1,q+1,L∞/bracketleftBig\n(1+CP)/ba∇dbl∇u/ba∇dblL∞(0,T;Lq+1(Ω))\n+CΩ\n1|u0|L1(Ω)+CΩ\n2√\nT/ba∇dblut/ba∇dblL2(0,T;L2(Ω))/bracketrightBig\n≤CΩ\nW1,q+1,L∞/bracketleftBig\n(1+CP)/ba∇dbl∇u/ba∇dblL∞(0,T;Lq+1(Ω))\n+CΩ\n1|u0|L1(Ω)+CΩ\n2T/ba∇dblut/ba∇dblL∞(0,T;L2(Ω))/bracketrightBig\n,(1.16)\nwhich we will apply when investigating (1.8).\nFrom (1.12) we can as well obtain\n|u(t)|L∞(Ω)≤CΩ\nW1,q+1,L∞/bracketleftBig\n(1+CP)|∇u0+/integraldisplayt\n0∇ut(s)ds|Lq+1(Ω)\n+CΩ\n1|u0|L1(Ω)+CΩ\n2/integraldisplayt\n0|ut(t)|L2(Ω)ds/bracketrightBig\n≤CΩ\nW1,q+1,L∞/bracketleftBig\n(1+CP)(|∇u0|Lq+1(Ω)+(tq/integraldisplayt\n0|∇ut|q+1\nLq+1(Ω)ds)1/q+1)\n+CΩ\n1|u0|L1(Ω)+CΩ\n2/integraldisplayt\n0|ut(t)|L2(Ω)ds/bracketrightBig\n,\nwhich leads to the estimate\n/ba∇dblu/ba∇dblL∞(0,T;L∞(Ω))≤CΩ\nW1,q+1,L∞/bracketleftBig\n(1+CP)(|∇u0|Lq+1(Ω)\n+Tq\nq+1/ba∇dbl∇ut/ba∇dblLq+1(0,T;Lq+1(Ω)))\n+CΩ\n1|u0|L1(Ω)+CΩ\n2T/ba∇dblut/ba∇dblL∞(0,T;L2(Ω))/bracketrightBig\n,(1.17)\nthat will be employed when dealing with the possible degeneracy of the coefficient\n1−2kuin (1.6) and (1.7). We will also frequently make use of Young’s inequality\nin the form\nab≤εas+C(ε,s)bs\ns−1(a,b>0, ε>0,1<s<∞), (1.18)\nwithC(ε,s) = (s−1)ss\ns−1ε−1\n1−s.\nWhen dealing with the q-Laplace damping term in the equations, the inequality (cf.\n[11])\n/a\\}b∇acketle{t|b|q−1b−|a|q−1a,b−a/a\\}b∇acket∇i}ht ≥0, a,b∈Rd, (1.19)\n56 V. NIKOLI ´C\nvalid for all q, will be of use as well.\n2.Westervelt’s equation in the formulation (1.6)and(1.7)\nWe will begin by looking at the problems (1.6) and (1.7) with β=γ= 0:\n\n\n(1−2ku)utt−c2∆u−bdiv/parenleftBig\n((1−δ)+δ|∇ut|q−1)∇ut/parenrightBig\n= 2k(ut)2in Ω×(0,T],\nc2∂u\n∂n+b((1−δ)+δ|∇ut|q−1)∂ut\n∂n=gon Γ×(0,T],\nαut+c2∂u\n∂n+b((1−δ)+δ|∇ut|q−1)∂ut\n∂n= 0 onˆΓ×(0,T],\n(u,ut) = (u0,u1) onΩ×{t= 0},(2.1)\nFollowing the approach in [2], we will first consider the equation where n onlinearity\nappears only in the damping term\n\n\nautt−c2∆u−bdiv/parenleftBig\n((1−δ)+δ|∇ut|q−1)∇ut/parenrightBig\n+fut\n= 0 in Ω ×(0,T]\nc2∂u\n∂n+b((1−δ)+δ|∇ut|q−1)∂ut\n∂n=gon Γ×(0,T],\nαut+c2∂u\n∂n+b((1−δ)+δ|∇ut|q−1)∂ut\n∂n= 0 onˆΓ×(0,T],\n(u,ut) = (u0,u1) onΩ×{t= 0},(2.2)\nand prove local well-posedness.\nProposition 2.1. LetT >0,c2,b>0,α≥0,δ∈(0,1),q≥1and assume that\n(i)•a∈L∞(0,T;L∞(Ω)),at∈L∞(0,T;L2(Ω)),0<a≤a(t,x)≤a,\n•f∈L∞(0,T;L2(Ω)),\n•g∈Lq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ)),\n•u0∈H1(Ω),u1∈L2(Ω),\nwith\n/ba∇dblf−1\n2at/ba∇dblL∞(0,T;L2(Ω))≤ˆb<min/braceleftBigb(1−δ)\n2(CΩ\nH1,L4)2,a\n4T(CΩ\nH1,L4)2/bracerightBig\n.(2.3)\nThen(2.2)has a weak solution\nu∈˜X:={v:v∈C(0,T;H1(Ω))∩C1(0,T;L2(Ω))\n∧vt∈Lq+1(0,T;W1,q+1(Ω))},(2.4)\nwhich is unique and satisfies the energy estimate\n/bracketleftBiga\n4−ˆb(CΩ\nH1,L4)2T−ǫ0/bracketrightBig\n/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω))+c2\n4/ba∇dbl∇u/ba∇dbl2\nL∞(0,T;L2(Ω))\n+/bracketleftBigb(1−δ)\n2−ˆb(CΩ\nH1,L4)2/bracketrightBig\n/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))+α\n2/ba∇dblut/ba∇dbl2\nL2(0,T;L2(ˆΓ))\n+/bracketleftBigbδ\n2−ǫ1/bracketrightBig\n/ba∇dbl∇ut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))(2.5)\n≤a\n2|u1|2\nL2(Ω)+c2\n2|∇u0|2\nL2(Ω)+1\n4ǫ0(Ctr\n1CΩ\n2)2/ba∇dblg/ba∇dbl2\nL1(0,T;W−q\nq+1,q+1\nq(Γ))LOCAL EXISTENCE RESULTS 7\n+C(ǫ1,q+1)(Ctr\n1(1+CP))q+1\nq/ba∇dblg/ba∇dblq+1\nq\nLq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ)),\nfor some constants\n0<ǫ0<a\n4−ˆb(CΩ\nH1,L4)2T,0<ǫ1<bδ\n2. (2.6)\nIf, in addition to (i),\n(ii)•f∈L∞(0,T;H1(Ω)),/ba∇dblf/ba∇dblL∞(0,T;H1(Ω))≤˜b,\n•g∈L∞(0,T;W−q\nq+1,q+1\nq(Γ)),gt∈Lq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ)),\n•u1∈W1,q+1(Ω),\nthen\nu∈X:=C1(0,T;W1,q+1(Ω))∩H2(0,T;L2(Ω)), (2.7)\nand satisfies the energy estimate\nµa−τ\n2/ba∇dblutt/ba∇dbl2\nL2(0,T;L2(Ω))+µ[b(1−δ)\n4−σ]/ba∇dbl∇ut/ba∇dbl2\nL∞(0,T;L2(Ω))\n+/bracketleftBiga\n4−(CΩ\nH1,L4)2ˆbT−ǫ0(µ+1)−µ1\n2τ(CΩ\nH1,L4)4˜b2T/bracketrightBig\n/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω))\n+/bracketleftBigb(1−δ)\n2−(CΩ\nH1,L4)2ˆb−µ(1\n2τ(CΩ\nH1,L4)4˜b2+c2)/bracketrightBig\n/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))\n+c2\n4(1−µc2\nσ)/ba∇dbl∇u/ba∇dbl2\nL∞(0,T;L2(Ω))+µ[bδ\n2(q+1)−η]/ba∇dbl∇ut/ba∇dblq+1\nL∞(0,T;Lq+1(Ω))\n+[bδ\n2−ǫ1(µ+1)]/ba∇dbl∇ut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))+α\n2/ba∇dblut/ba∇dbl2\nL2(0,T;L2(ˆΓ))\n+µα\n4/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(ˆΓ))\n≤C/parenleftBig\nCΓ(g)+|u1|2\nH1(Ω)+|∇u0|2\nL2(Ω)+|u1|q+1\nW1,q+1(Ω)+|u1|2\nL2(ˆΓ)/parenrightBig\n,(2.8)\nfor some sufficiently small constants µ,σ,τ,η> 0, some large enough C >0, and\nCΓ(g) =1/summationdisplay\ns=0/ba∇dblds\ndtsg/ba∇dbl2\nL1(0,T;W−q\nq+1,q+1\nq(Γ))+1/summationdisplay\ns=0/ba∇dblds\ndtsg/ba∇dblq+1\nq\nLq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ))\n+/ba∇dblg/ba∇dbl2\nL∞(0,T;W−q\nq+1,q+1\nq(Γ))+/ba∇dblg/ba∇dblq+1\nq\nL∞(0,T;W−q\nq+1,q+1\nq(Γ)).(2.9)\nProof.The weak form of (2.2) is given as/integraldisplay\nΩ/braceleftBig\nauttw+c2∇u·∇w+b/parenleftBig\n(1−δ)+δ|∇ut|q−1/parenrightBig\n∇ut·∇w/bracerightBig\ndx+α/integraldisplay\nˆΓutwdx\n=−/integraldisplayt\n0/integraldisplay\nΩfutwdx+/integraldisplay\nΓgwdx,∀w∈W1,q+1(Ω), (2.10)\nwith initial conditions ( u0,u1).\nWe will use the standardGalerkinmethod (see forinstanceSection 7 .2, [3] forthe\ncase of second-order linear hyperbolic equations and Section 2, [2] for the problem\n(2.2) with homogeneous Dirichlet boundary data), where we will first construct\napproximations of the solution, and then by obtaining energy estima tes guarantee\nweak convergence of these approximations.\n1. Smooth approximation of a,f, andg.Let us first introduce sequences\n(ak)k∈N, (fk)k∈Nand (gk)k∈Nwhich represent smooth in time approximations of a,\nf, andg:8 V. NIKOLI ´C\n•(ak)k∈N⊆C∞([0,T]×Ω)∩W1,∞(0,T;L2(Ω)),\nak→ainL∞(0,T;L∞(Ω)),ak,t→atinL∞(0,T;L2(Ω)),\n0<a≤ak(t,x)≤a,\n•(fk)k∈N⊆C∞((0,T)×Ω),fk→finL∞(0,T;L2(Ω)),\n•(gk)k∈N⊆C∞(0,T;W−q\nq+1,q+1\nq(Γ)),gk→ginLq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ)),\n• /ba∇dblfk−1\n2ak,t/ba∇dblL∞(0,T;L2(Ω))≤ˆb,\nand, for fixed k∈N, prove that there exists a solution u(k)of\n/integraldisplay\nΩ/braceleftBig\naku(k)\nttw+c2∇u(k)·∇w+b/parenleftBig\n(1−δ)+δ|∇u(k)\nt|q−1/parenrightBig\n∇u(k)\nt·∇w/bracerightBig\ndx\n+α/integraldisplay\nˆΓu(k)\ntwdx=−/integraldisplay\nΩfku(k)\ntwdx+/integraldisplay\nΓgwdx, ∀w∈W1,q+1(Ω),(2.11)\nwith initial conditions ( u0,u1).\n(a) Galerkin approximations. We start by proving existence and uniqueness\nof a solution for a finite-dimensional approximation of (2.11). We cho ose smooth\nfunctionswm=wm(x),m∈Nsuch that\n{wm}m∈Nis an orthonormal basis of L2\n˜ak(Ω),\n{wm}m∈Nis a basis of W1,q+1(Ω),\n{wm|ˆΓ}m∈Nis an orthonormal basis of L2(ˆΓ),\nwhereL2\n˜akis the weighted L2-space based on the inner product /a\\}b∇acketle{tf,g/a\\}b∇acket∇i}htL2\n˜ak(Ω):=\n/integraltext\nΩ˜akfgdx, with ˜ak=1\nT/integraltextT\n0ak(t)dt.\nNext, we construct a sequence of finite dimensional subspaces VnofL2\n˜ak(Ω)∩\nW1,q+1(Ω),\nVn= span{w1,w2,...,w n}.\nClearly,Vn⊆Vn+1,Vn⊆L2\n˜ak(Ω)∩W1,q+1(Ω) and/uniontext\nn∈NVn=W1,q+1(Ω).\nLet (u0,n)n∈N, (u1,n)n∈Nbe sequences such that\n•u0,n∈Vn,u0,n→u0inH1(Ω),\n•u1,n∈Vn,u1,n→u1inL2(Ω).\nWe can now consider a sequence of discretized versions of (2.11),\n/integraldisplay\nΩ/braceleftBig\naku(k)\nn,ttwn+c2∇u(k)\nn·∇wn+b/parenleftBig\n(1−δ)+δ|∇u(k)\nn,t|q−1/parenrightBig\n∇u(k)\nn,t·∇wn/bracerightBig\ndx\n+α/integraldisplay\nˆΓu(k)\nn,twndx=−/integraldisplay\nΩfku(k)\nn,twndx+/integraldisplay\nΓgkwndx,∀wn∈Vn,(2.12)\nwithu(k)\nn(t)∈Vnand initial conditions ( u0,n,u1,n). For each n∈N, we face an\ninitial value problem for a second order system of ordinary different ial equations\nwith coefficients and right hand side that are C∞functions of t. According to\nstandard existence theory for ordinary differential equations (c f. [12]), there exists\na unique solution u(k)\nn∈C∞(0,˜T,Vn) of (2.12) for some ˜T≤Tsufficiently small.\nBy employing the uniform energy estimates obtained below, we can co nclude that\n˜T=T.LOCAL EXISTENCE RESULTS 9\n(b) Lower energy estimate. Testing (2.12) with wn=u(k)\nn,t(t)∈Vnand inte-\ngrating with respect to time results in\n1\n2/bracketleftbigg/integraldisplay\nΩak/parenleftBig\nu(k)\nn,t/parenrightBig2\ndx+c2|∇u(k)\nn|2\nL2(Ω)/bracketrightbiggt\n0+α/integraldisplayt\n0/integraldisplay\nˆΓ|u(k)\nn,t|2dxds\n+b/integraldisplayt\n0/integraldisplay\nΩ/parenleftBig\n(1−δ)+δ|∇u(k)\nn,t|q−1/parenrightBig\n|∇u(k)\nn,t|2dxds\n=−/integraldisplayt\n0/integraldisplay\nΩ(fk−1\n2ak,t)/parenleftBig\nu(k)\nn,t/parenrightBig2\ndxds+/integraldisplayt\n0/integraldisplay\nΓgku(k)\nn,tdxds\n≤/ba∇dblfk−1\n2ak,t/ba∇dblL∞(0,T;L2(Ω))/integraldisplayt\n0|u(k)\nn,t|2\nL4(Ω)ds+/integraldisplayt\n0/integraldisplay\nΓgku(k)\nn,tdxds.(2.13)\nFor estimating the boundary integral appearing on the right side, w e will make use\nof (1.13) to obtain\n/integraldisplayt\n0/integraldisplay\nΓgku(k)\nn,tdxds≤/integraldisplayt\n0|u(k)\nn,t(s)|\nW1−1\nq+1,q+1(Γ)|gk(s)|\nW−q\nq+1,q+1\nq(Γ)ds\n≤Ctr\n1/integraldisplayt\n0|u(k)\nn,t(s)|W1,q+1(Ω)|gk(s)|\nW−q\nq+1,q+1\nq(Γ)ds\n≤Ctr\n1/integraldisplayt\n0/bracketleftBig\n(1+CP)|∇u(k)\nn,t(s)|Lq+1(Ω) (2.14)\n+CΩ\n2|u(k)\nn,t(s)|L2(Ω)/bracketrightBig\n|gk(s)|\nW−q\nq+1,q+1\nq(Γ)ds\n≤ǫ1/ba∇dbl∇u(k)\nn,t/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))+ǫ0/ba∇dblu(k)\nn,t/ba∇dbl2\nL∞(0,T;L2(Ω))\n+C(ǫ1,q+1)(Ctr\n1(1+CP))q+1\nq/ba∇dblgk/ba∇dblq+1\nq\nLq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ))\n+1\n4ǫ0(Ctr\n1CΩ\n2)2/ba∇dblgk/ba∇dbl2\nL1(0,T;W−q\nq+1,q+1\nq(Γ)),\nwithǫ0,ǫ1>0. By taking the essential supremum with respect to tin (2.13) and\nemploying the embedding H1(Ω)֒→L4(Ω), as well as the inequality (1.15), we\nobtain the estimate\n/bracketleftBiga\n4−ˆb(CΩ\nH1,L4)2T−ǫ0/bracketrightBig\n/ba∇dblu(k)\nn,t/ba∇dbl2\nL∞(0,T;L2(Ω))+c2\n4/ba∇dbl∇u(k)\nn/ba∇dbl2\nL∞(0,T;L2(Ω))\n+/bracketleftBigb(1−δ)\n2−ˆb(CΩ\nH1,L4)2/bracketrightBig\n/ba∇dbl∇u(k)\nn,t/ba∇dbl2\nL2(0,T;L2(Ω))+α\n2/ba∇dblu(k)\nn,t/ba∇dbl2\nL2(0,T;L2(ˆΓ))\n+/bracketleftBigbδ\n2−ǫ1/bracketrightBig\n/ba∇dbl∇u(k)\nn,t/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))(2.15)\n≤C(ǫ1,q+1)(Ctr\n1(1+CP))q+1\nq/ba∇dblgk/ba∇dblq+1\nq\nLq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ))+a\n2|u(k)\n1,n|2\nL2(Ω)\n+1\n4ǫ0(Ctr\n1CΩ\n2)2/ba∇dblgk/ba∇dbl2\nL1(0,T;W−q\nq+1,q+1\nq(Γ))+c2\n2|∇u(k)\n0,n|2\nL2(Ω).\nWe choose ǫ0,ǫ1small enough\n0<ǫ0<a\n4−ˆb(CΩ\nH1,L4)2T,0<ǫ1<bδ\n2, (2.16)10 V. NIKOLI ´C\nso that coefficients appearing in the estimate remain positive. As by a ssumption\ngk∈Lq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ)), we conclude that the sequence of Galerkin approx-\nimations/parenleftbig\nu(k)\nn/parenrightbig\nn∈Nis bounded in the Banach space\n˜X:={v:v∈C(0,T;H1(Ω))∩C1(0,T;L2(Ω))∧vt∈Lq+1(0,T;W1,q+1(Ω))}.\nIt follows from (2.15) that\n/parenleftbig\nu(k)\nn,t/parenrightbig\nn∈Nis uniformly bounded in L2(0,T;L2(Ω)), (2.17)\n/parenleftbig\n∇u(k)\nn,t/parenrightbig\nn∈Nis uniformly bounded in Lq+1(0,T;Lq+1(Ω)), (2.18)\n|∇u(k)\nn,t|q−1∇u(k)\nn,tis uniformly bounded in Lq+1\nq(0,T;Lq+1\nq(Ω)),and (2.19)\n/parenleftbig\nu(k)\nn,t|ˆΓ/parenrightbig\n∈Nis uniformly bounded in L2(0,T;L2(ˆΓ)), (2.20)\nwhich are all reflexive Banach spaces.\n(c) Convergence of Galerkin approximations. Due to (2.17)-(2.20) there\nexistsaweaklyconvergentsubsequenceof( u(k)\nn)n∈N, whichwestilldenote( u(k)\nn)n∈N,\nand au(k)such that\nu(k)\nn,t⇀u(k)\ntinL2(0,T;L2(Ω)), (2.21)\n∇u(k)\nn,t⇀∇u(k)\ntinLq+1(0,T;Lq+1(Ω)), (2.22)\n|∇u(k)\nn,t|q−1∇u(k)\nn,t⇀|∇u(k)\nt|q−1∇u(k)\ntinLq+1\nq(0,T;Lq+1\nq(Ω)), (2.23)\nu(k)\nn,t|ˆΓ⇀u(k)\nt|ˆΓinL2(0,T;L2(ˆΓ)). (2.24)\nOur task next is to prove that the weak limit u(k)solves (2.11). Fix k,m∈Nand\nletφm∈C∞(0,T,Vm)⊂Lq+1(0,T;W1,q+1(Ω)) withφm(T) = 0. For any n≥m,\nbyVm⊆Vnwe have\n/integraldisplayT\n0/integraldisplay\nΩ/braceleftBig\naku(k)\nttφm+c2∇u(k)·∇φm+b/parenleftBig\n(1−δ)+δ|∇u(k)\nt|q−1/parenrightBig\n∇u(k)\nt·∇φm\n+fku(k)\ntφm/bracerightBig\ndxds+α/integraldisplayT\n0/integraldisplay\nˆΓu(k)\ntφmdxds−/integraldisplayT\n0/integraldisplay\nΓgkφmdxds\n=−/integraldisplayT\n0/integraldisplay\nΩ[u(k)\nt−u(k)\nn,t]/parenleftBig\nakφm/parenrightBig\ntdxds−/integraldisplay\nΩ[u1−u1,n]ak(0)φm(0)dxds\n+c2/integraldisplayT\n0/integraldisplay\nΩ[∇u(k)−∇u(k)\nn]·∇φmdxds (2.25)\n+/integraldisplayT\n0/integraldisplay\nΩ[u(k)\nt−u(k)\nn,t]fkφmdxds+b(1−δ)/integraldisplayT\n0/integraldisplay\nΩ[∇u(k)\nt−∇u(k)\nn,t]·∇φmdxds\n+bδ/integraldisplayT\n0/integraldisplay\nΩ[|∇u(k)\nt|q−1∇u(k)\nt−|∇u(k)\nn,t|q−1∇u(k)\nn,t]·∇φmdxds\n+α/integraldisplayT\n0/integraldisplay\nˆΓ[u(k)\nt−u(k)\nn,t]φmdxds→0 asn→ ∞,\ndue to (2.21)-(2.24). Since/uniontext\nm∈NVmis dense in W1,q+1(Ω),u(k)indeed solves\n(2.11). By testing the problem (2.11) with u(k)\ntand proceding as in 1.(b) we can\nconclude that this weak limit satisfies the estimate (2.15) with u(k)\nnreplaced by\nu(k).LOCAL EXISTENCE RESULTS 11\n2.kkk→∞∞∞.Owing to the previous conclusion, we can find a weakly convergent\nsubsequence of ( u(k)), which we again denote ( u(k)), andu∈˜Xsuch that\nu(k)\nt⇀utinL2(0,T;L2(Ω)), (2.26)\n∇u(k)\nt⇀∇utinLq+1(0,T;Lq+1(Ω)), (2.27)\n|∇u(k)\nt|q−1∇u(k)\nn,t⇀|∇ut|q−1∇utinLq+1\nq(0,T;Lq+1\nq(Ω)),(2.28)\nu(k)\nt|ˆΓ⇀ut|ˆΓinL2(0,T;L2(ˆΓ)). (2.29)\nIt remains to show that usatisfies (2.10). For all w∈C∞(0,T;W1,q+1(Ω)) with\nw(T) = 0 we have\n/integraldisplayt\n0/integraldisplay\nΩ/braceleftBig\nauttw+c2∇u·∇w+b/parenleftBig\n(1−δ)+δ|∇ut|q−1/parenrightBig\n∇ut·∇w+futw/bracerightBig\ndxds\n+α/integraldisplayt\n0/integraldisplay\nˆΓutwdxds−/integraldisplayt\n0/integraldisplay\nΓgwdxds\n=−/integraldisplayt\n0/integraldisplay\nΩ[ut−u(k)\nt]/parenleftBig\naw/parenrightBig\ntdxds−/integraldisplayt\n0/integraldisplay\nΩu(k)\nt/parenleftBig\n[a−ak]w/parenrightBig\ntdxds\n−/integraldisplay\nΩu1/parenleftBig\n[a(0)−ak(0)]w(0)/parenrightBig\ndx+/integraldisplayt\n0/integraldisplay\nΩc2[∇u−∇u(k))]·∇wdxds\n+/integraldisplayt\n0/integraldisplay\nΩb(1−δ)[∇ut−∇u(k)\nt]·∇wdxds\n+/integraldisplayt\n0/integraldisplay\nΩbδ[|∇ut|q−1∇ut−|∇u(k)\nt|q−1∇u(k)\nt]·∇wdxds\n+/integraldisplayt\n0/integraldisplay\nΩ[ut−u(k)\nt]fwdxds +/integraldisplayt\n0/integraldisplay\nΩ[f−fk]u(k)\ntwdxds\n+α/integraldisplayt\n0/integraldisplay\nˆΓ[ut−u(k)\nt]wdxds−/integraldisplayt\n0/integraldisplay\nΓ[g−gk]wdxds→0 ask→ ∞,\nsince we demanded that ak→ainL∞(0,T;L∞(Ω)),ak,t→atinL∞(0,T;L2(Ω)),\nfk→finL∞(0,T;L2(Ω)) andgk→ginLq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ)). This relation\nproves that usolves (2.10). The weak limit then satisfies the estimate (2.5).\n3. Uniqueness. To confirm uniqueness, note that the difference ˆ u=u1−u2\nbetween any two weak solutions u1,u2of (2.2) is a weak solution of the problem\n\n\naˆutt−c2∆ˆu−b(1−δ)∆ˆut−bδdiv/parenleftBig\n|∇u1\nt|q−1∇u1\nt−|∇u2\nt|q−1∇u2\nt/parenrightBig\n+fˆut= 0,\nc2∂ˆu\n∂n+b(1−δ)∂ˆut\n∂n+bδ(|∇u1\nt|q−1∂u1\nt\n∂n−|∇u2\nt|q−1∂u2\nt\n∂n) = 0 on Γ ,\nαˆut+c2∂ˆu\n∂n+b(1−δ)∂ˆut\n∂n+bδ(|∇u1\nt|q−1∂u1\nt\n∂n−|∇u2\nt|q−1∂u2\nt\n∂n) = 0 on ˆΓ,\n(ˆu,ˆut)|t=0= (0,0).\n(2.30)\nMultiplication of (2.30) by ˆ utyields\n1\n2/bracketleftbigg/integraldisplay\nΩa(ˆut)2dx+c2|∇ˆu|2\nL2(Ω)/bracketrightbiggt\n0+b(1−δ)/integraldisplayt\n0/integraldisplay\nΩ|∇ˆut|2dxds+α/integraldisplayt\n0|ˆut|2\nL2(ˆΓ)ds12 V. NIKOLI ´C\n+/integraldisplayt\n0/integraldisplay\nΩ(f−1\n2at)(ˆut)2dxds≤0,\nsince due to the inequality (1.19) we have\nbδ/integraldisplayt\n0/integraldisplay\nΩ(|∇u1\nt|q−1∇u1\nt−|∇u2\nt|q−1∇u2\nt)·∇ˆutdxds≥0.(2.31)\nFrom here we conclude that ˆ ut= 0 and∇ˆu= 0 almost everywhere, which results in\nthesolutionbeingunique uptoanadditiveconstant. Theinitial condit ion ˆu|t=0= 0\nprovides us with uniqueness.\n4. Higher energy estimate. To obtain higher order estimate (2.8), we will test\n(2.12) with wn=u(k)\nn,tt(t)∈Vnand then combine the result with the lower order\nestimate (2.5) we derived previously. Multiplication by u(k)\nn,tt(t) and integrationwith\nrespect to time produces\n/integraldisplayt\n0/integraldisplay\nΩak(u(k)\nn,tt)2dxds+/bracketleftbiggb(1−δ)\n2|∇u(k)\nn,t|2\nL2(Ω)+bδ\nq+1|∇u(k)\nn,t|q+1\nLq+1(Ω)/bracketrightbiggt\n0\n+α\n2/bracketleftBig/integraldisplay\nˆΓ(u(k)\nn,t)2dx/bracketrightBigt\n0\n=c2/integraldisplayt\n0|∇u(k)\nn,t|2\nL2(Ω)ds−c2/bracketleftbigg/integraldisplay\nΩ∇u(k)\nn·∇u(k)\nn,tdx/bracketrightbiggt\n0+/integraldisplayt\n0/integraldisplay\nΓgku(k)\nn,ttdxds(2.32)\n−/integraldisplayt\n0/integraldisplay\nΩfku(k)\nn,tu(k)\nn,ttdxds.\nTo estimate the boundary integral on the right side, we employ (1.13 ) to obtain\n/integraldisplayt\n0/integraldisplay\nΓgku(k)\nn,ttdxds\n≤Ctr\n1|u(k)\nn,t(t)|W1,q+1(Ω)|gk(t)|\nW−q\nq+1,q+1\nq(Γ)−/integraldisplay\nΓgk(0)u(k)\nn,t(0)dx\n−/integraldisplayt\n0/integraldisplay\nΓgk,tu(k)\nn,tdxds\n≤η/ba∇dbl∇u(k)\nn,t/ba∇dblq+1\nL∞(0,T;Lq+1(Ω))+ǫ0/ba∇dblu(k)\nn,t/ba∇dbl2\nL∞(0,T;L2(Ω))\n+C(η,q+1)(Ctr\n1(1+CP))q+1\nq/ba∇dblgk/ba∇dblq+1\nq\nL∞(0,T;W−q\nq+1,q+1\nq(Γ))\n+1\n2ǫ0(Ctr\n1CΩ\n2)2/ba∇dblgk/ba∇dbl2\nL∞(0,T;W−q\nq+1,q+1\nq(Γ))+|u1,n|q+1\nW1,q+1(Ω)\n+C(1,q+1)(Ctr\n1|gk(0)|\nW−q\nq+1,q+1\nq(Γ))q+1\nq\n+C(ǫ1,q+1)(Ctr\n1(1+CP))q+1\nq/ba∇dblgk,t/ba∇dblq+1\nq\nLq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ))\n+ǫ1/ba∇dbl∇u(k)\nn,t/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))+1\n2ǫ0(Ctr\n1CΩ\n2)2/ba∇dblgk,t/ba∇dbl2\nL1(0,T;W−q\nq+1,q+1\nq(Γ)),(2.33)\nwhich together with\n/integraldisplayt\n0/integraldisplay\nΩfku(k)\nn,tu(k)\nn,ttdxds\n≤1\n2τ(CΩ\nH1,L4)4/ba∇dblfk/ba∇dbl2\nL∞(0,T;H1(Ω))/bracketleftBig\nT/ba∇dblu(k)\nn,t/ba∇dbl2\nL∞(0,T;L2(Ω))LOCAL EXISTENCE RESULTS 13\n+/ba∇dbl∇u(k)\nn,t/ba∇dbl2\nL2(0,T;L2(Ω))/bracketrightBig\n+τ\n2/ba∇dblu(k)\nn,tt/ba∇dbl2\nL2(0,T;L2(Ω)), (2.34)\nand taking esssup\n[0,T]in (2.32) leads to the estimate\na−τ\n2/ba∇dblu(k)\nn,tt/ba∇dbl2\nL2(0,T;L2(Ω))+/parenleftBigb(1−δ)\n4−σ/parenrightBig\n/ba∇dbl∇u(k)\nn,t/ba∇dbl2\nL∞(0,T;L2(Ω))\n+/parenleftBigbδ\n2(q+1)−η/parenrightBig\n/ba∇dbl∇u(k)\nn,t/ba∇dblq+1\nL∞(0,T;Lq+1(Ω))+α\n4/ba∇dblu(k)\nn,t/ba∇dbl2\nL∞(0,T;L2(ˆΓ))\n≤c2/ba∇dbl∇u(k)\nn,t/ba∇dbl2\nL2(0,T;L2(Ω))+σ|∇u1,n|2\nL2(Ω)+c4\n4σ(/ba∇dbl∇u(k)\nn/ba∇dbl2\nL∞(0,T;L2(Ω))\n+|∇u0,n|2\nL2(Ω))+1\n2τ(CΩ\nH1,L4)4˜b2[T/ba∇dblu(k)\nn,t/ba∇dbl2\nL∞(0,T;L2(Ω))\n+/ba∇dbl∇u(k)\nn,t/ba∇dbl2\nL2(0,T;L2(Ω))]+η/ba∇dbl∇u(k)\nn,t/ba∇dblq+1\nL∞(0,T;Lq+1(Ω))\n+ǫ1/ba∇dbl∇u(k)\nn,t/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))+ǫ0/ba∇dblu(k)\nn,t/ba∇dbl2\nL∞(0,T;L2(Ω))\n+1\n2ǫ0(Ctr\n1CΩ\n2)2/parenleftBig\n/ba∇dblgk/ba∇dbl2\nL∞(0,T;W−q\nq+1,q+1\nq(Γ))+/ba∇dblgk,t/ba∇dbl2\nL1(0,T;W−q\nq+1,q+1\nq(Γ))/parenrightBig\n+|u1,n|q+1\nW1,q+1(Ω)+C(1,q+1)(Ctr\n1|gk(0)|\nW−q\nq+1,q+1\nq(Γ))q+1\nq\n+b(1−δ)\n2|∇u1,n|2\nL2(Ω)+α\n2|u1,n|2\nL2(ˆΓ)+bδ\nq+1|∇u1,n|q+1\nLq+1(Ω)\n+(Ctr\n1(1+CP))q+1\nq/parenleftBig\nC(ǫ1,q+1)/ba∇dblgk,t/ba∇dblq+1\nq\nLq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ))\n+C(η,q+1)/ba∇dblgk/ba∇dblq+1\nq\nL∞(0,T;W−q\nq+1,q+1\nq(Γ))/parenrightBig\n.(2.35)\nSince there are terms on the right side in (2.35) which cannot be domin ated by\nthe terms on the left hand side, we need to also employ the lower estim ate (2.15).\nAdding (2.15) and µtimes (2.35) yields (2.8) with ureplaced by u(k)\nn, provided that\nwe choose\n0<τ <a,0<η<bδ\n2(q+1),0<σ<b(1−δ)\n4,\n0<µ<min/braceleftBigb(1−δ)\n2−(CΩ\nH1,L4)2ˆb\n1\n2τ(CΩ\nH1,L4)4˜b2+c2,a\n4−(CΩ\nH1,L4)2ˆbT−ǫ0\nǫ0+1\n2τ(CΩ\nH1,L4)4˜b2T,σ\nc2,bδ\n2−ǫ1\nǫ1/bracerightBig\n,(2.36)\nso that the coefficients in (2.8) are positive.\nAsbyassumption gk∈L∞(0,T;W−q\nq+1,q+1\nq(Γ))andgk,t∈Lq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ)),/parenleftbig\nu(k)\nn/parenrightbig\nn∈Nis a bounded sequence in\nX:=C1(0,T;W1,q+1(Ω))∩H2(0,T;L2(Ω)).\nWe further obtain\n/parenleftbig\nu(k)\nn,t/parenrightbig\n∈Nis uniformly bounded in L2(0,T;L2(Ω)), (2.37)\n/parenleftbig\n∇u(k)\nn,t/parenrightbig\n∈Nis uniformly bounded in Lq+1(0,T;Lq+1(Ω)), (2.38)\n|∇u(k)\nn,t|q−1∇u(k)\nn,tis uniformly bounded in Lq+1\nq(0,T;Lq+1\nq(Ω)) and (2.39)\n/parenleftbig\nu(k)\nn,t|ˆΓ/parenrightbig\n∈Nis uniformly bounded in L2(0,T;L2(ˆΓ)), (2.40)14 V. NIKOLI ´C\nwhich are reflexive Banach spaces.\nFrom here, after proceeding as in the step 1.(c) and 2, we can conc lude that (2.10)\nhas a unique solution u∈Xwhich satisfies the estimate (2.8). /square\nLet us now consider the boundary value problem (2.2) with an added lo wer order\nlinear damping term:\n\n\nautt−c2∆u−bdiv/parenleftBig\n((1−δ)+δ|∇ut|q−1)∇ut/parenrightBig\n+βut\n+fut= 0 in Ω ×(0,T),\nc2∂u\n∂n+b((1−δ)+δ|∇ut|q−1)∂ut\n∂n=gon Γ×(0,T),\nαut+c2∂u\n∂n+b((1−δ)+δ|∇ut|q−1)∂ut\n∂n= 0 onˆΓ×(0,T),\n(u,ut) = (u0,u1) onΩ×{t= 0},(2.41)\nwhereβ >0. This is a linearized version of the problem (1.6) with nonlinearity\nappearing only through the damping term. The additionaly introduce dβ−lower\norder term will allow us to remove restrictions on final time Tin the estimates (2.5)\nand (2.8). Indeed, by testing the equation with utand integrating with respect to\nspace and time, we obtain\n1\n2/bracketleftbigg/integraldisplay\nΩa(ut)2dx+c2|∇u|2\nL2(Ω)/bracketrightbiggt\n0+b/integraldisplayt\n0/integraldisplay\nΩ/parenleftBig\n(1−δ)+δ|∇ut|q−1/parenrightBig\n|∇ut|2dxds\n+β/integraldisplayt\n0/integraldisplay\nΩ|ut|2dxds+α/integraldisplayt\n0/integraldisplay\nˆΓ|ut|2dxds\n≤ˆb(CΩ\nH1,L4)2/integraldisplayt\n0|ut|2\nH1(Ω)ds+ǫ1/ba∇dbl∇ut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))\n+C(ǫ1,q+1)(Ctr\n1(1+CP))q+1\nq/ba∇dblg/ba∇dblq+1\nq\nLq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ))\n+ǫ0/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω)+1\n4ǫ0(Ctr\n1CΩ\n2)2/ba∇dblg/ba∇dbl2\nL1(0,T;W−q\nq+1,q+1\nq(Γ)),\nwhich leads to the lower order energy estimate\n(a\n4−ǫ0)/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω))+/parenleftBigb(1−δ)\n2−ˆb(CΩ\nH1,L4)2/parenrightBig\n/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))\n+/parenleftBigbδ\n2−ǫ1/parenrightBig\n/ba∇dbl∇ut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))+(β\n2−ˆb(CΩ\nH1,L4)2))/ba∇dblut/ba∇dbl2\nL2(0,T;L2(Ω))\n+c2\n4/ba∇dbl∇u/ba∇dbl2\nL∞(0,T;L2(Ω))+α\n2/ba∇dblut/ba∇dbl2\nL2(0,T;L2(ˆΓ))\n≤a\n2|u1|2\nL2(Ω)+c2\n2|∇u0|2\nL2(Ω)+1\n4ǫ0(Ctr\n1CΩ\n2)2/ba∇dblg/ba∇dbl2\nL1(0,T;W−q\nq+1,q+1\nq(Γ))\n+C(ǫ1,q+1)(Ctr\n1(1+CP))q+1\nq/ba∇dblg/ba∇dblq+1\nq\nLq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ)),(2.42)\nprovided that /ba∇dblf−1\n2at/ba∇dblL∞(0,T;L2(Ω))≤ˆb <min{β\n2(CΩ\nH1,L4)2,b(1−δ)\n2(CΩ\nH1,L4)2}and that\n0<ǫ0<a\n4, 0<ǫ1<bδ\n2.\nTesting with uttand adding µtimes the obtained estimate to (2.42) results in theLOCAL EXISTENCE RESULTS 15\nhigher order energy estimate valid for arbitrary time:\nµa−τ\n2/ba∇dblutt/ba∇dbl2\nL2(0,T;L2(Ω))+µ/parenleftBigb(1−δ)\n4−σ/parenrightBig\n/ba∇dbl∇ut/ba∇dbl2\nL∞(0,T;L2(Ω))\n+µ(a+µβ\n4−ǫ0(µ+1))/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω))+ˇb/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))\n+µα\n4/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(ˆΓ))+(bδ\n2−µ(ǫ1+1))/ba∇dbl∇ut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))\n+/parenleftBigβ\n2−(CΩ\nH1,L4)2ˆb−µ\n2τ(CΩ\nH1,L4)4˜b2/parenrightBig\n/ba∇dblut/ba∇dbl2\nL2(0,T;L2(Ω))\n+c2\n4/parenleftBig\n1−µc2\nσ/parenrightBig\n/ba∇dbl∇u/ba∇dbl2\nL∞(0,T;L2(Ω))+α\n2/ba∇dblut/ba∇dbl2\nL2(0,T;L2(ˆΓ))\n+µ(bδ\n2(q+1)−η)/ba∇dbl∇ut/ba∇dblq+1\nL∞(0,T;Lq+1(Ω))\n≤C/parenleftBig\nCΓ(g)+|u1|2\nH1(Ω)+|∇u0|2\nL2(Ω)+|u1|q+1\nW1,q+1(Ω)+|u1|2\nL2(ˆΓ)/parenrightBig\n,(2.43)\nwithˇb=b(1−δ)\n2−(CΩ\nH1,L4)2ˆb−µ(1\n2τ(CΩ\nH1,L4)4˜b2+c2), for some appropriately chosen\nC >0. Therefore we obtain:\nProposition 2.2. Letβ >0and the assumptions (i) in Proposition 2.1hold, with\n/ba∇dblf−1\n2at/ba∇dblL∞(0,T;L2(Ω))≤ˆb<min{β\n2(CΩ\nH1,L4)2,b(1−δ)\n2(CΩ\nH1,L4)2}.\nThen(2.41)has a unique weak solution in ˜X, with˜Xdefined as in (2.4), which\nsatisfies (2.42)for some sufficiently small constants ǫ0,ǫ1>0.\nIf, in addition to (i), the assumptions (ii) in Proposition 2.1are satisfied, then\nu∈X, withXdefined as in (2.7), andusatisfies the energy estimate (2.43)\nfor some sufficiently small constants ǫ0,ǫ1,µ,σ,τ > 0and some large enough C,\nindependent of T.\nWe continue with considering an equation with an added lower order no nlinear\ndamping term:\n\n\nautt−c2∆u−bdiv/parenleftBig\n((1−δ)+δ|∇ut|q−1)∇ut/parenrightBig\n+γ|ut|q−1ut+fut\n= 0 in Ω ×(0,T],\nc2∂u\n∂n+b((1−δ)+δ|∇ut|q−1)∂ut\n∂n=gon Γ×(0,T],\nαut+c2∂u\n∂n+b((1−δ)+δ|∇ut|q−1)∂ut\n∂n= 0 onˆΓ×(0,T],\n(u,ut) = (u0,u1) onΩ×{t= 0},(2.44)\nwithγ >0, which is motivated by the problem (1.7). Once we multiply (2.44) by\nutand integrate by parts, we produce\n1\n2/bracketleftbigg/integraldisplay\nΩa(ut)2dx+c2|∇u|2\nL2(Ω)/bracketrightbiggt\n0+b/integraldisplayt\n0/integraldisplay\nΩ/parenleftBig\n(1−δ)+δ|∇ut|q−1/parenrightBig\n|∇ut|2dxds\n+α/integraldisplayt\n0/integraldisplay\nˆΓ|ut|2dxds+γ/integraldisplayt\n0/integraldisplay\nΩ|ut|q+1dxds\n=/integraldisplayt\n0/integraldisplay\nΩ(f−1\n2at)(ut)2dxds+/integraldisplayt\n0/integraldisplay\nΓgutdxds.16 V. NIKOLI ´C\nWe will make use of the following inequality\n/integraldisplayt\n0/integraldisplay\nΓgutdxds≤ǫ0\n2/ba∇dblut/ba∇dblq+1\nLq+1(0,T;W1,q+1(Ω))\n+C(ǫ0\n2,q+1)(Ctr\n1/ba∇dblg/ba∇dbl\nLq+1\nq(0,T,W−q\nq+1,q+1\nq(Γ)))q+1\nq,(2.45)\nand, forq>1,\n/integraldisplayt\n0/integraldisplay\nΩ(f−1\n2at)(ut)2dxds≤/integraldisplayt\n0/parenleftBig/integraldisplay\nΩ|ut|q+1dx/parenrightBig2\nq+1/parenleftBig/integraldisplay\nΩ|f−1\n2at|q+1\nq−1/parenrightBigq−1\nq+1ds\n=/integraldisplayt\n0|ut|2\nLq+1(Ω)|f−1\n2at|\nLq+1\nq−1(Ω)ds\n≤ǫ0\n2/ba∇dblut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))+C(ǫ0\n2,q+1\n2)/ba∇dblf−1\n2at/ba∇dblq+1\nq−1\nLq+1\nq−1(0,T;Lq+1\nq−1(Ω)),\nto obtain lower order energy estimate\na\n4/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω))+c2\n4/ba∇dbl∇u/ba∇dbl2\nL∞(0,T;L2(Ω))+b(1−δ)\n2/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))\n+bδ−ǫ0\n2/ba∇dbl∇ut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))+(γ\n2−ǫ0)/ba∇dblut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))\n+α\n2/ba∇dblut/ba∇dbl2\nL2(ˆΓ)\n≤C(ǫ0\n2,q+1)(Ctr\n1/ba∇dblg/ba∇dbl\nLq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ)))q+1\nq+a\n2|u1|2\nL2(Ω) (2.46)\n+C(ǫ0\n2,q+1\n2)/ba∇dblf−1\n2at/ba∇dblq+1\nq−1\nLq+1\nq−1(0,T;Lq+1\nq−1(Ω))+c2\n2|∇u0|2\nL2(Ω),\nassuming that f,at∈Lq+1\nq−1(0,T;Lq+1\nq−1(Ω)) and 0 <ǫ0<γ\n2.\nFor obtaining higher order estimate, we multiply (2.44) with utt, integrate with\nrespect to space and time and make use of the estimate\n/integraldisplayt\n0/integraldisplay\nΓguttdxds≤η/ba∇dblut/ba∇dblq+1\nL∞(0,T;W1,q+1(Ω))+ǫ0\n2/ba∇dblut/ba∇dblq+1\nLq+1(0,T;W1,q+1(Ω))\n+C(η,q+1)(Ctr\n1/ba∇dblg/ba∇dbl\nL∞(0,T,W−q\nq+1,q+1\nq(Γ)))q+1\nq (2.47)\n+|u1|q+1\nW1,q+1(Ω)+C(1,q+1)(Ctr\n1|g(0)|\nW−q\nq+1,q+1\nq(Γ))q+1\nq\n+C(ǫ0\n2,q+1)(Ctr\n1/ba∇dblgt/ba∇dbl\nLq+1\nq(0,T,W−q\nq+1,q+1\nq(Γ)))q+1\nq.\nIn order to avoid dependence on time, we approach estimate (2.34) differently this\ntime: by employing the embedding Lq+1(Ω)֒→L4(Ω), valid for q≥3, we obtain\n/integraldisplayt\n0/integraldisplay\nΩfututtdxds≤1\n2τ(CΩ\nH1,L4)2(CΩ\nLq+1,L4)2/integraldisplayt\n0|f(s)|2\nH1(Ω)|ut(s)|2\nLq+1(Ω)ds\n+τ\n2/ba∇dblutt/ba∇dbl2\nL2(0,T;L2(Ω))\n≤ǫ0\n2/ba∇dblut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))+τ\n2/ba∇dblutt/ba∇dbl2\nL2(0,T;L2(Ω))\n+C(ǫ0\n2,q+1\n2)(1\n2τ(CΩ\nH1,L4CΩ\nLq+1,L4)2/ba∇dblf/ba∇dbl2\nL2(q+1)\nq−1(0,T;H1(Ω)))q+1\nq−1,LOCAL EXISTENCE RESULTS 17\nwhich, together with (2.46), leads to the higher order estimate\nµa−τ\n2/ba∇dblutt/ba∇dbl2\nL2(0,T;L2(Ω))+µ/parenleftBigb(1−δ)\n4−σ/parenrightBig\n/ba∇dbl∇ut/ba∇dbl2\nL∞(0,T;L2(Ω))\n+a\n4/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω))+c2\n4/parenleftBig\n1−µc2\nσ/parenrightBig\n/ba∇dbl∇u/ba∇dbl2\nL∞(0,T;L2(Ω))\n+µα\n4/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(ˆΓ))+/parenleftBigbδ\n2−ǫ0\n2(µ+1)/parenrightBig\n/ba∇dbl∇ut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))\n+/parenleftBigb(1−δ)\n2−µc2/parenrightBig\n/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))+α\n2/ba∇dblut/ba∇dbl2\nL2(0,T;L2(ˆΓ))\n+/parenleftBigγ\n2−ǫ0(µ+1)/parenrightBig\n/ba∇dblut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))\n+µ/parenleftBigbδ\n2(q+1)−η/parenrightBig\n/ba∇dbl∇ut/ba∇dblq+1\nL∞(0,T;Lq+1(Ω))\n+µ/parenleftBigγ\n2(q+1)−η/parenrightBig\n/ba∇dblut/ba∇dblq+1\nL∞(0,T;Lq+1(Ω))\n≤C/parenleftBig1/summationdisplay\ns=0/ba∇dblds\ndtsg/ba∇dblq+1\nq\nLq+1\nq(0,T,W−q\nq+1,q+1\nq(Γ))+/ba∇dblg/ba∇dblq+1\nq\nL∞(0,T,W−q\nq+1,q+1\nq(Γ))\n+/ba∇dblf−1\n2at/ba∇dblq+1\nq\nLq+1\nq(0,T;Lq+1\nq(Ω))+/ba∇dblf/ba∇dbl2(q+1)\nq−1\nL2(q+1)\nq−1(0,T;H1(Ω))+|u1|2\nH1(Ω)\n+|∇u0|2\nL2(Ω)+|u1|q+1\nW1,q+1(Ω)+|u1|2\nL2(ˆΓ)/parenrightBig\n,(2.48)\nassumingf∈L2(q+1)\nq−1(0,T;H1(Ω) and choosing τ,σ,η,µ> 0 to be sufficiently small.\nProposition 2.3. LetT >0,c2,b,γ >0,α≥0,δ∈(0,1),q>1and\n(i)•a∈L∞(0,T;L∞(Ω)),at∈Lq+1\nq−1(0,T;Lq+1\nq−1(Ω)),0<a≤a(t,x)≤a,\n•f∈Lq+1\nq−1(0,T;Lq+1\nq−1(Ω)),\n•g∈Lq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ)),\n•u0∈H1(Ω),u1∈L2(Ω).\nThen(2.44)has a unique weak solution in ˜X, with˜Xdefined as in (2.4), which\nsatisfies (2.46)for some 0<ǫ0<γ\n2.\nIf, in addition to (i), the following assumptions are satisfi ed\n(ii)•q≥3,\n•f∈L2(q+1)\nq−1(0,T;H1(Ω)),\n•g∈L∞(0,T;W−q\nq+1,q+1\nq(Γ)),gt∈Lq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ)),\n•u1∈W1,q+1(Ω),\nthen(2.44)has a unique weak solution in X, withXdefined as in (2.4), which\nsatisfies the energyestimate (2.48)for some sufficiently small constants µ,σ,τ,η> 0\nand some large enough C >0, independent of T.\nRemark2.4. Duetotheterms /ba∇dblf−1\n2at/ba∇dblq+1\nq−1\nLq+1\nq−1(0,T;Lq+1\nq−1(Ω))and/ba∇dblf/ba∇dbl2(q+1)\nq\nL2(q+1)\nq−1(0,T;H1(Ω))\nappearing on the right hand side in the estimate (2.48), we will not be a ble to prove\nlocal well-posedness of the problem (1.9) by employing this estimate. Instead, pro-\nvided the assumptions (ii) in Proposition 2.1 hold, we could proceed with the same\nestimates as in the proof of that proposition, and for evaluating bo undary integrals18 V. NIKOLI ´C\napply (2.45) and (2.47), to obtain the following energy estimate:\nµa−τ\n2/ba∇dblutt/ba∇dbl2\nL2(0,T;L2(Ω))+µ/parenleftBigb(1−δ)\n4−σ/parenrightBig\n/ba∇dbl∇ut/ba∇dbl2\nL∞(0,T;L2(Ω))\n+/parenleftBiga\n4−(CΩ\nH1,L4)2ˆbT−µ1\n2τ(CΩ\nH1,L4)4˜b2T/parenrightBig\n/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω))\n+c2\n4/parenleftBig\n1−µc2\nσ/parenrightBig\n/ba∇dbl∇u/ba∇dbl2\nL∞(0,T;L2(Ω))+µα\n4/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(ˆΓ))\n+ˇb/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))+/parenleftBigbδ\n2−ǫ0(µ+1)/parenrightBig\n/ba∇dbl∇ut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))\n+(γ\n2−ǫ0(µ+1))/ba∇dblut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))+α\n2/ba∇dblut/ba∇dbl2\nL2(0,T;L2(ˆΓ))\n+µ/parenleftBigbδ\n2(q+1)−η/parenrightBig\n/ba∇dbl∇ut/ba∇dblq+1\nL∞(0,T;Lq+1(Ω))\n+µ/parenleftBigγ\n2(q+1)−η/parenrightBig\n/ba∇dblut/ba∇dblq+1\nL∞(0,T;Lq+1(Ω))\n≤C/parenleftBig1/summationdisplay\ns=0/ba∇dblds\ndtsg/ba∇dblq+1\nq\nLq+1\nq(0,T,W−q\nq+1,q+1\nq(Γ))+/ba∇dblg/ba∇dblq+1\nq\nL∞(0,T,W−q\nq+1,q+1\nq(Γ))\n+|u1|2\nL2(ˆΓ)+|u1|2\nH1(Ω)+|∇u0|2\nL2(Ω)+|u1|q+1\nW1,q+1(Ω)/parenrightBig\n,(2.49)\nwithˇb=b(1−δ)\n2−(CΩ\nH1,L4)2ˆb−µ(1\n2τ(CΩ\nH1,L4)4˜b2+c2), for some appropriately chosen\nconstantsτ,σ,ǫ0,η,µ>0 and large enough C, independent of T.\nRelyingonProposition2.1, wecannowprovethelocalwell-posedness forthebound-\nary value problem (2.1).\nTheorem 2.5. Letc2,b >0,α≥0,δ∈(0,1),k∈R,q > d−1,q≥1,\ng∈L∞(0,T;W−q\nq+1,q+1\nq(Γ)),gt∈Lq+1\nq(0,T;H−q\nq+1,q+1\nq(Γ)). For anyT >0there\nis aκT>0such that for all u0,u1∈W1,q+1(Ω), with\nCΓ(g)+|u0|2\nL1(Ω)+|∇u0|2\nLq+1(Ω)+|u1|2\nH1(Ω)+|∇u0|2\nL2(Ω)+|u1|q+1\nW1,q+1(Ω)\n+|u1|2\nL2(ˆΓ)≤κ2\nT (2.50)\nthere exists a unique weak solution u∈ Wof(2.1), where\nW={v∈X:/ba∇dblvtt/ba∇dblL2(0,T;L2(Ω))≤m\n∧/ba∇dblvt/ba∇dblL∞(0,T;H1(Ω))≤m\n∧/ba∇dbl∇vt/ba∇dblLq+1(0,T;Lq+1(Ω))≤M\n∧(v,vt)|t=0= (u0,u1)},(2.51)\nwith\n2|k|CΩ\nW1,q+1,L∞/bracketleftBig\nmax{1+CP,CΩ\n1}κT+(1+CP)Tq\nq+1M+CΩ\n2Tm)/bracketrightBig\n<1,(2.52)\nandmandMsufficiently small, where CΓ(g)is defined as in (2.9).\nProof.We will carry out the proof by using a fixed point argument. We define an\noperator T:W →X,v/ma√sto→ Tv=u, whereusolves (2.10) with\na= 1−2kv, f=−2kvt. (2.53)LOCAL EXISTENCE RESULTS 19\nWe will show that assumptions of Proposition 2.1 are satisfied. Since v∈ W, and\nq>d−1 so we can make use of the embedding W1,q+1(Ω)֒→L∞(Ω), we have by\n(1.17)\n|2kv(x,t)| ≤2|k|CΩ\nW1,q+1,L∞/bracketleftBig\nmax{1+CP,CΩ\n1,}κT+(1+CP)Tq\nq+1M\n+CΩ\n2Tm/bracketrightBig\n,\nandat=−2kvt∈L∞(0,T;L2(Ω).\nIt follows that 0 <a= 1−a0<a<a= 1+a0, where\na0= 2|k|CΩ\nW1,q+1,L∞/bracketleftBig\nmax{1+CP,CΩ\n1}κT+(1+CP)Tq\nq+1M+CΩ\n2Tm/bracketrightBig\n.\nFurthermore,\n/ba∇dblf−1\n2at/ba∇dblL∞(0,T;L2(Ω))=/ba∇dblkvt/ba∇dblL∞(0,T;L2(Ω))≤ |k|m,\n/ba∇dblf/ba∇dblL∞(0,T;H1(Ω))= 2|k|/ba∇dblvt/ba∇dblL∞(0,T;H1(Ω))≤2|k|m.\nHence the higher order energy estimate (2.8) holds and by choosing m,M >0 such\nthat\n2|k|CΩ\nW1,q+1,L∞((1+CP)Tq\nq+1M+CΩ\n2Tm)<1,\nm<1\n|k|min/braceleftBigb(1−δ)\n2(CΩ\nH1,L4)2,a\n4T(CΩ\nH1,L4)2/bracerightBig\n,\nand making the bound κTon initial and boundary data small enough\nκT<1\nmax{1+CP,CΩ\n1}/parenleftBig1\n2|k|CΩ\nW1,q+1,L∞−(1+CP)Tq\nq+1M−CΩ\n2Tm/parenrightBig\n,\nκ2\nT≤1\nCmin/braceleftBig/parenleftBiga\n4−(CΩ\nH1,L4)2ˆbT−ǫ0(µ+1)−µ1\n2τ(CΩ\nH1,L4)4˜b2T/parenrightBig\nm2,\nµa−τ\n2m2,µ/parenleftBigb(1−δ)\n4−σ/parenrightBig\nm2,/parenleftBigbδ\n2−ǫ1(µ+1)/parenrightBig\nMq+1/bracerightBig\n,\nwe achieve that u∈ W, with constants ǫ0,ǫ1,τ,η,σ,µ chosen as in (2.16) and (2.36)\nandCas in (2.8).\nIn order to prove contractivity, consider vi∈ W,ui=Tvi∈ W,i= 1,2 and denote\nˆu=u1−u2,ˆv=v1−v2. Subtracting the equation (2.2) for u1andu2yields:\n\n\n(1−2kv1)ˆutt−c2∆ˆu−b(1−δ)∆ˆut−bδdiv/parenleftBig\n|∇u1\nt|q−1∇u1\nt−|∇u2\nt|q−1∇u2\nt/parenrightBig\n,\n= 2k(ˆvu2\ntt+v1\ntˆut+ ˆvtu2\nt) in Ω,\nc2∂ˆu\n∂n+b(1−δ)∂ˆut\n∂n+bδ(|∇u1\nt|q−1∂u1\nt\n∂n−|∇u2\nt|q−1∂u2\nt\n∂n) = 0 on Γ ,\nαˆut+c2∂ˆu\n∂n+b(1−δ)∂ˆut\n∂n+bδ(|∇u1\nt|q−1∂u1\nt\n∂n−|∇u2\nt|q−1∂u2\nt\n∂n) = 0 on ˆΓ,\n(ˆu,ˆut)|t=0= (0,0).(2.54)\nAfter testing (2.54) with ˆ utand making use of the inequality (2.31), we obtain\n1\n2/bracketleftBig/integraldisplay\nΩ(1−2kv1)(ˆut)2dx+c2|∇ˆu|2\nL2(Ω)/bracketrightBigt\n0+b(1−δ)/integraldisplayt\n0|∇ˆut|2\nL2(Ω)ds\n+α/integraldisplayt\n0/integraldisplay\nˆΓ|ˆut|2dxds20 V. NIKOLI ´C\n≤2|k|/integraldisplayt\n0/integraldisplay\nΩ(1\n2v1\nt(ˆut)2+ ˆvu2\nttˆut+ ˆvtu2\ntˆut)dxds,\nand therefore we have\n1\n2/bracketleftBig/integraldisplay\nΩ(1−2kv1)(ˆut)2dx+c2|∇ˆu|2\nL2(Ω)/bracketrightBigt\n0+b(1−δ)/integraldisplayt\n0|∇ˆut|2\nL2(Ω)ds\n+α/integraldisplayt\n0/integraldisplay\nˆΓ|ˆut|2dxds\n≤ |k|(CΩ\nH1,L4)2/parenleftBig\n/ba∇dblv1\nt/ba∇dblL∞(0,T;L2(Ω))/integraldisplayt\n0|ˆut|2\nH1(Ω)ds\n+/ba∇dblu2\ntt/ba∇dblL2(0,T;L2(Ω))[/ba∇dblˆv/ba∇dbl2\nL∞(0,T;H1(Ω))+/integraldisplayt\n0|ˆut|2\nH1(Ω)ds]\n+/ba∇dblu2\nt/ba∇dblL∞(0,T;L2(Ω))[/ba∇dblˆvt/ba∇dbl2\nL2(0,T;H1(Ω))+/integraldisplayt\n0|ˆut|2\nH1(Ω)ds]/parenrightBig\n.\nUtilizing the fact that v1,v2,u1,u2∈ Wand the inequalities /ba∇dbl∇ˆv/ba∇dbl2\nL∞(0,T,L2(Ω))≤\nT/ba∇dbl∇ˆvt/ba∇dbl2\nL2(0,T,L2(Ω)),/ba∇dblˆv/ba∇dbl2\nL∞(0,T,L2(Ω))≤T/ba∇dblˆvt/ba∇dbl2\nL2(0,T,L2(Ω))and/ba∇dblˆvt/ba∇dbl2\nL2(0,T;L2(Ω))≤\nT/ba∇dblˆvt/ba∇dbl2\nL∞(0,T;L2(Ω))leads to\n1−a0\n4/ba∇dblˆut/ba∇dbl2\nL∞(0,T;L2(Ω))+c2\n4/ba∇dbl∇ˆu/ba∇dbl2\nL∞(0,T;L2(Ω))+b(1−δ)\n2/ba∇dbl∇ˆut/ba∇dbl2\nL2(0,T;L2(Ω))\n≤ |k|(CΩ\nH1,L4)2m/parenleftBig\n3(T/ba∇dblˆut/ba∇dbl2\nL∞(0,T;L2(Ω))+/ba∇dbl∇ˆut/ba∇dbl2\nL2(0,T;L2(Ω)))+T2/ba∇dblˆvt/ba∇dbl2\nL∞(0,T;L2(Ω))\n+T/ba∇dbl∇ˆvt/ba∇dbl2\nL2(0,T;L2(Ω))+T/ba∇dblˆvt/ba∇dbl2\nL∞(0,T;L2(Ω))+/ba∇dbl∇ˆvt/ba∇dbl2\nL2(0,T;L2(Ω))/parenrightBig\n.\nIt follows that\n/parenleftBig1−a0\n4−3T|k|(CΩ\nH1,L4)2m/parenrightBig\n/ba∇dblˆut/ba∇dbl2\nL∞(0,T;L2(Ω))+c2\n4/ba∇dbl∇ˆu/ba∇dbl2\nL∞(0,T;L2(Ω))\n+/parenleftBigb(1−δ)\n2−3|k|(CΩ\nH1,L4)2m/parenrightBig\n/ba∇dbl∇ˆut/ba∇dbl2\nL2(0,T;L2(Ω))\n≤ |k|(CΩ\nH1,L4)2m(T+1)max {1,T}/parenleftBig\n/ba∇dblˆvt/ba∇dbl2\nL∞(0,T;L2(Ω))+/ba∇dbl∇ˆvt/ba∇dbl2\nL2(0,T;L2(Ω)))\nand altogether we have\nmin{1−a0\n4−3T|k|(CΩ\nH1,L4)2m,b(1−δ)\n2−3|k|(CΩ\nH1,L4)2m,c2\n4}|||u|||2\n≤ |k|(CΩ\nH1,L4)2m(T+1)max {1,T}|||v|||2,(2.55)\nwhere|||u|||2=/ba∇dblˆut/ba∇dbl2\nL∞(0,T;L2(Ω))+/ba∇dbl∇ˆut/ba∇dbl2\nL2(0,T;L2(Ω))+/ba∇dbl∇ˆu/ba∇dbl2\nL∞(0,T;L2(Ω)). We\nconcludefrom(2.55)that Tisacontractionwithrespecttothe norm |||·|||, provided\nthatmis sufficiently small. This, together with the self-mapping property an dW\nbeing closed, provides existence and uniqueness of a solution. /square\nRelying on Proposition 2.2 we can obtain local well-posedness for the p roblem\n(1.6)withβ >0. Sinceweneedtoavoiddegeneracyoftheterm1 −2kuandtherefore\nmake use of the estimate (1.17) to get the condition (2.52), we cann ot completely\navoid restriction on final time in the fully nonlinear equation. Inspect ing the proof\nof Theorem 2.5 immediately yields:\nTheorem 2.6. Letβ >0and the assumptions of Theorem 2.5hold. For any T >0\nthere is aκT>0such that for all u0,u1∈W1,q+1(Ω), with(2.50), there exists aLOCAL EXISTENCE RESULTS 21\nunique weak solution u∈ Wof(2.1), whereWis defined as in (2.51), with(2.52)\nandmandMsufficiently small.\nFor obtaining well-posedness for the problem (1.7) with γ >0, we cannot rely\non estimates in Proposition 2.3 to prove self-mapping of the fixed-po int operator\nT, instead we make use of (2.49); therefore restrictions on final tim e persist in the\nnonlinear equation. Analogously to Theorem 2.5 we obtain:\nTheorem 2.7. Letγ >0and the assumptions of Theorem 2.5hold. For any T >0\nthere is aκT>0such that for all u0,u1∈W1,q+1(Ω), with\n1/summationdisplay\ns=0/ba∇dblds\ndtsg/ba∇dblq+1\nq\nLq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ))+/ba∇dblg/ba∇dblq+1\nq\nL∞(0,T;W−q\nq+1,q+1\nq(Γ))+|u0|2\nL1(Ω)\n+|∇u0|2\nLq+1(Ω)+|u1|2\nH1(Ω)+|∇u0|2\nL2(Ω)+|u1|q+1\nW1,q+1(Ω)+|u1|2\nL2(ˆΓ)≤κ2\nT,(2.56)\nthere exists a unique weak solution u∈ Wof(2.1), whereWis defined as in (2.51),\nwith(2.52), andmandMsufficiently small.\n3.Acoustic-acoustic coupling\nWe will now consider the problem of an acoustic-acoustic coupling whic h can be\nmodeled by the equation with coefficients varying in space (1.10). We w ill make\nthe following assumptions on the coefficients in (1.10):\n\n\nλ,̺,b,δ,k ∈L∞(Ω),\n∃λ,λ,̺,̺: 0<λ≤λ(x)≤λ,0<̺≤̺(x)≤̺,in Ω,\n∃b,b,δ,δ: 0<b≤b(x)≤b,0<δ≤δ(x)≤δ<1 in Ω.(3.1)\nWecanagainfirstinspecttheproblemwithnonlinearitypresentonlyin thedamping\nterm:\n\n\nautt−div(1\n̺(x)∇u)−div/parenleftBig\nb(x)(((1−δ(x))+δ(x)|∇ut|q−1)∇ut/parenrightBig\n+fut= 0 in Ω ×(0,T],\n1\n̺(x)∂u\n∂n+b(x)((1−δ(x))+δ(x)|∇ut|q−1)∂ut\n∂n=gon Γ×(0,T],\nα(x)ut+1\n̺(x)∂u\n∂n+b(x)((1−δ(x))+δ(x)|∇ut|q−1)∂ut\n∂n= 0 onˆΓ×(0,T],\n(u,ut) = (u0,u1) onΩ×{t= 0}.\nAnalogously to Propositon 2.1 and Theorem 2.5 we obtain\nCorollary 3.1. Let the assumptions (3.1)and the assumptions (i) in Propositon\n2.1be satisfied, with\n/ba∇dblf−1\n2at/ba∇dblL∞(0,T;L2(Ω))≤ˆb<min/braceleftBigb(1−δ)\n2(CΩ\nH1,L4)2,a\n4T(CΩ\nH1,L4)2/bracerightBig\n.\nThen(3.2)has a weak solution u∈˜X, with˜Xdefined as in (2.4), which satisfies\nthe energy estimate\n/bracketleftBiga\n4−ˆb(CΩ\nH1,L4)2T−ǫ0/bracketrightBig\n/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω))+c2\n4/ba∇dbl∇u/ba∇dbl2\nL∞(0,T;L2(Ω))\n+/bracketleftBigb(1−δ)\n2−ˆb(CΩ\nH1,L4)2/bracketrightBig\n/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))22 V. NIKOLI ´C\n+/bracketleftBigbδ\n2−ǫ1/bracketrightBig\n/ba∇dbl∇ut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))+α\n2/ba∇dblu(k)\nn,t/ba∇dbl2\nL2(0,T;L2(ˆΓ))\n≤a\n2|u1|2\nL2(Ω)+c2\n2|∇u0|2\nL2(Ω)+1\n4ǫ0(Ctr\n1CΩ\n2)2/ba∇dblg/ba∇dbl2\nL1(0,T;W−q\nq+1,q+1\nq(Γ))\n+C(ǫ1,q+1)(Ctr\n1(1+CP))q+1\nq/ba∇dblg/ba∇dblq+1\nq\nLq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ)),\nfor some sufficiently small constants ǫ0,ǫ1>0.\nIf, additionally, assumptions (ii) of Proposition 2.1hold, thenu∈X, whereXis\ndefined as in (2.7), and satisfies the energy estimate (2.8), whereαis replaced with\nα,b(1−δ)withb(1−δ),bδwithbδandc2\n2(1−µc2\nσ)withc2\n4−µc4\n4σ, for some small\nenough constants τ,η,σ,µ> 0and some large enough C >0, independent of T.\nCorollary 3.2. Letg∈L∞(0,T;W−q\nq+1,q+1\nq(Γ)),gt∈Lq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ)),\nq > d−1,q≥1and assumptions (3.1)be satisfied. For any T >0there is a\nκT>0such that for all u0,u1∈W1,q+1(Ω), with(2.50), there exists a unique weak\nsolutionu∈ Wof(1.10), whereWis defined as in (2.51), with(2.52)with|k|\nreplaced by |k|L∞(Ω), andmandMsufficiently small.\n4.Westervelt’s equation in the formulation (1.8)\nWe begin with the problem (1.8) in the case β= 0:\n\n\n(1−2ku)utt−c2div(∇u+ε|∇u|q−1∇u)−b∆ut\n= 2k(ut)2in Ω×(0,T],\nc2∂u\n∂n+c2ε|∇u|q−1∂u\n∂n+b∂ut\n∂n=gon Γ×(0,T],\nαut+c2∂u\n∂n+c2ε|∇u|q−1∂u\n∂n+b∂ut\n∂n= 0 onˆΓ×(0,T],\n(u,ut) = (u0,u1) onΩ×{t= 0}.(4.1)\nWe will first study the problem with the nonlinearityappearing only thr ough damp-\ning:\n\n\nautt−c2div(∇u+ε|∇u|q−1∇u)−b∆ut+fut= 0 in Ω ×(0,T],\nc2∂u\n∂n+c2ε|∇u|q−1∂u\n∂n+b∂ut\n∂n=gon Γ×(0,T],\nαut+c2∂u\n∂n+c2ε|∇u|q−1∂u\n∂n+b∂ut\n∂n= 0 onˆΓ×(0,T],\n(u,ut) = (u0,u1) onΩ×{t= 0}.(4.2)\nProposition 4.1. LetT >0,c2,b,ε>0,α≥0,δ∈(0,1),q≥1and assume that\n•a∈L∞(0,T;L∞(Ω)),at∈L∞(0,T;L2(Ω)),0<a<a(x,t)<a,\n•f∈L∞(0,T;L2(Ω)),\n•g∈L2(0,T;H−1/2(Γ)),\n•u0∈W1,q+1(Ω),u1∈L2(Ω),\nwith\n/ba∇dblf−1\n2at/ba∇dblL∞(0,T;L2(Ω))≤ˆb<min{a\n4T(CΩ\nH1,L4)2,b\n2(CΩ\nH1,L4)2}.\nThen(4.2)has a weak solution\nu∈˜X:=C1(0,T;L2(Ω))∩C(0,T;W1,q+1(Ω))∩H1(0,T;H1(Ω)),(4.3)LOCAL EXISTENCE RESULTS 23\nwhich satisfies the energy estimate\n/bracketleftBiga\n4−(Ctr\n2)2τ−Tˆb(CΩ\nH1,L4)2/bracketrightBig\n/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω))+c2\n4/ba∇dbl∇u/ba∇dbl2\nL∞(0,T;L2(Ω))\n+/bracketleftBigb\n2−(Ctr\n2)2τ−ˆb(CΩ\nH1,L4)2/bracketrightBig\n/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))\n+c2ε\n2(q+1)/ba∇dbl∇u/ba∇dblq+1\nL∞(0,T;Lq+1(Ω))+α\n2/ba∇dblut/ba∇dbl2\nL2(0,T;L2(ˆΓ))\n≤1\n4τ(/ba∇dblg/ba∇dbl2\nL1(0,T;H−1/2(Γ))+/ba∇dblg/ba∇dbl2\nL2(0,T;H−1/2(Γ)))+a\n2|u1|2\nL2(Ω)\n+c2\n2|∇u0|2\nL2(Ω)+c2ε\nq+1|∇u0|q+1\nLq+1(Ω),(4.4)\nfor some constant\n0<τ <min/braceleftBigb−2ˆb(CΩ\nH1,L4)2\n2(Ctr\n2)2,a−4(CΩ\nH1,L4)2ˆbT\n4(Ctr\n2)2/bracerightBig\n.\nProof.The proof follows along the line of the standard Galerkin approximatio n\nmethod. Here we will focus on deriving the energy estimate.\nThe weak form of the problem is given as follows:\n/integraldisplay\nΩ/braceleftBig\nauttw+c2(∇u+ε|∇u|q−1∇u)·∇w+b∇ut·∇w/bracerightBig\ndx+α/integraldisplay\nˆΓutwdx\n=−/integraldisplayt\n0/integraldisplay\nΩfutwdx+/integraldisplay\nΓgwdx, ∀w∈W1,q+1(Ω).(4.5)\nTesting (4.5) with utand integrating with respect to space and time yields\n/bracketleftBig1\n2/integraldisplay\nΩa(ut)2dx+c2\n2|∇u|2\nL2(Ω)+c2ε\nq+1|∇u|q+1\nLq+1(Ω)/bracketrightBigt\n0\n+b/integraldisplayt\n0|∇ut(s)|2\nL2(Ω)ds+α/integraldisplayt\n0|ut(s)|2\nL2(ˆΓ)ds\n=/integraldisplayt\n0/integraldisplay\nΩ(f−1\n2at)(ut)2dxds+/integraldisplayt\n0/integraldisplay\nΓgutdxds\n≤ /ba∇dblf−1\n2at/ba∇dblL∞(0,T;L2(Ω))/integraldisplayt\n0|ut(s)|2\nL4(Ω)ds+/integraldisplayt\n0/integraldisplay\nΓgutdxds.(4.6)\nBy taking esssup\n[0,T]in (4.6) and making use of the embedding H1(Ω)֒→L4(Ω) and\nestimating the boundary integral in the following way\n/integraldisplayt\n0/integraldisplay\nΓgutdxds≤τ(Ctr\n2)2/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))+1\n4τ/ba∇dblg/ba∇dbl2\nL2(0,T;H−1/2(Γ))\n+τ(Ctr\n2)2/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω))+1\n4τ/ba∇dblg/ba∇dbl2\nL1(0,T;H−1/2(Γ)),\nwe obtain\na\n4/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω))+b\n2/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))+c2ε\n2(q+1)/ba∇dbl∇u/ba∇dblq+1\nL∞(0,T;Lq+1(Ω))\n+c2\n4/ba∇dbl∇u/ba∇dbl2\nL∞(0,T;L2(Ω))+α\n2/ba∇dblut/ba∇dbl2\nL2(0,T;L2(ˆΓ))\n≤ˆb(CΩ\nH1,L4)2/ba∇dblut/ba∇dbl2\nL2(0,T;H1(Ω))+τ(Ctr\n2)2(/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω))+/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω)))24 V. NIKOLI ´C\n+1\n4τ(/ba∇dblg/ba∇dbl2\nL2(0,T;H−1/2(Γ))+/ba∇dblg/ba∇dbl2\nL1(0,T;H−1/2(Γ)))+a\n2|u1|2\nL2(Ω)+c2\n2|∇u0|2\nL2(Ω)\n+c2ε\nq+1|∇u0|q+1\nLq+1(Ω),\nwhich leads to (4.4). /square\nIf we consider an equation with an added linear lower order damping te rm\n\n\nautt−c2div(∇u+ε|∇u|q−1∇u)−b∆ut+βut+fut= 0 in Ω ×(0,T],\nc2∂u\n∂n+c2ε|∇u|q−1∂u\n∂n+b∂ut\n∂n=gonΓ×(0,T],\nαut+c2∂u\n∂n+c2ε|∇u|q−1∂u\n∂n+b∂ut\n∂n= 0 onˆΓ×(0,T],\n(u,ut) = (u0,u1) onΩ×{t= 0},(4.7)\nwe will be able to obtain an energy estimate valid for arbitrary time:\nProposition 4.2. Letβ >0and the assumptions in Proposition 4.1hold with\n/ba∇dblf−1\n2at/ba∇dblL∞(0,T;L2(Ω))≤ˆb<1\n2(CΩ\nH1,L4)2min{b,β}.\nThen(4.7)has a weak solution in ˜X, defined as in (4.3), which satisfies the energy\nestimate\na\n4/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω))+/bracketleftBigb\n2−(Ctr\n2)2τ−ˆb(CΩ\nH1,L4)2/bracketrightBig\n/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))\n+/bracketleftBigβ\n2−(Ctr\n2)2τ−ˆb(CΩ\nH1,L4)2/bracketrightBig\n/ba∇dblut/ba∇dbl2\nL2(0,T;L2(Ω))+c2\n4/ba∇dbl∇u/ba∇dbl2\nL∞(0,T;L2(Ω))\n+c2ε\n2(q+1)/ba∇dbl∇u/ba∇dblq+1\nL∞(0,T;Lq+1(Ω))+α\n2/ba∇dblut/ba∇dbl2\nL2(0,T;L2(ˆΓ))\n≤1\n4τ/ba∇dblg/ba∇dbl2\nL2(0,T;H−1/2(Γ))+a\n2|u1|2\nL2(Ω)+c2\n2|∇u0|2\nL2(Ω)+c2ε\nq+1|∇u0|q+1\nLq+1(Ω),(4.8)\nfor some constant\n0<τ <min/braceleftBigb−2ˆb(CΩ\nH1,L4)2\n2(Ctr\n2)2,β−2(CΩ\nH1,L4)2ˆb\n2(Ctr\n2)2/bracerightBig\n.\nWe now proceed to the question of local existence of weak solutions for the problem\n(4.1).\nTheorem 4.3. Letc2,b >0,α≥0,δ∈(0,1),k∈R,q > d−1,q≥1,\ng∈L2(0,T;H−1/2(Γ)). For anyT >0there is aκT>0such that for all u0∈\nW1,q+1(Ω),u1∈L2(Ω)with\n/ba∇dblg/ba∇dbl2\nL2(0,T;H−1/2(Γ))+/ba∇dblg/ba∇dbl2\nL1(0,T;H−1/2(Γ))+|u1|2\nL2(Ω)+|∇u0|2\nL2(Ω)+|∇u0|q+1\nLq+1(Ω)\n+|u0|2\nL1(Ω)≤κ2\nT\nthere exists a weak solution u∈ Wof(4.1)where\nW={v∈˜X:/ba∇dblvt/ba∇dblL∞(0,T;L2(Ω))≤m\n∧/ba∇dbl∇vt/ba∇dblL2(0,T;L2(Ω))≤m\n∧/ba∇dbl∇v/ba∇dblL∞(0,T;Lq+1(Ω))≤M},(4.9)\nwith\n2|k|CΩ\nW1,q+1,L∞[CΩ\n1κT+(1+CP)M+CΩ\n2Tm]<1, (4.10)LOCAL EXISTENCE RESULTS 25\nandmandMare sufficiently small.\nProof.We define an operator T:W →˜X,v/ma√sto→ Tv=u, whereusolves (4.2) with\na= 1−2kv, f=−2kvt. (4.11)\nProposition 4.1 will allow us to prove that Tis a self-mapping. The assumptions\nof the proposition are satisfied, since for v∈ Wbecause of (1.16) we have\n0<a= 1−a0<a(x,t)<a= 1+a0,\nwhere\na0= 2|k|CΩ\nW1,q+1,L∞[CΩ\n1κT+(1+CP)M+CΩ\n2Tm],\nand by (4.9) /ba∇dblf−1\n2at/ba∇dblL∞(0,T;L2(Ω))=/ba∇dblkvt/ba∇dblL∞(0,T;L2(Ω))≤ |k|m. The energy\nestimate (4.4) holds and we can conclude that for any m,M >0 such that\n2|k|CΩ\nW1,q+1,L∞((1+CP)M+CΩ\n2Tm)<1,\nm≤1\n|k|min{a\n4(CΩ\nH1,L4)2,b\n2(CΩ\nH1,L4)2},\nand under the assumption on smallness of initial and boundary data\nκT<1\nCΩ\n1/parenleftBig1\n2|k|CΩ\nW1,q+1,L∞−(1+CP)M−CΩ\n2Tm/parenrightBig\n,\nκ2\nT≤min{1\nC(a\n4−Tˆb(CΩ\nH1,L4)2−(Ctr\n2)2τ)m2,\n1\nC(b\n2−ˆb(CΩ\nH1,L4)2−(Ctr\n2)2τ)m2,1\nCc2ε\n2(q+1)Mq+1},\nwhereC= max{1\n4τ,a\n2,c2\n2,c2ε\nq+1}, operator Tmaps into W.\nSinceWisclosedandboundedinthedualofaseparableBanachspace, Wisweakly-\nstar compact. Existence of solutions then results from a compact ness argument (see\nTheorem 6.1, [2]): the sequence of fixed point iterates undefined byun=Tun−1,\n(1−2kun−1)un\ntt−c2div(∇un+ǫ|∇un|q−1∇un)−b∆un\nt+βun\nt= 2kun−1\ntun\nt,\nwithu0chosen to be compatible with initial and boundary conditions, hasa we akly-\nstarconvergentsubsequencewhosew ∗-limit ¯uliesinW. Thislimitisaweaksolution\nof the problem since\n/integraldisplayT\n0/integraldisplay\nΩ/braceleftbig\n(1−2k¯u)¯uttφ+[c2(∇¯u+ε|∇¯u|q−1∇¯u)+b∇¯ut]·∇φ−2k(¯ut)2φ\n+βutφ/bracerightbig\ndxds+α/integraldisplayT\n0/integraldisplay\nˆΓ¯utφdxds−/integraldisplayT\n0/integraldisplay\nΓgφdxds\n=/integraldisplayT\n0/integraldisplay\nΩ/braceleftbig\n−¯ut((1−2k¯u)φ)t+[c2(∇¯u+ε|∇¯u|q−1∇¯u)+b∇¯ut]·∇φ−2k(¯ut)2φ\n+βutφ/bracerightbig\ndxds+α/integraldisplayT\n0/integraldisplay\nˆΓ¯utφdxds−/integraldisplayT\n0/integraldisplay\nΓgφdxds\n=/integraldisplayT\n0/integraldisplay\nΩ/braceleftBig\n−(¯u−un)t((1−2k¯u)φ)t+2kun\nt((¯u−un−1)φ)t\n−2k(¯ut−un\nt)¯utφ−2k(¯ut−un−1\nt)un\ntφ+β(¯ut−un\nt)φ+[c2∇(¯u−un)26 V. NIKOLI ´C\n+b∇(¯u−un)t]·∇φ+c2ε/integraldisplay1\n0|∇(un+σˆu)|q−3[|∇(un+σˆu)|2∇ˆu\n+(q−1)(∇(un+σˆu)·∇ˆu)∇(un+σˆu)]dσ·∇φ/bracerightBig\ndxds\n+α/integraldisplayT\n0/integraldisplay\nˆΓ(¯u−un)φdxds→0 ask→ ∞,\nfor anyφ∈C∞\n0((0,T)×Ω), where ˆu= ¯u−un. /square\nRelying on Proposition 4.2, we can also achieve short-time existence o f solutions\nfor the problem (1.8), with β >0. Due to the estimate (1.16) and therefore bound\n(4.10), the dependency on final time Tcannot be completely avoided.\nTheorem 4.4. Let the assumptions of Theorem 4.3hold andβ >0. For anyT >0\nthere is aκT>0such that for all u0∈W1,q+1(Ω),u1∈L2(Ω)with\n/ba∇dblg/ba∇dbl2\nL2(0,T;H−1/2(Γ))+|u1|2\nL2(Ω)+|∇u0|2\nL2(Ω)+|∇u0|q+1\nLq+1(Ω)+|u0|2\nL1(Ω)≤κ2\nT,\nthere exists a weak solution u∈ Wof(1.8), whereWis defined as in (4.9), with\n(4.10), andmandMare sufficiently small.\nNote that here, as in the case of homogeneous Dirichlet boundary c onditions, the\nuniqueness remains an open problem due to the presence of q−Laplace damping\nterm which hinders the derivation of higher order energy estimates . For details, the\nreader is refered to Remark 8, [2].\n5.Westervelt’s equation in the formulation (1.9)\nWe begin with investigations of the problem (1.9) in the case γ= 0:\n\n\nutt−c2\n1−2˜kut∆u−bdiv/parenleftBig\n((1−δ)+δ|∇ut|q−1)∇ut/parenrightBig\n= 0 in Ω ×(0,T],\nc2\n1−2˜kut∂u\n∂n+b((1−δ)+δ|∇ut|q−1)∂ut\n∂n=gon Γ×(0,T],\nαut+c2\n1−2˜kut∂u\n∂n+b((1−δ)+δ|∇ut|q−1)∂ut\n∂n= 0 onˆΓ×(0,T],\n(u,ut) = (u0,u1) onΩ×{t= 0}.(5.1)\nWe will once again first consider an equation with the nonlinearity only a ppearing\nthrough the damping term:\n\n\nutt−a∆u−bdiv/parenleftBig\n((1−δ)+δ|∇ut|q−1)∇ut/parenrightBig\n= 0 in Ω ×(0,T],\na∂u\n∂n+b((1−δ)+δ|∇ut|q−1)∂ut\n∂n=gonΓ×(0,T],\nαut+a∂u\n∂n+b((1−δ)+δ|∇ut|q−1)∂ut\n∂n= 0 onˆΓ×(0,T],\n(u,ut) = (u0,u1) onΩ×{t= 0}.(5.2)\nProposition 5.1. LetT >0,b>0,α≥0,δ∈(0,1)and assume that\n•a∈L2(0,T;L∞(Ω)),∇a∈L2(0,T;L2(Ω)),\n•g∈Lq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ)),\n•u0∈H1(Ω),u1∈L2(Ω),\n•q>d−1,q≥1,LOCAL EXISTENCE RESULTS 27\n/braceleftBiggˆb:=b(1−δ)\n2−T\n2/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))−(√\nT+1\n2)/ba∇dbla/ba∇dblL2(0,T;L∞(Ω))>0,\n˜b:=1\n4−2T(CΩ\nW1,q+1,L∞CΩ\n2)2/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))>0,forq>1,(5.3)\n\n\nˆb:=b\n2−(T\n2+(CΩ\nH1,L∞)2)/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))\n−(√\nT+1\n2)/ba∇dbla/ba∇dblL2(0,T;L∞(Ω))>0,\n˜b:=1\n4−T(CΩ\nH1,L∞)2/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))>0,forq= 1.(5.4)\nThen(5.2)has a weak solution\nu∈˜X:=C1(0,T;L2(Ω))∩W1,q+1(0,T;W1,q+1(Ω))}, (5.5)\nwhich, forq>1, satisfies the energy estimate\nˆb/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))+/parenleftBig\n˜b−ǫ0/parenrightBig\n/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω)\n+(bδ\n2−ǫ1)/ba∇dbl∇ut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))+α\n2/ba∇dblut/ba∇dbl2\nL2(0,T;L2(ˆΓ))\n≤/parenleftBig\n/ba∇dbla/ba∇dblL2(0,T;L∞(Ω))+/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))/parenrightBig1\n2|∇u0|2\nL2(Ω)+1\n2|u1|2\nL2(Ω)\n+C(ǫ1\n2,q+1\n2)T/parenleftBig\n(CΩ\nW1,q+1,L∞(1+CP))2/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))/parenrightBigq+1\nq−1\n+C(ǫ1\n2,q+1)(CΩ\nW1,q+1,L∞(1+CP))q+1\nq/ba∇dblg/ba∇dblq+1\nq\nLq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ))\n+1\n4ǫ0(Ctr\n1CΩ\n2)2/ba∇dblg/ba∇dbl2\nL1(0,T;W−q\nq+1,q+1\nq(Γ)),(5.6)\nand forq= 1satisfies\n/parenleftBig\nˆb−ǫ0/parenrightBig\n/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))+/parenleftBig\n˜b−ǫ0/parenrightBig\n/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω)+α\n2/ba∇dblut/ba∇dbl2\nL2(0,T;L2(ˆΓ))\n≤/parenleftBig\n/ba∇dbla/ba∇dblL2(0,T;L∞(Ω))+/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))/parenrightBig1\n2|∇u0|2\nL2(Ω)+1\n2|u1|2\nL2(Ω)\n+1\n4ǫ0(Ctr\n2)2(/ba∇dblg/ba∇dbl2\nL1(0,T;H−1/2(Γ))+/ba∇dblg/ba∇dbl2\nL2(0,T;H−1/2(Γ))),\nfor some sufficiently small constants ǫ0,ǫ1>0.\nProof.We will focus on acquiring crucial energy estimates. Testing the pro blem\nwithutand integrating with respect to space and time leads to\n1\n2/bracketleftBig\n|ut(s)|2\nL2(Ω)/bracketrightBigt\n0+/integraldisplayt\n0/parenleftBig\nb(1−δ)|∇ut(s)|2\nL2(Ω)\n+bδ|∇ut(s)|q+1\nLq+1(Ω)+α|ut(s)|2\nL2(ˆΓ)/parenrightBig\nds\n=/integraldisplayt\n0/integraldisplay\nΩ/parenleftBig\n−a∇ut·∇u−ut∇a·∇u/parenrightBig\ndxds+/integraldisplayt\n0/integraldisplay\nΓgutdxds\n≤/integraldisplayt\n0/parenleftBig\n|a(s)|L∞(Ω)|∇ut(s)|L2(Ω)+|∇a(s)|L2(Ω)|ut(s)|L∞(Ω)/parenrightBig\n·/bracketleftBig\n|∇u0|L2(Ω)+/radicalBigg\ns/integraldisplays\n0|∇ut(σ)|2\nL2(Ω)dσ/bracketrightBig\nds+/integraldisplayt\n0/integraldisplay\nΓgutdxds\n≤/parenleftBig\n/ba∇dbla/ba∇dblL2(0,t;L∞(Ω))/ba∇dbl∇ut/ba∇dblL2(0,t;L2(Ω))28 V. NIKOLI ´C\n+/ba∇dbl∇a/ba∇dblL2(0,t;L2(Ω))/radicalBigg/integraldisplayt\n0|ut(s)|2\nL∞(Ω)ds/parenrightBig\n·/bracketleftBig\n|∇u0|L2(Ω)+√\nt/ba∇dbl∇ut/ba∇dblL2(0,t;L2(Ω))/bracketrightBig\n+/integraldisplayt\n0/integraldisplay\nΓgutdxds (5.7)\n≤ /ba∇dbla/ba∇dblL2(0,T;L∞(Ω))/parenleftBig1\n2/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))+1\n2|∇u0|2\nL2(Ω)\n+√\nT/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))/parenrightBig\n+/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))/integraldisplayT\n0|ut(s)|2\nL∞(Ω)ds\n+/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))/parenleftBig1\n2|∇u0|2\nL2(Ω)+1\n2T/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))/parenrightBig\n+ǫ1\n2/ba∇dbl∇ut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))+ǫ0/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω))\n+C(ǫ1\n2,q+1)(Ctr\n1(1+CP))q+1\nq/ba∇dblg/ba∇dblq+1\nq\nLq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ))\n+1\n4ǫ0(Ctr\n1CΩ\n2)2/ba∇dblg/ba∇dbl2\nL1(0,T;W−q\nq+1,q+1\nq(Γ)),\nwhere we have applied (2.14) to estimate the boundary integral on t he right side.\nWe can make use of the embedding W1,q+1(Ω)֒→L∞(Ω) together with the in-\nequality (1.13) to obtain\n/integraldisplayT\n0|ut(s)|2\nL∞(Ω)ds≤(CΩ\nW1,q+1,L∞)2/integraldisplayT\n0|ut(s)|2\nW1,q+1(Ω)ds\n≤(CΩ\nW1,q+1,L∞)2/integraldisplayt\n0/parenleftBig\n(1+CP)|∇ut(s)|Lq+1(Ω)+CΩ\n2|ut(s)|L2(Ω)/parenrightBig2\nds\n≤2(CΩ\nW1,q+1,L∞)2(1+CP)2/integraldisplayt\n0|∇ut(s)|2\nLq+1(Ω)ds\n+2(CΩ\nW1,q+1,L∞CΩ\n2)2/ba∇dblut/ba∇dbl2\nL2(0,T;L2(Ω)),\nand then from (5.7), for q>1, we further get\n1\n2/bracketleftBig\n|ut|2\nL2(Ω)/bracketrightBigt\n0+/integraldisplayt\n0/parenleftBig\nb(1−δ)|∇ut(s)|2\nL2(Ω)+bδ|∇ut(s)|q+1\nLq+1(Ω)ds/parenrightBig\n+α/integraldisplayt\n0|ut(s)|2\nL2(ˆΓ)ds\n≤ /ba∇dbla/ba∇dblL2(0,T;L∞(Ω))/parenleftBig\n(√\nT+1\n2)/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))+1\n2|∇u0|2\nL2(Ω)/parenrightBig\n+ǫ1/ba∇dbl∇ut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))+ǫ0/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω))\n+C(ǫ1\n2,q+1\n2)T((CΩ\nW1,q+1,L∞(1+CP))2/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω)))q+1\nq−1 (5.8)\n+2/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))(CΩ\nW1,q+1,L∞CΩ\n2)2/ba∇dblut/ba∇dbl2\nL2(0,T;L2(Ω))\n+/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))/parenleftBig1\n2T/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))+1\n2|∇u0|2\nL2(Ω)/parenrightBig\n+C(ǫ1\n2,q+1)(Ctr\n1(1+CP))q+1\nq/ba∇dblg/ba∇dblq+1\nq\nLq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ))\n+1\n4ǫ0(Ctr\n1CΩ\n2)2/ba∇dblg/ba∇dbl2\nL1(0,T;W−q\nq+1,q+1\nq(Γ)),LOCAL EXISTENCE RESULTS 29\nfor someǫ0,ǫ1>0. By taking esssup\n[0,T]in (5.8) and making ǫ0andǫ1small enough\nwe gain (5.6). /square\nProposition 5.2. LetT >0,b>0,α≥0,δ∈(0,1)and assume that\n•a(t,x)≥a>0,\n•a∈L∞(0,T;L∞(Ω)),at∈L2(0,T;L2(Ω)),∇a∈L2(0,T;L4(Ω)),\n•g∈L∞(0,T;W−q\nq+1,q+1\nq(Γ)),gt∈Lq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ)),\n•u0∈W1,4(Ω),u1∈W1,q+1(Ω),\n•q≥3,\nwith\n˜a:=a\n4−1\n2/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))>0,\n˜b:=1\n4−/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))T(CΩ\nLq+1,L4CΩ\n2)2>0,(5.9)\nthen(5.2)has a weak solution\nu∈X:=C1(0,T;W1,q+1(Ω))∩H2(0,T;L2(Ω))}, (5.10)\nwhich satisfies the energy estimate\nµ/bracketleftBig1\n2−τ/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))/bracketrightBig\n/ba∇dblutt/ba∇dbl2\nL2(0,T;L2(Ω))\n+µb(1−δ)\n8/ba∇dbl∇ut/ba∇dbl2\nL∞(0,T;L2(Ω))+/bracketleftBig\n˜b−ǫ0(µ+1)/bracketrightBig\n/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω)\n+/bracketleftBigb(1−δ)\n2−µ/ba∇dbla/ba∇dblL∞(0,T;L∞(Ω))/bracketrightBig\n/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))\n+/bracketleftBig\nµbδ\n2(q+1)−η(2µ+1)/bracketrightBig\n/ba∇dbl∇ut/ba∇dblq+1\nL∞(0,T;Lq+1(Ω))+µα\n4/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(ˆΓ))\n+/bracketleftBig\n˜a−µ2\nb(1−δ)/ba∇dbla/ba∇dbl2\nL∞(0,T;L∞(Ω))/bracketrightBig\n/ba∇dbl∇u/ba∇dbl2\nL∞(0,T;L2(Ω))\n+/bracketleftBigbδ\n2−ǫ1(µ+1)/bracketrightBig\n/ba∇dbl∇ut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))+α\n2/ba∇dblut/ba∇dbl2\nL2(0,T;L2(ˆΓ))(5.11)\n≤C/parenleftBig\n(T/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω)))q+1\nq−1+/ba∇dbla/ba∇dblL∞(0,T;L∞(Ω))|∇u0|2\nL2(Ω)\n+(/ba∇dblat/ba∇dblL4/3(0,T;L2(Ω))+/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω)))|∇u0|2\nL4(Ω)\n+/ba∇dbla/ba∇dblL∞(0,T;L∞(Ω))/parenleftBig\n|∇u0|2\nL2(Ω)+|∇u1|L2(Ω)|∇u0|L2(Ω)/parenrightBig\n+((1\n2+T3/4)√\nT/ba∇dblat/ba∇dblL4/3(0,T;L2(Ω)))q+1\nq−1+(T2/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω)))q+1\nq−1\n+(T5/2/ba∇dblat/ba∇dblL2(0,T;L2(Ω)))q+1\nq−1+/ba∇dblat/ba∇dblL2(0,T;L2(Ω))√\nT|∇u0|2\nL4(Ω)\n+|u1|2\nH1(Ω)+|u1|q+1\nW1,q+1(Ω)+|u1|2\nL2(ˆΓ)+CΓ(g)/parenrightBig\n,\nfor some sufficiently small constants ǫ0,ǫ1,η,µ,τ > 0, some large enough C >0\nandCΓ(g)defined as in (2.9).\nProof.In order to obtain higher order estimate, we will multiply (5.2) first by ut,\nproceeding differently than in Proposition 5.1, and then by uttand combine the two\nobtained estimates. Multiplication by utand integration with respect to space and30 V. NIKOLI ´C\ntime produces\n1\n2/bracketleftBig\n|ut(s)|2\nL2(Ω)+|√a∇u|2\nL2(Ω)/bracketrightBigt\n0+/integraldisplayt\n0/parenleftBig\nb(1−δ)|∇ut(s)|2\nL2(Ω)\n+bδ|∇ut(s)|q+1\nLq+1(Ω)/parenrightBig\nds+α/integraldisplayt\n0|ut(s)|2\nL2(ˆΓ)ds\n=/integraldisplayt\n0/integraldisplay\nΩ/parenleftBig1\n2at|∇u|2−ut∇a·∇u/parenrightBig\ndxds+/integraldisplayt\n0/integraldisplay\nΓgutdxds\n≤/integraldisplayt\n0/integraldisplay\nΩ1\n2at|∇u|2dxds+/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))/parenleftBigT\n2/ba∇dblut/ba∇dbl2\nL∞(0,T;L4(Ω)) (5.12)\n+1\n2/ba∇dbl∇u/ba∇dbl2\nL∞(0,T;L2(Ω))/parenrightBig\n+ǫ1/ba∇dbl∇ut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))\n+C(ǫ1,q+1)(Ctr\n1(1+CP))q+1\nq/ba∇dblg/ba∇dblq+1\nq\nLq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ))\n+ǫ0/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω)+1\n4ǫ0(Ctr\n1CΩ\n2)2/ba∇dblg/ba∇dbl2\nL1(0,T;W−q\nq+1,q+1\nq(Γ)),\nfor someǫ0,ǫ1>0. We will make use of the embedding Lq+1(Ω)֒→L4(Ω) and\nYoung’s inequality (1.18) to estimate\n1\n2/integraldisplayt\n0/integraldisplay\nΩat|∇u|2dxds\n≤1\n2/ba∇dbl∇u/ba∇dbl2\nL∞(0,T;L4(Ω))/integraldisplayt\n0|at|L2(Ω)ds\n≤1\n2/ba∇dblat/ba∇dblL2(0,T;L2(Ω))√\nT/ba∇dbl∇u/ba∇dbl2\nL∞(0,T;L4(Ω))\n≤ /ba∇dblat/ba∇dblL2(0,T;L2(Ω))√\nT/bracketleftBig\nT2/ba∇dbl∇ut/ba∇dbl2\nL∞(0,T;L4(Ω))+|∇u0|2\nL4(Ω)/bracketrightBig\n≤ /ba∇dblat/ba∇dblL2(0,T;L2(Ω))√\nT/bracketleftBig\nT2(CΩ\nLq+1,L4)2/ba∇dbl∇ut/ba∇dbl2\nL∞(0,T;Lq+1(Ω))+|∇u0|2\nL4(Ω)/bracketrightBig\n≤η\n2/ba∇dbl∇ut/ba∇dblq+1\nL∞(0,T;Lq+1(Ω))+/ba∇dblat/ba∇dblL2(0,T;L2(Ω))√\nT|∇u0|2\nL4(Ω)\n+C(η\n2,q+1\n2)(/ba∇dblat/ba∇dblL2(0,T;L2(Ω))T5/2(CΩ\nLq+1,L4)2)q+1\nq−1,(5.13)\nfor someη>0 andq>1. We can also obtain\n/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))T\n2/ba∇dblut/ba∇dbl2\nL∞(0,T;L4(Ω))\n≤ /ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))T\n2(CΩ\nLq+1,L4)2/ba∇dblut/ba∇dbl2\nL∞(0,T;Lq+1(Ω))\n≤ /ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))T(CΩ\nLq+1,L4)2/bracketleftBig\nC2\nP/ba∇dbl∇ut/ba∇dbl2\nL∞(0,T;Lq+1(Ω))\n+(CΩ\n2)2/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω))/bracketrightBig\n≤η\n2/ba∇dbl∇ut/ba∇dblq+1\nL∞(0,T;Lq+1(Ω))+C(η\n2,q+1\n2)((CPCΩ\nLq+1,L4)2/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))T)q+1\nq−1\n+/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))T(CΩ\nLq+1,L4CΩ\n2)2/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω)),\nwhich together with (5.13) results in the following estimate:\n(˜b−ǫ0)/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω))+˜a/ba∇dbl∇u/ba∇dbl2\nL∞(0,T;L2(Ω))+α\n2/ba∇dblut/ba∇dbl2\nL2(0,T;L2(ˆΓ))LOCAL EXISTENCE RESULTS 31\n+b(1−δ)\n2/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))+(bδ\n2−ǫ1)/ba∇dbl∇ut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))\n≤η/ba∇dbl∇ut/ba∇dblq+1\nL∞(0,T;Lq+1(Ω))+/ba∇dblat/ba∇dblL2(0,T;L2(Ω))√\nT|∇u0|2\nL4(Ω)\n+C(η\n2,q+1\n2)((CPCΩ\nLq+1,L4)2/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))T)q+1\nq−1 (5.14)\n+C(η\n2,q+1\n2)(/ba∇dblat/ba∇dblL2(0,T;L2(Ω))T5/2(CΩ\nLq+1,L4)2)q+1\nq−1\n+1\n2/ba∇dbla/ba∇dblL∞(0,T;L∞(Ω))|∇u0|2\nL2(Ω)+1\n4ǫ0/ba∇dblg/ba∇dbl2\nL1(0,T;W−q\nq+1,q+1\nq(Γ))\n+1\n2|u1|2\nL2(Ω)+C(ǫ1,q+1)(Ctr\n1(1+CP)/ba∇dblg/ba∇dbl\nLq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ)))q+1\nq.\nTesting with uttyields\n/integraldisplayt\n0|utt(s)|2\nL2(Ω)ds+/bracketleftbiggb(1−δ)\n2|∇ut|2\nL2(Ω)+bδ\nq+1|∇ut|q+1\nLq+1(Ω)+α\n2|ut|2\nL2(ˆΓ)/bracketrightbiggt\n0\n=/integraldisplayt\n0/integraldisplay\nΩ(−a∇utt·∇u−utt∇a·∇u)dxds+/integraldisplayt\n0/integraldisplay\nΓguttdxds\n=/integraldisplayt\n0/integraldisplay\nΩ/parenleftbig\nat∇ut·∇u+a|∇ut|2−utt∇a·∇u/parenrightbig\ndxds−/bracketleftbigg/integraldisplay\nΩa∇ut·∇udx/bracketrightbiggt\n0\n+/integraldisplayt\n0/integraldisplay\nΓguttdxds\n≤/integraldisplayt\n0/parenleftbig\n|at(s)|L2(Ω)|∇ut(s)|L4(Ω)+|∇a(s)|L4(Ω)|utt(s)|L2(Ω)/parenrightbig\n·/bracketleftBig\n|∇u0|L4(Ω)+4/radicalBigg\ns3/integraldisplays\n0|∇ut(σ)|4\nL4(Ω)dσ/bracketrightBig\nds\n+/ba∇dbla/ba∇dblL∞(0,T;L∞(Ω))/parenleftBig\n|∇ut(t)|L2(Ω)|∇u(t)|L2(Ω)+|∇u1|L2(Ω)|∇u0|L2(Ω)\n+/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))/parenrightBig\n+/integraldisplayt\n0/integraldisplay\nΓguttdxds\n≤ /ba∇dblat/ba∇dblL4/3(0,T;L2(Ω))/parenleftbigg\n(1\n2+T3\n4)/ba∇dbl∇ut/ba∇dbl2\nL4(0,T;L4(Ω))+1\n2|∇u0|2\nL4(Ω)/parenrightbigg\n+/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))(τ/ba∇dblutt/ba∇dbl2\nL2(0,T;L2(Ω))+1\n2τ|∇u0|2\nL4(Ω)\n+1\n2τT3\n2/ba∇dbl∇ut/ba∇dbl2\nL4(0,T;L4(Ω)))+/ba∇dbla/ba∇dblL∞(0,T;L∞(Ω))/parenleftBig\n|∇u1|L2(Ω)|∇u0|L2(Ω)\n+/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))/parenrightBig\n+2\nb(1−δ)/ba∇dbla/ba∇dbl2\nL∞(0,T;L∞(Ω))|∇u(t)|2\nL2(Ω)\n+b(1−δ)\n8|∇ut(t)|2\nL2(Ω)+/integraldisplayt\n0/integraldisplay\nΓguttdxds,\nfor someτ >0. We can make use of Young’s inequality and the inequality\n(2.33) for the boundary integral together with the inequality /ba∇dbl∇ut/ba∇dblL4(0,T;L4(Ω))≤\nT1\n4/ba∇dbl∇ut/ba∇dblL∞(0,T;L4(Ω))≤T1\n4CΩ\nLq+1,L4/ba∇dbl∇ut/ba∇dblL∞(0,T;Lq+1(Ω))to obtain\n/integraldisplayt\n0|utt(s)|2\nL2(Ω)ds+/bracketleftbiggb(1−δ)\n2|∇ut|2\nL2(Ω)+bδ\nq+1|∇ut|q+1\nLq+1(Ω)+α\n2|ut|2\nL2(ˆΓ)/bracketrightbiggt\n032 V. NIKOLI ´C\n≤ /ba∇dblat/ba∇dblL4/3(0,T;L2(Ω))1\n2|∇u0|2\nL4(Ω)+η\n2/ba∇dbl∇ut/ba∇dblq+1\nL∞(0,T;Lq+1(Ω))\n+C(η\n2,q+1\n2)((1\n2+T3\n4)√\nT(CΩ\nLq+1,L4)2/ba∇dblat/ba∇dblL4/3(0,T;L2(Ω)))q+1\nq−1\n+/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))/parenleftBig\nτ/ba∇dblutt/ba∇dbl2\nL2(0,T;L2(Ω))+1\n2ǫ|∇u0|2\nL4(Ω)/parenrightBig\n+η\n2/ba∇dbl∇ut/ba∇dblq+1\nL∞(0,T;Lq+1(Ω))+C(η\n2,q+1\n2)(1\n2τT2(CΩ\nLq+1,L4)2/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω)))q+1\nq−1\n+/ba∇dbla/ba∇dblL∞(0,T;L∞(Ω))/parenleftBig\n|∇u1|L2(Ω)|∇u0|L2(Ω)+/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))/parenrightBig\n+2\nb(1−δ)/ba∇dbla/ba∇dbl2\nL∞(0,T;L∞(Ω))|∇u(t)|2\nL2(Ω)+b(1−δ)\n8|∇ut(t)|2\nL2(Ω)\n+η/ba∇dbl∇ut/ba∇dblq+1\nL∞(0,T;Lq+1(Ω))+C(η,q+1)(Ctr\n1(1+CP))q+1\nq/ba∇dblg/ba∇dblq+1\nq\nL∞(0,T;W−q\nq+1,q+1\nq(Γ))\n+ǫ0/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω))+1\n2ǫ0(Ctr\n1CΩ\n2)2/ba∇dblg/ba∇dbl2\nL∞(0,T;W−q\nq+1,q+1\nq(Γ))\n+|u1|q+1\nW1,q+1(Ω)+C(1,q+1)(Ctr\n1|g(0)|\nW−q\nq+1,q+1\nq(Γ)))q+1\nq\n+C(ǫ1,q+1)(Ctr\n1(1+CP))q+1\nq/ba∇dblgt/ba∇dblq+1\nq\nLq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ))\n+ǫ1/ba∇dbl∇ut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))+1\n2ǫ0(Ctr\n1CΩ\n2)2/ba∇dblgt/ba∇dbl2\nL1(0,T;W−q\nq+1,q+1\nq(Γ)),\nwhich, by taking essential supremum with respect to tand then adding µtimes\nobtained inequality to (5.14) results in the higher order estimate (5.1 1). /square\nLet us now consider the problem with the added lower order nonlinear damping\nterm\n\n\nutt−a∆u−bdiv/parenleftBig\n((1−δ)+δ|∇ut|q−1)∇ut/parenrightBig\n+γ|ut|q−1ut= 0 in Ω ×(0,T],\na∂u\n∂n+b((1−δ)+δ|∇ut|q−1)∂ut\n∂n=gonΓ×(0,T],\nαut+a∂u\n∂n+b((1−δ)+δ|∇ut|q−1)∂ut\n∂n= 0 onˆΓ×(0,T],\n(u,ut) = (u0,u1) onΩ×{t= 0},(5.15)\nwhereγ >0. This is a linearized version of (1.9), where nonlinearity appears\nonly through the damping terms. We can utilize the embedding W1,q+1(Ω)֒→\nL∞(Ω), Young’s inequality in the form (1.18) and estimate the boundary integral\nby employing (2.45), to obtain\n1\n2/bracketleftBig\n|ut(s)|2\nL2(Ω)/bracketrightBigt\n0+/integraldisplayt\n0/parenleftBig\nb(1−δ)|∇ut(s)|2\nL2(Ω)+bδ|∇ut(s)|q+1\nLq+1(Ω)\n+γ|ut(s)|q+1\nLq+1(Ω)+α|ut(s)|2\nL2(ˆΓ)/parenrightBig\nds\n≤ /ba∇dbla/ba∇dblL2(0,T;L∞(Ω))/parenleftBig√\nT/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))+1\n2/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω)) (5.16)\n+1\n2|∇u0|2\nL2(Ω)/parenrightBig\n+ǫ0/integraldisplayT\n0|ut(s)|q+1\nW1,q+1(Ω)dsLOCAL EXISTENCE RESULTS 33\n+C(ǫ0\n2,q+1\n2)T((CΩ\nW1,q+1,L∞)2/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω)))q+1\nq−1\n+/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))/parenleftBig1\n2T/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))+1\n2|∇u0|2\nL2(Ω)/parenrightBig\n+C(ǫ0\n2,q+1\n2)(Ctr\n1/ba∇dblg/ba∇dbl\nLq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ)))q+1\nq,\nfor someǫ0>0 andq>1,q>d−1. By taking esssup\n[0,T]in (5.16) we get\nˆb/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))+1\n4/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω))+α\n2/ba∇dblut/ba∇dbl2\nL2(0,T;L2(ˆΓ))\n+(bδ\n2−ǫ0)/ba∇dbl∇ut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))+(γ\n2−ǫ0)/ba∇dblut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))\n≤/parenleftBig\n/ba∇dbla/ba∇dblL2(0,T;L∞(Ω))+/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))/parenrightBig1\n2|∇u0|2\nL2(Ω)+1\n2|u1|2\nL2(Ω)\n+C(ǫ0\n2,q+1\n2)T((CΩ\nW1,q+1,L∞)2/ba∇dbl∇a/ba∇dblL∞(0,T;L2(Ω)))q+1\nq−1\n+C(ǫ0\n2,q+1\n2)(Ctr\n1/ba∇dblg/ba∇dbl\nLq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ)))q+1\nq,(5.17)\nfor some 0<ǫ0<1\n2min{bδ,γ}andˆb>0 defined as in (5.3).\nNote that the addition of the lower order damping term allows us to re move the\nsecond assumption in (5.3) on smallness of a.\nIn the case of q= 1 (andd= 1),usatisfies\n/parenleftBig\nˆb−ǫ0/parenrightBig\n/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))+1\n4/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω)\n+/parenleftBig\n˜b−ǫ0/parenrightBig\n/ba∇dblut/ba∇dbl2\nL2(0,T;L2(Ω))+α\n2/ba∇dblut/ba∇dbl2\nL2(0,T;L2(ˆΓ))\n≤/parenleftBig\n/ba∇dbla/ba∇dblL2(0,T;L∞(Ω))+/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))/parenrightBig1\n2|∇u0|2\nL2(Ω)+1\n2|u1|2\nL2(Ω)\n+1\n4ǫ0(Ctr\n2)2/ba∇dblg/ba∇dbl2\nL2(0,T;H−1/2(Γ)),(5.18)\nwhere˜b:=γ\n2−(CΩ\nH1,L∞)2/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))>0, andˆbis defined as in (5.4).\nTo obtain higher order estimate, we test the problem again by utand integrate with\nrespect to space and time to obtain\n1\n2/bracketleftBig\n|ut|2\nL2(Ω)+|√a∇u|2\nL2(Ω)/bracketrightBigt\n0+/integraldisplayt\n0/parenleftBig\nb(1−δ)|∇ut|2\nL2(Ω)+bδ|∇ut|q+1\nLq+1(Ω)\n+γ|ut|q+1\nLq+1(Ω)/parenrightBig\nds+α/integraldisplayt\n0/integraldisplay\nˆΓ|ut|2\nL2(ˆΓ)dxds\n=/integraldisplayt\n0/integraldisplay\nΩ/parenleftBig1\n2at|∇u|2−ut∇a·∇u/parenrightBig\ndxds+/integraldisplayt\n0/integraldisplay\nΓgutdxds (5.19)\n≤/integraldisplayt\n0/integraldisplay\nΩ1\n2at|∇u|2dxds+/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))/parenleftBigT\n2/ba∇dblut/ba∇dbl2\nL∞(0,T;L4(Ω))\n+1\n2/ba∇dbl∇u/ba∇dbl2\nL∞(0,T;L2(Ω))/parenrightBig\n+ǫ1/ba∇dblut/ba∇dblq+1\nLq+1(0,T;W1,q+1(Ω))ds\n+C(ǫ1,q+1\n2)(Ctr\n1/ba∇dblg/ba∇dbl\nLq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ)))q+1\nq.34 V. NIKOLI ´C\nTaking esssup\n[0,T]in (5.19) and making use of (5.13) and\n/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))T\n2/ba∇dblut/ba∇dbl2\nL∞(0,T;L4(Ω))\n≤η/ba∇dblut/ba∇dblq+1\nL∞(0,T;Lq+1(Ω))+C(η,q+1\n2)((CΩ\nLq+1,L4)2/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))T)q+1\nq−1,\nleads to the estimate\n1\n4/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω))+/parenleftBiga\n4−1\n2/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω)))/parenrightBig\n/ba∇dbl∇u/ba∇dbl2\nL∞(0,T;L2(Ω))\n+b(1−δ)\n2/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))+(bδ\n2−ǫ1)/ba∇dbl∇ut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))\n+(γ\n2−ǫ1)/ba∇dblut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))+α\n2/ba∇dblut/ba∇dbl2\nL2(0,T;L2(ˆΓ))\n≤η/ba∇dblut/ba∇dblq+1\nL∞(0,T;W1,q+1(Ω))+C(η,q+1\n2)(/ba∇dblat/ba∇dblL2(0,T;L2(Ω))T5/2(CΩ\nLq+1,L4)2)q+1\nq−1(5.20)\n+/ba∇dblat/ba∇dblL2(0,T;L2(Ω))√\nT|∇u0|2\nL4(Ω)+1\n2/ba∇dbla/ba∇dblL∞(0,T;L∞(Ω))|∇u0|2\nL2(Ω)\n+C(η,q+1\n2)(T(CΩ\nLq+1,L4)2/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω)))q+1\nq−1+1\n2|u1|2\nL2(Ω)\n+C(ǫ1\n2,q+1\n2)(Ctr\n1/ba∇dblg/ba∇dbl\nLq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ)))q+1\nq,\nfor someη>0.\nTesting with uttand proceeding as in the case of γ= 0, with the use of (2.47) for\nthe estimation of the boundary integral, results in the higher order energy estimate\nµ/parenleftBig1\n2−τ/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))/parenrightBig\n/ba∇dblutt/ba∇dbl2\nL2(0,T;L2(Ω))+1\n4/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω)\n+/parenleftBigbδ\n2−ǫ1(µ+1)/parenrightBig\n/ba∇dbl∇ut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))\n+/parenleftBigb(1−δ)\n2−µ/ba∇dbla/ba∇dblL∞(0,T;L∞(Ω))/parenrightBig\n/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))\n+/parenleftBig\nµbδ\n2(q+1)−µ(η+1)/parenrightBig\n/ba∇dbl∇ut/ba∇dblq+1\nL∞(0,T;Lq+1(Ω))+α\n2/ba∇dblut/ba∇dbl2\nL2(0,T;L2(ˆΓ))\n+/parenleftBig\n˜a−µ2\nb(1−δ)/ba∇dbla/ba∇dbl2\nL∞(0,T;L∞(Ω))/parenrightBig\n/ba∇dbl∇u/ba∇dbl2\nL∞(0,T;L2(Ω))\n+(γ\n2−µ(ǫ1+1))/ba∇dblut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω)+µb(1−δ)\n8/ba∇dbl∇ut/ba∇dbl2\nL∞(0,T;L2(Ω))(5.21)\n+/parenleftBig\nµγ\n2(q+1)−η(2µ+1)/parenrightBig\n/ba∇dblut/ba∇dblq+1\nL∞(0,T;Lq+1(Ω))+µα\n2/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(ˆΓ))\n≤C/parenleftBig\n(T/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω)))q+1\nq−1+/ba∇dbla/ba∇dblL∞(0,T;L∞(Ω))|∇u0|2\nL2(Ω)\n+(/ba∇dblat/ba∇dblL4/3(0,T;L2(Ω))+/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω)))|∇u0|2\nL4(Ω)\n+/ba∇dbla/ba∇dblL∞(0,T;L∞(Ω))/parenleftBig\n|∇u0|2\nL2(Ω)+|∇u1|L2(Ω)|∇u0|L2(Ω)/parenrightBig\n+((1\n2+T3/4)√\nT/ba∇dblat/ba∇dblL4/3(0,T;L2(Ω)))q+1\nq−1+|u1|2\nH1(Ω)+|u1|q+1\nW1,q+1(Ω)\n+(T2/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω)))q+1\nq−1+(T5/2/ba∇dblat/ba∇dblL2(0,T;L2(Ω)))q+1\nq−1+|u1|2\nL2(ˆΓ)\n+1/summationdisplay\ns=0/ba∇dblds\ndtsg/ba∇dblq+1\nq\nLq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ))+/ba∇dblg/ba∇dblq+1\nq\nL∞(0,T;W−q\nq+1,q+1\nq(Γ))/parenrightBigLOCAL EXISTENCE RESULTS 35\nwith ˜adefined in (5.9), for some sufficiently small constants ǫ1,η,µ,τ > 0 and some\nlarge enough C >0. Note that here the second assumption in (5.9) on smallness of\nawas not needed.\nProposition 5.3. LetT >0,b>0,α≥0,δ∈(0,1),γ >0and let the assump-\ntions in Proposition 5.1hold with\n\n\nb(1−δ)\n2−(√\nT+1\n2)/ba∇dbla/ba∇dblL2(0,T;L∞(Ω))−T\n2/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))>0,forq>1,\nˆb=b\n2−(T\n2+(CΩ\nH1,L∞)2)/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))−(√\nT+1\n2)/ba∇dbla/ba∇dblL2(0,T;L∞(Ω))>0,\n˜b=γ\n2−(CΩ\nH1,L∞)2/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))>0,forq= 1.\nThen(5.15)has a weak solution u∈˜X, with˜Xdefined as in (5.5), which satisfies\nthe energy estimate (5.17)forq>1and estimate (5.18)forq= 1.\nIf the assumptions in Proposition 5.2are satisfied with\n˜a=a\n4−1\n2/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))>0,\nthenu∈X, withXas in(5.10), and satisfies the energy estimate (5.21).\nWe will now proceed to investigate existence of solutions for (5.1).\nTheorem5.4. Letc2,b>0,α≥0,δ∈(0,1),˜k∈R,q≥3,g∈L∞(0,T;W−q\nq+1,q+1\nq(Γ)),\ngt∈Lq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ)). There exist κ >0,T >0such that for all\nu0∈W1,4(Ω),u1∈W1,q+1(Ω), with\nCΓ(g)+|∇u0|2\nL2(Ω)+|u1|2\nH1(Ω)+|u1|q+1\nW1,q+1(Ω)+|u1|2\nL2(ˆΓ)≤κ2\nthere exists a weak solution u∈ Wof(5.1)where\nW={v∈X:/ba∇dblvtt/ba∇dblL2(0,T;L2(Ω))≤m\n∧/ba∇dblvt/ba∇dblL∞(0,T;H1(Ω))≤m\n∧/ba∇dbl∇vt/ba∇dblL∞(0,T;Lq+1(Ω))≤M},(5.22)\nwithmandMsufficiently small, and CΓ(g)is defined as in (2.9).\nProof.We define an operator T:W →X,v/ma√sto→ Tv=uwhereusolves (5.2) with\na=c2\n1−2˜kvt. (5.23)\nFrom (1.14), we obtain for v∈ W\n/ba∇dbl2˜kvt/ba∇dblL∞(0,T;L∞(Ω))≤2|˜k|CΩ\nW1,q+1,L∞/bracketleftBig\n(1+CP)M+CΩ\n2m/bracketrightBig\n,\nand, assuming 2 |˜k|CΩ\nW1,q+1,L∞/bracketleftBig\n(1 +CP)M+CΩ\n2m/bracketrightBig\n<1, we can verify hypothesis\nof Proposition 5.2:\na(t,x)≥c2\n1+2|˜k|/ba∇dblvt/ba∇dblL∞(0,T;L∞(Ω))\n≥c2\n1+2|˜k|CΩ\nW1,q+1,L∞((1+CP)M+CΩ\n2m):=a,\n/ba∇dbla/ba∇dblL∞(0,T;L∞(Ω))≤c2\n1−2|˜k|/ba∇dblvt/ba∇dblL∞(0,T;L∞(Ω))36 V. NIKOLI ´C\n≤c2\n1−2|˜k|CΩ\nW1,q+1,L∞/bracketleftBig\n(1+CP)M+CΩ\n2m/bracketrightBig,\n/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))=/ba∇dbl2˜kc2\n(1−2˜kvt)2∇vt/ba∇dblL2(0,T;L4(Ω))\n≤2|˜k|c2\n(1−2|˜k|CΩ\nW1,q+1,L∞((1+CP)M+CΩ\n2m))2CΩ\nLq+1,L4√\nTM,\n/ba∇dblat/ba∇dblL4/3(0,T;L2(Ω))=/ba∇dbl2˜kc2\n(1−2˜kvt)2vtt/ba∇dblL4/3(0,T;L2(Ω))\n≤2|˜k|c2\n(1−2|˜k|CΩ\nW1,q+1,L∞((1+CP)M+CΩ\n2m))24√\nTm,\n/ba∇dblat/ba∇dblL2(0,T;L2(Ω))≤2|˜k|c2\n(1−2|˜k|CΩ\nW1,q+1,L∞((1+CP)M+CΩ\n2m))2m.\nIt follows that assumptions are satisfied provided m,M,κandTare sufficiently\nsmall such that\n2|˜k|CΩ\nW1,q+1,L∞/bracketleftBig\n(1+CP)M+CΩ\n2m/bracketrightBig\n<1,\n2|˜k|c2\n(1−2|˜k|CΩ\nW1,q+1,L∞((1+CP)M+CΩ\n2m))2CΩ\nLq+1,L4√\nTM\n≤c2\n2(1+2|˜k|CΩ\nW1,q+1,L∞((1+CP)M+CΩ\n2m)),and\n2|˜k|c2T3/2M(CΩ\nLq+1,L4)3(CΩ\n2)2\n(1−2|˜k|CΩ\nW1,q+1,L∞((1+CP)M+CΩ\n2m))2<1\n4.\nTherefore the energy estimate (5.11) is satisfied and we have\nµ/bracketleftBig1\n2−τ/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))/bracketrightBig\n/ba∇dblutt/ba∇dbl2\nL2(0,T;L2(Ω))+µb(1−δ)\n8/ba∇dbl∇ut/ba∇dbl2\nL∞(0,T;L2(Ω))\n+/bracketleftBigbδ\n2−ǫ1(µ+1)/bracketrightBig\n/ba∇dbl∇ut/ba∇dblq+1\nLq+1(0,T;Lq+1(Ω))+/bracketleftBig\n˜b−ǫ0(µ+1)/bracketrightBig\n/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(Ω)\n+/bracketleftBigb(1−δ)\n2−µ/ba∇dbla/ba∇dblL∞(0,T;L∞(Ω))/bracketrightBig\n/ba∇dbl∇ut/ba∇dbl2\nL2(0,T;L2(Ω))\n+/bracketleftBig\n˜a−µ2\nb(1−δ)/ba∇dbla/ba∇dbl2\nL∞(0,T;L∞(Ω))/bracketrightBig\n/ba∇dbl∇u/ba∇dbl2\nL∞(0,T;L2(Ω))+α\n2/ba∇dblut/ba∇dbl2\nL2(0,T;L2(ˆΓ))\n+/bracketleftBig\nµbδ\n2(q+1)−η(2µ+1)/bracketrightBig\n/ba∇dbl∇ut/ba∇dblq+1\nL∞(0,T;Lq+1(Ω))+µα\n4/ba∇dblut/ba∇dbl2\nL∞(0,T;L2(ˆΓ))\n≤˜C/parenleftBig\n(T√\nTM)q+1\nq−1+|∇u0|2\nL2(Ω)+|u1|2\nH1(Ω)+(T5/2M)q+1\nq−1\n+(4√\nTm+√\nTm+√\nTM)|∇u0|2\nL4(Ω)+|u1|q+1\nW1,q+1(Ω)+|u1|2\nL2(ˆΓ)\n+((1\n2+T3/4)T3/4m)q+1\nq−1+(T5/2m)q+1\nq−1+CΓ(g)/parenrightBig\n,\nfor some large enough ˜C, and hence if Tand the bound κare sufficiently small,\nand we choose mandMappropriately, Tis a self-mapping. Since Wis closed, we\nobtain existence of solutions through compactness argument. /squareLOCAL EXISTENCE RESULTS 37\nRelyingonProposition5.3, wecanobtainlocalexistenceofsolutionsf ortheproblem\n(1.9) withγ >0.\nTheorem 5.5. Let the assumptions of Theorem 5.4hold andγ >0. There exist\nκ>0,T >0such that for all u0∈W1,4(Ω),u1∈W1,q+1(Ω), with\n1/summationdisplay\ns=0/ba∇dblds\ndtsg/ba∇dblq+1\nq\nLq+1\nq(0,T;W−q\nq+1,q+1\nq(Γ))+/ba∇dblg/ba∇dblq+1\nq\nL∞(0,T;W−q\nq+1,q+1\nq(Γ)))\n+|∇u0|2\nL2(Ω)+|u1|2\nH1(Ω)+|u1|q+1\nW1,q+1(Ω)+|u1|2\nL2(ˆΓ)≤κ2\nthere exists a weak solution u∈ Wof(1.9), whereWis defined as in (5.22), and\nmandMare sufficiently small.\nDue to the presence of q−Laplace damping term, the derivation of energy es-\ntimates is possible only for multipliers of lower order (see Remark 4, [2]) and the\nquestion of uniqueness remains open.\nAcknowledgments. The author thanks Barbara Kaltenbacher for many fruitful\ndiscussions and comments. The financial support by the FWF (Aust rian Science\nFund) under grant P24970 is gratefully acknowledged.\nReferences\n[1] A. Bamberger, R. Glowinski and Q.H. Tran, A domain decomp osition method for the acoustic\nwave equation with discontinuous coefficients and grid chang e,SIAM J. Numer. Anal. ,34\n(1997), 603–639.\n[2] R. Brunnhuber, B. Kaltenbacher and P. Radu, Relaxation o f regularity for the Westervelt\nequation by nonlinear damping with application in acoustic -acoustic and elastic-acoustic cou-\npling,Evol. Eq. Control Theory , to appear\n[3] L. C. Evans, Partial Differential Equations , American Mathematical Society, Providence,\n1998.\n[4] M. F. Hamilton and D. T. Blackstock, Nonlinear Acoustics , Academic Press, New York,\n1997.\n[5] B. Kaltenbacher, Boundary observability and stabiliza tion for Westervelt type wave equations\nwithout interior damping, Applied Mathematics and Optimization /62 (2010), 381–410.\n[6] B. Kaltenbacher and I. Lasiecka, Global existence and ex ponential decay rates for the West-\nervelt equation, Discrete and Continuous Dynamical Systems Series S ,2(2009), 503–525.\n[7] B. Kaltenbacher, I. Lasiecka and S. Veljovi´ c, Well-pos edness and exponential decay for the\nWestervelt equation with inhomogeneous Dirichlet boundar y data, J. Escher et al (Eds):\nProgress in Nonlinear Differential Equations and Their Appl ications,60, (2011), 357–387.\n[8] B.Kaltenbacher and I. Lasiecka, Well-posedness of the W estervelt and the Kuznetsov equation\nwith nonhomogeneous Neumann boundary conditions, AIMS Proceedings , (2011).\n[9] M. Kaltenbacher, Numerical Simulations of Mechatronic Sensors and Actuator s, Springer,\nBerlin, 2004.\n[10] G. Leoni, A first course in Sobolev spaces , American Mathematical Society, Providence,\n2009.\n[11] P. Lindqvist, Notes on p-Laplace equation , Lecture notes, University of Jyv¨ askyl¨ a, 2006.\n[12] G. Teschl, Ordinary Differential Equations and Dynamical Systems , American Mathematical\nSociety, Providence, 2012.\n[13] P.J. Westervelt, Parametric acoustic array, The Journal of the Acoustic Society of America ,\n35(1963), 535–537.\nInsitut f ¨ur Mathematik, Universit ¨at Klagenfurt, Universit ¨atsstraße 65-57, 9020\nKlagenfurt am W ¨orthersee, Austria\nE-mail address :vanja.nikolic@aau.at"    },    {        "title": "1911.02775v2.Quantum_Oscillations_of_Gilbert_Damping_in_Ferromagnetic_Graphene_Bilayer_Systems.pdf",        "content": "arXiv:1911.02775v2  [cond-mat.mes-hall]  15 Apr 2020Quantum Oscillations of Gilbert Damping in Ferromagnetic/ Graphene Bilayer\nSystems\nYuya Ominato1and Mamoru Matsuo1,2\n1Kavli Institute for Theoretical Sciences, University of Ch inese Academy of Sciences, Beijing 100190, China and\n2CAS Center for Excellence in Topological Quantum Computati on,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n(Dated: April 16, 2020)\nWe study the spin dynamics of a ferromagnetic insulator on wh ich graphene is placed. We show\nthat the Gilbert damping is enhanced by the proximity exchan ge coupling at the interface. The\nmodulation of the Gilbert damping constant is proportional to the product of the spin-up and\nspin-down densities of states of graphene. Consequently, t he Gilbert damping constant in a strong\nmagnetic field oscillates as a function of the external magne tic field that originates from the Landau\nlevel structure of graphene. We find that a measurement of the oscillation period enables the\nstrength of the exchange coupling constant to be determined . The results theoretically demonstrate\nthat the ferromagnetic resonance measurements may be used t o detect the spin resolved electronic\nstructure of the adjacent materials, which is critically im portant for future spin device evaluations.\nIntroduction .—Graphene spintronics is an emergent\nfield aiming at exploiting exotic spin-dependent proper-\ntiesofgrapheneforspintronicsdevices[1]. Althoughpris-\ntinegrapheneisanon-magneticmaterial,therehavebeen\neffortstointroducemagnetismintographenetofindspin-\ndependent phenomena and to exploit its spin degrees of\nfreedom. Placing graphene on a magnetic substrate is\na reasonable way, which leads to magnetic proximity ef-\nfect and lifting of spin degeneracy [2, 3]. Subsequently,\nmagnetization was induced in graphene and spin depen-\ndent phenomena, such as the anomalous Hall effect [4, 5]\nand non-local spin transport [6, 7], were observed. In all\nthese experiments, a spin-dependent current was gener-\nated by an electric field. There is an alternative way to\ngenerate a spin current called spin pumping [8–12]. The\nproximity exchange coupling describes spin transfer at\nthe magnetic interface and a spin current is injected us-\ning ferromagnetic resonance (FMR) from ferromagnetic\nmaterials into the adjacent materials. The generation of\na spin current is experimentally detectable through both\nthe inverse spin Hall effect and modulation of the FMR,\nwhich were experimentally confirmed at magnetic inter-\nfaces between graphene and several magnetic materials\n[13–18].\nThe theory of spin transport phenomena at magnetic\ninterfaces has been formulated based on the Schwinger-\nKeldysh formalism [19], which is applicable to magnetic\ninterfaces composed of a variety of systems, such as a\nparamagnetic metal and a ferromagnetic insulator (FI)\n[20–23], a superconductor and FI [24, 25], and two FIs\n[26, 27]. The modulation of FMR has been investigated\nin several papers. The modulation of Gilbert damping\nwasfound to be proportionalto the imaginarypartofthe\ndynamical spin susceptibility [21, 23–25, 28, 29], which\nmeans that one can detect spin excitations and electronic\nproperties of adjacent materials through the FMR mea-\nsurements. This implies that the FMR measurements\nof FI/graphene bilayer systems allow us to access thespin-dependent properties of graphene in quantum Hall\nregime [30, 31]. However, the modulation of FMR at the\nmagnetic interface between a FI and graphene has not\nbeen investigated and the effect of Landau quantization\non the FMR signal is unclear.\nIn this work, we study the modified magnetization dy-\nnamics of a FI adjacent to graphene. Figure 1 (a) shows\naschematicofthe system. Microwavesareirradiatedand\nthe precession of localized spins is induced. Figure 1 (b)\nand (c) shows the electronic structure of graphene on the\nFI under aperpendicular magneticfield. The spin degen-\neracy is lifted by the exchange coupling at the interface\nand spin-split Landau levels are formed. The densities of\nstates for spin-up and spin-down are shown in the right\npanel; Landau level broadening is included. We find that\nthe modulation of Gilbert damping is proportionalto the\nproduct of the densities of states for spin-up and spin-\ndown, so that the FMR measurements may be used as\na probe of the spin-resolved densities of states. Owing\nto the peak structure of the density of states, the mod-\nulation of Gilbert damping exhibits peak structure and\nan oscillation as a function of Fermi level and magnetic\nfield, which reflects the Landau level structure. One may\ndetermine the exchange coupling constant by analyzing\nthe period of the oscillation.\nModel Hamiltonian .—The totalHamiltonian H(t)con-\nsists of three terms,\nH(t) =HFI(t)+HGr+Hex. (1)\nThe first term HFI(t) describes the bulk FI\nHFI(t) =/summationdisplay\nk/planckover2pi1ωkb†\nkbk−h+\nac(t)b†\nk=0−h−\nac(t)bk=0,(2)\nwhereb†\nkandbkdenote the creation and annihilation\noperators of magnons with momentum k. We assume a\nparabolic dispersion /planckover2pi1ωk=Dk2−/planckover2pi1γB, withγ(<0) the\nelectron gyromagnetic ratio. The coupling between the2\nMicrowaveB(a) System (b) spin splitting \nExchange \ncoupling\nB01230\n-1 \n-2 \n-3 \nDOSE\nup down(c) spin-split Landau level \nkx kyE\nE\nB123\n0-1 \n-2 \n-3 0\nkx kyE\nFIG. 1. (Color online) Schematic picture of the FMR measurem ent and the energy spectrum of graphene in a strong perpen-\ndicular magnetic field. (a) Graphene on a ferromagnetic insu lator substrate. The magnetic field perpendicular to graphe ne\nis applied and the microwave is irradiated to the FI. (b) The s pin degeneracy is lifted by the exchange coupling. (c) The\nperpendicular magnetic field leads to the spin-split Landau level structure. The density of states has a peak structure a nd the\nlevel broadening originating from disorder is included.\nmicrowave and magnons is given by\nh±\nac(t) =/planckover2pi1γhac\n2√\n2SNe∓iΩt, (3)\nwherehacand Ω are the amplitude and frequency of the\nmicrowave radiation, respectively, and Sis the magni-\ntude of the localized spin in the FI. The above Hamilto-\nnian is derived from a ferromagnetic Heisenberg model\nusing the Holstein-Primakoff transformation and the\nspin-wave approximation ( Sz\nk=S−b†\nkbk,S+\nk=√\n2Sbk,\nS−\n−k=√\n2Sb†\nk, whereSkis the Fourier transform of the\nlocalized spin in the FI).\nThe second term HGrdescribes the electronic states\naround the Kpoint in graphene under a perpendicular\nmagnetic field,\nHGr=/summationdisplay\nnXsεnc†\nnXscnXs, (4)\nwherec†\nnXsandcnXsdenote the creation and annihi-\nlation operators of electrons with Landau level index\nn= 0,±1,±2,···, guiding center X, and spin up s= +\nand spin down s=−. The eigenenergy is given by\nεn= sgn(n)√\n2e/planckover2pi1v2/radicalbig\n|n|B, (5)\nwherevis the velocity and the sign function is defined\nas\nsgn(n) :=\n\n1 (n >0)\n0 (n= 0)\n−1 (n <0). (6)\nIn the following, we neglect the Zeeman coupling be-\ntween the electron spin and the magnetic field because\nit is much smaller than the Landau-level separation and\nthe exchange coupling introduced below. In graphene,\nthere are two inequivalent valleys labelled KandK′. Inthis paper, we assume that the intervalley scattering is\nnegligible. This assumption is valid for an atomically flat\ninterface,whichisreasonablegiventherecentexperimen-\ntal setups [4, 17, 18]. Consequently, the valley degree\nof freedom just doubles the modulation of the Gilbert\ndamping.\nThe third term Hexis the exchange coupling at the\ninterface consisting of two terms\nHex=HZ+HT, (7)\nwhereHZdenotes the out-of-plane component of the\nexchange coupling and leads to the spin splitting in\ngraphene,\nHZ=−JS/summationdisplay\nnX/parenleftBig\nc†\nnX+cnX+−c†\nnX−cnX−/parenrightBig\n,(8)\nwithJthe exchange coupling constant. The z-\ncomponent of the localized spin is approximated as\n∝angbracketleftSz\nk∝angbracketright ≈S. The out-of-plane component HZis modeled\nas a uniform Zeeman-like coupling, although in general,\nHZcontains the effect of surface roughness, which gives\noff-diagonal terms. The Hamiltonian HTdenotes the in-\nplane component of the exchange coupling and describes\nspin transfer between the FI and graphene,\nHT=−/summationdisplay\nnX/summationdisplay\nn′X′/summationdisplay\nk/parenleftBig\nJnX,n′X′,ks+\nnX+,n′X′−S−\nk+h.c./parenrightBig\n,\n(9)\nwhereJnX,n′X′,kis the matrix element for the spin trans-\nfer processes and s+\nnX+,n′X′−is the spin-flip operator for\nthe electron spin in graphene.\nModulation of Gilbert Damping .—To discuss the\nGilbert damping, we calculated the time-dependent sta-\ntistical averageof the localized spin under the microwave\nirradiation. The first-order perturbation calculation\ngives the deviation from the thermal average,\nδ∝angbracketleftS+\nk=0(t)∝angbracketright=−h+\nac(t)GR\nk=0(Ω). (10)3\nThe retarded Green’s function is written as\nGR\nk(ω) =2S//planckover2pi1\nω−ωk+iαGω−(2S//planckover2pi1)ΣR\nk(ω),(11)\nwhere we have introduced the phenomenological dimen-\nsionlessdampingparameter αG, calledthe Gilbert damp-\ning constant, which originates from the magnon-phonon\nand magnon-magnon coupling, etc [32–34]. In this pa-\nper, we focus on the modulation of the Gilbert damping\nstemming from the spin transfer processes at the inter-\nface. The self-energy from the spin transfer processes at\nthe interface within second-order perturbation is given\nby\nΣR\nk(ω) =/summationdisplay\nnX/summationdisplay\nn′X′|JnX,n′X′,k|2χR\nn+,n′−(ω).(12)\nThe spin susceptibility is given by\nχR\nn+,n′−(ω) =fn+−fn′−\nεn+−εn′−+/planckover2pi1ω+i0,(13)\nwherefns= 1//parenleftbig\ne(εns−µ)/kBT+1/parenrightbig\nis the Fermi distribu-\ntion function and εns=εn−JSsis the spin-split Landau\nlevel. From the self-energy expression, one sees that the\nmodulation of the Gilbert damping reflects the property\nof the spin susceptibility of graphene. The modulation\nof the Gilbert damping under the microwave irradiation\nis given by [21, 23–25, 28, 29]\nδαK\nG=−2SImΣR\nk=0(ω)\n/planckover2pi1ω, (14)\nwhere the superscript Ksignifies the contribution from\ntheKvalley.\nTo further the calculation, we assume that the ma-\ntrix element JnX,n′X′,k=0is approximated by a constant\nJ0, including detail properties of the interface, that is,\nJnX,n′X′,k=0≈J0. Withthisassumption,theself-energy\nbecomes\nImΣR\nk=0(ω) =−|J0|2π/planckover2pi1ω/integraldisplay\ndε/parenleftbigg\n−∂f(ε)\n∂ε/parenrightbigg\nD+(ε)D−(ε),\n(15)\nwhereDs(ε) is the density of states for spin s=±\nDs(ε) =A\n2πℓ2\nB/summationdisplay\nn1\nπΓ\n(ε−εns)2+Γ2,(16)\nwith magnetic length ℓB=/radicalbig\n/planckover2pi1/(eB) and area of the\ninterface A. Here, we have introduced a constant Γ de-\nscribing level broadening arising from surface roughness\nand impurity scattering. This is the simplest approx-\nimation to include the disorder effect. The density of\nstates shows peaks at the Landau level, which is promi-\nnent when its separation exceeds the level broadening.\nLandau level (JS = 20  meV) δα G [δα 0   10-2 ]\n0.6\nB [T]1.0μ [meV] \n0.4 0.240 \n20 \n-20\n-400\n0.8s=-, n=0\ns=+, n=0123\n-1 \n-2 \n-3 Γ = 1  meV\nkBT = 1  meV (= 11  K)\n6\n03\n0.6\nB [T]1.0μ [meV] \n0.4 0.240 \n20 \n-20\n-400\n0.8\nFIG. 2. (Color online) Modulation of the Gilbert damping\nconstant δαGand spin-split Landau levels as a function of\nthe Fermi level µand the magnetic field B. The spin splitting\nJSis set to 20meV. In the left panel, δαGhas peaks at the\ncrossing points of spin-up and spin-down Landau levels. In\nthe right panel, the blue and red curves identify the spin-up\nand spin-down Landau levels, respectively.\nFinally, the modulation of the Gilbert damping constant\nδαGis derived as\nδαG= 2πgvS|J0|2/integraldisplay\ndε/parenleftbigg\n−∂f(ε)\n∂ε/parenrightbigg\nD+(ε)D−(ε),(17)\nwheregv= 2 denotes the valley degree of freedom.\nFrom this expression, one sees that the modulation of\nthe Gilbert damping is proportional to the product of\nthe densities of states for spin-up and spin-down. There-\nfore, combined with the density of states measurement,\nfor example, a capacitance measurement [35], the FMR\nmeasurement is used to detect the spin-resolved densities\nof states.\nFigure 2 shows the spin-split Landau levels and the\nmodulation of the Gilbert damping δαGas a function of\nthe Fermi level µand the magnetic field B. We use δα0\nas a unit of δαG\nδα0= 2πgvS|J0|2/parenleftbiggA\n2πℓ2\nB1\nmeV/parenrightbigg2\n.(18)\nWe note that δα0(∝B2) depends on the magnetic field.\nBoth the level broadening Γ and the thermal broadening\nkBTare set to 1meV, and JSis set to 20meV [2–4].\nδαGreflects the Landau level structure and has peaks at\ncrossing points of spin-up and spin-down Landau levels.\nThe peakpositions aredetermined bysolving εn+=εn′−\nand the inverse of the magnetic field at the peaks is given\nby\n1\nB=2e/planckover2pi1v2\n(2JS)2/parenleftBig/radicalbig\n|n|−/radicalbig\n|n′|/parenrightBig2\n. (19)\nThe peak structurebecomes prominent when the Landau\nlevel separation exceeds both level and thermal broaden-\ning.4\nΓ = 1 meV\nkBT = 1 meV (= 11 K)\n5 meV (= 57 K)\n10 meV (= 115 K)kBT = 1 meV (= 11 K)\nΓ = 1 meV \n2 meV \n4 meV μ = JS = 20 meV μ = JS = 20 meV (a) (b)\nΔ(1/B)\n3\n1/B [1/T]4δα G [δα 0   10 -2 ]\n2 18\n6\n4\n2\n0\n5 3\n1/B [1/T]4δα G [δα 0   10 -2 ]\n2 18\n6\n4\n2\n0\n5Δ(1/B)\nFIG. 3. (Color online) Quantum oscillation of the modulatio n\nof the Gilbert damping constant δαGas a function of the\ninverse of the magnetic field 1 /B. The Fermi level µand the\nmagnitude of the spin splitting JSare set to 20meV. (a)\nΓ = 1meV and δαGis plotted at several temperatures. (b)\nkBT= 1meV and δαGis plotted for several Γ’s. The period of\nthe oscillation ∆(1 /B) is indicated by double-headed arrows.\nFigure 3 shows the modulation of the Gilbert damping\nδαGas a function of the inverse of the magnetic field\n1/Bwith the Fermi level set to µ= 20meV, where the\nspin-down zeroth Landau level resides. δαGshows peak\nstructure and a periodic oscillatorybehavior. The period\nof the oscillation ∆(1 /B) is derived from Eq. (19) and is\nwritten as\n∆/parenleftbigg1\nB/parenrightbigg\n=2e/planckover2pi1v2\n(2JS)2. (20)\nThe above relation means that the magnitude of the spin\nsplitting JSis detectable by measuring the period of the\noscillation ∆(1 /B). For the peak structure to be clear,\nboth leveland thermalbroadeningmust to be sufficiently\nsmaller than the Landau level separation; otherwise, the\npeak structure smears out.\nDiscussion .—To observe the oscillation of Gilbert\ndamping, at least two conditions must be satisfied. First,\nthe well-separated landau levels have to be realized in\nthe magnetic field where the FMR measurements is fea-\nsible. Second, the FMR modulation caused by the ad-\njacent graphene have to be detectable. The graphene\nLandau levels are observed in recent experiments at 2T\n[36], andrecentbroadbandferromagneticresonancespec-\ntrometer enables the generation of microwaves with fre-\nquencies ≤40GHz and FMR measurements in a mag-\nnetic field ≤2T [37]. The modulation of the FMR\nlinewidth in Permalloy/Graphene [14, 16], yttrium iron\ngarnet/Graphene [17, 18] have been reported by sev-\neral experimental groups, although all of them were per-\nformed at room temperature. Therefore, the above two\nconditionsareexperimentallyfeasibleandourtheoretical\npredictions can be tested in an appropriate experimental\nsetup.Conclusion .—We have studied the modulation of the\nGilbert damping δαGin a ferromagnetic insulator on\nwhich graphene is placed. The exchange coupling at\nthe interface and the perpendicular magnetic field lead\nto the spin-split Landau levels in graphene. We showed\nthatδαGis proportional to the product of the densities\nof states for spin-up and spin-down electrons. Therefore,\nthe spin-resolved densities of states can be detected by\nmeasuring δαGand the total density of states. When the\nFermi level is fixed at a Landau level, δαGoscillates as a\nfunction of the inverse of the magnetic field. The period\nof the oscillation provides information on the magnitude\nof the spin splitting. Our main message is that the FMR\nmeasurement is a probe of spin-resolved electronic struc-\nture. In addition to spin current generation, one may use\ntheFMRmeasurementstodetectthe electronicstructure\nof adjacent materials.\nAcnowledgement WethankJ.Fujimoto, T.Kato,R.\nOhshima, and M. Shiraishi for helpful discussions. 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Duine†\nInstitute for Theoretical Physics, Utrecht University,\nLeuvenlaan 4, 3584 CE Utrecht, The Netherlands\n(Dated: October 31, 2018)\nStarting from a model that consists of a semiclassical spin c oupled to two leads we present a\nmicroscopic derivation of the Langevin equation for the dir ection of the spin. For slowly-changing\ndirection it takes on the form of the stochastic Landau-Lifs chitz-Gilbert equation. We give ex-\npressions for the Gilbert damping parameter and the strengt h of the fluctuations, including their\nbias-voltage dependence. At nonzero bias-voltage the fluct uations and damping are not related by\nthe fluctuation-dissipation theorem. We find, however, that in the low-frequency limit it is possible\nto introduce a voltage-dependent effective temperature tha t characterizes the fluctuations in the\ndirection of the spin, and its transport-steady-state prob ability distribution function.\nPACS numbers: 72.25.Pn, 72.15.Gd\nI. INTRODUCTION\nOne of the major challenges in the theoretical descrip-\ntion of various spintronics phenomena1, such as current-\ninduced magnetization reversal2,3,4,5and domain-wall\nmotion6,7,8,9,10,11,12, is their inherent nonequilibrium\ncharacter. In addition to the dynamics of the collective\ndegreeoffreedom, themagnetization, thenonequilibrium\nbehavior manifests itself in the quasi-particle degrees of\nfreedomthataredrivenoutofequilibriumbythe nonzero\nbias voltage. Due to this, the fluctuation-dissipation\ntheorem13,14cannot be applied to the quasi-particles.\nThis, in part, has led to controversysurrounding the the-\nory of current-induced domain wall motion15,16.\nEffective equations of motion for order-parameter\ndynamics that do obey the equilibrium fluctuation-\ndissipation theorem often take the form of Langevin\nequations, or their corresponding Fokker-Planck\nequations13,14,17. In the context of spintronics the rele-\nvant equation is the stochastic Landau-Lifschitz-Gilbert\nequationforthe magnetizationdirection18,19,20,21,22,23,24.\nIn this paper we derive the generalization of this equa-\ntion to the nonzero-current situation, for a simple\nmicroscopic model consisting of a single spin coupled\nto two leads via an onsite Kondo coupling. This model\nis intended as a toy-model for a magnetic impurity\nin a tunnel junction25,26,27. Alternatively, one may\nthink of a nanomagnet consisting of a collection of\nspins that are locked by strong exchange coupling.\nThe use of this simple model is primarily motivated\nby the fact that it enables us to obtain analytical\nresults. Because the microscopic starting point for\ndiscussing more realistic situations has a similar form,\nhowever, we believe that our main results apply quali-\ntatively to more complicated situations as well. Similar\nmodels have been used previously to explicitly study\nthe violation of the fluctuation-dissipation relation28, 1 1.1 1.2 1.3\n 0 0.1 0.2 0.3 0.4 0.5α/α0\n|e|V/µ\n 0 0.5 1 1.5 2 2.5 3\n 0  2  4  6  8  10Teff/T\n|e|V/(kB T)\nFIG. 1: Effective temperature as a function of bias voltage.\nThedashedlineshows thelarge bias-voltage asymptoticres ult\nkBTeff≃ |e|V/4 +kBT/2. The inset shows the bias-voltage\ndependence of the Gilbert damping parameter normalized to\nthe zero-bias result.\nand the voltage-dependence of the Gilbert damping\nparameter27. Starting from this model, we derive an\neffective stochastic equation for the dynamics of the\nspin direction using the functional-integral description\nof the Keldysh-Kadanoff-Baymnonequilibrium theory29.\n(For similar approaches to spin and magnetization\ndynamics, see also the work by Rebei and Simionato30,\nNussinov et al.31and Duine et al.32.) This formalism\nleads in a natural way to the path-integral formulation\nof stochastic differential equations33,34. One of the\nattractive features of this formalism is that dissipation\nand fluctuations enter the theory separately. This allows\nus to calculate the strength of the fluctuations even\nwhen the fluctuation-dissipation theorem is not valid.\nWe find that the dynamics of the direction of the spin\nis described by a Langevin equation with a damping ker-2\n,T\nSµL,T µR\nFIG. 2: Model system of a spin Sconnected to two tight-\nbinding model half-infinite leads. The chemical potential o f\nthe left lead is µLand different from the chemical potential\nof the right lead µR. The temperature Tof both leads is for\nsimplicity taken to be equal.\nnel and a stochastic magnetic field. We give explicit\nexpressions for the damping kernel and the correlation\nfunction of the stochastic magnetic field that are valid\nin the entire frequency domain. In general, they are not\nrelated by the fluctuation-dissipation theorem. In the\nlow-frequency limit the Langevin equation takes on the\nform ofthe stochasticLandau-Lifschitz-Gilbertequation.\nMoreover, in that limit it is always possible to introduce\nan effective temperature that characterizes the fluctua-\ntions and the equilibrium probability distribution for the\nspin direction. In Fig. 1 we present our main results,\nnamely the bias-voltage dependence of the effective tem-\nperature and the Gilbert damping parameter. We find\nthat the Gilbert damping constant initially varies lin-\nearly with the bias voltage, in agreement with the re-\nsult of Katsura et al.27. The voltage-dependence of the\nGilbert damping parameter is determined by the den-\nsity of states evaluated at an energy equal to the sum\nof the Fermi energy and the bias voltage. The effective\ntemperature is for small bias voltage equal to the actual\ntemperature, whereas for large bias voltage it is inde-\npendent of the temperature and proportional to the bias\nvoltage. This bias-dependence of the effective tempera-\nture is traced back to shot noise35.\nEffective temperatures for magnetization dynam-\nics have been introduced before on phenomenolog-\nical grounds in the context of thermally-assisted\ncurrent-driven magnetization reversal in magnetic\nnanopillars36,37,38. A current-dependent effective tem-\nperature enters in the theoretical description of these\nsystems because the current effectively lowers the energy\nbarrier thermal fluctuations have to overcome. In addi-\ntion to this effect, the presence of nonzero current alters\nthe magnetization noise due to spin current shot noise35.\nCovington et al.39interpret their experiment in terms of\ncurrent-dependent noise although this interpretation is\nstill under debate30. Foroset al.35also predict, using a\ndifferent model and different methods, a crossover from\nthermal to shot-noise dominated magnetization noise for\nincreasing bias voltage. Our main result in Fig. 1 is an\nexplicit example of this crossover for a specific model.\nThe remainderofthe paperis organizedasfollows. We\nstart in Sec. II by deriving the general Langevin equation\nforthe dynamicsofthe magneticimpurity coupledtotwo\nleads. In Sec. III and IV we discuss the low-frequency\nlimit in the absence and presence of a current, respec-\ntively. We end in Sec. V with our conclusions.II. DERIVATION OF THE LANGEVIN\nEQUATION\nWe use a model that consists of a spin Son a site\nthat is coupled via hopping to two semi-infinite leads, as\nshown in Fig. 2. The full probability distribution for the\ndirection ˆΩ of the spin on the unit sphere is written as a\ncoherent-state path integral over all electron Grassmann\nfield evolutions ψ∗(t) andψ(t), and unit-sphere paths\nS(t), that evolve from −∞totand back on the so-called\nKeldysh contour Ct. It is given by29\nP[ˆΩ,t] =/integraldisplay\nS(t)=ˆΩd[S]δ/bracketleftBig\n|S|2−1/bracketrightBig\nd[ψ∗]d[ψ]\n×exp/braceleftbiggi\n/planckover2pi1S[ψ∗,ψ,S]/bracerightbigg\n, (1)\nwhere the delta functional enforces the length constraint\nof the spin. In the above functional integral an inte-\ngration over boundary conditions at t=−∞, weighted\nby an appropriate initial density matrix, is implicitly in-\ncluded in the measure. We have not included boundary\nconditions on the electron fields, because, as we shall see,\nthe electron correlation functions that enter the theory\nafter integrating out the electrons are in practice conve-\nniently determined assumingthat the electronsareeither\nin equilibrium or in the transport steady state.\nThe action S[ψ∗,ψ,S] is the sum of four parts,\nS[ψ∗,ψ,S] =SL/bracketleftBig/parenleftbig\nψL/parenrightbig∗,ψL/bracketrightBig\n+SR/bracketleftBig/parenleftbig\nψR/parenrightbig∗,ψR/bracketrightBig\n+SC/bracketleftBig/parenleftbig\nψ0/parenrightbig∗,ψ0,/parenleftbig\nψL/parenrightbig∗,ψL,/parenleftbig\nψR/parenrightbig∗,ψR/bracketrightBig\n+S0/bracketleftBig/parenleftbig\nψ0/parenrightbig∗,ψ0,S/bracketrightBig\n. (2)\nWe describe the leads using one-dimensional non-\ninteracting electron tight-binding models with the action\nSL/R/bracketleftBig/parenleftBig\nψL/R/parenrightBig∗\n,ψL/R/bracketrightBig\n=\n/integraldisplay\nCtdt′\n\n/summationdisplay\nj,σ/parenleftBig\nψL/R\nj,σ(t′)/parenrightBig∗\ni/planckover2pi1∂\n∂t′ψL/R\nj,σ(t′)\n+J/summationdisplay\n/an}bracketle{tj,j′/an}bracketri}ht;σ/parenleftBig\nψL/R\nj,σ(t′)/parenrightBig∗\nψL/R\nj′,σ(t′)\n\n, (3)\nwhere the sum in the second term of this action is\nover nearest neighbors only and proportional to the\nnearest-neighbor hopping amplitude Jin the two leads.\n(Throughout this paper the electron spin indices are de-\nnoted byσ,σ′∈ {↑,↓}, and the site indices by j,j′.) The\ncoupling between system and leads is determined by the\naction\nSC[/parenleftbig\nψ0/parenrightbig∗,ψ0,/parenleftbig\nψL/parenrightbig∗,ψL,/parenleftbig\nψR/parenrightbig∗,ψR] =/integraldisplay\nCtdt′JC/summationdisplay\nσ/bracketleftBig/parenleftbig\nψL\n∂L,σ(t′)/parenrightbig∗ψ0\nσ(t′)+/parenleftbig\nψ0\nσ(t′)/parenrightbig∗ψL\n∂L,σ(t′)/bracketrightBig\n+3\n/integraldisplay\nCtdt′JC/summationdisplay\nσ/bracketleftBig/parenleftbig\nψR\n∂R,σ(t′)/parenrightbig∗ψ0\nσ(t′)+/parenleftbig\nψ0\nσ(t′)/parenrightbig∗ψR\n∂R,σ(t′)/bracketrightBig\n,\n(4)\nwhere∂L and∂R denote the end sites of the semi-infinite\nleft and right lead, and the fields/parenleftbig\nψ0(t)/parenrightbig∗andψ0(t) de-\nscribe the electrons in the single-site system. The hop-\nping amplitude between the single-site system and the\nleads is denoted by JC. Finally, the action for the sys-\ntem reads\nS0/bracketleftBig/parenleftbig\nψ0/parenrightbig∗,ψ∗,S/bracketrightBig\n=/integraldisplay\nCtdt′\n/summationdisplay\nσ/parenleftbig\nψ0\nσ(t′)/parenrightbig∗i/planckover2pi1∂\n∂t′ψ0\nσ(t′)\n−/planckover2pi1SA(S(t′))·dS(t′)\ndt′+h·S(t′)\n+∆/summationdisplay\nσ,σ′/parenleftbig\nψ0\nσ(t′)/parenrightbig∗τσ,σ′·S(t′)ψ0\nσ′(t′)\n.(5)\nThe second term in this action is the usual Berry phase\nfor spin quantization40, withA(S) the vector potential\nof a magnetic monopole\nǫαβγ∂Aγ\n∂Sβ=Sα, (6)\nwhere a sum over repeated Greek indices α,β,γ∈\n{x,y,z}is implied throughout the paper, and ǫαβγis\nthe anti-symmetric Levi-Civita tensor. The third term in\nthe action in Eq. (5) describes the coupling of the spin to\nan external magnetic field, up to dimensionful prefactors\ngiven by h. (Note that hhas the dimensions of energy.)\nThe last term in the action models the s−dexchange\ncoupling of the spin with the spin of the conduction elec-\ntrons in the single-site system and is proportional to the\nexchange coupling constant ∆ >0. The spin of the con-\nduction electronsisrepresentedbythe vectorofthe Pauli\nmatrices that is denoted by τ.\nNext, we proceed to integrate out the electrons using\nsecond-order perturbation theory in ∆. This results in\nan effective action for the spin given by\nSeff[S] =/integraldisplay\nCtdt′/bracketleftbigg\nS/planckover2pi1A(S(t′))·dS(t′)\ndt′+h·S(t′)\n−∆2/integraldisplay\nCtdt′′Π(t′,t′′)S(t′)·S(t′′)/bracketrightbigg\n. (7)\nThis perturbation theory is valid as long as the electron\nband width is much larger than the exchange interac-\ntion with the spin, i.e., J,JC≫∆. The Keldysh quasi-\nparticleresponsefunctionisgivenintermsoftheKeldysh\nGreen’s functions by\nΠ(t,t′) =−i\n/planckover2pi1G(t,t′)G(t′,t), (8)\nwhere the Keldysh Green’s function is defined by\niG(t,t′) =/angbracketleftBig\nψ0\n↑(t)/parenleftbig\nψ0\n↑(t′)/parenrightbig∗/angbracketrightBig\n=/angbracketleftBig\nψ0\n↓(t)/parenleftbig\nψ0\n↓(t′)/parenrightbig∗/angbracketrightBig\n.(9)We willgiveexplicit expressionsforthe responsefunction\nand the Green’s function later on. For now, we will only\nmake use of the fact that a general function A(t,t′) with\nits argumentson the Keldysh contour is decomposed into\nits analytic pieces by means of\nA(t,t′) =θ(t,t′)A>(t,t′)+θ(t′,t)A<(t,t′),(10)\nwhereθ(t,t′) is the Heaviside step function on the\nKeldysh contour. There can be also a singular piece\nAδδ(t,t′), but suchageneraldecompositionisnot needed\nhere. Also needed are the advanced and retarded com-\nponents, denoted respectively by the superscript ( −) and\n(+), and defined by\nA(±)(t,t′)≡ ±θ(±(t−t′))/bracketleftbig\nA>(t,t′)−A<(t,t′)/bracketrightbig\n,(11)\nand, finally, the Keldysh component\nAK(t,t′)≡A>(t,t′)+A<(t,t′), (12)\nwhich, as we shall see, determines the strength of the\nfluctuations.\nNext we write the forward and backward paths of the\nspin on the Keldysh contour, denoted respectively by\nS(t+) andS(t−), as a classical path Ω(t) plus fluctua-\ntionsδΩ(t), by means of\nS(t±) =Ω(t)±δΩ(t)\n2. (13)\nMoreover,it turns out to be convenient to write the delta\nfunctional, which implements the length constraintofthe\nspin, as a path integral over a Lagrange multiplier Λ( t)\ndefined on the Keldysh contour. Hence we have for the\nprobability distribution in first instance that\nP[ˆΩ,t] =/integraldisplay\nS(t)=ˆΩd[S]d[Λ]exp/braceleftbiggi\n/planckover2pi1Seff[S]+i\n/planckover2pi1SΛ[S,Λ]/bracerightbigg\n,\n(14)\nwith\nSΛ[S,Λ] =/integraldisplay\nCtdt′Λ(t′)/bracketleftBig\n|S(t′)|2−1/bracketrightBig\n.(15)\nWe then also have to split the Lagrange multiplier into\nclassical and fluctuating parts according to\nΛ(t±) =λ(t)±δλ(t)\n2. (16)\nNote that the coordinate transformations in\nEqs. (13) and (16) have a Jacobian of one. Before\nwe proceed, we note that in principle we are required\nto expand the action up to all orders in δΩ. Also note\nthat for some forward and backward paths S(t+) and\nS(t−) on the unit sphere the classical path Ωis not\nnecessarily on the unit sphere. In order to circumvent\nthese problems we note that the Berry phase term in\nEq. (5) is proportional to the area on the unit sphere\nenclosed by the forward and backward paths. Hence, in4\nthe semi-classical limit S→ ∞27,40paths whose forward\nand backward components differ substantially will be\nsuppressed in the path integral. Therefore, we take this\nlimit from now on which allows us to expand the action\nin terms of fluctuations δΩ(t) up to quadratic order. We\nwill see that the classical path Ω(t) is now on the unit\nsphere. We note that this semi-classical approximation\nis not related to the second-order perturbation theory\nused to derive the effective action.\nSplitting the paths in classical and fluctuation parts\ngives for the probability distribution\nP[ˆΩ,t] =/integraldisplay\nΩ(t)=ˆΩd[Ω]d[δΩ]d[λ]d[δλ]exp/braceleftbiggi\n/planckover2pi1S[Ω,δΩ,λ,δλ]/bracerightbigg\n,\n(17)\nwith the action, that is now projected on the real-time\naxis,\nS[Ω,δΩ,λ,δλ] =/integraldisplay\ndt/braceleftbigg\n/planckover2pi1SǫαβγδΩβ(t)dΩα(t)\ndtΩγ(t)\n+δΩα(t)hα+2δΩα(t)Ωα(t)λ(t)\n+δλ(t)/bracketleftbig\n|Ω(t)|2−1+|δΩ(t)|2/4/bracketrightbig/bracerightbigg\n−∆2/integraldisplay\ndt/integraldisplay\ndt′/braceleftBig\nδΩα(t)/bracketleftBig\nΠ(−)(t′,t)+Π(+)(t,t′)/bracketrightBig\nΩα(t′)/bracerightBig\n−∆2\n2/integraldisplay\ndt/integraldisplay\ndt′/bracketleftbig\nδΩα(t)ΠK(t,t′)δΩα(t′)/bracketrightbig\n. (18)\nFrom this action we observe that the integration over\nδλ(t) immediately leads to the constraint\n|Ω(t)|2= 1−|δΩ(t)|2\n4, (19)\nas expected. Implementing this constraint leads to terms\nof order O(δΩ3) or higher in the above action which we\nare allowed to neglect because of the semi-classical limit.\nFrom now on we can therefore take the path integration\noverΩ(t) on the unit sphere.\nThephysicalmeaningofthetermslinearandquadratic\ninδΩ(t) becomes clear after a so-called Hubbard-\nStratonovich transformation which amounts to rewrit-\ning the action that is quadratic in the fluctuations as\na path integral over an auxiliary field η(t). Performing\nthis transformation leads to\nP[ˆΩ,t] =/integraldisplay\nΩ(t)=ˆΩd[Ω]d[δΩ]d[η]d[λ]\n×exp/braceleftbiggi\n/planckover2pi1S[Ω,δΩ,λ,η]/bracerightbigg\n,(20)\nwhere the path integration over Ωis now on the unit\nsphere. The action that weighs these paths is given by\nS[Ω,δΩ,λ,η] =/integraldisplay\ndt/bracketleftbigg\n/planckover2pi1SǫαβγδΩβ(t)dΩα(t)\ndtΩγ(t)\n+δΩα(t)hα+2δΩα(t)Ωα(t)λ(t)+δΩα(t)ηα(t)/bracketrightbigg−∆2/integraldisplay\ndt/integraldisplay\ndt′/braceleftBig\nδΩα(t)/bracketleftBig\nΠ(−)(t′,t)+Π(+)(t,t′)/bracketrightBig\nΩα(t′)/bracerightBig\n+1\n2∆2/integraldisplay\ndt/integraldisplay\ndt′/bracketleftBig\nηα(t)/parenleftbig\nΠK/parenrightbig−1(t,t′)ηα(t′)/bracketrightBig\n.(21)\nNote that the inverse in the last term is defined as/integraltext\ndt′′ΠK(t,t′′)/parenleftbig\nΠK/parenrightbig−1(t′′,t′) =δ(t−t′).\nPerforming now the path integral over δΩ(t), we ob-\nserve that the spin direction Ω(t) is constraint to obey\nthe Langevin equation\n/planckover2pi1SǫαβγdΩβ(t)\ndtΩγ(t) =hα+2λ(t)Ωα(t)\n+ηα(t)+/integraldisplay∞\n−∞dt′K(t,t′)Ωα(t′),(22)\nwith the so-called damping or friction kernel given by\nK(t,t′) =−∆2/bracketleftBig\nΠ(−)(t′,t)+Π(+)(t,t′)/bracketrightBig\n.(23)\nNote that the Heaviside step functions in Eq. (11) appear\nprecisely such that the Langevin equation is causal. The\nstochastic magnetic field is seen from Eq. (21) to have\nthe correlations\n∝an}bracketle{tηα(t)∝an}bracketri}ht= 0 ;\n∝an}bracketle{tηα(t)ηβ(t′)∝an}bracketri}ht=iδαβ/planckover2pi1∆2ΠK(t,t′).(24)\nUsing the fact that Ω(t) is a unit vector within our semi-\nclassical approximation, the Langevin equation for the\ndirection of the spin ˆΩ(t) is written as\n/planckover2pi1SdˆΩ(t)\ndt=ˆΩ(t)×/bracketleftbigg\nh+η(t)+/integraldisplay∞\n−∞dt′K(t,t′)ˆΩ(t′)/bracketrightbigg\n,\n(25)\nwhich has the form of a Landau-Lifschitz equation with\na stochastic magnetic field and a damping kernel. In\nthe next sections we will see that for slowly-varying spin\ndirection we get the usual form of the Gilbert damping\nterm.\nSo far, we have not given explicit expressions for the\nresponsefunctionsΠ(±),K(t,t′). Todeterminethesefunc-\ntions, we assume that the left and right leads are in\nthermal equilibrium at chemical potentials µLandµR,\nrespectively. Although not necessary for our theoretical\napproachwe assume, for simplicity, that the temperature\nTof the two leads is the same. The Green’s functions for\nthe system are then given by41,42\n−iG<(ǫ) =A(ǫ)\n2/summationdisplay\nk∈{L,R}N(ǫ−µk) ;\niG>(ǫ) =A(ǫ)\n2/summationdisplay\nk∈{L,R}[1−N(ǫ−µk)] ;\nG≶,K(t−t′) =/integraldisplaydǫ\n(2π)e−iǫ(t−t′)//planckover2pi1G≶,K(ǫ),(26)\nwithN(ǫ) ={exp[ǫ/(kBT)]+1}−1the Fermi-Dirac dis-\ntribution function with kBBoltzmann’s constant, and\nA(ǫ) =i/bracketleftBig\nG(+)(ǫ)−G(−)(ǫ)/bracketrightBig\n, (27)5\nthe spectral function. Note that Eq. (26) has a particu-\nlarlysimpleformbecausewearedealingwithasingle-site\nsystem. The retarded and advanced Green’s functions\nare determined by\n/bracketleftBig\nǫ±−2/planckover2pi1Σ(±)(ǫ)/bracketrightBig\nG(±)(ǫ) = 1, (28)\nwithǫ±=ǫ±i0, and the retarded self-energy due to one\nlead follows, for a one-dimensional tight-binding model,\nas\n/planckover2pi1Σ(+)(ǫ) =−J2\nC\nJeik(ǫ)a, (29)\nwithk(ǫ) = arccos[ −ǫ/(2J)]/athe wave vector in the\nleads at energy ǫ, andathe lattice constant. The ad-\nvanced self-energy due to one lead is given by the com-\nplex conjugate of the retarded one.\nBefore proceeding we give a brief physical description\nof the above results. (More details can be found in\nRefs. [41] and [42].) They arise by adiabatically elim-\ninating (“integrating out”) the leads from the system,\nassuming that they are in equilibrium at their respective\nchemical potentials. This procedure reduces the problem\nto a single-site one, with self-energy corrections for the\non-site electron that describe the broadening of the on-\nsite spectral function from a delta function at the (bare)\non-site energy to the spectral function in Eq. (27). More-\nover, the self-energy corrections also describe the non-\nequilibrium occupation of the single site via Eq. (26)\nFor the transport steady-state we have that\nΠ(±),K(t,t′) depends only on the difference of the time\narguments. Using Eq. (8) and Eqs. (10), (11), and (12)\nwe find that the Fourier transforms are given by\nΠ(±)(ǫ)≡/integraldisplay\nd(t−t′)eiǫ(t−t′)//planckover2pi1Π(±)(t,t′)\n=/integraldisplaydǫ′\n(2π)/integraldisplaydǫ′′\n(2π)1\nǫ±+ǫ′−ǫ′′\n×/bracketleftbig\nG<(ǫ′)G>(ǫ′′)−G>(ǫ′)G<(ǫ′′)/bracketrightbig\n,(30)\nand\nΠK(ǫ) =−2πi/integraldisplaydǫ′\n(2π)/integraldisplaydǫ′′\n(2π)δ(ǫ+ǫ′−ǫ′′)\n×/bracketleftbig\nG>(ǫ′)G<(ǫ′′)+G<(ǫ′)G>(ǫ′′)/bracketrightbig\n.(31)\nIn the next two sections we determine the spin dynamics\nin the low-frequency limit, using these expressions to-\ngether with the expressions for G≶(ǫ). We consider first\nthe equilibrium case.\nIII. EQUILIBRIUM SITUATION\nIn equilibrium the chemical potentials of the two leads\nare equal so that we have µL=µR≡µ. Combining re-\nsults from the previous section, we find for the retardedand advanced response functions (the subscript “0” de-\nnotes equilibrium quantities) that\nΠ(±)\n0(ǫ) =/integraldisplaydǫ′\n(2π)/integraldisplaydǫ′′\n(2π)A(ǫ′)A(ǫ′′)\n×[N(ǫ′−µ)−N(ǫ′′−µ)]\nǫ±+ǫ′−ǫ′′.(32)\nThe Keldysh component of the response function is in\nequilibrium given by\nΠK\n0(ǫ) =−2πi/integraldisplaydǫ′\n(2π)/integraldisplaydǫ′′\n(2π)A(ǫ′)A(ǫ′′)δ(ǫ−ǫ′+ǫ′′)\n{[1−N(ǫ′−µ)]N(ǫ′′−µ)+N(ǫ′−µ)[1−N(ǫ′′−µ)]}.(33)\nThe imaginary part of the retarded and advanced re-\nsponse functions are related to the Keldysh component\nby means of\nΠK\n0(ǫ) =±2i[2NB(ǫ)+1]Im/bracketleftBig\nΠ(±)\n0(ǫ)/bracketrightBig\n,(34)\nwithNB(ǫ) ={exp[ǫ/(kBT)]−1}−1the Bose distribu-\ntion function. This is, in fact, the fluctuation-dissipation\ntheorem which relates the dissipation, determined as we\nshall see by the imaginary part of the retarded and\nadvanced components of the response function, to the\nstrength of the fluctuations, determined by the Keldysh\ncomponent.\nFor low energies, corresponding to slow dynamics, we\nhave that\nΠ(±)\n0(ǫ)≃Π(±)\n0(0)∓i\n4πA2(µ)ǫ . (35)\nWith this result the damping term in the Langevin equa-\ntion in Eq. (25) becomes\n/integraldisplay∞\n−∞dt′K(t,t′)ˆΩ(t′) =−/planckover2pi1∆2A2(µ)\n2πdˆΩ(t)\ndt,(36)\nwhere we have not included the energy-independent part\nofEq. (35) because it does not contribute to the equation\nof motion for ˆΩ(t). In the low-energy limit the Keldysh\ncomponent of the response function is given by\nΠK\n0(ǫ) =A2(µ)\niπkBT . (37)\nPutting all these results together we find that the dy-\nnamics of the spin direction is, as long as the two leads\nare in equilibrium at the same temperature and chemical\npotential,determinedbythestochasticLandau-Lifschitz-\nGilbert equation\n/planckover2pi1SdˆΩ(t)\ndt=ˆΩ(t)×[h+η(t)]−/planckover2pi1α0ˆΩ×dˆΩ(t)\ndt,(38)\nwith the equilibrium Gilbert damping parameter\nα0=∆2A2(µ)\n2π. (39)6\nUsing Eqs. (24), (37), and (39) we find that the strength\nof the Gaussian stochastic magnetic field is determined\nby\n∝an}bracketle{tηα(t)ηβ(t′)∝an}bracketri}ht= 2α0/planckover2pi1kBTδ(t−t′)δαβ.(40)\nNote that these delta-function type noise correlations\nare derived by approximating the time dependence of\nΠK(t,t′) by a delta function in the difference of the time\nvariables. This means that the noisy magnetic field η(t)\ncorresponds to a Stratonovich stochastic process13,14,17.\nThe stationary probability distribution function gen-\nerated by the Langevin equation in Eqs. (38) and (40) is\ngiven by the Boltzmann distribution18,19,20,21,22,23,24\nP[ˆΩ,t→ ∞]∝exp/braceleftBigg\n−E(ˆΩ)\nkBT/bracerightBigg\n, (41)\nwith\nE[ˆΩ] =−h·ˆΩ, (42)\nthe energy of the spin in the external field. It turns\nout that Eq. (41) holds for any effective field h=\n−∂E[ˆΩ]/∂ˆΩ, and in particular for the case that E[ˆΩ] is\nquadratic in the components of ˆΩ as is often used to\nmodel magnetic anisotropy.\nIt is important to realize that the equilibrium prob-\nability distribution has precisely this form because of\nthe fluctuation-dissipation theorem, which ensures that\ndissipation and fluctuations cooperate to achieve ther-\nmal equilibrium13,14. Finally, it should be noted that\nthis derivation of the stochastic Landau-Lifschitz-Gilbert\nequation from a microscopic starting point circumvents\nconcerns regarding the phenomenological form of damp-\ning and fluctuation-dissipation theorem, which is subject\nof considerable debate22,23.\nIV. NONZERO BIAS VOLTAGE\nIn this section we consider the situation that the chem-\nical potential of the left lead is given by µL=µ+|e|V,\nwith|e|V >0 the bias voltage in units of energy, and\nµ=µRthe chemical potential of the right lead. Using\nthe general expressions given for the response functions\nderived in Sec. II, it is easy to see that the imaginary\npart of the retarded and advanced components of the\nresponse functions are no longer related to the Keldysh\ncomponent by means of the fluctuation-dissipation theo-\nrem in Eq. (34). See also the work by Mitra and Millis28\nfor a discussion of this point. As in the previous section,\nwe proceed to determine the low-frequency behavior of\nthe response functions.\nUsing Eqs. (26), (27), and (30) we find that the re-\ntarded and advanced components of the response func-\ntion are given by\nΠ(±)(ǫ) =∓i\n8π/bracketleftbig\nA2(µ+|e|V)+A2(µ)/bracketrightbig\nǫ .(43)In this expression we have omitted the energy-\nindependent part andthe contribution followingfrom the\nprincipal-value part of the energy integral because, as we\nhave seen previously, these do not contribute to the final\nequation of motion for the direction of the spin. Follow-\ning the same steps as in the previous section, we find\nthat the damping kernel in the general Langevin equa-\ntion in Eq. (25) reduces to a Gilbert damping term with\na voltage-dependent damping parameter given by\nα(V) =∆2\n4π/bracketleftbig\nA2(µ+|e|V)+A2(µ)/bracketrightbig\n≃α0/bracketleftbigg\n1+O/parenleftbigg|e|V\nµ/parenrightbigg/bracketrightbigg\n. (44)\nThis result is physically understood by noting that the\nGilbert damping is determined by the dissipative part of\nthe response function Π(+)(ǫ). In this simple model, this\ndissipative part gets contributions from processes that\ncorrespond to an electron leaving or entering the system,\nto or from the leads, respectively. The dissipative part\nis in general proportional to the density of states at the\nFermi energy. Since the Fermi energy of left and right\nlead is equal to µ+|e|Vandµ, respectively, the Gilbert\ndamping has two respective contributions corresponding\nto the two terms in Eq. (44).\nNote that the result that the Gilbert damping param-\neter initially varies linearly with the voltage is in agree-\nment with the results of Katsura et al.27, although these\nauthorsconsideraslightlydifferentmodel. Inthe insetof\nFig. 1 we show the Gilbert damping parameter as a func-\ntion of voltage. The parameters taken are ∆ /J= 0.1,\nJC=J,µ/J= 1 andµ/(kBT) = 100.\nAlthoughwe cannolongermakeuseofthefluctuation-\ndissipation theorem, we are nevertheless able to deter-\nmine the fluctuations by calculating the low-energy be-\nhavioroftheKeldyshcomponentoftheresponsefunction\nin the nonzero-voltage situation. It is given by\nΠK(ǫ) =−i\n2/integraldisplaydǫ′\n(2π)A2(ǫ′){[N(µL−ǫ′)+N(µR−ǫ′)]\n×[N(ǫ′−µL)+N(ǫ′−µR)]}. (45)\nWe define an effective temperature by means of\nkBTeff(T,V)≡iΠK(ǫ)∆2\n2α(V). (46)\nThis definition is motivated by the fact that, as we\nmention below, the spin direction obeys the stochas-\ntic Landau-Lifschitz-Gilbert equation with voltage-\ndependentdampingandfluctuationscharacterizedbythe\nabove effective temperature43. From the expression for\nα(V) and ΠK(ǫ) we see that in the limit of zero bias\nvoltage we recover the equilibrium result Teff=T. In\nthe situation that |e|Vis substantially larger than kBT,\nwhichis usuallyapproachedin experiments, wehavethat\nkBTeff(T,V)≃|e|V\n4+kBT\n2, (47)7\nwhich in the limit that |e|V≫kBTbecomes indepen-\ndent of the actual temperature of the leads. In Fig. 1\nthe effective temperature as a function of bias voltage is\nshown, using the expression for ΠK(ǫ) given in Eq. (45).\nThe parameters are the same as before, i.e., ∆ /J= 0.1,\nJC=J,µ/J= 1 andµ/(kBT) = 100. Clearly the ef-\nfective temperature changes from Teff=Tat zero bias\nvoltagetotheasymptoticexpressioninEq.(47)shownby\nthe dashed line in Fig. 1. The crossover between actual\ntemperatureandvoltageasameasureforthefluctuations\nis reminiscent of the theory of shot noise in mesoscopic\nconductors44. This is not surprising, since in the single-\nsite model we use the noise in the equation of motion ul-\ntimately arises because of fluctuations in the number of\nelectronsin thesingle-sitesystem, andisthereforeclosely\nrelated to shot noise in the current through the system.\nForoset al.35calculate the magnetization noise arising\nfrom spin currentshot noisein the limit that |e|V≫kBT\nand|e|V≪kBT. In these limits our results are similar\nto theirs.\nWith the above definition of the effective temperature\nwefind that in the nonzerobiasvoltagesituationthe spin\ndirection obeys the stochastic Landau-Lifschitz-Gilbert\nequation, identical in form to the equilibrium case in\nEqs. (38) and (40), with the Gilbert damping parame-\nter and temperature replaced according to\nα0→α(V) ;\nT→Teff(T,V). (48)\nMoreover, the transport-steady-state probability distri-\nbution for the direction of the spinis a Boltzmann distri-\nbution with the effective temperature characterizing the\nfluctuations.\nV. DISCUSSION AND CONCLUSIONS\nWe have presented a microscopic derivation of the\nstochastic Landau-Lifschitz-Gilbert equation for a semi-\nclassical single spin under bias. We found that the\nGilbert damping parameter is voltage dependent and to\nlowest order acquires a correction linear in the bias volt-\nage, in agreement with a previous study for a slightly\ndifferent model27. In addition, we have calculated the\nstrength of the fluctuations directly without using the\nfluctuation-dissipation theorem and found that, in the\nlow-frequency regime, the fluctuations are characterized\nby a voltage and temperature dependent effective tem-\nperature.\nTo arrive at these results we have performed a low\nfrequency expansion of the various correlation functions\nthat enter the theory. Such an approximation is valid as\nlong as the dynamics is much slower than the times set\nby the other energy scales in the system such as temper-\nature and the Fermi energy. Moreover, in order for the\nleads to remain in equilibrium as the spin changes direc-\ntion, the processes in the leads that lead to equilibrationhave to be much faster than the precession period of the\nmagnetizationspin. Both these criteria are satisfied in\nexperiments with magnetic materials. In principle how-\never, the full Langevinequationderivedin Sec. II alsode-\nscribes dynamics beyond this low-frequency approxima-\ntion. The introduction of the effective temperature relies\non the low-frequency approximation though, and for ar-\nbitrary frequencies such a temperature can no longer be\nuniquely defined28.\nAn effective temperature for magnetization dynam-\nics has been introduced before on phenomenological\ngrounds36,37,38. Interestingly, the phenomenological ex-\npression of Urazhdin et al.36, found by experimentally\nstudying thermal activation of current-driven magneti-\nzation reversal in magnetic trilayers, has the same form\nas our expression for the effective temperature in the\nlarge bias-voltage limit [Eq. (47)] that we derived micro-\nscopically. Zhang and Li37, and Apalkov and Visscher38,\nhave, on phenomenological grounds, also introduced an\neffective temperature to study thermally-assisted spin-\ntransfer-torque-induced magnetization switching. In\ntheir formulation, however, the effective temperature is\nproportional to the real temperature because the current\neffectively modifies the energy barrier for magnetization\nreversal.\nForoset al.35consider spin current shot noise in the\nlargebias-voltagelimit andfind forsufficiently largevolt-\nage that the magnetization noise is dominated by shot\nnoise. Moreover, they also consider the low bias-voltage\nlimit and predict a crossover for thermal to shot-noise\ndominated magnetization fluctuations. Our main result\nin Fig. 1 provides an explicit example of this crossover\nfor a simple model system obtained by methods that are\neasily generalized to more complicated models. In the\nexperiments of Krivorotov et al.45the temperature de-\npendence of the dwell time of parallel and anti-parallel\nstates of a current-driven spin valve was measured. At\nlow temperatures kBT/lessorsimilar|e|Vthe dwell times are no\nlonger well-described by a constant temperature, which\ncould be a signature of the crossover from thermal noise\nto spin current shot noise. However, Krivorotov et al.\ninterpret this effect as due to ohmic heating, which is\nnot taken into account in the model presented in this\npaper, nor in the work by Foros et al.35. Moreover, in\nrealistic materials phonons provide an additional heat\nbath for the magnetization, with an effective tempera-\nture that may depend in a completely different manner\non the bias voltage than the electron heat-bath effec-\ntive temperature. Nonetheless, we believe that spin cur-\nrent shot noise may be observable in future experiments\nand that it may become important for applications as\ntechnological progress enables further miniaturization of\nmagnetic materials. Moreover, the formalism presented\nhereisanimportantstepinunderstandingmagnetization\nnoise from a microscopic viewpoint as its generalization\nto more complicated models is in principle straightfor-\nward. Possible interesting generalizations include mak-\ning one of the leads ferromagnetic (see also Ref. [46]).8\nSince spin transfer torques will occur on the single spin\nas a spin-polarized current from the lead interacts with\nthe single-spin system, the resulting model would be a\ntoy model for microscopically studying the attenuation\nof spin transfer torques and current-driven magnetiza-\ntion reversal by shot noise. Another simple and use-\nful generalization would be enlarging the system to in-\nclude more than one spin. The formalism presented here\nwould allow for a straightforward microscopic calcula-\ntion of Gilbert damping and adiabatic and nonadiabatic\nspin transfer torques which are currently attracting a lot\nof interest in the context of current-driven domain wall\nmotion6,7,8,9,10,11,12. The application of our theory in thepresentpaperis, in additiontoitsintrinsicphysicalinter-\nest, chosen mainly because of the feasibility of analytical\nresults. Theapplicationsmentionedabovearemorecom-\nplicated and analytical results may be no longer obtain-\nable. 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Buhrman,\nPhys. Rev. Lett. 93, 166603 (2004) .\n46New. J. Phys. 10, 013017 (2008)."    },    {        "title": "2309.14167v3.Ultrafast_Demagnetization_through_Femtosecond_Generation_of_Non_thermal_Magnons.pdf",        "content": "Ultrafast Demagnetization through Femtosecond Generation of Non-thermal Magnons\nMarkus Weißenhofer1, 2,∗and Peter M. Oppeneer1\n1Department of Physics and Astronomy, Uppsala University, P. O. Box 516, S-751 20 Uppsala, Sweden\n2Department of Physics, Freie Universit¨ at Berlin, Arnimallee 14, D-14195 Berlin, Germany\n(Dated: January 17, 2024)\nUltrafast laser excitation of ferromagnetic metals gives rise to correlated, highly non-equilibrium\ndynamics of electrons, spins and lattice, which are, however, poorly described by the widely-used\nthree-temperature model (3TM). Here, we develop a fully ab-initio parameterized out-of-equilibrium\ntheory based on a quantum kinetic approach–termed (N+2) temperature model –that describes\nmagnon occupation dynamics due to electron-magnon scattering. We apply this model to per-\nform quantitative simulations on the ultrafast, laser-induced generation of magnons in iron and\ndemonstrate that on these timescales the magnon distribution is non-thermal: predominantly high-\nenergy magnons are created, while the magnon occupation close to the center of the Brillouin zone\neven decreases, due to a repopulation towards higher energy states via a so-far-overlooked scattering\nterm. We demonstrate that the simple relation between magnetization and temperature computed\nat equilibrium does not hold in the ultrafast regime and that the 3TM greatly overestimates the\ndemagnetization. The ensuing Gilbert damping becomes strongly magnon wavevector dependent\nand requires a description beyond the conventional Landau-Lifshitz-Gilbert spin dynamics. Our ab-\ninitio -parameterized calculations show that ultrafast generation of non-thermal magnons provides\na sizable demagnetization within 200fs in excellent comparison with experimentally observed laser-\ninduced demagnetizations. Our investigation emphasizes the importance of non-thermal magnon\nexcitations for the ultrafast demagnetization process.\nI. INTRODUCTION\nThe discovery that magnetic order can be manipu-\nlated on sub-picosecond timescales by femtosecond laser\npulses [1–3] has fueled the emergence of intensive exper-\nimental and theoretical research efforts in the field of ul-\ntrafast magnetization dynamics. What makes this field\nparticularly interesting, apart from its technological po-\ntential in future memory and spintronic devices [4, 5], is\nthat many well-established physical paradigms cannot be\nsimply transferred from the equilibrium to the ultrafast\nregime, due to its highly non-equilibrium nature. Relat-\nedly, albeit more than 25 years of intense research, the\nunderlying mechanisms of ultrafast demagnetization are\nstill heavily debated [6–8]: while some works [9–14] lean\ntowards longitudinal excitations – i.e., the reduction of\nthe magnetic moment carried by each atom due to the de-\ncrease of exchange splitting – others [15–19] hint at trans-\nverse spin excitations – a reduction of the average magne-\ntization due to the mutual tilting of the moments carried\nby different atoms – as the main contribution. Non-local\ncontributions due to superdiffusive spin currents [20, 21]\nare relevant in certain situations [22–25]. However, it has\nbecome evident that they are most likely not the only\nmechanism of ultrafast demagnetization [26, 27].\nTheoretical models describing ultrafast magnetization\ndynamics typically rely on a separation of electronic,\nphononic and – if magnetization dynamics are to be con-\nsidered separately – spin degrees of freedom. Beaurepaire\net al. [1] introduced the three-temperature model (3TM)\nto explain the flow of the energy transferred by the laser\n∗markus.weissenhofer@fu-berlin.deby assuming that each subsystem is internally in thermal\nequilibrium and the system can hence be described by\nthree temperatures (for electrons, phonons and spins), to-\ngether with the respective distributions (Fermi-Dirac and\nBose-Einstein). However, it was pointed out in numer-\nous investigations that the distributions are non-thermal\non ultrafast timescales [28–37]. Also, the 3TM discards\ncompletely the transfer of angular momentum due to de-\nmagnetization, which, according to recent experiments\n[38, 39], appears to be primarily to the lattice.\nTransverse demagnetization is often studied using\natomistic spin dynamics simulations based on the\nstochastic Landau-Lifshitz-Gilbert (LLG) equation to-\ngether with an extended Heisenberg model [40–42], which\ncan successfully reproduce experimentally measured de-\nmagnetization curves [43, 44]. The stochastic LLG is\na Langevin-type equation with a coupling to a heat\nbath with given temperature via a single parameter, the\nGilbert damping parameter. This parameter includes all\npossible contributions – Fermi surface breathing, crystal\ndefects, coupling to phonons, s−dcoupling, etc. [45–52] –\nto damping and while it can in principle be obtained from\nab initio calculations, in practice it is typically taken from\nexperimental measurements of ferromagnetic resonance\n(FMR) [53]. On the one hand, this ensures the versatility\nof atomistic spin dynamics simulations, but on the other\nhand, it obscures the details of the underlying micro-\nscopic energy and angular momentum transfer processes\n- which are crucial for understanding the fundamentals\nof ultrafast demagnetization. For this reason, steps have\nbeen taken in recent years to explicitly consider the cou-\npling of spins to phonons [54–62] and electrons [63–65].\nAlso, due to the classical nature of the commonly used\nstochastic LLG, the equilibrium magnon occupations cal-arXiv:2309.14167v3  [cond-mat.mtrl-sci]  15 Jan 20242\nculated by it follow Rayleigh-Jeans rather than Bose-\nEinstein statistics, henceforth leading to the wrong tem-\nperature scaling of the magnetization [66, 67]. Implemen-\ntation of quantum statistics in the spin-dynamics simu-\nlations can however provide the correct low-temperature\nscaling of the magnetization [68, 69].\nIn this work, we investigate the laser-induced gener-\nation of magnons, the low energy transverse excitations\nof the spin system, due to electron-magnon scattering.\nWe develop a quantum kinetic approach, which will be\ntermed (N+2)-temperature model [(N+2)TM], to per-\nform quantitative simulations of the time evolution of\nthe non-thermal magnon dynamics in bcc iron. Being\nbased on ab initio parameters and considering also non-\nthermal magnon distributions, our work goes well beyond\nwhat has been done in Refs. [63, 64, 70] and the conven-\ntional 3TM. In addition, we show that the 3TM and its\nrelevant parameters can be obtained from our (N+2)TM\nand, with that, from ab initio calculations. Importantly,\nusing ab initio calculated input parameters, our quantum\nkinetic theory predicts a sizable and ultrafast demagne-\ntization of iron within 200 fs, in excellent agreement with\nexperiments [15].\nII. OUT-OF-EQUILIBRIUM MAGNON\nDYNAMICS MODEL\nTo describe the time evolution of the ultrafast non-\nthermal magnon occupation dynamics, we assume that\ntheir creation and annihilation is dominated by electron-\nmagnon scattering processes. In this work, we use the\nsp−dmodel [71, 72] to describe such processes. The\nbasic idea of both s−dmodel and sp−dmodel is the\nseparation of electrons in localized ( dband) electrons and\nitinerant ( sband or sandpbands) electrons. The mag-\nnetic moments of the delectrons make up the Heisenberg-\ntype [73] magnetic moments of constant length, the small\nenergy excitations of which are the magnons. The itin-\nerant electrons are described within a Stoner-type model\n[74]. While an unambiguous identification of spandd\nelectrons as localized and itinerant is strictly speaking\nnot possible, it has nonetheless been established in liter-\nature that these models provide a suitable framework for\nthe description of electron-spin interaction in many phe-\nnomena relevant for spintronics, e.g. magnetic relaxation\n[75–77], ultrafast demagnetization [63–65, 70, 78–80] and\nspin torques [81].\nWe assume local exchange between the itinerant and\nlocalized spins, as given by the Hamiltonian ˆHem∼PN\ni=1ˆsitin·ˆSloc\ni, with Nbeing the number of atoms,\nandˆsitinand ˆSloc\nithe spin operators for itinerant ( sp)\nelectrons and localized ( d) electrons at atom i. In sec-\nond quantization and second order in magnon variables(details in Method Section V A), the Hamiltonian reads\nˆHem≈ −∆X\nkν\u0010\nˆc†\nkν↑ˆckν↑−ˆc†\nkν↓ˆckν↓\u0011\n−∆r\n2\nSNX\nkνν′,q\u0010\nˆc†\nk+qν↑ˆckν′↓ˆb†\n−q+ ˆc†\nk+qν↓ˆckν′↑ˆbq\u0011\n+∆\nSNX\nkνν′,qq′\u0010\nˆc†\nk−q+q′ν↑ˆckν′↑−ˆc†\nk−q+q′ν↓ˆckν′↓\u0011\nˆb†\nqˆbq′.\n(1)\nHere, ∆ is the sp−dexchange parameter, Sis the ab-\nsolute value of the localized spins, kandqare vectors in\nreciprocal space, ˆ c(†)\nkνσis the fermionic electron annihila-\ntion (creation) operator for the itinerant electrons – with\nνbeing the band index and σ∈ {↑,↓}– and ˆb(†)\nqis the\nbosonic magnon annihilation (creation) operator. The\nfirst term in Equation (1) describes the spin-splitting of\nthe itinerant electrons due to the exchange with the lo-\ncalized magnetic moments, the second one the excitation\n(annihilation) of a magnon due to a spin flip process and\nthe third one the spin-conserving scattering of a magnon\nand an electron from one state to another. It is worth\nnoting that the second term leads to a transfer of both\nenergy and angular momentum (i.e., spin) – since it can\nchange the total number of magnons – while the third\nterm can only transfer energy. For this reason, this term\nwas discarded earlier works [63–65], however, our quanti-\ntative analysis reveals that the energy transferred by this\nterm can exceed the energy transferred by the term first\norder in magnon operators.\nWe complete our Hamiltonian H=ˆHe+ˆHm+ˆHem\nby considering ˆHe=P\nkνσεkνσˆc†\nkνσˆckνσand ˆHm=P\nqℏωqˆb†\nqˆbq, with εkνσ=εkν−2∆δσ↑being the mode\nand spin dependent electron energies that are calcu-\nlated from first-principles calculations and ℏωqbeing the\nmagnon energies. Note that we have absorbed the term\nzero-th order in magnon variables in Equation (1) in the\notherwise spin-independent ˆHe.\nNext, we use the Hamiltonian introduced above to\nconstruct a quantum kinetic approach for the descrip-\ntion of the out-of-equilibrium dynamics of electrons and\nmagnons. We define the rates of energy exchange be-\ntween both subsystems as\n˙Em=X\nqℏωq˙nq (2)\n˙Ee=X\nkνσεkνσ˙fkνσ=−X\nqℏωq˙nq. (3)\nwhere the dot represents temporal derivative and with\nthe electron ( fkνσ) and magnon ( nq) occupation num-\nbers. The equivalence in Equation (3) results from the\nconservation of total energy. The time derivatives of\nthe occupation numbers can be calculated by applying\nFermi’s golden rule to the scattering Hamiltonian (1).3\nTo simplify the calculations, we further assume a ther-\nmal electron distribution and can hence introduce a sin-\ngle electronic temperature Tethat relates to the occu-\npation of electronic states via the Fermi-Dirac distribu-\ntion. This allows us to apply and also extend (by in-\ncluding terms second order in the bosonic operators) the\nideas laid out in Allen’s seminal work on electron-phonon\ninteraction [82] to electron-magnon scattering, yielding\n˙nq=\u0002\nnBE(ωq, Te)−nq\u0003\nγq+P\nq′\u0002\n(nq+ 1)nq′nBE(ωq−\nωq′, Te)+(q↔q′)\u0003\nΓqq′, with nBE(ωq, Te) = [eℏωq\nkBTe−1]−1\nbeing the Bose-Einstein distribution evaluated at the\nelectron temperature. The scattering rates are given by\nγq=4π∆2\nSNωqI↑↓(Te)X\nkνν′δ(εF−εk−qν↑)δ(εF−εkν′↓),\n(4)\nΓqq′=2π∆2\nS2N2(ωq−ωq′)X\nσIσσ(Te)\n×X\nkνν′δ(εF−εk−q+q′νσ)δ(εF−εkν′σ),(5)\nwith εFbeing the Fermi energy. The functions Iσσ′(Te)\nhave the property lim Te→0Iσσ′(Te) = 1 and account\nfor the smearing of the Fermi-Dirac distribution at high\nelectron temperatures, similar to what has been derived\nfor electron-phonon scattering [35]. The expression for\nIσσ′(Te) and details of the derivation of Equations (4)–\n(5) are in the Method Section V A. Note that a com-\nparison with linear spin-wave theory in the framework\nof the Landau-Lifshitz-Gilbert equation [83] reveals that\nγq/ωq=αqcan be viewed as a mode-dependent Gilbert\ndamping parameter.\nDue to the assumption that the electron occupation\nnumbers follow the Fermi-Dirac distribution at all times,\nthe change in electron energy is determined by the\nchange in Te, i.e., ˙Ee=P\nkνσεkνσ(∂fkνσ/∂T e)˙Te=\nCe˙Te, with the electronic heat capacity Ce=P\nkνσεkνσ(∂fkνσ/∂T e). By additionally considering the\nabsorption of a laser pulse with power P(t) by the\nelectrons and a coupling of the electrons to a phonon\nheat bath as in the 2TM, we finally obtain our out-of-\nequilibrium magnon dynamics model:\n˙nq=h\nnBE(ωq, Te)−nqi\nγq\n+X\nq′h\n(nq+ 1)nq′nBE(ωq−ωq′, Te) + (q↔q′)i\nΓqq′,\n(6)\n˙Te=1\nCeh\n−X\nqℏωq˙nq+Gep(Tp−Te) +P(t)i\n, (7)\n˙Tp=−Gep\nCp(Tp−Te). (8)\nHere, Tp,Cpand Gepare the phonon temperature\nand heat capacity and electron-phonon coupling con-stant, respectively. Note that we do not consider di-\nrect magnon-phonon coupling, which has been shown\nto be a reasonable approximation for 3 dferromagnets\n[43, 44]. We would like to point out that the non-\nthermal magnon occupations nqcan be translated to\nmode-specific temperatures via the Bose-Einstein distri-\nbution, Tq:=ℏωq/(kBln(n−1\nq+ 1)). Based on this – and\nin distinction from the 3TM – we term the framework\nprovided by Equations (6)-(8) the (N+2)-temperature\nmodel ((N+2)TM). Below, we reveal by solving these\ncoupled equations numerically that they provide a vi-\nable framework to describe laser-induced ultrafast mag-\nnetization dynamics and the generation of non-thermal\nmagnons, going beyond the well-established 3TM.\nBefore doing so, we want to shortly discuss the relation\nbetween the (N+2)TM introduced here and the 3TM. Al-\nbeit their phenomenological nature, the 2TM ( TeandTp)\nand the 3TM ( Te,TpandTm) have been successfully ap-\nplied to explain a plethora of phenomena [84], perhaps\nmost prominently by Beaurepaire et al. to describe the\nultrafast demagnetization of Ni [1]. Allen [82] and Man-\nchon et al. [78] demonstrated that the 2TM and the 3TM\ncan be derived from a microscopic out-of-equilibrium ap-\nproach similar to the one used here. By assuming instan-\ntaneous relaxation of the magnon occupation numbers\nto the Bose-Einstein distribution with a single magnon\ntemperature Tm, our (N+2)TM reduces to the 3TM (in\nabsence of magnon-phonon coupling),\nCm˙Tm=Gem(Te−Tm),\nCe˙Te=Gem(Tm−Te) +Gep(Tp−Te) +P(t),\nCp˙Tp=Gep(Te−Tp),(9)\nwith the magnon heat capacity Cm=P\nqCq=P\nqℏωq(∂nq/∂T m) and the electron-magnon coupling\nconstant\nGem=X\nqCqh\nγq+X\nq′kBTm\nℏωqΓqq′i\n. (10)\nDetails of the derivation are found in Method Section\nV B. The above expression goes beyond what was derived\nin Ref. [78] by including terms second order in magnon\nvariables and allows us to determine the electron-magnon\ncoupling fully based on ab initio parameters. We would\nlike to point out that it can be extended further by going\nto higher order in the magnon variables.\nIII. RESULTS\nA. Magnon lifetimes and Gilbert damping\nWe apply the (N+2)TM model defined by Equations\n(6)-(8) to bcc iron. To obtain a full solution of the out-\nof-equilibrium dynamics, it is required to calculate ma-\nterial specific quantities. First, we estimate ∆ ≈0.75 eV\nfrom the band structure and with that we compute the4\nΓ H N Γ P H020406080100120magnon frequency (THz)\n100101102103\nlifetime (ps)\nFigure 1. Magnon dispersion of bcc iron with lifetimes γ−1\nq\ngiven as color code, shown along high-symmetry lines of the\nBZ. The lifetimes are due to the first-order contribution to\nthe electron-magnon scattering.\nquantities γq, Γqq′andIσσ′(Te), both using the full-\npotential linear augmented plane wave code ELK [85]\n(details can be found in the Method Section V C). For\nbcc iron it turns out that Iσσ′(Te) only scales weakly\nwith temperature and hence we use the low tempera-\nture limit Iσσ′(Te) = 1 hereinafter. The parameters gov-\nerning the magnon energies ℏωq=S(2d+P\njJij[1−\nexp(−iq·(rj−ri))]) were taken from earlier works: the\nexchange constants Jijare from first-principles calcula-\ntions [86] and the magneto-crystalline anisotropy energy\nd= 6.97µeV per atom is from experiments [87]. Further,\nwe used the saturation moment µs= 2.2µBand spin\nS= 2.2/2. Based on these parameters and the formulas\nderived above, we get Cm= 5.720×104Jm−3K−1and\nGem= 6.796×1017Wm−3K−1atTm= 300 K. Notably,\nthe term first order in magnon variables leads to a con-\ntribution to Gemthat is one order of magnitude smaller\nthan the second-order term. We further use the room-\ntemperature values Ce= 1.013×105Jm−3K−1,Cp=\n3.177×106Jm−3K−1andGep= 1.051×1018Wm−3K−1\nthat were obtained in Refs. [35, 37] from first-principles\ncalculations. The influence of a temperature dependent\nelectronic heat capacity Ceon the demagnetization is dis-\ncussed in the Supporting Information.\nBoth ωqand the inverse of γq, i.e., the lifetime of\nmagnons due to the contribution to electron-magnon\nscattering linear in the magnon variables, are shown in\nFigure 1 along high-symmetry lines of the Brillouin zone\n(BZ). It can readily be observed that the lifetimes of\nhigh-frequency magnons are drastically reduced as com-\npared to the low energy ones. The q-dependent lifetimes\ngive rise to mode-specific Gilbert damping αq(=ωq/γq).\nOur finding of mode-dependent Gilbert damping is con-\nsistent with experiments [88] and also with a recent field-\ntheory derivation [89]. The computed αqvalues, shown\nin Method Section V C, range between 1 .5×10−3and\n1.08×10−2. These values are close to the experimentally\nobtained ones (via FMR measurements) for Fe ranging\nfrom 1 .9×10−3to 7.2×10−3[90–95], however with a\n0 1 2 3300400500600700temperature T (K)(a)\n0 1 2 3\ntimet(ps)−1.0−0.50.0∆M/M 0(%) (b)Te\nTp\n/angbracketleftTq/angbracketrightT3TM\ne\nT3TM\np\nT3TM\nm\n0.0 0.2 0.4 0.6 0.8 1.0\nlaser fluence (mJ /cm2)0510demagnetization (%)\n(c)Figure 2. Laser-induced ultrafast non-equilibrium dynamics\nof iron calculated from an ab initio parameterized model.\n(a) Temporal evolution of electron temperature Te, phonon\ntemperature Tpand average magnon temperature ⟨Tq⟩=\n1/NP\nqTqobtained by the (N+2)TM (solid lines). The\nblue shaded region indicates the temperature range within\nwhich all magnon temperatures are contained. Dashed lines\nshow the results of the 3TM solved with ab initio calcu-\nlated input parameters. (b) Relative change of total mag-\nnetization of the localized magnetic moments ∆ M/M 0=P\nq(ninit\nq−nq)/(NS−P\nqninit\nq), with ninit\nq=nBE(ωq,300 K)\nbeing the occupation number before the laser pulse. (c) De-\nmagnetization max( |∆M/M 0|) versus laser fluence computed\nfor a ferromagnetic layer with a thickness of 20 nm. The dot-\nted line serves as a guide to the eye.\nsomewhat larger variation with qas compared to what\nwas reported in Ref. [83].\nWe note that the q-dependent Gilbert damping goes\nbeyond the conventional LLG description which assumes\none single damping parameter for all spin dynamics.\nMoreover, a further distinction between the current the-\nory and the LLG framework is that, in the latter, there\nis a single damping term that governs both the energy\nand angular momentum transfer [96], whereas the cur-\nrent theory has two terms [see Equation (1)], one that\ntransfers energy and angular momentum and one that\ntransfers only energy. As shown in the following, this 2nd\nterm is found to be important for non-thermal magnon5\ngeneration.\nB. Ultrafast dynamics\nBased on the above-given parameters, we calculate\nthe coupled out-of-equilibrium magnon, electron, and\nphonon dynamics induced by a Gaussian laser pulse\nP(t) = A/p\n2πζ2exp[−(t/ζ)2/2] with A= 9.619×\n107Jm−3andζ= 60 fs for N= 203magnon modes. Note\nthat this value of Atranslates to an absorbed fluence of\n0.19 mJ /cm2for a ferromagnetic layer with thickness of\n20 nm, which is a typical thickness in ultrafast demagne-\ntization experiments [1].\nFigure 2(a) depicts the time evolution of electron,\nphonon and average magnon temperature – together with\nthe temperature range of all magnon temperatures – cal-\nculated using the (N+2)TM. The electron temperature\nreaches a maximum of 685 K at around 52 fs after the\nmaximum of the laser pulse (located at t= 0) and con-\nverges to the phonon temperature in less than 1 .5 ps. The\nmaximum of the average magnon temperature of 520 K\nis reached only slightly after the electronic one at around\n136 fs, followed by a convergence to the electronic and\nphononic temperature to a final temperature of around\n329 K at 3 ps, in agreement with what can be estimated\nfrom the energy supplied by the laser pulse and the in-\ndividual heat capacities via ∆ T=A/(Cm+Ce+Cp) =\n28.8 K. Notably, the magnon temperatures still cover a\nrange of around 50 K at this point in time. Our results\nclearly demonstrate the shortcomings of the conventional\n3TM (shown as dotted lines): While the initial increase\nof temperatures is comparable to the (N+2)TM, magnon\nthermalization happens much faster in the 3TM.\nIn Figure 2(b), we show the laser-induced change in\nmagnetization (associated with the localized magnetic\nmoments) due to the creation of additional magnons. We\nobserve ultrafast transversal demagnetization of around\none percent in less than 300 fs, demonstrating that the\ntimescales obtained by our ab initio based calculations\nare in reasonable agreement with experimental measure-\nments (see, e.g., [15, 97–99]). Notably, the minimum\nof the magnetization and the maximum in the average\nmagnon temperature computed by the (N+2)TM are at\ndifferent points in time. Also, the drop in the (localized)\nmagnetization is much less pronounced than expected\nfrom the increase in average temperature: in thermal\nequilibrium, a temperature increase from 300 K to above\n500 K approximately leads a demagnetization of 20% for\niron [100]. These observations clearly demonstrate the\nshortcomings of the 3TM – where a thermal magnon dis-\ntribution at all times is assumed – and underline the im-\nportance of treating the full, non-thermal magnon distri-\nbution in the ultrafast regime.\nFigure 2(c) depicts the maximum of the demagnetiza-\ntion versus laser fluence for an iron layer of 20 nm. We\nfind a nonlinear dependence, which is a result of the non-\nlinearity of our (N+2)TM, and a substantial demagneti-\n0 2 4 6 8 10\ntimet(ps)0.850.900.951.00M/M 0\n(N+2)TM\nexperimentFigure 3. Comparison between experiment and the (N+2)TM\ntheory for ultrafast demagnetization in iron. The experimen-\ntal data (symbols) are those of Carpene et al. [15] and the\nsolid lines are calculated from the ab initio parameterized\n(N+2)TM.\nzation of around ten percent at 0 .95 mJcm−2. We note\nthat for high fluences, higher-order magnon-magnon scat-\ntering terms that are not included in the current model\ncould start to play a role.\nThe obtained amount of demagnetization and the mag-\nnetization decay time (below 200 fs) for this fluence are\ncomparable with experiments, which suggests that ultra-\nfast magnon excitation [15–17] provides a viable mech-\nanism for ultrafast laser-induced demagnetization. It\nis also consistent with time-resolved extreme ultraviolet\nmagneto-optical and photoemission investigations that\ndetected magnon excitations during ultrafast demagneti-\nzation of elemental ferromagnetic samples [18, 101].\nFor a more precise examination of the predictions\nof the (N+2)TM, we compare the calculated time-\ndependent demagnetization with experimental data for\nFe in Figure 3. The experimental data were measured\nby Carpene et al. [15] on a 7-nm thin film, using the\ntime-resolved magneto-optical Kerr effect for two differ-\nent pump laser fluences of 1 .5 mJcm−2and 3 mJcm−2.\nIn the calculations we used an absorbed laser fluence\nthat is about five times lower, as the exact value of the\nabsorbed fluence in experiments is difficult to estimate\n(due to influence of optical losses, sample reflection,\netc). Specifically, in the simulations we used absorbed\nlaser energies of 433 Jcm−3and 693 Jcm−3in a 7-nm Fe\nfilm. Figure 3 exemplifies that not only the amount of\ndemagnetization but also the full time dependence of\nthe demagnetization predicted by the (N+2)TM is in\nremarkable agreement with experiments.\nC. Non-thermal magnon dynamics\nNext, we analyze the non-thermal magnon dynam-\nics in more detail in Figure 4. There, we show the\nmagnon temperatures versus frequency (a) and along\nhigh-symmetry lines of the BZ (b) at different points in\ntime. The laser pulse primarily heats up high energy\nmagnons, while the temperature of low energy magnons6\n300500\n−250 fs\n300500\n0 fs\n300500temperature (K)\n250 fs\n300500\n500 fs\n0 20 40 60 80 100 120\nmagnon frequency (THz)300500\n1000 fs\nΓ H N Γ P H−200 fs−100 fs0 fs100 fs200 fs300 fs400 fs500 fs600 fs700 fs800 fs900 fs1000 fs\n300350400450500550600\ntemperature (K)(a) (b)\nFigure 4. Magnon temperatures of iron during ultrafast laser excitation at different points in time (w.r.t. the maximum of\nthe laser pulse) calculated from the ab initio parameterized (N+2)TM. (a) Magnon temperatures (dots) versus frequency. The\nsolid line indicates the electron temperature. (b) Magnon dispersion and their temperatures, depicted by the color code, shown\nalong high-symmetry lines of the BZ.\nbarely changes and even decreases slightly in the vicinity\nof the Γ point (the temperatures drop by up to around\n2.5 K). This surprising observation is caused by a\nredistribution of magnons from this region to other\nparts of the BZ due to the term second order in the\nmagnon operators in Equation (1); the effective second\norder scattering rate γ(2)\nq:=P\nq′Γqq′is negative for low\nmagnon frequencies (more details can be found in the\nMethod Section V C). It is also observed that although\nthe magnon temperatures reached after the laser pulse\nare generally higher at higher frequencies, however, there\nis not necessarily a monotonous increase of temperature\nwith frequency at all times: e.g., at 100 fs after the laser\npulse [Figure 4(b)], the temperatures at the points H, N,\nand P is higher than in between these points. Notably,\nthe position of the maximum magnon temperature in\nthe BZ also varies with time.\nD. Discussion\nDifferent physical mechanisms have been proposed for\nultrafast demagnetization in elemental 3 dferromagnets\n[9, 11, 15, 18]. The preeminent mechanisms are Elliott-\nYafet (EY) electron-phonon spin-flip scattering [11, 13]\nand ultrafast magnon generation [15]. In the former, a\nStoner-type picture is used to model the longitudinal re-\nduction of the atomic moment due to electron-phonon\nspin-flip scattering, whereas the latter is based on length-\nconserving transverse spin-wave excitations. Experimen-\ntal indications of electron-phonon scattering [38, 39] aswell as of electron-magnon scattering have been reported\n[18, 101].\nThe strength of the different demagnetization channels\nis an important issue in the on-going discussion on the\ndominant origin of ultrafast demagnetization [7]. Ab ini-\ntiocalculated quantities such as the EY spin-flip prob-\nability are essential to achieve reliable estimates [102–\n104]. Griepe and Atxitia [14] recently employed the\nmicroscopic 3TM [11] and obtained quantitative agree-\nment with measured demagnetizations for the elemental\n3dferromagnets. They compared the fitted EY spin-\nflip probability αsfwith ab initio calculated values [104]\nand found these to be in good agreement, in support of\nan electron-phonon mechanism of ultrafast demagnetiza-\ntion. A drawback of their employed approach is how-\never that only magnetization reducing spin flips are in-\ncluded. EY spin flips that increase the magnetization are\nalso possible and, including these would lead to a signifi-\ncantly smaller demagnetization amplitude [104]. This in\nturn would question again what amount of demagneti-\nzation is precisely due to EY electron-phonon spin flip\nscattering. Conversely, in our non-thermal magnon ap-\nproach we employ ab initio calculated quantities without\nfit parameter. We find that the ab initio predicted ultra-\nfast demagnetization agrees accurately with experiments,\nwhich provides a strong support for the prominence of the\nnon-thermal magnon channel to the ultrafast demagne-\ntization process.7\nIV. CONCLUSIONS\nWe have developed an ab initio parameterized quan-\ntum kinetic approach to study the laser-induced gen-\neration of magnons due to electron-magnon scattering,\nwhich we applied to iron. Our results clearly demon-\nstrate that on ultrafast timescales the magnon distribu-\ntion is non-thermal and that henceforth the simple re-\nlation between magnetization and temperature via the\nM(T) curves computed at equilibrium does not hold:\nsince predominantly high-energy magnons are excited the\nenergy transferred from the laser-excited electrons cre-\nates relatively few magnons and hence the demagneti-\nzation (proportional to the total number of magnons) is\nmuch less pronounced than expected from the increaseof the average magnon temperature. Notably, the num-\nber of magnons actually decreases near the center of the\nBrillouin zone, which is due to the scattering from low to\nhigh energy magnons by a previously neglected scattering\nterm that can transfer energy but not angular momen-\ntum. This term, which is not included in LLG simula-\ntions, is a crucial quantity for out-of-equilibrium magnon\ndynamics.\nOurab initio -based calculations of the induced demag-\nnetization in iron furthermore provide strong evidence\nthat non-thermal magnons are excited fast and lead to\na sizable demagnetization within 200 fs. The result-\ning time-dependent demagnetization agrees remarkably\nwell with experiments, which establishes the relevance of\nmagnon excitations for the process of ultrafast optically\ninduced demagnetization.\nV. METHOD\nA. Derivation of electron-magnon scattering rates\nIn this Method Section we derive the (N+2)TM for the description of non-thermal magnons from a microscopic\nHamiltonian for electron-magnon scattering. We start with a local sp−dmodel Hamiltonian,\nˆHem=−Jsp−dX\niδ(r−ri)ˆsitin·Sloc\ni, (11)\nwith Jsp−dbeing the sp−dvolume interaction energy, ˆsitin=ˆσbeing the spin operators of itinerant ( sandp)\nelectrons and Sloc\nibeing the localized ( d) spins located at ri. For now, we treat the latter as classical vectors. The\nexpectation value for a given spin wave function Ψ(r) is given by\n⟨ˆHem⟩=−Jsp−dX\niZ\nΨ†(r)δ(r−ri)ˆsitin·Sloc\niΨ(r)dr (12)\n=−Jsp−dX\niZ\nδ(r−ri)\u0000Ψ∗\n↑(r),Ψ∗\n↓(r)\u0001\b\nˆσxSx\ni+ ˆσySy\ni+ ˆσzSz\ni\t\u0012\nΨ↑(r)\nΨ↓(r)\u0013\ndr (13)\n=−Jsp−dX\niZ\nδ(r−ri)(\nΨ∗\n↑(r)Ψ↓(r)S−\ni+ Ψ∗\n↓(r)Ψ↑(r)S+\ni+ (Ψ∗\n↑(r)Ψ↑(r)−Ψ∗\n↓(r)Ψ↓(r))Sz\ni)\ndr.(14)\nHere, we have introduced S±\ni=Sx\ni±iSy\ni. Next, we perform a plane wave expansion of the wave functions (for a\nsingle band of itinerant electrons),\nΨσ(r) =1√\nVX\nkeik·rckσ, (15)\nand a Holstein-Primakoff transformation of the localized spins,\nS+\ni=p\n2S−b∗\nibibi, S−\ni=b∗\nip\n2S−b∗\nibi, Sz\ni=S−b∗\nibi, (16)\ntogether with introducing the Fourier transform of the magnon amplitudes\nb∗\ni=1√\nNX\nqe−iq·rib∗\nq, b i=1√\nNX\nqeiq·ribq. (17)8\nInsertion of (15)–(17) into (14) and keeping terms up to second order in magnon variables, we get\n⟨ˆHem⟩=−Jsp−d\nVX\niX\nkk′(r\n2S\nNX\nqe−i(k−k′+q)·ric∗\nk↑ck′↓b∗\nq+r\n2S\nNX\nqe−i(k−k′−q)·ric∗\nk↓ck′↑bq\n+Se−i(k−k′)·ri(c∗\nk↑ck′↑−c∗\nk↓ck′↓)−1\nNX\nqq′e−i(k−k′+q−q′)·ri(c∗\nk↑ck′↑−c∗\nk↓ck′↓)b∗\nqbq′) (18)\n=−Jsp−dSN\nVX\nk(c∗\nk↑ck↑−c∗\nk↓ck↓)−Jsp−dSN\nVX\nkqr\n2\nSN\u0010\nc∗\nk+q↑ck↓b∗\n−q+c∗\nk+q↓ck↑bq\u0011\n+Jsp−d\nVX\nkqq′\u0010\nc∗\nk−q+q′↑ck↑−c∗\nk−q+q′↓ck↓\u0011\nb∗\nqbq′.(19)\nFor multiple itinerant bands and in second quantization we obtain\nˆHem=−∆X\nkν(ˆc†\nkν↑ˆckν↑−ˆc†\nkν↓ˆckν↓)−∆r\n2\nSNX\nkνν′,q\u0010\nˆc†\nk+qν↑ˆckν′↓ˆb†\n−q+ ˆc†\nk+qν↓ˆckν′↑ˆbq\u0011\n+∆\nSNX\nkνν′,qq′\u0010\nˆc†\nk−q+q′ν↑ˆckν′↑−ˆc†\nk−q+q′ν↓ˆckν′↓\u0011\nˆb†\nqˆbq′.(20)\nwhere we have introduced ∆ =Jsp−dSN\nV. Note that due to the plane wave ansatz we have implicitly assumed that\nthe itinerant electrons are completely delocalized and interband scattering (from νtoν′̸=ν) fully contributes to the\nelectron-magnon scattering.\nNext, we use Fermi’s golden rule to get the change of the magnon occupation number nq=⟨ˆb†\nqˆbq⟩. Fermi’s golden\nrule computes the probability W(i→f) for a small perturbation term in the Hamiltonian, ˆH′(in our specific case,\nˆHem) via\nW(i→f) =2π\nℏ|⟨f|ˆH′|i⟩|2δ(Ef−Ei), (21)\nwhere |i⟩and|f⟩denote the initial and final state, respectively.\nWe start with the term first order in the magnon variables,\n˙n(1)\nq=W(nq→nq+ 1)−W(nq→nq−1)\n=2π\nℏ2∆2\nSNX\nkνν′\b\n(1−fk−qν↑)fkν′↓−(fk−qν↑−fkν′↓)nq\t\nδ(εkν′↓−εk−qν↑−ℏωq),(22)\nwith fkνσ=⟨ˆc†\nkνσˆckνσ⟩andεkνσandℏωqbeing the eigenenergies of electrons and magnons, respectively.\nHereinafter, we make the assumption that due to the fast equilibration processes for electrons, they always follow\nthe Fermi-Dirac distribution, fFD(εkνσ, Te) = [e(εkνσ−εF)/kBTe+ 1]−1,with a single electron temperature Te. Before\nwe continue we need the following relation,\nfFD(εkν′↓, Te)(1−fFD(εk−qν↑, Te))δ(εkν′↓−εk−qν↑−ℏωq) =\n(fFD(εk−qν↑, Te)−fFD(εkν′↓, Te))nBE(ωq, Te)δ(εkν′↓−εk−qν↑−ℏωq)(23)\nwith nBE(ωq, Te) = [eℏωq\nkBTe−1]−1being the Bose-Einstein distribution evaluated at the electron temperature. Now\nwe can simplify Equation (22), yielding\n˙n(1)\nq≈2π\nℏ2∆2\nSNX\nkνν′\u0002\nnBE(ωq, Te)−nq\u0003\n(fFD(εk−qν↑, Te)−fFD(εkν′↓, Te))δ(εkν′↓−εk−qν↑−ℏωq)\n=\u0002\nnBE(ωq, Te)−nq\u0003\nγq. (24)9\nWith γqbeing the linewidth – i.e., the inverse lifetime – of the magnon due to the first order contribution to electron-\nmagnon scattering. Following the ideas laid out by Allen [82] and Maldonado et al. [35], it can be computed as\nγq=2π\nℏ2∆2\nSNX\nkνν′[fFD(εk−qν↑, Te)−fFD(εkν′↓, Te)]δ(εkν′↓−εk−qν↑−ℏωq) (25)\n=2π\nℏ2∆2\nSNX\nkνν′Z\ndε δ(ε−εk−qν↑)Z\ndε′δ(ε′−εkν′↓)[fFD(ε, Te)−fFD(ε′, Te)]δ(ε′−ε−ℏωq) (26)\n≈2π\nℏ2∆2\nSNX\nkνν′δ(εF−εk−qν↑)δ(εF−εkν′↓)Z\ndεZ\ndε′[fFD(ε, Te)−fFD(ε′, Te)]δ(ε′−ε−ℏωq)g↑(ε)g↓(ε′)\ng↑(εF)g↓(εF)\n(27)\n≈2π\nℏ2∆2\nSNℏωqX\nkνν′δ(εF−εk−qν↑)δ(εF−εkν′↓)Z\ndε(−1)∂fFD(ε, Te)\n∂εg↑(ε)g↓(ε+ℏωq)\ng↑(εF)g↓(εF)(28)\n≈2π\nℏ2∆2\nSNℏωqX\nkνν′δ(εF−εk−qν↑)δ(εF−εkν′↓)Z\ndε(−1)∂fFD(ε, Te)\n∂εg↑(ε)g↓(ε)\ng↑(εF)g↓(εF)(29)\n=4π∆2\nNSωqX\nkνν′δ(εF−εk−qν↑)δ(εF−εkν′↓)I↑↓(Te) (30)\nwith εFbeing the Fermi energy, the spin-dependent density of states is gσ(ε) =P\nkνδ(ε−εkνσ) and the thermal\ncorrection factor given by\nIσσ′(Te) =Z\ndε(−1)∂fFD(ε, Te)\n∂εgσ(ε)g′\nσ(ε)\ngσ(εF)g′σ(εF). (31)\nIt is obvious that lim Te→0Iσσ′(Te) = 1. Note that we have used that the energy scale of magnons is much smaller\nthan the one of electrons, i.e., that ℏωq≪ε, ε′.\nThe contribution of the term second order in magnon variables to the occupation number can be calculated analogous\nand reads\n˙n(2)\nq=2π\nℏ\u0010∆\nSN\u00112X\nkνν′σ,q′n\n(nq+ 1)nq′\u0010\n(1−fFD(εk−q+q′νσ, Te))fFD(εkν′σ, Te)δ(ℏωq−ℏωq′+εk−q+q′νσ−εkν′σ)\u0011\n−\u0010\nq↔q′\u0011o\n(32)\n=2π\nℏ\u0010∆\nSN\u00112X\nkνν′σ,q′n\n(nq+ 1)nq′nBE(ωq−ωq′, Te)\u0010\nfFD(εk−q+q′νσ, Te)−fFD(εkν′σ, Te)\u0011\n×\nδ(ℏωq−ℏωq′+εk−q+q′νσ−εkν′σ)−\u0010\nq↔q′\u0011o(33)\n≈2π\nℏ\u0010∆\nSN\u00112X\nkνν′σ,q′n\n(nq+ 1)nq′nBE(ωq−ωq′, Te)(ℏωq−ℏωq′)δ(εF−εk−q+q′νσ)δ(εF−εkν′σ)Iσσ(Te)−\u0010\nq↔q′\u0011o\n(34)\n=2π\nℏ\u0010∆\nSN\u00112X\nq′n\n(nq+ 1)nq′nBE(ωq−ωq′, Te) +\u0000\nq↔q′\u0001oX\nkνν′σ(ℏωq−ℏωq′)δ(εF−εk−q+q′νσ)δ(εF−εkν′σ)Iσσ(Te)\n(35)\n=2π\nℏ\u0010∆\nSN\u00112X\nq′n\n(nq+ 1)nq′nBE(ωq−ωq′, Te) +\u0000\nq↔q′\u0001oX\nkνν′σ(ℏωq−ℏωq′)δ(εF−εk−q+q′νσ)δ(εF−εkν′σ)Iσσ(Te)\n(36)\n=X\nq′n\n(nq+ 1)nq′nBE(ωq−ωq′, Te) +\u0000\nq↔q′\u0001o\nΓqq′(Te)(37)\nwith\nΓqq′(Te) =2π\nℏ\u0010∆\nSN\u00112\n(ℏωq−ℏωq′)X\nσIσσ(Te)X\nkνν′δ(εF−εk−q+q′νσ)δ(εF−εkν′σ). (38)10\nB. Derivation of the three temperature model\nIn what follows, it is demonstrated that the three temperature model (3TM) can be obtained from the (N+2)-\ntemperature model derived in the main text,\n˙nq=h\nnBE(ωq, Te)−nqi\nγq+X\nq′h\n(nq+ 1)nq′nBE(ωq−ωq′, Te) + (q↔q′)i\nΓqq′,(39)\n˙Te=1\nCeh\n−X\nqℏωq˙nq+Gep(Tp−Te) +P(t)i\n, (40)\n˙Tp=−Gep\nCp(Tp−Te), (41)\nby assuming instantaneous relaxation of the magnon occupation numbers to the Bose-Einstein distribution with a\nsingle magnon temperature Tm, i.e., nq=nBE(ωq, Tm). For the sake of readability we rewrite nBE(ωq, Tm) =nq(Tm).\nWe start with the first order scattering term:\n˙n(1)\nq= [nq(Te)−nq(Tm)]γq≈(Te−Tm)∂nq(T)\n∂T\f\f\f\f\nT=Tmγq(Te) = (Te−Tm)Cqγq\nℏωq. (42)\nHere we have introduced the mode-dependent magnon heat capacity Cq=ℏωq∂nq(Tm)\n∂T.\nIn order to calculate the scattering term second order in the magnon variables, we first introduce the following\nrelation\n\u0000\nnq′(Tm) + 1\u0001\nnq(Tm) =\u0002\nnq′(Tm)−nq(Tm)\u0003\nnq−q′(Tm). (43)\nNow we calculate\n˙n(2)\nq=X\nq′\u0010\n(nq(Tm) + 1) nq′(Tm)nq−q′(Te) + (q↔q′)\u0011\nΓqq′ (44)\n=X\nq′\u0010\nnq′−q(Tm)nq−q′(Te)−(q↔q′)\u0011\n×\u0000\nnq(Tm)−nq′(Tm)\u0001\nΓqq′ (45)\n=X\nq′1\n2\u0012\ncoth\u0012ℏ(ωq′−ωq)\n2kBTe\u0013\n−coth\u0012ℏ(ωq′−ωq)\n2kBTm\u0013\u0013\u0000\nnq(Tm)−nq′(Tm)\u0001\nΓqq′ (46)\n≈X\nq′nq(Tm)−nq′(Tm)\nℏ(ωq′−ωq)kB(Te−Tm)Γqq′ (47)\n≈X\nq′∂nq(Tm)\n∂(ℏωq)kB(Tm−Te)Γqq′ (48)\n=X\nq′∂nq(T)\n∂T\f\f\f\f\nT=TmkBTm\nℏωq(Te−Tm)Γqq′ (49)\n=X\nq′CqkBTm\n(ℏωq)2(Te−Tm)Γqq′. (50)\nUsing the expressions for ˙ n(1)\nqand ˙n(2)\nq, the change in total energy of the magnons can then be calculated as\n∂Em\n∂t=∂Em\n∂Tm∂Tm\n∂t=X\nqℏωq∂nq(T)\n∂T|T=Tm\n| {z }\nCm∂Tm\n∂t= (Te−Tm)X\nqCq\u0010\nγq+X\nq′kBTm\nℏωqΓqq′\u0011\n.\n| {z }\nGem(51)\nWith that, the (N+2)TM transforms into the 3TM (in the absence of magnon-phonon coupling), which is given by\nCm˙Tm=Gem(Te−Tm),\nCe˙Te=Gem(Tm−Te) +Gep(Tp−Te) +P(t),\nCp˙Tp=Gep(Te−Tp).(52)11\nC.Ab initio calculations\nTo obtain a full solution of the (N+2)TM, it is necessary to compute the material specific quantities ∆, γq, Γqq′\nandIσσ(Te). For this purpose, we use the full-potential linear augmented plane wave code ELK [85].\nAs a first step, we determine the coupling parameter ∆ of the sp−dmodel, which sets the general scale of the\nelectron-magnon scattering. As shown in the main text, the first term (zeroth order in magnon variables) in the\nelectron-magnon scattering Hamiltonian reads ˆH(0)\nem=−∆P\nkν(ˆc†\nkν↑ˆckν↑−ˆc†\nkν↓ˆckν↓), with ν∈ {s, p}. Based on this,\n∆ can be estimated from the projected density of states (DOS), since it is one half of the spin-dependent energy\nsplitting of the s- and p-bands. In general, this splitting may vary for different electronic states. This is not accounted\nfor in the model used here, where instead a single parameter is used to model the spin splitting. We find, however,\nthat for bcc iron this is justified, since the shift in both s- and p-bands around the Fermi energy – the relevant\nregion for electron-magnon scattering – between spin up and down states is approximately constant with a value of\n∆≈0.75 eV, see left panel of Figure 5.\nNow we calculate the first and second order scattering rates using the formulas derived above,\nγq=4π∆2\nSNωqI↑↓(Te)X\nkνν′δ(εF−εk−qν↑)δ(εF−εkν′↓), (53)\nΓqq′=2π∆2\nS2N2(ωq−ωq′)X\nσIσσ(Te)X\nkνν′δ(εF−εk−q+q′νσ)δ(εF−εkν′σ). (54)\nThe calculation of both quantities requires a spin-dependent summation over the Fermi surface, analogous to what\nwas done in Ref. [103] for the evaluation of the spin-dependent Eliashberg function for electron-phonon scattering.\nAs in Ref. [103] we use a Gaussian broadening of the Dirac delta distributions by 0 .03 eV. Also, since we only include\nthe contribution of s- and p-states (indicated by ν, ν′) to the scattering, we have to project the Kohn-Sham states\n(indicated by n, n′) onto the spherical harmonics Ym\nlvia\nδ(εF−εkνσ)δ(εF−εk′ν′σ′) =X\nnn′Pnν\nkσPn′ν′\nk′σ′δ(εF−εknσ)δ(εF−εk′n′σ′), (55)\nwith Pnν\nkσbeing projector functions.\nThe functions Iσσ′(Te) describe corrections to the scattering rate at high electron temperatures and are given by\nIσσ′(Te) =Z\ndε(−1)∂fFD(ε, Te)\n∂εgσ(ε)g′\nσ(ε)\ngσ(εF)g′σ(εF), (56)\nwith gσ(ε) =P\nkνδ(ε−εkνσ) =P\nkνP\nnPnν\nkσδ(ε−εknσ) being the cumulative DOS of both s- and p-states. We\nfind that they increase monotonously with the electron temperature (see right panel of Figure 5). However, even\nfor temperature up to 2000 K, the Iσσ′(Te) functions are below two. Hence, we concluded that the approximation\nIσσ′= 1 is reasonable for the laser fluences – heating the electrons up to around 700 K – considered in the main text.\nFigure 6 depicts the numerically calculated scattering rates using Iσσ′= 1 and ∆ = 0 .75 eV as obtained above. In\nthe left panel, we show the scattering rate γqthat is first order in the magnon variables through color code on the\nmagnon dispersion. It is strictly positive and tends to increase with magnon frequency. The right panel shows the\neffective scattering rate γ(2)\nq=P\nq′Γqq′due to the scattering term second order in magnon variables. Notably, this\nquantity is negative for low frequencies and positive for high frequencies, indicating that it leads to a depopulation\nof magnons at low energies due a scattering from low to high energies (the total magnon number is kept constant).\nIn general, the values of the effective second order scattering rate are comparable to the one first order in magnon\nvariables. They are, however, distributed differently: e.g., for magnons close to the Γ point the second order scattering\nrate is by far the dominating one. This is the reason why, as demonstrated in the main text, a laser pulse can in fact\nlead to a cooling of low energy magnons, i.e., to a decrease of their occupation numbers.\nLastly, we show in Figure 7 the ab initio computed mode-dependent Gilbert damping, αq=ωq/γq. Interestingly,\nthe Gilbert damping αqis large ( ∼0.01) at the BZ center and at the high-symmetry points H, N and P at the\nBZ edge. There is also a noticeable directional anisotropy in the Gilbert damping for modes along Γ −H and Γ −P.\nWe emphasize that the Gilbert damping is here due to the electron-magnon scattering term that is first order in\nthe magnon variables. Other scattering mechanisms as phonon-magnon scattering could contribute further to the\nmode-specific Gilbert damping.12\n−10−5 0 5 10\nenergyε−εF(eV)−0.04−0.020.000.020.040.06projected DOS (eV−1)s\np\n500 1000 1500 2000\nelectron temperature Te(K)1.01.21.41.61.8thermal correction factorI↑↓\nI↑↑\nI↓↓\nFigure 5. Left: Projected spin-polarized DOS for bcc iron. Spin-minority density is shown by positive values, spin-majority\ndensity by negative values. The exchange splitting is 2∆ ≈1.5 eV in a large interval around the Fermi energy and for both s-\nandp-states. Right : Thermal correction factors Iσσ′versus electron temperature Tecalculated from the projected DOS.\nΓ H N Γ P H020406080100120magnon frequency (THz)\n12345\nscattering rate γq(THz)\nΓ H N Γ P H020406080100120magnon frequency (THz)\n−4−3−2−101\nscattering rate γ(2)\nq(THz)\nFigure 6. Magnon dispersion of bcc iron along high-symmetry lines of the Brillouin zone. The color coding describes ( left)\nthe scattering rates γqdue to the electron-magnon scattering term first order in magnon variables γqand ( right) the effective\nscattering rate γ(2)\nq=P\nq′Γqq′due to the term second order in magnon variables, calculated with Iσσ′= 1 and ∆ = 0 .75 eV.\nΓ H N Γ P H020406080100120magnon frequency (THz)\n0.0020.0040.0060.0080.010\nGilbert damping αq\nFigure 7. Calculated mode-specific Gilbert damping αq=ωq/γq, depicted by the color code on the magnon dispersion of bcc\niron. The mode-specific Gilbert damping αqis due to the electron-magnon scattering term first order in magnon variables.13\nACKNOWLEDGMENTS\nThe authors thank K. Carva for valuable discussions.\nThis work has been supported by the Swedish Re-\nsearch Council (VR), the German Research Foundation\n(Deutsche Forschungsgemeinschaft) through CRC/TRR\n227 “Ultrafast Spin Dynamics” (project MF, project-ID:\n328545488), and the K. and A. Wallenberg Foundation\n(Grant No. 2022.0079). 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Emmerich,\nR. Adam, C. Chen, H. C. Kapteyn, M. M. Murnane,\nL. Plucinski, D. Steil, B. Stadtm¨ uller, M. Cinchetti,\nM. Aeschlimann, C. M. Schneider, and S. Mathias, Sci.\nAdv. 3, e1602094 (2017).\n[102] D. Steiauf and M. F¨ ahnle, Phys. Rev. B 79, 140401\n(2009).\n[103] K. Carva, M. Battiato, and P. M. Oppeneer, Phys. Rev.\nLett. 107, 207201 (2011).\n[104] K. Carva, M. Battiato, D. Legut, and P. M. Oppeneer,\nPhys. Rev. B 87, 184425 (2013)."    },    {        "title": "1102.4551v1.Ab_initio_calculation_of_the_Gilbert_damping_parameter_via_linear_response_formalism.pdf",        "content": "arXiv:1102.4551v1  [cond-mat.mtrl-sci]  22 Feb 2011APS/123-QED\nAb-initio calculation of the Gilbert damping parameter via linear response formalism\nH. Ebert, S. Mankovsky, and D. K¨ odderitzsch\nUniversity of Munich, Department of Chemistry,\nButenandtstrasse 5-13, D-81377 Munich, Germany\nP. J. Kelly\nFaculty of Science and Technology and MESA+ Institute for Na notechnology,\nUniversity of Twente, P.O. Box 217, 7500 AE Enschede, The Net herlands\n(Dated: October 14, 2018)\nA Kubo-Greenwood-like equation for the Gilbert damping par ameterαis presented that is based\non the linear response formalism. Its implementation using the fully relativistic Korringa-Kohn-\nRostoker (KKR) band structure method in combination with Co herent Potential Approximation\n(CPA) alloy theory allows it to be applied to a wide range of si tuations. This is demonstrated with\nresults obtained for the bcc alloy system Fe xCo1−xas well as for a series of alloys of permalloy with\n5d transition metals. To account for the thermal displaceme nts of atoms as a scattering mechanism,\nan alloy-analogy model is introduced. The corresponding ca lculations for Ni correctly describe the\nrapid change of αwhen small amounts of substitutional Cu are introduced.\nPACS numbers: Valid PACS appear here\nI. INTRODUCTION\nThe magnetization dynamics that is relevant for the\nperformance of any type of magnetic device is in gen-\neral governed by damping. In most cases the magneti-\nzation dynamics can be modeled successfully by means\nof the Landau-Lifshitz-Gilbert (LLG) equation [1] that\naccounts for damping in a phenomenological way. The\npossibility to calculate the corresponding damping pa-\nrameter from first principles would open the perspective\nof optimizing materials for devices and has therefore mo-\ntivated extensive theoretical work in the past. This led\namong others to Kambersky’s breathing Fermi surface\n(BFS) [2] and torque-correlation model (TCM) [3], that\nin principle provide a firm basis for numerical investi-\ngations based on electronic structure calculations [4, 5].\nThe spin-orbit coupling that is seen as a key factor in\ntransferring energy from the magnetization to the elec-\ntronic degrees of freedom is explicitly included in these\nmodels. Most ab-initioresults havebeen obtained for the\nBFS model though the torque-correlation model makes\nfewer approximations [4, 6]. In particular, it in principle\ndescribes the physical processes responsible for Gilbert\ndamping over a wide range of temperatures as well as\nchemical(alloy)disorder. However,inpractice,likemany\nother models it depends on a relaxation time parame-\nterτthat describes the rate of transfer due to the vari-\nous types of possible scattering mechanisms. This weak\npoint could be removed recently by Brataas et al. [7]\nwho described the Gilbert damping by means of scatter-\ning theory. This development supplied the formal basis\nfor the first parameter-free investigations on disordered\nalloys for which the dominant scattering mechanism is\npotential scattering caused by chemical disorder [8].\nAs pointed out by Brataas et al. [7], their approach is\ncompletelyequivalenttoaformulationintermsofthelin-\nearresponseorKuboformalism. Thelatterrouteistakenin this communication that presents a Kubo-Greenwood-\nlike expression for the Gilbert damping parameter. Ap-\nplication of the scheme to disordered alloys demonstrates\nthat this approach is indeed fully equivalent to the scat-\ntering theory formulation of Brataas et al. [7]. In addi-\ntion a scheme is introduced to deal with the temperature\ndependence of the Gilbert damping parameter.\nFollowing Brataas et al. [7], the starting point of our\nscheme is the Landau-Lifshitz-Gilbert (LLG) equation\nfor the time derivative of the magnetization /vectorM:\n1\nγd/vectorM\ndτ=−/vectorM×/vectorHeff+/vectorM×/bracketleftBigg˜G(/vectorM)\nγ2M2sd/vectorM\ndτ/bracketrightBigg\n,(1)\nwhereMsis the saturation magnetization, γthe gyro-\nmagnetic ratio and ˜Gthe Gilbert damping tensor. Ac-\ncordingly, the time derivative of the magnetic energy is\ngiven by:\n˙Emag=/vectorHeff·d/vectorM\ndτ=1\nγ2˙/vector m[˜G(/vector m)˙/vector m] (2)\nin terms of the normalized magnetization /vector m=/vectorM/Ms.\nOntheotherhandtheenergydissipationoftheelectronic\nsystem ˙Edis=/angbracketleftBig\ndˆH\ndτ/angbracketrightBig\nis determined by the underlying\nHamiltonian ˆH(τ). Expanding the normalized magne-\ntization/vector m(τ), that determines the time dependence of\nˆH(τ) about its equilibrium value, /vector m(τ) =/vector m0+/vector u(τ), one\nhas:\nˆH=ˆH0(/vector m0)+/summationdisplay\nµ/vector uµ∂\n∂/vector uµˆH(/vector m0). (3)\nUsing the linear response formalism, ˙Ediscan be written2\nas [7]:\n˙Edis=−π/planckover2pi1/summationdisplay\nii′/summationdisplay\nµν˙uµ˙uν/angbracketleftBigg\nψi|∂ˆH\n∂uµ|ψi′/angbracketrightBigg/angbracketleftBigg\nψi′|∂ˆH\n∂uν|ψi/angbracketrightBigg\n×δ(EF−Ei)δ(EF−Ei′),(4)\nwhereEFis the Fermi energy and the sums run over all\neigenstates αof the system. Identifying ˙Emag=˙Edis,\none gets an explicit expression for the Gilbert damping\ntensor˜Gor equivalently for the damping parameter α=\n˜G/(γMs):\nαµν=−π/planckover2pi1γ\nMs/summationdisplay\nii′/angbracketleftBigg\nψi|∂ˆH\n∂uµ|ψi′/angbracketrightBigg/angbracketleftBigg\nψi′|∂ˆH\n∂uν|ψi/angbracketrightBigg\n×δ(EF−Ei)δ(EF−Ei′).(5)\nAn efficient way to deal with Eq. (5) is achieved by ex-\npressing the sum over the eigenstates by means of the\nretarded single-particle Green’s function Im G+(EF) =\n−π/summationtext\nα|ψα/angbracketright/angbracketleftψα|δ(EF−Eα). This leads for the parame-\nterαto a Kubo-Greenwood-like equation:\nαµν=−/planckover2pi1γ\nπMsTrace/angbracketleftBigg\n∂ˆH\n∂uµImG+(EF)∂ˆH\n∂uνImG+(EF)/angbracketrightBigg\nc(6)\nwith/angbracketleft.../angbracketrightcindicating a configurational average in case of\na disordered system (see below). Identifying ∂ˆH/∂uµ\nwith the magnetic torque Tµthis expression obviously\ngives the parameter αin terms of a torque-torque corre-\nlation function. However, in contrast to the conventional\nTCM the electronic structure is not represented in terms\nof Bloch states but using the retarded electronic Green\nfunction giving the present approach much more flexibil-\nity. As it corresponds one-to-one to the standard Kubo-\nGreenwood equation for the electrical conductivity, the\ntechniques developed to calculate conductivities can be\nstraightforwardly adopted to evaluate Eq. (6).\nThe most reliable way to account for spin-orbit cou-\npling as the source of Gilbert damping is to evaluate\nEq. (6) using a fully relativistic Hamiltonian within the\nframework of local spin density formalism (LSDA) [9]:\nˆH=c/vector α/vector p+βmc2+V(/vector r)+β/vector σ/vector mB(/vector r).(7)\nHereαiandβare the standard Dirac matrices and /vector pis\nthe relativistic momentum operator [10]. The functions\nVandBare the spin-averagedand spin-dependent parts\nrespectively of the LSDA potential. Eq. (7) implies for\nthe magnetic torque Tµoccurring in Eq. (6) the expres-\nsion:\nTµ=∂\n∂uµˆH=βBσµ. (8)\nThe Green’s function G+in Eq. (5) can be obtained in a\nvery efficient way by using the spin-polarized relativisticversion of multiple scattering theory [9] that allows us to\ntreat magnetic solids:\nG+(/vector rn,/vector rm′,E) =/summationdisplay\nΛΛ′Zn\nΛ(/vector rn,E)τnm\nΛΛ′(E)Zm×\nΛ′(/vector rm′,E)\n−/summationdisplay\nΛZn\nΛ(/vector r<,E)Jn×\nΛ′(/vector r>,E)δnm.(9)\nHere coordinates /vector rnreferring to the center of cell n\nhave been used with |/vector r<|=min(|/vector rn|,|/vector rn′|) and|/vector r>|=\nmax(|/vector rn|,|/vector rn′|). The four component wave functions\nZn\nΛ(/vector r,E) (Jn\nΛ(/vector r,E)) are regular (irregular) solutions to\nthe single-site Dirac equation for site nandτnm\nΛΛ′(E) is\nthe so-called scattering path operator that transfers an\nelectronic wave coming in at site minto a wave going\nout from site nwith all possible intermediate scattering\nevents accounted for coherently.\nUsing matrix notation, this leads to the following ex-\npression for the damping parameter:\nαµµ=g\nπµtot/summationdisplay\nnTrace/angbracketleftbig\nT0µ˜τ0nTnµ˜τn0/angbracketrightbig\nc(10)\nwith the g-factor 2(1 + µorb/µspin) in terms of the spin\nand orbital moments, µspinandµorb, respectively, the\ntotal magnetic moment µtot=µspin+µorb, and ˜τ0n\nΛΛ′=\n1\n2i(τ0n\nΛΛ′−τ0n\nΛ′Λ) and the energy argument EFomitted.\nThematrix elementsofthe torqueoperator Tnµareiden-\ntical to those occurring in the context of exchange cou-\npling [11] and can be expressed in terms of the spin-\ndependent part Bof the electronic potential with matrix\nelements:\nTnµ\nΛ′Λ=/integraldisplay\nd3rZn×\nΛ′(/vector r) [βσµBxc(/vector r)]Zn\nΛ(/vector r).(11)\nAs indicated above, the expressions in Eqs. (6) – (11)\ncan be applied straightforwardly to disordered alloys.\nIn this case the brackets /angbracketleft.../angbracketrightcindicate the necessary\nconfigurational average. This can be done by describ-\ning in a first step the underlying electronic structure\n(forT= 0 K) on the basis of the Coherent Potential\nApproximation (CPA) alloy theory. In the next step\nthe configurational average in Eq. (6) is taken follow-\ning the scheme worked out by Butler [12] when dealing\nwith the electrical conducting at T= 0 K or residual\nresistivity respectively, of disordered alloys. This im-\nplies in particular that so-called vertex corrections of the\ntype/angbracketleftTµImG+TνImG+/angbracketrightc− /angbracketleftTµImG+/angbracketrightc/angbracketleftTνImG+/angbracketrightcthat\naccount for scattering-in processes in the language of the\nBoltzmann transport formalism are properly accounted\nfor.\nThermal vibrations as a source of electron scattering\ncan in principle be accounted for by a generalization of\nEqs. (6) – (11) to finite temperatures and by includ-\ning the electron-phonon self-energy Σ el−phwhen calcu-\nlating the Greens function G+. Here we restrict our-\nselves to elastic scattering processes by using a quasi-\nstatic representation of the thermal displacements of the3\n0 0.1 0.2 0.3 0.4 0.50.6 0.7\nconcentration xCo0123456α(x) x10-3Expt\nTheory (CPA), bcc\nTheory (NL CPA)Fe-Co\nn(EF)\nn(EF) (sts./Ry)\n102030405060\n0\nFIG. 1: Gilbert damping parameter for bcc Fe xCo1−xas a\nfunction of Co concentration: full circles - the present res ults\nwithin CPA, empty circles - within non-local CPA (NL CPA),\nand full diamonds - experimental data by Oogane [14].\natoms from their equilibrium positions. We introduce\nan alloy-analogy model to average over a discrete set\nof displacements that is chosen to reproduce the ther-\nmal root mean square average displacement/radicalbig\n/angbracketleftu2/angbracketrightTfor\na given temperature T. This was chosen according to\n/angbracketleftu2/angbracketrightT=1\n43h2\nπ2mkΘD[Φ(ΘD/T)\nΘD/T+1\n4] with Φ(Θ D/T) the De-\nbye function, hthe Planck constant, kthe Boltzmann\nconstant and Θ Dthe Debye temperature [13]. Ignoring\nthe zero temperature term 1 /4 and assuming a frozen\npotential for the atoms, the situation can be dealt with\nin full analogy to the treatment of disordered alloys de-\nscribed above.\nThe approach described above has been applied to\nthe ferromagnetic 3d-transition metal alloy systems bcc\nFexCo1−x, fcc Fe xNi1−xand fcc Co xNi1−x. Fig. 1 shows\nas an example results for bcc Fe xCo1−xforx≤0.7. The\ncalculated damping parameter α(x) forT= 0 K is found\nin very good agreement with the results based on the\nscatteringtheoryapproach[8]demonstratingnumerically\nthe equivalence of the two approaches. An indispensable\nrequirement to achieve this agreement is to include the\nvertex corrections mentioned above. In fact, ignoring\nthem leads in some cases to completely unphysical re-\nsults. To check the reliability of the standard CPA, that\nimplies a single-site approximation when performing the\nconfigurationalaverage,weperformedcalculationsonthe\nbasis of the non-local CPA [15]. In this case four atom\ncluster have been used leading - apart from the very di-\nlute case - practically to the same results as the CPA. As\nfound before for fcc Fe xNi1−x[8] the theoretical results\nforαreproduce the concentration dependence of the ex-\nperimental data quite well but are found too low (see\nbelow). As suggested by Eq. (10) the variation of α(x)\nwith concentration xmay reflect to some extent the vari-\nation of the average magnetic moment µtotof alloy. As\nthe moments as well as the spin-orbit coupling strength\nof Fe and Co don’t differ too much, the variation of α(x)\nshould be determined in the concentrated regime primar-\nily by the electronic structure at the Fermi energy EF.As Fig. 1 shows, there is indeed a close correlation of the\ndensity of states n(EF) that may be seen as a measure\nfor the available relaxation channels.\nWhile the scattering and linear response approach are\ncompletely equivalent when dealing with bulk alloys the\nlatter allows us to perform the necessary configuration\naveragingin a much more efficient way. This allows us to\nstudy with moderate effort the influence of varying the\nalloy composition on the damping parameter α. Corre-\nsponding work has been done in particular using permal-\nloy as a starting material and adding transition metals\n(TM) [16] orrareearthmetals [17]. Fig. 2(top) showsre-\nsultsobtainedbysubstitutingFeandNiatomsinpermal-\nloy by 5d TMs. As found by experiment [16] αincreases\n00.05 0.1 0.15x01234 α x10-2\nTa W Re OsIr PtAu02468α x10-2\nTa W Re OsIr PtAu-0.300.30.60.9mspin5d (µB)WOs\nIr\nPt\nAuRe\nTa\nn5d(EF)\n5d spin moment\n061218\nn5d(EF) (sts./Ry)\nFIG. 2: Top: Change of the Gilbert damping parameter ∆ α\nw.r.t. permalloy(Py)forvariousPy/5dTM systemsasafunc-\ntion of 5d TM concentration; Middle: Gilbert damping pa-\nrameter αfor Py/5d TM systems with 10 % 5d TM content\nin comparison with experiment [16]; Bottom: spin magnetic\nmoment m5d\nspinand density of states n(EF) at the Fermi en-\nergy of the 5 dcomponent in Py/5d TM systems with 10 %\n5d TM content.\nin all cases nearly linearly with the 5d TM content. The\ntotal damping for 10 % 5d TM content shown in the\nmiddle panel of Fig. 2 varies roughly parabolically over\nthe 5d TM series. In contrast to the Fe xCo1−xalloys\nconsidered above, there is now an S-like variation of the\nmomentsµ5d\nspinover the series (Fig. 2, bottom), char-\nacteristic of 5d impurities in the pure hosts Fe and Ni\n[18, 19]. In spite of this behaviour of µ5d\nspinthe variation4\n00.050.10.15α(T)expt:    pure Ni\ntheory: pure Ni\n00.050.10.15α(T)expt:    Ni + 0.17 wt.%Cu\ntheory: Ni + 0.2 at.%Cu\n0 100 200 300 400 500\nTemperature (K)00.050.10.15α(T)expt: Ni + 5 wt.%Cu\ntheory: Ni + 5 at.%Cu\nFIG. 3: Temperature variation of Gilbert damping of pure\nNi and Ni with Cu impurities: present theoretical results vs\nexperiment [20]\nofα(x) seems again to be correlated with the density of\nstatesn5d(EF) (Fig. 2 bottom). Again the trend of the\nexperimental data is well reproduced by the theoretical\nones that are however somewhat too low.\nOne of the possible reasons for the discrepancy of the\ntheoretical and experimental results shown in Figs. 1 and\n2 might be the neglect of the influence of finite temper-\natures. This can be incorporated as indicated above by\naccounting for the thermal displacement of the atoms in\na quasi-static way and performing a configurational av-\nerage over the displacements using the CPA. This leads\neven for pure systems to a scattering mechanism and this\nwaytoafinite valuefor α. Correspondingresultsforpure\nNi are given in Fig. 3 that show in full accordance with\nexperiment a rapid decrease of αwith increasing tem-\nperature until a regime with a weak variation of αwith\nTis reached. This behavior is commonly interpreted as\na transition from conductivity-like to resistivity-like be-\nhaviour reflecting the dominance of intra- and inter-band\ntransition, respectively [4], that is related to the increase\nof the broadening of electron energy bands caused by the\nincrease of scattering events with temperature. Adding\nonly less than 1 at. % Cu to Ni, the conductivity-like\nbehavior at low temperatures is strongly reduced whilethe high temperature behavior is hardly changed. A fur-\nther increase of the Cu content leads to the impurity-\nscattering processes responsible for the band broaden-\ning dominating α. This effect completely suppresses the\nconductivity-likebehavior in the low-temperatureregime\nbecause of the increase of scattering events due to chem-\nical disorder. Again this is fully in line with the experi-\nmental data, providing a straightforward explanation for\ntheir peculiar variation with temperature and composi-\ntion.\nFromtheresultsobtainedforNionemayconcludethat\nthermal lattice displacements are only partly responsible\nfor the finding that the damping parameters obtained\nfor Py doped with the 5 dTM series, and Fe xCo1−xare\nsomewhatlowcomparedwith experiment. This indicates\nthat additional relaxation mechanisms like magnon scat-\ntering contribute. Again, these can be included at least\nin a quasi-static way by adopting the point of view of a\ndisordered local moment picture. This implies scatter-\ning due to random temperature-dependent fluctuations\nof the spin moments that can also be dealt with using\nthe CPA.\nInsummary, aformulationforthe Gilbert dampingpa-\nrameterαin terms of a torque-torque-correlation func-\ntion was derived that led to a Kubo-Greenwood-like\nequation. The scheme was implemented using the fully\nrelativistic KKR band structure method in combination\nwith the CPA alloy theory. This allows us to account for\nvarious types of scattering mechanisms in a parameter-\nfree way. Corresponding applications to disordered tran-\nsition metal alloys led to very good agreement with re-\nsults based on the scattering theory approach of Brataas\net al. demonstrating the equivalence of both approaches.\nThe flexibility and numerical efficiency of the present\nscheme was demonstrated by a study on a series of\npermalloy-5dTMsystems. Toinvestigatetheinfluenceof\nfinite temperatures on α, a so-called alloy-analogymodel\nwas introduced that deals with the thermal displacement\nof atoms in a quasi-static manner. Applications to pure\nNi gave results in very good agreement with experiment\nand in particular reproduced the dramatic change of α\nwhen Cu is added to Ni.\nAcknowledgments\nThe authors would like to thank the DFG for finan-\ncial support within the SFB 689 “Spinph¨ anomene in re-\nduzierten Dimensionen” and within project Eb154/23for\nfinancialsupport. PJKacknowledgessupportbyEUFP7\nICT Grant No. 251759 MACALO.\n[1] T. L. Gilbert, IEEE Transactions on Magnetics 40, 3443\n(2004).[2] V. Kambersky, Can. J. Phys. 48, 2906 (1970).\n[3] V. Kambersky, Czech. J. Phys. 26, 1366 (1976), URL5\nhttp://dx.doi.org/10.1007/BF01587621 .\n[4] K. Gilmore, Y. U. Idzerda, and M. D. Stiles,\nPhys. Rev. Lett. 99, 027204 (2007), URL\nhttp://link.aps.org/doi/10.1103/PhysRevLett.99.0272 04.\n[5] M. F¨ ahnle and D. Steiauf, Phys. Rev. B 73, 184427\n(2006).\n[6] V. Kambersky, Phys. Rev. B 76, 134416 (2007).\n[7] A. Brataas, Y. Tserkovnyak, and G. E. W.\nBauer, Phys. Rev. Lett. 101, 037207 (2008), URL\nhttp://link.aps.org/doi/10.1103/PhysRevLett.101.037 207.\n[8] A. A. Starikov, P. J. Kelly, A. Brataas,\nY. Tserkovnyak, and G. E. W. Bauer,\nPhys. Rev. Lett. 105, 236601 (2010), URL\nhttp://link.aps.org/doi/10.1103/PhysRevLett.105.236 601.\n[9] H. Ebert, in Electronic Structure and Physical Properties\nof Solids , edited by H. Dreyss´ e (Springer, Berlin, 2000),\nvol. 535 of Lecture Notes in Physics , p. 191.\n[10] M. E. Rose, Relativistic Electron Theory (Wiley, New\nYork, 1961).\n[11] H. Ebert and S. Mankovsky, Phys.\nRev. B 79, 045209 (2009), URL\nhttp://link.aps.org/doi/10.1103/PhysRevB.79.045209 .\n[12] W. H. Butler, Phys. Rev. B 31, 3260 (1985), URL\nhttp://link.aps.org/doi/10.1103/PhysRevB.31.3260 .[13] E. M. Gololobov, E. L. Mager, Z. V. Mezhevich, and\nL. K. Pan, phys. stat. sol. (b) 119, K139 (1983).\n[14] M. Oogane, T. Wakitani, S. Yakata, R. Yilgin, Y. Ando,\nA. Sakuma, and T. Miyazaki, Jap. J. Appl. Phys. 45,\n3889 (2006).\n[15] D. K¨ odderitzsch, H. Ebert, D. A. Rowlands, and\nA. Ernst, New Journal of Physics 9, 81 (2007), URL\nhttp://dx.doi.org/10.1088/1367-2630/9/4/081 .\n[16] J. O. Rantschler, R. D. McMichael, A. Castillo, A. J.\nShapiro, W. F. Egelhoff, B. B. Maranville, D. Pu-\nlugurtha, A. P. Chen, and L. M. Connors, J. Appl. Phys.\n101, 033911 (2007).\n[17] G. Woltersdorf, M. Kiessling, G. Meyer, J.-U. Thiele, a nd\nC. H. Back, Phys. Rev. Lett. 102, 257602 (2009), URL\nhttp://link.aps.org/doi/10.1103/PhysRevLett.102.257 602.\n[18] B. Drittler, N. Stefanou, S. Bl¨ ugel, R. Zeller, and\nP. H. Dederichs, Phys. Rev. B 40, 8203 (1989), URL\nhttp://link.aps.org/doi/10.1103/PhysRevB.40.8203 .\n[19] N. Stefanou, A. Oswald, R. Zeller, and P. H.\nDederichs, Phys. Rev. B 35, 6911 (1987), URL\nhttp://link.aps.org/doi/10.1103/PhysRevB.35.6911 .\n[20] S. M. Bhagat andP. Lubitz, Phys. Rev.B 10, 179 (1974)."    },    {        "title": "0811.2235v2.Intrinsic_Coupling_between_Current_and_Domain_Wall_Motion_in__Ga_Mn_As.pdf",        "content": "arXiv:0811.2235v2  [cond-mat.mes-hall]  27 Jun 2009Intrinsic Coupling between Current and Domain Wall Motion i n (Ga,Mn)As\nKjetil Magne Dørheim Hals, Anh Kiet Nguyen, and Arne Brataas\nDepartment of Physics, Norwegian University of Science and Technology, NO-7491, Trondheim, Norway\nWe consider current-induceddomain wall motion and, therec iprocal process, movingdomain wall-\ninduced current. The associated Onsager coefficients are exp ressed in terms of scattering matrices.\nUncommonly, in (Ga,Mn)As, the effective Gilbert damping coe fficientαwand the effective out-of-\nplane spin transfer torqueparameter βware dominated byspin-orbit interaction incombination wit h\nscattering off the domain wall, and not scattering off extrins ic impurities. Numerical calculations\ngiveαw∼0.01 andβw∼1 in dirty (Ga,Mn)As. The extraordinary large βwparameter allows\nexperimental detection of current or voltage induced by dom ain wall motion in (Ga,Mn)As.\nThe principle of giant magneto resistance is used to\ndetect magnetic information. Large currents in mag-\nnetic nanostructures can manipulate the magnetization\nvia spin transfer torques [1]. A deeper knowledge of the\ncoupled out-of-equilibrium quasi-particle and magnetiza-\ntion dynamics is needed to precisely control and utilize\ncurrent-induced spin transfer torques.\nThemagnetizationrelaxestowardsits equilibriumcon-\nfiguration by releasing magnetic moments and energy\ninto reservoirs. This friction process is usually described\nby the Gilbert damping constant αin the Landau-\nLifshitz-Gilbert (LLG) equation. Spins traversingamag-\nnetic domain wall exert an in-plane and an out-of-plane\ntorqueonthe wall[2]. In dirtysystems, when the domain\nwall is wider than the mean-free-path, the out-of-plane\ntorque, often denoted the non-adiabatic torque, is pa-\nrameterized by the so-called β-factor [2]. The Gilbert\ndamping coefficient α, the in-plane spin-transfer torque,\nand the out-of-plane torque coefficient βdetermine how\nthe magnetization is influenced by an applied current,\ne.g. the current-induced Walker domain wall drift veloc-\nity is proportional to β/α[2, 3, 4]. Scattering off impu-\nrities are important for αandβ[2, 3, 4]. Additionally,\ndomain wall scattering can contribute to αandβ. In\nballistic(Ga,Mn)As, intrinsic spin-orbit coupling causes\nsignificant hole reflection at the domain wall, even in the\nadiabatic limit when the wall is much thicker than the\nFermi wavelength [5]. This grossly increases the out-of-\nplane spin-transfer torque, and consequently the current-\ndriven domain wall mobility. So far, there are no inves-\ntigations on the effect of these domain wall induced hole\nreflections on the effective Gilbert damping constant α.\nExperimental(Ga,Mn)As samples aredirty sothat the\neffectofdisorderontheeffectiveGilbertdampingandthe\nout-of-plane spin transfer torque should be taken into\naccount. We find surprisingly that, in systems with a\nlargeintrinsic spin-orbitcoupling, domainwallscattering\ncontributesdominantlyto αandβeveninthedirtylimit.\nIntrinsic current-domain wall motion coupling is robust\nagainst impurity scattering.\nCurrent-induced domain-wall motion has been seen in\nmany experiments [3]. The reciprocaleffect, domain-wall\nmotion induced current, is currentlytheoreticallyinvesti-gated[6, 7], andseenexperimentally[8]. Aprecessingdo-\nmain wall induces a charge current in ferromagnetic met-\nals [6] similar to spin-pumping in layered ferromagnet-\nnormal metal systems [9]. For rigid domain wall motion,\nthe induced chargecurrentis proportionalto β/α[7]. We\nfind that βandβ/αin (Ga,Mn)As are so large that the\ncurrent, or equivalently, the voltage induced by a moving\ndomain wall is experimentally measurable.\nOnsager’s reciprocity relations dictate that response\ncoefficients of domain wall motion induced current and\ncurrent induced domain wall motion are related. In dirty\nsystems, these relations have been discussed in Ref. [7].\nRef.[7]alsousedthescatteringtheoryofadiabaticpump-\ning to evaluate the non-adiabatic spin-transfer torque in\nballistic systems without intrinsic spin-orbit interaction.\nWe first extend the pumping approach to (Ga,Mn)As\nwith strong intrinsic spin-orbit interaction, and second,\nalso evaluate the Onsager coefficient as a function of\nsample disorder. In determining all Onsager coefficients,\nmagnetization friction must be evaluated on the same\nfooting. To this end, we generalize the energy pump-\ning scattering theory of Gilbert damping [10] to domain\nwall motion. Our numerical calculation demonstrates,\nfor the first time, that domain wall scattering is typi-\ncally more important than impurity scattering for the\neffective domain wall motion friction in systems with a\nstrongintrinsicspin-orbitinteraction. OurnovelOnsager\nscattering approach can also be used to compute the ef-\nfective rigid domain wall motion αandβparameters in\nrealistic materials like Fe, Ni, Co, and alloysthereof from\nfirst-principles.\nLet us discuss in more detail the Onsager reciprocity\nrelations in our system. The magnetic field is a thermo-\ndynamic force for the magnetization since it can move\ndomainwalls. The electric field is athermodynamic force\nfor the charges as it induces currents. In systems where\ncharge carriers also carry spin, the magnetic and charge\nsystems are coupled. Through this coupling, the elec-\ntric field can move a domain wall and, vice versa, the\nmagnetic field can induce a current. This phenomenon,\nwhere the thermodynamic force of one system can induce\na flux in another system is well-known in thermodynam-\nics [11]: Assume a system described by the quantities2\n{qi},Xidenotes the thermodynamic force, and Jithe\nflux associated with the quantity qi. In linear response,\nJi=/summationtext\njLijXj, whereLijare the Onsager coefficients.\nOnsager’s reciprocity principle dictates Lij=ǫiǫjLji,\nwhereǫi= 1 (ǫi=−1) ifqiis even (odd) under time-\nreversal [11]. Fluxes and forces are not uniquely defined,\nbut the Onsager reciprocity relations are valid when the\nentropy generation is ˙S=/summationtext\niJiXi[11].\nWe first derive expressions for the Onsager coefficients\nand determine the Onsager reciprocity relations between\na charge current and a moving domain wall in terms\nof the scattering matrix. Subsequently, we derive the\nrelation between the Onsager coefficients and the effec-\ntive Gilbert damping parameter αwand the out-of-plane\ntorque parameter βwfor domain wall motion. Finally,\nwe numerically compute αwandβwfor (Ga,Mn)As.\nWe start the derivation of the Onsager coefficients in\nterms of the scattering matrix by assuming the following\nfree energy functional for the magnetic system\nF[M] =Ms/integraldisplay\ndr/parenleftbiggJ\n2[(∇θ)2+sin2(θ)(∇φ)2]+\nK⊥\n2sin2(θ)sin2(φ)−Kz\n2cos2(θ)−Hextcos(θ)/parenrightbigg\n,(1)\nwhereMs,JandHextare the saturation magnetization,\nspin-stiffness and external magnetic field, respectively,\nandKzandK⊥are magnetic anisotropy constants. The\nlocal magnetization angles θandφare defined with re-\nspect to the z- andx-axis, respectively. The system con-\ntains a Bloch wall rotating in the (transverse) x-zplane,\ncos(θ) = tanh([ y−rw]/λw), sin(θ) = 1/cosh([y−rw]/λw),\nwhererwis the position of the wall, and λwis the wall\nwidth. We assume the external magnetic field is lower\nthan the Walker threshold, so that the wall rigidly moves\n(˙φ= 0) with a constant drift velocity. In this case rw\nandφcompletely characterize the magnetic system, and\nλw=/radicalBig\nJ/(Kz+K⊥sin2(φ)) [4]. The current is along\nthey-axis.\nTheheatdissipatedperunittimefromachargecurrent\nJcis˙Q=Jc(VL−VR), where VL(VR) is the voltage in\nthe left (right) reservoir. Using the relation dS=dQ/T,\nthis implies an entropy generation ˙S=Jc(VL−VR)/T.\nThus,Xc≡(VL−VR)/Tis the thermodynamic force\ninducing the flux Jc. We assume the magnetic system\nto be at constant temperature, which means that the\nheat transported out of the magnetic system as the do-\nmain wall moves equals the loss of free energy. This\nimplies an entropy generation ˙S=˙Q/T=−˙F/T=\n(−∂F[rw,φ]/T∂rw) ˙rw=XwJw, where we have defined\nthe force Xw≡ −∂F[rw,φ]/T∂rwand flux Jw≡˙rw. Us-\ning Eq. (1), wefind Xw=−2AMsHext/T, whereAis the\nconductor’s cross-section. Fluxes are related to forces by\nJw=LwwXw+LwcXc (2)\nJc=LccXc+LcwXw, (3)whereLcc=GTandGis the conductance. Lww(Lwc)\ndetermine the induced domain wall velocity by an exter-\nnal magnetic field (a current). The induced current by a\nmoving domain wall caused by an external magnetic field\nHextis controlled by Lcw. Both charge and rware even\nunder time-reversal so that Lcw=Lwc[12].\nThe current induced by a moving domain wall is para-\nmetric pumping in terms of the scattering matrix [9]:\nJc,α=e˙rw\n2π/summationdisplay\nβ=1,2ℑm/braceleftbigg\nTr/bracketleftbigg∂Sαβ\n∂rwS†\nαβ/bracketrightbigg/bracerightbigg\n,(4)\nwhereSαβis the scattering matrix between transverse\nmodes in lead βto transverse modes in lead α. The\nsystem has two leads ( α,β∈ {1,2}). The trace is over\nall propagating modes at the Fermi energy EF. From\nEqs. (2) and (3) we find Jc=Lcw˙rw/Lww.\nWe considertransportwellbelowthe criticaltransition\ntemperature in (Ga,Mn)As, which is relatively low, and\nassume the energy loss in the magnetic system is trans-\nferred into the leads by holes. Generalizing Ref. [10] to\ndomain wall motion, this energy-flux is related to the\nscattering matrix:\nJE=¯h\n4πTr/braceleftbiggdS\ndtdS†\ndt/bracerightbigg\n=¯h˙r2\nw\n4πTr/braceleftbigg∂S\n∂rw∂S†\n∂rw/bracerightbigg\n.(5)\nFor a domain wall moved by an external magnetic field,\nwe then find that XwJw=J2\nw/Lww=JE/T. In sum-\nmary, the Onsager coefficients in Eq. (2) and Eq. (3) are\nLww=/parenleftbigg¯h\n4πTr/braceleftbigg∂S\n∂rw∂S†\n∂rw/bracerightbigg/parenrightbigg−1\n, (6)\nLcw=2e\n¯h/summationtext\nβ=1,2ℑm/braceleftBig\nTr/bracketleftBig\n∂Sαβ\n∂rwS†\nαβ/bracketrightBig/bracerightBig\nTr/braceleftBig\n∂S\n∂rw∂S†\n∂rw/bracerightBig ,(7)\nLcc=e2\nhTr/braceleftbig\nt†t/bracerightbig\n, (8)\nwheretis the transmission coefficient in the scatter-\ning matrix. We have omitted the temperature factor\nin the coefficients (6), (7), and (8) since it cancels with\nthe temperature factor in the forces, i.e.we transform\nL→L/TandX→TX. The Onsager coefficient ex-\npressions in terms of the scattering matrix are valid irre-\nspectiveofimpuritydisorderandspin-orbitinteractionin\nthe band structures or during scattering events, and can\ntreat transport both in ballistic and diffusive regimes.\nLet us compare the global Onsager cofficients (6), (7),\nand (8) with the local Onsager coefficients in the dirty\nlimit to gainadditionalunderstanding. In the dirtylimit,\nall Onsager cofficients become local and the magnetiza-\ntion dynamics can be described by the following phe-\nnomenological local LLG equation [2, 3]:\n˙m=−γm×Heff+αm×˙m\n−(1−βm×)(vs·∇)m, (9)3\nwheremis the magnetization direction, Heffis the\neffective magnetic field, γis the gyromagnetic ratio,\nvs=−¯hPj/(eS0),S0=Ms/γ,Msthe magnetization,\nαthe Gilbert damping constant, Pthe spin-polarization\nalong−mof the charge carriers [13], and βis the out-\nof-plane spin-transfer torque parameter. Substituting\na Walker ansatz into Eq. (9) gives below the Walker\nthreshold [4]: α˙rw/λw=−γHext−¯hβPj/(eS0λw). In\ndirty, local, systems this equation determines the rela-\ntion between the flux Jwand the forces XwandXc\nasLww=λw/(2AS0α) andLwc=−¯hβPG/(eαS0A),\nwhere we have used j=σ(VL−VR)/L, andG=σA/L.\nHere,Lis the length of the conductor, ethe electron\ncharge, and σthe conductivity. This motivates defining\nthe following dimensionless global coefficients:\nαw≡λw\n2AS0Lww, βw≡ −λwe\n2¯hPGLwc\nLww.\nαwis the effective Gilbert damping coefficient and βwis\nthe effective out-of-plane torque on the domain wall.\nWe will in the following investigate αwandβwfor\n(Ga,Mn)As by calculating the scattering matrix expres-\nsions in Eq. (6) and Eq. (7). We use the following Hamil-\ntonian to model quantum transport of itinerant holes:\nH=HL+h(r)·J+V(r). (10)\nHere,HLis the 4×4 Luttinger Hamiltonian (parame-\nterized by γ1andγ2) for zincblende semiconductors in\nthe spherical approximation, while h·Jdescribes the\nexchange interaction between the itinerant holes and the\nlocal magnetic moment of the Mn dopants. We introduce\nAnderson impurities as V(r) =/summationtext\niViδr,Ri, whereRiis\nthe position of impurity i,Viits impurity strength, and\nδthe Kroneckerdelta. More details about the model and\nthe numericalmethod used can be found in Refs. [14, 15].\nWe consider a discrete conductor with transverse di-\nmensions Lx= 23nm,Lz= 17nmand length Ly=\n400nm. The lattice constant is 1 nm, much less than the\ntypical Fermi wavelength λF∼8nm. The Fermi energy\nEF= 82meVis measured from the bottom of the lowest\nsubband. |h|= 0.5×10−20Jandγ1= 7. The typical\nmean-free path for the systems studied ranges from the\ndiffusive to the ballistic regime l∼23nm→ ∞, and we\nare in the metallic regime kFl≫1. The domain wall\nlength is λw= 40nm. The spin-density S0from the local\nmagneticmoments is S0= 10¯hx/a3\nGaAs,aGaAsthe lattice\nconstant for GaAs, and x= 0.05 the doping level[14].\nFig. 1a shows the computed effective Gilbert damping\ncoefficient αwversusλw/lfor (Ga,Mn)As containing one\nBloch wall. Note the relatively high αw∼5×10−3in\nthe ballistic limit. Additional impurities, in combination\nwith the spin-orbit coupling, assist in releasing energy\nand angular momentum into the reservoirs and increase\nαw. However, as shown in Fig. 1a, impurities contribute\nonly about 20% to αweven when the domain wall is two0 0.5 1 1.50246x 10−3\n00.5 11.5 22.5024x 10−3\n0 0.5 1 1.502468\n00.5 11.5 22.50123λw/l\nλw/lαw βw\nαw βwγ2\nγ2(a)\n(b)\nFIG. 1: (a): Effective Gilbert damping αwas function of\nλw/l, whereλwis the domain wall length and lis the mean\nfree path when γ2= 2.5. Here,λwis kept fixed, and lis\nvaried. Inset: αwas a function of spin-orbit coupling γ2for a\nclean system, l=∞.(b):βwas a function of λw/l, whereλw\nis the domain wall length and lis the mean free path when\nγ2= 2.5. Here,λwis kept fixed, and lis varied. Inset: βwas\na function of spin-orbit coupling γ2for a clean system, l=∞.\nIn all plots, line is guide to the eye.\ntimes longer than the mean free path. Due to the strong\nspin-orbit coupling, ballistic domain walls have a large\nintrinsic resistance [5] that survives the adiabatic limit.\nWhen itinerant holes scatter off the domain wall their\nmomentum changes and through the spin-orbit coupling\ntheir spin also changes. This is the dominate process for\nreleasing energy and magnetization into the reservoirs.\nThe saturated value αw∼6×10−3is of the same order\nas the estimates in Ref. [16] for bulk (Ga,Mn)As. The\ninset in Fig. 1a shows the domain-wall contribution to\nαwversus the spin-orbit coupling for a clean system with\nno impurities. αwmonotonicallydecreasesfor decreasing\nγ2and vanish for γ2→0. Since, λw/λF∼5, itinerant\nholes will, without spin-orbit coupling, traverse the do-4\nmain wall adiabatically.\nFig. 1b shows βwversusλw/l.βwdecreases with\nincreasing disorder strength. This somewhat counter\nintuitive result stem from the fact that domain walls\nin systems with spin-orbit coupling have a large intrin-\nsic domain wall resistance [5] which originates from the\nanisotropy in the distribution of conducting channels [5].\nThe reflected spins do not follow the magnetization of\nthe domain wall, and thereby cause a large out-of-plane\ntorque [2]. This causes the large βwin the ballistic limit.\nScalar, rotational symmetric impurities tend to reduce\nthe anisotropy in the conducting channels, and thereby\nreduce the intrinsic domain wall resistance and conse-\nquently reduce βw. Deeper into the diffusive regime, β\nsaturates. Here, the domain wall resistance and βware\nkept at high levels due to the increase in the spin-flip\nrate caused by impurity scattering. The saturated value\nisβ∼1. For even dirtier systems than a reasonable\ncomputing time allows, we expect a further increase in\nβw. In comparison, simple microscopic theories for fer-\nromagnetic metals where one disregards the spin-orbit\ncoupling in the band structure predict β∼0.001−0.01\n[2, 3, 4]. Similar to the Gilbert damping, in ballistic sys-\ntemsβwincreaseswith spin-orbit coupling because ofthe\nincreased domain wall scattering [5], see Fig. 1b inset.\nβwcan be measured experimentally by the induced\ncurrent or voltage from a domain wall moved by an ex-\nternal magnetic field as a function of the domain wall\nvelocity [7]. From the Onsager relations we have that\nJc=LcwXw. UsingXw=Jw/Lww, the induced current\nand voltage are [7]:\nJc=−2β¯hPG\neλw˙rw⇒V=−2βw¯hP\neλw˙rw.(11)\nAn estimate of the maximum velocity of a domain wall\nmoved by an external magnetic field below the Walker\ntreshold is ˙ rw∼10m/s[17]. With λw= 40nmand\nP= 0.66 this indicates an experimentally measurable\nvoltageV∼0.2µV.\nInconclusion, wehavederivedOnsagercoefficientsand\nreciprocity relations between current and domain wall\nmotion in terms of scattering matrices. In (Ga,Mn)As,\nwe find the effective Gilbert damping constant αw∼0.01\nand out-of-plane spin transfer torque parameter βw∼1.\nIn contrast to ferromagnetic metals, the main contribu-\ntions to αwandβwin (Ga,Mn)As are intrinsic, and in-\nduced by scattering off the domain wall, while impurity\nscattering is less important. The large βwparameter im-\nplies a measurable moving domain wall induced voltage.\nThis work was supported in part by the Re-search Council of Norway, Grants Nos. 158518/143 and\n158547/431, computing time through the Notur project\nand EC Contract IST-033749 ”DynaMax”.\n[1] J.C. Slonczewski,J. Magn. Magn. Mater. 159, L1 (1996);\nL. Berger,Phys. Rev. B 54, 9353 (1996).\n[2] G. Tatara, H. Kohno,Phys. Rev. Lett. 92,086601 (2004);\nS. Zhang, Z. Li,Phys. Rev. Lett. 93, 127204 (2004); S. E.\nBarnes, S. Maekawa,Phys. Rev. Lett. 95, 107204 (2005);\nA. Thiaville, Y.Nakatani, J. Miltat, Y.Suzuki,Europhys.\nLett.69, 990 (2005); Y. Tserkovnyak, H. J. Skadsem,\nA. Brataas, G. E. W. Bauer, Phys. Rev. B 74, 144405\n(2006); A. K. Nguyen, H. J. Skadsem, A. Brataas, Phys.\nRev. Lett. 98, 146602 (2007).\n[3] For reviews see e.g.D.C. Ralph, M.D. Stiles, J. Magn.\nMagn. Mater., 3201190 (2008).\n[4] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, J.\nMagn. Magn. Mater., 3201282 (2008).\n[5] A. K. Nguyen, R. V. Shchelushkin and A. Brataas, Phys.\nRev. Lett. 97, 136603 (2006); R. Oszwaldowski, J. A.\nMajewski and T. Dietl,Phys. Rev. B 74, 153310 (2006).\n[6] G. E. Volovik,J. Phys.C: Sol. State Phys. 20, L83(1987);\nS.E. BarnesandS.Maekawa,Phys.Rev.Lett. 98, 246601\n(2007); J.I. Ohe, A. Takeuchi, and G. Tatara, Phys. Rev.\nLett.99, 266603 (2007); S. A.Yang, D. Xiao, andQ. Niu,\ncond-mat/0709.1117.\n[7] R. A. Duine, Phys. Rev. B 77, 014409 (2008);\narXiv:0809.2201v1; Y. Tserkovnyak and M. Mecklen-\nburg,Phys. Rev. B 77, 134407 (2008).\n[8] S. A. Yang et al.,Phys. Rev. Lett. 102, 067201 (2009).\n[9] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n[10] A. Brataas, Y. Tserkovnyak and G.E.W Bauer, Phys.\nRev. Lett. 101, 037207 (2008).\n[11] S. R. de Groot, Thermodynamics of Irreversible Pro-\ncesses(North-Holland, Amsterdam, 1952).\n[12] The magnetic texture is inverted in this equation, i.e.\nLcw[m] =Lwc[−m].\n[13] We define the spin-polarization P=/angbracketleftJz/angbracketright=P\nj(ψ†\njJzψj)/N(m=−ˆ z), where the sum is over N\npropagating modes, ψjare the corresponding spinor-\nvalued wavefunctions, and Jzthe dimensionless angular\nmomentum operator.\n[14] T. Jungwirth, J. Sinova, J. Maˇ sek, J. Kuˇ cera, and A. H.\nMacDonald, Rev. Mod. Phys. 78, 809 (2006).\n[15] A. K. Nguyen, and A. Brataas, Phys. Rev. Lett. 101,\n016801 (2008).\n[16] J. Sinova et al., Phys. Rev. B 69, 085209 (2004); Y.\nTserkovnyak, G. A. Fiete and B. I. Halperin,Appl. Phys.\nLett.84, 5234 (2004); I. Garate and A. MacDonald,\narXiv:0808.3923v1.\n[17] A. Dourlat, V. Jeudy, A. Lemaitre and C. Gourdon,\nPhys. Rev. B 78, 161303(R) (2008)."    },    {        "title": "2301.12797v1.Investigation_of_Ultrafast_Demagnetization_and_Gilbert_Damping_and_their_Correlation_in_Different_Ferromagnetic_Thin_Films_Grown_Under_Identical_Conditions.pdf",        "content": "1  Investigation  of Ultrafast  Demagnetization  and Gilbert  Damping  and their  Correlation  in Different  \nFerromagnetic  Thin  Films  Grown  Under  Identical  Conditions1 \nSuchetana Mukhopadhyay, Sudip Majumder, Surya Narayan Panda, Anjan Barman*  \nDepartment  of Condensed  Matter  and Materials  Physics,  S.N. Bose  National  Center  for Basic  \nSciences,  Block -JD, Sector  III, Salt Lake,  Kolkata  700106,  India  \nEmail:  abarman@bose.res.in  \n \n \nAbstract  \nFollowing the demonstration of laser -induc ed ultrafast demagnetization in ferromagnetic nickel,  several  \ntheoretical  and phenomenological  propositions  have sought  to uncover  its underlying  physics.  In this work \nwe revisit the three temperature model (3TM) and the microscopic three  temperature  model  (M3TM)  to \nperform  a comparative  analysis  of ultrafast  demagnetization  in 20-nm-thick  cobalt,  nickel  and permalloy  \nthin films  measured  using  an all-optical  pump - probe  technique.  In addition  to the ultrafast  dynamics  at \nthe femtosecond  timescales,  the nano second  magnetization  precession  and damping  are recorded  at various  \npump  excitation  fluences  revealing  a fluence -dependent  enhancement  in both the demagnetization  times  \nand the damping  factors.  We confirm  that the Curie  temperature  to magnetic  moment  ratio of a given \nsystem acts as a figure of merit for the demagnetization time, while the demagnetization  times  and damping  \nfactors  show  an apparent  sensitivity  to the density  of states  at the Fermi  level for a given system.  \nFurther, from numerical simulations of the ultrafast demagnetization  based on both the 3TM and the M3TM, \nwe extract the reservoir coupling parameters that best  reproduce  the experimental  data and estimate  the \nvalue  of the spin flip scattering  probability  for each system. We discuss how the f luence -dependence of \ninter-reservoir coupling parameters  so extracted  may reflect  a role played  by nonthermal  electrons  in the \nmagnetization  dynamics  at low laser  fluences.  \n \n1. Introduction  \nOptical  excitation  of a magnetic  material  with a short  and intense  laser pulse  sets in motion  a chain  \nof microscopic  events  that result  in a macroscopic,  measurable  quenching  of the net magnetization.  \nSince  the processing  speed  of magnetic  storage  devices  is limited  by the maximum  speeds  at which  the \nmagnetization  can be manipulated,  the possibility  of controlling  magnetization  at sub-picosecond  timescales  \nby employing  ultrashort  laser pulses  holds  tremendous  potential  for applications  in spin-based  memory  and \nstorage  devices  with ultrafast  processing  speeds  including  laser -induced  opto-magnetism  and all-optical  \nswitching  [1–4] as well as THz spintronic  devices  [5–7]. Meanwhile,  the relaxation  timescales  of the laser -\ninduced  magnetization  precession  in ferromagnetic  thin films  associated  with the magnetic  damping  set \nfundamen tal limits  on magnetization  switching  and data transfer  rates  in spintronic  devices.     \nPicosecond  laser -induced  magnetization  quenching  in ferromagnetic  gadolinium  was reported  by \nVaterlaus  et al. in 1991,  who also reported  for the first time a character istic timescale  of 100±80  ps \n[8] associated  with the magnetization  loss. In this early  work,  the timescale  was identified  with the \ntimescale  of the spin-lattice  interactions  mediating  the disruption of magnetic ordering due to laser \nheating of the lattice,  though later works  contradicted  this interpretation  [9, 10]. In fact, by the mid-\n1990s,  it had begun  to be recognized that finer time resolution was necessary to probe this phenomenon \nto uncover a  fuller  picture  of the associated  relaxation  processes  occu rring  at sub-picosecond  timescales.  \nIn 1996,  Beaurepaire  et al. demonstrated  in a seminal  work  that faster  demagnetization  occurring at sub -\npicosecond timescales could be triggered by femtosecond laser pulses in a  nickel thin film [11].  The \nultrafast demag netization phenomenon went on to be demonstrated  in a wide  array  of ferromagnetic  \nsystems  [12–19], triggering  a flurry  of research  that continues  till date. A few years after the pioneering \nexperiments, Koopmans et al. characterized for the  first time the full magneto -optical response to \nfemtosecond laser pulses in a ferromagnet by  time-resolved measurements of Kerr ellipticity and \nrotation [13].  However, the observation of  nonmagnetic contributions to the Kerr rotation signal \nnaturally led some to question  whether  the ultrafast quenching in response to the laser excitation was \nindeed a magnetic phenomenon  [13, 14]. In 2003, Rhie et al.  used time -resolved photoemission  \n \n1First  submitted  in March  2022  2  spectroscopy to probe the collapse of the 3d exchange s plitting in nickel as an unambiguous signature of a \nphotoinduced demagnetization occurring over 300±70 fs [15]. This was soon followed by the first reports \nof an estimate of the characteristic timescale of femtosecond laser -induced demagnetization derived from \nquantum -mechanical principles [16].  \nOne of the major reasons behind this sustained interest is the need to achieve a complete understanding of \nthe microscopic mechanisms that underlie the ultrafast demagnetization phenomenon that have so far \nremained elusive. Much inquiry has been focused on how the laser -induced loss of magnetic order is \ncompensated for via the transfer of angular momentum of the spin system at the associated timescales. \nOver the years, several mechanisms have been proposed for explai ning the angular momentum \nconservation associated with the ultrafast demagnetization, which may be broadly categorized in two \ndistinct classes. One of these argues in favor of a dominant contribution to the demagnetization arising \nfrom nonlocal transport p rocesses driven by the laser pulse, such as superdiffusive transport of spin -\npolarized hot electrons [20, 21] or heat currents [22]. The other narrative relies on local spin -flip scattering \nprocesses occurring by the collision of excited electrons with imp urities, phonons, and magnons [17, 23, \n24]. In this category, electron -lattice and electron -spin relaxation mechanisms are accepted as the major \ndemagnetization channels in the picoseconds following the laser excitation [25], with several works \nattributing  a pivotal role to the material -specific spin -orbit interaction [23, 25 –27]. On the other hand, it is \npossible to model the demagnetization by adopting a thermal description, considering the laser excitation \nas a “heat” source that excites electrons close to the Fermi level instantaneously to very high energies, \nwhose eventual relaxation processes drive the magnetization loss. The laser excitation energy controlled \nby the applied laser pump fluence has a direct influence on the maximum electron temperatures  reached \nand thus plays a pivotal role in the ultrafast magnetization dynamics that follow as a result.  \nIn this work, employing an all -optical time -resolved magneto -optical Kerr effect (TR -MOKE) technique, \nwe investigate the ultrafast spin dynamics along w ith the nanosecond magnetization precession and \ndamping at different laser pump fluences in three ferromagnetic thin films: 20 -nm-thick cobalt, nickel and \npermalloy grown under uniform deposition conditions. Cobalt and nickel are elementary 3d ferromagneti c \ntransition metals well studied in the literature in both their elementary forms and as constituting elements \nin multilayered structures where their strong spin -orbit coupling mediates interfacial effects such as spin \npumping. Permalloy, an alloy comprise d of approximately 80% nickel and 20% iron, is a prototype \nmaterial for spintronic applications due to several desirable qualities such as low coercivity,   large  \npermeability  and  in  particular,   low  Gilbert  damping,   which  aids  in minimizing the  maximum power \nconsumption of devices and allows spin wave propagation on length scales of the order of device size. The \nsimultaneous investigation of the fluence -dependent modulation of ultrafast demagnetization and damping \nin all the above ferromagnetic thin films is motivated by the primary objectives of exploring the dominant \nmicroscopic processes underlying the demagnetization in these systems, correlating the observed \nmagnetization dynamics to the material properties of each system, and in the process  exploring their \ntunability with laser pump fluence. We set out to achieve this in two ways: (a) by modelling the ultrafast \ndemagnetization at various pump fluences using two theoretical models sharing a common link and (b) by \ncorrelating the laser -induced  changes in ultrafast demagnetization to the changes in the Gilbert damping \nfactor which characterizes the magnetization precession. TR -MOKE is a well established local  and  non -\ninvasive all -optical measurement technique tunable to an extremely fine time resolution limited only by \nthe laser pulse width and is thus well suited to our purpose.  \nFor the detailed analysis of ultrafast demagnetization in our samples, we extract the values of \ndemagnetization time and fast relaxation time for each sample using a p henomenological fitting function \nand record its systematic variation with the pump fluence. We subsequently model the demagnetization \ndata using two well known theoretical models: the phenomenological three -temperature model (3TM) and \nthe microscopic three  temperature model (M3TM), and thereby extract values for the microscopic \nparameters relevant for the demagnetization process and calculate the temporal evolution of the simulated \ntemperatures of the electron, spin and lattice systems within the first few picoseconds of the laser \nexcitation. Both these models assume a thermal picture to explain the initial electron temperature rise and \nsubsequent relaxation but differ significantly in their approach with regards to the treatment of the \nmagnetization. A syst ematic study implementing both models to analyze TR -MOKE data recorded under \nuniform experimental conditions on identically prepared samples is absent in the literature and will prove \ninstructive. Further, a thorough investigation and characterization of t he ultrafast demagnetization, \nmagnetic damping and their intercorrelation in three different systems deposited under identical deposition \nconditions has not been carried out before. For the purposes of a comparative analysis, it is  vital to study \nsamples g rown under the same deposition conditions.  The conductivity of the  substrate  can also directly  \ninfluence  demagnetization  time by promoting  or hindering  ultrafast spin transport processes [20, 28] while \nthe deposition technique determines the structural  properties of the samples that may indirectly affect the 3  demagnetization time.  Moreover,  since the experimentally obtained demagnetization timescale has been \nshown to be quite  sensitive to extrinsic factors such as the laser pulse duration and the spectral ba ndwidth \n[29],  it is useful to measure the demagnetization times for various thin films from experiments  performed  \nin the same  experimental  setup  under  near-identical  experimental  conditions.  \n \n2. Materials  and Methods  \n2.1 Sample  fabrication  \n20 nm -thick cobalt,  nickel and permalloy thin films were deposited by electron beam  evaporation \nunder an average base pressure of 1 ×10−7 Torr and at a very low deposition  rate of 0.2 Å/s chosen to \nachieve uniform deposition.  Each film was deposited on an  insulating  8 mm × 8 mm silicon  substrate  \ncoated  with  285-nm-thick  SiO 2. Subsequently, the films were capped in-situ with a 5 -nm-thick \nprotective gold layer (base  pressure  ~1 × 10−7 Torr,  deposition  rate 0.1 Å/s) to prevent  surface  \noxidation  of the ferromagnetic layer and protect it from possible degradation due to high power laser  \nexposure during the optical pump -probe measurements [ 30, 31]. The thickness of the capping  layer  was \nkept more  than three  times  smaller  than the optical  penetration  depth  of 400 nm light in gold.  After  \nthe deposition  of the capping  layer,  the surface  topography  of the samples  was investigated  by atomic  \nforce  microscopy  (AFM)  using  a Tap190Al -G tapping  mode  AFM  probe as shown in Figure 1 ( a). The \naverage surface roughness  Ra of the cobalt,  nickel and  permalloy  films  were  obtained  as 0.91 nm, 0.36 nm \nand 0.41 nm respectively.  Thus,  reasonably  low surface  roughn ess in the sub-nm range  was obtained  \nwhich  is comparable  for all the samples.  In addition,  the static  magneto -optical  Kerr effect  was used to \nstudy  the magnetic  hysteresis  of the deposited  samples  to confirm  their ferromagnetic  nature  (Figure  \n1(b)). \n \n 2.2 All-optical  measurement  of ultrafast  spin dynamics  \nMeasurement of laser -induced magnetization dynamics is carried out using a TR -MOKE  technique \nin two -color optical pump -probe arrangement using a 400 nm pump beam and 800  nm probe beam \nhaving a pump -probe cr oss-correlation width of ~100 fs [30] shown  schematically in Figure 1( c). A \ntypical TR -MOKE trace is shown in Figure 1( d), comprising  ultrafast  demagnetization  (Regime  I), fast \nremagnetization  (Regime  II) and damped  magnetization precession (Regime III).Th is technique enables \nthe direct observation of the  spin dynamics in the femto - and picosecond time domain. Femtosecond \npulses are generated  by an amplified laser system (Libra, Coherent Inc.)  employing a chirped -pulse \nregenerative  amplifier and a Ti:sapphi re laser oscillator (Coherent Inc.)  pumped by a neodymium -doped  \nyttrium lithium fluoride (Nd -YLF) laser.  The second harmonic of the amplified laser output  \n(wavelength = 400 nm, repetition rate = 1 kHz, pulse width >40 fs), generated through a  lithium  \ntribo rate (LBO)  nonlinear  crystal,  is used  for laser  excitation  of the ferromagnetic  thin films.  The \ntime-delayed fundamental beam (wavelength = 800 nm, repetition rate = 1 kHz, pulse width ~40 fs) is \nused to probe the ensuing magnetization dynamics.  In our  setup, different wavelengths are employed \nfor the pump and the probe pulse to eliminate the  possibility of state blocking effects arising from the \nuse of identical wavelengths for pumping  and probing [32]. A computer -controlled variable delay \ngenerator offers  precise control of the  delay time between pump and probe. Before commencing \nmeasurements on any sample, the  zero delay was carefully estimated by maximizing the transient \nreflectivity signal of a bare  silicon  substrate  placed  adjacent  to the sample  on the same  sample  holder.  \nTR-MOKE  experiments are performed with a non -collinear pump -probe geometry.  The pump beam,  \nfocused to a spot size of ~300 µm, is incident obliquely on the sample while the probe  beam, with a \nspot size of ~100 µm, is incident normal to the sample surface and aligned  to the center of the pump \nspot. The pump -probe spatial overlap on the sample was carefully  maintained. The choice of a \nrelatively smaller spot size of the probe beam as compared to the  pump beam facilitates optical \nalignment and ensures that the probe beam detects the local  magnetization changes from a part of the \nsample uniformly irradiated by the pump.  Before  reflection on the samples, the probe beam is polarized \northogonally to the linearly polarized  pump beam. After reflec tion, the Kerr -rotated probe beam is split \nand one part is fed directly  into a Si -photodiode to measure the time -resolved reflectivity signal.  The \nother part is fed  into an identical photodiode after passing through a Glan -Thompson polarizer adjusted \nto a small  angle  from  extinction  to minimize  optical  artifacts  in the Kerr  rotation  signal.  In this way,  \nsimultaneous  measurement  of the time-resolved  total reflectivity  and Kerr  rotation  signals is possible.  \nAn optical chopper operating at 373 Hz placed in th e path of the pump  beam provides a reference signal \nfor the digital signal processing lock -in amplifiers (Stanford  Research  Systems,  SR830)  which  record  \nthe modulated  signal  in a phase  sensitive  manner.  All experiments  were  carried  out under  ambient  \ncondit ions of temperature  and pressure.  4  3. Results  and Discussion  \n 3.1 Theoretical  models  for ultrafast  demagnetization  \nThe phenomenon  of optically  induced  ultrafast  demagnetization  starts  with the irradiation  of the \nmagnetic  sample  with  a brief  and intense  optical  laser  pulse,  exciting  electrons  momentarily a few \nelectron volts above the Fermi level.  Though the exact sequence of events  following  the initial  excitation  \nis difficult  to trace  due to the highly  nonequilibrium  conditions  created  by it, a qualitative  overview  of \nthe complete  demagnetization  process  is fairly  well   established.  The laser  excitation  generates  a \nnonthermal  pool of excited  electrons  which  thermalize  rapidly  within  several  femtoseconds  via \nelectron -electron  interactions.  Spin -dependent  scatter ing events  taking  place  during  this transient  regime  \nlead to a sharp  drop  in magnetization  observable  around  a few hundred  femtoseconds  in the \nexperimental  Kerr  rotation  signal.  Subsequently,  the thermalized  electrons  may release  their  excess  \nenergy  via a variety  of relaxation  channels,  such  as by excitation  of phonons  or magnons.  This results  in a \npartial  recovery  of the magnetization  beyond  which  heat dissipation  into the environment  promotes  further  \nrecovery  on a longer  timescale.  The 3TM  posits  that the thermodynamics  of the demagnetization  \nphenomenon  can be described  simply  by considering  energy  exchange  between  three  thermal  reservoirs  \n[33],  each of which  is assigned  a temperature:  the electrons  at temperature  Te, the lattice  at temperature  \nTl and the electronic  spin reservoir  at temperature  Ts. Since  the reservoirs  are in thermal  contact  and \nthe overall  process  is adiabatic,  equilibration  of the excited  electrons  with the spin and lattice  reservoirs  \nvia energy  transfer  may be described  by coupled  rate equations  in the following  manner:   \n \n \n \n   (1) \n \n \n \nwhere,  Ce, Cl and Cs denote the specific heats of the electron,  lattice and spin reservoirs  respectively,  \nwhile  Gel, Ges, and Gsl denote  the inter -reservoir  coupling  parameters.  The term  P (t) describes  the action  \nof the laser  pulse  as a source  term driving  the excitation  of the electron  reservoir  to high temperatures.  The \nthermal  diffusion  term  \n  describes  heat dissipation  occurring  via thermal  conduction  \nalong  the sample  thicknes s. Under  this description,  the observed  demagnetization  is attributed  to a \nrise in the spin temperature  Ts occurring  shortly  after  the electron  temperature  rise. The coupling  \nstrengths  between  the electron -spin and electron -lattice  subsystems  qualitatively  determine  the efficiency  \nof energy  transfer  between  them  and hence  influence  the timescales  associated  with the demagnetization \nand fast relaxation.  However, the 3TM is purely phenomenological and does  not explicitly consider any \nmicroscopic mechanisms un derlying the phenomenon it describes.  On the other  hand,  the M3TM  \nproposed  by Koopmans  et al. [23] provides  a “spin -projected”  perspective  [27] to explain  the ultrafast  \ntransfer  of angular  momentum  highlighting  the role of Elliot -Yafet  type ultrafast  spin-flip scattering  in \nthe demagnetization  process.  The initial  excitation  by the laser  pulse  disturbs  the electronic  subsystem  \nfrom  equilibrium  which  leads  to an imbalance  in spin-up and spin-down  scattering  rates,  resulting  in the \nobserved  loss of magnetic  order.  The process  is mediated  by spin-orbit  interactions  leading  to the \nformation  of hot spots  in the band  structure  where  spin-up and spin-down  channels  are intermixed.  An \nelectron  scattered  into these  hot spots  via a phonon - or impurity -mediated  scatteri ng will flip its spin with  \na finite  probability.  The individual  scattering  events  are characterized  by a parameter asf across  the  \nsample,  identified  with the probability  of spin-flip due to  electron -phonon  scattering.  The magnitude  \nof this parameter  will directly  depend  on the extent of  spin-orbit  coupling and  hence  is expected  to be \ncomparable  in materials with  similar  spin-orbit  coupling  strengths.  The M3TM  retains  the coupled  rate \nequations  for the electron  and lattice  temperatures,  similar  to the thermal description  provided  by the \n3TM.  However  the fundamental  difference  from  the 3TM  is that in the framework  of the M3TM,  the \nspin bath is formed  by a collection  of two-level  systems  obeying  Boltzmann  statistics.  Instead  of \nassigning  a temperature  to the spin bath,  the normalized  magnetization  is directly  calculated  from  the \nassociated  exchange  splitting.  The rate of change  of the magnetization,  derived  analytically \nconsidering an Elliot -Yafet scattering -driven demagnetization, is parametrized by  asf and c oupled to the \nelectron and the lattice subsystem temperatures. The assignment of a  characteristic temperature to the spin \nsubsystem is replaced in the M3TM by an evolution  equation  for the magnetization:   \n 5  \n        (2) \n \nThe quantity  R is a mate rial-specific  factor  which  influences  the demagnetization  rate and is \nproportional  to asfTc2/μat where  TC is the material Curie temperature and µat is the atomic magnetic \nmoment. Though the two models  differ in their approaches, one can immediately discern certain \nsimilarities in their domains of  validity.  Both models are appropriate only when th e nonlocal \nmechanisms driving ultrafast  demagnetization such as superdiffusive spin transport can be neglected. \nWe note here that it  has been  reported  that spin transport  is not a major  contributor  to the ultrafast  \ndemagnetization in transition metals [34] . We nevertheless use an insulating SiO 2-coated Si  substrate \nfor our samples to minimize spin transport effects such that analysis with the local  models described \nabove should suffice in our case. Any additional contributions arising from  the gold capping  layer  \nwould  be uniform  across  all samples  investigated  and therefore  unlikely  to impact the main results of our \ncomparative study.  Moreover, since the thickness of the  capping layer is much smaller than the \npenetration depth of both 400 nm and 800 nm light  in gold [35], the pump excitation fully penetrates \ndown to the magnetic layer ensuring that the  effect of direct laser excitation of the ferromagnet is \nprobed in our case.  Thus, we set up  numerical calculations based on the models described above in \norder  to extract microscopic  information  from  the experimentally  obtained  demagnetization  traces.  \n 3.2 Ultrafast  demagnetization  in cobalt,  nickel,  and permalloy  thin films  \nWe proceed by performing time -resolved measurements of the polar magneto -optical Kerr  effect in \nthe cobalt, nickel and permalloy thin film samples as a function of the laser fluence.  Measurements are \ncarried out under a strong enough external magnetic field kept constant at  around 2 kOe tilted at a small \nangle from the sample plane to saturate  the magnetization of  the samples. The pump fluence is varied \nbetween 0.8 -8.7 mJ/cm2 by varying the power of the  pump pulse.   The results are presented in Figure 2.   \nTo ascertain that the measured Kerr  signal  reflects  the true magnetization  dynamics  without any \nspurious  contribution  from  optical effects triggered in the initial stages of laser excitation, we also \nexamine the transient  reflectivity signal for each sample. The fluence dependent variation in the \nreflectivity can be  found in Figure S1 of the  Supplementary Materials, demonstrating that at any given \nfluence  the amplitude  of the reflectivity  signal  is negligibly  small  compared  to that of the  Kerr  rotation.  \nWe nevertheless restrict ourselves to the low fluence regime to avoid nonlinear  effects a nd sample \ndamage.  For all our experiments, the probe fluence is kept constant at a  value about half that of the \nlowest pump fluence used to prevent additional contribution to  the spin dynamics by probe excitation.  \nAs seen in Figure 2, the ultrafast demagne tization  completes within 1 ps for all three samples \nconsidered which is followed by a fast recovery of  the magnetization,  all observed  within  the \nexperimental  time window  of 4 ps. These  experimental traces clearly exhibit the “Type -I” or “one -step” \ndemagn etization expected for  transition metal thin films at room temperature and under low -to-\nmoderate pump fluence  [23]. The amplitude of the maximum quenching of the Kerr rotation signal \nincreases with the  laser  fluence,  allowing  us to rule out nonlinear  effec ts [36].  Closer  inspection  of the \ntraces  also reveals an increase of the time taken to demagnetize the samples with increasing fluence for  \nall three  samples.  To quantify  this increase,  we fit our demagnetization  traces  to a phenomenological  \nexpression  based on the 3TM  and valid  in the low laser  fluence  regime  [37]:   \n \n(3) \n \nwhere  Θ(t) is the Heaviside  step function,  δ(t) is the Dirac  delta  function  and Γ(t) is the \nGaussian laser pulse.  The constant A1 represents the value of the normalized  magnetization after \nremagnetization has completed and equilibrium between the electron,  spin and lattice  reservoirs  has \nbeen  re-established.  A2 is proportional  to the initial  rise in electron  temperature  and hence  to the \nmaximum  magnetization  quenching.  A3 represents  the magnitude of state -filling effects present during \nthe onset of the demagnetiz ation response,  which is negligible in our case.  τM and τE are the \ndemagnetization time and fast relaxation  time,  respectively.  Prior  to the fitting,  all the experimental  \ntraces  were  normalized  by hysteresis measurements of the Kerr rotation signal under the saturating \nmagnetic field in the  absence of la ser excitation.  We find that within the range of fluence values \nconsidered,  permalloy  exhibits  the largest  magnetization  quenching  of 54.6%,  followed  by a 23.7%  \nquenching achieved in nickel, while the magnetization of cobalt the least, only about 8% for  the largest  \napplied  fluence.  The demagnetization  occurs  at a characteristic  timescale  of 230-280 fs for cobalt, 160 -6  210 fs for nickel, and 220 -250 fs for permalloy, increasing with the  laser fluence.  This effect can be \nattributed to enhanced spin -fluctuation s at elevated spin  temperatures  for  higher  fluences  [38].      \nAt  a  fluence  of  4.8  mJ/cm2,   the  extracted  demagnetization  times  are 276.6  ± 3.41 fs for cobalt,  \n187.3  ± 2.89 fs for nickel  and 236.8  ± 2.45 fs for permalloy.  The timescale  for the magnetization  \nrecovery  τE also increases  with increasing pump fluence. The variation of these characteristic \ntimescales with laser fluence is  shown  in Figure  3. These  fluence -dependent  trends  in τM and τE hint at a \nspin-flip process -dominated ultrafast demagnetization in our stu died systems [23, 39, 40]. The values  of \nτM extracted  from  our experiments  lie within  the typical  range  of 100-300 fs consistent  with \nprevious reports of the ultrafast demagnetization times in these metals [17, 23], and are  too large  to \nrepresent  a superdi ffusive  transport -driven  demagnetization  [41].  \n For the 3TM and M3TM simulations, we choose a laser pump term given by \nproportional to  the pump fluence F and following a Gaussian temporal profile.  \nThe maximum rise of the  electron temperature a nd thus also the extent of demagnetization depends \nsensitively on this  term which is hence adjusted to reproduce the maximum quenching observed \nexperimentally.  We use a pulse width τp = 100 fs determined by the pump -probe cross -correlation in all  \ncalculations.  Intrinsic  to both the models  we consider  is the assumption  that electron  thermalization occurs \nextremely fast.  The thermal  diffusion term  can be neglected in our case since th e thicknesses of the films \nwe study are kept slightly greater than the optical penetration depth of 400 nm pump beam in  those films.  \nThis ensures uniform heating of the films in the vertical direction while also  avoiding laser penetration \ninto the substrat e in which case heat dissipation into the substrate  would have to be taken into account. \nBesides, the timescales associated with heat dissipation  are generally  tens to hundreds  of picoseconds,  \nmuch  longer  than the demagnetization  and fast relaxation  times,  and hence  unlikely  to significantly  \ninfluence  our observations  at these  timescales.  Since both models we consider are thermal in their \napproach, choosing correct  values for the reservoir specific heats is vital for a proper simulation of the \ndemagnetizati on. For the electronic  specific  heat Ce, we assume  a linear  dependence  on the electronic  \ntemperature Ce(Te) = γTe  derived from the Sommerfeld free -electron approximation where γ is determined \nby the electronic density of states at the Fermi level [42].  The value of γ for permalloy is approximated as \na weighted average of the individual γ values of nickel and iron  in permalloy. The lattice specific heat Cl is \ncalculated at each value of the lattice temperature  according to the following relation derived from Debye \ntheory:  \n(4) \n \nwhere NA is the  Avogadro’s  number,  kB is the Boltzmann’s  constant  and θD  is  the  Debye  temperature.  \nFinally,  we fix the spin specific  heat Cs to its value  at room  temperature  for the 3TM  calculations,  \nobtained  by subtracting  the electronic  and lattice  contributions  from  the experimental values of the total \nspecific heat found in the literature [43].  Considering a  spin-temperature -dependent form of Cs was not \nfound to significantly affect our conclusions  as described in Section IV of the Supplementary Materials.  \nThe fixed parameter set used in  our calculations have been listed in Table 1. To relate the experimental \ndemagnetization to  the temperature  of the spin subsystem  under  the 3TM  framework,  the spin \ntemperature  Ts is  mapped to the magnetization of the system via the Weiss’ mean field theory [44], \nwhich is  then fitted with the experimental magnetization traces to obtain the empirical inter -reservoir  \ncoupling parameters Gel and Ges consistent with the observed dynamics.  We neglect the  spin-lattice \ncoupling parameter Gsl for the 3TM simulations since in ferromagnetic transitio n metals the energy \ntransfer between electrons and lattice is far greater than that between  lattice  and spins  [45].  \n \n Table 1 . Fixed parameter set used in the calculations. Literature values have been used for all parameters listed   \n[30, 31, 34, 35].  \n \n For the 3TM simulations, we proceeded to extract Gel and Ges by first fitting the  demagnetization \ndata at the lowe st fluence to the model. However, fitting the higher fluence  data using identical values \nof the coupling parameters as extracted at the lowest fluence did  not result in a good match to our \nexperimental results.  The coupling parameters extracted  from the lo w fluence data led to an \noverestimation of the demagnetization time at the higher  fluences. It was seen that a 5 -10% increase in Sampl e Tc (K) θD (K) γ (Jm-3K-2) Cs (Jm-3K-1) at (B) \nNi 627  450  1065  3.07 × 105 0.62  \nCo 1388  445  714  1.59 × 105 1.72  \nPy 860  454  992  2.67 × 105 1.00  7  Ges from its value at its adjacent lower fluence  value rectifies the overestimation of the demagnetization \ntime.  On the other h and, the remagnetization dynamics is most sensitive to the Gel parameter so that the \noverall dynamics  is best reproduced  only  by adjusting  both  Gel and Ges. As shown  in Figure  4, the \nresulting  fit shows excellent agreement with the experimental data.  This exercise reveals the crucial role  \nplayed by the electron -spin relaxation channels in determining the timescale associated with  the initial \ndemagnetization while the magnetization recovery is primarily mediated by the  electron -lattice \ninteraction.  We also f ind that the mismatch between model and experiment  can be resolved by \nconsidering an increasing trend of Gel and Ges with pump fluence arising  from a faster demagnetization \nprocess for the same percentage quenching as compared to the  model predictions with in the studied \nfluence range. The values of the microscopic parameters  extracted from the least -squares fits with their \ncorresponding error bounds can be found in  Supplementary Tables S1 -S3. Since the exact values of the \ncoupling parameters extracted  from  the fits naturally  depend  on the values  chosen  for the fixed  \nparameters,  the interpretation  of the results  from  these  fits is best limited  to a comparative  one. \nFor the M3TM  simulations,  the demagnetization  traces  are fitted  directly  to Equation  2 yielding  \nGel and asf as fit parameters.  In this case,  asf plays  a role  in  determining  the  maximum extent and the \nassociated timescale of the demagnetization via the scaling factor R while Gel continues to influence \nmainly the magnetization recovery process.  How ever, the  demagnetization  time is less sensitive  to \nchanges  in asf than it is to Ges in the 3TM  case.  This results in somewhat higher values of Gel and a \nsharper rise with pump fluence than those  extracted  from  the 3TM  simulations,  in order  to compensate  \nfor the overestimation  of demagnetization time that results from the model if Gel  at the lowest fluence is \nused for all  the fits.  We have obtained an asf of ~0.02 for cobalt,  ~0.05-0.06 for nickel, and  ~0.03-0.06 \nfor permalloy.  The value of asf we have ext racted for nickel is an order of  magnitude lower than the value \nasf = 0.185 first reported by Koopmans et al. [23] but quite  close to the value of 0.08 reported by Roth et \nal. [39].  This discrepancy is expected, as the  artificially high value of 0.185 aros e due to an \noverestimation of the electronic specific heat in  the original  work,  avoided  here by considering  \nexperimentally  determined  γ values  reported  in the literature.  The observation  that asf [Co] <asf [Ni] is \nconsistent  with previous  reports  [23, 40]. \nWith  regard  to thermal  treatments  of ultrafast  demagnetization  like the 3TM  and M3TM,  there is \nsome contention centered on the role o f the nonthermal electrons generated within the  first 50-100 \nfemtoseconds  after laser excitation  [9, 48, 49]. Within  this time frame,  the excited  hot electrons  have not \nyet thermalized  and their energies  cannot  be described  by a Fermi -Dirac  distribution.  However,  both \nthe 3TM  and the M3TM  completely  neglect  the finite  electron  thermalization  time and consider  only the \nenergy  exchange  between  the thermalized  electron  population  with the lattice  and/or  the spin subsystem  as the \ninitial  nonequilibrium  distribu tion is assumed  to last for much  shorter  durations  (≳ 10 fs) than the \ntimescale  over which  demagnetization  typically  occurs  (≳ 100 fs). However,  strictly  speaking,  this \nassumption  is valid  only when  the thermalization  proceeds  very rapidly  such as can be ensured with  high \nfluence  excitations  [50].  In the low  fluence  regime,  electron thermalization  is slower,  hence  the influence  \nof nonthermal  electrons  on the measured  dynamics  may become  more  important  with decreasing  pump  \nfluence.  The increasing  of Gel with fluence  deduced  from our calculations may be an indication of a \ndecrease in the efficiency of electron -lattice  interactions  at low laser  fluences  when  nonthermal  electrons  \nprevail.  However,  the specific  values of Gel and Ges values we have obtained for th e different systems \nremain comparable to  values  previously  reported  for these  metals  [11, 23, 37]. \nAfter completion of the fitting routine,  we set up simulations using the optimized values of  the free \nparameters to obtain the temporal evolution of Te, Tl and Ts as a function of the  pump -probe delay.   The \ncalculated profiles of Te, Tl and Ts for four different values of the  pump fluence are presented in Figure \n5 for nickel, showing good agreement between the  results predicted by the 3TM and the M3TM. Simil ar \nprofiles for cobalt and permalloy are  given in Figs.  S2 and S3 of the Supplementary Materials.  As the \npump fluence is increased,  laser excitation deposits a greater amount of energy in the electronic \nreservoir which leads to  a proportionate increase in the initial electron temperature rise.  However, the \nmaximum  electron temperature reached for the highest applied pump fluence remains well below the  \nFermi  temperature  for the respective  material,  ensuring  the assumption  of the linear  temperature  \ndependence  of the electronic  specific  heat remains  valid  [42].  Given  the fundamentally different origins \nof the 3TM and M3TM, the close agreement between the  temperature profiles calculated using the two \nmodels and among the extracted values of the  electron -lattice coupling parameter Gel is remarkable. \nThis observation reflects that, given the  strength of the laser excitation determining the initial \ntemperature excursion of the electron  system,  the Gel  parameter common to both the 3TM and the \nM3TM framework plays a  pivotal  role in determining  the profile  of the ensuing  magnetization  changes.  8  This  also  explains why Gel is seen to follow a similar fluence -dependent trend for a given system in \nboth models.  While the electron temperature is coupled to the spin temperat ure via the Ges parameter  in \nthe 3TM,  it enters  into the evolution  equation  for the magnetization  via  Equation 2 in the M3TM case. \nFrom Figure 5, a qualitative difference in the temporal profiles  of the electron,  spin and lattice \ntemperatures is also appa rent.  Referring to the figures  corresponding  to the 3TM,  while  Te exhibits  a \nsharp  peak  within  the first few hundred  femtoseconds, Ts exhibits a shorter and broader peak at slightly \nlonger time delays.  The delayed peak in Ts occurs when excited electrons t ransfer energy to the spin \ndegrees of  freedom  [51] through  scattering  events  mediated  by the coupling  parameter  Ges, which  \nqualitatively determines the maximum spin temperature value [11].  These scattering events  can result in \nthe generation of magnons, St oner excitations and spin fluctuations which may  play a part in the \nobserved demagnetization [52].  Lastly, the lattice temperature gradually  increases over the first few \npicoseconds after excitation, reaching an equilibrium value at  which  it stays  nearly  constant  over \nseveral  picoseconds.  \nIn Figure  6 we present  a comparison  between  the experimental  demagnetization  curves  and calculated \nelectron temperature profiles for each of our three samples. Under the same applied  pump  fluence,  different  \namplitudes  of magnetization  quenching  and electron  temperature  rise are observed. It is clearly seen that \nnickel quenches more strongly than cobalt, an observation  which  can be explained  by band  structure  \neffects  and a more  efficient  laser -induced  electronic  excitation i n nickel owing to its much lower Curie \ntemperature [34, 40, 53], reflected in the  M3TM as a higher extracted value of the spin -flip probability  \nasf  in nickel.   On the other  hand, a lower electronic specific heat capacity in cobalt as compared to \nnickel [42] predictably  leads  to a much  greater  rise in its electron  temperature  for the same  pump  fluence.  \nIn the case of permalloy,  alloying  with iron can reduce  the electronic  density  of states  at the Fermi  level  \n as compared to nickel, decreasing γ. Because o f the larger electron temperatures reached as a  result, a \nvalue of asf comparable to that of nickel is sufficient to drive a much larger reduction  in the \nmagnetization.  Permalloy also shows a somewhat larger demagnetization time at a  given pump fluence \nas compared to nickel.  Since a standard TR -MOKE optical pump -probe  measurement  cannot  be used  to \nprobe  the demagnetization  response  in alloys  in an element -specific manner, interpreting this finding is \ndifficult. It has been reported previously  using element -specific measurements in a nickel -iron alloy that \nthe demagnetization of nickel  proceeds  faster  than that of iron [54].  Our observation  of a slower  \ndemagnetization  in permalloy may be reflective of this result.  Finally, the demagnetization times for the \nthree systems  at any given  pump  fluence  is seen to sensitively  depend  on the ratio  of TC/µat, \nconfirming  its role as a ‘figure  of merit’  (FOM)  for the demagnetization  time [23].  As shown  in \nFigure 8 ( a), the demagnetization proceeds slower for a lower value  of the FOM. Thus at a  given \nincident laser pump fluence, the demagnetization times are found to correlate with the  values  of this \nquantifying  parameter,  rather  than the material  Curie  temperature  directly.  \n \n 3.3 Precessional  dynamics  and correlation  of Gilbert damping  parameter  with ultrafast  demagnetization  \ntime \nTo gain further  insight  into the microscopic  mechanism  underlying  the ultrafast  \ndemagnetization,  we study also the precessional magnetization dynamics in our thin film  systems. A \ntypical time -resol ved Kerr rotation data showing the different temporal regimes of  laser -induced \nmagnetization dynamics is given in Figure 1( d). Regime I and II represent the  ultrafast  demagnetization  \nand magnetization  recovery  processes  respectively,  while  the oscillatory  signal  of regime  III represents  \nthe magnetization  precession  with the Gilbert  damping  characteristic  of the ferromagnetic  material.  To \nextract  the Gilbert  damping  factor  α, the background -subtracted oscillatory Kerr signal is first fitted \nwith a damped sinusoidal  function  given  by:  \n \n  \n (5) \n   \nwhere  f is the precessional frequency, τ is the relaxation time, and φ is the initial phase of oscillation. \nThe depende nce of f on applied bias magnetic field H is fitted to the Kittel formula relevant for \nferromagnetic systems:  \n \n \n (6) \n \nwhere  γ0 = gµB/ℏ is  the  gyromagnetic  ratio  and  Meff  is the effective  saturation  magnetization  \nof the probed  volume.  From the Kittel  fit, the values  of g and Meff can be extracted  as fit \nparameters.  While the value  of g is found  to be 2 ± 0.1 for all the samples,  the extracted  values  9  of Meff are 1280.2  ± 1.69 emu/cm3 for cobalt,  400.5  ± 11.46  emu/cm3 for nickel  and 737.1  ± 5.32 \nemu/cm3 for permalloy  (Section  V of the Supplementary  Materials).  \nSubsequently,  the effective  damping  factor  α is calculated  depending  on the extracted  value  of τ and Meff \nas:  \n \n(7) \n \nAlthough  the timescale  of precessional  dynamics  and magnetic  damping  in ferromagnetic  thin films differ  \nby many orders of magnitude,  similar underlying  phenomena may oft en determine  the characteristics of \neither process.  Through several independent investigations relating these  two distinct processes, the \ncorrelation between the demagnetization time τM and the Gilbert  damping  factor  α emerged  as a critical  \nindicator  of the dominant  microscopic  contribution  to the ultrafast  demagnetization.  Koopmans  et al. in \n2005  analytically  derived  an inversely  proportional  relationship  between  the Gilbert  damping  factor  and \nultrafast  demagnetization  time based  on quantum  mechanical  considerations  [16].  However,  the validity  of \nthe model  could  not be confirmed  experimentally  when  τM and α were  tuned  by transition  metal  and rare \nearth  doping  [18].  Later  it was proposed  that the Gilbert  damping  parameter  and ultrafast  \ndemagnetization  time can be either  directly  or inversely  proportional  to each other  depending on the \npredominant micr oscopic mechanism contributing to the magnetic damping  [55].  A proportional  dependence  \nwould  suggest  a dominating  conductivity -like or “breathing  Fermi  surface”  contribution  to the damping  \ndue to the scattering  of intraband  electrons  and holes,  while  an inverse  dependence  may arise in materials  \nwith a dominant  resistivity -like damping  arising  from  inter-band  scattering  processes.  Zhang  et al. later \nsuggested  that an inverse  correlation  between  Gilbert  damping  factor  and ultrafast  demagnetization  time \nmay arise in the presence  of spin transport,  which  provides  an additional  relaxation  channel  resulting  in an \nenhancement  of the magnetic  damping  along  with an acceleration  of the demagnetization  process  at the \nfemtosecond  timescale  [56].  In cases  where  local  spin-flip scattering  processes  are expected  to dominate  the \ndemagnetization  process,  the breathing  Fermi  surface  model  is valid  predicting  a proportional  relationship  \nbetween  τM and α. \nTo study  the correlation  between  ultrafast  demagnetization  and damping  in a system,  both τM and α \nhave to be systematically  varied.  For a given  material,  the exciting  pump  fluence  can be used to \nmodulate the demagnetization time.  As discussed in the previous section, we  observe  an increment  of the \ndemagnetization  time with the pump  fluence  for the systems  under  investigation.  An enhancement  in α \nis also observed  as the pump  fluence  is increased  from  0.8 mJ/cm2 to 8.7 mJ/cm2 for each sample,  \nassoci ated with a corresponding  decrease  in the precessional  relaxation  time.  Within  the experimental  \nfluence  range,  α is found  to be enhanced from 0.0089 to 0.0099 for cobalt, from 0.0346 to 0.0379 for nickel \nand from 0.0106 to  0.0119  for permalloy.  This enhanc ement  of the damping  can be attributed  to the \ntransient  rise in the system temperature as the laser pump fluence is increased [57].  As a laser -induced  \nincrease in electron temperature also drives the demagnetization process at the femtosecond  timescale,  one \ncan qualitatively  correlate  the demagnetization  time with the magnetic  damping at various excitation \nenergies even for a single sample.  Clearly,  a direct correlation  between demagnetization time and the Gilbert \ndamping factor is obtained in our samples b y measuring the magnetization dynamics at various pump \nfluences, as shown in Figure 7 for the  nickel thin film.  Thus, we infer that similar relaxation processes likely \ndrive the laser induced  transient  magnetization  dynamics  at the two different  timescales .   At a low \nfluence  value  of 2.1 mJ/cm2, the extracted  value  of α is the highest  for the nickel  film,  and relatively  \nlow for cobalt  and permalloy.  The observed  trend  in Figure  8 (b) can be directly  correlated  with the \nvalues  of the Sommerfeld  coefficient  γ given  in Table  1. This observation  is in accordance  with \nKamber sky’s  spin flip scattering  theory  [58],  wherein  α is directly  proportional  to the density  of states  at \nthe Fermi  level,  which  in turn governs  the value  of γ. \n \n \n4. Summary  \nTo summarize,  we have investigated  fluence -dependent  ultrafast  demagnetization  in identi cally -grown  \ncobalt,  nickel  and permalloy  thin films  using  an all-optical  time-resolved  magneto -optical Kerr effect \ntechnique.  We study both the femtosecond laser -induced ultrafast  demagnetization  and the magnetization  \nprecession  and damping  in each of our samples  for various  pump  excitation  fluences.  Using  a \nphenomenological  expression,  we extract  the values  of demagnetization  time and fast relaxation  time 10  for each of our samples  to show  that they exhibit  an increasing  trend  with applied  laser fluence  consi stent  \nwith a spin-flip scattering -dominated  demagnetization  in our samples.  Two distinct  ultrafast  demagnetization  \nmodels  can each reproduce  our experimental  demagnetization  traces  remarkably  well.  By directly  fitting  \nour experimental  data to numerical  solutions  of these  models,  we have extracted  values  for the electron -\nlattice  and electron -spin coupling  parameters  and an empirical  estimate  of the spin flip scattering  \nprobability  for each of our samples  and demonstrated  the fluence -dependent  variation  in calculated  \nelectron,  spin,  and lattice  temperatures  as a function  of the pump -probe  delay.  We conjecture  that the \nincreasing  trend  of the extracted  electron -lattice  coupling  parameter  with fluence  may indicate  a decrease  \nin the efficiency  of electron -lattice  interactions  in the low laser fluences  when  nonthermal  electrons may \nplay a role in influencing the magnetization dynamics.  We have compared the  experimental  and theoretical  \nresults  obtained  from  each of the systems  under  investigation  and confirmed  that the ratio of the Curie  \ntemperature  to the atomic  magnetic  moment  acts as a figure  of merit  sensitive  to the demagnetization  time \nfor these  systems.  Our study  demonstrates the mutual agreement between the phenomenological three \ntemperature model  and the microscopic  three  temperature  model  for the systems  we study  and highlights  the \npossibility  of a fluence -dependent  enhancement  in the inter-reservoir  interactions  in the low laser  fluence  \nregime.  Finally,  we have reported  from  our experiments  a direct  correla tion between  the ultrafast  \ndemagnetization  time and the enhancement  of the magnetic  damping  that transiently appears as a result of \nthe laser excitation, reflecting that both phenomena are  sensitive  to similar  underlying  microscopic  \nprocesses.  \n \n5. 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Xu, Transient  \nenhancement of magnetization damping in cofeb film via pul sed laser excitation, Applied  Physics  \nLetters  109 (4) (2016)  042401.  \n[58] V.  Kambersky´,   On  the  landau –lifshitz   relaxation   in  ferromagnetic  metals,    Canadian \nJournal  of Physics  48 (24) (1970)  2906 –2911.  14  \n \n \nFigure 1: ( a) Atomic force microscopy images  of the samples showing the surface topography. ( b) \nStatic -MOKE  hysteresis loops measured for each sample. ( c) A schematic of the samples used in our \nmeasurements and the  measurement  geometry.  The 400 nm pump  and 800 nm probe  are incident  non-\ncollinearly  on the sample  surface.  \n(d) Typical  TR-MOKE  data showing  different  temporal  regimes  of the magnetization  dynamics.  15   \n \n \nFigure  2: Ultrafast  demagnetization  traces  obtained  from  TR-MOKE  measurements  on (a) cobalt,  \n(b) nickel  and ( c) permalloy thin films at va rious pump fluences. The downward arrows represent \nincreased pump fluence.  The symbols  represent  experimental  data points  and the solid  lines  represent  \nfit to Equation  3. 16  \n \n \nFigure 3:  Variation of ( a) demagnetization time τM and ( b) fast relaxation time τE with the pump fluence \nfor 20-nm-thick cobalt, nickel and permalloy films. Both timescales show an increasing trend with \nincreasing pump  fluence.  \n \n \n \n \nFigure 4:  Sensitivity of the demagnetization traces to the various adju stable fit parameters.  \nSymbols  represent  data points  and solid  lines  represent  fits to the 3TM.  Data  shown  is for the \ncobalt  thin film.  For F =0.8  mJ/cm2, Gel=8.2  × 1017 Js−1m−3K−1 and Ges=6.9  × 1016 Js−1m−3K−1; \nfor F =2.1  mJ/cm2, Gel=1.1  × 1018 Js−1m−3K−1 and Ges=8 × 1017 Js−1m−3K−1. \n17   \n \n \nFigure  5: Temporal  evolution  of the reservoir  temperatures  of nickel  Te (blue),  Tl (red),  and Ts (green)  \ncalculated  using  the (a) phenomenological  three  temperature  model  and (b) the microscopic  three  \ntemperature  model.  \n \n \n \n \n \n \n \n \nFigure  6: Comparison  between  (a) the experimental  demagnetization  and (b) the simulated  electron  \ntemperature  profiles  for cobalt,  nickel  and permalloy  at a pump  fluence  of 4.8 mJ/cm2. \n18   \n \n \nFigure  7: (a) Fluence -dependent  variation  of precessional  dynamics  for the nickel  thin film.  \nSymbols  are experimental data points and the solid lines are fits to Equation 5.  (b) Variation of f \nwith pump fluence.  (c) Variation  of α with pump  fluence.  (d) Direct  correlation  between  τM and \nα. \n \n \n \n \n \n \n \n \nFigure 8:  (a) Inverse variation of the demagnetization time with the ratio of Curie temperature to \natomic  magnetic moment and ( b) inverse correlation of the demagnetization time with ma gnetic \ndamping across three  different  materials.  \n19   \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nSupplementary  Materials  S1  I. Transient  reflectivity  measurements  \nTime -resolved  transient  reflectivity  signal  is recorded  by TR-MOKE  measurements  \nsimultaneously with the Kerr rotation signal  quantifying the magnetic response.At any  given \nfluence, the maximum amplitude of the transient reflectivity change is about a  factor of 100 \nsmaller as compared to that of the Kerr rotation.  This validates that the  Kerr rotation signal \nreflects the genuine  magnetic response with negligible nonmagnetic  contribution.  \n \n \nFigure S1:  Time -resolved transient reflectivity measured by TR -MOKE for cobalt, nickel, and \npermalloy.  The upward  arrows  represent  increased  pump  fluence.  \n \n \n \nII. Calculation of electron, spin, an d lattice temperature profiles  \nThe calculated  temporal  evolution  of the reservoir  temperatures  for the 20-nm-thick  cobalt  and \npermalloy  thin films  are represented  in Fig. S2 and Fig. S3, respectively.  The electron  \ntemperature  shows  a rapid  and dramatic  increase  within  one picosecond  of the laser excitation,  \nbeyond  which  it decays  to an equilibrium  value  over a few picoseconds.  Over  this period,  the lattice  \nsubsystem  is gradually  heated  above  the equilibrium  temperature  while  the heating  of the spin \nsubsyste m directly  leads  to the demagnetization.  The maximum  electron,  spin and lattice  temperatures  \nincrease  with increasing  laser  fluence.  \nS2  \n \n \nFigure  S2: Electron,  lattice,  and spin temperature  profiles  for the cobalt  thin film calculated  \nusing  (a) the three tem perature model and (b) the microscopic three temperature model for \nvarious laser fluences  (blue:  Te, dashed,  red: Tl and dotted,  green:  Ts). \n \n \nFigure S3: Electron, lattice, and spin temperature profiles for the permalloy thin film calculated \nusing (a)  the three  temperature  model  and (b) the microscopic  three  temperature  model  for various  \nlaser  fluences).  \n \nIII. Extraction  of inter -reservoir  coupling  constants  \nThe least -squares fitting routines set up based on the three temperature model and the  microscopic  \nthree  temperature  model  return  the values  of the inter-reservoir  coupling  \nS3  constants  that best reproduce  the experimental  demagnetization  traces.  The microscopic  three  \ntemperature  model  additionally  returns  the value  of spin-flip probability asf associated with  the \nspin-flip scattering events which directly lead to the  magnetization quenching.  The extracted \nvalues of these microscopic parameters are  given in Tables S1 -S3. Fig. S4 shows a comparison of \nthe values of the electron -lattice  coupling parameter Gel as extracted from the fits to the three \ntemperature model and  the microscopic three temperature model.  For reasons discussed in the \nmain text, the  spin-lattice coupling parameter Gsl is taken to be zero for the three temperature \nmodel  simulations.  Fig. S5 shows  that,  when  Gsl  is  set  equal  to  3  ×  1016 Jm−1s−1m−3K−1 as \nconsidered in Phys.  Rev.  Lett.  76, 4250 (1996), the simulated trace  and calculated electron \ntemperature profile are nearly indistinguishable from the case in  which  Gsl is neglected.  \n \nFigure  S4: Values  of the electron -lattice  coupling  parameter  Gel extracted  from  fitting  to the three  \ntemperature  model  and microscopic  three  temperature  model.  \n \n \n \n \nFigure  S5:  (a)  Results  of three  temperature  modelling  and (b) calculated  reservoir  temperature  \nprofiles  for two values  of Gsl. \n \nPump fluence \n(mJ/cm2) Gel[3TM]  (×1018) \n(Js-1m-3K-1) Gel[M3TM]  (×1018) \n(Js-1m-3K-1) Ges(×1017) \n(Js-1m-3K-1) asf (×10-1) \n0.8 0.82±0.081  0.85±0.097  0.69±0.037  0.19±0.023  \n2.1 1.08±0.063  1.18±0.105  0.8±0.029  0.16±0.014  \n3.5 1.26±0.057  1.48±0.106  0.91±0.028  0.16±0.009  \n4.8 1.2±0.037  1.57±0.096  0.97±0.015  0.17±0.008  \n6.0 1.21±0.042  1.49±0.072  1.02±0.022  0.17±0.008  \n7.3 1.56±0.076  1.98±0.087  1.15±0.041  0.16±0.004  \n8.7 1.65±0.045  2.18±0.066  1.28±0.036  0.17±0.003  \nTabl e S1:  Microscopic parameters extracted from the 3TM and M3TM s imulations for cobalt.  \n \n \n \nS4   \nPump fluence \n(mJ/cm2) Gel[3TM]  (×1018) \n(Js-1m-3K-1) Gel[M3TM]  (×1018) \n(Js-1m-3K-1) Ges(×1017) \n(Js-1m-3K-1) asf (×10-1) \n0.8  1.29±0.154  1.41±0.172  2.1±0.182  0.53±0.03  \n2.1 1.52±0.061  1.74±0.077  2.84±0.087  0.55±0.012  \n3.5 1.53±0.0 51 1.77±0.069  2.93±0.076  0.56±0.011  \n4.8 1.65±0.047  1.9±0.06  2.86±0.064  0.51±0.008  \n6.0 1.49±0.036  1.73±0.039  2.74±0.048  0.57±0.007  \n7.3 1.58±0.028  1.88±0.033  2.87±0.038  0.59±0.006  \n8.7 1.58±0.048  1.77±0.039  2.77±0.062  0.62±0.008  \nTabl e S2:  Microscopic pa rameters extracted from the 3TM and M3TM simulations for nickel . \n \n \nPump fluence \n(mJ/cm2) Gel[3TM]  (×1018) \n(Js-1m-3K-1) Gel[M3TM]  (×1018) \n(Js-1m-3K-1) Ges(×1017) \n(Js-1m-3K-1) asf (×10-1) \n0.8 1.62±0.144  1.86±0.184  2.16±0.145  0.31±0.016  \n2.1 1.79±0.074  2.14± 0.084  2.34±0.073  0.34±0.007  \n3.5 2.05±0.081  2.46±0.065  2.65±0.084  0.39±0.005  \n4.8 2.08±0.068  2.55±0.038  2.81±0.071  0.45±0.006  \n6.0 2.02±0.053  2.5±0.065  2.72±0.058  0.49±0.007  \n7.3 2.07±0.083  2.67±0.052  2.87±0.089  0.56±0.009  \n8.7 2.11±0.084  2.71±0.069  2.87±0 .086 0.64±0.01  \nTabl e S1:  Microscopic parameters extracted from the 3TM and M3TM simulations for permalloy . \n \nIV. Temperature dependence of spin specific heat  \nConsidering the theory of Manchon et al. ( Phys.  Rev.  B 85, 064408 (2012)) as adopted  in Sci. \nRep. 10:6355 (2020), the specific heat Cs of the spin subsystem in the 3TM  framework  can be \ncalculated  as a function  of the spin temperature  Ts as: \n \n      (S1) \n \nFig. S6 below  shows  the trends  in the inter-reservoir  coupling  parameters  for two scenar ios: \nconsidering the  room temperature  value  of Cs(T0) or an explicit temperature  (and hence  time)  \ndependence  of Cs. \n \n \nFigure  S6: Fluence -dependent  variation  of (a) Gel and (b) Ges considering  two different  forms  \nfor the spin specific  heat Cs. \n \n \n \nS5  ∼ V. Bias fiel d dependence of precessional frequency and Gilbert damping  \nThe bias  magnetic  field-dependent variation of precessional dynamics in cobalt,  nickel and permalloy  \nis shown in Fig.  S7 - Fig. S9 for a pump fluence of 4.8 mJ/cm2. The oscillatory Kerr  signal is \nfitted with Eq. 5 of the main text to extract the precessional frequency f and relaxation time τ.   \nThe bias field dependence of the extracted frequency,  shown in Fig.  S7(b) for cobalt, is fitted to the \nKittel formula (Eq. 6) to extract the effective saturation  magnetization  Meff . Once  Meff have  been  \nderived,  the Gilbert  damping  factor  α is calcula ted using Eq. 7 of the main text. The extracted α \ndoes not show any systematic  variation  with  bias field.  \n \n \n \nFigure  S7: (a) Bias field -dependent  variation  of precessional  dynamics  for the cobalt  thin film.  \nSymbols  are experimental  data points  and the solid lines  are fits to Eq. 5 of the main  text. (b) \nVariation  of f with H. The solid  red line is a fit to the Kittel  formula.  (c) Variation  of τ with \nH. (d) Variation  of α with H. The solid  line is a guide  to the eye. \n \nVI. Pump fluence dependence of precessional frequency and Gilbert \ndamping  \nThe variation of precessional dynamics in cobalt, nickel and permalloy with the laser  pump \nfluence is shown in Fig. S10 for the cobalt and in Fig. S11 for the permalloy thin  film at a \nsaturating  field  of   2 kOe.  The oscilla tory Kerr  signal  is fitted  with Eq.  5 of the main  text to \nextract  the precessional  frequency  f and relaxation  time τ. With  increasing  pump  fluence,  the \nprecessional  frequency  decreases  and the magnetic  damping  is enhanced.  A direct  correlation  \nbetween  the demagnetization  time τM and α is observed.  S6  \n \n \nFigure  S8: (a) Bias field -dependent  variation  of precessional  dynami cs for the nickel  thin film.  \nSymbols  are experimental  data points  and the solid  lines  are fits to Eq. 5 of the main  text. (b) \nVariation  of f with H. The solid  red line is a fit to the Kittel  formula.  (c) Variation  of τ with \nH. (d) Variation  of α with H. The solid  line is a guide  to the eye. \n \n \nFigure  S9: (a) Bias field -dependent  variation  of precessional  dynamics  for the  permalloy  thin  \nfilm.  Symbols  are experimental  data points  and the solid  lines  are fits to Eq. 5 of the main  text. \n(b) Variation  of f with H. The solid  red line is a fit to the Kittel  formula.  (c) Variation  of τ  \nwith H.  (d) Variation  of α with H. The solid  line is a guide  to the eye. \nS7  \n \n \nFigure S10: (a) Fluence -dependent variation of precessional dynamics for the cobalt thin film. \nSymbo ls are experimental data points and the solid lines are fits to Eq.  5 of the main text.  (b) \nVariation of f with pump  fluence.  (c) Variation  of α with pump  fluence.  (d) Direct  correlation  \nbetween  τM and α. \n \n \n \nFigure S11:  (a) Fluence -dependent variation of precessional dynamics for the permalloy thin film.  \nSymbols  are experimental  data points  and the solid  lines  are fits to Eq. 5 of the main  text. (b) \nVariation  of f with pump  fluence.  (c) Variation  of α with  pump  fluence.  (d) Direct  correlation  \nbetween  τM and α. \n"    },    {        "title": "1605.01694v2.Theory_of_magnon_motive_force_in_chiral_ferromagnets.pdf",        "content": "Theory of magnon motive force in chiral ferromagnets\nUtkan G ung ord u\u0003and Alexey A. Kovalev\nDepartment of Physics and Astronomy and Nebraska Center for Materials and Nanoscience,\nUniversity of Nebraska, Lincoln, Nebraska 68588, USA\nWe predict that magnon motive force can lead to temperature dependent, nonlinear chiral damping\nin both conducting and insulating ferromagnets. We estimate that this damping can signi\fcantly\nin\ruence the motion of skyrmions and domain walls at \fnite temperatures. We also \fnd that in\nsystems with low Gilbert damping moving chiral magnetic textures and resulting magnon motive\nforces can induce large spin and energy currents in the transverse direction.\nPACS numbers: 85.75.-d, 72.20.Pa, 75.30.Ds, 75.78.-n\nEmergent electromagnetism in the context of spintron-\nics [1] brings about interpretations of the spin-transfer\ntorque [2, 3] and spin-motive force (SMF) [4{10] in terms\nof \fctitious electromagnetic \felds. In addition to pro-\nviding beautiful interpretations, these concepts are also\nvery useful in developing the fundamental understanding\nof magnetization dynamics. A time-dependent magnetic\ntexture is known to induce an emergent gauge \feld on\nelectrons [5]. As it turns out, the spin current gener-\nated by the resulting \fctitious Lorentz force (which can\nalso be interpreted as dynamics of Berry-phase leading\nto SMF) in\ruences the magnetization dynamics in a dis-\nsipative way [5, 6, 11{16], a\u000becting the phenomenologi-\ncal Gilbert damping term in the Landau-Lifshitz-Gilbert\n(LLG) [17] equation. Inadequacy of the simple Gilbert\ndamping term has recently been seen experimentally in\ndomain wall creep motion [18]. Potential applications\nof such studies include control of magnetic solitons such\nas domain walls and skyrmions [19{31], which may lead\nto faster magnetic memory and data storage devices\nwith lower power requirements [32{34]. Recently, phe-\nnomena related to spin currents and magnetization dy-\nnamics have also been studied in the context of energy\nharvesting and cooling applications within the \feld of\nspincaloritronics [35{39].\nMagnons, the quantized spin-waves in a magnet, are\npresent in both conducting magnets and insulating mag-\nnets. Treatment of spin-waves with short wavelengths as\nquasiparticles allows us to draw analogies from systems\nwith charge carriers. For instance, the \row of thermal\nmagnons generates a spin transfer torque (STT) [40{42]\nand a time-dependent magnetic texture exerts a magnon\nmotive force. According to the Schr odinger-like equa-\ntion which governs the dynamics of magnons in the adia-\nbatic limit [41], the emergent \\electric\" \feld induced by\nthe time-dependent background magnetic texture exerts\na \\Lorentz force\" on magnons, which in turn generates\na current by \\Ohm's law\" (see Fig. 1). Despite the sim-\nilarities, however, the strength of this feedback current\nhas important di\u000berences from its electronic analog: it is\ninversely proportional to the Gilbert damping and grows\n\u0003ugungordu@unl.eduwith temperature.\nIn this paper, we formulate a theory of magnon feed-\nback damping induced by the magnon motive force. We\n\fnd that this additional damping strongly a\u000bects the dy-\nnamics of magnetic solitons, such as domain walls and\nskyrmions, in systems with strong Dzyaloshinskii-Moriya\ninteractions (DMI). We also \fnd that the magnon mo-\ntive force can lead to magnon accumulation (see Fig. 1),\nnon-vanishing magnon chemical potential, and large spin\nand energy currents in systems with low Gilbert damp-\ning. To demonstrate this, we assume di\u000busive transport\nof magnons in which the magnon non-conserving relax-\nation time, \u001c\u000b, is larger compared to the magnon conserv-\ning one,\u001cm(\u001c\u000b\u001d\u001cm). For the four-magnon thermal-\nization,\u001cm=~=(kBT)(Tc=T)3, and for LLG damping,\n\u001c\u000b=~=\u000bkBT, this leads to the constraint \u000b(Tc=T)3\u001c1\n[43, 44].\nEmergent electromagnetism for magnons. We initially\nassume that the magnon chemical potential is zero. The\nvalidity of this assumption is con\frmed in the last sec-\ntion. E\u000bects related to emergent electromagnetism for\nmagnons can be captured by considering a ferromagnet\nwell below the Curie temperature. We use the stochastic\nLLG equation:\ns(1 +\u000bn\u0002)_n=n\u0002(He\u000b+h); (1)\nwheresis the spin density along n,He\u000b=\u0000\u000enF[n]\nis the e\u000bective magnetic \feld, F[n] =R\nd3rF(n) is the\nfree energy and his the random Langevin \feld. It is\nconvenient to consider the free energy density F(n) =\nJ(@in)2=2 +^Dei\u0001(n\u0002@in) +H\u0001n+Kun2\nzwhereJis\nthe exchange coupling, ^Dis a tensor which describes the\nDMI [47],H=Ms\u00160Haezdescribes the magnetic \feld,\nKudenotes the strength of uniaxial anisotropy, Msis\nthe saturation magnetization, Hais the applied magnetic\n\feld, and summation over repeated indices is implied. At\nsu\u000eciently high temperatures, the form of anisotropies is\nunimportant for the discussion of thermal magnons and\ncan include additional magnetostatic and magnetocrys-\ntalline contributions.\nLinearized dynamics of magnons can be captured by\nthe following equation [48]:\ns(i@t+ns\u0001At) =\u0002\nJ(@i=i\u0000ns\u0001[Ai\u0000Di=J])2+'\u0003\n ;\n(2)arXiv:1605.01694v2  [cond-mat.mes-hall]  13 Jul 20162\nμ/μ0\n-0.3-0.2-0.100.10.20.3\nμ/μ0\n-0.75-0.50-0.2500.250.500.75\nFIG. 1. (Color online) A moving magnetic texture, such as\na domain wall (top) or an isolated skyrmion (bottom), gen-\nerates an emergent electric \feld and accumulates a cloud of\nmagnons around it. In-plane component of nsand electric\n\feldEare represented by small colored arrows and large black\narrows, respectively. Magnon chemical potential \u0016is mea-\nsured in\u00160=\u0018~vD=J \u0001 for domain wall and \u00160=\u0018~vD=JR\nfor skyrmion, where \u0018is the magnon di\u000busion length. Soli-\ntons are moving along + xaxis with velocity v,nz=\u00061 at\nx=\u00071 for domain wall and nz= 1 at the center for the\nisolated hedgehog skyrmion. Material parameters for Co/Pt\n(J= 16pJ/m, D= 4mJ/m2,Ms= 1:1MA/m,\u000b= 0:03,\nat room temperature [45]) were used for domain wall lead-\ning to \u0001\u00197nm, and Cu 2OSeO 3parameters ( J= 1:4pJ/m,\nD= 0:17mJ/m2,s= 0:5~=a3,a= 0:5nm with\u000b= 0:01, at\nT\u001850K [46]) for skyrmion leading to R\u001950nm. System\nsize is taken to be 6\u0001 \u00022\u0001 for domain wall and 6 R\u00026Rfor\nskyrmion.\nwhere'absorbs e\u000bect of anisotropies, DMI and the\nmagnetic \feld,  =nf\u0001(e0\nx+ie0\ny) describes \ructu-\nationsnf=n\u0000nsq\n1\u0000n2\nfaround slow component\nns(jnj=jnsj= 1,ns?nf) in a rotated frame in\nwhiche0\nz=ns,Di=^Dei, andA\u0016\u0002\u0011 ^R@\u0016^RTcorre-\nsponds to the gauge potential with \u0016=x;y;z;t . Note\nthat in the rotated frame, we have n!n0=^Rnand\n@\u0016!(@\u0016\u0000A0\n\u0016\u0002) withA0\n\u0016\u0002= (@\u0016^R)^RT. In deriv-\ning Eq. 2, we assumed that the exchange interaction is\nthe dominant contribution and neglected the coupling\nbetween the circular components of  and ydue to\nanisotropies [41, 49].\nThe gauge potential in Eq. (2) leads to a reactive\ntorque in the LLG equation for the slow dynamics [50].\nAlternatively, one can simply average Eq. (1) over the\nfast oscillations arriving at the LLG equation with the\nmagnon torque term [40]:\ns(1 +\u000bns\u0002)_ns\u0000ns\u0002Hs\ne\u000b=~(j\u0001D)ns;(3)\nwhereHs\ne\u000b=\u0000\u000ensF[ns] is the e\u000bective \feld for theslow magnetization calculated at zero temperature [51],\nji= (J=~)hns\u0001(nf\u0002@inf)iis the magnon current and\nDi=@i+ (^Dei=J)\u0002is the chiral derivative [48, 52, 53].\nMagnon feedback damping. The magnon current jis\ninduced in response to the emergent electromagnetic po-\ntential, and can be related to the driving electric \feld\nEi=~ns\u0001(@tns\u0002Dins) by local Ohm's law j=\u001bE\nwhere\u001bis magnon conductivity. The induced elec-\ntric \feld Ecan be interpreted as a magnon generaliza-\ntion of the spin motive force [6]. The magnon feed-\nback torque\u001c=~\u001b(E\u0001D)nshas dissipative e\u000bect on\nmagnetization dynamics and leads to a damping tensor\n^\u000bemf=\u0011(ns\u0002Dins)\n(ns\u0002Dins) in the LLG equation\nwith\u0011=~2\u001b=s[54].\nA general form of the feedback damping should also\ninclude the contribution from the dissipative torque\n[40]. Here we introduce such \f-terms phenomenologically\nwhich leads to the LLG equation:\ns(1 +ns\u0002[^\u000b+^\u0000])_ns\u0000ns\u0002Hs\ne\u000b=\u001c; (4)\nwhere\u001cis the magnon torque term and we separated the\ndissipative ^ \u000band reactive ^\u0000 contributions:\n^\u000b=\u000b+^\u000bemf\u0000\u0011\f2Dins\nDins; (5)\n^\u0000 =\u0011\f[(ns\u0002Dins)\nDins\u0000Dins\n(ns\u0002Dins)];\nwhere in general the form of chiral derivatives in the \f-\nterms can be di\u000berent. Given that \fand\u000bare typically\nsmall for magnon systems, the term ^ \u000bemfwill dominate\nthe feedback damping tensor. An unusual feature of the\nchiral part of the damping is that it will be present even\nfor a uniform texture. While the DMI prefers twisted\nmagnetic structures, this can be relevant in the presence\nof an external magnetic \feld strong enough to drive the\nsystem into the ferromagnetic phase.\nIn conducting ferromagnets, charge currents also lead\nto a damping tensor of the same form where the strength\nof the damping is characterized by \u0011e=~2\u001be=4e2swith\n\u001beas the electronic conductivity [11, 13, 15], which\nshould be compared to \u0011in conducting ferromagnets\nwhere both e\u000bects are present. Since the magnon feed-\nback damping \u0011grows as/1=\u000b, the overall strength of\nmagnon contribution can quickly become dominant con-\ntribution in ferromagnets with small Gilbert damping.\nUnder the assumption that magnon scattering is domi-\nnated by the Gilbert damping such that the relaxation\ntime is given by \u001c\u000b= 1=2\u000b!, magnon conductivity is\ngiven by\u001b3D\u00181=6\u00192\u0015~\u000bin three-dimensions and \u001b2D\u0018\n1=4\u0019~\u000bin two-dimensions [40] where \u0015=p\n~J=skBTis\nthe wavelength of the thermal magnons. For Cu 2OSeO 3\nin ferromagnetic phase, we \fnd \u0011\u00192nm2. Similarly,\nfor a Pt/Co/AlO xthin \flm of thickness t= 0:6nm yield\n\u0011\u00191nm2at room temperature. This shows that the\nmagnon feedback damping can become signi\fcant in fer-\nromagnets with sharp textures and strong DMI.\nDomain wall dynamics. We describe the domain wall\npro\fle in a ferromagnet with DMI by Walker ansatz3\ntan(\u0012(x;t)=2) = exp(\u0006[x\u0000X(t)]=\u0001) whereX(t) and\n\u001e(t) denote the center position and tilting angle of the\ndomain wall [55], \u0001 =p\nJ=K 0is the domain wall width,\nK0=Ku\u0000\u00160M2\ns=2 includes the contributions from\nuniaxial anisotropy as well as the demagnetizing \feld\nand ^D=\u0000D(sin\r11 + cos\rez\u0002) contains DMI due to\nbulk and structure inversion asymmetries whose relative\nstrength is determined by \r. After integrating the LLG\nequation, we obtain the equations of motion for a domain\nwall driven by external perpendicular \feld [56]:\n\u0000XX_X=\u0001 + _\u001e=FX;\u0000\u001e\u001e_\u001e\u0000_X=\u0001 =F\u001e;(6)\nwhere \u0000XX=\u000b+\u0011(D=J)2sin2(\r+\u001e)=3 and \u0000\u001e\u001e=\n\u000b+\u0011[2=3\u00012+ (\u0019D=2J\u0001) cos(\r+\u001e) + (D=J)2cos2(\r+\n\u001e)] are dimensionless angle-dependent drag coe\u000ecients,\nFX=H=s andF\u001e= [Ksin 2\u001e+ sin(\r+\u001e)D\u0019=2\u0001]=sare\ngeneralized \\forces\" associated with the collective coordi-\nnatesXand\u001e,Kis the strength of an added anisotropy\ncorresponding, e.g., to magnetostatic anisotropy K=\nNx\u00160M2\ns=2 whereNxis the demagnetization coe\u000ecient.\nIn deriving these equations, we have neglected higher or-\nder terms in \u000band\f[57].\nTime-averaged domain wall velocity obtained from\nnumerical integration of the equations of motion for\na Co/Pt interface with Rashba-like DMI is shown in\nFig. 2. Thermal magnon wavelength at room tempera-\nture (\u00190:3nm) is much shorter than the domain wall size\n\u0001 =p\nJ=K 0\u00197nm, so the quasiparticle treatment of\nmagnons is justi\fed. We observe that damping reduces\nthe speed at \fxed magnetic \feld, and this e\u000bect is en-\nhanced with increasing DMI strength Dand diminishing\nthe Gilbert damping \u000b(see Fig. 2).\nAnother important observation is that in the presence\nof the feedback damping, the relation between applied\n\feld and average domain wall velocity becomes nonlin-\near. This is readily seen from steady state solution of the\nequations of motion before the Walker breakdown with\n\u001e=\u001e0which solves sin( \r+\u001e0)D\u0019=2s\u0001 =\u0000[H=s][\u000b+\n\u0011(D=J)2sin2(\r+\u001e0)=3]\u00001(noting that D=\u0001\u001dK,\nimplying a N\u0013 eel domain wall [56, 58]) and X=vt,\nleading to the cubic velocity-\feld relation ( sv=\u0001)(\u000b+\n\u0011[2sv=J\u0019 ]2=3) =Hfor \feld-driven domain wall motion.\nThe angle\u001e0also determines the tilting of Eas seen in\nFig. 1.\nSkyrmion dynamics. Under the assumption that the\nskyrmion retains its internal structure as it moves, we\ntreat it as a magnetic texture ns=ns(r\u0000q(t)) with\nq(t) being the time-dependent position (collective coor-\ndinate [61]) of the skyrmion. We consider the motion\nof a skyrmion under the temperature gradient r\u001f=\n\u0000rT=T, which exerts a magnon torque:\n\u001c= (1 +\fTns\u0002)(Lr\u001f\u0001D)ns; (7)\nwhereLis the spin Seebeck coe\u000ecient and \fTis the \\\f-\ntype\" correction. These are given by L3D\u0018kBT=6\u00192\u0015\u000b\nand\fT\u00193\u000b=2 in three-dimensions and L2D\u0018kBT=4\u0019\u000b\nand\fT\u0019\u000bin two-dimensions within the relaxation time\nD=4mJ/m2\nD=2mJ/m2\nD=1mJ/m2\n0 10 20 30 4002004006008001000\nμ0Ha(mT)〈X〉(m/s)Co/Pt\nD=1.5mJ/m2D=1mJ/m2D=0.5mJ/m2\n0 1 2 3 402004006008001000\nμ0Ha(mT)〈X〉(m/s)Pt/CoFeB/MgOFIG. 2. (Color online) Domain wall velocity as a func-\ntion of the magnetic \feld and varying strength of DMI for\nCo/Pt and Pt/CoFeB/MgO \flms. Solid (dashed) lines cor-\nrespond to dynamics at zero (room) temperature. We used\nmaterial parameters Ms= 1:1MA/m,J= 16pJ/m, K0=\n0:34MJ/m3,\u000b= 0:03 [45] for Co/Pt, and Ms= 0:43MA/m,\nJ= 31pJ/m, K0= 0:38MJ/m3,\u000b= 4\u000210\u00003[31, 59, 60] for\nPt/CoFeB/MgO.\napproximation [41, 42]. Multiplying the LLG equation\nEq. (4) withR\nd2r@qjns\u0001ns\u0002and substituting _ns=\n\u0000_qi@qins, we obtain the equation of motion for v=_q:\ns(W\u0000Qz\u0002)v+ (\fT\u0011D\u0000Qz\u0002)Lr\u001f=F:(8)\nAbove,W=\u00110\u000b+\u0011\u000b0can be interpreted as the con-\ntribution of the renormalized Gilbert damping, Q=R\nd2rns\u0001(@xns\u0002@yns)=4\u0019is the topological charge of the\nskyrmion,\u00110is the dyadic tensor, \u0011Dis the chiral dyadic\ntensor which is\u0018\u00110for isolated skyrmions and vanishes\nfor skyrmions in SkX lattice [48] (detailed de\fnitions of\nthese coe\u000ecients are given in the Supplemental Material\n[62]). The \\force\" term F=\u0000rU(q) due to the e\u000bec-\ntive skyrmion potential U(q) is relevant for systems with\nspatially-dependent anisotropies [63], DMI [64], or mag-\nnetic \felds. In deriving this equation, we only considered\nthe dominant feedback damping contribution ^ \u000bemfwhich\nis justi\fed for small \u000band\f. For temperature gradients\nand forces along the x-axis we obtain velocities:\nvx=\u0000L@x\u001f(Q2+W\fT\u0011D) +FxW\ns(Q2+W2);\nvy=\u0000L@x\u001fQ(\fT\u0011D\u0000W) +FxQ\ns(Q2+W2): (9)\nThe Hall angle de\fned as tan \u0012H=vy=vxis strongly af-\nfected by the renormalization of Wsince tan\u0012H=Q=W\nfor a \\force\" driven skyrmion and tan \u0012H\u0019(\fT\u00110\u0000\nW)=Qfor a temperature gradient driven skyrmion. Sim-\nilar to the domain wall velocity in Fig. 2, the Hall e\u000bect\nwill depend on the overall temperature of the system.\nWe \fnd that for a skyrmion driven by @x\u001f, the Hall an-\ngle\u0012Hmay \rip the sign in magnets with strong DMI\nas the temperature increases. We estimate this should\nhappen in Cu 2OSeO 3atT\u001850K using a typical radial\npro\fle for a rotationally symmetric skyrmion given by\nUsov ansatz cos( \u0012=2) = (R2\u0000r2)=(R2+r2) forr\u0014R\nandR\u00192\u0019J=D\u001952nm.\nMagnon pumping and accumulation. The motion of\nskyrmions induces a transverse magnon current across4\nthe sample. This e\u000bect can be quanti\fed by the av-\nerage magnon current due to magnon motive force per\nskyrmion:\nj=\u001bZ\nd2rE=\u0019R2= (v\u0002ez)4\u001b~2Q=R2: (10)\nThe current can only propagate over the magnon di\u000bu-\nsion length; thus, it can be observed in materials with\nlarge magnon di\u000busion length or small Gilbert damping.\nμ/μ0\n-1.5-1.0-0.500.51.01.5\nFIG. 3. (Color online) An array of moving skyrmions (only 3\nshown in the \fgure) induces a transverse current and accumu-\nlation of magnons along the edges . \u0016is obtained by numer-\nically solving the di\u000busion equation using material parame-\nters for Pt/CoFeB/MgO given in the caption of Fig. 2 with\nR= 35nm. System height and distance between skyrmion\ncenters are taken to be 3 R.\nSo far, we have assumed a highly compressible limit in\nwhich we disregard any build up of the magnon chem-\nical potential \u0016. In a more realistic situation the build\nup of the chemical potential will lead to magnon di\u000bu-\nsion. To illustrate the essential physics, we consider a\nsituation in which the temperature is uniform. For slow\nmagnetization dynamics in which magnons quickly estab-\nlish a stationary state (i.e. R=v\u001d\u001c\u000bfor skyrmions and\n\u0001=_X\u001d\u001c\u000bfor domain walls, which is satis\fed at high\nenough temperatures) we write a stationary magnon dif-\nfusion equation:\nr2\u0016=\u0016\n\u00182+r\u0001E; (11)\nwhere\u0018=\u0015=2\u0019\u000bis the magnon di\u000busion length and\nwe used the local Ohm's law \u0000r\u0016=j=\u001b\u0000E. Renor-\nmalization of magnon current in Eq. (3) then follows\nfrom solution of the screened Poisson equation j=\u001bE+(\u001b=4\u0019)rR\nd3r0(r0\u0001E)e\u0000jr\u0000r0j=\u0018=jr\u0000r0jin three dimen-\nsions andj=\u001bE+ (\u001b=2\u0019)rR\nd2r0(r0\u0001E)K0(jr\u0000r0j=\u0018)\nin two dimensions where K0is the modi\fed Bessel func-\ntion for an in\fnitely large system [65]. By analyzing\nthe magnon current due to magnon accumulation ana-\nlytically and numerically, we \fnd that renormalization\nbecomes important when the length associated with the\nmagnetic texture is much smaller than the magnon dif-\nfusion length.\nFinally, we numerically solve Eq. (11) for isolated soli-\ntons (see Fig. 1) and for an array of moving skyrmions\n(see Fig. 3). Given that the width of the strip in Fig. 3\nis comparable to the magnon di\u000busion length one can\nhave substantial accumulation of magnons close to the\nboundary. Spin currents comparable to the estimate in\nEq. (10) can be generated in this setup and further de-\ntected by the inverse spin Hall e\u000bect [66]. From Eq. (10),\nfor a skyrmion with R= 35nm moving at 10m/s in\nPt/CoFeB/MgO with D= 1:5mJ/m2[31], we obtain an\nestimate for spin current js=j~\u001810\u00007J/m2which\nroughly agrees with the numerical results. This spin cur-\nrent will also carry energy and as a result will lead to a\ntemperature drop between the edges.\nConclusion. We have developed a theory of magnon\nmotive force in chiral conducting and insulating ferro-\nmagnets. The magnon motive force leads to tempera-\nture dependent, chiral feedback damping. The e\u000bect of\nthis damping can be seen in the non-linear, temperature\ndependent behavior of the domain wall velocity. In ad-\ndition, observation of the temperature dependent Hall\nangle of skyrmion motion can also reveal this additional\ndamping contribution. 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Kovalev\nDepartment of Physics and Astronomy and Nebraska Center for Materials and Nanoscience,\nUniversity of Nebraska, Lincoln, Nebraska 68588, USA\nTHIELE’S EQUATION OF MOTION FOR SKYRMION\nWe consider the motion of a rotationally symmetric skyrmion under the influence of applied temperature gradient.\nAssuming that skyrmion drifts without any changes to its internal structure, ns(r,t) =ns(r−q(t)) where qis the\nposition of the skyrmion, we multiply the LLG equation with the operator/integraltext\nd2r∂qjns·ns×and integrate over the\nregion containing the skyrmion and obtain the following equation motion:\ns(W−Q/primez×)v+ (βTηD−Qz×)L∂χ=F. (1)\nwhere v=˙qis the skyrmion velocity, W=η0α+η(α0−β2α2),Lis the spin Seebeck coefficient, χ= 1/T,F=−∇U,\nQis the topological charge defined in the main text and Q/prime=Q−ηβα 1. In terms of the polar coordinates ( θ,φ) of\nns, the dyadic tensor η0and the chiral dyadic tensor ηDare given by\nη0=π/integraldisplayR\n0dr/parenleftbigg(r∂rθ)2+ sin2θ\nr/parenrightbigg\n, ηD=η0+D\nJπ/integraldisplayR\n0dr(sinθcosθ+r∂rθ), (2)\nThe damping terms αican be expanded in powers of D/J asαi=/summationtext\njαβi,Dj(D/J)j, whereαβi,Djis given by\nαβ0,D2=π/integraldisplayR\n0dr(r∂rθ)2cos2θ+ sin2θ\nr\nαβ0,D=π/integraldisplayR\n0dr(r∂rθ)r∂rθtanθcos2θ+ sin2θ\nr2\nαβ0,D0=π/integraldisplayR\n0dr(r∂rθ)22 sin2θ\nr3(3)\nαβ,D2=2π/integraldisplayR\n0dr(∂rθ) sinθ(cos2θ+ 1)\nαβ,D=4π/integraldisplayR\n0dr(∂rθ) sinθcos2θtanθ+r∂rθ\nr\nαβ,D0=2π/integraldisplayR\n0dr(∂rθ) sinθcos2θtan2θ+ (r∂rθ)2\nr2(4)\nαβ2,D2=π/integraldisplayR\n0dr/parenleftbiggcos2θsin2θ+ (r∂rθ)2\nr/parenrightbigg\nαβ2,D=2π/integraldisplayR\n0dr/parenleftbiggcos2θsin2θtanθ+ (r∂rθ)3\nr2/parenrightbigg\nαβ2,D0=π/integraldisplayR\n0dr/parenleftbiggcos2θsin2θtan2θ+ (r∂rθ)4\nr3/parenrightbigg\n(5)\nThe integrals can be evaluated by using an approximate radial profile for θ(r). Using Usov ansatz yields the values\nenumerated in Table I for αβi,Dj. Only the dominant terms are kept in the main text.2\n1 D/J (D/J)2\n1496π\n15R252π\n5R16π\n5\nβC\nR2472π\n15R16π\n3\nβ2 5056\n105R22804π\n105R464π\n105\nTABLE I. List of feedback damping coefficients αβi,Djfor a rotationally symmetric skyrmion using Usov ansatz cos( θ/2) =\n(R2−r2)/(R2+r2) forr≤Rand 0 forr > R . Rows correspond to α0,α1andα2, expanded in powers of D/J. Above,\nC≈449. Remaining parameters are given as η0= 16π/3,ηD=η0+ (4πR/3)(D/J).\nTRANSPORT COEFFICIENTS\nTexture-independent part of the transport coefficients can be obtained using the Boltzmann equation within the\nrelaxation-time approximation in terms of the integral [1, 2]\nJij\nn=1\n(2π)3/planckover2pi1/integraldisplay\nd/epsilon1τ(/epsilon1)(/epsilon1−µ)n(−∂/epsilon1f0)/integraldisplay\ndS/epsilon1vivj\n|v|(6)\nasσ=J0and Π =−J1/J0. Above,τ(/epsilon1) is the relaxation time, /epsilon1(k) =/planckover2pi1ωk,vi=∂ωk/∂ki,dS/epsilon1is the area d2k\ncorresponding to a constant energy surface with /epsilon1(k) =/epsilon1,f0is the Bose-Einstein equilibrium distribution. Under the\nassumption that the scattering processes are dominated by Gilbert damping, we set τ(/epsilon1)≈1/2αω. By evaluating the\nintegral after these substitutions, we obtain σ2D≈F−1/6π2λ/planckover2pi1αin three dimensions ( d= 3), where λ=/radicalbig\n/planckover2pi1J/skBT\nis the wavelength of the thermal magnons, F−1=/integraltext∞\n0d/epsilon1/epsilon1d/2e/epsilon1+x/(/epsilon1+x)(e/epsilon1+x−1)2∼1 evaluated at the magnon gap\nx=/planckover2pi1ω0/kBT. Similarly for d= 2, we obtain σ2D≈F−1/4π/planckover2pi1α.\nThe spin Seebeck coefficient Lis given by−/planckover2pi1σΠ = /planckover2pi1J1, for which we obtain L3D≈F0kBT/6π2λαin 3D and\nL2D≈F0kBT/4παin 2D, where F0=/integraltext∞\n0d/epsilon1/epsilon1d/2/(e/epsilon1+x−1)2∼1. Ford >2 and small x, the numerical factor F0\ncan be expressed in terms of Riemann zeta function and Euler gamma function as ζ(d/2)Γ(d/2 + 1) [3]. In the main\ntext, the numerical factors F−1andF0are omitted.\n[1] N. Ashcroft and N. Mermin, Solid State Physics (Saunders College, Philadelphia, 1976).\n[2] A. A. Kovalev and Y. Tserkovnyak, EPL (Europhysics Lett. 97, 67002 (2012).\n[3] R. K. Pathria, Statistical Mechanics (Butterworth-Heinemann, 1996), 2nd ed."    },    {        "title": "1111.1219v1.Tunable_magnetization_relaxation_in_spin_valves.pdf",        "content": "arXiv:1111.1219v1  [cond-mat.mes-hall]  4 Nov 2011Tunable magnetization relaxation in spin valves\nXuhui Wang∗and Aurelien Manchon\nPhysical Science & Engineering Division, KAUST, Thuwal 239 55-6900, Kingdom of Saudi Arabia\n(Dated: June 24, 2018)\nIn spin values the damping parameters of the free layer are de termined non-locally by the entire\nmagnetic configuration. In a dual spin valve structure that c omprises a free layer embedded be-\ntween two pinned layers, the spin pumping mechanism, in comb ination with the angular momentum\nconservation, renders the tensor-like damping parameters tunable by varying the interfacial and dif-\nfusive properties. Simulations based on the Landau-Lifshi tz-Gilbert phenomenology for a macrospin\nmodel are performed with the tensor-like damping and the rel axation time of the free layer mag-\nnetization is found to be largely dependent on while tunable through the magnetic configuration of\nthe source-drain magnetization.\nPACS numbers: 75.70.Ak, 72.25.Ba, 75.60.Jk, 72.25.Rb\nA thorough knowledge of magnetization relaxation\nholds the key to understand magnetization dynamics in\nresponse to applied fields1and spin-transfer torques.2,3\nIn the framework of Landau-Lifshitz-Gilbert (LLG) phe-\nnomenology, relaxation is well captured by the Gilbert\ndamping parameterthat is usuallycited asa scalarquan-\ntity. As pointed out by Brown half a century ago,4the\nGilbert damping for a single domain magnetic particle is\nin general a tensor.\nWhen a ferromagnetic thin film is deposited on a nor-\nmal metal substrate, an enhanced damping has been ob-\nserved ferromagnetic resonance experiments.5This ob-\nservation is successfully explained by spin pumping:6,7\nThe slow precession of the magnetization pumps spin\ncurrent into the adjacent normal metal where the dis-\nsipation of spin current provides a non-local mechanism\nto the damping. The damping enhancement is found\nto be proportional to spin mixing conductance, a quan-\ntity playing key roles in the magneto-electronic circuit\ntheory.7,8\nThepumped spincurrent Ip∝M×˙Misalwaysin the\nplaneformedbythefreelayermagnetizationdirection M\nand the instantaneous axis about which the magnetiza-\ntion precesses. Therefore, in a single spin valve, when M\nis precessingaround the source(drain) magnetization m,\nthe pumpingcurrentisalwaysinthe planeof mandM.9\nLet us assume an azimuth angle θbetween mandM. In\nsuch anin-plane configuration, the pumping current Ip\nhas a component Ipsinθthat is parallel to m. The spin\ntransfer torque acting on the source (drain) ferromagnet\nmis the component of spin current that is in the plane\nand perpendicular to m. To simplify the discussion, we\nconsider it to be completely absorbed by m. The lon-\ngitudinal (to m) component experiences multiple reflec-\ntion at the source (drain) contact, and cancels the damp-\ning torque by an amount proportional to Ipsin2θbut is\nstill aligned along the direction of M×˙M. Therefore\nthe total damping parameter has an angle θdependence\nbut still picks up a scalar (isotropic) form. This is the\nwell-known dynamic stiffness explained by Tserkovnyak\net al.9In the most general case, when the precessing axis\nof the free layer is mis-aligned with m, there is always anout-of-plane pumping torque perpendicular to the plane.\nIn the paradigmof Slonczewski, this out-of-plane compo-\nnent is not absorbed at the interface of the source (drain)\nferromagnetic nodes, while the conservation of angular\nmomentum manifests it as a damping enhancement that\nshows the tensor form when installed in the LLG equa-\ntion.\nStudies in lateral spin-flip transistors have suggested\na tensor form for the enhanced damping parameters.10\nIn spin valves, works based on general scattering the-\nory have discussed the damping in the framework of\nfluctuation-dissipation theorem11and shown that the\nGilbert dampingtensorcanbe expressedusingscattering\nmatrices,12thus enabling first-principle investigation.13\nBut explicit analytical expressions of the damping ten-\nsor, its dependence on the magnetic configuration as well\nas the material properties and particularly its impact on\nthe magnetization relaxation are largely missing.\nIn this paper, we investigate the Gilbert damping pa-\nrameters of the free layer in the so-called dual spin valve\n(DSV).14–16We analyze the origin of the damping tensor\nand derive explicit analytical expressions of its non-local\ndependence on the magnetic configuration and materials\nproperties. A generalization of our damping tensor to a\ncontinuous magnetic texture agrees well with the results\nin earlier works. Particularly, we show, in numeric sim-\nulations, that by tuning the magnetic configurations of\nthe entire DSV, the relaxation time of the free layer can\nbe increased or decreased.\nmLM mR\nReservoir Reservoir\nFIG. 1: A dual spin valve consists of a free layer (with magne-\ntization direction M) sandwiched by two fixed ferromagnetic\nlayers (with magnetization directions mLandmR) through\ntwo normal metal spacers. The fixed layer are attached to\nreservoirs.2\nTo analyze the spin and charge currents in a DSV, we\nemploy the magneto-electronic circuit theory and spin\npumping,7,8in combination with diffusion equations.17\nPillar-shaped metallic spin valves usually consist of\nnormal-metal ( N) spacers much shorter than its spin-flip\nrelaxation length, see for example Ref.[3,15]. To a good\napproximation, in the Nnodes, a spatially homogeneous\nspin accumulation is justified and the spin current ( Ii)\nconservation dictates/summationtext\niIi= 0 (where subscript iindi-\ncates the source of spin current).\nA charge chemical potential ( µ) and a spin accumula-\ntion (s) are assigned to every ForNnode. In a transi-\ntion metalferromagnet,astrongexchangefieldalignsthe\nspin accumulation to the magnetization direction. At ev-\neryF|Ninterface, the charge and spin currents on the N\nside are determined by the contact conductance and the\ncharge and spin distributions on both sides of the con-\ntact. For example, at the contact between the left lead\nferromagnet to the left normal metal N1, called L|N1\nthereafter, the currents are8\nIL=e\n2hGL[(µ1−µL)+PL(s1−sL)·mL],\nIL=−GL\n8π[2PL(µ1−µL)mL+(s1−sL)·mLmL\n+ηL(s1−s1·mLmL)]. (1)\nWe have used the notation G=g↑+g↓is the sum of\nthe spin- σinterface conductance gσ. The contact polar-\nisationP= (g↑−g↓)/(g↑+g↓). The ratio η= 2g↑↓/G\nis between the real part of the spin-mixing conductance\ng↑↓and the total conductance G. The imaginary part\nofg↑↓is usually much smaller than its real part, thus\ndiscarded.18The spin-coherence length in a transition\nmetalferromagnetisusuallymuchshorterthanthethick-\nness of the thin film,19which renders the mixing trans-\nmission negligible.7The precession of the free layer mag-\nnetization Mpumps a spin current Ip= (/planckover2pi1/4π)g↑↓\nFM×\n˙Minto the adjacent normal nodes N1andN2, which\nis given by the mixing conductance g↑↓\nFat theF|N1(2)\ninterface (normal metals spacers are considered identical\non both sides of the free layer).\nA back flow spin current at the F|N1interface reads\nI1=−GF\n8π[2PF(µ1−µF)M+(s1−sF)·MM\n+ηF(s1−s1·MM)] (2)\non theN1side. Therefore, a weak spin-flip scattering\ninN1demands IL+I1+Ip= 0, which is dictated by\nangularmomentum conservation. The sameconservation\nlaw rules in N2, whereIR+I2+Ip= 0.\nFor the ferromagnetic ( F) nodes made of transition\nmetals, the spin diffusion is taken into account properly.9\nIn a strong ferromagnet, any transverse components de-\ncay quickly due to the large exchange field, thus the\nlongitudinal spin accumulation sν=sνmν(withν=\nL,R,F) diffuses and decays exponentially at a length\nscale given by spin diffusion length ( λsd) as∇2\nxsν=sν/λsd. The difference in spin-dependent conductivty\nof majority and minority carriers is taken into account\nby enforcing the continuity of longitudinal spin current\nmν·Iν=−(D↑\nν∇xs↑\nν−D↓\nν∇xs↓\nν) at the every F|Nin-\nterface. We assume vanishing spin currents at the outer\ninterfaces to reservoirs.\nThe diffusion equations and current conservation de-\ntermine, self-consistently, the spin accumulations and\nspin currents in both NandFnodes . We are mainly\nconcerned with the exchange torque9T=−M×(IL+\nIR)×Macting on M. A general analytical formula is\nattainable but lengthy. In the following, we focus on two\nscenarios that are mostly relevant to the state-of-the-art\nexperiments in spin valves and spin pumping: (1) The\nfree layer has a strong spin flip (short λsd) and the thick-\nnessdF≥λsd, for which the permalloy (Py) film is an\nideal candidate;15(2) The free layer is a half metal, such\nas Co2MnSi studied in a recent experiment.20\nStrong spin flip in free layer. We assume a strong spin\nflip scattering in the free layer i.e., dF≥λsd. We leave\nthe diffusivity properties in the lead Fnodes arbitrary.\nThe total exchange torque is partitioned into two parts:\nAnisotropic part that is parallel to the direction of the\nGilbert damping M×˙Mand ananisotropic part that is\nperpendicular to the plane spanned by mL(R)andM(or\nthe projection of M×˙Mto the direction mL(R)×M),\ni.e.,\nT=/planckover2pi1g↑↓\nF\n4π(DL\nis+DR\nis)/parenleftBig\nM×˙M/parenrightBig\n+/planckover2pi1g↑↓\nF\n4πM×/bracketleftBig\n(DL\nanˆAL,an+DR\nanˆAR,an)˙M/bracketrightBig\n,(3)\nwhere the material-dependent parameters DL(R)\nisand\nDL(R)\nanare detailed in the Appendix A.\nMost interest is in the anisotropic damping described\nby a symmetric tensor with elements\nˆAij\nan=−mimj (4)\nwherei,j=x,y,z(we have omitted the lead index Lor\nR). The elements of ˆAanare given in Cartesian coordi-\nnates of the source-drain magnetization direction. The\nanisotropic damping appears as M׈Aan˙Mthat is al-\nways perpendicular to the free layer magnetization direc-\ntion, thus keeping the length of Mconstant.11It is not\ndifficult to show that when Mis precessing around m,\nthe anisotropic part vanishes due to ˆAan˙M= 0.\nWe generalizeEq.(4) toa continuousmagnetictexture.\nConsider here only one-dimensional spatial dependence\nand the extension to higher dimensions is straightfor-\nward. The Cartesian component of vector U≡M×\nˆAan˙MisUi=−εijkMjmkml˙Ml(whereεijkis the Levi-\nCivita tensor and repeated indices are summed). We as-\nsume the fixed layer and the free layer differ in space by\na lattice constant a0, which allows mk≈Mk(x+a0). A\nTaylor expansion in space leads to U=−a2\n0M×(ˆD˙M),\nwherethematrixelements ˆDkl= (∂xM)k(∂xM)landwe3\nhave assumed that the magnetization direction is always\nperpendicular to ∂xM. In this case, three vectors ∂xM,\nM×∂xMandMare perpendicular to each other. A\nrotation around Mbyπ/2 leavesMand˙Munchanged\nwhile interchanging ∂xMwithM×∂xM, we have\nˆDkl= (M×∂xM)k(M×∂xM)l, (5)\nwhich agrees with the so-called differential damping ten-\nsor Eq.(11) in Ref.[21].\nEq.(3) suggests that the total exchange torque on the\nfree layer is a linear combination of two independent ex-\nchange torques arsing from coupling to the left and the\nrightFnodes. This form arises due to a strong spin-\nflip scattering in the free layer that suppresses the ex-\nchange between two spin accumulations s1ands2in the\nNnodes. In the pursuit of a concise notation for the\nGilbert form, the exchange torque can be expressed as\nT=M×← →α˙Mwith a total damping tensor given by\n← →α=/planckover2pi1g↑↓\nF\n4π/parenleftBig\nDL\nis+DR\nis+DL\nanˆAL,an+DR\nanˆAR,an/parenrightBig\n.(6)\nThe damping tensor← →αis determined by the entire mag-\nnetic configuration of the DSV and particularly by the\nconductance of F|Ncontacts and the diffusive proper-\nties theFnodes.\nHalf metallic free layer . This special while experimen-\ntally relevant20case means PF= 1. Half-metallicity\nin combination with the charge conservation enforces a\nlongitudinal back flow that is determined solely by the\nbias current: The spin accumulations in Nnodes do\nnot contribute to the spin accumulation inside the free\nlayer, thus an independent contribution due to left and\nright leads is foreseen. We summarize the material spe-\ncific parameters in the Appendix A. When spin flip is\nweak in the source-drain ferromagnets, ξL≈0 leads to\nDis≈0. In this configuration, by taking a (parallel or\nanti-parallel)source-drainmagnetization direction as the\nprecessingaxis,thetotaldampingenhancementvanishes,\nwhich reduces to the scenario of ν= 1 in Ref.[9].\nMagnetization relaxation . To appreciate the impact of\nan anisotropic damping tensor on the magnetization re-\nlaxation, we perform a simulation, for the free layermag-\nnetization, using Landau-Lifshitz-Gilbert (LLG) equa-\ntion augmented by the tensor damping, i.e.,\ndM\ndt=−γM×Heff+α0M×dM\ndt\n+γ\nµ0MsVM×← →αdM\ndt.(7)\nα0is the (dimensionless) intrinsic Gilbert damping pa-\nrameter. Symbol γis the gyromagnetic ratio, Msis the\nsaturation magnetization, and Vis the volume of the\nfree layer. µ0stands for the vacuum permeability. The\ndynamics under the bias-driven spin transfer torque is\nnot the topic in this paper, but can be included in a\nstraightforward way.22We give in the Appendix B the\nexpressions of the bias-driven spin torques.We are mostly interested in the relaxation of the mag-\nnetization, instead of particular magnetization trajecto-\nries, in the presence of a tensor damping. The follow-\ning simulation is performed for the scenario where the\nfree layer has a strong spin flip, i.e., Case (1). We em-\nploy the pillar structure from Ref.[15] while consider-\ning the free layer (Py) to be 8nm thick (a thicker free\nlayer favors a better thermal stability.15) The source-\ndrain ferromagnets are cobalt (Co) and we expect the\nresults are valid for a larger range of materials selec-\ntions. The Py film is elliptic with three axes given by\n2a= 90 nm, 2 b= 35 nm,15andc= 8 nm. The de-\nmagnetizing factors Dx,y,zin the shape anisotropy en-\nergyEdem= (1/2)µ0M2\nsV/summationtext\ni=x,y,zDiM2\niareDx= 0.50,\nDy= 0.37 and Dz= 0.13. An external field Haleads\nto a Zeeman splitting EZee=−Vµ0MsHa·M. For Py\nfilms, we neglect the uniaxial anisotropy. The total free\nenergyET=EZee+Edemgives rise to an effective field\nHeff=−(1/VMsµ0)∂ET/∂M.\nThe spin-dependent conductivities in the bulk of Co\nandthe spin diffusion length λCo≈60nm aretaken from\nthe experimental data.24For Py, we take λPy≈4 nm.25\nTo have direct connection with experiments, the above\nmentioned bare conductance has to be renormalized by\nthe Sharvin conductance.26For Py/Cu the mixing con-\nductance, we take the value g↑↓\nFS−1≈15 nm−2,26which\ngivesRL(R)F≈1.0.\n56789100.950.960.970.980.991(a) Bz = 50 G; I.P. y−axis.\n  \n(y,y)(y,x)(x,x)(x,z)(y,z)(z,z)369(b) Relaxation time\n(x,x)\n(y,y)\n(z,z)\nFIG. 2: (Color online) Mzas a function of time (in ns) in\npresence of differentsource-drain magnetic configurations and\napplied fields. (a) The external magnetic field Bz= 50 Gauss\nis applied along z-axis. The blue (dashed), red (solid) and\nblack (dotted dash) curves correspond to source-drain magn e-\ntization in configurations ( y,y), (x,x), and (z,z) respectively.\n(b) Magnetization relaxation times (in the unit of ns)versu s\nsource-drain magnetic configurations at different applied fi eld\nalongz-axis.:Bz= 10 G (red /square),Bz= 50 G (blue /circlecopyrt),\nBz= 200 G (green ▽),Bz= 800 G (black ♦). Lines are a\nguide for the eyes. The initial position (I.P.) of the free la yer\nis taken along y-axis.\nThe relaxation time τris extracted from the sim-\nulations by demanding at a specific moment τrthe\n|Mz−1.0|<10−3, i.e., reaches the easy axis. In the\nabsence of bias, panel (a) of Fig.2 shows the late stage\nof magnetization relaxation from an initial position ( y-4\naxis) in the presence of an tensor damping, under various\nsource-drain (SD) magnetic configurations. The results\nare striking: Under the same field, switching the SD con-\nfigurations increases or decreases τr. In panel (b), the\nextracted relaxation times τrversus SD configurations\nunder various fields are shown. At low field Bz= 10 G\n(red/square), when switching from ( z,z) to (y,y),τris im-\nproved from 8.0 ns to 6.3 ns, about 21%. At a higher\nfieldBz= 800 G (black ♦), the improvement is larger\nfrom 5.2 at ( z,z) to 3.6 at ( y,y), nearly 31%. To a large\ntrend, the relaxation time improvement is more signifi-\ncant at higher applied fields.\nIn conclusion, combining conservation laws and\nmagneto-electronic circuit theory, we have analyzed the\nGilbert damping tensor of the free layer in a dual spin\nvalve. Analytical results of the damping tensor as func-\ntions of the entire magnetic configuration and material\nproperties are obtained. Numerical simulations on LLG\nequation augmented by the tensor damping reveal a tun-\nable magnetization relaxation time by a strategic selec-\ntion of source-drain magnetization configurations. Re-\nsults presented in this paper open a new venue to the\ndesign and control of magnetization dynamics in spin-\ntronic applications.\nX.Wang is indebted to G. E. W. Bauer, who has\nbrought the problem to his attention and offered invalu-\nable comments.\nAppendix A: Material dependent parameters\nIn this paper, RL(R)F≡g↑↓\nL(R)/g↑↓\nFis the mixing con-\nductance ratio and χL(R)≡mL(R)·M. The diffusivity\nparameter ξL(R)=φL(R)(1−P2\nL(R))/ηL(R), where for the\nleftFnode\nφL=1\n1+(σ↑\nL+σ↓\nL)λLe2\n4hSσ↑\nLσ↓\nLtanh(dL/λL)GL(1−P2\nL)(A1)\nwherehthe Planck constant, Sthe area of the thin film,\nethe elementary charge, λLthe spin diffusion length, dL\nthe thickness of the film, and σ↑(↓)the spin-dependent\nconductivity. φRisobtainedbysubstituting all LbyRin\nEq.(A1). Parameter ξFis given by ξF= (1−P2\nF)φF/ηF\nwith\nφF=1\n1+(σ↑\nF+σ↓\nF)λFe2\n4hSσ↑\nFσ↓\nFGF(1−P2\nF).(A2)\nThe material dependent parameters as appearing in the\ndamping tensor Eq.(6) are: (1) In the case of a strongspin flip in free layer,\nDL(R)\nis=RL(R)F\nLL(R)F/bracketleftBig\nξL(R)RL(R)F+ξL(R)ξF(1−χ2\nL(R))\n+ξF(1−χ2\nL(R))χ2\nL(R)/bracketrightBig\n,\nDL(R)\nan=RL(R)F\nLL(R)F(ξL(R)−1)[ξF(1−χ2\nL(R))+RL(R)F]\n1+RL(R)F,\nLL(R)F=(1+RL(R)Fχ2\nL)ξF(1−χ2\nL(R))\n+RL(R)F/bracketleftBig\n(1−χ2\nL(R))(1+ξL(R)ξF)\n+ξL(R)RL(R)F+ξFχ2\nL(R)/bracketrightBig\n; (A3)\n(2)In the case of a half metallic free layer\nDL(R)\nis=RL(R)FξL(R)\n(1−χ2\nL(R))+ξL(R)(χ2\nL+RL(R)F),\nDL(R)\nan=RL(R)F\n1+RL(R)F\n×ξL(R)−1\n(1−χ2\nL(R))+ξL(R)(χ2\nL+RL(R)F).(A4)\nAppendix B: Bias dependent spin torques\nThe full analytical expression of bias dependent spin\ntorques are rather lengthy. We give here the expres-\nsions, under a bias current I, for symmetric SD fer-\nromagnets (i.e., φL=φR=φthusξL=ξR=ξ)\nwith parallelor anti-parallelmagnetization direction. (1)\nWith a strong spin flip in the free layer, the parallel\nSD magnetization leads to vanishing bias-driven torque\nT(b)\n⇑⇑= 0; WhentheSDmagnetizationsareanti-parallelly\n(i.e.,mL=−mR≡m),\nT(b)\n⇑⇓=I/planckover2pi1Pφ\ne(1+R)L/bracketleftbig\n(ξF+RξFχ2+R)(1−χ2)\n+R(R+ξF(1−χ2)+χ2)/bracketrightbig\nmF×(m×mF).\n(B1)\n(2) When the free layer is half metallic, for symmetric\nSD ferromagnets , T(b)\n⇑⇑= 0 and\nT(b)\n⇑⇓=I/planckover2pi1\neφP\n(1−ξ)(1−χ2)+ξ(χ2+R)mF×(m×mF).\n(B2)\n∗Electronic address: xuhui.wang@kaust.edu.sa\n1L. D. Landau and E. M. Lifshitz, Statistical Physics ,Part\n2(Pergamon, Oxford, 1980); T. L. Gilbert, IEEE. Trans.\nMag.40, 2443 (2004).2J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996);\nL. Berger, Phys. Rev. B 54, 9353 (1996).\n3E. B. Myers, et al., Science 285, 867 (1999); J. A. Katine,\net al., Phys. Rev. Lett. 84, 3149 (2000); S. I. Kiselev, et5\nal., Nature (London) 425, 380 (2003).\n4W. F. Brown, Phys. Rev. 130, 1677 (1963).\n5Mizukami et al., Jpn. J. Appl. Phys. 40, 580 (2001); J.\nMagn. Mater. Magn. 226, 1640 (2001).\n6Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002); Phys. Rev. B 66, 224403\n(2002).\n7Y. Tserkovnyak, et al., Rev. Mod. Phys. 77, 1375 (2005).\n8A. Brataas, G. E. W. Bauer, and P. J. Kelly, Phys. Rep.\n427, 157 (2005).\n9Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B67, 140404(R) (2003).\n10X. Wang, G. E. W. Bauer, and A. Hoffmann, Phys. Rev.\nB73, 054436 (2006).\n11J. Foros, et al., Phys. Rev. B 78, 140402(R) (2008); Phys.\nRev. B79, 214407 (2009).\n12A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys.\nRev. Lett. 101, 037207 (2008); arXiv:1104.1625.\n13A.A.Starikov, et al., Phys.Rev.Lett. 105, 236601 (2010).\n14L. Berger, J. Appl. Phys. 93, 7693 (2003).\n15G. D. Fuchs, et al., Appl. Phys. Lett. 86, 152509 (2005).16P. Bal´ aˇ z, M. Gmitra, and J. Barna´ s, Phys. Rev. B 80,\n174404 (2009); P. Yan, Z. Z. Sun, and X. R. Wang, Phys.\nRev. B83, 174430 (2011).\n17T. Valet and A. Fert, Phys. Rev. B 48, 7099 (1993); A. A.\nKovalev, A. Brataas, and G. E. W. Bauer Phys. Rev. B\n66, 224424 (2002).\n18K. Xia,et al., Phys. Rev. B 65, 220401(R) (2002).\n19M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407\n(2002).\n20H. Chudo, et al., J. Appl. Phys. 109, 073915 (2011).\n21S. Zhang and S. -L. Zhang, Phys. Rev. Lett. 102, 086601\n(2010).\n22J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 72,\n014446 (2005).\n23J. Osborn, Phys. Rev. 67, 351 (1945).\n24J. Bass and W. P. Pratt, J. Magn. Magn. Mater. 200, 274\n(1999).\n25A. Fert and L. Piraux, J. Magn. Magn. Mater. 200, 338\n(1999).\n26G. E. W. Bauer, et al., Phys. Rev. B 67, 094421 (2003)."    },    {        "title": "2209.14179v1.Unidirectional_magnetic_coupling.pdf",        "content": "Unidirectional magnetic coupling\nH. Y. Yuan,1R. Lavrijsen,2and R. A. Duine1, 2\n1Institute for Theoretical Physics, Utrecht University,\nPrincetonplein 5, 3584 CC Utrecht, The Netherlands\n2Department of Applied Physics, Eindhoven University of Technology,\nP.O. Box 513, 5600 MB Eindhoven, The Netherlands\n(Dated: September 29, 2022)\nWe show that interlayer Dzyaloshinskii-Moriya interaction in combination with non-local Gilbert\ndamping gives rise to unidirectional magnetic coupling. That is, the coupling between two magnetic\nlayers | say the left and right layer | is such that dynamics of the left layer leads to dynamics of\nthe right layer, but not vice versa. We discuss the implications of this result for the magnetic sus-\nceptibility of a magnetic bilayer, electrically-actuated spin-current transmission, and unidirectional\nspin-wave packet generation and propagation. Our results may enable a route towards spin-current\nand spin-wave diodes and further pave the way to design spintronic devices via reservoir engineering.\nIntroduction. | Non-reciprocal transmission of elec-\ntrical signals lies at the heart of modern communication\ntechnologies. While semi-conductor diodes, as an exam-\nple of an electronic component that underpins such non-\nreciprocity, have been a mature technology for several\ndecades, new solutions are being actively pursued [1, 2].\nSuch research is spurred on by the emergence of quan-\ntum technologies that need to be read out electrically\nbut should not receive unwanted back-action from their\nelectronic environment.\nComplementary to these developments, spintronics has\nsought to control electronic spin currents and, more\nrecently, spin currents carried by spin waves | i.e.,\nmagnons | in magnetic insulators [3]. Devices that im-\nplement non-reciprocal spin-wave spin currents have been\nproposed [4{7]. Most of these proposals rely on dipolar\ninteractions [8{11] or Dzyaloshinskii-Moriya interactions\n(DMI) [12{16]. Other proposals involve the coupling of\nthe spin waves to additional excitations such that the spin\nwaves are endowed with non-reciprocity. Examples are\nthe coupling of the spin waves to magnetoelastic, optical,\nand microwave excitations [17{22].\nMost of these proposals have in common that they con-\nsider spin-wave dispersions that are asymmetric in wave\nvector. For example, due to the DMI spin waves at one\nparticular frequency have di\u000berent wave numbers and ve-\nlocities for the two di\u000berent directions. There are there-\nfore spin waves travelling in both directions. This may\nbe detrimental for some applications. For example, one\nwould like to shield quantum-magnonic technologies from\nspin-current noise [23], and completely quench the spin-\ncurrent transmission in one of the two directions along a\nwire.\nHere, we propose a set-up that realizes unidirectional\nmagnetic coupling between two magnetic layers or be-\ntween two magnetic moments. The ingredients are DMI\nand dissipative coupling between the two layers or mo-\nments. The dissipative coupling takes the form of a non-\nlocal Gilbert damping and may arise, for example, from\nthe combined action of spin pumping and spin transfer.Then, one magnet emits spin current when it precesses,\nwhich is absorbed by the other. The resulting dissipa-\ntive coupling turns out to, for certain parameters, pre-\ncisely cancel the DMI in one direction. As a result, an\nexcitation of one of the magnets leads to magnetization\ndynamics of the other, but not vice versa. This yields\nspin-wave propagation that is truly uni-directional: for\nspeci\fc direction and magnitude of the external \feld, all\nspin waves travel in one direction only.\nMinimal model. | Let us start with the minimal set-\nup that demonstrates the unidirectional coupling. We\n\frst consider two identical homogeneous magnetic lay-\ners that are coupled only by an interlayer DMI with\nDzyaloshinskii vector Dand by interlayer spin pumping\n(see Fig. 1). The magnetization direction in the layers\nis denoted by mi, wherei2f1;2glabels the two lay-\ners. We also include an external \feld H. The magnetic\nenergy is given by\nE[m1;m2] =D\u0001(m1\u0002m2)\u0000\u00160MsH\u0001(m1+m2);(1)\nwhereMsis the saturation magnetization of both layers\nand\u00160is the vacuum susceptibility. The magnetization\ndynamics of layer 1 is determined by the Landau-Lifshitz-\nGilbert (LLG) equation\n@m1\n@t=\r\nMsm1\u0002\u000eE\n\u000em1+\u000bnlm1\u0002@m2\n@t; (2)\nwhere\ris the gyromagnetic ratio and \u000bnlcharacterizes\nthe strength of the non-local damping that in this set-up\nresults from the combination of spin pumping and spin\ntransfer torques, as described in the introduction. The\nequation of motion for the magnetization dynamics of the\nsecond layer is found by interchanging the labels 1 and\n2 in the above equation. Working out the e\u000bective \feldsarXiv:2209.14179v1  [cond-mat.mes-hall]  28 Sep 20222\nFM\nFM\nm2\nm1xy\nz, H, D\nFIG. 1. Schematic of two magnetic moments coupled by an\ninterlayer DMI and by interlayer spin pumping. The dynam-\nics of m1induces the motion of m2, but not vice versa for\nappropriate parameters.\n\u000eE=\u000emiyields\n@m1\n@t=\r\nMsm1\u0002(m2\u0002D\u0000\u00160MsH) +\u000bnlm1\u0002@m2\n@t;\n(3a)\n@m2\n@t=\r\nMsm2\u0002(D\u0002m1\u0000\u00160MsH) +\u000bnlm2\u0002@m1\n@t;\n(3b)\nwhere the sign di\u000berence in e\u000bective-\feld contribution\nfrom the DMI stems from the asymmetric nature of the\nDMI. We show now that depending on the magnitude and\ndirection of the e\u000bective \feld, this sign di\u000berence leads\nfor one of the layers to cancellation of the torques due\nto interlayer DMI and non-local damping. As the can-\ncellation does not occur for the other layer, and because\nthe DMI and non-local damping are the mechanisms that\ncouple the layers in the model under consideration, this\nleads to uni-directional magnetic coupling.\nTaking the external \feld to be much larger than the\ninterlayer DMI, i.e., \u00160jHj\u001djDj=Ms, and taking \u000bnl\u001c\n1, we may replace @mi=@tby\u0000\r\u00160mi\u0002Hon the right-\nhand side of Eqs. (3) because the external \feld then is\nthe dominant contribution to the precession frequency.\nFor the \feld H=D=\u000bnl\u00160Ms, one then \fnds that\n@m1\n@t=\u0000\r\n\u000bnlMsm1\u0002D; (4a)\n@m2\n@t=2\r\nMsm2\u0002(D\u0002m1)\u0000\r\n\u000bnlMsm2\u0002D:(4b)\nHence, the coupling between the two magnetic layers is\nunidirectional at the \feld H=D=\u000bnl\u00160Ms: the magne-\ntization dynamics of layer 1 leads to dynamics of layer\n2 as evidenced by Eq. (4b), but not vice versa as im-\nplied by Eq. (4a). This one-way coupling is reversed by\nchanging the direction of the \feld to \u0000Hor the sign of\nthe non-local coupling \u000bnl.\nMagnetic susceptibility. | Let us now take into ac-\ncount the Gilbert damping within the layers, exchange,\nand anisotropies and discuss the in\ruence of the unidi-\nrectional coupling on the magnetic susceptibility. Theenergy now reads\nE[m1;m2] =\u0000Jm1\u0001m2+D\u0001(m1\u0002m2)\n\u0000\u00160MsH\u0001(m1+m2)\u0000K\n2\u0000\nm2\n1;z+m2\n2;z\u0001\n;(5)\nwith the constant Kcharacterizing the strength of the\nanisotropy and Jthe exchange. We shall focus on the\nferromagnetic coupling ( J >0) without loss of generality.\nThe LLG equation now becomes\n@m1\n@t=\r\nMsm1\u0002@E\n@m1+\u000bm1\u0002@m1\n@t\n+\u000bnlm1\u0002@m2\n@t; (6)\nwith\u000bthe Gilbert damping constant of each layer, and\nwhere the equation for the second layer is obtained from\nthe above by interchanging the labels 1 and 2. We\ntake the external \feld in the same direction as the\nDzyaloshinskii vector and D=D^z,H=H^z, while\n\u00160MsH;K\u001dD, so that the magnetic layers are aligned\nin the ^z-direction. Linearizing the LLG equation around\nthis direction we write mi= (mi;x;mi;y;1)Tand keep\nterms linear in mi;xandmi;y. Writing\u001ei=mi;x\u0000imi;y,\nwe \fnd, after Fourier transforming to frequency space,\nthat\n\u001f\u00001(!)\u0012\u001e1(!)\n\u001e2(!)\u0013\n= 0: (7)\nTo avoid lengthy formulas, we give explicit results below\nfor the case that J= 0, while plotting the results for\nJ6= 0 in Fig. 2. The susceptibility tensor \u001fij, or magnon\nGreen's function, is given by\n\u001f(!) =1\n((1 +i\u000b)!\u0000!0)2\u0000(\rD=Ms)2\u0000\u000b2\nnl!2)\n\u0002\u0012(1 +i\u000b)!\u0000!0i(\rD=Ms\u0000\u000bnl!)\n\u0000i(\rD=Ms+\u000bnl!) (1 +i\u000b)!\u0000!0\u0013\n;(8)\nwith!0=\r(\u00160H+K=Ms) the ferromagnetic-resonance\n(FMR) frequency of an individual layer. The poles of\nthe susceptibility determine the FMR frequencies of the\ncoupled layers and are, for the typical case that \u000b;\u000b nl\u001c\n1, given by\n!\u0006=!r;\u0006\u0000i\u000b!r;\u0006; (9)\nwith resonance frequency\n!r;\u0006=\r(\u00160H+K=Ms\u0006D=Ms): (10)\nWhen\r\u00160H= (1\u0007\u000bnl)D=(\u000bnlMs)\u0000K=Ms\u0019\nD=(\u000bnlMs)\u0000K=Mswe have for J= 0 that\u001f12(!r;\u0006) = 0\nwhile\u001f21(!r;\u0006)6= 0, signalling the non-reciprocal cou-\npling. That is, the excitation of layer 1 by FMR leads\nto response of magnetic layer 2, while layer 1 does not\nrespond to the excitation of layer 2. For opposite direc-\ntion of \feld the coupling reverses: the excitation of layer3\n|χ21(J=0)|\n|χ21(J=0.5D)|\n|χ21(J=15D)|\n|χ12(J=0)|\n|χ12(J=0.5D)|\n|χ12(J=15D)|\n0.96 0.98 1.00 1.02 1.040100200300400\nω/ωH\nFIG. 2. Magnetic susceptibilities of two magnetic layers as a\nfunction of frequency at di\u000berent exchange couplings. !H\u0011\n\r(\u00160H+K=M s). The resonance frequencies are located at\nthe peak positions. The parameters are D=! H= 0:001;\u000bnl=\n0:001;\u000b= 0:002.\n2 by FMR leads in that case to response of magnetic\nlayer 1, while layer 2 does not respond to the excitation\nof layer 1. As is observed from Fig. 2, for \fnite but\nsmallJ\u001cD, the coupling is not purely unidirectional\nanymore but there is still a large non-reciprocity. For\nJ\u001dD, this non-reciprocity is washed out.\nElectrically-actuated spin-current transmission. | In\npractice, it may be challenging to excite the individual\nlayers independently with magnetic \felds, which would\nbe required to probe the susceptibility that is determined\nabove. The two layers may be more easily probed inde-\npendently by spin-current injection/extraction from ad-\njacent contacts. Therefore, we consider the situation that\nthe two coupled magnetic layers are sandwiched between\nheavy-metal contacts (see Fig. 3(a)). In this set-up, spin\ncurrent may be transmitted between the two contacts\nthrough the magnetic layers.\nFollowing the Green's function formalism developed by\nZheng et al. [24], the spin-current from the left (right)\nlead to its adjacent magnetic layer is determined by the\ntransmission function of the hybrid system T12(T21)\ngiven by\nTij(!) = Trh\n\u0000i(!)G(+)(!)\u0000j(!)G(\u0000)(!)i\n: (11)\nHere,G(+)(!) is the retarded Green's function for\nmagnons in contact with the metallic leads that is de-\ntermined by Dyson's equation\u0002\nG(+)\u0003\u00001(!) =\u001f\u00001(!)\u0000\n\u0006(+)\n1(!)\u0000\u0006(+)\n2(!), where the retarded self energy\n~\u0006(+)\ni(!) accounts for the contact with the metallic lead\ni. These self energies are given by\n~\u0006(+)\n1(!) =\u0000i~\u000b0\n1\u0012\n!0\n0 0\u0013\n; (12)and\n~\u0006(+)\n2(!) =\u0000i~\u000b0\n2\u00120 0\n0!\u0013\n: (13)\nThe rates for spin-current transmission from the heavy\nmetal adjacent to the magnet iinto it, are given by\n\u0000i(!) =\u00002Imh\n\u0006(+)\ni(!)i\n=~. The couplings \u000b0\ni=\n\rRe[g\"#\ni]=4\u0019Msdiare proportional to the real part of the\nspin-mixing conductance per area g\"#\nibetween the heavy\nmetal and the magnetic layer i, and further depend on\nthe thickness diof the magnetic layers. Finally, the ad-\nvanced Green's function is G(\u0000)(!) =\u0002\nG(+)\u0003y.\nIn the analytical results below, we again restrict our-\nselves to the case that J= 0 for brevity, leaving the\ncaseJ6= 0 to plots. Using the above ingredients,\nEq. (11) is evaluated. Taking identical contacts so that\n\u000b0\n1=\u000b0\n2\u0011\u000b0, we \fnd that\nT12=4(\u000b0)2!2(\rD=Ms+\u000bnl!)2\njC(!)j2; (14)\nwhile\nT21=4(\u000b0)2!2(\rD=Ms\u0000\u000bnl!)2\njC(!)j2; (15)\nwith\nC(!) = [!H\u0000(1 +i(\u000b\u0000\u000bnl+\u000b0))!]\u0001\n[!H\u0000(1 +i(\u000b+\u000bnl+\u000b0))!]\u0000(\rD=Ms)2:(16)\nFrom the expression for C(!) it is clear that, since\n\u000b;\u000b nl;\u000b0\u001c1, the transmission predominantly occurs\nfor frequencies equal to the resonance frequencies !r;\u0006\nfrom Eq. (9). Similar to the discussion of the suscepti-\nbilities, we have for \felds \r\u00160H=D=\u000b nl\u0000K=Msthat\nthe transmission T12(!=D=\u000b nl)6= 0, while T21(!=\nD=\u000b nl) = 0. As a result, the spin-current transmis-\nsion is unidirectional at these \felds. For the linear spin-\nconductances Gij, given byGij=R\n~!(\u0000N0(~!))Tij(!),\nwe also have that G126= 0, while G21= 0. Here,\nN(~!) = [e~!=kBT\u00001]\u00001is the Bose-Einstein distri-\nbution function at thermal energy kBT. For the oppo-\nsite direction of external \feld we have G12= 0, while\nG216= 0. Like in the case of the susceptibility discussed\nin the previous section, a \fnite but small exchange cou-\npling makes the spin current transport no longer purely\nunidirectional, while maintaining a large non-reciprocity\n(see Fig. 3(b)).\nSpin-wave propagation. | Besides the unidirectional\ncoupling of two magnetic layers, the above results may\nbe generalized to a magnetic multilayer, or, equivalently,\nan array of coupled magnetic moments that are labeled\nby the index isuch that the magnetization direction of\nthei-th layer is mi. This extension allows us to engi-\nneer unidirectional spin-wave propagation as we shall see4\nm1\nm2\nLead Lead FM FM(a)\n(b)\nT21(J=0)\nT21(J=0.5D)\nT21(J=15D)\nT12(J=0)\nT12(J=0.5D)\nT12(J=15D)\n0.96 0.98 1.00 1.02 1.040.0000.0020.0040.0060.008\nω/ωH\nFIG. 3. (a) Schematic of the system that the two coupled\nmagnetic layers are sandwiched between two heavy-metal con-\ntacts. (b) Transmission of the hybrid system as a function of\nfrequency.\nbelow. We consider the magnetic energy\nE[m] =X\nk[D\u0001(mk\u0002mk+1)\u0000\u00160MsH\u0001mk];(17)\nand \fnd | within the same approximations as for our\ntoy model above | for the magnetization dynamics that\n@mk\n@t=2\r\nMsmk\u0002(D\u0002mk\u00001)\u0000\r\n\u000bnlMsmk\u0002D;(18)\nfor the \feld H=D=\u000bnl\u00160Ms. This shows that for these\n\felds the magnetic excitations travel to the right | cor-\nresponding to increasing index k| only. The direction\nof this one-way propagation is reversed by changing the\nmagnetic \feld to \u0000Hor by changing the sign of the non-\nlocal damping.\nTo study how spin waves propagate in an array of cou-\npled magnetic moments described by the Hamiltonian in\nEq. (17). We start from the ground state mk= (0;0;1)T\nand perturb the left-most spin ( k= 0) to excite spin\nwaves. Since the dynamics of this spin is not in\ruenced\nby the other spins for the \feld H=D=\u000b nl\u00160Ms, its\nsmall-amplitude oscillation can be immediately solved\nas\u001e0(t) =\u001e0(t= 0) exp(\u0000i!0t\u0000\u000b!0t) with\u001ek=\nmk;x\u0000imk;yas used previously. The dynamics of the\nspins to the right of this left-most spin is derived by solv-\ning the LLG equation (18) iteratively, which yields\n\u001ek(t) =\u001e0(t= 0)e\u0000i!0te\u0000\u000b!0t\nk!(\u00002\u000bnl!0t)k;(19)wherek= 0;1;2;:::N\u00001.\nTo guarantee the stability of the magnetization dynam-\nics, the dissipation matrix of the N-spin system should\nbe negative-de\fnite, which imposes a constraint on the\nrelative strength of Gilbert damping and non-local damp-\ning, i.e.,\u000b > 2\u000bnlcos\u0019\nN+1. For an in\fnitely-long chain\nN!1 , we have\u000b>2\u000bnl. Physically, this means that\nthe local dissipation of a spin has to be strong enough to\ndissipate the spin current pumped by its two neighbors.\nFor a spin chain with \fnite number of spins, \u000b= 2j\u000bnljis\nalways su\u000ecient to guarantee the stability of the system.\nTaking this strength of dissipation simpli\fes Eq. (19) to\n\u001ek(t) =\u001e0(t= 0)e\u0000it=(\u000b\u001c)e\u0000t=\u001c\nk!(\u0000t=\u001c)k; (20)\nwhere\u001c\u00001=\u000b!0is the inverse lifetime of the FMR\nmode. This spatial-temporal pro\fle of spins is the same\nas a Poisson distribution with both mean and variance\nequal to\u001b=t=\u001cexcept for a phase modulation, and it\ncan be further approximated as a Gaussian wavepacket\non the time scale t\u001d\u001c, i.e.\n\u001e(x) =\u001e0(t= 0)e\u0000it=(\u000b\u001c)\np\n2\u0019\u001be\u0000(x\u0000\u001b)2\n2\u001b: (21)\nSuch similarity suggests that any local excitation of the\nleft-most spin will generate a Gaussian wavepacket prop-\nagating along the spin chain. The group velocity of the\nmoving wavepacket is v=a=\u001c, whereais the distance\nbetween the two neighboring magnetic moments. The\nwidth of the wavepacket spreads with time as ap\nt=\u001c,\nwhich resembles the behavior of a di\u000busive particle. Af-\nter su\u000eciently long time, the wavepacket will collapse.\nOn the other hand, the excitation is localized and can-\nnot propagate when the right-most spin ( k=N\u00001) is\nexcited, because its left neighbor, being in the ground\nstate, has zero in\ruence on its evolution. These results\ndemonstrate the unidirectional properties of spin-wave\ntransport in our magnetic array.\nDiscussion, conclusion, and outlook. | We have\nshown that the ingredients for unidirectional coupling be-\ntween magnetic layers or moments are that they are cou-\npled only by DMI and non-local Gilbert damping. While\nin practice it may be hard to eliminate other couplings,\nthe DMI and non-local coupling need to be su\u000eciently\nlarger than the other couplings to observe unidirectional\ncoupling.\nThere are several systems that may realize the unidi-\nrectional coupling we propose. A \frst example is that of\ntwo magnetic layers that are coupled by a metallic spacer.\nSuch a spacer would accommodate non-local coupling via\nspin pumping and spin transfer. For a spacer that is\nmuch thinner than the spin relaxation length, we \fnd,\nfollowing Refs. [25{27], that \u000bnl=\r~Re[~g\"#]=4\u0019dMs,\nwith ~g\"#the spin-mixing conductance of the interface\nbetween the magnetic layers and the spacer, dthe thick-\nness of the magnetic layers. For simplicity, we took the5\nmagnetic layers to have equal properties. The two mag-\nnetic layers may be coupled by the recently-discovered\ninterlayer DMI [28, 29], tuning to a point (as a function\nof thickness of the spacer) where the ordinary RKKY ex-\nchange coupling is small. We estimate \u000bnl= 4:5\u000210\u00003\nford= 20 nm, Re[~g\"#] = 4:56\u00021014\n\u00001m\u00002and\nMs= 1:92\u0002105A=m (YIGjPt). The required mag-\nnetic \feld for unidirectional magnetic coupling is then\naround 4.5 T for D= 1 mT. Another possible platform\nfor realizing the unidirectional coupling is the system of\nFe atoms on top of a Pt substrate that was demonstrated\nrecently [30]. Here, the relative strength of the DMI and\nexchange is tuned by the interatomic distance between\nthe Fe atoms. Though not demonstrated in this experi-\nment, the Pt will mediate non-local coupling between the\natoms as well. Hence, this system may demonstrate the\nunidirectional coupling that we proposed.\nThe non-local damping is expected to be generically\npresent in any magnetic material and does not require\nspecial tuning, though it may be hard to determine its\nstrength experimentally. Hence, an attractive implemen-\ntation of the unidirectional coupling would be a magnetic\nmaterial with spins that are coupled only via DMI, with-\nout exchange interactions. While such a material has\nto the best of our knowledge not been discovered yet,\nit is realized transiently in experiments with ultrafast\nlaser pulses [31]. Moreover, it has been predicted that\nhigh-frequency laser \felds may be used to manipulate\nDMI and exchange, even to the point that the former is\nnonzero while the latter is zero [32, 33].\nPossible applications of our results are spin-wave and\nspin-current diodes and magnetic sensors, where a weak\n\feld signal can be ampli\fed and transported through\nthe unidirectional coupling to the remote site to be read\nout without unwanted back-action. Finally, we remark\nthat the unidirectional magnetic coupling that we pro-\npose here may be thought of as reservoir engineering, cf.\nRef. [34]. In our proposal, the reservoir is made up by the\ndegrees of freedom that give rise to the non-local damp-\ning, usually the electrons. We hope that this perspective\nmay pave the way for further reservoir-engineered mag-\nnetic systems\nAcknowledgements. | It is a great pleasure to\nthank Mathias Kl aui and Thomas Kools for discus-\nsions. 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X 5,021025 (2015)."    },    {        "title": "1709.04401v1.Life_span_of_blowup_solutions_to_semilinear_wave_equation_with_space_dependent_critical_damping.pdf",        "content": "arXiv:1709.04401v1  [math.AP]  13 Sep 2017Life-span of blowup solutions to semilinear wave equation\nwith space-dependent critical damping\nMasahiro Ikeda∗and Motohiro Sobajima†\nAbstract. This paper is concerned with the blowup phenomena for initia l value problem of semilinear wave\nequation with critical space-dependent damping term\n∂2\ntu(x,t)−∆u(x,t)+V0|x|−1∂tu(x,t)=|u(x,t)|p,(x,t)∈RN×(0,T),\nu(x,0)=εf(x), x∈RN,\n∂tu(x,0)=εg(x), x∈RN,(DW: V0)\nwhere N≥3,V0∈[0,(N−1)2\nN+1),fandgare compactly supported smooth functions and ε > 0 is a small\nparameter. The main result of the present paper is to give a so lution of (DW: V0) and to provide a sharp\nestimate for lifespan for such a solution whenN\nN−1<p≤pS(N+V0), where pS(N) is the Strauss exponent\nfor (DW:0). The main idea of the proof is due to the technique o f test functions for (DW:0) originated by\nZhou–Han (2014, MR3169791). Moreover, we find a new threshol d value V0=(N−1)2\nN+1for the coefficient of\ncritical and singular damping |x|−1.\nMathematics Subject Classification (2010): Primary: 35L70.\nKey words and phrases : wave equation with singular damping, Small data blow up, St rauss exponent, Critical and subcrit-\nical case, the Gauss hypergeometric functions .\n1 Introduction\nIn this paper we consider the blowup phenomena for initial va lue problem of semilinear wave equation\nwith scale-invariant damping term of space-dependent type as follows:\n∂2\ntu(x,t)−∆u(x,t)+a(x)∂tu(x,t)=|u(x,t)|p,(x,t)∈RN×(0,T),\nu(x,0)=εf(x), x∈RN,\n∂tu(x,0)=εg(x), x∈RN,(1.1)\nwhere N≥3,a(x)=V0|x|−1(V0≥0),ε > 0 is a small parameter and f,gare smooth nonnegative\nfunctions satisfying g/nequivalence0 with\nsupp( f,g)⊂B(0,R0)={x∈RN;|x|≤R0}\nfor some R0>0. Note that by taking uλ(x,t)=λ2\np−1u(λx,λt) withλ=R0, we can always assume R0=1\nwithout loss of generality.\nThe study of blowup phenomena for (1.1) with N=3 and V0=0 was initially started by F. John in\n[5] for 1<p<1+√\n2. Strauss conjectured in [ 9] that the number p0(N) given by the positive root of\nthe quadratic equation\n(N−1)p2−(N+1)p−2=0\n∗Department of Mathematics, Faculty of Science and Technolo gy, Keio University, 3-14-1 Hiyoshi,\nKohoku-ku, Yokohama, 223-8522, Japan /Center for Advanced Intelligence Project, RIKEN, Japan, E- mail:\nmasahiro.ikeda@keio.jp/masahiro.ikeda@riken.jp\n†Department of Mathematics, Faculty of Science and Technolo gy, Tokyo University of Science, 2641 Yamazaki, Noda-shi,\nChiba, 278-8510, Japan, E-mail: msobajima1984@gmail.com\n1is the threshold for dividing the following two situations: blowup phenomena at a finite time for arbitrary\nsmall initial data and global existence of small solutions. The conjecture of Strauss was completely\nsolved until Yordanov–Zhang [ 11] and Zhou [ 12].\nAfter that the lifespan of solutions to nonlinear wave equat ions ((1.1) with V0=0) with small initial\ndata has been considered by many authors. If 1 <p<p0(N), then by Sideris [ 8] and Di Pomponio–\nGeorgiev [ 2] we have the two-sided estimates for lifespan of solution wi th small initial data as\ncε2p(p−1)\n2+(N+1)p−(N−1)p2+δ≤LifeSpan( u)≤Cε2p(p−1)\n2+(N+1)p−(N−1)p2\nwith arbitrary small δ>0. For the critical case p=p0(N), Takamura–Wakasa [ 10] succeeded in proving\nsharp upper bound of lifespan\nexp[cε−p(p−1)]≤LifeSpan( u)≤exp[Cε−p(p−1)]\nfor remaining case N=4, and by [ 10] the study of the lifespan for blowup solutions to nonlinear wave\nequations with small data has been completed (for the other c ontributions see e.g, [ 10] and its references\ntherein). In the connection to the previous paper, we have to remark that Zhou–Han [ 13] gave a short\nproof for verifying the sharp upper bound of lifespan by usin g an estimate established in [ 11] and a kind\nof test functions including the Gauss hypergeometric funct ions.\nIn this paper, we mainly deal with the problem (1.1) with N≥3 and V0>0. Because of the strong\nsingularity of damping term at the origin, the study of (1.1) has not been considered so far. Since the\nproblem has a scaling-invariant structure, one can expect t hat some threshold for V0appears.\nThe first purpose of this paper is to clarify the local wellpos edness of (1.1) for 1 <p<N−2\nN−4in\nsolutions in H2(RN). The second is to show an upper bound of the lifespan of solut ions to (1.1) with\nrespect to small parameter ε > 0 and to pose a threshold number for V0dividing completely di fferent\nsituations.\nThe first assertion of this paper is for local wellposedness o f (1.1).\nProposition 1.1. Let N≥3, V0≥0and\n1<p<∞ if N=3,4\n1<p<N−2\nN−4if N≥5.\nFor every (f,g)∈H2(RN)∩H1(RN)andε > 0, there exist T=T(/ba∇dblf/ba∇dblH2,/ba∇dblg/ba∇dblH1,ε)>0and a unique\nstrong solution of (1.1) in the following class:\nu∈ST=C2([0,T];L2(RN))∩C1([0,T];H1(RN))∩C([0,T];H2(RN))\nMoreover, one has for every t ≥0,\nsupp u(t)⊂B(0,R0+t).\nDefinition 1.1. We denote LifeSpan( u) as the maximal existence time for solution of (1.1), that is ,\nLifeSpan( u)=sup{T>0 ;u∈ST&uis a solution of (1.1) in (0 ,T)}\nDefinition 1.2. We introduce the following quadratic polynomial\nγ(n;p)=2+(n+1)p−(n−1)p2\n2and denote p0(n) as the positive root of the quadratic equation γ(n;p)=0 as in Introduction. We also\nput\nV∗=(N−1)2\nN+1\nand for areas for ( p,V0) as follows:\nΩ0={(p,V0) ;p=p0(N+V0),0≤V0<V∗} (1.2)\nΩ1=/braceleftigg\n(p,V0) ; max/braceleftigg\np0(N+2+V0),2\nN−1−V0/bracerightigg\n≤p<p0(N+V0),0≤V0<V∗/bracerightigg\n(1.3)\nΩ2=/braceleftigg\n(p,V0) ;2(N+1)\nN+1+V0<p<2\nN−1−V0,(N+1)(N−2)\nN+2<V0<V∗/bracerightigg\n(1.4)\nΩ3=/braceleftigg\n(p,V0) ; max/braceleftiggN\nN−1,N+3+V0\nN+1+V0/bracerightigg\n<p<max/braceleftigg\np0(N+2+V0),2(N+1)\nN+1+V0/bracerightigg/bracerightigg\n(1.5)\nV0\npV∗\np0(N) p0(N+2)N+1\nN−1N\nN−1Ω1Ω2\nΩ3Ω0\nFigure 1: the regions Ω0,Ω1,Ω2andΩ3\nNow we are in a position to state our main result in this paper a bout upper bound of lifespan of\nsolutions to ( 1.1).\nTheorem 1.2. LetN\nN−1<p<∞if N=3,4andN\nN−1<p<N−2\nN−4if N≥5. Fix (f,g)satisfying\nf≥0, g≥0, g/nequivalence0andsupp( f,g)⊂B(0,1). Let uεbe the solution of (1.1) in Proposition 1.1 with the\nparameterε>0. If(p,V0)∈Ω0∪Ω1∪Ω2∪Ω3, then LifeSpan( uε)<∞. Moreover, one has\nLifeSpan( uε)≤exp[Cε−p(p−1)] if(p,V0)∈Ω0,\nC′\nδε−2p(p−1)/γ(N+V0;p)−δif(p,V0)∈Ω1,\nC′′\nδε−2(p−1)\n2N−(N−1+V)p−δif(p,V0)∈Ω2,\nC′′′\nδε−1−δif(p,V0)∈Ω3,(1.6)\n3whereδcan be chosen arbitrary small and C δ, C′\nδ, C′′\nδand C′′′\nδare positive constants which depend on\nall parameters without ε.\nRemark 1.1.We emphasize the following two facts. The proof of [ 13] depends on an estimate established\nby [11] (for detail, see [ 11, (2,5’)]), however, our proof does not depend on that. The pr oof of Theorem\n1.2 can be applicable to weaker solutions of (1.1) belonging toC([0,T));H1(RN)∩Lp((0,T)×RN).\nRemark 1.2.Taking the threshold value V0=V∗formally, we have\nγ(N+V∗;p)=2(1+N p)/parenleftigg\n1−N−1\nN+1p/parenrightigg\nand therefore p0(N+V∗)=N+1\nN−1=1+2\nN−1. On the one hand, critical exponent for the blowup phenomena\nfor the problem\n∂2\ntu(x,t)−∆u(x,t)+/an}b∇acketle{tx/an}b∇acket∇i}ht−α∂tu(x,t)=|u(x,t)|p,(x,t)∈RN×(0,T),\nu(x,0)=εf(x), x∈Ω,\n∂tu(x,0)=εg(x), x∈Ω,(1.7)\nis given by pF(α)=1+2\nN−α,α∈[0,1) which is so-called Fujita exponent (see e.g., Ikehata–To dorova–\nYordanov [ 4] and also Ikeda–Ogawa [ 3]). We formally put again a threshold value α=1. Then one can\nfind\np0(N+V0)=pF(1).\nThe left-hand side comes from the blowup phenomena for nonli near wave equation and the right-hand\nside comes from the one for nonlinear heat equation. In this c onnection, we would conjecture that if\nV0>V∗, then the threshold of blowup phenomena is given by the Fujit a exponent pF(1).\nRemark 1.3.If (p,V0)∈Ω0∪Ω1, then Theorem 1.2 seems to give a sharp lifespan of solutions to (1.1)\nwith small initial data. In the case ( p,V0)∈Ω3, we cannot derive the estimates for lifespan with ε−τ\nwithτless than one. So the estimate in Ω3seems not to be sharp. For the case ( p,V0)∈Ω2the effect of\ndiffusion structure seems to appear in the estimate.\nThe present paper is organized as follows. In Section 2, we fir st give existence and uniqueness of\nlocal-in-time solutions to (1.1) if p≤N−2\nN−4by using the standard semigroup properties. In Section 3, we\nconstruct special solutions of linear wave equation with an ti-damping term−V0\n|x|∂tu. In this point we use\nthe idea due to [ 13] (they only considered the case V0=0), which will be a test function for proving\nblowup phenomena. In Section 4, we prove blowup phenomena by dividing two cases p<p0(N+V0)\nandp=p0(N+V0).\n2 Local solvability of nonlinear wave equation with singula r damping\nIn this section we construct a solution of (1.1) with initial data belonging to H2(RN)×H1(RN). To do so,\nwe first treat the linear problem\n∂2\ntu(x,t)−∆u(x,t)+a(x)∂tu(x,t)=0,(x,t)∈RN×(0,∞),\nu(x,0)=u0(x), ∂ tu(x,0)=u1(x), x∈RN.(2.1)\n42.1 C0-Semigroup for linear wave equation with singular damping\nNow we start with the usual N-dimensional Laplacian\nAu:=−∆u,D(A)=H2(RN).\nWe note that Aism-accretive in L2(RN), that is, ( I+A)D(A)=L2(RN) and (−∆u,u)≥0 for every\nu∈H2(RN). SetH=H1(RN)×L2(RN) and\nA=/parenleftigg0−1\nA0/parenrightigg\n,D(A)=H2(RN)×H1(RN)\nand\nB=/parenleftigg0 0\n0|x|−1/parenrightigg\n,D(B)=H1(RN)×{v∈L2(RN) ;|x|−1v∈L2(RN)}\nand put\nAκ=A+κB,D(Aκ)=D(A)∩D(B)\nThen in view of Hille–Yosida theorem, we have the following C0-semigroup onH(see e.g., Pazy [ 7]).\nLemma 2.1. Let N≥3. For every V≥0,1\n2I+Aκis m-accretive inH. Therefore−Aκgenerates a C 0-\nsemigroup{Tκ(t)}t≥0onH. Moreover, if supp( u0,u1)⊂B(0,R), then supp[ Tκ(t)(u0,u1)]⊂B(0,R+t).\nProof. By Hardy’s inequality we have D(A)⊂D(B). This means that D(Aκ)=D(A)∩D(B)=D(A).\n(Accretivity) By interation by parts, we have\n/parenleftigg\nA/parenleftiggu\nv/parenrightigg\n,/parenleftiggu\nv/parenrightigg/parenrightigg\nH=/parenleftigg/parenleftigg−v\n−∆u/parenrightigg\n,/parenleftiggu\nv/parenrightigg/parenrightigg\nH=/integraldisplay\nRN/parenleftig\n−vu−∇v·∇u−(∆u)v/parenrightig\ndx=−/integraldisplay\nRNuv dx.\nSinceκBis clearly accretive, we have the accretivity of Aκ.\n(Maximality) LetF=(f,g)∈H. ThenλU+AU+κBU=Fis equivalent to the system\nλu−v=f, λ v−∆u+κ|x|−1v=g.\nSubstituting v=λu−f, we see that\nλ2u−∆u+λκ|x|−1u=g+λf+k|x|−1f=fλ,κ.\nTaking /tildewideu(y)=u(λ−1y) and/tildewidestfλ,κ(y)=fλ,κ(λ−1y) yields that\n/tildewideu−∆/tildewideu+k|y|−1/tildewideu=λ−2/tildewidestfλ,κ.\nThis is nothing but the resolvent problem of the Schr¨ odinge r operator with positive Coulomb potentials.\nTherefore there exists /tildewideuλ,k∈H2(RN) such that\n/tildewideuλ,k−∆/tildewideuλ,k+k|y|−1/tildewideuλ,k=λ−2/tildewidestfλ,k.\nPutting uλ,k(x)=/tildewideuλ,k(λ2x)∈H2(RN), we obtain\nλ2uλ,k−∆uλ,k+λk|x|−1uλ,k=fλ,k=g+λf+k|x|−1f.\nFinally setting vλ,k=λuλ,k−f∈H1(RN), we obtain ( λI+A+kB)(uλ,k,vλ,k)=(f,g).\nThe finite propagation property follows from the standard ar gument for wave equation with regular\ndamping term. The proof is complete. /square\n52.2 Local solvability of nonlinear problem\nWe consider (1.1) with initial data ( u(0),∂tu(0))=U0=(u0,u1)∈H2(RN)×H1(RN) which is equivalent\nto the following problem\n∂t/parenleftiggu(t)\nv(t)/parenrightigg\n+Aκ/parenleftiggu(t)\nv(t)/parenrightigg\n=/parenleftigg0\nN(u(t),v(t))/parenrightigg\nwithN(u,v)=(0,|u|p). Here we construct the corresponding mild solution in H2(RN)×H1(RN) given\nby/parenleftiggu(t)\nv(t)/parenrightigg\n=Tκ(t)/parenleftiggu0\nu1/parenrightigg\n+/integraldisplayt\n0Tκ(t−s)/parenleftigg0\nN(u(s),v(s))/parenrightigg\nds,\nwhere{Tκ(t)}t≥0is determined in Lemma 2.1.\nLemma 2.2. (i) The following metric space\nXT=/braceleftig\nU∈C([0,T];H)∩L∞([0,T];D(Aκ)) ; sup\n0<t<T/ba∇dblU(t)/ba∇dblD(Aκ)≤M/bracerightig\n,M:=2(/ba∇dblU0/ba∇dblD(Aκ)+1)\nwith the distance\nd(U1,U2) :=max\n0≤t≤T/vextenddouble/vextenddouble/vextenddoubleU(t)−U2(t)/vextenddouble/vextenddouble/vextenddoubleH,U1,U2∈XT.\nis complete.\n(ii)Ifsupp( u0,u1)⊂B(0,R), then the following metric space\nYT=/braceleftig\nU∈C([0,T];H)∩L∞([0,T];D(Aκ)) ; sup\n0<t<T/ba∇dblU(t)/ba∇dblD(Aκ)≤M,supp( u(t),v(t))⊂B(0,R+t)/bracerightig\nwith the same distance d is also complete.\nProof. Take a Cauchy sequence {Un}n∈NinXT. The completeness of C([0,T];H) yields that there exists\nU∞∈C([0,T];H) such that\nUn→U∞strongly in C([0,T];H) as n→∞.\nMoreover, we can subtract a subsequence Unjand/tildewideU∈L∞([0,T];D(Aκ)) such that\nsup\n0<t<T/ba∇dbl/tildewideU(t)/ba∇dblD(Aκ)≤M\nand\nUnj→/tildewideU∗-weakly in L∞([0,T];D(Aκ)) as j→∞.\nTherefore we have U∞=/tildewideU. Therefore XTis a complete metric space. If supp Un(t)⊂{x;|x|≤R+t}, then\nby strong convergence we have supp U∞(t)⊂{x;|x|≤R+t}. This means that YTis also complete. /square\nLemma 2.3. There exists T 0such thatΨ:XT0→XT0andΨ:YT0→YT0are both well-defined, and Ψ\nis contractive in X T0and in Y T0.\nProof. First observe that by finite propagation property in Lemma 2. 1, we can deduce supp ΨU(t)⊂\nB(0,R+t) when supp U0⊂B(0,R). Since XTandYTare endowed with the same distance, It su ffices to\nprove the assertion for XT.\nWe recall that the norms in D(Aκ) and in H2(RN)∩H1(RN) are equivalent.\n6(well-defined) IfU=(u,v)∈XT, then\n/ba∇dblN(U(t))/ba∇dbl2\nD(Aκ)≤C2\nκ/ba∇dblN(U(t))/ba∇dbl2\nH2×H1\n=C2\nκ/vextenddouble/vextenddouble/vextenddouble|u(t)|p/vextenddouble/vextenddouble/vextenddouble2\nH1\n=C2\nκ/integraldisplay\nRN/parenleftig\n|u(t)|p+|∇(|u(t)|p)|2/parenrightig\ndx\n=C2\nκ/parenleftigg/integraldisplay\nRN|u(t)|2pdx+p2/integraldisplay\nRN|u(t)|2(p−1)|∇u(t)|2dx/parenrightigg\n≤C2\nκ/parenleftigg/integraldisplay\nRN|u(t)|N(p−1)dx/parenrightigg2\nN/parenleftigg/integraldisplay\nRN|u(t)|2N\nN−2dx/parenrightigg1−2\nN\n+p2C2\nκ/parenleftigg/integraldisplay\nRN|u(t)|N(p−1)dx/parenrightigg2\nN/parenleftigg/integraldisplay\nRN|∇u(t)|2N\nN−2dx/parenrightigg1−2\nN\n.\nSince p≤N−2\nN−4, we have N(p−1)≤(1\n2−2\nN) and therefore\n/ba∇dblN(U(t))/ba∇dblD(Aκ)≤C′\nκ/ba∇dblu(t)/ba∇dblH2\n≤C′′\nκ/ba∇dblU(t)/ba∇dblH2×H1\n≤C′′\nκ/ba∇dblU(t)/ba∇dblD(Aκ)\n≤C′′\nκM.\nTherefore we have for 0 ≤t≤T≤log 2,\n/ba∇dblΨU(t)/ba∇dblD(Aκ)≤/ba∇dblTκ(t)U0/ba∇dblD(Aκ)+/integraldisplayt\n0/ba∇dblTκ(t−s)[N(U(s))]/ba∇dblD(Aκ)ds\n≤et/2/ba∇dblU0/ba∇dblD(Aκ)+/integraldisplayt\n0e(t−s)/2/ba∇dblN(U(s))/ba∇dblD(Aκ)ds\n≤√\n2/ba∇dblU0/ba∇dblD(Aκ)+√\n2C′′\nκMt.\nTherefore there exists T1∈(0,log 2] such that sup0<t<T1/ba∇dblΨU(t)/ba∇dblD(Aκ)≤M.\nNext we prove continuity of ΨUon [0,T]. For 0≤t1<t2≤T,\n/ba∇dblΨU(t1)−ΨU(t2)/ba∇dblH≤/ba∇dbl[Tκ(t1)−Tκ(t2)]U0/ba∇dblH\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt1\n0T(t1−s)N(U(s))ds−/integraldisplayt2\n0T(t2−s)N(U(s′))ds′/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleH\n≤|t1−t2|/ba∇dblA kU0/ba∇dblH\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt1\n0[I−T(t2−t1)]T(t1−s)N(U(s))ds/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleH+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt2\nt1T(t2−s)N(U(s′))ds′/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleH\n≤|t1−t2|/ba∇dblA kU0/ba∇dblH\n+eT/2|t2−t1|/integraldisplayt1\n0/ba∇dblAκN(U(s))/ba∇dblHds+eT/2/integraldisplayt2\nt1/ba∇dblN(U(s′))/ba∇dblHds′\n≤M′|t1−t2|.\n7(contractivity) IfU1=(u1,v1),U2=(u2,v2)∈XT, then\n/ba∇dblN(U1(t))−N(U2(t))/ba∇dbl2\nH=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftigg0\n|u1(t)|p−|u2(t)|p/parenrightigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nH\n≤p2/integraldisplay\nRN(|u1(t)|+|u2(t)|)2(p−1)|u1(t)−u2(t)|2dx\n≤p2/vextenddouble/vextenddouble/vextenddouble|u1(t)|+|u2(t)|/vextenddouble/vextenddouble/vextenddouble2(p−1)\nLN(p−1)/ba∇dblu1(t)−u2(t)/ba∇dbl2\nL2N\nN−2\n≤C(/ba∇dblu1(t)/ba∇dblH2+/ba∇dblu2(t)/ba∇dblH2)2(p−1)/ba∇dblu1(t)−u2(t)/ba∇dbl2\nH1\n≤(C′)2M2(p−1)/ba∇dblU1(t)−U2(t)/ba∇dbl2\nH.\nThis implies that\n/ba∇dblΨU1(t)−ΨU2(t)/ba∇dblH≤/integraldisplayt\n0/ba∇dblTκ(t−s)[N(U1(s))−N(U2(s))]/ba∇dblHds\n≤C′Mp−1eT/2/integraldisplayt\n0/ba∇dblU1(s)−U2(s)/ba∇dblHds.\nConsequently, taking T0∈(0,T1] satisfying\n2C′Mp−1TeT/2≤1,\nwe obtain d(ΨU1,ΨU2)≤1\n2d(U1,U2), that is,Ψis contractive in XT0and also in YT0./square\nProof of Proposition 1.1. By Lemma 2.3, we can find a unique fixed point U∞ofΨinYT0. Moreover,\ncombining the previous arguments implies\n/ba∇dblN(U∞(t1))−N(U∞(t2))/ba∇dblH≤C′Mp−1/ba∇dblU∞(t1)−U∞(t2)/ba∇dblH\n≤C′′Mp|t1−t2|.\nThusN(U∞(·)) is Lipschitz continuous on [0 ,T]. By [ 7, Corollary 4.2.11(p.109)], we verify that Ut+\nAκU=F(U∞) has a unique strong solution U∗\n∞given by\nU∗\n∞(t)=Tκ(t)U0+/integraldisplayt\n0Tκ(t−s)N(U∞(s))ds,t∈[0,T0].\nSince U∞is a fixed point ofΨ, we obtain U∗\n∞(t)=U∞(t) for t∈[0,T0]. Since U∞is a strong solution,\nwe have∂tU∞=−AκU∞+N(U∞)∈L∞(0,T;H) a.e. on [0,T]. This gives us that\n∂tu=v∈L∞(0,T;H2(RN))∩W1,∞(0,T;H1(RN))∩C([0,T];H1(RN)),\nand\n∂tv=∆u−κ\n|x|v+|u|p∈L∞(0,T;H1(RN))∩C([0,T];L2).\nHence u∈C2([0,T];L2(RN)) is nothing but a strong solution of\n∂2\ntu−∆u+κ\n|x|∂tu=|u|p\non [0,T]. Uniqueness of local solutions is due to a proof similar to t he contractivity of Ψand the finite\npropagation property follows from the use of YT0. /square\n83 Special solutions of linear damped wave equation\nIn this section we construct special solutions of linear dam ped wave equation which will be test functions\nfor proving blowup properties.\nThe following function plays an essential role in the proof o f upper bound of lifespan of solutions to\n(1.1). Similar test functions appear in Zhou–Han [ 13].\nDefinition 3.1. Forβ>0, set\nΦβ(x,t)=(|x|+t)−βF/parenleftigg\nβ,N−1+V0\n2,N−1;2|x|\n2+t+|x|/parenrightigg\n,\nwhere F(a,b,c;z) is the Gauss hypergeometric function given by\nF(a,b,c;z)=∞/summationdisplay\nn=0(a)n(b)n\n(c)nzn\nn!\nwith ( d)0=1 and ( d)n=/producttextn\nk=1(d+k−1) for n∈N. (For further properties of F(·,·,·;z), see e.g., Chaper\n8 in Beals–Wong [ 1]).\nFor the reader’s convenience we would give a derivation the G auss hypergeometric function from the\nwave equation.\nLemma 3.1. Forβ>0,Φβsatisfies the wave equation with the anti-damping term\n∂2\ntΦβ−∆Φβ−V0\n|x|∂tΦβ=0,inQ={(x,t)∈RN×(0,∞) ;|x|<2+t}.\nProof. We can putΦ(x,t)=Φβ(x,t−2) for t>0. We start with the desired equation\n∂2\ntΦ(x,t)−∆Φ(x,t)−V0\n|x|∂tΦ(x,t)=0,in{(x,t)∈RN×(0,∞) ;|x|<t}. (3.1)\nPut\nu(x,t)=(|x|+t)−βϕ/parenleftigg2|x|\n|x|+t/parenrightigg\n,\nThen setting z=2|x|\n|x|+t=2−2t\n|x|+t, we have|x|+t=2t\n2−zand therefore\nu(x,t)=(2t)−β(2−z)βϕ(z).\nObserving that\n∂z\n∂t=−2|x|\n(|x|+t)2=−z(2−z)\n2t,\nwe have\n∂tu=−2β(2t)−β−1(2−z)βϕ(z)+(2t)−β[−β(2−z)β−1ϕ(z)+(2−z)βϕ′(z)]∂z\n∂t\n=−2β(2t)−β−1(2−z)βϕ(z)−(2t)−β−1[−β(2−z)β−1ϕ(z)+(2−z)βϕ′(z)]z(2−z).\n=−(2t)−β−1(2−z)β+1/bracketleftig\nβϕ(z)+zϕ′(z)/bracketrightig\n9and also\n∂2\ntu=(2t)−β−2(2−z)β+2/bracketleftig\n(β+1)(βϕ(z)+zϕ′(z))+z(βϕ(z)+zϕ′(z))′/bracketrightig\n=(2t)−β−2(2−z)β+2/bracketleftig\nβ(β+1)ϕ(z)+2(β+1)zϕ′(z)+z2ϕ′′(z)/bracketrightig\n.\nOn the other hand, for radial derivative, we see from∂z\n∂r=2t\n(|x|+t)2=(2−z)2\n2tthat\n∂ru=(2t)−β/bracketleftig\n−β(2−z)β−1ϕ(z)+(2−z)βϕ′(z)/bracketrightig∂z\n∂r\n=(2t)−β−1(2−z)β+1/bracketleftig\n−βϕ(z)+(2−z)ϕ′(z)/bracketrightig\nand\n∂2\nru=(2t)−β−2(2−z)β+2/bracketleftig\n(β+1)βϕ(z)−2(β+1)(2−z)ϕ′(z)+(2−z)2ϕ′′(z)/bracketrightig\n.\nCombining these equalities andt\nr(2−z)−1=1\nz, we obtain\n0=/parenleftigg\n∂2\ntu−∂2\nru−N−1\nr∂ru+V0\nr∂tu/parenrightigg\n(2t)β+2(2−z)−β−2\n=β(β+1)ϕ(z)+2(β+1)zϕ′(z)+z2ϕ′′(z)−(β+1)βϕ(z)−2(β+1)(2−z)ϕ′(z)+(2−z)2ϕ′′(z)\n−N−1\nr(2t)(2−z)−1/bracketleftig\n−βϕ(z)+(2−z)ϕ′(z)/bracketrightig\n−V0\nr(2t)(2−z)−1/bracketleftig\n−βϕ(z)−zϕ′(z)/bracketrightig\n=−4(1−z)ϕ′′(z)+4(β+1)ϕ′(z)+2β(N−1+V0)\nzϕ(z)+2(N−1)\nz/bracketleftig\n−(2−z)ϕ′(z)/bracketrightig\n−2V0ϕ′(z)\n=−4\nz/bracketleftigg\n(1−z)zϕ′′(z)+/bracketleftigg\nN−1−/parenleftigg\n1+β+N−1+V0\n2/parenrightigg\nz/bracketrightigg\nϕ′(z)−β(N−1+V0)\n2ϕ(z)/bracketrightigg\n.\nThis is nothing but the Gauss hypergeometric di fferential equation\nz(1−z)ϕ′′(z)+(c−(1+a+b)z)ϕ′(z)−abϕ(z)=0\nwith\n(a,b,c)=/parenleftigg\nβ,N−1+V0\n2,N−1/parenrightigg\n.\nThis implies that ϕ(z)=F(β,N−1+V0\n2,N−1;z). /square\nLemma 3.2. (i) For everyβ>0and(x,t)∈Q,\n∂tΦβ(x,t)=−βΦβ+1(x,t).\n(ii)If0<β<N−1−V0\n2, then there exists a constant c β>0such that for every (x,t)∈Q,\ncβ(2+t)−β≤Φβ(x,t)≤c−1\nβ(2+t)−β.\n(iii)Ifβ>N−1−V0\n2, then there exists a constant c′\nβ>0such that for every (x,t)∈Q,\ncβ(2+t)−β/parenleftigg\n1−|x|\nt+2/parenrightiggN−1−V0\n2−β\n≤Φβ(x,t)≤c−1\nβ(2+t)−β/parenleftigg\n1−|x|\nt+2/parenrightiggN−1−V0\n2−β\n.\n10Proof. (i)In view of the proof of Lemma 3.1, we have\n∂tΦβ(x,t)=−(2t)−β−1(2−z)β+1[βϕ(z)+zϕ′(z)]\nwith s=2|x|\n2+t+|x|. It suffices to show that\nβϕ(z)+zϕ′(z)=βF/parenleftigg\nβ+1,N−1+V0\n2,N−1;z/parenrightigg\n,z∈(0,1). (3.2)\nPutψ(z)=βϕ(z)+zϕ′(z) for z∈[0,1). Then by the definition of F(·,·,·;z), we haveψ(0)=β. On the\nother hand, we see from the gauss hypergeometric equation wi tha=β,b=N−1+V0\n2andc=N−1 that\n(1−z)ψ′(z)=(1−z)/parenleftig\n(β+1)ϕ′(z)+zϕ′′(z)/parenrightig\n=(β+1)(1−z)ϕ′(z)+z(1−z)ϕ′′(z)\n=(a+1)(1−z)ϕ′(z)−(c−(1+a+b)z)ϕ′(z)+abϕ(z)\n=(a+1−c)ϕ′(z)+bzϕ′(z)+abϕ(z)\n=(a+1−c)ϕ′(z)+bψ(z)\nand therefore (1−z)ψ′(z)−bψ(z)=(a+1−c)ϕ′(z). The definition of ψyields\nz(1−z)ψ′(z)−bzψ(z)=(a+1−c)zϕ′(z)\n=(a+1−c)ψ(z)−(a+1−c)aϕ(z)\nDifferentiating the above equality, we have\nz(1−z)ψ′′(z)+(1−(2+b)z)ψ′(z)−bψ(z)=(a+1−c)ψ′(z)−(a+1−c)aϕ′(z)\n=(a+1−c)ψ′(z)−a/parenleftig\n(1−z)ψ′(z)−bψ(z)/parenrightig\n.\nHence we have z(1−z)ψ′′(z)+(c−(2+a+b)z)ψ′(z)+(a+1)bψ(z)=0. Since N≥2, all bound solutions\nof this equation near 0 can be written by ψ(z)=hF(a+1,b,c;z) with h∈R. Combining the initial value\nψ(0)=β, we obtain (3.2).\nThe remaining assertions (ii)and(iii)are a direct consequence of the integral representation for mula\nF(a,b,c,z)=1\nB(c,c−a)/integraldisplay1\n0sa−1(1−s)c−a−1(1−zs)−bds,0≤z<1\nwhen c>0 and c−a>0. The proof is complete. /square\n4 Proof of blowup phenomena\nIn this section we prove upper bound of the lifespan of soluti ons to (1.1) and its dependence of εunder\nthe condition 0≤V0<V∗=(N−1)2\nN+1.\n114.1 Preliminaries for showing blowup phenomena\nWe first state a criterion for derivation of upper bound for li fespan.\nLemma 4.1. Let H∈C2([σ0,∞)be nonnegative function.\n(i)Assume that there exists positive constants c ,C,C′such that\nC[H(σ)]p≤H′′(σ)+cH′(σ)\nσ\nwith H (σ)≥εpCσ2and H′(σ)≥εpCσ. Then H blows up before σ=C′′ε−p−1\n2for some C′′>0.\n(ii)Assume that there exists positive constants c ,C,C′such that\nCσ1−p[H(σ)]p≤H′′(σ)+2H′(σ)\nwith H (σ)≥εpCσand H′(σ)≥εpC. Then H blows up before σ=C′′ε−p(p−1)for some C′′>0.\nProof. The assertion follows from [ 13, Lemma 2.1] with the argument in [ 13, Section 3]. /square\nWe focus our eyes to the following functionals.\nDefinition 4.1. Forβ∈(0,N−1−V0\n2), define the following three functions\nGβ(t) :=/integraldisplay\nRN|u(x,t)|pΦβ(x,t)dx,t≥0,\nHβ(t) :=/integraldisplayt\n0(t−s)(2+s)Gβ(t)ds,t≥0,\nJβ(t) :=/integraldisplayt\n0(2+s)−3Hβ(t)ds,t≥0.\nNote that we can see from Lemma 3.2 (ii)thatGβ(t)≈(2+t)−β/ba∇dblu(t)/ba∇dblp\nLp(RN).\nLemma 4.2. If uεis a solution of (1.1) in Proposition 1.1 with parameter ε >0, then Jβdoes not blow\nup until LifeSpan( uε).\nProof. It follows from the embedding H2(RN)→Lp(RN) (given by Gagliardo-Nirenberg-Sobolev in-\nequalities) that/ba∇dbluε(t)/ba∇dblLpis continuous on [0 ,LifeSpan( uε)) and also Gβ(t). This means that Jβ(t) is finite\nfor all t∈[0,LifeSpan( uε)). /square\nLemma 4.3. For everyβ>0and t≥0,\n(2+t)2Jβ(t)=1\n2/integraldisplayt\n0(t−s)2Gβ(s)ds.\nProof. This can be verified by integration by parts twice, by noting t hat\nd\nds/parenleftig\n(t−s)2(1+s)−1/parenrightig\n=2(1+t)2\n(1+s)3.\n/square\n12Lemma 4.4. Let u be a solution of (1.1) . Then for every β>0and t≥0,\nεEβ,0+εEβ,1t+/integraldisplayt\n0(t−s)Gβ(s)ds=/integraldisplay\nRNu(x,t)Φβ(x,t)dx+2β/integraldisplayt\n0/integraldisplay\nRNu(x,s)Φβ+1(x,s)dx ds\n+V0/integraldisplayt\n0/integraldisplay\nRN1\n|x|u(x,t)Φβ(x,t)dx ds, (4.1)\nwhere\nEβ,0=/integraldisplay\nRNf(x)Φβ(x,0)dx>0.\nEβ,1=/integraldisplay\nRNg(x)Φβ(x,0)dx+β/integraldisplay\nRNf(x)Φβ+1(x,0)dx+V0/integraldisplay\nRN1\n|x|f(x)Φβ(x)dx>0.\nProof. By the equation in (1.1) we see from integration by parts that\nGβ(t)=/integraldisplay\nRN/parenleftigg\n∂2\ntu(t)−∆u(t)+V0\n|x|∂tu(t)/parenrightigg\nΦβ(t)dx\n=/integraldisplay\nRN/parenleftigg\n∂2\ntu(t)+V0\n|x|∂tu(t)/parenrightigg\nΦβ(t)dx−/integraldisplay\nRNu(t)(∆Φβ(t))dx.\nUsing Lemma 3.1, we have\nGβ(t)=/integraldisplay\nRN/parenleftigg\n∂2\ntu(t)+V0\n|x|∂tu(t)/parenrightigg\nΦβ(t)dx−/integraldisplay\nRNu(t)/parenleftigg\n∂2\ntΦβ(t)−V0\n|x|∂tΦβ(t)/parenrightigg\ndx\n=d\ndt/bracketleftigg/integraldisplay\nRN/parenleftig\n∂tu(t)Φβ(t)−u(t)∂tΦβ(t)/parenrightig\ndx+V0/integraldisplay\nRN1\n|x|u(t)Φβ(t)dx/bracketrightigg\n.\nNoting that Lemma 3.2 (i)(the formula ∂tΦβ=−βΦβ+1), we have\nεEβ,1+/integraldisplayt\n0Gβ(s)ds=/integraldisplay\nRN/parenleftig\n∂tu(t)Φβ(t)−u(t)∂tΦβ(t)/parenrightig\ndx+V0/integraldisplay\nRN1\n|x|u(t)Φβ(t)dx\n=d\ndt/bracketleftigg/integraldisplay\nRNu(t)Φβ(t)dx/bracketrightigg\n+2β/integraldisplay\nRNu(t)Φβ+1(t)dx+V0/integraldisplay\nRN1\n|x|u(t)Φβ(t)dx.\nIntegrating it again, we obtain (4.1). /square\nThe following lemma makes sense when2\nN−1−V0<p0(N+V0) which is equivalent to 0 ≤V0<(N−1)2\nN+1.\nLemma 4.5. AssumeN\nN−1<p<∞and0≤V0<(N−1)2\nN+1.(i)Let q>1satisfy max{p,2\nN−1−V0}<q<∞\nand put\nβ=N−1−V0\n2−1\nq∈/parenleftigg\n0,N−1−V0\n2/parenrightigg\n.\nThen there exists a positive constant C 1>0such that\nεEβ,0+εEβ,1t+/integraldisplayt\n0(t−s)Gβ(s)ds≤C1/bracketleftigg\n/ba∇dblu(t)/ba∇dblLp(2+t)N\np′−β+/integraldisplayt\n0/ba∇dblu(s)/ba∇dblLp(2+s)N\np′−N−1−V0\n2−1\np′ds/bracketrightigg\n.\n(ii)If p>2\nN−1−V0, then setting β0=N−1−V0\n2−1\np∈/parenleftig\n0,N−1−V0\n2/parenrightig\n,one has\n/integraldisplayt\n0(t−s)Gβ(s)ds≤C1/bracketleftigg\n/ba∇dblu(t)/ba∇dblLp(2+t)N\np′−β+/integraldisplayt\n0/ba∇dblu(s)/ba∇dblLp(2+s)N\np′−β0−1(log(2+s))1\np′ds/bracketrightigg\n.\n13Proof. By Lemma 4.4 with finite propagation property, we have\nεEβ,0+εEβ,1t+/integraldisplayt\n0(t−s)Gβ(s)ds=Iβ,1(t)+2βIβ,2(t)+V0Iβ,3(t).\nwhere\nIβ,1(t)=/integraldisplay\nB(0,1+t)u(x,t)Φβ(x,t)dx\nIβ,2(t)=/integraldisplayt\n0/parenleftigg/integraldisplay\nB(0,1+t)u(x,s)Φβ+1(x,s)dx/parenrightigg\nds\nIβ,3(t)=/integraldisplayt\n0/parenleftigg/integraldisplay\nB(0,1+t)1\n|x|u(x,s)Φβ(x,s)dx/parenrightigg\nds.\nUsing Lemma 3.2 (ii), we have\nIβ,1(t)≤/parenleftigg/integraldisplay\nB(0,1+t)|u(x,t)|pdx/parenrightigg1\np/parenleftigg/integraldisplay\nB(0,1+t)Φβ(x,t)p′dx/parenrightigg1\np′\n≤c−1\nβN−1\np′|SN−1|1\np′/ba∇dblu(t)/ba∇dblLp(2+t)N\np′−β.\nand\nI′\nβ,3(t)≤/parenleftigg/integraldisplay\nB(0,1+t)|u(x,t)|pdx/parenrightigg1\np/parenleftigg/integraldisplay\nB(0,1+t)1\n|x|p′Φβ(x,t)p′dx/parenrightigg1\np′\n≤c−1\nβ(N−p′)−1\np′|SN−1|1\np′/integraldisplayt\n0/ba∇dblu(t)/ba∇dblLp(2+t)N\np′−β−1ds.\nNoting that β+1=N−1−V0\n2−1\nq′, we see from Lemma 3.2 (iii)that\nI′\nβ,2(t)≤/parenleftigg/integraldisplay\nB(0,1+t)|u(x,t)|pdx/parenrightigg1\np/parenleftigg/integraldisplay\nB(0,1+t)Φβ+1(x,t)p′dx/parenrightigg1\np′\n≤(c′\nβ)−1/ba∇dblu(t)/ba∇dblLp(2+t)−β−1/integraldisplay\nB(0,1+t)/parenleftigg\n1−|x|\nt+2/parenrightigg(N−1−V0\n2−β−1)p′\ndx1\np′\n=(c′\nβ)−1|SN−1|1\np′/ba∇dblu(t)/ba∇dblLp(2+t)−β−1/integraldisplay1+t\n0/parenleftbigg\n1−r\nt+2/parenrightbigg−p′\nq′\nrN−1dr1\np′\n=(c′\nβ)−1|SN−1|1\np′/ba∇dblu(t)/ba∇dblLp(2+t)N\np′−β−1/integraldisplay1\n1\n2+tρ−p′\nq′(1−ρ)N−1dρ1\np′\n≤(c′\nβ)−1|SN−1|1\np′/parenleftiggp′\nq′−1/parenrightigg1\np′\n/ba∇dblu(t)/ba∇dblLp(2+t)N\np′−β−1+1\nq′−1\np′.\nThus we have\nεEβ,0+εEβ,1t+/integraldisplayt\n0(t−s)Gβ(s)ds≤C1/bracketleftigg\n/ba∇dblu(t)/ba∇dblLp(2+t)N\np′−β+/integraldisplayt\n0/ba∇dblu(s)/ba∇dblLp(2+s)N\np′−β−1+1\nq′−1\np′ds/bracketrightigg\n.\nBy the definition of βwe have the first desired inequality. The second is verified by noticing q′/p′=1\nin the previous proof. /square\n144.2 Proof of Theorem 1.2 for subcritical case max{N\nN−1,N+3+V0\nN+1+V0}<p<p0(N+V0)\nProof. Fixq>pas the following way:\n1\nq∈/parenleftigg\n0,N−1−V0\n2/parenrightigg\n∩/parenleftigg(N−1+V)p−(N+1+V)\n2,(N+1+V0)p−(N+3+V0)\n2/parenrightigg\n.\nThe above set is not empty when ( p,V0)∈Ω1∪Ω2∪Ω3; note that for respective cases we can take\n1\nq=1\np−δ if (p,V0)∈Ω1,\nN−1−V0\n2−δ if (p,V0)∈Ω2,\n(N+1+V0)p−(N+3+V0)\n2−δif (p,V0)∈Ω3\nwith arbitrary small δ>0. Moreover, this condition is equivalent to\nq>p, β=N−1−V0\n2−1\nq>0,&λ=γ(N+V0;p)\n2p−1\np+1\nq∈(0,p−1).\nThen we see by Lemma 4.5 (i)that\nεEβ,0+εEβ,1t+/integraldisplayt\n0(t−s)Gβ(s)ds≤C′\n1/bracketleftigg\nGβ(t)1\np(2+t)N−β\np′+/integraldisplayt\n0Gβ(s)1\np(2+s)N−β\np′−1\nq−1\np′ds/bracketrightigg\n.(4.2)\nObserve that\nN−β\np′−1\nq−1\np=1\np(p−1−λ)>0,\nIntegrating (4.2) over [0 ,t], we deduce\nεEβ,0t+εEβ,1\n2t2+1\n2/integraldisplayt\n0(t−s)2Gβ(s)ds\n≤C′\n1/bracketleftigg/integraldisplayt\n0Gβ(s)1\np(2+s)N−β\np′ds+/integraldisplayt\n0(t−s)Gβ(s)1\np(2+s)N−β\np′−1\nq−1\np′ds/bracketrightigg\n≤C′\n1/parenleftigg/integraldisplayt\n0(2+s)Gβ(s)ds/parenrightigg1\np/parenleftigg/integraldisplayt\n0(2+s)(N−β\np′−1\np)p′ds/parenrightigg1\np′\n+/parenleftigg/integraldisplayt\n0(t−s)p′(2+s)(N−β\np′−1\nq−1\np)p′−1ds/parenrightigg1\np′\n≤C2/parenleftigg/integraldisplayt\n0(2+s)Gβ(s)ds/parenrightigg1\np/bracketleftbigg\n(2+t)N−β\np′−1\np+1\np′+(2+t)1+N−β\np′−1\nq−1\np/bracketrightbigg\n≤2C2/parenleftigg/integraldisplayt\n0(2+s)Gβ(s)ds/parenrightigg1\np\n(2+t)1+p−1−λ\np.\nWe see from the definition of Hβthat\n(2C2)−p(2+t)1+λ−2p/parenleftigg\nεEβ,0t+εEβ,1\n2t2/parenrightiggp\n≤H′\nβ(t).\nHence\nH′\nβ(t)≥C4εp(2+t)1+λ,t≥1.\n15Integrating it over [0 ,t], we have for t≥2,\nHβ(t)≥/integraldisplayt\n0H′\nβ(s)ds≥/integraldisplayt\n1H′\nβ(s)ds≥C4εp/integraldisplayt\n1(2+s)λds≥C4εp\n4(2+λ)(2+t)2+λ.\nWe see from the definition of Iβthat for t≥2,\nJ′\nβ(t)=(2+s)−3Hβ(t)≥C4εp\n4(2+λ)(2+t)−1+λ.\nand for t≥4,\nJβ(t)=/integraldisplayt\n2J′\nβ(s)ds≥C4εp\n8λ(2+λ)(2+t)λ.\nOn the other hand, we see from Lemma 4.3 that\n(2C2)−p[Jβ(t)]p≤(2+t)2−λJ′′\nβ(t)+3(2+t)1−λJ′\nβ(t)].\nMoreover, setting Jβ(t)=/tildewideJβ(σ),σ=2\nλ(2+t)λ\n2, we see\n(2+t)1−λ\n2J′\nβ(t)=/tildewideJ′\nβ(σ),(2+t)2−λJ′′\nβ(t)+2−λ\n2(2+t)1+λJ′\nβ(t)=/tildewideJ′′\nβ(σ).\nThen\nC−p\n5[/tildewideJβ(σ)]p≤/tildewideJ′′\nβ(σ)+4+λ\nλσ−1/tildewideJ′\nβ(σ), σ≥σ0=2\nλ2λ\n2,\n/tildewideJ′\nβ(σ)≥C6εpσ, σ≥σ1=2\nλ4λ\n2,\n/tildewideJβ(σ)≥C6εpσ2, σ≥σ2=2\nλ6λ\n2.\nConsequently, by Lemma 4.1 (i)we see that /tildewideJβblows up before C7ε−p−1\n2and then, Jβblows up before\nC7ε−p−1\nλ. By virtue of Lemma 4.2, we have LifeSpan( uε)≤C7ε−p−1\nλ.\nFinally, we remark that if ( p,V0)∈Ω1, then we can take 1 /q=1/p−δfor arbitrary small δ>0 and\nthenλ=γ(N+V0;p)/(2p)−1\np+1\nq=γ(N+V0;p)/(2p)−δ. This implies that\nLifeSpan u≤C7ε−2p(p−1)\nγ(N+V0;p)−δ′\n.\nfor arbitrary small δ′>0. The proof is complete. /square\n4.3 Proof of Theorem 1.2 for critical case p=p0(N+V0)\nProof. In this case we set\nβδ=N−1−V0\n2−1\np+δ∈/parenleftigg\n0,N−1−V0\n2/parenrightigg\n.\nThen by Lemma 4.5 (ii)withβ=βδ,\nεEβ,0+εEβ,2t≤C1/bracketleftigg\n/ba∇dblu(t)/ba∇dblLp(2+t)N\np′−β+/integraldisplayt\n0/ba∇dblu(s)/ba∇dblLp(2+s)N\np′−β0−1ds/bracketrightigg\n≤K1/bracketleftigg/parenleftig\nGβ0(t)/parenrightig1\np(2+t)N−β0\np′+(β0−β)+/integraldisplayt\n0/parenleftig\nGβ0(t)/parenrightig1\np(2+s)N−β0\np′−1ds/bracketrightigg\n.\n16Noting thatN−β0\np′=1+1\npand integrating it over [0 ,t], we have\nεEβδ,0t+εEβδ,1\n2t2≤K1/bracketleftigg/integraldisplayt\n0/parenleftig\nGβ0(s)/parenrightig1\np(2+s)1+1\np+(β0−βδ)ds+/integraldisplayt\n0(t−s)/parenleftig\nGβ0(t)/parenrightig1\np(2+s)1\npds/bracketrightigg\n≤K1/parenleftigg/integraldisplayt\n0Gβ0(s)(1+s)ds/parenrightigg1\np/parenleftigg/integraldisplayt\n0(2+s)p′+(β0−βδ)p′ds/parenrightigg1\np′\n+/parenleftigg/integraldisplayt\n0(t−s)p′ds/parenrightigg1\np′\n≤K2/parenleftigg/integraldisplayt\n0Gβ0(s)(1+s)ds/parenrightigg1\np\n(2+t)1+1\np′.\nBy the definition of Hβ0, we have for t≥1,\nH′\nβ0(t)≥K−p\n2εp/parenleftigg\nEβδ,0t+Eβδ,1\n2t2/parenrightiggp\n(2+t)1−2p≥K3εp(2+t)\nand then for t≥2,\nHβ0(t)≥/integraldisplayt\n1˜G′\nβ0(s)ds≥K4εp(2+t)2.\nOn the other hand, by Lemma 4.5 (ii)we have\n/integraldisplayt\n0(t−s)Gβ0(s)ds≤C1/bracketleftigg\n/ba∇dblu(t)/ba∇dblLp(2+t)N\np′−β0+/integraldisplayt\n0/ba∇dblu(s)/ba∇dblLp(2+s)N\np′−β0−1(log(2+s))1\np′ds/bracketrightigg\n.\nNotingN−β0\np′=1+1\npagain and integrating it over [0 ,t], we have\n1\n2/integraldisplayt\n0(t−s)Gβ0(s)ds\n≤K′\n1/bracketleftigg/integraldisplayt\n0Gβ0(s)1\np(2+s)N−β0\np′ds+/integraldisplayt\n0(t−s)Gβ0(s)1\np(2+s)N−β0\np′−1(log(2+s))1\np′ds/bracketrightigg\n≤K′\n1H′\nβ0(t)1\np/bracketleftigg/integraldisplayt\n0(2+s)p′ds+/integraldisplayt\n0(t−s)p′log(2+s)ds/bracketrightigg\n≤K′\n2H′\nβ0(t)1\np(2+t)1+1\np′(log(2+t))1\np′.\nAs in the proof of subcritial case, we deduce\n(K′\n2)−p(log(2+t))1−pJβ0(t)p≤H′\nβ0(t)(1+t)−1\n≤(2+t)2J′′\nβ0(t)+3(2+t)J′\nβ0(t).\nHere we take Jβ0(t)=/tildewideJβ0(σ) withσ=log(2+t). Since\n(2+t)J′\nβ0(t)=/tildewideJ′\nβ0(σ),(2+t)2J′\nβ0(t)+(2+t)J′′\nβ0(t)=/tildewideJ′′\nβ0(σ),\nwe obtain for σ≥σ0:=log 2,\n(K′\n2)−pσ1−p/tildewideJβ0(σ)p≤/tildewideJ′′\nβ0(σ)+2/tildewideJ′\nβ0(σ).\n17Moreover, we have for σ≥σ1=log 4,\n/tildewideJ′\nβ0(σ)=(2+t)J′\nβ0(t)\n=(2+t)−2Hβ0(t)\n≥K4εp\nand therefore for σ≥σ2=2 log 4,\n/tildewideJβ0(σ)≥K4\n2εpσ.\nApplying Lemma 4.1 (ii)we deduce that /tildewideJβ0blows up before σ=K5ε−p(p−1). Then by definition Jβ0\nblows up before exp[ K5ε−p(p−1)]. Consequently, using Lemma 4.2, we obtain\nLifeSpan u≤exp[K5ε−p(p−1)].\nThe proof is complete. /square\nRemark 4.1.In particular. in the proof of Theorem 1.2 with p=p(N+V0), we have used two kind of\nauxiliary parameters 1 /q=N−1−V0\n2−1\npand 1/q=N−1−V0\n2−1\np+δ. The first choice is for deriving lower\nbound of the functional Jβ0and the second is for deriving di fferential inequality for Jβ0. The first choice\nis essentially different from the idea of Yordanov–Zhang [ 11] to prove the lower bound of a functional.\nAcknowedgements\nThis work is partially supported by Grant-in-Aid for Young S cientists Research (B) No.16K17619 and\nby Grant-in-Aid for Young Scientists Research (B) No.15K17 571.\nReferences\n[1] R. Beals, R. Wong, “Special functions,” A graduate text. Cambridge Studies in Advanced Mathe-\nmatics 126, Cambridge University Press, Cambridge, 2010.\n[2] S. Di Pomponio, V . Georgiev, Life-span of subcritical semilinear wave equation , Asymptot. Anal.\n28(2001), 91–114.\n[3] M. Ikeda, T. Ogawa, Lifespan of solutions to the damped wave equation with a crit ical nonlinearity ,\nJ. Differential Equations 261(2016), 1880–1903.\n[4] R. Ikehata, G. Todorova, B. Yordanov, Critical exponent for semilinear wave equations with space -\ndependent potential , Funkcial. Ekvac. 52(2009), 411–435.\n[5] F. John, Blow-up of solutions of nonlinear wave equations in three sp ace dimensions , Manuscripta\nMath. 28(1979), 235–268.\n[6] N.A. Lai, H. Takamura, K. Wakasa, Blow-up for semilinear wave equations with the scale invari ant\ndamping and super-Fujita exponent , J. Differential Equations 263(2017), 5377–5394.\n[7] A. Pazy, A. “Semigroups of linear operators and applicat ions to partial differential equations,” Ap-\nplied Mathematical Sciences 44, Springer-Verlag, New York, 1983.\n18[8] T.C. Sideris, Nonexistence of global solutions to semilinear wave equati ons in high dimensions , J.\nDifferential Equations 52(1984), 378–406.\n[9] W.A. Strauss, Nonlinear scattering theory at low energy , J. Funct. Anal. 41(1981), 110–133.\n[10] H. Takamura, K. Wakasa, The sharp upper bound of the lifespan of solutions to critica l semilinear\nwave equations in high dimensions , J. Differential Equations 251(2011), 1157–1171.\n[11] B. Yordanov, Q.S. Zhang, Finite time blow up for critical wave equations in high dimen sions , J.\nFunct. Anal. 231(2006), 361–374.\n[12] Y . Zhou, Blow up of solutions to semilinear wave equations with criti cal exponent in high dimen-\nsions , Chin. Ann. Math. Ser. B 28(2007), 205–212.\n[13] Y . Zhou, W. Han, Life-span of solutions to critical semilinear wave equatio ns, Comm. Partial Dif-\nferential Equations 39(2014), 439–451.\n19"    },    {        "title": "1807.06164v1.On_the_blow_up_for_critical_semilinear_wave_equations_with_damping_in_the_scattering_case.pdf",        "content": "arXiv:1807.06164v1  [math.AP]  17 Jul 2018ON THE BLOW-UP FOR CRITICAL SEMILINEAR WAVE\nEQUATIONS WITH DAMPING IN THE SCATTERING CASE\nKYOUHEI WAKASA AND BORISLAV YORDANOV\nAbstract. We consider the Cauchy problem for semilinear wave equation s\nwith variable coefficients and time-dependent scattering da mping in Rn, where\nn≥2. It is expected that the critical exponent will be Strauss’ number p0(n),\nwhich is also the one for semilinear wave equations without d amping terms.\nLai and Takamura [7] have obtained the blow-up part, togethe r with the\nupper bound of lifespan, in the sub-critical case p < p0(n). In this paper, we\nextend their results to the critical case p=p0(n). The proof is based on [16],\nwhich concerns the blow-up and upper bound of lifespan for cr itical semilinear\nwave equations with variable coefficients.\n1.Introduction\nWe study the blow-up problem for critical semilinar wave equations wit h vari-\nable coefficients and scattering damping depending on time. The pert urbations of\nLaplacian are uniformly elliptic operators\n∆g=n/summationdisplay\ni,j=1∂xigij(x)∂xj\nwhose coefficients satisfy, with some α >0,the following:\n(1.1) gij∈C1(Rn),|∇gij(x)|+|gij(x)−δij|=O(e−α|x|) as|x| → ∞.\nThe admissible damping coefficients are a∈C([0,∞)), such that\n(1.2) ∀t≥0a(t)≥0 and/integraldisplay∞\n0a(t)dt <∞.\nForn≥2 andp >1, we consider the Cauchy problem\nutt−∆gu+a(t)ut=|u|p, x∈Rn, t >0, (1.3)\nu|t=0=εu0, ut|t=0=εu1, x∈Rn, (1.4)\nwhereu0, u1∈C∞\n0(Rn) andε >0 is a small parameter. Our results concern only\nthe critical case p=p0(n) with Strauss’ exponent defined in (1.5) below.\nLet us briefly review previous results concerning (1.3) with gij=δijand various\ntypes of damping a. Whena(t) = 1, Todorova and Yordanov [13] showed that the\nsolution of (1.3) blows up in finite time if 1 < p < p F(n), where pF(n) = 1+2 /n\nis the Fujita exponent known to be the critical exponent for the se milinear heat\nequation. The same work also obtained global existence for p > pF(n). Finally,\nZhang [20] established the blow-up in the critical case p=pF(n).\nThe other typical example of effective damping is a(t) =µ/(1+t)βwithµ >0\nandβ∈R. When−1< β <1, Lin, Nishihara and Zhai [9] obtained the expected\nDate: November 15, 2021.\n12 KYOUHEI WAKASA AND BORISLAV YORDANOV\nblow-up result, if 1 < p≤pF(n), and global existence result, if p > pF(n); see also\nD’Abbicco, S.Lucente and M.Reissig [2].\nIn the caseof critical decay β= 1, there areseveralworksabout finite time blow-\nup and global existence. Wakasugi [17] showed the blow-up, if 1 < p≤pF(n) and\nµ >1 or 1< p≤1+2/(n+µ−1) and 0 < µ≤1. Moreover, D’Abbicco [1] verified\nthe global existence, if p > pF(n) andµsatisfies one of the following: µ≥5/3 for\nn= 1,µ≥3 forn= 2 and µ≥n+2 forn≥3. An interesting observation is that\nthe Liouville substitution w(x,t) := (1+ t)µ/2u(x,t) transforms the damped wave\nequation (1.3) into the Klein-Gordon equation\nwtt−∆w+µ(2−µ)\n4(1+t)2w=|w|p\n(1+t)µ(p−1)/2.\nThus, one expects that the critical exponent for µ= 2 is related to that of the semi-\nlinear wave equation. D’Abbicco, Lucente and Reissig [3] have actua lly obtained\nthe corresponding blow-up result, if 1 < p < p c(n) := max {pF(n), p0(n+2)}and\n(1.5) p0(n) :=n+1+√\nn2+10n−7\n2(n−1)\nis the so-called Strauss exponent, the positive root of the quadra tic equation\n(1.6) γ(p,n) := 2+( n+1)p−(n−1)p2= 0.\nTheir work also showed the existence of global classical solutions fo r smallε >0, if\np > pc(n) and either n= 2 orn= 3 and the data are radially symmetric. Finally,\nwe mention that our original equations (1.3) is related to semilinear wa ve equations\nin the Einstein-de Sitter spacetime considered by Galstian & Yagdjian [4].\nWe recall that p0(n) in (1.5) is the critical exponent for the semilinear wave\nequation conjectured by Strauss [11]. The hypothesis has been ve rified in several\ncases; see [16] and the references therein. A related problem is to estimate the\nlifespan, or the maximal existence time Tεof solutions to (1.3), (1.4) in the energy\nspaceC([0,Tε),H1(Rn))∩C1([0,Tε),L2(Rn)).\nLai, Takamura and Wakasa [8] have obtained the blow-up part of St rauss’ con-\njecture, together with an upper bound of the lifespan Tε, for (1.3), (1.4) in the\ncasen≥2, 0< µ <(n2+n+ 2)/2(n+ 2) and pF(n)≤p < p0(n+ 2µ). Later,\nIkeda and Sobajima [5] were able to replace these conditions by less restrictive\n0< µ <(n2+n+2)/(n+2) and pF(n)≤p≤p0(n+µ). In addition, they have\nderived an upper bound on the lifespan. Tu and Lin [14], [15] have impro ved the\nestimates of Tεin [5] recently.\nForβ≤ −1, the long time behavior of solutions to (1.3), (1.4) is quite different.\nWhenβ=−1, Wakasugi [18] has obtained the global existence for exponents\npF(n)< p < n/ [n−2]+, where\n[n−2]+:=/braceleftbigg∞ forn= 1,2,\nm/(n−2) for n≥3.\nIkeda and Wakasugi [6] have proved that the global existence ac tually holds for any\np >1 whenβ <−1.\nForβ >1, we expect the critical exponent to be exactly the Strauss expo nent.\nIn fact, Lai and Takamura [7] have shown that certain solutions of (1.3), (1.4) blow\nup in finite time when 1 < p < p 0(n). Moreover, Liu and Wang [10] have just\nobtained the global existence results for n= 3,4 andp > p0(n) on asymptotically\nEuclidean manifolds.3\nIfTεdenotes the lifespan of these solutions, then [7] have also given the upper\nboundTε≤Cε−2p(p−1)/γ(p,n)forn≥2 and 1< p < p 0(n). This result is probably\nsharp, since Takamura [12] proved the same type of estimate in the sub-critical case\nof Strauss’ conjecture for semilinear wave equations without dam ping. However,\nboth the conjecture and lifespan bound remained open problems in t he critical case\np=p0(n).\nThe purpose of this paper is to verify the blow-up for p=p0(n) and to give a\nproof that extends to more general damping, including a(t)∼(1+t)−βwithβ >1.\nWe also succeed to derive an exponential type upper bound on the lif espanTε,\nwhich is the same as that of the Strauss conjecture in the conserv ative critical case.\nSuch results are consistent with our knowledge of the linear problem corresponding\nto (1.3), (1.4); Wirth [19] has shown that energy space solutions sc atter, that is\napproach solutions to the free wave equations, as t→ ∞.\nTheorem 1.1. Letn≥2,p=p0(n)anda(t)satisfy (1.2). Assume that both\nu0∈H1(Rn)andu1∈L2(Rn)are nonnegative, do not vanish identically and have\nsupports in the ball {x∈Rn:|x| ≤R0}, whereR0>1.\nIf (1.3) has a solution (u,ut)∈C([0,Tε),H1(Rn)×L2(Rn)), such that\n(1.7) supp( u,ut)⊂ {(x,t)∈Rn×[0,Tε) :|x| ≤t+R},\nwithR≥R0, thenTε<∞. Moreover, there exist constants ε0=ε0(u0,u1,n,p,R,a )\nandK=K(u0,u1,n,p,R,a ), such that\n(1.8) Tε≤exp/parenleftBig\nKε−p(p−1)/parenrightBig\nfor0< ε≤ε0.\nRemark 1.2.The lifespan estimates is the same as that of the Strauss conjectu re\nin the critical case of semilinear wave equations without damping. For details, see\nthe introduction in [16]. We also note that Liu and Wang [10] have obta ined the\nsharp lower bound of the lifespan, Tε≥exp(cε−2) ifn= 4 and p=p0(4) = 2.\nOur proof is based on the approach of Wakasa and Yordanov [16]. Av eraging the\nsolutionwith respecttoasuitabletestfunction, wederiveasecond -orderdissipative\nODE which corresponds to equation (1.3). The key point is to establis h lower\nbounds for the fundamental system of solutions to this ODE; see L emma 2.3. As a\nconsequence, we can follow [16] and obtain the same nonlinear integ ral inequality.\nThe final blow-up argument also repeats the iteration argument of [16].\n2.Test Functions\nSimilarly to the proof of [16], we first consider the following elliptic prob lem:\n(2.1) ∆ gϕλ=λ2ϕλ, x∈Rn,\nwhereλ∈(0,α/2].Asλ|x| → ∞, theseϕλ(x) are asymptotically given by ϕ(λx),\nwithϕbeing the standard radial solution to the unperturbed equation ∆ ϕ=ϕ:\n(2.2) ϕ(x) =/integraldisplay\nSn−1ex·ωdSω∼cn|x|−(n−1)/2e|x|,|x| → ∞.\nWe recall the following result about the existence and main propertie s ofϕλ.\nLemma 2.1. Letn≥2. There exists a solution ϕλ∈C∞(Rn)to (2.1), such that\n(2.3) |ϕλ(x)−ϕ(λx)| ≤Cαλθ, x∈Rn, λ∈(0,α/2],\nwhereθ∈(0,1]andϕ(x) =/integraltext\nSn−1ex·ωdSω∼cn|x|−(n−1)/2e|x|, cn>0,as|x| → ∞.4 KYOUHEI WAKASA AND BORISLAV YORDANOV\nMoreover, ϕλ(·)−ϕ(λ·)is a continuous L∞(Rn)valued function of λ∈(0,α/2]\nand there exist positive constants D0, D1andλ0, such that\n(2.4) D0/an}b∇acketle{tλ|x|/an}b∇acket∇i}ht−(n−1)/2eλ|x|≤ϕλ(x)≤D1/an}b∇acketle{tλ|x|/an}b∇acket∇i}ht−(n−1)/2eλ|x|, x∈Rn,\nholds whenever 0< λ≤λ0.\nProof.See Lemma 2.2 in [16]. /square\nGivenλ0∈(0,α/2] andq >0, we also introduce the auxiliary functions\nξq(x,t) =/integraldisplayλ0\n0e−λ(t+R)coshλtϕλ(x)λqdλ, (2.5)\nηq(x,t,s) =/integraldisplayλ0\n0e−λ(t+R)sinhλ(t−s)\nλ(t−s)ϕλ(x)λqdλ, (2.6)\nfor (x,t)∈Rn×Rands∈R.Useful estimates are collected in the next lemma.\nLemma 2.2. Letn≥2. There exists λ0∈(0,α/2], such that the following hold:\n(i) if0< q,|x| ≤Rand0≤t, then\nξq(x,t)≥A0,\nηq(x,t,0)≥B0/an}b∇acketle{tt/an}b∇acket∇i}ht−1;\n(ii) if0< q,|x| ≤s+Rand0≤s < t, then\nηq(x,t,s)≥B1/an}b∇acketle{tt/an}b∇acket∇i}ht−1/an}b∇acketle{ts/an}b∇acket∇i}ht−q;\n(iii) if(n−3)/2< q,|x| ≤t+Rand0< t, then\nηq(x,t,t)≤B2/an}b∇acketle{tt/an}b∇acket∇i}ht−(n−1)/2/an}b∇acketle{tt−|x|/an}b∇acket∇i}ht(n−3)/2−q.\nHereA0andBk,k= 0,1,2,are positive constants depending only on α,qandR,\nwhile/an}b∇acketle{ts/an}b∇acket∇i}ht= 3+|s|.\nProof.See Lemma 3.1 in [16]. /square\nThe following lemma plays a key role in the proof of Theorem1.1.\nLemma 2.3. Letλ >0and introduce the ordinary differential operators\nLa=∂2\nt+a(t)∂t−λ2, L∗\na=∂2\ns−∂sa(s)−λ2.\nThe fundamental system of solutions {y1(t,s;λ),y2(t,s;λ)}, defined through\nLay1(t,s;λ) = 0, y 1(s,s;λ) = 1, ∂ty1(s,s;λ) = 0,\nLay2(t,s;λ) = 0, y 2(s,s;λ) = 0, ∂ty2(s,s;λ) = 1,\ndepends continuously on λand satisfies the following estimates, for t≥s≥0:\n(i)y1(t,s;λ)≥e−/bardbla/bardblL1coshλ(t−s),\n(ii)y2(t,s;λ)≥e−2/bardbla/bardblL1sinhλ(t−s)\nλ.\nMoreover, the conjugate equations and initial conditions h old:\n(iii)L∗\nay2(t,s;λ) = 0,\n(iv)y1(t,0;λ) =a(0)y2(t,0;λ)−∂sy2(t,0;λ).\nProof.See Section 4. /square5\n3.Proof of Theorem 1.1\nLetube a weak solution to problem (1.3), defined below, and ηq(x,t,t) be a test\nfunction, defined in Section 2, with q >−1. We will show that\n(3.1) F(t) =/integraldisplay\nRnu(x,t)ηq(x,t,t)dx\nsatisfies a nonlinear integral inequality which implies finite time blow-up. Our\ndefinition of weak solutions is standard: ( u,ut)∈C([0,Tε),H1(Rn)×L2(Rn)) and\n∀φ∈C∞\n0(Rn×[0,Tε)) andt∈(0,Tε)\n/integraldisplay\nus(x,t)φ(x,t)dx−/integraldisplay\nus(x,0)φ(x,0)dx\n−/integraldisplayt\n0/integraldisplay\n(us(x,s)φs(x,s)−g(x)∇u(x,s)·∇φ(x,s)−a(s)us(x,s)φ(x,s))dxds\n=/integraldisplay∞\n0/integraldisplay\n|u(x,s)|pφ(x,s)dxds.\nIn the next result, however, it will be more convenient to work with\n/integraldisplay\n(us(x,t)φ(x,t)−u(x,t)φs(x,t)+a(t)u(x,t)φ(x,t))dx\n−/integraldisplay\n(us(x,0)φ(x,0)−u(x,0)φs(x,0)+a(0)u(x,0)φ(x,0))dx (3.2)\n+/integraldisplayt\n0/integraldisplay\nu(x,s)(φss(x,s)−∆gφ(x,s)−(a(s)φ(x,s))s)dxds\n=/integraldisplay∞\n0/integraldisplay\n|u(x,s)|pφ(x,s)dxds,\nwhich follows from integration by parts. We can also use φ∈C∞(Rn×[0,Tε)),\nsinceu(·,s) is compactly supported for every s.\nProposition 3.1. Let the assumptions in Theorem 1.1 be fulfilled and q >−1.\n(3.3)/integraldisplay\nRnu(x,t)ηq(x,t,t)dx\n≥εe−/bardbla/bardblL1/integraldisplay\nRnu0(x)ξq(x,t)dx+εe−2/bardbla/bardblL1t/integraldisplay\nRnu1(x)ηq(x,t,0)dx\n+e−2/bardbla/bardblL1/integraldisplayt\n0(t−s)/integraldisplay\nRn|u(x,s)|pηq(x,t,s)dxds\nfor allt∈(0,Tε).\nProof.We will apply (3.2) to φ(x,s) =ϕλ(x)y2(t,s;λ), which satisfies\nφss(x,s)−∆gφ(x,s)−(a(s)φ(x,s))s= 0,\na(0)φ(x,0)−φs(x,0) =ϕλ(x)y1(t,0;λ),6 KYOUHEI WAKASA AND BORISLAV YORDANOV\nfrom Lemma 2.3 ( iii) and (iv), respectively. Then we obtain\n/integraldisplay\nu(x,t)ϕλ(x)dx=εy1(t,0;λ)/integraldisplay\nu0(x)ϕλ(x)dx\n+εy2(t,0;λ)/integraldisplay\nu1(x)ϕλ(x)dx\n+/integraldisplayt\n0y2(t,s;λ)/parenleftbigg/integraldisplay\n|u(x,s)|pϕλ(x)dx/parenrightbigg\nds,\nwhere the initial conditions are determined by (1.4) and the pair {y1,y2}is defined\nin Lemma 2.3. Making use of estimates ( i) and (ii) in this lemma, we have that\n/integraldisplay\nu(x,t)ϕλ(x)dx≥εe−/bardbla/bardblL1cosh(λt)/integraldisplay\nu0(x)ϕλ(x)dx\n+εe−2/bardbla/bardblL1sinhλt\nλ/integraldisplay\nu1(x)ϕλ(x)dx\n+e−2/bardbla/bardblL1/integraldisplayt\n0sinhλ(t−s)\nλ/parenleftbigg/integraldisplay\n|u(x,s)|pϕλ(x)dx/parenrightbigg\nds.\nThe lower bound (3.3) follows from multiplying the aboveinequality by λqe−λ(t+R),\nintegrating on [0 ,λ0] and interchanging the order of integration between λandx.\nRecalling definitions (2.5) and (2.6) for ξqandηq, we complete the proof. /square\nSimilarly to the proof of Proposition 4.2. in [16], we obtain the convenien t\niteration frame by using Lemma 2.2.\nProposition 3.2. Suppose that the assumptions in Theorem 1.1 are fulfilled and\nchooseq= (n−1)/2−1/p.IfF(t)is defined in (3.1), there exists a positive constant\nC=C(n,p,q,R,a ), such that\n(3.4) F(t)≥C\n/an}b∇acketle{tt/an}b∇acket∇i}ht/integraldisplayt\n0t−s\n/an}b∇acketle{ts/an}b∇acket∇i}htF(s)p\n(log/an}b∇acketle{ts/an}b∇acket∇i}ht)p−1ds\nfor allt∈(0,Tε).\nThe finite time blow-up and lifespan estimate (1.8) can now be derived f ollowing\nSections 4 and 5 in [16].\n4.Proof of Lemma 2.3\nLet us recall that λ >0 andLa= (d/dt)2+a(t)d/dt−λ2.There exists a pair of\nC2-solutions {y1(t,s;λ),y2(t,s;λ)}which depends continuously on λand satisfies\nLay1= 0, y 1(s,s;λ) = 1, y′\n1(s,s;λ) = 0,\nLay2= 0, y 2(s,s;λ) = 0, y′\n2(s,s;λ) = 1,\nfort≥s≥0.We will show that {y1(t,s;λ),y2(t,s;λ)}behaves similarly to the\nfundamental system of L0, that is {coshλ(t−s),λ−1sinhλ(t−s)}, ast−s→ ∞7\nandλ→0.Our proof gives two-sided bounds and relies only on three identities:\n(y′\n1eA(t))′=λ2y1eA(t),whereA(t) =/integraldisplayt\n0a(r)dr, (4.1)\n/parenleftbigg\ny1eA(t)−/integraldisplayt\nsa(r)y1eA(r)dr/parenrightbigg′′\n=λ2y1eA(t), (4.2)\ny′\n2y1−y2y′\n1=eA(s)−A(t)or/parenleftbiggy2\ny1/parenrightbigg′\n=eA(s)−A(t)\ny2\n1. (4.3)\nTo verify claim ( i), we observe that y1(t0,s;λ) = 0 at some t0> sleads to a\ncontradiction: if t0is the first such number, then y1(t,s;λ)≥0 fort∈[s,t0] and\n(4.1) imply that\ny′\n1(t,s;λ)eA(t)=λ2/integraldisplayt\nsy1(r,s;λ)eA(r)dr≥0,soy′\n1(t,s;λ)≥0 fort∈[s,t0].\nHence,y1(t,s;λ) is increasing on [ s,t0] and 0 = y1(t0,s;λ)≥y1(s,s;λ) = 1 can\nnot hold. The positivity of y1(t,s;λ) also yields, through (4.1), the positivity of its\nderivative: y′\n1(t,s;λ)≥0 for all t≥s.\nWe can now derive an upper bound on y1using that y′′\n1=λ2y1−ay′\n1≤λ2y1\nandy1(s,s;λ) = 1,y′\n1(s,s;λ) = 0:\n(4.4) y1(t,s;λ)≤coshλ(t−s), t≥s.\nThe lower bound on y1is a consequence of (4.2) and the positivity of y1(t,s;λ):\n/parenleftbigg\ny1eA(t)−/integraldisplayt\nsa(r)y1eA(r)dr/parenrightbigg′′\n≥λ2/parenleftbigg\ny1eA(t)−/integraldisplayt\nsa(r)y1eA(r)dr/parenrightbigg\n.\nCombining this inequality with the initial values at t=s,\n/parenleftbigg\ny1eA(t)−/integraldisplayt\nsa(r)y1eA(r)dr/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=s=eA(s),\nd\ndt/parenleftbigg\ny1eA(t)−/integraldisplayt\nsa(r)y1eA(r)dr/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=s= 0,\nwe obtain that\ny1(t,s;λ)eA(t)−/integraldisplayt\nsa(τ)y1eA(τ)dτ≥eA(s)coshλ(t−s).\nAfter simplifying,\n(4.5) y1(t,s;λ)≥eA(s)−A(t)coshλ(t−s), t≥s.\nSinceA(s)−A(t)≥ −/ba∇dbla/ba∇dblL1and (4.4) holds, claim ( i) follows:\n(4.6) cosh λ(t−s)≥y1(t,s;λ)≥e−/bardbla/bardblL1coshλ(t−s).\nTo check claim ( ii), we combine (4.4), (4.5) and identity (4.3):\neA(t)−A(s)\ncosh2λ(t−s)≥/parenleftbiggy2\ny1/parenrightbigg′\n≥eA(s)−A(t)\ncosh2λ(t−s).\nUsingA(s)−A(t)≥ −/ba∇dbla/ba∇dblL1and integration on [ s,t], we have that\ne/bardbla/bardblL1\nλtanhλ(t−s)≥y2(t,s;λ)\ny1(t,s;λ)≥e−/bardbla/bardblL1\nλtanhλ(t−s).8 KYOUHEI WAKASA AND BORISLAV YORDANOV\nThe final result follows from (4.6):\ne/bardbla/bardblL1sinhλ(t−s)\nλ≥y2(t,s;λ)≥e−2/bardbla/bardblL1sinhλ(t−s)\nλ.\nFinally, we will show the equalities in ( iii) and (iv).Sety1(t) :=y1(t,0;λ) and\ny2(t) :=y2(t,0;λ). It easy to see that\ny2(t,s,λ) =y1(t)y2(s)−y1(s)y2(t)\ny′\n1(s)y2(s)−y1(s)y′\n2(s)= (y1(s)y2(t)−y1(t)y2(s))eA(s).\nThus, we can calculate ∂i\nsy2(t,s,λ) withi= 1,2 as follows:\n∂sy2(t,s;λ) = (y′\n1(s)y2(t)−y1(t)y′\n2(s))eA(s)+a(s)y2(t,s;λ),\n(4.7)\n∂2\nsy2(t,s;λ) = (y′′\n1(s)y2(t)−y1(t)y′′\n2(s))eA(s)\n+a(s)(y′\n1(s)y2(t)−y1(t)y′\n2(s))eA(s)+∂s(a(s)y2(t,s;λ)).\nNoticing that yi(s) withi= 1,2 satisfy the differential equation Layi(s) = 0, we\nget (iii).\nTo derive ( iv), we just set s= 0 in (4.7) for ∂sy2(t,s;λ) and use the initial\nconditions for yi(s). The proof is complete.\nReferences\n[1] M.D’Abbicco, The threshold of effective damping for semilinear wave equat ions, Mathemat-\nical Methods in Applied Sciences., 38(2015) 1032-1045.\n[2] M.D’Abbicco, S.Lucente and M.Reissig, Semi-linear wave equations with effective damping ,\nChin. Ann. Math. Ser. B, 34(2013), 345-380.\n[3] M.D’Abbicco, S.Lucente and M.Reissig, A shift in the Strauss exponent for semilinear wave\nequations with a not effective damping , J. Differential Equations., 259(2015), 5040-5073.\n[4] A.Galstian and K.Yagdjian, Finite lifespan of solutions of the semilinear wave equatio n in\nthe Einstein-de Sitter spacetime , preprint, arXiv:1612.09536v2.\n[5] M.Ikeda and M.Sobajima, Life-span of solutions to semilinear wave equation with tim e-\ndependent critical damping for specially localized initia l data, Mathematische Annalen,\n(2018), 1-24.\n[6] M.Ikeda and Y.Wakasugi, Global well-posedness for the semilinear wave equation wit h time\ndependent damping in the overdamping case , accepted for publication in Proceedings of the\nAmerican Mathematical Society, arXiv:1708.08044v2.\n[7] N.-A.,Lai and H.Takamura, Blow-up for semilinear damped wave equations with sub-Stra uss\nexponent in the scattering case , Nonlinear Analysis TMA 168(2018), 222-237.\n[8] N.-A.Lai, H.Takamura and K. Wakasa, Blow-up for semilinear wave equations with the scale\ninvariant damping and super-Fujita exponent , J. Differential Equations, 263(2017), 5377-\n5394.\n[9] J.Lin, K.Nishihara and J.Zhai, Critical exponent for the semilinear wave equation with tim e-\ndependent damping , Discrete and Continuous Dynamical Systems - Series A., 32(2012),\n4307-4320.\n[10] M.Liu and C.Wang Global existence of semilinear damped wave equations in rel ation with\nthe Strauss conjecture , preprint, arXiv:1807.05908.\n[11] W.A.Strauss, Nonlinear scattering theory at low energy , J. Funct. Anal., 41(1981), 110-133.\n[12] H.Takamura, Improved Kato’s lemma on ordinary differential inequality a nd its application\nto semilinear wave equations , Nonlinear Analysis TMA 125(2015), 227-240.\n[13] G.Todorova and B.Yordanov, Critical exponent for a nonlinear wave equation with dampin g,\nJ. Differential Equations, 174(2001) 464-489.\n[14] Z.Tu and J.Lin, A note on the blowup of scale invariant damping wave equation with sub-\nStrauss exponent , preprint, arXiv:1709.00866.\n[15] Z.Tu and J.Lin, Life-Span of Semilinear Wave Equations with Scale-invaria nt Damping:\nCritical Strauss Exponent Case , preprint, arXiv:1711.00223.9\n[16] K.Wakasa and B.Yordanov, Blow-up of solutions to critical semilinear wave equations with\nvariable coefficients , preprint, arXiv:1807.02772.\n[17] Y.Wakasugi, Critical exponent for the semilinear wave equation with sca le invariant damping ,\nFourier analysis, 375-390, Trends Math., Birkh¨ auser/Spr inger, Cham, (2014).\n[18] Y.Wakasugi, Scaling variables and asymptotic profiles for the semilinea r damped wave equa-\ntion with variable coefficients , J. Math. Anal. Appl., 447(2017), 452-487.\n[19] J.Wirth, Wave equations with time-dependent dissipation. I. Non-eff ective dissipation , J.\nDifferential Equations, 222(2006), 487-514.\n[20] Q.S.Zhang, A blow-up result for a nonlinear wave equation with damping: the critical case ,\nC. R. Math. Acad. Sci. Paris, S´ er. I 333 (2001) 109-114.\nDepartment of Mathematics, Faculty of Science and Technolo gy, Tokyo University\nof Science, 2641 Yamazaki, Noda-shi, Chiba, 278-8510, Japan\nOffice of International Affairs, Hokkaido University, Kita 15 , Nishi 8, Kita-ku,\nSapporo, Hokkaido 060-0815, Japan and Institute of Mathemat ics, Sofia\nE-mail address :wakasakyouhei@ma.noda.tus.ac.jp\nE-mail address :byordanov@oia.hokudai.ac.jp"    },    {        "title": "1507.08227v2.Spin_dynamics_and_relaxation_in_the_classical_spin_Kondo_impurity_model_beyond_the_Landau_Lifschitz_Gilbert_equation.pdf",        "content": "Spin dynamics and relaxation in the classical-spin Kondo-impurity model beyond the\nLandau-Lifschitz-Gilbert equation\nMohammad Sayad and Michael Pottho\u000b\nI. Institut f ur Theoretische Physik, Universit at Hamburg, Jungiusstra\u0019e 9, 20355 Hamburg, Germany\nThe real-time dynamics of a classical spin in an external magnetic \feld and locally exchange\ncoupled to an extended one-dimensional system of non-interacting conduction electrons is studied\nnumerically. Retardation e\u000bects in the coupled electron-spin dynamics are shown to be the source\nfor the relaxation of the spin in the magnetic \feld. Total energy and spin is conserved in the\nnon-adiabatic process. Approaching the new local ground state is therefore accompanied by the\nemission of dispersive wave packets of excitations carrying energy and spin and propagating through\nthe lattice with Fermi velocity. While the spin dynamics in the regime of strong exchange coupling\nJis rather complex and governed by an emergent new time scale, the motion of the spin for\nweakJis regular and qualitatively well described by the Landau-Lifschitz-Gilbert (LLG) equation.\nQuantitatively, however, the full quantum-classical hybrid dynamics di\u000bers from the LLG approach.\nThis is understood as a breakdown of weak-coupling perturbation theory in Jin the course of time.\nFurthermore, it is shown that the concept of the Gilbert damping parameter is ill-de\fned for the\ncase of a one-dimensional system.\nPACS numbers: 75.78.-n, 75.78.Jp, 75.60.Jk, 75.10.Hk, 75.10.Lp\nI. INTRODUCTION\nThe Landau-Lifshitz-Gilbert (LLG) equation1{3has\noriginally been considered to describe the dynamics of\nthe magnetization of a macroscopic sample. Nowadays it\nis frequently used to simulate the dynamics of many mag-\nnetic units coupled by exchange or magnetostatic interac-\ntions, i.e., in numerical micromagnetics.4The same LLG\nequation can be used on an atomistic level as well.5{9For\na suitable choice of units and for several spins Sm(t) at\nlattice sites m, it has the following structure:\ndSm(t)\ndt=Sm(t)\u0002B+X\nnJmnSm(t)\u0002Sn(t)\n+X\nn\u000bmnSm(t)\u0002dSn(t)\ndt: (1)\nIt consists of precession terms coupling the spin at site\nmto an external magnetic \feld Band, via exchange\ncouplingsJmn, to the spins at sites n. Those pre-\ncession terms typically have a clear atomistic origin,\nsuch as the Ruderman-Kittel-Kasuya-Yoshida (RKKY)\ninteraction10{12which is mediated by the magnetic po-\nlarization of conduction electrons. The non-local RKKY\ncouplingsJmn=J2\u001fmnare given in terms of the ele-\nments\u001fmnof the static conduction-electron spin suscep-\ntibility and the local exchange Jbetween the spins and\nthe local magnetic moments of the conduction electrons.\nOther possibilities comprise direct (Heisenberg) exchange\ninteractions, intra-atomic (Hund's) couplings as well as\nthe spin-orbit and other anisotropic interactions. The re-\nlaxation term, on the other hand, is often assumed as lo-\ncal,\u000bmn=\u000emn\u000b, and represented by purely phenomeno-\nlogical Gilbert damping constant \u000bonly. It describes the\nangular-momentum transfer between the spins and a usu-\nally unspeci\fed heat bath.On the atomistic level, the Gilbert damping must be\nseen as originating from microscopic couplings of the\nspins to the conduction-electron system (as well as to\nlattice degrees of freedom which, however, will not be\nconsidered here). There are numerous studies where the\ndamping constant, or tensor, \u000bhas been computed nu-\nmerically from a more fundamental model including elec-\ntron degrees of freedom explicitly13{15or even from \frst\nprinciples.16{21All these studies rely on two, partially\nrelated, assumptions: (i) The spin-electron coupling J\nis weak and can be treated perturbatively to lowest or-\nder, i.e., the Kubo formula or linear-response theory is\nemployed. (ii) The classical spin dynamics is slow as\ncompared to the electron dynamics. These assumptions\nappear as well justi\fed but they are also necessary to\nachieve a simple e\u000bective spin-only theory by eliminat-\ning the fast electron degrees of freedom.\nThe purpose of the present paper is to explore the\nphysics beyond the two assumptions (i) and (ii). Us-\ning a computationally e\u000ecient formulation in terms of\nthe electronic one-particle reduced density matrix, we\nhave set up a scheme by which the dynamics of classi-\ncal spins coupled to a system of conduction electrons can\nbe treated numerically exactly. The theory applies to ar-\nbitrary coupling strengths and does not assume a separa-\ntion of electron and spin time scales. Our approach is a\nquantum-classical hybrid theory22which may be charac-\nterized as Ehrenfest dynamics, similar to exact numerical\ntreatments of the dynamics of nuclei, treated as classical\nobjects, coupled to a quantum system of electrons (see,\ne.g., Ref. 23 for an overview). Some other instructive ex-\namples of quantum-classical hybrid dynamics have been\ndiscussed recently.24,25\nThe obvious numerical advantage of an e\u000bective spin-\nonly theory, as given by LLG equations of the form (1),\nis that in solving the equations of motion there is only\nthe time scale of the spins that must be taken care of. AsarXiv:1507.08227v2  [cond-mat.mes-hall]  28 Nov 20152\ncompared to our hybrid theory, much larger time steps\nand much longer propagation times can be achieved. Op-\nposed to ab-initio approaches16,17,26we therefore con-\nsider a simple one-dimensional non-interacting tight-\nbinding model for the conduction-electron degrees of free-\ndom, i.e., electrons are hopping between the nearest-\nneighboring sites of a lattice. Within this model ap-\nproach, systems consisting of about 1000 sites can be\ntreated easily, and we can access su\u000eciently long time\nscales to study the spin relaxation. An equilibrium state\nwith a half-\flled conduction band is assumed as the ini-\ntial state. The subsequent dynamics is initiated by a\nsudden switch of a magnetic \feld coupled to the classical\nspin. The present study is performed for a single spin,\ni.e., we consider a classical-spin Kondo-impurity model\nwith antiferromagnetic local exchange coupling J, while\nthe theory itself is general and can be applied to more\nthan a single or even to a large number of spins as well.\nAs compared to the conventional (quantum-spin)\nKondo model,27,28the model considered here does not\naccount for the Kondo e\u000bect and therefore applies to sit-\nuations where this is absent or less important, such as\nfor systems with large spin quantum numbers S, strongly\nanisotropic systems or, as considered here, systems in a\nstrong magnetic \feld. To estimate the quality of the\nclassical-spin approximation a priori is di\u000ecult.29{31For\none-dimensional systems, however, a quantitative study\nis possible by comparing with full quantum calculations\nand will be discussed elsewhere.32\nThere are di\u000berent questions to be addressed: For\ndimensional reasons, one should expect that linear-\nresponse theory, even for weak J, must break down at\nlong times. It will therefore be interesting to compare\nthe exact spin dynamics with the predictions of the LLG\nequation for di\u000berent J. Furthermore, the spin dynamics\nin the long-time limit can be expected to be sensitively\ndependent on the low-energy electronic structure. We\nwill show that this has important consequences for the\ncomputation of the damping constant \u000band that\u000bis\neven ill-de\fned in some cases. An advantage of a full\ntheory of spin and electron dynamics is that a precise\nmicroscopic picture of the electron dynamics is available\nand can be used to discuss the precession and relaxation\ndynamics of the spin from another, namely from the elec-\ntronic perspective. This information is in principle exper-\nimentally accessible to spin-resolved scanning-tunnelling\nmicroscope techniques33{36and important for an atom-\nistic understanding of nano-spintronics devices.37,38We\nare particularly interested in the physics of the system\nin the strong- Jregime or for a strong \feld Bwhere the\ntime scales of the spin and the electron dynamics become\ncomparable. This has not yet been explored but could\nbecome relevant to understand real-time dynamics in re-\nalizations of strong- JKondo-lattice models by means of\nultracold fermionic Yb quantum gases trapped in optical\nlattices.39,40\nThe paper is organized as follows: We \frst introduce\nthe model and the equations of motion for the exactquantum-classical hybrid dynamics in Sec. II and discuss\nsome computational details in Sec. III. Sec. IV provides\na comprehensive discussion of the relaxation of the clas-\nsical spin after a sudden switch of a magnetic \feld. The\nreversal time as a function of the interaction and the \feld\nstrength is analyzed in detail. We then set the focus on\nthe conduction-electron system which induces the relax-\nation of the classical spin by dissipation of energy. In Sec.\nV, the linear-response approach to integrate out the elec-\ntron degrees of freedom is carefully examined, including a\ndiscussion of the additional approximations that are nec-\nessary to re-derive the LLG equation and the damping\nterm in particular. Sec. VI summarizes the results and\nthe main conclusions.\nII. MODEL AND THEORY\nWe consider a classical spin SwithjSj= 1=2, which is\ncoupled via a local exchange interaction of strength Jto\nthe local quantum spin si0at the sitei0of a system of N\nitinerant and non-interacting conduction electrons. The\nconduction electrons hop with amplitude \u0000T(T > 0)\nbetween non-degenerate orbitals on nearest-neighboring\nsites of aD-dimensional lattice, see Fig. 1. Lis the\nnumber of lattice sites, and n=N=L is the average\nconduction-electron density.\nThe dynamics of this quantum-classical hybrid\nsystem22is determined by the Hamiltonian\nH=\u0000TX\nhiji;\u001bcy\ni\u001bcj\u001b+Jsi0S\u0000BS: (2)\nHere,ci\u001bannihilates an electron at site i= 1;:::;L with\nspin projection \u001b=\";#, andsi=1\n2P\n\u001b\u001b0cy\ni\u001b\u001b\u001b\u001b0ci\u001b0is\nthe local conduction-electron spin at i, where\u001bdenotes\nthe vector of Pauli matrices. The sum runs over the\ndi\u000berent ordered pairs hijiof nearest neighbors. Bis\nan external magnetic \feld which couples to the classical\nspin.\nTo be de\fnite, an antiferromagnetic exchange coupling\nJ > 0 is assumed. If Swas a quantum spin with\nS= 1=2, Eq. (2) would represent the single-impurity\nKondo model.27,28However, in the case of a classical spin\nconsidered here, there is no Kondo e\u000bect. The semiclas-\nsical single-impurity Kondo model thus applies to sys-\ntems where a local spin is coupled to electronic degrees\nof freedom but where the Kondo e\u000bect absent or sup-\npressed. This comprises the case of large spin quantum\nnumbersS, or the case of temperatures well above the\nKondo scale, or systems with a ferromagnetic Kondo cou-\nplingJ <0 where, for a classical spin, we expect a qual-\nitatively similar dynamics as for J >0.\nWe assume that initially, at time t= 0, the clas-\nsical spinS(t= 0) has a certain direction and that\nthe conduction-electron system is in the corresponding\nground state, i.e., the conduction electrons occupy the\nlowestNone-particle eigenstates of the non-interacting3\nHamiltonian Eq. (2) for the given S=S(t= 0) up to\nthe chemical potential \u0016. A non-trivial time evolution is\ninitiated if the initial direction of the classical spin and\nthe direction of the \feld Bare non-collinear.\nTo determine the real-time dynamics of the electronic\nsubsystem, it is convenient to introduce the reduced one-\nparticle density matrix. Its elements are de\fned as ex-\npectation values,\n\u001aii0;\u001b\u001b0(t)\u0011hcy\ni0\u001b0ci\u001bit; (3)\nin the system's state at time t. Att= 0 we have\u001a(0) =\n\u0002(\u0016\u0000T(0)). The elements of \u001a(0) are given by\n\u001ai\u001b;i0\u001b0(0) =X\nkUi\u001b;k\u0002(\u0016\u0000\"k)Uy\nk;i0\u001b0; (4)\nwhere \u0002 is the step function and where Uis the uni-\ntary matrix diagonalizing the hopping matrix T(0), i.e.,\nUyT(0)U=\"with the diagonal matrix \"given by the\neigenvalues of T(0). The hopping matrix at time tis\ncan be read o\u000b from Eq. (2). It comprises the physical\nhopping and the contribution resulting from the coupling\nterm. Its elements are given by\nTi\u001b;i0\u001b0(t) =\u0000T\u000ehii0i\u000e\u001b\u001b0+\u000eii0\u000ei0i0J\n2(S(t)\u001b)\u001b\u001b0:(5)\nHere\u000ehii0i= 1 ifi;i0are nearest neighbors and zero else.\nThere is a closed system of equations of motion for the\nclassical spin vector S(t) and for the one-particle density\nmatrix\u001a(t). The time evolution of the classical spin is\ndetermined via ( d=dt)S(t) =fS;Hclass:gby the classical\nHamilton function Hclass:=hHi. This equation of mo-\ntion is the only known way to consistently describe the\ndynamics of quantum-classical hybrids (see Refs. 22,41,42\nand references therein for a general discussion). The\nPoisson bracket between arbitrary functions AandBof\nthe spin components is given by,43,44\nfA;Bg=X\n\u000b;\f;\r\"\u000b\f\r@A\n@S\u000b@B\n@S\fS\r; (6)\nwhere the sums run over x;y;z and where \"\u000b\f\ris the\nfully antisymmetric \"-tensor. With this we \fnd\nd\ndtS(t) =Jhsi0it\u0002S(t)\u0000B\u0002S(t): (7)\nThis is the Landau-Lifschitz equation where the expec-\ntation value of the conduction-electron spin at i0is given\nby\nhsi0it=1\n2X\n\u001b\u001b0\u001ai0\u001b;i0\u001b0(t)\u001b\u001b0\u001b; (8)\nand where Jhsi0itacts as an e\u000bective time-dependent\ninternal \feld in addition to the external \feld B.\nS(t)JTi0BFIG. 1: (Color online) Classical spin S(t) coupled via an an-\ntiferromagnetic local exchange interaction of strength Jto a\nsystem of conduction electrons hopping with nearest-neighbor\nhopping amplitude Tover the sites of a one-dimensional lat-\ntice with open boundaries. The spin couples to the central\nsitei0of the system and is subjected to a local magnetic \feld\nof strength B.\nThe equation of motion for hsiitreads as\nd\ndthsiit=\u000eii0JS(t)\u0002hsiit\n+Tn:n:X\nj1\n2iX\n\u001b\u001b0(hcy\ni\u001b\u001b\u001b\u001b0cj\u001b0it\u0000c.c.);(9)\nwhere the sum runs over the nearest neighbors of i. The\nsecond term on the right-hand side describes the coupling\nof the local conduction-electron spin to its environment\nand the dissipation of spin and energy into the bulk of\nthe system (see below). Apparently, the system of equa-\ntions of motion can only be closed by considering the\ncomplete one-particle density matrix Eq. (3). It obeys a\nvon Neumann equation of motion,\nid\ndt\u001a(t) = [T(t);\u001a(t)] (10)\nas is easily derived, e.g., from the Heisenberg equation of\nmotion for the annihilators and creators.\nAs is obvious from the equations of motion, the real-\ntime dynamics of the quantum-classical Kondo-impurity\nmodel on a lattice with a \fnite but large number of sites L\ncan be treated numerically exactly (see also below). Nev-\nertheless, the model comprises highly non-trivial physics\nas the electron dynamics becomes e\u000bectively correlated\ndue to the interaction with the classical spin. In addi-\ntion, the e\u000bective electron-electron interaction mediated\nby the classical spin is retarded: electrons scattered from\nthe spin at time twill experience the e\u000bects of the spin\ntorque exerted by electrons that have been scattered from\nthe spin at earlier times t0<t.\nIII. COMPUTATIONAL DETAILS\nEqs. (5), (7), (8) and (10) represent a coupled non-\nlinear system of \frst-order ordinary di\u000berential equations\nwhich can be solved numerically. By blocking up the\nvon Neumann equation (10), the di\u000berential equations\nare written in a standard form _y=f(y(t);t), wherey(t)\nis a high-dimensional vector, such that an explicit Runge-\nKutta method can be applied. A high-order propagation4\ntechnique is used45which provides the numerically exact\nsolution up to 6-th order in the time step \u0001 t. For a typ-\nical system consisting of about L= 103sites this implies\nthat\u0018106coupled equations are solved.\nWe consider a one-dimensional system with open\nboundaries consisting of L= 1001 sites and a local per-\nturbation at the central site i0of the system, see Fig. 1.\nFor a half-\flled tight-binding conduction band the Fermi\nvelocityvF= 2Troughly determines the maximum speed\nof the excitations and de\fnes a \\light cone\".46,47This\nmeans that \fnite-size e\u000bects due to scattering at the sys-\ntem boundaries become relevant after a propagation time\ntmax\u0018500 (in units of 1 =T). A time step \u0001 t= 0:1 is\nusually su\u000ecient for reliable numerical results up to tmax,\ni.e., about 5000 time steps are performed. The compu-\ntational cost is moderate, and calculations can be per-\nformed in a few hours on a standard desktop computer.\nAssuming, for example, that B= (0;0;B), the Hamil-\ntonian is invariant under rotations around the zaxis. It\nis then easily veri\fed that not only the length of the spin\njSj= 1=2 is conserved but also the total number of con-\nduction electrons,\nNtot=X\ni\u001bhcy\ni\u001bci\u001bi; (11)\nthez-component of the total spin,\nStot;z=Sz+X\nihsizi; (12)\nas well as the total energy,\nEtot=hHi= tr (\u001a(t)T(t))\u0000BS(t): (13)\nDespite the fact that the model does not include a di-\nrect (e.g., Coulomb) interaction among the conduction\nelectrons, the average occupation numbers of the basis of\none-particle states in which the hopping matrix T(t) is\ndiagonal at time tarenotconserved. This is due to the\ne\u000bective retarded interaction mediated by the classical\nspin. Hence, the system is not integrable, unlike a free\nfermion gas. The conservation of the above-mentioned\nglobal observables serves as a sensitive check for the ac-\ncuracy of the numerical procedure.\nIV. COUPLED SPIN AND ELECTRON\nDYNAMICS\nA. Spin relaxation\nFig. 2 shows the real-time dynamics of the classical\nspin forJ= 1. Energy and time units are \fxed by the\nnearest-neighbor hopping T= 1 throughout the paper.\nInitially, for t= 0, the spin is oriented (almost) antipar-\nallel to the external local \feld B= (0;0;B) withB= 1,\ni.e., initially Sx(0) =1\n2sin#,Sy(0) = 0,Sz(0) =\u00001\n2cos#\n-0.5-0.3-0.10.10.30.5-0.5-0.3-0.10.10.30.5-0.5-0.3-0.10.10.30.5\nxyz\n10−1100101102\ntimet-0.50-0.250.000.250.50/vectorSSx\nSzFIG. 2: (Color online) Real-time dynamics of the classical\nspin. Upper panel: Bloch sphere representation. Lower panel:\nxandzcomponent of S(t) (jSj= 1=2). Calculations for\nexchange coupling J= 1 and \feld strength B= 1 and for\na system of L= 1001 sites. ( L= 1001 is kept \fxed for the\nrest of the paper). Energy and time units are \fxed by the\nnearest-neighbor hopping T= 1.\nwhere a non-zero but small polar angle #=\u0019=50 is nec-\nessary to slightly break the symmetry of the initial state\nand to start the dynamics.\nFor the same setup, the Landau-Lifschitz-Gilbert equa-\ntion would essentially predict two e\u000bects: \frst, a preces-\nsion of the classical spin around the \feld direction with\nLarmor frequency !L=Bz, and second, a relaxation\nof the spin to the equilibrium state with Sparallel to\nB=Bezfort!1 . Both e\u000bects are also found in\nthe full dynamics of the quantum-classical hybrid model.\nThe frequency of the oscillation of Sx(t) that is seen in\nFig. 2 is!L, andSzis reversed after a few hundred time\nunits. The precessional motion is easily explained by the\ntorque on the spin exerted by the \feld according to Eq.\n(7). The explanation of the damping e\u000bect is more in-\nvolved:\nEven for the high \feld strength considered here, the\nspin dynamics is slow as compared to the characteristic\nelectronic time scale such that it could be reasonable to\nassume the electronic system being in its instantaneous5\n10\u0000210\u00001100101102timet3.083.103.123.14~S(t)h~si0it\u0000(t)~B\u0000(t)J=46810J=20xy\nFIG. 3: (Color online) Time dependence of the angle \r(t)\nenclosed by S(t) andhsi0itforB= 0:1 and di\u000berent Jas\nindicated.\nground state at any instant of time and corresponding\nto the con\fguration of the classical spin. This, however,\nwould imply that the expectation value hsi0itof the local\nconduction-electron moment at i0is always strictly par-\nallel toS(t) and, hence, there would not be any torque\nmediated by the exchange coupling JonS(t).\nIn fact, the direction of hsi0itis always somewhat be-\nhind the \\adiabatic direction\", i.e., behind \u0000S(t): This\nis shown in Fig. 3 for a \feld of strength B= 0:1 where\nthe spin dynamics is by a factor 10 slower, compared to\nFig. 2, and for di\u000berent stronger exchange couplings J.\nEven in this case the process is by no means adiabatic,\nand the angle \r(t) betweenS(t) andhsi0itis close to but\nclearly smaller than \r=\u0019at any instant of time. This\nnon-adiabaticity results from the fact that the motion\nof the classical spin a\u000bects the conduction electrons in\na retarded way, i.e., it takes a \fnite time until the local\nconduction-electron spin hsi0itati0reacts to the motion\nof the classical spin.\nThis retardation e\u000bect results in a torque Jhsi0it\u0002\nS(t)6= 0 exerted on the classical S(t) in the +zdirection\nwhich adds to the torque due to Band which drives the\nspin to its new equilibrium direction. Hence, retardation\nis the physical origin of the Gilbert spin damping.\nWith increasing time, the deviation of \r(t) from the\ninstantaneous equilibrium value \r=\u0019increases in mag-\nnitude, i.e., the direction of hsi0itis more and more be-\nhind the adiabatic direction, and the torque increases. Its\nmagnitude is at a maximum at the same time when the\noscillatingx;ycomponents of S(t) are at a maximum\n(see Fig. 2). The z-component of the torque does not\nvanish before the spin has reached its new equilibrium\npositionS(t)/ez.\nFig. 3 shows results for di\u000berent J. Generally, non-\nadiabatic e\u000bects show up if the typical time scale of the\ndynamics is faster than the relaxation time, i.e., the time\nnecessary to transport the excitation away from the lo-\ncationi0where it is created initially. Roughly this time\nscale is set by the inverse hopping 1 =T. One therefore\nexpects that, for \fxed T, a stronger Jimplies a stronger\n0 20 40 60 80 100\nexchange coupling J050100150200250300350400reversal time τ100∗τ1/τ2\nτ1\nτ2FIG. 4: (Color online) Time for a spin reversal Sz=1\n2cos(\u0019\u0000\n#) =\u00001\n2cos(#)!Sz=1\n2cos(#) for#=#1=\u0019=50 (reversal\ntime\u001c1) and for#=#2=\u0019=25 (reversal time \u001c2). Calcula-\ntions as a function of Jfor \fxedB= 0:1.\nretardation of the conduction-electron dynamics. The re-\nsults for di\u000berent Jshown in Fig. 3 in fact show that the\nmaximum deviation of \r(t) from the adiabatic direction\n\r=\u0019increases with increasing J(for very strong Jthe\ndynamics becomes much more complicated, see below).\nThis results in a stronger torque on S(t) inzdirection\nand thus in a stronger damping. The picture is also qual-\nitatively consistent with the LLG equation as the Gilbert\ndamping constant \u000bincreases with J.\nThe spin (almost) reverses its direction after a \fnite\nreversal time \u001cwhich is shown in Fig. 4 as a function\nofJ. Calculations have been performed for an initial\ndirection of the classical spin with Sz(0) =\u0000(1=2) cos#\nwith two di\u000berent polar angles #1;2. If#is su\u000eciently\nsmall, the results for di\u000berent #are expected to di\u000ber by\naJandBindependent constant factor only. As Fig. 4\ndemonstrates, the ratio \u001c1=\u001c2is in fact nearly constant.\nFor weakJand up to coupling strengths of about J.30,\nwe \fnd that the reversal time decreases with increasing\nJ. With increasing J, the retardation e\u000bect increases, as\ndiscussed above, and the stronger damping results in a\nshorter reversal time.\n0 2 4 6 8 10\n1/B0100200300400reversal time τ\nJ= 2\nJ= 4\nFIG. 5: (Color online) Reversal time \u001c1as function of Bfor\n\fxedJ= 2 andJ= 4 as indicated.6\nThe prediction of the LLG equation for the reversal\ntime of a single spin \u001ccan be derived analytically48and\nis given by\n\u001c/1 +\u000b2\n\u000b1\nBln\f\f\f1=2\u0000Sz(0)\n1=2 +Sz(0)\f\f\f: (14)\nHowever, down to the smallest Jfor which\u001ccan be cal-\nculated reliably, our results for the full spin dynamics do\nnot scale as \u001c/1=J2as one would expect for weak J\nassuming that \u000b/J2(see discussion in Sec. V B).\nTheBdependence of the reversal time is shown in Fig.\n5. With increasing \feld strength, the classical spin S(t)\nprecesses with a higher Larmor frequency !L\u0019Baround\nthezaxis. Hence, non-adiabatic e\u000bects increase. The\nstronger the \feld, the more delayed is the precessional\nmotion of the local conduction-electrons spin hsi0it. This\nresults in a stronger torque in + zdirection exerted on\nthe classical spin. Therefore, the relaxation is faster and\nthe reversal time \u001csmaller. For weak and intermediate\n\feld strengths, \u001cis roughly proportional to 1 =B. This is\nconsistent with the prediction of the LLG equation, see\nEq. (14).\nIn the limit of very strong \felds one would expect an\nincrease of the reversal time with increasing Bsince the\n\feld term will eventually dominate the dynamics, i.e.,\nonly the precessional motion survives which implies a di-\nverging reversal time. In fact, for a \feld strength exceed-\ning a critical strength Bc, which depends on J, there is\nno full relaxation any longer, and \u001c=1. This strong-\nBregime cannot be captured by the LLG equation and\ndeserves further studies.\nThe strong- Jregime is interesting as well. For cou-\npling strengths exceeding J\u001930 the reversal time in-\ncreases withJ(see Fig. 4). Eventually, the reversal time\nmust even diverge. This is obvious as the dynamics is\ndescribed by a simple two-spin model in the limit J=1\nwhich cannot show spin relaxation. The corresponding\nequations of motion are obtained by from Eqs. (7) and\n(9) by setting t= 0 andB= 0:\nd\ndtS(t) =Jhsi0it\u0002S(t);\nd\ndthsi0it=JS(t)\u0002hsi0it (15)\nNote that we have jhsi0itj= 1=2 forJ!1 . The equa-\ntions are easily solved by exploiting the conservation of\nthe total spin Stot=S(t) +hsi0it. Both spins precess\nwith constant frequency !0=JStotaroundStot. Their\ncomponents parallel to Stotare equal, and their compo-\nnents perpendicular to Stotare anti-parallel and of equal\nlength.\nHowever, the two-spin dynamics of the J=1limit\nis not stable against small perturbations. Fig. 6 shows\nthe classical spin dynamics of the full model (with T=\n1 andB= 0:1) for a very strong but \fnite coupling\nJ= 100. Here, the motion of the classical spin gets\nvery complicated as compared with the highly regular\n-0.50.00.5\n/vectorSx\ny\nz\n-2.00.02.0\nJ/angbracketleft/vector si0/angbracketright×/vectorS\n-0.50.00.5\n−/vectorB×/vectorS\n10−1100101102\ntimet3.103.153.20\nγ=/negationslash(/vectorS/vector s0)FIG. 6: (Color online) Real-time dynamics in the strong- J\nregime:J= 100 and B= 0:1.First panel: Classical spin\nS(t).Second panel: Torque on S(t) due to the exchange\ninteraction. Third panel: Torque on S(t) due to the \feld\nterm. Fourth panel: Angle enclosed by S(t) andhsi0it\nbehavior in the weak- Jregime (cf. Fig. 2). In particular,\nthez-component of S(t) is oscillating on nearly the same\nscale as the xandycomponents. This characteristic\ntime scale \u0001 t\u00194 of the oscillation corresponds to a\nfrequency!\u00191:5 which di\u000bers by more than an order\nof magnitude from both, the Larmor frequency B= 0:1\nand from the exchange-coupling strength J= 100. Note\nthat the oscillation of Sz(t) is actually the reason for the\nambiguity in the determination of the precise reversal\ntime and gives rise to the error bars in Fig. 4 for strong\nJ.\nWe attribute the complexity of the dynamics to the\nfact that the torque due to the \feld term and the torque\ndue to the exchange coupling are of comparable magni-\ntudes, see second and third panel in Fig. 6. It is interest-\ning that even for strong J, where one would expect S(t)\nandhsi0itto form a tightly bound local spin-zero state,\nthere is actually a small deviation from perfect antiparal-\nlel alignment, i.e., \r(t)6=\u0019, as can be seen in the fourth\npanel of Fig. 6. This results in a \fnite z-component of the\ntorque onS(t) which leads to a very fast reversal with\nSz(t)\u0019+0:5 at timet\u00192:1. Contrary to the weak-\ncoupling limit, however, the z-component of the torque\nchanges sign at this point and drives the spin back to the7\n\u0000z-direction. At t\u00193:2, however, the z-component of\nS(t) once more reverses its direction. Here, the torque\ndue to the exchange coupling vanishes completely as S(t)\nandhsi0itare perfectly antiparallel (see the \frst zero of\n\r(t) in the fourth panel). The motion continues due to\nthe non-zero \feld-induced torque. This pattern repeats\nseveral times. S(t) mainly oscillates within a plane in-\ncluding and slowly rotating around the z-axis.\nEventually, there is a perfect relaxation of the classical\nspin for large tbut in a very di\u000berent way as compared to\nthe weak-coupling limit. While the deviation from \r=\u0019\nis small at any instant of time as for weak J, the most\napparent di\u000berence is perhaps that the new ground state\nis approached with an oscillating behavior of \r(t) around\n\r=\u0019, i.e.,hsi0itmay run behind or ahead ofS(t) as\nwell.\nLet us summarize at this point the main di\u000berences\nbetween the quantum-classical hybrid and the e\u000bective\nLLG dynamics of the classical spin: For weak JandB,\nthe qualitative behavior, precessional motion and relax-\nation, is the same in both approaches. Quantitatively,\nhowever, the LLG equation is inconsistent with the ob-\nservedJdependence of the reversal time when assuming\n\u000b/J2. TheBdependence of 1 =\u001cis linear as expected\nfrom the LLG approach. For strong J, the spin dynamics\nqualitatively di\u000bers from LLG dynamics and gets more\ncomplicated with a new time scale emerging. Absence of\ncomplete relaxation, as observed in the strong- Blimit, is\nalso not accessible to an e\u000bective spin-only theory.\nB. Energy dissipation\nTo complete the picture of the relaxation dynamics\nof the classical spin, the discussion should also comprise\nthe dynamics of the electronic degrees of freedom. The\nspin relaxation must be accompanied by a dissipation of\nenergy and spin into the bulk of the electronic system\nsince the total energy and the total spin are conserved\nquantities, see Eqs. (12) and (13), while conservation of\nthe total particle number, Eq. (11), is trivially ensured by\nthe particle-hole symmetric setup considered here where\nthe average conduction-electron number at every site is\ntime-independent:P\n\u001bhni\u001bit= 1.\nThe total energy is given by Etot=hHi, see Eq. (13),\nand is a sum over di\u000berent contributions, Etot=EB(t)+\nEhop(t)+Eint(t), namely the interaction energy with the\n\feld\nEB(t) =\u0000BS(t); (16)\nthe kinetic (hopping) energy of the conduction-electron\nsystem\nEhop(t) =\u0000TX\nhijiX\n\u001bhcy\ni\u001bcj\u001bit; (17)\nand the exchange-interaction energy\nEint(t) =Jhsi0itS(t): (18)\n-0.50.00.5\nEB\n-1.2724-1.2718-1.2712energyEhop/L\nEtot/L\n-0.68-0.67-0.66\nEint\n100101102\ntimet−1.30−1.25−1.20−1.15\nei0±1\nei0±3\nei0±2ei0±50ei0±100FIG. 7: (Color online) Di\u000berent contributions to the total\nenergy as functions of time for J= 5 andB= 1. Top panel :\nEB, energy of the classical spin in the external \feld. Second\npanel :Ehop=L, kinetic (hopping) energy of the conduction-\nelectron system per site, and Etot=L, total energy per site.\nThird panel :Eint, exchange-interaction energy. The fourth\npanel shows the time-dependence of the total-energy density\nat di\u000berent distances from the site i0.\nThe time dependence of those contributions is shown in\nFig. 7 forJ= 5 andB= 1.\nThe top panel of Fig. 7 shows that the system re-\nleases the interaction energy j2BSjof the classical spin\nin the external \feld by aligning the spin to the \feld di-\nrection. In the long-time limit, this energy is stored in\nthe conduction-electron system: The average kinetic en-\nergy per site ( L= 1001) increases by the same amount\nas shown in the second panel (note the di\u000berent scales).\nThe exchange-interaction energy changes with time but\nis the same for t= 0 andt!1 (see third panel). The\ntotal energy is constant (second panel).\nThe relaxation of the classical spin in the external \feld\nBimplies an energy \row away from the site i0into the\nbulk of the conduction-electron system such that locally ,\nin the vicinity of i0the system is in its new ground state.\nTo discuss this energy \row, it is convenient to consider\nthe total energy as a lattice sum Etot=P\niei(t) over the8\ni0−500 i0 i0+ 500\nsites0255075100125150175200225250time\n-1.3-1.2-1.1-1.0-0.9-0.8\ntotal energy density ei\nFIG. 8: (Color online) Spatiotemporal evolution of the total-\nenergy density ei(t), as de\fned in Eq. (19). Calculation for\nJ= 5,B= 1.\ntotal energy \\density\" de\fned as\nei(t) =\u0000Tn:n:(i)X\njX\n\u001bhcy\ni\u001bcj\u001bit+\u000eii0(Jhsi0it\u0000B)S(t);\n(19)\nwhere the sum over jruns over the nearest neighbors of\nsitei. The time dependence of the energy density in the\nvicinity ofi0and at distances 50 and 100 is shown in the\nfourth panel of Fig. 7.\nAt any site in the conduction-electron system, the\nenergy density increases from its ground-state value,\nreaches a maximum and eventually relaxes to the en-\nergy density of the new ground state. Since the latter\nis just the ground state with the reversed classical spin,\nthe new ground-state energy density is the same as in\nthe initial state at t= 0. As can be seen in the fourth\npanel of Fig. 7, there is also a slight spatial oscillation\nof the ground-state energy density which just re\rects the\nFriedel oscillations around the impurity at i0.\nComplete relaxation means that the excitation energy\nis completely removed from the vicinity of i0and trans-\nported into the bulk of the system. That this is in fact\nthe case can be seen by comparing the energy density at\ndi\u000berent distances from i0. It is also demonstrated by\nFig. 8 which visualizes the energy-current density which\nsymmetrically points away from i0: The total energy of\nthe excitation \rowing through each pair of sites i0\u0006\u0001iis\nconstant, i.e., the time-integrated energy \rux,R\ndtei(t)\nis the same for all i.\nAs is seen in Fig. 7 (fourth panel) and Fig. 8, there is\na considerable dispersion of the excitation wave packet\ncarrying the energy. For example, at i0+ 100 it takes\nmore than four times longer, as compared to i0+ 1, un-\ntil most of the excitation has passed through (note the\nlogarithmic time scale).\nThe broadening of the wave packet, due to dispersion,\nis asymmetric and bound by an upper limit for the speed\nof the excitation which is roughly set by the Fermi veloc-\nityvF= 2T. This Lieb-Robinson bound46,47determines\ni0\u0000500i0i0+5 0 0sites050100150200250300350400450500time \u0000 \u0000 \u0000 \u0000 \u0000-0.06-0.030.000.030.06hszii\ni0\u0000500i0i0+5 0 0sites050100150200250300350400450500time \u0000 \u0000 \u0000 \u0000 \u0000-0.04-0.020.000.020.04hsxiisx\nszFIG. 9: (Color online) Spatiotemporal evolution of the total-\nspin densityhsiit(upper panel: x-component, lower panel z-\ncomponent) for J= 5 andB= 0:1. See Fig. 10 for snapshots\nat times indicated by the arrows.\nthe \\light cone\" seen in Fig. 8.\nC. Spin dissipation\nThe same upper speed limit, given by the Fermi ve-\nlocity of the conduction-electron system, is also seen in\nthe spatiotemporal evolution of the conduction-electron\nspin densityhsiit. This is shown in Fig. 9 for a di\u000berent\nmagnetic \feld strength B= 0:1 where the classical spin\ndynamics is slower. Apparently, the wave packet of exci-\ntations emitted from the impurity not only carries energy\nbut also spin. It symmetrically propagates away from i0\nand, att\u0019300, reaches the system boundary where it\nis re\rected perfectly. Up to t= 500 there is hardly any\ne\u000bect visible in the local observables close to i0that is\na\u000bected by the \fnite system size.\nSnapshots of the conduction-electron spin dynamics\nare shown in Fig. 10 for the initial state at t= 0 and for\nstates at four later times t >0 which are also indicated\nby the arrows in Fig. 9. At t= 0 the conduction-electron\nsystem is in its ground state for the given initial direction\nof the classical spin. The latter basically points into the\n\u0000zdirection, apart from a small positive x-component\n(#=\u0019=50) which is necessary to break the symmetry9\n-0.0030.0000.003\n-0.0030.0000.003\n-0.0030.0000.003/angbracketleft/vector si/angbracketright\n-0.0030.0000.003sx sz\ni0−100i0−60i0−20-0.0030.0000.003\nt= 0 t= 80t= 60 t= 100sz sx\ni0i0+ 100 i0+ 500\nsites\nt= 250\nFIG. 10: (Color online) Snapshots the of conduction-electron magnetic moments hsiitat di\u000berent times tas indicated on the\nright and by the corresponding arrows in Fig. 9. Red lines: z-components ofhsiit. Blue lines: xcomponents. The pro\fles are\nperfectly symmetric to the impurity site i=i0but displayed up to distances ji\u0000i0j\u0014100 on the left-hand side and up to the\nsystem boundary, ji\u0000i0j\u0014500, on the right-hand side. Parameters J= 5;B= 0:1.\nof the problem and to initiate the dynamics. This tiny\ne\u000bect will be disregarded in the following.\nFrom the perspective of the conduction-electron sys-\ntem, the interaction term JSsi0acts as a local external\nmagnetic \feld JSwhich locally polarizes the conduction\nelectrons at i0. SinceJis antiferromagnetic, the local\nmomenthsi0ipoints into the + zdirection. At half-\flling,\nthe conduction-electron system exhibits pronounced an-\ntiferromagnetic spin-spin correlations which give rise to\nan antiferromagnetic spin-density wave structure aligned\nto thezaxis att= 0, see \frst panel of Fig. 10.\nThe total spin Stot= 0 att= 0, i.e., the classical\nspinSis exactly compensated by the total conduction-\nelectron spinhstoti=P\nihsii=\u0000Sin the ground state.\nThis can be traced back to the fact that for a D= 1-\ndimensional tight-binding system with an odd number of\nsitesL, withN=Land with a single static magnetic\nimpurity, there is exactly one localized state per spin pro-\njection\u001b, irrespective of the strength of the impurity po-\ntential (here given by JS= 0:5Jez). The number of \"\none-particle eigenstates therefore exceeds the number of\n#states by exactly one.\nSince the energy of the excitation induced by the ex-\nternal \feldBis completely dissipated into the bulk, the\nstate of the conduction-electron system at large t(but\nshorter than t\u0019500 where \fnite-size e\u000bects appear)\nmust locally, close to i0, resemble the conduction-electron\nground state for the reversed spin S= +0:5ez. This\nimplies that locally all magnetic moments hsiitmust re-\nverse their direction. In fact, the last panel in Fig. 10(left) shows that the new spin con\fguration is reached\nfort= 250 at sites with distance ji\u0000i0j.100, see\ndashed line, for example. For later times the spin con-\n\fguration stays constant (until the wave packet re\rected\nfrom the system boundaries reaches the vicinity of i0).\nThe reversal is almost perfect, e.g., hsi0it=0= 0:2649!\nhsi0it\u0015250=\u00000:2645. Deviations of the same order\nof magnitude are also found at larger distances, e.g.,\ni=i0\u0000100. We attribute those tiny e\u000bects to a weak de-\npendence of the local ground state on the non-equilibrium\nstate far from the impurity at t= 250, see right part of\nthe last panel in Fig. 10.\nThe other panels in Fig. 10 demonstrate the mecha-\nnism of the spin reversal. At short times (see t= 60,\nsecond panel) the perturbation of the initial equilibrium\ncon\fguration of the conduction-electron moments is still\nweak. For t= 80 andt= 100 one clearly notices the\nemission of the wave packet starting. Locally, the an-\ntiferromagnetic structure is preserved (see left part) but\nsuperimposed on this, there is an additional spatial struc-\nture of much longer size developing. This \fnally forms\nthe wave packet which is emitted from the central re-\ngion. Its spatial extension is about \u0001 \u0019300 as can be\nestimated for t= 250 (last panel on the right) where\nit covers the region 200 .i.500. The same can be\nread o\u000b from the upper part of Fig. 9. Assuming that\nthe reversal of each of the conduction-electron moments\ntakes about the same time as the reversal of the classi-\ncal spin, \u0001 is roughly given by the reversal time times\nthe Fermi velocity and therefore strongly depends on J10\nandB. For the present case, we have \u001c1\u0019150=Twhich\nimplies \u0001\u0019150\u00022 = 300 in rough agreement with the\ndata.\nIn the course of time, the long-wave length structure\nsuperimposed on the short-range antiferromagnetic tex-\nture develops a node. This can be seen for t= 100 and\ni\u001940 (fourth panel, see dashed line). The node marks\nthe spatial border between the new (right of the node,\ncloser toi0) and the original antiferromagnetic structure\nof the moments and moves away from i0with increasing\ntime.\nAt a \fxed position i, the reversal of the conduction-\nelectron moment hsiittakes place in a similar way as\nthe reversal of the classical spin (see both panels in Fig.\n9 for a \fxed i). During the reversal time, its xandy\ncomponents undergo a precessional motion while the z\ncomponent changes sign. Note, however, that during the\nreversaljhsiijgets much larger than its value in the initial\nand in the \fnal equilibrium state.\nV. EFFECTIVE CLASSICAL SPIN DYNAMICS\nA. Perturbation theory\nEqs. (7) and (9) do not form a closed set of equations of\nmotion but must be supplemented by the full equation of\nmotion (10) for the one-particle conduction-electron den-\nsity matrix. This implies that the fast electron dynamics\nmust be taken into account explicitly even if the spin\ndynamics is much slower. Hence, there is a strong mo-\ntivation to integrate out the conduction-electron degrees\nof freedom altogether and to take advantage from a much\nlarger time step within a corresponding spin-only time-\npropagation method. Unfortunately, a simple e\u000bective\nspin-only action can be obtained in the weak-coupling\n(small-J) limit only.13,14This weak-coupling approxima-\ntion is also implicit to all e\u000bective spin-only approaches\nthat consider the e\u000bect of conduction electrons on the\nspin dynamics.49\nIn the weak- Jlimit the electron degrees of freedom can\nbe eliminated in a straightforward way by using standard\nlinear-response theory:50We assume that the initial state\natt= 0 is given by the conduction-electron system in its\nground state or in thermal equilibrium and an arbitrary\nstate of the classical spin. This may be realized formally\nby suddenly switching on the interaction J(t) at time\nt= 0, i.e.,J(t) =J\u0002(t) and by switching the local \feld\nfrom some initial value Biniatt= 0 to a \fnal value B\nfort >0. The response of the conduction-electron spin\nati0and timet >0 (hsi0it= 0 fort= 0) due to the\ntime-dependent perturbation J(t)S(t) is\nhsi0it=JZt\n0dt0\u0005(ret)(t;t0)\u0001S(t0) (20)\nup to linear order in J. Here, the free ( J= 0) local\nretarded spin susceptibility of the conduction electrons\u0005(ret)(t;t0) is a tensor with elements\n\u0005(ret)\n\u000b\f(t;t0) =\u0000i\u0002(t\u0000t0)h[s\u000b\ni0(t);s\f\ni0(t0)]i; (21)\nwhere\u000b;\f =x;y;z . Using this in Eq. (7), we get an\nequation of motion for the classical spins only,\nd\ndtS(t) =S(t)\u0002B\n\u0000J2S(t)\u0002Zt\n0dt0\u0005(ret)(t\u0000t0)\u0001S(t0) (22)\nwhich is correct up to order J2.\nThis represents an equation of motion for the classical\nspin only. It has a temporally non-local structure and\nincludes an e\u000bective interaction of the classical spin at\ntimeS(t) with the same classical spin at earlier times\nt0< t. In the full quantum-classical theory where the\nelectronic degrees of freedom are taken into account ex-\nactly, this retarded interaction is mediated by a non-\nequilibrium electron dynamics starting at site i0and time\nt0and returning back to the same site i0at timet > t0.\nHere, for weak J, this is replaced by the equilibrium and\nhomogeneous-in-time conduction-electron spin suscepti-\nbility \u0005(ret)(t\u0000t0). Compared with the results of the\nfull quantum-classical theory, we expect that the pertur-\nbative spin-only theory breaks down after a propagation\ntimet\u00181=Jat the latest.\nUsing Wick's theorem,50the spin susceptibility is eas-\nily expressed in terms of the greater and the lesser equi-\nlibrium one-particle Green's functions, G>\nii;\u001b\u001b0(t;t0) =\n\u0000ihci\u001b(t)cy\ni\u001b0(t0)iandG<\nii;\u001b\u001b0(t;t0) =ihcy\ni\u001b0(t0)ci\u001b(t)i, re-\nspectively:\n\u0005(ret)\n\u000b\u000b0(t\u0000t0) = \u0002(t\u0000t0)1\n2\n\u0002Im tr 2\u00022h\n\u001b\u000bG>\ni0i0(t;t0)\u001b\u000b0G<\ni0i0(t0;t)i\n:(23)\nAssuming that the conduction-electron system is charac-\nterized by a real, symmetric and spin-independent hop-\nping matrix Tij(as given by the \frst term of Eq. (2)),\nG>andG<are unit matrices with respect to the spin\nindices. They are easily expressed as explicit functions\nofT(see Ref. 51, for example). With tr ( \u001b\u000b\u001b\u000b0) = 2\u000e\u000b\u000b0,\nwe \fnd\n\u0005(ret)\n\u000b\u000b0(t\u0000t0) = \u0002(t\u0000t0)\u000e\u000b\u000b0Im\n\u0002 \ne\u0000iT(t\u0000t0)\n1 +e\u0000\f(T\u0000\u0016)!\ni0i0 \ne\u0000iT(t0\u0000t)\ne\f(T\u0000\u0016)+ 1!\ni0i0(24)\nfor a conduction-electron system at inverse temperature\n\fand chemical potential \u0016.\nUsing Eq. (24) we have computed the spin suscepti-\nbility for the ground state ( \f=1) of the conduction-\nelectron system. This \fxes the kernel in the integro-\ndi\u000berential equation Eq. (22) which is solved numeri-\ncally by standard techniques.52We again consider the11\n10−1100101102\ntimet-0.50-0.250.000.250.50/vectorSJ= 1.0Sz\nSx\nFIG. 11: (Color online) Components Sx(t) andSz(t) of the\nclassical spin after a sudden switch of the \feld from xto\nzdirection at t= 0. Calculations for B= 1 andJ= 1.\nSolid lines: results of the linear-response dynamics, Eq. (22).\nDashed lines: results of the exact quantum-classical dynamics\nforL= 1001.\nsystem displayed in Fig. 1 with a single classical spin\ncoupled via Jto the central site i0of a chain consisting\nofL= 1001 sites. Finite-size artifacts do not show up\nbeforetmax= 500.\nFigs. 11 and 12 show the resulting linear-response dy-\nnamics of the classical spin after preparing the initial\nstate of the system with the classical spin pointing into\n+xdirection while B= (0;0;B). The external magnetic\n\feld induces a precessional motion of the classical spin:\nthere is a rapid oscillation of its xcomponent (and of its\nycomponent, not shown) with frequency !\u0019B(blue\nlines). Damping is induced by dissipation of energy and\nspin: for large times, the zcomponent aligns to the ex-\nternal \feld (red lines).\nFor weak coupling, up to J= 1 (Fig. 11), there is an al-\nmost perfect agreement between the results of the exact\nquantum-classical dynamics (full lines) and the linear-\nresponse theory (dashed lines) up to the maximum prop-\nagation time tmax= 500. We note that, compared to\nthe full theory, there is a tiny deviation of the linear-\nresponse result for the zcomponent of S(t) visible in\nFig. 11 for times t&10. Hence, on this level of accuracy,\nt1\u001910 sets the time scale up to which the linear-response\ntheory is valid. This may appear surprising as this im-\npliest1J= 10 for the \\small\" dimensionless parameter\nof the perturbation theory. One has to keep in mind,\nhowever, that even if the perturbation is \\strong\", its\ne\u000bects can be rather moderate since only non-adiabatic\nterms\u0018S(t)\u0002S(t0) contribute in Eq. (22).\nForJ= 3, see Fig. 12, damping of the classical spin\nsets in much earlier. Visible deviations of the linear-\nresponse theory from the full dynamics already appear\non a time scale that is almost two orders of magnitude\nsmaller as compared to the case J= 1. A simple rea-\nsoning based on the argument that the dimensionless\nexpansion parameter is t1Jfails as this disregards the\nstrong enhancement of retardation e\u000bects with increas-\n10−1100101\ntimet-0.50-0.250.000.250.50/vectorSJ= 3.0Sz\nSxFIG. 12: (Color online) The same as Fig. 11 but for a di\u000berent\nexchange coupling constant J= 3.\ningJ, which have been discussed in Sec. IV A. These\ne\u000bects make the perturbation much more e\u000bective, i.e.,\nlead to a torque, which is exerted by the conduction elec-\ntrons on the classical spin, growing stronger than linear\ninJ.\nWe conclude that linear-response theory is highly at-\ntractive formally as it provides a tractable spin-only e\u000bec-\ntive theory. On the other hand, substantial discrepancies\ncompared to the full (non-perturbative) theory show up\nas soon as damping e\u000bects become stronger. Note that,\nwith increasing time, these deviations must diminish and\ndisappear eventually since both, the full and the e\u000bec-\ntive theory, predict a fully relaxed spin state for t!1\n{ see lower panel of Fig. 12, for example. At least for\nsimple systems with a single classical spin, as considered\nhere, this implies that the e\u000bective theory provides qual-\nitatively reasonable results.\nB. Landau-Lifschitz-Gilbert equation\nTo derive the LLG equation, the linear-response the-\nory must be further simpli\fed:14,53As is obvious from\nEq. (22), the spin-susceptibility \u0005(ret)(t\u0000t0) can be inter-\npreted as an e\u000bective retarded self-interaction of the spin.\nWe assume that the electron dynamics is much faster\nthan the spin dynamics. On the time scale of the spin\ndynamics, the self-interaction then takes place almost in-\nstantaneously, i.e., the memory kernel \u0005(ret)\n\u000b\u000b0(t\u0000t0) =\n\u000e\u000b\u000b0\u0005(ret)(t\u0000t0) in Eq. (22) is peaked at t0\u0019t. We can\ntherefore approximate S(t0)\u0019S(t) + (t0\u0000t)_S(t) under\nthe integral in Eq. (22). This immediately yields:\ndS(t)\ndt=S(t)\u0002B+\u000b(t)S(t)\u0002dS(t)\ndt; (25)\nwhere\n\u000b(t) =J2Zt\n0d\u001c\u001c\u0005(ret)(\u001c) (26)12\nafter substituting t07!\u001c=t\u0000t0.\nEq. (25) takes the form of the standard LLG equation\nfor a single classical spin [cf. Eq. (1)] if \u000b(t) is replaced\nby\n\u000b\u0011lim\nt!1\u000b(t) =J2Z1\n0dtt\u0005(ret)(t): (27)\nNote that this is a necessary step to arrive at a constant\ndamping parameter which can again be justi\fed by not-\ning that \u0005(ret)(t) is peaked at t= 0.\nBefore proceeding, let us stress that Eq. (27) is, or\nis equivalent to, the standard expression used for com-\nputing the Gilbert damping constant in various studies:\nAfter Fourier transformation,\n\u0005(ret)(!) =Z\ndtei!t\u0005(ret)(t); (28)\none ends up with\n\u000b=\u0000iJ2@\n@!\u0005(ret)(!)\f\f\f\n!=0=J2@\n@!Im \u0005(ret)(!)\f\f\f\n!=0;\n(29)\nwhich has also been derived, e.g., in Refs. 14,54 in dif-\nferent contexts. The frequency-dependent spin correla-\ntion \u0005(ret)(!) in Eq. (29) can be obtained explicitly as\nthe Fourier transform of \u0005(ret)(t) given by Eq. (24). A\nstraightforward calculation yields:\n\u000b=\u0019\n2J2Z\nd!df(!)\nd!Aloc(!)Aloc(!); (30)\nwheref(!) = 1=(exp(\f!)+1) is the Fermi function, and\nAloc(!) =L\u00001X\nk\u000e(!+\u0016\u0000\"(k)) (31)\nforL!1 is the local one-particle spectral function.\nHere we have assumed periodic boundary conditions, i.e.,\nspatial homogeneity, such that the hopping matrix Tis\ndiagonalized by Fourier transformation:\nTij=X\nkUik\"(k)Uy\nkj(32)\nwithUik=L\u00001=2exp(ikRi). The eigenvalues of T\nare given by the tight-binding dispersion \"(k) of the\nconduction-electron Bloch band.\nWithin our simple tight-binding model for the\nconduction-electron system, Eqs. (29) and (30) are equiv-\nalent with Kambersky's breathing Fermi-surface the-\nory, related torque-correlation models and scattering\ntheory and have frequently been used for ab initio as\nwell as model computations of the Gilbert damping\nconstant.15,18,20,21,55{58\nLet us remark that Eq. (30) demonstrates that \u000b<0.\nThis results from the convention for the coupling of the\nmagnetic \feld to the spin, namely H=H(B= 0)\u0000BS\n[see Eq. 2)], which has been adopted here. As a conse-\nquence, the precession of S(t) aroundBis described by\naleft-hand helix, _S=S(t)\u0002B, and thus \u000bmust be\nnegative to describe damping.\n10−1100101102103\ntimet−0.10−0.050.000.050.10t·Πret(t)\nΠret(t)FIG. 13: (Color online) Local retarded spin susceptibility of\nthe conduction electrons \u0005(ret)(t) (red line) and t\u0005(ret)(t)\n(blue) as obtained from Eq. (33) for a one-dimensional\nconduction-electron system with L= 10000 sites and peri-\nodic boundary conditions.\nC. Ill-de\fned Gilbert damping\nThe above discussion shows that Eq. (27) represents\nthe fundamental de\fnition of the Gilbert damping con-\nstant\u000band that the limit t!1 is crucial to recover the\nLLG equation in its standard form. The existence of the\nlong-time limit, however, decisively depends on the long-\ntime behavior of the retarded spin-correlation function\n\u0005(ret)(t). Starting from Eq. (24), this is easily computed\nas\n\u0005(ret)(t) = \u0002(t)Im1\nL2X\nk;pe\u0000i\"(k)t\n1 +e\u0000\f(\"(k)\u0000\u0016)ei\"(p)t\ne\f(\"(p)\u0000\u0016)+ 1:\n(33)\nFig. 13 gives an example for the time-dependence of\n\u0005(ret)(t). The calculations have been done at half-\flling,\n\f=1forL= 104sites. We note that the susceptibility\nis in fact peaked at t= 0. For long times, it oscillates\nwith frequency !\u0005= 4 and decays as 1 =t. This is an im-\nportant observation as it implies that the limit in Eq. (27)\ndoes not exist and that, therefore, the damping constant\n\u000bis ill-de\fned.\nTo analyze the physical origin of the divergent integral,\nwe rewrite the spin susceptibility in Eq. (33) as\n\u0005(ret)(t) = \u0002(t)Im[A(unocc)\nloc(\u0000t)A(occ)\nloc(t)]: (34)\nIts long-time behavior is governed by the long-time be-\nhavior of the Fourier transform\nA(occ;unocc)\nloc(t) =Z\nd!ei!tA(occ;unocc)\nloc(!) (35)\nof the occupied, A(occ)\nloc(!)\u0011f(!)Aloc(!), and of the un-\noccupied,A(unocc)\nloc(!)\u0011f(!)Aloc(!), part of the spectral\ndensity, see Eq. (31).\nFor functions with smooth !dependence, the Fourier\ntransform generically drops to zero exponentially fast if13\nt!1 . A power-law decay, however, is obtained if there\nare singularities of A(occ;unocc)\nloc(!). We can distinguish\nbetween van Hove singularities, which are, e.g., of the\nform/\u0002(!\u0000!0)(!\u0000!0)k(withk>\u00001), and the step-\nlike singularity/\u0002(!\u0000!0) (i.e.k= 0), arising in the\nzero-temperature limit at !0= 0 due to the Fermi func-\ntion. Generally, a singularity of order kgives rise to the\nasymptotic behavior A(occ;unocc)\nloc(t)/t\u00001\u0000k, apart from\na purely oscillatory factor ei!0t. For the present case,\nthe van Hove singularities of A(occ;unocc)\nloc(!) at\u0006!0= 2\nexplain, via Eq. (34) the oscillation of \u0005(ret)(t) with fre-\nquency!\u0005= 2!0= 4.\nGenerally, the location of the van Hove singularity\non the frequency axis, i.e. !0, determines the oscilla-\ntion period while the decay of \u0005(ret)(t) is governed by\nthe strength of the singularity. Consider, as an exam-\nple, the zero-temperature case and assume that there are\nno van Hove singularities. The sharp Fermi edge im-\npliesA(occ;unocc)\nloc(t)/t\u00001, and thus \u0005(ret)(t)/t\u00002. The\nGilbert-damping constant is well de\fned in this case.\nThe strength of van Hove singularities depends on\nthe lattice dimension D.59For a one-dimensional lat-\ntice, we have van Hove singularities with k=\u00001=2, and\nthus \u0005(ret)(t)/t\u00001, consistent with Fig. (13). Here,\nthe strong van Hove singularity dominates the long-time\nasymptotic behavior as compared to the weaker Fermi-\nedge singularity. For D= 3, we have k= 1=2 and\n\u0005(ret)(t)/t\u00003if\f <1while for\f=1the Fermi-\nedge dominates and \u0005(ret)(t)/t\u00002. TheD= 2 case is\nmore complicated: The logarithmic van Hove singularity\n/lnj!jleads to \u0005(ret)(t)/t\u00002. This, however, applies\nto cases o\u000b half-\flling only. At half-\flling the van Hove\nand the Fermi-edge singularity combine to a singularity\n/\u0002(!) lnj!jwhich gives \u0005(ret)(t)/ln2(t)=t2. For \fnite\ntemperatures, we again have \u0005(ret)(t)/t\u00002.\nThe existence of the integral Eq. (27) depends on the\nt!1 behavior and either requires a decay as \u0005(ret)(t)/\nt\u00003or faster, or an asymptotic form \u0005(ret)(t)/ei!0t=t2\nwith an oscillating factor resulting from a non-zero po-\nsition!06= 0 of the van Hove singularity. For the one-\ndimensional case, we conclude that the LLG equation\n(with a time-independent damping constant) is based on\nan ill-de\fned concept. Also, the derivation of Eqs. (29)\nand (30) is invalid in this case as the !derivative and the\ntintegral do not commute. This conclusion might change\nfor the case of interacting conduction electrons. Here one\nwould expect a regularization of van Hove singularities\ndue to a \fnite imaginary part of the conduction-electron\nself-energy.\nVI. CONCLUSIONS\nHybrid systems consisting of classical spins coupled\nto a bath of non-interacting conduction electrons rep-\nresent a class of model systems with a non-trivial real-\ntime dynamics which is numerically accessible on longtime scales. Here we have considered the simplest vari-\nant of this class, the Kondo-impurity model with a clas-\nsical spin, and studied the relaxation dynamics of the\nspin in an external magnetic \feld. As a fundamental\nmodel this is interesting of its own but also makes con-\ntact with di\u000berent \felds, e.g., atomistic spin dynamics in\nmagnetic samples, spin relaxation in spintronics devices,\nfemto-second dynamics of highly excited electron systems\nwhere local magnetic moments are formed due to electron\ncorrelations, and arti\fcial Kondo systems simulated with\nultracold atoms in optical lattices.\nWe have compared the coupled spin and electron dy-\nnamics with the predictions of the widely used Landau-\nLifshitz-Gilbert equation which is supposed to cover the\nregime of weak local exchange Jand slow spin dynamics.\nFor the studied setup, the LLG equation predicts a rather\nregular time evolution characterized by spin precession,\nspin relaxation and eventually reversal of the spin on a\ntime scale\u001cdepending on J(and the \feld strength B).\nWe have demonstrated that this type of dynamics can be\nrecovered and understood on a microscopic level in the\nmore fundamental quantum-classical Kondo model. It is\ntraced back to a non-adiabatic dynamics of the electron\ndegrees of freedom and the feedback of the electronic sub-\nsystem on the spin. It turns out that the spin dynamics\nis essentially a consequence of the retarded e\u000bect of the\nlocal exchange. Namely, the classical spin can be seen as\na perturbation exciting the conduction-electron system\nlocally. This electronic excitation propagates and feeds\nback to the classical spin, but at a later time, and thereby\ninduces a spin torque.\nWe found that this mechanism drives the relaxation of\nthe system to its local ground state irrespective of the\nstrength of the local exchange J. As the microscopic dy-\nnamics is fully conserving, the energy and spin of the\ninitial excitation which is locally stored in the vicinity of\nthe classical spin, must be dissipated into the bulk of the\nsystem in the course of time. This dissipation could be\nuncovered by studying the relaxation process from the\nperspective of the electron degrees of freedom. Dissipa-\ntion of energy and spin takes place through the emission\nof a dispersive spin-polarized wave packet propagating\nthrough the lattice with the Fermi velocity. In this pro-\ncess the local conduction-electron magnetic moment at\nany given distance to the impurity undergoes a reversal,\ncharacterized by precession and relaxation, similar to the\nmotion of the classical spin.\nThe dynamics of the classical spin can be qualitatively\nvery di\u000berent from the predictions of the LLG equation\nfor strongJ. In this regime we found a complex mo-\ntion characterized by oscillations of the angle between\nthe classical spin S(t) and the local conduction-electron\nmagnetic moment at the impurity site hsi0iaround the\nadiabatic value \r=\u0019which takes place on an emergent\nnew time scale.\nIn the weak- Jlimit, the classical spin dynamics is qual-\nitatively predicted correctly by the LLG equation. At\nleast partially, however, this must be attributed to the14\nfact that the LLG approach, by construction, recovers\nthe correct \fnal state where the spin is parallel to the\n\feld. In fact, quantitative deviations are found during\nthe relaxation process. The LLG approach is based on\n\frst-order perturbation theory in Jand on the additional\nassumption that the classical spin is slow. To pinpoint\nthe source of the deviations, we have numerically solved\nthe integro-di\u000berential equation that is obtained in \frst-\norder-in-Jperturbation theory and compared with the\nfull hybrid dynamics. The deviations of the perturbative\napproach from the exact dynamics are found to gradually\nincrease with the propagation time (until the proximity\nto the \fnal state enforces the correct long-time asymp-\ntotics). This is the expected result as the dimensionless\nsmall parameter is Jt. However, with increasing Jthe\ntime scale on which perturbation theory is reliable de-\ncreases much stronger than 1 =Jdue to a strong enhance-\nment of retardation e\u000bects which make the perturbation\nmore e\u000bective and produce a stronger torque.\nGenerally, the perturbation can be rather ine\u000bective in\nthe sense that it produces a torque /S(t)\u0002S(t0) which\nis very weak if the process is nearly adiabatic. This ex-\nplains that \frst-order perturbation theory and the LLG\nequation is applicable at all for couplings of the order of\nhoppingJ\u0018T. For the present study this can also be\nseen as a fortunate circumstance since the regime of very\nweak couplings J\u001cTis not accessible numerically. In\nthis case the spin-reversal time scale gets so large that\nthe propagation of excitations in the conduction-electron\nsubsystem would by a\u000bected by backscattering from the\nedges of the system which necessarily must be assumed\nas \fnite for the numerical treatment.\nFor the one-dimensional lattice studied here, a di-\nrect comparison between LLG equation and the exact\nquantum-classical theory is not meaningful as the damp-\ning constant \u000bis ill-de\fned in this case. We could argue\nthat the problem results from the strength of the vanHove singularities in the conduction-electron density of\nstates which dictates the long-time behavior of the mem-\nory kernel of the integro-di\u000berential equation which is\ngiven by the equilibrium spin susceptibility. As the type\nof the van Hove singularity is characteristic for all sys-\ntems of a given dimension, we can generally conclude that\nthe LLG approach reduces to a purely phenomenological\nscheme in the one-dimensional case. However, it is an\nopen question, which will be interesting to tackle in the\nfuture, if this conclusion is still valid for systems where\nthe Coulomb interaction among the conduction electrons\nis taken into account additionally.\nThere are more interesting lines of research which are\nbased on the present work and could be pursued in the\nfuture. Those include systems with more than a single\nspin where, e.g., the e\u000bects of a time-dependent and\nretarded RKKY interaction can be studied additionally.\nWe are also working on a tractable extension of the the-\nory to account for longitudinal \ructuations of the spins\nto include time-dependent Kondo screening, and the\ncompetition with RKKY coupling, on a time-dependent\nmean-\feld level. 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It is shown that such resonances can\nbe manipulated by tuning up either the coe\u000ecient of the fractional damping or the\norder of the corresponding fractional derivatives.\n1 Introduction\nThe simplest oscillating system (a harmonic oscillator ) can be modeled by a mass at the\nend of a spring which slides back and forth without friction. The motion is characterized by\nthe natural frequency of oscillation !0and the total stored energy E(which is a constant\nof motion and de\fnes the amplitude of oscillation) [1]. Actual oscillating systems present\nsome loss of energy due to friction forces so that the amplitude of their oscillations is\na decreasing function of time. However, the oscillations can be driven to avoid their\ndamping down by the action of a repetitive force F(t) on the system. Such a system\nis called driven damped oscillator [2]. An special excitation of the system arises when\nthe frequency of the applied force matches the natural frequency of the oscillator since\nthe spectral energy distribution takes its maximum value. The phenomenon, known as\nresonance , is a subject of study in classical mechanics, electromagnetism, optics, acoustics\nand quantum mechanics, among other physical theories [3].\nThe present work is addressed to the study of the driven damped oscillator in the con-\ntext of fractional calculus [4{6]. That is, the second-order di\u000berential equation associated\nwith the Newtonian law of motion for a damped oscillator that is driven by an external\nforce will be substituted by a fractional di\u000berential equation of order 2 \u000b, with 0<\u000b\u00141.\nSpecial emphasis will be placed on the resonance phenomenon.\n2 Fractional Oscillator with fractional damping\nGiven an oscillator of natural frequency !0, the general expression for the displacement\nof the mass can be expressed as the integral equation [8,9]\nx(t) =x0+It( _x0) +!2\n0It[It(x(t))]; (1)\n1arXiv:1706.08596v1  [physics.class-ph]  15 Jun 2017wherex0and _x0are constants of integration, and\nIt(x(t)) :=Z\nx(t)dt (2)\nrepresents the Riemman time-integration of x(t). The fractional generalization of (1) is\nperformed in two steps. First we replace Itwith the Riemman-Liouville fractional time-\nintegral operator I\u000b[4{6], and!0with!\u000b\n0. The latter for consistency of units. Then we\nhave\nx(t) =x0+I\u000b( _x0) +!2\u000b\n0I\u000b[I\u000b(x(t))];0<\u000b\u00141: (3)\nNow, a fractional di\u000berential form of (3) can be obtained by applying twice the time-\nfractional derivative operator of Caputo\nD\u000bD\u000bx(t) +!2\u000bx(t) = 0 (4)\n(for details about the operator D\u000bsee, e.g., [5]). Let us introduce a `fractional damping'\nwhich is proportional to the fractional time-derivative of the position D\u000bx(t). That is\nD\u000bD\u000bx(t) + 2\f\u000bD\u000bx(t) +!2\u000bx(t) = 0: (5)\nOne can show that the solution of this last equation is of the form\nx(t) =x0t\u0000\u000b\np\n\f2\u000b\u0000!2\u000b\n0[E\u000b;1\u0000\u000b(\u0000\n\u0000t\u000b)\u0000E\u000b;1\u0000\u000b(\u0000\n+t\u000b)]\n+2\f\u000bx0+x(\u000b)\n0p\n\f2\u000b\u0000!2\u000b\n0[E\u000b;1(\u0000\n\u0000t\u000b)\u0000E\u000b;1(\u0000\n+t\u000b)];(6)\nwhere\nE\u000b;\f(z) =1X\nk=0zk\n\u0000(\u000bk+\f);Re(\u000b)>0;Re(\f)>0; z2C; (7)\nis the Mittag-Le\u000fer function [7], and\n\n\u0006=\f\u000b\u0006q\n\f2\u000b\u0000!2\u000b\n0:\n3 Driven fractional oscillator with fractional damp-\ning\nLet us add a driving force at the right hand side of Eq. (5), we have\nD\u000bD\u000bx(t) +\f\u000bD\u000bx(t) +!2\u000bx(t) =f0cos (!t+\u001e); (8)\nwheref0is a real constant. Applying the Laplace transform Land solving for X(s) =\nL[x(t)] we arrive at the expression\nX(s) =f0\u0014scos\u001e\u0000!sin\u001e\n(s2+!2) (s2\u000b+ 2\f\u000bs\u000b+!2\u000b\n0)\u0015\n+x0s2\u000b\u00001+\u0010\nx0+ 2\f\u000bx(\u000b)\n0\u0011\ns\u000b\u00001\ns2\u000b+ 2\f\u000bs\u000b+!2\u000b\n0: (9)\n2Makingf0= 0 we see that the second term in (9) corresponds to transient oscillations\nbecause there is no force present which can ensure their predominance; the corresponding\ninverse Laplace transform has been evaluated in the previous section. In turn, for the\ninverse Laplace transform of the \frs term one has\nx(t) =f0\n2\u0019ilim\nT!1Zg+iT\ng\u0000iTest\u0014scos\u001e\u0000!sin\u001e\n(s2+!2) (s2\u000b+ 2\f\u000bs\u000b+!2\u000b\n0)\u0015\nds: (10)\nThe integrand contains a branch point at s= 0 and simple poles at s=\u0006i!, and\ns= (\n\u0006e\u0006i\u0019)1=\u000b. Following [10] we \fnd that x(t) is given as the sum of three contributions:\nx1(t),x2(t) andx3(t). The \frst one results from the calculation of (10) along the Hankel-\nBromwich path shown in Fig. 1 of Ref. [10], we obtain\nx1(t) =f0\n\u0019Z1\n0e\u0000rt[rcos\u001e +!sin\u001e ]\u0002\nr2\u000bsin(2\u0019\u000b) + 2\f\u000br\u000bsin(\u0019\u000b)\u0003\n(r2+!2) [r4\u000b+ 4\f2\u000br2\u000b+!4\u000b+ 4\f\u000br3\u000bcos(\u000b\u0019) + 4\f\u000br\u000b!2\u000b\n0+ 2r2\u000b!2\u000b\n0cos(2\u000b\u0019)]dr:\n(11)\nThe latter expression vanishes as t! 1 . On the other hand, the sum of residues\nassociated with the poles s= (\n\u0006e\u0006i\u0019)1=\u000bgives\nx2(t) =2f0et\r+cos(\u0019=\u000b)\n\r\u000b\u00001\n+(\r\u000b\n\u0000\u0000\r\u000b\n+)\"\n!2\r+cos\u001ecos\u0000\nt\r+sin(\u0019=\u000b)\u0000\u0019\n\u000b(\u000b\u00002)\u0001\n+\r3\n+cos\u001ecos (t\r+sin(\u0019=\u000b)\u0000\u0019)\n!4+\r4\n++ 2!2\r2\n+cos (2\u0019=\u000b)#\n+2f0et\r\u0000cos(\u0019=\u000b)\n\r\u000b\u00001\n\u0000(\r\u000b\n+\u0000\r\u000b\n\u0000)\"\n!2\r\u0000cos\u001ecos\u0000\nt\r\u0000sin(\u0019=\u000b)\u0000\u0019\n\u000b(\u000b\u00002)\u0001\n+\r3\n\u0000cos\u001ecos (t\r\u0000sin(\u0019=\u000b)\u0000\u0019)\n!4+\r4\n\u0000+ 2!2\r2\n\u0000cos (2\u0019=\u000b)#\n+2f0et\r+cos(\u0019=\u000b)\n\r\u000b\u00001\n+(\r\u000b\n\u0000\u0000\r\u000b\n+)\"\n!3sin\u001ecos\u0000\nt\r+sin(\u0019=\u000b)\u0000\u0019\n\u000b(\u000b\u00001)\u0001\n+!2\r2\n+sin\u001ecos\u0000\nt\r+sin(\u0019=\r)\u0000\u0019\n\u000b(\u000b+ 1)\u0001\n!4+\r4\n++ 2!2\r2\n+cos (2\u0019=\u000b)#\n+2f0et\r\u0000cos(\u0019=\u000b)\n\r\u000b\u00001\n\u0000(\r\u000b\n+\u0000\r\u000b\n\u0000)\"\n!3sin\u001ecos\u0000\nt\r\u0000sin(\u0019=\u000b)\u0000\u0019\n\u000b(\u000b\u00001)\u0001\n+!2\r2\n\u0000sin\u001ecos\u0000\nt\r\u0000sin(\u0019=\r)\u0000\u0019\n\u000b(\u000b+ 1)\u0001\n!4+\r4\n\u0000+ 2!2\r2\n\u0000cos (2\u0019=\u000b)#\n;\n(12)\nwhere\r\u0006= \n1=\u000b\n\u0006. The term x2(t) is parameterized by the order of the Caputo operator\nD\u000b, the coe\u000ecient \fof the fractional damping, and the natural frequency !0; the ap-\npropriate combination of these three parameters produces x2!0 ast!1 . Further\ndetails will be reported elsewhere. On the other hand, the term associated with the poles\ns=\u0006i!is of the form\nx3(t) =Acos(!t+\u000e); (13)\nwhere the amplitude and phase are respectively given by\nA=f0q\n!4\u000b+!4\u000b\n0+ 2!2\u000b!2\u000b\n0cos (\u0019\u000b\u00002\u001e) + 4!\u000b\f\u000b\u0002\n!\u000b\f\u000b+ (!2\u000b+!2\u000b\n0) cos\u0000\u0019\u000b\n2\u00002\u001e\u0001\u0003\n!4\u000b+!4\u000b\n0+ 2!2\u000b!2\u000b\n0cos (\u0019\u000b) + 4!\u000b\f\u000b\u0002\n!\u000b\f\u000b+ (!2\u000b+!2\u000b\n0) cos\u0000\u0019\u000b\n2\u0001\u0003;(14)\nand\n\u000e= arctan\"\n\u0000!2\u000bsin (\u0019\u000b\u0000\u001e) +!2\n0sin (\u001e) + 2\f\u000b!\u000bsin\u0000\u0019\u000b\n2\u0001\n!2\u000bcos (\u0019\u000b\u0000\u001e) +!2\n0cos (\u001e) + 2\f\u000b!\u000bcos\u0000\u0019\u000b\n2\u0001#\n: (15)\n3As in the conventional case, the amplitude Aof the oscillations dictated by x3(t) is\nproportional to the amplitude f0of the driving force. At zero frequency !(i.e., for a\nconstant driving force), the quotient \u0003 = A=f 0becomes!\u00002\u000b\n0which, in turn, reproduces\nthe (low frequencies) Newtonian result for \u000b= 1. At very high frequencies we \fnd\n\u0003\u0019!\u00002\u000b, so that the external driving force is dominant. On the other hand, at !=!0\nwith\fand\u001e\fxed, we \fnd that \u0003 is as larger as \u000bapproaches the value \u000b= 1 and\nbecomes smaller for \u000b2(0;1=2). That is, given the fractional damping parameter \f, the\nfractional system behaves as an underdamped oscillator for \u000b!1, and as an overdamped\none if\u000bapproximates 1 =2 from above.\nFigure 1: (Color online) The quotient \u0003 = A=f 0of the amplitude of oscillation Ade\fned in (14) and\nthe amplitude f0of the driving force introduced in (8) with \u001e= 0,!0= 1,\f= 0:1!0, for\u000b= 0:99\n(dotted-black), \u000b= 0:95 (solid-red), and \u000b= 0:90 (dashed-blue).\nThe behavior of \u0003 is depicted in Fig. 1 as a function of the frequency !with\u001e,!0\nand\f\fxed, and for di\u000berent values of \u000b. For\u000b= 1 (i.e., for the Newtonian case) we\n\fnd that \u0003 reaches its maximum value when the driving force oscillates at the natural\nfrequency!0, as expected. Such a large response of the system to the driving force is\nthe \fngerprint of a resonance. Notice however that the maximum decreases and shifts to\nthe left as\u000bdecreases. That is, for \u000b.1 the resonance occurs at a frequency !which\nis lower than !0. The latter means that the resonances can be controlled by \fxing the\nfractional-damping parameter \fand tuning up the order of the fractional derivative D\u000b.\nThe same holds if one \fxes the value of \u000band adjust the fractional-damping parameter \f.\n4 Conclusions\nUsing fractional calculus one \fnds that the classical harmonic oscillator is a\u000bected by an\n`intrinsic damping' [8, 9], such a damping is also present in the quantum-fractional case\n[11]. The response of the classical fractional oscillator to the presence of a driving force has\nbeen already studied in e.g. [10]. In this paper we have presented some preliminary results\nof our study on a driven fractional oscillator which is a\u000bected by a fractional damping\nof the form \f\u000bD\u000bx(t), withD\u000bthe Caputo time-derivative operator and 0 < \u000b\u00141. In\n4particular, we have shown that the resonance phenomenon can be controlled by tuning up\neither the coe\u000ecient \fof the fractional-damping or the order \u000bof the Caputo operator.\nFurther results will be reported elsewhere.\n5 Acknowledgments\nF.O.R. acknowledges the funding received through a CONACyT scholarship.\nReferences\n[1] French A P, Vibrations and waves , W W Norton, New York, 1971\n[2] Taylor J R, Classical Mechanics University Science Books 2005\n[3] Rosas-Ortiz O, Fern\u0013 andez Garc\u0013 \u0010a N, Cruz y Cruz S, AIP Conf. Proc. 1077 (2008)\n31\n[4] Miller K S and Ross B, An introduction to the fractional calculus and fractional\ndi\u000berential equations , John Wiley, New York, 1993\n[5] Podlubny I, Fractional Di\u000berential Equations , Academic Press, New York, 1999\n[6] Hilfer R, Applications of Fractional Calculus in Physics , World Scienti\fc, New York,\n2000\n[7] Erd\u0013 elyi A, Higher Transcendental Functions , Vol. III, McGraw-Hill, New York, 1955\n[8] Narahari A B N, Hanneken J W, Enck T and Clarke T, Physica A 297(2001) 361\n[9] Olivar-Romero F, A \frst approach to the fractional quantum mechanics , M.Sc. Thesis\n(in Spanish), Physics Department, Cinvestav, M\u0013 exico City, 2014\n[10] Narahari A B N, Hanneken J W and Clarke T, Physica A 309(2002) 275\n[11] Olivar-Romero F and Rosas-Ortiz O, J. Phys. Conf. Ser. 698(2016) 012025\n5"    },    {        "title": "1706.03942v1.Uniform_energy_decay_for_wave_equations_with_unbounded_damping_coefficients.pdf",        "content": "arXiv:1706.03942v1  [math.AP]  13 Jun 2017Uniform Energy Decay for Wave Equations with\nUnbounded Damping Coefficients\nRyo IKEHATA∗\nDepartment of Mathematics, Graduate School of Education\nHiroshima University\nHigashi-Hiroshima 739-8524, Japan\nHiroshi TAKEDA†\nDepartment of Intelligent Mechanical Engineering, Faculty of Engin eering\nFukuoka Institute of Technology\nFukuoka 811-0295, Japan\nOctober 14, 2018\nAbstract\nWeconsiderthe Cauchyproblemforwaveequationswith unbounded dampingcoefficients\ninRn. For a general class of unbounded damping coefficients, we derive u niform total energy\ndecay estimates together with a unique existence result of a weak s olution. In this case we\nnever impose strongassumptions such as compactness ofthe sup port of the initial data. This\nmeans that we never rely on the finite propagation speed property of the solutions, and we\ntry to deal with an essential unbounded coefficient case. One of ou r methods comes from an\nidea developed in [9].\n1 Introduction\nWe consider the mixed problem for wave equations with a local ized damping in Rn(n≥1)\nutt(t,x)−∆u(t,x)+a(x)ut(t,x) = 0,(t,x)∈(0,∞)×Rn, (1.1)\nu(0,x) =u0(x), ut(0,x) =u1(x), x∈Rn, (1.2)\nwhere (u0,u1) are initial data chosen as:\nu0∈H2(Rn), u1∈H1(Rn),\nand\nut=∂u\n∂t, utt=∂2u\n∂t2,∆ =n/summationdisplay\nj=1∂2\n∂x2\nj, x= (x1,···,xn).\nNote that solutions and/or functions considered in this pap er are all real valued except for\nseveral parts concerning the Fourier transform.\n∗Corresponding author: ikehatar@hiroshima-u.ac.jp\n†h-takeda@fit.ac.jp\nKeywords and Phrases: Unbounded damping; Wave equation; Ca uchy problem; Weighted initial data; Mul-\ntiplier method; Fourier analysis; Total energy decay, Weak solutions.\n2010 Mathematics Subject Classification. Primary 35L05; Se condary 35B35, 35B40.\n1Concerning the decay or non-decay property of the total or lo cal energy to problem (1.1)-\n(1.2) withx-dependent variable damping coefficients, many research man uscripts are already\npublished by Alloui-Ibrahim-Khenissi [1], Bouclet-Royer [2], Daoulatli [5], Ikehata [7], Ikehata-\nTodorova-Yordanov[10],Joly-Royer [11],Kawashita[13], Khader[12],Matsumura[16], Mochizuki\n[18], Mochizuki-Nakazawa [19], Nakao [21], Nishiyama [23] , Nishihara [22], Radu-Todorova-\nYordanov [24], Sobajima-Wakasugi [26], Todorova-Yordano v [28], Uesaka [29] and Wakasugi\n[30], Zhang [32] and the references therein. However, we sho uld emphasize that those cases\nare quite restricted to the bounded damping coefficient case, i.e.,a∈L∞(Rn). This condition\nseems to be essential to get the uniqueexistence of mild or we ak solutions to problem (1.1)-(1.2).\nSo, when we do not assume the boundedness of the coefficient a(x), a natural question arises\nwhether one can construct a unique weak or mild solution u(t,x) to problem (1.1)-(1.2) together\nwith some decay property of the total energy or not. Quite rec ently Sobajima-Wakasugi [27]\nhave announced an interesting result from the viewpoint of t he diffusion phenomenon of the\nsolution to the equation (1.1) together with a unique global existence result. The mixed prob-\nlem to the equation (1.1) in [27] is considered in the exterio r domain of a bounded obstacle,\nand the Dirichlet null boundary condition is treated. They t reated typically a(x) =a0|x|αwith\na0>0 andα >0. But, their results require a stronger assumption such as t he compactness\nof the support of the initial data. This implies that they hav e to rely on the finite speed of\npropagation property (FSPP for short) of the solution, so th at in an essential meaning, their\nframework seems to be still restricted to the bounded dampin g coefficient case for each t≥0.\nIn the treatment of the unbounded coefficient a(x), it seems important and interesting not to\nassume such FSPP. Additionally, they essentially used rath er stronger regularity condition on\nthe initial data such that [ u0,u1]∈(H2∩H1\n0)×H1\n0. In connection with this topic, D’Abbicco\n[3] (and for a more general class, see D’Abbicco-Ebert [4] an d Reissig [25]) and Wirth [31] have\never studied t-dependent unbounded damping coefficient case:\nutt(t,x)−∆u(t,x)+b(1+t)αut(t,x) =f(u),\nwhereα∈[0,1] andb>0. Therefore, in the x-dependent unbounded coefficient case, a unique\nexistence of the solution itself together with some decay pr operty of the total energy are com-\npletely open in the framework of non-compactly supported in itial data class. When Sobajima-\nWakasugi [27] constructs a unique solution, they have used d irectly the well-known result due\nto Ikawa [6]. That means their result is still under the alrea dy known framework. In our result\nto be announced, we have to discuss how we should construct a w eak solution itself because no\nany well-known theories can be applied directly. This is a ma in difficulty in our result.\nIn this connection, originally Komatsu [14] first proposed t his open question in his Master\nthesis in January 2016 such that for the unbounded damping co efficient case a /∈L∞(Rn),\ncan one construct a global in time solution? Unfortunately, Komatsu [14] could not solve his\nproblem before finishing Master course. This paper gives an a nswer to his problem.\nNow, let us start with introducing our new result. Before sta ting our result, we shall give\nthe following only one assumption ( A) on the damping coefficient a(x), and the definition of the\nsolution to be constructed.\n(A)a∈C(Rn), and there exists a constant V0>0 such that 0 <V0≤a(x) (∀x∈Rn).\nDefinition. A function u: [0,∞)×Rn→Ris called as the weak solution if it satisfies\n/integraldisplay∞\n0/integraldisplay\nRnu(t,x)(φtt(t,x)−∆φ(t,x)−a(x)φt(t,x))dxdt\n=/integraldisplay\nRnu1(x)φ(0,x)dx−/integraldisplay\nRnu0(x)φt(0,x)dx+/integraldisplay\nRna(x)u0(x)φ(0,x)dx\n2for anyφ∈C∞\n0([0,∞)×Rn).\nOur new result reads as follows.\nTheorem 1.1 Letn≥3and assume (A). If the initial data [u0,u1]∈(H2(Rn)∩L1(Rn))×\n(H1(Rn)∩L1(Rn))further satisfies a(·)u0∈L1(Rn)∩L2(Rn), then there exists a unique weak\nsolutionu∈L∞(0,∞;H1(Rn))∩W1,∞(0,∞;L2(Rn))to problem (1.1)-(1.2)satisfying\n/ba∇dblu(t,·)/ba∇dbl2≤CI2\n00(1+t)−1, Eu(t)≤CI2\n00(1+t)−2,\nwith some constant C >0, where\nI00:=/parenleftBig\n/ba∇dblu0/ba∇dbl2+/ba∇dbl∇u0/ba∇dbl2+/ba∇dbla(·)u0/ba∇dbl2\n1+/ba∇dbla(·)u0/ba∇dbl2+/ba∇dblu1/ba∇dbl2+/ba∇dblu1/ba∇dbl2\n1/parenrightBig1/2.\nRemark 1.1 If one considers the mixed problem (1.1)-(1.2) with the Diri chlet null boundary\ncondition on the smooth exterior domain, one can treat the tw o dimensional case. In that case,\nwe need to assume the logarithmic type weight condition such that/ba∇dbllog(B|x|)(a(·)u0+u1)/ba∇dbl<\n+∞on the initial data with some constant B >0. This has a close relation to the place where\none obtains Lemma 2.1 below in the two dimensional exterior d omain case. For more detail, one\ncan refer the reader to [9].\nRemark 1.2 The assumption [ u0,u1]∈H2(Rn)×H1(Rn) is not essential. It will be used\nonly to justify the integration by parts in the course of the p roof. That condition seems not\nto be so rare (cf. [19]). One may be able to generalize to the mo re weak case such that\n[u0,u1]∈H1(Rn)×L2(Rn), however, for simplicity we do not deal with such case.\nExample. As the typical unbounded example for a(x), we can choose a(x) := (1+ |x|2)α\n2with\nα∈[0,∞),a(x) :=e|x|, and so on.\nThis paper is organized as follows. In section 2 we shall prov e Theorem 1.1 by relying on a\nmultiplier method which was introduced in [9]. In section 3 w e give several remarks and open\nproblems. Section 4 is devoted to the appendix to check the kn own result.\nNotation. Throughout this paper, /ba∇dbl · /ba∇dblqstands for the usual Lq(Rn)-norm. For simplicity of\nnotation, in particular, we use /ba∇dbl·/ba∇dblinstead of /ba∇dbl·/ba∇dbl2. Furthermore, we denote /ba∇dbl·/ba∇dblH1as the usual H1-norm.\nTheL2-inner product is denoted by ( f,g) :=/integraldisplay\nRnf(x)g(x)dxforf,g∈L2(Rn). The total energy Eu(t)\ncorresponding to the solution u(t,x) of (1.1) is defined by\nEu(t) :=1\n2(/ba∇dblut(t,·)/ba∇dbl2+/ba∇dbl∇u(t,·)/ba∇dbl2),\nwhere\n|∇f(x)|2:=n/summationdisplay\nj=1|∂f(x)\n∂xj|2.\nThe weighted L1-space and its norm /ba∇dbl·/ba∇dbl1,mcan be defined as\nf∈L1,m(Rn)⇔f∈L1(Rn),/ba∇dblf/ba∇dbl1,m:=/integraldisplay\nRn(1+|x|)m|f(x)|dx<+∞.\nThe subspace Xn\nmofL1,mis defined by\nXn\nm:={f∈L1,m(Rn) :/integraldisplay\nRnf(x)dx= 0}.\n3And also, for the Hilbert space Xwe define a class of vector valued continuous functions C0([0,∞);X) as\nfollows:f∈C0([0,∞);X)ifandonlyif f∈C([0,∞);X)andtheclosureoftheset {t∈[0,∞);/ba∇dblf(t)/ba∇dblX/ne}ationslash=\n0}is compact in [0 ,∞).\nOn the other hand, we denote the Fourier transform of f(x) byˆf(ξ) := (1\n2π)n\n2/integraldisplay\nRne−ix·ξf(x)dxas\nusual with i:=√−1, and we define the usual convolution by\n(f∗g)(x) :=/integraldisplay\nRnf(x−y)g(y)dy.\n2 Proof of Theorem 1.1.\nInthecourse of theproof, thenextinequality concerningth eFourier image of theRiesz potential\nplays an alternative rolefortheHardyinequality. Thiscom es from [8, Proposition2.1]. Itshould\nbeemphasizedthat this inequality holdseven inthelow dime nsional caseif wecontrol theweight\nparameterγ.\nProposition 2.1 Letn≥1andγ∈[0,1].\n(1)Iff∈L2(Rn)∩L1,γ(Rn), andθ∈[0,n\n2), then there exists a constant C=Cn,θ,γ>0such\nthat\n/integraldisplay\nRn|ˆf(ξ)|2\n|ξ|2θdξ≤C/parenleftbigg\n/ba∇dblf/ba∇dbl2\n1,γ+|/integraldisplay\nRnf(x)dx|2+/ba∇dblf/ba∇dbl2/parenrightbigg\n.\n(2)Iff∈L2(Rn)∩Xn\nγ, andθ∈[0,γ+n\n2), then it is true that\n/integraldisplay\nRn|ˆf(ξ)|2\n|ξ|2θdξ≤C(/ba∇dblf/ba∇dbl2\n1,γ+/ba∇dblf/ba∇dbl2)\nwith some constant C=Cn,θ,γ>0.\nToconstructaglobalweaksolutionwefirstdefineasequenceo ftheweaksolutions {u(m)(t,x)}\n(m∈N) to the approximated problem below:\nu(m)\ntt(t,x)−∆u(m)(t,x)+am(x)u(m)\nt(t,x) = 0,(t,x)∈(0,∞)×Rn, (2.1)\nu(m)(0,x) =u0(x), u(m)\nt(0,x) =u1(x), x∈Rn, (2.2)\nwheream∈C(Rn) can be chosen to satisfy\nam(x) =/braceleftBigga(x) (|x| ≤m)\nV0 (|x|>m+1),(2.3)\nand\nV0≤am(x)≤a(x), am(x)→a(x)as m→ ∞(pointwise )x∈Rn(2.4)\nfor eachx∈Rn.\n4Now let us consider the problem (2.1)-(2.2) with initial dat a [u0,u1]∈H2(Rn)×H1(Rn).\nThen, for each m∈Nsinceam∈C(Rn)∩L∞(Rn) it is well known that the Cauchy prob-\nlem (2.1)-(2.2) has a unique strong solution u(m)∈C([0,∞);H2(Rn))∩C1([0,∞);H1(Rn))∩\nC2([0,∞);L2(Rn)) satisfying the energy identity:\nEu(m)(t)+/integraldisplayt\n0/integraldisplay\nRnam(x)|u(m)\ns(s,x)|2dxds=E(0), (2.5)\nwhere\nE(0) :=1\n2(/ba∇dblu1/ba∇dbl2+/ba∇dbl∇u0/ba∇dbl2).\nTo begin with we prove the following crucial estimate. The le mma below is a combination\nof the method introduced in [9] (= the modified Morawetz metho d [20]) and Proposition 2.1.\nLemma 2.1 Letn≥3. Under the same assumptions as in Theorem 1.1, the(unique)solution\nu(m)(t,x)to problem (2.1)-(2.2) satisfies\n/ba∇dblu(m)(t,·)/ba∇dbl2+/integraldisplayt\n0/integraldisplay\nRnam(x)|u(m)(s,x)|2dxds\n≤C/parenleftBig\n/ba∇dbla(·)u0/ba∇dbl2\n1+/ba∇dbla(·)u0/ba∇dbl2+/ba∇dblu1/ba∇dbl2\n1+/ba∇dblu1/ba∇dbl2/parenrightBig\n=:CI2\n0\nwith a constant C >0, whereCis independent of m.\nProof.The original idea comes from [9]. For the solution u(m)(t,x) to problem (2.1)-(2.2), one\nintroduces an auxiliary function\nW(t,x) :=/integraldisplayt\n0u(m)(s,x)ds.\nThenW(t,x) satisfies\nWtt−∆W+am(x)Wt=am(x)u0+u1,(t,x)∈(0,∞)×Rn, (2.6)\nW(0,x) = 0, Wt(0,x) =u0(x), x∈Rn. (2.7)\nMultiplying (2 .6) byWtand integrating over [0 ,t]×Rnwe get\n1\n2(/ba∇dblWt(t,·)/ba∇dbl2+/ba∇dbl∇W(t,·)/ba∇dbl2)+/integraldisplayt\n0/ba∇dbl/radicalBig\nam(·)Ws(s,·)/ba∇dbl2ds\n=1\n2/ba∇dblu0/ba∇dbl2+/integraldisplayt\n0(am(·)u0+u1,Ws(s,·))ds. (2.8)\nNext one uses (1) of Proposition 2.1 with θ= 1 andγ= 0, the Plancherel theorem and the\nCauchy-Schwarz inequality to obtain a series of inequaliti es below:\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0(am(·)u0+u1,Ws(s,·))ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0d\nds(am(·)u0+u1,W(s,·))ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRn(am(x)u0(x)+u1(x))W(t,x)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRn\nξ(/hatwidest(amu0)(ξ)+ ˆu1(ξ))ˆW(t,ξ)dξ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n5≤/integraldisplay\nRn\nξ|/hatwidest(amu0)(ξ)+ ˆu1(ξ)|\n|ξ|(|ξ||ˆW(t,ξ)|)dξ\n≤/parenleftBigg/integraldisplay\nRn\nξ|/hatwidest(amu0)(ξ)+ ˆu1(ξ)|2\n|ξ|2dξ/parenrightBigg1/2/parenleftBigg/integraldisplay\nRn\nξ|ξ|2|ˆW(t,ξ)|2dξ/parenrightBigg1/2\n≤/integraldisplay\nRn\nξ|/hatwidest(amu0)(ξ)+ ˆu1(ξ)|2\n|ξ|2dξ+1\n4/integraldisplay\nRn\nξ|ξ|2|ˆW(t,ξ)|2dξ\n≤C/ba∇dblamu0+u1/ba∇dbl2\n1+/ba∇dblamu0+u1/ba∇dbl2+C/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRn(am(x)u0(x)+u1(x))dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+1\n4/ba∇dbl∇W(t,·)/ba∇dbl2\n≤C/parenleftbigg\n/ba∇dblam(·)u0/ba∇dbl2\n1+/ba∇dblu1/ba∇dbl2\n1+/ba∇dblam(·)u0/ba∇dbl2+/ba∇dblu1/ba∇dbl2+|/integraldisplay\nRn(am(x)u0(x)+u1(x))dx|2/parenrightbigg\n+1\n4/ba∇dbl∇W(t,·)/ba∇dbl2\n(2.9)\nwith some constant C >0. Combining (2 .8) with (2.9) we can derive\n1\n2/ba∇dblWt(t,·)/ba∇dbl2+1\n4/ba∇dbl∇W(t,·)/ba∇dbl2+/integraldisplayt\n0/integraldisplay\nRnam(x)|Ws(s,x)|2dxds\n≤C/parenleftbigg\n/ba∇dblam(·)u0/ba∇dbl2\n1+/ba∇dblu1/ba∇dbl2\n1+/ba∇dblam(·)u0/ba∇dbl2+/ba∇dblu1/ba∇dbl2+|/integraldisplay\nRn(am(x)u0(x)+u1(x))dx|2/parenrightbigg\nwith some constant C >0. Sinceam(x)≤a(x), in the case when n≥3 one has\n1\n2/ba∇dblWt(t,·)/ba∇dbl2+1\n4/ba∇dbl∇W(t,·)/ba∇dbl2+/integraldisplayt\n0/integraldisplay\nRnam(x)|Ws(s,x)|2dxds\n≤C/parenleftbigg\n/ba∇dbla(·)u0/ba∇dbl2\n1+/ba∇dblu1/ba∇dbl2\n1+/ba∇dbla(·)u0/ba∇dbl2+/ba∇dblu1/ba∇dbl2+|/integraldisplay\nRn(a(x)|u0(x)|+|u1(x)|)dx|2/parenrightbigg\n.\nWe easily see that the constant Cin the above estimates is independent of m. Thus, one has\nthe desired estimate because of the fact Wt=u(m). ✷\nLemma 2.2 Under the same assumptions as in Theorem 1.1, the(unique)solutionu(m)(t,x)\nto problem (2.1)-(2.2) satisfies\n(1+t)Eu(m)(t)≤E(0)(1+1\nV0+1\n2ε)+1\n2(u1,u0) =:I2\n1,(t≥0),\n/integraldisplayt\n0Eu(m)(s)ds≤I2\n1,(t≥0),\nwith some small constant ε>0, whereCis independent of m.\nProof.For simplicity of notation, we use w(t,x) in place of u(m)(t,x), i.e.,w(t,x) satisfies the\nequation below:\nwtt(t,x)−∆w(t,x)+am(x)wt(t,x) = 0,(t,x)∈(0,∞)×Rn, (2.10)\nw(0,x) =u0(x), wt(0,x) =u1(x), x∈Rn. (2.11)\nNote thatEw(t) =Eu(m)(t) satisfies (2.5). Then, since\nd\ndt{(1+t)Ew(t)} ≤Ew(t),\n6it follows from (2.5) that\n(1+t)Ew(t)≤E(0)+1\n2/integraldisplayt\n0/ba∇dblws(s,·)/ba∇dbl2ds+1\n2/integraldisplayt\n0/ba∇dbl∇w(s,·)/ba∇dbl2ds\n≤E(0)+1\n2V0/integraldisplayt\n0/integraldisplay\nRnam(x)|ws(s,x)|2dxds+1\n2/integraldisplayt\n0/ba∇dbl∇w(s,·)/ba∇dbl2ds\n≤E(0)+1\n2V0E(0)+1\n2/integraldisplayt\n0/ba∇dbl∇w(s,·)/ba∇dbl2ds. (2.12)\nOn the other hand, by multiplying both sides of (2.10) by w(t,x) it follows that\nd\ndt(wt(t,·),w(t,·)) +/ba∇dbl∇w(t,·)/ba∇dbl2+1\n2d\ndt/integraldisplay\nRnam(x)|w(t,x)|2dx=/ba∇dblwt(t,·)/ba∇dbl2,(2.13)\nso that by integrating both sides over [0 ,t] one has\n/integraldisplayt\n0/ba∇dbl∇w(s,·)/ba∇dbl2ds+1\n2/integraldisplay\nRnam(x)|w(t,x)|2dx\n=/integraldisplayt\n0/ba∇dblws(s,·)/ba∇dbl2ds−(wt(t,·),w(t,·)) +(u1,u0)\n≤1\nV0/integraldisplayt\n0/integraldisplay\nRnam(x)|ws(s,x)|2dxds+1\n2ε/ba∇dblwt(t,·)/ba∇dbl2+ε\n2/ba∇dblw(t,·)/ba∇dbl2+(u1,u0)\n≤1\nV0/integraldisplayt\n0/integraldisplay\nRnam(x)|ws(s,x)|2dxds+1\nεEw(t)+ε\n2V0/integraldisplay\nRnam(x)|w(t,x)|2dx+(u1,u0)\n≤1\nV0E(0)+1\nεE(0)+ε\n2V0/integraldisplay\nRnam(x)|w(t,x)|2dx+(u1,u0),\nwhere we have just used (2.5) and the Cauchy-Schwarz inequal ity with some positive parameter\nε>0. This implies\n/integraldisplayt\n0/ba∇dbl∇w(s,·)/ba∇dbl2ds+1\n2(1−ε\nV0)/integraldisplay\nRnam(x)|w(t,x)|2dx\n≤E(0)(1\nV0+1\nε)+(u1,u0). (2.14)\nBychoosing ε>0sufficiently small, from(2.13) and(2.14) onecanget thedes iredtwo estimates.\nIn this final check, because of (2.6) we have to make the follow ing estimate once more:\n/integraldisplayt\n0/ba∇dblwt(s,·)/ba∇dbl2ds≤1\nV0/integraldisplay\nRnam(x)|ws(s,x)|2dxds≤1\nV0E(0).\n✷\nLemma 2.3 Under the same assumptions as in Theorem 1.1, the(unique)solutionu(m)(t,x)\nto problem (2.1)-(2.2) satisfies\n(1+t)2Eu(m)(t)≤E(0)+1\nV0(E(0)+I2\n1)+(u1,u0)+1\n2/integraldisplay\nRna(x)|u0(x)|2dx+C\n2I2\n0+I2\n1\nε=:I2\n2,\n(1+t)/ba∇dblu(m)(t,·)/ba∇dbl2≤2(V0−ǫ)−1{(u1,u0)+1\n2/integraldisplay\nRna(x)|u0(x)|2dx+C\n2I2\n0+1\nV0(E(0)+I2\n1)+I2\n1\nε}=:I2\n3\nwith some constant C >0, whereCis independent of m.\n7Proof.We use the same notation as in the proof of Lemma 2.2. Then, by m ultiplying both sides\nof (2.11) by (1+ t)w, and integrating it over [0 ,t]×Rnone can arrive at the important identity:\n1\n2/ba∇dblu0/ba∇dbl2+/integraldisplayt\n0(1+s)/ba∇dbl∇w(s,·)/ba∇dbl2ds+(1+t)\n2/integraldisplay\nRnam(x)|w(t,x)|2dx\n=−(1+t)(wt(t,·),w(t,·)) +(u1,u0)+1\n2/ba∇dblw(t,·)/ba∇dbl2+1\n2/integraldisplayt\n0/integraldisplay\nRnam(x)|w(s,x)|2dxds\n+1\n2/integraldisplay\nRnam(x)|u0(x)|2dx+/integraldisplayt\n0(1+s)/ba∇dblws(s,·)/ba∇dbl2ds. (2.15)\nNow, by using the Cauchy-Schwarz inequality and Lemma 2.2 we can first estimate\n−(1+t)(wt(t,·),w(t,·))≤(1+t)\n2ε/ba∇dblwt(t,·)/ba∇dbl2+ε\n2(1+t)/ba∇dblw(t,·)/ba∇dbl2\n≤1+t\nεEw(t)+ε\n2V0(1+t)/integraldisplay\nRnam(x)|w(t,x)|2dx\n≤I2\n1\nε+ε\n2V0(1+t)/integraldisplay\nRnam(x)|w(t,x)|2dx. (2.16)\n(2.16) and (2.17) imply\n1\n2/ba∇dblu0/ba∇dbl2+/integraldisplayt\n0(1+s)/ba∇dbl∇w(s,·)/ba∇dbl2ds+(1+t)\n2(1−ε\nV0)/integraldisplay\nRnam(x)|w(s,x)|2dx\n≤(u1,u0)+1\n2/integraldisplay\nRna(x)|u0(x)|2dx+1\n2/ba∇dblw(t,·)/ba∇dbl2+1\n2/integraldisplayt\n0/integraldisplay\nRnam(x)|w(s,x)|2dxds\n+I2\n1\nε+/integraldisplayt\n0(1+s)/ba∇dblws(s,·)/ba∇dbl2ds. (2.17)\nOn the other hand, it follows from (2.5) and Lemma 2.2 that\n/integraldisplayt\n0(1+s)/ba∇dblws(s,·)/ba∇dbl2ds≤1\nV0/integraldisplayt\n0(1+s)/integraldisplay\nRnam(x)|ws(s,x)|2dxds\n=−1\nV0/integraldisplayt\n0(1+s)E′\nw(s)ds=−1\nV0(1+t)Ew(t)+1\nV0E(0)+1\nV0/integraldisplayt\n0Ew(s)ds\n≤1\nV0(E(0)+I2\n1). (2.18)\nLet us finalize the proof of Lemma 2.3. To do so, we rely on the in equality:\nd\ndt{(1+t)2Ew(t)} ≤2(1+t)Ew(t),\nso that\n(1+t)2Ew(t)≤E(0)+/integraldisplayt\n0(1+s)/ba∇dblws(s,·)/ba∇dbl2ds+/integraldisplayt\n0(1+s)/ba∇dbl∇w(s,·)/ba∇dbl2ds (2.19)\nBecause of Lemma 2.1, (2.17) with small ε>0 and (2.18) one has\n(1+t)2Ew(t)≤E(0)+2\nV0(E(0)+I2\n1)+(u1,u0)+1\n2/integraldisplay\nRna(x)|u0(x)|2dx+C\n2I2\n0+I2\n1\nε=:I2\n2.\n8Concerning the fast L2-decay estimate, we use (2.5) and (2.17) with small ε>0 to have\n(1+t)\n2(1−ε\nV0)V0/integraldisplay\nRn|w(t,x)|2dx≤(1+t)\n2(1−ε\nV0)/integraldisplay\nRnam(x)|w(t,x)|2dx\n≤(u1,u0)+1\n2/integraldisplay\nRna(x)|u0(x)|2dx+1\n2/ba∇dblw(t,·)/ba∇dbl2+1\n2/integraldisplayt\n0/integraldisplay\nRnam(x)|w(s,x)|2dxds\n+I2\n1\nε+/integraldisplayt\n0(1+s)/ba∇dblws(s,·)/ba∇dbl2ds. (2.20)\nThe result follows from (2.20) and (2.18) and Lemma 2.1 by cho osingε>0 small enough. ✷\nProof of Theorem 1.1. From Lemma 2.3 we first notice that {u(m)}is a bounded sequence in\nL∞(0,∞;H1(Rn)) and inL∞(0,∞;L2(Rn)). Furthermore, {u(m)\nt}is also a bounded sequence\ninL∞(0,∞;L2(Rn)). Therefore, there exist a subsequence {u(µ)}of the original one {u(m)},\nand a function u=u(t,x)∈L∞(0,∞;H1(Rn)) satisfying ut∈L∞(0,∞;L2(Rn)) such that\nu(µ)→u(weakly∗)in L∞(0,∞;H1(Rn)) (µ→ ∞), (2.21)\nu(µ)\nt→ut(weakly∗)in L∞(0,∞;L2(Rn)) (µ→ ∞), (2.22)\nu(µ)→u(weakly∗)in L∞(0,∞;L2(Rn)) (µ→ ∞). (2.23)\nBy multiplying both sides of the approximated equation (2.1 ) withmreplaced by µthe test\nfunctionφ∈C∞\n0([0,∞)×Rn), one can get the following weak form of the problem (2.1)-(2 .2)\nwith the help of the integration by parts:\n/integraldisplay∞\n0/integraldisplay\nRnu(µ)(t,x)(φtt(t,x)−∆φ(t,x)−aµ(x)φt(t,x))dxdt\n=/integraldisplay\nRnu1(x)φ(0,x)dx−/integraldisplay\nRnu0(x)φt(0,x)dx+/integraldisplay\nRnaµ(x)u0(x)φ(0,x)dx. (2.24)\nNow, it follows from (2.23) that as µ→ ∞,\n/integraldisplay∞\n0/integraldisplay\nRnu(µ)(t,x)(φtt(t,x)−∆φ(t,x))dxdt→/integraldisplay∞\n0/integraldisplay\nRnu(t,x)(φtt(t,x)−∆φ(t,x))dxdt.(2.25)\nFurthermore, because of the Lebesgue dominated convergenc e theorem one can get\n/integraldisplay\nRnaµ(x)u0(x)φ(0,x)dx→/integraldisplay\nRna(x)u0(x)φ(0,x)dx(µ→ ∞). (2.26)\nOn the other hand, for each fixed φ∈C∞\n0([0,∞)×Rn), if we take µ∈Nlarge enough, then it\nfollows from the compact support condition on φ(t,x) that\n/integraldisplay∞\n0/integraldisplay\nRnu(µ)(t,x)aµ(x)φt(t,x)dxdt=/integraldisplay∞\n0/integraldisplay\nRnu(µ)(t,x)a(x)φt(t,x)dxdt.\nSo, sincea(·)φt(t,·)∈L1([0,∞);L2(Rn)), it follows from (2.23) that\n/integraldisplay∞\n0/integraldisplay\nRnu(µ)(t,x)aµ(x)φt(t,x)dxdt→/integraldisplay∞\n0/integraldisplay\nRnu(t,x)a(x)φt(t,x)dxdt(µ→ ∞).(2.27)\nTherefore, by taking µ→ ∞in (2.24), by means of (2.25)-(2.27) one can check that the li mit\nfunctionu(t,x) is the weak solution to problem (1.1)-(1.2).\n9Finally, let us check decay estimates of the energy and L2-norm of solutions such that\nEu(t)≤C(1+t)−2,/ba∇dblu(t,·)/ba∇dbl2≤C(1+t)−1(2.28)\nwith some constant C >0.\nFor this end, one first remarks that for each φ∈L1(0,∞;C∞\n0(Rn)), it follows from the\nintegration by parts that\n/integraldisplay∞\n0(∂u(µ)\n∂xj(t,·),φ(t,·))dt=−/integraldisplay∞\n0(u(µ)(t,·),∂φ\n∂xj(t,·))dt,\nso that we can have\nlimµ→∞/integraldisplay∞\n0(∂u(µ)\n∂xj(t,·),φ(t,·))dt=−/integraldisplay∞\n0(u(t,·),∂φ\n∂xj(t,·))dt.\nSinceu∈L∞([0,∞);H1(Rn)), it follows from the integration by parts again one can get\nlimµ→∞/integraldisplay∞\n0(∂u(µ)\n∂xj(t,·),φ(t,·))dt=/integraldisplay∞\n0(∂u\n∂xj(t,·),φ(t,·))dt.\nBy density of L1(0,∞;C∞\n0(Rn)) intoL1(0,∞;L2(Rn)), for each j= 1,2,···,nit is true that\n∂u(µ)\n∂xj→∂u\n∂xj(weakly∗)in L∞(0,∞;L2(Rn)) (µ→ ∞). (2.29)\nNow, let us finalize the proof of Theorem 1.1. First of all, we p repare the following basic lemma.\nLemma 2.4 Assume that a sequence {vm} ⊂L∞(0,∞;L2(Rn)satisfies\nvm→v(weakly∗)in L∞(0,∞;L2(Rn)) (m→ ∞),\nfor somev∈L∞(0,∞;L2(Rn)), and\n/ba∇dblvm(t,·)/ba∇dbl ≤C(1+t)−γ\nwith some constants C >0, andγ >0. Then, it is also true that\n/ba∇dblv(t,·)/ba∇dbl ≤C(1+t)−γ.\nProof.Takeψ∈C0([0,∞);L2(Rn)), and set wm(t,x) := (1+t)γvm(t,x). Then, since\n(1+t)γψ(t,x)∈L1(0,∞;L2(Rn)), by assumption it follows that\nlimm→∞/integraldisplay∞\n0/integraldisplay\nRnwm(t,x)ψ(t,x)dxdt= limm→∞/integraldisplay∞\n0/integraldisplay\nRnvm(t,x)((1+t)γψ(t,x))dxdt\n=/integraldisplay∞\n0/integraldisplay\nRnv(t,x)((1+t)γψ(t,x))dxdt=/integraldisplay∞\n0/integraldisplay\nRn((1+t)γv(t,x))ψ(t,x)dxdt.\nBecause of the density of C0([0,∞);L2(Rn)) intoL1(0,∞;L2(Rn)) again (cf. Miyadera [17,\nTheorem 15.3]), we have\nwm= (1+t)γvm→(1+t)γv(weakly∗)in L∞(0,∞;L2(Rn)) (m→ ∞),\n10so that it follows that\n/ba∇dbl(1+t)γv(t,·)/ba∇dbl ≤liminfm→∞/ba∇dbl(1+t)γvm/ba∇dblL∞(0,∞;L2(Rn))≤C\nfor eacht≥0. This implies the desired statement. ✷\nProof of Theorem 1.1 completed. It follows from Lemma 2.3 that\n/ba∇dbl∂u(µ)(t,·)\n∂xj/ba∇dbl ≤√\n2I2(1+t)−1,/ba∇dblu(µ)\nt(t,·)/ba∇dbl ≤√\n2I2(1+t)−1.\nThus, it follows from (2.29) and (2.22) and Lemma 2.4 that\n/ba∇dbl∂u(t,·)\n∂xj/ba∇dbl ≤√\n2I2(1+t)−1, (2.30)\n/ba∇dblut(t,·)/ba∇dbl ≤√\n2I2(1+t)−1. (2.31)\nFurthermore, from Lemma 2.3 one also has\n/ba∇dblu(µ)(t,·)/ba∇dbl ≤I3(1+t)−1/2(t≥0).\nTherefore, (2.23) and Lemma 2.4 imply\n/ba∇dblu(t,·)/ba∇dbl ≤I3(1+t)−1/2,(t≥0). (2.32)\n(2.30)-(2.32) imply the desired estimates (2.28) of Theore m 1.1. Note that all quantities Ij\n(j= 0,1,2,3) can be absorbed into CI00defined in Theorem 1.1 with some constant C >0.\nIn this connection, we should remark the following relation because of the Cauchy-Schwarz\ninequality:\n/integraldisplay\nRna(x)|u0(x)|2dx≤/radicalBigg/integraldisplay\nRna(x)2|u0(x)|2dx/radicalBigg/integraldisplay\nRn|u0(x)|2dx.\nA uniqueness argument is standard, so we shall omit its detai l. ✷\n3 Remark on the low dimensional case.\nLet us give some remarks on the low dimensional case and sever al open problems.\n(I)When one gets the result for n= 2, we may use (2) of Proposition 2.1 with θ= 1 and\nγ∈(0,1] to get the similar estimate to (2.9), which is an essential part of proof. But, in this\ncase we have to assume a stronger assumption such that\n/integraldisplay\nR2(a(x)u0(x)+u1(x))dx= 0. (3.1)\nUnfortunately, this assumption (3.1) is not hereditary to t he approximate solution, i.e.,\n/integraldisplay\nR2(am(x)u0(x)+u1(x))dx= 0\nis not necessarily true. So, it is completely open to get the s imilar result in the low dimensional\ncase (i.e.,n= 1,2).\n11(II)One can not treat the associated nonlinear equation suc h that\nutt(t,x)−∆u(t,x)+a(x)ut(t,x) =f(u(t,x)). (3.2)\nThis is because one encounters lack of some compactness argu ment when we want to get the\nlimit such that (see Lions [15, (1.42)])\nf(u(m))→f(u)weakly∗in L∞([0,∞);X)\n(asm→ ∞), whereXis a Banach space.\nIn the unbounded coefficient case there are still many obstacl es to be overcome. In this sense,\nthe dampedwave equation with unboundedcoefficient and non-c ompactly supportedinitial data\nseem to be quite difficult to be treated.\n(III)We have another idea to deal with the two dimensional ca se, that is, the idea to rely on\nthe well-known inequality:/integraldisplay\nR2|ˆf(ξ)|2\n|ξ|2dξ≤C/ba∇dblf/ba∇dbl2\nH1, (3.3)\nin place of (2) of Proposition 2.1, where H1(R2) is the so called Hardy space. Unfortunately, in\nthis case we also encounter the same problem as the heredity o f the property a(·)u0∈ H1(R2)\ntoam(·)u0∈ H1(R2).\n4 Appendix.\nIn this section, let us check the density of L1(0,∞;C∞\n0(Rn)) intoL1(0,∞;L2(Rn)).\nTakef∈L1(0,∞;L2(Rn)) and for each L>0 set\nfL(t,x) =/braceleftBiggf(t,x) (|x| ≤L)\n0 ( |x|>L).\nNext set\nh(t) =\n\ne−1\n1−t2(|t|<1)\n0 ( |t| ≥1),\nandρ(x) :=h(|x|2)/parenleftbigg/integraldisplay\nRnh(|x|2)dx/parenrightbigg−1\n, and forε>0 define\nρε(x) :=1\nεnρ(x\nε).\nUnder these preparations, we define an approximating sequen ce of the original function fby\nfε,L(t,x) := (fL(t,·)∗ρε)(x).\nThen it is standard to check that fε,L(t,·)∈C∞\n0(Rn) for eacht≥0 andL>0, and\n/ba∇dblfε,L(t,·)/ba∇dbl ≤ /ba∇dblfL(t,·)/ba∇dbl ≤ /ba∇dblf(t,·)/ba∇dbl,\nso thatfε,L∈L1(0,∞;L2(Rn)) for each L >0 andε >0. Furthermore, for each L >0 and\nt≥0 it is known that\n/ba∇dblfε,L(t,·)−fL(t,·)/ba∇dbl →0 (ε→0).\n12Now, for an arbitrary fixed η>0, chooseM >0 so large such that\n/ba∇dblfM−f/ba∇dblX<η\n2, (4.1)\nwhereX:=L1(0,∞;L2(Rn)), and/ba∇dbl·/ba∇dblXis the standard norm of X:\n/ba∇dblg/ba∇dblX:=/integraldisplay∞\n0/ba∇dblg(t,·)/ba∇dbldt.\nNext, for such a fixed M >0, by taking ε >0 sufficiently small, if one applies the Lebesgue\nconvergence theorem, one can get\n/ba∇dblfε,M−fM/ba∇dblX<η\n2. (4.2)\nTherefore, it follows from (4.1) and (4.2) with such large M >0 and small ε>0 that\n/ba∇dblfε,M−f/ba∇dblX≤ /ba∇dblfε,M−fM/ba∇dblX+/ba∇dblfM−f/ba∇dblX<η\n2+η\n2=η,\nwhich implies the density of L1(0,∞;C∞\n0(Rn)) intoL1(0,∞;L2(Rn)). ✷\nAcknowledgement. The work of the firstauthor (R. IKEHATA) was supportedin part by Grant-\nin-Aid for Scientific Research 15K04958 of JSPS. The work of t he second author (H. 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Part 2, 1 87-206, Contemp. Math.,\n554, Israel Math. Conf. Proc., Amer. Math. Soc., Providence , RI, 2011.\n[26] M. Sobajima and Y. Wakasugi, Diffusion phenomena for the w ave equation with space-\ndependent damping in an exterior domain, J. Diff. Eqns 261 (201 6), 5690-5718.\n14[27] M. Sobajima and Y. Wakasugi, Diffusion phenomena for the w ave equation with space-\ndependent damping term growing at infinity, arXiv:1704.076 50v1, 25, April 2017.\n[28] G. Todorova and B. Yordanov, Weighted L2-estimates of dissipative wave equations with\nvariable coefficients, J. Diff. Eqns 246 (2009), 4497-4518.\n[29] H. Uesaka, The total energy decay of solutions for the wa ve equation with a dissipative\nterm, J. Math. Kyoto Univ. 20 (1980), 57-65.\n[30] Y. Wakasugi, On diffusion phenomena for the linear wave eq uation with space-dependent\ndamping, J. Hyperbolic differ. Equ. 11 (2014), 795-819.\n[31] J. Wirth, Wave equations with time-dependent dissipat ion II. Effective dissipation, J. Diff.\nEqns 232 (2007), 74-103.\n[32] Z. Zhang, Fast decay of solutions for wave equations wit h localized dissipation on noncom-\npact Riemannian manifolds, Nonlinear Analysis: Real World Appl. 27 (2016), 246-260.\n15"    },    {        "title": "0904.1455v1.Evaluating_the_locality_of_intrinsic_precession_damping_in_transition_metals.pdf",        "content": "arXiv:0904.1455v1  [cond-mat.mtrl-sci]  9 Apr 2009Evaluating the locality of intrinsic precession damping in transition metals\nKeith Gilmore1,2and Mark D. Stiles1\n1Center for Nanoscale Science and Technology\nNational Institute of Standards and Technology,\nGaithersburg, MD 20899-6202\n2Maryland Nanocenter, University of Maryland,\nCollege Park, MD 20742-3511\n(Dated: December 4, 2018.)\nThe Landau-Lifshitz-Gilbert damping parameter is typical ly assumed to be a local quantity, in-\ndependent of magnetic configuration. To test the validity of this assumption we calculate the\nprecession damping rate of small amplitude non-uniform mod e magnons in iron, cobalt, and nickel.\nAt scattering rates expected near and above room temperatur e, little change in the damping rate is\nfound as the magnon wavelength is decreased from infinity to a length shorter than features probed\nin recent experiments. This result indicates that non-loca l effects due to the presence of weakly\nnon-uniform modes, expected in real devices, should not app reciably affect the dynamic response of\nthe element at typical operating temperatures. Conversely , at scattering rates expected in very pure\nsamples around cryogenic temperatures, non-local effects r esult in an order of magnitude decrease\nin damping rates for magnons with wavelengths commensurate with domain wall widths. While this\nlow temperature result is likely of little practical import ance, it provides an experimentally testable\nprediction of the non-local contribution of the spin-orbit torque-correlation model of precession\ndamping. None of these results exhibit strong dependence on the magnon propagation direction.\nMagnetization dynamics continues to be a techno-\nlogically important, but incompletely understood topic.\nHistorically, field induced magnetization dynamics have\nbeen described adequately by the phenomenological\nLandau-Lifshitz (LL) equation [1]\n˙m=−|γM|m×H+λˆm×(m×H),(1)\nor the mathematically equivalent Gilbert form [2, 3].\nEquation 1 accounts for the near equilibrium dynamics\nof systems in the absence of an electrical current. γM\nis the gyromagnetic ratio and λis the phenomenological\ndamping parameter, which quantifies the decay of the\nexcited system back to equilibrium. The LL equation\nis a rather simple approximation to very intricate dy-\nnamic processes. The limitations of the approximations\nentering into the LL equation are likely to be tested by\nthe next generation of magnetodynamic devices. While\nmanygeneralizationsforthe LLequationarepossible, we\nfocus on investigatingthe importance of non-local contri-\nbutions to damping. It is generally assumed in both ana-\nlyzing experimental results and in performing micromag-\nnetic simulations that damping is a local phenomenon.\nWhile no clearexperimental evidence exists to contradict\nthis assumption, the possibility that the damping is non-\nlocal – that it depends, for example, on the local gradient\nofthemagnetization–wouldhaveparticularimplications\nfor experiments that quantify spin-current polarization\n[4], for storage [5] and logic [6] devices based on using\nthis spin-current to move domain-walls, quantifying vor-\ntex [7] and mode [8] dynamics in patterned samples, and\nthe behavior of nano-contact oscillators [9, 10].\nWhile several viable mechanisms have been proposed\nto explain the damping process in different systems\n[11, 12, 13, 14, 15, 16, 17], werestrictthescopeofthispa-\nper to investigating the degree to which the assumptionof local damping is violated for small amplitude dynam-\nics within pure bulk transition metal systems where the\ndominant source of damping is the intrinsic spin-orbit in-\nteraction. For such systems, Kambersk´ y’s [14] spin-orbit\ntorque-correlationmodel, which predicts a decay rate for\nthe uniform precession mode of\nλ0=π¯hγ2\nM\nµ0/summationdisplay\nnm/integraldisplay\ndk/vextendsingle/vextendsingleΓ−\nnm(k)/vextendsingle/vextendsingle2Wnm(k),(2)\nhasrecentlybeen demonstratedtoaccountforthe major-\nity of damping [18, 19]. The matrix elements |Γ−\nnm(k)|2\nrepresent a scattering event in which a quantum of the\nuniformmodedecaysintoasinglequasi-particleelectron-\nhole excitation. This annihilation of a magnon raises the\nangularmomentum of the system, orienting the magneti-\nzation closer to equilibrium. The excited electron, which\nhas wavevector kand band index m, and the hole, with\nwavevector kand band index n, carry off the energy and\nangular momentum of the magnon. This electron-hole\npair is rapidly quenched through lattice scattering. The\nweighting function Wnm(k) measures the rate at which\nthe scattering event occurs. The very short lifetime of\nthe electron-hole pair quasiparticle (on the order of fs at\nroom temperature) introduces significant energy broad-\nening (several hundred meV). The weighting function,\nwhich is a generalization of the delta function appearing\nin a simple Fermi’s golden rule expression, quantifies the\nenergy overlap of the broadened electron and hole states\nwith each other and with the Fermi level.\nEquation 2, which has been discussed extensively\n[14, 18, 19, 20], considers only local contributions to the\ndamping rate. Non-local contributions to damping may\nbe studied through the decay of non-uniform spin-waves.\nAlthough recent efforts have approached the problem of2\nnon-localcontributionsto the dissipationofnon-collinear\nexcited states [21, 22] the simple step of generalizing\nKambersk´ y’s theory to non-uniform mode magnons has\nnot yet been taken. We fill this obvious gap, obtaining a\ndamping rate of\nλq=π¯hγ2\nM\nµ0/summationdisplay\nnm/integraldisplay\ndk/vextendsingle/vextendsingleΓ−\nnm(k,k+q)/vextendsingle/vextendsingle2Wnm(k,k+q)\n(3)\nfor a magnon with wavevector q. We test the impor-\ntance of non-local effects by quantifying this expression\nfor varying degrees of magnetic non-collinearity (magnon\nwavevector magnitude). The numerical evaluation of\nEq. 3 for the damping rate of finite wavelength magnons\nin transition metal systems, presented in Fig. 1, and the\nensuing physical discussion form the primary contribu-\ntion of this paper. We find that the damping rate ex-\npected inverypuresamplesatlowtemperatureisrapidly\nreduced as the magnon wavevector |q|grows, but the\ndamping rate anticipated outside of this ideal limit is\nbarely affected. We provide a simple band structure ar-\ngument to explain these observations. The results are\nrelevant to systems for which the non-collinear excita-\ntion may be expanded in long wavelength spin-waves,\nprovided the amplitude of these waves is small enough to\nneglect magnon-magnon scattering.\nCalculations for the single-mode damping constant\n(Eq. 3) as a function of electron scattering rate are pre-\nsented in Fig. 1 for iron, cobalt, and nickel. The Gilbert\ndamping parameter α=λ/γMis also given. Damp-\ning rates are given for magnons with wavevectors along\nthe bulk equilibrium directions, which are /angbracketleft100/angbracketrightfor Fe,\n/angbracketleft0001/angbracketrightfor Co, and /angbracketleft111/angbracketrightfor Ni. Qualitatively and quan-\ntitatively similar results were obtained for other magnon\nwavevector directions for each metal. The magnons re-\nportedoninFig.1constitutesmalldeviationsofthemag-\nnetization transverse to the equilibrium direction with\nwavevectormagnitudes between zero and 1 % of the Bril-\nlouin zone edge. This wavevector range corresponds to\nmagnon half-wavelengths between infinity and 100 lat-\ntice spacings, which is 28.7 nm for Fe, 40.7 nm for Co,\nand 35.2 nm for Ni. This range includes the wavelengths\nreported by Vlaminck and Bailleul in their recent mea-\nsurement of spin polarization [4].\nResults for the three metals are qualitatively similar.\nThemoststrikingtrendisadramatic,orderofmagnitude\ndecreaseofthe damping rate at the lowestscatteringrate\ntested as the wavevector magnitude increases from zero\nto 1 % of the Brillouin zone edge. This observation holds\nin each metal for every magnon propagation direction\ninvestigated. For the higher scattering rates expected in\ndevices at room temperature there is almost no change\nin the damping rate as the magnon wavevector increases\nfrom zero to 1 % of the Brillouin zone edge in any of the\ndirections investigated for any of the metals.\nTo understand the different dependences of the damp-\ning rate on the magnon wavevector at low versus high\nscattering rates we first note that the damping rate10121013101410151091010\n0.010.1λ (s-1)\nγ (s-1)\n α 1081090.01λ (s-1) \n α1090.01 1.0 0.1\n0.01\n λ (s-1)\n αhγ (eV)\n10-310-3\n10-30.0\n0.001\n0.003\n0.005\n0.01\n0.0\n0.001\n0.003\n0.005\n0.01\n0.0\n0.001\n0.003\n0.005\n0.01Fe\nCo\nNiq\nq\nq\nFIG. 1: Damping rates versus scattering rate. The preces-\nsion damping rates for magnons in iron, cobalt, and nickel\nare plotted versus electron scattering rate for several mag non\nwavevectors. A dramatic reduction in damping rate is ob-\nserved at the lowest scattering rates. The Landau-Lifshitz λ\n(Gilbert α) damping parameter is given on the left (right)\naxes. Electron scattering rate is given in eV on the top axis.\nMagnon wavevector magnitudes are given in units of the Bril-\nlouin zone edge and directions are as indicated in the text.\n(Eqs. 2 & 3) is a convolution of two factors: the torque\nmatrix elements and the weighting function. The ma-\ntrix elements do not change significantly as the magnon\nwavevector increases, however, the weighting function\ncan change substantially. The weighting function\nWnm(k,k+q)≈An,k(ǫF)Am,k+q(ǫF) (4)3\n789101112\n789101112\n Energy  (eV)\nH F P\nFIG. 2: Partial band structure of bcc iron. The horizontal\nblack line indicates the Fermi level and the shaded region\nrepresents the degree of spectral broadening. The solid dot is\na hypothetical initial electron state while the open circle is a\npotential final scattering state. (Initial and final state wa ve-\nvector separations are exaggerated for clarity of illustra tion.)\nThe intraband magnon decay rate diminishes as the energy\nseparation of the states exceeds the spectral broadening.\ncontains a product of the initial and final state electron\nspectral functions\nAn,k(ǫ) =1\nπ¯hγ\n(ǫ−ǫn,k)2+(¯hγ)2, (5)\nwhichareLorentziansinenergyspace. Thespectralfunc-\ntion for state |n,k/angbracketright, which has nominal band energy ǫn,k,\nis evaluated within a verynarrowrangeofthe Fermi level\nǫF. The width of the spectral function ¯ hγis given by\nthe electron scattering rate γ= 1/2τwhereτis the\norbital lifetime. (The lifetimes of all orbital states are\ntaken to be equal for these calculations and no specific\nscattering mechanism is implied.) The weighting func-\ntion restricts the electron-hole pair generated during the\nmagnon decay to states close in energy to each other and\nnear the Fermi level. For high scattering rates, the elec-\ntron spectral functions are significantly broadened and\nthe weighting function incorporates states within an ap-\npreciablerange(severalhundredmeV) ofthe Fermi level.\nFor low scattering rates, the spectral functions are quite\nnarrow (only a few meV) and both the electron and hole\nstate must be very close to the Fermi level.\nThe second consideration useful for understanding the\nresults of Fig. 1 is that the sum in Eqs. 2 & 3 can be\ndivided into intraband ( n=m) and interband ( n/negationslash=m)\nterms. For the uniform mode, these two contributions\ncorrespond to different physical processes with the intra-\nband contributiondominatingatlowscatteringratesand\nthe interband terms dominating at high scattering rates\n[14, 18, 19, 20].101210131014101510-3\n10-3\n10-4\n1071081090.01 0. 11\n0.01\n 0.0\n 0.001\n 0.003\n 0.005\n 0.01λintra,inte r (s-1)\nγ (s-1)q in ( π/a)interband\nintraband\n αintra,inter hγ (eV)Increasing q\nFIG. 3: Intraband and interband damping contributions in\niron. Theintrabandandinterbandcontributionstothedamp -\ning rate of magnons in the /angbracketleft100/angbracketrightdirection in iron are plot-\nted versus scattering rate for several magnitudes of magnon\nwavevector. Magnitudes are given in units of the Brillouin\nzone edge.\nFor intraband scattering, the electron and hole occupy\nthe sameband and must haveessentiallythe sameenergy\n(within ¯hγ). The energy difference between the electron\nand hole states may be approximated as ǫn,k+q−ǫn,k≈\nq·∂ǫn,k/∂k. The generation of intraband electron-hole\npairs responsible for intraband damping gets suppressed\nasq·∂ǫn,k/∂kbecomes largecomparedto ¯ hγ. Unless the\nbands are very flat at the Fermi level there will be few lo-\ncations on the Fermi surface that maintain the condition\nq·∂ǫn,k/∂k<¯hγfor low scattering rates as the magnon\nwavevectorgrows. (See Fig. 2). Indeed, at low scattering\nrates when ¯ hγis only a few meV, Fig. 3 shows that the\nintraband contribution to damping decreases markedly\nwith only modest increase of the magnon wavevector.\nSince the intraband contribution dominates the inter-\nband term in this limit the total damping rate also de-\ncreases sharply as the magnon wavevector is increased\nfor low scattering rates. For higher scattering rates, the\nelectronspectralfunctionsaresufficientlybroadenedthat\nthe overlap of intraband states does not decrease appre-\nciably as the states are separated by finite wavevector\n(q·∂ǫn,k/∂k<¯hγgenerally holds over the Fermi sur-\nface). Therefore, the intraband contribution is largelyin-\ndependentofmagnonwavevectorathighscatteringrates.\nThe interband contribution to damping involves scat-\ntering between states in different bands, separated by the\nmagnon wavevector q. Isolating the interband damping\ncontribution reveals that these contributions are insensi-\ntive to the magnon wavevector at higher scattering rates\nwhere they form the dominant contribution to damp-\ning (see Fig. 3). To understand these observations we\nagain compare the spectral broadening ¯ hγto the quasi-\nparticle energy difference ∆m,k+q\nn,k=ǫm,k+q−ǫn,k. The\nquasiparticle energy difference may be approximated as4\n∆m,k\nn,k+q·∂∆m,k\nn,k/∂k. The interband energy spacings\nare effectively modulated by the product of the magnon\nwavevector and the slopes of the bands. At high scatter-\ning rates, when the spectral broadening exceeds the ver-\ntical band spacings, this energy modulation is unimpor-\ntant and the damping rate is independent of the magnon\nwavevector. At low scattering rates, when the spec-\ntral broadening is less than many of the band spacings,\nthis modulation can alter the interband energy spacings\nenough to allow or forbid generation of these electron-\nhole pairs. For Fe, Co, and Ni, this produces a modest\nincrease in the interband damping rate at low scattering\nrates as the magnon wavevector increases. However, this\neffect is unimportant to the total damping rate, which\nremains dominated by the intraband terms at low scat-\ntering rates.\nLastly, we describe the numerical methods employed\nin this study. Converged ground state electron densities\nwere first obtained via the linear-augmented-plane-wave\nmethod. The Perdew-Wang functional for exchange-\ncorrelation within the local spin density approximation\nwas implemented. Many details of the ground state den-\nsity convergence process are given in [23]. Densities were\nthen expanded into Kohn-Sham orbitals using a scalar-\nrelativistic spin-orbit interaction with the magnetiza-\ntion aligned along the experimentally determined mag-\nnetocrystalline anisotropy easy axis. The Kohn-Sham\nenergies were artificially broadened through the ad hoc\nintroduction of an electron lifetime. Matrix elements of\nthe torque operator Γ−= [σ−,Hso] were evaluated sim-\nilarly to the spin-orbit matrix elements [24]. ( σ−is the\nspin lowering operator and Hsois the spin-orbit Hamil-\ntonian.) The product of the matrix elements and the\nweightingfunction wereintegratedover k-spaceusingthe\nspecial points method with a fictitious smearing of the\nFermi surface for numerical stability. Convergence wasobtained by sampling the full Brillouin zone with 1603\nk-points for Fe and Ni, and 1602x 91 points for Co.\nIn summary, we have investigated the importance of\nnon-local damping effects by calculating the intrinsic\nspin-orbit contribution to precession damping in bulk\ntransition metal ferromagnets for small amplitude spin-\nwaveswith finite wavelengths. Results ofthe calculations\ndo not contradict the common-practice assumption that\ndamping is a local phenomenon. For transition metals,\nat scattering rates corresponding to room temperature,\nwe find that the single-mode damping rate is essentially\nindependent of magnon wavevector for wavevectors be-\ntween zero and 1 % of the Brillouin zone edge. It is not\nuntil low temperatures in the most pure samples that\nnon-local effects become significant. At these scatter-\ning rates, damping rates decrease by as much as an or-\nder of magnitude as the magnon wavevector is increased.\nThe insensitivity of damping rate to magnon wavevector\nat high scattering rates versus the strong sensitivity at\nlow scattering rates can be explained in terms of band\nstructure effects. Due to electron spectral broadening at\nhigh scattering rates the energy conservation constraint\nduring magnon decay is effectively relaxed, making the\ndamping rate independent of magnon wavevector. The\nminimal spectral broadening at low scattering rates –\nseenonlyinverypureandcoldsamples–greatlyrestricts\nthe possible intraband scattering processes, lowering the\ndamping rate. The prediction of reduced damping at low\nscattering rates and non-zero magnon wavevectors is of\nlittle practical importance, but could provide an accessi-\nble test of the torque-correlationmodel. Specifically, this\nmight be testable in ferromagnetic semiconductors such\nas (Ga,Mn)As forwhich manyspin-waveresonanceshave\nbeen experimentally observed at low temperatures [25].\nThis work has been supported in part through NIST-\nCNST / UMD-Nanocenter cooperative agreement.\n[1] L.LandauandE. Lifshitz, Phys.Z.Sowjet. 8, 153 (1935).\n[2] T. L. Gilbert, Armour research foundation project No.\nA059, supplementary report, unpublished (1956).\n[3] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n[4] V. Vlaminck and M. Bailleul, Science 322, 410 (2008).\n[5] S.S.P. Parkin, M. Hayashi, and L. Thomas, Science 320,\n190 (2008).\n[6] M. Hayashi, L. Thomas, R. Moriya, C. Rettner, and\nS.S.P. Parkin, Science 320, 209 (2008).\n[7] B.E. Argyle, E. Terrenzio, and J.C. Slonczewski,\nPhys. Rev. Lett. 53, 190 (1984).\n[8] R.D. McMichael, C.A. Ross, and V.P. Chuang,\nJ. Appl. Phys. 103, 07C505 (2008).\n[9] S. Kaka, M.R. Pufall, W.H. Rippard, T.J. Silva,\nS.E. Russek, and J.A. Katine, Nature 437, 389 (2005).\n[10] F.B. Mancoff, N.D. Rizzo, B.N. Engel, and S. Tehrani,\nNature437, 393 (2005).\n[11] B. Heinrich, D. Fraitov´ a, and V. Kambersk´ y,\nPhys. Stat. Sol. 23, 501 (1967).\n[12] V. Kambersk´ y, Can. J. Phys. 48, 2906 (1970).[13] V. Korenman and R.E. Prange, Phys. Rev. B 6, 2769\n(1972).\n[14] V. Kambersk´ y, Czech. J. Phys. B 26, 1366 (1976).\n[15] J. Sinova, T. Jungwirth, X. Liu, Y. Sasaki, J.K. Furdyna ,\nW.A. Atkinson, and A.H. MacDonald, Phys. Rev. B 69,\n085209 (2004).\n[16] Y. Tserkovnyak, A. Brataas, and G.E.W. Bauer,\nPhys. Rev. Lett. 88, 117601 (2002).\n[17] M. Zwierzycki, Y. Tserkovnyak, P.J. Kelly, A. Brataas,\nand G.E.W. Bauer, Phys. Rev. B 71, 064420 (2005).\n[18] K. Gilmore, Y.U. Idzerda, and M.D. Stiles,\nPhys. Rev. Lett. 99, 027204 (2007).\n[19] V. Kambersk´ y, Phys. Rev. B 76, 134416 (2007).\n[20] K.Gilmore, Y.U.Idzerda, andM.D.Stiles, J.Appl.Phys .\n103, 07D303 (2008).\n[21] M. F¨ ahnle and D. Steiauf, Phys. Rev. B 73, 184427\n(2006).\n[22] J. Foros, A.Brataas, Y.Tserkovnyak,andG.E.W. Bauer,\nPhys. Rev. B 78, 140402(R) (2008).\n[23] L.F. Mattheiss and D.R. Hamann, Phys. Rev. B 33, 8235\n(1986).\n[24] M.D. Stiles, S.V. Halilov, R.A. Hyman, and A. Zangwill,\nPhys. Rev. B 64, 104430 (2001).\n[25] S.T.B. Goennenwein, T. Graf, T. Wassner, M.S. Brandt,M. Stutzmann, A. Koeder, S. Frank, W. Schoch, and\nA. Waag, Journal of Superconductivity 16, 75 (2003)."    },    {        "title": "1406.2491v2.Influence_of_Ta_insertions_on_the_magnetic_properties_of_MgO_CoFeB_MgO_films_probed_by_ferromagnetic_resonance.pdf",        "content": "arXiv:1406.2491v2  [cond-mat.mtrl-sci]  13 Aug 2014Influence ofTa insertions onthe magneticproperties ofMgO /CoFeB/MgOfilms probed by\nferromagnetic resonance\nMariaPatriciaRouelliSabino,Sze TerLim,andMichaelTran\nDataStorage Institute,Agency for Science, Technology and Research, 5Engineering Drive 1,117608 Singapore\n(Dated: April 3,2018)\nAbstract We show by vector network analyzer ferromagnetic resonance measurements that low Gilbert damping α,\ndownto0.006,canbeachievedinperpendicularlymagnetize dMgO/CoFeB/MgOthinfilmswithultrathininsertionsofTa\nintheCoFeBlayer. AlthoughincreasingthenumberofTainse rtionsallowsthickerCoFeBlayerstoremainperpendicular ,\nthe effective areal magnetic anisotropy does not improve withmore insertions, whichcome withan increase in α.\nPerpendicularmagnetic anisotropy (PMA) is the key to furth er downscaling of spin transfer torque magnetoresistive\nrandommemorydevices,asitallowstwokeyrequirementstob esatisfied: lowcriticalcurrent Ic0andhighthermalstability\n∆, the latter of which is proportional to the energy barrier Ebbetween the two stable magnetic states. The spin torque\nswitching efficiency, defined as Eb/Ic0, is commonly used as a metric to account for both requirement s. For a Stoner-\nWohlfarthmodel,itisgivenby[1]( /planckover2pi1/4e)·(η/α),whereαistheGilbertdampingparameter,and ηisthespinpolarization\nfactor, which is related to the tunnel magnetoresistance ra tio (TMR) byη=[TMR(TMR+2)]1/2/[2(TMR+1)]. It thus\nbecomes evident that for high switching e fficiency, one has to decrease αwhile keeping TMR high. Magnetic tunnel\njunctions(MTJs)basedonCoFeB /MgOsystemsarewell knownto providehighTMR[2] andhaverec entlybeenshown\nto possess PMA, which is attributed to the CoFeB /MgO interface.[3] A Ta layer is usually placed adjacent to th e CoFeB\nto induce the proper crystallization necessary for PMA and h igh TMR [4]. In Ta /CoFeB/MgO systems, however, spin\npumping to the Ta increases α. [5] Moreover, the CoFeB layer also needs to be ultrathin (ty pically less than 1.5nm) in\nordertoexhibitPMA.[3]Toimprovethethermalstabilityas devicesarescaleddowntosmallerdiameters,increasingth e\neffectivearealanisotropyenergydensity Kef ftisdesired.\nOne approach to address these issues is the use of double-MgO structures, i.e., those in which both the barrier layer\nand cappinglayer straddlingthe free layerare made of MgO. I mproved Ic0and/or∆have beenreportedin devicesusing\ndoubleMgOfreelayers. [6,7,8,9]Theimprovementintherma lstabilityisattributedtotheadditionalCoFeB /MgOinter-\nface,whereaslower Ic0isassociatedwithlow α. Indeed,αdownto0.005hasbeenmeasuredinin-planeMgO /FeB/MgO\nfilms,[10] which agrees with device measurements.[11] The s tacks investigated in these damping studies, however, did\nnot have the Ta layer used in practical free layers with perpe ndicular anisotropy.[9, 12] In addition, although the inte r-\nfacial anisotropy in the out-of-plane devices measured by T sunegi et al.[11] can be as high as 3.3 mJ /m2, the effective\nperpendicularanisotropywasratherlow ( Kef ft≈0.04mJ/m2) relativetothat ofa Ta /CoFeB/MgOstack[3]. In thiswork,\nweexploretheinfluenceofTainsertionswithintheCoFeBlay erofMgO/CoFeB/MgOfilmsbymagnetometryandvector\nnetwork analyzer ferromagnetic resonance (VNA-FMR) measu rements. The insertion of extremely thin Ta layers (0.3\nnm) inside the CoFeB layer aids crystallization, allowing a larger total CoFeB thickness to remain perpendicular, [13]\nwith aneffectivearealanistropycomparableto thatofTa /CoFeB/MgO.\nTwo sample series were deposited by magnetron sputtering on SiO2substrates with seed layers of Ta 5 /TaN 20/Ta 5\nin an ultrahigh vacuum environment (all thicknesses in nm). The stack configurations of the two sample series are: (1)\nMgO 3/CoFeB 1.0/Ta 0.3/CoFeB 0.5 - 1.5/MgO 3 (“single-insertion”) and (2) MgO 3 /CoFeB 1.0/Ta 0.3/CoFeB 0.5 -\n1.5/Ta0.3/CoFeB1.0/MgO3(“double-insertion”),wheretheCoFeBcompositionis Co40Fe40B20(at%). TheTainsertion\nlayer thickness is in the regime allowing strong ferromagne tic coupling between the CoFeB layers. [16] Two other\nsample serieswere grownasreferences: (a)MgO3 /CoFeB1.0-2.5/MgO3(“zero-insertion”),and(b)seed /CoFeB1.0-\n2.5/MgO3(“single-MgO”).Forallthedouble-MgOsamples,anult rathinCoFeBlayerbelowthebottomMgOlayerwas\nalsodepositedforgoodMgOgrowth. Weconfirmedfromseparat emeasurementsthatthislayerdoesnotcontributetothe\nmagneticsignal. All sampleswerecappedwith15nmofTa forp rotectionandwereannealedpost-growthat 300◦Cfor1\nh in vacuum. Although3 nm MgO is too thick for practical use in MTJs, it was chosen to ensure continuityof the MgO\nlayers and lessen the influence of the layersbeyondit.[10, 1 4, 15] (Measurementsof similar sampleswith 1 nm of MgO\nonbothsidesofthe magneticlayeryieldedthesametrends.)\nMagnetization measurements were performed using an altern ating gradient magnetometer (AGM). The PMA im-\nproveswith doublingof the CoFeB /MgO interface and with increasingnumber of Ta insertions, n, as shown in Fig. 1(a)\nfor samples with a similar total nominal CoFeB thickness tnom≈2.5 nm. We also confirmed that we cannot obtain a\n1perpendicular easy axis in double-MgO structures without T a insertions.[17] The double-insertion sample, on the othe r\nhand,exhibitslargeout-of-planeremanenceas shownin the inset of Fig. 1(a). A coercivefield less than0.01T (inset) is\ntypicalofCoFeBfilmswithPMA[18,16].\n/s49 /s50 /s51 /s52/s49/s50/s51/s52/s53\n/s45/s48/s46/s48/s49 /s48/s46/s48/s49\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s32/s115/s105/s110/s103/s108/s101/s32/s77/s103/s79\n/s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s48\n/s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s49\n/s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s50/s77/s47/s65/s32/s40/s65 /s109/s50\n/s47/s109/s50\n/s41/s32/s120/s32/s49/s48/s45/s54\n/s116\n/s110/s111/s109/s32/s40/s110/s109/s41/s48/s40/s98/s41\n/s32/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40/s97/s46/s117/s46/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s70/s105/s101/s108/s100/s32/s40/s84/s41/s116\n/s110/s111 /s109/s32 /s32/s50/s46/s53/s110/s109/s40/s97/s41\nFigure 1: (Color online) (a) Out-of-planeAGM loops for samp les with single-MgO (red squares), zero-insertion(purple\ninvertedtriangles),single-insertion(bluecircles),an ddouble-insertion(greentriangles),withtotalnominalC oFeBthick-\nnesstnom≈2.5nm. Insetshowsalow-fieldout-of-planeloopforthesame doubleinsertionsample. (b)Magneticmoment\nperunitarea( M/A)asafunctionofthetotalnominalCoFeBthicknessforallsa mpleseries. Linearfitsareshownassolid\nlines.MSandtMDLcanbeextractedfromtheslopeand xintercept,respectively,andaresummarizedinTable 1.\nIt is known that Ta can create a magnetically dead layer (MDL) when it is near a magnetic layer.[19] We plot the\nmagneticmomentperarea against tnom[Fig.1(b)]to obtainthethicknessoftheMDL foreachseries fromthexintercept\nofalinearfit. TheresultsaresummarizedinTable1,alongwi ththeMSvaluesobtainedfromtheslope. WefindanMDL\nthickness tMDLof 0.24±0.09 nm for n=1 and 0.7±0.1 nm for n=2. These thicknesses are similar to the total Ta\ninsertionthicknessintherespectiveseries,andareconsi stentwiththepictureofCoFeBintermixingwithTatoproduc ea\nmagneticallydeadvolume. The tMDLvalueforthesingle-MgOsamples(0.26 ±0.08nm)agreeswithvaluesfoundinthe\nliterature.[14, 20, 21] On the other hand, no dead layer was f ound for the zero-insertionsamples, which is similar to the\nresultsinRef. [19].\nTable1: SummaryofMagneticProperties\nSeries tMDL(nm) MS(MA/m) Ki(mJ/m2) Kv(MJ/m3)\nsingle MgO 0 .26±0.08 1.51±0.08 1.61±0.07−0.29±0.08\ndouble MgO, n=0 0.04±0.06 1.29±0.05 0.91±0.09 0.37±0.05\ndouble MgO, n=1 0.24±0.09 1.12±0.05 2.18±0.08−0.34±0.04\ndouble MgO, n=2 0.7±0.1 1.10±0.04 2.4±0.1−0.25±0.03\nVNA-FMR was used to measure the e ffective anisotropy field and damping parameter of the samples . In the VNA-\nFMR setup, the samples were placed face down on a coplanar wav eguide and situated in a dc magnetic field of up to\n1.2 T applied perpendicular to the film plane. The transmission scattering parameter S21was measured at a specific\nfrequency while the dc field was swept. For each sweep, the rea l and imaginary parts of the resonance response were\nfitted simultaneouslyusingthecomplexsusceptibilityequ ation\nχ(H)=Mef f(H−Mef f+i∆H\n2)\n(H−Mef f)2−/parenleftBig2π/planckover2pi1f\ngµB/parenrightBig2+i∆H(H−Mef f)(1)\nwherefis the frequencyofthe ac field, Mef f=MS−H⊥\nK,∆His the fullwidth at half-maximum, H⊥\nKis the anisotropy\nfield perpendicular to the plane, gis the spectroscopic splitting factor, µBis the Bohr magneton, and /planckover2pi1is the reduced\nPlanck’s constant. Nonmagnetic contributions to the S21parameter and a linear time-dependentdrift of the instrume nts\nwere taken into account during the fit. We note that only one re sonance peak is observed within the range studied. A\nrepresentativefitofthesusceptibilitydataisshowninFig .2(a)foradouble-insertionsamplewith tnom=2.5nm. Inusing\nEq.1,a valueof g=2isfirst assumedtoobtainvaluesfor Mef fand∆H, whichdoesnotaffectthe finalresult.\nForeachfrequency,a resonancefield\nµ0Hres(f)=2π/planckover2pi1\ngµBf+µ0Mef f (2)\naccording to Kittel’s equation is calculated and plotted ag ainst the frequency, as shown in Fig. 2(c). A linear fit, now\nwithgandMef fas fitting parameters, is then performed. The e ffective anisotropyenergydensity Kef fcan be calculated\n2from the effective anisotropy field HKef f(=−Mef f) asKef f=HKef fMS/2, noting that a positive anisotropy constant\ncorrespondsto aperpendiculareasyaxis.\nToobtainα,we performa linearfit ofthe measuredFMRlinewidthasafunc tionofthefrequencyto\nµ0∆H(f)=4π/planckover2pi1α\ngµBf+µ0∆H0 (3)\nwhere∆H0is the inhomogeneous linewidth broadening, and the value of gused is the fitted value from Eq. 2. We\nnote that two-magnon scattering contributions to the linew idth are eliminated owing to the perpendicular measurement\nconfiguration[22]. Such a fit is shown in Fig. 2(c). Only data p oints taken well beyond the saturation field for each\nsamplewereusedinthefit, andasymptoticanalysisasdescri bedinRef. [23]fortheaccessiblefrequencyrangewasalso\nperformed.\nWe define an effective thickness tef f=tnom−tMDLand show the calculated Kef ftef f(to which Ebis proportional)\nfor both sample series in Fig. 3(a). The x-axis error bars originate from the fitting error in obtainin gtMDL. We find that\nfortef f>2 nm,double-insertionsampleshavehigher Kef ftef fthan single-insertionsamplesforthe same tef f. However,\nthe maximum Kef ftef fachieved for both the single- and double-insertion series d oes not significantly exceed Kef ftef f\nmeasuredin ourthinnestsingle-MgOsample( tef f=1.0nm),similar tothatobservedin MTJmeasurements.[7]\nTounderstandthisfurther,we considerthedi fferentcontributionsto Kef ftef f,whichisgivenby\nKef ftef f=Ki+(Kv−µ0M2\nS\n2)tef f (4)\nwhereKiis the total interfacial anisotropy constant, including al l CoFeB/MgO interfaces; Kvis the volume anisotropy\nconstant; and the demagnetizing energy is given by the M2\nSterm. We assume that any interfacial anisotropy from the\nTa/CoFeB interface is negligible.[24] Kiis commonly derived from the yintercept of a linear Kef ftef fversustef ffit,\nwhereasKvcan be calculated from the slope if MSis known. Because it is possible that for CoFeB thicknesses b elow\n1.0 nm,Kiis degraded because of Ta reaching the CoFeB /MgO interface,[25] we consider only the linear region of the\ncurve during the fit. The calculated values, given in Table 1, demonstrate that the absence of a Ta insertion leads to the\nlowest value of Ki(0.91±0.09mJ/m2), explaining why n=0 samples did not exhibit a perpendiculareasy axis. On the\nother hand, Kiforn=1 (2.18±0.08mJ/m2) andn=2 (2.4±0.1mJ/m2) are both larger than the single-MgO series\n(1.61±0.07mJ/m2), as would be expected from the additional PMA from the secon d CoFeB/MgO interface. However,\nthe anisotropy per interface did not double with the additional CoFeB /MgO, which may be attributed to the di fferent\ndegrees of crystallization for single- and double-MgO samp les. An indication of better crystallization into CoFe in th e\nsingle-MgO series is its higher MS.Kvis negative and does not vary appreciably in samples where Ta is present, in\ncontrast to the positive value found for zero-insertion sam ples. The role of Ta with regard to Kvis not yet understood,\nas previouslypointed out by Sinha et al. [20], and a detailed study of the amount, proximity,and profile of Ta would be\nnecessarytoclarifythesee ffects.\nTurningourattentionto α,weidentifyasingle-MgOsample( tef f≈0.8nm)andasingle-insertionsample( tef f≈1.3\nnm)withacomparable Kef ftef f≈0.2mJ/m2. Weimmediatelynoticethat αforthesingle-insertionsampleisaroundtwo\ntimeslowerthanthatforthe single-MgOsample.\nThis dramaticdecrease in αmay be attributedto the suppressionof spin pumpingby the Mg O layers straddlingboth\nsides of the precessing magnet.[10, 26] Indeed, measuremen ts of zero-insertion samples show no thickness dependence\n[purple dashed line in Fig. 3(b)] and a low mean value of α=0.0035±0.0002 comparable to the bulk damping of\nCo40Fe40B20[5,27]. However,adecreasein αwithincreasing tef fcanstillbeseeninboththesingle-anddouble-insertion\nseries. One possible reason is the alloying of CoFeB and Ta, a s Ta is known to readily intermix with CoFeB [21], and\nhigher damping may be expected from CoFeBTa alloys [28]. The relative percentage of CoFeBTa alloy decreases with\nincreasing CoFeB thickness, coinciding with the αdecrease. This picture is also consistent with the jump in αfrom\nsingle-to double-insertionsamples, i.e.,thereismoreCo FeBTa alloybecausetherearemoreTa insertions. [11, 10]\nItmayalsobepossiblethatspinpumpingtotheTainsertionl ayeroccurs,asinthecaseofthePdinterlayerinCoFe /Pd\nmultilayers[29]. The complexityof oursystem, however,pr eventsusfromusinga simple multilayermodel. One reason\nisthatthemiddleCoFeBlayer(inthedouble-insertioncase )mayhavedifferentpropertiesfromtheCoFeBlayersadjacent\nto MgO, because CoFeB crystallizes from the MgO interface, [ 30] with which the middle CoFeB has no contact. The\ndegree of Ta intermixing also depends on the deposition orde r and will vary across the structure.[19] At this point, we\ncannot discriminate the mechanism behind the damping behav ior. It may be worthwhile to study the use of CoFeBTa\nalloysasinterlayerstopossiblyhavemorecontroloverthe amountanddistributionofTa inthestack.[31]\nInconclusion,wehavedemonstratedPMAandlowdampingindo uble-MgOstructures. AthinTainsertionlayerwas\nfound to significantly increase the PMA - no perpendicular ea sy axis was realized in our MgO /Co40Fe40B20/MgO films\nwithout Ta - and adding more insertionsallowed thicker CoFe B layers to remain perpendicular. However, the maximum\nKef ftef findouble-MgOsamplesiscomparableonlywiththatofthesin gle-MgOsampleforthisCoFeBcomposition.[9]\nOn the other hand, αfor double MgO films increases with the number of insertions b ut is still lower than that of single\n3MgO films for the entire range. Considering both trends with n, we find that the optimal stack in the range of samples\nwe studiedis a double-MgO, n=1 sample with tnom=1.75nm,which exhibits Kef ftef f=0.27mJ/m2at a low damping\nvalueof0.006.\nAcknowledgement\nWe expressgratitudeforsupportfromtheA*STARGraduateAc ademySINGAProgram.\nReferences\n[1] J. Z. Sun, S. L. Brown, W. Chen, E. A. Delenia, M. C. Gaidis, J. Harms, G. Hu, Xin Jiang, R. Kilaru, W. Kula,\nG.Lauer,L.Q.Liu,S.Murthy,J.Nowak,E.J.Oâ ˘A´ZSullivan,S.S.P.Parkin,R.P.Robertazzi,P.M.Rice,G.Sa ndhu,\nT.Topuria,andD.C. Worledge: Phys.Rev.B88,104426(2013) .\n[2] S. Yuasa: J. Phys.Soc.Jpn.77,031001(2008).\n[3] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and\nH. Ohno: Nat.Mater. 9,721(2010).\n[4] D. C. Worledge, G. Hu, David W. Abraham, J. Z. Sun, P. L. Tro uilloud, J. Nowak, S. Brown, M. C. Gaidis, E. J.\nO’Sullivan,andR. P.Robertazzi: Appl.Phys.Lett.98,0225 01(2011).\n[5] X. Liu,W.Zhang,M.J. Carter,andG. 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Yakushiji,A.Fukushima,H.Kubota,M.Konoto,andS. Yuasa: Appl.Phys.Express6,113006(2013).\n[13] V. B. Naik,H.Meng,andR. Sbiaa: AIP Advances,2,042182 (2012).\n[14] D. D. Lam, F. Bonell, S. Miwa, Y. Shiota, K. Yakushiji, H. Kubota, T. Nozaki, A. Fukushima, S. Yuasa, and\nY. Suzuki: J. Kor.Phys.Soc.62,1461(2013).\n[15] M. P.R. G.Sabino,S.T.Lim,andM.Tran: (unpublished).\n[16] V. Sokalski,M.T.Moneck,E.Yang,andJ.-G. Zhu: Appl.P hys.Lett 101,072411(2012).\n[17] H. Sato,M.Yamanouchi,S.Ikeda,S. Fukami,F. Matsukur a,andH.Ohno: IEEETrans.Mag.49,4437(2013).\n[18] G. Malinowski, K. C. Kuiper, R. Lavrijsen, H. J. M. Swagt en, and B. Koopmans: Appl. Phys. Lett. 94, 102501\n(2009).\n[19] S. Y. Jang,C.-Y. You,S. H.Lim,andS. R.Lee.: J. Appl.Ph ys.109,013901(2011).\n[20] J. Sinha, M. Hayashi, A. J. Kellock, S. Fukami, M. Yamano uchi, H. Sato, S. Ikeda, S. Mitani, S.-H. Yang, S. S. P.\nParkin,andH.Ohno: Appl.Phys.Lett.102,242405(2013).\n[21] Y.-H. Wang, W.-C. Chen, S.-Y. Yang, K.-H. Shen, C. Park, M.-J. 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Noshiro,C. Yoshida, Y. Yamazaki, A. Taka hashi, Y. Iba, A. Hatada, M. Nakabayashi,T. Takenaga,\nM. Aoki,andT.Sugii.IEDM,29.1.1(2012).\n5/s50/s48/s48 /s50/s50/s48 /s50/s52/s48/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48\n/s40/s99/s41/s82/s101/s115/s111/s110/s97/s110/s99/s101/s32/s70/s105/s101/s108/s100/s32/s40/s109 /s84/s41/s32\n/s32\n/s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48\n/s76/s105/s110/s101/s119/s105/s100/s116/s104/s32/s40/s109 /s84/s41/s32/s68/s97/s116/s97 /s32\n/s32/s70/s105/s116\n/s70/s114/s101/s113/s117/s101/s110/s99/s121 /s32/s40/s71/s72/s122/s41/s32/s68/s97/s116/s97\n/s32/s70/s105/s116/s83\n/s50/s49/s32/s82/s101/s97/s108/s32/s40/s97/s46/s117/s46/s41\n/s40/s97/s41\n/s40/s98/s41/s83\n/s50/s49/s32/s73/s109/s97/s103/s46/s32/s40/s97/s46/s117/s46/s41\n/s65/s112/s112/s108/s105/s101/s100/s32/s70/s105/s101/s108/s100/s32/s40/s109/s84/s41\nFigure 2: (a) Real and (b) imaginary parts of the S21parameter obtained from VNA-FMR measurements for a double-\ninsertion sample with tnom=2.5 nm at 12 GHz while a perpendiculardc magnetic field is swept. The lines are fits to an\nexpressionusingEq.1, takingnonmagneticcontributionst oS21anda lineardriftintoaccount. (c)Field-sweptlinewidth\nandresonancefieldsforthesamesampleasafunctionoffrequ ency. Thelinearfitsdescribedinthetextareusedtoextract\nHKef f(=−Mef f)andα.\n/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s48 /s49 /s50 /s51/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s115/s105/s110/s103/s108/s101/s32/s77/s103/s79\n/s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s48\n/s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s49\n/s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s50/s75\n/s101/s102/s102/s116\n/s101/s102/s102/s32/s40/s109/s74/s47/s109/s50\n/s41\n/s40/s97/s41/s40/s49/s48/s45/s51\n/s41\n/s116\n/s101/s102/s102/s32/s40/s110/s109/s41/s40/s98/s41\nFigure3: (a) Kef ftef fand(b)αversuseffectiveCoFeBthickness tef fobtainedfromfield-sweptVNA-FMRmeasurements\nfor all sample series. Solidlines in (a)are linear fits. Purp ledashedline in (b) correspondsto the mean αvalueaveraged\noverall zero-insertionsamples,whichwasfoundtobeconst antwithinerroracrosstheentirethicknessrangestudied.\n6"    },    {        "title": "2310.15945v1.Frictional_weakening_of_a_granular_sheared_layer_due_to_viscous_rolling_revealed_by_Discrete_Element_Modeling.pdf",        "content": "Springer Nature 2021 L ATEX template\nFrictional weakening of a granular sheared layer due to\nviscous rolling revealed by Discrete Element Modeling\nAlexandre Sac-Morane1,2*, Manolis Veveakis1†and Hadrien Rattez2†\n1*Multiphysics Geomechanics Lab, Duke University, Durham, 27708, NC, USA.\n2Institute of Mechanics, Materials and Civil Engineering, UCLouvain, Louvain-la-Neuve,\n1348, Belgium.\n*Corresponding author. E-mail: alexandre.sac-morane@uclouvain.be;\nContributing authors: manolis.veveakis@duke.edu; hadrien.rattez@uclouvain.be;\n†These authors contributed equally to this work.\nAbstract\nConsidering a 3D sheared granular layer through a discrete element modeling, it is well known the\nrolling resistance influences the macro friction coefficient. Even if the rolling resistance role has been\ndeeply investigated previously because it is commonly used to represent the shape and the roughness\nof the grains, the rolling viscous damping coefficient is still not studied. This parameter is rarely used\nor only to dissipate the energy and to converge numerically. This paper revisits the physical role of\nthose coefficients with a parametric study of the rolling friction and the rolling damping at different\nshear speeds and different confinement pressures. It has been observed the damping coefficient induces\na frictional weakening. Hence, competition between the rolling resistance and the rolling damping\noccurs. Angular resistance aims to avoid grains rolling, decreasing the difference between the angular\nvelocities of grains. Whereas, angular damping acts in the opposite, avoiding a change in the difference\nbetween the angular velocities of grains. In consequence, grains stay rolling and the sample toughness\ndecreases. This effect must be considered to not overestimate the frictional response of a granular layer.\nKeywords: Discrete element method, Rolling parameter, Sheared layer friction, Granular materials\n1 Introduction\nAccurately measuring or calculating the frictional\nstrength of granular sheared layers is of paramount\nimportance across all fields of granular media-\nrelated sciences, including earthquakes and fault\nmechanics [1, 2], landslides [3], and debris flows\n[4] to name but a few. It is very well accepted\nnowadays that the calculation of a macroscopic\nproperty like the frictional coefficient of granular\nmedia is the result of grain-to-grain interactions atthe micro-scale [5, 6]. Therefore, in order to accu-\nrate capture those effects and homogenize them\nto the friction coefficient of a layer, higher order\nanalytical and numerical approaches need to be\nconsidered [7–9].\nOne of the most well accepted approaches in\ndirect modeling of granular media is the Discrete\nElement Method [10], which has been designed\nto consider those interactions between grains [11].\nThe method started with a simple linear contact\nlaw[12, 13], but since then contact laws have been\nmodified by [14]: (i) considering the grain crushing\n1arXiv:2310.15945v1  [physics.geo-ph]  24 Oct 2023Springer Nature 2021 L ATEX template\n2 Rolling relaxation controls friction weakening\n[15, 16], (ii) investigating the effect of the pres-\nsure solution [17–19], (iii) exploring the effect of\nthe healing [20, 21], (iv) appreciating the influence\nof the cohesion in the matter [22–24] or (v) the\ncohesion induced by the pore fluid [25, 26] among\nothers. Also, it allows some to focus on the tem-\nperature influence, identifying the pressurization\nof the pore fluid [27, 28] and grain melting [29, 30]\nas the main phenomena driving the evolution of\nthe frictional strength of a fault zone during large\ncrustal events.\nThe present work is motivated by previous\nexperiments made on antigorite [31], and the main\ngoal is to assess the influence on global behav-\nior of contact laws and parameters values. Even\nthough different relevant outputs for dense gran-\nular flows are reviewed by the French research\ngroup Groupement de Recherche Milieux Divis´ es\n(GDR MiDi) [32], we are focused in this paper\non the macro friction coefficient at steady state.\nAs such, we introduce and explore the influence\nof the rolling resistance between grains to the\nmacroscopic strength of a granular sheared layer.\nExperimental results [33, 34], and numerical ones\n[35–37] have highlighted that grain rolling has a\nreal impact on the sample behavior with many\nrolling models being formulated since [38, 39].\nIn the literature the elastic-plastic spring dash-\npot model is identified as the benchmark for this\nresponse [40, 41] and has been extended to con-\nclude that: (i) rolling helps the formation of shear\nbands and decrease the sample strength [42–45];\n(ii) the stress-dilatancy curves are modified [46–\n49] when accounting for the rolling resistance\ncoming from intragranular friction [50, 51] and\nroughness [52–54].\nHowever, the computational cost of these\napproaches is not negligible, and especially if\ngrains clusters [55], superquadric particles [56] or\neven polyhedral shapes [57–59] are assumed to\napproximate the shape, those simulations become\nquickly computationally costly. Because of this\nfact, geometric laws for the rolling friction have\nbeen developed [60, 61] allowing for simulations to\nkeep using round particles with a rolling resistance\nstemming from an equivalent shape. However, the\nintroduced angular damping influence is not well\nconstrained, hence being neglected in most of the\nDEM simulations only used for stability reasons\n[38, 40] rather than for physical robustness [62] In\nthis work we revisit the physical role of the rollingresistance in a granular sheared layer and perform\na parametric study over the rolling friction and the\nrolling viscous damping coefficients to understand\nbetter their influence on the macroscopic friction\ncoefficient of a granular sheared layer.\n2 Theory and formulation\nThe Discrete Element Model (DEM) is an\napproach developed by Cundall & Strack [11] to\nsimulate granular materials at the particles level.\nThe foundation of this method is to consider\ninside the material the individual particles and\ntheir interactions explicitly. Newton’s laws (linear\nand angular momentum) are used to compute the\nmotion of the grains, as follows:\nm∂vi\n∂t=m×gi+X\nfi (1)\nI×ρ∂ωi\n∂t=X\nϵijkfjRk+X\nMi (2)\nwhere mis the particle mass, vthe particle\nvelocity, gthe gravity acceleration, fthe contact\nforces, Ithe moment of inertia, ρthe particle den-\nsity, ωthe angular velocity, ϵijkthe Levi-Civita\nsymbol, Rthe radius, Mthe contact moments.\nConsidering two particles with radii R1and\nR2, the interaction between particles is computed\nonly if the distance δbetween grains satisfies the\nfollowing inequality:\nδ < R1+R2(3)\nOnce contact is detected between grains 1 and\n2, interactions (force and moment) are computed\nfrom relative motions ∆ uand ∆ ωas:\n∆ui=u1\ni−u2\ni+ϵijk\u0000\nR1\njθ1\nk−R2\njθ2\nk\u0001\n(4)\n∆ωi=ω1\ni−ω2\ni (5)\nwhere uis the particle displacement, θthe\nangular displacement and ωthe angular velocity\nof the grain.\nThe contact models between cohesionless par-\nticles obey the Hertz contact theory [63]. Normal,\ntangential and angular models are shown at figure\n1 and described in the following.Springer Nature 2021 L ATEX template\nRolling relaxation controls friction weakening 3\nFig. 1 The contact between two particles obeys to normal,\ntangential and rolling elastic-plastic spring-dashpot laws.\nAs the contact can happen between particles\nwith different properties, some equivalent param-\neters need to be defined. The equivalent radius R∗\nand equivalent mass m∗are defined at equations\n6 and 7 with an harmonic mean.\n1\nR∗=1\nR1+1\nR2(6)\n1\nm∗=1\nm1+1\nm2(7)\nThe equivalent Young modulus Y∗and shear\nmodulus G∗are defined at equations 8 and 9 with\nan harmonic mean adjusted by Poisson’s ratio.\n1\nY∗=1−ν12\nY1+1−ν22\nY2\n⇐⇒Y∗=Y\n2(1−ν2)(8)\n1\nG∗=2(2−ν1)(1 + ν1)\nY1+2(2−ν2)(1 + ν2)\nY2\n⇐⇒G∗=Y\n4(2−ν)(1 + ν)\n(9)\nThe equivalent moment of inertia I∗is defined\nat equation 10 with an harmonic mean of the dif-\nferent moments of inertia displaced at the contact\npoint.\n1\nI∗=1\nI1+m1R12+1\nI2+m2R22 (10)\nNormal model\nThe normal force is formulated as:\nfn=kn∆n−γnvn (11)\nThe reaction is divided into a spring part\nand a damping part, with the normal stiffness knformulated as:\nkn=4\n3Y∗p\nR∗∆n (12)\nFollowing the Hertz contact theory, this parameter\ndepends mainly on the normal overlap ∆ n. Thus,\nthe normal force is not linear with respect to the\noverlap. The normal stiffness depends also on the\nequivalent Young modulus Y∗and the equivalent\nradius R∗defined before. The normal damping γn\nis null in this paper because the restitution coef-\nficient eis taken at the value 1. This choice have\nbeen made to focus on the influence of the rolling\ndamping.\nTangential model\nThe tangential force is formulated to verify the\nCoulomb friction law defined on the friction coef-\nficient between particle µpand the normal force\nfn:\nft=kt∆t−γtvt≤µpfn (13)\nThe reaction is divided into a spring part and\na damping part. The tangential stiffness ktis\nformulated as:\nkt= 8G∗p\nR∗∆n (14)\nFollowing the Hertz contact theory, this parameter\ndepends mainly on the normal overlap ∆ n. Thus,\nthe tangential force is not linear with respect to\nthe overlap. The tangential stiffness depends also\non the equivalent shear modulus G∗and the equiv-\nalent radius R∗defined before. The tangential\ndamping γtis also null in this paper because the\nrestitution coefficient eis taken at the value 1.\nAngular model\nA lot of angular models could be applied but\nan elastic-plastic spring-dashpot model is used\nbecause it is the most accurate choice. Hence, the\nmodel allows energy dissipation during relative\nrotation and provides packing support for static\nsystem, two main functions to verify in a par-\nticulate system [38]. The reaction moment Mis\nformulated as:\nM=Mk+Md(15)Springer Nature 2021 L ATEX template\n4 Rolling relaxation controls friction weakening\nThis reaction is divided into a spring part Mk\nand a damping part Mddefined at equations 16\nand 19:\nMk\nt+∆t=Mk\nt−kr∆θ≤Mm(16)\nThe incremental angle ∆ θis obtained by a\ntime integration of the angular velocity ∆ ω×\ndt. The angular stiffness kris formulated at\nequation 17 by considering a continuously dis-\ntributed system of normal and tangential spring\nat the interface [38], [62].\nkr= 2,25knµ2\nrR∗2(17)\nThe rolling friction coefficient µris introduced.\nThis variable is a dimensionless parameter defined\nas [38]:\nµr=tan(β) (18)\nThe angle βrepresents the maximum angle of\na slope on which the rolling resistance moment\ncounterbalances the moment due to gravity on\nthe grain, see figure 2. The influence of µris\ninvestigated in this paper.\nFig. 2 Definition of the rolling resistance coefficient µr=\ntan(β).\nMd\nt+∆t=\u001a\n−Cr∆ωifMk\nt+∆t< Mm\n0 if Mk\nt+∆t=Mm (19)\nIt appears the damping part Mdis defined\nwith a rolling viscous damping parameter Cr\nformulated at equation 20.\nCr= 2ηrp\nIrkr (20)\nThe rolling viscous damping coefficient ηr\nis introduced. This variable is a dimensionless\nparameter and its influence is investigated in this\npaper. Moreover, it appears the rolling viscous\ndamping parameter depends on the rolling stiff-\nness krand so on the rolling friction coefficient\nµr.As described by equations 16 and 19, the\nspring and damping parts are restricted by a plas-\ntic behavior, the rolling of particles. The rolling\nstarts when the spring part reaches the plastic\nlimit Mmdefined at equation 21.\nMm=µrR∗fn (21)\nThis limit depends on the equivalent radius,\nthe rolling friction coefficient and the normal force.\nOnce rolling occurs, the reaction from the angular\nspring takes the value of Mmand the damping\nelement is deleted.\n3 Numerical model\nThe simulation setup is illustrated at figure 3. The\nbox is a 0 ,004m×0,006m×0,0024mregion.\nFaces x and z are under periodic conditions. The\nsize of domain has been chosen to respect a suf-\nficient number of grain over the different axis.\nOn the axis x, the shearing direction, there are\nlx/d50= 4/0.26 = 15 particles. On the axis z, the\nminor direction, there are lz/d50= 2.4/0.26 = 10\nparticles. On the axis y, the size allows the parti-\ncles generation shown at figure 4. Sizes have been\nminimized to reduce the number of grains and so\nthe computational cost, main trouble with DEM\nsimulations. The gravity is not considered because\nits effect stays negligible under the vertical pres-\nsure applied.\nFig. 3 The simulation box with triangle plates and peri-\nodic faces.\nThe simulation, made by the open-source soft-\nware LIGGGHTS [64], is in several steps illus-\ntrated at figure 4:\n1. The box, bottom and top triangle plates are\ncreated. The triangle pattern represents the\nroughness of the plates with a geometry simi-\nlar to experimental tests [65, 66]. The specific\nsize of the triangle is defined to be 1 .5 times\nthe largest particle diameter.Springer Nature 2021 L ATEX template\nRolling relaxation controls friction weakening 5\n2. 2500 particles are generated following the dis-\ntribution presented in table 1 equivalent to the\none used in [21]. This number of particles allows\nto get 17 particles on the height where the\nresidual shear band sizes for different distribu-\ntions was assumed to be between 9 ×d50and\n16×d50[67].\n3. Top plate applies vertical stress of 10 MPa\nby moving following the y axis. This plate is\nfree to move vertically to verify this confin-\ning and allow volume change. The value of the\nvertical stress has been chosen from previous\nexperiments and numerical simulations [68–70].\n4. The sample is sheared by moving the bot-\ntom plate at the speed of 100 µm/s until\n100% strain. This step is then repeated at the\nspeed of 300 and 1000 µm/s. Those veloci-\nties have been chosen from previous numerical\nsimulations [70] and in-situ estimations [71] to\nminimize the computational cost but still to\nrepresent real sheared layers.\nFig. 4 The simulation is in multiple steps : creation of\nthe box and particles, application of the normal force and\nshearing.\nRadius Percentage Number of particles\nR1 = 0 ,2mm 14% 2500\nR2 = 0 ,15mm 29%\nR3 = 0 ,1mm 57%\nTable 1 Distribution used described by discrete radius,\npercentage of the mass and total number of grains.\nThen, the influence of the vertical pressure is\ninvestigated. The same set-up is used except the\nvertical pressure ( P= 1MPa and = 100 MPa ).\nThe rolling friction coefficient is constant µr=\n0,5 and the rolling viscous damping coefficient\nchanges ηr= 0,25, 0,5 or 0 ,75.\nThe different parameters needed for the DEM\nsimulation are presented in table 2 and the value\nhas been chosen from previous articles to represent\nrock material [19, 31, 44, 70]. We can notice the\ntime step dtmust verify the Rayleigh conditionVariable Short\nNameValue\nSimulation variables\nTime step dt 1,5e−6s\nHeight of the\nsampleh 0,005 m\nShear rate γ′2−6−20%\nContact stiffness\nnumberκ 400\nInertial number I 10−6−10−5\nMechanical variables\nDensity ρ 2000000 kg/m3\nYoungs modulus Y 70GPa\nPoissons ratio ν 0,3\nRestitution coef-\nficiente 1\nRolling friction\ncoefficientµr 0−0,25−0,5−0,75−1\nRolling vis-\ncous damping\ncoefficientηr 0−0,25−0,5−0,75\nFriction\ncoefficientµp 0,5\nTable 2 DEM parameters used during simulations.\n[63, 72, 73] defined as:\ndtR=π×r×p\nρ/G\n0,1631×ν+ 0,8766(22)\nWith every computing test, the main prob-\nlem is the running time. The time step dtmust\nbe selected considering the number of particles,\nthe computing power, the stability of the sim-\nulation and the time scale of the test. In our\ncase, we are looking for a 102seconds term. If we\ninclude the default value into equation 22 the time\nstep is around 10−8second and the running time\nskyrockets. To answer this we can easily change\nthe density ρand the shear modulus G. We will\nsee those parameters are included in two dimen-\nsionless numbers defined at the equation 23: the\ncontact stiffness number κ[74–76] and the inertial\nnumber I[32, 75].\nκ=\u0010\nY\nP(1−ν2)\u00112/3\nThe contact stiffness number\nI=γ′d50p\nρ/P The inertial number\n(23)\nWhere γ′=vshear/his the shear rate, his\nthe height of the sample during the shear, d50the\nmean diameter, Pthe pressure applied.\nIt appears the constitutive law is sensitive to κ\nbecause grains are not rigid enough ( κ≤104) [74].Springer Nature 2021 L ATEX template\n6 Rolling relaxation controls friction weakening\nBy this fact, it becomes not possible to change the\nYoung modulus Y(and so the shear modulus G).\nThe inertial number Irepresents the behavior of\nthe grains, which can be associated with solids,\nliquids or gases [77]. This dimensionless parame-\nter does not affect the constitutive law if the flow\nregime is at critical state ( I≤10−3) [32, 75]. In\nconclusion, the density ρcan be modified, if we\nstay under the condition I≤10−3, to increase the\ntime step and solve our computing problem.\n4 Results and discussion\nRolling Parameters Influence\nA parametric study has been done on the\nrolling friction coefficient µrand the rolling vis-\ncous damping coefficient ηr. As figure 5 shows,\nthe macro friction coefficient is plotted following\nthe shear strain. This coefficient µis computed\nby considering µ=Fy/Fx, where Fy(resp. Fx) is\nthe component following the y-axis (resp. x-axis)\nof the force applied on the top plate. Because of\nthe granular aspect, there is a lot of oscillation.\nTo reduce this noise, at least 3 simulations are run\nby a set of parameters ( µr,ηr) and a mean curve\nis computed. Moreover, only the steady-state is\nconsidered and an average value is estimated.\nFig. 5 Example of µ-γcurve (dotted lines mark the dif-\nferent velocity steps 100 - 300 - 1000 µm/s).\nThe comparison of the macro friction coeffi-\ncient with different parameters set is highlighted\nat figure 6. It appears there is an increase of the\nsheared layer friction coefficient with the rolling\nresistance µruntil a critical point depending on\nthe rolling damping ηr. This reduction of the\nstiffness with the rolling damping is not easy to\nunderstand at the first point. The larger is the\ndamping, the stiffer should be the system. Twomain questions should be answered: why does the\nfriction coefficient decrease with the rolling resis-\ntance if there is damping? Why is the reduction\nlarger with the damping value?\nvshear = 100 µm/s\nvshear = 300 µm/s\nvshear = 1000 µm/s\nFig. 6 Evolution of the macro friction coefficient with\nthe rolling friction coefficient µrand the rolling viscous\ndamping coefficient ηrat different shear speeds with P=\n10MPa .\nFigure 7 helps to understand the behavior. It\nshows the rotation of particle (in red) during four\ndifferent cases. We can notice that the fewer rota-\ntions there are, the stiffer the system will be. It\nappears the number of rolling particle increases\nwith the rolling resistance, see the three first plots\nof figure 7. The decrease of the friction coefficient\nis explained by particles rolling. Moreover, it is\nshown at the second and last plot of figure 7 that\ndamping increases the number of rolling particles\nand so the friction coefficient is reduced.Springer Nature 2021 L ATEX template\nRolling relaxation controls friction weakening 7\nµr= 0,25−ηr= 0,5 µr= 0,5−ηr= 0,5\nµr= 1−ηr= 0,5 µr= 0,5−ηr= 0\nFig. 7 Slide of sample highlighting grain rotation (rolling\nis in red) for several cases.\nA focus on the model equations must be done\nat relation 24 to understand better those observa-\ntions (the input rolling parameters are emphasized\nin red). First, it appears the increment of the\nspring ∆ Mk\nrdepends on µ2\nrwhile the plastic limit\nµrR∗fndepends only on µr. There is a square\nfactor between those values. Thus, this plastic\nlimit, and so grain rolling, is reached easier with\na larger rolling resistance µrfor a same angular\ndisplacement θ.\nMm=µrR∗fn\n∆Mk\nr=−2,25knµr2R∗2∆θ\nMd\nr,t+∆t=\n\nif|Mk\nr,t+∆t|< µrR∗fn:\n−2ηrµr√2,25Irknω\nif|Mk\nr,t+∆t|=µrR∗fn:\n0(24)\nConcerning the damping, it avoids the vari-\nation of the angular position (∆ ω→0) during\nthe elastic phase. As we have seen before, the\nmain part of the sample is at the plastic phase\nand particles roll. So, it is as the damping acts\nin opposition of the angular spring, keeping grain\ninto the plastic phase. We can notice that we have\ndecided in this paper to shut down the damping\nmoment when the angular plastic limit is reached\n(see equation 24).\nIn this way, we can understand better the\nreduction of the friction coefficient with the rolling\nstiffness µrif damping is active. We can noticethere is no decrease but an increase of the friction\ncoefficient in the case of no damping. In absence\nof this one, the angular spring can act normally.\nThe larger the rolling parameter is, the stiffer the\nglobal sample is.\nThis observation can be used if we decide to\ntrack particle size distribution and grain shape\nevolution. Experiments and simulations have high-\nlighted that particles tend to become less or\nmore [78–80] rounded under large deformation.\nThe work of Buscarnera and Einav, extending\nthe Continuum Breakage Mechanics, reconciles\nthose conflicting observations [81]. Shape descrip-\ntors are converging to attractors. The evolution of\nthe aspect ratio α, related to the grain morphol-\nogy, is plotted following a breakage parameter B\n(B= 0 means unbroken state and B= 1 complete\nbreakage) and the stress σ. It appears that from\ndifferent initial values the aspect ratio converges\nto the the same value, the attractor.\nSoftening and hardening behavior can be\nunderstood thank to relations between shape and\nrolling friction coefficient, the evolution of the\ngrain shape and our work. For example, if the\naspect ratio decreases during the test, particles\nbecome less rounded, the rolling friction coefficient\nincreases, the sample toughness evolves depending\non the position of the critical point (softening or\nhardening).\nParticle Size Distribution Influence\nAs equation 24 shows, the plastic moment\ndepends on the equivalent radius R∗. Appendix A\nnotices smallest particles tend to roll more than\nthe others. The particle size distribution should\ninfluence the macro friction coefficient. Results\nfrom this observation are described at appendix\nB. This one highlights the fact that macro fric-\ntion coefficient evolves with mean diameter d50.\nThere is an increase of the residual value with this\nparameter. Unfortunately, other simulations must\nbe done to interpolate an accurate tendency. In\nfacts, the number of particles (and so the height\nof the original sample) must be increased. Thus,\nparticle size distribution with larger mean diame-\nterd50can provide a sufficient number of particle\nto reduce the DEM noise.\nVertical Pressure InfluenceSpringer Nature 2021 L ATEX template\n8 Rolling relaxation controls friction weakening\nMoreover, equation 24 highlights the plastic\nmoment depends on normal forces fn(and so on\nvertical pressure P). Figure 8 illustrates the evolu-\ntion of the macro friction coefficient with different\nvertical pressure Pand rolling viscous damping\ncoefficient ηr.\nvshear = 100 µm/s\nvshear = 300 µm/s\nvshear = 1000 µm/s\nFig. 8 Evolution of the macro friction coefficient with the\nrolling viscous damping coefficient ηrand the vertical pres-\nsurePat different shear speeds with µr= 0,5.\nIt appears the critical point defined previ-\nously (point before which the friction weakening\nappears) depends on the vertical pressure. This\nbehaviour has already been observed [70, 82]. It\nappears the importance of the rolling increases\nwith the vertical pressure. The mean angular\nvelocity ωzon the z-axis have been computed over\nconfigurations P−ηr−vshear at table 3. It is\nhighlighted that the vertical pressure Pfavours\nthe grain rolling. That is why a friction weakening\noccurs with this parameter.vshear = 100 µm/s\nηr= 0,25 0,50 0,75\n1MPa 0,3 2,2 33,7\n10MPa 20,3 455,6 1056,2\n100MPa 2384,57467,811774 ,1\nvshear = 200 µm/s\nηr= 0,25 0,50 0,75\n1MPa 0,6 3,6 34,1\n10MPa 23,9 426,6 1034,5\n100MPa 2327,37564,811685 ,2\nvshear = 1000 µm/s\nηr= 0,25 0,50 0,75\n1MPa 1,3 4,9 29,7\n10MPa 28,1 342,1 1023,3\n100MPa 2070,07483,611759 ,4\nTable 3 The mean angular velocity ωzon the z-axis for\ndifferent configurations ηr−Pat different shear speeds.\nSpeed Influence\nFigure 9 highlights the shearing speed influ-\nence on the system. It is the same results as\nbefore but plotted in another way. It appears there\nis no speed effect visible in most simulations as\nthe friction coefficient keeps the same value. It\nis not surprising that no speed effects are spot-\nted because there are no other parameters except\nthe damping parameter which depends on speed\nor time. A speed influence is nevertheless noticed\nfor cases where the friction coefficient starts to\ndecrease with rolling resistance (for example the\ncaseµr= 0,5 and ηr= 0,5 at figure 9). As shown\nat figure 7 for this set, few particles (in orange or in\nwhite) are still not rolling during this critical step.\nThe damping value is not large enough to cancel\nthe effect of the spring and few grains are in the\nelastic phase. The damping creates so in this case\na speed influence. If the damping value is larger,\nwe have seen particles tend to be all in the plastic\nphase. If it is lower, the damping is negligible or\nnull. In both cases, the speed effect disappears.\n5 Conclusion\nIn this paper, we have considered granular materi-\nals into a plane shear configuration to investigate\nthe effect of the rolling resistance and damping on\nthe macroscopic friction coefficient. Thank numer-\nical DEM simulations, the relation between those\nparameters becomes clearer. It appears :Springer Nature 2021 L ATEX template\nRolling relaxation controls friction weakening 9\nηr= 0\nηr= 0,25\nηr= 0,5\nηr= 0,75\nFig. 9 Evolution of the sample friction coefficient with the\nshear speed and the rolling friction coefficient µrat differ-\nent rolling viscous damping coefficients with P= 10 MPa .\n1. In the no damping case, the sample stiffness\nincreases with the rolling resistance.2. The consideration of the rolling damping intro-\nduces a critical point. For a constant damping\nvalue, the sample stiffness increases the rolling\nparameter until this critical point is reached.\nThen, the stiffness starts to decrease until a\nresidual value. Hence, the damping tend to act\nagainst the spring and grains roll.\n3. No visible speed effects have been highlighted\nexcept at critical point. For the same rolling\nresistance value : (i) When the damping param-\neter is not large enough, the angular spring is\nthe main element and no speed dependency is\nspotted, (ii) when the damping parameter is\ntoo large, all grains are in the plastic phase\n(roll) and the residual value is reached and\n(iii) when the damping parameter is at critical\nvalue, there is no main element in the rolling\nmodel, speed dependency occurs.\n6 Acknowledgements\nComputational resources have been provided by\nthe Consortium des ´Equipements de Calcul Inten-\nsif (C ´ECI), funded by the Fonds de la Recherche\nScientifique de Belgique (F.R.S.-FNRS) under\nGrant No. 2.5020.11 and by the Walloon Region.\nSupport by the CMMI-2042325 project is also\nacknowledged.\n7 Statements and\nDeclarations\nConflict of interest The authors declare that\nthey have no conflict of interest.\nAppendix A Distribution of\nthe rolling with\nthe radius\nFigure A1 highlights the distribution of the rolling\nin the sample with µr= 0,5,ηr= 0,5 and\nvshear = 100 µm/s . It appears smallest particles\nare rolling more than largest particles. The obser-\nvation stays qualitative and should be studied\nquantitatively to estimate better the behaviour.Springer Nature 2021 L ATEX template\n10 Rolling relaxation controls friction weakening\nFig. A1 Distribution of the rolling in the sample (blue is\nsmall rolling, red is large rolling).\nAppendix B Influence of the\nParticle Size\nDistribution\nDuring the creation of a fault zone, the parti-\ncles are crushed and the particle size distribution\nevolves greatly. Moreover, the figure A1 from the\nappendix A highlights the fact that small parti-\ncles tend to roll more than large ones (decreasing\nthus the sample toughness). To investigate how\nthis evolution can affect the frictional behavior\nof a fault, the Particle Size Distribution (PSD)\nis described by two main parameters: the mean\ndiameter d50and the fractal dimension Ddefined\nat equation B1 [83–85].\nN=r−D(B1)\nwith Nis the number of particle of radius r.\nSimulations have been run at the shear speed\nof 1000 µm/s during one step. Two rolling resis-\ntance µrare investigated 0 ,2 (smooth particles)\nand 1 ,4 (rough particles) and the rolling damping\nηris not considered (= 0). The different PSD used\nare described in table B1 and results are shown in\nfigure B2.\nIn the case of smooth particle, it seems curves\nare really similar. The only exception is the case\nD 1 D50 0,26 where a peak appears. More dif-\nferences can be appreciated in the case of rough\nparticles. A peak can be shown at the start of the\nshear movement. This peak varies with the PSD,\nfor example, the D 2,6 D50 0,4 one is more pro-\nnounced than the D 1 D50 0,4 one. It has already\nbeen observed that the peak tends to increase with\nthe fractal dimension [67].\nPSD has a significant influence on the residual\nfriction coefficient if the rolling resistance becomes\nlarge. In fact, this parameter seems to stay aroundPSD\nNameRadius Percentage Number of\nparticles\nD 1 R1 = 0 ,25mm 27% 750\nD50 0,4 R2 = 0 ,21mm 32%\nR3 = 0 ,16mm 41%\nD 2 R1 = 0 ,25mm 22% 750\nD50 0,4 R2 = 0 ,21mm 31%\nR3 = 0 ,17mm 47%\nD 2,6 R1 = 0 ,25mm 20% 750\nD50 0,4 R2 = 0 ,21mm 30%\nR3 = 0 ,17mm 50%\nD 1 R1 = 0 ,2mm 22% 2500\nD50 0,26 R2 = 0 ,14mm 30%\nR3 = 0 ,09mm 48%\nD 2 R1 = 0 ,2mm 14% 2500\nD50 0,26 R2 = 0 ,15mm 29%\nR3 = 0 ,1mm 57%\nD 2,6 R1 = 0 ,2mm 12% 2500\nD50 0,26 R2 = 0 ,15mm 25%\nR3 = 0 ,11mm 63%\nTable B1 Distribution used described by discrete\nradius, percentage of the mass and total number of grains\nin the case of smooth and rough particles.\nµr= 0.2\nµr= 1.4\nFig. B2 Evolution of the µwith the strain for several\nPSD.\nthe value of 0,5 for smooth particles. Whereas,\nin the case of rough particles, values from tests\nwith a mean diameter d50= 0,4mm are larger\n(≈0.8) than the ones obtained from tests with\nd50= 0,26mm(≈0.7).\nIn conclusion, it appears the rolling resistance\nhas an influence on the friction coefficient via\nthe PSD (and especially via the mean diame-\nterd50). More tests must be done to appreciateSpringer Nature 2021 L ATEX template\nRolling relaxation controls friction weakening 11\nresults. Hence, curves obtained from PSD with\nd50= 0.4mm are really noisy because there are\nnot enough grains (the sample height have been\nconserved).\nReferences\n[1] Myers, R., Aydin, A.: The evolution of faults\nformed by shearing across joint zones in sand-\nstone. J. of Struct. Geol. 26, 947-966 (2004)\nhttps://doi.org/10.1016/j.jsg.2003.07.008\n[2] Poulet, T., Veveakis, M., Herwegh, M., Bucking-\nham, T., Regenauer-Lieb, K.: Modeling episodic fluid-\nrelease events in the ductile carbonates of the Glarus\nthrust. Geophys. Res. 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It is shown that the spin-pumping-induced dynamic coupling between \nFM layers modifies the linearized Gilbert equations in a way that replaces the scalar Gilbert damping \nconstant with a nonlocal matrix of Cartesian dampi ng tensors. The latter are shown to be methodically \ncalculable from a matrix algebra solution of the Valet- Fert transport equations. As an example, explicit \nanalytical results are obtained for a 5-layer (spi n-valve) of form NM/FM/NM'/FM/NM. Comparisons \nwith earlier well known results of Tserkovnyak et al. for the related 3-layer FM/NM/FM indicate that \nthe latter inadvert ently hid the tensor character of the damping, and instea d singled out the diagonal \nelement of the local damping tens or along the axis normal to the plane of the two magnetization \nvectors. For spin-valve devices of  technological interest, the influen ce of the tensor components of the \ndamping on thermal noise or spin -torque critical currents are st rongly weighted by the relative \nmagnitude of the elements of the nonlocal, anisotropic stiffness-fiel d tensor-matrix, and for in-plane \nmagnetized spin-valves are generally more sensitive to the in-plane element of the damping tensor.  I. INTRODUCTION \n       For purely scientific r easons, as well as technological applica tions such as magnetic field sensors \nor dc current tunable microwave oscillator s, there is significant present interest1 in the magnetization \ndynamics in current-perpendicular-to-plane (CPP)  metallic multilayer devices comprising multiple \nferromagnetic (FM) films separated by normal meta l (NM) spacer layers. The phenomenon of spin-\npumping, described earlier by Tserkovnyak et al.2,3 introduces an additional source of dynamic \ncoupling, either between the magnetization of a single FM layer and its NM elect ronic environment, or \nbetween two or more FM layers as mediated through their NM spacers. In the former case,2 the effect \ncan resemble an enhanced magnetic damping of an individual FM layer, whic h has important practical \napplication for substantially increasing the spin-t orque critical currents of CPP spin-valves employed \nas giant-magnetoresistiv e (GMR) sensors for read head applications.4 Considered in this paper is a \ngeneral treatment in the case of two or more FM layers in a CPP stack, where it will be shown in Sec. \nII that spin-pumping modifies the linearized equations of motion in a way that replaces a scalar \ndamping constant with a nonlocal matrix of Cartesian damping tensors.5 Analytical results for the case \nof a 5-layer spin-valve stack of the form NM/FM/NM'/FM/NM are discussed in de tail in Sec. III, and \nare in Sec. IV compared and contrasted with the early well-know n results of Tserkovnyak et al..3, as \nwell as some very recent results of that author and colleagues.6 In the case of CPP-GMR devices of \ntechnological interest, the relativ e importance of the different elements of the damping tensor on \ninfluencing measureable thermal fluctuations or spin-t orque critical currents is shown to be strongly \nweighted by the  anisotropic nature of the stiffness-field tensor-matrix.  \n \nII. SPIN-PUMPING AN D TENSOR DAMPING \n       As discussed by Tserkovnyak et al,2,3 the spin current pumpI flowing into the normal metal (NM) \nlayer at an FM/NM interface (Fig. 1) due to the spin-pumping effect is  described the expression  \n \n⎥⎦⎤\n⎢⎣⎡− ×π=↑↓ ↑↓\ndtdgdtdgm mm IˆIm )ˆˆ( Re4pumph                                                     (1) \n \nwhere  is a dimensionless mixing conductance, and m is the unit magnetization vector. In this \npaper,  for any ferromagnetic (FM) layer is treated as  a uniform macrospin. A restatement of (1) in \nterms more natural to Valet-Fert↑↓g ˆ\nmˆ\n7form of transport equations is di scussed in Appendix A. With the \nnotational conversion , where A is the cross sectional area of the film stack, \nequation (1), for the case , simplifies to  pump pump) 2 / (J I A eh− →\n↑↓ ↑↓> > g gIm Re \ninterface NM/FM for \" \" interface, FM/NM for \" \"ReIm,ˆ ˆˆ) 2 / (\n22\npump\n+ −≡ ε⎟\n⎠⎞⎜\n⎝⎛ε + ×π≅↑↓↑↓\n↑↓rr\ndtd\ndtd\nre h e m mm Jm                                       (2) \n \nwhere  is the inverse mixing conductance (with  dimensions of resistance-area), and  \nis the well known inverse conductance quantum ↑↓ ↑↓≅ r rRe22 /e h\n) k 9 . 12 (Ω≅ . In the present notation, all spin current \ndensities  have the same dimensions as electron charge current density , and for conceptual \nsimplicity are defined with a parallel (i.e., ) rather than anti-parallel alignment with \nmagnetization . Positive J is defined as electrons flowing to the right (along  in Fig. 1.).  spinJeJ\nm Jˆ ˆspin+ =\nmˆ yˆ+\n       For a FM layer sandwiched by tw o NM layers in which the FM layer is the  layer  of a \nmultilayer  thj ) 0 (≥j\nfilm stack, spin-pumping contributions at the  interface, i.e., either left or right thi ) (j i= ) 1 (+=j i  \nFM-NM interfaces, (2) can be expressed as  \n \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛ε + ×−=\n↑↓−\n+ =dtd\ndtd\nr ej\nij\nj\nij i\nj j im m\nm Jˆ ˆ\nˆ) 1 (\n2pump\n1 ,h                                           (3) \n \n      The physical picture to now be invoked is that of small  (thermal) fluctuations of m about \nequilibrium  giving rise to the  terms in (2). Since ˆ\n0ˆm dt d/ˆm 1ˆ≡m , the three vector components of \n and/or are not linearly independent. To remove this  interdependency, as well as higher order mˆ dt d/ˆmFig. 1 Cross section cartoon of an N-layer multilayer stack with N-1 interior nterfaces of FM-NM or NM-NM type, \nsuch as found in CPP-GMR pillars sandwiched between conductive leads of much larger cross section. In the \nexample shown, the jth layer is FM, sandwiched by NM layers, with spin pumping contributions at the ith (NM/FM) \nand ( i+1)st (FM/NM) interfaces located at iy y=  and 1+iy (with i=j  for the labeling scheme shown).  j=0j-1j=1\ni=0 i=1 i-1j\ni+1j+1-Jpump\ni Jpump\ni+1\ni=jNM NM FMmj\nj=N-1z\ny\nlead lead\ni=N N-1terms in (3) it thus is useful to work in a primed coordinate system where , through use of a \n Cartesian rotation matrix  such that 0ˆ ˆm z=′\n3 3× )ˆ(0mℜ m mˆ ˆ′⋅ℜ= .8 To first order in linearly independent  \nquantities , ) , (y xm m′ ′ m m m′⋅ ℜ + =~ˆ ˆ0 , where , and where ℜ ⎟⎠⎞⎜⎝⎛≡′\n′′′′\nymxmm~ denotes the  matrix from \nthe first two (i.e., x and y ) columns of 2 3×\nℜ. Replacing z m′→ˆ ˆ0 , and _ˆ×′z  with matrix multiplication, \nthe linearized  form of (3) becomes \n \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n′′\n⋅⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nε− ε⋅ ℜ−=\n′′−\n+ = ↑↓ dt m ddt m d\nr e2h\ny jx j\nii\nj\nij i\nj j i//\n11 ~ ) 1 ( pump\n1 ,J                                  (4) \n \n     Using the present sign convention, j i s j A t Mm S ˆ/ ) (γ = is the spin angular momentum of the  \nFM layer with saturation magnetization-thickness product , and  is the gyromagnetic ratio. \nTaking thj\nj st M) ( 0> γ\nsM=M  as constant, it follows by angu lar momentum conservation that3 \n \n∑+\n=−× × − = ⇔γ1\nˆ ˆ ) 1 (21 ˆ ) (NMj\nj ij i jj i j j j s\ne dtd\nA dtd t M\nm J mS mh                                         (5) \n \nis the contribution to  due to the net transverse  spin current entering the FM layer (Fig. 1). \nIn (5),  denotes the spin-current density in the NM layer at the  FM-NM interface. Taking the \ncross product  on both sides of (5), transforming to pr imed coordinates by matrix-multiplying by \n, and employing similar linearization as to obtain (4), one finds to first order that dt dj/ˆmthj\nNM\niJthi\n×mˆ\nTℜ = ℜ−1\n \n⎥⎥\n⎦⎤\n⎢⎢\n⎣⎡\n− ≡ Δ ℜ ⋅⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛−=′\n×′ ∑+\n=−⋅1spin NM) 1 (~\n0 11 0\n21ˆj\nj iij i\nj jj\nje dtd\nAJ JS\nzT h                                     (6) \n \nwhere Tℜ~ is the  matrix transpose of ℜ 3 2×~. By definition, 0 ˆ~\n0= ⋅ ℜj jmT. \n      The quantities  in (6) are not known a priori , but must be determined after solution of the \nappropriate transport equations (e.g. , Appendix B). Even in the absence of charge current flow (i.e., \n as considered here, the  are nonzero due to the set of  in (4) which appear as \nsource terms in the boundary conditions (A 9) at each FM-NM interface. Given the linear relation of \n(4), one can now apply linear superposition to express spin\njJΔ\n) 0=eJspin\njJΔpump\niJ \n∑ ∑+\n=↑↓ ↑↓ ↑↓≡′\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nε− ε⋅ ℜ ⋅ = Δ1spin 1\n21 1,11 ~ 1\n2k\nk ii k kk\nk jk\nkjr r dtdC\nremJt h                                (7) \n \nin terms of the set of 3-D dimensionless Cartesian tensor jkCt\n. The jkCt\n are convenient for formal \nexpressions such as (9), or for analytical work in  algebraically simple cases, such as exampled in \nSec.III. However, they are also subject to met hodical computation. For the kth magnetic layer, the \ncolumns of each  are the dimensionless vectors simultaneously obtainable \nfor all magnetic layers j from a matrix solutionrd nd st3 or , 2 , 1jkCtspin\njJΔ\n9 of the Valet-Fert7 transport equatio ns with nonzero \ndimensionless spin-pump vectors )ˆor ,ˆ,ˆ)( / ( ) 1 (pump\n1 ,z y x J↑↓ ↑↓ −\n+ =− =i kk i\nk k ir r .  \n      To include spin currents via (5) into  the magnetization dynamics, the conventional Gilbert \nequations of motion for  can be amended as ) (ˆtm\n \ndtd\nA t M dtd\ndtdj\nj sj\nj j j ij S m\nm H mm 1\n) (ˆ\nˆ ) ˆ(ˆG eff γ+ × α + × γ − =                                     (8) \n \nwhere  is the usual (scalar) Gilbert damping paramete r. From (6) and (7), one can deduce that the \nrightmost term in  (8) will scale linearly with G\njα\ndt d/m′, as does the conventiona l Gilbert damping term. \nCombining these terms together after applying the analogous linearization procedure to (8) as was \ndone in going from (5) to (6), one obtains  \n \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nε− ε⋅ ℜ ⋅ ⋅ ℜ ⋅⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n− πγ= α′α′+ δ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nαα≡ α′′⋅ α′−′⋅ − ⋅ ℜ γ =′\n×′\n↑↓∑\n11 ~ ~\n0 11 0 2 /\n) 4 (00] ) ˆ(~[ ˆ\n2\npumppumpeff eff\nGG\nk jk j\nj j sjkjk jk\njj\njkk\nkjk j j j j jj\nj\nC\nre h\nt Mdtd\ndtd\nt h tt tt\nTT mm H m Hm\nz\n                      (9) \nwhere Kronecker delta k j k jjk jk ≠=δ= = δ if 0 and , if 1 . \n      In (9),  is a 2-dimensional Cartesian \"damping tensor \" expressed in a coordinate system where \n, while  is a \"nonlocal tensor\" spanning two such  coordinate systems. This formalism \nfollows naturally from the lineariza tion of the equations of motion for non-collinear macrospins, and is \nparticularly useful for describing the influence of \"tensor damping\" on the thermal fluctuations and/or j jα′t\nj jz m′=′ ˆ ˆ0 j k j≠α′tspin-torque critical currents of su ch multilayer film structures (e.g., as  described further in Sec. IV.). \nDue to the spin-pumping contribution pump\njkα′t, the four individual  (with v u\njk′ ′α′ y x v u′′=′ ′ or , ) are in \ngeneral nonzero with , reflecting the true tensor na ture of the damping in this \ncircumstance, which is additionally nonl ocal between magnetic layers (i.e., ). The are \nsomewhat arbitrary to the extent that one may replace y y\njkx x\njk′ ′ ′ ′α′≠ α′\n0≠ α′′ ′\n≠v u\nj jkv u\njk′ ′α′\n2~ ~ℜ ⋅ ℜ ↔ ℜ in (9), where 2ℜ is the \nmatrix representation of any rotation about the 2 2× z′ˆaxis. \n      It is perhaps tempting to contemplate an \"inverse linearization\" of (9 ) to obtain a 3D nonlinear \nGilbert equation with a fully 3D damping tensor T\nk jk j jkℜ ⋅ α′⋅ ℜ = αt t. However, (9) has a null zˆ′ \ncomponent, and contains no information rega rding the heretofore undefined quantities  or . \nFor local, isotropic/scal ar Gilbert damping, one can independen tly argue on spatial symmetry grounds \nthat . However, the analogous extension is  not so obviously available for z u\njk′ ′α′z z\njk′ ′α′\nG G G α = α′= α′′ ′ ′ ′u u z z pump\njkα′t, \ngiven the intrinsically nonlocal, anisotropic natu re of spin-pumping. The proper general equation \nremains that of (8), with the rightmost term  given by that in (5), or its equivalent.  \n \nIII. EXAMPLE: FIVE LAYER SYSTEM \n \n \n   Fig. 2 shows a 5-layer system with 2 FM layers resembling a CPP-GMR spin-valve, to be used as a Fig. 2. Cartoon of a prototypical 5-layer CPP-GMR stack (leads not shown) with two FM layers (#1 and #3),\nsandwiching a central NM spacer layer (#2 ), and with outer NM cap layers (#0 and #4). For discussion purpose\n  \nprototype. Although the full genera lization is straightforward, the material properties and layer s\ndescribed in text, the magnetization vectors 1ˆm and 3ˆm can be considered to lie in the film plane ( zx- plane). m1\nNMj=0 j=1 23 4\nFM NM' FM NMm3z\nxθ\ny\ny1y0 y2y3y4y5thickness will be assumed symmetric about the centr al (#2) normal metal spacer layer, which will \nadditionally be taken to have  a large spin-diffusion length  (with  the thickness of the  \nlayer), such that the \"ballistic\" approximation (B3) applies. The inverse mixing conductances \nwill also be assumed to be real. Using the outer boundary conditions described by (B5), one \nfinds for the FM-NM interfaces at  that 2 2t l> >>jtthj\n↑↓\n=4 1-ir\n, and4 1y y y=\n \n] )) / hyp( ( [ˆ\nNMFM NM\n1 1pump\n1\n3 , 1 4 , 1l t l r rrJi\nj i iρ + ≡ ′+ =\n↑↓ ↑↓↑↓\n= =Jm J                                  (10a) \n \nFM\nNMFM\n4 1 1 4 21] )) / hyp( ( [J l t l r r Vi ρ + ≡′= Δ −=                                        (10b) \n \nwhere , , and subscript \"NM\" refers to either outer layer 0 or 4. In  (10) and below, \n are used interchangeably. Inside FM la yer 3, (B1,2) have solution expressible as 4 1r r=↑↓↑↓=4 1r r\nj j0ˆ ˆm m↔\n \n3 1 1 33 3 3 3spin\n33 3 3 3 4 3 3\n] ) / tanh( ) /[( ] ) / tanh( ( [() / ) sinh(( ) / ) cosh(( [ ) /( 1 ) () / ) cosh(( 2 ) / ) sinh(( 2 ) (\nFM FM FMFM FM FMFM FM\nA l t r l l t l r Bl y y B l y y A l y Jl y y B l y y A y y y V\n′+ ρ ρ +′− =− + − ρ =− + − = ≤ ≤ Δ\n                   (11) \n \nwhere the expression for  follows from (10b). Subscript \"FM\" re fers to either layers 1 or 3. The \nboundary conditions (A5) and (A9) applied to the FM/NM boundary at 3B\n3y y= yield  \n \n) ( ˆ]ˆ ) /( )[ ( ) 2 (pump\n3spin\n2 2 3 3spin\n2 3 2 2 2 3 21\nFM J J m m J V − + ⋅ = ρ − = −↑↓ ↑↓r l A r r BΔ               (12) \n \nwhere , . The \"ballistic\" values 3 2r r=↑↓↑↓=3 2r r2VΔ  and  are constant inside  central layer 2. \nUsing (11) to eliminate coefficient  in (12), the latter may be rewritten as spin\n2J\n3B\n \n⎥⎦⎤\n⎢⎣⎡\nρ ′+ρ +′+ − ≡− ⋅ ⋅ + = −\n↑↓\n↑↓↑↓\nFMFM\n)] /( ) / [tanh( 1)) / tanh( (\n21] )ˆ ˆ 2 1 [(\n11\n2 2\n2pump\n3spin\n2 3 3 2 2 21\nl l t rl t l rr r\nrqq r J J m m VTt\nΔ\n                                    (13) \n \nwhere  is the 3-D identity tensor, and  denotes the 3-D tensor formed from the vector outer -\nproduct of  with itself.  1tT\n3 3ˆ ˆm m⋅\n3ˆm     Working through the equivalent comput ations applied now to the NM/FM interface at 2y y= , one \nfinds  the analogous result: \n \n] )ˆ ˆ 2 1 [(pump\n2spin\n2 1 1 2 2 21J J m m V − ⋅ ⋅ + = +↑↓ Tq rt\nΔ                                      (14) \n \nEliminating  between (13) and (14) derives the remaining needed result for : 2VΔspin\n2J\n \n1\n3 3 1 1pump\n3pump\n2 21 spin\n2)]ˆ ˆ ˆ ˆ ( 1 [ ), (−⋅ + ⋅ + ≡ + ⋅ = ⋅T Tm m m m J J J q Q Qtt t\n                      (15) \n \ntreating tensor Qt\n as the  matrix inverse of the [ ]-bracketed te nsor in (15). Using (10a) and (15) to \ncompute , then additional use of (4) and (6), allow computation of the 3 3×\nNM\n4 1-=iJjkCt\n defined in (7): \n \n) / 1 / 1 ( / 1 ; 2 / , /, 1\n2 1 212 131 13 33 11\n↑↓ ↑↓ ↑↓ ↑↓↑↓ ↑↓ ↑↓+ = ≡′≡− = = + = =\nr r r r r b r r aQ b C C Q b a C Cttttttt\n                               (16) \n \n      For explicit evaluation of pump\njkα′t, it is convenient to assume a choice of 3 , 1~\n=ℜj for which 3 1ˆ ˆy y′=′ , \nsuch that  and  lie in the  plane. To simplify the inte rmediate algebra to obtain Q03ˆm01ˆm z x′ ′-t\n from \n(15), one can consider \"in-plane\" magnetizations (Fig.2), taking z mˆ ˆ03=, and  in the x-z plane \n( ). This allows a particularly easy determination of 01ˆm\nθ = ⋅cosˆ ˆ01z mjℜ~ for which : y y yˆ ˆ ˆ3 1=′=′\n \n0 , ;0 1 0sin 0 cos ~\n3 1 3 , 1 = θ θ = θ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛ θ − θ= ℜ=j j\njT                                       (17) \n \nUsing (16) and (17) with (9) allows explicit solution for the pump\njkα′t: \n \nθ + +θ + −= =\nθ + +θ + += =⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n+ δ− δ + δ\nπγ= α′\n↑↓\n2 231 132 22\n33 112\npump\nsin 2 1cos ) 2 1 (,\nsin 2 1cos 100 ) 1 2 ( 2 /\n) 4 (\nq qqd d\nq qq qd dd b ab a\nre h\nt M jk jkjk jk\nj j sjkh t\n                              (18) \n \nTaking , (18) holds for arbi trary orientation of  and , provided the flexibility \nin choosing the 03 01ˆ ˆ cos m m⋅ = θ01ˆm03ˆm\n3 , 1~\n=ℜj is used to maintain 3 1ˆ ˆy y′=′ . However, for multilayer film stacks with three or \nmore magnetic layers with magnetizations  that do not all lie in a singl e plane, it wi ll generally be \nthe case that some of the off-diagonal elements of the j0ˆm\npump\njkα′t will be nonzero.   \n \n \nIV. DISCUSSION \n \n      It is perhaps instructive to compare and contrast the results of (9) and (18). with the prior results of \nRef. 3. The latter are for a a trilaye r stack, corresponding most  directly to taking ∞→ ρNM  in the \npresent model, whereby . It is also effectively equivalent to the 5-layer case with \ninsulating outer boundaries in the limit , whereby  but  due to \nperfect cancellation by the spin current reflected from the  boundaries without intervening spin-\nflip scattering. Either  way, it corresponds to  in (10) and  in (16) and (18).  0NM\n4 , 1pump\n4 , 1= == = i iJ J\n0 ) / (NM→ l t 0pump\n4 , 1≠=iJ 0NM\n4 , 1→=iJ\n5 , 0=iy\n∞ →′ ′↑↓\n1 1,r r 0→a\n      However, a more interesting difference is that Ref. 3 treats  as stationary (hence , \nand  as undergoing a perfectly circular  precession about  with a possibly large cone angle 3ˆm ) 0pump\n3= J\n1ˆm3ˆm θ. \nBy contrast, the present analysis treats  and  equally as quasi-stationary vectors which undergo \nsmall but otherwise random fluctuations about their equilibrium positions  and , with \n. To further elucidate this distinction, one can assume the aforementioned physical \nmodel of Ref. 3, and reanalyz e that situation in terms of the present formalism. With \n, and by explicitly inserting the condition (e.g., from (3)) that , an \nexplicit solution of (15) can be expressed in the form: 1ˆm3ˆm\n01ˆm03ˆm\nθ = ⋅cos ˆ ˆ03 03m m\npump\n3 3 0 /J m = = dt d 0 ˆ1pump\n2≡ ⋅m J\n \n]ˆ\ncos ) 1 (ˆ) 1 ( ˆ cos[3pump\n2 2 2 23 12\npump\n2 212NMm Jm mJ J ⋅\nθ − ++ − θ+ =\nq qq q q                                   (19) \n \nCombining (19) with the earlier re sult from (5) and then (3) (with ) 0=ε , it is readily found that  \n \ndtd\nq qq q\nre h\nt Mq qq q\nt M et M e dtd\nA t M dtd\nsss s\n1\n2 2 22\n22\n11 32 2 2pump\n2 3 pump\n2 1\n12 1\n11\n1\n11\n1\nˆ\ncos ) 1 (sin ) 1 (12 /\n) 8 (ˆ ˆ\ncos ) 1 () ˆ)( 1 (ˆ\n) ( 4ˆ\n) ( 21ˆ\n) (ˆˆNM\nmm mJ mJ mJ mSmmm\n⎥⎥\n⎦⎤\n⎢⎢\n⎣⎡\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nθ − +θ +−πγ− =⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n×\nθ − +⋅ ++ ×γ− =×γ− = ×γ⇔ ×\n↑↓hhh\n                       (20) \n The last result in (20) uses  from (3), and the fact that pump\n2J θ = ×sin ˆ ˆ1 3m m , and that  and \n are parallel  vectors in the case of steady circular  precession of  about a fixed . It is the \ndirect equivalent of Eq. (9) of Ref. 3 with the identification dt d/ˆ1m\n1 3ˆ ˆm m×1ˆm3ˆm\n) 1 /(+⇔ν q q . \n     Although the final expression in (20) is azimuthally invariant with vector orientation of , it is \nmost convenient to compare it with  (18) at that instant where  is \"in-plane\" as shown in Fig. 2. At \nthat orientation,1ˆm\n1ˆm\ndt m d dt dm dt dy y / / /ˆ1 1 1′= → m , and it is immediately confirmed from (9) and (18) \n(with ) that the [ ]-term in (20) is simply the tensor element 0→ay y′′α′11 of pump\n11α′t. It is now seen that \nthe analysis of Ref. 3 happens to mask the tensor  nature of the spin-pump damping by its restricting \nattention a specific form of the mo tion of the magnetization vectors, which in this case singles out the \nsingle diagonal element of the pump\n11α′t tensor along the axis perpendicular to the plane formed by \nvectors  and . The very recent results of Ref. 6 do addr ess this deficiency of generality, and \nreveal the tensor nature of 1ˆm3ˆm\npump\n11α′t with specific results for ππ=θ and , 2 / , 0 . The present Sec. III \nadditionally includes the nonlocal tensors pump\n31pump\n13α′= α′t t, as well as diagonal terms jkaδ in (18) \n(and the variation in parameter q) when it is not the case that )/ hyp( ) (NM NM NM FM NM l t l r ρ<<-  in \nboundary condition (B4). The latter condition will likely apply in the case of the technological \nimportant example of CPP-GMR spin-valves.  \n      Speaking of such, two important practical i ssues for these devices involve thermal magnetic noise \nand spin-torque induced oscillat ions. As described previously8, an explicit linearization of the effH  \nterm in (9) about equilibrium state  that is a minimum of the free energy 0ˆm E leads to the following \nmatrix form of the linearized Gilber t equation including spin-pumping (with : )0=eJ\n \nmA t M\npE\nmH H Hp p p\nGp p\nDt p t HdtdG D\nj s\nj\njjk jk j jk\nkj\njk j j jkkj k jk j\njkj\njkkj k jk j\njkj j j j\nkk jkk\nkjk jk\nΔ≡∂∂\nΔ−≡ℜ ⋅ ⋅ ℜ ≡ ′\n∂∂\n− δ ⋅ ≡⎥\n⎦⎤\n⎢\n⎣⎡\nγα′− α′\n+ δγ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛−≡′\nγα′+ α′\n≡′⋅ ℜ ≡ ′= ⋅′⋅′+′⋅′+′ ∑ ∑\n) (\n,ˆ)ˆ( 1~ ~,ˆ1 ) ˆ(2 0 11 0,2) (~) ( ) (\n0 eff\n0eff\n0 eff\n0 0\nmmHmH\nH mh h mm\nt t t tt ttt ttt tt\nTT\n                           (21) \n where the  are small perturbation fields. The form of ) (tjhjkDt\n′ and jkG′t\n in (21) is chosen so that they \nretain the original delineation8 as symmetric and antisymmetic tensors regardless of the symmetry of \n. By use of a fixed  \"reference-moment\" jkα′tmΔ in the definition of , the \"stiffness-field\" tensor-\nmatrix  is symmetric positive-definite, and eff\njH\nv k u jv u\njk m m E H′ ′′ ′′∂′∂ ∂ ∝ ′ / ∑ ⋅ − =δj j j j sδ t M A E m h ) ( \n∑′⋅′Δ − =j j j m m h has the proper conjugate form so that (21) are now ready to directly apply \nfluctuation-dissipation expressions  specifically suited to such linear matrix equations of motion.8 \nTreating the fields  now as thermal fluctuat ion fields driving the ) (tjh′ ) (tjm′-fluctuations,  \n \nv u\njkB\nh hv u\njkB\nv k u j DmT kS DmT kh hv k u j′ ′\n′ ′′ ′\n′ ′ ′\nΔ= ω′⇔ τ δ′\nΔ= 〉′τ′〈′ ′2) ( ) (2) 0 ( ) (                                 (22) \n \nare the time-correlation or  cross power spectral density (PSD) F ourier transform pairs. Through their \nrelationship described in (21), the nonlocal, tensor nature of  the spin-pumping contribution pump\njkα′t to \njkα′t is directly translated into those of the FM FM2 2N N×  system \"damping tensor-matrix\" v u\njkD′ ′′↔′Dt\n, \nwhere  is the number of FM layers in the multila yer film stack. The cross-PSD tensor-matrix FMN\n) ( ) ( ω′↔ ω′′ ′′ ′ ′v k u jm mSm mSt\n for the m-fluctuations can then be expressed as′8  \n \n1)] ( [ ) () ( ) ( )] ( ) ( [ ) (\n−′ ′ ′ ′\n′+′ω −′≡ ω′ω′⋅′⋅ ω′→ ω′− ω′\nΔ ω= ω′\nD G HS Sh h m m\nttt tttt t t t\nim iT kB\nχχ χ χ χ@ @\n                           (23) \n \nwhere  is the complex susceptibility tensor-matrix for the ) (ω′χt} , {h m′′ system, and ) (ω′@χt its \nHermitian transpose. It has been theoretically argued10 that (22), and thus the second expression in (23), \nremain valid when , despite spin-torque contributions to  resulting in an asymmetric  0≠eJeff\njH H′t\n \n(e.g., see (25)) that violates the condition of therma l equilibrium implicitly a ssumed for the fluctuation \ndissipation relations.  \n     Since  is in general a fully nonlocal with anisotropic/tensor character, any additional tensor \nnature of H′t\nDt\n will likely be altered or muted as to the influence on the detectable -fluctuations. As an \nexample, one can again consider th e situation depicted in Fig. 2, applied to the case of a CPP-GMR \nspin-valve with typical in-plane  magnetization. The device's output noise PSD will reflect fluctuations m′in . Taking  to again play the simplifying role of an  ideal fixed (or pinned) reference layer \n(i.e., ), the PSD will be proportional to . As was also shown \npreviously,3 1ˆ ˆm m⋅3ˆm\n0 /ˆ3→dt dm ) ( sin1 12ω′θ′ ′′x xm mS\n11 it follows from (23) (and assuming azimuthal symmetry 011 11=′=′′ ′ ′′ x y y xH H ) that  \n \nx x y y y y x x x x y yy y x x y y x x\nsB\nm m\nH H H HH H\ntA MT kSx x\n′ ′′ ′ ′ ′ ′ ′ ′ ′′ ′′ ′ ′′′′′′\n′ ′\n′α′+′α′= ω Δ′ ′ γ = ωω Δ ω + ω − ωω α′+ ω′ ′ α′γ≅ ω′′\n11 11 11 11 11 11 02 2 2\n022\n112\n0 11 11 11\n1 ) ( ) () / (\n) (2) (1 1                         (24) \n \ntreating . The tensor influence of the  is seen to be weighted by the relative size \nof the stiffness-field matrix elements . For the thin film geometries 111 11 < < < α′α′′ ′′ ′ y y x x u u′ ′α′11\nv vH′ ′′11 A t< <  typical of such \ndevices, out-of-plane demagnetization fi eld contribution typically result in  that are an order of \nmagnitude larger than . Since y yH′ ′′11\nx xH′ ′′11x x y y ′ ′ ′′α′≤ α′11 11 from (18), it follows that the linewidth ωΔ and the \nPSD  in the spectral range of practical interest will both be expected to be determined \nprimarily by . ) (01 1ω ≤ ω ′′ ′′x xm mS\nx x′ ′α′11\n     A similar circumstance also applies to the im portant problem of critical currents for spin-torque \nmagnetization excitation in CPP-GMR spin valves with 0≠eJ . Consider the same example as above, \nagain treating  as stationary, and seeking nontri vial solutions of (21) (with of the form \n. Summarizing results obtainable from (5), (8), and (21) 3ˆm ) 0 ) (=′th\nste t−∝′) (1m\n \n0 detˆ\n) (2 /\n11 11 1111 11 111 2\n10eff\n1eff\n1NM\n=⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nα′ ′−′ ′ −′′+′ α′ ′−′× + =\n′ ′ ′ ′ ′ ′′ ′ ′ ′ ′ ′=\ny y y y x yy x x x x xsJ\ns H s Hs H s Ht Me\nem J H Hh\n                                           (25) \n \nwhere ,  as in (18),and where  in (25) is now the solution of the transport \nequations with  but . The cross-product form of the spin-torque contribution to  \nexplicitly yields an asymmetr ic/nonreciprocal contribution γ =′ /s sv u′ ′α′11 eJ∝NM\n2J\n0pump= J 0≠eJeff\n1H\nex y y xJ H H∝′−′′′′′\n11 11 to . The critical \ncurrent density is that value of  where  becomes negative. Given th e basic stability criterion \nthat , the spin-torque critical  condition from (25) can be expressed as  Ht\n′\neJ sRe\n0 det11>′Ht \nx y y x y y x x x x y yH H H H′′′′′′′ ′ ′ ′ ′ ′′−′=′α′+′α′11 11 11 11 11 11                                            (26) \n \nLike for thermal noise, the spin-torque critical  point should again be determined primarily by  for \nin-plane magnetized CPP-GMR spin-valves with typical x x′ ′α′11\nx x y yH H′ ′ ′′′> >′11 11. This simply reflects the fact \nthat the (quasi-uniform) modes of thermal fluctuati on or critical-point spin-torque oscillation tend to \nexhibit rather \"elliptical\", mostly in-plane motion when x x y yH H′ ′ ′′′> >′11 11. This is obviously different \nthan the steady, pure circular precession described in Ref. 3, which contrastingly highlights the \ninfluence of , along with its inte resting, additional y y′ ′α′11θ-dependence.  \n \nAPPENDIX A \n \n     The well known \"circuit theory\" formulation12 of the boundary conditions for the electron charge \ncurrent density  and the (dimensionally equiva lent) spin current density  at a FM/NM interface \ncan (taking ) be expressed as  eJspin\nNMJ\nm V ˆFM FM VΔ =Δ\n) ˆ )( ( ) )( (FM NM FM NM 21V G G V V G G Je Δ − ⋅ − + − + =↓ ↑ ↓ ↑m VΔ                        (A1) \n \n)ˆ ( Im ) ˆ ˆ( Reˆ)] ˆ )( ( ) )( [(\nNM NMFM NM FM NM NM 21 spin\nm V m V mm m V J\n× + × × +Δ − ⋅ + + − − =\n↑↓ ↑↓↓ ↑ ↓ ↑\nΔ ΔΔ\nG GV G G V V G G\n                      (A2) \nin terms of spin-indepe ndent electric potential V and accumulation VΔ (Δμe= ). Setting 0=eJ  in \n(A1) and substituting into (A2), one obtains in the limit the result 0 Im→↑↓G\n \n)ˆ ˆ( ˆ) ˆ (2\nNM FM NM NM0spinm V m m m V J × × + Δ − ⋅\n+=↑↓\n↓ ↑↓ ↑\n=Δ Δ G V\nG GG G\neJ                           (A3) \n \nComparing with Eq. (4) of Tserkovnyak et al.3 (with )sμΔ⇔V  and remembering the present \nconversion of , one immediately make s the identification spin 1 spin\nNM NM ) 2 / (I J−− ↔ e Ah\n \n↑↓ ↑↓= G e h A g) 2 / ( 22                                                      (A4) \n \nrelating dimensionless  in (1) to , the conventional mixing conductance (per area). ↑↓g↑↓G\n     The common approximations that  inside all FM layers, and that longitudinal  spin \ncurrent density is conserved at  FM/NM interfaces, yields the us ual interface boundary condition  m J ˆspin spin\nFM FM J=spin spin\nFM NMˆJ= ⋅m J                                                                     (A5) \nSolving for from (A2) then leads (with (A1)) to a second scalar boundary condition: m Jˆspin\nNM⋅\nspin\nFM FM NM4 4J\nG GG GJ\nG GG GV Ve ↓ ↑↓ ↑\n↓ ↑↓ ↑−−+= −                                               (A6) \nEquation (A6) is identical in form with the standard (collinear) Valet-Fert model,7 and immediately \nyields the following identifications \n↓ ↑↓ ↑\n↓ ↑↓ ↑\n+−= γ+=\nG GG G\nG GG Gr ,\n4                                                         (A7) \nfor the conventional Valet-Fert interface parameters .  γandr\n     The three vector terms on th e right of (A2) are mutually orthogona l. Working in a rotated (primed) \ncoordinate system where , (A1) and (A2) can be similarly inverted to solve for the three \ncomponents of the vector m z′=′ˆˆ\n)ˆ (FM NM m V ′Δ −′ VΔ  in terms of , , and . A final transformation \nback to the original (umprimed) coordinate s yields the vector interface-boundary condition spin\nNMJ′spin\nFMJeJ\n \n) / /( ) 2 / ( ) 2 /( 1ˆ Im Reˆ] ) Re [( ) ˆ (\n2spin spin spin\n21\nNM NM FM FM NM\nA g e h G rr r J r J r r Ve\n↑↓ ↑↓ ↑↓↑↓ ↑↓ ↑↓\n= ≡× + + γ − − = Δ − J m J m m VΔ\n          (A8) \n \nCombined with (A4), the last relation in (A8) yields (2). Equation (A8) is a generalization of Valet-\nFert to the non-collinear case.  \n     As noted by Tserkovnyak et al.,3 boundary conditions (A3) do not di rectly include spin-pumping \nterms, but instead involve only \"backflow\" terms  in the NM layer. With spin-pumping \nphysically present,  arises as the response to the spin accumulation back spin\nNM NM J J↔\nback\nNMJNMVΔ created by . It \nfollows that , where  is henceforth the total spin current in the NM layer. \nThus, including spin-pumping in Va let-Fert transport equations is then a matter of replacing \n in (A8). The modified form of (A 8), for a FM/NM interface, becomes: pumpJ\npump spin back\nNM NM J J J− =spin\nNMJ\npump spin spin\nNM NM J J J− →\n \n) (ˆ Im ) ( Reˆ] ) Re [( ) ˆ (\npump spin pump spinspin\n21\nNM NMFM FM NM\nJ J m J Jm m V\n− × + − +γ − − = Δ −\n↑↓ ↑↓↑↓\nr rJ r J r r Ve Δ\n                       (A9) \n \nFor an NM/FM interface, the sign is flippe d on the left sides of (A6) and (A9).  \n APPENDIX B \nFor 1-D transport (flow along the y-axis), the quasi-static Valet-Fert7 (drift-diffusion, quasi-static) \ntransport equations can be written as9 \n \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n∂∂+∂∂βρ==⎥\n⎦⎤\n⎢\n⎣⎡\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n∂∂⋅ β +∂∂\nρ=∂∂=\n∂∂\ny yVy yVJy l ye\nVm JVmV V\nΔΔ Δ Δ\n21ˆ1with along0 ˆ21 1,\nspin2 22\n                                      (B1) \n \nwhere = bulk resistivityρ13, l = spin diffusion length, and β = bulk/equilibrium spin current \npolarization in FM layers (  in NM layers). The solution for any one layer has the form 0≡ β\n \nm B m AB A V m V\nˆ ,ˆ : layers FM for,ˆ/ /\n21\nB Ae e C y J Vl y l y\ne\n= =+ = ⋅ β − + ρ =−Δ Δ                              (B2) \n \nIn the case where  film thickness, one may employ an al ternative \"ballistic\" approximation: > > >l\n \nC V , = = = ,spinB J A V Δ                                                        (B3) \n \nIt is not necessary to solve for the V and/or the C-coefficients using (A6) if only  and  are \nrequired. The remaining coefficients are determin ed by the interface boundary conditions (A5), (A6,7) \nand (A9), and external boundary conditions at the outer two surfaces of the film stack.  VΔspinJ\n     Regarding the latter, one approximation is to treat the external \"leads\" (with quasi-infinite cross \nsection) as equilibrium reservoirs and set 0 ) (, 0→==N i y y VΔ  at the outermost (i =0, N) lead-stack \ninterfaces of an N -layer stack (Fig. 1). The complimentary approximation is of an insulating boundary, \nwith . . For the case (such as in Sec. III) where the outer ( j=0, N-1) layers are \nNM, and the adjacent inner ( j=1, N-2) layers are FM, it is  readily found using (B1) and (B2) that  0 ) (, 0spin→==N i y y J\n \nNM NM) / hyp( ) ( 21 , 0 1 , 1 i j j N j N i l t l J V− = − =ρ ± = Δ                                       (B4) \n \nwhere hyp( ) = tanh( ) or coth( ) for equipotential,  or insulating boundaries, respectively. Combining \n(B4) with (A9), and neglecting ↑↓rIm , one finds for 0=eJ  that  \n ) / hyp( ) (ˆ)] / hyp( ) ( [\npump1 , 0 1 , 1 21\nFM NMFM FM\nj j j ii i\ni ii j j N j i N i\nl t l rrJJ l t l r V\nρ ++ =ρ + = Δ ±\n↑↓↑↓− = − =\nJm J                                 (B5) \n \nACKNOWLEDGMENT \n \nThe author would like to thank Y. Tserkovnya k for bringing Ref. 6 to his attention. \n \nREFERENCES \n \n1. D.C. Ralph and M.D. Stiles, J. Magn. Magn. Mat. 320, 1190 (2008) and many re ferences therein. \n2. Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B, 66, 224403 (2002). \n3. Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B, 67, 140404 (2003). \n4. S. Maat, N. Smith, M. J. Carey, and J.  R. Childress, Appl. Phys. Lett., 93, 103506 (2008). \n5. J. Foros, A. Brataas, Y. Tserkovnyak,  and G. E. W.Bauer, Phys. Rev. B 78, 140402 (2008). This \npaper describes a damping mechanism distinct from  Refs. 6, or this work, where nonlocal/tensor \nproperties arise from a strong magnetization gr adient in a single FM film or wire. \n6. J. Foros, A. Brataas, G. E. W.Bauer,  and Y. Tserkovnyak, arXiv:con-mat/0902.3779.  \n7. T. Valet and A. Fert, Phys. Rev. B, 48, 7099 (1993). \n8. N. Smith, J. Appl. Phys. 92, 3877 (2002); N. Smith, J. Magn. Magn. Mater. 321, 531 (2009) \n9. N. Smith, J. Appl. Phys., 99, 08Q703, (2006). \n10. R. Duine, A.S. Nunez, J. Si nova, A.H. MacDonald, Phys. Rev. B 75, 214420 (2007) \n11. N. Smith, J. Appl. Phys. 90, 5768 (2001). \n12. A. Brataas, Y. V. Nazarov, and G.  E. W. Bauer, Eur. Phys. J. B 22, 99, (2001). \n13. Some poor choice of words in the appendix of Ref. 9 c onfused the bulk resistivity, ρ, with the \nValet-Fert7 parameter . *ρ\n \n \n "    },    {        "title": "2103.04787v3.Emerging_magnetic_nutation.pdf",        "content": "Emerging magnetic nutation\nPascal Thibaudeau1,\u0003and Stam Nicolis2\n1CEA, DAM, Le Ripault, BP 16, F-37260, Monts, FRANCE\n2Institut Denis Poisson, Universit\u0013 e de Tours, Universit\u0013 e d'Orl\u0013 eans,\nCNRS (UMR7013), Parc de Grandmont, F-37200, Tours, FRANCEy\n(Dated: August 4, 2021)\nNutation has been recognized as of great signi\fcance for spintronics; but justifying its presence\nhas proven to be a hard problem. In this paper we show that nutation can be understood as\nemerging from a systematic expansion of a kernel that describes the history of the interaction of\na magnetic moment with a bath of colored noise. The parameter of the expansion is the ratio of\nthe colored noise timescale to the precession period. In the process we obtain the Gilbert damping\nfrom the same expansion. We recover the known results, when the coe\u000ecients of the two terms\nare proportional to one another, in the white noise limit; and show how colored noise leads to\nsituations where this simple relation breaks down, but what replaces it can be understood by the\nappropriate generalization of the \ructuation{dissipation theorem. Numerical simulations of the\nstochastic equations support the analytic approach. In particular we \fnd that the equilibration\ntime is about an order of magnitude longer than the timescale set by the colored noise for a wide\nrange of values of the latter and we can identify the presence of nutation in the non-uniform way\nthe magnetization approaches equilibrium.\nI. INTRODUCTION\nRecent progress in spintronics has led to the search\nfor processes and materials that can realize ever shorter\nswitching times for the magnetization{and this has\nopened a window to a r\u0013 egime, where nutation e\u000bects of\nthe average magnetization cannot be ignored. How to\ntake them into account becomes, therefore, of practical\ninterest [1]. However how to describe the emergence and\nthe relevance of nutation from \frst principles, as mag-\nnetic moments interact with a bath, has been and re-\nmains a challenging problem. One reason is that it is by\nno means obvious how to extract its properties from the\ninteraction with the bath.\nFor magnetic materials a common way of describing\nthe e\u000bects of the bath is by the so{called Gilbert damping\nmechanism [2]. What the two e\u000bects have in common is\nthe vector nature of the bath; where they di\u000ber is in how\nthis gets imprinted on the magnetization pro\fle in each\ncase.\nProviding a microscopic picture of how Gilbert damp-\ning may appear has long been recognized as an out-\nstanding question and there have been many attempts\nfor explaining how it may occur. However whether there\nmight be any relation with the e\u000bects of nutation has\nonly received attention. [3], where both were assumed to\nbe present and certain consequences for ultrafast switch-\ning were set forth. In that case, though, it was assumed\nthat the magnetic moment was not in interaction with a\nstochastic bath in full generality: the latter was present\nonly indirectly, through the deterministic Gilbert term.\nIn this paper we shall show that the nutation term and\nthe Gilbert term can both be obtained as well{de\fned\n\u0003pascal.thibaudeau@cea.fr\nystam.nicolis@lmpt.univ-tours.frcontributions from a systematic expansion of the equa-\ntions of motion of a magnetic moment, interacting with a\nvector bath, whose stochastic component is drawn from\ncolored noise. The expansion parameter can be identi\fed\nas the ratio of the timescale of the colored noise to the\nprecession frequency of the Larmor motion, here simply\nreduced to a Zeeman \feld only; this simpli\fes the calcu-\nlations, without any loss of generality.\nThe equation of motion of a classical or quantum mag-\nnetic moment, in the presence of this external \feld, is\nof \frst order in the dynamical variables and describes\nprecession [4]. This equation implies, in particular, some\nnon-trivial conservation laws: the norm of the magnetic\nmoment is conserved and, for a constant external \feld,\nso is the component along it [3, 5].\nWhen the magnetic moment interacts with a bath the\nconservation laws take the form of \ructuation{dissipation\nrelations, that describe the fact that the magnetic mo-\nment is in equilibrium with the bath. Indeed, the pro-\nposal in ref. [6] describes Gilbert damping and nutation\nas successive, relativistic, corrections to the dynamics of\na spin, in equilibrium with a quantum bath.\nIn the present paper we wish to explore the scenario,\nwhere a magnetic moment is in equilibrium with a vector\nbath, described by colored noise. The correlation time\nof the noise sets the short{time scale, so the relativistic\nexpansion of ref. [6] can be identi\fed as the expansion in\npowers of the ratio of the correlation time to the period\nof precession.\nWe \fnd that it is possible to recover both, Gilbert\ndamping and nutation of the magnetic moment, as terms\nin such an expansion.\nIt should be kept in mind that what is the \\most ap-\npropriate\" equation of motion (eom), that represents the\nmotion of a collection of interacting magnetic moments,\nis still the subject of intense debate, that goes back a\nlong time. Landau-Lifshitz [7] and Gilbert [2] introducedarXiv:2103.04787v3  [cond-mat.stat-mech]  3 Aug 20212\nan eom that described exclusively transverse damping,\nwhereas Bloch considered an eom that described exclu-\nsively longitudinal damping for the coarse{grained (spa-\ntial) average of the magnetization of the interacting mag-\nnetic moments. These equations have been extensively\nused to interpret measurements of spin relaxation and\nprovide a phenomenological viewpoint both for the origin\nof the e\u000bective \feld that de\fnes the precession axis for\nthe average magnetization, as well as for the origin of the\ndamping, whose e\u000bects can be reduced to a small number\nofdamping constants . A lot of attempts have been made\nto provide a microscopic foundation for the equation of\nmotion and, in particular, for accounting for the degrees\nof freedom that are behind the damping e\u000bects of the\nmagnetization [8{11]. There have been many arguments\nabout intrinsic and extrinsic e\u000bects, without, however,\nany insight into how these might be distinguished clearly\nin an invariant way.\nFor example, some authors [8, 12] considered a phe-\nnomenological theory describing one classical spin, em-\nbedded in a medium, that acts as a bath. This ap-\nproach leads to the well-known Landau-Lifshitz equation\n(resp. Bloch equation) in several limiting cases, i.e. in\nthe high temperature limit. The origin of the bath, that\ndescribes the \ructuations of the average magnetization,\nwas not spelled out, and only its role as an external ther-\nmostat (here called \ructuostat more generally) was as-\nsumed. This approach highlights that the damping is\nthen a consequence of memory e\u000bects, i.e. non-local in\ntime. Memory e\u000bects in their own right, were investi-\ngated theoretically in refs [13]. Depending on the form\nof the memory kernel involved, it was found that these\ncan lead to a compensation or even to an overcompensa-\ntion of the damping, since called \\Gilbert damping\".\nMore recently and based on both a quantized spin and\nenvironment Hamiltonian, Anders et.al derive a general\nspin operator equation of motion that describes three-\ndimensional precession and damping and consistently ac-\ncount for e\u000bects arising from memory, coloured noise\nand quantum statistics [14]. This reveals clearly reso-\nnant Lorentzian system{reservoir couplings that allow a\nsystematic comparison of dynamics between Ohmic and\nnon{Ohmic regimes. The quantized spin+reservoir prob-\nlem was also addressed before [15], \frst in a attempt to\njustify the form taken by the classical Landau-Lifshitz-\nBloch equation, without recognizing immediately the\nbene\ft of keeping, as long as possible, the memory ker-\nnel form induced by the motion equation of the quantum\nspin density operator.\nIn these approaches, it is implicitly assumed that the\nsystem does reach an equilibrium state, i.e. a state that is\ninvariant under global time translations. How the system\ncan, indeed, attain such a state has become a subject of\nconsiderable interest in the domain of glassy systems (for\nmagnetic systems these are known as \\spin glasses\"; an\nexample of the vast literature is refs. [16, 17]). One way\ncan be described as due to inertia , i.e. that the medium is\nnot in\fnitely rigid [18]. This implies that the magneticresponse depends not only on the magnetization itself,\nbut on its velocity, as well, and, therefore, the equation\nof motion is of second order in time [19, 20].\nThe corresponding equations of motion can be iden-\nti\fed with those of an Euler top, in a time{dependent\nexternal \feld, i.e. a torque. It is the impossibility of pro-\nviding a local description of the dynamics in terms of one\nset of \frst order equations that leads to non{local e\u000bects.\nThese can be captured by the so-called \\atomistic spin\ndynamics\"{as implemented, for instance, by Bhattachar-\njeeet al. [21]. A particular motivation was of capturing\nprocesses in the femtosecond regime by including the mo-\nment of inertia. They derived a generalized equation of\nmotion for the magnetization dynamics in the semiclassi-\ncal limit, which is non-local in both space and time. Con-\nsequently, they recovered a generalized Landau-Lifshitz-\nGilbert equation, which includes the moment of inertia\nand a second derivative of the magnetization in time.\nGoing further with this idea, Pervishko et al. [22] pro-\nposed an alternative derivation of the Gilbert damping in\na tensor form, within a mean-\feld approach. In this for-\nmalism, the itinerant electronic subsystem is considered\nin the presence of a nonequilibrium, classical magneti-\nzation \feld. When this \feld is su\u000eciently smooth and\nslow on the scales determined by the mean free path and\nscattering rate of the conduction electrons, the induced\nnonlocal spin polarization can be approximated using a\nlinear response Ansatz, thereby showing that the damp-\ning parameter emerges due to the coupling to the itiner-\nant subsystem. They derive a Kubo-St\u0014 reda formula for\nthe components of the Gilbert damping tensor and illus-\ntrate its relevance for the two-dimensional Rashba fer-\nromagnet, that can be realized at the interface between\nnonmagnetic and ferromagnetic layers. They argue that\nthis approach can be further applied to identify prop-\nerly the tensor structure of the Gilbert damping for more\ncomplicated model systems and real materials.\nMore recently, Mondal et al. [23] identi\fed the Gilbert\ndamping and nutation terms as \frst- and second-order\nrelativistic e\u000bects respectively, arising from the Foldy|\nWouthuysen transformation of a Dirac particle (that in-\ncludes spin\u00001\n2) motion under external \felds, embedded\nin a material medium.\nWhat is particularly striking in all these approaches is\nthat, while all end up with a description of Gilbert damp-\ning and of the torque that drives nutation, they seem\nto allow considerable ambiguity about the relative sign\nbetween the Gilbert damping and the nutation torque\ncontribution.\nWhile the microscopic origins of both Gilbert damping\nand magnetic inertia are still under debate, this uncer-\ntainty re\rects a fundamental issue, that deserves closer\nscrutiny.\nWe wish to report on our e\u000borts to resolve this am-\nbiguity. We shall show that the coupling of a magnetic\nmoment to a vector bath of colored noise is su\u000ecient for\ndescribing the emergence of both Gilbert damping and\nnutation, along with the relative sign; in addition, it pro-3\nvides a well{de\fned route to equilibrium. The parameter\nthat controls the relative signi\fcance of these e\u000bects is\nthe ratio of the colored noise timescale to the precession\nperiod. This is where the vector nature of the bath is of\nrelevance.\nThe plan of our paper is as follows: In section II we\ndescribe our model for a magnetic moment in a vector\nbath. In section III we provide representative solutions\nof the equations of motion obtained by numerical inte-\ngration and show how Gilbert damping and nutation can\nbe unambiguously identi\fed. In section IV we present\nour conclusions and ideas for further inquiry.\nII. MAGNETIC MOMENT IN A BATH\nConsider the spatial average of the magnetization M\nof a block of magnetic material. The \\reduced\" mag-\nnetization,m\u0011M=Ms, depends only on time and its\ndynamics can be described by its precession about an\ne\u000bective \feld, which can be written as the sum of two\nvectors!0(t) +\u000e!(t).!0(t) is de\fned, in turn, as the\nsum of the external magnetic \feld, applied on the mag-\nnetic system, and of the magnetic \feld, produced by the\naverage magnetization of the surrounding medium, i.e.\nthe reaction \feld.\n\u000e!(t) is a stochastic \feld, and is characterized phe-\nnomenologically by a single relaxation time \u001c. It de-\nscribes the \ructuations of the magnetic response of the\nmedium, in which the magnetic block is found.\nWe can describe the equilibrium of the magnetic\nblock with the medium, by the statement that h\u000e!i\u0011\n\r\u00160Ms\u001f\u00001hmi= \n s\u001f\u00001hmi, where\u001fis the susceptibil-\nity (not a function of time), and \ris the gyromagnetic ra-\ntio. Here the average is taken over the realizations of the\nsurrounding medium, considered as a bath. This state-\nment means that the expectation value of the \ructuating\n\feld at equilibrium is aligned with and proportional to\nthe expectation value of the magnetization [24]. When\n\u001f;which is identi\fed as the cumulant of the spin-spin\nfunction, depends explicitly on time, a convolution be-\ntween the \ructuating \feld and the magnetization has to\nbe used [25].\nThis procedure focuses on the \"relevant\" degrees of\nfreedom, labelled by mand sets them apart from the\n\"irrelevant\" variables, labelled by \u000e!. We are not inter-\nested, in the following, in the microscopic mechanisms\nthat may produce the e\u000bects of these variables [26, 27],\njust on their collective dynamics on the \\relevant\" de-\ngrees of freedom. (It has been proposed [28, 29] that the\nsymmetry that expresses the property that the physics\nshould not depend on how the \\dynamical\" from the de-\ngrees of freedom, that can de\fne the \\bath\", are chosen,\nis supersymmetry.)\nThese considerations can be expressed mathematicallyas follows:\ndm\ndt= (!0+\u000e!)\u0002m (1)\nd\u000e!\ndt=\u00001\n\u001c\u0000\n\u000e!\u0000\ns\u001f\u00001m\u0001\n+ \ns\u0011 (2)\nwhere\u0011is a random \feld, with ultra{local Gaussian cor-\nrelations, that describes the bath, which will be taken as\nthermal, in what follows, concretely:\nh\u0011I(t)i= 0 (3)\nh\u0011I(t)\u0011J(t0)i= 2D\u000eIJ\u000e(t\u0000t0) (4)\nwhereI;Jare the indices of the vector components. D\nis the amplitude of the noise and provides the de\fnition\nof the temperature T, through the Boltzmann{Einstein\nrelation,D/kBT=~, thereby expressing the \ructuation-\ndissipation theorem, for the bath. That the tempera-\nture is well{de\fned is ensured by the property that the\nnoise \feld\u0011(t) is drawn from a stationary stochastic pro-\ncess, i.e. enjoys global time translation invariance. Equa-\ntions (1) and (2) were \frst de\fned in [12] and evaluated\nin atomistic spin simulations [30]. It should be stressed\nthat this does not imply that the 2{point function of the\nmagnetic moment will have a simple dependence on the\ntemperature, due to the fact that its \ructuations, gener-\nically, will not be Gaussian [31].\nEq. (1) is purely transverse, therefore, the norm of mis\nconserved, ifm\u0001_m= 0,(d=dt)(jjmjj2) = 0. The latter\nrelation is, of course, true, in the absence of the bath; it\ndoes require, however, another de\fnition in its presence,\nsince the derivative is a singular quantity [32, 33]. Such a\nde\fnition can be obtained from the so-called Schwinger{\nDyson identities [34], namely as\n\u001c\nm\u0001dm\ndt\u001d\n= 0: (5)\nThe \feld\u000e!is de\fned by the stochastic di\u000berential\nequation (SDE) in eq. (2). Its solution can be shown to be\nan Ornstein-Uhlenbeck process [35]. Therefore, m(t) be-\ncomes a stochastic process, as well; moreover, the noise,\nthat enters additively in the equation for \u000e!, becomes\nmultiplicative for m(t); which implies that its correla-\ntion functions acquire a non{trivial dependence on the\ntemperature, de\fned through the bath. This is, often de-\nscribed as a \\breakdown\" of the \ructuation{dissipation\ntheorems [36, 37]. However, what this, simply, means is\nthat the non-linearities induce a non{trivial, but quite\ntransparent, dependence of the noise on the dynamics of\nthe magnetization; the two are, just, intertwined in a way\nthat is more subtle than hitherto acknowledged. Indeed,\nthis can be understood in terms of the variables that can\nresolve the dynamics of the bath, as an expression of\nreparametrization invariance in the space of \felds.\nFirst, suppose for simplicity that the system is not in\ncontact with the bath; \u0011(t) is absent from eq. (2). Then,\neqs.(1,2) de\fne the dynamics of a deterministic system4\nand can be explicitly solved: First of all, equation (2)\ncan be solved for \u000e!in terms ofm(t):\n\u000e!=\ns\u001f\u00001\n\u001cZt\n\u00001e\u0000t\u0000t0\n\u001cm(t0)dt0\n=\ns\u001f\u00001\n\u001cZ1\n0e\u0000u\n\u001cm(t\u0000u)du(6)\nThe equation (6) can then be introduced in eq. (1) to\nproduce an integral-di\u000berential equation for m:\ndm\ndt=\u0012\n!0+\ns\u001f\u00001\n\u001cZ1\n0e\u0000u\n\u001cm(t\u0000u)du\u0013\n\u0002m(7)\nThe integral highlights the dependence of the solution on\nthe full history of the magnetization, prior to time t, as\nwell as the putative e\u000bects of the damping induced by\nthe memory kernel with a characteristic time \u001c, de\fned\nby eq. (2). Indeed one of the purposes of this paper is to\nprovide an intrinsic de\fnition of such damping e\u000bects in\nan invariant way.\nIn order to \fnd approximate solutions, it is useful to\nexpandmin a Taylor series about some reference time t\nand exchange the sum and the integral. Assuming that\nFubini's theorem holds [38], we thus \fnd\n\u000e!=\ns\u001f\u00001\n\u001c1X\nn=0(\u00001)n\nn!dnm\ndtnZ1\n0e\u0000u\n\u001cundu\n= \n s\u001f\u000011X\nn=0(\u00001)n\nn!dnm\ndtn\u001cn\u0000(1 +n)\n= \n s\u001f\u000011X\nn=0(\u0000\u001c)ndnm\ndtn:(8)\nOf course, it is by no means obvious either that this series\nconverges, or that it is even legitimate to exchange sum\nand integral; we shall try to provide a posteriori checks\nthat are sensitive to these issues.\nWe shall now try to interpret the properties of the\nmagnetization, that are sensitive to our truncating the\nseries at a given order. When the sum stops at n= 1,\neq.(1) takes the form\ndm\ndt\u0019\u0012\n!0+ \ns\u001f\u00001m\u0000\ns\u001f\u00001\u001cdm\ndt\u0013\n\u0002m\n=!0\u0002m+\u000bm\u0002dm\ndt(9)\nwhere\u000b\u0011\ns\u001c\u001f\u00001can be, therefore, identi\fed as the\nGilbert damping constant, and eq. (9) is the eom writ-\nten in the standard Gilbert form [2]. This expression\nfor\u000bappears consistent with other forms reported in\nthe literature [25, 39]. It is therefore not surprising that\neventually the tensor character of both the inverse of the\nsusceptibility \u001fand the relaxation time \u001cproduces a ten-\nsor damping parameter \u000b, a feature already reported in\nferromagnetic metals assuming a torque-torque correla-\ntion model [40{42] in the highly anisotropic scattering\nregime of magnons.Upon including the n= 2 term, the equation for the\nmagnetization takes the form\ndm\ndt\u0019\u0012\n!0+ \ns\u001f\u00001m\u0000\ns\u001f\u00001\u001cdm\ndt\u0013\n\u0002m\n+ \ns\u001f\u00001\u001c2d2m\ndt2\u0002m\n=!0\u0002m+\u000bm\u0002\u0012dm\ndt\u0000\u001cd2m\ndt2\u0013\n:(10)\nThe term proportional to m\u0002(d2m=dt2) can be in-\nterpreted as describing the \"nutation\" of the magnetiza-\ntion [43]. It should be noted, at this point that this is\nthe \frst term that is, manifestly, symmetric under time{\nreversal. An issue of considerable interest is that of the\nrelative sign of the coe\u000ecients of the terms in the equa-\ntion of motion. Let us note that the sign of the inertial\ndamping (last term) seems to be opposite to the sign of\nthe usual damping term (second term), which is in agree-\nment with the theory of dampened magnetostriction, \frst\nintroduced by Suhl [44]. This is in contrast with refer-\nence [20], where the signs of the two damping terms are\nthe same. However the microscopic description in the\ntwo cases is completely di\u000berent.\nWhat we have thus shown is that both, the Gilbert\ndamping and the nutation term can be deduced as the\nconsequence of the coupling of a magnetic moment to an\nexternal \feld, upon taking into account the coupling to\nthe bath self{consistently.\nThis constitutes the central result of the paper.\nIn this deterministic situation and because \u001c >0, the\ndivergence of the volume of the phase space is negative{it\nshrinks, due to dissipation. At this level of truncation the\nexistence of an equilibrium state for the magnetization\nthat is unique and is described by a point is obvious.\nWhat is by no means obvious is what happens when the\nnon{local e\u000bects, described by the higher order terms,\nare taken into account.\nSetting this issue aside, for the moment, let us now\ntake into account the bath, at this approximation.\nWhen the noise \feld \u0011is present, \u000e!becomes a\nstochastic \feld, which contains an extra term. This term\ntakes into account the noise \feld in the memory kernel\nas follows:\n\u000e!\n\ns=\u001f\u000011X\nn=0(\u0000\u001c)ndnm\ndtn+Z1\n0e\u0000u\n\u001c\u0011(t\u0000u)du(11)\nLet us call \n(t)\u0011R1\n0e\u0000u\n\u001c\u0011(t\u0000u)du, the extra stochastic\n\feld. Equations (3),(4) imply that the random \feld \nhas the following properties:\nh\nI(t)i= 0 (12)\nh\nI(t)\nJ(t0)i=D\u001c\u000e IJe\u0000jt\u0000t0j\n\u001c (13)\nwhich means that it describes colored noise! This im-\nplies, in turn, for the magnetization that its correlation5\nfunctions are, generically, those of a centered and colored\nnoise stochastic process, and not of a white noise process,\nas it is usually assumed. We shall show now show that the\napproximations involved in the truncation to second or-\nder, i.e. including the nutation term, are self{consistent\nby solving the equations (1) numerically.\nIII. NUMERICAL RESULTS\nIn order to check on how the signature of the Gilbert\ndamping and that of nutation, produced by the \ructuat-\ning \feld\u000e!is imprinted in the magnetization pro\fle, we\nsolve the coupled equations (1) numerically.\nPrecession can be readily identi\fed as the rotation of\nthe magnetization around a given axis.\nNutation is the additional e\u000bect produced on the mag-\nnetization by the motion of this axis with time.\nUpon averaging over the realizations of the noise, if\nhmiis a constant vector at equilibrium, then h\u000e!ialso\nbecomes a constant vector, proportional to hmi. That\nmeans that the magnetization spins \frst around !0at\nshort times and then settles to spinning around !0+h\u000e!i\nat long times. But the torque produced at that time is\n!0\u0002hmi, because of the proportionality between h\u000e!i\nandhmi. As a consequence, only during a transient time,\nwhen\u000e!strongly varies, can the motion of the average\nmagnetization be strongly a\u000bected.\nAll these features can be read o\u000b \fgure 1, that displays\nboth the motion of the average mand\u000e!taken over\nmore than 1000 realizations of the noise, and for di\u000berent\nvalues of the correlation time of the noise, \u001c.\n-1-0.500.51<m>τ=5\n01234<δω>\n-1-0.500.51<m>\nτ=0.5\n01234<δω>\n0.1 1 10\ntime t-0.500.51<m>\nτ=0.1\n0.1 1 10\ntime t-3-2-101234<δω>\nFigure 1. (color online) Dynamics of the average magneti-\nzation (left panels) and \ructuation \feld (right panels) for\na varying\u001cparameter. Conditions are !0=<0;0;2\u0019 >,\nD= 50, \n s\u001f\u00001=\u0019,m(0) =<1;0;0>,\u000e!(0) =<1;0;0>.\nComponents of x,y,zare in black, red and green respectively.\nThe norm is displayed in blue.The equations (1) are integrated globally with an ex-\nplicit 4thorder Runge-Kutta algorithm and a variable\nstepping scheme, with only a renormalization of the mag-\nnetization at each step, in order to produce a precession\nand nutation motion consistent on the S2sphere. Better\nsymplectic algorithms [45], that preserve the structure of\nthe equations of motion, can be used, but they do not\na\u000bect the conclusions drawn.\nWhat we observe here is that the average magneti-\nzationhmitends to align with the e\u000bective \feld along\nthez-axis, by producing a dampened motion and a wrig-\ngling movement of the hmzicomponent, which is charac-\nteristic of a high frequency nutation e\u000bect, because of\nthe \fnite values of \u001cand \n s\u001f\u00001. When the suscep-\ntibility\u001fis decreased, while keeping all the other pa-\nrameters \fxed, the internal precession \feld, coming from\nthe \ructuations, dominates the natural precession \feld\n!0, that increases the precession pulsation. We observe\nthat increasing \u001cand reducing Msdoes indeed enhance\nthe e\u000bects of the nutation term. Moreover when \u001cis\nlarge, the di\u000busive term, that is generated by the noise,\ndominates the motion of the magnetization. A conse-\nquence is thathm(t)icannot stay constant even if, for\nall values of \u001c,hm:mi= 1 by construction. When the\ntime is long enough to capture the growing main com-\nponent of the magnetization then hmialigns itself on\nh\u000e!i. When\u001cis small, in the transient regime, the \ruc-\ntuating \feldh\u000e!icannot be su\u000eciently dampened and\nfollows more closely the dynamics of the magnetization.\nFor low values of the noise amplitude, the dynamics of\nthe averageh\u000e!iis insensitive to the noise amplitude\nand its leading motion is described by \u001ch\u000e!i\u0019\u000bhmi.\nWhen\u001ctakes values of O(1=!0), the leading motion\nofhmiis given by the Gilbert equation of precession\ndhmi=dt\u0019(!0\u0000\u000bhdmi=dt)\u0002hmi, that produces in\nreturn a dampened motion of h\u000e!i.\nOne conclusion of this study is identifying the appro-\npriate dimensionless combinations. Our results moti-\nvate de\fning the dimensionless quantities X0\u0011\u001c!0and\nX\u0011\u001c\u000e!. In terms of these te equations of motion 1\nand 2 take the form (upon de\fning x=t=\u001c)\ndm\ndx= (X0+X)\u0002m\ndX\ndx=\u0000(X\u0000\u000bm) + \u0001\u0011(14)\nwhere \u0001\u0011\ns\u001candh\u0011i(x)\u0011j(x0)i= 2D=\u001c\u000e ij\u000e(x\u0000x0). In\nthe particular case where X0=0, i.e. when no external\ntorque acts on the magnetization and with \u000b6= 0, then\nhmi\u0011m(0) is a constant of motion.\nUpon averaging over the noise realizations, the dynam-\nics ofhXiis given byhXi= (X0\u0000\u000bm(0))e\u0000x+\u000bm(0).\nThus the \ructuating \feld at equilibrium is given by\nhXi1=\u000bm(0) and no torque acts on the magnetiza-\ntion, keeping it constant over time.\nThe \fgure 2 displays the dampened motion of mand\nXas a function of the dimensionless time, for two con-\n\fgurations : \n sD= 0, i.e. without thermal noise, and6\n\nsD= 50, for the same external \feld X0=<0;0;\u0019 > .\nThe longitudinal behavior of the average magnetization\nis clearly visible by the decrease of the average norm\nkhmik. Moreover because the average eom for Xis in-\ndependent of the noise amplitude, as depicted, this is\nnot the case for the average magnetization, because there\nhX\u0002mi6=hXi\u0002hmi.\n0 10 20 30-1-0.500.51<m>ΩsD=0\nΩsD=50\n0 10 20 30\nt/τ-0.4-0.200.20.40.60.81<τδω>\nFigure 2. (color online) Dynamics of the average magneti-\nzation (up panel) and \ructuation \feld (down) for a varying\n\nsDparameter. Conditions are \u001c!0=<0;0;\u0019 > ,\u000b= 1,\nm(0) =<1;0;0>,\u001c\u000e!(0) =<0;0;0>. Components of x,\ny,zare in black, red and green respectively. The norm is\ndisplayed in blue.\nIV. CONCLUSIONS AND OUTLOOK\nIn this paper, we have shown that the mechanism of\nGilbert damping of the precession, as well as the e\u000bects of\nnutation can be understood in terms of an e\u000bective inter-\naction between magnetic moments and the \ructuationsof their e\u000bective \felds, when the latter are described by\ncolored noise in a systematic expansion in powers of the\nratio of the correlation time of the noise to the period\nde\fned by the precession torque.\nWe have identi\fed a relation between the Gilbert\ndamping parameter and the static (or spectral) inverse\nsusceptibility of the material, with the contribution to\na characteristic relaxation time, that can be assigned to\nmagnon scattering mechanisms, in the relaxation time\napproximation.\nIt is stressed that if it were possible to perform mea-\nsurements that could resolve the contribution of the nu-\ntation loops, as they are superimposed on the usual pre-\ncession motion of the magnetic moments, it would be\npossible to \fnd which processes provide the dominant\ncontribution leading to inertial damping, as recently been\nreported [1].\nThe relative sign between the Gilbert damping and the\ninertia term is negative as a consequence of the fact that\nthese two terms represent successive contributions of the\nTaylor expansion. Therefore studies that assume that\nthese terms have the same sign make additional assump-\ntions, that it would be very interesting to spell out.\nThe results obtained here relied on the equations of\nmotion alone. To better understand the space of states\nof the magnetization, it will be useful to adapt the tech-\nniques used in ref. [37] and to to understand the micro-\nscopic degrees of freedom that can de\fne the bath in an\ninvariant way it is necessary to implement the program\nthat is sketched in ref. [46].\nSince the magnetization vector naturally evolves ac-\ncording to Nambu mechanics, it will, also, be interest-\ning to understand how Nambu mechanics may accom-\nmodate Gilbert damping and nutation. Gilbert damping\nhas been studied, in this context, already, using di\u000berent\ntools, in refs. [5].\nOf course probing how the truncation to n= 2 breaks\ndown and how it may be completed remains to be under-\nstood.\nWe hope to report on progress on these issues in future\nwork.\n[1] K. Neeraj, N. Awari, S. Kovalev, D. Polley,\nN. Zhou Hagstr om, S. S. P. K. Arekapudi, A. Semisalova,\nK. Lenz, B. Green, J.-C. Deinert, I. Ilyakov, M. Chen,\nM. Bawatna, V. Scalera, M. d'Aquino, C. Serpico,\nO. Hellwig, J.-E. Wegrowe, M. Gensch, and S. Bonetti,\nNature Physics , 1 (2020).\n[2] T. L. Gilbert and J. M. Kelly, in Conference on Mag-\nnetism and Magnetic Materials (Pittsburgh, PA, 1955)\npp. 253{263.\n[3] I. Makhfudz, E. Olive, and S. Nicolis, Appl. Phys. Lett.\n117, 132403 (2020).\n[4] C. Cohen-Tannoudji, B. Diu, F. Lalo e, and S. R. Hemley,\nQuantum Mechanics , A Wiley-Interscience Publication\n(Wiley [u.a.], New York, NY, 1993).[5] P. Thibaudeau, T. Nussle, and S. Nicolis, J. Magn. Magn.\nMater. 432, 175 (2017).\n[6] R. Mondal, M. Berritta, A. K. Nandy, and P. M. Oppe-\nneer, Phys. 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Baglai1,2, Olle Eriksson2,3, and Dmitry Yudin1\n1ITMO University, Saint Petersburg 197101, Russia\n2Department of Physics and Astronomy, Uppsala University, Box 516, SE-75 121 Uppsala, Sweden\n3School of Science and Technology, ¨Orebro University, SE-701 82 ¨Orebro, Sweden\nABSTRACT\nA keen interest towards technological implications of spin-orbit driven magnetization dynamics requests a proper theoretical\ndescription, especially in the context of a microscopic framework, to be developed. Indeed, magnetization dynamics is so far\napproached within Landau-Lifshitz-Gilbert equation which characterizes torques on magnetization on purely phenomenological\ngrounds. Particularly, spin-orbit coupling does not respect spin conservation, leading thus to angular momentum transfer to\nlattice and damping as a result. This mechanism is accounted by the Gilbert damping torque which describes relaxation of the\nmagnetization to equilibrium. In this study we work out a microscopic Kubo-St ˇreda formula for the components of the Gilbert\ndamping tensor and apply the elaborated formalism to a two-dimensional Rashba ferromagnet in the weak disorder limit. We\nshow that an exact analytical expression corresponding to the Gilbert damping parameter manifests linear dependence on the\nscattering rate and retains the constant value up to room temperature when no vibrational degrees of freedom are present\nin the system. We argue that the methodology developed in this paper can be safely applied to bilayers made of non- and\nferromagnetic metals, e.g., CoPt.\nIntroduction\nIn spite of being a mature field of research, studying magnetism and spin-dependent phenomena in solids still remains one of\nthe most exciting area in modern condensed matter physics. In fact, enormous progress in technological development over the\nlast few decades is mainly held by the achievements in spintronics and related fields1–11. However the theoretical description of\nmagnetization dynamics is at best accomplished on the level of Landau-Lifshitz-Gilbert (LLG) equation that characterizes\ntorques on the magnetization. In essence, this equation describes the precession of the magnetization, mmm(rrr;t), about the effective\nmagnetic field, HHHeff(rrr;t), created by the localized moments in magnetic materials, and its relaxation to equilibrium. The latter,\nknown as the Gilbert damping torque12, was originally captured in the form ammm\u0002¶tmmm, where the parameter adetermines the\nrelaxation strength, and it was recently shown to originate from a systematic non-relativistic expansion of the Dirac equation13.\nThus, a proper microscopic determination of the damping parameter a(or, the damping tensor in a broad sense) is pivotal to\ncorrectly simulate dynamics of magnetic structures for the use in magnetic storage devices14.\nFrom an experimental viewpoint, the Gilbert damping parameter can be extracted from ferromagnetic resonance linewidth\nmeasurements15–17or established via time-resolved magneto-optical Kerr effect18, 19. In addition, it was clearly demonstrated\nthat in bilayer systems made of a nonmagnetic metal (NM) and a ferromagnet material (FM) the Gilbert damping is drastically\nenhanced as compared to bulk FMs20–24. A strong magnetocrystalline anisotropy, present in CoNi, CoPd, or CoPt, hints\nunambiguously for spin-orbit origin of the intrinsic damping. A first theoretical attempt to explain the Gilbert damping\nenhancement was made in terms of sdexchange model in Ref.25. Within this simple model, magnetic moments associated with\nFM layer transfer angular momentum via interface and finally dissipate. Linear response theory has been further developed\nwithin free electrons model26, 27, while the approach based on scattering matrix analysis has been presented in Refs.28, 29. In\nthe latter scenario spin pumping from FM to NM results in either backscattering of magnetic moments to the FM layer or\ntheir further relaxation in the NM. Furthermore, the alternative method to the evaluation of the damping torque, especially\nin regard of first-principles calculations, employs torque-correlation technique within the breathing Fermi surface model30.\nWhile a direct estimation of spin-relaxation torque from microscopic theory31, or from spin-wave spectrum, obtained on the\nbasis of transverse magnetic field susceptibility32, 33, are also possible. It is worth mentioning that the results of first-principles\ncalculations within torque-correlation model34–38and linear response formalism39, 40reveal good agreement with experimental\ndata for itinerant FMs such as Fe, Co, or Ni and binary alloys.\nLast but not least, an intensified interest towards microscopic foundations of the Gilbert parameter ais mainly attributed\nto the role the damping torque is known to play in magnetization reversal41. In particular, according to the breathing Fermi\nsurface model the damping stems from variations of single-particle energies and consequently a change of the Fermi surfacearXiv:1807.07897v2  [cond-mat.mes-hall]  21 Nov 2018z\nyx\nFM\nNMFigure 1. Schematic representation of the model system: the electrons at the interface of a bilayer, composed of a\nferromagnetic (FM) and a nonmagnetic metal (NM) material, are well described by the Hamiltonian (1). We assume the\nmagnetization of FM layer depicted by the red arrow is aligned along the zaxis.\nshape depending on spin orientation. Without granting any deep insight into the microscopic picture, this model suggests that\nthe damping rate depends linearly on the electron-hole pairs lifetime which are created near the Fermi surface by magnetization\nprecession. In this paper we propose an alternative derivation of the Gilbert damping tensor within a mean-field approach\naccording to which we consider itinerant subsystem in the presence of nonequilibrium classical field mmm(rrr;t). Subject to the\nfunction mmm(rrr;t)is sufficiently smooth and slow on the scales determined by conduction electrons mean free path and scattering\nrate, the induced nonlocal spin polarization can be approached within a linear response, thus providing the damping parameter\ndue to the itinerant subsystem. In the following, we provide the derivation of a Kubo-St ˇreda formula for the components of\nthe Gilbert damping tensor and illustrate our approach for a two-dimensional Rashba ferromagnet, that can be modeled by\nthe interface between NM and FM layers. We argue that our theory can be further applied to identify properly the tensorial\nstructure of the Gilbert damping for more complicated model systems and real materials.\nMicroscopic framework\nConsider a heterostructure made of NM with strong spin-orbit interaction covered by FM layer as shown in Fig. 1, e.g., CoPt.\nIn general FMs belong to the class of strongly correlated systems with partially filled dorforbitals which are responsible\nfor the formation of localized magnetic moments. The latter can be described in terms of a vector field mmm(rrr;t)referred to as\nmagnetization, that in comparison to electronic time and length scales slowly varies and interacts with an itinerant subsystem.\nAt the interface (see Fig. 1) the conduction electrons of NM interact with the localized magnetic moments of FM via a certain\ntype of exchange coupling, sdexchange interaction, so that the Hamiltonian can be written as\nh=p2\n2m+a(sss\u0002ppp)z+sss\u0001MMM(rrr;t)+U(rrr); (1)\nwhere first two terms correspond to the Hamiltonian of conduction electrons, on condition that the two-dimensional momentum\nppp= (px;py) =p(cosj;sinj)specifies electronic states, mis the free electron mass, astands for spin-orbit coupling strength,\nwhile sss= (sx;sy;sz)is the vector of Pauli matrices. The third term in (1) is responsible for sdexchange interaction with the\nexchange field MMM(rrr;t) =Dmmm(rrr;t)aligned in the direction of magnetization and Ddenoting sdexchange coupling strength. We\nhave also included the Gaussian disorder, the last term in Eq. (1), which represents a series of point-like defects, or scatterers,\nhU(rrr)U(rrr0)i= (mt)\u00001d(rrr\u0000rrr0)with the scattering rate t(we set ¯h=1throughout the calculations and recover it for the final\nresults).\nSubject to the norm of the vector jmmm(rrr;t)j=1remains fixed, the magnetization, in broad terms, evolves according to (see,\ne.g., Ref.42),\n¶tmmm=fff\u0002mmm=gHHHeff\u0002mmm+csss\u0002mmm; (2)\nwhere fffcorresponds to so-called spin torques. The first term in fffdescribes precession around the effective magnetic field\nHHHeffcreated by the localized moments of FM, whereas the second term in (2) is determined by nonequilibrium spin density of\nconduction electrons of NM at the interface, sss(rrr;t). It is worth mentioning that in Eq. (2) the parameter gis the gyromagnetic\nratio, while c= (gmB=¯h)2m0=dis related to the electron g\u0000factor ( g=2), the thickness of a nonmagnetic layer d, with mBand\nm0standing for Bohr magneton and vacuum permeability respectively. Knowing the lesser Green’s function, G<(rrrt;rrrt), one\ncan easily evaluate nonequilibrium spin density of conduction electrons induced by slow variation of magnetization orientation,\nsm(rrr;t) =\u0000i\n2Tr\u0002\nsmG<(rrrt;rrrt)\u0003\n=Qmn¶tmn+:::; (3)\n2/8where summation over repeated indexes is assumed ( m;n=x;y;z). The lesser Green’s function of conduction electrons\ncan be represented as G<=\u0000\nGK\u0000GR+GA\u0001\n=2, where GK,GR,GAare Keldysh, retarded, and advanced Green’s functions\nrespectively.\nKubo-St ˇreda formula\nWe further proceed with evaluating Qmnin Eq. (3) that describes the contribution to the Gilbert damping due to conduction\nelectrons. In the Hamiltonian (1) we assume slow dynamics of the magnetization, such that approximation MMM(rrr;t)\u0019\nMMM+(t\u0000t0)¶tMMMwith MMM=MMM(rrr;t0)is supposed to be hold with high accuracy,\nH=p2\n2m+a(sss\u0002ppp)z+sss\u0001MMM+U(rrr)+(t\u0000t0)sss\u0001¶tMMM; (4)\nwhere first four terms in the right hand side of Eq. (4) can be grouped into the Hamiltonian of a bare system, H0, which\ncoincides with that of Eq. (1), provided by the static magnetization configuration MMM. In addition, the expression (4) includes the\ntime-dependent term V(t)explicitly, as the last term. In the following analysis we deal with this in a perturbative manner. In\nparticular, the first order correction to the Green’s function of a bare system induced by V(t)is,\ndG(t1;t2) =Z\nCKdtZd2p\n(2p)2gppp(t1;t)V(t)gppp(t;t2); (5)\nwhere the integral in time domain is taken along a Keldysh contour, while gppp(t1;t2) =gppp(t1\u0000t2)[the latter accounts for the\nfact that in equilibrium correlation functions are determined by the relative time t1\u0000t2] stands for the Green’s function of the\nbare system with the Hamiltonian H0in momentum representation. In particular, for the lesser Green’s function at coinciding\ntime arguments t1=t2\u0011t0, which is needed to evaluate (3), one can write down,\ndG<(t0;t0) =i\n2¥Z\n\u0000¥de\n2pZd2p\n(2p)2n\ngR\npppsm¶g<\nppp\n¶e\u0000¶gR\nppp\n¶esmg<\nppp+g<\npppsm¶gA\nppp\n¶e\u0000¶g<\nppp\n¶esmgA\npppo\n¶tMm; (6)\nwhere m=x;y;z, while gR,gA, and g<are the bare retarded, advanced, and lesser Green’s functions respectively. To derive the\nexpression (6) we made use of Fourier transformation gppp=Rd(t1\u0000t2)gppp(t1\u0000t2)eie(t1\u0000t2)and integration by parts.\nTo finally close up the derivation we employ the fluctuation-dissipation theorem according to which g<(e) = [gA(e)\u0000\ngR(e)]f(e), where f(e) = [eb(e\u0000m)+1]\u00001stands for the Fermi-Dirac distribution with the Fermi energy m. Thus, nonequi-\nlibrium spin density of conduction electrons (3) within linear response theory is determined by Qmn=Q(1)\nmn+Q(2)\nmn, where\nQ(1)\nmn=1\n4Trh\nsm¥Z\n\u0000¥de\n2pZd2p\n(2p)2n¶gR\nppp\n¶esngR\nppp\u0000gR\npppsn¶gR\nppp\n¶e+gA\npppsn¶gA\nppp\n¶e\u0000¶gA\nppp\n¶esngA\npppo\nf(e)i\n; (7)\nwhich involves the integration over the whole Fermi sea, and\nQ(2)\nmn=1\n4Trh\nsm¥Z\n\u0000¥de\n2p\u0012\n\u0000¶f(e)\n¶e\u0013Zd2p\n(2p)2n\ngR\npppsngR\nppp+gA\npppsngA\nppp\u00002gR\npppsngA\npppoi\n; (8)\nwhich selects the integration in the vicinity of the Fermi level. Generally, the form of Qmnbelongs to the class of Kubo-St ˇreda\nformula, and, in essence, represents the response to the external stimulus in the form of ¶tMn. We can immediately establish a\nquantitative agreement between the result given by Eq. (8) and the previous studies within a Kubo formalism40, 43–46which allow\na direct estimation within the framework of disordered alloys. Formally, the expression (7) corresponds to the so-called St ˇreda\ncontribution. Such a term was originally identified in Ref.47when studying quantum-mechanical conductivity. Notably, in\nEq. (7) each term represents the product of either retarded or advanced Green’s functions. In this case the poles of the integrand\nfunction are positioned on the same side of imaginary plane, making disorder correction smaller in the weak disorder limit (see,\ne.g., Ref.48). Meanwhile, having no classical analog this contribution appears to be important enough when the spectrum of the\nsystem is gapped and the Fermi energy is placed exactly in the gap47. It is worth mentioning that the contribution due to Eq. (7)\nhas never been discussed in this context before. In the meantime, Kubo-St ˇreda expression for the components of the Gilbert\ndamping tensor has been addressed from the perspective of first-principles calculations49and current-induced torques50.\n3/8Results and discussion\nLet us apply the formalism developed in the previous section to a prototypical model: we work out the Gilbert damping tensor\nfor a Rashba ferromagnet with the magnetization mmm=zzzaligned along the zaxis. In the limit of weak disorder the Green’s\nfunction of a bare system can be expressed as\ngR\nppp(e) =\u0000\ne\u0000H0\u0000SR\u0001\u00001=e\u0000eppp+id+a(sss\u0002ppp)z+(D+ih)sz\n(e\u0000eppp+id)2\u0000a2p2\u0000(D+ih)2; (9)\nwhere eppp=p2=(2m)is the electron kinetic energy. We put the self-energy SRdue to scattering off scalar impurities into Eq. (9),\nwhich is determined from SR=\u0000i(d\u0000hsz)(see, e.g., Ref.51). In particular, for jej>jDjwe can establish that d=1=(2t)\nandh=0 in the weak disorder regime to the leading order.\nWithout loss of generality, in the following we restrict the discussion to the regime m>jDj, which is typically satisfied with\nhigh accuracy in experiments. As previously discussed, the contribution owing to the Fermi sea, Eq. (7), can in some cases be\nignored, while doing the momentum integral in Eq. (8) results in,\n1\nmtZd2p\n(2p)2gR\nppp(e)sssgA\nppp(e) =D2\nD2+2ersss+Dd\nD2+2er(sss\u0002zzz)+D2\u0000er\nD2+2er(sss\u0002zzz)\u0002zzz; (10)\nwhere r=ma2. Thus, thanks to the factor of delta function d(e\u0000m) =\u0000¶f(e)=¶e, to estimate Q(2)\nmnat zero temperature one\nshould put e=min Eq. (10). As a result, we obtain,\nQ(2)\nmn=\u00001\n4pm\nD2+2mr0\n@2tmr D 0\n\u0000D 2tmr 0\n0 0 2 tD21\nA: (11)\nMeanwhile, to properly account the correlation functions which appear when averaging over disorder configuration one has\nto evaluate the so-called vertex corrections, which from a physical viewpoint makes a distinction between disorder averaged\nproduct of two Green’s function, hgRsngAidis, and the product of two disorder averaged Green’s functions, hgRidissnhgAidis, in\nEq. (8). Thus, we further proceed with identifying the vertex part by collecting the terms linear in dexclusively,\nGGGs=Asss+B(sss\u0002zzz)+C(sss\u0002zzz)\u0002zzz; (12)\nprovided A=1+D2=(2er),B= (D2+2er)Dd=(D2+er)2, and C=D2=(2er)\u0000er=(D2+er). To complete our derivation\nwe should replace snin Eq. (8) by Gs\nnand with the aid of Eq. (10) we finally derive at e=m,\nQ(2)\nmn=0\n@Qxx Qxy 0\n\u0000QxyQxx\n0 0\u0000mtD2=(4pmr)1\nA: (13)\nWe defined Qxx=\u0000mtmr=[2p(D2+mr)]andQxy=\u0000mD(D2+2mr)=[4p(D2+mr)2], which unambiguously reveals that\naccount of vertex correction substantially modifies the results of the calculations. With the help of Eqs. (3), (11), and (13) we\ncan write down LLG equation. Slight deviation from collinear configurations are determined by xandycomponents ( mxand\nmyrespectively, so that jmxj;jmyj\u001c1). The expressions (11) and (13) immediately suggest that the Gilbert damping at the\ninterface is a scalar, aG,\n¶tmmm=˜gHHHeff\u0002mmm+aGmmm\u0002¶tmmm; (14)\nwhere the renormalized gyromagnetic ratio and the damping parameter are,\n˜g=g\n1+cDQxy;aG=\u0000cDQxx\n1+cDQxy\u0019\u0000cDQxx: (15)\nIn the latter case we make use of the fact that mc\u001c1for the NM thickness d\u0018100mm — 100 nm. In Eq. (14) we have\nredefined the gyromagnetic ratio g, but we might have renormalized the magnetization instead. From physical perspective,\nthis implies the fraction of conduction electrons which become associated with the localized moment owing to sdexchange\ninteraction. With no vertex correction included one obtains\naG=mc\n2p¯htmrD\nD2+2mr; (16)\n4/8t=1ns\nt=10ns\nD=0.2meV\nD=0.3meV\nD=1meV\n501001502002503000.0000.0010.0020.0030.004\nT,KaGFigure 2. Gilbert damping, obtained from numerical integration of Eq. (8), shows almost no temperature dependence\nassociated with thermal redistribution of conduction electrons. Dashed lines are plotted for D=1meV for t=1andt=10ns,\nwhereas solid lines stand for D=0:2, 0:3, and 1 meV for t=100 ns.\nwhile taking account of vertex correction gives rise to a different result,\naG=mc\n2p¯htmrD\nD2+mr: (17)\nTo provide a quantitative estimate of how large the St ˇreda contribution in the weak disorder limit is, on condition that m>jDj,\nwe work out Q(1)\nmn. Using ¶gR=A(e)=¶e=\u0000[gR=A(e)]2and the fact that trace is invariant under cyclic permuattaions we conclude\nthat only off-diagonal components m6=ncontribute. While the direct evaluation results in Q(1)\nxy=3mD=[2(D2+2mr)]in the\nclean limit. It has been demonstrated that including scattering rates dandhdoes not qualitatively change the results, leading to\nsome smearing only52.\nInterestingly, within the range of applicability of theory developed in this paper, the results of both Eqs. (16) and (17)\ndepend linearly on scattering rate, being thus in qualitative agreement with the breathing Fermi surface model. Meanwhile, the\nlatter does not yield any connection to the microscopic parameters (see, e.g., Ref.53for more details). To provide with some\nquantitative estimations in our simulations we utilize the following set of parameters. Typically, experimental studies based on\nhyperfine field measurements equipped with DFT calculations54reveal the sdStoner interaction to be of the order of 0.2 eV ,\nwhile the induced magnetization of s-derived states equals 0.002–0.05 (measured in the units of Bohr magneton, mB). Thus,\nthe parameter of sdexchange splitting, appropriate for our model, is D\u00180.2–1 meV . In addition, according to first-principles\nsimulations we choose the Fermi energy m\u00183 eV . The results of numerical integration of (8) are presented in Fig. 2 for several\nchoices of sdexchange and scattering rates, t. The calculations reveal almost no temperature dependence in the region up to\nroom temperature for any choice of parameters, which is associated with the fact that the dominant contribution comes from the\nintegration in a tiny region of the Fermi energy. Fig. 2 also reveal a non-negligible dependence on the damping parameter with\nrespect to both Dandt, which illustrates that a tailored search for materials with specific damping parameter needs to address\nboth the sdexchange interaction as well as the scattering rate. From the theoretical perspective, the results shown in Fig. 2\ncorrespond to the case of non-interacting electrons with no electron-phonon coupling included. Thus, the thermal effects are\naccounted only via temperature-induced broadening which does not show up for m>jDj.\nConclusions\nIn this paper we proposed an alternative derivation of the Gilbert damping tensor within a generalized Kubo-St ˇreda formula.\nWe established the contribution stemming from Eq. (7) which was missing in the previous analysis within the linear response\ntheory. In spite of being of the order of (mt)\u00001and, thus, negligible in the weak disorder limit developed in the paper, it should\nbe properly worked out when dealing with more complicated systems, e.g., gapped materials such as iron garnets (certain half\nmetallic Heusler compounds). For a model system, represented by a Rashba ferromagnet, we directly evaluated the Gilbert\ndamping parameter and explored its behaviour associated with the temperature-dependent Fermi-Dirac distribution. In essence,\nthe obtained results extend the previous studies within linear response theory and can be further utilized in first-principles\ncalculations. We believe our results will be of interest in the rapidly growing fields of spintronics and magnonics.\n5/8References\n1.Žuti´c, I., Fabian, J. & Das Sarma, S. Spintronics: Fundamentals and applications. Rev. Mod. Phys. 76, 323 (2004).\n2.Bader, S. D. & Parkin, S. S. P. 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Rev.\nLett.117, 046601 (2016).\n52.Nunner, T. S., et al. Anomalous Hall effect in a two-dimensional electron gas. Phys. Rev. B 76, 235312 (2007).\n53.Eriksson, O., Bergman, A., Bergqvist, L. & Hellsvik, J. Atomistic spin dynamics: Foundations and applications (Oxford\nUniversity Press, Oxford, 2017).\n54.Brooks, M. S. S. & Johansson, B. Exchange integral matrices and cohesive energies of transition metal atoms. J. Phys. F:\nMet. Phys. 13, L197 (1983).\n7/8Acknowledgements\nA.A.P. acknowledges the support from the Russian Science Foundation Project No. 18-72-00058. O.E. acknowledges support\nfrom eSSENCE, the Swedish Research Council (VR), the foundation for strategic research (SSF) and the Knut and Alice\nWallenberg foundation (KAW). D.Y . acknowledges the support from the Russian Science Foundation Project No. 17-12-01359.\nAuthor contributions statement\nD.Y . conceived the idea of the paper and contributed to the theory. A.A.P. wrote the main manuscript text, performed numerical\nanalysis and prepared figures 1-2. M.I.B. and O.E. contributed to the theory. All authors reviewed the manuscript.\nAdditional information\nCompeting interests The authors declare no competing interests.\n8/8"    },    {        "title": "1503.01027v1.Large_Deviations_for_the_Langevin_equation_with_strong_damping.pdf",        "content": "Large deviations for the Langevin equation with strong\ndamping\u0003\nSandra Cerraiy, Mark Freidlinz\nDepartment of Mathematics\nUniversity of Maryland\nCollege Park, MD 20742\nUSA\nAbstract\nWe study large deviations in the Langevin dynamics, with damping of order \u000f\u00001and\nnoise of order 1, as \u000f#0. The damping coe\u000ecient is assumed to be state dependent.\nWe proceed \frst with a change of time and then, we use a weak convergence approach\nto large deviations and their equivalent formulation in terms of the Laplace principle,\nto determine the good action functional.\nSome applications of these results to the exit problem from a domain and to the\nwave front propagation for a suitable class of reaction di\u000busion equations are considered.\n1 Introduction\nFor every\u000f>0, let us consider the Langevin equation\n8\n>><\n>>:q\u000f(t) =b(q\u000f(t))\u0000\u000b(q\u000f(t))\n\u000f_q\u000f(t) +\u001b(q\u000f(t))_B(t);\nq\u000f(0) =q2Rd;_q\u000f(0) =p2Rd:(1.1)\nHereB(t) is ar-dimensional standard Wiener process, de\fned on some complete stochastic\nbasis (\n;F;fFtg;P). In what follows, we shall assume that bis Lipschitz continuous and\n\u000band\u001bare bounded and continuously di\u000berentiable, with bounded derivative. Moreover,\n\u001bis invertible and there exist two constants 0 <\u000b 0<\u000b 1such thata0\u0014\u000b(q)\u0014\u000b1, for all\nq2Rd. Equation (1.1) can be rewritten as the following system in R2d\n8\n>><\n>>:_q\u000f(t) =p\u000f(t); q\u000f(0) =q2Rd;\n_p\u000f(t) =b(q\u000f(t))\u0000\u000b(q\u000f(t))\n\u000f_q\u000f(t) +\u001b(q\u000f(t))_B(t); p\u000f(0) =p2Rd;\n\u0003Key words : Large deviations, Laplace principle, over damped stochastic di\u000berential equations\nyPartially supported by the NSF grant DMS 1407615.\nzPartially supported by the NSF grant DMS 1411866.\n1arXiv:1503.01027v1  [math.PR]  3 Mar 2015and, due to our assumptions on the coe\u000ecients, for any \u000f>0,T >0 andk\u00151, the system\nabove admits a unique solution z\u000f= (q\u000f;p\u000f)2Lk(\n;C([0;T];R2d)), which is a Markov\nprocess.\nNow, if we do a change of time and de\fne q\u000f(t) :=q\u000f(t=\u000f),t\u00150, we have\n8\n><\n>:\u000f2q\u000f(t) =b(q\u000f(t))\u0000\u000b(q\u000f(t)) _q\u000f(t) +p\u000f\u001b(q\u000f(t)) _w(t);\nq\u000f(0) =q2Rd;_q\u000f(0) =p\n\u000f2Rd;(1.2)\nwherew(t) =p\u000fB(t=\u000f),t\u00150, is another Rr-valued Wiener process, de\fned on the same\nstochastic basis (\n ;F;fFtg;P).\nIn the present paper, we are interested in studying the large deviation principle for\nequation (1.2), as \u000f#0. Namely, we want to prove that the family fq\u000fg\u000f>0satis\fes a\nlarge deviation principle in C([0;T];H), with the same action functional Iand the same\nnormalizing factor \u000fthat describe the large deviation principle for the \frst order equation\n_g\u000f(t) =b(g\u000f(t))\n\u000b(g\u000f(t))+p\u000f\u001b(g\u000f(t))\n\u000b(g\u000f(t))_w(t); g\u000f(0) =q2Rd: (1.3)\nIn particular, as shown in Section 4, this implies that the asymptotic behavior of the\nexit time from a basin of attraction for the over damped Langevin dynamics(1.1) can be\ndescribed by the quasi potential Vassociated with I, as well as the asymptotic behavior of\nthe solutions of the degenerate parabolic and elliptic problems associated with the Langevin\ndynamics.\nMoreover, in Section 4, we will show how these results allow to prove that in reaction-\ndi\u000busion equations with non-linearities of KPP type, where the transport is described by\nthe Langevin dynamics itself, the interface separating the areas where u\u000fis close to 1 and\nto 0, as\u000f#0, is given in terms of the action functional I, as in the classical case, when the\nvanishing mass approximation is considered.\nIn [8] and [3], the system\n8\n><\n>:\u0016q\u0016;\u000f(t) =b(q\u0016;\u000f(t))\u0000\u000b(q\u0016;\u000f(t)) _q\u0016;\u000f(t) +p\u000f\u001b(q\u0016;\u000f(t)) _w(t);\nq\u0016;\u000f(0) =q2Rd; _q\u0016;\u000f(0) =p\n\u000f2Rd;(1.4)\nfor 0<\u0016;\u000f<< 1, has been studied, under the crucial assumption that the friction coe\u000ecient\n\u000bis independent of q.\nIt has been proven that, in this case, the so-called Kramers-Smoluchowski approxi-\nmation holds, that is for any \fxed \u000f > 0 the solution q\u0016;\u000fof system (1.4) converges in\nL2(\n;C([0;T];Rd)), as\u0016#0, tog\u000f, the solution of the \frst order equation (1.3). More-\nover, it has been proven that, if V\u0016(q;p) is the quasi-potential associated with the family\nfq\u0016;\u000fg\u000f>0, for\u0016>0 \fxed, then\nlim\n\u0016!0inf\np2RdV\u0016(q;p) =V(q);\n2whereVis the quasi-potential associated with the action-functional I.\nIn [9], equation (1.4) with non constant friction \u000bhas been considered and it has been\nshown that in this case the situation is considerably more delicate. Actually, the limit of\nq\u0016;\u000ftog\u000fhas only been proven via a previous regularization of the noise, which has led\nto the convergence of q\u0016;\u000fto the solution ~ g\u000fof the \frst order equation with Stratonovich\nintegral.\nFinally, we would like to mention that in the recent paper [12], by Lyv and Roberts, an\nanalogous problem has been studied for the stochastic damped wave equation in a bounded\nregular domain D\u001aRd, withd= 1;2;3,\n8\n>>><\n>>>:\u000f@2u(t;x)\n@t2= \u0001u(t;x) +f(u(t;x))\u0000@u(t;x)\n@t+\u000f\u000b@w(t;x)\n@t\nu(t;x) = 0; x2@D; u (0;x) =u0(x);@u(0;x)\n@t=v0;(x)\nwhere\u000f >0 is a small parameter, the friction coe\u000ecient is constant ( \u000b= 1),w(t;x) is\na smooth cylindrical Wiener process and fis a cubic non-linearity. By using the weak\nconvergence approach, the authors show that the family fu\u000fg\u000f>0satis\fes a large deviation\nprinciple in C([0;T];L2(D)), with normalizing factor \u000f2\u000band the same action functional\nthat describes the large deviation principle for the stochastic parabolic equation.\nAs mentioned above, in the present paper we are dealing with the case of non-constant\nfriction\u000band\u0016=\u000f2. Dealing with a non-constant friction coe\u000ecient turns out to be\nimportant in applications, as it allows to describes new e\u000bects in reaction-di\u000busion equations\nand exit problems (see section 4). Here, we will study the large deviation principle for\nequation(1.2) by using the approach of weak convergence (see [1] and [2]) and we will show\nthe validity of the Laplace principle, which, together with the compactness of level sets, is\nequivalent to the large deviation principle.\nAt this point, it is worth mentioning that one major di\u000eculty here is handling the\nintegralZt\n0exp\u0012\n\u0000Zt\ns\u000b(q\u000f(r))dr\u0013\n\u001b(q\u000f(s))dw(s);\nand proving that it converges to zero, as \u000f#0, inL1(\n;C([0;T];Rd)). Actually, as \u000bis non-\nconstant, the integral above cannot be interpreted as an It^ o's integral and in our estimates\nwe cannot use It^ o's isometry. Nevertheless, due to the regularity of q\u000f(t), we can consider\nthe integral above as a pathwise integral, and with appropriate integrations by parts, we\ncan get the estimates required to prove the Laplace principle.\n2 The problem and the method\nWe are dealing here with the equation\n8\n><\n>:\u000f2q\u000f(t) =b(q\u000f(t))\u0000\u000b(q\u000f(t)) _q\u000f(t) +p\u000f\u001b(q\u000f(t)) _w(t);\nq\u000f(0) =q2Rd;_q\u000f(0) =p\n\u000f2Rd;(2.1)\n3Herew(t),t\u00150, is ar-dimensional Brownian motion and the coe\u000ecients b,\u001band\u000bsatisfy\nthe following conditions.\nHypothesis 1. 1. The mapping b:Rd!Rdis Lipschitz-continuous and the map-\nping\u001b:Rd!L(Rr;Rd)is continuously di\u000berentiable and bounded, together with its\nderivative. Moreover, the matrix \u001b(q)is invertible, for any q2Rd, and\u001b\u00001:Rd!\nL(Rr;Rd)is bounded.\n2. The mapping \u000b:Rd!Rbelongs toC1\nb(Rd)and\ninf\nx2Rd\u000b(x) =:\u000b0>0: (2.2)\nIn view of the conditions on the coe\u000ecients \u000b,band\u001bassumed in Hypothesis 1, for\nevery \fxed \u000f>0, equation (2.6) admits a unique solution z\u000f= (q\u000f;p\u000f)2Lk(0;T;Rd), with\nT >0 andk\u00151.\nNow, for any predictable process utaking values in L2([0;T];Rr), we introduce the\nproblem\n_gu(t) =b(gu(t))\n\u000b(gu(t))+\u001b(gu(t))\n\u000b(gu(t))u(t); gu(0) =q2Rd: (2.3)\nThe existence and uniqueness of a pathwise solution guto problem (2.3) in C([0;T];Rd)\nis an immediate consequence of the conditions on the coe\u000ecients b,\u001band\u000bthat we have\nassumed in Hypothesis 1.\nIn what follows, we shall denote by Gthe mapping\nG:L2([0;T];Rr)!C([0;T];Rd); u7!G(u) =gu:\nMoreover, for any f2C([0;T];Rd) we shall de\fne\nI(f) =1\n2inf\u001aZT\n0ju(t)j2dt:f=G(u); u2L2([0;T];Rr)\u001b\n;\nwith the usual convention inf ;= +1. This means that\nI(f) =1\n2ZT\n0\f\f\f\f\u000b(f(s))\u001b\u00001(f(s))\u0012\n_f(s)\u0000b(f(s))\n\u000b(f(s))\u0013\f\f\f\f2\nds; (2.4)\nfor allf2W1;2(0;T;Rd).\nIf we denote by g\u000fthe solution of the stochastic equation\n_g\u000f(t) =b(g\u000f(t))\n\u000b(g\u000f(t))+p\u000f\u001b(g\u000f(t))\n\u000b(g\u000f(t))_w(t); g\u000f(0) =q2Rd; (2.5)\nwe have that Iis the large deviation action functional for the family fg\u000fg\u000f>0in the space\nof continuous trajectories C([0;T];Rd) (for a proof see e.g. [11]). This means that the\nlevel setsfI(f)\u0014cgare compact in C([0;T];Rd), for anyc>0, and for any closed subset\nF\u001aC([0;T];Rd) and any open set G\u001aC([0;T];Rd) it holds\nlim sup\n\u000f!0+\u000flogP(g\u000f2F)\u0014\u0000I(F);\nlim inf\n\u000f!0+\u000flogP(g\u000f2G)\u0015\u0000I(G);\n4where, for any subset A\u001aC([0;T];Rd), we have denoted\nI(A) = inf\nf2AI(f):\nThe main result of the present paper is to prove that in fact the family of solutions q\u000f\nof equation (1.2) satis\fes a large deviation principle with the same action functional Ithat\ndescribes the large deviation principle for the family of solutions g\u000fof equation (2.5). And,\ndue to the fact that q\u000f(t) =q\u000f(\u000ft),t\u00150, this allows to describe the behavior of the over\ndamped Langevin dynamics (1.1) (see Section 4 for all details).\nTheorem 2.1. Under Hypothesis 1, the family of probability measures fL(q\u000f)g\u000f>0, in the\nspace of continuous paths C([0;T];Rd), satis\fes a large deviation principle with action func-\ntionalI.\nIn order to prove Theorem 2.1, we follow the weak convergence approach, as developed\nin [1], (see also [2]). To this purpose, we need to introduce some notations. We denote by\nPTthe set of predictable processes in L2(\n\u0002[0;T];Rr), and for any T > 0 and\r >0, we\nde\fne the sets\nS\r\nT=\u001a\nf2L2(0;T;Rd) :ZT\n0jf(s)j2ds\u0014\r\u001b\nA\r\nT=\b\nu2PT:u2S\r\nT;P\u0000a.s.\t\n:\nNext, for any predictable process utaking values in L2([0;T];Rr), we denote by qu\n\u000f(t)\nthe solution of the problem\n8\n><\n>:\u000f2qu\n\u000f(t) =b(qu\n\u000f(t))\u0000\u000b(qu\n\u000f(t)) _qu\n\u000f(t) +p\u000f\u001b(qu\n\u000f(t)) _w(t) +\u001b(qu\n\u000f(t))u(t);\nqu\n\u000f(0) =q2Rd;_qu\n\u000f(0) =p\n\u000f2Rd:(2.6)\nAs well known, for any \fxed \u000f >0 and for any T > 0 andk\u00151, this equation admits a\nunique solution qu\n\u000finLk(\n;C([0;T];Rd)).\nBy proceeding as in the proof of [2, Theorem 4.3], the following result can be proven.\nTheorem 2.2. Letfu\u000fg\u000f>0be a family of processes in S\r\nTthat converge in distribution, as\n\u000f#0, to someu2S\r\nT, as random variables taking values in the space L2(0;T;Rd), endowed\nwith the weak topology.\nIf the sequencefqu\u000f\u000fg\u000f>0converges in distribution to gu, as\u000f#0, in the space of contin-\nuous paths C([0;T];Rd), then the family fL(q\u000f)g\u000f>0satis\fes a large deviation principle in\nC([0;T];Rd), with action functional I.\nActually, as shown in [2], the convergence of qu\u000f\u000ftoguimplies the validity of the\nLaplace principle with rate functional I. This means that, for any continuous mapping\n\u0003 :C([0;T];Rd)!Rit holds\nlim\n\u000f!0\u0000\u000flogEexp\u0012\n\u00001\n\u000f\u0003(q\u000f)\u0013\n= inf\nf2C([0;T];Rd)( \u0003(f) +I(f) ):\nAnd, as the level sets of Iare compact, this is equivalent to say that fL(q\u000f)g\u000f>0satis\fes a\nlarge deviation principle in C([0;T];Rd), with action functional I.\n53 Proof of Theorem 2.1\nAs we have seen in the previous section, in order to prove Theorem 2.1, we have to show\nthat iffu\u000fg\u000f>0is a family of processes in S\r\nTthat converge in distribution, as \u000f#0, to\nsomeu2 S\r\nT, as random variables taking values in the space L2(0;T;Rd), endowed with\nthe weak topology, then the sequence fqu\u000f\u000fg\u000f>0converges in distribution to gu, as\u000f#0, in\nthe spaceC([0;T];Rd).\nIn view of the Skorohod representation theorem, we can rephrase such a condition in\nthe following way. On some probability space ( \u0016\n;\u0016F;\u0016P), consider a Brownian motion \u0016 wt,\nt\u00150, along with the corresponding natural \fltration f\u0016Ftgt\u00150. Moreover, consider a family\noff\u0016Ftg-predictable processes f\u0016u\u000f;\u0016ug\u000f>0inL2(\u0016\n\u0002[0;T];Rd), taking values in S\r\nT,\u0016P-a.s.,\nsuch that the joint law of (\u0016 u\u000f;\u0016u;\u0016w), under \u0016P, coincides with the joint law of ( u\u000f;u;w ), under\nP, and such that\nlim\n\u000f!0\u0016u\u000f= \u0016u; \u0016P\u0000a.s. (3.1)\nasL2(0;T;R)-valued random variables, endowed with the weak topology. Let \u0016 q\u0016u\u000f\u000fbe the\nsolution of a problem analogous to (2.6), with uandwreplaced respectively by \u0016 u\u000fand \u0016w.\nThen, we have to prove that\nlim\n\u000f!0\u0016q\u0016u\u000f\u000f=g\u0016u; \u0016P\u0000a.s.\ninC([0;T];Rd). In fact, we will prove more. Actually, we will show that\nlim\n\u000f!0\u0016Esup\nt2[0;T]j\u0016q\u0016u\u000f\u000f(t)\u0000g\u0016u(t)j= 0: (3.2)\nIn order to prove (3.2), we will need some preliminary estimates. For any \u000f >0, we\nde\fne the process\nH\u000f(t) =p\u000fe\u0000A\u000f(t)Zt\n0eA\u000f(s)\u001b(qu\n\u000f(s))dw(s); t\u00150: (3.3)\nLemma 3.1. Under Hypothesis 1, for any T > 0,k\u00151and\r > 0, there exists \u000f0>0\nsuch that for any u2S\r\nTand\u000f2(0;\u000f0]\nsup\ns\u0014tEjH\u000f(t)jk\u0014ck;\r(T)(jqjk+jpjk+ 1)\u000f3k\n2+ck\u000fk\n2tk\n2e\u0000k\u000b0t\n\u000f2: (3.4)\nMoreover, we have\nEsup\nt2[0;T]jH\u000f(t)j\u0014p\u000fc\r(T)(1 +jqj+jpj): (3.5)\nProof. Equation (2.6) can be rewritten as the system\n8\n><\n>:_qu\n\u000f(t) =pu\n\u000f(t); qu\n\u000f(0) =q\n\u000f2_pu\n\u000f(t) =b(qu\n\u000f(t))\u0000\u000b(qu\n\u000f(t))pu\n\u000f(t) +p\u000f\u001b(qu\n\u000f(t)) _w(t) +\u001b(qu\n\u000f(t))u(t); pu\n\u000f(0) =p\n\u000f:\n6Thus, if for any 0 \u0014s\u0014tand\u000f>0 we de\fne\nA\u000f(t;s) :=1\n\u000f2Zt\ns\u000b(qu\n\u000f(r))dr; A\u000f(t) :=A\u000f(t;0);\nwe have\npu\n\u000f(t) =1\n\u000fe\u0000A\u000f(t)p+1\n\u000f2Zt\n0e\u0000A\u000f(t;s)b(qu\n\u000f(s))ds\n+1\n\u000f2Zt\n0e\u0000A\u000f(t;s)\u001b(qu\n\u000f(s))u\u000f(s)ds+1\n\u000f2H\u000f(t):(3.6)\nIntegrating with respect to t, this yields\nqu\n\u000f(t) =q+1\n\u000fZt\n0e\u0000A\u000f(s)pds+1\n\u000f2Zt\n0Zs\n0e\u0000A\u000f(s;r)b(qu\n\u000f(r))drds\n+1\n\u000f2Zt\n0Zs\n0e\u0000A\u000f(s;r)\u001b(qu\n\u000f(r))u\u000f(r)drds +1\n\u000f2Zt\n0H\u000f(s)ds:(3.7)\nThanks to the Young inequality, this implies that for any t2[0;T]\njqu\n\u000f(t)j\u0014jqj+\u000fjpj+cZt\n0(1 +jqu\n\u000f(s)j)ds+Zt\n0ju\u000f(s)jds+1\n\u000f2Zt\n0jH\u000f(s)jds\n\u0014c\r(T)(jqj+\u000fjpj+ 1) +1\n\u000f2Zt\n0jH\u000f(s)jds+Zt\n0jqu\n\u000f(s)jds;\nand from the Gronwall lemma we can conclude that\njqu\n\u000f(t)j\u0014c\r(T) (1 +jqj+jpj) +c(T)1\n\u000f2Zt\n0jH\u000f(s)jds:\nThis implies that for any k\u00151\njqu\n\u000f(t)jk\u0014ck;\r(T)(jqjk+jpjk+ 1) +ck;\r(T)\u000f\u00002kZt\n0jH\u000f(s)jkds; \u000f2(0;1]: (3.8)\nNow, due to (3.6), we have\njpu\u000f(t)j\u00141\n\u000fe\u0000\u000b0t\n\u000f2jpj+1\n\u000f2Zt\n0e\u0000\u000b0(t\u0000s)\n\u000f2(1 +jqu\n\u000f(s)j)ds\n+1\n\u000f2Zt\n0e\u0000\u000b0(t\u0000s)\n\u000f2ju\u000f(s)jds+1\n\u000f2jH\u000f(t)j;\nso that, thanks to (3.8), for any \u000f2(0;1] we get\njpu\n\u000f(t)j\u00141\n\u000fe\u0000\u000b0t\n\u000f2jpj+c\r(T)(jqj+jpj+ 1) +1\n\u000f2Zt\n0e\u0000\u000b0(t\u0000s)\n\u000f2ju\u000f(s)jds+c(T)1\n\u000f2jH\u000f(t)j:\n(3.9)\n7As well known, if f2C1([0;t]) andg2C([0;t]), then the Stiltjies integral\nZt\n0f(s)dg(s); t\u00150;\nis well de\fned and, if g(0) = 0, the following integration by parts formula holds\nZt\n0f(s)dg(s) =Zt\n0(g(t)\u0000g(s))h0(s)ds+g(t)h(0); t\u00150: (3.10)\nNow, the mapping\n[0;+1)!L(Rr;Rd); s7!eA\u000f(s)\u001b(qu\n\u000f(s));\nis di\u000berentiable, P-a.s., so that the stochastic integral in (3.3) is in fact a pathwise integral.\nIn particular, we can apply formula (3.10), with\nh(s) =eA\u000f(s)\u001b(qu\n\u000f(s)); g (s) =w(s);\nand we get\nH\u000f(t) =p\u000fZt\n0(w(t)\u0000w(s))e\u0000A\u000f(t;s)\u0012\u000b(qu\n\u000f(s))\n\u000f2+\u001b0(qu\n\u000f(s))pu\n\u000f(s)\u0013\nds\n+p\u000fw(t)e\u0000A\u000f(t)\u001b(q):(3.11)\nThanks to (3.9), this yields for any \u000f2(0;1]\njH\u000f(t)j\u0014cp\u000fZt\n0jw(t)\u0000w(s)je\u0000\u000b0(t\u0000s)\n\u000f2\n\u000f2\u0000\n1 +\u000f2jpu\n\u000f(s)j\u0001\nds+cp\u000fjw(t)je\u0000\u000b0t\n\u000f2\n\u0014c\r(T)(jqj+jpj+ 1)p\u000fZt\n\u000f2\n0jw(t)\u0000w(t\u0000\u000f2s)je\u0000\u000b0sds\n+p\u000fc\r(T)Zt\n\u000f2\n0jw(t)\u0000w(t\u0000\u000f2s)je\u0000\u000b0sjH\u000f(t\u0000\u000f2s)jds+cp\u000fjw(t)je\u0000\u000b0t\n\u000f2;\nand hence, for any k\u00151, we have\njH\u000f(t)jk\u0014ck;\r(T)(jqjk+jpjk+ 1)\u000fk\n2Zt\n\u000f2\n0jw(t)\u0000w(t\u0000\u000f2s)jke\u0000\u000b0sds\n+\u000fk\n2ck;\r(T)Zt\n\u000f2\n0jw(t)\u0000w(t\u0000\u000f2s)jke\u0000\u000b0sjH\u000f(t\u0000\u000f2s)jkds+ck\u000fk\n2jw(t)jke\u0000k\u000b0t\n\u000f2:\n8By taking the expectation, due to the independence of jw(t)\u0000w(t\u0000\u000f2s)jwithjH\u000f(t\u0000\u000f2s)j\nandRt\u0000\u000f2s\n0jH\u000f(r)jkdr, this implies that for any \u000f2(0;1]\nEjH\u000f(t)jk\u0014ck;\r(T)(jqjk+jpjk+ 1)\u000f3k\n2Zt\n\u000f2\n0sk\n2e\u0000\u000b0sds\n+\u000f3k\n2ck;\r(T)Zt\n\u000f2\n0sk\n2e\u0000\u000b0sEjH\u000f(t\u0000\u000f2s)jkds+ck\u000fk\n2tk\n2e\u0000k\u000b0t\n\u000f2\n\u0014ck;\r(T)(jqjk+jpjk+ 1)\u000f3k\n2+ck\u000fk\n2tk\n2e\u0000k\u000b0t\n\u000f2+\u000f3k\n2ck;\r(T) sup\ns\u0014tEjH\u000f(s)jk:\nTherefore, if we pick \u000f02(0;1] such that\n\u000f3k\n2ck;\r(T)<1\n2;\nwe get (3.4).\nNow, let us prove (3.5). From (3.11), we have\njH\u000f(t)j\u0014p\u000fcsup\nt2[0;T]jw(t)j\u0012\n1 +Zt\n0e\u0000\u000b0(t\u00002)\n\u000f2jpu\u000f\u000f(s)jds\u0013\n\u0014p\u000fcsup\nt2[0;T]jw(t)j \n1 +\u000f\u0012Zt\n0jpu\u000f\u000f(s)j2ds\u00131\n2!\n;\nand hence\nEsup\nt2[0;T]jH\u000f(t)j\u0014p\u000fc(T) \n1 +\u000f\u0012\nEZt\n0jpu\u000f\u000f(s)j2ds\u00131\n2!\n:\nThanks to (3.9), as a consequence of the Young inequality, we get\nZt\n0jpu\u000f\u000f(s)j2ds\u0014c\r(T)(1 +jqj2+jpj2) +1\n\u000f4c(T)Zt\n0jH\u000f(s)j2ds; (3.12)\nso that\nEsup\nt2[0;T]jH\u000f(t)j\u0014p\u000fc\r(T)(1 +jqj+jpj) +1p\u000fc(T)\u0012Zt\n0EjH\u000f(s)j2ds\u00131\n2\n:\nTherefore, (3.5) follows from (3.4).\nLemma 3.2. Under Hypothesis 1, for any T >0,k\u00151and\r >0there exists \u000f0>0such\nthat for any u2S\r\nTand\u000f2(0;\u000f0)we have\nEsup\nt2[0;T]jqu\n\u000f(t)jk\u0014ck;\r(T)(jqjk+jpjk+ 1)\u000f\u0000k\n2+ck;\r(T)\u000f2\u00003k\n2: (3.13)\n9Proof. Estimate (3.13) follows by combining together (3.4) and (3.8).\nNow, we are ready to prove (3.2), that, in view of Theorem 2.2, implies Theorem 2.1.\nTheorem 3.3. Letfu\u000fg\u000f>0be a family of predictable processes in S\r\nTthat converge P-a.s.,\nas\u000f#0, to someu2S\r\nT, with respect to the weak topology of L2(0;T;Rd). Then, we have\nlim\n\u000f!0Esup\nt2[0;T]jqu\u000f\u000f(t)\u0000gu(t)j= 0: (3.14)\nProof. Integrating by parts in (3.7), we obtain\nqu\u000f\u000f(t) =q+Zt\n0b(qu\u000f\u000f(s))\n\u000b(qu\u000f\u000f(s))ds+Zt\n0\u001b(qu\u000f\u000f(s))\n\u000b(qu\u000f\u000f(s))u\u000f(s)ds+R\u000f(t);\nwhere\nR\u000f(t) =p\n\u000fZt\n0e\u0000A\u000f(s)ds\u00001\n\u000b(qu\u000f\u000f(t))Zt\n0e\u0000A\u000f(t;s)b(qu\u000f\u000f(s))ds+p\u000fZt\n0\u001b(qu\u000f\u000f(s))\n\u000b(qu\u000f\u000f(s))dw(s)\n+Zt\n0\u0012Zs\n0e\u0000A\u000f(s;r)b(qu\u000f\u000f(r))dr\u00131\n\u000b2(qu\u000f\u000f(s))hr\u000b(qu\u000f\u000f(s));pu\u000f\u000f(s)ids\n\u00001\n\u000b(qu\u000f\u000f(t))H\u000f(t) +Zt\n01\n\u000b2(qu\u000f\u000f(s))H\u000f(s)hr\u000b(qu\u000f\u000f(s));pu\u000f\u000f(s)ids=:6X\nk=1Ik\n\u000f(t):\nThis implies that\nqu\u000f\u000f(t)\u0000gu(t) =Zt\n0\u0014b(qu\u000f\u000f(s))\n\u000b(qu\u000f\u000f(s))\u0000b(gu(s))\n\u000b(gu(s))\u0015\nds+Zt\n0\u0014\u001b(qu\u000f\u000f(s))\n\u000b(qu\u000f\u000f(s))\u0000\u001b(gu(s))\n\u000b(gu(s))\u0015\nu\u000f(s)ds\n+Zt\n0\u001b(gu(s))\n\u000b(gu(s))[u\u000f(s)\u0000u(s)]ds+R\u000f(t):\n(3.15)\nDue to the Lipschitz-continuity and the boundedness of the functions \u001band 1=\u000b, we have\nthat\u001b=\u000b is bounded and Lipschitz continuous. Then, as u\u000f2S\r\nT, we obtain\njqu\u000f\u000f(t)\u0000gu(t)j2\n\u0014c\f\f\f\fZt\n0\u001b(gu(s))\n\u000b(gu(s))[u\u000f(s)\u0000u(s)]ds\f\f\f\f2\n+cjR\u000f(t)j2+c(T)Zt\n0jqu\u000f\u000f(s)\u0000gu(s)j2ds\n+c(T)Zt\n0jqu\u000f\u000f(s)\u0000gu(s)j2ds Zt\n0ju\u000f(s)j2ds+ sup\ns2[0;t]jgu(s)j2!\n\u0014c\f\f\f\fZt\n0\u001b(gu(s))\n\u000b(gu(s))[u\u000f(s)\u0000u(s)]ds\f\f\f\f2\n+cjR\u000f(t)j2+c\r(T)Zt\n0jqu\u000f\u000f(s)\u0000gu(s)j2ds:\n10By the Gronwall lemma, this allows to conclude that\nsup\nt2[0;T]jqu\u000f\u000f(t)\u0000gu(t)j\n\u0014c\r(T) sup\nt2[0;T]\f\f\f\fZt\n0\u001b(gu(s))\n\u000b(gu(s))[u\u000f(s)\u0000u(s)]ds\f\f\f\f+c\r(T) sup\nt2[0;T]jR\u000f(t)j:(3.16)\nNow, for any \u000f>0, we de\fne\n\u0000\u000f(t) =Zt\n0\u001b(gu(s))\n\u000b(gu(s))[u\u000f(s)\u0000u(s)]ds:\nFor any 0<s<t we have\n\u0000\u000f(t)\u0000\u0000\u000f(s) =Zt\ns\u001b(gu(r))\n\u000b(gu(r))[u\u000f(r)\u0000u(r)]dr;\nso that, as u\u000fanduare both in S\r\nT,\nj\u0000\u000f(t)\u0000\u0000\u000f(s)j\u0014c\rp\nt\u0000s; \u000f> 0:\nAs \u0000\u000f(0) = 0, this implies that the family of continuous functions is f\u0000\u000fg\u000f>0is equibounded\nand equicontinuous, so that, by the Ascoli-Arzel\u0012 a theorem, there exists \u000fn#0 andv2\nC([0;T];Rd) such that\nlim\nn!0sup\nt2[0;T]j\u0000\u000fn(t)\u0000v(t)j= 0;P\u0000a.s.\nOn the other hand, as (3.1) holds, for any h2Rdwe have\nlim\n\u000f!0h\u0000\u000f(t);hi= lim\n\u000f!0\u001c\nu\u000f\u0000u;\u001b(gu(\u0001))\n\u000b(gu(\u0001))h\u001d\nL2(0;T;Rd)= 0;\nso that we can conclude that v= 0 and\nlim\n\u000f!0Esup\nt2[0;T]j\u0000\u000f(t)j= 0:\nThanks to (3.16), this implies that\nlim sup\n\u000f!0Esup\nt2[0;T]jqu\u000f\u000f(t)\u0000gu(t)j\u0014clim sup\n\u000f!0Esup\nt2[0;T]jR\u000f(t)j;\nso that (3.14) follows if we show that\nlim\n\u000f!0Esup\nt2[0;T]jR\u000f(t)j= 0: (3.17)\nWe have\njI1\n\u000f(t)j=jpj\n\u000f\f\f\f\fZt\n0e\u0000A\u000f(s)ds\f\f\f\f\u0014cjpj\u000f\u00001Zt\n0e\u0000\u000b0s\n\u000f2ds\u0014cjpj\u000f: (3.18)\n11Moreover\njI2\n\u000f(t)j=1\nj\u000b(qu\u000f\u000f(t))j\f\f\f\fZt\n0e\u0000A\u000f(t;s)b(qu\u000f\u000f(s))ds\f\f\f\f\n\u0014cZt\n0e\u0000\u000b0(t\u0000s)\n\u000f2(1 +jqu\u000f\u000f(s)j)ds\u0014c\u000f2 \n1 + sup\nt2[0;T]jqu\u000f\u000f(t)j!\n:\nThanks to (3.13), this implies\nEsup\nt2[0;T]jI2\n\u000f(t)j\u0014c\r(T)(jpj+jqj+ 1)\u000f3\n2; \u000f2(0;1]: (3.19)\nNext\nEsup\nt2[0;T]jI3\n\u000f(t)j=p\u000fEsup\nt2[0;T]\f\f\f\fZt\n0\u001b(qu\u000f\u000f(s))\n\u000b(qu\u000f\u000f(s))dw(s)\f\f\f\f\u0014c(T)p\u000f: (3.20)\nConcerning I4(t), we have\njI4\n\u000f(t)j=\f\f\f\fZt\n0\u0012Zs\n0e\u0000A\u000f(s;r)b(qu\u000f\u000f(r))dr\u00131\n\u000b2(qu\u000f\u000f(s))hr\u000b(qu\u000f\u000f(s));pu\u000f\u000f(s)ids\f\f\f\f\n\u0014\u000f2c \n1 + sup\nt2[0;T]jqu\u000f\u000f(t)j!Zt\n0jpu\u000f\u000f(s)jds;\nso that, due to (3.13) we obtain\nEsup\nt2[0;T]jI4\n\u000f(t)j\u0014\u000f2c\r(T)(jqj+jpj+ 1)\u000f\u00001\n2\u0012\nEZt\n0jpu\u000f\u000f(s)j2ds\u00131\n2\n:\nAs a consequence of (3.4) and (3.12), this yields\nEsup\nt2[0;T]jI4\n\u000f(t)j\u0014\u000fc\r(T)(jqj2+jpj2+ 1); \u000f2(0;\u000f0]: (3.21)\nConcerning I5\n\u000f(t), according to (3.5) we have\nEsup\nt2[0;T]jI5\n\u000f(t)j\u0014cEsup\nt2[0;T]jH\u000f(t)j\u0014p\u000fc\r(T)(1 +jqj+jpj): (3.22)\nFinally, it remains to estimate I6\n\u000f(t). We have\njI6\n\u000f(t)j=\f\f\f\fZt\n01\n\u000b2(qu\u000f\u000f(s))H\u000f(s)hr\u000b(qu\u000f\u000f(s));pu\u000f\u000f(s)ids\f\f\f\f\u0014cZt\n0jH\u000f(s)jjpu\u000f\u000f(s)jds;\nso that\nEsup\nt2[0;T]jI6\n\u000f(t)j\u0014c\u0012ZT\n0EjH\u000f(s)j2dsZT\n0Ejpu\u000f\u000f(s)j2ds\u00131\n2\n:\n12By using (3.12), this gives\nEsup\nt2[0;T]jI6\n\u000f(t)j\u0014c\r(T)(1 +jqj+jpj)\u0012ZT\n0EjH\u000f(s)j2ds\u00131\n2\n+1\n\u000f2ZT\n0EjH\u000f(s)j2ds;\nso that, from (3.4) we get\nEsup\nt2[0;T]jI6\n\u000f(t)j\u0014\u000fc\r(T)(1 +jqj+jpj); \u000f2(0;\u000f0]:\nThis, together with (3.18), (3.19), (3.20), (3.21) and (3.22), implies (3.17) and (3.14) follows.\n4 Some applications and remarks\nLetGbe a bounded domain in Rd, with a smooth boundary @G. We consider here the exit\nproblem for the process q\u000f(t) de\fned as the solution of equation (1.1). For every \u000f>0 we\nde\fne\n\u001c\u000f:= minft\u00150 :q\u000f(t)=2Gg; \u001c\u000f:= minft\u00150 :q\u000f(t)=2Gg;\nwhereq\u000f(t) =q\u000f(t=\u000f) is the solution of equation (2.6). It is clear that\n\u001c\u000f=1\n\u000f\u001c\u000f; q\u000f(\u001c\u000f) =q\u000f(\u001c\u000f):\nIn what follows, we shall assume that the dynamical system\n_q(t) =b(q(t)); t\u00150; (4.1)\nsatis\fes the following conditions.\nHypothesis 2. The pointO2Gis asymptotically stable for the dynamical system (4.1)\nand for any initial condition q2Rd\nlim\nt!1q(t) =O:\nMoreover, we have\nhb(q);\u0017(q)i>0; q2@G;\nwhere\u0017(q)is the inward normal vector at q2@G.\nNow, we introduce the quasi-potential associated with the action functional Ide\fned\nin (2.4)\nV(q) = infn\nI(f); f2C([0;T];Rd); f(0) =O; f (T) =q; T > 0o\n=1\n2inf(ZT\n0\f\f\f\f\u000b(f(s))\u001b\u00001(f(s))\u0012\n_f(s)\u0000b(f(s))\n\u000b(f(s))\u0013\f\f\f\f2\nds; f (0) =O; f (T) =q; T > 0)\n:\nIt is easy to check that, under our assumptions on \u000b(q), the quasi-potential Vcoincides\nwith\n1\n2inf\u001aZT\n0\f\f\f\u001b\u00001(f(s))\u0010\n_f(s)\u0000\u000b(f(s))b(f(s))\u0011\f\f\f2\nds; f (0) =O; f (T) =q; T > 0\u001b\n:(4.2)\n13Theorem 4.1. Under Hypotheses 1 and 2, for each q2fq2G:V(q)\u0014V0gandp2Rd,\nwe have\nlim\n\u000f!0\u000flogE(q;p)\u001c\u000f= lim\n\u000f!0\u000flogE(q;p)\u001c\u000f=V0; (4.3)\nand\nlim\n\u000f!0\u000flog\u001c\u000f= lim\n\u000f!0\u000flog\u001c\u000f=V0;in probability ; (4.4)\nwhere\nV0:= min\nq2@GV(q):\nMoreover, if the minimum of Von@Gis achieved at a unique point q?2@G, then\nlim\n\u000f!0q\u000f(\u001c\u000f) = lim\n\u000f!0q\u000f(\u001c\u000f) =q?: (4.5)\nProof. First, note that q\u000f(t) is the \frst component of the 2 d-dimensional Markov process\nz\u000f(t) = (q\u000f(t);p\u000f(t)). Because of the structure of the p-component of the drift of this process\nand our assumptions on the vector \feld b, starting from ( q;p)2R2d, the trajectory of z\u000f(t)\nspends most of the time in a small neighborhood of the point q=Oandp= 0, with\nprobability close to 1, as 0 < \u000f << 1. From time to time, the process z\u000f(t) deviates from\nthis point and, as proven in Theorem 2.1, the deviations of q\u000f(t) are governed by the large\ndeviation principle with action functional I, de\fned in (2.4). This allows to prove the\nvalidity of (4.3), (4.4) and (4.5) in the same way as Theorems 4.41, 4.42 and 4.2.1 from [11]\nare proven. We omit the details.\nAs an immediate consequence of (4.2) and [11, Theorem 4.3.1], we have the following\nresult.\nTheorem 4.2. Assumea(q) :=\u001b(q)\u001b?(q) =Iand\u000b(q)b(q) =\u0000rU(q) +l(q), for any\nq2Rd, for some smooth function U:Rd!Rhaving a unique critical point (a minimum)\natO2Rdand such that\nhrU(q);l(q)i= 0; q2Rd:\nThen\nV(q) = 2U(q); q2Rd:\nFrom Theorems 4.1 and 4.2, it is possible to get a number of results concerning the\nasymptotic behavior, as \u000f#0, of the solutions of the degenerate parabolic and the elliptic\nproblems associated with the di\u000berential operator L\u000fde\fned by\nL\u000fu(q;p) =1\n2dX\ni;j=1ai;j(q)@2u\n@pi@pj(q;p) +\u0012\nb(q)\u00001\n\u000f\u000b(q)p\u0013\n\u0001rpu(q;p) +p\u0001rqu(q;p):\nAssume now that the dynamical system (4.1) has several asymptotically stable attrac-\ntors. Assume, for the sake of brevity, that all attractors are just stable equilibriums O1,\nO2,. . . ,Ol. Denote byEthe set of separatrices separating the basins of these attractors, and\nassume the setEto have dimension strictly less than d. Moreover, let each trajectory q(t),\n14starting at q02RdnE, be attracted to one of the stable equilibriums Oi,i= 1;:::;l , as\nt!1 . Finally, assume that the projection of b(q) on the radius connecting the origin in\nRdand the point q2Rdis directed to the origin and its length is bounded from below by\nsome uniform constant \u0012>0 (this condition provides the positive recurrence of the process\nz\u000f(t) = (q\u000f(t);p\u000f(t)),t\u00150).\nIn what follows, we shall denote\nV(q1;q2)\n=1\n2infZT\n0\f\f\f\f\u000b(f(s))\u001b\u00001(f(s))\u0012\n_f(s)\u0000b(f(s))\n\u000b(f(s))\u0013\f\f\f\f2\nds; f (0) =q1; f(T) =q2; T > 0\t\nand\nVij=V(Oi;Oj); i;j2f1;:::;lg:\nIn a generic case, the behavior of the process ( q\u000f(t);p\u000f(t)), on time intervals of order\nexp(\u0015\u000f\u00001),\u0015 > 0 and 0< \u000f << 1, can be described by a hierarchy of cycles as in [11]\nand [6]. The cycles are de\fned by the numbers Vij. For (almost) each initial point qand a\ntime scale\u0015, these numbers de\fne also the metastable state Oi?,i?=i?(q;\u0015), whereq\u000f(t)\nspends most of the time during the time interval [0 ;exp(\u0015\u000f\u00001)]. Slow changes of the \feld\nb(q) and/or of the damping coe\u000ecient \u000b(q) can lead to stochastic resonance (compare with\n[7]).\nConsider next the reaction di\u000busion equation in Rd\n8\n><\n>:@u\n@t(t;q) =Lu(t;q) +c(q;u(t;q))u(t;q);\nu(0;q) =g(q); q2Rd; t> 0:(4.6)\nHereLis a linear second order uniformly elliptic operator, with regular enough coe\u000ecients.\nLetq(t) be the di\u000busion process in Rdassociated with the operator L. The Feynman-Kac\nformula says that ucan be seen as the solution of the problem\nu(t;q) =Eqg(q(t)) expZt\n0c(q(s);u(t\u0000s;q(s))ds: (4.7)\nReaction-di\u000busion equations describe the interaction between particle transport de\fned\nbyq(t) and reaction which consists of multiplication (if c(q;u)>0) and annihilation (if\nc(q;u)<0) of particles. In classical reaction-di\u000busion equations, the Langevin dynamics\nwhich describes a di\u000busion with inertia is replaced by its vanishing mass approximation. If\nthe transport is described by the Langevin dynamics itself, equation (4.6) should be replaced\nby an equation in R2d. Assuming that the drift is equal to zero ( b(q) = 0), and the damping\nis of order\u000f\u00001, as\u000f#0, this equation has the form8\n>>>>>>>><\n>>>>>>>>:@u\u000f\n@t(t;q;p ) =1\n2dX\ni;j=1ai;j(q)@2u\u000f\n@pi@pj(q;p)\u00001\n\u000f\u000b(q)p\u0001rpu\u000f(q;p) +p\u0001rqu\u000f(q;p)\n+c(q;u\u000f(t;q;p ))u\u000f(t;q;p ); t> 0;(q;p)2R2d;\nu\u000f(0;q;p) =g(q)\u00150;(q;p)2R2d:(4.8)\n15Now, we de\fne\nR(t;q) = sup\u001aZt\n0c(f(s);0)ds\u0000It(f) :f(0) =q; f(t)2G0\u001b\n;\nwhere\nIt(f) =1\n2Zt\n0\u000b2(f(s))a\u00001(f(s))_f(s)\u0001_f(s)ds;\nandG0= suppfg(q); q2Rdg.\nDe\fnition 4.3. 1. We say that Condition (N) is satis\fed if R(t;x)can be characterized,\nfor anyt>0andx2\u0006t=fq2Rd; R(t;q) = 0g, as\nsup\u001aZt\n0c(f(s);0)ds\u0000It(f); f(0) =q; f(t)2G0; R(t\u0000s;f(s))\u00140;0\u0014s\u0014t\u001b\n:\n2. We say that the non-linear term f(q;u) =c(q;u)uin equation (4.8) is of KPP\n(Kolmogorov-Petrovskii-Piskunov) type if c(q;u)is Lipschitz-continuous, c(q;0)\u0015\nc(q;u)>0, for any 0<u< 1,c(q;1) = 0 andc(q;u)<0, for anyu>1.\nTheorem 4.4. Let the non-linear term in (4.8) be of KPP type. Assume that Condition\n(N) is satis\fed and assume that the closure of G0= suppfg(q); q2Rdgcoincides with the\nclosure of the interior of G0. Then,\nlim\n\u000f!0u\u000f(t=\u000f;q;p ) = 0;ifR(t;q)<0; (4.9)\nand\nlim\n\u000f!0u\u000f(t=\u000f;q;p ) = 1;ifR(t;q)>0; (4.10)\nso that equation R(t;q) = 0 inR2dde\fnes the interface separating the area where u\u000f, the\nsolution of (4.8) , is close to 1and to 0, as\u000f#0.\nProof. If we de\fne u\u000f(t;q;p ) =u\u000f(t=\u000f;q;p ), the analog of (4.7) yields\nu\u000f(t;q;p ) =E(q;p)g(q\u000f(t)) exp\u00121\n\u000fZt\n0c(q\u000f(s);u\u000f(t\u0000s;q\u000f(s);p\u000f(s))ds\u0013\n; (4.11)\nwherez\u000f(t) = (q\u000f(s);p\u000f(s)) is the solution to equation (2.6). By taking into account our\nassumptions on c(q;u), we derive from (4.11)\nu\u000f(t;q;p )\u0014E(q;p)g(q\u000f(t)) exp\u00121\n\u000fZt\n0c(q\u000f(s);0)ds\u0013\n:\nTheorem 2.1 and the Laplace formula imply that the right hand side of the above inequality\nis logarithmically equivalent , as \u000f#0, to exp\u00001\n\u000fR(t;q)\u0001\nand this implies (4.9).\nIn order to prove (4.10), \frst of all one should check that if R(t;q) = 0, then for each\n\u000e>0\nu\u000f(t;q;p )\u0015exp\u0012\n\u00001\n\u000f\u000e\u0013\n; (4.12)\n16when\u000f>0 is small enough. This follows from (4.11) and Condition (N), if one takes into\naccount the continuity of c(q;u). The strong Markov property of the process ( q\u000f(t);p\u000f(t))\nand bound (4.12) imply (4.10) (compare with [5]).\nConsider, as an example, the case c(q;0) =c= const. Then\nR(t;q) =ct\u0000inffIt(f); f(0) =q; f(t)2G0g:\nThe in\fmum in the equality above coincides with\n1\n2t\u001a2(q;G 0);\n(see, for instance, [5] for a proof), where \u001a(q1;q2),q1;q22Rd, is the distance in the\nRiemaniann metric\nds=\u000b(q)vuutdX\ni;j=1ai;j(q)dqidqj:\nThis implies that the interface moves according to the Huygens principle with the constant\nspeedp\n2c, if calculated in the Riemannian metric ds.\nIf\u000b(q) = 0 in a domain G1\u001aRd, the points of G1should be identi\fed. The Riemaniann\nmetric in Rdinduces now, in a natural way, a new metric ~ \u001ain this space with identi\fed\npoints. The motion of the interface, in this case, can be described by the Huygens principle\nwith constant velocityp\n2cin the metric ~ \u001a.\nIfc(q;0) is not constant, the motion of the interface, in general, cannot be described\nby a Huygens principle. Actually, the motion can have jumps and other speci\fc features\n(compare with [5]).\nFinally, if the Condition (N) is not satis\fed, the function R(t;q) should be replaced by\nanother one. De\fne\n~R(t;q) = sup\u001a\nmin\n0\u0014a\u0014t\u0012Za\n0c(f(s);0)ds\u0000Ia(f)\u0013\n:f(0) =q; f(t)2G0\u001b\n:\nThe function ~R(t;q) is Lipschitz continuous and non-positive and if Condition (N) is satis-\n\fed, then\n~R(t;q) = minfR(t;q);0g:\nBy proceeding as in [6], it is possible to prove that\nlim\n\u000f!0u\u000f(t=\u000f;q;p ) = 0;ifR(t;q)<0;\nand\nlim\n\u000f!0u\u000f(t=\u000f;q;p ) = 1;\nif (t;q) is in the interior of the set f(t;q) :t>0; q2Rd;~R(t;q) = 0g.\nFinally, we would like to mention a few generalizations.\n171. The arguments that we we have used in the proof of Theorem 2.1, can be used to\nprove the same result for the equation\n8\n>><\n>>:q\u000f(t) =b(q\u000f(t))\u0000\u000b(q\u000f(t))\n\u000f_q\u000f(t) +1\n\u000f\f\u001b(q\u000f(t))_B(t);\nq\u000f(0) =q2Rd;_q\u000f(0) =p2Rd;\nfor any\f <1=2. As a matter of fact, with the very same method we can show that\nalso in this case the family fq\u000fg\u000f>0satis\fes a large deviation principle in C([0;T];Rd)\nwith action functional Iand with normalizing factor \u000f1\u00002\f.\n2. The damping can be assumed to be anisotropic. This means that the coe\u000ecient \u000b(q)\ncan be replaced by a matrix \u000b(q), with all eigenvalues having negative real part.\n3. Systems with strong non-linear damping can be considered. Namely, let ( q\u000f;p\u000f) be the\ntime-inhomogeneous Markov process corresponding to the following initial-boundary\nvalue problem for a degenerate quasi-linear equation on a bounded regular domain\nG\u001aRd\n8\n>>>>>>>>><\n>>>>>>>>>:@u\u000f(t;p;q )\n@t=1\n2dX\ni;j=1ai;j(q)@2u\u000f(t;q;p )\n@pi@pj+b(q)\u0001rpu\u000f(t;q;p )\n\u0000\u000b(q;u\u000f(t;q;p ))\n\u000fp\u0001rpu\u000f(t;q;p ) +p\u0001rqu\u000f(t;q;p ):\nu\u000f(0;q;p) =g(q); u\u000f(t;q;p )jq2@G= (q);\nExistence and uniqueness of such degenerate problem, under some mild conditions,\nfollows from [4, Chapter 5]. The non-linearity of the damping leads to some pecu-\nlarities in the exit problem and in metastability. In particular, in the generic case,\nmetastable distributions can be distributions among several asymptotic attractors and\nthe limiting exit distributions may have a density (see [10]).\nReferences\n[1] P. Dupuis, R. Ellis, A weak convergence approach to the theory of large\ndeviations , Wiley Series in Probability and Statistics, John Wiley and Sons, Inc.,\nNew York, 1997.\n[2] M. Bou\u0013 e, P. Dupuis, A variational representation for certain functionals of Brownian\nmotion , Annals of Probability 26 (1998), no. 4, 1641{1659.\n[3] Z. Chen, M.I. Freidlin, Smoluchowski-Kramers approximation and exit problems\nStochastics and Dynamycs 5 (2005), pp. 569{585.\n[4] M.I. Freidlin, Functional integration and partial differential equations ,\nAnnals of Mathematics Studies, 109, Princeton University Press, 1985.\n18[5] M.I. Freidlin, Limit theorems for large deviations and reaction-di\u000busion equations , An-\nnals of Probability 13 (1985), pp. 639{675.\n[6] M.I. Freidlin, Coupled reaction-di\u000busion equations , Annals of Probability 19 (1991),\npp. 29{57.\n[7] M.I. Freidlin, Quasi-deterministic approximation, metastability and stochastic reso-\nnance , Physica D 137 (2000), pp. 333{352.\n[8] M.I. Freidlin, Some remarks on the Smoluchowski-Kramers approximation , Journal of\nStatistical Physics 117, pp. 617{634, 2004.\n[9] M.I. Freidlin, W. Hu, Smoluchowski-Kramers approximation in the case of variable\nfriction , Journal of Mathematical Sciences, 179 (2011), pp. 184{207.\n[10] M.I. Freidlin, L. Koralov, Nonlinear stochastic perturbations of dynamical systems and\nquasi-linear parabolic PDE's with a small parameter , Probability Theory andRelated\nFields 147 (2010), pp. 273{301.\n[11] M.I. Freidlin, A.D. Wentzell, Random perturbations of dynamical systems ,\nThird Edition, Springer, Heidelberg, 2012.\n[12] Y. Lyv, A.J. Roberts, Large deviation principle for singularly perturbed stochastic\ndamped wave equations , Stochastic Analysis and Applications, 32 (2014), pp. 50-60.\n19"    },    {        "title": "1501.00444v1.Inertia__diffusion_and_dynamics_of_a_driven_skyrmion.pdf",        "content": "Inertia, diffusion and dynamics of a driven skyrmion\nChristoph Sch ¨utte,1Junichi Iwasaki,2Achim Rosch,1and Naoto Nagaosa2, 3,\u0003\n1Institut f ¨ur Theoretische Physik, Universit ¨at zu K ¨oln, D-50937 Cologne, Germany\n2Department of Applied Physics, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan\n3RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n(Dated: January 5, 2015)\nSkyrmions recently discovered in chiral magnets are a promising candidate for magnetic storage devices\nbecause of their topological stability, small size ( \u00183\u0000100nm), and ultra-low threshold current density ( \u0018\n106A/m2) to drive their motion. However, the time-dependent dynamics has hitherto been largely unexplored.\nHere we show, by combining the numerical solution of the Landau-Lifshitz-Gilbert equation and the analysis of a\ngeneralized Thiele’s equation, that inertial effects are almost completely absent in skyrmion dynamics driven by\na time-dependent current. In contrast, the response to time-dependent magnetic forces and thermal fluctuations\ndepends strongly on frequency and is described by a large effective mass and a (anti-) damping depending\non the acceleration of the skyrmion. Thermal diffusion is strongly suppressed by the cyclotron motion and is\nproportional to the Gilbert damping coefficient \u000b. This indicates that the skyrmion position is stable, and its\nmotion responds to the time-dependent current without delay or retardation even if it is fast. These findings\ndemonstrate the advantages of skyrmions as information carriers.\nPACS numbers: 73.43.Cd,72.25.-b,72.80.-r\nI. INTRODUCTION\nMass is a fundamental quantity of a particle determining its\nmechanical inertia and therefore the speed of response to ex-\nternal forces. Furthermore, it controls the strength of quantum\nand thermal fluctuations. For a fast response one usually needs\nsmall masses and small friction coefficients which in turn lead\nto large fluctuations and a rapid diffusion. Therefore, usually\nsmall fluctuations and a quick reaction to external forces are\nnot concomitant. However a “particle” is not a trivial object\nin modern physics, it can be a complex of energy and mo-\nmentum, embedded in a fluctuating environment. Therefore,\nits dynamics can be different from that of a Newtonian parti-\ncle. This is the case in magnets, where such a “particle” can\nbe formed by a magnetic texture1,2. A skyrmion3,4is a rep-\nresentative example: a swirling spin texture characterized by\na topological index counting the number of times a sphere is\nwrapped in spin space. This topological index remains un-\nchanged provided spin configurations vary slowly, i.e., dis-\ncontinuous spin configurations are forbidden on an atomic\nscale due to high energy costs. Therefore, the skyrmion is\ntopologically protected and has a long lifetime, in sharp con-\ntrast to e.g. spin wave excitations which can rapidly decay.\nSkyrmions have attracted recent intensive interest because of\ntheir nano-metric size and high mobility5–14. Especially, the\ncurrent densities needed to drive their motion ( \u0018106A/m2)\nare ultra small compared to those used to manipulate domain\nwalls in ferromagnets ( \u00181011\u000012A/m2)15–19.\nThe motion of the skyrmion in a two dimensional film can\nbe described by a modified version of Newton’s equation. For\nsufficiently slowly varying and not too strong forces, a sym-\nmetry analysis suggests the following form of the equations\nof motion,\nG\u0002_R+\u000bD_R+mR+\u000b\u0000\u0002R=Fc+Fg+Fth:(1)\nHere we assumed translational and rotational invariance of\nthe linearized equations of motion. The ‘gyrocoupling’ G=G^e?is an effective magnetic field oriented perpendicular to\nthe plane,\u000bis the (dimensionless) Gilbert damping of a sin-\ngle spin,\u000bDdescribes the friction of the skyrmion, mits mass\nandRits centre coordinate. \u0000parametrizes a peculiar type of\ndamping proportional to the acceleration of the particle. We\nname this term ‘gyrodamping’, since it describes the damping\nof a particle on a cyclotron orbit (an orbit with R/G\u0002_R),\nwhich can be stronger ( \u0000parallel to G) or weaker (antipar-\nallel to G) than that for linear motion. Our main goal will\nbe to describe the influence of forces on the skyrmion arising\nfrom electric currents ( Fc), magnetic field gradients (Fg) and\nthermal fluctuations (Fth).\nBy analyzing the motion of a rigid magnetic structure\nM(r;t) =M0(r\u0000R(t))forstatic forces, one can obtain\nanalytic formulas for G;\u000bD;FcandFgusing the approach\nof Thiele19–22,24. In Ref. [25], an approximate value for the\nmass of a skyrmion was obtained by simulating the motion of\na skyrmion in a nanodisc and by estimating contributions to\nthe mass from internal excitations of the skyrmion.\nFor rapidly changing forces, needed for the manipulation of\nskyrmions in spintronic devices, Eq. (1) is however not suffi-\ncient. A generalized version of Eq. (1) valid for weak but also\narbitrarily time-dependent forces can be written as\nG\u00001(!)V(!) =Fc(!) +Fg(!) +Fth(!) (2)\n=Sc(!)vs(!) +Sg(!)rBz(!) +Fth(!)\nHereV(!) =R\nei!t_R(t)dtis the Fourier transform of the\nvelocity of the skyrmion, vs(!)is the (spin-) drift velocity\nof the conduction electrons, directly proportional to the cur-\nrent,rBz(!)describes a magnetic field gradient in frequency\nspace. The role of the random thermal forces, Fth(!), is spe-\ncial as their dynamics is directly linked via the fluctuation-\ndissipation theorem to the left-hand side of the equation, see\nbelow. The 2\u00022matrix G\u00001(!)describes the dynam-\nics of the skyrmion; its small- !expansion defines the terms\nwritten on the left-hand side of Eq. (1). One can expectarXiv:1501.00444v1  [cond-mat.str-el]  2 Jan 20152\nFIG. 1: When a skyrmion is driven by a time dependent external\nforce, it becomes distorted and the spins precess resulting in a de-\nlayed response and a large effective mass. In contrast, when the\nskyrmion motion is driven by an electric current, the skyrmion ap-\nproximately flows with the current with little distortion and preces-\nsion. Therefore skyrmions respond quickly to rapid changes of the\nelectric current.\nstrongly frequency-dependent dynamics for the skyrmion be-\ncause the external forces in combination with the motion of\nthe skyrmion can induce a precession of the spin and also ex-\ncite spinwaves in the surrounding ferromagnet, see Fig. 1.\nWe will, however, show that the frequency dependence of the\nright-hand side of the Eq. (2) is at least as important: not only\nthe motion of the skyrmion but also the external forces excite\ninternal modes. Depending on the frequency range, there is\nan effective screening or antiscreening of the forces described\nby the matrices Sc(!)andSg(!). Especially for the current-\ndriven motion, there will be for all frequencies an almost exact\ncancellation of terms from G\u00001(!)andSc(!). As a result\nthe skyrmion will follow almost instantaneously any change\nof the current despite its large mass.\nIn this paper, we study the dynamics of a driven skyrmion\nby solving numerically the stochastic Landau-Lifshitz-Gilbert\n(LLG) equation. Our strategy will be to determine the param-\neters of Eq. (2) such that this equation reproduces the results\nof the LLG equation. Section II introduces the model and\noutlines the numerical implementation. Three driving mecha-\nnisms are considered: section III studies the diffusive motion\nof the skyrmion due to thermal noise, section IV the skyrmion\nmotion due to time-dependent magnetic field gradient and sec-\ntion V the current-driven dynamics. We conclude with a sum-\nmary and discussion of the results in Sec. VI.\nII. MODEL\nOur study is based on a numerical analysis of the stochastic\nLandau-Lifshitz-Gilbert (sLLG) equations27defined by\ndMr\ndt=\rMr\u0002[Be\u000b+b\r(t)]\n\u0000\r\u000b\nMMr\u0002(Mr\u0002[Be\u000b+b\r(t)]):(3)\nHere\ris the gyromagnetic moment and \u000bthe Gilbert damp-\ning;Be\u000b=\u0000\u000eH[M]\n\u000eMris an effective magnetic field created by\nthe surrounding magnetic moments and b\r(t)a fluctuating,stochastic field creating random torques on the magnetic mo-\nments to model the effects of thermal fluctuations, see below.\nThe Hamiltonian H[M]is given by\nH[M] =\u0000JX\nrMr\u0001\u0000\nMr+aex+Mr+aey\u0001\n\u0000\u0015X\nr\u0000\nMr\u0002Mr+aex\u0001ex+Mr\u0002Mr+aey\u0001ey\u0001\n\u0000B\u0001X\nrMr (4)\nWe useJ= 1 ,\r= 1 ,jMrj= 1 ,\u0015= 0:18Jfor the\nstrength of the Dzyaloshinskii-Moriya interaction and B=\n(0;0;0:0278J)for all plots giving rise to a skyrmion with a\nradius of about 15lattice sites, see Appendix A. For this pa-\nrameter set, the ground state is ferromagnetic, thus the single\nskyrmion is a topologically protected, metastable excitation.\nTypically we simulate 100\u0002100spins for the analysis of dif-\nfusive and current driven motion and 200\u0002200spins for the\nforce-driven motion. For these parameters lattice effects are\nnegligible, see appendix B. Typical microscopic parameters\nused, areJ= 1meV (this yields Tc\u001810K) which we use to\nestimate typical time scales for the skyrmion motion.\nFollowing Ref. 27, we assume that the field bfl\nr(t)is gen-\nerated from a Gaussian stochastic process with the following\nstatistical properties\n\nbfl\nr;i(t)\u000b\n= 0\n\nbfl\nr;i(t)bfl\nr0;j(s)\u000b\n= 2\u000bkBT\n\rM\u000eij\u000err0\u000e(t\u0000s) (5)\nwhereiandjare cartesian components and h:::idenotes\nan average taken over different realizations of the fluctuating\nfield. The Gaussian property of the process stems from the in-\nteraction of Mrwith a large number of microscopic degrees\nof freedom (central limit theorem) which are also responsi-\nble for the damping described by \u000b, reflecting the fluctuation-\ndissipation theorem. The delta-correlation in time and space\nin Eq. (5) expresses that the autocorrelation time and length of\nbfl\nr(t)is much shorter than the response time and length scale\nof the magnetic system.\nFor a numerical implementation of Eq. (3) we follow\nRef. 27 and use Heun’s scheme for the numerical integration\nwhich converges quadratically to the solution of the general\nsystem of stochastic differential equations (when interpreted\nin terms of the Stratonovich calculus).\nFor static driving forces, one can calculate the drift veloc-\nity_Rfollowing Thiele20. Starting from the Landau-Lifshitz\nGilbert equations, Eq. (3), we project onto the translational\nmode by multiplying Eq. (3) with @iMrand integrating over\nspace21–23.\nG=~M0Z\ndr n\u0001(@xn\u0002@yn)\nD=~M0Z\ndr(@xn\u0001@xn+@yn\u0001@yn)=2\nFc=G\u0002vs+\fDvs;\nFg=MsrB; M s=M0Z\ndr(1\u0000nz) (6)3\nwhere nis the direction of the magnetization, M0the lo-\ncal spin density, vsthe (spin-) drift velocity of the conduc-\ntion electrons proportional to the electric current, and Ms\nis the change of the magnetization induced by a skyrmion\nin a ferromagnetic background. The ’gyrocoupling vector’\nG= (0;0;G)TwithG=\u0006~M04\u0019is given by the winding\nnumber of the skyrmion, independent of microscopic details.\nIII. THERMAL DIFFUSION\nRandom forces arising from thermal fluctuations play a de-\ncisive role in controlling the diffusion of particles and there-\nfore also the trajectories R(t)of a skyrmion. To obtain R(t)\nand corresponding correlation functions we used numerical\nsimulations based on the stochastic Landau-Lifshitz-Gilbert\nequation27. These micromagnetic equations describe the dy-\nnamics of coupled spins including the effects of damping\nand thermal fluctuations. Initially, a skyrmion spin-texture\nis embedded in a ferromagnetic background. By monitoring\nthe change of the magnetization, we track the center of the\nskyrmion R(t), see appendix A for details.\nOur goal is to use this data to determine the matrix G\u00001(!)\nand the randomly fluctuating thermal forces, Fth(!), which\ntogether fix the equation of motion, Eq. (2), in the presence\nof thermal fluctuations ( rBz=vs= 0). One might worry\nthat this problem does not have a unique solution as both the\nleft-hand and the right-hand side of Eq. (2) are not known\na priori. Here one can, however, make use of the fact that\nKubo’s fluctuation-dissipation theorem26constraints the ther-\nmal forces on the skyrmion described by Fthin Eq. (2) by\nlinking them directly to the dissipative contributions of G\u00001.\nOn averagehFth= 0i, but its autocorrelation is proportional\nto the temperature and friction coefficients. In general it is\ngiven by\nhFi\nth(!)Fj\nth(!0)i=kBT[G\u00001\nij(!) +G\u00001\nji(\u0000!)]2\u0019\u000e(!+!0):\n(7)\nFor small ! one obtainshFx\nth(!)Fx\nth(!0)i =\n4\u0019kBT\u000bD\u000e(!+!0)while off-diagonal correla-\ntions arise from the gyrodamping hFx\nth(!)Fy\nth(!0)i=\n4\u0019i!kBT\u000b\u0000\u000e(!+!0). Using Eq. (7) and demand-\ning furthermore that the solution of Eq. (2) reproduces\nthe correlation function h_Ri(t)_Rj(t0)i(or, equivalently,\nh(Ri(t)\u0000Rj(t0))2i) obtained from the micromagnetic\nsimulations, leads to the condition26\nGij(!) =1\nkBTZ1\n0\u0002(t\u0000t0)h_Ri(t)_Rj(t0)i (8)\nei!(t\u0000t0)d(t\u0000t0):\nWe therefore determine first in the presence of thermal fluc-\ntuations (rBz=vs= 0) from simulations of the stochastic\nLLG equation (3) the correlation functions of the velocities\nand use those to determine Gij(!)using Eq. (8). After a sim-\nple matrix inversion, this fixes the left-hand side of the equa-\ntion of motion, Eq. (2), and therefore contains all information\n0 5 10 15 20 25t ωp00.511.522.5 <ΔR2>α=0.01\nα=0.05\nα=0.1\nα=0.15\nα=0.2FIG. 2: Time dependence of the correlation function\n(Ri(t0+t)\u0000Ri(t0))2\u000b\nforT= 0:1Jand different values\nof the Gilbert damping \u000b(!p=B= 0:0278Jis the frequency for\ncyclotron motion).\non the (frequency-dependent) effective mass, gyrocoupling,\ndamping and gyrodamping of the skyrmion. Furthermore, the\ncorresponding spectrum of thermal fluctuations is given by\nEq. (7).\nFig. 2 showsh(\u0001R)2it=h(Rx(t0+t)\u0000Rx(t0))2i. As\nexpected, the motion of the skyrmion is diffusive: the mean\nsquared displacement grows for long times linearly in time\nh(\u0001R)2it= 2Dt, whereDis the diffusion constant. Usu-\nally the diffusion constant of a particle grows when the fric-\ntion is lowered26. For the skyrmion the situation is opposite:\nthe diffusion constant becomes small for the small friction,\ni.e., small Gilbert damping \u000b. This surprising observation has\nits origin in the gyrocoupling G: in the absence of friction\nthe skyrmion would be localized on a cyclotron orbit. From\nEq. (1), we obtain\nD=kBT\u000bD\nG2+ (\u000bD)2(9)\nThe diffusion is strongly suppressed by G. As in most materi-\nals\u000bis much smaller than unity while D\u0018G , the skyrmion\nmotion is characterized both by a small diffusion constant\nand a small friction. Such a suppressed dynamics has also\nbeen shown to be important for the dynamics of magnetic\nvortices28. For typical parameters relevant for materials like\nMnSi we estimate that it takes 10\u00006sto10\u00005sfor a skyrmion\nto diffusive over an average length of one skyrmion diameter.\nTo analyze the dynamics on shorter time scales we show in\nFig. 3 four real functions parametrizing G\u00001(!): a frequency-\ndependent mass m(!), gyrocouplingG(!), gyrodamping\n\u000b\u0000(!)and dissipation strength \u000bD(!)with\nG\u00001(!) =\u0012\n\u000bD(!)\u0000i!m(!)\u0000G(!) +i\u000b!\u0000(!)\nG(!)\u0000i!\u000b\u0000(!)\u000bD(!)\u0000i!m(!)\u0013\nFor!!0one obtains the parameters of Eq. (1). All pa-\nrameters depend only weakly on temperature, Gandmare ap-\nproximately independent of \u000b, while the friction coefficients4\n0 1 2 3 4 5\nω / ωp04812 α\nthermal diffusion\n0 1 2 3 4 5\nω / ωp00.51-G / 4 π\n0 1 2 3 4 5\nω / ωp0100200\nα Γ0 1 2 3 4 5\nω / ωp050100\nm\ncurrent driven motion\nforce driven motion\nFIG. 3: Dissipative tensor \u000bD, massm, gyrocouplingGand gyro-\ndamping\u000b\u0000as functions of the frequency !for the diffusive motion\natT= 0:1(solid lines). They differ strongly from the “apparent”\ndynamical coefficients (see text) obtained for the force driven (red\ndashed line) and current driven motion (green dot-dashed line). We\nuse\u000b= 0:2,\f= 0:1. The error bars reflect estimates of systematic\nerrors arising mainly from discretization effects, see appendix B.\n00.05 0.1 0.15 0.2α01234 αT=0.15\nT=0.2\n00.05 0.1 0.15 0.2α00.51-G / 4 π\n00.05 0.1 0.15 0.2α010203040\nα Γ00.05 0.1 0.15 0.2α0255075100\nmT=0.05\nT=0.1\nFIG. 4: Dissipative strength \u000bD, massm, gyrocouplingGand gy-\nrodamping\u000b\u0000as functions of the Gilbert damping \u000bfor different\ntemperatures T.\n\u000bDand\u000b\u0000are linear in \u000b, see Fig. 4. In the limit T!0,\nG(!!0)takes the value\u00004\u0019, fixed by the topology of the\nskyrmion15,20.\nBoth the gyrodamping \u0000and and the effective mass m\nhave huge numerical values. A simple scaling analysis of the\nLandau-Lifshitz-Gilbert equation reveals that both the gyro-\ncouplingGandDare independent of the size of the skyrmion,\nwhile \u0000andmare proportional to the area of the skyrmion,\nand frequencies scale with the inverse area, see appendix\nB. For the chosen parameters (the field dependence is dis-cussed in the appendix B), we find m\u00190:3N\ripm0and\n\u000b\u0000\u0019\u000b0:7N\ripm0, wherem0=~2\nJa2is the mass of a sin-\ngle flipped spin in a ferromagnet ( 1in our units) and we have\nestimated the number of flipped spins, N\rip, from the total\nmagnetization of the skyrmion relative to the ferromagnetic\nbackground. As expected the mass of skyrmions grows with\nthe area (consistent with an estimate29formobtained from the\nmagnon spectrum of skyrmion crystals), the observation that\nthe damping rate \u000bDis independent of the size of skyrmions\nis counter-intuitive. The reason is that larger skyrmions have\nsmoother magnetic configurations, which give rise to less\ndamping. For realistic system parameters J= 1meV (which\nyields a paramagnetic transition temperature TC\u001810K, but\nthere are also materials, i.e. FeGe, where the skyrmion lattice\nphase is stabilised near room-temperature16) anda= 5 ˚A and\na skyrmion radius of 200 ˚A one finds a typical mass scale of\n10\u000025kg.\nThe sign of the gyrodamping \u000b\u0000is opposite to that of the\ngyrocouplingG. This implies that \u000b\u0000describes not damp-\ning but rather antidamping: there is less friction for cyclotron\nmotion of the skyrmion than for the linear motion. The nu-\nmerical value for the antidamping turns out to be so large\nthatDm+ \u0000G<0. This has the profound consequence that\nthe simplified equation of motion shown in Eq. (1) cannot be\nused: it would wrongly predict that some oscillations of the\nskyrmion are not damped, but grow exponentially in time due\nto the strong antidamping. This is, however, a pure artifact\nof ignoring the frequency dependence of G\u00001(!), and such\noscillations do not grow.\nFig. 3 shows that the dynamics of the skyrmion has a strong\nfrequency dependence. We identify the origin of this fre-\nquency dependence with a coupling of the skyrmion coordi-\nnate to pairs of magnon excitations as discussed in Ref. 31.\nMagnon emission sets in for ! > 2!pwhere!p=Bis the\nprecession frequency of spins in the ferromagnet (in the pres-\nence of a bound state with frequency !b, the onset frequency\nis!p+!b, Ref. 31). This new damping channel is most ef-\nficient when the emitted spin waves have a wavelength of the\norder of the skyrmion radius.\nAs a test for this mechanism, we have checked that only\nthis high-frequency damping survives for \u000b!0. In Fig. 5\nwe show the frequency dependent damping \u000bD(!)for various\nbare damping coefficients \u000b. For small!it is proportional to\n\u000bas predicted by the Thiele equation. For !>2!p, however,\nan extra dampling mechanism sets in: the skyrmion motion\ncan be damped by the emission of pairs of spin waves. This\nmechanism is approximately independent of \u000band survives\nin the\u000b!0limit. This leads necessarily to a pronounced\nfrequency dependence of the damping and therefore to the ef-\nfective mass m(!)which is related by the Kramers-Kronig\nrelationm(!) =1\n!R1\n\u00001\u000bD(!0)\n!0\u0000!d!0\n\u0019to\u000bD(!). Note also that\nthe large\u000bindependent mass m(!!0)is directly related to\nthe\u000bindependent damping mechanism for large !. Also the\nfrequency dependence of m(!)andG(!)can be traced back\nto the same mechanism as these quantities can be computed\nfrom\u000bD(!)and\u000b\u0000(!)using Kramers-Kronig relations. For\nlarge frequencies, the effective mass practically vanishes and5\n0 1 2 3 4 5\nω/ωp05101520αD(ω)α=0.05\nα=0.1\nα=0.2\nFIG. 5: Effective damping, \u000bD(!)for\u000b= 0:2,0:1and0:05.\n0 1 2 3 4 5\nω / ωp-400-2000200400Mz\ntot600Re/Im SgRe Sg11\nRe Sg21\nIm Sg11\nIm Sg21\nFIG. 6: Dynamical coupling coefficients for the force driven motion\n(\u000b= 0:2). In the static limit everything but the real part of the diago-\nnal vanishes. R eS11\ng(!)however approaches the total magnetization\nMz\ntotas expected. The error bars reflect estimates of systematic er-\nrors, see appendix B.\nthe ‘gyrocoupling’ Gdrops by a factor of a half.\nIV . FORCE-DRIVEN MOTION\nNext, we study the effects of an oscillating magnetic field\ngradient rBz(t)in the absence of thermal fluctuations. As\nthe skyrmion has a large magnetic moment Mz\ntotrelative to the\nferromagnetic background, the field gradient leads to a force\nacting on the skyrmion. In the static limit, the force is exactly\ngiven by\nFg(!!0) =Mz\ntotrBz: (10)\nUsing G\u00001(!)determined above, we can calculate how the\neffective force Sg(!)rBz(!)(see Eq. 2) depends on fre-\nquency. Fig. 6 shows that for !!0one obtains the expectedresultSg(!!0) =\u000eijMz\ntot, while a strong frequency de-\npendence sets in above the magnon gap, for !&!p. This\nis the precession frequency of spins in the external magnetic\nfield.\nIn general, both the screening of forces (parametrized\nbySg(!)) and the internal dynamics (described by\nG\u00001(!)) determines the response of skyrmions, V(!) =\nG(!)Sg(!)rBz(!). Therefore it is in general not possi-\nble to extract, e.g., the mass of the skyrmion as described by\nG\u00001(!)from just a measurement of the response to field gra-\ndients. It is, however, instructive to ask what “apparent” mass\none obtains, when the frequency dependence of Sg(!)is ig-\nnored. We therefore define the “apparent” dynamics G\u00001\na(!)\nbyGa(!)Sg(!= 0) = G(!)Sg(!). The matrix elements\nofG\u00001\na(!)are shown in Fig. 3 as dashed lines. The appar-\nent mass for gradient-driven motion, for example, turns out\nto be more than a factor of three smaller then the value ob-\ntained from the diffusive motion clearly showing the impor-\ntance of screening effects on external forces. The situation\nis even more dramatic when skyrmions are driven by electric\ncurrents.\nV . CURRENT-DRIVEN MOTION\nCurrents affect the motion of spins both via adiabatic and\nnon-adiabatic spin torques30. Therefore one obtains two types\nof forces on the spin texture even in the static limit19–22,24.\nThe effect of a time-dependent, spin-polarized current on\nthe magnetic texture can be modelled by supplementing the\nright hand side of eq. (3) with a spin torque term TST,\nTST=\u0000(vs\u0001r)Mr+\f\nM[Mr\u0002(vs\u0001r)Mr]:(11)\nThe first term is called the spin-transfer-torque term and is\nderived under the assumption of adiabaticity: the conduction-\nelectrons adjust their spin orientation as they traverse the mag-\nnetic sample such that it points parallel to the local magnetic\nmoment Mrowing toJHandJsd. This assumptions is justi-\nfied as the skyrmions are rather large smooth objects (due to\nthe weakness of spin-orbit coupling). The second so called \f-\nterm describes the dissipative coupling between conduction-\nelectrons and magnetic moments due to non-adiabatic effects.\nBoth\u000band\fare small dimensionless constants in typical ma-\nterials. From the Thiele approach one obtains the force\nFc(!!0) =G\u0002vs+\fDvs: (12)\nFor a Galilei-invariant system one obtains \u000b=\f. In this\nspecial limit, one can easily show that an exact solution of the\nLLG equations in the presence of a time-dependent current,\ndescribed by vs(t)is given by M(r\u0000Rt\n\u00001vs(t0)dt0)pro-\nvided, M(~ r)is a static solution of the LLG equation for vs=\n0. This implies that for \u000b=\f, the skyrmion motion exactly\nfollows the external current, _R(t) =vs(t). Using Eq. (2),\nthis implies that for \u000b=\fone has G\u00001(!) =Sc(!). Defin-\ning the apparent dynamics, as above, Ga(!)Sc(!= 0) =\nG(!)Sc(!)one obtains a frequency independent G\u00001\na(!) =6\n0 1 2 3 4 5\nω / ωp-4 π-10-50β D(0)510Re/Im ScIm Sc11\nIm Sc21Re Sc11\nRe Sc21\nFIG. 7: Dynamical coupling coefficients (symbols) for the current-\ndriven motion ( \u000b= 0:2,\f= 0:1,J= 1,\u0015= 0:18J,B= 0:0278 ).\nThese curves follow almost the corresponding matrix elements of\nG\u00001(!)shown as dashed lines. A deviation of symbols and dashed\nline is only sizable for Re S11\nc.\n0 1 2 3 4 5\nω/ωp-5051015 mα=0.2,β=0\nα=0.2,β=0.1\n0 1 2 3 4 5\nω/ωp0510\nα Γα=0.2,β=0.15\nα=0.2,β=0.19\nα=0.2,β=0.3\nFIG. 8: Mass m(!)and gyrodamping \u000b\u0000(!)as functions of the\ndriving frequency !for the current-driven motion. Note that both M\nand\u0000vanish for\u000b=\f.\nSc(!= 0) =\fD 1\u0000i\u001byG: the apparent effective mass and\ngyrodamping are exactly zero in this limit and the skyrmion\nfollows the current without any retardation. For \u000b6=\f, the\nLLG equations predict a finite apparent mass. Numerically,\nwe find only very small apparent masses, ma\nc/\u000b\u0000\f, see\ndot-dashed line in upper-right panel of Fig. 3, where the case\n\u000b= 0:2,\f= 0:1is shown. This is anticipated from the anal-\nysis of the\u000b=\fcase: As the mass vanishes for \u000b=\f= 0,\nit will be small as long as both \u000band\fare small. Indeed\neven for\u000b6=\fthis relation holds approximately as shown\nin Fig. 7. The only sizable deviation is observed for Re S11\nc\nfor which the Thiele equation predicts Re S11\nc(!!0) =\fD\nwhile Re G\u0000111(!!0) =\u000bDas observed numerically.\nA better way to quantify that the skyrmion follows the cur-rent even for \u000b6=\falmost instantaneously is to calculate\nthe apparent mass and gyrodamping for current driven mo-\ntion, where only results for \u000b= 0:2and\f= 0:1have been\nshown. As these quantities vanish for \u000b=\f, one can ex-\npect that they are proportional to \u000b\u0000\fat least for small \u000b;\f.\nThis is indeed approximately valid at least for small frequen-\ncies as can be seen from Fig. 8. Interestingly, one can even\nobtain negative values for \f > \u000b (without violating causal-\nity). Most importantly, despite the rather large values for \u000b\nand\fused in our analysis, the apparent effective mass and\ngyrodamping remain small compared to the large values ob-\ntained for force-driven motion or the intrinsic dynamics. This\nshows that retardation effects remain tiny when skyrmions are\ncontrolled by currents.\nVI. CONCLUSIONS\nIn conclusion, we have shown that skyrmions in chiral mag-\nnets are characterised by a number of unique dynamical prop-\nerties which are not easily found in other systems. First, their\ndamping is small despite the fact that skyrmions are large\ncomposite objects. Second, despite the small damping, the\ndiffusion constant remains small. Third, despite a huge iner-\ntial mass, skyrmions react almost instantaneously to external\ncurrents. The combination of these three features can become\nthe basis for a very precise control of skyrmions by time-\ndependent currents.\nOur analysis of the skyrmion motion is based on a two-\ndimensional model where only a single magnetic layer was\nconsidered. All qualitative results can, however, easily be\ngeneralized to a film with NLlayers. In this case, all terms\nin Eq. (1) get approximately multiplied by a factor NLwith\nthe exception of the last term, the random force, which is en-\nhanced only by a factorpNL. As a consequence, the diffu-\nsive motion is further suppressed by a factor 1=pNLwhile\nthe current- and force-driven motion are approximately unaf-\nfected.\nAn unexpected feature of the skyrmion motion is the an-\ntidamping arising from the gyrodamping. The presence of\nantidamping is closely related to another important property\nof the system: both the dynamics of the skyrmion and the ef-\nfective forces acting on the skyrmion are strongly frequency\ndependent.\nIn general, in any device based on skyrmions a combination\nof effects will play a role. Thermal fluctuations are always\npresent in room-temperature devices, the shape of the device\nwill exert forces13,14and, finally, we have identified the cur-\nrent as the ideal driving mechanism. In the linear regime, the\ncorresponding forces are additive. The study of non-linear\neffects and the interaction of several skyrmions will be impor-\ntant for the design of logical elements based on skyrmions and\nthis is left for future works. As in our study, we expect that\ndynamical screening will be important in this regime.7\n 30\n 40\n 50\n 60\n 70  30 40 50 60 70 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004\nFIG. 9: Skyrmion density based on the normalized z-component of\nthe magnetization.\nAcknowledgments\nThe authors are greatful for insightful discussions with K.\nEverschor and Markus Garst. Part of this work was funded\nthrough the Institutional Strategy of the University of Cologne\nwithin the German Excellence Initiative” and the BCGS. C.S.\nthanks the University of Tokyo for hospitality during his re-\nsearch internship where part of this work has been performed.\nN.N. was supported by Grant-in-Aids for Scientific Research\n(No. 24224009) from the Ministry of Education, Culture,\nSports, Science and Technology (MEXT) of Japan, and by the\nStrategic International Cooperative Program (Joint Research\nType) from Japan Science and Technology Agency. J.I. is sup-\nported by Grant-in-Aids for JSPS Fellows (No. 2610547).\nAppendix A: Definition of the Skyrmion’s centre coordinate\nIn order to calculate the Green’s function, Eq. (3), one\nneeds to calculate the velocity-velocity correlation function.\nTherefore it is necessary to track the skyrmion position\nthroughout the simulation. Mostly two methods have been\nused so far for this25: (i) tracking the centre of the topological\ncharge and (ii) tracking the core of the Skyrmion (reversal of\nmagnetization).\nThe topological charge density\n\u001atop(r) =1\n4\u0019^ n(r)\u0001(@x^ n(r)\u0002@y^ n(r)) (A1)\nintegrates to the number of Skyrmions in the system. There-\nfore for our case of a single Skyrmion in the ferromagnetic\nbackground this quantity is normalized to 1. The center of\ntopological charge can therefore be defined as\nR=Z\nd2r\u001atop(r)r (A2)\nFor the case of finite temperature this method can, however,\nnot be used directly. Thermal fluctuations in the ferromagnetic\nbackground far away from the skyrmion lead to a large noise\nto this quantity which diverges in the thermodynamic limit.\nA similar problem arises when tracking the center using the\nmagnetization of the skyrmion.One therefore needs a method which focuses only on the\nregion close to the skyrmion center. To locate the skyrmion,\nwe use thez-component of the magnetization but take into ac-\ncount only points where Mz(r)<\u00000:7(the magnetization of\nthe ferromagnetic background at T= 0is+1). We therefore\nuse\n\u001a(r) = (1\u0000Mz(r)) \u0002[\u0000Mz(r)\u00000:7] (A3)\nwhere \u0002[x]is the theta function. A first estimate, Rest=RV,\nfor the radius is obtained from\nRA=R\nAr\u001a(r)d2rR\nA\u001a(r)d2r(A4)\nby integrating over the full sample volume V.Restis noisy\ndue to the problems mentioned above but for the system\nsizes simulated one nevertheless obtains a good first esti-\nmate for the skyrmion position. This estimate is refined by\nusing in a second step for the integration area only D=\b\nr2R2jjr\u0000Restj<r\t\nwhereris choosen to be larger\nthan the radius of the skyrmion core (we use r= 1:3p\nN<=\u0019,\nwhereN<is the number of spins with Mz<\u00000:7). Thus\nwe obtain a reliable estimate, R=RD, not affected by spin\nfluctuations far away from the skyrmion. From the resulting\nR(t), one can obtain the velocity-velocity correlation function\nhVi(t0+t)VJ(t0)i.\nAppendix B: Scaling invariance\nTo obtain analytic insight into the question of how param-\neters depend on system size and to check the numerics for\nlattice artifacts it is useful to perform a scaling analysis of the\nsLLG equations and the effective equations of motion for the\nskyrmion.\nWe investigate a scaling transformation, where the radius\nof the skyrmion is enlarged by a factor \u0011,M(r)!~M(r) =\nM(r=\u0011).\nFor the scaling analysis, we use the continuum limit of\nEq. (4) (setting a= 1)\nH[M] =Z\nd2r\u0014J\n2(rM)2+\u0015M\u0001r\u0002 M\u0000B\u0001M\u0015\nThe three terms scale with \u00110,\u0011and\u00112, respectively. To ob-\ntain a larger skyrmion, we therefore have to rescale \u0015!\u0015=\u0011\nandB!B=\u00112. This implies that the Be\u000bterm in the sLLG\nequation scales with 1=\u00112and therefore also the time axis has\nto be rescaled, t!\u00112t, implying that all time scales are a\nfactor of\u00112longer and all frequencies a factor 1=\u00112smaller.\nSimiliarly, the driving current, i.e. vsis reduced by a factor\n\u0011\u00001while the temperature remains unscaled. This implies that\nwhen M(r;t)is a solution for a given value of \u0015,Bandvs\nandG(!)the corresponding velocity-correlation function of\nthe skyrmion, then M(r=\u0011;t=\u00112)is a solution for \u0015=\u0011,B=\u00112,\nvs=\u0011with correlation function G(!\u00112).\nAccordingly, the !!0limit and therefore the gyrocou-\nplingG, the friction constant \u000bDand the diffusion constant of8\n0.03 0.035 0.04\nBz00.511.5 α D\n0.03 0.035 0.04\nBz00.51-G / 4 π\n0.03 0.035 0.04\nBz05101520\nα Γ0.03 0.035 0.04\nBz020406080\nm\nFIG. 10: Field dependence of the dissipative tensor \u000bD, the massm,\nthe gyrocouplingGand the gyrodamping \u000b\u0000forJ= 1,\u0015= 0:18J.\nThe main effect is that both mand\u000b\u0000shrink when the size of the\nskyrmion shrinks with increasing Bz.\nthe skyrmion are independent of\u0011, consistent with the analyt-\nical formulas eq. (6). In contrast, the mass of the skyrmion,\nm, and the gyrodamping \u000b\u0000scale with\u00112. They are therefore\nproportional to the number of spins constituting the skyrmion\nconsistent with our numerical findings. When one does, how-\never, change the external magnetic field for fixed strength\u0015\nof the DM interaction, the internal structure of the skyrmion\nand therefore also G(!)change quantitatively in a way not\npredictable by scaling. In Fig. 10 we therefore show the B-dependence of \u000bD,m,Gand\u000b\u0000. For increasing Bthe size of\nthe skyrmion shrinks. From our previous analysis, it is there-\nfore not surprising that both the effective mass mand the gy-\nrodamping\u000b\u0000shrink substantially while \u000bDandGare less\naffected . Quantitatively, this change of mand\u000b\u0000is how-\never not proportional to the number of spins constituting the\nskyrmion.\nWe have tested numerically the scaling properties for \u0011=\n1:2and find that all features are quantitatively reproduced.\nSmall variations on the level of a few percent do, however,\noccur reflecting the typical size of features arising from the\ndiscretization of the continuum theory. A conservative esti-\nmate of such systematic discretization effects for the diffusive\nmotion is given by the error bars in Figs. 3 (all statistical er-\nrors are smaller than the thickness of the line). For the field-\ndriven motion (Fig. 3 and Fig. 6) spatial discretization effects\nlead to a different source of errors. For very small field gradi-\nents and high frequencies the displacement of the skyrmion is\nmuch smaller than the lattice spacing and the response is af-\nfected by a tiny pinning of the skyrmion to the discreet lattice.\nFor larger gradients, however, nonlinear effects set in and for\nsmall frequencies the skyrmion starts to approach the edge of\nthe simulated area. In Fig. 3, we therefore used for the force-\ndriven motionrB= 0:0005 for!<2!pandrB= 0:0015\nfor! >2!p. Error bars have been estimated from variations\nof the numerical values when rBwas varied from 0:0001 to\n0:0015 . For the current-driven motion errors are so tiny that\nthey are not shown.\n\u0003Electronic address: nagaosa@riken.jp\n1Hubert, A. & Sch ¨afer, R. Magnetic Domains: The Analysis of\nMagnetic Microstructures (Springer, Berlin, 1998).\n2Malozemoff, A. P. & Slonczewski, J.C. Magnetic Domain Walls\nin Bubble Materials (Academic Press, New York, 1979).\n3Skyrme, T. H. R. A Non-Linear Field Theory. Proc. Roy. 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Magnon-skyrmion scattering in chiral\nmagnets. arxiv 1405.1568 (2014)."    },    {        "title": "1805.06535v3.Stabilization_rates_for_the_damped_wave_equation_with_Hölder_regular_damping.pdf",        "content": "arXiv:1805.06535v3  [math.AP]  28 Nov 2018Stabilization rates for the damped wave equation with\nH¨ older-regular damping\nPerry Kleinhenz\nAbstract\nWe study the decay rate of the energy of solutions to the damped w ave equation\nin a setup where the geometric control condition is violated. We cons ider damping\ncoefficients which are 0 on a strip and vanish like polynomials, xβ. We prove that the\nsemigroup cannot be stable at rate faster than 1 /t(β+2)/(β+3)by producing quasimodes\nof the associated stationary damped wave equation. We also prove that the semigroup\nis stable at rate at least as fast as 1 /t(β+2)/(β+4). These tworesults establish an explicit\nrelation between the rate of vanishing of the damping and rate of de cay of solutions.\nOur result partially generalizes a decay result of Nonnemacher in whic h the damping\nis an indicator function on a strip.\n1 Introduction\nLetM= (M,g) be a Riemannian manifold. Fix some W∈L∞(M),W≥0. We study the\nasymptotic behavior as t→ ∞of solutions to the damped wave equation\n\n\n∂2\ntu−∆u+W(x)∂tu= 0 inM×R+\n(u,∂tu)|t=0= (u0,u1) inM\nu|∂M= 0 if ∂M/ne}ationslash=∅.(1)\nThe quantity of particular interest is the energy\nE(u,t) =1\n2/parenleftBig\n||∇u(·,t)||2\nL2+||∂tu(·,t)||2\nL2/parenrightBig\n. (2)\nIn this paper we will work on the square M= [−b,b]×[−b,b] or torusM=T2, which we\nparametrize by ( x,y). We will give detailed proofs in the case of the square then s how how\nthose results can be applied to the torus.\nFor some fixed β≥0 anda,σ>0, such that a+σ<b, we study damping W∈L∞(M)\nof the form\nW(x,y) =\n\n0 |x|<a\n(|x|−a)βa<|x|<a+σ\nc(|x|)>0a+σ<|x|<b,(3)\n1In particular note that the damping is invariant in the ydirection.\nRemark. The particular form of c(x) does not affect our result so long as c(x)∈\nL∞(a+σ,b) and it is uniformly bounded away from 0. However choosing a csuch thatW\nis smooth for |x|>aandW=C >0 for|x|>a+2σis perhaps the most interesting case\nin the context of existing results.\nDefinition 1. Letf(t) be a function such that f(t)→0 ast→ ∞. We say that (1) is\nstable at rate f(t) if there exists a constant C >0 such that for all ( u0,u1)∈(H2(M)∩\nH1\n0(M))×H1\n0(M) (orH2(M)×H1(M) if∂M=∅) ifusolves (1) with ( u0,u1) as Cauchy\ndata then\nE(u,t)≤Cf(t)2/parenleftBig\n||u0||2\nH2(M)+||u1||2\nH1(M)/parenrightBig\nfor allt>0.\nOur main result is\nTheorem 1.1. For allε>0, withWas in(3)the equation (1)cannot be stable at rate\nt−β+2\nβ+3−ε.\nMore precisely if we use m1(t) to denote the best possible ffor which definition 1 holds\nthen this result along with Lemma 4.6 of [AL+14] and Proposition 3 of [BD08] show that\nm1(t)≥C/(1+t)β+2\nβ+3for someC >0, where we use the notation of [BD08].\nWe also show that\nTheorem 1.2. ForWas in(3)withβ >0the equation (1)is stable at rate\nt−β+2\nβ+4.\nAgain using Lemma 4.6 of [AL+14] and Proposition 3 of [BD08] this shows m1(t)≤\nC/(1+t)β+2\nβ+4for someC >0.\nThe decay of energy of the damped wave equation is a well studi ed question. The\nstrongestpossibledecay isuniformstabilization, whichi sdefinedastheexistence of F(t)→\n0 ast→ ∞such that, for all usolving (1) with initial data in H1(M)×L2(M)\nE(u,t)≤F(t)E(u,0), t≥0.\nIt was established by [RT75] that uniform stabilization occ urs withF=Ce−κtfor\nsomeκ,C >0, when∂M=∅,W∈C0(M) and supp Wsatisfies the geometric control\ncondition (GCC). We recall that a set Usatisfies the GCC if there exists T >0 such that\nfor every geodesic γonMof lengthT,γ∩U/ne}ationslash=∅. The reverse implication, that uniform\nstabilization with any Fimplies supp Wsatisfies the GCC, was shown in [Ral69]. These\nresults were extended to the case M/ne}ationslash=∅by [CBR92] and [BG97]. This in turn guarantees\nthat when uniform stabilization occurs one can always let F=Ce−κtfor someκ,C >0.\nFor a finer discussion of when uniform stabilization occurs f orL∞damping see [BG18].\n2A natural next question to ask is what occurs when the GCC does not hold for supp W.\nBecause of the necessity of the GCC for uniform stabilizatio n, as soon as it does not hold\nwe must adjustthe kindof decay wehopefor. Thenextbest thin gis thestability definedin\nDefinition 1, which comes from [Leb96] and requires initial d ata with an additional spatial\nderivative.\nIn [Bur98], the author showed that the energy of a solution to (1) decays at least\nlogarithmically ( f(t) = 1/log(2 +t)) as soon as the damping W(x)≥c >0 on some\nopen, nonempty set. In [Leb96] the author gave explicit exam ples of domains on which\nf(t) = 1/log(2+t) is the exact decay rate, in particular when M=S2and{W >0}does\nnot intersect a neighborhood of the equator. For related wor k see also [LR97].\nIn the case of the square when the damped region contains a ver tical strip, [LR05]\nestablished a decay rate of f(t) = (log(t)/t)1/2. This was expanded to the case of partially\nrectangular domains when {W >0}contains a neighborhood of the nonrectangular part\nin [BH07]. Additionally in [BH07] a relation between vanish ing rate of the damping and\ndecay rate for the damped wave equation was established.\nThese results were refined by [AL+14]. The authors established a decay rate of f(t) =\n1/t1/2for the damped wave equation in a more general setting, so lon g as the associated\nSchr¨ odinger equation is controllable. This includes the c ase of not identically vanishing\ndamping on the 2 dimensional square (or torus) as a consequen ce of [Jaf90] (resp. [Mac10],\n[BZ12]).\nContinuing in the case of the 2 dimensional square [AL+14] also show that the system\ncan not be stable at rate f(t) = 1/t1+εfor anyε>0, when{W >0}does not satisfy GCC,\n(this condition is referred to as the GCC being strongly viol ated). They further show the\nexistence of a smooth damping coefficient which strongly viol ates the GCC for which the\nenergy decays at rate f(t) = 1/t1−εfor anyε>0.\nIn an appendix to [AL+14], Nonnenmacher shows that when the damping is the in-\ndicator function on a strip on the square or torus the system c annot be stable at rate\nf(t) = 1/t2/3+ε, for anyε>0. The complementary result was shown in [Sta17] to estab-\nlishf(t) = 1/t2/3as the exact rate of decay when damping is a strip on the square or torus.\nThe difference in behavior between the smooth and discontinuo us damping led the authors\nof [AL+14] to pose the problem of establishing an explicit relation between the vanishing\nrate of the damping and the decay rate.\nAnexplicitrelation wasestablishedby[LL17]inaslightly differentsetting. Theauthors\nstudy the case of a general manifold in which the damping is su pported everywhere but a\nflat subtorus. In the 2 dimensional case this is an example of t he GCC not holding but also\nnot being strongly violated. The damping is required to be in variant along this subtorus\nand must satisfy W(x)≤C|x|βnear where it vanishes. When this is the case the authors\nshow that the system cannot be stable at rate f(t) = 1/tβ+2\nβ+ε, for anyε >0. They also\nshow that if the vanishing behavior of the damping is further limited toC−1\n1|x|β≤W(x)≤\nC1|x|βthe system is stable at exactly the rate f(t) = 1/tβ+2\nβ(See also [BZ15]).\n3Note that in [LL17] decreasing βcorrespondsto faster vanishing(i.e. less regular damp-\ning) which produces faster decay, which is counter to the beh avior exhibited in [AL+14],\n[BH07] and our own result, namely that faster vanishing (i.e . less regular damping) pro-\nduces slower decay. However the dynamics in 2 dimensions in t he two situations are\ndifferent, with only one undamped orbit in the former as oppose d to a whole family in the\nlatter.\nOur paper provides a partial answer to the problem posed by [A L+14]. We establish an\nexplicit relation between the rate of vanishing of the dampi ng and the stability rate of the\nsystem, in a case where the GCC is strongly violated on the squ are orT2. Our work also\npartially extends that of Nonnenmacher in the appendix to [A L+14], which agrees with\nour first theorem when β= 0, although that result follows from the existence of modes\nof the stationary equation where we produce quasimodes. Our two results provide further\nevidence for the fact, discussed in [AL+14], [LL17], [BH07], that once the support of the\ndamping is fixed the rate of vanishing of the damping is the mos t significant feature when\ndetermining the decay rate.\nWe note that our second result improves that of Theorem 1.2 of [BH07], which gives a\ndecay rate of f(t) = 1/tβ/(β+4)(see also [AL+14] Theorem 2.6). We also note that there\nis a gap between our two results. Closing this gap would be an i nteresting area for further\nwork.\nIn the next section we outline the proof of Theorem 1.1. Secti ons 3, 4 and 5 contain\nthe details of the proof. Section 6 contains the proof of Theo rem 1.2.\nAcknowledgements The author would like to thank Jared Wunsch for his advice\nand guidance. The author would also like to thank Matthieu L´ eautaud for comments on\nan early manuscript. The author would also like to thank the r eferees for their prompt,\ndetailed and constructive comments. This research was supp orted in part by the National\nScience Foundation grant ”RTG: Analysis on manifolds” at No rthwestern University.\n2 Outline of Proof of Theorem 1.1\nTo prove Theorem 1.1 we rely on the following result from [AL+14] (Proposition 2.4) and\n[BT10] which relates energy decay of the damped wave equatio n to resolvent estimates of\nthestationary dampedwave equation. Withthisresultitiss ufficienttoproducesufficiently\ngood quasimodes, to do so we reduce the problem to one dimensi on and then the interval\n[0,b]. We will then show that we can use solutions to a related prob lem, the complex\nabsorbing potential, on the half line to produce the desired quasimodes. We finally show\nthat suchsolutionsof thecomplex absorbingpotential prob lemexistbyproducingsolutions\non (0,a) and (a,∞) separately, the latter following from a rescaling argumen t, we are able\nto glue these solutions together via a compatibility condit ion which we satisfy via the\nimplicit function theorem.\n4Proposition 2.1. Fixα, if there exist sequences {qj} ∈C,{uj} ∈H2(M)∩H1\n0(M),(or\nH2(M)if∂M=∅) and some j0∈N, such that for all j >|j0|\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle−∆uj+iqjW(x)uj−q2\njuj/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(M)≤C\n|Re(qj)|2/α/parenleftBig\n||uj||2\nH1(M)+|qj|2||uj||2\nL2(M)/parenrightBig\n,(4)\nand\n|qj| → ∞,|Im(qj)| ≤C\n|Re(qj)|1/α, (5)\nthen for all ε>0the system (1)is not stable at rate\n1/tα+ε.\nRemark. Although Proposition 2.1 has ||uj||2\nH1on the right hand side of (4) the\nquasimodes ujwe will apply it to satisfy a stronger inequality, namely\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle−∆uj+iqjW(x)uj−q2\njuj/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(M)≤C\n|Re(qj)|2||uj||2\nL2(M). (6)\nThestrengthoftheconclusionweobtainfromapplyingPropo sition2.1tothesequasimodes\nis instead limited by the qjfor which we have such an estimate due to (5).\nNotethatproducingsequences qjandujwhichsatisfythehypothesesofthisproposition\nwithα=β+2\nβ+3proves Theorem 1.1.\nWe will make two simplifications before proceeding. First we will reducethe problem to\nobtaining quasimodes of an ordinary differential equation on [−b,b]. We will then further\nrestrict our attention to the same equation on [0 ,b]. After making these simplifications\nwe will introduce three key parameters and the complex absor bing potential problem on\n(0,∞), solutions of which we will use to produce our desired quasi modes.\nFor the first simplification note that for any sequence of inte gersmjif/tildewideujis a sequence\nof functions on [ −b,b] which satisfy\n\n\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−∂2\nx/tildewideuj+iqjW/tildewideuj+/parenleftbigg\n4π2m2\nj\nb2−q2\nj/parenrightbigg\n/tildewideuj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(−b,b)≤C\n|Re(qj)|2||/tildewideuj||2\nL2(−b,b)asj→ ∞\n/tildewideuj(x) = 0|x|=b,\n(7)\nthenuj(x,y) =/tildewideuj(x)sin/parenleftBig2πmjy\nb/parenrightBig\nsatisfy (6). Therefore it is enough for us to find functions\nwhich satisfy (7) with qjwhich satisfy (5) with α=β+2\nβ+3.\nThe second simplification we make is to limit our attention to [0,b] from [−b,b]. Since\nour damping is even, if we find integers mjand functions /tildewideujon [0,b] which satisfy\n\n\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−∂2\nx/tildewideuj+iqjW/tildewideuj+/parenleftbigg\n4π2m2\nj\nb2−q2\nj/parenrightbigg\n/tildewideuj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,b)≤C\n|Re(qj)|2||/tildewideuj||2\nL2(0,b)asj→ ∞\n/tildewideuj(x) = 0x=b\n/tildewideuj(0) = 0 or/tildewideu′\nj(0) = 0(8)\n5we can extend the /tildewideujto−b≤x<0 by setting/tildewideuj(−x) =−/tildewideuj(x) (or/tildewideuj(−x) =/tildewideuj(x) resp.)\nand the resulting functions satisfy (7). Therefore it is eno ugh for us to find functions which\nsatisfy (8) with qjwhich satisfy (5) with α=β+2\nβ+3.\nBefore we introduce the complex absorbing potential we intr oduce three new parame-\nters. Leth∈[0,1) be a small parameter which will be sent to 0 and haveb\n2πh2∈N. Let\nlbe a bounded parameter, when working in the case u(0) = 0 we impose l∈Zand when\nu′(0) = 0 we impose l+1\n2∈Zbut otherwise leave lfree in relation to h. We define\nλh=πlh\na+Chh(β+4)/(β+2)Ch=Oh(1)∈C. (9)\nWe will eventually specify Chmore completely in Section 4. When appropriate we will\nrefer to sequences of these parameters as hj,λhj,ljandChj, where we emphasize that λhj\nandChjdepend onhj.\nNow we introduce the complex absorbing potential problem on (0,∞)\n/braceleftBigg\n0 =−h2∂2\nxv+i(x−a)β\n+v−λ2\nhv\nv(0) = 0 orv′(0) = 0.(10)\nIn order to relate this to (8) we make an ansatz for relations b etween the parameters.\nIfvjare a sequence of solutions of (10) for some hj,lj,λhj,Chj, we define qjandmjas\nfollows\nmj=b\n2πh2\nj∈N (11)\nqj=1\nh2\nj+λ2\nhj\n2=1\nh2\nj+π2l2\njh2\nj\na2+2Chjπlj\nah(2β+6)/(β+2)\nj +C2\nhjh(2β+8)/(β+2)\nj .\nNote that in this regime\nRe(qj) =1\nh2\nj+O(h2\nj) Im(qj) =2Im(Chj)πlj\nah(2β+6)/(β+2)\nj +O(h(2β+8)/(β+2)\nj ),\nso\nqj→ ∞and|Im(qj)| ≤C\n|Re(qj)|(β+3)/(β+2)asj→ ∞.\nAs we will see shortly, solutions of (10) in this regime satis fy the inequality in (8) but\nnot necessarily the boundary condition at x=b. In order to ensure they do we multiply\nthese solutions by a cutoff function which is 0 in a neighborho od ofb. We will see the\nresulting functions still satisfy the inequality in (8) as t he solutions of (10) in this regime\nhave rapid decay on the support of the potential (see Lemma 3. 1), which is exactly where\nerrors introduced by the cutoff function appear.\n6Remark We note also that it is because we are working with solutions t o (10) on the\nhalf-line that we will only produce quasimodes rather than r eal modes. It is necessary for\nus to work on the half-line in order to perform a rescaling tha t allows us to set h= 0, if we\nwere working on a finite interval the rescaling would make the interval depend on hwhich\nour approach is not well adapted to address.\nFixδ>0 such that a+σ<b−2δ; we define φ∈C∞(0,∞) to satisfy\nφ(x) =/braceleftBigg\n1x<b−2δ\n0x>b−δ(12)\nProposition 2.2. FixM >0, let{vj} ∈H2(0,∞)be a sequence of solutions of (10)with\neigenvalues\nλhj=πljhj\na+Chjh(β+4)/(β+2)\nj, Chj=O(1)∈C,\nwhere|lj| ≤Mandhj→0asj→ ∞andb\n2πh2\nj∈N. Set\nuj(x) =φ(x)vj(x).\nThen forjlarge enough so that hj<σβ/2the functions ujwithqj,mjas defined in (11)\nsatisfy(8)and(5).\nIt remains to be seen that we can indeed find solutions to the co mplex absorbing\npotential problem with eigenvalues of this form.\nTheorem 2.3. For alll∈Z,(orl+1\n2∈Z), there exists h0>0andK >0such that for\nallh∈(0,h0), there exists a Chwith|Ch|<Kandv∈H2(0,∞)∩H1\n0(0,∞),v/ne}ationslash= 0(resp.\nH2(0,∞)) satisfying (10)withv(0) = 0(resp.v′(0) = 0) withλgiven by (9).\nUsing Theorem 2.3 we obtain a sequence {vj}of solutions of (10) which satisfy the hy-\npotheses of Proposition 2.2 which in turn produces sequence s which satisfy the hypotheses\nof Propositions 2.1 which in turn proves Theorem 1.1.\nRemark. It is straightforward to extend these results to the case M=T2. We\nparametrize T2by [−b,b]×[−b,b] with parallel edges identified. Thus it is enough to show\nthat the quasimodes we produced on the square satisfy period ic boundary conditions and\nare thus functions on the torus. Our quasimodes are of the for m\nu(x,y) =vj(x)φ(x)sin/parenleftbigg2πmjy\nb/parenrightbigg\n,\nso it is straightforward to see they satisfy periodic bounda ry conditions in yandx(as\nu(x,y)≡0 for|x−b|<δand|b+x|<δ).\nWe will prove Proposition 2.2 in Section 3, we will then prove Theorem 2.3 in Section\n4. We prove a necessary estimate in Section 5. We finally prove Theorem 1.2 in Section 6.\n73 Proof of Proposition 2.2\nWe begin by stating an estimate necessary for the proof.\nLemma 3.1. Letv∈H1(0,∞)be a solution of (10)with eigenvalue λ=O(h)and letφ\nbe as in (12). Fixs∈Rthen forh<σβ/2for allNthere exists CN,s>0such that\n||φv||2\nHs\nh(a+σ,b)≤CN,shN||φv||2\nL2(0,b). (13)\nThis will be proved in Section 5 using the semiclassical elli pticity of −h2∂2\nx+i(x−a)β\n+−\nλ2\nhon (a+σ/4,b).\nProof of Proposition 2.2. We have a sequence vjof solutions of\n0 =−h2\nj∂2\nxvj+i(x−a)β\n+vj−λ2\nhjvj, x∈(0,∞),\nwith\nλhj=πlhj\na+O/parenleftBig\nh(β+4)/(β+2)\nj/parenrightBig\n, hj→0 asj→ ∞.\nIt is clear that uj=φvjhasuj(b) =φ(b)vj(b) = 0. Recalling (11) and the subsequent\ndiscussionqj,mjsatisfy (5). It remains to be seen that ujsatisfies the inequality in (8).\nBy (11) and (10) ujsatisfies\n−∂2\nxuj+iqjW(x)uj+/parenleftBigg\n4π2m2\nj\nb2−q2\nj/parenrightBigg\nuj\n=φ/parenleftBigg\niλ2\nhj\n2(x−a)β\n+vj−λ4\nhj\n4vj/parenrightBigg\n−φ′′vj−2φ′v′\nj+iqj/parenleftBig\nW(x)−(x−a)β\n+/parenrightBig\nφvj.\nThus\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−∂2\nxuj+iqjW(x)uj+/parenleftBigg\n4π2m2\nj\na2−q2\nj/parenrightBigg\nuj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,b)≤λ4\nhj\n4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(x−a)β\n+uj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,b)+λ8\nhj\n16||uj||2\nL2(0,b)+/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ′′vj/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,b)\n+4/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ′v′\nj/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,b)+|qj|2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(W−(x−a)β\n+)φvj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,b).\nSinceb−δ>a+σ, by Lemma 3.1 for any N >0\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ′′vj/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,b)+/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ′v′\nj/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,b)≤C\nh2||φvj||H2\nh(b−δ,b)≤CNhN||φvj||2\nL2(0,b).\nFurthermore by Lemma 3.1 for any N >0\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(W−(x−a)β\n+)φvj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,b)≤C||φvj||2\nL2(a+σ,b)dx≤CNhN||φvj||2\nL2(0,b).\n8Therefore\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−∂2\nxuj+iqj(x−a)β\n+uj+/parenleftBigg\n4π2m2\nj\nb2−q2\nj/parenrightBigg\nuj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,b)≤Cλ4\nhj\n4||uj||2\nL2(0,b)+λ8\nhj\n16||uj||2\nL2(0,b)+CNhN||uj||2\nL2(0,b)\n≤C\n|Re(qj)|2||uj||2\nL2(0,b).\nWhere we used that |Re(qj)|= 1/h2\nj+O(h2\nj) andλhj=O(hj).\nWe now show that there are solutions of (10) with the desired e igenvalues.\n4 Proof of Theorem 2.3\nFrom this point on we focus on solutions of (10) on (0 ,∞). We begin by considering the\ncases when v(0) = 0 orv′(0) = 0, but focus on the case v(0) = 0 for the bulk of the section.\nWe then explain how the proof changes for v′(0) = 0.\nIn order to produce solutions to (10) with the desired eigenv alues we will solve it on\n(0,a) and (a,∞) separately. That is given solutions vl,vr∈H2of (10) on (0 ,a) and (a,∞)\nrespectively, with the same values of λhandh, if there exists B∈Csuch that\n/braceleftBigg\nvl(a) =Bvr(a)\nv′\nl(a) =Bv′\nr(a),(14)\nthen\nv=/braceleftBigg\nvl(x)x<a\nBvr(x)a<x,\nsolves (10) on (0 ,∞) with the same λhandhandv∈H2(0,∞)∩H1\n0(0,∞) (orH2(0,∞)\nifv′\nl(0) = 0) . We will refer to equations (14) as the compatibility condition.\nWe will explicitly solve (10) on (0 ,a). We will then use a rescaling of the equation on\n(a,∞) and the implicit function theorem to show that the compatib ility condition can be\nsatisfied when λhis of the form (9) and his small enough.\nOn (0,a) (10) is solved by\nvl(x) =eiλh(x−a)/h+Ref(λh,h)e−iλh(x−a)/h,\nwhere we choose Ref( λh,h) to ensure the boundary condition at 0 is satisfied. That is\nRef(λh,h) =/braceleftBigg\n−e−2iλha/hv(0) = 0,\ne−2iλha/hv′(0) = 0.\nWe will work now specify to the case where v(0) = 0 and work through it in detail and\nthen summarize how the proof changes for v′(0) = 0.\n9We now rescale the equation on ( a,∞). IfFsolves\n/braceleftBigg\n0 =−F′′(x)+/parenleftBig\nixβ−λ2\nh\nh2β/(β+2)/parenrightBig\nF(x)x∈(0,∞)\nF′(0) = 1,(15)\nthenvr(x) =F(h−2/(β+2)(x−a)) solves (10) on ( a,∞) withv′\nr(a) =h−2/(β+2). This follows\nimmediately from the definition of Fand (10).\nRemark. This rescaling is necessary; in order to show the compatibil ity condition can\nbe satisfied for all hin a neighborhood of 0 we will apply the implicit function the orem and\nso must be able to set h= 0. This can not be done in a satisfactory way with solutions o f\n(10); however for λhof the form (9), (15) is well defined at h= 0 as\nλ2\nh\nh2β/(β+2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nh=0=1\nh2β/(β+2)Ch2+O(h(2β+6)/(β+2))/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nh=0= 0.\nBefore we apply the implicit function theorem we first establ ish an inequality for u∈\nH1(0,∞) using some facts about the Neumann spectrum of −∂2\nx+xβon (0,∞). We then\nuse this inequality to establish the existence and uniquene ss ofH2(0,∞) solutions to (15)\nand to show the boundary value F(0) is bounded away from 0 uniformly in the spectral\nparameter. We then explain how we can use the implicit functi on theorem to satisfy the\ncompatibility condition and make use of these properties of Fto do so.\nLetσN(−∂2\nx+xβ) be the spectrum of −∂2\nx+xβon (0,∞) with Neumann boundary\nconditions, and let\n/tildewiderλ1= infσN(−∂2\nx+xβ).\nLemma 4.1.\n/tildewiderλ1>0\nand\n/tildewiderλ1||u||2\nL2(0,∞)≤/vextendsingle/vextendsingle/vextendsingle/vextendsingleu′/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,∞)+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglexβ/2u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,∞), (16)\nfor allu∈H1(0,∞).\nProof.The Neumann spectrum is discrete (this follows for instance from [HS12] Theorems\n5.10 and 10.7) so we know that /tildewiderλ1is the lowest eigenvalue of the operator. We also know\nthat the spectrum doesn’t contain 0, since\n||∂xu||2\nL2(0,∞)+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglexβ/2u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,∞)>0,\nfor nontrivial u∈H1. Thus/tildewiderλ1>0.\nThe inequality follows immediately from the variational pr inciple for the spectrum of\nself adjoint operators (see [HS12] Corollary 12.2).\n10Lemma 4.2. For any |η|</tildewiderλ1, there exists a unique H2(0,∞)solution of\n/braceleftBigg\n0 =−F′′(x)+ixβF(x)−ηF(x)\nF′(0) = 1.(17)\nFurthermore if we let F(0,η)be the value of this function at x= 0there exists C >0such\nthat for all |η| ≤/tildewiderλ1\n2\n1/C≤ |F(0,η)| ≤C.\nandF(0,η)is holomorphic in ηon that same neighborhood.\nProof.We first show the existence of a solution. Let ψ∈C∞\n0(0,∞) withψ′(0) = 1,ψ(0) =\n0. DefineQ(η,ψ) as\nQ(η,ψ) :=ψ′′−ixβψ+ηψ.\nNow letJsolve /braceleftBigg\n−J′′+ixβJ−ηJ=Q\nJ′(0) = 0,(18)\nand note that F=ψ+Jsolves (17). We will apply the Lax-Milgram theorem to show th e\nexistence of solutions to (18).\nLet\nH=H1(0,∞)∩x−β/2L2(0,∞),\nand define the norm\n||u||2\nH=||u||2\nH1(0,∞)+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglexβ/2u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2(0,∞),\nnoting that His a Hilbert space with this norm.\nWe define the sesquilinear form B:H×H→C\nB[u,v] =ˆ∞\n0u′¯v′+ixβu¯v−ηu¯v.\nFor anyu,v∈H\n|B[u,v]| ≤ˆ\n|u′||v′|+xβ|u||v|+|η||u||v| ≤C||u||H||v||H.\nFurthermore for u∈H⊂H1using (16)\n|B[u,u]| ≥ˆ\n|u′|2+xβ|u|2−|η||u|2dx≥/parenleftbigg\n1−|η|\n/tildewiderλ1/parenrightbiggˆ\n|u′|2+xβ|u|2dx≥C||u||2\nH.\nTherefore by Lax-Milgram for any Q∈Hthere exists a unique J∈Hsuch that\nB[J,v] =ˆ\nQ¯vdx,\n11for allv∈H.\nTherefore there exists an F∈Hsolving (17) given by F=J+ψ.\nNow to show that F(0,η) is holomorphic in ηwe restate the result of our application\nof Lax-Milgram. We have shown that when |η|</tildewiderλ1, for allQ∈Hthere exists a unique\nJ∈Hsuch that /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(−∂2\nx+ixβ−η)−1Q/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nH=||J||H≤C||Q||H.\nTherefore( −∂2\nx+ixβ−η)isbijective withaboundedinversefor |η|</tildewiderλ1. Thustheresolvent\n(−∂2\nx+ixβ−η)−1exists for |η|</tildewiderλ1and by [HS12] Theorem 1.2 it is holomorphic in η\nthere as well.\nNow recall that the trace operator T:H1(0,∞)→Ris linear and continuous on H.\nThusTJ=J(0,η) is also holomorphic in η. Recall that F(0,η) =J(0,η)+ψ(0) =J(0,η),\nsoF(0,η) is holomorphic in η.\nTo see that Fis unique assume otherwise, so there exists F1∈H1(0,∞) which also\nsolves (17) with F1(0)/ne}ationslash=F(0). SetF2=F1−F, thenF′\n2(0) = 0,F2∈H1(0,∞) and\n0 =−∂2\nxF2+ixβF2−ηF2.\nMultiply both sides by ¯F2and integrate then integrate by parts\n0 =ˆ∞\n0|∂xF2|2+ixβ|F2|2−η|F2|2dx.\nThen using (16)\n0≥ˆ\n|∂xF2|2+xβ|F2|2−|η||F2|2dx≥/parenleftbigg\n1−|η|\n/tildewiderλ1/parenrightbiggˆ\n|∂xF2|2+xβ|F2|2dx>0,\nsinceF2∈H1(0,∞). The inequality is strict since |η|</tildewiderλ1, which is a contradiction.\nTo establish the stated regularity of Fnote that the equation (17) is elliptic and the\nleft hand side is 0 so in fact F∈H∞(0,∞) and thusF∈H2(0,∞) in particular.\nNow we show that F(0,η) is bounded away from 0 and ∞. The upper bound follows\nimmediately by the holomorphy of F(0,η) and the boundedness of η.\nTo see|F(0,η)|>1/Cit is enough to show that F(0,η)/ne}ationslash= 0 sinceFis continuous\nand we are considering ηin a compact set. Assume otherwise, so F(0,η0) = 0 for some\nη0∈C,|η0| ≤/tildewiderλ1/2. Multiply both sides of (17) by ¯Fthen integrate and integrate by parts\n0 =ˆ∞\n0|∂xF|2+ixβ|F|2−η0|F|2dx.\nThen using (16) (noting that here F∈H1\n0⊂H1)\n0≥ˆ\n|∂xF|2+xβ|F|2−|η0||F|2dx≥/parenleftbigg\n1−|η0|\n/tildewiderλ1/parenrightbiggˆ\n|∂xF|2+xβ|F|2dx>0,\nWe note that the inequality is strict since |η0|</tildewiderλ1, which is a contradiction.\n12Now we introduce µh∈C. We use it to clarify the dependence of λhonhand will\nimplicitly solve for it in terms of hin order to show that the compatibility condition can\nbe satisfied. If F0is the Dirichlet data of the unique H2solution of (15) for h=λh= 0,\nwe setC1=A1+µhso our definition of λhin (9) becomes\nλh=πlh\na+A1h(β+4)/(β+2)+µhh(β+4)/(β+2),\nwherel∈Zand\nA1=πlF0\na2.\nNow takevr(x) =F(h−2/(β+2)(x−a)),whereFis the unique H2solution of (15) with\nthe aboveλh. Recalling the explicit form of vl(x), the compatibility condition becomes\niλh\nh(1−Ref(µh,h)) =Bh−2/(β+2)\n1+Ref(µh,h) =BF(0,µh,h),\nwhereF(0,µh,h) denotes the Dirichlet data of Fand Ref(µh,h) = Ref(λh,h) and both are\nwritten this way to emphasize their dependence on µhandh. Divide the top equation by\nthe bottom\n/parenleftbiggπli\na+A1ih2/(β+2)+iµh2/(β+2)/parenrightbigg/parenleftBig\n1+exp(−2πil−2A1iah2/(β+2)−2iaµhh2/(β+2))/parenrightBig\nF(0,µh,h)\n=h−2/(β+2)/parenleftBig\n1−exp(−2πil−2A1iah2/(β+2)−2iaµhh2/(β+2))/parenrightBig\n.\nNow Taylor expand the exponentials for small h\n/parenleftbiggπli\na+A1ih2/(β+2)+iµhh2/(β+2)/parenrightbigg/parenleftBig\n2−2A1iah2/(β+2)−2iaµhh2/(β+2)+g(h)/parenrightBig\nF(0,µh,h)\n=h−2/(β+2)/parenleftBig\n2A1iah2/(β+2)+2iaµhh2/(β+2)−g(h)/parenrightBig\n,\nwhereg(h) is the remainder term from the Taylor expansion with g(h) =O(h4/(β+2)).\nSo in order to prove Theorem 2.3 it is enough to establish that for allhnear 0∈[0,∞)\nthere exists µh∈Csuch that the following function has a zero at ( µh,h);\nG(µ,h) =/parenleftbiggπli\na+A1ih2/(β+2)+iµh2/(β+2)/parenrightbigg/parenleftBig\n2−2A1iah2/(β+2)−2iaµh2/(β+2))+g(h)/parenrightBig\nF(0,µ,h)\n−2A1ia−2iaµ+g(h)h−2/(β+2). (19)\nTo do so we apply the implicit function theorem with weak regu larity hypotheses to solve\nforµh. We recall the implicit function theorem as stated in Theore m 11.1 of [LS90] (pp.\n166) can be applied if there exists some h0such that\n131.G(0,0) = 0\n2.Gis continuous on [0 ,1]×[0,h0)\n3.DµGexists and is continuous on [0 ,1]×[0,h0)\n4.DµG(0,0) is invertible.\nTo begin we see immediately that\nG(0,0) =2πli\naF0−2A1ia=2πli\naF0−2πli\naF0= 0.\nIn the following two lemmas we show that points 2 and 3, and 4 ar e satisfied.\nLemma 4.3. There exists h0>0such thatGas defined in (19)is continuous and∂\n∂µG\nexists and is continuous on {µ∈C;|µ|<1}×[0,h0).\nProof.To see that Gis continuous it will be enough to show that F(0,µ,h) is continuous\nas the other terms in G(µ,h) are clearly continuous in µandh. Similarly to see that∂\n∂µG\nexists and is continuous it is enough to see that∂\n∂µF(0,µ,h) exists and is continuous in µ\nandh.\nWe recall that F(0,µh,h) is the Dirichlet data for the L2solution of (17) with spectral\nparameter\nη=λ2\nh\nh2β/(β+2)=π2l2h4/(β+2)+(2A1πl+2πlµ)h6/(β+2)\na+(A2\n1+µ2+A1µ)h8/(β+2).\nBy Lemma 4.2 the Dirichlet data for the L2solution is holomorphic in |η|</tildewiderλ1. Thisηis\na sum of functions which are jointly continuous in µandhand continuously differentiable\ninµ, thereforeF(0,µ,h) is as well.\nFurthermore since |µ|<1 and there is a positive power of hin each term of ηthere\nexists some h0such that |η|</tildewiderλ1/2 for|µ|<1 andh∈[0,h0).\nLemma 4.4. WithGdefined as in (19)\n∂\n∂µG(µ,h)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nh=0,µ=0/ne}ationslash= 0,\nand thus is invertible.\nProof.Note\nG(µ,0) =2πli\naF(0,µ,0)−2A1ai−2aiµ\nso\n∂\n∂µG(µ,0) =2πli\na/parenleftbigg∂\n∂µF(0,µ,0)/parenrightbigg\n−2ai.\n14Whenh= 0 the equation Fsolves is 0 = −F′′(x)+ixβF(x).Therefore when h= 0 there\nis no dependence on µso∂\n∂µF(0,µ,h= 0) = 0. Thus\n∂\n∂µG(µ= 0,h= 0) =−2ai/ne}ationslash= 0.\nNow that we have shown we can apply the implicit function theo rem we conclude the\nproof of Theorem 2.3. That is given some lwe leth0be as in the proof of Lemma 4.3 and\nletK=π|F0|\na2+1. Then we choose an h∈(0,h0) and use the implicit function theorem to\nproduce aµh∈[0,1] such that G(µh,h) = 0 and |µh|<1. We setCh=πF0\na2+µhwe are\nthen able to solve (10) on (0 ,a) and (a,∞) forλh=πlh\na+A1h(β+4)/(β+2)+µhh(β+4)/(β+2)\nsuch that the compatibility conditions are satisfied giving us a solution v∈H2(0,∞)∩\nH1\n0(0,∞),v/ne}ationslash= 0 with |Ch|<Kandλof the appropriate form.\n4.1 Case u′(0) = 0\nWe now discuss how these proofs change when u′(0) = 0. When this is the case\nRef(λh,h) =e−2iλha/h.\nBecause of this we take l+ 1/2∈Zrather than l∈Z. This changes the specific steps\ntaken when going from the compatibility condition to the defi nition ofGin (19) but the\neventual definition of Gis the same. The specific form of lis otherwise not used so the\nremaining proofs in this section hold unchanged.\n5 Proof of Lemma 3.1\nProof.To show this result we consider our operator Ph=−h2∂2\nx+i(x−a)β\n+−λ2\nhon\n(a+σ/4,b). We note that the coefficients are smooth away from aand so it makes sense\nto look at the pricipal symbol\n|ξ|2+i(x−a)β\n+.\nIt is straightforward to see from this that Phis semiclassically elliptic on ( a+σ/4,b).\nRecall in our definition of φthat we required a+σ<b−2δand in (12) that φsatisfies\nφ(x) =/braceleftBigg\n1x<b−2δ\n0b−δ<x.\nWe now define ψ∈C∞(0,b) satisfying\nψ=/braceleftBigg\n0x<a+σ/2\n1a+σ<x<b,\n15so thatφψ∈Ψ0\nh(0,b) withWFh(φψ)⊂ellh(Ph).\nIfvis asolution of Phv= 0, by standardsemiclassical elliptic estimates (see fori nstance\n[Zwo12] Theorem 7.1) there exists χ∈C∞\n0(a+σ/4,b) such that for all s∈R\n||φψv||Hs\nh(a+σ/2,b)≤O(h∞)||χv||L2(a+σ/4,b)\nIn particular\n||φv||Hs\nh(a+σ,b)≤O(h∞)||v||L2(0,b).\nIt remains to show that\n||v||L2(0,b)/lessorsimilar||φv||L2(0,b).\nTo proceed we multiply both sides of (10) by ¯ vthen integrate and integrate by parts\n0 =h2ˆ∞\n0|∂xv|2dx+iˆ∞\n0(x−a)β\n+|v|2dx−λ2\nhˆ∞\n0|v|2dx.\nTake the imaginary part of both sides and rearrange\nˆ∞\n0(x−a)β\n+|v|2dx= Im(λ2\nh)ˆ∞\n0|v|2dx≤O(h2)ˆ∞\n0|v|2dx.\nFurthermore\nˆ∞\na+σσβ|v|2dx≤ˆ∞\na+σ(x−a)β\n+|v|2dx≤ˆ∞\n0(x−a)β\n+|v|2dx.\nSo ˆ∞\na+σ|v|2dx≤h2\nσβˆ∞\n0|v|2dx.\nAdd´∞\n0|v|2dxto both sides and rearrange\n/parenleftbigg\n1−h2\nσβ/parenrightbiggˆ∞\n0|v|2dx≤ˆa+σ\n0|v|2dx.\nNotice that ( b−2δ,b)⊂(0,∞) and (0,a+σ)⊂(0,b−2δ) so\n/parenleftbigg\n1−h2\nσβ/parenrightbiggˆb\nb−2δ|v|2dx≤/parenleftbigg\n1−h2\nσβ/parenrightbiggˆ∞\n0|v|2dx≤ˆa+σ\n0|v|2dx≤ˆb−2δ\n0|v|2dx.\nTherefore for h<σβ/2\nˆb\n0|v|2dx=ˆb−2δ\n0|v|dx+ˆb\nb−2δ|v|2dx≤/parenleftbigg\n1+σβ\nσβ−h2/parenrightbiggˆb−2δ\n0|v|2dx≤Cˆb\n0|φv|2dx.\n166 Proof of Theorem 1.2\nIn this section we establish a rate of decay of the energy by ad apting and improving\nSection 3 of [BH07] for our particular setup. That result and our result rely heavily on an\nobservability result by Burq and Zworski (Proposition 6.1) in [BZ04].\nTo prove Theorem 1.2 we again rely on Proposition 2.4 of [AL+14] (see also [BT10])\nwhich we state a variant of.\nProposition 6.1. If there exists C >0andq0≥0such that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle(−∆+iqW(x)−q2)−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nL2→L2≤C|q|1\nα−1(20)\nfor allq∈R,|q| ≥q0then(1)is stable at rate 1/tα.\nProof of Theorem 1.2. Consideru∈H2(M) solving\n(−∆+iqW−q2)u=f∈L2(M), u|∂M= 0, q≫1. (21)\nLet 0≤χ∈C∞\n0(R) be a cutoff function with χ= 0 for|x| ≥2 andχ= 1 for|x| ≤1.\nNow letχq=χ(qγW(x)) withγ=β\nβ+2. Note that χqis identically 0 for |x|>a+σ/4 for\nqlarge enough since W >0 on [a+σ,b]. Because of this χqand it’s derivatives are only\nsupported where Whas the form ( |x|−a)β\n+. Furthermore on the support of χ′\nq(x)\nW(x)∼1\nqγ, q≫1. (22)\nRemark. The proof in [BH07] uses an analogous setup with γ= 1. The key change we\nmake is to set γ=β\nβ+2. The rest of our argument is similar to the proof in [BH07] but we\ndetail it for the convenience of the reader and to explain why this value of γis ideal.\nThe function χqustill vanishes on ∂M(or if∂M=∅it still satisfies the periodicity\ncondition) and satisfies on M\n(−∆−q2)χqu=χqf+χ′′\nqu−2∂x(χ′\nqu)−iqW(x)χqu. (23)\nWe apply Proposition 6.1 of [BZ04] to this equation choosing the control region ωx=\n[a+σ/4,a+σ/2] and setting ω:=wx×[−b,b] to obtain\n||χqu||2\nL2≤C/parenleftBig/vextendsingle/vextendsingle/vextendsingle/vextendsingleχqf+χ′′\nqu−2∂x(χ′\nqu)−iqW(x)χqu/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nH−1\nxL2y(M)/parenrightBig\n+||χqu|ω||2\nL2(ω)\n≤C/parenleftBig\n||χqf||2\nL2+/vextendsingle/vextendsingle/vextendsingle/vextendsingleχ′′\nqu/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2+/vextendsingle/vextendsingle/vextendsingle/vextendsingleχ′\nqu/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2+||qWχqu||2\nL2/parenrightBig\n. (24)\nWe emphasize that χqvanishes on ω, which allowed us to drop that term. We now estimate\nthe remaining terms on the right hand side. Using that Wis exactly ( |x| −a)β\n+on the\nsupport ofχqand its derivatives we obtain the following bound on the deri vative ofχq,\n|χ′\nq|=|qγχ′\nq(qγW(x))W′(x)| ≤Cqγ/β, (25)\n17and similarly\n|χ′′\nq| ≤Cq2γ/β.\nNote that on the support of χ′\nqandχ′′\nqthe damping Wis smooth, so this computation is\nvalid for all β >0.\nNow write\nχ′\nqu=χ′\nqW1/2u\nW1/2,\nthen using (25) and (22)/vextendsingle/vextendsingle/vextendsingle/vextendsingleχ′\nq\nW1/2/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤qγ(1/2+1/β),\nand consequently/vextendsingle/vextendsingle/vextendsingle/vextendsingleχ′\nqu/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2≤Cqγ(1+2/β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleW1/2u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2. (26)\nWe estimate the L2norm ofχ′′\nquin a similar way\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleχ′′\nqu/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2≤O(1)qγ(1+4/β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleW1/2u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2. (27)\nFinally a similar argument shows\n||qWχqu||2\nL2≤O(1)q2−γ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleW1/2u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2. (28)\nThe smaller the terms on the right of (26), (27) and (28) are th e stronger the resolvent\nestimate is. Because of this we would like to minimize\nmax{2−γ,γ(1+4/β),γ(1+2/β)}.\nThis is attained when\n2−γ=γ(1+4/β),\ni..e.γ=β/(β+2). Therefore (26), (27), (28) along with (24) give\n||χqu||2\nL2≤O(1)/parenleftbigg\n||f||2\nL2+q(β+4)/(β+2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleW1/2u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2/parenrightbigg\n.\nNow note that pairing (21) with ¯ u\n|q|ˆ\nW|u|2dx≤ ||f||L2||u||L2. (29)\nTherefore\n||χqu||2\nL2≤O(1)/parenleftBig\n||f||2\nL2+q2/(β+2)||f||L2||u||L2/parenrightBig\n.\n18It remains to control the L2norm of (1 −χq)u. To do so we remark that 1 −χqis supported\nin the set where W≥1/qγ. Using (29) again\n||(1−χq)u||2\nL2≤qγˆ\n(1−χq)W|u|2dx≤qγ−1||f||L2||u||L2.\nTherefore\n||u||2\nL2≤O(1)/parenleftBig\n||f||2\nL2+q1/(β+2)||f||L2||u||L2/parenrightBig\nand thus we obtain\n||u||L2≤O(1)q2/(β+2)||f||L2, q∈R,|q| ≫1,\nwhich along with Proposition 6.1 gives stability at the stat ed rate.\nReferences\n[AL+14] N. Anantharaman, M. L´ eautaud, et al. Sharp polynomial d ecay rates for the\ndamped wave equation on the torus. Anal. PDE , 7(1):159–214, 2014.\n[BD08] C. J. K. Batty and T. Duyckaerts. Non-uniform stabili ty for boundedsemi-groups\non banach spaces. Journal of Evolution Equations , 8(4):765–780, Nov 2008.\n[BG97] N. Burq and P. G´ erard. Condition n´ ecessaire et suffis ante pour la contrˆ olabilit´ e\nexacte des ondes. Comptes Rendus de l’Acadˆ emie des Sciences - Series I - Math-\nematics, 325(7):749 – 752, 1997.\n[BG18] N. Burq and P. G´ erard. Stabilisation of wave equatio ns on the torus with rough\ndampings. arXiv preprint arXiv:1801.00983 , 2018.\n[BH07] N. Burq and M. Hitrik. Energy decay for damped wave equ ations on partially\nrectangular domains. Mathematical Research Letters , 14(1):35–47, 2007.\n[BT10] A. Borichev and Y. Tomilov. Optimal polynomial decay of functions and operator\nsemigroups. Mathematische Annalen , 347(2):455–478, 2010.\n[Bur98] N. Burq. D´ ecroissance de l’´ energie locale de l’´ e quation des ondes pour le probl` eme\next´ erieur et absence de r´ esonance au voisinage du r´ eel. Acta Math. , 180(1):1–29,\n1998.\n[BZ04] N.BurqandM.Zworski. Geometriccontrol intheprese nceofablack box. Journal\nof the American Mathematical Society , 17(2):443–471, 2004.\n[BZ12] N. Burq and M. Zworski. Control for Schr¨ odinger oper ators on tori. Mathematical\nResearch Letters , 19, 06 2012.\n19[BZ15] N. Burq and C. Zuily. Laplace eigenfunctions and damp ed wave equation on prod-\nuct manifolds. Applied Mathematics Research eXpress , 2015(2):296–310, 2015.\n[CBR92] G. Lebeau C. Bardos and J. Rauch. Sharp sufficient cond itions for the obser-\nvation, control, and stabilization of waves from the bounda ry.SIAM Journal on\nControl and Optimization , 30(5):1024–1065, 1992.\n[HS12] P. Hislop and I. Sigal. Introduction to spectral theory: With applications to\nSchr¨ odinger operators , volume 113. Springer Science & Business Media, 2012.\n[Jaf90] S. Jaffard. Contrˆ ole interne exact des vibrations d’ une plaque rectangulaire.\n(Internal exact control for the vibrations of a rectangular plate).Port. Math. ,\n47(4):423–429, 1990.\n[Leb96] G. Lebeau. Equation des ondes amorties. In Algebraic and Geometric Methods\nin Mathematical Physics: Proceedings of the Kaciveli Summe r School, Crimea,\nUkraine, 1993 , pages 73–109. Springer Netherlands, Dordrecht, 1996.\n[LL17] M. L´ eautaud and N. Lerner. Energy decay for a locally undamped wave equation.\nAnnales de la facult des sciences de Toulouse Sr.6 , 26(1):157–205, 2017.\n[LR97] G. Lebeau and L. Robbiano. Stabilisation de l´ equati on des ondes par le bord.\nDuke Math. J. , 86(3):465–491, 1997.\n[LR05] Z. Liu and B. Rao. Characterization of polynomial dec ay rate for the solution\nof linear evolution equation. Zeitschrift f¨ ur angewandte Mathematik und Physik\nZAMP, 56(4):630–644, 2005.\n[LS90] L.H. Loomis and S. Sternberg. Advanced Calculus . Jones and Bartlett Publishers,\nInc., 1990.\n[Mac10] F. Maci` a. High-frequency propagation for the Schr ¨ odinger equation on the torus.\nJ. Funct. Anal. , 258(3):933–955, 2010.\n[Ral69] J. Ralston. Solutions of the wave equation with loca lized energy. Communications\non Pure and Applied Mathematics , 22(6):807–823, 1969.\n[RT75] J. Rauch and M. Taylor. Exponential decay of solution s to hyperbolic equations\nin bounded domains. Indiana Univ. Math. J. , 24:7986, 1975.\n[Sta17] R. Stahn. Optimal decay rate for the wave equation on a square with constant\ndamping on a strip. Zeitschrift f¨ ur angewandte Mathematik und Physik , 68(2):36,\n2017.\n[Zwo12] M. Zworski. Semiclassical analysis , volume 138. American Mathematical Soc.,\n2012.\n20"    },    {        "title": "1703.07310v2.Using_rf_voltage_induced_ferromagnetic_resonance_to_study_the_spin_wave_density_of_states_and_the_Gilbert_damping_in_perpendicularly_magnetized_disks.pdf",        "content": "Using rf voltage induced ferromagnetic resonance to study the spin-wave density of states and the\nGilbert damping in perpendicularly magnetized disks\nThibaut Devolder\u0003\nCentre de Nanosciences et de Nanotechnologies, CNRS, Univ. Paris-Sud,\nUniversit ´e Paris-Saclay, C2N-Orsay, 91405 Orsay cedex, France\n(Dated: September 18, 2018)\nWe study how the shape of the spinwave resonance lines in rf-voltage induced FMR can be used to extract the\nspin-wave density of states and the Gilbert damping within the precessing layer in nanoscale magnetic tunnel\njunctions that possess perpendicular magnetic anisotropy. We work with a field applied along the easy axis to\npreserve the cylindrical symmetry of the uniaxial perpendicularly magnetized systems. We first describe the\nexperimental set-up to study the susceptibility contributions of the spin waves in the field-frequency space. We\nthen identify experimentally the maximum device size above which the spinwaves confined in the free layer can\nno longer be studied in isolation as the linewidths of their discrete responses make them overlap into a continuous\ndensity of states. The rf-voltage induced signal is the sum of two voltages that have comparable magnitudes: a\nfirst voltage that originates from the linear transverse susceptibility and rectification by magneto-resistance and a\nsecond voltage that arises from the non-linear longitudinal susceptibility and the resultant time-averaged change\nof the exact micromagnetic configuration of the precessing layer. The transverse and longitudinal susceptibility\nsignals have different dc bias dependences such that they can be separated by measuring how the device rectifies\nthe rf voltage at different dc bias voltages. The transverse and longitudinal susceptibility signals have different\nlineshapes; their joint studies in both fixed field-variable frequency, or fixed frequency-variable field configura-\ntions can yield the Gilbert damping of the free layer of the device with a degree of confidence that compares\nwell with standard ferromagnetic resonance. Our method is illustrated on FeCoB-based free layers in which\nthe individual spin-waves can be sufficiently resolved only for disk diameters below 200 nm. The resonance\nline shapes on devices with 90 nm diameters are consistent with a Gilbert damping of 0:011. A single value\nof the damping factor accounts for the line shape of all the spin-waves that can be characterized. This damp-\ning of 0.011 exceeds the value of 0.008 measured on the unpatterned films, which indicates that device-level\nmeasurements are needed for a correct evaluation of dissipation.\nThe frequencies of the magnetization eigenmodes of\nmagnetic body reflect the energetics of the magnetization.\nAs a result the frequency-based methods – the ferromag-\nnetic resonances (FMR)1and more generally the spin-wave\nspectroscopies– are particularly well designed for the metrol-\nogy of the various magnetic interactions. In particular, mea-\nsuring the Gilbert damping parameter \u000bthat describes the\ncoupling of the magnetization dynamics to the thermal bath,\nspecifically requires high frequency measurements. There are\ntwo main variants of these resonance techniques. The so-\ncalled conventional FMR and its modern version the vector\nnetwork analyzer2(VNA)-FMR are established technique to\nharness the coupling of microwave photons to the magneti-\nzation eigenmodes to measure to anisotropy fields1, demag-\nnetizing fields, exchange stiffness3, interlayer exchange4and\nspin-pumping5, most often at film level. More recent meth-\nods, like the increasingly popular spin-transfer-torque-(STT)-\nFMR, are developed6to characterize the magnetization dy-\nnamics of magnetic bodies embodied in electrical devices pos-\nsessing a magneto-resistance of some kind.\nIn conventional FMR or VNA-FMR, the community is well\naware that the line shape of a resonance is more complicated\nthan simple arguments based on the Landau-Lifshitz-Gilbert\nequation would tell. There are for instance substantial contri-\nbutions from microwave shielding effects7(”Eddy currents”)\nfor conductive ferromagnetic films8or ferromagnetic films in\ncontact with (or capacitively coupled to) a conductive layers.\nA hint to these effect is for instance to compare the lineshapes8\nfor the quasi-uniform precession mode and the first perpen-\ndicular standing spin wave modes that occur in different res-onance conditions. Note that the experimental lineshapes are\nalready complex in VNA-FMR despite the fact that the dy-\nnamics is induced by simple magnetic fields supposedly well\ncontrolled.\nIn contrast, STT-FMR methods rely on torques [spin-orbit\ntorques (SOT)9or STT] that have less hindsight that magnetic\nfields or that are the targeted measurements. These torques are\nrelated to the current across the device and the experimental\nanalysis generally assumes that this current is in phase with\nthe applied voltage. This implicitly assumes that the sam-\nple is free of capacitive and inductive responses, even at the\nmicrowave frequencies used for the measurement. A careful\nanalysis is thus needed when the STT-FMR methods analyze\nthe phase of the device response to separate the contribution of\nthe different torques6,10,11. Besides, the quasi-uniform mode\nis often the sole to be analyzed despite that fact that the line\nshapes of the higher frequency modes can be very different10.\nFinally, an external field is generally applied in a direction\nthat is not a principal direction of the magnetization energy\nfunctional12. While this maximizes the signal, this unfortu-\nnately makes numerical simulation unavoidable to model the\nexperimental responses.\nWith the progress in MTJ technologies, much larger\nmagneto-resistance are now available13, such that signals can\nbe measured while maintaining sample symmetries, for in-\nstance with a static field applied collinearly to the magne-\ntization. In addition, high anisotropy materials can now be\nincorporated in these MTJs. This leads to a priori much\nmore uniform magnetic configurations in which analytical de-\nscriptions are more likely to apply. In this paper, we revisitarXiv:1703.07310v2  [cond-mat.mtrl-sci]  4 Sep 20172\nrf-voltage induced FMR in a situation where the symmetry\nis chosen so that all torques should yield a priori the same\ncanonical lineshape for all spinwaves excited in the system.\nWe use PMA MTJ disks of sizes 500 nm, on which a quasi-\ncontinuum of more that 20 different spin-wave modes can be\ndetected, down to sizes of 60 nm where only a few discrete\nspinwave modes can be detected. We discuss the lineshapes\nof the spin-wave signals with the modest objective of deter-\nmining if at least the Gilbert damping of the dynamically ac-\ntive magnetic layer can be reliably extracted. We show that\nthe linear transverse susceptibility and the non-linear longitu-\ndinal susceptibilities must both be considered when a finite dc\nvoltage is applied through the device. We propose a method-\nology and implement it on a nanopillars made with a stan-\ndard MgO/FeCoB/MgO free layer system in which we obtain\na Gilbert damping of 0:011\u00060:0003 . This exceeds the value\nof 0.008 measured on the unpatterned film, which indicates\nthat device-level measurements are needed for a correct eval-\nuation of dissipation.\nThe paper is organized as follows:\nThe first section lists the experimental considerations, includ-\ning the main properties of the sample, the measurement set-\nup and the mathematical post-processing required for an in-\ncreased sensitivity. The second section discusses the origins\nof the measured resonance signals and their main properties.\nThe third section describes how the device diameter affects the\nspin-wave signals in rf-voltage-induced ferromagnetic reso-\nnance. The last section describes how the voltage bias depen-\ndence of the spinwave resonance signals can be manipulated\nto extract the Gilbert damping of the dynamically active mag-\nnetic layer. After the conclusion, an appendix details the main\nfeatures of the spectral shapes expected in ideal perpendicu-\nlarly magnetized systems.\nI. EXPERIMENTAL CONSIDERATIONS\nA. Magnetic tunnel junctions samples\nWe implement our characterization technique on the sam-\nples described in detail in ref. 14. They are tunnel junctions\nwith an FeCoB-based free layer and a hard reference system\nbased on a well compensated [Co/Pt]-based synthetic antifer-\nromagnet. All layers have perpendicular magnetic anisotropy\n(PMA). The perpendicular anisotropy of the thick ( t= 2nm)\nfree layer is ensured by a dual MgO encapsulation and an\niron-rich composition. After annealing, the free layer has an\nareal moment of Mst\u00191:8mA and an effective perpendic-\nular anisotropy field \u00160(Hk\u0000Ms)= 330 mT. Before pat-\ntering, standard ferromagnetic resonance measurements in-\ndicated a Gilbert damping parameter of the free layer being\n\u000b= 0:008. Depending on the size of the patterned device,\nthe tunnel magnetoresistance (TMR) is 220 to 250%, for a\nstack resistance-area product is RA = 12 \n:\u0016m2. The de-\nvices are circular pillars with diameters varied from 60 to 500\nnm. The materials, processing and device rfcircuitry were\noptimized for fast switching14spin-transfer-torque magnetic\nrandom access memories (STT-MRAM15) ; the quasi-static\ndcRF50 ΩPulse modulation ac, 50 kHz\n50 ΩVac ~ 40 mVLI75AamplifierMSG3697synthetiser\nK2400sourcemeter\nSR830lock-inamplifier×100-10 dB attenuation~ 0 dBm~ 14 µAdc10 mVdcMTJ device300 nm~700 ΩFIG. 1. (Color online). Sketch of the experimental set-up with an\n300\u0002300\u0016m2optical micrograph of the device circuitry. The given\nnumbers are the typical experimental parameters for a 300 nm diam-\neter junction. Inset: resistance versus out-of-plane field hysteresis\nloop for a device with 300 nm diameter.\ndcswitching voltage is \u0019600mV . In the present report, the\napplied voltages shall never exceed 100 mV to minimize spin-\ntransfer-torque effects. The fields will always be applied along\n(z) which is the easy magnetization axis. The sample will be\nmaintained in the antiparallel (AP) state.\nB. Measurement set-up\nThe pillars are characterized in a set-up (Fig. 1) inspired\nfrom spin-torque diode experiments6but an electrical band-\nwidth increased to 70 GHz. The objective is to identify the\nregions in theffrequency, fieldgspace in which the magneti-\nzation is responding in a resonant manner. The device is at-\ntacked with an rfvoltageVrf. A 10 dB attenuator is inserted\nat the output port of the synthesizer to improve its impedance\nmatching so as to avoid standing waves in the circuit. This im-\nproves the frequency flatness of the amplitude of the stimulus\narriving at the device. To ease the detection of the sample’s re-\nsponse, the rfvoltage is pulse-modulated at an acfrequency\n!ac=(2\u0019) = 50 kHz (Fig. 1). The current passing through\nthe MTJ has thus frequency components at the two sidebands\n!rf\u0006!ac. The acvoltage which appears across the device\nis amplified and analyzed by a lock-in amplifier. We shall\ndiscuss the origin of this acvoltage in section II. Optionally,\nthe device is biased using a dcsourcemeter supplying Vdcand\nmeasuringIdc.\nFigure 2 shows a representative map of thedVac\ndHzresponse\nobtained on a pillar of diameter 300 nm with Vdc= 10 mV.\nAs positive fields are parallel to the free layer magnetization,\nthe spin waves of the free layer appear with a positive fre-\nquency versus field slope, expected to be the gyromagnetic3\n⦰ 300 nm\nFIG. 2. Field derivative of the rectified voltagedVac\ndHzin the\nffrequency-fieldgparameter space for a 300 nm diameter device in\nthe AP state when the field is parallel to the free layer magnetization.\nThe linear features with positive (resp. negative) slopes correspond\nto free layer (resp. reference layers) confined spin-wave modes.\nBlack and white colors correspond to signals exceeding \u00060:01V/T.\nThe one-pixel high horizontal segments are experimental artefacts\ndue to transient changes of contact resistances.\nratio\r0of the free layer material (see appendix). Conversely,\nthe reference layer eigenmodes appear with a negative slope,\nexpectedly\u0000\r0, where this time \r0is gyromagnetic ratio of\nthe reference layer material combination. Working in the AP\nstate is thus a convenient way to easily distinguish between\nthe spinwaves of the free layer and of the reference layers.\nNote that the gyromagnetic ratios \r0of the free layer mode\nand the reference layer modes differ slightly owing to their\ndifference chemical nature. The free layer has a Land ´e factor\ng= 2:085\u00060:015where the error bar is given by the precision\nof the field calibration; the reference layer modes are consis-\ntent with a 1.2% larger gyromagnetic ratio. The accuracy of\nthis latter number is limited only by the signal-to-noise ratio\nin the measurement of the reference layer properties. Looking\nat Fig. 2, one immediately notices that the linewidths of the\nreference layer modes are much broader than that of the free\nlayer. While the linewidh of the reference layer modes will not\nbe analyzed here, we mention that this increased linewidth is\nto be expected for reference layers that contain heavy metals\n(Pt, Ru) with large spin-orbit couplings, hence larger damping\nfactors16.C. Experimental settings\nIn practice, we choose an applied field interval of\n[\u0000110;110mT]that is narrow enough to stay in a state whose\nresistance is very close to that of the remanent AP state. The\nfrequency!rf=(2\u0019)is varied from 1 to 70 GHz; we gener-\nally could not detect signals above 50 GHz. The practical\nfrequency range 2\u0019\u000250GHz=\r0\u00191:6T is much wider that\nour accessible field range. For wider views of the experimen-\ntal signals (for instance when the spin-wave density of states\nis the studied thing), we shall thus prefer to plot them versus\nfrequency than versus field. The response is recorded pixel by\npixel in in theffrequency, fieldgspace. The typical pixel size\nisf\u000eHz\u0002\u000efg=f1 mT\u000250 MHzg. The field and frequency\nresolutions are thus comparable (indeed 2\u0019\u0002\u000ef=\r 0= 1:7\nmT).\nD. Signal conditioning\n1. Mathematical post-treatments\nFinally, despite all our precautions to suppress the rectify-\ning phenomena that do not originate from magnetization dy-\nnamics, we have to artificially suppress the remaining ones.\nThis was done by mathematical differentiation, and we gener-\nally plotdVac\ndfordVac\ndHzin the experimental figures (Figs. 2-5).\n2. Dynamic range improvement by self-conformal averaging\nA special procedure (Fig. 3) is applied when a better signal\nto noise ratio is desired while the exact signal lineshape and\namplitudes are not to meant to be looked at. This procedure\nharnesses the fact that the normalized shape of the sample’s\nresponse is essentially self-conformal when moving across a\nline withd!\ndHz=\r0in theffrequency, fieldgparameter space\n(see appendix). The procedure consists in calculating the fol-\nlowing primitive:\ns(f0) =1\n2\r0HmaxzZ\ncontourdVac\ndHzdf ; (1)\nin which the integration contour is the segment linking the\npoints (\u0000Hmax\nz;f0\u0000\r0Hmax\nz) and (Hmax\nz;f0+\r0Hmax\nz)\nin theffield, frequencygparameter space. Such contours ap-\npear as pixel columns in Fig. 3(b). This primitive (eq. 1) is\nefficient to reveal the free layer spin-wave modes that yield an\notherwise too small signal. For instance when only 7 modes\ncan be detected in single field spectra [Fig. 3(a)], the aver-\naging procedure can increase this number to typically above\n25. The averaging procedure is also effective in suppressing\nthe signals of the reference layer as these laters average out\nover a contour designed for the free layer mode when in the\nAP state. However as the linewidth of the free layer modes\nis proportional to the frequency, it is not constant across the\ncontour; the higher signal to noise ratio is thus unfortunately4\nf\u0000\u00000Hz2⇡Hz(b)(a)\nFIG. 3. (Color online). Illustration of the dynamic range improve-\nment by self-conformal averaging (section I D 2). The procedure is\nimplemented on a 300 nm diameter device to evidence the free layer\nmodes. Bottom panel: field derivative of the acsignal in the rotated\nframe in which the modes withdf\ndHz=\r0\n2\u0019should appear as vertical\nlines. Top panel: comparison of a single field frequency scan (red)\nwith the average over all scans as performed in the !=\r0Hzdi-\nrection. Note that the signal of the lowest frequency mode (which\ncorresponds to the quasi-uniform precession) disappears near zero\nfield, at 5 mT (see the apparent break in the middle of the most left\nline in the bottom panel).\nobtained at the expense of a distorted (and unphysical) line-\nshape. Note also that this procedure can not be applied to the\nquasi-uniform precession mode as will be explained in section\nII D 2).\nII. ORIGIN AND NATURE OF THE RECTIFIED SIGNAL\nLet us now discuss the origin of the demodulated acvolt-\nage. In this section, we assume that the reference layer mag-\nnetization is static but not necessarily uniformly magnetized.\nWe can thus express any change of the resistance by writing\n\u000eR=\u000eR\n\u000eM\u000eM where\u000ehas to be understood as a functional\nderivative with respect to the free layer magnetization distri-\nbution.\nA. The two origins of the rectified signals\nTheacsignal can contain two components V1;acandV2;ac\nof different physical origins17. The first component is the\n’standard’ STT-FMR signal: the pulse-modulated rfcurrent is\nat the frequency sidebands !rf\u0006!acand it rectifies to acany\noscillation of the resistance \u000eRrfoccurring at the frequency\n!rf. We simply have V1;ac=\u000eRrf\u0002i!rf\u0006!ac.\nThe second acsignal (V2;ac) is related to the change of the\ntime-averaged resistance due to the population of spinwavescreated when the rfcurrent is applied12. Indeed the time-\naveraged magnetization distribution is not the same when the\nrfisonoroff. This change of resistance \u000eRaccan revealed by\nthe (optional) dccurrentIdcpassing through the sample, i.e.\nV2;ac=\u000eRac\u0002Idc.\nNote that a third rectification channel18can be obtained by\na combination of spin pumping and inverse spin Hall effect in\nin-plane magnetized systems19. This third rectification chan-\nnel yields symmetric lorentzian lines when applied to PMA\nsystems in out-of-plane applied fields (see eq. 23 in ref. 18).\nBesides, the spin-pumping is known to be largely suppressed\nby the MgO tunnel barrier20, such that we will consider that\nwe can neglect this third rectification channel from now on. In\nsummary, we have:\nV1;ac=Vrf\nR+ 50\u000eR\n\u000eM\u000eMrf and (2)\nV2;ac=Vdc\nR+ 50\u000eR\n\u000eM\u000eMac (3)\nThis has important consequences.\nB. Compared signal amplitudes in the P and AP states\nThe first important consequence of Eq. 2 and 3 is that the\nsignal amplitude depends on the nature of the micromagnetic\nconfiguration. As intuitive, both V1;acandV2;acscale with\nhow much the instantaneous device resistance depends on its\ninstantaneous micromagnetic configuration. This is expressed\nby the sensitivity factor\u000eR\n\u000eMwhich is essentially a magneto-\nresistance. We expect no signal when the resistance is insen-\nsitive to the magnetization distribution at first order (i.e. when\n\u000eR\n\u000eM\u00110).\nIn our samples, the shape of the hysteresis loop (Fig. 1)\nseems to indicate that the free layer magnetization is very uni-\nform when in the Parallel state. Consistently, the experimental\nrectified signal were found to be weak signals when in the P\nstate. Conversely, there is a pronounced curvature in the AP\nbranch of the R(Hz)hysteresis loop (see one example in the\ninset of Fig. 1). This indicates that the resistance is much de-\npendent on the exact magnetization configuration when in the\nAP state. Consistently, this larger\u000eR\n\u000eMin the AP state is proba-\nbly the reason why the rectified signal is much easier to detect\nin the AP state for our samples.\nC. Bias dependence of the rectified signals\nThe second important consequence of Eqs. 2-3 concerns the\ndependence of the rectified acsignalsV1;acandV2;acon the\ndcandrfstimuli. As\u000eMrfscales with the applied rftorque\naccording to a linear transverse susceptibility ( <e(\u001fxx), see\nappendix),V1;acis expected to scale with the rfpowerVrf2\n(see Eq. 2) independently from the dcbias, i.e. we have\nV1;ac/Vrf2:5\nIn contrast,\u000eMacis related to a longitudinal susceptibility and\nis thus quadratic with the rftorques (see appendix). Using\nEq. 3, we thus expect the following bias dependence\nV2;ac/Vrf2Vdc:\nD. Peculiarities of the quasi-uniform precession (QUP) mode\nThe last important consequence of Eqs. 2-3 concerns\nspecifically the quasi-uniform precession (QUP) mode that\nshows a peculiar acsignal.\n1. Quasi-absence of STT-FMR like signal for the quasi-uniform\nprecession mode\nIn the idealized macrospin case (see appendix) the uniform\nprecession is perfectly circular with no rfvariation of Mzat\nany order. If the fixed layer was uniformly magnetized along\nexactly (z), this would lead V1;ac= 0 such that the signal of\nthe QUP mode would be given by purely V2;ac. This qual-\nitatively ’pure V2;accharacter’ is confirmed experimentally\nby the fact that the signal of the QUP mode systematically\nchanges sign with Vdcin our sample series (not shown).\n2. Strong dependence of the QUP signal amplitude with the\napplied field\nIn addition, the experimental signal of the quasi-uniform\nmode is found to disappear at low fields [see Figs. 2, 3(b) and\n4(a, b)] exactly at the apex of the AP branch of the hysteresis\nloop (Fig. 1), i.e. for the field leading specifically todR\ndHz= 0.\nMoreover, the amplitude of the acsignal of the QUP mode ap-\npears to be essentially linearly correlated with the loop slope\ndR\ndHz(not shown). For instance, the V2;acof the QUP mode\nchanges sign when the applied field crosses the apex of the\nR(Hz)loops (Fig. 1).\nThe reason stems probably from the sensitivity factor\u000eR\n\u000eM\nand its correlation with the loop slopedR\ndHz; in some sense, a\nlarge loop slope should translate in a large sensitivity factor.\nWhile a numerical evaluation of this correlation goes beyond\nthe scope of this paper, we stress that if the magnetization was\nperfectly uniform there would be a one-to-one correlation be-\ntween loop slopedR\ndHzand magnetoresistance sensitivity fac-\ntor\u000eR\n\u000eM. This trend remains qualitatively true for the QUP\nmode. Indeed as the hysteresis loop is monitoring the spatial\naverage of the magnetization, it is more insightful for the uni-\nform mode than for any other (higher order) modes whose dy-\nnamic profiles spatially average to essentially zero21; the cor-\nrelation betweendR\ndHzand\u000eR\n\u000eMis thus expected to be maximal\nfor quasi-uniform changes of the magnetization configuration.\nWhile this property – the disappearance of the QUP mode\nsignal whendR\ndHz= 0 – can be used to distinguish the QUP\nmode from the higher order spin waves, the pronounced field\ndependence of the QUP signal complicates the analysis, as it\nprevents to conveniently analyze the field derivative of the acsignal (I D 1). In the remainder of this paper we shall focus on\nonly higher order modes to avoid such difficulties.\nE. Signals for non-uniform spin-waves\nBefore analyzing the spin-wave density of states (section\nIII), let us comment on the amplitude of the STT-FMR-like\nsignalV1;acfor the non-uniform spin-waves. In the perpen-\ndicular magnetization state, these spin-waves have a circular\nprecession22. By symmetry, the resistance is not expected to\nchange during a period of circular precession when in the per-\nfect collinear cases and for radial spin waves maintaining the\ncylindrical symmetry of the system. In other words, when\nthe dynamical magnetization of the eigenmode maintains the\ncylindrical symmetry and when the free and reference layers\nequilibrium magnetizations follow ~Mfree\u0002~Mref=~0every-\nwhere in the (xy) plane, with \u0002being the conventional vec-\ntor product) the device resistance is not expected to oscillate.\nWhile we can not identify to what extent we depart from this\nideal situation, we speculate that this perfect collinearity does\nnot happen in practice at least because of finite thermal fluctu-\nations. The effect of thermal fluctuations on the device resis-\ntance is not averaged out for non-uniform spin-waves, while\nit could be essentially averaged out for the QUP mode ana-\nlyzed earlier. In practice a finite variation of the resistance\n\u000eRrf6= 0 is always present during a precession period for a\nnon-uniform spin-wave. This provides a finite sensitivity to\nany spin-wave mode. This resistance variation at !rfhas the\nspectral shape of a transverse susceptibility term <e(\u001fxx)(see\nappendix).\nIII. SPIN WA VE DENSITY OF STATES AGAINST\nLATERAL CONFINEMENT\nAny reliable analysis of a spectral lineshape or linewidth re-\nquires to determine priorly how many spin-waves contribute\nto the lineshape under study. Therefore, before discussing the\nlineshapes of the individual spin-wave modes, let us determine\nhow the lateral confinement influences the measured rectified\nsignal. The impact of the device diameter on the spectral sig-\nnals is reported in Fig. 4.\nA. Spin-waves within the references layers\nFig. 4 indicates that the modes of the reference layers have\nfrequencies that are almost not affected by the device diam-\neter. This fact is related to the well compensated character\nof the synthetic antiferromagnet that composes the reference\nlayers. Indeed the internal demagnetizing fields compensate\nto some extent, such that they do not influence the frequency\nof the acoustical mode of a SAF as much as the anisotropies\nand the interlayer exchange couplings do.6\nSignal Spectral Peak-to-peak Full Width Zero crossings\nshape separation at Half Maximum separation\nExpected signals and their stimulus dependence:\nV1;ac/V2\nrf <e(\u001fxx) 2\u000b! - -\nV2;ac/V2\nrfVdc \u0001Mz - 2\u000b! -\nSignal extraction procedure from experiments:\nd\ndHzVexp\n1;acestimated fromdVexp\nac\ndHz\f\f\f\f\nVdc=0d<e(\u001fxx)\nd!- - 2\u000b!\nd\ndHzVexp\n2;acestimated from\"\ndVexp\nac\ndHz\f\f\f\f\nVdc6=0\u0000dVexp\nac\ndHz\f\f\f\f\nVdc=0#\nd=m(\u001fxx)\nd!2p\n3\u000b! - -\nTABLE I. Summary of the expected lineshapes and linewidths for the different signals that can be encountered in rf-voltage-induced FMR\nexperiments. An hyphen is inserted when the concept is not applicable.\nB. Spin-waves within the free layer\nConversely, the frequencies of the modes of the free layer\nare strongly affected by the device diameter (Fig. 4). First,\nthe modes are pushed to higher frequencies as the device\nis shrunk. At remanence, the lowest frequency mode is at\nfQUP= 12:3GHz for a diameter of 500 nm; it reaches 19.5\nGHz for 60 nm devices (not shown). Second, the frequency\nspacing between the free layer modes increases substantially\nwhen downsizing the device.\nThe first effect – increase of fQUPat downscaling – is in-\ndicative of a dependence of some effective fields with the\ndevice diameter. Among the effective fields, the only ones\nthat vary with the diameter are the exchange fields (positive\ncontribution to the frequency fQUPif magnetization is non-\nuniform), the demagnetizing fields (positive contribution to\nthe frequency fQUPat downscaling) and the local effective\nanisotropy fields in case some process damages alter locally\nthe interface anisotropy at the perimeter of the free layer (neg-\native contribution to the frequency fQUPat downscaling) or\nalter the local magnetization of the rim of the free layer (pos-\nitive contribution to the frequency fQUPat downscaling). The\nexchange fields are related to the non uniformities of either the\nstatic configuration –the fact that the AP state is not perfectly\nuniform as inferred previously from the loop in Fig. 1– or non\nuniformities of the dynamic magnetization –i.e. the fact that\nthe quasi-uniform mode is not a strictly uniform mode–. If\nthe frequency increase was due to the sole demagnetizing ef-\nfects, it could be estimated from the demagnetizing factors of\ndisks23which areNz\u00191\u0000(3\u0019=8)t=a, wheretandaare\nthe thickness and radius of the free layer. However a fQUP\nagainst 1=aplot (not shown) has a perceivable curvature near\nall sizes; an unwise linear fit through fQUPagainst the ex-\npected\rNzwould give a slope of \u00192:5T, which is obvi-\nously too large for the magnetization of the free layer. This\nindicates that the sole change of the global shape anisotropy\nwith the device diameter is insufficient to account for the in-crease offQUPat downscaling: exchange contributions or non\nuniformities induced by process damages also contribute to\nthe frequencies. Exchange contributions should not contribute\nfor the largest devices, however even for those devices the ex-\nperimental frequencies are larger than the ones expected from\nglobal shape anisotropy only, which argues for some process\ndamages. Since the TMR is almost independent of the de-\nvice size14, we can reasonably assume that the MgO/FeCoB\ninterface is not substantially affected by the patterning and\nthat consequently the interface anisotropy is essentially pre-\nserved at the rim of the free layer. We conclude that part of\nthe increase of fQUPat downscaling is due to a reduced mag-\nnetization (magnetically ”dead” or weak zone) near the edges\nof the free layer. This interpretation is probably very much\nstack and process technology dependent, hence it should not\nbe considered as general.\nThe second effect – the increased frequency spacing be-\ntween the modes at small diameters – is the expected effect\nof the confinement of the spin waves and the resulting in-\ncrease of the exchange contribution to the mode frequencies17.\nThe eigenmodes of perpendicularly magnetized circular disks\nare well understood and can be described analytically in a\nsemi-quantitative manner21,24–28. The frequency spacing be-\ntween the lowest frequency modes scales with \r0HJwhere\nHJ=2Ak2\n\u00160MSis a generalized exchange field with Athe ex-\nchange stiffness. The effective wavevector kis reminiscent of\nthe lateral confinement and reads k2= (u2\n2\u0000u2\n1)=a2\u00199=a2\nwhereu1andu2are the first zeros of the first and second\nBessel functions21. The lowest frequency spinwave modes\ncan be resolved only if their frequency spacing is comparable\nor greater than their linewidth 2\u000b\r0(Hz+Hk\u0000Ms+HJ)\n(see appendix).\nThis condition can be used to define a critical device diam-\neter:\na2\ncrit=9A\n\u000b\u00160MS(Hz+Hk\u0000Ms)(4)\nFor large devices with a\u001dacritwe expect to observe a quasi-7\ncontinuum of overlapping modes above fQUP, while discrete\nnon-overlapping modes are anticipated in the opposite limit.\nTypical parameters of an FeCoB-based free layer include a\nmagnetization of \u00160Ms= 1:2T and a Gilbert damping of29\n\u000b= 0:01. From the quasi-uniform mode frequency, we can\nget our effective anisotropy which is Hk\u0000Ms= 330 kA/m.\nIf the exchange stiffness of the free layer was bulk-like (i.e.\nA= 22 pJ/m) like in ref. 17, the critical diameter would\nbe2acrit= 444 nm. In practice the small frequency spac-\nings between the modes of our samples indicates that the ex-\nchange stiffness of our free layer is in the range of 6-7 pJ/m\ni.e. well below the bulk value. This estimate of the exchange\nstiffness was deduced assuming perfectly pinned boundary\nconditions for the spin-waves at the device edge, which is a\nquestionable30assumption. However the exchange stiffness is\nanyway weak in the free layer and this can be also qualita-\ntively seen directly from the spin wave spectroscopy: indeed\nthe frequency spacing of the lowest modes of the reference\nlayer system is typically twice larger that the frequency spac-\ning of the lowest modes of the free layer [see for instance\nFig. 3(b)]. While the reason for this small value of the free\nlayer exchange stiffness is not entirely clear, we emphasize\nthat having such a small exchange stiffness is not uncommon\nin magnetic systems that comprise only a small number of\natomic layers, starting for instance from 2 pJ/m for a single\nlayer of iron31. Anyway with these parameters, we expect\na clear separation of the lowest frequency modes at rema-\nnence provided that the device diameter is much smaller than\n2acrit= 250 nm.\nIn practice for 300 and 500 nm devices a fine structure\ncan still be detected in the spin-wave density of states [see\nFig. 4(c)] but it is hard to count the modes and guess their\nfrequencies out of this fine structure. In the remainder of this\npaper, we shall thus only consider devices of diameter less\nthan 200 nm, in which the different spin-wave modes can be\nunambiguously resolved [see Fig. 4(e-f)].\nIV . LINESHAPE EVOLUTIONS WITH BIAS AND\nEXTRACTION OF THE GILBERT DAMPING\nLet us now compare the shapes of the experimental rectified\nsignal with those expected (see appendix). For that purpose\nwe harness the different bias dependences (Table I) of the rec-\ntified signals V1;acandV2;acto isolate each of them in the\nexperimental signal Vac. We identify V1;acto the experimen-\ntal curveVacmeasured at Vdc= 0, and we construct an esti-\nmate ofV2;acby subtracting Vacmeasured at Vdc= 0 from\nthat measured at Vdc6= 0 (see Table I). Note because of this\nsubtraction, any dependence of the spin-wave frequency with\nthedcvoltage will prevent the measurement of the voltage de-\npendence of the damping factor or of the voltage dependence\nof the exciting torques.\nWe illustrate this procedure in Fig. 5 in which we plot the\nfield derivatives of the so-calculated V1;acandV2;acrectified\nsignals in both fixed field or fixed frequency experimental con-\nditions for a device of diameter 90 nm. We center the curves\non the second lowest frequency eigenmode since it provides\n(a)(b)\n(c)(e)(f)(d)FIG. 4. (Color online). Dependence of the field derivative of the ac\nvoltage over the device diameter. Top panels: full spectral depen-\ndence in the window [6;40GHz]\u0002[\u000090;90mT]for 90 nm (left)\nand 500 nm (right) diameter devices. Bottom panels: frequency de-\npendence of the field derivative of the acvoltage after self-conformal\naveraging for various device sizes. The signal amplitudes have been\nnormalized to ease their comparison.\nthe largest signals and it is reasonably separated from both the\nquasi-uniform precession mode and from the other high order\nmodes. The obtained experimental V1;accurves [see Fig. 5\n(a) and (d)] have the expected line shapes (see appendix) with\na negative peak surrounded by two tiny positive halos (areas\nshaded in red). The separation between the two zero crossings\nis9:5\u00060:5mT or 285\u000610MHz. The obtained experimental\nV2;accurves also have the expected line shape of the deriva-\ntive of a Lorentzian distribution (see appendix). As expected,\nthe sign of the response changes with the dcbias voltage. The\nseparation between the positive and negative maxima of the\ndistribution are 6:1\u00060:5mT and 170\u000610MHz.\nThese four different ways of measuring the linewidths\n[Fig. 5 (a, d, b, e)] are consistent with a free layer damping\nof\u000b= 0:011\u00060:0003 . Indeed this value of damping would\npredict linewidths of 2\u000bf= 295 MHz and 2\u00160\u000b!=\r 0=\n9:54mT (materialized as black bars in [Fig. 5 (a, d) and (d)])\nand1:15\u000bf= 171 MHz or 1:15\u000b!=\r 0= 6:21mT (material-\nized as blue bars in [Fig. 5 (b, c, e, f)]).\nThis proposed value of damping is also consistent with the\nlinewidths of higher order spin waves that appear at larger fre-\nquencies but with a lower signal. This is illustrated in fig. 6\nwhere a comparison is drawn between the values of the damp-\ning estimated for each applied field from the second (and most\nintense) mode and from the third mode for a device of diame-\nter 80 nm. The used procedure is a direct fit of the experimen-8\ntal lineshapes to the derivative of Eq. 7 with the damping, the\nresonance frequency and the signal amplitude as free parame-\nters. For this specific device, the estimates of the damping pa-\nrameter are subjected to a random error of standard deviation\n0.0018 around a mean value of \u000b= 0:0119 . It is interesting\nto note that the non-local contributions32,33to the damping ex-\npected for the relatively large wavevectors of the second and\nthird spin-wave modes seem to be too small to be observed\nin our samples. Note that as mentioned earlier, the same pro-\ncedure can in principle not be applied to the quasi-uniform\nmode since it exhibits a strong dependence of the mode am-\nplitude with the field which invalidates the procedure to some\nextent. For the sake of completeness of this paper, we have\nanyway fitted the experimental QUP lineshapes with the field\nderivative of Eq. 8; this is not possible near zero field, as the\ncorresponding signal vanishes. The value of the Gilbert damp-\ning that would be illegitimately deduced would be 0:01, i.e.\n20% lower than the correct value. Besides, the estimates from\nthe QUP mode would exhibit a substantially larger spread in\nthe fit results [compare the histograms in in fig. 6(b)]. For\nthese two reasons, we consider that the reliable estimate of the\ndamping is the one extracted from the non-uniform modes.\nAbove 100 mV of dcbias, the amplitude of the constructed\nexperimental V2;acstart to depart from proportionality with\nVdcand a frequency shift is observed, as expected when dc\nfield-like spin torques are applied. This comes with by a dis-\ntortion of the line shape, probably linked to the modification of\nthe spin waves lifetimes by spin-transfer torque as commonly\nobserved in in-plane magnetized MTJs34.\nV . SUMMARY AND CONCLUSIONS\nIn this paper, we have studied how to use rf-voltage-\ninduced ferromagnetic resonance to study the spin-wave den-\nsity of states and the Gilbert damping in perpendicularly mag-\nnetized disks embodied in magnetic tunnel junctions. We have\napplied the field along the easy axis to preserve the cylindrical\nsymmetry of the magnetization energy functional. The inter-\nest of this configuration is that all the current-induced torques\nthat potentially excite the dynamics yield the same type of\nsusceptibility spectral shape. Additionally, this configuration\nis the sole in which the applied field and the frequency play\nsimilar roles near FMR so that consistency crosschecks be-\ntween variable-field and variable-frequency experiments can\nbe performed to reveal and suppress potential experimental\nartefacts.\nWorking in a situation in which the fixed layer and the free\nlayer are oppositely magnetized in a convenient way to clas-\nsify the spin-waves according to their hosting layer, as the two\nsub-systems have opposite eigenmode frequency-versus-field\nslopes. The dcbias dependence of the signal of the quasi-\nuniform mode is peculiar and can be used to ambiguously\nidentify the free layer quasi-uniform mode in the manifold of\nspin-waves. The non-uniform (higher order) spin-waves are\neasier to analyze, as their amplitudes weakly depend on the\napplied field so that field differentiation can be used safely for\nbackground subtraction. Optionally, the dynamic range of theexperiment can be improved by self-conformal averaging of\nthe resonance spectra.\nThe unambiguous identification of the spin-wave frequen-\ncies requires devices that are sufficiently small to avoid that\nthe spinwave modes overlap into a quasi-continuous density\nof states. The critical device size is set by the exchange\nstiffness, the damping, the magnetization and the effective\nanisotropy field. In practice, device diameters below 200 nm\nare needed in our low-damped FeCoB-based PMA system.\nFor each spin-wave mode, the rf-voltage-induced spin-\nwave spectra contain contributions from two different phys-\nical mechanisms. The first one is the standard STT-FMR-like\nsignal, whose spectral shape is a linear transverse susceptibil-\nity term. It is independent from the dcvoltage applied across\nthe MTJ. The second one is a variation of the time-averaged\nmagnetic configuration when the rfvoltage is applied. It is\nproportional to the dcvoltage applied across the MTJ and it\nhas the spectral shape of a non-linear longitudinal susceptibil-\nity. The bias dependence can be used to separate these two\nsignals. The analysis of their spectral shape yields the Gilbert\ndamping within the precessing layer. A single value of the\ndamping factor is found to account for the lineshapes of all\nstudied spin-waves.\nThe spectra of rf-voltage-induced rectified voltages for a\nvanishing dcvoltage bias are in principle sufficient to get the\nGilbert damping of the dynamically active layer. However\nas microwave methods are prone to artefacts, a consistency\ncheck exploiting the bias dependence of the resonance spectra\nis useful for a consolidation of the numerical estimation of the\ndamping.\nThis work was supported in part by the Samsung Global\nMRAM Innovation Program, who provided also the samples.\nCritical discussions with Vladimir Nikitin, Jean-Paul Adam,\nJoo-V on Kim and Paul Bouquin are acknowledged.\nVI. APPENDIX: SUSCEPTIBILITIES IN AN IDEALIZED\nPMA FILM\nIn this appendix, our aim is to determine the transverse and\nlongitudinal microwave susceptibility versus frequency fand\nstatic fieldHzfor a PMA film as a response to an harmonic\ntransverse field hxcos(!t)~ ex. We shall write the equations\nwith this transverse field hxbut any other effect that yields a\ntorque possessing a component transverse to the static mag-\nnetization will yield similar lineshapes. This includes current-\ninduced Oersted-Ampere fields but also Slonczewski STT and\nfield-like STT as soon as the reference layer magnetization\n~Mrefis not strictly collinear with that of the free layer ~Mfree.\nThe susceptibility tensor will be used to deduce the shape of\nthe line expected in rf-voltage-induced FMR, as summarized\nin Table I. Throughout this appendix, we assume a dcfield\nHzperfectly perpendicular to the plane and a free layer mag-\nnetization~M=Mz~ ez+mx~ ex+my~ ey, where the transverse\nterms are assumed small and written as complex numbers in\nthe frequency space. For conveniency, we will use the nota-\ntionH0=Hz+Hk\u0000Ms. We shall also write the frequencies\nin field units and define !0=!=\r 0. This is meant to empha-9\nddHzVexp2,a cforVdc= 100 mVddHzVexp1,a cforVdc=0\nddHzVexp2,a cforVdc=\u0000100 mV\nFIG. 5. (Color online). Rectified acsignals versus frequency at fixed\napplied field (left panels) or versus field at fixed frequency (right\npanels). The plotted data ared\ndHzV1;ac(black, panels a and d) and\nd\ndHzV2;ac(blue, panels b, c, e and f) as estimated according to the\nformulas of Table I. The arbitrary vertical scale is the same for all\npanels. The dotted black lines are separated by 9.5 mT and 270 MHz.\nThe dotted blue lines are separated by 6.1 mT or 170 MHz. These\ndotted lines correspond to the expected linewidth for a damping of\n0.011. The panel (b) is measured for a device different from that of\nthe other panels.\nsize the fact that the generalized field H0and the generalized\nfrequency!0play very similar roles in the FMR of perpendic-\nularly magnetized macrospin when near resonnance. We shall\nsystematically assume that \u000b\u001c1and only keep the lowest\norder of the damping terms in the equations. In sections VI A\n-VI C we make the macrospin approximation. This approxi-\nmation is discussed in section VI D.\nA. Transverse linear susceptibility\nFollowing the usual procedure, we project the linearized\nLandau-Lifshitz-Gilbert equation along ~ exet~ ey:\n\u001a\nH0(my+\u000bmx) +imx!0=hxMs\u000b\nH0(mx\u0000\u000bmy)\u0000imy!0=hxMs\nWe then invert this system of equations to get the susceptibil-\nitiesmx=\u001fxxhxandmy=\u001fyxhx. They are:\n\u001fxx=MsH0\nH02\u0000!02+ 2i\u000bH0!0(5)\nand\n\u001fyx=\u0000iMs!0\nH02\u0000!02+ 2i\u000bH0!0(6)\n-100- 500 5 01 000.0000.0050.0100.0150.0200\n.0080 .0100 .0120 .014Third modeSecond modeOccurrence (a. u.)D\namping parameterQuasi-uniform mode(\nb)DampingF\nield (mT)(a)FIG. 6. (Color online). Statistical view of the Gilbert damping\nparameters obtained from the fitting of the three lowest spin-wave\nresonances of a device of 90 nm diameter. For each applied field,\nthe spectral line shapes of the quasi-uniform mode is fitted using\nd\ndHzV2;acfunction (red symbols) while the lineshapes of the two\nnext modes are fitted withd\ndHzV1;acfunction (blue and black). The\npanel (a) gathers the values of the damping parameters that best fit\neach spectrum recorded at a given applied field. Panel (b) displays\nthe histogram of the distribution of these estimates of the Gilbert\ndamping for the non-uniform modes (dashed-dotted line histogram\nand its blue Gaussian guide to the eye, of half width 0.0009) and\na Gaussian fit (red curve) of the distribution for the quasi-uniform\nmode, of half width 0.0015.\nSeveral points are worth to remind:\n(i) The dctransverse susceptibilities are \u001fdc\nxx=Ms=H0and\n\u001fdc\nyx= 0.\n(ii) The in-phase transverse susceptibility \u001fxxis peaked at\nthe FMR condition !0=H0. It reaches\u001fFMR\nxx =\u00001\n2i\u000b\u001fdc\nxx.\n(iii) When near the FMR condition, we have \u001fyx\u0019\u0000i\u001fxx.\nHence the two transverse components of the magnetization\nare in quadrature and the forced precession is essentially\ncircular. Note that this holds true despite the fact that\nthe pumping field is linearly polarized (i.e. along (x)only).\nIt would also remain true for other (e.g STT) pumping torques.\nFrom Eq. 5 we deduce the classical expressions for the real\nand imaginary parts of the transverse susceptibility:\n<e(\u001fxx) =MsH0(H02\u0000!02)\n4\u000b2H02!02+ (H02\u0000!02)2(7)\n=m(\u001fxx) =\u00002\u000bMs!0H02\n4\u000b2H02!02+ (H02\u0000!02)2(8)\nThe lineshapes given by the above expressions are shown\nin Fig. 7. Their main properties are summarized in Table I.10\nB. Longitudinal non-linear susceptibility\nLet us now express the non-linear change of the longitu-\ndinal magnetization \u0001Mzthat occurs due to the precession.\nThis can be viewed as an rf-induced reduction of the rema-\nnence. Using the circularity of the precession near the FMR\nresonance and the conservation of the magnetization norm to\nsecond order in mx;y, one gets:\n\u0001Mz\u0019jjhxjj2\n2MSM2\nsH02\n[H02\u0000!02]2+ 4\u000b2H02!02(9)\nwherejjhxjjis the (constant) amplitude of the applied rffield.\nIt is worth noticing that the longitudinal loss of the magneti-\nzation is stationary (constant in time) despite the fact that the\nmagnetization precesses continuously.\nC. Lineshapes and linewidths of the susceptibiliies and their\nderivatives\nThe different susceptibility expressions are plotted in Fig. 7.\nThe lineshape of the functions \u0001Mzand\u001fxxare essentially\ndetermined by their denominator which are the fast varying\nfunctions of eqs. 8 and 9. As \u001fxxand\u0001Mzare two signatures\nof the same resonance process, their denominators are equal\n(see eqs. 8 and 9) and a simple algebra confirms that =m(\u001fxx)\nand\u0001Mzlead to the same frequency or field linewidths (Half\nWidth at Half Maximum) which are:\n\u0001!= 2\u000b!or equivalently, \u0001Hz= 2\u000bH0(10)\nNote that \u0001!(resp. \u0001Hz) is also the frequency (resp. field)\nspacing between the positive maximum and negative maxi-\nmum of<e(\u001fxx)about the FMR condition.\nWe stress that this linewidth is different from that obtained\nby conventional FMR in which people examine the derivative\nof the absorption signald=m(\u001fxx)\ndHzversusHz. The peak-to-\npeak separation of the conventional FMR signal is \u0001Hz=\n2p\n3\u000bH0=2p\n3\u000b!0. The factor 2=p\n3is 1.1547. The spectral\nshapes of<e(\u001fxx)and\u0001Mzas deduced above are to be used\nin the main part of the paper to describe respectively V1;ac\nandV2;ac(Table I).\nD. Linewidth beyond the macrospin approximation\nSome words of caution are needed as the calculations done\nso far the appendix are for the uniform precession mode of a\nmacrospin, while most of our experimental results were ob-\ntained on the higher order (non-uniform) spin-waves. When\nmodeling non-uniform spin-waves, one needs to take into ac-\ncount additional exchange and dipole-dipole terms.\nFor non-uniform spin-waves in perpendicularly magnetized\nsystem, the exchange fields related to the non uniformity of\nthe dynamical magnetizations mxandmycan be added to H0\nto form a new generalized field ~H=H+Hk\u0000Ms+2Ak2\n\u00160MS,\n/s48/s46/s57/s48 /s48/s46/s57/s53 /s49/s46/s48/s48 /s49/s46/s48/s53 /s49/s46/s49/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s45/s49/s48/s48/s46/s57/s48 /s48/s46/s57/s53 /s49/s46/s48/s48 /s49/s46/s48/s53 /s49/s46/s49/s48\n/s77\n/s122\n/s82/s101 /s40 /s41\n/s112/s112/s32/s82/s101/s115/s111/s110/s97/s110/s99/s101/s45/s108/s105/s107/s101\n/s71/s101/s110/s101/s114/s97/s108/s105/s122/s101/s100/s32/s102/s105/s101/s108/s100/s32/s111/s114/s32/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40 /s39/s32/s111/s114/s32/s72/s39/s41/s70/s87 /s72/s77/s100/s100/s73/s109 /s40 /s41\n/s100/s100 /s77\n/s122\n/s111/s114/s100/s100/s82/s101 /s40 /s41\n/s73/s109 /s40 /s41/s32\n/s32/s76/s111/s115/s115/s45/s108/s105/s107/s101/s111/s114FIG. 7. (Color online). Transverse susceptibilities, longitudinal sus-\nceptibility and their derivatives in the PMA macrospin model. The\ncurves are plotted for \u000b= 0:02and a resonance condition of unity.\nThe responses amplitudes have been normalized to ease the compar-\nison between the different lineshapes. The black horizontal segment\nin the bottom panel is the FWHM of =m(\u001fxx)and\u0001Mzor equiva-\nlently the peak-to-peak separation of <e(\u001fxx)(also sketched as the\nblack square dots). The blue segment sketches the peak-to-peak sep-\naration ofd=m(\u001fxx)\nd!0 (also sketched as the empty blue square dots).\nThe area shaped in red is the positive halo that surrounds the main\n(negative) peak ind<e(\u001fxx)\nd!0.\nwherekis a generalized wavevector21,26,28. The additional\nexchange contributions act on mxandmyon equal footing,\nhence they maintain the circularity of the precession. The\nGilbert linewidth for a non-uniform spin-wave is simply35,36:\n\u0001!=\u000b!k@!k\n\r0@~H(11)\nWhere this equation holds as ~His a circular term. As a result\nof Eq. 11 the proportionality of an eigemode linewidth to its\nfrequency (Eq. 10) is not broken by the exchange contribu-\ntions in non-uniform spin-waves.\nConversely, the directional nature of the dipole-dipole in-\nteraction is such that the related effective fields act differently\non the two dynamical magnetizations mxandmyfor spin-\nwaves having a non-radial character. As a result the dynamic\ndemagnetizing fields can not be simply added to the circu-\nlar precession H0term and they induce some ellipticity of\nthe precession of the non-uniform spin-waves. Dipole-dipole\ninteractions make the eigenmode frequency non-linear with\nthe field (see Eq. 52 in ref. 22). For the lowest lying non-\nuniform spin-wave the dipole-dipole stiffness field is MSkt=2\nwithk\u0019\u0019=a. It is negligible against the generalized field\n~Hfor the thickness used in practice in PMA systems meant\nfor spin-torque applications. The precession stays therefore\nessentially circular for all the modes observed experimentally11\nhere; we will thus consider that it is legitimate to use Eq. 10\nand deduce the damping from the ratio of the half frequency\nlinewidth to the eigenmode frequency.\nFinally, we would like to mention that our method is notrestricted to the PMA materials only: it should hold when the\nconsidered spin-waves are quasi-circular. In particular, this is\nthe case of exchange-dominated spin waves in in-plane mag-\nnetized systems37.\n\u0003thibaut.devolder@u-psud.fr\n1C. Kittel, Physical Review 73, 155 (1948).\n2C. Bilzer, T. Devolder, P. Crozat, C. Chappert, S. Cardoso, and\nP. P. Freitas, Journal of Applied Physics 101, 074505 (2007).\n3T. Kubota, J. Hamrle, Y . Sakuraba, O. Gaier, M. Oogane,\nA. Sakuma, B. Hillebrands, K. Takanashi, and Y . 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Vansteenkiste, B. V . Waeyenberge, A. N. Kuchko,\nV . V . Kruglyak, and H. Fangohr, Physical Review B 92, 054430\n(2015).\n34S. Petit, C. Baraduc, C. Thirion, U. Ebels, Y . Liu, M. Li, P. Wang,\nand B. Dieny, Physical Review Letters 98, 077203 (2007).\n35A. N. Slavin and P. Kabos, IEEE Transactions on Magnetics 41,\n1264 (2005).\n36J.-V . Kim, Physical Review B 73, 174412 (2006).\n37M. Dvornik and V . V . Kruglyak, Physical Review B 84, 140405\n(2011)."    },    {        "title": "1709.07483v1.Low_Gilbert_Damping_Constant_in_Perpendicularly_Magnetized_W_CoFeB_MgO_Films_with_High_Thermal_Stability.pdf",        "content": "1 \n Low Gilbert Damping Constant in Perpendicularly Magnetized \nW/CoFeB/MgO Films with High Thermal Stability  \nDustin M. Lattery1†, Delin Zhang2†, Jie Zhu1, Jian-Ping Wang2*, and Xiao jia Wang1* \n1Department of Mechanical  Engineering, University of Minnesota, Minneapolis, MN 55455, USA  \n2Department of Electrical and Computer Engineering , University of Minnesota, Minneapolis, MN \n55455, USA   \n†These authors contributed equally to this work.  \n*Corres ponding a uthor s: wang4940@umn.edu  & jpwang@umn.edu   \n \nAbstract:  Perpendicular magnetic materials with low damping constant and high thermal stability \nhave great potential for realizing high -density, non -volat ile, and low -power consumption \nspintronic devices, which can  sustain operation reliability for high processing temperatures. In this \nwork, we study the Gilbert damping constant ( α) of perpendicularly magnetized W/CoFeB/MgO \nfilms with a high perpendicular magnetic anisotropy (PMA) and superb thermal stability. The α of \nthese PMA films annealed at different temperatures ( Tann) is determined via an all -optical T ime-\nResolved Magneto -Optical Kerr Effect method. We find that α of these W/CoFeB/MgO PMA \nfilms decreases with increasing Tann, reaches a minimum of α = 0.016 at Tann = 350 °C, and then \nincreases to 0.024 after post -annealing at 400 °C. The minimum α observed at 350 °C is \nrationalized by two competing effects  as Tann becomes higher : the enhanced crystallization of \nCoFeB and dead -layer growth occurring at the two interfaces of the CoFeB layer. We further \ndemonstrate that α of the 400 °C -annealed W/CoFeB/MgO film is comparable to that of a reference \nTa/CoFeB/MgO PMA fil m annealed at 300 °C , justif ying the enhanced thermal stability of the W -\nseeded CoFeB films.  2 \n I. INTRODUCTION  \nSince the first demonstration of perpendicular magnetic tunnel junctions with \nperpendicular magnetic anisotropy (PMA) Ta/CoFeB/MgO stacks  [1], interfacial PMA materials \nhave been extensively studied as promising candidates for ultra -high-density and low -power \nconsumption spintronic devices, including spin -transfer -torque magnetic rando m access memory \n(STT -MRAM)  [2,3] , electrical -field induced magnetization switching  [4-6], and spin -orbit torque \n(SOT) devices  [7-9]. An interfacial PMA stack typically consists of a thin ferromagnetic layer \n(e.g., CoFeB) sandwiched between a heavy metal layer ( e.g., Ta) and an oxide layer ( e.g., MgO). \nThe heavy metal layer interface  with the ferromagnetic layer  is responsible for the spin Hall effect , \nwhich is  favorable  for SOT and skyrmion devices  [10,11] . The critical switching current ( Jc0) \nshould be minimized to decrease the power consumption  of perpendicular STT -MRAM and SOT \ndevices . Reducing  Jc0 requires  the exploration of  new materials with low Gilbert damping  constant \n(α), large spin Hall angle ( θSHE), and large effective anisotropy ( Keff) [12,13] .  \nIn addition, spintronic devices need to sustain operation reliability for processing \ntemperatures as high as 400 °C  for their integration with existing CMOS fabrication technologies,  \nproviding the standard  back -end-of-line process compatibility  [14]. Based on this requirement, the \nmagnetic properties of a PMA material shou ld be thermally stable at annealing temperature s (Tann) \nup to  400 °C. Unfortunately, Ta/CoFeB/MgO  PMA films commonly used in spintronic devices \ncannot survive with Tann higher than 350 °C, due to Ta diffusion or CoFeB oxidation at the \ninterfaces  [15,16] . The diffusion of Ta atoms can act as scattering  sites to increase the spin -flip \nprobability  [17] and lead to a higher Gilbert Damping constant ( α), a measure of the energy \ndissipation from the magnetic precession into phonons or magnons  [18].  3 \n Modifying the composition of thin-film stack s can prevent heavy metal diffusion , which  \nis beneficial to both lowering  α and improving thermal stability  [19]. Along this line , new  \ninterfacial PMA stacks have been developed, such as  Mo/CoFeB/MgO, to circumvent the \nlimitation on device processing temperatures  [20,21] . While Mo/CoFeB/MgO films can indeed \nexhibit PMA at temperatures higher than 400 °C , they cannot be used for SOT devices due to the \nweak spin Hall effect of the Mo layer  [20,21] . Recently, W/CoFeB/MgO PMA thin films have \nbeen proposed because of their PMA property at high post -annealing temperature  [22], and  the \nlarge spin Hall angle of the W laye r (θSHE  ≈ 0.30)  [23], which is twice that of a Ta layer \n(θSHE  ≈ 0.12 ~ 0.15)  [9]. While  there have been a few scattered studies demonstrating the promise \nof fabricating SOT devices using the W/CoFeB/MgO stacks, special attention  has been given to \ntheir PMA properties and functionalities as SOT devic es [24]. A systematic investigation is lacking \non the effect of Tann on α of W/CoFeB/MgO PMA thin films.  \n \nII. SAMPLE PREPERATION AND MAGNETIC CHARACTERIZATION  \nIn this work, we grow a series of W(7)/Co 20Fe60B20(1.2)/MgO(2)/Ta(3) thin films on \nSi/SiO 2(300) substrates (thickness in nanometers) with a magnetron sputtering system \n(<5×1 08 Torr). These films are then post -annealed at varying temperatures (Tann = 250 ~ 400  °C) \nwithin a high -vacuum furnace (<1×106 Torr)  and their magnetic properties and damping constants \nas a function of Tann are systematically investigated . For comparison, a reference sample of \nTa(7)/Co 20Fe60B20(1.2)/MgO(2)/Ta(3) is also prepared to examine the effect of seeding layer to \nthe damping constant of these PMA films. The saturation magnetization ( Ms) and anisotropy  of \nthese films are measured with the Vibrating Sample Magnetometer (VSM)  module of a Physical 4 \n Property Measurement System . Figure  1 plots the magnetic hysteresis  loops and  associated  \nmagnetic properties extracted from VSM measurements.  \n \n \nFigure 1. Room temperature magnetic hysteresis loops of W/CoFeB/MgO PMA thin films post -\nannealed at ( a) 250 °C, (b) 300 °C, (c) 350 °C, and ( d) 400 °C. Black and red curves  denote external \nmagnetic field ( Hext) applied along and perpendicular to the film plane, res pectively . (e-g) Plots of \nthe saturation magnetization ( Ms), effective interfacial anisotropy ( Keff × t), and interfacial \nanisotropy ( Ki) as functions of Tann. \n \nWith the increase of Tann, Ms for the W/ CoFeB /MgO films decreases from ~780  to \n~630  emu/cm3 [Fig. 1 (e)]. The effective interfacial anisotropy  [(Keff × t) depicted in Fig. 1 (f)] \nshows a n increasing trend  with Tann (from ~0.18  to ~0.34 erg/cm2 when Tann increases from 250 to \n350 °C) and saturates at Tann = 350 °C. The positive values of Keff × t suggest that these \nW/CoFeB /MgO films maintain high PMA properties at elevated temperatures including 400  °C, \ndemonstrating their enhanced thermal stability compared to Ta/ CoFeB /MgO films that can only \nsustain PMA up to 350 ° C. Removing the influence  of the demagnetization energy from Keff × t 5 \n results in the  interfacial anisotropy ( Ki), which changes from 0.6 to 0.7 erg/cm2 with the increase \nof Tann up to 350  °C and then decreases to ~0.6 erg/cm2 at Tann = 400  °C [Fig. 1(f)]. Details about \nthe determination of Ki and Keff are provided in Section S1 of the Supp lemental  Material  (SM).  \n \n \nFigure 2 . The dead -layer extraction results. ( a), (b), (c), and ( d) represent the series of samples \nannealed at 250, 300, 350, and 400 C respectively. The tdead value is the extrapolated x-axis \nintercept from the linear fitting of the thickness -dependent saturation magnetization area product \n(Ms×t). \n \nWe attribute the decrease of Ki at high Tann to the growth of a dead layer at the CoFeB  \ninterfaces, which becomes prominent at higher Tann. To quantitatively determine the thickness of \nthe dead  layer as Tann increases, we prepare four sets of PMA stacks of \nW(7)/CoFeB( t)/MgO(2)/Ta( 3). One set contains five stacks with varying thicknesses for the \nCoFeB layer ( t = 1.2, 1.5, 1.8, 2.2, and 2.5 nm) and is post -annealed at a fixed Tann. Four Tann of 6 \n 250, 300, 350, and 400 °C  are used for four sets of the PMA stacks, respectively. The anne aling \nconditions are the same as those for the W(7)/CoFeB(1.2)/MgO(2)/Ta(3) samples for discussed \npreviously. We measure the magnetic hysteresis loops  of these samples using VSM and plot their \nsaturation magnetization area product (\nsMt ) as a function of film thickness ( t) in Fig. 2. Linear \nextrapolati on of the \nsMt  data provides the dead -layer thickness, at which the magnetization \nreduces to zero  as illustrate d by the x-axis intercept in Fig. 2.  \n \nIII. TR-MOKE MEASUREM ENTS  \nThe magnetization dynamics of these PMA thin films are determined using the all-optical  \nTime-Resolved Magneto -Optical Kerr Effect (TR -MOKE) method  [25-29]. This  pump -probe  \nmethod utilizes ultra -short laser pulses to thermally demagnetize the sample  and probe  the \nresulting Kerr rotation angle ( θK). In the polar -MOKE configuration,  θK is proportional to the \nchange of the out-of-plane  component of magnetization [Mz in Fig. 3(a)] [30]. Details  of the TR -\nMOKE setup are provided  in Section S2 of the SM.  \nThe TR-MOKE signal is fitted to the equation \n//\nK sin 2t C tA Be D ft e      , \nwhere A, B, and C are the offset, amplitude, and exponential decaying constant of the thermal \nbackground , respectively . D denotes the amplitude  of oscillations , f is the resonance  frequency , φ \nis a phase shift (related to the demagnetization process), and  is the relaxation time of \nmagnetization precession.  Directly from TR -MOKE measurements, an effective damping constant \n(αeff) can be extracted based on the relationship αeff = 1/(2πf). However, αeff is not an intrinsic \nmaterial property; rather, it depends on measurement conditions, such as the applied field direction \n[θH in Fig. 3(a)], the magnitude of the applied field ( Hext), and inhomogeneities of the sample ( e.g. \nlocal variation in the magnetic properties of the sample)  [31,32] .  7 \n  \n \nFigure  3. (a) Definition of the parameters and angles used in TR -MOKE experiments. The red \ncircle indicates the magnetization precession. θ is the equilibrium direction of the magnetization . \nθK is measured by the probe beam at a given time delay (Δ t). (b) The TR -MOKE data (open \nsymbols) and model fitting of  θK (black curves) for the 400 °C sample at 76° , for varying Hext from \n2.0 to 20 kOe.  \n \n \nTo obtain the Gilbert damping constant, the inhomogene ous contribution needs to be \nremoved from αeff, such that the remaining value of damping is a n intrinsic  material property  and \nindependent of the measurement conditions. To determine the inhomogeneous broadening in the 8 \n sample, the effective anisotropy field (\nk,eff eff s 2/ H K M ) needs to be pre -determined from either \n(1) the magnetic hysteresis loops; or (2) the fitting results of  f vs. Hext obtained from TR -MOKE.  \nThe resonance frequency, f, can be related to Hext through the Smit -Suhl approach  by identifying \nthe second derivatives of the total magnetic free energy, which combines a Zeeman energy, an \nanisotropy energy, and a demagnetization energy [33-35]. For a perpen dicularly magnetized thin \nfilm,  f is defined by Eqs. ( 1-4) [35]. \n12 f H H\n\n,      (1) \n 2\n1 ext H k,eff cos cos H H H      \n,     (2) \n  2 ext H k,eff cos cos 2 H H H      \n,     (3) \n  ext H k,eff2 sin sin 2HH  \n.             (4) \nThis set of equations permits calculat ion of  f with the material gyromagnetic ratio ( ), Hext, \nθH, Hk,eff, and the angle between the equilibrium magnetization di rection and the surface normal \n[θ, determined by Eq. ( 4)]. The measured values of f as a function of Hext can be fitted to Eq. (1) \nby treating and Hk,eff as fitting parameters. To minimize the fitting errors resulting from the \ninhomogeneous broadening effect that is pronounced at the low fields, we use measured \nfrequencies  at high fields ( Hext > 10 kOe) to determine Hk,eff. \nWith a known value of Hk,eff , the Gilbert damping constant of the sample can be determined \nthrough a fitting of the inverse relaxation time (1/ ) to Eq. (5). The two terms of Eq. (5) take into \naccount , respectively,  contributions from the intrinsic Gilbert damping of the materials (first term) \nand inhomogeneous broadening (second term)  [31]:  \n 1 2 k,eff\nk,eff1 1 1\n22dH H HdH   \n,     (5) 9 \n where H1 and H2 are related to the curvature of the magnetic free energy surface as defined by \nEqs. (2) and (3) [35,36] . The second term on the right side of Eq. (5) capture s the inhomogeneous \neffect by attributing it to a spatial variation in the magnetic properties (Δ Hk,eff), analogous  to the \nlinewidth broadening effect in F erromagnetic Resonance  measurements  [37]. The magnitude of \nk,eff/d dH\n can be calculated once the relationship of ω vs. Hext is determined with a numerical \nmethod. Both α and Δ Hk,eff (the inhomogeneous term related to the amount of spatial variation in \nHk,eff) are determined via the fitting of the measured 1/  based on Eq. ( 5). In this way, we can \nuniquely extract the field -independent  α, as an intrinsic material property, from the ef fective \ndamping ( αeff), which is directly obtained from TR -MOKE and dependent on Hext. \nIt should be noted here that the inhomogeneous broadening of the magnetization precession \nis presumably due to the multi -domain structure of the materials, which becomes  negligible in the \nhigh-field regime ( Hext >> Hk,eff) as the magnetization direction  of multiple magnetic domains  \nbecomes uniform. This is also reflected by the fact that the derivative in the second term of Eq.  (5) \napproaches zero for the high -field regim e [38]. \n \nIV. RESULTS AND DISCUSSION  \nThe measurement method is validated  by measuring  the Tann = 400 C at multiple angles \n(θH) of the external magnetic field direction.  By repeating this meas urement at varying  θH, we can \nshow that α is an intrinsic material property , independent of θH. Figure 4(a) plots the resonance \nfrequenc ies derived from TR -MOKE and model fittings for the 400 °C sample at two field \ndirections ( θH = 76° and 89° ).  For the data acquired at θH = 89°, a minimum f occurs at Hext ≈ Hk,eff. \nThis corresponds to the smallest amplitude of magnetization precession, when the equilibrium \ndirection of the magnetization is aligned with the applied field direction at the magnitud e of Hk,eff 10 \n [35]. The dip at this local minimum  diminishes when θH decreases, as reflected by the comparison \nbetween the red ( θH = 89°) and blue ( θH = 76°) lines in Fig. 4(a). With the Hk,eff extracted from the \nfitting of frequency data with θH = 89°, we generate the plot of theoretically predicted f vs. Hext \n[θH = 76° theory, blue line in Fig. 4(a)], which agrees well with experimental data [open square s \nin Fig. 4(a)].  \n \n \nFigure 4. (a) Measured f vs. Hext results for the 400 C sample at θH = 89° (open circles) and \nθH = 76° (open squares) and corresponding modeling at θH = 89° (red line) and θH = 76° (blue line) . \n(b) The measured inverse of relaxation time  (1/) at θH = 89° (open symbols) and the fitting of 1/  \nbased on Eq. (5) (dotted line). For reference, the first term of 1/  in Eq. (5) is also plotted (solid \nline), which accounts for the contribution from the Gilbert damping  only. (c) αeff as a function of \nHext for θH = 89° (red circles). Black circles are the extracted Gilb ert damping, which is \nindependent of Hext. The black dotted line shows the average of this extracted damping; ( d) and ( e) \ndepict similar plots of 1/  and damping constants for θH = 76°. Error bars in ( b) through ( e) come \nfrom the uncertainty in the mathematical fitting.  11 \n  \nThe inverse relaxation time (1/ should also have a minimum  value near Hk,eff for θH = 89° \nif the damping was purely from  Gilbert damping [as shown by the solid lines in Figs.  4(b) and \n4(d)]; however, the measured data do  not follow this trend. Adding the inhomogeneous term \n[dotted lines in Figs. 4(b) and 4(d)] more accurately describes the field  dependen ce of the measured \n1/[open symbols  in Figs. 4(b) and 4(d)] It should be noted that the dip of the predicted 1/ occurs \nwhen the frequency derivative term in Eq.  (5) approaches  zero; however,  this is not captured by \nthe measurement due to the finite interval over which we vary  Hext. Figures 4(c) and 4(e) depict  \nthe field -dependent effective damping ( αeff) and the Gilbert damping ( α) as the intrinsic material’s \nproperty obtained from fitting the measured 1/.  \nWith the knowledge that the value of α extracted with this method is the intrinsic material \nproperty, we repeat this data reduction technique for the annealed W/CoFeB /MgO samples \ndiscussed  in Fig. 1. The symbols in Fig. 5 represent  the resonance frequency and damping \nconstants (both effective damping and Gilbert damping) for all samples measured at θH ≈ 90°. The \nfittings for the resonance frequency [red lines , from Eq.  (1)] are also shown to demonstrate  the \ngood agreement between our TR -MOKE measurement and theoretical prediction. The \nuncertainties of f, , and Hk,eff are calculated from the least-square s fitting uncertainty and the \nuncertainty of measuring Hext with the Hall sensor.  12 \n \n \nFigure 5 . Results for f (a-d) and αeff (e-h, on a log scale) for individual samples. For comparison, \nthe Gilbert damping constant α is also plotted by subtracting the inhomogeneous terms from αeff. \nThe dashed line in (e -h) indicat es the average α. All samples are measured at θH = 90° except for \nthe 400 C sample ( θH = 89°).  \n \nThe summary of the anisotropy and damping measured via TR -MOKE is shown in Fig. 6. \nFigure 6(a) plots Hk,eff obtained from VSM (black open circle s) and TR -MOKE  (blue open \nsquares) , both of which exhibit a monotonic increasing trend as Tann becomes higher.  \nDiscrepancies in Hk,eff from these two methods can be attributed to the difference in the size of the \nprobing region , which is highly localized in TR -MOKE but sample -averaged in VSM. Since Hk,eff \ndetermined  from  TR-MOKE is obtained from fitting the measured frequency for a localized region, \nwe expect these values  more consistently  describ e the magnetization precession th an those \nobtained from VSM. The increase in Hk,eff  with Tann can be partially attributed to the crystallization 13 \n of the CoFeB layer  [32]. For temperatures higher than 350 °C, this increasing trend  of Hk,eff begins \nto lessen, presumably  due to the diffusion of W atoms into the CoFeB layer , which is more \npronounced at higher Tann. The W diffusion process  is also responsible for the decrease in Ms of \nthe CoFeB layer as Tann increases  [Fig. 1 (e)]. Subsequently, the  decrease in Ms leads to a further -\nreduced demagnetizing energy  and thus a  larger Hk,eff.  \nSimilar observation of Ms has been reported in literature for Ta/CoFeB/MgO PMA \nstructures and attribute d to the growth of a dead layer  at the heavy metal/CoFeB inter face [1]. \nFigure 6(b) summarizes tdead as a function of  Tann with tdead increas ing from 0.17 to 0.53 nm as Tann \nchanges fr om 250 to 400  C, as discussed in Section II.  \n \n \nFigure 6. Summary of the magnetic properties of W -seeded CoFeB as a function of Tann. (a) The \ndependence of Hk,eff on Tann obtained from both the VSM (black open circles) and TR-MOKE \nfitting (blue  open squares ). (b) The dependence of dead -layer thickness on Tann. (c) Damping \nconstants as a function of Tann. The minim um damping constant of α = 0.016 occurs at 350°C. The \nvalues for the all samples are obtained from measurements at θH = ~90°. For comp arison, α of the  \nreference Ta/CoFeB/MgO PMA sample annealed at 300 °C is also shown as a red triangle in ( c). 14 \n  \nFigure  6(c) depicts the dependence of α on Tann, which first decreases with Tann, reaches a \nminimum of 0.016 at 350 °C, and then increases as Tann rises to 400 °C. Similar trends have been \nobserved for Ta/CoFeB/MgO previously (minimum α at Tann = 300 °C) [32]. We speculate that \nthis dependence of damping on Tann is due to two competing effects: (1) the inc rease in \ncrystallization in the CoFeB layer with Tann which reduces the damping, and (2) the growth of a \ndead layer, which results from the diffusion of W and B atoms and is prominent at higher Tann. At \nTann = 400 °C, the dead -layer formation leads to a la rger damping presumably due to an increase \nin scattering sites (diffused atoms) that contribute to spin -flip events, as described by the Elliot -\nYafet relaxation mechanisms [17]. The observation that our W -seeded samples still sustain \nexcellent PMA properties at Tann = 400 °C confirms  their enhanced thermal stability, compared \nwith Ta/CoFeB/MgO stacks which fail at Tann = 350 °C or higher.  \nThe damping constants are comparable for the W/CoFeB/MgO and Ta/CoFeB/MgO films \nannealed at 300 °C, both of which are higher than that of the W/ CoFeB /MgO PMA with the \noptimal Tann of 350 °C.  Nevertheless,  our work focuses on the enhanced thermal stability of W -\nseeded CoFeB PMA films while still maintaining a relatively low damping constant. Such an \nadvantage enables W -seeded CoFeB layers to be viable and promising alternatives to \nTa/CoFeB/MgO , which is currently widely used in spintronic devices.  \n \nV. CONCLUSION  \nIn summary, we deposit a series of W -seeded CoFeB  PMA films with varying annealing \ntemperatures up to 400  °C and conduct ultrafast all -optical TR -MOKE measurements to study \ntheir magnetization precession dynamics. The Gilbert damping, as a n intrinsic  material property, \nis proven to be independent of meas urement conditions, such as the amplitudes and directions of 15 \n the applied field. The damping constant varies with Tann, first decreasing and then increasing, \nleading to a minimum of α = 0.016 for the sample anneale d at 350  °C. Due to the  dead -layer \ngrowth , the damping constant slightly increases to α = 0.024 at Tann = 400  °C, comparable to the \nreference Ta/ CoFeB /MgO PMA film annealed at 300°C, which demonstrates the improved \nenhanced thermal stability of W/ CoFeB /MgO over the Ta/ CoFeB /MgO structures. This strongly \nsuggests the great potential of W/ CoFeB /MgO PMA material systems for future spintronic device \nintegration that requires materials to sustain a processing temperature as high as 400 °C.  \n  16 \n Acknowledgements  \nThis work is supported by C -SPIN  (award #: 2013 -MA-2381) , one of six centers of STARnet, a \nSemiconductor Research Corporation program, sponsored by MARCO and DARPA. The authors \nwould like to thank Prof.  Paul Crowell and Dr. Changjiang Liu for valuable discussions.  \n \nSupplemental  Materials  Available:  Complete description of the TR -MOKE mea surement \nmethod, the angular dependent result summary , and the  description for determining the  interface \nanisotropy.  \n \n  17 \n References  \n[1] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. \nHayakawa, F. Matsukura, and H. Ohno, A perpendicular -anisotropy CoFeB -MgO magnetic tunnel \njunction, Nat. Mater. 9, 721 (2010).  \n[2] H. Sato, E. C. I. Enobio, M. 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B 92, 224402, 224402 (2015).  \n "    },    {        "title": "2212.01029v4.Equivalence_between_the_energy_decay_of_fractional_damped_Klein_Gordon_equations_and_geometric_conditions_for_damping_coefficients.pdf",        "content": "arXiv:2212.01029v4  [math.AP]  23 Aug 2023EQUIVALENCE BETWEEN THE ENERGY DECAY OF\nFRACTIONAL DAMPED KLEIN–GORDON EQUATIONS AND\nGEOMETRIC CONDITIONS FOR DAMPING COEFFICIENTS\nKOTARO INAMI AND SOICHIRO SUZUKI\nAbstract. We consider damped s-fractional Klein–Gordon equations on Rd,\nwheresdenotes the order of the fractional Laplacian. In the one-di mensional\ncased= 1, Green (2020) established that the exponential decay for s≥2 and\nthe polynomial decay of order s/(4−2s) hold if and only if the damping coeffi-\ncient function satisfies the so-called geometric control co ndition. In this note,\nwe show that the o(1) energy decay is also equivalent to these conditions in\nthe case d= 1. Furthermore, we extend this result to the higher-dimens ional\ncase: the logarithmic decay, the o(1) decay, and the thickness of the damp-\ning coefficient are equivalent for s≥2. In addition, we also prove that the\nexponential decay holds for 0 < s <2 if and only if the damping coefficient\nfunction has a positive lower bound, so in particular, we can not expect the\nexponential decay under the geometric control condition.\n1.Introduction\nWe consider the following fractional damped Klein–Gordon equations onRd:\n(1.1) utt(t,x)+γ(x)ut(t,x)+(−∆+1)s/2u(t,x) = 0,(t,x)∈R≥0×Rd,\nwheres >0, and 0 ≤γ∈L∞(Rd). Here we note that γutrepresents the damping\nforce and the operator ( −∆+1)s/2is defined by the Fourier transform on L2(Rd);\n(−∆+1)s/2u:=F−1(|ξ|2+1)s/2Fu, ξ∈Rd.\nWe recast the equation ( 1.1) as an abstract first-order equation for U= (u,ut):\nUt=AγU,Aγ=/parenleftbigg0 I\n−(−∆+1)s/2−γ(x)/parenrightbigg\n, (1.2)\nthenAγgenerates a C0-semigroup ( etAγ)t≥0onHs/2(Rd)×L2(Rd) (see [4]). Here\nthe Sobolev space Hr(Rd) is defined by\nHr(Rd):=/braceleftbigg\nu∈L2(Rd) :/bardblu/bardbl2\nHr=/integraldisplay\nRd(|ξ|2+1)r|Fu(ξ)|2dξ <∞/bracerightbigg\n.\nIn this paper, we discuss the decay rate of the energy\nE(t):=/bardbletAγ(u(0),ut(0))/bardblHs/2×L2=/parenleftbigg/integraldisplay\nRd(|(−∆+1)s/4u(t,x)|2+|ut(t,x)|2)dx/parenrightbigg1/2\n.\n2020Mathematics Subject Classification. 35L05, 42A38.\nThe first author was supported by JST SPRING Grant Number JPMJ SP2125 , the Inter-\ndisciplinary Frontier Next-Generation Researcher Progra m of the Tokai Higher Education and\nResearch System .\nThe second author was supported by Japan Society for the Prom otion of Science (JSPS)\nKAKENHI Grant Number JP20J21771 and JP23KJ1939.\n12 K. INAMI AND S. SUZUKI\nBy standard calculus, we have E(t) =E(0) ifγ≡0 and the exponential energy\ndecay ifγ≡C >0. In recent works, the intermediate case, that is, the case that\nγ= 0 on a large set is studied:\nDefinition 1.1. We say that Ω ⊂Rdsatisfies the Geometric Control Condition\n(GCC) if there exist L >0 and 0< c≤1 such that for any line segments l∈Rdof\nlengthL, the inequality\nH1(Ω∩l)≥cL\nholds, where H1denotes the one-dimensional Hausdorff measure.\nBurq and Joly [ 2] proved that if γis uniformly continuous and {γ≥ε}satisfies\n(GCC) for some ε >0, then we have the exponential energy decay in the non-\nfractional case s= 2. After that, Malhi and Stanislavova [ 6] pointed out that\n(GCC) is also necessary for the exponential decay in the one-dimen sional case\nd= 1:\nTheorem 1.2 ([6, Theorem 1]) .Letd= 1, lets= 2, and let 0≤γ∈L∞(R)be\ncontinuous. Then the following conditions are equivalent:\n(1.3)There exists ε >0such that the upper level set {γ≥ε}satisfies (GCC).\n(1.4)There exist C,ω >0such that whenever (u(0),ut(0))∈H1(R)×L2(R),\nE(t)≤Cexp(−ωt)E(0)\nholds for any t≥0.1\n(1.5) lim\nt→+∞/bardbletAγ/bardblH2×H1→H1×L2= 0.\nNote that for 0 ≤γ∈L∞(R), the condition ( 1.3) is also equivalent to that there\nexistsR >0 such that\ninf\na∈R/integraldisplaya+R\na−Rγ(x)dx >0.\nIn another paper [ 7], Malhi and Stanislavova introduced the fractional equation\n(1.1) and showed that if γis periodic, continuous and not identically zero, then\nwe have the exponential decay for any s≥2 and the polynomial decay of order\ns/(4−2s) for any 0 < s <2 in the case d= 1.\nRemark. Nonzero periodic functions satisfy (GCC) in the case d= 1, but it is not\ntrue in the higher-dimensional case d≥2. Wunsch [ 10] showed that continuous\nperiodic damping gives the polynomial energy decay of order 1 /2 for the non-\nfractional equation in the case d≥2. In addition, recently another proof and\nan extension to fractional equations of Wunsch’s result were obta ined by T¨ aufer\n[9] and Suzuki [ 8], respectively. Note that these results for periodic damping are\nestablished by reducing to estimates on the torus Td. Indeed, there are numerous\nstudies on bounded domains; see references in [ 2] and [3], for example.\nGreen [4] improved results of Malhi and Stanislavova as follows:\nTheorem 1.3 ([4, Theorem 1]) .Letd= 1, lets >0and let0≤γ∈L∞(R). Then\nthe following conditions are equivalent:\n1To be precise, the exponential decay estimate given in [ 6, Theorem 1] is a little weaker:\nE(t)≤Cexp(−ωt)/bardbl(u(0),ut(0))/bardblH2×H1. However, this is because the Gearhart–Pr¨ uss theorem\nin their paper ([ 6, Theorem 2]) is stated incorrectly. Using the theorem corre ctly (see Theorem\n3.1), one can obtain the exponential decay estimate E(t)≤Cexp(−ωt)E(0) as in ( 1.4).ENERGY DECAY OF DAMPED KLEIN–GORDON EQUATIONS 3\n(1.3)There exists ε >0such that the upper level set {γ≥ε}satisfies (GCC).\n(1.6)There exist C,ω >0such that whenever (u(0),ut(0))∈Hs(R)×Hs/2(R),\nE(t)≤/braceleftBigg\n(1+t)−s\n4−2s/bardbl(u(0),ut(0))/bardblHs×Hs/2if0< s <2,\nCexp(−ωt)E(0) ifs≥2\nholds for any t≥0.\nIn comparison with the result of [ 7], which states that ( 1.6) holds if γis peri-\nodic, continuous and not identically zero, Theorem 1.3refines this result by giving\na necessary and sufficient condition for ( 1.6). Furthermore, Theorem 1.3also im-\nproves the ( 1.3)⇐⇒(1.4) part of Theorem 1.2by extending it to fractional\nequations and removing the continuity of γ, but on the other hand, it lacks the\n(1.5) =⇒(1.3),(1.4) part. One of our goal is to recover this part for fractional\nequations:\nTheorem 1.4. Letd= 1, lets >0, and let 0≤γ∈L∞(R). Then the following\nconditions are equivalent:\n(1.3)There exists ε >0such that the upper level set {γ≥ε}satisfies (GCC).\n(1.6)There exist C,ω >0such that whenever (u(0),ut(0))∈Hs(R)×Hs/2(R),\nE(t)≤/braceleftBigg\nC(1+t)−s\n4−2s/bardbl(u(0),ut(0))/bardblHs×Hs/2if0< s <2,\nCexp(−ωt)E(0) ifs≥2\nholds for any t≥0.\n(1.7) lim\nt→+∞/bardbletAγ/bardblHs×Hs/2→Hs/2×L2= 0.\nWe also give the following result, which says that we cannot expect th e expo-\nnential decay for 0 < s <2 under (GCC).\nTheorem 1.5. Letd≥1, let0< s <2, and let 0≤γ∈L∞(Rd). Then there exist\nC,ω >0such that whenever (u(0),ut(0))∈Hs/2(Rd)×L2(Rd),\nE(t)≤Cexp(−ωt)E(0)\nholds for any t≥0if and only if essinfRdγ >0.\nNote that the “if” part easily follows by reducing to the constant da mping case,\nso we will prove the “only if” part. Furthermore, we extend Theore m1.4to the\nhigher-dimensional case d≥2 using a notion of thickness , which is equivalent to\n(GCC) in the case d= 1:\nDefinition 1.6. We say that a set Ω ⊂Rdisthickif there exists R >0 such that\ninf\na∈Rdmd(Ω∩(a+[−R,R]d))>0\nholds, where mddenotes the d-dimensional Lebesgue measure.\nThen we have the following result:\nTheorem 1.7. Letd≥2, lets≥2, and let 0≤γ∈L∞(Rd). Then the following\nconditions are equivalent:\n(1.8)There exists ε >0such that the upper level set {γ≥ε}is thick.4 K. INAMI AND S. SUZUKI\n(1.9)There exists C >0such that whenever (u(0),ut(0))∈Hs(Rd)×Hs/2(Rd),\nE(t)≤C\nlog(e+t)/bardbl(u(0),ut(0))/bardblHs×Hs/2\nholds for any t≥0.\n(1.10) lim\nt→+∞/bardbletAγ/bardblHs×Hs/2→Hs/2×L2= 0.\nThe implication ( 1.8) =⇒(1.9) is a generalization of the result given by Burq\nand Joly [ 2]. They established ( 1.9) under the so-called network control condition,\nwhich is stronger than ( 1.8). Also, similarly to the case d= 1, the condition ( 1.8)\nis equivalent to that there exists R >0 such that\ninf\na∈R/integraldisplay\na+[−R,R]dγ(x)dx >0.\nFinally, we explain the organizationof this paper. In Sections 2,3, and4, we will\ngive proofs of Theorems 1.4,1.5, and1.7, respectively. To prove these theorems,\nwe use a kind of uncertainty principle and results of the C0semigroup theory.\n2.Proof of Theorem 1.4\nTo prove this theorem, we use the following result by Batty, Boriche v, and\nTomilov [ 1]:\nTheorem 2.1 ([1, Theorem 1.4]) .LetAbe a generator of a bounded C0-semigroup\n(etA)t≥0on a Banach space X, andλ∈ρ(A). Then the following are equivalent:\n(2.1)σ(A)∩iR=∅,\n(2.2) lim t→∞/bardbletA(λ−A)−1/bardblB(X)= 0.\nIn the case A=Aγ, forλ∈ρ(Aγ), the map ( λ−Aγ)−1:Hs/2(R)×L2(R)→\nHs(R)×Hs/2(R) is surjective. Thus, we have:\nLemma 2.2 ([6, Corollary 2]) .For the semigroup etAγof the Cauchy problem\n(1.2), the following are equivalent:\n(2.3)σ(Aγ)∩iR=∅,\n(2.4) lim t→∞/bardbletAγ/bardblHs×Hs/2→Hs/2×L2= 0.\nProof of Theorem 1.4.It is enough to show that ( 1.7) =⇒(1.3), since (1.3)⇐⇒\n(1.6) is already known by Green [ 4] (Theorem 1.3) and (1.6) =⇒(1.7) is triv-\nial. Suppose that ( 1.7) holds, that is, lim t→+∞/bardbletAγ/bardblHs×Hs/2→Hs/2×L2= 0. By\nLemma2.2, we have iR⊂ρ(Aγ). This implies that for each λ∈R, there exists\nsomec0>0 such that\nc0/bardblU/bardblHs/2×L2≤ /bardbl(Aγ−iλI)U/bardblHs/2×L2\nholds for any U∈Hs(R)×Hs/2(R). Letting u∈L2(Rd) andU= ((−∆ +\n1)−s/4u,iu), we obtain\n2c0/bardblu/bardbl2\nL2≤ /bardbl((−∂xx+1)s/4−λ)u/bardbl2\nL2+/bardbl((−∂xx+1)s/4−λ+iγ)u/bardbl2\nL2\n≤3/bardbl((−∂xx+1)s/4−λ)u/bardbl2\nL2+2/bardblγu/bardbl2\nL2.ENERGY DECAY OF DAMPED KLEIN–GORDON EQUATIONS 5\nNow we consider the case λ= 1. Let u∈Hs/2(R) satisfy supp /hatwideu⊂[−D,D] for\nsomeD >0, which is chosen later. For such u, we have\n/bardbl((−∂xx+1)s/4−1)u/bardbl2\nL2=/integraldisplayD\n−D/bracketleftBig\n(|ξ|2+1)s/4−1/bracketrightBig2\n|/hatwideu(ξ)|2dξ\n≤/bracketleftBig\n(D2+1)s/4−1/bracketrightBig2\n/bardblu/bardbl2\nL2.\nHence, taking D >0 small enough, we get some c >0 such that\nc/bardblu/bardblL2≤ /bardblγu/bardblL2\nholds for any u∈Hs/2(R) satisfying supp /hatwideu⊂[−D,D]. Fixf∈ S(R)\\ {0}\nsuch that supp /hatwidef⊂[−D,D] and write fa(x):=f(x−a) for each a∈R, so that\n/hatwidefa(ξ) =eiaξ/hatwidef(ξ). Then, for each a∈RandR >0, we have\n0< c/bardblf/bardblL2=c/bardblfa/bardblL2≤ /bardblγfa/bardblL2=/parenleftBigg/integraldisplay\n[a−R,a+R]+/integraldisplay\n[a−R,a+R]c/parenrightBigg\n|γ(x)fa(x)|2dx.\nThe second integralgoesto 0 as R→+∞sinceγis bounded and |fa|2is integrable,\nandthisconvergenceisuniformwithrespectto a. Furthermore,forthefirstintegral,\nwe have/integraldisplaya+R\na−R|γ(x)fa(x)|2dx≤ /bardblγ/bardblL∞/bardblf/bardbl2\nL∞/integraldisplaya+R\na−Rγ(x)dx,\nsinceγandfare bounded and /bardblfa/bardblL∞=/bardblf/bardblL∞. Thus, there exists R >0 such\nthat\ninf\na∈R/integraldisplaya+R\na−Rγ(x)dx >0\nholds, which is equivalent to ( 1.3). /square\n3.Proof of Theorem 1.5\nThis section is based on the proof of Theorem 2 in Green [ 4]. To prove this\ntheorem, we use the classical semigroup result by Gearhart, Pr¨ u ss, and Huang:\nTheorem 3.1 (Gearhart–Pr¨ uss–Huang) .LetXbe a complex Hilbert space and let\n(etA)t≥0be a bounded C0-semigroup on Xwith infinitesimal generator A. Then\nthere exist C,ω >0such that\n/bardbletA/bardbl ≤Cexp(−ωt)\nholds for any t≥0if and only if iR⊂ρ(A)andsupλ∈R/bardbl(iλ−A)−1/bardblB(X)<∞.\nProof of Theorem 1.5.We will prove the contraposition of the “only if” part of\nTheorem 1.5, that is, if the energy decays exponentially and essinf x∈Rdγ(x) = 0\nholds, then s≥2. By the Gearhart–Pr¨ uss–Huang theorem and the exponential\ndecay, there exists c0>0 such that\nc0/bardblU/bardbl2\nHs/2×L2≤ /bardbl(Aγ−iλI)U/bardbl2\nHs/2×L2\nholds for any U∈Hs/2(Rd)×L2(Rd) and any λ∈R. Letting u∈L2(Rd) and\nU= ((−∆+1)−s/4u,iu), we obtain\n2c0/bardblu/bardbl2\nL2≤ /bardbl((−∆+1)s/4−λ)u/bardbl2\nL2+/bardbl((−∆+1)s/4−λ+iγ)u/bardbl2\nL2\n≤3/bardbl(−∆+1)s/4−λ/bardbl2\nL2+2/bardblγu/bardbl2\nL2.6 K. INAMI AND S. SUZUKI\nNow letu∈L2(Rd) satisfy\nsupp/hatwideu⊂ {ξ∈Rd:|(|ξ|2+1)s/4−λ| ≤K}=:Aλ(K)\nfor some K, which is chosen later. For such u, we have\n/bardbl((−∆+1)s/4−λ)u/bardbl2\nL2=/integraldisplay\nAλ(K)[(|ξ|2+1)s/4−λ]2|/hatwideu(ξ)|2dξ\n≤K2/bardblu/bardbl2\nL2.\nHence, taking K >0 small enough, we get some c >0 such that\n(3.1) c/bardblu/bardbl2\nL2≤ /bardblγu/bardbl2\nL2\nholds for any u∈L2(Rd) satisfying supp /hatwideu⊂Aλ(K) with some λ∈R.\nWe prove s≥2 by contradiction. Assume that s <2. In this case, the thickness\nof the annulus Aλ(K) is unbounded with respect to λ:\nlim\nλ→∞/vextendsingle/vextendsingle/vextendsingle/vextendsingle/radicalBig\n(λ+K)4/s−1−/radicalBig\n(λ−K)4/s−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle= lim\nλ→∞λ4/s−1\nλ2/s=∞.\nThus, the inequality ( 3.1) holds for any u∈L2(Rd) such that supp /hatwideuis compact.\nTo see this, notice that there exist a∈Rdandλ∈Rsatisfying a+supp/hatwideu⊂Aλ(K)\nfor such u. Therefore, letting ua(x):=eia·xu(x), we have\nc/bardblu/bardbl2\nL2=c/bardblua/bardbl2\nL2≤ /bardblγua/bardbl2\nL2=/bardblγu/bardbl2\nL2\nsince supp/hatwiderua=a+supp/hatwideu⊂Aλ(K).\nNow note that Eε:={x∈Rd:γ(x)< ε}has a positive measure for any ε >0,\nsince essinf x∈Rdγ(x) = 0. For each ε >0, we take a subset Fε⊂Eεsuch that\n0< md(Fε)<∞. TakeR,ε >0 arbitrarily and set\nfε:=χFε//radicalbig\nmd(fε), gR,ε:=F−1χB(0,R)Ffε,\nwhereχΩdenotes the indicator function of Ω ⊂Rd. By the definition, we have\nsupp/hatwidestgR,ε⊂B(0,R) andgR,ε→fεasR→ ∞inL2(Rd). Therefore, applying the\ninequality ( 3.1) togR,ε, we get\nc/bardblgR,ε/bardblL2≤ /bardblγgR,ε/bardblL2\n≤ /bardblγfε/bardblL2+/bardblγ(gR,ε−fε)/bardblL2\n=/parenleftbigg1\nmd(Fε)/integraldisplay\nFε|γ(x)|2dx/parenrightbigg1/2\n+/bardblγ(gR,ε−fε)/bardblL2\n≤ε+/bardblγ(gR,ε−fε)/bardblL2.\nTaking the limit as R→+∞, we obtain\n0< c=c/bardblfε/bardblL2≤ε.\nThis is a contradiction since ε >0 is arbitrary. /square\n4.Proof of Theorem 1.7\nThe proof of ( 1.10) =⇒(1.8) is similar to that of ( 1.7) =⇒(1.3) in Section\n2, and the implication ( 1.9) =⇒(1.10) is trivial. Therefore, we will show that\n(1.8) =⇒(1.9). We use a kind of the uncertainty principle to obtain a certain\nresolvent estimate for the fractional Laplacian:ENERGY DECAY OF DAMPED KLEIN–GORDON EQUATIONS 7\nTheorem 4.1 ([5, Theorem 3]) .LetΩ⊂Rdbe thick. Then there exist a constant\nC >0such that for each R >0, the inequality\n/bardblf/bardblL2(Rd)≤Cexp(CR)/bardblf/bardblL2(Ω)\nholds for any f∈L2(Rd)satisfying supp/hatwidef⊂B(0,R).\nIn orderto obtain the logarithmic energydecay ( 1.9), we use the following result.\nTheorem 4.2 ([2, Theorem 5.1]) .LetAbe a maximal dissipative operator (and\nhence generate the C0-semigroup of contractions (etA)t≥0) in a Hilbert space X.\nAssume that iR⊂ρ(A)and there exists C >0such that\n/bardbl(A−iλI)−1/bardblB(X)≤CeC|λ|\nholds for any λ∈R. Then, for each k >0, there exists Ck>0such that\n/bardbletA(I−A)−k/bardblB(X)≤Ck\n(log(e+t))k\nholds for any t≥0.\n4.1.Resolvent estimate. The proof of these propositions are based on [ 4].\nProposition 4.3. Lets≥1andΩ⊂Rdbe thick. Then there exist C,c >0such\nthat for all f∈L2(Rd)and allλ≥0,\ncexp(−Cλ)/bardblf/bardbl2\nL2(Rd)≤ /bardbl((−∆+1)s/2−λ)f/bardbl2\nL2(Rd)+/bardblf/bardbl2\nL2(Ω).\nProof of Proposition 4.3.LetAλ:={ξ∈Rd:|(|ξ|2+ 1)1/2−λ1/s| ≤1}. Since\nAλ⊂B(0,λ+2) and Ω is thick, Theorem 4.1implies that there exists C >0 such\nthat\n(4.1) /bardblf/bardblL2(Rd)≤Cexp(Cλ)/bardblf/bardblL2(Ω)\nholds for any λ≥0 and any f∈L2(Rd) satisfying supp /hatwidef⊂Aλ. Next, we set a\nprojection Pλ:=F−1χAλF, whereχAλdenotes the indicator function of Aλ. Then,\nsincePλfsatisfies the inequality ( 4.1) for each f∈L2(Rd), we obtain\n/bardblf/bardbl2\nL2(Rd)=/bardblPλf/bardbl2\nL2(Rd)+/bardbl(I−Pλ)f/bardbl2\nL2(Rd)\n≤Cexp(Cλ)/bardblPλf/bardbl2\nL2(Ω)+/bardbl(I−Pλ)f/bardbl2\nL2(Rd)\n=Cexp(Cλ)/bardblf−(I−Pλ)f/bardbl2\nL2(Ω)+/bardbl(I−Pλ)f/bardbl2\nL2(Rd)\n≤2Cexp(Cλ)/bardblf/bardbl2\nL2(Ω)+2Cexp(Cλ)/bardbl(I−Pλ)f/bardbl2\nL2(Ω)+/bardbl(I−Pλ)f/bardbl2\nL2(Rd)\n≤2Cexp(Cλ)/bardblf/bardbl2\nL2(Ω)+(2Cexp(Cλ)+1)/bardbl(I−Pλ)f/bardbl2\nL2(Rd).\nAlso, by Lemma 1 in [ 4], we have\nc/bardbl(I−Pλ)f/bardbl2\nL2(Rd)≤ /bardbl((−∆+1)s/2−λ)f/bardbl2\nL2(Rd)\nfor some c >0 independent with λ. Therefore, we conclude that\n/bardblf/bardbl2\nL2(Rd)≤Cexp(Cλ)/bracketleftBig\n/bardbl((−∆+1)s/2−λ)f/bardbl2\nL2(Rd)+/bardblf/bardbl2\nL2(Ω)/bracketrightBig\n.\n/square\nProposition 4.4. Lets≥2and assume that Ω⊂Rdis thick. Then there exist\nC,c >0such that for all U= (u1,u2)∈Hs(Rd)×Hs/2(Rd)and allλ∈R,\ncexp(−C|λ|)/bardblU/bardbl2\nHs/2(Rd)×L2(Rd)≤ /bardbl(A0−iλI)U/bardbl2\nHs/2(Rd)×L2(Rd)+/bardblu2/bardbl2\nL2(Ω).8 K. INAMI AND S. SUZUKI\nProof of Proposition 4.4.ForU= (u1,u2)∈Hs(Rd)×Hs/2(Rd), we set\n/parenleftbigg\nw1\nw2/parenrightbigg\n=/parenleftbigg\n(−∆+1)s/4−i\n(−∆+1)s/4i/parenrightbigg/parenleftbigg\nu1\nu2/parenrightbigg\n.\nBy the parallelogram law, we obtain\n/bardblw1/bardbl2\nL2(Rd)+/bardblw2/bardbl2\nL2(Rd)= 2/bardblU/bardbl2\nHs/2(Rd)×L2(Rd).\nMoreover, we have\n/bardbl(A0−iλI)U/bardbl2\nHs/2×L2=/bardbl(−∆+1)s/2(−iλu1+u2)/bardbl2\nL2+/bardbl−(−∆+1)s/2u1−iλu2/bardbl2\nL2\n=/bardbl−λw1+w2\n2+(−∆+1)s/2w1−w2\n2/bardbl2\nL2\n+/bardbl−(−∆+1)s/2w1+w2\n2+λw1−w2\n2/bardbl2\nL2\n=/bardblλw1−(−∆+1)s/2w1/bardbl2\nL2+/bardblλw2+(−∆+1)s/2w2/bardbl2\nL2.\nForλ≥0, applying Proposition 4.3tow1withs/2, we have\n2cexp(−Cλ)/bardblU/bardbl2\nHs/2×L2\n=cexp(−Cλ)(/bardblw1/bardbl2\nL2+/bardblw2/bardbl2\nL2)\n≤ /bardbl((−∆+1)s/4−λ)w1/bardbl2\nL2+/bardblw1/bardbl2\nL2(Ω)+cexp(−Cλ)/bardblw2/bardbl2\nL2\n≤ /bardbl((−∆+1)s/4−λ)w1/bardbl2\nL2+2/bardblw1−w2/bardbl2\nL2(Ω)+c/bardblw2/bardbl2\nL2\n≤ /bardbl((−∆+1)s/4−λ)w1/bardbl2\nL2+c/bardbl((−∆+1)s/4+λ)w2/bardbl2\nL2+8/bardblu2/bardbl2\nL2(Ω)\n≤c/bardbl(A0−iλI)U/bardbl2\nHs/2×L2+8/bardblu2/bardbl2\nL2(Ω).\nForλ <0, we get the same inequality replacing the role of w1withw2. /square\n4.2.Energy decay. Finally we prove ( 1.8) =⇒(1.9). By the assumption ( 1.8),\nΩ ={γ≥ε}is thick for some ε >0. Therefore, by Proposition 4.4, we have\ncexp(−C|λ|)/bardblU/bardbl2\nHs/2×L2≤ /bardbl(A0−iλI)U/bardbl2\nHs/2×L2+/bardblu2/bardbl2\nL2(Ω)\n≤2/bardbl(Aγ−iλI)U/bardbl2\nHs/2×L2+(2+ε−2)/bardblγu2/bardbl2\nL2(Ω).\nSinceA0is skew-adjoint, we obtain\nRe/a\\}bracketle{t(Aγ−iλI)U,U/a\\}bracketri}ht= Re/a\\}bracketle{t(A0−iλI)U,U/a\\}bracketri}ht−/a\\}bracketle{tγu2,u2/a\\}bracketri}ht=−/bardbl√γu2/bardbl2\nL2.\nBy the Cauchy–Schwarz inequality, we have\nD/bardblγu2/bardbl2\nL2≤ /bardblγ/bardblL∞/bardbl√γu2/bardbl2\nL2≤D2/bardblγ/bardbl2\nL∞/bardbl(Aγ−iλ)U/bardbl2\nHs/2×L2\nδ+δ/bardblU/bardbl2\nHs/2×L2.\nfor anyD,δ >0. Taking D= 2+ε−2andδ=cexp(−C|λ|)/2, we obtain\ncexp(−C|λ|)/bardblU/bardbl2\nHs/2×L2\n≤2/bardbl(Aγ−iλI)U/bardbl2\nHs/2×L2+(2+ε−2)/bardblγu2/bardbl2\nL2(Ω)\n≤2/bardbl(Aγ−iλI)U/bardbl2\nHs/2×L2+(2+ε−2)2/bardblγ/bardbl2\nL∞\ncexp(−C|λ|)/bardbl(Aγ−iλI)U/bardbl2\nHs/2×L2\n+1\n2cexp(−C|λ|)/bardblU/bardbl2\nHs/2×L2.ENERGY DECAY OF DAMPED KLEIN–GORDON EQUATIONS 9\nBy this inequality, we have\ncexp(−C|λ|)/bardblU/bardbl2\nHs/2×L2≤ /bardbl(Aγ−iλI)U/bardbl2\nHs/2×L2,\nhere the constants c,Cmay differ from the previous ones. Applying Theorem 4.2\nwithk= 1, we conclude that ( 1.9) holds.\nAcknowledgment\nThe authors would like to thank Professor Mitsuru Sugimoto for valu able dis-\ncussions.\nReferences\n[1] C. J. K. Batty, A. Borichev, and Y. Tomilov, Lp-tauberian theorems and Lp-rates for en-\nergy decay , J. Funct. Anal. 270(2016), no. 3, 1153–1201, DOI 10.1016/j.jfa.2015.12.003.\nMR3438332\n[2] N. Burq and R. Joly, Exponential decay for the damped wave equation in unbounded domains ,\nCommun. 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Nachr. 293(2020), no. 2, 363–375, DOI 10.1002/mana.201800417. MR406 3985\n[8] S. Suzuki, The uncertainty principle and energy decay estimates of the fractional Klein-\nGordon equation with space-dependent damping , arXiv:2212.02481 (2022).\n[9] M. T¨ aufer, Controllability of the Schr¨ odinger equation on unbounded domains without geo-\nmetric control condition , ESAIM Control Optim. Calc. Var. 29(2023), Paper No. 59, 11,\nDOI 10.1051/cocv/2023037. MR4621418\n[10] J. Wunsch, Periodic damping gives polynomial energy decay , Math. Res. Lett. 24(2017),\nno. 2, 571–580, DOI 10.4310/MRL.2017.v24.n2.a15. MR36852 85\n(Kotaro Inami) Graduate School of Mathematics, NagoyaUniversity, Furocho, Chikusa-\nku, Nagoya, Aichi, 464-8602, Japan\nEmail address :m21010t@math.nagoya-u.ac.jp\n(SoichiroSuzuki) Department of Mathematics, Chuo University, 1-13-27, Kasug a, Bunkyo-\nku, Tokyo, 112-8551, Japan\nEmail address :soichiro.suzuki.m18020a@gmail.com"    },    {        "title": "2002.06858v2.Self_similar_shrinkers_of_the_one_dimensional_Landau_Lifshitz_Gilbert_equation.pdf",        "content": "Self-similar shrinkers of the one-dimensional\nLandau–Lifshitz–Gilbert equation\nSusana Gutiérrez1and André de Laire2\nAbstract\nThe main purpose of this paper is the analytical study of self-shrinker solutions of the\none-dimensional Landau–Lifshitz–Gilbert equation (LLG), a model describing the dynamics\nfor the spin in ferromagnetic materials. We show that there is a unique smooth family of\nbackward self-similar solutions to the LLG equation, up to symmetries, and we establish\ntheir asymptotics. Moreover, we obtain that in the presence of damping, the trajectories of\nthe self-similar profiles converge to great circles on the sphere S2, at an exponential rate.\nIn particular, the results presented in this paper provide examples of blow-up in finite\ntime, where the singularity develops due to rapid oscillations forming limit circles.\nKeywords and phrases: Landau–Lifshitz–Gilbert equation, self-similar expanders, backward\nself-similar solutions, blow up, asymptotics, ferromagnetic spin chain, heat flow for harmonic\nmaps, quasi-harmonic sphere.\n2010Mathematics Subject Classification: 82D40; 35C06; 35B44; 35C20; 53C44; 35Q55;\n58E20; 35K55.\n82D40;35C06;35B44; 35C20;53C44;35Q55;58E20;35K55\n1 Introduction\n1.1 The Landau–Lifshitz–Gilbert equation: self-similar solutions\nIn this paper we continue the investigation started in [32, 33] concerning the existence and prop-\nerties of self-similar solutions for the Landau–Lifshitz–Gilbert equation (LLG). This equation\ndescribes the dynamics for the magnetization or spin in ferromagnetic materials [43, 27] and is\ngiven by the system of nonlinear equations\n∂tm=βm×∆m−αm×(m×∆m), (LLG)\nwherem= (m 1,m2,m3) :RN×I−→S2is the spin vector, I⊂R,β≥0,α≥0,×denotes\nthe usual cross-product in R3, and S2is the unit sphere in R3. This model for ferromagnetic\nmaterials constitutes a fundamental equation in the magnetic recording industry [53]. The\nparameters β≥0andα≥0are, respectively, the so-called exchange constant and Gilbert\ndamping, and take into account the exchange of energy in the system and the effect of damping\non the spin chain. By considering a time-scaling, one can assume without loss of generality that\nthe parameters αandβsatisfy\nα∈[0,1]andβ=/radicalbig\n1−α2.\n1School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom.\nE-mail: s.gutierrez@bham.ac.uk\n2Univ. Lille, CNRS, UMR 8524, Inria - Laboratoire Paul Painlevé, F-59000 Lille, France.\nE-mail: andre.de-laire@univ-lille.fr\n1arXiv:2002.06858v2  [math.AP]  21 May 2020From a purely mathematical point of view, the LLG equation is extremely interesting since\nit interpolates between two fundamental geometric evolution equations, the Schrödinger map\nequation and the heat flow for harmonic maps, via specific choices of the parameters involved.\nPrecisely, we recall that in the limit case α= 1(and, consequently, β= 0), (LLG) reduces to\nthe heat flow for harmonic maps onto S2,\n∂tm−∆m=|∇m|2m, (HFHM)\nand, ifα= 0(no damping), it reduces to the Schrödinger map equation\n∂tm=m×∆m. (SM)\nWhen 0<α< 1, (LLG) is of parabolic type. We refer the reader to [40, 29, 32, 33, 12, 14, 15, 13]\nand the references therein for more details and surveys on these equations.\nA natural question, that has proved relevant to the understanding of the global behavior\nof solutions and formation of singularities, is whether or not there exist solutions which are\ninvariant under scalings of the equation. In the case of the LLG equation it is straightforward\nto see that the equation is invariant under the following scaling: If mis a solution of (LLG),\nthenmλ(t,x) =m(λx,λ2t), for any positive number λ, is also a solution. Associated with this\ninvariance, a solution mof (LLG) defined on I=R+orI=R−is called self-similar if it is\ninvariant under rescaling, that is\nm(x,t) =m(λx,λ2t),∀λ>0,∀x∈RN,∀t∈I.\nFixingT∈Rand performing a translation in time, this definition leads to two types of self-\nsimilar solutions: A forward self-similar solution or expander is a solution of the form\nm(x,t) =f/parenleftbiggx√\nt−T/parenrightbigg\n,for (x,t)∈RN×(T,∞), (1.1)\nand a backward self-similar solution or shrinker is a solution of the form\nm(x,t) =f/parenleftbiggx√\nT−t/parenrightbigg\n,for (x,t)∈RN×(−∞,T), (1.2)\nfor some profile f:RN−→S2. In this manner, expanders evolve from a singular value at time\nT, while shrinkers evolve towards a singular value at time T.\nSelf-similar solutions have received a lot of attention in the study of nonlinear PDEs because\nthey can provide important information about the dynamics of the equations. While expanders\nare related to non-uniqueness phenomena, resolution of singularities and long time description\nof solutions, shrinkers are often related to phenomena of singularity formation (see e.g. [26, 18]).\nOn the other hand, the construction and understanding of the dynamics and properties of self-\nsimilar solutions also provide an idea of which are the natural spaces to develop a well-posedness\ntheory that captures these often very physically relevant structures. Examples of equations for\nwhichself-similarsolutionshavebeenstudiedinclude,amongothers,theNavier–Stokesequation,\nsemilinear parabolic equations, and geometric flows such as Yang–Mills, mean curvature flow\nand harmonic map flow. We refer to [37, 48, 36, 51, 5] and the references therein for more\ndetails.\nAlthough the results that will be presented in this paper relate to self-similar shrinkers of\nthe one-dimensional LLG equation (that is, to solutions m:R×I−→S2of LLG), for the sake\nof context we describe some of the most relevant results concerning maps from RN×IintoSd,\nwithN≥2andd≥2. In this setting one should point out that the majority of the works in the\n2literature concerning the study of self-similar solutions of the LLG equation are confined to the\nheat flow for harmonic maps equation, i.e. α= 1. In the case when α= 1, the main works on the\nsubject restrict the analysis to corotational maps taking values in Sd, which reduces the analysis\nof (HFHM) to the study of a second order real-valued ODE. Then tools such as the maximum\nprinciple or the shooting method can be used to show the existence of solutions. We refer to\n[19, 21, 23, 7, 8, 6, 22] and the references therein for more details on such results for maps taking\nvalues in Sd, withd≥3.Recently, Deruelle and Lamm [17] have studied the Cauchy problem\nfor the harmonic map heat flow with initial data m0:RN→Sd, withN≥3andd≥2, where\nm0is a Lipschitz 0-homogeneous function, homotopic to a constant, which implies the existence\nof expanders coming out of m0.\nWhen 0<α≤1, the existence of self-similar expanders for the LLG equation was recently\nestablished by the authors in [33]. This result is a consequence of a well-posedness theorem for\nthe LLG equation considering an initial data m0:RN→S2in the space BMO of functions of\nbounded mean oscillation. Notice that this result includes in particular the case of the harmonic\nmap heat flow.\nAs mentioned before, in the absence of damping ( α= 0), (LLG) reduces to the Schrödinger\nmap equation (SM), which is reversible in time, so that the notions of expanders and shrinkers\ncoincide. For this equation, Germain, Shatah and Zeng [24] established the existence of ( k-\nequivariant) self-similar profiles f:R2→S2.\n1.2 Goals and statements of main results\nThe results of this paper aim to advance our understanding of self-similar solutions of the one-\ndimensional LLG equation. In order to contextualize and motivate our results, we continue to\nprovide further details of what is known about self-similar solutions in this context.\nIn the 1d-case, when α= 0, (SM) is closely related to the Localized Induction Approximation\n(LIA), and self-similar profiles f:R→S2were obtained and analyzed in [34, 35, 41, 10]. In the\ncontext of LIA, self-similar solutions constitute a uniparametric family of smooth solutions that\ndevelop a singularity in the shape of a corner in finite time. For further work related to these\nsolutions, including the study of their continuation after the blow-up time and their stability,\nwe refer to the reader to [4, 3]. At the level of the Schrödinger map equation, these self-similar\nsolutions provide examples of smooth solutions that develop a jump singularity in finite time.\nIn the general case α∈[0,1], the analytical study of self-similar expanders of the one-\ndimensional (LLG) was carried out in [32]. Here, it was shown that these solutions are given by\na family of smooth profiles {fc,α}c,α, and that the corresponding expanders are associated with\na discontinuous (jump) singular initial data. We refer to [32, 33] for the precise statement of this\nresult, and the stability of these solutions, as well as the qualitative and quantitative analysis\nof their dynamics with respect to the parameters candα.\nIt is important to notice that in the presence of damping ( α > 0), since the LLG equation\nis not time-reversible, the notion of expander is different from that of shrinker. It is therefore\nnatural to ask the following question: What can be said about shrinker solutions for the one-\ndimensional LLG equation?\nAnswering this question constitutes the main purpose of this paper. Precisely, our main goals\nare to establish the classification of self-similar shrinkers of the one-dimensional LLG equation\nof the form (1.2) for some profile f:R→S2, and the analytical study of their properties. In\nparticular, we will be especially interested in studying the dynamics of these solutions as ttends\nto the time of singularity T, and understanding how the dynamical behavior of these solutions\nis affected by the presence of damping. Since, as it has been already mentioned, the case α= 0\n3has been previously considered in the literature (see [4, 31]), in what follows we will assume that\nα∈(0,1].\nIn order to state our first result, we observe that if mis a solution to (LLG) of the form (1.2)\nfor some smooth profile f, thenfsolves the following system of ODEs\nxf/prime\n2=βf×f/prime/prime−αf×(f×f/prime/prime),onR, (1.3)\nwhich recasts as\nαf/prime/prime+α|f/prime|2f+β(f×f/prime)/prime−xf/prime\n2= 0,onR, (1.4)\ndue to the fact that ftakes values in S2.\nIn the case α∈(0,1), it seems unlikely to be able to find explicit solutions to (1.4), and even\ntheir existence is not clear (see also equation (1.16)). Nevertheless, surprisingly we can establish\nthe following rigidity result concerning the possible weak solutions to (1.4) (see Section 2 for the\ndefinition of weak solution).\nTheorem 1.1. Letα∈(0,1]. Assume that fis a weak solution to (1.4). Thenfbelongs to\nC∞(R;S2)and there exists c≥0such that|f/prime(x)|=ceαx2/4, for allx∈R.\nTheorem 1.1 provides a necessary condition on the possible (weak) solutions of (1.4): namely\nthe modulus of the gradient of any solution mustbeceαx2/4, for somec≥0. We proceed now to\nestablish the existence of solutions satisfying this condition for any c>0(notice that the case\nwhenc= 0is trivial).\nTo this end, we will follow a geometric approach that was proven to be very fruitful in similar\ncontexts (see e.g. [41, 46, 42, 34, 16]), including the work of the authors in the study of expanders\n[32]. As explained in Subsection 3.1, this approach relies on identifying fas the unit tangent\nvectorm:=fof a curveXminR3parametrized by arclength. Thus, assuming that fis\na solution to (1.4) and using the Serret–Frenet system associated with the curve Xm, we can\ndeduce that the curvature and the torsion are explicitly given by\nk(x) =ceαx2/4,andτ(x) =−βx\n2, (1.5)\nrespectively, for some c≥0(see Subsection 3.1 for further details). In particular, we have\n|m/prime(x)|=k(x) =ceαx2/4, in agreement with Theorem 1.1. Conversely, given c≥0and denoting\nmc,αthe solution of the Serret–Frenet system\n\n\nm/prime(x) =k(x)n(x),\nn/prime(x) =−k(x)m(x) +τ(x)b(x),\nb/prime(x) =−τ(x)n(x),(1.6)\nwith curvature and torsion as in (1.5), and initial conditions (w.l.o.g.)\nm(0) = (1,0,0),n(0) = (0,1,0),b(0) = (0,0,1), (1.7)\nwe obtain a solution to (1.4). Moreover, we can show that the solutions constructed in this\nmanner provide, up to symmetries, all the solutions to (1.3). The precise statement is the\nfollowing.\nProposition 1.2. The set of nonconstant solutions to (1.3)is{Rmc,α:c >0,R∈SO(3)},\nwhereSO(3)is the group of rotations about the origin preserving orientation.\n4The above proposition reduces the study of self-similar shrinkers to the understanding of the\nfamilyofself-similarshrinkersassociatedwiththeprofiles {mc,α}c,α. Thenextresultsummarizes\nthe properties of these solutions.\nTheorem 1.3. Letα∈(0,1],c > 0,T∈Randmc,αbe the solution of the Serret–Frenet\nsystem(1.6)with initial conditions (1.7),\nk(x) =ceαx2/4andτ(x) =−/radicalbig\n1−α2x\n2.\nDefine\nmc,α(x,t) =mc,α/parenleftbiggx√\nT−t/parenrightbigg\n, t<T. (1.8)\nThen we have the following statements.\n(i) The function mc,αbelongs toC∞(R×(−∞,T);S2), solves(LLG)fort∈(−∞,T), and\n|∂xmc,α(x,t)|=c√\nT−teαx2\n4(T−t),\nfor all (x,t)∈R×(−∞,T).\n(ii) The components of the profile mc,α= (m1,c,α,m2,c,α,m3,c,α)satisfy that m1,c,αis even,\nwhilem2,c,αandm3,c,αare odd.\n(iii) There exist constants ρj,c,α∈[0,1], Bj,c,α∈[−1,1],andφj,c,α∈[0,2π),forj∈{1,2,3},\nsuch that we have the following asymptotics for the profile mc,α= (m1,c,α,m2,c,α,m3,c,α)\nand its derivative:\nmj,c,α(x) =ρj,c,αcos(cΦα(x)−φj,c,α)−βBj,c,α\n2cxe−αx2/4\n+β2ρj,c,α\n8csin(cΦα(x)−φj,c,α)/integraldisplay∞\nxs2e−αs2/4ds+β\nα5c2O(x2e−αx2/2),(1.9)\nand\nm/prime\nj,c,α(x) =−cρj,c,αsin(cΦα(x)−φj,c,α)eαx2/4\n+β2ρj,c,α\n8cos(cΦα(x)−φj,c,α)eαx2/4/integraldisplay∞\nxs2e−αs2/4ds+β\nα5cO(x2e−αx2/4),(1.10)\nfor allx≥1, where\nΦα(x) =/integraldisplayx\n0eαs2\n4ds.\nMoreover, the constants ρj,c,α,Bj,c,α, andφj,c,αsatisfy the following identities\nρ2\n1,c,α+ρ2\n2,c,α+ρ2\n3,c,α= 2, B2\n1,c,α+B2\n2,c,α+B2\n3,c,α= 1andρ2\nj,c,α+B2\nj,c,α= 1, j∈{1,2,3}.\n(iv) The solution mc,α= (m 1,c,α,m2,c,α,m3,c,α)satisfies the following pointwise convergences\nlim\nt→T−(mj,c,α(x,t)−ρj,c,αcos/parenleftbigcΦα/parenleftbigx√\nT−t/parenrightbig−φj,c,α/parenrightbig= 0,ifx>0,\nlim\nt→T−(mj,c,α(x,t)−ρ−\nj,c,αcos/parenleftbigcΦα/parenleftbig−x√\nT−t/parenrightbig−φj,c,α/parenrightbig= 0,ifx<0,(1.11)\nforj∈{1,2,3}, whereρ−\n1,c,α=ρ1,c,α,ρ−\n2,c,α=−ρ2,c,αandρ−\n3,c,α=−ρ3,c,α.\n5(v) For anyϕ∈W1,∞(R;R3), we have\nlim\nt→T−/integraldisplay\nRmc,α(x,t)·ϕ(x)dx= 0.\nIn particular, mc,α(·,t)→0ast→T−, as a tempered distribution.\nIt is important to remark that Theorem 1.3 provides examples of (smooth) solutions to the 1d-\nLLG equation that blow up in finite time. In order to see this, let us first recall that the existence\nof smooth solutions to (LLG) on short times can be established as in the case of the heat flow\nfor harmonic maps [45], using that (LLG) is a strongly parabolic system [30, 2]. In particular, in\nthe one-dimensional case, for any initial condition m0∈C∞(R,S2), there exists a maximal time\n0< T max≤∞such that (LLG) admits a unique, smooth solution m∈C∞(R×[0,Tmax);S2).\nMoreover, if Tmax<∞, then\nlim\nt→T−\nmax/bardbl∂xm(·,t)/bardblL∞(R)=∞.\nNext, observe that for any c>0andT∈R, the solution of the initial value problem associated\nwith (LLG) and with initial condition mc,α(·)at timeT−1is given by mc,αin Theorem 1.3,\nfort∈[T−1,T), and blows up at time T. Indeed, from (i)in Theorem 1.3 , we have that\nlim\nt→T−|∂xmc,α(x,t)|= lim\nt→T−c√\nT−teαx2\n4(T−t)=∞,\nforc>0and for allx∈R.\nNotice also that from the asymptotics in part (iii)and the symmetries of the profile estab-\nlished in part (ii), we obtain a precise description of the fast oscillating nature of the blow up of\nthe solution (1.8) given in Theorem 1.3. In this setting, we observe that part (iii)of the above\ntheorem provides the asymptotics of the profile mc,αat infinity, in terms of a fast oscillating\nprincipal part, plus some exponentially decaying terms. Notice that for the integral term in\n(1.9), we have (see e.g. [1])\n/integraldisplay∞\nxs2e−αs2/4ds=2xe−αx2/4\nα/parenleftBig\n1 +2\nαx2−4\nα2x4+···/parenrightBig\n,asx→∞,\nand that using the asymptotics for the Dawson’s integral [1], we also get\nΦα(x) =2eαx2/4\nαx/parenleftBig\n1 +2\nαx2+12\nα2x4+···/parenrightBig\n,asx→∞.\nIt is also important to mention that the big- Oin the asymptotics (1.9) does not depend on the\nparameters, i.e. there exists a universal constant C > 0, such that the big- Oin (1.9) satisfies\n|O(x2e−αx2/2)|≤Cx2e−αx2/2,for allx≥1.\nIn this manner, the constants multiplying the big- Oare meaningful and in particular, big- O\nvanishes when β= 0(i.e.α= 1).\nIn Figure 1 we have depicted the profile mc,αforα= 0.5andc= 0.5, where we can see their\noscillating behavior. Moreover, the plots in Figure 1 suggest that the limit sets of the trajectories\nare great circles on the sphere S2whenx→±∞. This is indeed the case. In our last result we\nestablish analytically that mc,αoscillates in a plane passing through the origin whose normal\nvector is given by B+\nc,α= (B1,c,α,B2,c,α,B3,c,α), andB−\nc,α= (−B1,c,α,B2,c,α,B3,c,α)asx→+∞\nandx→−∞, respectively.\n6m1m2m3\nB+\nc,α\nm1m2m3\n-1.0-0.50.00.51.0-1.0-0.50.00.51.0\nm1m2\nFigure 1: Profile mc,αforc= 0.5andα= 0.5. The figure on the left depicts profile for x∈R+\nand the normal vector B+\nc,α≈(−0.72,−0.3,0.63). The figure on the center shows the profile for\nx∈R; the angle between the circles C±\nc,αisϑc,α≈1.5951. The figure on the right represents the\nprojection of limit cycles C±\nc,αon the plane.\nTheorem 1.4. Using the constants given in Theorem 1.3, let P±\nc,αbe the planes passing through\nthe origin with normal vectors B+\nc,αandB−\nc,α= (−B1,c,α,B2,c,α,B3,c,α), respectively. Let C±\nc,αbe\nthe circles in R3given by the intersection of these planes with the sphere, i.e. C±\nc,α=P±\nc,α∩S2.\nThen the following statements hold.\n(i) For all|x|≥1, we have\ndist(mc,α(x),C±\nc,α)≤30√\n2β\ncα2|x|e−αx2/4. (1.12)\nIn particular\nlim\nt→T−dist(mc,α(x,t),C+\nc,α) = 0,ifx>0,\nlim\nt→T−dist(mc,α(x,t),C−\nc,α) = 0,ifx<0.(1.13)\n(ii) Letϑc,α= arccos(1−2B2\n1,c,α)be the angle between the circles C±\nc,α. Forc≥β√π/√α, we\nhave\nϑc,α≥arccos/parenleftBigg\n−1 +2πβ2\nc2α/parenrightBigg\n. (1.14)\nIn particular\nlimc→∞ϑc,α=π,for allα∈(0,1], and lim\nα→1ϑc,α=π,for allc>0.(1.15)\nThe above theorem above establishes the convergence of the limit sets of the trajectories\nof the profile mc,αto the great circles C±\nc,αas shown in Figure 1. Moreover, (1.12) gives us\nan exponential rate for this convergence. In terms of the solution mc,αto the LLG equation,\nTheorem 1.4 provides a more precise geometric information about the way that the solution\nblows up at time T, as seen in (1.13). The existence of limit circles for related ferromagnetic\nmodels have been investigated for instance in [52, 9] but to the best of our knowledge, this is the\nfirst time that this type of phenomenon has been observed for the LLG equation. In Figure 1\ncan see that ϑc,α≈1.5951forα= 0.5andc= 0.5, where we have chosen the value of csuch\nthat the angle is close to π/2.\nFinally, (1.14) and (1.15) in Theorem 1.4 provide some geometric information about behavior\nof the limit circles with respect to the parameters candα. In particular, formulae (1.15) states\n7that the angle between the limiting circles C+\nc,αandC−\nc,αisπasc→∞, for fixedα∈(0,1], and\nthe same happens as α→1, for fixedc>0. In other words, in these two cases the circles C±\nc,α\nare the same (but differently oriented).\n1.3 Comparison with the limit cases α= 0andα= 1\nIt is well known that the Serret–Fenet system can be written as a second-order differential\nequation. Forinstance, if (m,n,b) = (mj,nj,bj)3\nj=1isasolutionof (1.5)–(1.6), usingLemma3.1\nin [32], we have that new variable\ngj(s) =e1\n2/integraltexts\n0k(σ)ηj(σ)dσ,withηj(x) =nj(x) +ibj(x)\n1 +mj(x),\nsatisfies the equation, for j∈{1,2,3},\ng/prime/prime\nj(x)−x\n2(α+iβ)g/prime\nj(x) +c2\n4eαx2/2gj(x) = 0. (1.16)\nThen, in the case α= 1, it easy to check (see also Remark 3.3) that the profile is explicitly given\nby the plane curve\nmc,1(x) = (cos(cΦ1(x)),sin(cΦ1(x)),0). (1.17)\nIn particular, we see that the asymptotics in Theorem 1.3 are satisfied with\nρ1,c,1= 1, ρ 2,c,1= 1, ρ 3,c,1= 0, φ 1,c,1= 0, φ 2,c,1= 3π/2, φ 3,c,1∈[0,2π).\nThe caseα= 0is more involved, but using (1.16), the solution {mc,0,nc,0,bc,0}of the system\n(1.6)canstillbeexplicitlydeterminedintermsofconfluenthypergeometricfunctions. Thisleads\nto the asymptotics [34, 32, 20]\nmc,0(x) =Ac−2c\nxBccos/parenleftBigg\nx2\n4+c2ln(x) +π\n2/parenrightBigg\n+O/parenleftbigg1\nx2/parenrightbigg\n, (1.18)\nasx→∞, for some vectors Ac∈S2andBc∈R3. In particular, we see that mc,0(x)converges\nto some vector Ac, asx→∞. Hence, there is a drastic change in the behavior of the profile\nin the cases α= 0andα > 0: In the first case mc,0converges to a point at infinity, while in\nthe second case (1.12) tells us that mc,αconverges to a great circle. In this sense, there is a\ndiscontinuity in the behavior of mc,αatα= 0.\nAlso, from equation (1.16), we can formally deduce that the difference between the expanders\nand shrinkers corresponds to flipping the sign in the parameters α→−αandβ→−β. Notice\nthat the exponential coefficient in (1.16) is proportional to the square of the curvature, given by\nce−αx2/4for the skrinkers, and ceαx2/4for the expanders. We used equation (1.16) (with flipped\nsigns) to obtain the asymptotics of the expanders in [32], relying on the fact the exponential\nterm in equation vanishes as x→∞. However, the exponential grow in the case of skrinkers in\n(1.16) changes the behavior of the solution and we cannot use the methods introduced in [32].\nGoing back to Theorem 1.3, it is seems very difficult to get asymptotics for the constants\nin (1.9). Our strategy for the constants appearing in the asymptotics for the expanders in [32]\nrelied on obtaining uniform estimates and using continuity arguments. In particular, using the\nfact that the constants in (1.18) are explicit, we were able to get a good information about the\nconstants in the asymptotics when αwas close to 0. Due to the above mentioned discontinuity\n8ofmc,αatα= 0, it seems unlikely that the use of continuity arguments will provide information\nfor the constants in the asymptotics for the shrinkers.\nFinally, let us also remark that we cannot use continuation arguments to find the behavior\nof the circles for csmall. This is expected since m0,α(x) = (1,0,0)for allx∈R, whenc= 0\n(see (4.6)). In Section 4 we give some numerical simulations for csmall.\nStructure of the paper. The outline of this paper is the following. In Section 2, we study\n(1.4) as an elliptic quasilinear system and prove the rigidity result Theorem 1.1. By using the\nSerret–Frenet system, we prove there existence and uniqueness of solution, up to a rotation, in\nSection 3. We also use this system to obtain the asymptotics of the self-similar profiles. Finally,\nSection 4 is devoted to the proof of Theorem 1.4.\n2 Rigidity result. Theorem 1.1\nThe purpose of this section is to prove the rigidity result stated in Theorem 1.1 concerning\n(weak) solutions of the system\nxf/prime\n2=βf×f/prime/prime−αf×(f×f/prime/prime),onR. (2.1)\nWe start by introducing the notion of weak solution of the above system. To this end, we first\nobserve that the system (2.1) recasts as\nαf/prime/prime+α|f/prime|2f+β(f×f/prime)/prime−xf/prime\n2= 0, (2.2)\nusing the following vector identities for a (smooth) function fwith|f|= 1:\nf×f/prime/prime= (f×f/prime)/prime,\n−f×(f×f/prime/prime) =f/prime/prime+|f/prime|2f.(2.3)\nWe prefer to use the formulation (2.2) since it is simpler to handle in weak sense. Indeed, we\nsay thatf= (f1,f2,f3)∈H1\nloc(R,S2)is aweak solution to the system (2.2) if\n/integraldisplay\nR/parenleftBig\n−αf/prime·ϕ/prime+α|f/prime|2f·ϕ−β(f×f/prime)·ϕ/prime−x\n2f/primeϕ/parenrightBig\ndx= 0, (2.4)\nfor allϕ= (ϕ1,ϕ3,ϕ3)∈C∞\n0(R).\nUsing (2.3), we can recast (2.2) as,\nαf/prime/prime\n1+α|f/prime|2f1+β(f2f/prime/prime\n3−f3f/prime/prime\n2)−x\n2f/prime\n1= 0, (2.5a)\nαf/prime/prime\n2+α|f/prime|2f2+β(f3f/prime/prime\n1−f1f/prime/prime\n3)−x\n2f/prime\n2= 0, (2.5b)\nαf/prime/prime\n3+α|f/prime|2f3+β(f1f/prime/prime\n2−f2f/prime/prime\n1)−x\n2f/prime\n3= 0. (2.5c)\nThus we see that the weak formulation (2.4) can be written as\n/integraldisplay\nRA(f(x))f/prime(x)·ϕ/prime(x) =/integraldisplay\nRG(x,f,f/prime)ϕ(x),for allϕ∈C∞\n0(R), (2.6)\nwith\nA(u) =\nα−βu3βu2\nβu3α−βu1\n−βu2βu1α,\nandG(x,u,p) =\nαu1|p|2−xp1\n2\nαu2|p|2−xp2\n2\nαu3|p|2−xp3\n2\n,\n9whereu= (u1,u2,u3)andp= (p1,p2,p3). We want now to invoke the regularity theory for\nquasilinear elliptic system (see [39, 25]). To verify that the system is indeed uniformly elliptic,\nwe can easily check that\nA(u)ξ·ξ=α|ξ|2,for allξ,u∈R3.\nIn addition, Ghas quadratic growth on bounded domains, i.e.\n|G(x,u,p)|≤√\n3(M|p|2+R|p|),\nfor all|u|≤Mand|x|≤R. Since a weak solution fto (2.6) belongs by definition to H1\nloc(R;S2),\nwe have by the Sobolev embedding theorem that fis Hölder continuous with |f(x)|= 1.\nTherefore we can apply the results in Theorem 1.2 in [25] (see also Lemma 8.6.3 in [38] or\nTheorem 2.4.3 in [49] for detailed proofs), to conclude that f∈H2\nloc(R)∩W1,4\nloc(R), and so\nthatf∈C1,γ\nloc(R), for someγ∈(0,1). We get that G(x,f(x),f/prime(x))belongs toC0,γ\nloc(R), which\nallows us to invoke the Schauder regularity theory (see e.g. Theorem A.2.3 in [38]) to infer that\nf∈C2,γ\nloc(R). This implies that G(x,f(x),f/prime(x))belongs toC1,γ\nloc(R), as well as the coefficients of\nA(u), so the Schauder estimates yield that f∈C3,γ\nloc(R). By induction, we this argument shows\nthatf∈C∞(R).\nWe are now in position to complete the proof of Theorem 1.1. Indeed, let first remark that\ndifferentiating the relation |f|2= 1, we have the identities\nf·f/prime= 0, (2.7)\nf·f/prime/prime=−|f/prime|2. (2.8)\nBy taking the cross product of fand (2.2), and using (2.3), we have\nβf/prime/prime+β|f/prime|2f−α(f×f/prime)/prime+x\n2f×f/prime= 0. (2.9)\nThus, by multiplying (2.2) by α, (2.9) byβ, and recalling that α2+β2= 1, we get\nf/prime/prime+|f/prime|2f−x\n2(αf/prime−βf×f/prime) = 0.\nTaking the scalar product of this equation and f/prime, the identity (2.7) allow us to conclude that\n1\n2(|f/prime|2)/prime−αx\n2|f/prime|2= 0. (2.10)\nIntegrating, we deduce that there is a constant C≥0such that|f/prime|2=Ceαx2/2. This completes\nthe proof of Theorem 1.1.\nWe conclude this section with some remarks.\nRemark 2.1. A similar result to the one stated in Theorem 1.1 also holds for the expanders\nsolutions. Precisely, any weak solution to (2.1), withxf/prime/2replaced by−xf/prime/2in the l.h.s., is\nsmooth and there exists c≥0such that|f/prime(x)|=ce−αx2/4, for allx∈R.\nRemark 2.2. Let us mention that in the case α= 1, a nonconstant solution u:RN→Sdto\nequation\n∆u+|∇u|2u−x·∇u\n2= 0,onRN, (2.11)\n10is usually called quasi-harmonic sphere , since it corresponds to the Euler–Lagrange equations of\na critical point of the (so-called) quasi-energy [44]\nEquasi(u) =/integraldisplay\nRN|∇u(y)|2e−|y2|/4dy.\nIt has been proved in [19] the existence of a (real-valued) function hsuch that\nu(x) =/parenleftBigx\n|x|sin(h(|x|)),cos(h(|x|)/parenrightBig\nis a solution to (2.11)with finite quasi-energy for 3≤N=d≤6. In addition, there is no\nsolution of this form if d≥7[8]. Both results are based on the analysis of the second-order\nODE associated with h. We refer also to [21] for a generalization of the existence result for\nN≥3of other equivariant solutions to (2.11). In the case N= 1andd= 2, the solution to\n(2.11)is explicitly given by (1.17), and its associated quasi-energy is infinity, as remarked in\n[54].\n3 Existence, uniqueness and properties\n3.1 Existence and uniqueness of the self-similar profile. Proposition 1.2\nIn the previous section we have shown that any solution to the profile equation\nαm/prime/prime+α|m/prime|2m+β(m×m/prime)/prime−xm/prime\n2= 0, (3.1)\nis smooth and that there is c≥0such that\n|m/prime(x)|=ceαx2/4,for allx∈R. (3.2)\nWe want to give now the details about how to construct such a solution by using the Serret–\nFrenet frame, which will correspond to the profile mc,αin Theorem 1.3. The idea is to identify\nmas the tangent vector to a curve in R3, so we first recall some facts about curves in the space.\nGivenm:R→S2a smooth function, we can define the curve\nXm(x) =/integraldisplayx\n0m(s)ds, (3.3)\nso thatXmis smooth, parametrized by arclenght, and its tangent vector is m. In addition,\nif|m/prime|does not vanish on R, we can define the normal vector n(x) =m/prime(x)/|m/prime(x)|and the\nbinormal vector b(x) =m(x)×n(x). Moreover, we can define the curvature and torsion of Xm\nask(x) =|m/prime(x)|andτ(x) =−b/prime(x)·n(x). Since|m(x)|2= 1,for allx∈R, we have that\nm(x)·n(x) = 0, for allx∈R, that the vectors {m,n,b}are orthonormal and it is standard to\ncheck that they satisfy the Serret–Frenet system\n\n\nm/prime=kn,\nn/prime=−km+τb,\nb/prime=−τn.(3.4)\nLet us apply this construction to find a solution to (3.1). We define curve Xmas in (3.3), and\nremark that equation (3.1) rewrites in terms of {m,n,b}as\nx\n2kn=β(k/primeb−τkn)−α(−k/primen−kτb).\n11Therefore, from the orthogonality of the vectors nandb, we conclude that the curvature and\ntorsion ofXmare solutions of the equations\nx\n2k=αk/prime−βτkandβk/prime+αkτ= 0,\nthat is\nk(x) =ceαx2\n4andτ(x) =−βx\n2, (3.5)\nfor somec≥0. Of course, the fact that k(x) =ceαx2/4is in agreement with the fact that we\nmust have|m/prime(x)|=ceαx2/4.\nNow, given α∈[0,1]andc>0, consider the Serret–Frenet system (3.4) with curvature and\ntorsion function given by (3.5) and initial conditions\nm(0) = (1,0,0),n(0) = (0,1,0),b(0) = (0,0,1). (3.6)\nThen, by standard ODE theory, there exists a unique global solution {mc,α,nc,α,bc,α}in\n(C∞(R;S2))3, and these vectors are orthonormal. Also, it is straightforward to verify that\nmc,αis a solution to (3.1) satisfying (3.2).\nThe above argument provides the existence of solutions in the statement of Proposition 1.2.\nWe will now complete the proof of Proposition 1.2 showing the uniqueness of such solutions, up\nto rotations.\nTo this end, assume that ˜mis a weak nontrivial solution to (3.1). By Theorem 1.1, ˜mis\ninC∞(R,S2)and there exists c >0such that|˜m/prime(x)|=ceαx2/4, for allx∈R. Following the\nabove argument, the curve X˜m(defined in (3.3)), has curvature ceαx2/4and torsion−βx/2.\nSince the curve Xmc,αassociated with mc,α, andX˜mhave the same curvature and torsion,\nusing fundamental theorem of the local theory of space curves (see e.g. Theorem 1.3.5 in [47]),\nwe conclude that both curves are equal up to direct rigid motion, i.e. there exist p∈R3and\nR∈SO(3)such thatX˜m(x) =R(Xmc,α(x))+p, for allx∈R3. By differentiating this identity,\nwe finally get that ˜m=Rmc,α, which proves the uniqueness of solution, up to a rotation, as\nstated in Proposition 1.2.\n3.2 Asymptotics of the self-similar profile\nThe rest of this section is devoted to establish properties of the family of solutions {mc,α}c,α,\nfor fixedα∈(0,1]andc >0. Due to the self-similar nature of these solutions, this analysis\nreduces to study the properties of the associated profile mc,α, or equivalently, of the solution\n{mc,α,nc,α,bc,α}of the Serret–Frenet system (3.4) with curvature and torsion given in (3.5),\nand initial conditions (3.6).\nIt is important to mention that the recovery of the properties of the trihedron {m,n,b},\nand in particular of the profile m, from the knowledge of its curvature and torsion is a difficult\nquestion. This can be seen from the equivalent formulations of the Serret–Frenet equation in\nterms of a second-order complex-valued highly non-linear EDO, or in terms of a complex-valued\nRiccati equation (see e.g. [11, 50, 42, 32]). For this reason, the integration of the trihedron can\noften only be done numerically, rather than analytically.\nSince the Serret–Frenet equations are decoupled, we start by analyzing the system for the\n12scalarfunctionsmc,α,nc,αandbc,α\n\n\nm/prime\nc,α(x) =ceαx2\n4nc,α(x),\nn/prime\nc,α(x) =−ceαx2\n4mc,α(x)−βx\n2bc,α(x),\nb/prime\nc,α(x) =βx\n2nc,α(x),(3.7)\nwithinitialconditions (mc,α,bc,α,nc,α)(0),thatwesupposeindependentof candα,andsatisfying\nmc,α(0)2+bc,α(0)2+nc,α(0)2= 1.\nThen by ODE theory, the solution is smooth, global and satisfies\nmc,α(x)2+bc,α(x)2+nc,α(x)2= 1,for allx∈R. (3.8)\nMoreover, the solution depends continuously on the parameters c>0andα∈(0,1].\nTo study the behavior of the solution of the system (3.7), we need some elementary bounds\nfor the non-normalized complementary error function.\nLemma 3.1. Letγ∈(0,1]. The following upper bounds hold for x>0\n/integraldisplay∞\nxe−γs2ds≤1\n2γxe−γx2and/integraldisplay∞\nxse−γs2ds=1\n2γe−γx2. (3.9)\nAlso, forγ∈(0,1]andx≥1,\n/integraldisplay∞\nxs2e−γs2ds≤x\nγ2e−γx2,and/integraldisplay∞\nxs3e−γs2ds≤x2\nγ2e−γx2. (3.10)\nProof.We start recalling some standard bounds the complementary error function (see e.g. [1,\n28])\nxe−x2\n2x2+ 1≤/integraldisplay∞\nxe−s2ds≤e−x2\n2x,forx>0. (3.11)\nThe first formula in (3.9) follows by scaling this inequality. The second formula in (3.9) follows\nby integration by parts.\nTo prove the first estimate in (3.10), we use integration by parts and (3.9) to show that\n/integraldisplay∞\nxs2e−γs2ds=xe−γx2\n2γ+1\n2γ/integraldisplay∞\nxe−γs2ds≤e−γx2/parenleftbiggx\n2γ+1\n4γ2x/parenrightbigg\n≤xe−γx2/parenleftbigg1\n2γ+1\n4γ2/parenrightbigg\n,∀x≥1.\nSinceγ∈(0,1], we haveγ2≤γand thus we conclude the estimate for the desired integral. The\nsecond inequality in (3.10) easily follows from the identity\n/integraldisplay∞\nxs3e−γs2ds=1 +γx2\n2γ2e−γx2,∀x∈R,\nnoticing that 1 +γx2≤x2(1 +γ)≤2x2, sincex≥1andγ∈(0,1].\nNow we can state a first result on the behavior of {mc,α,nc,α,bc,α}.\n13Proposition 3.2. Letα∈(0,1]andc>0, and define\nΦα(x) =/integraldisplayx\n0eαs2\n4ds.\nThen the following statements hold.\ni) For allx∈R,\nbc,α(x) =Bc,α+βx\n2ce−αx2/4mc,α(x) +β\n2c/integraldisplay∞\nx/parenleftBigg\n1−αs2\n2/parenrightBigg\ne−αs2/4mc,α(s)ds,(3.12)\nwhere\nBc,α=bc,α(0)−β\n2c/integraldisplay∞\n0/parenleftBigg\n1−αs2\n2/parenrightBigg\ne−αs2/4mc,α(s)ds. (3.13)\nIn particular, for all x≥1\n|bc,α(x)−Bc,α|≤6β\ncαxe−αx2/4. (3.14)\nii) Setting wc,α=mc,α+inc,α, for allx∈R, we have\nwc,α(x) =e−icΦα(x)/parenleftBig\nWc,α−βx\n2ceicΦα(x)−αx2/4bc,α(x)\n−β\n2c/integraldisplay∞\nxeicΦα(s)−αs2/4/parenleftbigβs2\n2nc,α(s) +/parenleftbig1−αs2\n2/parenrightbigbc,α(s)/parenrightbigds/parenrightBig\n,(3.15)\nwhere\nWc,α=wc,α(0) +β\n2c/integraldisplay∞\n0eicΦα(s)−αs2/4/parenleftbigβs2\n2nc,α(s) +/parenleftbig1−αs2\n2/parenrightbigbc,α(s)/parenrightbigds.(3.16)\nIn particular, for all x≥1,\n|wc,α(x)−e−icΦα(x)Wc,α|≤10β\ncα2xe−αx2/4. (3.17)\nFurthermore, the limiting values Bc,αandWc,αare separately continuous functions of (c,α)for\n(c,α)∈(0,∞)×(0,1].\nProof.For simplicity, we will drop the subscripts candαif there is no possible confusion. From\n(3.7), we get\nb(x)−b(0) =/integraldisplayx\n0b/prime(s)ds=β\n2c/integraldisplayx\n0se−αs2\n4m/prime(s)ds\n=β\n2c/parenleftBig\nxe−αx2\n4m(x)−/integraldisplayx\n0/parenleftbig1−αs2\n2/parenrightbige−αs2\n4m(s)ds/parenrightBig\n,(3.18)\nwhere we have used integration by parts. Notice that/integraltext∞\n0(1−αs2/2)e−αs2/4m(s)dsis well-\ndefined, since α∈(0,1]andmis bounded. Therefore, the existence of B:= limx→∞b(x)follows\nfrom (3.18). Moreover,\nB:=b(0)−β\n2c/integraldisplay∞\n0/parenleftBigg\n1−αs2\n2/parenrightBigg\ne−αs2/4m(s)ds.\nFormula (3.12) easily follows from integrating b/primefromx∈Rto∞and arguing as above.\n14To prove (3.14), it is enough to observe that by Lemma 3.1, for x≥1and0<α≤1,\n/integraldisplay∞\nxe−αs2/4ds≤2\nαxe−αx2\n4≤2\nαxe−αx2\n4,and/integraldisplay∞\nxs2e−αs2/4ds≤16\nα2xe−αx2\n4.(3.19)\nSettingw=m+inand using (3.7), we obtain that wsatisfies the ODE\nw/prime+iceαx2/4w=−iβx\n2b(x), (3.20)\nor, equivalently,/parenleftBig\neicΦα(x)w/parenrightBig/prime\n=−iβx\n2b(x)eicΦα(x). (3.21)\nIntegrating (3.21) from 0tox>0, and writing\neicΦα(x)=−i\nc/parenleftBig\neicΦα(x)/parenrightBig/prime\ne−αx2/4,\nintegrating by parts, and using once again (3.7), we get\neicΦα(x)w(x) =w(0)−β\n2cxb(x)eicΦα(x)−αx2/4\n+β\n2c/integraldisplayx\n0eicΦα(s)−αs2/4/parenleftBigβ\n2s2n(s) + (1−αs2\n2)b(s)/parenrightBig\nds.\nSinceα∈(0,1], from the above identity it follows the existence of\nW:= limx→∞eicΦα(x)w(x),\nand formula (3.16) for W.\nFormula (3.15) now follows from integrating (3.21) from x >0to∞and arguing as in the\nprevious lines. The estimate in (3.17) can be deduced as before, since the bounds in (3.19) imply\nthat\n|wc,α(x)−e−icΦα(x)Wc,α|≤β\n2cxe−αx2/4/parenleftbigg\n1 +16(α+β)\n2α2+2\nα/parenrightbigg\n≤10β\ncα2xe−αx2/4,\nwhere we used that α+β≤2andα≤1.\nTo see that the limiting values Bc,αandWc,αgiven by (3.13) and (3.16) are continuous\nfunctions of (c,α), for (c,α)∈(0,∞)×(0,1], we recall that by standard ODE theory, the func-\ntionsmc,α(x),nc,α(x)andbc,α(x)are continuous functions of x,candα. Then, the dominated\nconvergence theorem applied to the formulae (3.13) and (3.16) yield the desired continuity.\nRemark 3.3. As mentioned before, the shrinkers of the 1d-harmonic heat flow can be computed\nexplicitly, because if α= 1, the system (1.5)-(1.6)-(1.7)can be solved easily. Indeed, in this case\nβ= 0, so that we obtain\nmc,1(x) = (cos(cΦ1(x)),sin(cΦ1(x)),0),\nnc,1(x) = (−sin(cΦ1(x)),cos(cΦ1(x)),0),\nbc,1(x) = (0,0,1),\nfor allx∈R.\nIn order to obtain a better understanding of the asymptotic behavior of {mc,α,nc,α,bc,α},\nwe need to exploit the oscillatory character of the function eicΦα(s)in the integrals (3.12) and\n(3.15). In our arguments we will use the following two lemmas.\n15Lemma 3.4. Let0<α≤1. Forσ∈R\\{0}andx∈R, the limit\n/integraldisplay∞\nxseiσΦα(s)ds:= limy→∞/integraldisplayy\nxseiσΦα(s)ds\nexists. Moreover, for all x≥1,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞\nxseiσΦα(s)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤11x\n|σ|αe−αx2/4, (3.22)\nand/integraldisplay∞\nxseiσΦα(s)ds=ix\nσeiσΦα(x)−αx2/4+O/parenleftBigg\nx2\nσ2e−αx2/2/parenrightBigg\n. (3.23)\nProof.Letx∈Rand takey≥x. Then, integrating by parts,\n/integraldisplayy\nxseiσΦα(s)ds=1\niσ/integraldisplayy\nxs(eiσΦα(s))/primee−αs2/4ds\n=s\niσeiσΦα(s)−αs2/4/vextendsingle/vextendsingle/vextendsingle/vextendsingley\nx−1\niσ/integraldisplayy\nxeiσΦα(s)−αs2/4/parenleftbig1−αs2\n2/parenrightbigds. (3.24)\nThe existence of the improper integral/integraltext∞\nxseiσΦα(s)dsfollows taking the limit as ygoes to∞\nin the above formula, and bearing in mind that α>0. The estimate (3.22) follows from (3.19)\nand the fact that x≥1and0<α≤1. Finally, integrating by parts once more, we have\niσ/integraldisplay∞\nxeiσΦα(s)−αs2/4/parenleftbig1−αs2\n2/parenrightbigds=−eiσΦα(x)−αx2/2/parenleftbig1−αx2\n2/parenrightbig−/integraldisplay∞\nxeiσΦα(s)−αs2/2/parenleftbigα2s3\n2−2αs/parenrightbigds.\nHence, using Lemma 3.1 and (3.24), we obtain (3.23).\nLemma 3.5. Letσ∈R\\{0},γ∈R,α>0and set ˜γ=γ+α/4. If0<˜γ≤1, then forx≥1,\n/integraldisplay∞\nxeiσΦα(s)−γs2ds=O/parenleftBigg\ne−˜γx2\n|σ|/parenrightBigg\n,/integraldisplay∞\nxseiσΦα(s)−γs2ds=O/parenleftBigg\nxe−˜γx2\n|σ|˜γ)/parenrightBigg\n,\n/integraldisplay∞\nxs2eiσΦα(s)−γs2ds=O/parenleftBigg\nx2e−˜γx2\n|σ|˜γ/parenrightBigg\n.(3.25)\nProof.Forn∈{0,1,2}, we set\nIn=/integraldisplay∞\nxsneiσΦα(s)−γs2ds.\nIn=1\niσ/parenleftbigg\n−xneiσΦα(x)−˜γx2−/integraldisplay∞\nxeiσΦα(s)−˜γs2/parenleftBig\nnsn−1−2˜γsn+1/parenrightBig\nds/parenrightbigg\n.\nThen the desired asymptotics follow from Lemma 3.1.\nUsing previous lemmas, we can now improve the asymptotics in Proposition 3.2 and obtain\nexplicitly the term decaying as e−αx2/4(multiplied by a polynomial).\nCorollary 3.6. With the same notation as in Proposition 3.2, the following asymptotics hold\nforx≥1\nbc,α(x) =Bc,α+βx\n2ce−αx2/4Re(e−icΦα(x)Wc,α) +β\nc2α3O(x2e−αx2/2), (3.26)\nwc,α(x) =e−icΦα(x)/parenleftBig\nWc,α−βBc,α\n2cxeicΦα(x)−αx2/4+iβ2Wc,α\n8c/integraldisplay∞\nxs2e−αs2/4ds/parenrightBig\n(3.27)\n+β\nc2α5O(x2e−αx2/2).\n16Proof.As usual, we drop the subscripts candαin the rest of the proof. Recalling that w=\nm+in, we have from (3.17),\nm= Re(e−icΦα(x)W) +β\ncα2O(xe−αx2/4).\nThus, replacing in (3.12),\nb(x) =B+βx\n2ce−αx2/4Re(e−icΦα(x)W) +β2\nc2α2O(x2e−αx2/2) +Rb(x),(3.28)\nwith\nRb(x) =β\n2cRe/parenleftBigg\nW/integraldisplay∞\nx/parenleftbig1−αs2\n2/parenrightbige−icΦα(s)−αs2/4ds+/integraldisplay∞\nx/parenleftbig1−αs2\n2/parenrightbigO/parenleftbigse−αs2/2\ncα2/parenrightbigds/parenrightBigg\n.\nByusingLemmas 3.1and3.5toestimatethefirstandsecondintegrals, respectively, weconclude\nthat\nRb(x) =β\nc2α3O/parenleftbigx2e−αx2/2/parenrightbig. (3.29)\nBy putting together (3.28) and (3.29), we obtain (3.26). To establish (3.27) we integrate (3.21)\nfromx≥1and∞, and use (3.26) and Lemma 3.1 to get\neicΦw(x)−W=I1(x) +I2(x) +I3(x) +β2\nc2α5O(xe−αx2/2), (3.30)\nwith\nI1(x) =iβB\n2/integraldisplay∞\nxseicΦα(s)ds, I 2(x) =iβ2W\n8c/integraldisplay∞\nxs2e−αs2/4ds,and\nI3(x) =iβ2¯W\n8c/integraldisplay∞\nxs2e2icΦα(s)−αs2/4ds,\nwhere we have used that Re(z) = (z+ ¯z)/2. The conclusion follows invoking again Lemmas 3.1,\n3.4 and 3.5.\nIn Figure 2 we depict the first components of the trihedron {mc,α,nc,α,bc,α}forc= 0.5and\nα= 0.5, andx>0. As described in Corollary 3.6 (recall that wc,α=mc,α+inc,α), in the plots\nin Figure 2 one can observe that, while both m1,c,αandb1,c,αoscillate highly for large values of\nx>0, the component b1,c,αconverges to a limit B1,c,α≈−0.72asx→+∞.\n246810\n-1.0-0.50.51.0\n(i)m1,c,α\n (ii)n1,c,α\n246810\n-1.0-0.8-0.6-0.4-0.2 (iii)b1,c,α\nFigure 2: Functions m1,c,α,n1,c,αandb1,c,αforc= 0.5andα= 0.5onR+. The limit at infinity\nin (iii) isB1,c,α≈−0.72.\n173.3 Proof of Theorem 1.3\nFor simplicity, we will drop the subscripts candαin the proof of Theorem 1.3.\nProof of Theorem 1.3. Let{m,n,b}be the solution of the Serret–Frenet system (3.4)–(3.5)\nwith initial condition (3.6). By ODE theory, we have that the solution {m,n,b}is smooth, is\nglobal, and satisfies\n|m(x)|=|n(x)|=|b(x)|= 1,for allx∈R, (3.31)\nand the orthogonality relations\nm(x)·n(x) =m(x)·b(x) =n(x)·b(x) = 0,for allx∈R. (3.32)\nDefine\nm(x,t) =m/parenleftbiggx√\nT−t/parenrightbigg\n, t<T.\nThen, it is straightforward to check that mis a smooth solution of the profile equation (3.1),\nand consequently msolves (LLG) for t∈(−∞,T). Moreover, using the Serret–Frenet system\n(3.4)–(3.5), we have\n|∂xm(x,t)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle1√\nT−tm/prime/parenleftbiggx√\nT−t/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle=c√\nT−teαx2\n4(T−t).\nThis shows part (i).\nNotice also that, since {mj,nj,bj}forj∈{1,2,3}solves (3.7) with initial condition (1,0,0),\n(0,1,0)and(0,0,1)respectively, the relation (3.8) is satisfied, i.e.\nm2\nj(x) +n2\nj(x) +b2\nj(x) = 1,for allx∈R, j = 1,2,3. (3.33)\nPart(ii)follows from the uniqueness given by the Cauchy–Lipschitz theorem for the solution\nof (3.1)–(3.5) with initial condition (3.6), and the invariance of (3.1)–(3.5)–(3.6) under the\ntransformations\nm(x) = (m1(x),m2(x),m3(x))→(m1(−x),−m2(−x),−m3(−x)),\nn(x) = (n1(x),n2(x),n3(x))→(−n1(−x),n2(−x),n3(−x)),\nb(x) = (b1(x),b2(x),b3(x))→(−b1(−x),b2(−x),b3(−x)),\nSettingWj=ρjeiφj, withρj≥0andφj∈[0,2π], the asymptotics for mjand it derivative\nm/prime\nj, forj∈{1,2,3}in part(iii)are a direct consequence of the asymptotic behavior of the\nprofile established in Proposition 3.2 and Corollary 3.6 (recall also that m/prime(x) =ceαx2\n4n(x)and\nw=m+in). In particular the following limits exist:\nWj:= limx→∞eicΦα(x)(mj+inj)(x), Bj:= limx→∞bj(x).\nNotice also that the relations in (3.33) implies that ρ2\nj+B2\nj= 1, and the fact that bis unitary\nimplies that B2\n1+B2\n2+B2\n3= 1, and thatρ2\n1+ρ2\n2+ρ2\n3= 2.\nThe convergence in part (iv)is an immediate consequence of the definition of min terms of\nthe profilem, and the asymptotics established in part (iii). The relations between ρ−\njandρj,\nforj∈{1,2,3}follow from the parity relations for the profile mestablished in part (ii).\nIt remains to prove part (v). Forϕ∈W1,∞(R), bearing in mind (3.15) and (3.17), it suffices\nto show that\nlim\nt→0+/integraldisplay\nRe−icΦα(x/√\nt)ϕ(x)dx= 0,and lim\nt→0+/integraldisplay\nR/parenleftbig1 +|x|√\nt/parenrightbige−αx2\n4t|ϕ(x)|dx= 0.(3.34)\n18The second limit is a direct consequence of following the explicit computations:\n/integraldisplay\nRe−αx2\n4tdx=/radicalbiggπt\nα,and/integraldisplay\nR|x|e−αx2\n4tdx=4t\nα. (3.35)\nFor the first limit in (3.34), we integrate by parts to obtain\nic/integraldisplay\nRe−icΦα(x/√\nt)ϕ(x)dx=−√\nt/integraldisplay\nR/parenleftbige−icΦα(x/√\nt))/primee−αx2\n4tϕ(x)dx\n=/integraldisplay\nRe−icΦα(x/√\nt)−αx2\n4t/parenleftbigg√\ntϕ/prime(x)−αx\n2√\ntϕ(x)/parenrightbigg\ndx.\nSince|ϕ|and|ϕ/prime|are bounded on R, the conclusion follows again by using (3.35).\n4 Limiting behaviour of the trajectories of the profiles\nIn this section we prove Theorem 1.4. In what follows we denote C±\nc,αthe great circleCc,α=\nP±\nc,α∩S2, withP±\nc,αbeing the planes passing through the origin with (unitary) normal vectors\nB+\nc,α= (B1,c,α,B2,c,α,B3,c,α)andB−\nc,α= (−B1,c,α,B2,c,α,B3,c,α),\ngiven by Theorem 1.3.\nWe will show that the trajectories of the profiles mc,αconverge to the great circles C±\nc,αas\nx→±∞, and study the behavior of the limit circles C±\nc,αwith respect to the parameters cand\nα, analyzing the angle ϑc,α∈[0,π]between their normal vectors B+\nc,αandB−\nc,α.\nWe start this section stating a corollary of Theorem 1.3 that will be used in what follows.\nPrecisely, recalling that w=m+in, from (3.17) and using the constants defined in Theorem 1.3,\nwe have the following\nCorollary 4.1. Forx≥1andj∈{1,2,3}, we have\nmj,c,α(x) =ρj,c,αcos(cΦα(x)−φj) +Rj(x), nj,c,α(x) =−ρj,c,αsin(cΦα(x)−φj) +˜Rj(x),\nfor some functions Rjand ˜Rjsatisfying the bounds |Rj(x)|,|˜Rj(x)|≤10β/(cα2)xe−αx2/4.\nWe will also use the two lemmas below in the proof of Theorem 1.4. The first one establishes\nsome relations between the constants appearing in the asymptotics of the profile mc,αthat can\nbe deduced by using some geometric properties of the Serret–Frenet system.\nLemma 4.2. The constants given by Theorem 1.3 satisfy the following identities\nB1,c,α=ρ2,c,αρ3,c,αsin(φ3,c,α−φ2,c,α), B2,c,α=ρ1,c,αρ3,c,αsin(φ1,c,α−φ3,c,α),\nB3,c,α=ρ1,c,αρ2,c,αsin(φ2,c,α−φ1,c,α),(4.1)\nand\nB1,c,αρ1,c,αeiφ1,c,α+B2,c,αρ2,c,αeiφ2,c,α+B3,c,∞ρ3,c,αeiφ3,c,α= 0, (4.2)\nρ2\n1,c,αe2iφ1,c,α+ρ2\n2,c,αe2iφ2,c,α+ρ2\n3,c,αe2iφ3,c,α= 0. (4.3)\n19Proof.Dropping the subscripts candα, using the relation b=m×nand the asymptotics in\nCorollary 4.1, we get for x≥1,\nb1(x) =−ρ2ρ3/parenleftbigcos(cΦα(x)−φ2) sin(cΦα(x)−φ3)−cos(cΦα(x)−φ3) sin(cΦα(x)−φ2)/parenrightbig+o(1),\nb2(x) =−ρ3ρ1/parenleftbigcos(cΦα(x)−φ3) sin(cΦα(x)−φ1)−cos(cΦα(x)−φ1) sin(cΦα(x)−φ3)/parenrightbig+o(1),\nb3(x) =−ρ1ρ2/parenleftbigcos(cΦα(x)−φ1) sin(cΦα(x)−φ2)−cos(cΦα(x)−φ2) sin(cΦα(x)−φ1),/parenrightbig+o(1),\nwhereo(1)is a function of x, which depends on the parameters αandc, that converges to 0as\nx→∞. Noticing that, for k,j∈{1,2,3},\n−cos(cΦα(x)−φj) sin(cΦα(x)−φk) + cos(cΦα(x)−φk) sin(cΦα(x)−φj) = sin(φk−φj),\nwe obtain\nb1(x) =ρ2ρ3sin(φ3−φ2) +o(1),\nb2(x) =ρ1ρ3sin(φ1−φ3) +o(1),\nb3(x) =ρ1ρ2sin(φ2−φ1) +o(1).\nLettingx→∞, in the above identities we obtain (4.1).\nTo establish the other identities, we first recall that the vectors m,nandbsatisfy the\nfollowing relations\nm·n=n·b=b·m= 0and|m|=|n|=|b|= 1.\nNow, from the asymptotics in Corollary 4.1 and the identity\n(m+in)·b= 0,\nwe have\nb1(x)ρ1e−i(cΦα(x)−φ1)+b2(x)ρ2e−i(cΦα(x)−φ2)+b3(x)ρ3e−i(cΦα(x)−φ3)=o(1),\nfori∈{1,2,3}. Dividing by e−icΦα(x)and letting x→∞, we obtain formula (4.2). Finally,\nformula (4.3) follows easily using a similar argument, bearing in mind the orthogonality relation\nm·n= 0,and that 2 cos(y) sin(y) = Im(e2iy), fory∈R.\nRemark 4.3. Although we do not use (4.3)in this work, this relation could be helpful in estab-\nlishing further properties of the solutions.\nNext we study the angle between the normal vectors to the great circles C±\nc,α, given by\nϑc,α= arccos(2B2\n1,c,α−1),withϑc,α∈[0,π]. We have the following.\nLemma 4.4. Forc≥β√π/√α, we have\nϑc,α≥arccos/parenleftBigg\n−1 +2πβ2\nc2α/parenrightBigg\n. (4.4)\nProof.Using the formula (3.12) in Proposition 3.2 for b1,c,αwithx= 0, we get\n|b1,c,α(0)−B1,c,α|≤β\n2c/integraldisplay∞\n0/parenleftBigg\n1 +αs2\n2/parenrightBigg\ne−αs2/4.\n20Noticing that b1,c,α(0) = 0, and that\n/integraldisplay∞\n0/parenleftBigg\n1 +αs2\n2/parenrightBigg\ne−αs2/4=2√π√α,\nwe conclude that\n2B2\n1,c,α≤2πβ2\nc2α.\nSincec≥β√π/√α, we have−1 + 2πβ2/(c2α)∈[−1,1], so we can use that the function arccos\nis decreasing to obtain (4.4).\nWe continue to prove Theorem 1.4.\nProof of Theorem 1.4. As usual, we omit the subscripts candαwhen there is no confusion.\nIn view of the symmetries established in Theorem 1.3, it is enough to prove the theorem for\nx→∞. We start noticing that |Bc,α|= 1, so that the distance between mandP+is given by\ndist(m(x),P+) =|m1(x)B1+m2(x)B2+m3(x)B3|.\nTo compute the leading term, we notice that using (4.2), we have\n3/summationdisplay\nj=1ρjBjcos(cΦα(x)−φj) = Re/parenleftBigg\neicΦα(x)/parenleftBigg3/summationdisplay\nj=1ρjBje−iφj/parenrightBigg/parenrightBigg\n= 0.\nThus the estimates in Corollary 4.1 give us\ndist(m(x),P+))≤30β\ncα2xe−αx2/4,forx≥1, (4.5)\nLet us fix x≥1and letQ(x)be the orthogonal projection of m(x)on the planeP+, so\nthat dist(m(x),P+) = dist(m(x),Q(x)). We also set C(x)∈C+such that dist(m(x),C+) =\ndist(m(x),C(x)). By the Pythagorean theorem,\n|Q(x)|2= 1−dist(m(x),Q(x))2\nand\ndist(m(x),C(x))2= dist(m(x),Q(x))2+ (1−|Q(x)|)2.\nTherefore\ndist(m(x),C(x))2= 2−2(1−dist(m(x),Q(x))2)1/2,\nand using the elementary inequality√1−y≥1−y, fory∈[0,1], we get\ndist(m(x),C(x))≤√\n2 dist(m(x),Q(x)).\nCombing this estimate with (4.5), we conclude that\ndist(m(x),C+)) = dist(m(x),C(x)))≤√\n2 dist(m(x),P+))≤30√\n2β\ncαxe−αx2/4.\nThe limits in (1.13) follow at once using the definition of mc,αin (1.8) (recall that ϑc,α∈[0,π]).\nThe statement in (ii)is an immediate consequence of Lemma 4.4. This finishes the proof of\nTheorem 1.4.\n21The limit limc→∞ϑc,α=πin part (ii) of Theorem 1.4 helps us to understand the angle\nbetween the great circles as c→∞. However, the behavior of the limit circles C±\nc,αforcsmall\nis much more involved. We conclude this section with some reflections on the behavior of the\nlimit circlesC±\nc,αwhencis small.\nLet us remark that when c= 0, the explicit solution to (1.6)–(1.7) is given by\nm0,α(x) = (1,0,0),\nn0,α(x) = (0,cos(βx2/4),−sin(βx2/4)),\nb0,α(x) = (0,sin(βx2/4),cos(βx2/4)).(4.6)\nThus we see that in this limit case, there is a change in the behavior of the solution: There is\nno limit circle and the vector b0,αdoes not have a limit at infinity. On the other hand, we know\nthat (mc,α(x),nc,α(x),bc,α(x))are continuous with respect to the to c,αandx. Therefore,\nlim\nc→0b1,c,α(x) =b1,0,α(x) = 0,for allx∈R. (4.7)\nOf course, we cannot conclude from (4.7) estimates for B1,c,α, ascgoes to 0. For this reason,\nwe performed some numerical simulations for different values of csmall. In Figures 3 and 4, we\nshow one of these simulations, in the case α= 0.5andc= 0.01, where we see that m1,c,α≈1\nandb1,c,α≈0on[0,2]in agreement with (4.6) and the continuous dependence on c. On the\nother hand, for x≥8.5, we haveb1,c,α(x)≈−1.\nHowever, it seems difficult to infer from our simulations the behavior of Bc,αascgoes to zero.\nFor instance, we have computed numerically Bc,α, and it is not clear that this quantity converges\nforcsmall. Forinstance, wehaveobtained B1,c,α=−0.99215, forc= 10−12,B1,c,α=−0.992045,\nforc= 10−14, andB1,c,α=−0.991965, forc= 10−16.\n2468100.20.40.60.81.0\n(i)m1,c,α\n246810\n-1.0-0.8-0.6-0.4-0.2 (ii)b1,c,α\nFigure 3: Functions m1,c,αandb1,c,αforc= 0.01andα= 0.5. The limit at infinity in (ii) is\nB1,c,α≈−0.996417.\nAcknowledgements. S. Gutiérrez was partially supported by ERCEA Advanced Grant 2014\n669689 - HADE. The Université de Lille also supported S. Gutiérrez’s research visit during July\n2018 through their Invited Research Speaker Scheme. 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Chinese\nUniv. Ser. B , 17(2):164–170, 2002.\n26"    },    {        "title": "1211.0492v1.Dynamic_Spin_Injection_into_Chemical_Vapor_Deposited_Graphene.pdf",        "content": "Page 1 of 13 \n Dynamic Spin Injection into Chemical Vapor Deposited  \nGraphene  \nA. K. Patra1,a), S. Singh2,a), B. Barin2, Y. Lee3, J.-H. Ahn3, E. del Barco2,b), E. R. \nMucciolo2 and B. Özyilmaz1,4,5,6 ,b) \n1Department of Physics, National University of Singapore, 2 Science Dri ve 3, Singapore 117542  \n2Department of Physics, University of Central Florida, Orlando, Florida USA, 32816  \n3 School of Advanced Materials Science & Engineering, SKKU Advanced Institute of \nNanotechnology  (SAINT) , Sungkyunkwan University, Suwon, Republic  Kore a 440746  \n4NanoCore, 4 Engineering Drive 3, National University of Singapore, Singapore 117576  \n5Graphene Research Center, National University of Singapore, Singapore 117542  \n6NUS Graduate School for Integrative Sciences and Engineering (NGS), National Univer sity of \nSingapore, Singapore 117456  \n \nWe demonstrate dynamic spin injection into chemical vapor deposition (CVD)  grown \ngraphen e by s pin pumping from permalloy  (Py) layer s. Ferromagnetic resonance \nmeasurements at room temperature reveal a strong enhancement of the  Gilbert damping \nat the Py/graphene interface , exceeding that observed in even Py/platinum  interfaces.  \nSimilar results are als o shown on Co/graphene layers . This enhancement  in the Gilbert \ndamping is understood as the consequence of spin pumping at t he interface d riven by \nmagnetization dynamics .  Our observation s suggest a  strong enhancement of spin -orbit \ncoupling in CVD graphene , in agreement with earlier spin valve measurements.    \na)A. K. Patra and S. Singh contributed equally to this work.  \nb) Autho rs to whom correspondence should be addressed. Electronic mail s: delbarco@physics.ucf.edu  and \nphyob@nus.edu.sg  \n \nPACS numbers: 72.25. -b, 76.50.+g  Page 2 of 13 \n In spintronic s, where the electron’s  spin degree of freedom , rather than its charge , is \nemployed to process in formation , the efficient generation of the large spin current s stands as  a \nkey requirement for future spintronic device s and applications. Several approaches to generate \npure spin currents  have been proposed and  are being widely investigated, namely, non-local spin \ninjection  [1], spin Hall effect  [2-4], and spin pumping  [5,6]. Among these , spin pumping offers  \nthe advantage  of producing  spin current s over large ( mesoscopic ) areas [7-13] at \nferromagnet ic/non-magnetic  (FM/NM)  interface s. In addition, dynamical  spin pumping is \ninsensitive to a potential impedance mismatch at the FM/NM interface  [14], a problem  \nubiquitous  in the non -local spin injection approach.  Dynamical s pin pumping consists of  \ngenerating pure spin current (i.e., with no net charge current) away from  a ferromagnet  into a \nnon-magnetic material,  induced by the  coherent precession of the magnetization  upon application \nof microwave stimuli of frequency matching the  ferromagnetic resonance (FMR)  of the system  \n[15]. Since pure spin currents carry awa y spin angular momentum, in an FMR experiment the \ntransfer of angular momentum from the FM into the NM layer results in an  enhance ment of the \nGilbert damping in the ferromagnet [5-15]. Most  studies of dynamical spin pumping on FM/NM  \ninterfaces have made us e of Pt and Pd NM layers, since the large spin -orbit coupling in these \nsystems enables the conversion of the injected spin current into an electric voltage  across the N M \nlayer, a phenomenon known as i nverse spin Hall effect  (ISHE) . Recently, spin pumping h as been \nexperimentally demonstrated in FM/semiconductor interfaces (e.g. , GaAs  [13] and p -type Si  \n[15]). However, there is no experimental report on spin pumping in FM/graphene interface s, \nthough graphene  [16] (a two -dimensional layer  of carbon atoms ), pos sesses unique electronic \nproperties (e.g. high mobility  and gate-tunable charge carrier , among others ), and stands as an \nexcellent material for spin transport due to its large spin coherence length [17]. Page 3 of 13 \n In this Letter we report experimental FM R studies of Py and Co  films and  polycrystalline \ngraphene grown by chemical vapor deposition on Cu foils [ 18,19] (henceforth, Co/Gr and Py/Gr , \nrespectively ) performed in  a broad -band microwave coplanar waveguide (CPW)  spectrometer . \nThe observation of a remarkable  broadening of the FMR absorption peaks in the Py/ Gr (88%)  \nand Co/Gr (133%) films demonstrate a strong  increase  of the Gilbert damping in the FM layer  \ndue to spin pumping at the FM/Gr interface and the consequent loss of angular momentum  \nthrough spin inject ion into the CVD graphene  layer . To account for such a remarkable absorption  \nof angular momentum , the spin orbit coupling in CVD graphene must be orders of magnitude \nlarge r than what is predicted for pristine, exfoliated graphene . \nTo prepare the FM/Gr sample s, single layer CVD grown graphene [18, 19] was first \ntransferred onto a Si substrate with 300 nm thick SiO 2 layer . The sample was then anneal ed in a \nH2/Ar environment at 300 °C for 3 hour to remove all organic residue s. For the Py layer we chose \nNi80Fe20, a material extensively  used for magnetic thin film  studies because  of its low \nmagneto crystalline anisotropy  and its insensitivity to strain.  The FM layer  (Py/14nm, Co/15nm)  \nwas deposited on top of the graphene layer lying over the SiO 2/Si substrate by electron-beam \nevaporat ion at a base pressure of 310-7 Torr. For the purpose of FMR comparison experiments, \na control FM film of the same thickness was deposited simultaneously on the same SiO 2 wafer in \nan area where graphene was no t present . The schematic of th e FM/Gr samples  is shown in \nFig. 1-a, together with the Raman spectr um of the  CVD  graphene  before the deposition of Py  \n(Fig. 1-c). The high intens ity of the  2D peak , when  compared to the G peak , and the  weakness of \nthe D peak , suggests that graphene is sin gle layer and of high quality ( i.e. low degree of \ninhomogeneity/defects ).   Page 4 of 13 \n \n \nFig. 1 : (Color online) (a) Schematic of the FM/Gr film sample. (b) Schematic of the FMR measurement \nsetup, with the sample placed up -side-down on top of the micro -CPW . (c) Raman spectrum of CVD \ngraphene.  \nFMR measurements were carried out at room temperature with a high -frequency \nbroadband (1 -50 GHz) micro -coplanar -waveguide ( -CPW) [20] using the flip -chip method [ 21-\n23], by which the sample is placed up -side-down covering the cen tral part of the CPW  (as shown \nin Fig. 1-b), where the transmission line is constricted to increase the density of the microwave \nfield and enhance sensitivity . The CPW was covered with a 100nm -thick insulating layer of \nPMMA resist, hardened by electron bea m exposure, to avoid any influence of the CPW, made \nout of gold, on the sample dynamics. A 1.5  Tesla rotatable  electromagnet was employed to vary \nthe applied field direction from the in -plane ( = 0o) to normal -to-the film plane (  = 90o) \ndirections . Fig. 2-a shows t he angular dependence of the FMR field measured at 10  GHz for both \nPy and Py/Gr films. The rotation plane is chosen to keep the dc magnetic field , H, perpendicular \nto the microwave field felt by the sample at all times, as shown in Fig. 1-b. The resonance fiel d \nincreases as the magnetic field is directed away from the film plane (i.e. increasing ), as \nexpected for a thin film ferromagnet with in-plane shape magneto -anisotropy.  The angular \ndependence of the  FMR field ( HR) can be fitted using the resonance frequency condition given \nby the Smit and Beljers formula [ 23,24], Page 5 of 13 \n \n21HH,                                                                       (1)  \nwhere \nf2  is the angular frequency, \n/Bg  the gyromagnetic ratio, and H1 and H2 are \ngiven by  \nH1=Hcos(-)-4Meffsin2\nH2=Hcos(-)4Meffcos22K2\nMssin2\n ,                                       (2) \nwhere  is the magnetization  angle , \n 2\n2 1 cos 4 2 4 4S S S eff MK MK M M   is the \neffective demagnetization field,  Ms is the saturation magnetization, and K1 and K2 are the first \nand second order anisotropy energies, respectively . The best fit s to the data in Fig. 2-a are given \nby the parameters shown  in the third  column of Table 1 , together with the corresponding \nparameters extracted from equivalent measurements on the Co and Co/Gr films (not shown) . \n \n \nFig. 2 : (color online) (a) Angular dependence of the FMR fi elds measured on both Py and Py/Gr samples \nat f = 10 GHz with the dc magnetic field, H, applied in a plane perpendicular to the microwave field \ngenerated by  the CPW at the sample position. (b) In-plane frequency dependence  of the FMR fields for \nboth Py and  Py/Gr samples. The intercepts with the x-axis give the effective demagnetizing fields of the \nsamples.  Page 6 of 13 \n It is useful to study the resonant behavior by applying the magnetic field at  = 0o \n(parallel configuration) and   = 90 (perpendicular configuration),  since the frequency behavior \nof the FMR fields are given respectively by,  \n       \n\n\n\n\n \n\n\n\n,//,2\n//\n4) 4 (\neff Reff R R\nM HM HH\n ,             (3) \nwhere \nBg  is the gyromagnetic ratio , \n1 //,4 4A S eff H M M   , \n2 1 ,4 4A A S eff H H M M \n, with \nS A MK H1 12  and \nS A MK H2 24  the first and second \norder anisotropy fields, respectively , which relate to su rface, interface and/or magnetoelastic \nanisotropy. Note that K 1 > 0 (>> K2) provides out -of-plane anisotropy, competing with the in -\nplane shape anisotropy . Consequently, a graphical representation of the in - and out -of-plane \nfrequency response of the FMR f ields , conforming to Eq . (3), results in a linear behavior from \nwhich the slope and intercept with the magnetic field axis give  and the effective \ndemagnetiz ation field s, respectively.  The results obtained for the Py and Py/Gr samples are \nshown in Fig. 2-b and 2-c, and the extracted parameters are listed  in the third  column of Table  1, \ntogether with those extracted from the Co and Co/Gr . Note that the anisotropy fields depend on \nthe selection of the saturation magnetization, w ith theoretical values MS,Py = 9.27 kG (attending \nto a 20/80 -Ni/Fe ratio  and assuming identical densities) , and MS,Co = 17.59 kG. For the Co and \nCo/Gr films, the effective saturation magnetization ( Meff = 17.7 kG) is similar to the one \nexpected from theo ry, hence there is negligible ou t-of-plane anisotropy (K 1 ~ 0), in agreement \nwith previous studies [ 25]. The situation is different in the case of the Py and Py/Gr, where the \nsmall Py anisotropy field HA1 = 1.98 kG grows significantly in the Py/Gr ( HA1 = 3.60 kG), \nsuggesting an increase of the Py surface anisotropy due to the presence of the graphene layer ( i.e. Page 7 of 13 \n interface effect). Nevertheless, the magnetization remains in the plane of the film  for all samples . \nTheory  Sample  HR vs.  , f Damping  Changes  \nPy: Ni80Fe20 \ngeff = 2.10  \nFeFe\nS NiNi\nSFe\nSNi\nS\neffg M g MM Mg\n2.0 8.02.0 8.0\n\n \n \nMs = 9.27 kG \nFe\nSNi\nS M M Ms 2.0 8.0\n \n \nwith \n21.2 094.6  NiNi\nS g kG M\n \n0.2 016.22  FeFe\nS g kG M\n \n Py g = 2.110 \n = 0.0 113 \nG = 0.311 GHz \nK1 increases  \n(interface)  \n \nDamping increases \nby ~88% Meff = 7.30 kG \nH1 = 1.98 kG \nK1 = 0.73106 erg/cc  \nPy/Gr  g = 2.107 \n = 0.0 213 \nG = 0.585 GHz Meff,// = 5.70 kG \nHA1 = 3.60 kG \nK1 = 1.32106 erg/cc  \nCo \ng = 2.145 \nMs = 17.59  kG Co g = 2. 149  = 0.0 210 \nG = 1.11 GHz (no K1)  \nDamping increases \nby ~133% Meff = 17.7 kG \nCo/Gr  g = 2. 149  = 0.0 489 \nG = 2.59 GHz Meff = 17.5 kG \n \nTABLE  I: Parameters extracted from the analysis of the data reported in this work.  \n \nWe now focus on the FMR linewidth and its frequency dependence when the magnetic \nfield is applied parallel to the film (  = 0o), from which information about the Gilbert damping \n(i.e., spin relaxation dynamics)  can be directly extracted. The inset to  Fig. 3 shows a field \nderivate of the CPW S 21 transmission parameter obtained when exciting the FMR at 10  GHz in \nboth Py and Py/Gr samples , with HR = 1.28 kG and 1.55 kG, respectively . The peak -to-peak \ndistance repre sents the linewidth , H, of the FMR, whose behavior as a function of frequency is \nshown for both samples in the main panel of Fig. 3. A remarkable in crease of the FMR linewidth \nby 88% is observed in the Py/Gr sample , and even higher (133%) in the Co/ Gr fil ms. The change \nin the linewidth  must be attributed to a substantial enhancement of the Gilbert damping in the Page 8 of 13 \n FM film due to the influence of the graphene directly underneath.  The frequency dependence of \nthe FMR linewidth can be written as a contribution f rom two parts : \nf H H\n\n34\n0\n,                                                               ( 4) \nwhere  is the parameter of the Gilbert damping \nSM G . The first term, \n0H , accounts for \nsample -dependent in homogeneous broadening of the linewidth and is independent of frequenc y, \nwhile the second term represents the dynamical  broadening of the FMR  and scales linearly with \nfrequency .  \n \n \nFig. 3: (color online) Frequency dependence of the FRM linewidth for Py , Py/Gr , Co and Co/Gr  films \nobtained with the magnetic field applied at  = 0 (in-plane configuration). The inset shows the field \nderivatives of CPW S 21 transmission parameter (at 10  GHz)  of the Py and Py/Gr samples , from which the \nlinewidth, H, is calculated as the peak -to-peak distance.  \n Page 9 of 13 \n As observed in Fig. 3, the measured l inewidth for  both FM and FM/Gr samples increase s \nlinearly with frequency, with negligible  inhomogen eous broadening,  indicat ing that damping in \nthe FM film can be properly  explained by the phenomenological Landau-Lifshitz -Gilbert \ndamp ing model . A similar br oadening of the FMR linewidth is observed in both samples when \nthe field is applied perpendicular to the plane, excluding frequency -dependence inhomogeneous \nbroadening (e.g. two -magnon scattering produced by changes in morphology of the FM surface \n[26]), as its possible source. By fitting the data in Fig. 3 to Eq. (4) (using \n0H  = 0), the damping \nparameter s  and G are determined and given in the fourth  column of Table  1 for all studied \nsamples. The Gilbert damping increases  substantially in the FM/Gr films  as a result of the \nincreased linewidth, when compared to the values  obtained in the FM sample s (which are \ncomparable  with values given in the literature  for similar Py  and Co  films  [9,22]). This is our key \nfinding.  Remarkably , the change in the damping para meter  in the Py/Gr sample \n(\nPy GrPy /  = 0.01 ) is even  more pronounced  than those observed in  Py/Pt systems, in \nwhich the thick (when compared to graphene) heavy transition metal  Pt layer provides the  large \nspin-orbit coupling necessary to absorb (i.e. , relax ) the spin accumulation  pumped away from the \nferromagnet.  The efficiency of spin injection is usually cataloged by means of the interfacial \nspin-mixing conductance, which is proportional to the additional damping parameter, , as \nfollows:   \n FMSdMg4\n,                                                               (5) \ngiving \ng = 5.261019 m-2 for our Py/Gr sample  with the thickness of the Py film dFM = 14 nm. \nThe Py/Gr  value is substantially larger than those found in other Py/NM systems with a metallic Page 10 of 13 \n NM layer, e.g. , \ng = 2.191019 m-2 in Py(Ni 81Fe19:10nm)/Pt(10nm)  [9] or \ng = 2.11019 m-2 in \nPy(Ni 80Fe20:15nm)/Pt(15nm)  [11]. Note that in the cited experiments, the spin -diffusion length \nof the non -magnetic layer (~10  nm for Pt) is smaller than the layer thickness. This is sign ificant \nsince it explains how the Pt layer is capable of dissipating the spin accumulation generated by the \ndynamical spin pumping , and account for the loss of angular momentum in the Py . In the case of \ngraphe ne, the enhancement of the damping parameter  is more complicated to understand. In a \nstandard FM/NM metallic system , the spin current injected in to the NM layer decays mainly \nperpendicularly to the interface [ 27], causing the enhancement of the damping parameter to \ndepend on the ratio between the layer  thickness and the spin -diffusion length in the NM . \nHowever, graphene  has effectively zero thickness  and, at least theoretically , a very weak intrinsic \nspin-orbit coupling. Therefore, the spin current must  decay in a FM/Gr film parallel and not \nperpendicul ar to the interface . Furthermore, some sizable spin -orbit coupling must exist in CVD \ngraphene  films . The lat ter may also explain the generally observed very short spin relaxation \ntimes in lateral CVD graphene spin valves  [28]. Recently, small levels of hyd rogen  [29] and \ncopper adatoms [30] have been predicted to lead to a strong e nhancement of the spin-orbit \ncoupling, bringing it into  meV range. Cu adat oms are  certainly likely to be present  in the CVD  \nsamples utilized in our experiments, pointing at a possi ble explanation for the large spin \npumping effect observed in our FM/Gr films .  \n  \n \n Page 11 of 13 \n  \nReferences:  \n[1] F. J. Jadema, A. T. Filip, and B. J. van Wees: Nature  (London) 410, 345  (2001)  \n[2] 2.T. Kimura, Y. Ot ani, T. Sato, S. Takahashi, and S. Maekawa, Phys. Rev. Lett.  98, 156601 \n(2007).  \n[3] S. O. Valenzuela and M. Tinkham, Nature (London ) 442, 176 (2006 ). \n[4] T. Seki, Y. Hasegawa, S. Mitani, S. Takahashi, H. Imamura, S. Maekawa, J. 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"    },    {        "title": "1910.13107v1.Perpendicular_magnetic_anisotropy_in_Pt_Co_based_full_Heusler_alloy_MgO_thin_films_structures.pdf",        "content": "1 \n Perpendicular magnetic anisotropy in Pt/Co -based full Heusler alloy /MgO  thin films \nstructures  \n \nM.S. Gabor1a), M. Nasui1, A. Timar -Gabor2  \n1Center for Superconductivity, Spintronics and Surface Science, Physics and Chemistry \nDepartment, Technical University of Cluj -Napoca, Str. Memorandumului , 400114 Cluj -\nNapoca , Romania  \n2 Interdisciplinary Research Institute on Bio -Nano -Sciences, Babeș -Bolyai  University, \nStr. Treboniu Laurean, 400271, Cluj -Napoca, Romania  \n  \n \nAbstract  \nPerpendicular magnetic anisotropy  (PMA)  in ultrathin magnetic structures  is a key ingredient for the \ndevelopment of electrically controlled spintronic devices. Due to their relatively large spin -polarization , \nhigh Curie temperature  and low Gilbert damping t he Co -based full Heusler alloys are of special \nimportance  from  a scientific and application s point of view. Here, we study the mechanisms responsible \nfor the PMA in Pt/Co -based full Heusler alloy /MgO  thin films structures . We show that the  ultrathin  \nHeusler films exhibit  strong PMA even in the absence of magnetic annealing. By means of ferromagne tic \nresonance experiments, we demonstrate that the effective magnetization  shows a two -regime behavior  \ndepending on the thickness of the Heusler layers . Using  Auger spectroscopy measurements,  we evidence \ninterdiffusion at the underlayer/Heusler interface  and the formation of an interfacial CoFe -rich layer which \ncauses  the two -regime behavior. In the case of the ultrathin films, th e interfacial CoFe -rich layer promotes \nthe strong PMA through the electronic hybridization of the metal alloy  and oxygen  orbitals across the \nferromagnet /MgO interface . In addition,  the interfacial CoFe -rich layer  it is also generating  an increase  of \nthe Gilbert damping  for the ultrathin films  beyond the spin -pumping effect . Our results  illustrate  that the \nstrong PMA is not an intrinsic property of the Heusler/MgO interface but  it is actively influenced by the \ninterdiffusion, which can be tuned by a proper choice of the underlayer material, as we show for the case \nof the Pt, Ta and Cr underlayers.  \n \n \n                                                 \na) mihai.gabor@phys.utcluj.ro  2 \n  \n \nIntroduction  \n \nUltrathin films structures showing perpendicular magnetic anisotropy (PMA) are under intensive \nresearch for the development of electrically controlled spintronic devices. Particularly, current induced \nspin–orbit torques (SOTs) in heavy -metal/ferromagnet (FM) heterostructures showing PMA are used to \ntrigger  the magnetization switching1,2. Besides , the antisymmetric  interfacial Dzyaloshinskii -Moriya3,4 \ninteraction ( iDMI)  in similar  PMA  architectures , if strong enough, can lead to the formation of special \nchiral structures like skyrmions5 which are  drivable  by electrical currents6. In the case of  the spin transfer \ntorque magnet ic random -access  memories (STT -MRAMs), considered a s a poten tial replacement for the \nsemiconductor -based  ones, the use of strong PMA materials is required for increased thermal stability7, \nwhile high spin polarization and low Gilbert damping is needed  to obtain large magnetoresistive ratios \nand efficient curre nt induced STT switching8,9.  \nCobalt-based full Heusler alloys are a special class of ferromagnetic materials that attract an increased \nscientific interest , since the ir theoretical  prediction of  half-metallicity10. These compounds are described \nby the formula Co2YZ, where Y is a transition metal,  or a mixture of two trans ition metals , and Z is a main \ngroup , or a mixture of two main group  sp element s. Large magnetoresistive ratios are experimentally \ndemonstrated in certain Co2YZ based in -plane11-18 and out -of-plane19,20 magnetized magnetic tunnel \njunctions  (MTJs)  and r elative ly low Gilbert dampin g parameters were  determined for some compounds21-\n28. Furthermore , PMA was evidenced for  Co2FeAl/ MgO19,29-33, Co 2FeAl 0.5Si0.5/MgO34-37, \nCo2FeSi/MgO38,39 or Co2FexMn 1-xSi/MgO40,41 structures with different  non-magnetic  underla yers. In \nsome of the cases, an annealing stage was necessary to induce PMA, while  for the other the perpendicular \nmagnetiz ation  was achieved even in the as -deposited state. The origin of the strong  PMA in th is type of \nstructures is still under debate . It could be related to both the oxidation at the Heusler/MgO interface42,43 \nand to the spin -orbit interaction effects at the heavy -metal underlayer/Heusler interface44,45. Moreover, it \nwas recently pointed out that in t he case of Co 2FeAl/MgO the diffusion of Al towards the MgO layer \nduring annealing  plays an important role for the  stabiliz ation of  the PMA46,47. The precise knowledge and \ncontrol of the mechanisms responsible for PMA is essential in order to be able to develop viable  spintronic \napplication s. Therefore, i n this paper, we  study  the underlying physics governing the  PMA for Co2FeAl, \nCo2FeAl 0.5Si0.5, Co2FeSi and Co2Fe0.5Mn 0.5Si Heusler alloy thin films  sandwiched between Pt and  MgO \nlayers. We show that b elow  a certain  critical thickness all the Heusler films  show strong PMA even in the 3 \n absence of magnetic annealing . Additionally, using ferromagnetic resonance experiments , we demonstrate \nthat, depending on the thickness  of the Heusler layers,  the effective perpendicular magnetic anisotropy \nshows a  two-regime  behavior.  After excluding other possible mechanism s, we evidence  using Auger \nspectroscopy measu rements, that the diffusion of the lighter elements towards the Pt underlayer and the \nformation of an interfacial CoFe -rich layer causes the  two-regime behavior . In the case of the ultrathin \nfilms , this interfacial CoFe -rich layer  promotes the  strong  PMA through the hybridization of the [Co,Fe] \n3𝑑𝑧2 and O 2𝑝𝑧 orbital s at the interface  and is also responsible  for the increased Gilbert  damping . Our \nstudy reveals that the strong PMA is not intrinsic to the Heusler/MgO interface . It is strongly  influenced \nby the interdiffusion  and can be adjusted  by a proper choice of the underlayer  material , as we show for \nthe case of the Pt, Ta and Cr  underlayers.  \n \nExperimental   \n \nAll the samples studied here were grown at room temperature on thermally oxidized silicon substrates \nin a magnetron sputtering system having a base pressure lower than 2×10-8 Torr.  The main samples have \nthe following structure: Si/SiO 2//Ta (3 nm )/Pt (4 nm)/ FM (0.8-10 nm)/MgO (1 nm)/ Ta (3 nm), where FM \nstands for  Co2FeAl (CFA), Co 2FeAl 0.5Si0.5 (CFAS), Co 2FeSi (CFS), Co 2Fe0.5Mn 0.5Si (CFMS) or CoFeB \n(CFB) , depending on the sample. Additional samples were grown,  and their  structure will be discussed \nlater in the text. The metallic layers were deposited  by dc sputtering  under an argon pressure of 1 mTorr, \nwhile the MgO layer was grown by rf sputtering under an argon pressure of 10 mTorr. The Heusler alloy s \nthin films were sputtered  from stoichiometric targets. The 3 nm thick Ta buffer layer was grown directly \non the substrate to minimize  the roughness and to facilitate the (111) texturing of the  upper Pt  layer. The \n1 nm thick MgO layer was deposited to induce perpendicular m agnetic anisotropy on the Heusler thin \nfilm7. An additional  3 nm th ick Ta capping layer was sputtered  to protect the samples from oxidation due \nto air exposure. The structure of the samples was characterized by x -ray diffraction (XRD) using a four -\ncircle diffractometer. The static magnetic properties have been investigated using a Vibrating Sample \nMagnetometer (VSM) , while  the dynamic magnetic properties by using a TE(011) cavity  Ferromagnetic \nResonance (FMR)  setup working in X -band  (9.79 GHz ). Auger spectra have been recorded  in derivative \nmode, using  a cylindrical mirror analyzer spectrometer working at an electron beam energy of 3keV.  \nDepth profile analysis have been performed by successive  recording of the Auger spectra and Ar ion \nsputter -etching of the surface of the samples by  using a relatively low  ion energy of 600 eV.    \n 4 \n  \n \nResults and discussions  \n \nFigure 1 (a) shows  2θ/ω x-ray diffraction patterns recorded for four representative  Pt (4 nm)/  Co2YZ \n(10 nm)/  MgO (1 nm)  samples . Irrespective of the Heusler composition, the patterns show the (111) and \n(222) peaks  belonging to the Pt  layer , the (022) peak  arising from the Heusler films  and the  (001) peak of \nthe Si substrate . This indicates that the Pt layer has a (111) out -of-plane texture, while the Heusler films \nare (011) out -of-plane textured.  Laue oscillations are observable around the (111) Pt reflection which \nconfirms the good crystalline quality for the Pt films48. Moreover, ϕ -scan measurements (not shown here) \nindicate that both the Pt underlayer and the Heusler fil ms have no in -plane texturing but  show an  in-plane \nisotropic distribution  of the crystallites . No peak belon ging to the Ta capping layer was observed, \nindicating  that the film is in an amorphous or nanocrystalline state.  \nThe static magnetic properties of our films were characterized by VSM measurements . Figure 2 show s \nhysteresis loops measured with the magnetic field applied perpendicular  to the plane of the samples, for \nrepresentative Heusler films thickness es. In order to remove th e substrate diamagnetic contribution,  we \nfitted the large field data with a linear function and extract ed the linear slope  from the raw data.  Regardless  \nof the ir composition , all the Heusler films show a similar behavior. Above a critical spin-reorientation \ntransition thickness  the samples show in-plane magnetic anisotropy . This is indicated by the shape of the \nhysteresis loop s in Fig. 2  (a)–(d), which is typical for a hard axis of magnetization, showing a continuous \nrotation of the magnetization up to saturation.  Below th is critical thickness, the samples show PMA , which \nis attested by  the square shaped hysteresis loops in Fig. 2 (e) –(h). We also determined the saturation \nmagnetization  (𝑀𝑆) and the effective thickness es of the ferr omagnetic layers  using hysteresis loop \nmeasurements  and the procedure described in 31.  The effective thicknesses of the ferromagnetic layers \nare used throughout  the paper and the 𝑀𝑆 is found  to be  790 ± 70 emu/cm3, 660 ± 50 emu/cm3, 935 ± 75 \nemu/cm3 and 895 ± 75 emu/cm3 for CFS, CFMS, CFAS and CFA samples, respectively.   \nIn order to get more insights on the magnetic anisotropy properties of our films, we have performed \nFMR measurements  with the magnetic field applied at diffe rent θH angles  (defined in the inset of Fig. 4) \nwith respect to the normal direction of the layers . Figure 3 shows typical FMR spectra for various fie ld \nangle s recorded  for a 2.4 nm thick Pt/CFAS sample . We define the resonance field HR as the intersection \nof the spectr um with the base line, and the linewidth HPP as the distance between the positive and negative \npeaks of the spectrum. Figure 4 shows the θH dependence of the HR and of the linewidth HPP for the 2.4 5 \n nm thick Pt/CFAS sample. In order to extract the relevant FMR parameters, we analyzed the θH \ndependence of the FMR spectrum using  a model in which the total energy per unit volume is given by  \n𝐸=−𝑀𝑆𝐻cos(𝜃𝐻−𝜃𝑀)+2𝜋𝑀𝑆2cos2𝜃𝑀−𝐾⊥cos2𝜃𝑀,                             (1) \nwhere the first term is the Zeeman energy, the second term is the demagnetizing energy, and the last term \nis the magnetic anisotropy energy.  The 𝑀𝑆 is the saturation magnetization, 𝜃𝐻 and 𝜃𝑀 are the field and \nmagnetization angles defined in the inset of Fig. 4, and the 𝐾⊥ is the effective perpendicular magnetic \nanisotropy constant. From eq. 1 and the Landau -Lifshitz -Gilbert equation, one can der ive the resonance \ncondition as49 \n(𝜔\n𝛾)2\n=𝐻1×𝐻2,                                                                (2)  \nwhere 𝜔 is the angular frequency of the microwave , 𝛾 is the gyromagnetic ratio , given by 𝛾=𝑔𝜇𝐵ℏ where \n𝑔 is the  Landé  g-factor, 𝜇𝐵 is the Bohr magneton and ℏ is the reduced Plan ck constant , and with 𝐻1 and \n𝐻2 given by   \n𝐻1=𝐻𝑅cos(𝜃𝐻−𝜃𝑀)−4𝜋𝑀effcos2𝜃𝑀,                                             (3) \n𝐻2=𝐻𝑅cos(𝜃𝐻−𝜃𝑀)−4𝜋𝑀effcos2𝜃𝑀,                                           (4) \nwhere 4𝜋𝑀eff is the effective magnetization defined as  4𝜋𝑀eff=4𝜋𝑀𝑆−2𝐾⊥𝑀𝑆⁄ and 𝐻𝑅 is the \nresonance field.  For each value of  𝜃𝐻, the 𝜃𝑀 at resonance is calculated from the energy minimum \ncondition 𝜕𝐸 𝜕𝜃𝑀=0 ⁄ . Hence , the 𝐻𝑅 dependence on 𝜃𝐻 can be fitted by Eq. (2)-(4) using 4𝜋𝑀eff and \n𝑔 as adjustable parameters . A typical fit curve is s hown in Fig. 4(a).  \n Figure 5 shows the 𝑔 factor depen dence on the thickness of the Heusler layers for samples with \ndifferent Heusler layer composition s. Depending on the thickness , two regimes are discernable . For \nrelatively  large thickness es, above 2.5-3 nm, the  𝑔 factor shows rather  constant value s between 2.07 and \n2.11, depending on the type of the Heusler layer . For lower thickness es, 𝑔 shows a monotonous decrease, \nregardless  of the Heusler  layer composition. This is an interface effect and it is usually attributed to the \nfact that at the interfaces , due to the symmetry breaking, the orbital motion is n o longer entirely  quenched  \nand will contribute to the gyromagnetic ratio50,51. Another possibility, which cannot be exclude d in our \ncase, is the reduction of the 𝑔 factor due to intermixing between the ferromagnetic  Heusler layer and non -\nmagnetic materials at the interfaces50. \n Figure 6 shows the effective magnetization 4𝜋𝑀eff dependence on the inverse thickness of the    \nferromagnetic layer for samples with different composition s. It is to be mentioned  that the 4𝜋𝑀eff was \ndetermined from FMR experiments only for samples with in-plane magnetic anisotropy (positive 4𝜋𝑀eff). \nIn the case of ultrathin samples showing perpendicular magnetic anisotropy  (negative 4𝜋𝑀eff), due to the 6 \n strong linewidth enhancement , it was not possible to obtain reliable resonance curves.  Therefore, in this \ncase the 4𝜋𝑀eff was estimated from VSM measurements.  Generally, it is considered  that the effective \nperpendicular magnetic anisotropy co nstant  𝐾⊥ can be written as the sum of a volume  (𝐾𝑉), which includes \nmagneto -crystalline and strain related anisotropies, and a surface (𝐾𝑆) contribution : 𝐾⊥=𝐾𝑉+𝐾𝑆𝑡⁄, \nwhere 𝑡 is the thickness of the ferromagnetic layer . Thus, the effective magnetization can be written as  \n \n4𝜋𝑀eff=(4𝜋𝑀𝑆−2𝐾𝑉\n𝑀𝑆)−2𝐾𝑆\n𝑀𝑆1\n𝑡.                                                             (5) \n \nThe above relation implies a linear dependence of  the effective magnetization on the inverse thickness of \nthe ferromagnetic layer.  However, as shown in Fig.6  the Heusler samples do not show a single linear \ndependence for the  entire  thickness range, but two regimes above and below a  certain critical thickness.  \nUsing the 𝑀𝑆 values determined from VSM measurements and b y fitting the experimental data in the large  \nthickness regime to eq. (5) , we extract a surface  anisotropy constant 𝐾𝑆 for the CFA  and CFAS of 0.24  ± \n0.03 erg/ cm2 and 0.22 ± 0.02 erg/cm2 and a volume contribution 𝐾𝑉 of (1.27 ± 0. 69)×106 erg/cm3 and \n(1.51 ± 0.7)×106 erg/cm3, respectively. In the case of the CFMS and CFS, the 𝐾𝑆 was negligible small \nwithin the error bars  and the 𝐾𝑉 was found to be (0.44 ± 0.38)×106 erg/cm3 and ( 0.51 ± 0. 4)×106 erg/cm3, \nrespectively. Using the as extracted values of th e anisotropy constants , we can calculate , for example,  in \nthe case of the CFAS samples a spin-reorientation tra nsition thickness of around 0.55 nm. This is clearly \nnot in agreement with the experimental data , as seen from Fig. 2 and 6 , already  a 1 nm thick CFAS film \nshows strong PMA  and it is spontaneous perpendicular ly magnetized . This is a consequence of the fact \nthat the  1 nm thick CFAS film  falls within  the second anisotropy regime below the critical thickness . The \noccurrence  of this second anisotropy regime with larger effective perpendicular magnetic anisotropy can \nhave several explanations.  For such thin films one must always consider the possible influences of the \nsurface roughness. If the roughness is relatively large , an in -plane demagnetization field will develop at \nthe edges of the terraces  which will reduce the shape anisotropy and fav or perpendicular magneti zation . \nThis is equivalent  to the  emergence  of an additional dipolar surface anisotropy contribution52. The \nroughness is  a parameter which is not easily quantifiable  experimentally  in such thin  multilayer structures. \nHowever, it is reasonable to expect to be comparable  for similar  heterostructures in which the Heusler \nalloy film is replaced with a CFB layer . Atomic force microscopy topography images (not shown here) \nrecorded for heterostructure w ith CFB and CFAS layers are featureless and show a similar RMS \nroughness.  As such, if the low thickness  anisotropy  regime is due to the roughness it must be observable \nalso in the case of CFB samples. However, this is not the case , as shown in Fig. 6, the CFB samples show 7 \n a single linear behavior for the whole range of  thickness . Fitting the data to eq. (5), allowed us to extract  \nfor CFB samples  a surface anisotropy contribution 𝐾𝑆 of 0.79 ± 0.04 erg/cm2 and a  negligible small 𝐾𝑉 \nvolume contribution , in line with previous reports7,53. These findings suggest  that the roughness is not \nresponsible for the two regimes behavior observed in the case of the Heusler samples.  \nAnother possible  physical mechanism  which can explain the presence of the two regimes is  the strain \nvariation  due to coherent –incoherent  growth transition54,55. Within this model, below the critical thickness , \nthe ferromagnetic layer grows uniformly strained in order to account for the lattice  misfit with the adjacent \nlayer s. Above the critical thickness, the strains are partially relaxed through the formation  of misfit \ndislocations.  The changes in the magnetoelastic anisotropy contributions corresponding to this structural \ntransition  can be responsible for t he presence of the two regimes54,55. This scenario is likely in the case of \nthe Heusler samples , since both the bottom Pt layer  the upper Heusler film  grow  out-of-plane textured.  In \norder to test this hypothesis, we have deposited two additional sets of samples. The first set consisted  of \nSi/SiO 2//Ta ( 6 nm)/ CFAS  (tCFAS)/MgO (1 nm)/Ta (3 nm)  samples . The motivation to grow this type of \nsamples was to obtain Heusler films with no out -of-plane texturing. Indeed, x -ray diffra ction measurement  \n[Fig. 1(b)], performed on a Ta/CFAS sample with a Heusler layer thickness of 10 nm , did not indicate the \npresence of any diffraction peak s, except  for the one belonging to the Si substrate. This suggest that both \nthe Ta and the CFAS films are either nanocrystalline or amorphous . Thus , in this type of structure we do \nnot expect the presence of the coherent –incoherent growth transition. The second set of samples consisted \nof epitaxial  MgO  (001)//Cr ( 4 nm)/CFAS  (tCFAS)/MgO (1 nm)/Ta (3 n m) structures . The x -ray diffraction \nmeasurement [Fig. 1 (b)], performed on a Cr/CFAS sample with a Heusler layer thickness of 10 nm, \nindicate s the exclusive presence of the (001) type reflections from the MgO substrate and the Cr and CFAS \nlayers . This confirms the epitaxial growth of the stacks, except for the Ta capping layer, which is \namorphous. Having in view the epitaxial growth we might expect for these samples a possible coherent –\nincoherent growth transition, eventually at higher CFAS thick nesses having in view the relative low \nmismatch between the CFAS lattice and the 45° in-plane rotated Cr lattice  (0.7%). The effective \nmagnetization dependence on the inverse thickness of the ferromagnetic layer for amorphous Ta/CFAS \nand epitaxial Cr/CFAS samples alongside with the Pt/CFAS samples is shown in Fig. 7 . Is to be mentioned \nthat in the for the Ta/CFAS samples the PMA  was obtained for thicknesses  below 1.6 nm , while for the \nCr/CFAS the PMA was not achieved even for thicknesses down to 1 nm. Interestingly, in the case of the \nepitaxial Cr/CFAS samples , for which one might expect possible coherent –incoherent growth transition, \na single linear behavior for the whole thickness range is observ ed. In the case amorphous  Ta/CFAS \nsamples, for which the coherent –incoherent growth transition is not expected, a two-regimes behavior  can 8 \n be distinguished . Although we cannot rule a possible coherent -incoherent growth transition  at larger \nthicknesses, t he results indicate this mechanism is not responsible for the two regimes behavior that we \nobserve  at relatively low thicknesses and other mechanism s must be at play.  \n By fitting the high thickness regime data from Fig.7  to eq. (5)  we extracted for Ta/CFAS  samples a \nsurface anisotropy contribution 𝐾𝑆 of 0.27 ± 0.08 erg/cm2. The volume contribution,  𝐾𝑉, was determine d \nto the be negligible small, as expected for untextured films. Remarkably, the 𝐾𝑆 for the Ta/CFAS samples \nis similar within the error bar s to one obtained for  the Pt/CFAS samples.  Moreover, even in the low \nthickness regime the 𝐾𝑆 might be assumed  similar for the two sets of samples. However, we must  consider  \nthe large uncertainty  having in view the  sparse  data points available for fitting  in the low thickness regime. \nEven so, the clear difference between the two sets of samples is that the Ta/CFAS one shows a  larger \ncritical thickness  (around 2.4 nm ) that separates  the two anisotropy regimes , as compared t o the Pt/CFAS  \none (around 1.5 nm) . This suggests that the possible mechanism responsible for the two -regime behavior \nmight be related to the atomic diffusion at the  Pt/CFAS and  Ta/CFAS interface s. It is well known that Ta \nis prone to diffusion of light elements56. Therefore , a larger critical thickness for the Ta/CFAS samples \nwill imply a larger  atomic  diffusion at the Ta/CFAS interface  compared to the Pt/CFAS one .  \nTo test th e hypothesis  of the interdiffusion , we performed Auger electron spectroscopy (AES) analyses \non th e three set of samples : Pt/CFAS, Cr/CFAS and Ta/CFAS. AES is a surface sensitive technique which \ncan give information about the chemical composition of the surface  with a depth detection limit of 1 -2 \nnm. We started from 10 nm thick CFAS layer samples  and first Ar ion etched the CFAS films down to 4 \nnm thickness and recorded the AES spectra.  Subsequently, t he Ar ion etching and AES spectra recording \nwas repeated  in steps of 1 nm until reaching the underlaye r/CFAS interface.  The etching rate of CFAS \nwas previously calibrated using  ex-situ x-ray reflectometry measurements.  Figure 8 (a) shows two spectra \nrecorded for the Pt/CFAS sample, one after etching the  CFAS layer down to 4 nm (Pt/CFAS 4 nm) and \nthe other  one after etching the CFAS layer down to 1 nm of thickness (Pt/CFAS 1 nm). In the case of the \nPt/CFAS 4 nm spectr um the peaks of Co and Fe are visible alongside with the peaks fro m Al and Si. The \ninset of Fig. 8(a) depicts a n enlargement of the Pt/CF AS 4 nm spectrum around the peaks of Al and Si. \nThe amplitude of the Co and Fe peaks is m uch larger than the amplitude of the of Al and Si ones. This is \ndue to the higher  concentration and higher  Auger relative sensitivity of the  Co and Fe compared to the Al \nand Si. In the case of the Pt/CFAS 1 nm the spectrum  shows the peaks from Co and Fe , with a lower \namplitude,  and the peaks from the Pt underlayer. The presence of the Co and Fe peaks together with the \nPt peaks is not surprising . It is owed to the possible interdiffusion layer at the interface  and to the finite \ndepth resolution  of the  AES which probes both the CFAS layer and the Pt underlayer. The Al and Si peaks 9 \n are not observable , which can be associated  to the relatively low amplitude of the Al and Si falling below \nthe detection limit of the measurement. To test this possibility,  we acquired  Auger  spectra in a narrow \nenergy window around the Al peak , using a longer acquisition time and averaging 10 spectra for e ach \nrecoded  spectrum.  We selected the Al peak and not the Si one because of its larger ampli tude. These \nspectra  recorded for the Pt/CFAS, Ta/CFAS and  Cr/CFAS samples after etching the CFAS layer down to \n4, 3, 2 and 1 nm are shown in Fig.8  (b)-(d). In the case of the Pt/CFAS sample the Al peak is observable \nfor CFAS thickness es down to 1 nm , while in the case of Ta /CFAS  for thicknesses down to 2 nm. \nInterestingly, in the case of the Cr/CFAS sample the Al peak is visible even for a CFAS thickness of 1  \nnm, although with lower amplitude. These findings suggest th at at the underlayer/CFAS interface there is \na diffusion of the light er elements (Al and most likely also Si)  towards the underlayer, with different \ndegree, depe nding  on the nature of the underla yer. As shown schematically in Fig. 8, due to th is lighter \nelements diffusion a CoFe -rich layer form s at the underlayer/CFAS interface. The extent  of the CoFe rich \nlayer depends on the nature of the underlayer . It has the largest thickness for the Ta underlayer (between \n2 and 3 nm) , it is decreasing for the Pt underlayer (between 1 and 2 nm) and it is most likely non -existing \nor extremely thin (below 1 nm) in the case of the Cr underlayer.  \nThe presence of th e CoFe r ich layer agrees  with our findings concerning the occurrence  of the high \nand the low effective PMA  regime s depend ing on the thickness of the Heusler layer. In the case of  the \nPt/CFAS /MgO  samples , the low effective PMA  regime  occurs  for a CFAS layer thickness above  1.6 nm. \nIn this case, the bottom interface consists of Pt/CoFe -rich layer, while the top one of CFAS/MgO . In \nprinciple, both interfaces could contribute to PMA  through Co-O hybridization in the case of the Co-\nterminated CFA S/MgO interface43 or through the d –d hybridization between the  spin-split Co 3d bands \nand the Pt layer 5d bands with large  spin-orbit  coupling44,45. However, their contribution to PMA is small \nand, as we previously mentioned,  would not stabilize perpendicular magnetization except for extremely \nthin CFAS layer s. In the case  of the  high effective  PMA regime (below 1.6 nm) , the bottom interface is \nsimilar consisting of Pt/CoFe -rich layer  and will contribute negligibl y to PMA . However, the top interfa ce \nis now constituted of CoFe -rich layer/MgO  and will induce strong PMA through the hybridization of the \n[Co,Fe] 3𝑑𝑧2 and O 2𝑝𝑧 orbitals42. The premise that the strong PMA is induced by the CoFe -rich \nlayer/MgO interface  is also consistent with our observat ions regarding the dependence of the magnetic \nanisotropy on the nature of the underlayer. As seen in Fig. 7, i n the case of the Cr/CFAS samples, where \nno CoFe -rich layer was evidenced, there is only one anisotropy regime with a relatively low effective \nPMA . In the case of the Ta/CFAS sa mples, the high effective PMA regime is present starting from a larger \nCFAS thickness , as compared to de case of Pt/CFAS samples , which is in agreement  with the thicker 10 \n CoFe -rich layer observed for the Ta/CFAS relative to the Pt/CFAS ones.  It is to be mentioned that in the \ncase of Ru/CFA/MgO and Cr/CFA/Mg O annealed samples  Al diffusion towards the MgO but not towards \nthe underlayer was previously observed46,47. The lack of Al diffusion towards the Cr underlayer is in \nagreement with our findings. In the case of the aforementioned studies, the thermal  annealing of the \nsamples was necessary  to facilitate the Al diffusion  and to achieve strong PMA. In our case, for the Pt and \nTa underlayer , we attain  strong PMA in the low thickness regime without the need of thermal annealing. \nThis indicates that for the Pt and Ta underlayer s the [Al,Si] diffusion takes place during the growth of the \nCFAS film, which results in the formation of the interfacial CoFe -rich layer  directly during deposition . A \nfurther deposition of MgO on this CoFe -rich layer will generate the strong PMA  through the hybridizati on \nof the [Co,Fe] 3𝑑𝑧2 and O 2𝑝𝑧 orbitals42. Having in view the si milar behavior  of the magnetic anisotropy \nfor the CFA, CFAS, CFMS and CFS Heusler alloys thin films that we study here , it is reasonable to \nassume that in all the cases there is  a diffusion of the lighter elements  (Al, Si) towards the Pt underlayer \nand the formation of the CoFe -rich interfacial  layer , which , when MgO is deposited on top,  will give rise \nto the strong PMA in the low thickness regime.  \nWe now discuss the thickness  dependence of the Gilbert damping parameter extracted from  the θH \ndependence of the linewidth HPP. It is known  that generally the linewidth is given by a sum o f extrinsic  \nand intrinsic contr ibution  as49,57-59: \n𝐻PP=𝐻PPint+𝐻PPext,                                                                (5)  \n𝐻PPint=𝛼(𝐻1+𝐻2)|d𝐻𝑅\nd(𝜔𝛾⁄)|,                                                       (6) \n𝐻PPext=|d𝐻𝑅\nd(4𝜋𝑀eff)|Δ(4𝜋𝑀eff)+|d𝐻𝑅\nd𝜃𝐻|Δ𝜃𝐻+Δ𝐻TMS,                            (7) \nwhere, 𝛼 is the intrinsic Gilbert dampi ng parameter and the three  terms in equation (7) are the linewidth \nenhancement due to the anisotropy distribution , due to deviation from planarity of the films  and due to the \ntwo-magnon scattering. In the case of our films, the θH dependence of the linewidth HPP is well fitted \nusing only the intrinsic contribution  and the extrinsic enhancement due to the anisotropy distribution. For \nthis, |d𝐻𝑅d(𝜔𝛾⁄) ⁄ |  and |d𝐻𝑅d(4𝜋𝑀eff) ⁄ |  are numerically calculated using Eqs. (1)-(4) and the HPP vs. \nθH experimental dependence is fi tted to Eq. (5) using 𝛼 and Δ(4𝜋𝑀eff) as adjustable parameters49. An \nexample of a  fit curve is depicted  in Fig. 4(b)  for the case of the 2.4 nm thick Pt/CFAS sample . Figure 9 \nshows the 𝛼 dependence on the inverse ferromagnetic layer  (1/t) thickness for the Pt/CFA, Pt/CFS, \nPt/CFMS , Pt/CF AS and Pt/CFB samples.  We will first discuss the case of CFB, where  a linear dependence \nis observed . The linear increase of the Gilbert damping  parameter with 1/t is expected  and it is due to the \nangular momentum loss due to the spin pumping effect in the Pt layer. In this type of structures it  was 11 \n shown60 that the total damping is given by 𝛼=𝛼0+𝛼SP𝑡⁄, where  𝛼0 is the Gilbert damping of the \nferromagnetic film and 𝛼SP is due to the spin pu mping effect.  By linear fitting the data in Fig. 9  we obtain \na Gilbert damping parameter for the CFB of 0.0028 ± 0.0003 , in agreement with other reports61,62. In the \ncase of the CFAS films, the linear dependence  is observed only for the large thickness region  and by fitting \nthis data we obtain a Gilbert da mping parameter of 0.00 53 ± 0.00 12, consistent wit h previously reported \nvalue s for relative ly thick er films25. The low thickness data deviates from the  linear dependence . This \nbehavior is similar for all the other studied Heusler films, with the low thickness deviation being even \nmore pronounced.  The strong increase of the damping can be related to  the [Al,Si] diffusion and the  \nformation of the interfacial CoFe -rich layer. Since the [Al,Si]  diffusion  is more important for thinner films , \nit will have a stronger impact on the chemical composition relative to the thicker ones. The relatively  small \ndamping of the Co based full Heusl er alloys is a consequence of the ir specific electronic structure21. \nConsequently , deviations from the  correct stoichiometry , which is expected to have a n important  effect \non the electronic structure , will lead to a strong increase of the damping, as shown, for example, by ab-\ninitio calculation in the case of Al deficient CFA films47. Therefore, the increase of the damping beyond \nthe spin pumping effect for the thinner  Heusler films is explained by  the interfacial CoFe -rich layer \nformat ion.  \n \nConclusions  \n \nWe have studied the mechanism s responsible for PMA in the case of  Co2FeAl, Co 2FeAl 0.5Si0.5, Co 2FeSi \nand Co 2Fe0.5Mn 0.5Si Heusler alloy thin films sandwiched between Pt and MgO layers.  We showed that \nthe ultrathin Heusler films exhibit  strong PMA irrespective of their composition. The effective \nmagnetization displays  a two -regime behavior depending on the thickness of the Heusler layers.  The two -\nregime behavior is generated  by the formation of a n CoFe -rich layer at the underlayer/Heusler  interface \ndue to the interdiffusion. The strong PMA observed in the case of the ultrathin films can be explained by \nthe electronic hybridization of the CoFe -rich metal lic layer  and oxygen orbitals across the \nferromagne t/MgO interface . The formation of the  interfacial  CoFe -rich layer  causes the increase of the \nGilbert damping  coefficient  beyond the spin pumping for the ultrathin Heusler films. Our results illustrate \nthat the strong PMA is not an intrinsic property of the  Heusler/MgO interface,  but it is actively influenced \nby the interdiffusion, which can be tuned by a proper choice of the underlayer material.  \n \n  12 \n FIG. 1. (a) 2θ/ω x-ray diffraction patterns recorded for four representative  Pt/Co 2YZ/MgO samples having \na thickness of the Heusler layer of 10 nm.  The patterns show the (111) and (222) peaks belonging to the \nPt layer , the (022) peak from the Heusler films and the (001) peak of the Si substrate.  (b) 2θ/ω x-ray \ndiffraction patterns for the Ta/CFAS (10 nm)/MgO and Cr/CFAS (10 nm)/MgO samples indicating the \namorphous or epitaxial  growth of the CFAS layer , respectively.  \n13 \n FIG. 2. Hysteresis  loops measured with the magnetic field applied perpendicular to the plane of the \nsamples . Depending on the thickness of the Heusler layers, the samples show  in-plane magnetic anisotropy \n(a)-(d) or perpendicular magnetic anisotropy (e) -(h).  \n  \n14 \n FIG. 3. Typical  FMR spectra measured at 9.79 GHz  for different θH field angles for a 2.4 nm thick \nPt/CFAS sample.  \n15 \n FIG. 4. (a) Resonance field HR and (b) linewidth HPP dependence on the θH field angle for a 2.4 nm thick \nPt/CFAS sample . The inset s hows a schematic of the measurement geometry. The points  stand for  \nexperimental data while the lines represent  the result of the theoretical fits, as described in text.  \n \n  \n16 \n FIG. 5. \ng factor dependence on the thickness of the Heusler layers for samples with different Heusler layer \ncomposition.  \n \n  \n  \n17 \n FIG. 6. The effective magnetization \n4effM dependence on the inverse thickness of the  ferromagnetic \nlayer for samples with different composition s. The points are experimental data while the lines are linear \nfits. In the case of the Heusler samples two linear fits correspond to the two anisotropy regimes.   \n \n18 \n FIG. 7. The effective magnetization  \n4effM dependence on the inverse thickness of the ferromagnetic \nlayer for amorphous Ta/CFAS and epitaxial Cr/CFAS samples.  The data for Pt/CFAS is also shown for \ncomparison. The points are experimental data while the lines are linear fits.  \n \n19 \n FIG. 8. (a) AES spectra recoded for the Pt/CFAS sample  after etching the CFAS layer down to 4 and 1 \nnm, respectively . The inset shows a zoom around de Al and Si peaks. AES spectra recorded around the \nAl peak  after etching the CFAS layer down to 4, 3, 2 and 1 nm for the (b) Pt/CFAS, (c) Ta/CFAS and \n(d) Cr/CFAS samples.  Schematic representation of the [Al,Si] diffusion to wards the underlayer and the \ninterfacial CoFe -rich layer formation . \n \n \n20 \n FIG. 9. Gilbert damping parameter ( 𝛼) dependence on the inverse ferromagnetic layer (1/t) thickness for \nthe Pt/CFA, Pt/CFAS , Pt/CFMS, Pt/CFS and Pt/CFB samples.  The points are experim ental data while \nthe lines are linear fits for  Pt/CFB and Pt/CFAS samples. 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Tiusan,  \nJournal of Physics D: Applied Physics 51 (4), 0 45002 (2018).  \n \n "    },    {        "title": "1708.07296v1.Nonlinear_network_dynamics_for_interconnected_micro_grids.pdf",        "content": "Nonlinear network dynamics for\ninterconnected micro-grids\nDario Bauso\u0003\nJune 7, 2021\nAbstract\nThis paper deals with transient stability in interconnected micro-grids. The\nmain contribution involves i) robust classi\fcation of transient dynamics for di\u000berent\nintervals of the micro-grid parameters (synchronization, inertia, and damping); ii)\nexploration of the analogies with consensus dynamics and bounds on the damping\ncoe\u000ecient separating underdamped and overdamped dynamics iii) the extension to\nthe case of disturbed measurements due to hackering or parameter uncertainties.\nKeywords: Synchronization; consensus; Nonlinear control; Transient stability.\n1 Introduction\nThis paper investigates transient stability of interconnected micro-grids. First we develop\na model for a single micro-grid combining swing dynamics and synchronization, inertia\nand damping parameters. We focus on the main characteristics of the transient dynamics\nespecially the insurgence of oscillations in underdamped transients. The analysis of the\ntransient dynamics is then extended to multiple interconnected micro-grids. By doing this\nwe relate the transient characteristics to the connectivity of the graph. We also investigate\nthe impact of the disturbed measurements (due to hackering or parameter uncertainties)\non the transient.\n1.1 Main theoretical \fndings\nThe contribution of this paper is three-fold. First, for the single micro-grid we identify\nintervals for the parameters within which the behavior of the transient stability has similar\ncharacteristics. This shows robustness of the results and extends the analysis to cases\nwhere the inertia, damping and synchronization parameters are uncertain. In particular\nwe prove that underdamped dynamics and oscillations arise when the damping coe\u000ecient\n\u0003Dario Bauso is with the Department of Automatic Control and Systems Engineering, The University\nof She\u000eeld, Mappin Street She\u000eeld, S1 3JD, United Kingdom, and with the Dipartimento di Ingegneria\nChimica, Gestionale, Informatica, Meccanica, Universit\u0012 a di Palermo, V.le delle Scienze, 90128 Palermo,\nItaly.fd.bauso@sheffield.ac.uk garXiv:1708.07296v1  [math.OC]  24 Aug 2017is below a certain threshold which we calculate explicitly. The threshold is obtained as\nfunction of the product between the inertia coe\u000ecient and the synchronization parameter.\nSecond, for interconnected micro-grids, under the hypothesis of homogeneity, we prove\nthat the transient stability mimics a consensus dynamics and provide bounds on the\ndamping coe\u000ecient for the consensus value to be overdamped or underdamped. This\nresult is meaningful as it sheds light on the insurgence of topology-induced oscillations.\nThese bounds depend on the topology of the grid and in particular on its maximum\nconnectivity, namely, the maximum number of links over all the nodes of the network.\nWe also observe that the consensus value changes dramatically with increasing damping\ncoe\u000ecient. This implies that the micro-grid, if working in islanding mode, can synchronize\nto a frequency which deviates from the nominal one of 50 Hz. This \fnding extends to\nsmart-grids with di\u000berent inertia but same ratio between damping and inertia coe\u000ecient.\nThird, we extend the analysis to the case where both frequency and power \row mea-\nsurements are subject to disturbances. Using a traditional technique in nonlinear analysis\nand control we isolate the nonlinearities in the feedback loop, and analyze stability under\nsome mild assumptions on the nonlinear parameters. The obtained result extends also to\nthe case where the model parameters like synchronization coe\u000ecient, inertia and damping\ncoe\u000ecients are uncertain. This adds robustness to our \fndings and proves validity of the\nresults even under modeling errors.\nTo corroborate our theoretical \fndings a case study from the Nigerian distribution\nnetwork is discussed.\n1.2 Related literature\nThis study leverages on previous contributions of the authors in [2] and [3]. In [2] the\nauthor studies \rexible demand in terms of a population of smart thermostatically con-\ntrolled loads and shows that the transient dynamics can be accommodated within the\nmean-\feld game theory. In [3] the author extends the analysis to uncertain models in-\nvolving both stochastic and deterministic (worst-case) analysis approaches. The analysis\nof interconnected micro-grids builds on previous studies provided in [5]. Here the authors\nlink transient stability in multiple electrical generators to synchronization in a set of cou-\npled Kuramoto oscillators. The connection between Kuramoto oscillators and consensus\ndynamics is addressed in [7]. A game perspective on Kuramoto oscillators is in [8], where\nit is shown that the synchronization dynamics admits an interpretation as game dynam-\nics with equilibrium points corresponding to Nash equilibria. The observed deviation of\nthe consensus value from the nominal mains frequency in the case of highly overdamped\ndynamics can be linked to ine\u000eciency of equilibria as discussed in [9]. This study has\nbene\fted from some graph theory tools and analysis e\u000eciently and concisely exposed in\n[4]. The model used in this paper, which combines swing dynamics with synchronization,\ninertia and damping parameters has been inspired by [6]. The numerical analysis has\nbeen conducted using data provided in [1].\nThis paper is organized as follows. In Section 2, we model a single micro-grid. In\nSection 3, we turn to multiple interconnected micro-grids. In Section 4, we analyze the\nimpact of measurement disturbances. In Section 5, we provide numerical studies on the\nNigerian grid. Finally, in Section 6, we provide conclusions.2 Model of a single micro-grid\nConsider a single micro-grid connected to the network, refer to it as the ith micro-grid.\nLet us denote by Pithe power \row into the ith micro-grid. Also let fibe the frequency\ndeviation of micro-grid iandfja virtual signal representing the frequency of the mains.\nBy applying dc approximation, the power Pievolves according to\n_Pi=Tij(fj\u0000fi) =Tijeij; (1)\nwhereTijis the synchronizing coe\u000ecient. This coe\u000ecient is obtained as the inverse of the\ntransmission reactance between micro-grid iandj. In other words, the power Pidepends\non the frequency error eij=fj\u0000fi. The physical intuition of this is that in response to\na positive error we have power injected into the ith micro-grid from the jth micro-grid.\nVice versa, a negative error induces power from micro-grid itoj.\nThe dynamics for fifollows a traditional swing equation\n_fi=\u0000Di\nMifi+Pi\nMi; (2)\nwhereMiandDiare the inertia and damping constants of the ith micro-grid, respectively.\nBy denoting fi=x(i)\n1,Pi=x(i)\n2,fj=x(j)\n1, and by considering fjas an exogenous input\nto theith micro-grid, the dynamics of the ith micro-grid reduces to the following second-\norder system\"\n_x(i)\n1\n_x(i)\n2#\n=\u0014\u0000Di\nMi1\nMi\n\u0000Tij0\u0015\"\nx(i)\n1\nx(i)\n2#\n+\u00140\nTij\u0015\nx(j)\ni: (3)\nFigure 1 shows the block representation and corresponding transfer function of the\ndynamical system (3).\nfj +\n\u0000eijTij\ns1\nMis+Difi Pi\nFigure 1: Block representation of the ith micro-grid.\nTheorem 1 Dynamics (3) is asymptotically stable. Furthermore, let Di>2p\nTijMithen\nthe origin is an asymptotically stable node. Vice versa, if Di<2p\nTijMithen the origin\nis an asymptotically stable spiral.\nProof. For the \frst part, stability derives from Tr(A) =\u0000Di\nMi, whereTr(A) is the trace\nof matrixAand from \u0001( A) =Tij\nMi>0, where \u0001( A) is the determinant of matrix A. Let\nus recall that stability depends on the eigenvalues of Aand that the expression of the\neigenvalues is given by\n\u00151;2=Tr(A)\u0006p\nTr(A)2\u00004\u0001(A)\n2\n=1\n2\u0010\n\u0000Di\nMi\u0006q\n(Di\nMi)2\u00004Tij\nMi\u0011\n:(4)Tr(A)2<4\u0001(A)\nsaddle pointsTr(A)2>4\u0001(A)\nunstable nodes a.s. nodesa.s. spirals unst. spirals\nTr(A)\u0001(A)\nFigure 2: Classi\fcation of equilibrium points.\nAs the trace Tr(A) is strictly negative and the determinant \u0001( A) is strictly positive, then\nthis corresponds to any point in the fourth quadrant in Fig 2, which characterizes stable\nsystems.\nAs for the rest of the proof, we know that if Di>2p\nTijMithenTr(A)2>4\u0001(A)\nand the origin is an asymptotically stable node. This corresponds to any point in the\nfourth quadrant in Fig 2 outside the parabolic curve, whereby the system is stable and\nno oscillations occur. The parabolic curve identi\fes the set of points for which Tr(A)2=\n4\u0001(A).\nThe last case is when Di<2p\nTijMiwhich implies Tr(A)2<4\u0001(A) and therefore\nthe origin is an asymptotically stable spiral. This corresponds to any point in the fourth\nquadrant in Fig 2, inside the parabolic curve whereby the system is stable but oscillations\nmay occur due to imaginary parts in the eigenvalues.\nThe above theorem sheds light on the role of the di\u000berent parameters in the transient\nstability of the micro-grid.\nExample 1 In particular, let the synchronization coe\u000ecient be Tij= 1 and the inertia\ncoe\u000ecient be M= 1 and investigate the role of the damping coe\u000ecient D. From (4) the\ntransient dynamics is determined by the eigenvalues \u00151;2=\u0000Di\u0006p\nD2\ni\u00004\n2:We can conclude\nthat\n\u000fifD > 2all eigenvalues are real and negative and no oscillations arise. The\nslowest eigenmode is determined by the smallest (in modulus) eigenvalue, which\nis\u0000Di+p\nD2\ni\u00004\n2.\n\u000fDi\u000berently, if D\u00142we have complex eigenvalues given by \u00151;2=\u0000Di\n2\u0006ip\nD2\ni\u00004\n2\nand we observe damped oscillations. The damping factor depends on the real part\nRe(\u00151;2) =\u0000Di\n2while oscillation frequencies are related to the imaginary part\nIm(\u00151;2) =p\nD2\ni\u00004\n2.\nExample 2 In this example we set the damping coe\u000ecient D= 1 and the inertia coe\u000e-\ncientM= 1 and investigate the role of the synchronization coe\u000ecient Tij. Again, from(4), the eigenvalues governing the transient dynamics are \u00151;2=\u00001\u0006p\n1\u00004Tij\n2:Then we\nhave the following cases:\n\u000fifTij<1\n4the eigenvalues are all real and negative and we observe no oscillations.\nThe transient is dominated by the slowest eigenmode, which in turn is determined\nby the smallest (in modulus) eigenvalue, i.e.\u00001+p\n1\u00004Tij\n2.\n\u000fUnlikewise, if Tij>1\n4the eigenvalues are complex and given by \u00151;2=\u00001\n2\u0006ip\n1\u00004Tij\n2\nin correspondence to which the transient dynamics shows damped oscillations. The\ndamping factor is determined by the real part Re(\u00151;2) =\u00001\n2and the oscillation\nfrequencies are determined by the imaginary part Im(\u00151;2) =p\n1\u00004Tij\n2.\nThe above theorem and examples identify intervals for the parameters within which\nthe behavior of the transient stability is unchanged. This provides robustness to our\nresults and extend the analysis to cases where the inertia, damping and synchronization\nparameters are uncertain.\n3 Multiple interconnected micro-grids\nLet us now consider a network G= (V;E) of interconnected smart-grids, where Vis the set\nof nodes, and Eis the set of arcs. Figure 3 displays an example of interconnection topology.\nNodes represent smart-grids units and arcs represent power lines interconnections. We\nuse shades of gray to emphasize di\u000berent levels of connectivity of the smart-grids. The\nconnectivity of a grid is indicated by the degree of the node. We recall that for undirected\ngraphs the degree of a node is number of links with an extreme in node i. We denote by\ndithe degree of node i.\nFigure 3: Graph topology indicating smart-grids and interconnections.\nBuilding on model (3) developed for the single grid, we derive the following macroscopicdynamics for the whole grid:\n2\n666666664_x(1)\n1...\n_x(n)\n1\n_x(1)\n2...\n_x(n)\n23\n777777775=2\n666666664\u0000D1\nM1::: 01\nM1::: 0\n0... 0 0...0\n0:::\u0000Dn\nMn0:::1\nMn\n\u0000T11::: T 1n 0...0\n......\nTn1:::\u0000Tnn 0::: 03\n7777777752\n666666664x(1)\n1...\nx(n)\n1\nx(i)\n2...\nx(n)\n23\n777777775:\nIn the above set of equations, the block matrix\nL:=2\n64T11:::\u0000T1n\n...\n\u0000Tn1::: Tnn3\n75\nis the graph-Laplacian matrix. Given a weighted graph its components L= [lij]i;j2f1;:::;ng\nare given by\nlij=\u001a\u0000Tij ifi6=j;P\nh=1;h6=iTihifi=j:(5)\nNote that given a Laplacian matrix, its row-sums are zero, its diagonal entries are non-\nnegative, and its non-diagonal entries are nonpositive. The above set of equations can be\nrewritten in compact form as follows\n\u0014_X1\n_X2\u0015\n=\"\n\u0000Diag\u0010\nDi\nMi\u0011\nDiag\u0010\n1\nMi\u0011\n\u0000L 0#\n|{z }\nA\u0014X1\nX2\u0015\n: (6)\nWe also recall that L= [lij]i;j2f1;:::;ngwhere for an unweighted and undirected graph we\nhave\nlij=8\n<\n:\u00001 if (i;j) is an edge and not self-loop ;\nd(i) ifi=j;\n0 otherwise.(7)\nWe are ready to establish the next result. Let us denote by spanf1g=f\u00182Rn:\n9\u00112Rs:t:\u0018 =\u00111g. Furthermore, let the following consensus set be de\fned as\nC=f\u00182Rn:\u00182spanf1g;min\njxj(0)\u0014\u0018\u0014max\njxj(0)g:\nTheorem 2 Let a network of homogeneous micro-grids be given, and set Di=Dfor all\ni. LetMi= 1 for alli, andTij= 1 for any (i;j)2E. Then dynamics (6) describes a\nconsensus dynamics, i.e.,\nlim\nt!1Xi(t) =x\u0003\ni2C; i = 1;2:\nFurthermore, let D >p\u00004\u0016ithen the consensus value vector (x\u0003\n1;x\u0003\n2)Tis an asymptot-\nically stable node. Vice versa, if D <p\u00004\u0016ithen (x\u0003\n1;x\u0003\n2)Tis an asymptotically stable\nspiral.Proof. Let us start by \fnding the roots of det(\u0015I\u0000A). To this purpose, we recall that\nfor any generic block matrix it holds\ndet(\u0014A B\nC D\u0015\n) =det(DA\u0000BC);ifBD=DB. (8)\nThen, from the above we have\ndet(\u0015I\u0000A) =det\u0010\"\n\u0015I+Diag\u0010\nDi\nMi\u0011\n\u0000Diag\u0010\n1\nMi\u0011\nL \u0015 I#\u0011\n=det(\u00152I+\u0015I\u0001Diag (Di\nMi) +Diag (1\nMi)\u0001L):(9)\nUnder the homogeneity assumption Di=Dfor alli, we have\ndet\u0010\n\u00152I+\u0015I\u0001Diag\u0010\nDi\nMi\u0011\n+Diag\u0010\n1\nMi\u0011\n\u0001L\u0011\n=det\u0010\n(\u00152+\u0015D)I+L\u0011\n=Qn\ni=1\u0010\n(\u00152+\u0015D)\u0000\u0016i\u0011\n;(10)\nwhere\u0016iis theith eigenvalue of\u0000L. The roots of (10) can be obtained by solving\n\u00152+\u0015D\u0000\u0016i= 0 from which we have\n\u0015+\ni=\u0000D+p\nD2+4\u0016i\n2; \u0015\u0000\ni=\u0000D\u0000p\nD2+4\u0016i\n2: (11)\nFrom the above, after noting that the real part of the eigenvalues is negative, we can\nconclude that system (6) is asymptotically stable.\nRemark 1 The result stated in the above theorem applies also to the case where the micro-\ngrids have di\u000berent inertia but the same ratio D=Di\nMifor alli2V. In this case we need\nto consider the Laplacian matrix of the corresponding weighted graph ~L=Diag (1\nMi)Land\nthe associated eigenvalues.\nWe next recall some properties of the Laplacian spectrum and use such properties to\ninvestigate the insurgence of topology-induced oscillations. The maximal eigenvalue ~ \u0016n\nof a symmetric Laplacian matrix L=LTinRn\u0002nsatis\fes the following lower and upper\nbounds which are degree-dependent:\ndmax\u0014~\u0016n\u00142dmax; (12)\nwhere the maximum degree is dmax = maxi21;:::;ndi[4, Chapter 6]. We also observe that\nthe eigenvalues appearing in (11) refer to the negative Laplacian, and therefore we have\n\u0016i=\u0000~\u0016ifor every eigenvalue \u0016iof the negative Laplacian \u0000Land ~\u0016iof the Laplacian L.\nCorollary 1 The following properties hold:\u000fAll eivenvalues \u0015+\ni;\u0015\u0000\nifori= 1;:::;n are real and negative if D\u0015p8dmax;\n\u000fThere exists at least one complex eigenvalue/eigenmode if D\u0014p4dmax:\nCorollary 2 Given a chain topology of n\u00153nodes, for which dmax = 2 the following\nproperties hold:\n\u000fAll eivenvalues \u0015+\ni;\u0015\u0000\nifori= 1;:::;n are real and negative if D\u00154;\n\u000fThere exists at least one complex eigenvalue/eigenmode if D\u00142p\n2:\n3.1 Example of two interconnected micro-grids\nIn this section, we specialize the above results to the case of two interconnected micro-\ngrids. The interconnection topology is a chain one with two nodes, and the maximal\ndegree isdmax = 1. A graph representation is displayed in Fig. 4.\n+\n\u0000eji\nTij\ns1\nMjs+Djfj Pj fj+\n\u0000eij\nTij\ns1\nMis+Difi Pi\nFigure 4: Block representation of two interconnected micro-grids.\nDynamics (6) can be rewritten as\n2\n6664_x(i)\n1\n_x(j)\n1\n_x(i)\n2\n_x(j)\n23\n7775=2\n664\u0000D1\nM101\nM10\n0\u0000D2\nM201\nM2\n\u0000T11T12 0 0\nT12\u0000T110 03\n7752\n6664x(i)\n1\nx(j)\n1\nx(i)\n2\nx(j)\n23\n7775: (13)\nThe Laplacian of the weighted graph is given by\n~L=Diag (1\nMi)L=\u00141\nM10\n01\nM2\u0015\u00141\u00001\n\u00001 1\u0015\n: (14)\nAssumingD=D1\nM1=D2\nM2, from Corollary 1 we infer that\n\u000fAll eivenvalues \u0015+\ni;\u0015\u0000\nifori= 1;:::;n are real and negative if D\u0015p\n8;\n\u000fThere exists at least one complex eigenvalue/eigenmode if D\u00142:\nIn other words, if the ratio between the damping coe\u000ecient and the inertia of each micro-\ngrid is greater thanp\n8 then we certainly have an overdamped dynamics, and observe no\novershoots and no oscillations. Di\u000berently, if the ratio between the damping coe\u000ecient\nand the inertia of each micro-grid is less than 2 then we certainly have an underdamped\ndynamics, and observe overshoots and oscillations.4 Absolute stability\nIn this section we extend the analysis to the case where both frequency and power \row\nmeasurements are subject to disturbances. Using a traditional technique in nonlinear\nanalysis and control we isolate the nonlinearities in the feedback loop, and analyze stability\nunder some mild assumptions on the nonlinear parameters.\nLikewise in the previous section we consider two interconnected micro-grids, and as-\nsume that each micro-grid can be described in terms of power \row Piand frequency fi.\nAssuming disturbed measurements on fi, the evolution of the power \row is given by\n_Pi=Tij(fj\u0000 (fi)) =Tij~eij; (15)\nwhereTijis the synchronizing coe\u000ecient as in the previous section and where the new\nterm (:) is a sector nonlinearity satisfying the following assumption.\nAssumption 1 Function (fi;t)is time-varying and satis\fes the sector condition\n0\u0014 (fi)\u0014~kfi:\nNow, the power Pidepends on a disturbed measure of the frequency error ~ eij:=fj\u0000 (fi).\nThe dynamics for fistill follows a traditional swing equation, which now involves\ndisturbed measurements of the frequency  (fi) and of the power \row  (Pi):\n_fi=\u0000Di\nMi (fi) + (Pi)\nMi+!: (16)\nIn the above model, MiandDiare the inertia and damping constants of the ith micro-grid,\nrespectively. Similarly to the previous section, we denote fi=x(i)\n1,Pi=x(i)\n2,fj=x(j)\n1,\nand by considering fjas an exogenous input to micro-grid i, the dynamics of micro-grid\nireduces to the following second-order system\n_x=\"\n_x(i)\n1\n_x(i)\n2#\n=\u0014\u0000Di\nMi1\nMi\n\u0000Tij0\u0015\n|{z}\nA\"\n (x(i)\n1)\n (x(i)\n2)#\n+\u00141 0\n0Tij\u0015\n|{z}\nB\u0014!\nx(j)\ni\u0015\n:(17)\nThe block system of the ith micro-grid, which admits the state space representation\n(17), is displayed in Fig. 5.\nBuilding on the Kalman-Yakubovich-Popov lemma, absolute stability is linked to\nstrictly positive realness of Z(s) =I+KG(s) whereK=~kIandG(s) is the transfer\nfunction of linear part of system (17) which is obtained as G(s) =CT[sI\u0000A]\u00001Bwhere\nwe set\nA=\u0014\u0000Di\nMi1\nMi\n\u0000Tij0\u0015\n; B =\u00141 0\n0Tij\u0015\n; C =\u00141 0\n0 1\u0015\n:\nWe recall from Theorem 1 that matrix Ais Hurwitz.fj+\n\u0000eTij\ns1\nMis+Difi\n (fi;t)Pi!\nFigure 5: Block system representing micro-grid i.\nThe idea now is to isolate the nonlinearities in the feedback loop and introduce a\nnew variable for them, say  . Let us \frst obtain the transfer function associated to the\ndynamical system (17):\nG(s) =CT[sI\u0000A]\u00001B\n=1\ns(s+Di\nMi)+Tij\nMi\u0014s1\nMi\n\u0000Tijs+Di\nMi\u0015\n\u0001\u00141 0\n0Tij\u0015\n=1\ns(s+Di\nMi)+Tij\nMi\"\nsTij\nMi\n\u0000Tij(s+Di\nMi)Tij#\n=1\n\u0001(sI\u0000A)\"\nsTij\nMi\n\u0000Tij(s+Di\nMi)Tij#\n;(18)where \u0001(sI\u0000A) =s(s+Di\nMi) +Tij\nMi. Then, for Z(s) we obtain\nZ(s) =I+KG(s)\n=\u00141 0\n0 1\u0015\n+k\ns(s+Di\nMi)+Tij\nMi\"\nsTij\nMi\n\u0000Tij(s+Di\nMi)Tij#\n=\u00141 0\n0 1\u0015\n+k\n\u0001(sI\u0000A)\"\nsTij\nMi\n\u0000Tij(s+Di\nMi)Tij#\n=1\n\u0001(sI\u0000A)\"\nks+ \u0001(sI\u0000A) kTij\nMi\n\u0000kTijk(s+Di\nMi)Tij+ \u0001(sI\u0000A)#\n=1\ns(s+Di\nMi)+Tij\nMi\n\u0001\u0014\nks+s(s+Di\nMi) +Tij\nMikTij\nMi\n\u0000kTij k(s+Di\nMi)Tij+s(s+Di\nMi) +Tij\nMi\u0015\n=1\ns(s+Di\nMi)+Tij\nMi\n\u0001\u0014\ns2+ (Di\nMi+k)s+Tij\nMikTij\nMi\n\u0000kTij s2+ (Di\nMi+kTij)s+Tij\nMi+kTijDi\nMi\u0015\n:\nNote that matrix Ais Hurwitz. This implies that also Z(s) is Hurwitz as the poles of\nZ(s) coincide with the eigenvalues of A. We use this in the proof of absolute stability of\nthe dynamical system (17) established next.\nTheorem 3 Let the dynamical system (17) be given where Ais Hurwitz. Furthermore,\nlet us consider the sector nonlinearities as in Assumption 1. Then, Z(s)is strictly positive\nreal and system (17) is absolutely stable.\nProof. We \frst prove that Z(s) is strictly positive real. For this to be true, the following\nconditions must hold true:\n\u000fZ(s) is Hurwitz, namely the poles of all entries of the matrix Z(s) have negative\nreal parts;\n\u000fZ(j!) +Z(\u0000j!)>0;8!2R;\n\u000fZ(1) +ZT(1)>0.\nFor the \frst condition note that Z(s) is Hurwitz as its poles are the roots of s(s+\nDi\nMi) +Tij\nMi= 0, which coincide with the values obtained in (4) and which we rewrite\nhere for convenience: \u00151;2=1\n2\u0010\n\u0000Di\nMi\u0006q\n(Di\nMi)2\u00004Tij\nMi\u0011\n. As for the second condition,Z(j!) +Z(\u0000j!)>0;8!2R, let us obtain for Z(j!) andZ(\u0000j!) the following\nexpressions:\nZ(j!) =1\nTij\nMi\u0000!2+Di\nMij!\n\u0001\u0014Tij\nMi\u0000!2+ (Di\nMi+k)j! kTij\nMi\n\u0000kTijTij\nMi+kTijDi\nMi\u0000!2+ (Di\nMi+kTij)j!\u0015\n:\nZ(\u0000j!) =1\nTij\nMi\u0000!2\u0000Di\nMij!\n\u0001\u0014Tij\nMi\u0000!2\u0000(Di\nMi+k)j! kTij\nMi\n\u0000kTijTij\nMi+kTijDi\nMi\u0000!2\u0000(Di\nMi+kTij)j!\u0015\n;\nBy combining the expressions above for Z(j!) andZ(\u0000j!) we then obtain\nZ(j!) +Z(\u0000j!) =1\n\u0001(j!I\u0000A)\u0001(\u0000j!I\u0000A)\u0001(\u0000j!I\u0000A)\n\u0001\u0014Tij\nMi\u0000!2+ (Di\nMi+k)j! kTij\nMi\n\u0000kTijTij\nMi+kTijDi\nMi\u0000!2+ (Di\nMi+kTij)j!\u0015\n+\u0001(j!I\u0000A)\n\u0001\u0014Tij\nMi\u0000!2\u0000(Di\nMi+k)j! kTij\nMi\n\u0000kTijTij\nMi+kTijDi\nMi\u0000!2\u0000(Di\nMi+kTij)j!\u0015\n=1\u0010\nTij\nMi\u0000!2\u00112\n\u0000\u0010\nDi\nMij!\u00112\u0014z11z12\nz21z22\u0015\n;(19)\nwhere we set z11andz22as follows:\nz11= 2[!4\u00002!2Tij\nMi+!2Di\nMi(Di\nMi+k) + (Tij\nMi)2]\n= 2[(!2\u0000Tij\nMi)2+!2Di\nMi(Di\nMi+k)];\nz22= 2[!4\u0000!2(Tij\nMi+kTijDi\nMi) +!2Di\nMi(Di\nMi+kTij)\n\u0000!2Tij\nMi+Tij\nMi(Tij\nMi+kTijDi\nMi) +!2Di\nMi(Di\nMi+kTij)\n= 2[(!2\u0000Tij\nMi)2\u0000!2kTijDi\nMi\n+Tij\nMikTijDi\nMi+!2Di\nMi(Di\nMi+kTij)]\n= 2[(!2\u0000Tij\nMi)2+!2(Di\nMi)2+Tij\nMikTijDi\nMi]:From the above equation we then have\nZ(j!) +Z(\u0000j!) =1\u0010\nTij\nMi\u0000!2\u00112\n\u0000\u0010\nDi\nMij!\u00112\n\u0001\u0014\n2[(!2\u0000Tij\nMi)2+!2Di\nMi(Di\nMi+k)]\nz21\nz12\n2[(!2\u0000Tij\nMi)2+!2(Di\nMi)2+Tij\nMikTijDi\nMi]\u0015\n>0;for all!:\nThe last inequality follows from the trace of the above matrix being positive. To see\nthis note that\nz11+z22\n= 2[(!2\u0000Tij\nMi)2+!2Di\nMi(Di\nMi+k)]\n+2[(!2\u0000Tij\nMi)2+!2(Di\nMi)2+Tij\nMikTijDi\nMi]>0:(20)\nAs for the third condition, namely Z(1) +ZT(1)>0, we have that\nlim!!1z12= lim!!1z21= 0;\nlim!!1z11= lim!!1z22= 2:(21)\nThen we obtain that Z(1) +ZT(1) = 2I>0. We can conclude that also the third\ncondition is veri\fed.\nNow we wish to show that there exists a Lyapunov function V(x) =xT\bx, where\n\b = [\bij]2R2\u00022is symmetric. After di\u000berentiation with respect to time and using (17)\nwe obtain_V(t;x) = _xT\bx+xT\bx\n=xTAT\bx+xT\bAx\u0000 TBT\bx\u0000xT\bB \n= [x1x2]\u0014\u0000Di\nMi1\nMi\n\u0000Tij0\u0015T\u0014\b11\b12\n\b21\b22\u0015\u0014x1\nx2\u0015\n+[x1x2]\u0014\b11\b12\n\b21\b22\u0015\u0014\u0000Di\nMi1\nMi\n\u0000Tij0\u0015\u0014x1\nx2\u0015\n\u0000[ 1 2]\u00141 0\n0Tij\u0015\u0014\b11\b12\n\b21\b22\u0015\u0014x1\nx2\u0015\n\u0000[x1x2]\u0014\b11\b12\n\b21\b22\u0015\u00141 0\n0Tij\u0015\nB\u0014 1\n 2\u0015\n;\nwhere we denote  (t;y) = [ 1 2]T. From Assumption 1 and the property of \frst and\nthird sector nonlinearities we have \u00002 T( \u0000Ky)\u00150. Furthermore, from symmetry ofmatricesPandK=~kI, the time derivative of the candidate Lyapunov function can be\nrewritten as\n_V(t;x)\u0014xT(AT\b +PA)x\u00002xT\bB \u00002 T( \u0000Ky)\n=xT(AT\b + \bA)x\u00002xT\bB + 2 TKCx\u00002 T \n=xT(AT\b + \bA)x+ 2xT(CTK\u0000\bB) \u00002 T \n= [x1x2] \u0014\u0000Di\nMi1\nMi\n\u0000Tij0\u0015T\u0014\b11\b12\n\b21\b22\u0015\n+\u0014\b11\b12\n\b21\b22\u0015\u0014\u0000Di\nMi1\nMi\n\u0000Tij0\u0015\u0013\u0014x1\nx2\u0015\n+2[x1x2]\u0012\u0014k0\n0k\u0015\n\u0000\u0014\b11\b12\n\b21\b22\u0015\u00141 0\n0Tij\u0015\u0013\u0014 1\n 2\u0015\n\u00002[ 1 2]\u0014 1\n 2\u0015\n:\nThe right-hand side of the above inequality is negative if there exist matrices \u0005 2R2\u00022\nand a positive scalar \u000fsuch that\n\u001aAT\b + \bA=\u0000\u0005T\u0005\u0000\u000f\b;\n\bB=CTK\u0000p\n2\u0005T;(22)\nor in explicit form\n\u0014\u0000Di\nMi1\nMi\n\u0000Tij0\u0015T\u0014\b11\b12\n\b21\b22\u0015\n+\u0014\b11\b12\n\b21\b22\u0015\u0014\u0000Di\nMi1\nMi\n\u0000Tij0\u0015\n=\u0000\u0014\u000511\u000512\n\u000521\u000522\u0015T\u0014\u000511\u000512\n\u000521\u000522\u0015\n\u0000\u000f\u0014\b11\b12\n\b21\b22\u0015\n;\n\u0014\b11\b12\n\b21\b22\u0015\u00141 0\n0Tij\u0015\n=\u0014k0\n0k\u0015\n\u0000p\n2\u0014\u000511\u000521\n\u000512\u000522\u0015\n;\nBy introducing the solutions of the above in terms of \b, \u0005 and \u000f, the time derivativeof the candidate Lyapunov function can be rewritten as\n_V(t;x)\u0014\u0000\u000fxT\bx\u0000xT\u0005T\u0005x+ 2p\n2xT\u0005T \u00002 T \n=\u0000\u000fxT\bx\u0000[\u0005x\u0000p\n2 ]T[\u0005x\u0000p\n2 ]\n\u0014\u0000\u000fxT\bx\n\u0000\u000f[x1x2]\u0014\b11\b21\n\b12\b22\u0015\u0014x1\nx2\u0015\n:\nIt is well known that from the Kalman-Yakubovich-Popov lemma, there exist solutions in\nterms of \b, \u0005, and \u000fsatisfying the above set of matrix equalities, as the transfer function\nZ(s) is positive real and this concludes our proof.\nRemark 2 The above theorem has been obtained under the hypothesis that both frequency\nand power \row measurements are subject to disturbances. The same result extend straight-\nforwardly also to the case where the model parameters Tij,MiandDiare uncertain.\n5 Simulations\nThis section provides simulation studies to corroborate the theoretical results developed\nin the previous sections. The analysis is based on open source data relating to a part of\nthe Nigerian grid obtained from [1]. The data set shows the one-line diagram of part of\nthe distribution network including the geographical location of generators and load buses.\nFigure 6 displays the one-line diagram with the geographical names.\nYobe Borno\nAdamawa\nTarabaGombeBauchiPlateauKadunaKanoJigawa Katsina\nFigure 6: One-line diagram of part of Nigerian grid [1].\nFrom the one-line diagram we obtain the graph representation showing the intercon-\nnection between bus loads as in Fig. 7. The graph is characterized by 11 nodes and 10\narcs. Most nodes have degree 1 or 2 except for Gombe, and Kano which have degree 4,\nand 3, respectively. The graph is undirected, i.e. the in\ruence of smart-grid ionjis\nbidirectional.YobeBorno\nAdamawa\nTarabaGombeBauchiPlateauKadunaKanoJigawa Katsina\nFigure 7: Graph representation of part of Nigerian grid [1].\nThe numerical studies involve two sets of simulations. The \frst set of simulations has\nbeen conducted considering the following normalized parameters: number of smart-grids\nn= 11, damping constant D= 1;3;6 for three consecutive runs of simulations; Inertial\nconstantM= 1; Synchronizing coe\u000ecient T= 1; Horizon window involving N= 500\niterations; Step size dt=:01. The parameters and the dynamics is normalized in an\ninterval [0;1]. For instance the initial state of each grid is a randomized bidimensional\nvector in the interval [0 ;1]. To simulate periodic disturbances, the initial state is reini-\ntialized every 10 sec. To obtain realistic plots we rescale the state variable around 50 Hz\nfor the frequency and 30 MWh for the power \row. Figure 8 displays the evolution of the\nfrequency of each smart-grid. Frequencies are measured in Hz and are centered around\n50 Hz which is the nominal value. Oscillations remain within 1% of the nominal value,\ni.e., in the interval [49 :95;50:05]. From top to bottom we consider an increasing damping\nconstantD= 1;3;6 which re\rects in damped oscillations and smaller time constants.\nFigure 9 displays the evolution of the power \rows in each smart-grid. Power \rows are\nmeasured in MWh and are centered around the nominal value of 30 MWh. From the plots\nwe observe that oscillations remain within 3 :3% of the nominal value, i.e., in the interval\n[29:00;31:00] MWh. From top to bottom the damping constant is increasing and equal\ntoD= 1;3;6. Note that the maximal degree of the network is dmax = 4 and therefore\nforD= 1;3 we have D <p4dmax = 4 and oscillations emerge as we have complex\neigenvalues for \u0015+\ni;\u0015\u0000\ni;forA. Unlikewise for D= 6 it holds D >p8dmax =p\n32 and\ntherefore no oscillations and no complex eigenvalues emerge. The maximal eigenvalue of\nthe Laplacian has been obtained as ~ \u0016n= 5:1748.\nIn a second set of simulations, we isolate one smart-grid from the rest of the power\nnetwork and investigate the transient response under disturbances in the measurement\nof frequency and power. Such disturbances are modeled using the paradigm developed\nin Section 4. In particular we consider a \frst and third quadrant nonlinearity in the\nfeedback loop. The function is periodic and we take for it the expression  (t) = 1 +\nsin(\u0018ft) in [0;2], wherefis the frequency, tis time, and \u0018is a factor increasing the\nperiodicity of the oscillation. For the second set of simulations we consider the following\nnormalized parameters: number of smart-grids n= 1, damping constant D= 1; Inertial\nconstantM= 1; Synchronizing coe\u000ecient T= 1; periodicity factor \u0018= 1;5;10 for three\nconsecutive runs of simulations; Horizon window involving N= 1000 iterations; Step sizeFigure 8: Time series of smart-grids frequencies in Hz.\ndt=:01; The initial state of each grid is a randomized bidimensional vector in the interval\n[0;1]. Both variables are rescaled around 50 Hz for the frequency and 30 MWh for the\npower \row. Figure 10 displays the time evolution of the frequency of each smart-grid\n(left) and power \row (right). As in the previous simulation example, frequencies are\nmeasured in Hz and are centered around 50 Hz which is the nominal value. We observe\nthat oscillations remain within 1% of the nominal value, i.e., in the interval [49 :95;50:05].\nFrom top to bottom the damping constant is D= 1;3;5 and this implies a higher damping,\nsmaller time constants, and faster convergence. Power \rows are measured in MWh and\nare centered around the nominal value of 30 MWh. The plots show that oscillations\nremain within 3 :3% of the nominal value, i.e., in the interval [29 :00;31:00] MWh. From\ntop to bottom the damping constant is increasing and equal to D= 1;3;5.\n6 Discussion and conclusions\nFor single and multiple interconnected micro-grids, we have studied transient stability,\nnamely the capability of the micro-grids to remain in synchronism even under cyber-\nattacks or model uncertainties. First we have showed that transient dynamics can be\nrobustly classi\fed depending on speci\fc intervals for the micro-grid parameters, such asFigure 9: Time series of smart-grids power \rows in MWh.\nsynchronization, inertia, and damping parameters. We have then turned to study the\nanalogies with consensus dynamics. We have obtained bounds on the damping coe\u000ecient\nwhich determine wether the network dynamics is underdamped or overdamped. Such\na result is meaningful as in the case of underdamped dynamics we observe oscillation\naround the consensus value, whereas in the case of underdamped dynamics we observe\na deviation of the consensus value from the nominal mains frequency. The bounds are\nlinked to the connectivity of the network. We have also extended the stability analysis to\nthe case of disturbed measurements due to hackering or parameter uncertainties. Using\ntraditional nonlinear analysis and the Kalman-Yakubovich-Popov lemma we have \frst\nisolated the nonlinear terms in the feedback loop and have showed that nonlinearities do\nnot compromise the stability of the system.\nThere are three key directions for future work. First we wish to relax constraints\non the nature of the disturbances. Indeed here we have assumed that such disturbances\ncan be modeled using \frst and third quadrant nonlinearities. Such nonlinearities tends\nto vanish around the equilibrium points. In reality, disturbances due to hackering can\nimpact the systems even at the equilibrium, thus leading to synchronization de\fciency.\nA second direction involves the analysis of the impact of stochastic disturbances on the\ntransient stability. Concepts like stochastic stability, stability of moments, and almost\nsure stability will be used to classify the resulting stochastic transient dynamics. Finally,Figure 10: Time series of smart-grids power \rows in MWh.\na third direction involves the extension to the case of a single or multiple heterogeneous\npopulations of micro-grids. In this context we will try to gain a better insights on scal-\nability properties and emergent behaviors. The latter is a terminology used in complex\nnetwork theory to address macroscopic phenomena arising from microscopic behavioral\npatterns.\nReferences\n[1] F. K. Ariyo, M. O. Omoigui. Investigation of Nigerian 330 kV Electrical Network\nwith Distributed Generation Penetration { Part I: Basic Analyses. Electrical and\nElectronic Engineering , Scienti\fc & Academic Publishing, 3(2), 49{71, 2013.\n[2] F. Bagagiolo, D. Bauso. Mean-\feld games and dynamic demand management in\npower grids. Dynamic Games and Applications , 4(2), 155{176, 2014.\n[3] D. Bauso. Dynamic demand and mean-\feld games, IEEE Transactions on Automatic\nControls , in press 10.1109/TAC.2017.2705911[4] F. Bullo. Lectures on Network Systems , Version 0.95, 2017,\nhttp://motion.me.ucsb.edu/book-lns.\n[5] F. D or\rer, F. Bullo. Synchronization and Transient Stability in Power Networks and\nNonuniform Kuramoto Oscillators. SIAM Journal on Control Optimization , 50(3),\n1616{1642, 2012.\n[6] T. Namerikawa, N. Okubo, R. Sato, Y. Okawa, M. Ono. Real-Time Pricing Mecha-\nnism for Electricity Market With Built-In Incentive for Participation. IEEE Trans-\nactions on Smart Grid , 6(6), 2714{2724, 2015.\n[7] R. Olfati-Saber, J. A. Fax, R. M. Murray. Consensus and Cooperation in Networked\nMulti-Agent Systems. Proceedings of the IEEE , vol. 95, no. 1, pp. 215{233, 2007.\n[8] H. Yin, P. G. Mehta, S. P. Meyn, U. V. Shanbhag, Synchronization of Coupled\nOscillators is a Game. IEEE Transactions on Automatic Control, 57(4) (2012) 920{\n935.\n[9] H. Yin, P. G. Mehta, S. P. Meyn, U. V. Shanbhag. On the E\u000eciency of Equilibria in\nMean-Field Oscillator Games, Dynamic Games and Applications, 4(2) (2014) 177{\n207."    },    {        "title": "2303.11128v1.Nonlinear_Damping_and_Field_aligned_Flows_of_Propagating_Shear_Alfvén_Waves_with_Braginskii_Viscosity.pdf",        "content": "Draft version March 21, 2023\nTypeset using L ATEX default style in AASTeX631\nNonlinear Damping and Field-Aligned Flows of Propagating Shear Alfv\u0013 en Waves with Braginskii\nViscosity\nAlexander J. B. Russell1\n1School of Science and Engineering,\nUniversity of Dundee,\nDundee, DD1 4HN, Scotland, UK\nABSTRACT\nBraginskii MHD provides a more accurate description of many plasma environments than classical\nMHD since it actively treats the stress tensor using a closure derived from physical principles. Stress\ntensor e\u000bects nonetheless remain relatively unexplored for solar MHD phenomena, especially in nonlin-\near regimes. This paper analytically examines nonlinear damping and longitudinal \rows of propagating\nshear Alfv\u0013 en waves. Most previous studies of MHD waves in Braginskii MHD considered the strict\nlinear limit of vanishing wave perturbations. We show that those former linear results only apply to\nAlfv\u0013 en wave amplitudes in the corona that are so small as to be of little interest, typically a wave\nenergy less than 10\u000011times the energy of the background magnetic \feld. For observed wave ampli-\ntudes, the Braginskii viscous dissipation of coronal Alfv\u0013 en waves is nonlinear and a factor around 109\nstronger than predicted by the linear theory. Furthermore, the dominant damping occurs through the\nparallel viscosity coe\u000ecient \u00110, rather than the perpendicular viscosity coe\u000ecient \u00112in the linearized\nsolution. This paper develops the nonlinear theory, showing that the wave energy density decays with\nan envelope (1+ z=Ld)\u00001. The damping length Ldexhibits an optimal damping solution, beyond which\ngreater viscosity leads to lower dissipation as the viscous forces self-organise the longitudinal \row to\nsuppress damping. Although the nonlinear damping greatly exceeds the linear damping, it remains\nnegligible for many coronal applications.\nKeywords: Alfv\u0013 en waves (23), Solar corona (1483), Solar coronal heating (1989), Solar coronal holes\n(1484), Solar wind (1534), Magnetohydrodynamics (1964), Space plasmas (1544), Plasma\nastrophysics (1261), Plasma physics (2089)\n1.INTRODUCTION\nAlfv\u0013 enic waves are a ubiquitous feature of natural plasmas, including the solar corona (Tomczyk et al. 2007; De\nPontieu et al. 2007; Lin et al. 2007; Okamoto et al. 2007) and solar wind (Coleman 1967; Belcher & Davis 1971).\nIn solar physics, these waves contain su\u000ecient energy to heat the open corona and accelerate the fast solar wind\n(McIntosh et al. 2011), and they damp signi\fcantly within a solar radius above the surface (Bemporad & Abbo 2012;\nHahn et al. 2012; Hahn & Savin 2013; Hahn et al. 2022). How these Alfv\u0013 enic waves damp in astrophysical and space\nplasmas is an important question that has remained open for almost a century (see early papers by Alfv\u0013 en 1947 and\nOsterbrock 1961; modern reviews by De Moortel & Browning 2015, Arregui 2015 and Van Doorsselaere et al. 2020;\nand historical perspectives by Russell 2018 and De Moortel et al. 2020).\nMost theoretical knowledge about solar Alfv\u0013 enic waves is based on \\classical\" magnetohydrodynamics (MHD), a\nmathematical framework that originated from intuitive coupling of Maxwell's equations and Euler equations of inviscid\nhydrodynamics (Hartmann 1937; Alfv\u0013 en 1942, 1943, 1950; Batchelor 1950) and became widely adopted in large part\ndue to its success providing insight into diverse natural phenomena (see e.g. Priest 2014). However, classical MHD is\nCorresponding author: Alexander J. B. Russell\na.u.russell@dundee.ac.ukarXiv:2303.11128v1  [astro-ph.SR]  20 Mar 20232 Russell\nStart: General MHD with Braginskii  πSet geometryLinearise in  , includes setting    in  .b/B0h=̂zπ  removed. Dissipation by   only.η0η2Full calculation of QRatio of heating terms:  Q0/Q2Linearise in  , setting  . (ωiτi)−2η1,η2≪η0Dissipation by  . Nonlinear in  .η0b/B0\nVanishing waves (bB0)2≪(ωiτi)−2Coronal regime (bB0)2≫(ωiτi)−2\nFigure 1. Schematic paths of reasoning. The vertical branch gives priority to smallness of the wave amplitude and concludes\nthat damping is a linear process governed by the perpendicular viscosity coe\u000ecient \u00112, e.g. §8 of Braginskii (1965). The\nhorizontal branch gives priority to the smallness of \u00112=\u00110, leading to nonlinear damping via \u00110. This paper follows the diagonal\nbranch, which includes deriving the validity condition for the two outcomes. Nonlinear damping via \u00110is appropriate for most\ncoronal applications.\none member of a larger family of plasma descriptions, some of which o\u000ber a more complete description of the plasma.\nThis paper analytically examines Alfv\u0013 en wave damping in the more general framework of Braginskii MHD, which\nunlike classical MHD, retains the anisotropic viscous stress tensor.\nA number of authors, including §8 of Braginskii (1965), have previously investigated viscous damping of Alfv\u0013 en waves\nin the linear limit of vanishingly small wave amplitude. When priority is given to smallness of the wave amplitude,\nthe problem becomes framed as a matter of how anisotropic viscosity a\u000bects velocities that are perpendicular to the\nmagnetic \feld (the direction of which is treated as unchanging). With this approximation, damping is determined by\nthe \\perpendicular\" viscosity coe\u000ecient \u00112, which is extremely small in the corona. It was thus originally concluded\nthat viscous damping is very weak for coronal Alfv\u0013 en waves unless they have very short wavelengths. This path of\nreasoning is shown as the vertical branch in Fig. 1.\nThere is, however, another way to view the problem. Viscous damping of Alfv\u0013 en waves can alternatively be considered\nwith priority given to the largeness of parallel viscosity coe\u000ecient \u00110. Given that \u00112=\u00110&10\u000011is typical in the corona\n(Hollweg 1985), even a very small component of vparallel to the total magnetic \feld Bwould be expected to produce\nmajor departures from linear theory. This path of reasoning is shown as the horizontal branch in Fig. 1.\nThe second viewpoint of the problem takes impetus from the observation that (unless wave amplitudes vanish\nentirely) Alfv\u0013 en waves do have a non-zero velocity component parallel to the total magnetic \feld. Two e\u000bects contribute\nto this, which are separated if one expands V\u0001B=V\u0001(b+B0), where B0is the equilibrium magnetic \feld and bis\nthe magnetic perturbation. First, V\u0001bis non-zero for an Alfv\u0013 en wave, since the velocity perturbation perpendicular\ntoB0is aligned with the magnetic \feld perturbation b. In other words, de\rection of the magnetic \feld from its\nequilibrium direction implies there is a non-zero velocity component parallel to the total magnetic \feld. Second, in\ncompressible plasma, the magnetic pressure of the magnetic \feld perturbation drives a nonlinear ponderomotive \row\nparallel to the equilibrium magnetic \feld (e.g. Hollweg 1971). The ponderomotive \row makes V\u0001B0nonzero as well.Nonlinear Alfv \u0013en Waves with Viscosity Tensor 3\nBoth of these e\u000bects allow for the possibility of nonlinear viscous damping via the large parallel viscosity coe\u000ecient\n\u00110.\nWith the bene\ft of modern observations (e.g. McIntosh et al. 2011; Morton et al. 2015), it is known that normalised\nwave amplitudes b=B 0\u0018V=vA\u00180:1 are typical for the base of an open coronal \feld region, for example. The\n\\smallness\" of the square of this ratio is very modest in comparison to the extreme largeness of \u00110=\u00112. Thus, it is\nlikely from the outset that viscous damping of Alfv\u0013 en waves will be a nonlinear process governed by \u00110and the wave\namplitude. This paper provides mathematical evidence that this heuristic analysis holds true, along with detailed\nexamination of the consequences.\nVarious previous studies have explored e\u000bects of Braginskii viscosity on MHD waves since Braginskii (1965). In\nsolar physics, the e\u000bect of linearized Braginskii viscosity was revisited from the 1980s to the mid-1990s through the\nlens of phase mixing and resonant absorption, with the aim of determining how including the viscosity tensor modi\fes\nthese scale-shortening processes and their heating properties. At the time, it was common practice in solar MHD\nwave theory to work with linearized equations. Thus, due to linearization, Steinolfson et al. (1986), Hollweg (1987),\nRuderman (1991), Ofman et al. (1994) and Erdelyi & Goossens (1995) obtained analytical and numerical results that\nstrictly apply to Alfv\u0013 en waves of vanishing amplitude.\nIn adjacent \felds, the e\u000bect of anisotropic viscosity on MHD waves has also been investigated with an eye on\nMHD turbulence and the solar wind. Of particular note, Montgomery (1992) advocated that Braginskii viscosity is\nimportant in hot tenuous plasmas, that in many circumstances it should be treated using parallel ion viscosity, and\nthat plasma motions may self-organise to suppress damping. He further applied these ideas to anisotropy in MHD\nturbulence, on the basis that a quasi-steady turbulence is composed of the undamped modes. Quantitative elaboration\nin Montgomery (1992) was based on a linear normal mode analysis, which captures linear damping of magnetoacoustic\nwaves by parallel viscosity, but excludes nonlinear viscous damping of Alfv\u0013 en waves. The conclusion that a linearized\nstress tensor damps Alfv\u0013 en waves only negligibly, while damping magnetoacoustic waves signi\fcantly, was further\nreinforced by related work by Oughton (1996, 1997).\nSimilar ideas to ours regarding the importance of nonlinearity were advocated by Nocera et al. (1986), who modelled\nAlfv\u0013 en waves subject to the \u00110part of the Braginskii viscous stress tensor, retaining the leading-order nonlinear terms\nin the wave perturbations. Consistent with the argument above, their calculations found that coronal Alfv\u0013 en waves\ndamp nonlinearly by parallel viscosity. The current paper complements and extends the previous analysis by Nocera\net al. (1986), with the goal of producing a comprehensive understanding of the nonlinear damping and \feld-aligned\n\rows of propagating shear Alfv\u0013 en waves with Braginskii viscosity.\nA limitation of the mathematical techniques used in this paper is that they exclude certain other nonlinear e\u000bects\nthat may be important in plasmas, such as nonlinear interactions between waves. Numerical investigations will be\nrequired in future to verify the analytical theory presented here, compare the relative importance of viscous damping\nand other nonlinear e\u000bects such as parametric decay instability, and consider interactions between nonlinear processes\nin Braginskii MHD.\nThis paper is organised as follows. §2 provides scienti\fc background on single-\ruid Braginskii MHD and its\nrelationship to other single-\ruid plasma models. §3 quantitatively examines Alfv\u0013 en wave heating by the full Braginskii\nviscous stress tensor, demonstrating the importance of nonlinear \u00110terms and compressibility, and obtaining the wave\ndecay properties for the weakly viscous limit using energy principles. In §4, we argue that in highly viscous limit,\nviscous heating is suppressed by self-organisation of the ponderomotive \row, which implies that viscosity strongly\nalters the \feld-aligned \row associated with Alfv\u0013 en waves in this regime. §5 further strengthens the analysis, using\nmultiple scale analysis to obtain the decay properties without restrictions on the Alfv\u0013 enic Reynolds number, assuming\nthe framework of Braginskii MHD. The paper \fnishes with discussion in §6 and summary of main conclusions in §7.\n2.BRAGINSKII MHD\nBraginskii MHD is an important plasma description that treats anisotropic viscosity and thermal conduction using\nrigorous closure from physical principles. This section provides a short primer on single-\ruid Braginksii MHD, its\nconnection with pressure (or temperature) anisotropy, and its relation to classical MHD and the CGL double-adiabatic\nequations.\nAs is described in various plasma textbooks, \ruid variables can be rigorously and robustly de\fned as velocity\nmoments of the underlying particle distribution functions. Transport equations for each particle species are then4 Russell\nderived by taking moments of the kinetic Boltzmann equation, and combined to obtain the single \ruid equations.\nRecommended presentations can be found in Schunk & Nagy (2009) Chapter 7 and the Appendix of Spitzer (1962).\nAssuming quasi-neutrality, and conservation of mass, momentum and energy, this process yields the mass continuity\nequation,\n@\u001a\n@t+r\u0001(\u001aV) = 0; (1)\nmomentum equation\n\u001aDV\nDt=\u0000r\u0001P+\u001aG+j\u0002B; (2)\nenergy equation,\nD\nDt\u00123\n2p\u0013\n+5\n2pr\u0001V=\u0000\u0019:rV\u0000r\u0001 q+j\u0001(E+V\u0002B); (3)\nhigher-order transport equations if required, and the generalized Ohm's law.\nThe pressure tensor Pthat appears in Eq. (2) is the most fundamental representation of the internal forces associated\nwith thermal motions of particles. It is symmetric, so it represents six degree of freedom. The momentum equation\ncan also be reformulated by introducing the scalar pressure and stress tensor as\np=1\n3Trace ( P) =1\n3P\u000b\u000b; \u0019 \u000b\f=P\u000b\f\u0000p\u000e\u000b\f; (4)\nwhere\u000e\u000b\fis the Kronecker delta. So de\fned, the stress tensor \u0019is symmetric and traceless. These de\fnitions gives\nthe replacement\u0000r\u0001P=\u0000rp\u0000r\u0001\u0019.\nDeriving transport equations by moment taking meets with a fundamental closure problem: the transport equation\nfor each \ruid variable depends on a higher-order variable, producing an in\fnite regress unless the system can be\nclosed by other considerations. The method of closure is therefore a major distinguishing feature between di\u000berent\n\ruid models for plasmas. It is also a major source of validity caveats. Various di\u000berent methods of closure produce\ngoverning equations that conserve mass, momentum and energy, since these properties are already built into Eqs. (1){\n(3). However, the di\u000berent models discussed below disagree on the internal forces and heating, and can therefore\nproduce di\u000berent behaviors.\nClassical MHD (Hartmann 1937; Alfv\u0013 en 1942, 1943; Batchelor 1950) corresponds to a closure treatment in which the\nstress tensor and the heat \row vector are dropped from Eqs. (2) and (3). Dropping the stress tensor can be justi\fed\nwhen particle collisions or other forms of particle scattering such as wave-particle interactions are frequent enough\nthat the pressure tensor remains very close to isotropic. The resulting MHD equations are valid for many situations,\nfor instance modelling static equilibria, or dynamic situations in which the divergence of the stress tensor remains\nsmall compared to the Lorentz force. It is nonetheless a truncation since higher order variables are set to zero rather\nthan approximated. Furthermore, collisionality in environments such as the solar corona is low enough that the stress\ntensor can become signi\fcant for various dynamic phenomena, including MHD waves.\nBraginskii MHD uses a less restrictive method of closure. As is detailed by Braginskii (1965), when the collisional\nmean free path is signi\fcantly shorter than length scales over which \ruid quantities vary, the heat \row vector takes\nthe form of an anisotropic thermal conduction, and the stress tensor takes the form of an anisotropic viscosity. Closure\ncan therefore be achieved by expressing qand\u0019in terms of lower-order \ruid variables, which are traditionally derived\nusing methods similar to Chapman & Cowling (1939) or Grad (1949).\nThe anisotropy inherent in qand\u0019can be appreciated heuristically, by considering the helical motion of charged\nparticles in magnetized plasmas. The mean free path parallel to the magnetic \feld is the same as for unmagnetized\nplasmas, implying that transport parallel to the magnetic \feld is the same as for unmagnetized plasmas. Meanwhile,\nthe mean free path perpendicular to the magnetic \feld is the gyroradius, which is typically much less than the mean\nfree path parallel to the magnetic \feld, which supresses perpendicular transport. Hence both thermal conduction and\nviscous stresses are anisotropic with respect to the magnetic \feld direction, often extremely so.\nThe full Braginskii stress tensor, used in §3, involves \fve viscosity coe\u000ecients. A useful simpli\fcation, used in §5,\nis that for strong magnetizations, \n i\u001ci\u001d1, the parallel \u00110coe\u000ecient greatly exceeds the other viscosity coe\u000ecients.\nHence, one can often simplify by neglecting the smaller coe\u000ecients (although, as shown in §3 it can be necessary to\nretain other viscosity coe\u000ecients if length scales are highly anisotropic). In this simpli\fcation, one has the followingNonlinear Alfv \u0013en Waves with Viscosity Tensor 5\ncovariant expressions for parallel viscosity (Lifshitz & Pitaevskii 1981; Hollweg 1986):\n\u0019\u000b\f=\u00003\u00110\u0012\nh\u000bh\f\u0000\u000e\u000b\f\n3\u0013\u0012\nh\u0016h\u0017\u0000\u000e\u0016\u0017\n3\u0013\n@\u0016V\u0017;\nQvisc= 3\u00110\u0012\u0012\nh\u000bh\f\u0000\u000e\u000b\f\n3\u0013\n@\u000bV\f\u00132\n;(5)\nwhere h=B=jBjis the unit vector in the direction of the magnetic \feld. These expressions are di\u000berent to the\nisotropic viscosity that appears in the Navier-Stokes equations, owing to the anisotropy introduced by the magnetic\n\feld.\nParallel viscosity is closely related to pressure anisotropy. As pointed out by Chew et al. (1956), when \n i\u001ci\u001d1 the\nparticle Lorentz force makes the pressure tensor gyrotropic, giving it the form\nP\u000b\f=p?\u000e\u000b\f+ (pjj\u0000p?)h\u000bh\f: (6)\nThis is a signi\fcant simpli\fcation, since the six degrees of freedom of a general pressure tensor have been replaced\nwith two variables, pjjandp?. The de\fnitions in Eqs. (4) then yield p= (pjj+ 2p?)=3 and\n\u0019\u000b\f= (pjj\u0000p?)\u0012\nh\u000bh\f\u0000\u000e\u000b\f\n3\u0013\n: (7)\nEquation (7) shows that pressure anisotropy has an equivalent stress tensor, which is proportional to pjj\u0000p?.\nFurthermore, Eqs. (5) and (7) both have the form \u0019\u000b\f\u0018(h\u000bh\f\u0000\u000e\u000b\f=3), so equivalence of the stress tensors reduces\nto equivalence of the scalar factors in the two equations. An illuminating analysis of the conditions under which\nthey converge has been written by Hollweg (1985, 1986), the most important condition being that collisions (or other\nprocesses such wave-particle interactions) relax the pressure anisotropy driven by velocity gradients to an extent that\nthe pressure is only weakly anisotropic. Classical MHD, for comparison, assumes that pressure anisotropy can be\nneglected altogether.\nFor low collisionality, the quasistatic approximation in Braginskii MHD ceases to be valid and strong pressure\nanisotropy may develop. Under these conditions, separate evolution equations can be derived for pjjandp?(Chew\net al. 1956; Hollweg 1986). However, the closure problem rears its head again, because those equations depend on the\nheat \row vector. A simple approach to obtaining a closed system is to ignore the heat \row vector, thus obtaining\nthe CGL double adiabatic equations (Chew et al. 1956), which are commonly used for collisionless plasma. More\nsophisticated approaches also exist that solve for the evolution of the pressure anisotropy or the evolution of the stress\ntensor, retaining the heat \row vector and closing by other means. The works by Balescu (1988); Schunk & Nagy\n(2009); Zank (2014); Hunana et al. (2019a,b, 2022) provide further reading on this topic.\nSummarising, there exists a family of adjacent (sometimes overlapping) single-\ruid models for plasmas. The most\nappropriate choice for a particular problem and/or context depends on the collisionality. When MHD timescales are\ngreater than the ion collision time, Braginskii MHD provides rigorous closure and treats the internal forces and heat\n\row more accurately than classical MHD.\n3.ALFV \u0013EN WAVE HEATING BY BRAGINSKII VISCOSITY\n3.1. Model\nWe quantitatively examine the viscous dissipation for an Alfv\u0013 en wave, which is a transverse wave polarized so that the\nmagnetic perturbation is perpendicular to the equilibrium magnetic \feld and the wavevector. Setting the equilibrium\nmagnetic \feld in the z-direction, the magnetic perturbation in the x-direction and the wavevector in the yz-plane, we\nconsider a total magnetic \feld of the form\nB=b(y;z;t )ex+B0ez: (8)\nThis ansatz automatically satis\fes r\u0001B= 0. For the velocity \feld we assume the form\nV=Vx(y;z;t )ex+Vz(y;z;t )ez: (9)6 Russell\nTheVxis the dominant velocity component. In linearized theory it would be the only component of V. Additionally,\nwe have explicitly included a higher-order Vzterm that represents the nonlinear ponderomotive \row parallel to the\nequilibrium magnetic \feld, which is driven by gradients of the magnetic pressure perturbation b2=2\u00160associated with\na \fnite-amplitude Alfv\u0013 en wave (e.g. Hollweg 1971). The Vzterm can be dropped when the plasma is incompressible\n(see §3.3), however it is required for a nonlinear treatment of compressible plasma and a\u000bects the wave heating via the\nparallel viscosity coe\u000ecient \u00110(as remarked in §1). The expression for Vzin classical MHD is given later in Eq. (29).\nIn a full solution, derivatives of b2=2\u00160with respect to ygive rise to an additional nonlinear y-component of V,\nwhich in turn produces a nonlinear y-component of B. These terms are not shown explicitly in Eqs. (8) and (9). Such\nterms were included by Nocera et al. (1986) and appear not to a\u000bect our main conclusions, provided the perpendicular\nwavelength of the Alfv\u0013 en wave is su\u000eciently large.\nThe viscous force is determined from the viscous stress tensor \u0019\u000b\fby\nFvisc;\u000b =\u0000@\u0019\u000b\f\n@x\f; (10)\nand the viscous heating rate is determined using\nQvisc=\u0000\u0019\u000b\f@V\u000b\n@x\f; (11)\nwhere\u000b2fx;y;zg,\f2fx;y;zg, thex\fare components of the position vector, V\u000bare components of Vand repeated\nindices imply summation in the Einstein convention.\nA vital point is that the viscous stress tensor depends on the direction of the magnetic \feld given by the unit vector\nh=B=jBj, which for our Alfv\u0013 en wave model in Eq. (8) has\nhx=bp\nB2\n0+b2; hy= 0; hz=B0p\nB2\n0+b2; (12)\nwithh2\nx+h2\nz= 1. Our analysis di\u000bers from many past works by considering hx6= 0 and identifying the dominant\nheating contribution at the end, as opposed to setting hx= 0 before evaluating the damping e\u000bect on Alfv\u0013 en waves.\nApplying formulas from §4 of Braginskii (1965) (equivalent matrix expressions are given by Hogan 1984), the stress\ntensor is related to \fve viscosity coe\u000ecients by\n\u0019\u000b\f=\u00002X\ni=0\u0011iWi\u000b\f+4X\ni=3\u0011iWi\u000b\f: (13)\nThe gyroviscous \u00113and\u00114terms do not contribute to heating, so evaluating the heating rate Qviscrequires\nW0\u000b\f=3\n2\u0012\nh\u000bh\f\u00001\n3\u000e\u000b\f\u0013\u0012\nh\u0016h\u0017\u00001\n3\u000e\u0016\u0017\u0013\nW\u0016\u0017;\nW1\u000b\f=\u0012\n\u000e?\n\u000b\u0016\u000e?\n\f\u0017+1\n2\u000e?\n\u000b\fh\u0016h\u0017\u0013\nW\u0016\u0017;\nW2\u000b\f=\u0000\n\u000e?\n\u000b\u0016h\fh\u0017+\u000e?\n\f\u0017h\u000bh\u0016\u0001\nW\u0016\u0017;(14)\nwhere\u000e\u000b\fis the Kronecker delta,\n\u000e?\n\u000b\f=\u000e\u000b\f\u0000h\u000bh\f; (15)\nand the rate of strain tensor is\nW\u000b\f=@V\u000b\n@x\f+@V\f\n@x\u000b\u00002\n3\u000e\u000b\fr\u0001V: (16)Nonlinear Alfv \u0013en Waves with Viscosity Tensor 7\nFor the shear Alfv\u0013 en wave geometry described by Eq. (9), the Witensors become\nW0=\u0000\nhxhz@zVx+\u00002\n3\u0000h2\nx\u0001\n@zVz\u00010\nB@3h2\nx\u00001 0 3hxhz\n0\u00001 0\n3hxhz0 3h2\nz\u000011\nCA; (17)\nW1=hx(hz@zVx\u0000hx@zVz)0\nB@\u0000h2\nz0hxhz\n0 1 0\nhxhz0\u0000h2\nx1\nCA+ (hx@yVx\u0000hz@yVz)0\nB@0hz0\nhz0\u0000hx\n0\u0000hx01\nCA; (18)\nW2=\u0000\u0000\n1\u00002h2\nx\u0001\n@zVx\u00002hxhz@zVz\u00010\nB@2hxhz0 1\u00002h2\nx\n0 0 0\n1\u00002h2\nx0\u00002hxhz1\nCA+ (hx@yVx+hz@yVz)0\nB@0hx0\nhx0hz\n0hz01\nCA; (19)\nThe viscous heating rate with hx6= 0 retained is thus\nQvisc=\u00110\n3\u0000\n3hxhz@zVx+ (2\u00003h2\nx)@zVz\u00012+\u00111h2\nx(hz@zVx\u0000hx@zVz)2+\u00111(hz@yVx\u0000hx@yVz)2\n+\u00112\u0000\u0000\n1\u00002h2\nx\u0001\n@zVx\u00002hxhz@zVz\u00012+\u00112(hx@yVx+hz@yVz)2:(20)\n3.2. Two small parameters\nAs anticipated in §1 (e.g. Fig. 1), two parameters determine the relative importance of individual terms in Eq. (20).\nThe \frst small parameter is ( b=B 0)2, the ratio of the wave's magnetic energy density to the energy density of the\nbackground magnetic \feld, which enters through hxandhz. In the modern era, extensive observations of coronal\nMHD waves (Nakariakov & Verwichte 2005; De Moortel & Nakariakov 2012) allow ( b=B 0)2to be quanti\fed with good\ncertainty, directly from resolved wave observations or indirectly from spectral line widths. For example, Morton et al.\n(2015) studied waves at the base of a coronal open \feld region using both approaches and reported a wave speed vA=\n400 km s\u00001and wave motions at v= 35 km s\u00001. Both measurements are consistent with earlier \fndings for coronal\nholes and the quiet Sun (e.g. McIntosh et al. 2011). From observations like these, h2\nx\u0018(b=B 0)2\u0018(v=vA)2\u001810\u00002.\nThe second small parameter is (\n i\u001ci)\u00002, which sets the viscosity coe\u000ecients \u00111and\u00112relative to\u00110. The value of\n\ni\u001cican vary signi\fcantly in the corona, but if magnetic null points are excluded one obtains values similar to the\nestimates made by Hollweg (1985), who found 3 :4\u0002105for a solar active region and 7 :2\u0002105near the base of a\ncoronal hole. We therefore expect (\n i\u001ci)\u00002.10\u000011under common conditions, and \u00111and\u00112simplify to\n\u00112=6408\n5125(\ni\u001ci)\u00002\u00110; \u0011 1=1\n4\u00112: (21)\nThe numerical coe\u000ecients in Eq. (21) are obtained in the limit (\n i\u001ci)\u00002!0, e.g. from Eq. (73) of Hunana et al.\n(2022). They are approximate for \fnite (\n i\u001ci)\u00002but have a high degree of accuracy because the corrections to the\ncoe\u000ecients are of the order of (\n i\u001ci)\u00002.10\u000011. Inspecting Eq. (21) and considering (\n i\u001ci)\u00002.10\u000011, the\u00112and\u00111\ncoe\u000ecients are both vastly smaller than \u00110.\nThe smallness of ( b=B 0)2and the smallness of (\n i\u001ci)\u00002compete to make di\u000berent terms dominate the viscous heating.\nIf one tries to simplify Eq. (20) by setting ( b=B 0)2to zero, then hx= 0 andVz= 0 givesQvisc=\u00112(@zVx)2+\u00111(@yVx)2,\nas obtained by Braginskii (1965). On the other hand, if one tries to simplify by \frst taking (\n i\u001ci)\u00002to zero then only\n\u00110terms remain, suggesting a di\u000berent conclusion. Thus, the quantitative results recover the two branches shown in\nFig. 1. To correctly determine the damping under coronal conditions, one must carefully compare terms in the full\nEq. (20), bearing in mind that there are two small parameters, which we do now (diagonal branch in Fig. 1).\n3.3. Heating rate for incompressible plasma\nThe analysis for incompressible plasma is relatively straightforward, which makes it a natural starting point for\ndiscussion. The assumption of incompressibility is appropriate for liquid metals or high-beta plasmas, but not, we\nnote, for the corona. The use of coronal wave amplitudes and magnetizations in this section is therefore intended to\nbe instructive only, with the compressible \fnite-beta treatment that follows later in this paper being required to treat\nthe corona.8 Russell\nIn the incompressible case, r\u0001V= 0 applied to our Alfv\u0013 en wave geometry implies Vz= 0. Thus Eq. (20) with\n\u00111=1\n4\u00112simpli\fes to\nQvisc=\u001a\n3\u00110h2\nxh2\nz+\u00112\u0012\u0000\n1\u00002h2\nx\u00012+1\n4h2\nxh2\nz\u0013\u001b\n(@zVx)2+\u00112\u00121\n4h2\nz+h2\nx\u0013\n(@yVx)2: (22)\nThe terms involving @zVxset the viscous dissipation due to wavelengths parallel to the equilibrium magnetic \feld,\nand we \frst ask whether dissipation due to parallel wavelengths is dominated by the linear \u00112contribution that has\nbeen widely recognised since Braginskii (1965), or the nonlinear \u00110contribution. The ratio of the two terms inside the\ncurly brackets in Eq. (22) is\n3h2\nx\u00110\n\u00112\u00141\u0000h2\nx\n(1\u00002h2x)2+1\n4h2x(1\u0000h2x)\u0015\n\u00193h2\nx\u00110\n\u00112\u00192:4h2\nx(\ni\u001ci)2; (23)\nwhere the \frst step simpli\fes using h2\nx\u001c1 (for the observed value of h2\nx\u001910\u00002, retaining the terms in the square\nbracket increases the ratio by 2.8%, so this approximation is both accurate and conservative) and the substitution for\n\u00110=\u00112is by Eq. (21). For the coronal wave amplitudes and \n i\u001civalues noted in §3.2, this ratio exceeds 109, with\nthe nonlinear damping via \u00110dominating the heating rate by that factor. In other words, the viscous dissipation of\nAlfv\u0013 en waves via derivatives aligned with the equilibrium magnetic \feld is a factor 109stronger than predicted by\nlinear theory.\nWe now evaluate the role of derivatives perpendicular to the equilibrium magnetic \feld by comparing the nonlinear\n\u00110term in Eq. (22) to the term involving @yVx. The ratio of these heating rate terms is\n12h2\nx\u00110\n\u00112\u0012\u0015?\n\u0015jj\u00132\u00141\u0000h2\nx\n1 + 3h2x\u0015\n\u001912h2\nx\u00110\n\u00112\u0012\u0015?\n\u0015jj\u00132\n\u00199:6h2\nx(\ni\u001ci)2\u0012\u0015?\n\u0015jj\u00132\n; (24)\nwhere\u0015?and\u0015jjare the wavelengths perpendicular and parallel to the equilibrium magnetic \feld (for the observed\nvalue ofh2\nx\u001910\u00002, the approximation of the terms in h2\nxis accurate to 3.9%). For the coronal parameters noted\nabove, if\u0015?\u0019\u0015jjthen the nonlinear \u00110term again dominates by a factor that exceeds 109. For smaller transverse\nwavelengths, the nonlinear \u00110term dominates whenever \u0015?&10\u00005\u0015jj. If one considers a wave speed of 400 km s\u00001\nand a frequency of 3 mHz, consistent with the observations by Morton et al. (2015), the condition that the nonlinear \u00110\nterm dominates becomes \u0015?&800 m. Given that CoMP has imaged Alfv\u0013 enic waves using 3 Mm pixels, this condition\nappears to be met by a very large margin, making the nonlinear \u00110dissipation dominant over the \u00112linear dissipation.\nThe purpose of deriving Eq. (22) and the ratios on the left hand sides of Eq. (23) and (24) such that they include\nall appearances of hxandhzis that they can be evaluated exactly for a given value of hx. This makes it explicit that\nour conclusions are insensitive to the precise value of hx, only that the value of hxis broadly consistent with coronal\nobservations. While that approach is most comprehensive, the same conclusions can also be reached by separately\nsimplifying each term in Eq. (22) using h2\nx\u001c1 andh2\nz= 1\u0000h2\nx\u00191 to obtain the less cumbersome formula\nQvisc=\b\n3\u00110h2\nx+\u00112\t\n(@zVx)2+\u00111(@yVx)2; (25)\nand comparing terms to reach the same conclusions.\n3.4. Compressible plasma with large Re\nUnder typical coronal conditions, the thermal pressure is too small to prevent compression of the plasma by nonlinear\nmagnetic pressure forces, thus a nonlinear Vzdevelops that is known as the ponderomotive \row (Hollweg 1971). This\n\row component a\u000bects the viscous heating rate via the parallel viscosity coe\u000ecient \u00110, hence compressible theory is\nrequired for nonlinear viscous damping of Alfv\u0013 en waves in plasma.\nWe de\fne the Alfv\u0013 enic Reynolds number as\nRe =\u001avA\nkjj\u00110: (26)\nThis dimensionless parameter di\u000bers from the traditional Reynolds number since it refers to the Alfv\u0013 en speed vA=\nB=p\u00160\u001ainstead of a typical \ruid velocity. This distinction mirrors that between the Lundquist number and magnetic\nReynolds number in resistive MHD. Justi\fcation for de\fning Re according to Eq. (26) will be found in the detailedNonlinear Alfv \u0013en Waves with Viscosity Tensor 9\nmathematical solutions in §5, in which it is found to be a natural parameter of the system (also see Nocera et al.\n1986).\nIn this section, the ponderomotive Vzwill be related to Vxusing expansions in the amplitude of the primary wave\n\felds. Several assumptions are used to accomplish this. First, we make use of the result that a travelling wave solution\npropagating in the positive z-direction has\n@\n@t\u0011\u0000vA@\n@z; (27)\nwherevAis the wave speed. For simplicity it is assumed that derivatives of background quantities are su\u000eciently weak\nto play a higher-order role on the dynamics. We also simplify here by replacing full treatment of thermal conduction\nwith two thermodynamic cases: adiabatic and isothermal. Finally, it is assumed that Re is large enough that viscous\nforces can be neglected at leading order when evaluating Vz, which makes it possible to obtain an algebraic relationship\nbetweenVzandb2. This assumption will be removed for §5, in which the e\u000bect of viscous forces on VxandVzis\nincluded.\nThex-components of the momentum and induction equations are una\u000bected by the ponderomotive \row at leading\norder in the wave amplitude. From them one recovers the Wal\u0013 en relation for propagating Alfv\u0013 en waves, b=B 0=\u0000Vx=vA.\nAt leading order, the z-component of the momentum equation is\n\u001a0@Vz\n@t+@\n@z\u0012\n\u000ep+b2\n2\u00160\u0013\n= 0; (28)\nwhere the viscous force has been neglected since we currently consider the limit of large Re. Using Eq (27) and\nintegrating yields an algebraic relationship between Vz,\u000epandb2. In an adiabatic treatment, the energy equation\nyields\u000ep=\rp0Vz=vA, hence we obtain\nVz\nvA=1\n2(1\u0000\f)\u0012b\nB0\u00132\n; (29)\nwhere\n\f=\u0012cs\nvA\u00132\n=\r\n2\u0012p\nB2\n0=2\u00160\u0013\n: (30)\nIn an isothermal treatment, the ideal gas law p=\u001aRT yields\u000ep=p 0=\u000e\u001a=\u001a 0=Vz=vA. This does not change the\nform of Eq. (29); instead, the isothermal case is recovered simply by setting \r= 1 in the de\fnition of \f.\nDe\fning\fas the square of the ratio of the sound speed ( cs=p\n\rp0=\u001a0) to the Alfv\u0013 en speed di\u000bers slightly from the\nconvention of de\fning \fas the ratio of thermal pressure to magnetic pressure, due to the factor \r=2, which is 5/6 for\nan adiabatic monoatomic gas, and 1 =2 for an isothermal model. De\fning \fas the speed ratio squared leads to cleaner\nmathematics for many MHD wave problems, including this one, and it has therefore become established practice in\nMHD wave theory.\nThe\f= 1 singularity in Eq. (29) arises because cs=vAimplies resonance between the Alfv\u0013 en wave and an acoustic\nwave, which resonantly transfers energy between the waves. In this speci\fc case, Eq. (27) does not apply because it\ndoes not account for evolution due to resonance. Similarly, Eq. (28) assumes that Vz\u001cvAto simplify the convective\nderivative, and the solution in Eq. (29) does not satisfy this condition in the immediate vicinity of \f= 1. The\f= 1\nresonance and the singularity in Eq. (29) are not of concern for most coronal applications, which typically have \f <0:2,\nbut there are special cases in which it is of interest, such as waves propagating towards coronal magnetic nulls or across\nthe\f= 1 layer in the lower solar atmosphere. Russell et al. (2016) have previously applied such nonlinear resonant\ncoupling to the problem of sunquake generation by magnetic \feld changes during solar \rares.\nDi\u000berentiating Eq. (29) and employing the relation b=B 0=\u0000Vx=vAyields\n@\u000bVz=\u00001\n(1\u0000\f)hx\nhz@\u000bVx; (31)\nwhich can be used to eliminate Vzfrom Eq. (20). To leading order in h2\nxin each viscosity coe\u000ecient, we \fnd\nQvisc=\b\nC\u00110h2\nx+\u00112\t\n(@zVx)2+\u00111(@yVx)2; (32)\nwhere\nC=1\n3\u00121\u00003\f\n1\u0000\f\u00132\n: (33)10 Russell\nSome special cases are noteworthy. The incompressible results of §3.3 are recovered for \f!1 , which gives Vz!0\nandC!3. Similarly, the cold plasma solution is recovered by setting \f= 0, which gives Vz= (b=B 0)2=2 and\nC= 1=3.\nThe nonlinear heating rate for cold plasma ( \f= 0) is a factor nine smaller than for incompressible plasma ( \f!1 ),\nwhich demonstrates the importance of compressibility for this problem. Furthermore, C(\f) is monotonically decreasing\nbetween\f= 0 and\f= 1=3. Since 0< \f < 1=3 for most coronal applications, the dissipation rate due to nonlinear\nBraginskii viscosity in these environments is reduced compared to the cold plasma solution. For example, given \f= 0:1,\nthe heating rate is approximately 60% of the value for cold plasma. It is therefore evident that compressibility and\n\fnite-beta e\u000bects must be treated when assessing viscous dissipation of Alfv\u0013 en waves.\nAnother important feature is that Chas a zero for \f= 1=3. This is one circumstance in which\nVz\nvA=3\n4\u0012Vx\nvA\u00132\n=3\n4\u0012b\nB0\u00132\n; (34)\nwhich causes cancellation within the \u00110contribution to Qvisc. That a particular organisation of Vz=vAcan suppress\nnonlinear viscous dissipation is an important novel \fnding that §4 explores further in the context of low Re.\nThe \fnal feature of Cis the singularity at \f= 1. As noted earlier in this section, cs=vAimplies that the Alfv\u0013 en\nwave is in resonance with a sound wave, which transfers energy between the Alfv\u0013 en wave and the sound wave. Caution\nis needed around the resonance, since resonant energy transfer cannot be described using Eq. (27), which was used to\nderive Eq. (33).\nEvaluation of ratios of heating terms from Eq. (32) proceeds as for the comparison in Sec. 3.3, but with \u00110multiplied\nbyC=3. The top-level conclusions remain intact: heating by the Braginskii viscous stress tensor is dominated by an\n\u00110term that is nonlinear in the wave amplitude, and for coronal values, heating due to the nonlinear \u00110term is many\norders of magnitude larger than the heating due to the linear \u00111and\u00112terms.\n3.5. What wave amplitude is linear?\nAn important implication of the preceding analysis is that nonlinear e\u000bects become signi\fcant for anisotropic viscosity\nat far lower wave amplitudes than they do for other terms in the MHD equations. Linearizing the Braginskii viscous\nstress tensor is only appropriate when h2\nx\u0018(b=B 0)2\u001c(\ni\u001ci)\u00002, which in the corona corresponds to a requirement\nthat the wave energy density is less than 10\u000011times the energy density of the background magnetic \feld, far too\nsmall to be relevant to coronal energetics. Waves that have small enough amplitudes to be governed by linear viscous\ndamping theory would be unobservable and have no e\u000bect on the coronal energy balance. Thus, for coronal Alfv\u0013 en\nwaves, viscosity must be treated nonlinearly in the wave amplitude, as well as anisotropically due to the magnetic \feld.\nInterestingly, this linearization condition is far more stringent than the linearization condition for other terms in the\nMHD equations, whereby h2\nxis normally compared to unity. The extreme di\u000berence in these linearization conditions\nis due to the large \u00110=\u00112ratio produced by the strong magnetization.\n3.6. Damping scales for large Re (energy derivation)\nIt is of major interest to know the time and length scales over which waves damp. This section provides a relatively\nsimple derivation of the decay scales for nonlinear viscous damping of propagating shear Alfv\u0013 en waves for large Re,\nusing energy principles.\nDropping the \u00111and\u00112terms from Eq. (32), the heating rate due to the \u00110parallel viscosity coe\u000ecient for large Re\nis\nQvisc=\u00110\n3\u00121\u00003\f\n1\u0000\f\u00132\u0012b\nB0\u00132\n(@zVx)2: (35)\nA wave energy decay time can be de\fned according to \u001cd=hEwi=hQvisci, whereh:idenotes the time average over\na wave period, and Ewis the wave energy density. The corresponding decay length is Ld=vA\u001cd.\nFor forward propagating Alfv\u0013 en waves, Ew\u0019\u001a0V2\nx, andVx=acos(\u001e) where\u001e=kjj(z\u0000vAt). Hence,\nEw=\u001a0a2(1 + cos(2\u001e))\n2: (36)\nSimilarly, using Eq. (35) with ( b=B 0)2\u0019(Vx=VA)2,\nQvisc=\u00110k2\njj\n3v2\nA\u00121\u00003\f\n1\u0000\f\u00132\na4(1\u0000cos(4\u001e))\n8; (37)Nonlinear Alfv \u0013en Waves with Viscosity Tensor 11\nwhich give the fast time averages\nhEwi=\u001a0a2\n2; (38)\nhQi=\u00110k2\njja4\n24v2\nA\u00121\u00003\f\n1\u0000\f\u00132\n: (39)\nThe wave energy decay scales are therefore\n\u001cd=12\u001a0\n\u00110k2\njj(a=vA)2\u00121\u0000\f\n1\u00003\f\u00132\n; (40)\nLd=12\u001a0vA\n\u00110k2\njj(a=vA)2\u00121\u0000\f\n1\u00003\f\u00132\n: (41)\nEquations (40) and (41) show that waves with larger kjj(equivalently, higher frequencies) are damped on shorter\nscales. We also remark that since Lddepends on the amplitude of Vx(the constant a), the decay envelope is non-\nexponential. The damping properties are elaborated on more fully in §5, in which the assumption of large Re is\nremoved and the functional form of the wave envelope is determined.\n4.SELF-ORGANISED VISCOUS FLOW\nIn the limit Re!0, the viscous force in the z-component of the momentum equation risk becoming extremely large,\nunless the \row self-organises to prevent this. Correspondingly, in the limit Re !0, strong dissipation will prevent\nwaves from propagating, unless Vzis determined by viscosity. One can therefore expect self-organisation of the \row\npattern for Alfv\u0013 en waves in highly viscous plasma (small values of Re), which is a concept previously advanced by\nMontgomery (1992).\nTo investigate quantitatively, we analyze the highly magnetised regime \n i\u001ci\u001d1, simplifying the stress tensor\nand heating rate by retaining only the \u00110parallel viscosity coe\u000ecient. Inspecting Eq. (5), components of \u0019\u000b\fare\nproportional to ( h\u0016h\u0017\u0000\u000e\u0016\u0017=3)@\u0016V\u0017, andQviscis proportional to the square of this expression. Applying the shear\nAlfv\u0013 en wave geometry of Eqs. (8) and (9) and simplifying by h2\nx\u001c1,\n\u0012\nh\u0016h\u0017\u0000\u000e\u0016\u0017\n3\u0013@V\u0017\n@x\u0016\u0019hx@Vx\n@z+2\n3@Vz\n@z: (42)\nViscous forces and heating can be suppressed, allowing Alfv\u0013 en wave propagation, if the \row self-organises to keep this\nexpression close to zero. Using the Alfv\u0013 en wave relation to substitute hx\u0019b=B 0\u0019\u0000Vx=vAand integrating, we \fnd\nthat for small Re\nVz\nvA=3\n4\u0012Vx\nvA\u00132\n=3\n4\u0012b\nB0\u00132\n: (43)\nThe relation speci\fed by Eq. (43) appeared previously in the di\u000berent context of §3.4, where it was seen that viscous\ndissipation of Alfv\u0013 en waves in high Re plasma is suppressed for the special case of \f= 1=3. The \row pattern required\nto produce cancellation within the \u00110part ofQviscis independent of Re and \f, but it occurs for di\u000berent reasons\nin the two cases: in §3.4 it arose as a special case of ponderomotive \row with \fnite \f; when Re is small, it occurs\nbecause of self-organisation through viscous forces.\nThis novel result demonstrates that decay scales and other properties derived in §3 should not be extrapolated to\nsmall Re. Instead, we expect that as Re !0, the viscous force organises the \row such that Vzobeys Eq. (43), for\nwhich dissipation is suppressed by cancellation within the \u00110part ofQvisc.\n5.MULTIPLE SCALE ANALYSIS\nSection 3 used methods of analysis based on heating rates and energy principles. Section 5 now takes a complementary\napproach of solving the full set of governing equations using multiple scale analysis, to reinforce the results of §3,\nextend to general Re by including the e\u000bect of the viscous force on V, and obtain additional results including the\nfunctional form of the nonlinear decay.12 Russell\n5.1. Comparison to Nocera et al. (1986)\nWe preface the multiple scale analysis part of this paper with some remarks about related calculations by Nocera\net al. (1986). Their work and ours both concentrate on \u00110viscosity as the main source of wave damping, treating this\nnonlinearly in the wave amplitude (the horizontal branch of Figure 1). Also in common, both treat ponderomotive\nand \fnite\fe\u000bects.\nThe previous work of Nocera et al. (1986) derived a version of the viscous stress tensor that includes the leading\norder e\u000bect of hx6= 0 in the \u00110term. Terms in the viscosity tensor were then compared, concluding like our §3\n(but by di\u000berent arguments) that the nonlinear \u00110term exceeds contributions from other viscosity coe\u000ecients when\n(b=B 0)2\u001d(\ni\u001ci)\u00002(their Eq. (3.13)). The two studies thus agree on the dominance of nonlinear \u00110viscosity.\nNocera et al. (1986) then found a decay length using the following strategy. A self-consistent perturbation ordering\nwas introduced, then the x-components of the momentum and induction equations were combined to obtain a single\nequation for Vx, which at linear order is a wave equation. Next, all variables apart from Vxwere eliminated from the\nleading-order nonlinear term. Finally, they concluded from a stability analysis that waves with k?= 0 are damped\nnonlinearly, with a decay time that has the same form as our Eq. (40) (their Eq. (5.7), given in terms of normalised\nvariables).\nThe detailed derivation that follows in §5.2 draws inspiration from the framework developed by Nocera et al. (1986).\nWe have also taken the opportunity to make several changes that we regard as improvements, most importantly:\n1. Nocera et al. (1986) assumed that the fast time average of Vzis zero, which necessitated adding a non-zero\nconstant of integration to Vz. By contrast, we will set the constant of integration to zero, which is the only choice\nfor which an Alfv\u0013 en wave driver switching on at one boundary does not unphysically send an instantaneous signal\nto in\fnity. Additional support for our choice comes from simulations of nonlinear longitudinal \rows produced\nby Alfv\u0013 en waves (e.g. McLaughlin et al. 2011), which are consistent with the constraint used in our work.\n2. The stability analysis in §5 Nocera et al. (1986) is replaced with a multiple scale analysis of the type covered in\nChapter 11 of Bender & Orszag (1978).\n3. Nocera et al. (1986) made their wave envelope a function of z+vAt. We treat the envelope as time-independent\nand thus explicitly investigate damping of a propagating wave with respect to distance.\n4. Our derivation provides the envelope of Vxas well as the decay length.\n5. Our solution is valid for general Re, whereas Nocera et al. (1986) solved for the decay scales in the low-viscosity\nlimit of high Re only.\nEqually, Nocera et al. (1986) treated cases that we do not, including the possibility of k?large enough for coupling\nbetween the Alfv\u0013 en and fast modes to alter the wave properties (referred to in their paper as the case of phase mixed\nwaves).\n5.2. Detailed solution\n5.2.1. Geometry and perturbations\nWe assume the Alfv\u0013 en wave geometry of Eqs. (8) and (9), set @=@y\u00110 to concentrate on waves without short\nperpendicular scales, and introduce density and pressure perturbations \u000e\u001aand\u000eptogether with a self-consistent\nperturbation ordering that has Vx=vA\u0018b=B 0\u0018\u000f1=2andVz=vA\u0018\u000e\u001a=\u001a 0\u0018\u000ep=p 0\u0018\u000f. The viscosity \u00110and\nbackground quantities B0,\u001a0andp0are treated as locally homogeneous for simplicity.\n5.2.2. Nonlinear wave equation\nStarting from the ideal induction equation,\n@B\n@t=r\u0002(V\u0002B); (44)\nwe have@b\n@t\u0000B0@Vx\n@z=\u0000@\n@z(bVz) (exact); (45)\nwhere the linear terms have been grouped on the left hand side and the nonlinear term on the right hand side.Nonlinear Alfv \u0013en Waves with Viscosity Tensor 13\nThe momentum equation is\n\u001a\u0012@V\u000b\n@t+ (V\u0001r)V\u000b\u0013\n=\u0000@\n@x\u000b\u0012\np+B2\n2\u00160\u0013\n+1\n\u00160(B\u0001r)B\u000b\u0000@\u0019\u000b\f\n@x\f: (46)\nWhen\u00110contributions dominate the viscous force, Eqs. (13) and (17) give\n\u0000\u0019xz= 3\u00110b\nB0\u0012b\nB0@Vx\n@z+2\n3@Vz\n@z\u0013\n+O(\u000f5=2) (47)\nso thex-component of Eq. (46) becomes\n@Vx\n@t\u0000B0\n\u00160\u001a0@b\n@z=\u0000\u000e\u001a\n\u001a0@Vx\n@t\u0000Vz@Vx\n@z+3\u00110\n\u001a0@\n@z\u0012b\nB0\u0014b\nB0@Vx\n@z+2\n3@Vz\n@z\u0015\u0013\n+O(\u000f5=2); (48)\nwhere linear terms and nonlinear terms have again been placed on opposite sides of the equation.\nTaking the time derivative of Eq. (48) and using Eq. (45) to eliminate bfrom the linear terms,\n\u0012@2\n@t2\u0000v2\nA@2\n@z2\u0013\nVx=\u0000v2\nA@2\n@z2\u0012b\nB0Vz\u0013\n\u0000@\n@t\u0012\u000e\u001a\n\u001a0@Vx\n@t+Vz@Vx\n@z\u0013\n+3\u00110\n\u001a0@2\n@t@z\u0012b\nB0\u0014b\nB0@Vx\n@z+2\n3@Vz\n@z\u0015\u0013\n+O(\u000f5=2):\n(49)\nInterpreting Eq. (49), the linear terms (on the left hand side) correspond to a wave equation with wave speed vA. The\nleading nonlinear terms (those shown explicitly on the right hand side) include the leading-order e\u000bect of the anisotropic\nviscosity, which enters at the same order as the leading nonlinear terms that appear in perturbative nonlinear theory\nof ideal Alfv\u0013 en waves.\nNext, we eliminate band\u000e\u001afrom theO(\u000f3=2) nonlinear terms in Eq. (49). Equations (45) and (48) are solved at\nlinear order by the Alfv\u0013 en wave relation\nb\nB0=\u0006Vx\nvA+O(\u000f3=2): (50)\nWe choose the negative sign so waves travel in the positive zdirection, giving\nb\nB0=\u0000Vx\nvA+O(\u000f3=2): (51)\nThe travelling wave behaviour of the linear solution together with assumption that the wave envelope changes over a\ndistance controlled by the leading order nonlinear terms in Eq. (49) allows replacement\n@\n@t=\u0000vA@\n@z+O(\u000f): (52)\nThe density perturbation is governed by the mass continuity equation\n@\u001a\n@t+r\u0001(\u001aV) = 0; (53)\nwhich gives for our shear Alfv\u0013 en wave\n@\u000e\u001a\n@t=\u0000\u001a0@Vz\n@z+O(\u000f2): (54)\nThen, using Eq. (52) and integrating,\n\u000e\u001a\n\u001a0=Vz\nvA+O(\u000f2): (55)\nThe constant of integration has been set to zero, for reasons discussed in §5.1.\nUsing these results, Eq. (49) becomes\n\u0012@2\n@t2\u0000v2\nA@2\n@z2\u0013\nVx=vA@2\n@z2(VxVz) +3\u00110\n\u001a0@2\n@t@z\u0012Vx\nvA\u0014Vx\nvA@Vx\n@z\u00002\n3@Vz\n@z\u0015\u0013\n+O(\u000f5=2): (56)14 Russell\nNow that the problem has been reduced to the two variables VxandVz, it is convenient to make the \u000fdependence\nexplicit by introducing dimensionless variables vandwde\fned by\nVx(z;t) =\u000f1=2vAv(z;t); Vz(z;t) =\u000fvAw(z;t): (57)\nExpressing Eq. (56) in the dimensionless variables vandw, and dropping the non-explicit higher order terms from the\nright hand side, we seek solutions to\n\u0012@2\n@t2\u0000v2\nA@2\n@z2\u0013\nv=\u000f\u0012\nv2\nA@2\n@z2(vw) +\u00110\n\u001a0@2\n@t@z\u0012@v3\n@z\u00002v@w\n@z\u0013\u0013\n: (58)\n5.2.3. Multiple scale analysis\nEquation (58) is now solved using multiple scale analysis (e.g. Bender & Orszag 1978). Applying this technique, one\nintroduces a new variable Z=\u000fzthat de\fnes a long length scale, and the perturbation expansions\nv(z;t) =v0(z;Z;t ) +\u000fv1(z;Z;t ) +::: (59)\nw(z;t) =w0(z;Z;t ) +\u000fw1(z;Z;t ) +:::: (60)\nDerivatives are treated using the chain rule as though zandZare independent variables and setting d Z=dz=\u000f.\nThus,\n@v\n@z=@v0\n@z+\u000f\u0012@v0\n@Z+@v1\n@z\u0013\n+O(\u000f2); (61)\n@2v\n@z2=@2v0\n@z2+\u000f\u0012\n2@2v0\n@Z@z+@2v1\n@z2\u0013\n+O(\u000f2); (62)\nwith equivalent expressions for derivatives of w.\nSubstituting into Eq. (58), collecting \u000f0terms, and thus solving the homogeneous wave equation\n\u0012@2\n@t2\u0000v2\nA@2\n@z2\u0013\nv0= 0; (63)\nobtains d'Alembert's solution\nv0(z;Z;t ) =f(z\u0000vAt;Z) +g(z+vAt;Z): (64)\nFor forward propagating waves, the function gis zero.\nThe corresponding w0is obtained by integrating the z-component of the momentum equation, Eq. (46), which gives\nVz\nvA=1\n\u001a0v2\nA\u0012\n\u000ep+b2\n2\u00160+\u0019zz\u0013\n+O(\u000f2): (65)\nFrom Eqs. (13) and (17), we have\n\u0000\u0019zz= 2\u00110\u0012b\nB0@Vx\n@z+2\n3@Vz\n@z\u0013\n+O(\u000f2): (66)\nA substitution for the pressure perturbation \u000epis obtained by integrating the energy equation,\n@p\n@t+V\u0001rp+\rpr\u0001V= (\r\u00001)Qvisc: (67)\nThe viscous heating Qviscis of orderO(\u000f2), so integration gives the adiabatic relation\n\u000ep\np0=\rVz\nvA+O(\u000f2): (68)\nAlternatively, one can consider isothermal conditions using \u000ep=p 0=Vz=vAfrom the ideal gas law, which is recovered\nfrom Eq. (68) by setting \r= 1.Nonlinear Alfv \u0013en Waves with Viscosity Tensor 15\nUsing Eqs. (66) and (68), and eliminating bterms using Eq. (51), Eq. (65) can be expressed as\n\u0012\n1\u0000\f+4\n3\u00110\n\u001a0vA@\n@z\u0013\nw=\u00121\n2+\u00110\n\u001a0vA@\n@z\u0013\nv2; (69)\nwhere\f= (cs=vA)2and dropped terms are O(\u000f). Whenvandware expanded according to Eqs. (59) and (60), an\nequation identical to (69) connects w0andv0.\nInspecting Eq. (69), it is evident that obtaining wfor a known vin general requires solving a \frst order linear partial\ndi\u000berential equation. In the limit where the viscous terms can be neglected, the problem simpli\fes to the algebraic\nw=v2=2(1\u0000\f) relation used in Sec. 3.4. Similarly, when the viscous terms dominate, one obtains the w= (3=4)v2\nrelation for viscously self-organised parallel \row discussed in Sec. 3.4. For the detailed solution in this section, we\nretain the complete set of forces that determine Vz, solving the full Eq. (69).\nSolution is facilitated by considering the special case where v0oscillates sinusoidally in time. For the rest of this\nderivation we therefore set\nv0=A(Z)ei\u001e+A\u0003(Z)e\u0000i\u001e; \u001e =kjj(z\u0000vAt); (70)\nwhereA(Z)2Cand\u0003denotes the complex conjugate. Representing Ain polar form,\nA(Z) =R(Z)ei\u0012(Z); (71)\nEq. (70) is equivalent to\nv0(z;Z;t ) = 2R(Z) cos(kjj(z\u0000vAt) +\u0012(Z)): (72)\nFrom inspection, 2 R(Z) is the local amplitude and \u0012(Z) is a phase shift. We have found the complex form in Eq. (70)\nmore convenient to work with in the following.\nWe now solve for the corresponding w0. Noting that\nv2\n0=A2e2i\u001e+ 2jAj2+A\u00032e\u00002i\u001e; (73)\nwherejAj2=AA\u0003, we seek a solution of the form\nw0=De2i\u001e+D\u0003e\u00002i\u001e+ ^w0: (74)\nSubstituting into Eq. (69), terms in e0give\n^w0=jAj2\n1\u0000\f; (75)\nwhile terms in e2i\u001eand e\u00002i\u001eindependently give\nD=\u000bA2; \u000b =1\n2(1 + 4ikjj\u00110=\u001a0vA)\n(1\u0000\f+ (8=3)ikjj\u00110=\u001a0vA): (76)\nThe solution for w0can also be expressed without complex numbers. Making explicit the real and imaginary parts\nof\u000b=\u000br+i\u000bi, we have the real constants\n\u000br=1\n2(1\u0000\f+ (32=3)(Re)\u00002)\n((1\u0000\f)2+ (64=9)(Re)\u00002); (77)\n\u000bi=2\n3(1\u00003\f)(Re)\u00001\n((1\u0000\f)2+ (64=9)(Re)\u00002); (78)\nwhere Re = \u001a0vA=kjj\u00110consistent with Eq. (26). It is then easily shown that\nw0\n2R(Z)2=\u000brcos(2(kjj(z\u0000vAt) +\u0012(Z)))\u0000\u000bisin(2(kjj(z\u0000vAt) +\u0012(Z))) +1\n2(1\u0000\f): (79)\nTo deduceR(Z) and\u0012(Z), we return to analysing Eq. (58). The \u000f1terms in Eq. (58) give the inhomogeneous partial\ndi\u000berential equation\n\u0012@2\n@t2\u0000v2\nA@2\n@z2\u0013\nv1= 2v2\nA@2v0\n@Z@z+v2\nA@2\n@z2(v0w0) +\u00110\n\u001a0@2\n@t@z\u0012@v3\n0\n@z\u00002v0@w0\n@z\u0013\n: (80)16 Russell\nThev0andw0terms drive v1, and the solution for v1will have a secular contribution (i.e. one or more terms that grow\nrelative to corresponding solutions of the homogeneous equation) if terms on the right hand side resonate with the\nsolution to the undriven wave equation. In the speci\fc case where v0is given by Eq. (70), secular terms in the solution\nforv1will restrict the domain for which v0is a valid approximation if the right hand side of Eq. (80) contains ei\u001eor\ne\u0000i\u001eterms. The central idea in multiple scale analysis is to solve for the A(Z) that makes the resonance disappear,\nmakingv0a durable approximation for v.\nUsing Eqs. (70) and (74){(76), ei\u001eterms vanish from the right hand side of Eq. (80) if and only if\n1\nAdA\ndZ=\u0000kjjjAj2\n2\u0014\ni\u0012\n\u000b+1\n1\u0000\f\u0013\n+kjj\u00110\n\u001a0vA(3\u00004\u000b)\u0015\n: (81)\nThe same condition also removes the e\u0000i\u001eterms.\nChanging to polar form, Eq. (71) implies\n1\nAdA\ndZ=1\nRdR\ndZ+id\u0012\ndZ: (82)\nHence, the real and imaginary parts of Eq. (81) yield the real ordinary di\u000berential equations\ndR\ndZ=\u0000\u00141\n2R3; (83)\nd\u0012\ndZ=\u00142R2; (84)\nwhere\n\u00141=kjj\u0012kjj\u00110\n\u001a0vA(3\u00004\u000br)\u0000\u000bi\u0013\n; (85)\n\u00142=\u0000kjj\n2\u0012\n\u000br+1\n1\u0000\f\u0000kjj\u00110\n\u001a0vA4\u000bi\u0013\n: (86)\nEqs. (83) and (84) govern the local amplitude and phase drift of the Alfv\u0013 en wave respectively ( c.f.Eq. (72)).\nOur main interest is in R(Z), which determines how the waves decay. Equation (83) is a separable \frst order\ndi\u000berential equation. The solution is\nR(Z) =R(0)p\n1 +\u00141R(0)2Z: (87)\nFor\u00141>0 the wave envelope decays non-exponentially, over a damping length that is inversely proportional to the\nsquare of the initial wave amplitude. Having obtained R(Z), the solution for \u0012(Z) is obtained by directly integrating\nEq. (84). Using Eq. (87),\n\u0012(Z) =\u0012(0) +\u00142\n\u00141ln\f\f1 +\u00141R(0)2Z\f\f: (88)\n5.2.4. Solution in original variables\nHaving ensured corrections to v\u0019v0remain of order \u000f\u0018(b=B 0)2\u001c1, the multiple scale analysis is concluded by\nusingv0as the approximation for v. Returning to the original variables,\nVx(z;t) =a0p\n1 +z=Ldcos\u0000\nkjj(z\u0000vAt) + (\u00142=\u00141) lnj1 +z=Ldj+\u00120\u0001\n; (89)\nwherea0is the amplitude of Vx(0;t),\u00120sets the initial phase of the wave (at z= 0,t= 0) and\nLd=4\n\u00141(a0=vA)2(90)\nis the decay length. Using Eqs. (77), (78) and (85),\n\u00141=kjj\n3Re(1\u00003\f)2\n(1\u0000\f)2+ (64=9)(Re)\u00002; (91)Nonlinear Alfv \u0013en Waves with Viscosity Tensor 17\nwhere Re is the Reynolds number for the wave, as de\fned by Eq. (26). Thus,\nLd=12Re\nkjj(a0=vA)2(1\u0000\f)2+ (64=9)(Re)\u00002\n(1\u00003\f)2: (92)\nIf one neglects the Re\u00002term in the numerator of Eq. (92), then Ldagrees exactly with the formula in Eq. (41) that\nwe derived from energy principles. The formula for Ldin Eq. (92) is more general since it was derived without direct\nassumptions about the value of Re = kjj\u00110=\u001a0vA, although the multiple scale analysis requires that the combination\nof parameters kjj,a2\n0and Re are such that waves damp over a signi\fcantly longer scale than the wavelength.\n5.3. Non-exponential decay and interpretation of damping length\nAs a general principle, the Alfv\u0013 en wave energy density Ew=\u001aV2\nxdecays more rapidly than the perturbation Vx, due\nto the quadratic power. For exponential decay this is re\rected in a factor two di\u000berence in the respective e\u00001decay\nlengths. For the non-exponential decay produced by nonlinear viscous damping, the situation is handled di\u000berently.\nThe sameLddescribesVxandEw, however they have di\u000berent functional forms. The velocity amplitude decays as\n(1 +z=Ld)\u00001=2(see Eq. (89)), while the wave energy density decays as (1 + z=Ld)\u00001. Therefore, over a distance Ld,\nthe velocity amplitude reduces by a factorp\n2 and the energy density halves.\n5.4. Inclusion of thermal conduction\nThe multiple scale analysis can also be modi\fed to include explicit thermal conduction. Since thermal conduction\nis highly anisotropic, we include the parallel thermal conduction, setting the heat \row vector to\nq=\u0000Kjj(h\u0001rT)h; (93)\nwhereKjjis the coe\u000ecient of parallel thermal conduction, and temperature T=p=\u001aRwhereRis the gas constant.\nThe energy equation with heat \row is\n@p\n@t+V\u0001rp+\rpr\u0001V= (\r\u00001)(Qvisc\u0000r\u0001 q); (94)\nwhich replaces Eq. (67).\nIt follows that \u000ep=p 0is related to vz=vAby the partial di\u000berential equation\n\u0012\n1 + \u0003@\n@z\u0013\u000ep\np0=\u0012\n\r+ \u0003@\n@z\u0013Vz\nvA+O(\u000f2): (95)\nwhere\n\u0003 =(\r\u00001)Kjj\n\u001a0RvA(96)\nis a conductive length scale. In the limit of weak thermal conduction, \u0003 !0 gives\u000ep=p 0=\rVz=vA, recovering the\nadiabatic case treated above. Similarly, for strong thermal conduction, \u0003 !1 gives\u000ep=p 0=Vz=vA, recovering the\nisothermal case.\nIntroducing a dimensionless pressure variable cde\fned by\u000ep=\u000fp0c(z;t), expanding c(z;t) =c0(z;Z;t )+\u000fc1(z;Z;t )+\n:::and setting c0=Ce2i\u001e+C\u0003e2i\u001e+ ^c0, terms in e0in Eq. (95) yield ^ c0=\r^w0, and terms in e2i\u001eyieldC= \u0000D,\nwhere\n\u0000 =\r+ 2ikjj\u0003\n1 + 2ikjj\u0003: (97)\nSolving further, an equation D=\u000bA2analogous to Eq. (76) is obtained but with \freplaced by the complex-valued\n\u0000p0=\u001a0v2\nAin the formula for \u000b. Meanwhile, (75) and (81) are unchanged, retaining the real-valued \f=\rp0=\u001a0v2\nA. The\nwave amplitude is therefore governed by results identical to Eqs. (83) and (85), with the aforementioned change in the\nde\fnition of \u000b.18 Russell\n6.DISCUSSION\n6.1. Optimum damping\nInspecting Eq. (92), the formula for Ldkjjhas a minimum with respect to the Alfv\u0013 enic Reynolds number at Re =\n8=(3j1\u0000\fj). Thus, shear Alfv\u0013 en waves with kjj= 3\u001a0vAj1\u0000\fj=3\u00110are damped in the fewest number of wavelengths,\nwhich we refer to as optimum damping. The optimally damped waves have\nLd\n\u0015jj=32\n\u0019j1\u0000\fj\n(a0=vA)2(1\u00003\f)2: (98)\nWhen\f\u001c1, the right hand side of Eq. (98) is approximately ten divided by the square of the normalised wave\namplitude. Hence, while nonlinear viscous damping can in principle damp Alfv\u0013 en waves in a small number of wave-\nlengths, this requires large amplitudes a=vA\u00181, or\f\u00181. For a more typically encountered amplitudes a=vA\u001810\u00001\nand low\f, one \fndsLd=\u0015jj&1000, making nonlinear viscous damping negligible for many coronal situations.\n6.2. Viscous self-organisation\nThe suppression of nonlinear viscous damping for small Re (highly viscous plasma) does not mean that viscous\ne\u000bects are unimportant in this regime. On the contrary, nonlinear damping is suppressed for small R ebecause viscous\nforces organise the parallel \row associated with the Alfv\u0013 en wave to approach the relationship Vz=vA= (3=4)(Vx=vA)2.\nThis modi\fcation of the parallel \row plays a crucial role in avoiding signi\fcant nonlinear damping in highly viscous\nplasma, when modelled using Braginskii MHD.\n6.3. Validity constraints\nThroughout this paper, we have assumed that \f6= 1 to avoid resonant wave coupling. This condition holds\nthroughout most of the corona, so it is appropriate for our primary applications. Additionally, the multiple scale\nanalysis in §5 uses\u000f\u0018(Vx=vA)2\u0018(b=B 0)2as a small parameter, one consequence of which is that the nonlinearly\ndamping occurs over a distance considerably greater than the parallel wavelength. As noted in §3.2, transverse coronal\nwaves are observed in open-\feld regions with \u000f\u001810\u00002, making weakly nonlinear theory appropriate for such situations.\nObtaining nonlinear viscous solutions in the resonant and strongly nonlinear regimes nonetheless remain interesting\nfuture challenges for plasma theory.\nApplicability of this paper's results to physical problems is also constrained to conditions under which Braginskii\nMHD can be rigorously applied. As discussed in §2, the traditional derivation of Braginskii MHD assumes that the\ncollisional mean free path is less than the macroscopic scales. Comparing the mean free path parallel to the magnetic\n\feld to the parallel wavelength, this condition can be given as kjj\u0015mfp<1, where\u0015mfp=vTi\u001ci,vTi=p\nkBT=miand\n\u001ciis the ion collision time. Using the formula (Braginskii 1958, 1965; Hollweg 1985)\n\u00110= 0:96nkBT\u001ci; (99)\nand the de\fnition of Re in Eq. (26), one can show that\nkjj\u0015mfp<1,\f1=2Re =\u001acs\nkjj\u00110>1: (100)\nIn other words, Braginskii MHD requires that the Reynolds number based on the sound speed is greater than unity. One\nshould therefore be cautious about applying small Alfv\u0013 enic Reynolds number results such as viscous self-organisation\nto real low- \fplasmas.\n6.4. Formulas for applications\nFor applications to real plasmas, the following formulas are convenient. In cases where the parallel viscosity coe\u000ecient\nis determined by Coulomb collisions,\n\u00110= 0:96nkBT\u001ci=22\n\u0015C\u000210\u000017T5=2; (101)\nwhere this formula is stated in S.I. units with Tin kelvin, and \u0015Cis the Coulomb logarithm (e.g. Hollweg 1985). The\nReynolds number de\fned in Eq. (26) can then be expressed as\nRe = 5:8\u00021020\u0015CB2f\u00001T\u00005=2; (102)Nonlinear Alfv \u0013en Waves with Viscosity Tensor 19\nalso in S.I. units, where f=vAkjj=2\u0019is the wave frequency. This formula makes explicit the dependences on\nfrequency, magnetic \feld strength and temperature. The Alfv\u0013 enic Reynolds number is smallest when the plasma has\nhigh temperature and low magnetic \feld strength, and for higher frequency waves. Finally, we express the damping\nlength in Eq. (92) as a function of frequency and the mean square velocity\nV2\nx\u000b\n=a2\n0=2, which gives\nLd=3\n\u0019v3\nARe\nfhV2xi(1\u0000\f)2+ (64=9)(Re)\u00002\n(1\u00003\f)2: (103)\n6.5. Waves in a coronal open-\feld region\nOutgoing transverse waves in the magnetically open solar corona contain su\u000ecient energy to heat the open corona and\naccelerate the fast solar wind (McIntosh et al. 2011; Morton et al. 2015), and they are observed to damp signi\fcantly\nwithin a solar radius above the Sun's surface (Bemporad & Abbo 2012; Hahn et al. 2012; Hahn & Savin 2013; Hahn\net al. 2022). Heating at these altitudes is also thought to be important for producing the observed rapid acceleration of\nthe fast solar wind (Habbal et al. 1995; McKenzie et al. 1995). The problem of how the outgoing waves damp has not\nbeen conclusively solved, although one leading hypothesis is turbulent cascade driven by interactions with downgoing\nAlfv\u0013 en waves (Hollweg 1986; Heyvaerts & Priest 1992; Matthaeus et al. 1999; Cranmer et al. 2007; Verdini et al. 2010;\nMiki\u0013 c et al. 2018) produced either by re\rection from density inhomogeneities (van Ballegooijen & Asgari-Targhi 2016;\nPascoe et al. 2022) or by parametric decay instability (Galeev & Oraevskii 1963; Derby 1978; Goldstein 1978; Shoda\net al. 2019; Hahn et al. 2022).\nHere, we demonstrate that Braginskii viscosity does not cause signi\fcant damping of Alfv\u0013 en waves at the altitudes\nat which the traditional derivation of Braginskii MHD holds. For concreteness, we consider the Sun's northern polar\nopen-\feld region on 27 March 2012, using observational values reported by Morton et al. (2015). Enhanced wave\npower was present around f= 5 mHz, which suggests Alfv\u0013 enic waves produced by p-modes (Morton et al. 2019). We\nwill calculate damping lengths for this frequency, noting that Re and Lddepend on f, withLd\u0018f\u00002in the limit\nof high Re. Morton et al. (2015) inferred that the Alfv\u0013 en speed was nearly constant with vA= 400 km s\u00001on their\ndomain ofr= 1:05 to 1:20R\f. For temperature, we set T= 1:6\u0002106K, the formation temperature of the Fe XIII\nlines used by the CoMP instrument, which implies the proton thermal speed VTi=p\nkBT=miis 115 km s\u00001. Hence,\nin for an isothermal equation of state \f= 0:083 and\f1=2= 0:29. For the wave velocity amplitude, Morton et al.\n(2015) recommended that the reported non-thermal line width should be used, which varies with altitude.\nStarting with lowest altitude observed by Morton et al. (2015), r= 1:05R\f, we setn= 1014m\u00003,B= 2\u000210\u00004T\nand take the rms value of Vxas 35 km s\u00001. We therefore \fnd \u0015C= 19 and Re = 28. Since \f1=2Re = 8>1, Braginksii\nMHD applies and we evaluate Ld= 4:2\u0002108km\u0011600R\f.\nAtr= 1:20, we set n= 1013m\u00003,B= 6\u000210\u00005T and take the rms value of Vxas 50 km s\u00001. The observed\nparameters therefore give \u0015C= 21 and Re = 2 :7. Since\f1=2Re = 0:8\u00191, this altitude is close to the maximum at\nwhich the assumptions by which Braginskii MHD is traditionally derived remains valid (for this particular open \feld\nregion, and assuming Eq. (101)). Evaluating the damping length here returns Ld= 4:2\u0002107km\u001161R\f.\nWe conclude that Braginskii viscosity does not cause signi\fcant wave damping below r= 1:2R\f, which is consistent\nwith observational results that Alfv\u0013 enic wave amplitudes in coronal holes follow ideal WKB scaling out to around this\naltitude (Cranmer & van Ballegooijen 2005; Hahn & Savin 2013).\nBetween the altitudes we have examined, Ldreduces by an order of magnitude. If one were to extrapolate using high\nRe or incompressible results, it would appear that viscous damping becomes important near the altitudes at which the\nwaves are observed to damp. We are cautious about making such an assertion for two reasons. First, as discussed in\n§6.1, our results show that for Re <8=(3j1\u0000\f)) the damping length in a Braginskii MHD model increases again as the\n\feld-aligned \row self organises to supress viscous damping. Secondly, as the plasma becomes increasingly collisionless\n(\f1=2Re<1) the traditional derivation of Bragniskii MHD falters.\nIntriguingly, it may be signi\fcant that the onset of wave damping broadly coincides with the altitude at which\nBraginskii MHD can no longer be con\fdently applied if one invokes the \u00110expression for Coulomb collisions given in\nEq. (101). This correspondence is suggestive that the wave damping observed in coronal holes may involve collisionless\nand heat \row e\u000bects not found in the most common \ruid models.\n6.6. Future work\nThe present types of analyses should be extended in future to other types of propagating transverse MHD waves.\nThe nonlinear longitudinal \row that accompanies propagating torsional Alfv\u0013 en waves di\u000bers from its counterpart for20 Russell\npropagating shear Alfv\u0013 en waves (Vasheghani Farahani et al. 2011) and it will be of interest to investigate how this\ndi\u000berence a\u000bects the nonlinear viscous damping. It is similarly desirable to determine how nonlinear viscosity a\u000bects\npropagating kink waves (Edwin & Roberts 1983).\nFor propagating shear Alfv\u0013 en waves, viscous damping appears most promising near the cs=vAsingularity, which\nmust be treated using di\u000berent methods to those used in this paper. The solar wind frequently has \f\u00181, while\f= 1\noccurs in the lower solar atmosphere and in the vicinity of coronal nulls points. Hence, this case is of considerable\nphysical interest. One challenge for application to magnetic nulls is that the magnetic \feld unit vector h=B=jBj\nis not de\fned at the null itself, so one must be careful to evaluate the Braginskii stress tensor using appropriate\ncalculations, e.g. see recent discussion by MacTaggart et al. (2017).\nA further challenge is to develop a theory of nonlinear viscous damping applicable to strongly nonlinear waves with\namplitudes b\u0018B0and greater. The results of the multiple-scale analysis in §5 are rigorous only for the weakly\nnonlinear case, in which \u000f\u0018(b=B 0)2can be treated as a small parameter and it is assumed that the damping length is\nsigni\fcantly longer than the wavelength. Strongly nonlinear Alfv\u0013 en waves with b\u0018B0are a feature of the solar wind,\nand while the low collisionality of the solar wind means that Braginskii MHD may not be an appropriate framework\nfor that application, extending the current work to strongly nonlinear waves remains an interesting problem.\nThere is a diverse collection of MHD wave problems beyond wave damping for which viscous e\u000bects are likely\nto be signi\fcant. Prime among these are nonlinear phenomena involving Alfv\u0013 en waves, for which the nonlinear\nviscosity tensor enters the equations at the same order as the e\u000bect of interest. For example, standing Alfv\u0013 en waves\ndrive signi\fcantly stronger \feld-aligned \rows than occur for propagating waves because standing Alfv\u0013 en waves create\ninhomogeneous time-averaged magnetic pressure. There could also be signi\fcant value in investigating how viscosity\nmodi\fes wave interactions, including Alfv\u0013 en wave collisions and parametric decay instability (Galeev & Oraevskii 1963;\nDerby 1978; Goldstein 1978), which are central to leading hypotheses of wave heating in the magnetically open solar\ncorona.\nFinally, we point to the continuing need for basic plasma physics research to provide increasingly rigorous derivation\nand validation of the appropriate \ruid equations for weakly collisional and collisionless plasma, in the face of the\nclosure problem summarised in §2. As discussed in §2 and 6.3, Braginskii MHD breaks down at higher altitudes\nin the corona as the plasma becomes increasingly collisionless (see Eqs. (100) and (102)). The CGL double-adiabatic\nequations and other models that evolve the stress tensor may provide a more suitable framework in these conditions.\nHunana et al. (2019a,b, 2022) provide recent discussions of such models and their limitations. Alternatively, it may\nbe necessary for the solar waves community to more widely adopt non-\ruid plasma models. However, tractability of\nkinetic models remains a limiting factor, especially in light of the large separations between kinetic and macroscopic\nscales that are characteristic of the Sun's corona. Eloquent comments on these matters can be found in Montgomery\n(1996).\n7.CONCLUSIONS\nThis paper has investigated the properties of propagating shear Alfv\u0013 en waves subject to the nonlinear e\u000bects of the\nBraginskii viscous stress tensor. The main points are as follows:\n1. For many plasma environments, including the low-altitude solar corona, Braginskii MHD provides a more accurate\ndescription of plasma than classical MHD does, by rigorously treating the stress tensor and thermal conduction.\nStress tensor e\u000bects nonetheless remain relatively unexplored for many solar MHD phenomena.\n2. The dominant viscous e\u000bects for propagating shear Alfv\u0013 en waves are nonlinear in the wave amplitude and occur\nthrough the \\parallel\" viscosity coe\u000ecient, \u00110. Theoretical results based on linearizing the stress tensor with\nrespect to the wave amplitude are only valid for amplitudes satisfying ( b=B 0)2\u001c(\ni\u001ci)\u00002. Such waves would\nbe energetically insigni\fcant under normal coronal conditions, hence nonlinear treatment is required.\n3. Compressibility and pressure a\u000bect the nonlinear \feld-aligned \row associated with shear Alfv\u0013 en waves, hence\nthey impact nonlinear wave damping. Both must be included to produce accurate coronal results.\n4. Braginskii viscosity damps propagating shear Alfv\u0013 en waves nonlinearly, such that the primary wave \felds band\nVxdecay as (1 + z=Ld)\u00001=2, where the decay length\nLd=12Re\nkjj(a0=vA)2(1\u0000\f)2+ (64=9)(Re)\u00002\n(1\u00003\f)2:Nonlinear Alfv \u0013en Waves with Viscosity Tensor 21\nHere,a0is the initial velocity amplitude of the wave, \f= (cs=vA)2and Re =\u001avA=kjj\u00110is the Alfv\u0013 enic Reynolds\nnumber of the wave. The energy density decays as (1 + z=Ld)\u00001.\n5. Optimal damping (the minimum normalised damping length kjjLd) is obtained when Re = 8 =(3j1\u0000\fj). For low\n\fplasma and ( a0=vA).10\u00001, one \fnds Ld=\u0015jj&1000, indicating that nonlinear viscous damping is negligible\nfor many coronal situations.\n6. The asymptotic behaviour that Ld!1 in the highly viscous regime Re !0 is attributed to self-organisation\nof the parallel \row by viscous forces such that Vz=vA\u0019(3=4)(Vx=vA)2, which suppresses dissipation.\n7. Applicability of the Braginskii MHD solutions to real plasmas is constrained by the traditional derivation of\nBraginskii MHD assuming that kjj\u0015mfp<1 which is equivalent to \f1=2Re =\u001acs=kjj\u00110>1. In other words,\nBraginskii MHD requires that the Reynolds number based on the sound speed is greater than unity. We therefore\nrecommend that only the damping results for large Alfv\u0013 enic Reynolds number should be applied to real coronal\nplasma, using the simpli\fed formula Ld= 12Re(1\u0000\f)2=(kjj(a0=vA)2(1\u00003\f)2)) that has been derived in this\npaper by two di\u000berent techniques: energy principles and multiple scale analysis.\n8. Application to transverse waves observed in a polar open-\feld region concludes that nonlinear Braginskii viscosity\ndoes not cause signi\fcant damping of the waves at the altitudes at which the assumptions by which Braginskii\nMHD is traditionally derived remain valid ( r.1:2R\ffor the considered region and wave properties). Intrigu-\ningly, the observed onset of wave damping broadly coincides with the altitude at which Braginskii MHD can no\nlonger be con\fdently applied if one invokes the \u00110expression for Coulomb collisions given in Eq. (101).\nThis work was prompted by and bene\fted from conversations with Paola Testa, Bart De Pontieu, Vanessa Polito,\nGraham Kerr, Mark Cheung, Wei Liu, David Graham, Joel Allred, Mats Carlsson, Iain Hannah and Fabio Reale\nduring a research visit to LMSAL funded by ESA's support for the IRIS mission (August 2018) and meetings of\nInternational Space Science Institute (Bern) International Team 355 (November 2018). I am grateful to Peter Cargill,\nAndrew Wright, Bart De Pontieu and Paola Testa for comments on an early draft (June 2019), and Declan Diver for\nencouragment to explore connections with pressure anisotropy. I thank the reviewer for considered and constructive\nsuggestions, and several unnamed individuals for comments that also improved the manuscript. 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Moreover, when the damping\nand the coupling coefficients are sufficiently large, the globa l random attractor\nis a one-dimensional random horizontal curve regardless of the strength of\nthe noises, and the system has a rotation number, which impli es that the\noscillators in the system tend to oscillate with the same fre quency eventually\nandthereforethesocalled frequencylockingissuccessful . Theresultsobtained\nin this paper generalize many existing results on the asympt otic dynamics for\na single second order noisy oscillator to systems of coupled second order noisy\noscillators. They show that coupled damped second order osc illators with\nlarge damping have similar asymptotic dynamics as the limit ing coupled first\norder oscillators as the damping goes to infinite and also tha t coupled damped\nsecondorderoscillators havesimilarasymptoticdynamics astheirproperspace\ncontinuous counterparts, which are of great practical impo rtance.\nKeywords : Coupled second order oscillators; white noises; random at tractor;\nrandom horizontal curve; rotation number; frequency locki ng\nAMS Subject Classification : 60H10, 34F05, 37H10.\n1 Introduction\nThis paper is devoted to the study of the asymptotic dynamics of the following system of second\norder oscillators driven by additive noises:\nd˙uj+αduj+K(Au)jdt+βg(uj)dt=fjdt+ǫjdWj, (1.1)\nwherej∈Zd\nN:={j= (j1,...,jd)∈Zd: 1≤j1,...,jd≤N},ujis a scalar unknown function of\ntandu= (u1,u2,···,uNd)⊤,αandKare positive constants, Ais anNd×Ndmatrix and ( Au)j\nstands for the jth component of the vector Au,β∈R,gis a periodic function, fjandǫjare\n1The first author is partially supported by NSF grant DMS-0907 752\n2The third author is supported by National Natural Science Fo undation of China under Grant 10771139, and\nthe Innovation Program of Shanghai Municipal Education Com mission under Grant 08ZZ70\n1constants, and {Wj(t)}j∈Zd\nNare independent two-sided real-valued Wiener processes. M oreover,\nAandgsatisfy\n(HA)Ais anNd×Ndnonnegative definite symmetric matrix with eigenvalues den oted byλi,\ni= 0,1,...,Nd−1satisfying that\n0 =λ0< λ1≤ ··· ≤λNd−1,\nλ0is algebraically simple, and (1,...,1)⊤∈RNdis an eigenvector corresponding to λ0= 0.\n(HG)g∈C1(R,R)has the following properties\ng(x+κ) =g(x),|g(x)| ≤c1,|g′(x)| ≤c2,∀x∈R,\nwherec1>0,c2>0andκ >0is the smallest positive period of g.\nSystem (1.1) appears in many applied problems including Jos ephson junction arrays and\ncoupled pendula (see [11], [13], [22], [27], etc.). Physica lly,αin (1.1) represents the damping of\nthe system and Kis the coupling coefficient of the system. (1.1) then represen ts a system of\nNdcoupled damped oscillators independently driven by white n oises.\nSystem (1.1) also arises from various spatial discretizati ons of certain damped hyperbolic\npartial differential equations. For example, the Nd×NdmatrixAin (1.1) includes the dis-\ncretization of negative Laplace operator −∆ with Neumann or periodic boundary conditions\ndefined as follows:\n(Au)j= (Au)(j1,j2,...,jd)=1\nh2/bracketleftbig\n2duj−u(j1+1,j2,...,jd)−u(j1,j2+1,...,jd)−···−u(j1,j2,...,jd+1)\n−u(j1−1,j2,...,jd)−u(j1,j2−1,...,jd)−···−u(j1,j2,...,jd−1)/bracketrightbig\n,\nwith Neumann boundary condition\nu(j1,...,ji−1,0,ji+1,...,jd)=u(j1,...,ji−1,1,ji+1,...,jd),\nu(j1,...,ji−1,N+1,ji+1,...,jd)=u(j1,...,ji−1,N,ji+1,...,jd)\nor periodic boundary condition\nu(j1,...,ji−1,0,ji+1,...,jd)=u(j1,...,ji−1,N,ji+1,...,jd),\nu(j1,...,ji−1,N+1,ji+1,...,jd)=u(j1,...,ji−1,1,ji+1,...,jd)\nforj= (j1,...,jd)∈Zd\nNandi= 1,...,d. Thus, (1.1) with uj=u(j1h,···,jih,···,jdh)\n(h=L/N),Abeing as above, fj=f,ǫj=ǫ, andWj=Wis a spatial discretization of the\nfollowing problem\nd˙u+αdu−K∆udt+βg(u)dt=fdt+ǫdW,inU×R+(1.2)\nwith Neumann boundary condition or periodic boundary condi tion, i.e.,\n∂u\n∂n= 0 on ∂U×R+\nor\nu|Γj=u|Γj+d,∂u\n∂xj/vextendsingle/vextendsingle/vextendsingle\nΓj=∂u\n∂xj/vextendsingle/vextendsingle/vextendsingle\nΓj+d, j= 1,...,d,\n2where Γ j=∂U∩{xj= 0}, Γj+d=∂U∩{xj=L},j= 1,...,dandU=/producttextd\ni=1(0,L). Note that\nifg(u) = sinu, (1.2) is the so called damped sine-Gordon equation, which i s used to model, for\ninstance, the dynamics of a continuous family of junctions ( see [25]).\nTwo of the main dynamical aspects about coupled oscillators and damped wave equations\nconsidered in the literature are the existence and structur e of global attractors and the phe-\nnomenon of frequency locking. A large amount of research has been carried out toward these\ntwo aspects for a variety of systems related to (1.1). See for example, [17, 18, 20, 22, 28] for\nthe study of coupled oscillators with constant or periodic e xternal forces; [12, 19, 21, 25, 26]\nfor the study of the deterministic damped sine-Gordon equat ion; [8, 16, 23] for the study of\ncoupled oscillators driven by white noises; and [10, 24, 29] for the study of stochastic damped\nsine-Gordon equation. Many of the existing works focus on th eexistence of global attractors and\nthe estimate of the dimension of the global attractors. In [8 , 23, 24], the existence and structure\nof random attractors of stochastic oscillators and stochas tic damped wave equations are studied.\nIn particular, the asymptotic dynamics of a single second or der noisy oscillator, i.e., (1.1) with\nN= 1, is studied in [23]. The author of [23] proved the existenc e of a random attractor which is\na family of horizontal curves and the existence of a rotation number which implies the frequency\nlocking. In [8], the authors considered a class of coupled fir st order oscillators driven by white\nnoises. Among those, the existence of a one-dimensional ran dom attractor and the existence of\na rotation number are proved in [8]. The system of coupled firs t order oscillators considered in\n[8] is of the form\nduj+K(Au)jdt+βg(uj)dt=fjdt+ǫdWj, j∈Zd\nN. (1.3)\nNote that, by resealing the time variable by t→t\nα, (1.1) becomes\n1\nαd˙uj+duj+K(Au)jdt+βg(uj)dt=fjdt+ǫdWj, j∈Zd\nN. (1.4)\nHence, (1.3) can be formally viewed as the limiting system of (1.1) as the damping coefficient α\ngoes toinfinite. In[24], theauthorsinvestigated theexist ence andstructureof randomattractors\nof damped sine-Gordon equations of the form (1.2) with Neuma nn boundary condition, which\nis a space continuous counterpart of (1.1) as mentioned abov e.\nHowever, many important dynamical aspects including the ex istence of global attractor and\nthe occurrence of frequency locking have been hardly studie d for coupled second order oscillators\nof the form (1.1) driven by white noises. It is of great intere st to investigate the extent to\nwhich the existing results on asymptotic dynamics of a singl e second order noisy oscillator may\nbe generalized to systems of coupled second order noisy osci llators. Thanks to the relations\nbetween (1.1) and (1.3) and between (1.1) and (1.2), it is als o of great interest to explore the\nsimilarity and difference between the dynamics of coupled dam ped second order oscillators and\nits limiting coupled first order oscillators as the damping c oefficient goes to infinite and between\nthe dynamics of coupled damped second order oscillators and their proper space continuous\ncounterparts. The objective of this paper is to carry out a st udy along this line. In particular,\nwe study the asymptotic or global dynamics of (1.1), includi ng the existence and structure of\nglobal attractor in proper phase space and the success of fre quency locking.\nIn order to do so, as usual, we first change (1.1) to some system of coupled first order random\nequations. Assume N≥2 andd≥1 (N= 1reduces tothesingle noisy oscillator case considered\nin [23]). Let u= (uj)j∈Zd\nN,g(u) = (g(uj))j∈Zd\nN,f= (fj)j∈Zd\nN,W(t) = (ǫjWj(t))j∈Zd\nN. Then,\n3(1.1) can be written as the following matrix form,\nd˙u+αdu+KAudt+βg(u)dt=fdt+dW(t). (1.5)\nLet\nΩj= Ω0={ω0∈C(R,R) :ω0(0) = 0}\nequipped with the compact open topology, Fj=B(Ω0) be the Borel σ-algebra of Ω 0andPjbe\nthe corresponding Wiener measure for j∈Zd\nN. Let Ω =/producttext\nj∈Zd\nNΩj,Fbe the product σ-algebra\non Ω and Pbe the induced product Wiener measure. Define ( θt)t∈Ron Ω via\nθtω(·) =ω(·+t)−ω(t), t∈R.\nThen, (Ω ,F,P,(θt)t∈R) is an ergodic metric dynamical system (see [1]). Consider t he Ornstein-\nUhlenbeck equation,\ndz+zdt=dW(t), z∈RNd. (1.6)\nLetz(θtω) = (zj(θtω))j∈Zd\nNbe the unique stationary solution of (1.6) (see [1, 2, 9] for t he\nexistence and various properties of z(·)). Letv= ˙u−z(θtω). We obtain the following equivalent\nsystem of (1.5),/braceleftBigg\n˙u=v+z(θtω),\n˙v=−KAu−αv+f−βg(u)+(1−α)z(θtω).(1.7)\nTo study the global dynamics of (1.1), it is therefore equiva lent to study the global dynamics\nof (1.7). Observe that the natural phase space for (1.7) is E:=RNd×RNdwith the standard\nEuclidean norm. Thanks to the presence of the damping, it is e xpected that (1.7) possesses a\nglobal attractor in certain sense. However, due to the uncon trolled component of the solutions\nalong the direction of the eigenvectors of the linear operat or in the right of (1.7) corresponding\nto the zero eigenvalue, there is no bounded attracting sets i nEwith the standard Euclidean\nnorm, which will lead to nontrivial dynamics. There is also s ome additional difficulty if one\nstudies (1.7) in Ewith the standard Euclidean norm due to the zero limit of some eigenvalues\nof the linear operator in the right of (1.7) as α→ ∞. The later difficulty does not appear for\ncoupled first order oscillators studied in [8] and for a singl e noisy oscillator considered in [23].\nWe will overcome the difficulty by using some equivalent norm o nEand considering (1.7) in\nsome proper quotient space of Eand prove the existence of a global random attractor as well\nas the existence of a rotation number of (1.7).\nTo be more precise, let\nC=/parenleftbigg0I\n−KA−αI/parenrightbigg\n. (1.8)\nBy simple matrix analysis, the eigenvalues of Care given by (see [14, 20] for example)\nµ±\ni=−α±√\nα2−4Kλi\n2, i= 0,1,...,Nd−1. (1.9)\nNote that µ+\n0= 0, which requires some special consideration for the solut ions along the direction\nof the eigenvector η0= (1,...,1,0,...,0)⊤corresponding to µ+\n0. We overcome this difficulty by\nconsidering (1.7) in the cylindrical space E1/κη0Z×E2, whereE1= span{η0},E2is the space\n4spanned by all the eigenvectors corresponding to non-zero e igenvalues of C(see section 4 for\ndetails). We then prove\n(1)For any α >0andK >0, system (1.1)possesses a global random attractor (which is\nunbounded along the one-dimensional space E1and bounded along the one-codimensional space\nE2) (see Theorem 4.2, Corollary 4.3 and Remark 4.4).\nIt is expected physically that when the damping coefficient α→ ∞, the dynamics of (1.7)\nbecomes simpler or the structure of the global attractor of ( 1.7) becomes simpler. However,\nµ+\ni→0 asα→ ∞fori= 1,2,···,Nd−1, which gives rise to some difficulty for studying\nthe structure of the global attractor in Ewith the standard Euclidean norm. We introduce\nan equivalent norm on Eto overcome this difficulty (see section 3 for the introductio n of the\nequivalent norm, the choice of such equivalent norm was first discovered in [15]) and prove\n(2)WhenαandKare sufficiently large, the global random attractor of (1.1)is a one-dimensional\nrandom horizontal curve (see Theorem 5.3 and Corollary 5.4), a nd the rotation number (see Def-\ninition 6.1) of (1.1)exists (see Theorem 6.3 and Corollary 6.4).\nNote that roughly a real number ρ∈Ris called the rotation number of (1.1) or (1.7) if for\nany solution {uj(t)}j∈Zd\nNof (1.1), the limit lim t→∞uj(t)\ntexists almost surely for any j∈Zd\nNand\nlim\nt→∞uj(t)\nt=ρfora.e. ω∈Ω and j= 1,2,···,Nd\n(see Definition 6.1 and the remark after Definition 6.1). Henc e if (1.1) has a rotation number,\nthen the oscillators in the system tend to oscillate with the same frequency eventually and\ntherefore the so called frequency locking is successful.\n(1) and (2) above are the main results of the paper. They make a n important contribution\nto the understanding of coupled second order oscillators dr iven by noises. Property (1) shows\nthat system (1.1) is dissipative along the one-codimension al space E2. By property (2), the\nasymptotic dynamics of (1.1) with sufficiently large αandKis one dimensional regardless of\nthe strength of noise. Property (2) also shows that all the so lutions of (1.1) tend to oscillate\nwith the same frequency eventually almost surely and hence f requency locking is successful in\n(1.1) provided that αandKare sufficiently large.\nThe results obtained in this paper generalize many existing results on the asymptotic dynam-\nics for a single damped noisy oscillator to systems of couple d damped noisy oscillators. They\nshow that coupled damped second order oscillators with larg e damping have similar asymptotic\ndynamics as the limiting coupled first order oscillators as t he damping goes to infinite and hence\none may use coupled first order oscillators to analyze qualit ative properties of coupled second\norder oscillators with large damping, which is of great prac tical importance. They also show\nthat coupled damped second order oscillators have similar a symptotic dynamics as their proper\nspace continuous counterparts and hence one may use finitely many coupled oscillators to study\nqualitative properties of damped wave equations, which is o f great practical importance too.\nThe rest of the paper is organized as follows. In section 2, we present some basic concepts and\nproperties for general random dynamical systems. In sectio n 3, we provide some basic settings\nabout (1.1) and show that it generates a random dynamical sys tem. We prove in section 4 the\nexistence of a global random attractor of the random dynamic al system φgenerated by (1.1) for\nanyα >0 andK >0. We show in section 5 that the global random attractor of φis a random\nhorizontal curve and show in section 6 that (1.1) has a rotati on number, respectively, provided\nthatαandKare sufficiently large.\n52 Random Dynamical Systems\nIn this section, we collect some basic knowledge about gener al random dynamical system (see\n[1, 4] for details). Let ( X,d) be a complete and separable metric space with Borel σ-algebra\nB(X).\nDefinition 2.1. Acontinuous random dynamical system over (Ω ,F,P,(θt)t∈R)is a(B(R+)×\nF ×B(X),B(X))-measurable mapping\nϕ:R+×Ω×X→X,(t,ω,x)/ma√sto→ϕ(t,ω,x)\nsuch that the following properties hold:\n(1)ϕ(0,ω,x) =xfor allω∈Ω;\n(2)ϕ(t+s,ω,·) =ϕ(t,θsω,ϕ(s,ω,·))for alls,t≥0andω∈Ω;\n(3)ϕ(t,ω,x)is continuous in xfor every t≥0andω∈Ω.\nFor given x∈XandE,F⊂X, we define\nd(x,F) = inf\ny∈Fd(x,y)\nand\ndH(E,F) = sup\nx∈Ed(x,F).\ndH(E,F) is called the Hausdorff semi-distance fromEtoF.\nDefinition 2.2. (1) A set-valued mapping ω/ma√sto→D(ω) : Ω→2Xis said to be a random set\nif the mapping ω/ma√sto→d(x,D(ω))is measurable for any x∈X. Ifω/ma√sto→d(x,D(ω))is\nmeasurable for any x∈XandD(ω)is closed (compact) for each ω∈Ω, thenω/ma√sto→D(ω)\nis called a random closed (compact) set . A random set ω/ma√sto→D(ω)is said to be bounded\nif there exist x0∈Xand a random variable R(ω)>0such that\nD(ω)⊂ {x∈X:d(x,x0)≤R(ω)}for allω∈Ω.\n(2) A random set ω/ma√sto→D(ω)is called tempered provided that for some x0∈XandP-a.e.\nω∈Ω,\nlim\nt→∞e−βtsup{d(b,x0) :b∈D(θ−tω)}= 0for allβ >0.\n(3) A random set ω/ma√sto→B(ω)is said to be a random absorbing set if for any tempered random\nsetω/ma√sto→D(ω), there exists t0(ω)such that\nϕ(t,θ−tω,D(θ−tω))⊂B(ω)for allt≥t0(ω), ω∈Ω.\n(4) A random set ω/ma√sto→B1(ω)is said to be a random attracting set if for any tempered random\nsetω/ma√sto→D(ω), we have\nlim\nt→∞dH(ϕ(t,θ−tω,D(θ−tω),B1(ω)) = 0for allω∈Ω.\n(5) A random compact set ω/ma√sto→A(ω)is said to be a global random attractor if it is a random\nattracting set and ϕ(t,ω,A(ω)) =A(θtω)for allω∈Ωandt≥0.\n6Theorem 2.3. Letϕbe a continuous random dynamical system over (Ω,F,P,(θt)t∈R). If there\nis a random compact attracting set ω/ma√sto→B(ω)ofϕ, thenω/ma√sto→A(ω)is a global random attractor\nofϕ, where\nA(ω) =/intersectiondisplay\nt>0/uniondisplay\nτ≥tϕ(τ,θ−τω,B(θ−τω)), ω∈Ω.\nProof.See [1, 4].\n3 Basic Settings\nIn this section, we give some basic settings about (1.1) and s how that it generates a random\ndynamical system.\nFirst, let Y= (u,v)⊤andF(θtω,Y) = (z(θtω),f−βg(u) +(1−α)z(θtω))⊤. System (1.7)\ncan then be written as\n˙Y=CY+F(θtω,Y), (3.1)\nwhereCis as in (1.8).\nRecall that z(θtω) = (zj(θtω))j∈Zd\nNis the unique stationary solution of (1.6). Note that the\nrandom variable |zj(ω)|is tempered and the mapping t/ma√sto→zj(θtω) isP-a.s. continuous (see\n[1, 2]). More precisely, there is a θt-invariant ˜Ω⊂Ω withP(˜Ω) = 1 such that t/ma√sto→zj(θtω) is\ncontinuous for ω∈˜Ω andj∈Zd\nN. We will consider (1.7) or (3.1) for ω∈˜Ω and write ˜Ω as Ω\nfrom now on.\nLetE=RNd×RNdandFω(t,Y) :=F(θtω,Y), thenFω(·,·) :R×E→Eis continuous\nintand globally Lipschitz continuous in Yfor each ω∈Ω. By classical theory of ordinary\ndifferential equations concerning existence and uniqueness of solutions, for each ω∈Ω and any\nY0∈E, (3.1) has a uniqueness solution Y(t,ω,Y0),t≥0, satisfying\nY(t,ω,Y0) =eCtY0+/integraldisplayt\n0eC(t−s)F(θsω,Y(s,ω,Y0))ds, t≥0. (3.2)\nMoreover, it follows from [1] that Y(t,ω,Y0) is measurable in ( t,ω,Y0). Hence (3.1) generates a\ncontinuous random dynamical system on E,\nY:R+×Ω×E→E,(t,ω,Y0)/ma√sto→Y(t,ω,Y0). (3.3)\nDefine a mapping φ:R+×Ω×E→Eby\nφ(t,ω,φ0) =Y(t,ω,Y0(ω))+(0,z(θtω))⊤, (3.4)\nwhereφ0= (u0,u1)⊤∈EandY0(ω) = (u0,u1−z(ω))⊤. Then φis a continuous random\ndynamical system associated with the problem (1.1) on E.\nRecall that the eigenvalues of Care given by (see [14, 20] for example)\nµ±\ni=−α±√\nα2−4Kλi\n2, i= 0,1,...,Nd−1. (3.5)\nBy (3.5), Chas at least two real eigenvalues 0 and −αwith eigenvalues η0= (1,...,1,0,...,0)⊤,\nη−1= (1,...,1,−α,...,−α)⊤∈E, respectively. Let E1= span{η0},E−1= span{η−1},E11=\nE1+E−1andE22=E⊥\n11, the orthogonal complement space of E11inE, thenE=E11⊕E22.\n7To control the unboundedness of solutions in the direction o fη0, we will study (3.1) in the\ncylindrical space E1/κη0Z×E2, whereE2=E−1⊕E22(see Section 4 for details).\nObserve that the Lipschitz constant of Fwith respect to YinEwith the standard Euclidean\nnorm is independent of α >0. Butµ+\ni→0 asα→ ∞fori≥1, which gives rise to some\ndifficulty for the investigation of (3.1) in Ewith the standard Euclidean norm. To overcome the\ndifficulty, we introduce a new norm which is equivalent to the s tandard Euclidean norm on E.\nHere, we only collect some results about the new norm (see [15 , 20] for details).\nDefine two bilinear forms on E11andE22, respectively. For Yi= (ui,vi)⊤∈E11,i= 1,2, let\n/an}bracketle{tY1,Y2/an}bracketri}htE11=α2\n4/an}bracketle{tu1,u2/an}bracketri}ht+/an}bracketle{tα\n2u1+v1,α\n2u2+v2/an}bracketri}ht, (3.6)\nwhere/an}bracketle{t·,·/an}bracketri}htdenotes the inner product on RNd, and for Yi= (ui,vi)⊤∈E22, i= 1,2, let\n/an}bracketle{tY1,Y2/an}bracketri}htE22=/an}bracketle{tKAu1,u2/an}bracketri}ht+(α2\n4−δKλ1)/an}bracketle{tu1,u2/an}bracketri}ht+/an}bracketle{tα\n2u1+v1,α\n2u2+v2/an}bracketri}ht,(3.7)\nwhereδ∈(0,1]. It is easy to check that the Poincar´ e-type inequality\n/an}bracketle{tAu,u/an}bracketri}ht ≥λ1/bardblu/bardbl2,∀Y= (u,v)⊤∈E22\nholds (see [20] for example), where /bardbl·/bardblis the standard Euclidean norm on RNd. Thus (3.7) is\npositive definite. For any Yi=Y(1)\ni+Y(2)\ni∈E,i= 1,2, where Y(1)\n1,Y(1)\n2∈E11,Y(2)\n1,Y(2)\n2∈E22,\nwe define\n/an}bracketle{tY1,Y2/an}bracketri}htE=/an}bracketle{tY(1)\n1,Y(1)\n2/an}bracketri}htE11+/an}bracketle{tY(2)\n1,Y(2)\n2/an}bracketri}htE22. (3.8)\nLemma 3.1 ([20]).(1)(3.6)and(3.7)define inner products on E11andE22, respectively.\n(2)(3.8)defines an inner product on E, and the corresponding norm /bardbl·/bardblEis equivalent to the\nstandard Euclidean norm on E.\n(3) In terms of the inner product /an}bracketle{t·,·/an}bracketri}htE,E1andE11are orthogonal to E−1andE22, respec-\ntively.\n(4) In terms of the norm /bardbl·/bardblE, the Lipschitz constant LFofFwith respect to Ysatisfies\nLF=2c2|β|\nα, (3.9)\nwherec2is as in(HG).\nNote that E2is orthogonal to E1andE=E1⊕E2. Denote by PandQ(=I−P) the\nprojections from EintoE1andE2, respectively. Set\na=α\n2−/vextendsingle/vextendsingle/vextendsingleα\n2−δKλ1\nα/vextendsingle/vextendsingle/vextendsingle. (3.10)\nLemma 3.2. (1) For any Y∈E2,/an}bracketle{tCY,Y/an}bracketri}htE≤ −a/bardblY/bardbl2\nE.\n(2)/bardbleCtQ/bardblE≤e−atfort≥0.\n(3)eCtPY=PYforY∈E,t≥0.\n8Proof.(1) and (2) follow from similar arguments as in Lemma 2.3 and C orollary 2.4 in [19].\nLet us show (3). For Y∈E, sincePY∈E1andd\ndteCtPY=eCtCPY= 0, we have eCtPY=\neC0PY=PY.\nBy Lemma 3.2 (2), the constant ain (3.10) describes the exponential decay rate of eCt|QE\nin the new norm. By Lemma 3.1 (4), LFtends to 0 as α→ ∞with respect to the new norm,\nwhich essentially helps to overcome the difficulty induced fr om the fact that µ+\ni→0 asα→0\nfori≥1.\nThe following lemma will be needed to take care unboundednes s of the solutions along the\ndirection of the eigenvectors corresponding to µ+\n0.\nLemma 3.3. Letp0=κη0∈E(κis the smallest positive period of g). The random dynamical\nsystemYdefined in (3.3)isp0-translation invariant in the sense that\nY(t,ω,Y0+p0) =Y(t,ω,Y0)+p0, t≥0, ω∈Ω, Y0∈E.\nProof.SinceCp0= 0 and F(t,ω,Y) isp0-periodic in Y,Y(t,ω,Y0) +p0is a solution of (3.1)\nwith initial data Y0+p0. Thus,Y(t,ω,Y0)+p0=Y(t,ω,Y0+p0).\nBy (3.3) and Lemma 3.3, φis alsop0-translation invariant.\nLemma 3.4. For any ǫ >0, there is tempered random variable ˜r(ω)>0such that\n/bardblz(θtω)/bardbl ≤eǫ|t|˜r(ω)for allt∈R, ω∈Ω, (3.11)\nwhere˜r(ω)satisfies\ne−ǫ|t|˜r(ω)≤˜r(θtω)≤eǫ|t|˜r(ω), t∈R, ω∈Ω. (3.12)\nProof.Forj∈Zd\nN, since|zj(ω)|is a tempered random variable and the mapping t/ma√sto→ln|zj(θtω)|\nisP-a.s. continuous, itfollowfromProposition4.3.3in[1]th atforany ǫj>0thereisantempered\nrandom variable rj(ω)>0 such that\n1\nrj(ω)≤ |zj(ω)| ≤rj(ω),\nwhererj(ω) satisfies, for P-a.e.ω∈Ω,\ne−ǫj|t|rj(ω)≤rj(θtω)≤eǫj|t|rj(ω), t∈R. (3.13)\nLetr(ω) = (rj(ω))j∈Zd\nN,ω∈Ω and take ǫj=ǫ,j∈Zd\nN, then we have\n/bardblz(θtω)/bardbl ≤/parenleftBigg/summationdisplay\nj∈Zd\nNe2ǫ|t|r2\nj(ω)/parenrightBigg1\n2\n=eǫ|t|/bardblr(ω)/bardbl, t∈R, ω∈Ω.\nLet ˜r(ω) =/bardblr(ω)/bardbl,ω∈Ω. Then (3.11) is satisfied and (3.12) is trivial from (3.13).\n94 Existence of Random Attractor\nIn this section, we study the existence of a random attractor . We assume that p0=κη0∈E1\nandδ∈(0,1] is such that a >0, where ais as in (3.10). We remark in the end of this section\nthat such δalways exists.\nBy Lemma 3.3 and the fact that Chas a zero eigenvalue, we will define a random dynamical\nsystemYon some cylindrical space induced from the random dynamical systemYonE. Then\nby properties of Yrestricted on E2, we can prove the existence of a global random attractor of\nY. Thus, we can say that Yhas a global random attractor which is unbounded along E1and\nbounded along E2. Now, we define Y.\nLetT1=E1/p0ZandE=T1×E2, wherep0Z={kp0:k∈Z}. ForY0∈E, letY0:=\nY0(modp0), whichisan element of E. Notethat, byLemma3.3, Y(t,ω,Y0+kp0) =Y(t,ω,Y0)+\nkp0,∀k∈Zfort≥0,ω∈Ω andY0∈E. With this, we define Y:R+×Ω×E→Eby setting\nY(t,ω,Y0) =Y(t,ω,Y0) (modp0), (4.1)\nwhereY0=Y0(modp0). ThenY:R+×Ω×E→Eis a random dynamical system. Similarly,\nthe random dynamical system φdefined in (3.4) also induces a random dynamical system Φon\nE. By (3.3), (3.4) and (4.1), Φis defined by\nΦ(t,ω,Φ0) =Y(t,ω,Y0)+ ˜z(θtω) (modp0), t≥0, ω∈Ω, (4.2)\nwhereΦ0=φ0(modp0), ˜z(θtω) = (0,z(θtω))⊤andY0=Φ0−˜z(ω) (modp0).\nRecall that PandQ(=I−P) are the projections from EintoE1andE2, respectively.\nDefinition 4.1. Letω∈ΩandR: Ω→R+be a random variable. A random pseudo-ball\nω/ma√sto→B(ω) inEwith random radius ω/ma√sto→R(ω)is a set of the form\nω/ma√sto→B(ω) ={b∈E:/bardblQb/bardblE≤R(ω)}.\nFurthermore, a random set ω/ma√sto→B(ω)∈Eis called pseudo-tempered provided that ω/ma√sto→QB(ω)\nis a tempered random set in E, i.e., for P-a.e.ω∈Ω,\nlim\nt→∞e−βtsup{/bardblQb/bardblE:b∈B(θ−tω)}= 0for allβ >0.\nClearly, any random pseudo-ball ω/ma√sto→B(ω) inEhas the form ω/ma√sto→E1×QB(ω), where\nω/ma√sto→QB(ω)isarandomball in E2. Thenthemeasurability of ω/ma√sto→B(ω)istrivial. ByDefinition\n4.1, ifω/ma√sto→B(ω) is a random pseudo-ball in E, thenω/ma√sto→B(ω) (modp0) is random bounded\nset inE. And if ω/ma√sto→B(ω) is a pseudo-tempered random set in E, thenω/ma√sto→B(ω) (modp0) is\na tempered random set in E.\nWe next show the existence of a global random attractor of the induced random dynamical\nsystemYdefined in (4.1).\nTheorem 4.2. Letα >0andK >0. Then the induced random dynamical system Ydefined\nin(4.1)has a global random attractor ω/ma√sto→A0(ω).\nProof.Forω∈Ω, we obtain from (3.2) that\nY(t,ω,Y0(ω)) =eCtY0(ω)+/integraldisplayt\n0eC(t−s)F(θsω,Y(s,ω,Y0(ω)))ds. (4.3)\n10The projection of (4.3) on E2is\nQY(t,ω,Y0(ω)) =eCtQY0(ω)+/integraldisplayt\n0eC(t−s)QF(θsω,Y(s,ω,Y0(ω)))ds. (4.4)\nBy replacing ωbyθ−tω, it follows from (4.4) that\nQY(t,θ−tω,Y0(θ−tω)) =eCtQY0(θ−tω)+/integraldisplayt\n0eC(t−s)QF(θs−tω,Y(s,θ−tω,Y0(θ−tω)))ds.\nIt then follows from Lemma 3.2 and Q2=Qthat\n/bardblQY(t,θ−tω,Y0(θ−tω))/bardblE\n≤e−at/bardblQY0(θ−tω)/bardblE+/integraldisplayt\n0e−a(t−s)/bardblF(θs−tω,Y(s,θ−tω,Y0(θ−tω)))/bardblEds.(4.5)\nBy Lemma 3.4 with ǫ=a\n2and the equivalence of /bardbl·/bardblEand/bardbl·/bardblonE, there is a M1>0 such\nthat\n/bardblF(θs−tω,Y(s,θ−tω,Y0(θ−tω)))/bardblE\n≤M1/bardblF(θs−tω,Y(s,θ−tω,Y0(θ−tω)))/bardbl\n=M1/parenleftBig\n/bardblz(θs−tω)/bardbl2+/bardblf−βg(Yu(s,θ−tω))+(1−α)z(θs−tω)/bardbl2/parenrightBig1\n2\n≤M1/parenleftBig\n(3α2−6α+4)/bardblz(θs−tω)/bardbl2+3/bardblf/bardbl2+3β2c2\n1Nd/parenrightBig1\n2\n≤a1ea\n2(t−s)˜r(ω)+a2,\nwhereYusatisfies Y(s,θ−tω,Y0(θ−tω)) = (Yu(s,θ−tω),Yv(s,θ−tω))⊤,a1=M1√\n3α2−6α+4\nanda2=M1/radicalbig\n3/bardblf/bardbl2+3β2c2\n1Nd. We then find from (4.5) that\n/bardblQY(t,θ−tω,Y0(θ−tω))/bardblE≤e−at/bardblQY0(θ−tω)/bardblE+2a1\na(1−e−a\n2t)˜r(ω)+a2\na(1−e−at).\nNow for ω∈Ω, define\nR0(ω) =4a1\na˜r(ω)+2a2\na.\nThen, for any pseudo-tempered random set ω/ma√sto→B(ω) inEand any Y0(θ−tω)∈B(θ−tω), there\nis aTB(ω)>0 such that for t≥TB(ω),\n/bardblQY(t,θ−tω,Y0(θ−tω))/bardblE≤R0(ω), ω∈Ω,\nwhich implies\nY(t,θ−tω,B(θ−tω))⊂B0(ω) for all t≥TB(ω), ω∈Ω,\nwhereω/ma√sto→B0(ω) is the random pseudo-ball centered at origin with random ra diusω/ma√sto→R0(ω).\nNote that ω/ma√sto→R0(ω) is a random variable since ω/ma√sto→˜r(ω) is a random variable, then the\nmeasurability of random pseudo-ball ω/ma√sto→B0(ω) is trivial from Definition 4.1.\nForω∈Ω, letB(ω) =B(ω) (modp0) andB0(ω) =B0(ω) (modp0), we then have\nY(t,θ−tω,B(θ−tω))⊂B0(ω) for all t≥TB(ω), ω∈Ω,\n11whereTB(ω) =TB(ω) forω∈Ω, i.e.,ω/ma√sto→B0(ω) is the random absorbing set of Y. Moreover,\nω/ma√sto→B0(ω) is bounded and closed, hence compact in E, it then follows from Theorem 2.3 that\nYhas a global random attractor ω/ma√sto→A0(ω), where\nA0(ω) =/intersectiondisplay\nt>0/uniondisplay\nτ≥tY(τ,θ−τω,B0(θ−τω)), ω∈Ω.\nThis completes the proof.\nCorollary 4.3. Letα >0andK >0. Then the induced random dynamical system Φdefined\nin(4.2)has a global random attractor ω/ma√sto→A(ω), whereA(ω) =A0(ω)+ ˜z(ω) (modp0)for all\nω∈Ω.\nProof.It follows from (4.2) and Theorem 4.2.\nRemark 4.4. (1) For any α >0andK >0, there is a δ∈(0,1]such that α2>2δKλ1which\nimpliesa >0, whereais as in(3.10)andλ1is the smallest positive eigenvalue of A.\n(2) We say that the random dynamical system Y(orφ) has a global random attractor in the\nsense that the induced random dynamical system Y(orΦ) has a global random attractor,\nand we will say that Y(orφ) has a global random attractor directly in the sequel. We\ndenote the global random attractor of Yandφbyω/ma√sto→A0(ω)andω/ma√sto→A(ω)respectively.\nIndeed,ω/ma√sto→A0(ω)andω/ma√sto→A(ω)satisfy\nA0(ω) =A0(ω) (modp0),A(ω) =A(ω) (modp0), ω∈Ω.\nHence a global random attractor of Y(orφ) is unbounded along the one-dimensional space\nE1and bounded along the one-codimensional space E2.\n(3) Observe the global attractors of many dissipative syste ms related to (1.1)is one-dimensional\n(see [8, 17, 18, 19, 20, 21, 23, 24, 26]). Similarly, we expect that the random attractor\nω/ma√sto→A(ω)ofφis one-dimensional for each ω∈Ωprovided that αis sufficiently large. We\nprove that this is true in the next section.\n(4) By (2), the system (1.1)is dissipative along E2(i.e. it possesses a global random attractor\nwhich is bounded along E2). In section 6, we will show that (1.1)with sufficiently large\nαandKalso has a rotation number and hence all the solutions tend to oscillate with the\nsame frequency eventually.\n5 One-dimensional Random Attractor\nIn this section, we apply the invariant and inertial manifol d theory, in particular, the theory\nestablishedin[7]toshowthat therandomattractor ofthera ndomdynamicalsystem φgenerated\nby (1.1) is one-dimensional (more precisely, is a horizonta l curve) provided that αandKare\nsufficiently large (see Remark 4.4 (2) for the random attracto r). This method has been applied\nby Chow, Shenand Zhou[8] to systems of coupled firstorder noi sy oscillators and by Shen, Zhou\nand Shen [24] to the stochastic damped sine-Gordon equation . The reader is referred to [3, 5]\nfor the theory and application of inertial manifold theory f or stochastic evolution equations.\nAssume that p0=κη0anda >4LF(see (3.10) for aand (3.9) for LF). Note that the\ncondition a >4LFindicates that the exponential decay rate of eCt|QEin the norm /bardbl · /bardblEis\n12larger than four times the Lipschitz constant of Fin the norm /bardbl·/bardblE. It will be seen at the end\nof this section that the condition a >4LFcan be satisfied provided that αandKare sufficiently\nlarge.\nDefinition 5.1. Suppose {Φω}ω∈Ωis a family of maps from E1toE2andn∈N. A family of\ngraphsω/ma√sto→ℓ(ω)≡ {(p,Φω(p)) :p∈E1}is said to be a randomnp0-period horizontal curve if\nω/ma√sto→ℓ(ω)is a random set and {Φω}ω∈Ωsatisfy the Lipshitz condition\n/bardblΦω(p1)−Φω(p2)/bardblE≤ /bardblp1−p2/bardblEfor allp1,p2∈E1, ω∈Ω\nand the periodic condition\nΦω(p+np0) = Φω(p)for allp∈E1, ω∈Ω.\nNote that for any ω∈Ω,ℓ(ω) is a deterministic np0-period horizontal curve. When n= 1,\nwe simply call it a horizontal curve.\nLemma 5.2. Leta >4LF. Suppose that ω/ma√sto→ℓ(ω)is a random np0-period horizontal curve in\nE. Then, ω/ma√sto→Y(t,ω,ℓ(ω))is also a random np0-period horizontal curve in Efor allt >0.\nMoreover, ω/ma√sto→Y(t,θ−tω,ℓ(θ−tω))is a random np0-period horizontal curve for all t >0.\nThe proof of Lemma 5.2 is similar to that of Lemma 4.2 in [24]. W e hence omit it here.\nChooseγ∈(0,a\n2) such that\n2c2|β|\nα/parenleftBigg\n1\nγ+1\na−2γ/parenrightBigg\n<1, (5.1)\nwhere2c2|β|\nαis the Lipschitz constant of F(see (3.9)). We remark at the end of this section that\nsuch aγexists provided that αandKare sufficiently large. The main result in this section is\nas follows.\nTheorem 5.3. Assume that a >4LFand there is γ∈(0,a\n2)such that (5.1)holds. Then the\nglobal random attractor ω/ma√sto→A0(ω)of the random dynamical system Yis a random horizontal\ncurve.\nProof.Since equation (3.1) can be viewed as a deterministic system with a random parameter\nω∈Ω, we write it here as (3.1) ωforω∈Ω. We first show that for any fixed ω∈Ω, (3.1) ωhas\na one-dimensional attracting invariant manifold W(ω).\nIn order to do so, for fixed ω∈Ω, let\nW(ω) ={Y0∈E|Y(t,ω,Y0) exists for t≤0 and sup\nt≤0/bardbleγtY(t,ω,Y0)/bardblE<∞}.\nWe prove that W(ω) is a one-dimensional attracting invariant manifold of (3. 1)ω.\nFirst of all, by the definition of W(ω), it is clear that for any t∈R,\nY(t,ω,W(ω)) =W(θtω),\n13that is,{W(ω)}ω∈Ωis invariant. By the variation of constant formula, Y0∈W(ω) if and only if\nthere is˜Y(t) with˜Y(0) =Y0, supt≤0/bardbleγt˜Y(t)/bardblE<∞,\n˜Y(t) =eCtξ+/integraldisplayt\n0eC(t−s)PFω(s,˜Y(s))ds+/integraldisplayt\n−∞eC(t−s)QFω(s,˜Y(s))ds, t≤0,(5.2)\nandY(t,ω,Y0) =˜Y(t), where Fω(t,Y) =F(θtω,Y) andξ=P˜Y(0)∈E1. ForH: (−∞,0]→E\nsuch that supt≤0/bardbleγtH(t)/bardblE<∞, define\n(LH)(t) =/integraldisplayt\n0eC(t−s)PH(s)ds+/integraldisplayt\n−∞eC(t−s)QH(s)ds, t≤0.\nThen\nsup\nt≤0/bardbleγt(LH)(t)/bardblE≤(1\nγ+1\na−γ)sup\nt≤0/bardbleγtH(t)/bardblE≤/parenleftBigg\n1\nγ+1\na−2γ/parenrightBigg\nsup\nt≤0/bardbleγtH(t)/bardblE,\nwhich means that /bardblL/bardbl ≤1\nγ+1\na−2γ. Thus, Theorem 3.3 in [7] shows that for any ξ∈E1, equation\n(5.2) has a unique solution ˜Yω(t,ξ) satisfying supt≤0/bardbleγt˜Yω(t,ξ)/bardblE<∞. Let\nh(ω,ξ) =Q˜Yω(0,ξ) =/integraldisplay0\n−∞e−CsQFω(s,˜Yω(s,ξ))ds, ω∈Ω.\nThen,\nW(ω) ={ξ+h(ω,ξ) :ξ∈E1}\nandW(ω) is a one dimensional invariant manifold of (3.1) ω. Furthermore, for any ǫ∈(0,γ), by\nLemma 3.4, we have\n/bardblh(θ−tω,ξ)/bardblE≤a1\na−ǫ˜r(ω)eǫt+a2\na, t≥0, (5.3)\nwherea1,a2is the same as in the proof Theorem 4.2.\nTo show the attracting property of W(ω), we prove for each given ω∈Ω the existence of a\nstable foliation {Ws(Y0,ω) :Y0∈W(ω)}of the invariant manifold W(ω) of (3.1) ω. Consider\nthe following integral equation\nˆY(t) =eCtη+/integraldisplayt\n0eC(t−s)Q/parenleftBig\nFω(s,ˆY(s)+Yω(s,ξ+h(ω,ξ)))\n−Fω(s,Yω(s,ξ+h(ω,ξ)))/parenrightBig\nds\n+/integraldisplayt\n∞eC(t−s)P/parenleftBig\nFω(s,ˆY(s)+Yω(s,ξ+h(ω,ξ)))\n−Fω(s,Yω(s,ξ+h(ω,ξ)))/parenrightBig\nds, t≥0,(5.4)\nwhereξ+h(ω,ξ)∈W(ω),η=QˆY(0)∈E2andYω(t,ξ+h(ω,ξ)) :=Y(t,ω,ξ+h(ω,ξ)),t≥0\nis the solution of (3.1) with initial data ξ+h(ω,ξ) for fixed ω∈Ω. Theorem 3.4 in [7] shows\nthat for any ξ∈E1andη∈E2, equation (5.4) has a unique solution ˆYω(t,ξ,η) satisfying\nsupt≥0/bardbleγtˆYω(t,ξ,η)/bardblE<∞and for any ξ∈E1,η1, η2∈E2,\nsup\nt≥0eγt/bardblˆYω(t,ξ,η1)−ˆYω(t,ξ,η2)/bardblE≤M2/bardblη1−η2/bardblE, (5.5)\n14whereM2=1\n1−2c2|β|\nα/parenleftbig\n1\nγ+1\na−2γ/parenrightbig. Let\nˆh(ω,ξ,η) =ξ+PˆYω(0,ξ,η)\n=ξ+/integraldisplay0\n∞e−CsP/parenleftBig\nFω(s,ˆYω(s,ξ,η)+Yω(s,ξ+h(ω,ξ)))\n−Fω(s,Yω(s,ξ+h(ω,ξ)))/parenrightBig\nds.\nThen,Ws(ω,ξ+h(ω,ξ)) ={η+h(ω,ξ)+ˆh(ω,ξ,η) :η∈E2}is a foliation of W(ω) atξ+h(ω,ξ).\nObserve that\nˆYω(t,ξ,η)+Yω(t,ξ+h(ω,ξ))−Yω(t,ξ+h(ω,ξ))\n=ˆYω(t,ξ,η)\n=eCt(η+h(ω,ξ)+ˆh(ω,ξ,η)−ξ−h(ω,ξ))\n+/integraldisplayt\n0eC(t−s)/parenleftBig\nFω(s,ˆYω(s,ξ,η)+Yω(s,ξ+h(ω,ξ)))\n−Fω(s,Yω(s,ξ+h(ω,ξ)))/parenrightBig\nds(5.6)\nand\nYω(t,η+h(ω,ξ)+ˆh(ω,ξ,η))−Yω(t,ξ+h(ω,ξ))\n=eCt(η+h(ω,ξ)+ˆh(ω,ξ,η)−ξ−h(ω,ξ))\n+/integraldisplayt\n0eC(t−s)/parenleftBig\nFω(s,Yω(s,η+h(ω,ξ)+ˆh(ω,ξ,η)))\n−Fω(s,Yω(s,ξ+h(ω,ξ)))/parenrightBig\nds.(5.7)\nComparing (5.6) with (5.7), we find that\nˆYω(t,ξ,η) =Yω(t,η+h(ω,ξ)+ˆh(ω,ξ,η))−Yω(t,ξ+h(ω,ξ)), t≥0.(5.8)\nIn addition, if η= 0, then by the uniqueness of solution of (5.4), ˆYω(t,ξ,0)≡0 fort≥0, which\ntogether with (5.5) and (5.8) shows that\nsup\nt≥0eγt/bardblYω(t,η+h(ω,ξ)+ˆh(ω,ξ,η))−Yω(t,ξ+h(ω,ξ))/bardblE≤M2/bardblη/bardblE (5.9)\nfor anyξ∈E1andη∈E2. Therefore, {Ws(Y0,ω) :Y0∈W(ω)}is a stable foliation of the\ninvariant manifold W(ω) of (3.1) ωand then W(ω) is a one-dimensional attracting invariant\nmanifold of (3.1) ω.\nNext we show that A0(ω) =W(ω) andA0(ω) is a random horizontal curve. Let ω/ma√sto→B(ω) be\nany pseudo-tempered random set in E. For any t >0 andY0∈B(θ−tω), there is ξ(θ−tω,Y0)∈\nE1such that\nY0∈Ws(θ−tω,ξ(θ−tω,Y0)+h(θ−tω,ξ(θ−tω,Y0))).\n15Letη(θ−tω) = supY0∈B(θ−tω)/bardblQY0−h(θ−tω,ξ(θ−tω,Y0))/bardblE. Then by (5.3) and (5.9),\n/bardblY(t,θ−tω,Y0)−Y(t,θ−tω,ξ(θ−tω,Y0)+h(θ−tω,ξ(θ−tω,Y0)))/bardblE\n≤M2e−γtη(θ−tω)\n→0 ast→ ∞,\nwhich implies that for ω∈Ω,\ndH(Y(t,θ−tω,B(θ−tω)),W(ω))→0 ast→ ∞.\nTherefore,\nA0(ω) =W(ω) forω∈Ω.\nMoreover, for any random horizontal curve ω/ma√sto→ℓ(ω) inEcontained in some pseudo-tempered\nrandom set,\ndH(Y(t,θ−tω,ℓ(θ−tω)),A0(ω))→0 ast→ ∞\nfor every ω∈Ω, which means that lim t→∞Y(t,θ−tω,ℓ(θ−tω))⊂A0(ω). Since A0(ω) is one-\ndimensional, we have for ω∈Ω,\nA0(ω) = lim\nt→∞Y(t,θ−tω,ℓ(θ−tω)).\nIt then follows from Lemma 5.2 that ω/ma√sto→A0(ω) is a random horizontal curve.\nCorollary 5.4. Assume that a >4LFand there is γ∈(0,a\n2)such that (5.1)holds. Then the\nrandom attractor ω/ma√sto→A(ω)of the random dynamical system φis a random horizontal curve.\nProof.It follows from Corollary 4.3, Remark 4.4 and Theorem 5.3.\nRemark 5.5. At the beginning of this section, we assume that a >4LF. Sincea=α\n2−|α\n2−δKλ1\nα|\nandLF=2c2|β|\nα, we can take α,Ksatisfyingα\n2−/vextendsingle/vextendsingle/vextendsingleα\n2−δKλ1\nα/vextendsingle/vextendsingle/vextendsingle>8c2|β|\nα, whereλ1is the smallest\npositive eigenvalue of A. On the other hand, we need some γ∈(0,a\n2)such that (5.1)holds.\nNote that\nmin\nγ∈(0,a\n2)/parenleftBigg\n1\nγ+1\na−2γ/parenrightBigg\n=/parenleftBigg\n1\nγ+1\na−2γ/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nγ=a\n2+√\n2=(√\n2+1)2\na,\nwhich implies that there exist α,Ksatisfying\nα\n2−/vextendsingle/vextendsingle/vextendsingleα\n2−δKλ1\nα/vextendsingle/vextendsingle/vextendsingle>2c2|β|(√\n2+1)2\nα>8c2|β|\nα. (5.10)\nIndeed, let c= 2c2|β|(√\n2+1)2, then for any α >√\n2candK >c\nλ1, there is a δ >0satisfying\nc\nKλ1< δ <min/braceleftBigα2−c\nKλ1,1/bracerightBig\nsuch that (5.10)holds.\n166 Rotation Number\nIn this section, we study the phenomenon of frequency lockin g, i.e., the existence of a rotation\nnumber of the coupled second order oscillators with white no ises (1.1).\nDefinition 6.1. The coupled second order system with white noises (1.1)is said to have a rota-\ntion number ρ∈Rif, forP-a.e.ω∈Ωand each φ0= (u0,u1)⊤∈E, the limit limt→∞Pφ(t,ω,φ0)\nt\nexists and\nlim\nt→∞Pφ(t,ω,φ0)\nt=ρη0,\nwhereη0is the basis of E1.\nNote that the rotation number is considered here by restrict ingφonE1, sinceφis dissipative\nonE2and limits likewise in Definition 6.1 vanish. From (3.4), we h ave\nPφ(t,ω,φ0)\nt=PY(t,ω,Y0(ω))\nt+P(0,z(θtω))⊤\nt, (6.1)\nwhereφ0= (u0,u1)⊤andY0(ω) = (u0,u1−z(ω))⊤. By Lemma 2.1 in [9], it is easy to prove that\nlimt→∞P(0,z(θtω))⊤\nt= (0,0)⊤. Thus, it sufficient to prove the existence of the rotation num ber\nof the random system (3.1).\nLet us show a simple lemma which will be used. For any pi=siη0∈E1,i= 1,2, we define\np1≤p2ifs1≤s2.\nThen we have\nLemma 6.2. Suppose that a >4LF. Letℓbe any deterministic np0-periodic horizontal curve\n(ℓsatisfies the Lipschitz and periodic condition in Definition 5 .1). For any Y1, Y2∈ℓwith\nPY1≤PY2, there holds\nPY(t,ω,Y1)≤PY(t,ω,Y2)fort >0, ω∈Ω. (6.2)\nThe proof of this lemma is similar to that of Lemma 6.3 in [24]. We then omit it here. We\nnow have the main result in this section.\nTheorem 6.3. Leta >4LF. Then the rotation number of (3.1)exists.\nProof.By the random dynamical system Ydefined in (4.1), we define the corresponding skew-\nproduct semiflow Θt: Ω×E→Ω×Efort≥0 by setting\nΘt(ω,Y0) = (θtω,Y(t,ω,Y0)).\nObviously, (Ω ×E,F ×B,(Θt)t≥0) is a measurable dynamical system, where B=B(E) is the\nBorelσ-algebra of E. It also can be verified that there is a measure µon Ω×Esuch that\n(Ω×E,F ×B, µ,(Θt)t≥0) becomes an ergodic metric dynamical system (see [6]). Note that\nPY(t,ω,Y0)\nt=PY0\nt+1\nt/integraldisplayt\n0PF(θsω,Y(s,ω,Y0))ds.\n17SinceF(θsω,Y(s,ω,Y0)+kp0) =F(θsω,Y(s,ω,Y0)),∀k∈Z, wecanidentify F(θsω,Y(s,ω,Y0))\nwithF(θsω,Y(s,ω,Y0)) and write\nF(θsω,Y(s,ω,Y0)) =F(θsω,Y(s,ω,Y0)).\nThus,\nPY(t,ω,Y0)\nt=PY0\nt+1\nt/integraldisplayt\n0PF(θsω,Y(s,ω,Y0))ds\n=PY0\nt+1\nt/integraldisplayt\n0F(Θs(ω,Y0))ds,(6.3)\nwhereF=P◦F∈L1(Ω×E,F × B, µ). Lett→ ∞in (6.3), lim t→∞PY0\nt= (0,0)⊤and by\nErgodic Theorems in [1], there exist a constant ρ∈Rsuch that\nlim\nt→∞1\nt/integraldisplayt\n0F(Θs(ω,Y0))ds=ρη0,\nwhich means\nlim\nt→∞PY(t,ω,Y0)\nt=ρη0.\nforµ-a.e.(ω,Y0)∈Ω×E. Thus, there is Ω∗⊂Ω withP(Ω∗) = 1 such that for any ω∈Ω∗, there\nisY∗\n0(ω)∈Esuch that\nlim\nt→∞PY(t,ω,Y∗\n0(ω))\nt=ρη0.\nBy Lemma 3.3, we have that for any n∈Nandω∈Ω∗,\nlim\nt→∞PY(t,ω,Y∗\n0(ω)±np0)\nt= lim\nt→∞PY(t,ω,Y∗\n0(ω))±np0\nt=ρη0. (6.4)\nNow for any ω∈Ω∗and any Y∈E, there is n0(ω)∈Nsuch that\nPY∗\n0(ω)−n0(ω)p0≤PY≤PY∗\n0(ω)+n0(ω)p0\nand there is a n0(ω)p0-periodic horizontal curve l0(ω) such that Y∗\n0(ω)−n0(ω)p0,Y,Y∗\n0(ω)+\nn0(ω)p0∈l0(ω). Then by Lemma 6.2, we have\nPY(t,ω,Y∗\n0(ω)−n0(ω)p0)≤PY(t,ω,Y)≤PY(t,ω,Y∗\n0(ω)+n0(ω)p0),\nwhich together with (6.4) implies that for any ω∈Ω∗and any Y∈E,\nlim\nt→∞PY(t,ω,Y)\nt=ρη0.\nConsequently, for any a.e. ω∈Ω and any Y∈E,\nlim\nt→∞PY(t,ω,Y)\nt=ρη0.\nThe theorem is thus proved.\nCorollary 6.4. Assume that a >4LF. Then the rotation number of the coupled second order\nsystem with white noises (1.1)exists.\nProof.It follows from (6.1) and Theorem 6.3.\n18References\n[1] L.Arnold, Random dynamical systems, Springer Monograp hs in Mathematics, Springer-\nVerlag, Berlin, 1998.\n[2] P.W. Bates, K. Lu and B. Wang, Random attractors for stoch astic reaction-diffusion equa-\ntions on unbounded domains, J. Diff. Eq. 246(2009), 845-869.\n[3] A. Bensoussan, and F. 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The exact analytical expressions \nof frequency and linewidth of  the magnetic  nutation resonance are deduced from the frequency dependent susceptibility \ndetermined by the i nertial Landau -Lifshitz -Gilbert equation.  The study  shows that the dependence of nutation linewidth on the \nGilbert precess ion damping  has a minimum , which becomes more expressive with increas e of the applied magnetic field.  \n \nPACS numbers: 76.50.+g, 78.47.jp, 75.50. -y \n \nI. INTRODUCTION  \nRecently, the effects of inertia in the spin dynamics of \nferromag nets were reported to cause nutation resonance  [1-\n12] at frequencies  higher  than the conventional ferromagnetic \nresonance . It was shown  that inertia  is responsible for the \nnutation , and that this type of motion should be considered  \ntogether with magnetization precession in the applied \nmagnetic field . Nutation in ferromagnets was confirmed \nexperimentally only recently  [2], since  nutation and \nprecession operate at substantially different time scales , and \nconventi onal microwave ferromagnetic resonance (FMR) \nspectroscopy techniques do not easily reach the high -\nfrequency (sub-Terahertz) regime re quired to observe the \ninertia effect which in addition yields a much weaker signal . \nSimilar to any other oscillatory system, t he magnetiza tion \nin a ferromagnet has resonant frequencies  usually studied by  \nferromagnetic resonance  [13,14]. The resonant \neigenfrequency is determined by  the magnetic parameters  of \nthe material  and applied magnetic field . Including inertia of \nthe magnetization in th e model description shows that  nutation \nand precession are complementary to each other  and several \nresonances can be generated . In this Letter , we concentrate on \nthe investigation of the resonance characteristics of nutation.  \nThe investigation of nutation is connected to the progress \nmade  in studies of the  spin dynamics at ultrashort  time \nscales  [15,16] . These successes led to the rapi d development \nof a new scientific field , the so -called ultrafast magnetism  [17-\n25]. The experimental as well as theoretical investigation of \nthe inertial  spin dynamics is at the very beginning , although it \nmight be of significance for future high speed spintronics \napplications  including ultrafast magnetic switching .  Besides nutation driven by magnetization inertia, several  \nother origins  of nutation have been reported . Transient \nnutations (Rabi oscillations) have been widely investigate d in \nnuclear magnetic resonance  [26] and electron spin \nresonance  [27-29], they were  recently addressed in \nferromagnets  [30]. A complex dynamics and Josephson \nnutation of a local spin \n1/ 2s  as well as large spin cluster \nembedded in the tunnel junction between ferromagnetic leads \nwas shown to occur  due to  a coupling to Josephson \ncurrent  [31-33]. Low-frequency nutation was observ ed in \nnanomagnets exhibiting a non -linear FMR  with the large -\nangle precession of magnetization where the onset of spin \nwave instabilities can be delay ed due to geometric \nconfinement  [34]. Nutation dynamics due to inertia of \nmagnetization in ferromagnetic thin films was observed for \nthe first time by Neeraj et al.  [2]. \nThe microscopic derivation of the magnetization  inertia \nwas performed in  ref. [3-7]. A relation between the Gilbert \ndamping constant  and the inertia l regime  characteristic time  \nwas elaborated in ref. [3]. The exchange interaction, damping, \nand moment of inertia can be calculated from first principles \nas shown in  [7]. The study of inertia spin dynamics  with a \nquantum approach  in metallic ferromagnets  was performed \nin [8]. In addition, nutation  was theoretically analyzed as a \npart of magnetization dynamics in ferromagnetic \nnanostructure  [9,10] and nanoparticles  [11]. Despite these \nadvances, exact analytical expressions for the high-frequency \nsusceptibility including inertia had not been derived yet.  \nIn [35], the inertial regime  was introduced in the \nframework of the mesoscopic nonequilibrium \nthermodynamics theory , and it was shown to be responsible \nfor the nutation superimposed on the precession of \nmagnetization . Wegrowe and Ciornei  [1] discussed  the  \n2 \n equivalence between the inertia in the dynamics of uniform \nprecession and a spinning top within the framework of the  \nLandau –Lifshitz –Gilbert equation generalized to the inertial \nregime. This equation was  studied  analytical ly and \nnumerical ly [12,36]. Although the se reports provide \nnumerical tools for obtaining resonance characteristics, the \ncomplexity of the numerical solution of differential equations \ndid not allow to estimate the nutation frequency and linewidth \naccurately . Also in a recent  remarkable paper  [37] a novel \ncollective excitation  – the nutation wave  – was reported,  and \nthe dispersion characteristics  were derived wit hout discussion \nof the nutation  resonance lineshapes  and intensities.  \nThus, at present, there is a necessity to study  the resonance \nproperties of nutation in ferromagnet s, and this paper is \ndevoted to this study. We performed the investigation based \non the Landau -Lifshitz -Gilbert equation with  the addition al \ninertia term  and provide an  analytical solution.   \nIt is well known that the Landau -Lifshitz -Gilbert equation \nallow s finding the susceptibility as the ratio between  the time-\nvarying magnetization and the time-varying driving magnetic \nfield (see for exampl e [38,39] and references therein). This \nsusceptibility describes  well the magnetic response of a \nferromagnet in the linear regime, that is  a small cone angle of \nthe precession . In this description , the ferromagnet usually is \nplaced in a  magnetic field  big enough to align all  atomic \nmagnetic moments along the field , i.e., the ferromagnet is in \nthe saturated state and the  magnetization  precess es. The \napplied  driving magnetic field  allows one to obser ve FMR  as \nsoon as the driving field frequency coincides with  \neigenfrequency of precession. Using the expression for \nsusceptibility, one can elaborate  such resonance \ncharacteristics as eigenfrequency and linewidth.  We will \npresent similar expressions for the dynamic susceptibility, \ntaking nutation into account.  \nII. SUSCEPTIBILITY  \nThe ferromagnet  is subjected to a uniform bias magnetic \nfield \n0H  acting along the z -axis and being strong enough to \ninitiate  the magnetic saturation  state. The small time -varying \nmagnetic field \nh  is superimposed on the bias field. The \ncoupling between impact and response, taking into account \nprecession, damping, and nutation, is given by the Inertial  \nLandau -Lifshitz -Gilbert ( ILLG)  equation  \n \n2\n2\n0,effd d d\ndt M dt dt       M M MMH   (1) \nwhere \n  is the gyromagnetic ratio, \nM  the magnetization  \nvector , \n0M  the magnetization at saturation, \neffH  the vector \nsum of all magnetic fields, external and internal, acting upon \nthe magnetization , \n the Gilbert damping , and \n  the inertial \nrelaxatio n time. For simplicity, we assume that the \nferromagnet is infinite, i.e. there is no demagnetization  correction , with negligible magnetocrystalline anisotropy , and \nonly the  externally applied field s contribute  to the total  field. \nThus, the bias magnetic field \n0H  and signal field \nh  are \nincluded in \neffH . We assume that the signal is small \n0,hH\n hence the magnetization is directed along \n0.H  \nOur interest is to study the correlated  dynamics of nutation \nand precession  simultaneously;  therefore we write the \nmagnetization  and magnetic field  in the general ized form  \nusing  the Fourier transformation  \n \n 01ˆ  ,\n2itt M z d e\n\n\n Mm   (2) \n \n 01ˆ  ,\n2it\nefft H z d e\n\n\n Hh   (3) \nwhere \nˆz  is the unit vector along the z -axis. If we  substitute  \nthese expressions  in the ILLG equation  and neglect the small \nterms, it leads to  \n \n\n \n 00\n211 \n22\nˆˆ\nˆˆ .i t i td i e d e\nM z H z\ni z z   \n\n   \n   \n\n     \n    \n        m\nhm\nmm   (4) \nBy performing the Fourier transform and changing the order \nof integration , equation  (4) becomes   \n \n\n\n \n00\n21 2\n1 2\nˆˆ\nˆˆ ,it\nitd dt i e\nd dt e\nM z H z\ni z z\n  \n\n   \n    \n\n \n\n\n   \n \n     \n       \nm\nhm\nmm   (5) \nwhere the integral representation of the Dirac delta  function  \ncan be found. With the delta function,  the equation (5)\nsimplifies to  \n \n  \n 00\n2ˆˆ\nˆˆ .i M z H z\ni z z     \n      \n   m h m\nmm   (6) \nBy projecting to Cartesian coordinates and  introducing the \ncircular variables  for positive and negative circular \npolarization  \n,xy m m im  \n,xy h h ih one obtains  \n \n  \n 2\n20,\n0,HM\nHMm m i m m h\nm m i m m h   \n\n        \n          \n        (7) \nwhere \n0 H H  is the precession frequency and \n0.M M\n The small -signal susceptibility follows  from \nthese equations :  \n3 \n  \n2\n2,\n,\n.M\nH\nM\nHim\nih\n\n   \n\n  \n   \n\n  \n  \n\n   (8) \nIt is seen that the susceptibility (8) is identical with  the \nsusceptibility for LLG equation , if one drops  the inertial term , \nthat is  \n0.  \nLet us separate dispersive and d issipative  parts  of the \nsusceptibility \n,i       \n \n \n 2\n2,\n,\n,\n,MH\nM\nMH\nMD\nD\nD\nD\n\n   \n\n   \n\n\n\n\n\n\n  \n\n\n   (9) \n \n 2 2 4 3\n2 2 22\n22 , 1   \n     \n    H H HD   (10) \n \n 2 2 4 3\n2 2 22\n22 . 1   \n     \n    H H HD   (11) \n \nThe frequency dependence of the dissipative parts of \nsusceptibilit ies \n and \n  is shown in the Fig. 1. The plus \nand minus subscripts correspond to right -hand and left -hand \ndirection of rotation. Since the denominators \nD  and \nD  are \nquartic polynomials, four critical points  for either  \n or \n \ncan be expected . Two of them  that are extrema with  a clear \nphysical meaning are plotted.  In Fig. 1(a) the extremum , \ncorresponding  to FMR  at \n0 H H  is shown . Due to the \ncontribution  of nutation , the frequency and linewidth of this \nresonance are slightly different from the ones of usual FMR . \nThe resonance occurs  for right -hand precession, i.e. positive \npolarization.   \nIn Fig. 1(b) the nutation resonance  possessing negative \npolarization is presented.  Note  that the polarizations of \nferromagnetic and nutation resonances are reverse d. \nIII. APPROXI MAT ION FOR  NUTATION \nFREQUENCY  \nLet us  turn to the description  of an approximation  of the \nnutation resonance frequency. If we equate the denominator \nD\n to zero, solve the resulting equation, we obtain the \napproximation from the real part of  the roots. This is reasonable , since the numerator of \n  is the linear function of \n\n , and we  are interested in \n1.  Indeed , the equation  \n \n 2 2 4 3 2 2\n202 2 1\n2H\nHH      \n     \n    (12) \nhas four roots that are complex conjugate in pairs  \n \n1,221 1 4 2,2H\nFMRiiw   \n       (13) \n \n1,221 1 4 2.2H\nNiiw   \n       (14) \n \nFIG. 1. (Color online) (a) The FMR peak with nutation. (b) \nThe nutation resonance. The calculation was performed for \n1/ 2 28  GHz T ,\n \n00 1 T, M  \n00 100 mT, H  \n0.0065\n and \n1110  s.  \n \nOne should choose  the same sign from the \n  symbol  in each \nformula , simultaneously . The real part of expression  (13) \ngives the approximate frequency for FMR , but in negative \nnumbers, so the sign should be inversed . The approximate \nfrequency of FMR in positive numbers can be derived from \nequation \n0. D  The approximate nutation frequency  is \nobtained  by the real part of  the expression (14). One takes half \nthe sum of two conjugate roots  \n1,2,Nw neglect s the high \n  \nterms , and obtains  the nutation resonance frequency  \n \n1 1 2\n.2NH\nw \n\n   (15) \nNote  that the expression of \nNw  is close  to the approximation  \ngiven in  [36] at \n1/ , H    namely  \n \nweak\nnu1\n.H \n\n   (16) \nThe similarity of both approximations b ecomes clear , if we \nperform a Taylor series expansion  and return to the notation \n,H\n \n\n2\n2 2 3 3 3\n2\nweak\nnu\n2 2 3 3 31 1 2 1\n2 2 4\n1,4\n1 1\n2\n1.18\n6H HH\nN\nH\nH HH\nHw\nO\nO  \n \n    \n   \n       \n\n   \n\n \n \n4 \n IV. PRECISE  EXPRESSIONS FOR  FREQUENCY \nAND LINEWIDTH OF NUT ATION  \nThe analytical approach proposed in this Letter yields  \nprecise values of the frequency  of nutation resonance  and the \nfull width at half maximum (FWHM) of the peak . The \nfrequency is found by extremum, when the derivative of the \ndissipative part of susceptibilities (9) is zero \n \n0.\n   (17) \nIt is enough to determine  zeros of the n umerator of th e \nderivative , that are given by  \n \n 2 2 4 3 2 2 23 4 2 1 0.HH                (18) \nLet us use Ferrari's solution  for this quartic equation  and \nintroduce the notation:  \n \n22\n2\n2\n2\n2\n3\n23\n24\n343\n4,\n2 1,  \n3,8\n,28,\n3.16 25,\n6rH\nrH\nrr\nr r\nr r r\nrr\nr r r r\nr\nr rrr\nr\nr\nrC\nE\nC\nC\nCEcA\nB\nBaA A\nBBbAA\nBB\nA AA\n\n \n\n\n\n\n\n\n  \n\n   (19) \nIn Ferrari's method , one should determine a root of the nested \ndepressed cubic equation . In the investigated case , the root is \nwritten  \n \n5,6r\nr r ray U V     (20) \nwhere  \n \n32\n3\n2\n32,27 4 2\n,3\n12\n1,\n.3 108 8r r r\nr\nr\nr\nr\nr\nrr\nrr\nr r rP Q QU\nPVU\nPc\nQaa\nabc   \n\n \n     (21) \nThus, the precise value of the nutation frequency is given by  \n \n2\n42\n2 13 2 .2 2rr r\nN\nr\nr\nrr\nrry\nA\nbaa\nay\nyB  \n   \n   (22) \nThe performed analysis shows that approximate value of \nnutation resonance frequency is close to precise value.  The linewidth of the nutation resonance is found at  a half \npeak height. If one denotes the maximum by \n,N X \n the equation  which determines \nfrequencies at half magnitude  is \n \n 2 2 4 3 2 2\n212 2 12\n2 0.H\nH H MX      \n      \n\n   (23) \nWe repeat the procedure for finding solu tions with Ferrari's \nmethod  introducing the  new notations  \n \n 22\n2\n2\n2\n2\n3\n23\n24\n3 4 21\n2\n,\n12 1 ,2\n1\n2\n3,8\n,28\n3.16 2,\n56,\n4lw\nlw\nlw\nlw\nlw\nlw lw\nlw\nlw lw\nlw lw lw\nlw\nlw lw\nlw lw lw lw lw lw\nlw\nlw lwH\nHM\nH\nlw\nlw\nlw lwA\nB\nBaA A\nBBbAA\nB B B D\nA AX\nX\nCX\nDX\nEX\nC\nCD\nA\nCEcA A\n\n \n \n\n\n\n\n\n\n  \n\n\n\n  \n  \n\n\n   (24) \nA root of the nested depressed cubic equation \nlwy  must be \nfound in the same way as provided in (20) with the \ncorresponding replacement of variables, i.e. subscript r is \nreplaced by lw. The difference between two adjacent roots \ngives the nutation linewidth  \n \n23 2 .\n2lw\nN lw lw\nlw lwbay\na y   \n   (25) \nThe explicit expression for the linewidth can be written using \nthe equations  (19)-(25). \n \n \nFIG. 2. (Color online) The dependence of the nutation \nlinewidth on the inertial relaxat ion time for \n00 100 mT, H\n \n00 1 T, M  and \n0.0065.  \n \n5 \n  \nThe effect of the inertial relaxation time on the nutation \nlinewidth is shown in Fig.  2. One can see that increasing \ninertial relaxation time leads to narrowing of the linewidth. \nThis behavior is expected and is consistent with the traditional \nview that decreasing of losses results in narrowing of \nlinewidth.  \n \n \nFIG. 3. (Color online) The dependence of nutation \nresonance linewidth on precession Gilbert damping \nparameter at different magnetic fields \n0H  for \n00 1 T M\nand \n1110  s.  \n \nSince the investig ated oscillatory system implemen ts \nsimultaneous two types of  motions , it is of interest to study the \ninfluence of the Gilbert precession  damping parameter \n  on \nthe nutation  resonance linewidth. The result is presented in \nFig. 3 and is valid for ferromagnets with vanishing anisotropy.  \nOne sees that the dependence of \nN  on \n  shows a \nminimum that becomes more expressive with increasing  bias \nmagnetic field.  In other words, t he linewidth is parametrized \nby the magnitude of field. This non-trivial  behavior of \nlinewidth relates with the nature of th is oscillatory system, \nwhich performs  two coupled  motions.  \nTo elucidate the non-trivial behavior , one can consider  the \nsusceptibility  (9) in the same way as it is usually performed \nfor the forced harmonic oscillator with damping  [40]. For this \noscillator , the linewidth can be direct ly calculated from the \ndenominator of the response expression once the driving \nfrequency is equal to eigenfreq uency. In the investigated case \nof magnetization with inertia , the response expression is (9) \nwith denominator s (10) and (11) written as  \n \n 2 2 4 3\n2 2 22\n21 . 2H H HD   \n      \n   \n   (26) \nSince the applied magnetic field is included in this expression  \nas \n0,H H the linewi dth depends on the field.  \nThe obtained result can be generalized to a fin ite sample \nwith magnetocrystalline anisotropy with method of effective \ndemagnetizing factors  [41,42] . In this case the bias magnetic field \n0H  denotes an external field and in the final expressions \nthis field should be replaced by \n 0 0 0ˆˆ ,i a d NN  H H M  \nwhere \nˆ\naN  is the anisotropy demagnetizing tensor and \nˆ\ndN  is \nthe shape demagnetizing tensor.  \nV. CONCLUSION  \nIn summary, we derived a general analytical expression for \nthe linewidth and f requency of nutation resonance in \nferromagnets, depending on magnetization, the Gilbert \ndamping, the inertial relaxation time and applied magnetic \nfield.  We show the nutation linewidth can be tuned by the \napplied magnetic field , and this tunability breaks the direct \nrelation between losses and the linewidth. This for example  \nleads to the appearance of a minimum  in the nutation \nresonance linewidth for the damping parameter  around  \n0.15.\n The obtained results are valid for ferromagnets with \nvanishing anisotropy . \nACKNOWLEDGEMENTS  \nWe thank  Benjamin Zingsem for helpful discussions . 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A 64, 968 (1951).  \n "    },    {        "title": "2009.11244v1.Remark_on_the_exponential_decay_of_the_solutions_of_the_damped_wave_equation.pdf",        "content": "arXiv:2009.11244v1  [math.AP]  23 Sep 2020REMARK ON THE EXPONENTIAL DECAY OF THE\nSOLUTIONS OF THE DAMPED WAVE EQUATION\nGIOVANNI CIMATTI\nAbstract. A condition which guaranties the exponential decay of the so lu-\ntions of the initial-boundary value problem for the damped w ave equation is\nproved. A method for the effective computability of the coeffic ient of expo-\nnential decay is also presented.\n1.introduction\nLet Ω be a bounded and open subset of RNand Γ its boundary. Consider the\ninitial-boundary value problem\n(1.1) utt−∆u+aut= 0 in Ω ×(0,∞)\n(1.2) u= 0 on Γ ×[0,∞)\n(1.3) u(x,0) =uo(x) forx∈Ω\n(1.4) ut(x,0) =u1(x) forx∈Ω,\nwhereais a positive constant, uo(x) andu1(x) are given functions satisfying the\nusual regularity and compatibility conditions implied in the assumption w e make\nthatu(x,t) is a classical solution of problem (1.1)-(1.4). P.H. Rabinowitz has\nproven [1] that the ”energy” i.e./integraltext\nΩ|∇u(x,t)|2dxassociated with the solution of\n(1.1)-(1.4) decays exponentially as t→ ∞. A different proof of this fact, important\nin applications, has been given by R. Temam [2], [3] and by R Temam and J.M .\nGhidaglia in [4]. In this paper we consider the initial-boundary value prob lem\n(1.1)-(1.4) with the more general equation\n(1.5) utt−∆u+σ(x,t)ut= 0 in Ω ×(0,∞),\nwhereσ(x,t) is supposed to be continuous and positive in Ω ×(0,∞). In general\nthere is no exponential decay, as the following example shows. Let N= 1, Ω=(1 ,2)\nandσ(x,t) =2t\nx2−3x+2. The solution is now given by u(x,t) =t(x2−3x+ 2) for\nwhich there is no exponential decay. We prove, with fully elementary means, and\n2010Mathematics Subject Classification. 35B35.\nKey words and phrases. Damped wave equation, Gronwall’s lemma, Exponential decay , Coef-\nficient of exponential decay .\n12 GIOVANNI CIMATTI\nwithout using the semigroup’s theory, that a sufficient condition for having the\nexponential decay is\n(1.6) 0 < σ0≤σ(x,t)≤σ1<∞,\nwhereσ0= infσ(x,t) andσ1= supσ(x,t). We also give an effective way to\ncompute the decay exponent in terms of the parameters σ0,σ1,λ1, whereλ1is the\nfirst eigenvalue of the laplacian in Ω with zero boundary conditions.\n2.exponential decay\nLetu(x,t) be a regular solution of (1.5), (1.2), (1.3) and (1.4) with σ(x,t) sat-\nisfying (1.6). Define\n(2.1) v(x,t) =ut(x,t)+ǫu(x,t) with 0 < ǫ≤σ0.\nWe have\n(2.2) utt=vt−ǫv+ǫ2u.\nSubstituting (2.2) in (1.5) we obtain\n(2.3) vt+(σ−ǫ)v+(ǫ2−ǫσ)u−∆u= 0.\nLet us multiply (2.3) by v. We arrive at\n(2.4)/integraldisplay\nΩvtvdx+/integraldisplay\nΩ(σ−ǫ)v2dx−/integraldisplay\nΩv∆udx+/integraldisplay\nΩ(ǫ2−ǫσ)uvdx= 0.\nSincev= 0 on Γ ×[0,∞), integrating by parts in (2.4) we have\n(2.5)1\n2d\ndt/integraldisplay\nΩv2dx+/integraldisplay\nΩ(σ−ǫ)v2dx+/integraldisplay\nΩ∇v·∇udx−ǫ/integraldisplay\nΩ(σ−ǫ)uvdx= 0.\nBy (2.1) ∇v=∇ut+ǫ∇u, hence from (2.5),\n(2.6)1\n2d\ndt/integraldisplay\nΩ/parenleftbig\n|∇u|2+v2/parenrightbig\ndx+/integraldisplay\nΩ(σ−ǫ)v2dx+ǫ/integraldisplay\nΩ|∇u|2dx−ǫ/integraldisplay\nΩ(σ−ǫ)uvdx= 0.\nRecalling (2.1) we have\n(2.7)/integraldisplay\nΩ(σ−ǫ)v2dx≥/integraldisplay\nΩ(σ0−ǫ)v2dx≥0.\nTo control from below the last integral in (2.6) we estimate from abo veǫ/integraltext\nΩ(σ−\nǫ)uvdx. Letλ1>0 be the first eigenvalue of the laplacian with zero boundary\ncondition on Γ. Using the Cauchy-Schwartz and Poincar inequalities w e obtain,\nwithη >0,\n(2.8) ǫ/integraldisplay\nΩ(σ−ǫ)uvdx≤ǫ(σ1−ǫ)\n2η/integraldisplay\nΩ|v|2dx+ǫ(σ1−ǫ)η\n2λ1/integraldisplay\nΩ|∇u|2dx.\nChanging sign in (2.8) and substituting in (2.6) we obtainTHE EXPONENTIAL DECAY FOR THE DAMPED WAVE EQUATION 3\n(2.9)\n1\n2d\ndt/integraldisplay\nΩ/parenleftbig\n|∇u|2+v2/parenrightbig\ndx+/bracketleftbigg\nǫ+ǫ(ǫ−σ1)η\n2λ1/bracketrightbigg/integraldisplay\nΩ|∇u|2dx+/bracketleftbigg\nσ0−ǫ+ǫ(ǫ−σ1)\n2η/bracketrightbigg/integraldisplay\nΩ|v|2dx≤0.\nDefining\n(2.10) f(ǫ,η,σ1,λ1) =ǫ+ǫ(ǫ−σ1)η\n2λ1, g(ǫ,η,σ0,σ1) =σ0−ǫ+ǫ(ǫ−σ1)\n2η\n(2.9) becomes\n(2.11)\n1\n2d\ndt/integraldisplay\nΩ/parenleftbig\n|∇u|2+v2/parenrightbig\ndx+f(ǫ,η,σ1,λ1)/integraldisplay\nΩ|∇u|2dx+g(ǫ,η,σ0,σ1)/integraldisplay\nΩ|v|2dx≤0.\nWe wish to find couples ( ǫ,η) satisfying the conditions 0 < ǫ < σ 0and 0< ηsuch\nthat for every choice of the parameters σ0,σ1,λ1,f(ǫ,η,σ,λ1) andg(ǫ,η,s0,σ1) are\npositive and equal. Among these couples we must find the ( ǫ∗,η∗) which gives to\nf(andg) the greatest possible value. This will permits to apply to (2.11) the\nGronwall’s inequality. The exponential decay with the best decay exp onent then\nfollows easily. Let us consider in the plane ǫ,η, (ǫ >0) the families of curves\n(2.12) f(ǫ,η,σ1,λ1)−g(ǫ,η,σ0,σ1) = 0.\nSinceǫλ1>0, (2.12) is equivalent to\n(2.13) ǫ(ǫ−σ1)η2+2λ1(2ǫ−σ0)η+λ1ǫ(σ1−ǫ) = 0.\nSolving (2.13) with respect to ηwe obtain two branches of solutions. The one of\ninterest to us is\n(2.14) η=/radicalbig\n(σ0−2ǫ)2λ2\n1+ǫ2(σ1−ǫ)2λ1+(2ǫ−σ0)λ1\nǫ(σ1−ǫ).\nInserting (2.14) in f(ǫ,η,σ1,λ1) (or ing(ǫ,η,σ0,σ1)) we obtain\n(2.15) F(ǫ,σ0,σ1,λ1) =σ0\n2−/radicalbig\n(σ0−2ǫ)2λ2\n1+ǫ2(σ1−ǫ)2λ1\n2λ1.\nWe study\n(2.16) α=F(ǫ,σ0,σ1,λ1)\nas a function of ǫdepending of the three parameters σ0,σ1,λ1. We have\nLemma 2.1. Letσ1> σ0>0andλ1>0. If¯ǫis a solution of the equation\nF(ǫ,σ0,σ1,λ1) = 0, then¯ǫ < σ0.\nProof.By contradiction, suppose ¯ ǫ=σ0+γ2,γ/negationslash= 0 is a solution. We have\n(2.17)σ2\n0λ2\n1= (−σ0−2γ2)2λ2\n1+(σ0+γ2)2(σ1−σ0−γ2)2λ1>(σ0+2γ2)2λ2\n1> σ2\n0λ2\n1.\nOn the other hand, if ¯ ǫ=σ0, we have4 GIOVANNI CIMATTI\n(2.18) σ2\n0λ2\n1=σ0λ2\n1+σ2\n0(σ1−σ0)2λ1> σ2\n0λ2\n1.\n/square\nFor every values of the parameters ǫ= 0 is a solution of F(ǫ,σ0,σ1,λ1) = 0.\nMoreover, lim ǫ→±∞F(ǫ,σ0,σ1,λ1) =−∞and\n(2.19) F′(ǫ,σ0,σ1,λ1) =2(σ0−2ǫ)λ1−ǫ(σ1−ǫ)2+ǫ2(σ1−ǫ)\n2/radicalbig\n(σ0−2ǫ)2λ2\n1+ǫ2(σ1−ǫ)2λ1.\nWe have F′(0,σ0,σ1,λ1) = 1 for every value of the parameters σ0,σ1andλ1.\nTherefore, in a small interval (0 ,β),β >0Fis positive for every value of the\nparameters. To study completely F′we make the following elementary discussion.\nThe cubic expression\n(2.20) −2ǫ3+3σ1ǫ2−(4λ1+σ2\n1)ǫ+2λ1σ0\nhas the same sign of F′(ǫ,σ0,σ1,λ1). On the other hand, D= 3σ2\n1−24λ1is\nthe discriminant of the derivative of (2.20) Thus, if D <0, (2.19) is always strictly\ndecreasing. As a consequence, F′(ǫ,σ0,σ1,λ1) vanishes in exactly one point ¯ ǫand\n0<¯ǫ. This implies that F(ǫ,σ0,σ1,λ1), which always vanishes for ǫ= 0 and is\npositive immediately to the right of ǫ= 0, has a positive absolute maximum in\nǫ∗and vanishes in ǫ= ¯ǫ >¯ǫ∗. IfD= 0 a bifurcation occurs, and, when D >0,\nF(ǫ,σ0,σ1,λ1)hastworelativemaxima’sandonerelativeminimum. Themaximum\nimmediately to the right of ǫ= 0 is certainly positive, the second maximum may\nor may not be positive. Let ǫ∗be the point of absolute maximum of F(ǫ,σ0,σ1).\nDefine\n(2.21) α∗=F(ǫ∗,σ0,σ1,λ1)\nand\n(2.22) η∗=/radicalbig\n(σ0−2ǫ∗)2λ2\n1+ǫ∗2(σ1−ǫ∗)2λ1+(2ǫ∗−σ0)λ1\nǫ∗(s1−ǫ∗).\nWith this choice of ǫandηwe havef(ǫ∗,η∗,σ1,λ1) =g(ǫ∗,η∗,σ0,σ1) =α∗>0.\nTherefore, (2.11) becomes\n(2.23)1\n2d\ndt/integraldisplay\nΩ/parenleftbig\n|∇u|2+v2/parenrightbig\ndx+α∗/integraldisplay\nΩ(|∇u|2+v2)dx≤0.\nUsing the Gronwall’s inequality, we obtain\n(2.24)/integraldisplay\nΩ/parenleftbig\n|∇u|2+v2/parenrightbig\ndx≤/integraldisplay\nΩ(|∇u(x,0)|2+|v(x,t)|2)dx e−2α∗t.\nThe right hand side of (2.24) can be computed in terms of the initial an d bound-\nary data satisfied by u(x,t). For, from (1.3) we obtain ∇u(x,0) =∇u0(x). More-\nover, since v(x,t) =ut(x,t)+ǫ∗u(x,t), we have v(x,0) =u1(x)+ǫ∗u0(x). HenceTHE EXPONENTIAL DECAY FOR THE DAMPED WAVE EQUATION 5\n/integraldisplay\nΩ|∇u|2dx≤/integraldisplay\nΩ|∇u0(x)|2+|u1(x)+ǫ∗u0(x)|2dx e−2α∗t.\nThis proves the exponential decay.\nRemark . For the regular solutions of the non-linear equation\n(2.25) utt−∆u+m(u)ut= 0 in Ω ×(0,∞)\nthe exponential decay can be obtained, with minor changes, if we ke ep the initial\nand boundary conditions (1.2), (1.3) and (1.4) and assume m∈C0(R1) and 0<\nm0≤m(u)≤m1. To see that, we simply define σ(x,t) =m(u(x,t)) where u(x,t)\nis the regular solution of the non linear problem (2.25), (1.2), (1.3) an d (1.4)\nReferences\n1. P.H. Rabinowitz, Periodic solutions of nonlinear hyperb olic partial differential equations,\nComm. Pure Appl. Math., 22, 145-205, (1967).\n2. R. Temam, Infinite-Dimensional Dynamical Systems in Mech anics and Physics, Springer-\nVerlag, (1988).\n3. R. Temam, Behaviour at time t= 0 of the solutions of semilinear evolutions equations, Jou r.\nDiff. Eqs., 43, 73-92, (1982).\n4. Ghidaglia and R. Temam, Attractors for damped nonlinear h yperbolic equations, J. Math.\nPures Appl., 66, 273-319, (1987).\nDepartment of Mathematics, Largo Bruno Pontecorvo 5, 56127 Pisa Italy\nE-mail address :cimatti@dm.unipi.it"    },    {        "title": "1510.03571v2.Nonlocal_torque_operators_in_ab_initio_theory_of_the_Gilbert_damping_in_random_ferromagnetic_alloys.pdf",        "content": "arXiv:1510.03571v2  [cond-mat.mtrl-sci]  19 Nov 2015Nonlocal torque operators in ab initio theory of the Gilbert damping in random\nferromagnetic alloys\nI. Turek∗\nInstitute of Physics of Materials, Academy of Sciences of th e Czech Republic, ˇZiˇ zkova 22, CZ-616 62 Brno, Czech Republic\nJ. Kudrnovsk´ y†and V. Drchal‡\nInstitute of Physics, Academy of Sciences of the Czech Repub lic,\nNa Slovance 2, CZ-182 21 Praha 8, Czech Republic\n(Dated: July 5, 2018)\nWe present an ab initio theory of theGilbert dampingin substitutionally disorder ed ferromagnetic\nalloys. The theory rests on introduced nonlocal torques whi ch replace traditional local torque\noperators in the well-known torque-correlation formula an d which can be formulated within the\natomic-sphereapproximation. Theformalism is sketchedin asimpletight-bindingmodel andworked\nout in detail in the relativistic tight-binding linear muffin -tin orbital (TB-LMTO) method and\nthe coherent potential approximation (CPA). The resulting nonlocal torques are represented by\nnonrandom, non-site-diagonal and spin-independent matri ces, which simplifies the configuration\naveraging. The CPA-vertex corrections play a crucial role f or the internal consistency of the theory\nand for its exact equivalence to other first-principles appr oaches based on the random local torques.\nThis equivalence is also illustrated by the calculated Gilb ert damping parameters for binary NiFe\nand FeCo random alloys, for pure iron with a model atomic-lev el disorder, and for stoichiometric\nFePt alloys with a varying degree of L1 0atomic long-range order.\nPACS numbers: 72.10.Bg, 72.25.Rb, 75.78.-n\nI. INTRODUCTION\nThe dynamics of magnetization of bulk ferromagnets,\nutrathin magnetic films and magnetic nanoparticles rep-\nresents an important property of these systems, espe-\ncially in the context of high speed magnetic devices for\ndata storage. While a complete picture of magnetization\ndynamics including, e.g., excitation ofmagnonsand their\ninteraction with other degrees of freedom, is still a chal-\nlenge for the modern theory of magnetism, remarkable\nprogresshas been achieved during the last years concern-\ning the dynamics of the total magnetic moment, which\ncan be probed experimentally by means of the ferromag-\nnetic resonance1or by the time-resolved magneto-optical\nKerr effect.2Time evolution of the macroscopic magne-\ntization vector Mcan be described by the well-known\nLandau-Lifshitz-Gilbert (LLG) equation3,4\ndM\ndt=Beff×M+M\nM×/parenleftbigg\nα·dM\ndt/parenrightbigg\n,(1)\nwhereBeffdenotes an effective magnetic field (with the\ngyromagnetic ratio absorbed) acting on the magnetiza-\ntion,M=|M|, and the quantity α={αµν}denotes\na symmetric 3 ×3 tensor of the dimensionless Gilbert\ndamping parameters ( µ,ν=x,y,z). The first term in\nEq. (1) defines a precession of the magnetization vector\naround the direction of the effective magnetic field and\nthe second term describes a damping of the dynamics.\nThe LLG equation in itinerant ferromagnets is appropri-\nate for magnetization precessions very slow as compared\nto precessions of the single-electron spin due to the ex-\nchange splitting and to frequencies of interatomic elec-\ntron hoppings.A large number of theoretical approaches to the\nGilbert damping has been workedout during the last two\ndecades; here we mention only schemes within the one-\nelectron theory of itinerant magnets,5–20where the most\nimportant effects of electron-electron interaction are cap-\ntured by means of a local spin-dependent exchange-\ncorrelation (XC) potential. These techniques can be\nnaturally combined with existing first-principles tech-\nniques based on the density-functional theory, which\nleads to parameter-freecalculations of the Gilbert damp-\ning tensor of pure ferromagnetic metals, their ordered\nand disordered alloys, diluted magnetic semiconductors,\netc. One part of these approaches is based on a static\nlimit of the frequency-dependent spin-spin correlation\nfunction of a ferromagnet.5–8,15,16Other routes to the\nGilbert damping employ relaxations of occupation num-\nbers of individual Bloch electron states during quasi-\nstatic nonequilibrium processes or transition rates be-\ntween different states induced by the spin-orbit (SO)\ninteraction.9–12,14,20The dissipation of magnetic energy\naccompanying the slow magnetization dynamics, evalu-\nated within a scattering theory or the Kubo linear re-\nsponse formalism, leads also to explicit expressions for\nthe Gilbert damping tensor.13,17–19Most of these formu-\nlations yield relations equivalent to the so-called torque-\ncorrelation formula\nαµν=−α0Tr{Tµ(G+−G−)Tν(G+−G−)},(2)\ninwhich thetorqueoperators Tµareeither duetothe XC\nor SO terms of the one-electron Hamiltonian. In Eq. (2),\nwhich has a form of the Kubo-Greenwood formula and is\nvalid for zero temperature of electrons, the quantity α0is\nrelated to the system magnetization (and to fundamental2\nconstants and units used, see Section IIB), the trace is\ntaken over the whole Hilbert space of valence electrons,\nandthesymbols G±=G(EF±i0)denotetheone-particle\nretarded and advanced propagators (Green’s functions)\nat the Fermi energy EF.\nImplementation of the above-mentioned theories in\nfirst-principles computational schemes proved opposite\ntrends of the intraband and interband contributions to\nthe Gilbert damping parameter as functions of a phe-\nnomenological quasiparticle lifetime broadening.7,11,12\nThese qualitative studies have recently been put on a\nmore solid basis by considering a particular mechanism\nof the lifetime broadening, namely, a frozen temperature-\ninduced structural disorder, which represents a realistic\nmodel for a treatment of temperature dependence of the\nGilbert damping.21,22This approach explained quanti-\ntatively the low-temperature conductivity-like and high-\ntemperature resistivity-like trends of the damping pa-\nrameters of iron, cobalt and nickel. Further improve-\nmentsofthemodel, includingstatictemperature-induced\nrandomorientationsoflocalmagneticmoments, haveap-\npeared recently.23\nTheab initio studies have also been successful in re-\nproduction and interpretation of values and concentra-\ntion trends of the Gilbert damping in random ferromag-\nnetic alloys, such as the NiFe alloy with the face-centered\ncubic (fcc) structure (Permalloy)17,22and Fe-based al-\nloys with the body-centered cubic (bcc) structure (FeCo,\nFeV,FeSi).19,22,24Otherstudiesaddressedalsotheeffects\nof doping the Permalloy and bcc iron by 5 dtransition-\nmetal elements19,20,22and of the degree of atomic long-\nrange order in equiconcentration FeNi and FePt alloys\nwith the L1 0-type structures.20Recently, an application\nto halfmetallic Co-based Heusler alloys has appeared as\nwell.25The obtained results revealed correlations of the\ndamping parameter with the density of states at the\nFermienergyandwiththesizeofmagneticmoments.22,24\nIn a one-particle mean-field-like description of a ferro-\nmagnet, the total spin is not conserved due to the XC\nfield and the SO interaction. The currently employed\nformsofthetorqueoperators Tµinthe torque-correlation\nformula (2) reflect these two sources; both the XC- and\nthe SO-induced torques are local and their equivalence\nfor the theory of Gilbert damping has been discussed\nby several authors.15,16,26In the case of random alloys,\nthis equivalence rests on a proper inclusion of vertex cor-\nrections in the configuration averaging of the damping\nparameters αµνas two-particle quantities.\nThe purpose of the present paper is to introduce an-\nother torque operator that can be used in the torque-\ncorrelationformula(2) andto discussits properties. This\noperatoris due to intersiteelectronhopping andit is con-\nsequently nonlocal; in contrast to the local XC- and SO-\ninduced torques which are random in random crystalline\nalloys, the nonlocal torque is nonrandom, i.e., indepen-\ndent on the particular configuration of a random alloy,\nwhich simplifies the configuration averaging of Eq. (2).\nWe show that a similar nonlocal effective torque appearsin the fully relativistic linear muffin-tin orbital (LMTO)\nmethod in the atomic-sphere approximation (ASA) used\nrecently for calculations of the conductivity tensor in\nspin-polarized random alloys.27,28Here we discuss theo-\nretical aspects of the averaging in the coherent-potential\napproximation (CPA)29,30and illustrate the developed\nab initio scheme byapplicationsto selected binaryalloys.\nWe also compare the obtained results with those of the\nLMTO-supercell technique17and with other CPA-based\ntechniques, the fully relativisticKorringa-Kohn-Rostoker\n(KKR) method19,22and the LMTO method with a sim-\nplified treatment of the SO interaction.20\nThe paper is organized as follows. The theoretical for-\nmalism is contained in Section II, with a general discus-\nsion of various torque operators and results of a simple\ntight-binding model presented in Section IIA. The fol-\nlowingSection IIB describes the derivation of the LMTO\ntorque-correlation formula with nonlocal torques; tech-\nnical details are left to Appendix A concerning linear-\nresponse calculations with varying basis sets and to Ap-\npendix B regarding the LMTO method for systems with\na tilted magnetization direction. Selected formal proper-\nties of the developed theory are discussed in Section IIC.\nApplications of the developed formalism can be found\nin Section III. Details of numerical implementation are\nlisted in Section IIIA followed by illustrating examples\nforsystemsofthreedifferent kinds: binarysolidsolutions\nof 3dtransition metals in Section IIIB, pure iron with a\nsimple model of random potential fluctuations in Section\nIIIC, and stoichiometric FePt alloys with a partial long-\nrange order in Section IIID. The main conclusions are\nsummarized in Section IV.\nII. THEORETICAL FORMALISM\nA. Torque-correlation formula with alternative\ntorque operators\nThe torque operators Tµentering the torque-\ncorrelation formula (2) are closely related to compo-\nnents of the time derivative of electron spin. For spin-\npolarized systems described by means of an effective\nSchr¨ odinger-Paulione-electronHamiltonian H, actingon\ntwo-componentwavefunctions, thecompletetimederiva-\ntive of the spin operator is given by the commutation re-\nlationtµ=−i[σµ/2,H], where ¯ h= 1 is assumed and σµ\n(µ=x,y,z) denote the Pauli spin matrices. Let us write\nthe Hamiltonian as H=Hp+Hxc, whereHpincludes all\nspin-independent terms and the SO interaction (Hamil-\ntonian of a paramagnetic system) while Hxc=Bxc(r)·σ\ndenotes the XC term due to an effective magnetic field\nBxc(r). The complete time derivative (spin torque) can\nthen be written as tµ=tso\nµ+txc\nµ, where\ntso\nµ=−i[σµ/2,Hp], txc\nµ=−i[σµ/2,Hxc].(3)\nAs discussed, e.g., in Ref. 15, the use of the complete\ntorquetµinthetorque-correlationformula(2)leadsiden-3\ntically to zero; the correct Gilbert damping coefficients\nαµνfollow from Eq. (2) by using either the SO-induced\ntorquetso\nµ, or the XC-induced torque txc\nµ. Note that only\ntransverse components (with respect to the easy axis of\nthe ferromagnet)of the vectors tsoandtxcare needed for\nthe relevant part of the Gilbert damping tensor (2).\nThe equivalence of both torque operators (3) for the\nGilbert damping can be extended. Let us consider a sim-\nple system described by a model tight-binding Hamilto-\nnianH, written now as H=Hloc+Hnl, where the first\ntermHlocis a lattice sum of local atomic-like terms and\nthe nonlocal second term Hnlincludes all intersite hop-\nping matrix elements. Let us assume that all effects of\nthe SO interaction and XC fields are contained in the\nlocal term Hloc, so that the hopping elements are spin-\nindependent and [ σµ,Hnl] = 0. (Note that this assump-\ntion, often used in model studies, is satisfied only ap-\nproximatively in real ferromagnets with different widths\nof the majority and minority spin bands.) Let us write\nexplicitly Hloc=/summationtext\nR(Hp\nR+Hxc\nR), whereRlabelsthe lat-\ntice sites and where Hp\nRcomprises the spin-independent\npart and the SO interaction of the Rth atomic poten-\ntial while Hxc\nRis due to the local XC field of the Rth\natom. The operators Hp\nRandHxc\nRact only in the sub-\nspace of the Rth site; the subspaces of different sites\nare orthogonal to each other. The total spin operator\ncan be written as σµ/2 = (1/2)/summationtext\nRσRµ, where the local\noperator σRµis the projection of σµon theRth sub-\nspace. Let us assume that each term Hp\nRis spherically\nsymmetric and that Hxc\nR=Bxc\nR·σR, where the effec-\ntive field Bxc\nRof theRth atom has a constant size and\ndirection. Let us introduce local orbital-momentum op-\neratorsLRµand their counterparts including the spin,\nJRµ=LRµ+ (σRµ/2), which are generators of local\ninfinitesimal rotations with respect to the Rth lattice\nsite, and let us define the corresponding lattice sums\nLµ=/summationtext\nRLRµandJµ=/summationtext\nRJRµ=Lµ+(σµ/2). Then\nthe local terms Hp\nRandHxc\nRsatisfy, respectively, commu-\ntation rules [ JRµ,Hp\nR] = 0 and [ LRµ,Hxc\nR] = 0. By using\nthe above assumptions and definitions, the XC-induced\nspin torque (3) due to the XC term Hxc=/summationtext\nRHxc\nRcan\nbe reformulated as\ntxc\nµ=−i/summationdisplay\nR[σRµ/2,Hxc\nR] =−i/summationdisplay\nR[JRµ,Hxc\nR] (4)\n=−i/summationdisplay\nR[JRµ,Hp\nR+Hxc\nR] =−i[Jµ,Hloc]≡tloc\nµ.\nThe last commutator defines a local torque operator tloc\nµ\ndue to the local part of the Hamiltonian Hlocand the op-\neratorJµ,incontrasttothespinoperator σµ/2inEq.(3).\nLet us define the complementary nonlocal torque tnl\nµdue\nto the nonlocal part of the Hamiltonian Hnl, namely,\ntnl\nµ=−i[Jµ,Hnl] =−i[Lµ,Hnl], (5)\nand let us employ the fact that the complete time deriva-\ntive of the operator Jµ, i.e., the torque ˜tµ=−i[Jµ,H] =\ntloc\nµ+tnl\nµ, leads identically to zero when used in Eq. (2).This fact implies that the Gilbert damping parame-\nters can be also obtained from the torque-correlation\nformula with the nonlocal torques tnl\nµ. These torques\nare equivalent to the original spin-dependent local XC-\nor SO-induced torques; however, the derived nonlocal\ntorques are spin-independent, so that commutation rules\n[tnl\nµ,σν] = 0 are satisfied.\nInordertoseetheeffect ofdifferent formsofthe torque\noperators, Eqs. (3) and (5), we have studied a tight-\nbinding model of p-orbitals on a simple cubic lattice with\nthe ground-state magnetization along zaxis. The local\n(atomic-like) terms of the Hamiltonian are specified by\nthe XC term bσRzand the SO term ξLR·σR, which\nare added to a random spin-independent p-level at en-\nergyǫ0+DR, whereǫ0denotes the nonrandom center of\nthep-band while the random parts DRsatisfy configu-\nration averages /an}bracketle{tDR/an}bracketri}ht= 0 and /an}bracketle{tDR′DR/an}bracketri}ht=γδR′Rwith\nthe disorder strength γ. The spin-independent nonlocal\n(hopping) part of the Hamiltonian has been confined to\nnonrandom nearest-neighbor hoppings parametrized by\ntwoquantities, W1(ppσhopping) and W′\n1(ppπhopping),\nsee, e.g., page 36 of Ref. 31. The particular values have\nbeen set to b= 0.3,ξ= 0.2,EF−ǫ0= 0.1,γ= 0.05,\nW1= 0.3 andW′\n1=−0.1 (the hoppings were chosen\nsuch that the band edges for ǫ0=b=ξ=γ= 0 are±1).\nTheconfigurationaverageofthe propagators /an}bracketle{tG±/an}bracketri}ht=¯G±\nand of the torque correlation (2) was performed in the\nself-consistentBornapproximation(SCBA)includingthe\nvertex corrections. Since all three torques, Eqs. (3) and\n(5), are nonrandom operators in our model, the only rel-\nevant component of the Gilbert damping tensor, namely\nαxx=αyy=α, could be unambiguously decomposed in\nthe coherent part αcohand the incoherent part αvcdue\nto the vertex corrections.\nThe results are summarized in Fig. 1 which displays\nthe torque correlation α/α0as a function of the SO cou-\nplingξ(Fig. 1a) and the XC field b(Fig. 1b). The total\nvalueα=αcoh+αvcis identical for all three forms of\nthe torque operator, in contrast to the coherent parts\nαcohwhich exhibit markedly different values and trends\nas compared to each other and to the total α. This re-\nsult is in line with conclusions drawn by the authors of\nRef. 15, 16, and 26 proving the importance of the ver-\ntex corrections for obtaining the same Gilbert damping\nparameters from the SO- and XC-induced torques. The\nonly exception seems to be the case of the SO splitting\nmuch weaker than the exchange splitting, where the ver-\ntex corrections for the SO-induced torque can be safely\nneglected, see Fig. 1a. This situation, encountered in\n3dtransition metals and their alloys, has been treated\nwith the SO-induced torque on an ab initio level with ne-\nglectedvertexcorrectionsinRef. 11and12. Onthe other\nhand, the use of the XC-induced torque calls for a proper\nevaluation of the vertex corrections; their neglect leads\ntoquantitativelyandphysicallyincorrectresultsasdocu-\nmented by recent first-principles studies.19,22The vertex\ncorrectionsareindispensablealsoforthe nonlocaltorque,\nin particular for correct vanishing of the total torque cor-4\n 0 2 4\n 0  0.1  0.2torque  correlation\nspin-orbit  coupling(a)\ntotcoh-nl\ncoh-xc\ncoh-so\n 0 2 4\n 0  0.1  0.2  0.3torque  correlation\nexchange  field(b)\ntotcoh-nl\ncoh-xc\ncoh-so\nFIG. 1. (Color online) The torque correlation α/α0, Eq. (2),\nin a tight-binding p-orbital model treated in the SCBA as\na function of the spin-orbit coupling ξ(a) and of the ex-\nchange field b(b). The full diamonds display the total torque\ncorrelation (tot) and the open symbols denote the coherent\ncontributions αcoh/α0calculated with the SO-induced torque\n(coh-so), the XC-induced torque (coh-xc), Eq. (3), and the\nnonlocal torque (coh-nl), Eq. (5).\nrelation both in the nonrelativistic limit ( ξ→0, Fig. 1a)\nand in the nonmagnetic limit ( b→0, Fig. 1b).\nFinally, let us discuss briefly the general equivalence of\nthe SO- and XC-induced spin torques, Eq. (3), in the\nfully relativistic four-component Dirac formalism.32,33\nThe Kohn-Sham-Dirac Hamiltonian can be written as\nH=Hp+Hxc, whereHp=cα·p+mc2β+V(r) and\nHxc=Bxc(r)·βΣ, wherecis the speed of light, mde-\nnotes the electron mass, p={pµ}refers to the momen-\ntum operator, V(r) is the spin-independent part of the\neffective potential and the α={αµ},βandΣ={Σµ}\narethe well-known4 ×4matricesofthe Diractheory.34,35Then the XC-induced torque is txc=Bxc(r)×βΣ,\nwhich is currently used in the KKR theory of the Gilbert\ndamping.19,22The SO-induced torque is tso=p×cα,\ni.e., it is given directly by the relativistic momentum ( p)\nand velocity ( cα) operators. One can see that the torque\ntsoislocalbutindependent oftheparticularsystemstud-\nied. A comparison of both alternatives, concerning the\ntotal damping parameters as well as their coherent and\nincoherent parts, would be desirable; however, this task\nis beyond the scope of the present study.\nB. Effective torques in the LMTO method\nIn ourab initio approach to the Gilbert damping, we\nemploy the torque-correlation formula (2) with torques\nderived from the XC field.15,19,22The torque operators\nare constructed by considering infinitesimal deviations of\nthe direction of the XC field of the ferromagnet from its\nequilibrium orientation, taken asa reference state. These\ndeviations result from rotations by small angles around\naxesperpendiculartothe equilibrium direction ofthe XC\nfield; componentsofthetorqueoperatorarethengivenas\nderivatives of the one-particle Hamiltonian with respect\nto the rotation angles.36\nFor practical evaluation of Eq. (2) in an ab initio tech-\nnique (such as the LMTO method), one has to consider\na matrix representation of all operators in a suitable or-\nthonormal basis. The most efficient techniques of the\nelectronic structure theory require typically basis vectors\ntailored to the system studied; in the present context,\nthis leads naturally to basis sets depending on the angu-\nlar variables needed to define the torque operators. Eval-\nuation of the torque correlation using angle-dependent\nbases is discussed in Appendix A, where we prove that\nEq. (2) can be calculated solely from the matrix ele-\nments of the Hamiltonian and their angular derivatives,\nsee Eq. (A7), whereas the angular dependence of the ba-\nsis vectorsdoes not contribute directly to the final result.\nThe relativistic LMTO-ASA Hamiltonian matrix for\nthe reference system in the orthogonal LMTO represen-\ntation is given by37–39\nH=C+(√\n∆)+S(1−γS)−1√\n∆, (6)\nwhere the C,√\n∆ andγdenote site-diagonal matrices\nof the standard LMTO potential parameters and Sis\nthe matrix of canonical structure constants. The change\nof the Hamiltonian matrix Hdue to a uniform rotation\nof the XC field is treated in Appendix B; it is sum-\nmarized for finite rotations in Eq. (B7) and for angu-\nlar derivatives of Hin Eqs. (B8) and (B9). The resol-\nventG(z) = (z−H)−1of the LMTO Hamiltonian (6)\nfor complex energies zcan be expressed using the auxil-\niary resolvent g(z) = [P(z)−S]−1, which represents an\nLMTO-counterpart of the scattering-path operator ma-\ntrix of the KKR method.32,33The symbol P(z) denotes\nthe site-diagonal matrix of potential functions; their an-\nalytic dependence on zand on the potential parameters5\ncan be found elsewhere.27,37The relation between both\nresolvents leads to the formula28\nG+−G−=F(g+−g−)F+, (7)\nwhere the same abbreviation F= (√\n∆)−1(1−γS) as in\nEq. (B8) was used and g±=g(EF±i0) .\nThe torque-correlation formula (2) in the LMTO-ASA\nmethod follows directly from relations (A7), (B8), (B9)\nand (7). The components of the Gilbert damping tensor\n{αµν}in the LLG equation (1) can be obtained from a\nbasic tensor {˜αµν}given by\n˜αµν=−α0Tr{τµ(g+−g−)τν(g+−g−)},(8)\nwhere the quantities\nτµ=−i[Jµ,S] =−i[Lµ,S] (9)\ndefine components of an effective torque in the LMTO-\nASA method. The site-diagonal matrices JµandLµ\n(µ=x,y,z) are Cartesian components of the total and\norbital angular momentum operator, respectively, see\ntext aroundEqs.(B8) and (B9). The tracein (8) extends\nover all orbitals of the crystalline solid and the prefactor\ncan be written as α0= (2πMspin)−1, where Mspinde-\nnotes the spin magnetic moment of the whole crystal in\nunits of the Bohr magneton µB.15,19,22\nLet us discuss properties of the effective torque (9).\nIts form is obviously identical to the nonlocal torque (5).\nThe matrix τµis non-site-diagonal, but—for a random\nsubstitutional alloy on a nonrandom lattice—it is non-\nrandom (independent on the alloy configuration). More-\nover, it is given by a commutator of the site-diagonal\nnonrandom matrix Jµ(orLµ) and the LMTO structure-\nconstantmatrix S. Thesepropertiespointtoacloseanal-\nogy between the effective torque and the effective veloci-\nties in the LMTO conductivity tensor based on a concept\nof intersite electron hopping.27,28,40Let us mention that\nexisting ab initio approaches employ random torques,\neither the XC-induced torque in the KKR method19,22\nor the SO-induced torque in the LMTO method.20An-\nother interesting property of the effective torque τµ(9)\nis its spin-independence which follows from the spin-\nindependence of the matrices LµandS.\nThe explicit relation between the symmetric tensors\n{αµν}and{˜αµν}canbeeasilyformulatedfortheground-\nstate magnetization along zaxis; then it is given simply\nbyαxx= ˜αyy,αyy= ˜αxx, andαxy=−˜αxy. These\nrelations reflect the fact that an infinitesimal deviation\ntowards xaxis results from an infinitesimal rotation of\nthe magnetization vector around yaxis and vice versa.\nNote that the other components of the Gilbert damp-\ning tensor ( αµzforµ=x,y,z) are not relevant for the\ndynamics of small deviations of magnetization direction\ndescribed by the LLG equation (1). For the ground-\nstate magnetization pointing along a general unit vector\nm= (mx,my,mz), one has to employ the Levi-Civita\nsymbolǫµνλin order to get the Gilbert damping tensorαas\nαµν=/summationdisplay\nµ′ν′ηµµ′ηνν′˜αµ′ν′, (10)\nwhereηµν=/summationtext\nλǫµνλmλ. The resultingtensor(10) satis-\nfies the condition α·m= 0 appropriate for the dynamics\nof small transverse deviations of magnetization.\nThe application to random alloys requires configura-\ntion averaging of ˜ αµν(8). Since the effective torques τµ\nare nonrandom, one can write a unique decomposition\nof the average into the coherent and incoherent parts,\n˜αµν= ˜αcoh\nµν+ ˜αvc\nµν, where the coherent part is expressed\nby means of the averaged auxiliary resolvents ¯ g±=/an}bracketle{tg±/an}bracketri}ht\nas\n˜αcoh\nµν=−α0Tr{τµ(¯g+−¯g−)τν(¯g+−¯g−)}(11)\nand the incoherent part (vertex corrections) is given as a\nsum of four terms, namely,\n˜αvc\nµν=−α0/summationdisplay\np=±/summationdisplay\nq=±sgn(pq)Tr/an}bracketle{tτµgpτνgq/an}bracketri}htvc.(12)\nIn this work, the configuration averaging has been done\nin the CPA. Details concerning the averaged resolvents\ncan be found, e.g., in Ref. 39 and the construction of the\nvertex corrections for transport properties was described\nin Appendix to Ref. 30.\nC. Properties of the LMTO torque-correlation\nformula\nThe damping tensor (8) has been formulated in the\ncanonical LMTO representation. In the numerical im-\nplementation, the well-known transformation to a tight-\nbinding(TB)LMTOrepresentation41,42isadvantageous.\nThe TB-LMTO representation is specified by a diag-\nonal matrix βof spin-independent screening constants\n(βR′ℓ′m′s′,Rℓms=δR′Rδℓ′ℓδm′mδs′sβRℓin a nonrelativis-\ntic basis) and the transformation of all quantities be-\ntween both LMTO representations has been discussed in\nthe literature for pure crystals42as well as for random\nalloys.28,39,43The same techniques can be used in the\npresent case together with an obvious commutation rule\n[Jµ,β] = [Lµ,β] = 0. Consequently, the conclusions\ndrawn are the same as for the conductivity tensor:28the\ntotal damping tensor (8) as well as its coherent (11) and\nincoherent (12) parts in the CPA are invariant with re-\nspect to the choice of the LMTO representation.\nIt should be mentioned that the central result, namely\nthe relations(8) and (9), is not limited to the LMTO the-\nory, but it can be translatedinto the KKRtheory aswell,\nsimilarly to the conductivity tensor in the formalism of\nintersite hopping.40The LMTO structure-constant ma-\ntrixSandtheauxiliaryGreen’sfunction g(z)willbethen\nreplacedrespectivelybytheKKRstructure-constantma-\ntrix and by the scattering-path operator.32,33Note, how-6\never, that the total ( Jµ) and orbital ( Lµ) angular mo-\nmentum operatorsin the effective torques (9) will be rep-\nresented by the same matrices as in the LMTO theory.\nLet us mention for completeness that the present\nLMTO-ASA theory allows one to introduce effective lo-\ncal (but random) torques as well. This is based on the\nfact that only the Fermi-level propagators g±defined by\nthe structure constant matrix Sand by the potential\nfunctions at the Fermi energy, P=P(EF), enter the\nzero-temperature expression for the damping tensor ˜ αµν\n(8). Since the equation of motion ( P−S)g±= 1 implies\nimmediately S(g+−g−) =P(g+−g−) and, similarly,\n(g+−g−)S= (g+−g−)P, one can obviously replace the\nnonlocal torques τµ(9) in the torque-correlation formula\n(8) by their local counterparts\nτxc\nµ= i[P,Jµ], τso\nµ= i[P,Lµ].(13)\nThese effective torques are represented by random, site-\ndiagonal matrices; the τxc\nµandτso\nµcorrespond, respec-\ntively, to the XC-induced torque used in the KKR\nmethod22and to the SO-induced torque used in the\nLMTO method with a simplified treatment of the SO-\ninteraction.20In the case of random alloys treated in the\nCPA, the randomness of the local torques (13) calls for\nthe approach developed by Butler44for the averaging of\nthe torque-correlationcoefficient (8). One can provethat\nthe resulting damping parameters ˜ αµνobtained in the\nCPA with the local and nonlocal torques are fully equiv-\nalent to each other; this equivalence rests heavily on a\nproper inclusion of the vertex corrections45and it leads\nto further important consequences. First, the Gilbert\ndamping tensor vanishes exactly for zero SO interaction,\nwhich follows from the use of the SO-induced torque τso\nµ\nand from the obvious commutation rule [ P,Lµ] = 0 valid\nfor the spherically symmetric potential functions (in the\nabsence of SO interaction). This result is in agreement\nwith thenumericalstudyofthe toymodel inSectionIIA,\nsee Fig. 1a for ξ= 0. On an ab initio level, this prop-\nerty has been obtained numerically both in the KKR\nmethod22and in the LMTO method.26Second, the XC-\nandSO-inducedlocaltorques(13)withintheCPAareex-\nactly equivalent as well, as has been indicated in a recent\nnumerical study for a random bcc Fe 50Co50alloy.26In\nsummary, the nonlocaltorques(9) andboth localtorques\n(13) can be used as equivalent alternatives in the torque-\ncorrelation formula (8) provided that the vertex correc-\ntions are included consistently with the CPA-averaging\nof the single-particle propagators.\nIII. ILLUSTRATING EXAMPLES\nA. Implementation and numerical details\nThe numerical implementation of the described the-\nory and the calculations have been done with similar\ntools as in our recent studies of ground-state46and 3 6 9\n 0  10  20  30103 α\nε  (µRy)fcc Ni80Fe20bcc Fe80Co20\n(x 10)\nFIG. 2. The Gilbert damping parameters αof random fcc\nNi80Fe20(full circles) and bcc Fe 80Co20(open squares) alloys\nas functions of the imaginary part of energy ε. The values of\nαfor the Fe 80Co20alloy are magnified by a factor of 10.\ntransport27,28,47properties. The ground-state magne-\ntization was taken along zaxis and the selfconsistent\nXC potentials were obtained in the local spin-density ap-\nproximation (LSDA) with parametrization according to\nRef. 48. The valence basis comprised s-,p-, andd-type\norbitalsand the energyargumentsforthe propagators¯ g±\nand the CPA-vertex corrections were obtained by adding\na tiny imaginary part ±εto the real Fermi energy. We\nhave found that the dependence of the Gilbert damping\nparameter on εis quite smooth and that the value of\nε= 10−6Ry is sufficient for the studied systems, see\nFig. 2 for an illustration. Similar smooth dependences\nhavebeenobtainedalsoforotherinvestigatedalloys,such\nas Permalloy doped by 5 delements, Heusler alloys, and\nstoichiometric FePt alloys with a partial atomic long-\nrange order. In all studied cases, the number Nofk\nvectors needed for reliable averaging over the Brillouin\nzone (BZ) was properly checked; as a rule, N∼108in\nthe full BZ was sufficient for most systems, but for di-\nluted alloys (a few percent of impurities), N∼109had\nto be taken.\nB. Binary fcc and bcc solid solutions\nThe developed theory has been applied to random bi-\nnary alloys of 3 dtransition elements Fe, Co, and Ni,\nnamely, to the fcc NiFe and bcc FeCo alloys. The most\nimportant results, including a comparison to other exist-\ningab initio techniques, are summarized in Fig. 3. One\ncan see a good agreement of the calculated concentration\ntrends of the Gilbert damping parameter α=αxx=αyy\nwith the results of an LMTO-supercell approach17and\nof the KKR-CPA method.22The decrease of αwith in-7\n 0 4 8 12\n 0  0.2  0.4  0.6103 α\nFe  concentrationfcc  NiFe(a)\nthis work\nLMTO-SC\n 0 2 4 6\n 0  0.2  0.4  0.6103 α\nCo  concentrationbcc  FeCo(b)\nthis work\nKKR-CPA\nFIG. 3. (Color online) The calculated concentration depen-\ndences of the Gilbert damping parameter αfor random fcc\nNiFe (a) and bcc FeCo (b) alloys. The results of this work\nare marked by the full diamonds, whereas the open circles\ndepict the results of other approaches: the LMTO supercell\n(LMTO-SC) technique17and the KKR-CPA method.22\ncreasingFecontentintheconcentratedNiFealloyscanbe\nrelatedto the increasingalloymagnetization17andto the\ndecreasing strength of the SO-interaction,20whereas the\nbehaviorinthedilutelimitcanbeexplainedbyintraband\nscattering due to Fe impurities.11,12,14In the case of the\nFeCo system, the minimum of αaround 20% Co, which\nis also observed in room-temperature experiments,49,50\nis related primarily to a similar concentration trend of\nthe density of states at the Fermi energy,22though the\nmaximum of the magnetization at roughly the same alloy\ncomposition51might partly contribute as well.\nA more detailed comparison of all ab initio results is\npresented in Table I for the fcc Ni 80Fe20random alloy\n(Permalloy). The differences in the values of αfrom the\ndifferenttechniquescanbe ascribedtovarioustheoretical\nfeatures and numericaldetails employed, such asthe sim-TABLE I. Comparison of the Gilbert damping parameter α\nfor the fcc Ni 80Fe20random alloy (Permalloy) calculated by\nthe present approach and by other techniques using the CPA\nor supercells (SC). The last column displays the coherent pa rt\nαcohof the total damping parameter according to Eq. (11).\nThe experimental value corresponds to room temperature.\nMethod α αcoh\nThis work, ε= 10−5Ry 4 .9×10−31.76\nThis work, ε= 10−6Ry 3 .9×10−31.76\nKKR-CPAa4.2×10−3\nLMTO-CPAb3.5×10−3\nLMTO-SCc4.6×10−3\nExperimentd8×10−3\naReference 22.\nbReference 20.\ncReference 17.\ndReference 49.\nplified treatment of the SO-interaction in Ref. 20 instead\nof the fully relativistic description, or the use of super-\ncells in Ref. 17 instead of the CPA. Taking into account\nthat calculated residual resistivities for this alloy span a\nwide interval between 2 µΩcm, see Ref. 27 and 52, and\n3.5µΩcm, see Ref. 17, one can consider the scatter of the\ncalculated values of αin Table I as little important. The\ntheoretical values of αare smaller systematically than\nthe measured values, typically by a factor of two. This\ndiscrepancy might be partly due to the effects of finite\ntemperatures as well as due to additional structural de-\nfects of real samples.\nA closer look at the theoretical results reveals that the\ntotal damping parameters αareappreciablysmallerthan\nthemagnitudesoftheircoherentandvertexparts,seeTa-\nble I for the case of Permalloy. This is in agreement with\ntheresultsofthemodelstudyinSectionIIA;similarcon-\nclusions about the importance of the vertex corrections\nhave been done with the XC-induced torques in other\nCPA-based studies.19,22,26The present results prove that\nthis unpleasant feature of the nonlocal torques does not\nrepresent a serious obstacle in obtaining reliable values\nof the Gilbert damping parameter in random alloys. We\nnote that the vertex corrections can be negligible in ap-\nproaches employing the SO-induced torques, at least for\nsystems with the SO splittings much weaker than the XC\nsplittings,12such as the binary ferromagnetic alloys of 3 d\ntransition metals,26see also Section IIA.\nC. Pure iron with a model disorder\nAs it has been mentioned in Section I, the Gilbert\ndamping of pure ferromagnetic metals exhibits non-\ntrivial temperature dependences, which have been re-\nproduced by means of ab initio techniques with vari-\nous levels of sophistication.11,12,21,23In this study, we\nhave simulated the effect of finite temperatures by intro-8\n 0 3 6 9\n0123402040103 α\nρ  (µΩcm)\n103 δ2  (Ry2)bcc  Fe\nFIG. 4. (Color online) The calculated Gilbert damping pa-\nrameter α(full squares) and the residual resistivity ρ(open\ncircles) of pure bcc iron as functions of δ2, where δis the\nstrength of a model atomic-level disorder.\nducing static fluctuations of the one-particle potential.\nThe adopted model of atomic-level disorderassumes that\nrandom spin-independent shifts ±δ, constant inside each\natomic sphere and occurring with probabilities 50% of\nbothsigns,areaddedtothenonrandomselfconsistentpo-\ntential obtained at zero temperature. The Fermi energy\niskeptfrozen,equaltoitsselfconsistentzero-temperature\nvalue. This model can be easily treated in the CPA; the\nresulting Gilbert damping parameter αof pure bcc Fe as\na function of the potential shift δis plotted in Fig. 4.\nThecalculateddependence α(δ) isnonmonotonic, with\na minimum at δ≈30 mRy. This trend is in a qualita-\ntive agreement with trends reported previously by other\nauthors, who employed phenomenological models of the\nelectron lifetime11,12as well as models for phonons and\nmagnons.21,23The origin of the nonmonotonic depen-\ndenceα(δ) has been identified on the basis of the band\nstructure of the ferromagnetic system as an interplay be-\ntween the intraband contributions to α, dominating for\nsmall values of δ, and the interband contributions, domi-\nnating for large values of δ.7,11,12Since the present CPA-\nbased approach does not use any bands, we cannot per-\nform a similar analysis.\nThe obtained minimum value of the Gilbert damping,\nαmin≈10−3(Fig. 4), agrees reasonably well with the\nvalues obtained by the authors of Ref. 11, 12, 21, and\n23. This agreement indicates that the atomic-level dis-\norder employed here is equivalent to a phenomenological\nlifetime broadening. For a rough quantitative estimation\nof the temperature effect, one can employ the calculated\nresistivity ρof the model, which increases essentially lin-\nearly with δ2, see Fig. 4. Since the metallic resistivity\ndue to phonons increases linearly with the temperature\nT(for temperatures not much smaller than the Debye\ntemperature), one can assume a proportionality between 10 20 30 40\n0 0.5 1152535103 α\nDOS(EF)  (states/Ry)\nLRO  parameter  SL10  FePt\nFIG. 5. (Color online) The calculated Gilbert damping pa-\nrameter α(full squares) and the total DOS (per formula unit)\nat the Fermi energy (open circles) of stoichiometric L1 0FePt\nalloys as functions of the LRO parameter S.\nδ2andT. The resistivity of bcc iron at the Curie tem-\nperature TC= 1044 K due to lattice vibrations can be\nestimated around 35 µΩcm,23,53which sets an approx-\nimate temperature scale to the data plotted in Fig. 4.\nHowever, a more accurate description of the temperature\ndependence of the Gilbert damping parameter cannot be\nobtained, mainly due to the neglected true atomic dis-\nplacements and the noncollinearity of magnetic moments\n(magnons).23\nD. FePt alloys with a partial long-range order\nSince important ferromagnetic materials include or-\ndered alloys, we address here the Gilbert damping in sto-\nichiometric FePt alloys with L1 0atomic long-range order\n(LRO). Their transport properties47and the damping\nparameter20have recently been studied by means of the\nTB-LMTO method in dependence on a varying degree\nof the LRO. These fcc-based systems contain two sublat-\nticeswith respectiveoccupationsFe 1−yPtyandPt 1−yFey\nwherey(0≤y≤0.5) denotes the concentration of anti-\nsite atoms. The LRO parameter S(0≤S≤1) is then\ndefined as S= 1−2y, so that S= 0 corresponds to the\nrandom fcc alloy and S= 1 corresponds to the perfectly\nordered L1 0structure.\nThe resulting Gilbert damping parameter is displayed\nin Fig. 5 as a function of S. The obtained trend with a\nbroadmaximumat S= 0andaminimumaround S= 0.9\nagrees very well with the previous result.20The values of\nαin Fig. 5 are about 10% higher than those in Ref. 20,\nwhich can be ascribed to the fully relativistic treatment\nin the present study in contrast to a simplified treatment\nof the SO interaction in Ref. 20. The Gilbert damping9\nin the FePt alloys is an order of magnitude stronger than\nin the alloys of 3 delements (Section IIIB) owing to the\nstronger SO interaction of Pt atoms. The origin of the\nslow decrease of αwith increasing S(for 0≤S≤0.9)\ncan be explained by the decreasing total density of states\n(DOS) at the Fermi energy, see Fig. 5, which represents\nan analogy to a similar correlation observed, e.g., for bcc\nFeCo alloys.22\nAll calculated values of αshown in Fig. 5, correspond-\ning to 0 ≤S≤0.985, are appreciably smaller than\nthe measured one which amounts to α≈0.06 reported\nfor a thin L1 0FePt epitaxial film.54The high measured\nvalue of αmight be thus explained by the present cal-\nculations by assuming a very small concentration of an-\ntisites in the prepared films, which does not seem too\nrealistic. Another potential source of the discrepancy\nlies in the thin-film geometry used in the experiment.\nMoreover, the divergence of αin the limit of S→1\n(Fig. 5) illustrates a general shortcoming of approaches\nbased on the torque-correlation formula (2), since the\nzero-temperature Gilbert damping parameter of a pure\nferromagnet should remain finite. A correct treatment\nof this case, including the dilute limit of random alloys\n(Fig. 3), must take into account the full interacting sus-\nceptibility in the presence of SO interaction.15,55Pilotab\ninitiostudies in this direction have recently appeared for\nnonrandom systems;56,57however, their extension to dis-\nordered systems goes far beyond the scope of this work.\nIV. CONCLUSIONS\nWe have introduced nonlocal torques as an alterna-\ntive to the usual local torque operators entering the\ntorque-correlation formula for the Gilbert damping ten-\nsor. Within the relativistic TB-LMTO-ASA method,\nthis idea leads to effective nonlocal torques as non-site-\ndiagonal and spin-independent matrices. For substitu-\ntionally disordered alloys, the nonlocal torques are non-\nrandom, which allows one to develop an internally con-\nsistent theory in the CPA. The CPA-vertex corrections\nproved indispensable for an exact equivalence of the non-\nlocal nonrandom torques with their local random coun-\nterparts. The concept of the nonlocal torques is not lim-\nited to the LMTO method and its formulation both in\na semiempirical TB theory and in the KKR theory is\nstraightforward.\nThe numerical implementation and the results for bi-\nnary solid solutions show that the total Gilbert damping\nparameters from the nonlocal torques are much smaller\nthan magnitudes of the coherent parts and of the ver-\ntex corrections. Nevertheless, the total damping param-\neters for the studied NiFe, FeCo and FePt alloys compare\nquantitatively very well with results of other ab initio\ntechniques,17,20,22which indicates a fair numerical sta-\nbility of the developed theory.\nThe performed numerical study of the Gilbert damp-\ning in pure bcc iron as a function of an atomic-level dis-order yields a nonmonotonic dependence in a qualitative\nagreementwith the trends consisting of the conductivity-\nlike and resistivity-like regions, obtained from a phe-\nnomenological quasiparticle lifetime broadening7,11,12or\nfrom the temperature-induced frozen phonons21,22and\nmagnons.23Future studies should clarify the applicabil-\nity of the introduced nonlocal torques to a full quanti-\ntative description of the finite-temperature behavior as\nwell as to other torque-related phenomena, such as the\nspin-orbit torques due to applied electric fields.58,59\nACKNOWLEDGMENTS\nThe authors acknowledge financial support by the\nCzech Science Foundation (Grant No. 15-13436S).\nAppendix A: Torque correlation formula in a matrix\nrepresentation\nIn this Appendix, evaluation of the Kubo-Greenwood\nexpression for the torque-correlation formula (2) is dis-\ncussed in the case of the XC-induced torque operators\nusing matrix representations of all operators in an or-\nthonormal basis that varies due to the varying direc-\ntion of the XC field. All operators are denoted by a\nhat, in order to be distinguished from matrices repre-\nsenting these operators in the chosen basis. Let us con-\nsider a one-particle Hamiltonian ˆH=ˆH(θ1,θ2) depend-\ning on two real variables θj,j= 1,2, and let us denote\nˆT(j)(θ1,θ2) =∂ˆH(θ1,θ2)/∂θj. In our case, the variables\nθjplay the role of rotation angles and the operators ˆT(j)\nare the corresponding torques. Let us denote the resol-\nvents of ˆH(θ1,θ2) at the Fermi energy as ˆG±(θ1,θ2) and\nlet us consider a special linear response coefficient (argu-\nmentsθ1andθ2are omitted here and below for brevity)\nc= Tr{ˆT(1)(ˆG+−ˆG−)ˆT(2)(ˆG+−ˆG−)} (A1)\n= Tr{(∂ˆH/∂θ1)(ˆG+−ˆG−)(∂ˆH/∂θ2)(ˆG+−ˆG−)}.\nThis torque-correlation coefficient equals the Gilbert\ndamping parameter (2) with the prefactor ( −α0) sup-\npressed. For its evaluation, we introduce an orthonormal\nbasis|χm(θ1,θ2)/an}bracketri}htand represent all operators in this ba-\nsis. This leads to matrices H(θ1,θ2) ={Hmn(θ1,θ2)},\nG±(θ1,θ2) ={(G±)mn(θ1,θ2)}andT(j)(θ1,θ2) =\n{T(j)\nmn(θ1,θ2)}, where\nHmn=/an}bracketle{tχm|ˆH|χn/an}bracketri}ht,(G±)mn=/an}bracketle{tχm|ˆG±|χn/an}bracketri}ht,\nT(j)\nmn=/an}bracketle{tχm|ˆT(j)|χn/an}bracketri}ht=/an}bracketle{tχm|∂ˆH/∂θj|χn/an}bracketri}ht,(A2)\nand, consequently, to the response coefficient (A1) ex-\npressed by using the matrices (A2) as\nc= Tr{T(1)(G+−G−)T(2)(G+−G−)}.(A3)\nHowever, in evaluation of the last expression, atten-\ntion has to be paid to the difference between the ma-\ntrixT(j)(θ1,θ2) and the partial derivative of the matrix10\nH(θ1,θ2) with respect to θj. This difference follows from\nthe identity ˆH=/summationtext\nmn|χm/an}bracketri}htHmn/an}bracketle{tχn|, which yields\nT(j)\nmn=∂Hmn/∂θj+/summationdisplay\nk/an}bracketle{tχm|∂χk/∂θj/an}bracketri}htHkn\n+/summationdisplay\nkHmk/an}bracketle{t∂χk/∂θj|χn/an}bracketri}ht, (A4)\nwhere we employed the orthogonality relations\n/an}bracketle{tχm(θ1,θ2)|χn(θ1,θ2)/an}bracketri}ht=δmn. Their partial derivatives\nyield\n/an}bracketle{tχm|∂χn/∂θj/an}bracketri}ht=−/an}bracketle{t∂χm/∂θj|χn/an}bracketri}ht ≡Q(j)\nmn,(A5)\nwhere we introduced elements of matrices Q(j)={Q(j)\nmn}\nforj= 1,2. Note that the matrices Q(j)(θ1,θ2) reflect\nexplicitlythe dependenceofthebasisvectors |χm(θ1,θ2)/an}bracketri}ht\nonθ1andθ2. The relation (A4) between the matrices\nT(j)and∂H/∂θ jcan be now rewritten compactly as\nT(j)=∂H/∂θ j+[Q(j),H]. (A6)\nSince the last term has a form of a commutator with the\nHamiltonianmatrix H, theuseofEq.(A6)intheformula\n(A3) leads to the final matrix expression for the torque\ncorrelation,\nc= Tr{(∂H/∂θ 1)(G+−G−)(∂H/∂θ 2)(G+−G−)}.(A7)\nThe equivalence of Eqs. (A3) and (A7) rests on the rules\n[Q(j),H] = [EF−H,Q(j)] and (EF−H)(G+−G−) =\n(G+−G−)(EF−H) = 0 and on the cyclic invariance of\nthe trace. It is also required that the matrices Q(j)are\ncompatible with periodic boundary conditions used in\ncalculations of extended systems, which is obviously the\ncase for angular variables θjrelated to the global changes\n(uniform rotations) of the magnetization direction.\nThe obtained result means that the original response\ncoefficient (A1) involving the torques as angular deriva-\ntives of the Hamiltonian can be expressed solely by us-\ning matrix elements of the Hamiltonian in an angle-\ndependent basis; theangulardependence ofthebasisvec-\ntors does not enter explicitly the final torque-correlation\nformula (A7).\nAppendix B: LMTO Hamiltonian of a ferromagnet\nwith a tilted magnetic field\nHere we sketch a derivation of the fully relativis-\ntic LMTO Hamiltonian matrix for a ferromagnet with\nthe XC-field direction tilted from a reference direction\nalong an easy axis. The derivation rests on the form of\nthe Kohn-Sham-Dirac Hamiltonian in the LMTO-ASA\nmethod.37–39The symbols with superscript 0 refer to the\nreferencesystem,thesymbolswithoutthissuperscriptre-\nfer to the system with the tilted XC field. The operators\n(Hamiltonians, rotation operators) are denoted by sym-\nbols with a hat. The spin-dependent parts of the ASApotentials due to the XC fields are rigidly rotated while\nthe spin-independent parts are unchanged, in full anal-\nogy to the approach employed in the relativistic KKR\nmethod.19,22\nThe ASA-Hamiltonians of both systems are given by\nlattice sums ˆH0=/summationtext\nRˆH0\nRandˆH=/summationtext\nRˆHR, where\nthe individual site-contributions are coupled mutually by\nˆHR=ˆURˆH0\nRˆU+\nR, whereˆURdenotesthe unitaryoperator\nof a rotation (in the orbital and spin space) around the\nRth lattice site which brings the local XC field from its\nreference direction into the tilted one. Let |φ0\nRΛ/an}bracketri}htand\n|˙φ0\nRΛ/an}bracketri}htdenote, respectively, the phi and phi-dot orbitals\nof the reference Hamiltonian ˆH0\nR, then\n|φRΛ/an}bracketri}ht=ˆUR|φ0\nRΛ/an}bracketri}ht,|˙φRΛ/an}bracketri}ht=ˆUR|˙φ0\nRΛ/an}bracketri}ht(B1)\ndefine the phi and phi-dot orbitals of the Hamiltonian\nˆHR. The orbital index Λ labels all linearly indepen-\ndentsolutions(regularattheorigin)ofthespin-polarized\nrelativistic single-site problem; the detailed structure of\nΛ can be found elsewhere.37–39Let us introduce further\nthe well-known empty-space solutions |K∞,0\nRN/an}bracketri}ht(extending\nover the whole real space), |Kint,0\nRN/an}bracketri}ht(extending over the\ninterstitial region), and |K0\nRN/an}bracketri}htand|J0\nRN/an}bracketri}ht(both trun-\ncated outside the Rth sphere), needed for the definition\nof the LMTOs of the reference system.41,42,60Their in-\ndexN, which defines the spin-spherical harmonics of the\nlarge component of each solution, can be taken either in\nthe nonrelativistic ( ℓms) form or in its relativistic ( κµ)\ncounterpart. We define further\n|ZRN/an}bracketri}ht=ˆUR|Z0\nRN/an}bracketri}htforZ=K∞, K, J. (B2)\nIsotropyofthe emptyspaceguaranteesrelations(for Z=\nK∞,K,J)\n|ZRN/an}bracketri}ht=/summationdisplay\nN′|Z0\nRN′/an}bracketri}htUN′N,\n|Z0\nRN/an}bracketri}ht=/summationdisplay\nN′|ZRN′/an}bracketri}htU+\nN′N, (B3)\nwhereU={UN′N}denotes a unitary matrix represent-\ning the rotation in the space of spin-spherical harmonics\nand where U+\nN′N≡(U+)N′N= (UNN′)∗= (U−1)N′N;\nthe matrix Uis the same for all lattice sites Rsince we\nconsider only uniform rotations of the XC-field direction\ninside the ferromagnet. The expansion theorem for the\nenvelope orbital |K∞,0\nRN/an}bracketri}htis\n|K∞,0\nRN/an}bracketri}ht=|Kint,0\nRN/an}bracketri}ht+|K0\nRN/an}bracketri}ht\n−/summationdisplay\nR′N′|J0\nR′N′/an}bracketri}htS0\nR′N′,RN,(B4)\nwhereS0\nR′N′,RNdenote elements of the canonical\nstructure-constant matrix (with vanishing on-site ele-\nments,S0\nRN′,RN= 0) of the reference system. The use\nof relations (B3) in the expansion (B4) together with an\nabbreviation\n|Kint\nRN/an}bracketri}ht=/summationdisplay\nN′|Kint,0\nRN′/an}bracketri}htUN′N (B5)11\nyields the expansion of the envelope orbital |K∞\nRN/an}bracketri}htas\n|K∞\nRN/an}bracketri}ht=|Kint\nRN/an}bracketri}ht+|KRN/an}bracketri}ht\n−/summationdisplay\nR′N′|JR′N′/an}bracketri}ht(U+S0U)R′N′,RN,(B6)\nwhereUandU+denote site-diagonal matrices with el-\nementsUR′N′,RN=δR′RUN′Nand (U+)R′N′,RN=\nδR′RU+\nN′N. Note the same form of expansions (B4) and\n(B6), with the orbitals |Z0\nRN/an}bracketri}htreplaced by the rotated or-\nbitals|ZRN/an}bracketri}ht(Z=K∞,K,J), with the interstitial parts\n|Kint,0\nRN/an}bracketri}htreplacedbytheirlinearcombinations |Kint\nRN/an}bracketri}ht, and\nwith the structure-constant matrix S0replaced by the\nproduct U+S0U.\nThe non-orthogonal LMTO |χ0\nRN/an}bracketri}htfor the reference\nsystem is obtained from the expansion (B4), in which all\norbitals|K0\nRN/an}bracketri}htand|J0\nRN/an}bracketri}htare replaced by linear com-\nbinations of |φ0\nRΛ/an}bracketri}htand|˙φ0\nRΛ/an}bracketri}ht. A similar replacement of\nthe orbitals |KRN/an}bracketri}htand|JRN/an}bracketri}htby linear combinations of\n|φRΛ/an}bracketri}htand|˙φRΛ/an}bracketri}htin the expansion (B6) yields the non-\northogonal LMTO |χRN/an}bracketri}htfor the system with the tilted\nXC field. The coefficients in these linear combinations—\nobtained from conditions of continuous matching at the\nsphere boundaries and leading directly to the LMTO po-\ntentialparameters—areidenticalforboth systems, asfol-\nlows from the rotationrelations (B1) and (B2). For these\nreasons, the only essential difference between both sys-\ntems in the construction of the non-orthogonal and or-\nthogonal LMTOs (and of the accompanying Hamiltonian\nand overlap matrices in the ASA) is due to the difference\nbetween the matrices S0andU+S0U.\nAs a consequence, the LMTO Hamiltonian matrix in\nthe orthogonalLMTO representationfor the system with\na tilted magnetizationis easilyobtained fromthat forthe\nreference system, Eq. (6), and it is given by\nH=C+(√\n∆)+U+SU(1−γU+SU)−1√\n∆,(B7)\nwhere the C,√\n∆ andγare site-diagonal matrices of\nthe potential parameters of the reference system and\nwhere we suppressed the superscript 0 at the structure-\nconstant matrix Sof the reference system. Note that\nthe dependence of Hon the XC-field direction is con-\ntained only in the similarity transformation U+SUof\nthe original structure-constant matrix Sgenerated by\nthe rotation matrix U. For the rotation by an angle\nθaround an axis along a unit vector n, the rotation\nmatrix is given by U(θ) = exp( −in·Jθ), where the\nsite-diagonal matrices J≡(Jx,Jy,Jz) with matrix\nelements Jµ\nR′N′,RN=δR′RJµ\nN′N(µ=x,y,z) reduce\nto usual matrices of the total (orbital plus spin) angu-\nlar momentum operator. The limit of small θyields\nU(θ)≈1−in·Jθ, which leads to the θ-derivative of\nthe Hamiltonian matrix (B7) at θ= 0:\n∂H/∂θ= i(F+)−1[n·J,S]F−1, (B8)\nwhere we abbreviated F= (√\n∆)−1(1−γS) andF+=\n(1−Sγ)[(√\n∆)+]−1. Since the structure-constant matrixSis spin-independent, the total angular momentum op-\neratorJin (B8) canbe replacedbyits orbitalmomentum\ncounterpart L≡(Lx,Ly,Lz), so that\n∂H/∂θ= i(F+)−1[n·L,S]F−1.(B9)\nThe relations (B8) and (B9) are used to derive the\nLMTO-ASA torque-correlation formula (8).\nAppendix C: Equivalence of the Gilbert damping in\nthe CPA with local and nonlocal torques\n(Supplemental Material)\n1. Introductory remarks\nThe problem of equivalence of the Gilbert damping\ntensor expressed with the local (loc) and nonlocal (nl)\ntorques can be reduced to the problem of equivalence of\nthese two expressions:\nαloc=α0Tr/an}bracketle{t(g+−g−)[P,K](g+−g−)[P,K]/an}bracketri}ht\n=α0/summationdisplay\np=±/summationdisplay\nq=±sgn(pq)Tr/an}bracketle{tgp[P,K]gq[P,K]/an}bracketri}ht\n=α0/summationdisplay\np=±/summationdisplay\nq=±sgn(pq)βloc\npq, (C1)\nand\nαnl=α0Tr/an}bracketle{t(g+−g−)[K,S](g+−g−)[K,S]/an}bracketri}ht\n=α0/summationdisplay\np=±/summationdisplay\nq=±sgn(pq)Tr/an}bracketle{tgp[K,S]gq[K,S]/an}bracketri}ht\n=α0/summationdisplay\np=±/summationdisplay\nq=±sgn(pq)βnl\npq. (C2)\nThe symbols Tr and /an}bracketle{t.../an}bracketri}htand the quantities α0,g±,P\nandShavethe samemeaning as in the main text and the\nquantity Ksubstitutes any of the operators (matrices)\nJµorLµ. Note that owing to the symmetric nature of\ntheoriginaldampingtensors,theanalysiscanbeconfined\nto scalar quantities αlocandαnldepending on a general\nsite-diagonal nonrandom operator K. The choice of K=\nKµin (C1) and (C2) produces the diagonal elements of\nboth tensors, whereas the choice of K=Kµ±Kνfor\nµ/ne}ationslash=νleads to all off-diagonal elements. The quantities\nβloc\npqandβnl\npqare expressions of the form\nβloc= Tr/an}bracketle{tg1(P1K−KP2)g2(P2K−KP1)/an}bracketri}ht,\nβnl= Tr/an}bracketle{tg1[K,S]g2[K,S]/an}bracketri}ht, (C3)\nwhere the g1andg2replace the gpandgq, respectively.\nFor an internal consistency of these and following expres-\nsions, we have also introduced P1=P2=P.\nThis supplement contains a proof of the equivalence\nofβlocandβnland, consequently, of αlocandαnl. The\nCPA-average in βnlwith a nonlocal nonrandom torque\nhas been done using the theory by Velick´ y29as worked\nout in detail within the present LMTO formalism by\nCarva et al.30whereas the averaging in βlocinvolving\na local but random torque has been treated using the\napproach by Butler.4412\n2. Auxiliary quantities and relations\nSincethenecessaryformulasoftheCPAinmultiorbital\ntechniques30,44are little transparent, partly owing to the\ncomplicated indices of two-particle quantities, we employ\nhere a formalism with the lattice-site index Rkept but\nwith all orbital indices suppressed.\nThe Hilbert spaceis a sum ofmutually orthogonalsub-\nspaces of individual lattice sites R; the corresponding\nprojectors will be denoted by Π R. A number of rele-\nvant operators are site-diagonal, i.e., they can be written\nasX=/summationtext\nRXR, where the site contributions are given\nbyXR= ΠRX=XΠR= ΠRXΠR. Such operators\nare, e.g., the random potential functions, Pj=/summationtext\nRPj\nR,\nand the nonrandom coherent potential functions Pj=/summationtext\nRPj\nR, wherej= 1,2. The operator Kin (C3) is site-\ndiagonal as well, but its site contributions KRwill not\nbe used explicitly in the following.\nAmong the number ofCPA-relationsfor single-particle\nproperties, we will use the equation of motion for the\naverage auxiliary Green’s functions ¯ gj(j= 1,2),\n¯gj(Pj−S) = (Pj−S)¯gj= 1, (C4)\nas well as the definition of random single-site t-matrices\ntj\nR(j= 1,2) with respect to the effective CPA-medium,\ngiven by\ntj\nR= (Pj\nR−Pj\nR)[1+ ¯gj(Pj\nR−Pj\nR)]−1.(C5)\nThe operators tj\nRare site-diagonal, being non-zero only\nin the subspace of site R. The last definition leads to\nidentities\n(1−t1\nR¯g1)P1\nR=P1\nR+t1\nR(1−¯g1P1\nR),\nP2\nR(1−¯g2t2\nR) =P2\nR+(1−P2\nR¯g2)t2\nR,(C6)\nwhich will be employed below together with the CPA-\nselfconsistency conditions /an}bracketle{ttj\nR/an}bracketri}ht= 0 (j= 1,2).\nFor the purpose of evaluation of the two-particle aver-\nages in (C3), we introduce several nonrandom operators:\nf12= ¯g1K−K¯g2, ζ12= ¯g1[K,S]¯g2,(C7)\nand a site-diagonal operator γ12=/summationtext\nRγ12\nR, where\nγ12=P1K−KP2, γ12\nR=P1\nRK−KP2\nR.(C8)\nBy interchanging the superscripts 1 ↔2 in (C7) and\n(C8), one can also get quantities f21,ζ21,γ21andγ21\nR;\nthis will be implicitly understood in the relations below\nas well. The three operators f12,ζ12andγ12satisfy a\nrelation\nf12+ζ12+¯g1γ12¯g2= 0, (C9)\nwhich can be easily proved from their definitions (C7)\nand (C8) and from the equation of motion (C4). An-\nother quantityto be used in the followingis a nonrandomsite-diagonal operator ϑ12related to the local torque and\ndefined by\nϑ12\nR=/an}bracketle{t(1−t1\nR¯g1)(P1\nRK−KP2\nR)(1−¯g2t2\nR)/an}bracketri}ht,\nϑ12=/summationdisplay\nRϑ12\nR. (C10)\nIts site contributions can be rewritten explicitly as\nϑ12\nR=γ12\nR+/an}bracketle{tt1\nR(f12+ ¯g1γ12\nR¯g2)t2\nR/an}bracketri}ht.(C11)\nThe last relation follows from the definition (C10), from\ntheidentities(C6)andfromtheCPA-selfconsistencycon-\nditions. Moreover,the site contributions ϑ12\nRandγ12\nRsat-\nisfy a sum rule\nγ12\nR=/summationdisplay\nR′′/an}bracketle{tt1\nR¯g1γ12\nR′¯g2t2\nR/an}bracketri}ht+/an}bracketle{tt1\nRζ12t2\nR/an}bracketri}ht+ϑ12\nR,(C12)\nwhere the prime at the sum excludes the term with R′=\nR. This sum rule can be proved by using the definitions\nofζ12(C7) and γ12\nR(C8) and by employing the previous\nrelation for ϑ12\nR(C11) and the equation of motion (C4).\nThe treatment of two-particle quantities requires the\nuse of a direct product a⊗bof two operators aandb.\nThis is equivalent to the concept of a superoperator, i.e.,\na linear mapping defined on the vector space of all linear\noperators. In this supplement, superoperators are de-\nnoted by an overhat, e.g., ˆ m. In the present formalism,\nthe direct product of two operators aandbcan be iden-\ntified with a superoperator ˆ m=a⊗b, which induces a\nmapping\nx/ma√sto→ˆmx= (a⊗b)x=axb, (C13)\nwherexdenotes an arbitrary usual operator. This defi-\nnition leads, e.g., to a superoperator multiplication rule\n(a⊗b)(c⊗d) = (ac)⊗(db). (C14)\nIn the CPA, the most important superoperators are\nˆw12=/summationdisplay\nR/an}bracketle{tt1\nR⊗t2\nR/an}bracketri}ht (C15)\nand\nˆχ12=/summationdisplay\nRR′′\nΠR¯g1ΠR′⊗ΠR′¯g2ΠR(C16)\nwhere the prime at the double sum excludes the terms\nwithR=R′. The quantity ˆ w12represents the irre-\nducible CPA-vertex and the quantity ˆ χ12corresponds to\narestrictedtwo-particlepropagatorwithexcludedon-site\nterms. By using these superoperators, the previous sum\nrule (C12) can be rewritten compactly as\n(ˆ1−ˆw12ˆχ12)γ12= ˆw12ζ12+ϑ12,(C17)\nwhereˆ1 = 1⊗1 denotes the unit superoperator.13\nLet us introduce finally a symbol {x;y}, wherexand\nyare arbitrary operators, which is defined by\n{x;y}= Tr(xy). (C18)\nThissymbolissymmetric, {x;y}={y;x}, linearinboth\narguments and it satisfies the rule\n{(a⊗b)x;y}={x;(b⊗a)y},(C19)\nwhich follows from the cyclic invariance of the trace. An\nobvious consequence of this rule are relations\n{ˆw12x;y}={x; ˆw21y},\n{ˆχ12x;y}={x; ˆχ21y}, (C20)\nwhere ˆw21and ˆχ21are defined by (C15) and (C16) with\nthe superscript interchange 1 ↔2.\n3. Expression with the nonlocal torque\nThe configuration averaging in βnl(C3), which con-\ntains the nonrandom operator [ K,S], leads to two terms\nβnl=βnl,coh+βnl,vc, (C21)\nwhere the coherent part is given by\nβnl,coh= Tr{¯g1[K,S]¯g2[K,S]}(C22)\nand the vertex corrections can be compactly written as30\nβnl,vc={(ˆ1−ˆw12ˆχ12)−1ˆw12ζ12;ζ21},(C23)\nwith all symbols and quantities defined in the previous\nsection. The coherent part can be written as a sum of\nfour terms,\nβnl,coh=βnl,coh\nA+βnl,coh\nB+βnl,coh\nC+βnl,coh\nD,\nβnl,coh\nA= Tr{S¯g1KS¯g2K},\nβnl,coh\nB= Tr{¯g1SK¯g2SK},\nβnl,coh\nC=−Tr{¯g1KS¯g2SK},\nβnl,coh\nD=−Tr{S¯g1SK¯g2K}, (C24)\nwhich can be further modified using the equation of mo-\ntion (C4) and its consequences, e.g., S¯gj=Pj¯gj−1. For\nthe first term βnl,coh\nA, one obtains:\nβnl,coh\nA= Tr{P1¯g1KP2¯g2K}+Tr{KK}\n−Tr{KP2¯g2K}−Tr{P1¯g1KK}.(C25)\nThe last three terms do not contribute to the sum over\nfour pairs of indices ( p,q), where p,q∈ {+,−}, in\nEq. (C2). For this reason, they can be omitted for the\npresent purpose, which yields expressions\n˜βnl,coh\nA= Tr{P1¯g1KP2¯g2K},\n˜βnl,coh\nB= Tr{¯g1P1K¯g2P2K}, (C26)where the second relation is obtained in the same way\nfrom the original term βnl,coh\nB. A similar approach can\nbe applied to the third term βnl,coh\nC, which yields\nβnl,coh\nC=−Tr{¯g1KP2¯g2P2K}\n+Tr{¯g1KP2K}+Tr{¯g1KSK}.(C27)\nThe last term does not contribute to the sum over four\npairs (p,q) in Eq. (C2), which leads to expressions\n˜βnl,coh\nC= Tr{¯g1KP2K}−Tr{¯g1KP2¯g2P2K},\n˜βnl,coh\nD= Tr{P1K¯g2K}−Tr{P1¯g1P1K¯g2K},(C28)\nwhere the second relation is obtained in the same way\nfrom the original term βnl,coh\nD. The sum of all four con-\ntributions in (C26) and (C28) yields\n˜βnl,coh=˜βnl,coh\nA+˜βnl,coh\nB+˜βnl,coh\nC+˜βnl,coh\nD\n= Tr{¯g1KP2K}+Tr{P1K¯g2K}\n+Tr{¯g1γ12¯g2γ21}, (C29)\nwhere weused the operators γ12andγ21defined by (C8).\nThe total quantity βnl(C21) is thus equivalent to\n˜βnl=˜βnl,coh+βnl,vc\n= Tr{¯g1KP2K}+Tr{P1K¯g2K}\n+Tr{¯g1γ12¯g2γ21}+βnl,vc,(C30)\nwhere the tildes mark omission of terms irrelevant for the\nsummation over ( p,q) in Eq. (C2).\n4. Expression with the local torque\nThe configuration averagingin βloc(C3), involving the\nrandom local torque, leads to a sum of two terms:44\nβloc=βloc,0+βloc,1, (C31)\nwhere the term βloc,0is given by a simple lattice sum\nβloc,0=/summationdisplay\nRβloc,0\nR,\nβloc,0\nR= Tr/angbracketleftbig\n¯g1(1−t1\nR¯g1)(P1\nRK−KP2\nR)\nׯg2(1−t2\nR¯g2)(P2\nRK−KP1\nR)/angbracketrightbig\n,(C32)\nsee Eq. (76) of Ref. 44, and the term βloc,1can be written\nin the present formalism as\nβloc,1={ˆχ12(ˆ1−ˆw12ˆχ12)−1ϑ12;ϑ21},(C33)\nwhich corresponds to Eq. (74) of Ref. 44. The definitions\nof ˆw12and ˆχ12aregivenby(C15)and(C16), respectively,\nand ofϑ12andϑ21by (C10).\nThe quantity βloc,0\nR(C32) gives rise to four terms,\nβloc,0\nR=QR,A+QR,B+QR,C+QR,D, (C34)\nQR,A= Tr/an}bracketle{t¯g1(1−t1\nR¯g1)P1\nRK¯g2(1−t2\nR¯g2)P2\nRK/an}bracketri}ht,\nQR,B= Tr/an}bracketle{tP1\nR¯g1(1−t1\nR¯g1)KP2\nR¯g2(1−t2\nR¯g2)K/an}bracketri}ht,\nQR,C=−Tr/an}bracketle{tP1\nR¯g1(1−t1\nR¯g1)P1\nRK¯g2(1−t2\nR¯g2)K/an}bracketri}ht,\nQR,D=−Tr/an}bracketle{t¯g1(1−t1\nR¯g1)KP2\nR¯g2(1−t2\nR¯g2)P2\nRK/an}bracketri}ht,14\nwhich will be treated separately. The term QR,Acan\nbe simplified by employing the identities (C6) and the\nCPA-selfconsistency conditions. This yields:\nQR,A=UR,A+VR,A, (C35)\nUR,A= Tr{¯g1P1\nRK¯g2P2\nRK},\nVR,A= Tr/an}bracketle{t¯g1t1\nR(1−¯g1P1\nR)K¯g2t2\nR(1−¯g2P2\nR)K/an}bracketri}ht\n=VR,A1+VR,A2+VR,A3+VR,A4,\nVR,A1= Tr/an}bracketle{t¯g1t1\nRK¯g2t2\nRK/an}bracketri}ht,\nVR,A2= Tr/an}bracketle{t¯g1t1\nR¯g1P1\nRK¯g2t2\nR¯g2P2\nRK/an}bracketri}ht,\nVR,A3=−Tr/an}bracketle{t¯g1t1\nR¯g1P1\nRK¯g2t2\nRK/an}bracketri}ht,\nVR,A4=−Tr/an}bracketle{t¯g1t1\nRK¯g2t2\nR¯g2P2\nRK/an}bracketri}ht.\nA similar procedure applied to QR,Byields:\nQR,B=UR,B+VR,B, (C36)\nUR,B= Tr{P1\nR¯g1KP2\nR¯g2K},\nVR,B= Tr/an}bracketle{t(1−P1\nR¯g1)t1\nR¯g1K(1−P2\nR¯g2)t2\nR¯g2K/an}bracketri}ht\n=VR,B1+VR,B2+VR,B3+VR,B4,\nVR,B1= Tr/an}bracketle{tt1\nR¯g1Kt2\nR¯g2K/an}bracketri}ht,\nVR,B2= Tr/an}bracketle{tP1\nR¯g1t1\nR¯g1KP2\nR¯g2t2\nR¯g2K/an}bracketri}ht,\nVR,B3=−Tr/an}bracketle{tt1\nR¯g1KP2\nR¯g2t2\nR¯g2K/an}bracketri}ht,\nVR,B4=−Tr/an}bracketle{tP1\nR¯g1t1\nR¯g1Kt2\nR¯g2K/an}bracketri}ht.\nThe term QR,Crequires an auxiliary relation\nP1\nR¯g1(1−t1\nR¯g1)P1\nR=P1\nR(¯g1P1\nR−1)\n+P1\nR−(1−P1\nR¯g1)t1\nR(1−¯g1P1\nR),(C37)\nthat follows from a repeated use of the identities (C6).\nThis relation together with the CPA-selfconsistency lead\nto the form:\nQR,C=UR,C+VR,C, (C38)\nUR,C= Tr{P1\nR(1−¯g1P1\nR)K¯g2K}\n−Tr/an}bracketle{tP1\nRK¯g2(1−t2\nR¯g2)K/an}bracketri}ht,\nVR,C=−Tr/an}bracketle{t(1−P1\nR¯g1)t1\nR(1−¯g1P1\nR)K¯g2t2\nR¯g2K/an}bracketri}ht\n=VR,C1+VR,C2+VR,C3+VR,C4,\nVR,C1=−Tr/an}bracketle{tt1\nRK¯g2t2\nR¯g2K/an}bracketri}ht,\nVR,C2=−Tr/an}bracketle{tP1\nR¯g1t1\nR¯g1P1\nRK¯g2t2\nR¯g2K/an}bracketri}ht,\nVR,C3= Tr/an}bracketle{tt1\nR¯g1P1\nRK¯g2t2\nR¯g2K/an}bracketri}ht,\nVR,C4= Tr/an}bracketle{tP1\nR¯g1t1\nRK¯g2t2\nR¯g2K/an}bracketri}ht.\nA similar procedure applied to QR,Dyields:\nQR,D=UR,D+VR,D, (C39)\nUR,D= Tr{¯g1KP2\nR(1−¯g2P2\nR)K}\n−Tr/an}bracketle{t¯g1(1−t1\nR¯g1)KP2\nRK/an}bracketri}ht,\nVR,D=−Tr/an}bracketle{t¯g1t1\nR¯g1K(1−P2\nR¯g2)t2\nR(1−¯g2P2\nR)K/an}bracketri}ht\n=VR,D1+VR,D2+VR,D3+VR,D4,\nVR,D1=−Tr/an}bracketle{t¯g1t1\nR¯g1Kt2\nRK/an}bracketri}ht,\nVR,D2=−Tr/an}bracketle{t¯g1t1\nR¯g1KP2\nR¯g2t2\nR¯g2P2\nRK/an}bracketri}ht,\nVR,D3= Tr/an}bracketle{t¯g1t1\nR¯g1KP2\nR¯g2t2\nRK/an}bracketri}ht,\nVR,D4= Tr/an}bracketle{t¯g1t1\nR¯g1Kt2\nR¯g2P2\nRK/an}bracketri}ht,Let us focus now on U-terms in Eqs. (C35 – C39). The\nsecond terms in UR,C(C38) and UR,D(C39) do not con-\ntribute to the sum over four pairs ( p,q) in Eq. (C1),\nso that the original UR,CandUR,Dcan be replaced by\nequivalent expressions\n˜UR,C= Tr{P1\nR(1−¯g1P1\nR)K¯g2K},\n˜UR,D= Tr{¯g1KP2\nR(1−¯g2P2\nR)K}.(C40)\nThe sum of all U-terms for the site Ris then equal to\n˜UR=UR,A+UR,B+˜UR,C+˜UR,D\n= Tr{P1\nRK¯g2K}+Tr{¯g1KP2\nRK}\n+Tr{¯g1γ12\nR¯g2γ21\nR}, (C41)\nwhereγ12\nRandγ21\nRare defined in (C8), and the lattice\nsum of all U-terms can be written as\n/summationdisplay\nR˜UR= Tr{P1K¯g2K}+Tr{¯g1KP2K}\n+/summationdisplay\nRTr{¯g1γ12\nR¯g2γ21\nR}. (C42)\nThe summation of V-terms in Eqs. (C35 – C39) can be\ndone in two steps. First, we obtain\nVR,1=VR,A1+VR,B1+VR,C1+VR,D1\n= Tr/an}bracketle{tt1\nRf12t2\nRf21/an}bracketri}ht,\nVR,2=VR,A2+VR,B2+VR,C2+VR,D2\n= Tr/an}bracketle{tt1\nR¯g1γ12\nR¯g2t2\nR¯g2γ21\nR¯g1/an}bracketri}ht,\nVR,3=VR,A3+VR,B3+VR,C3+VR,D3\n= Tr/an}bracketle{tt1\nR¯g1γ12\nR¯g2t2\nRf21/an}bracketri}ht,\nVR,4=VR,A4+VR,B4+VR,C4+VR,D4\n= Tr/an}bracketle{tt1\nRf12t2\nR¯g2γ21\nR¯g1/an}bracketri}ht, (C43)\nwhere the operators f12andf21have been defined in\n(C7). Second, one obtains the sum of all V-terms for the\nsiteRas\nVR=VR,1+VR,2+VR,3+VR,4 (C44)\n= Tr/an}bracketle{tt1\nR(f12+ ¯g1γ12\nR¯g2)t2\nR(f21+ ¯g2γ21\nR¯g1)/an}bracketri}ht.\nThe lattice sums of all U- andV-terms lead to an expres-\nsion equivalent to the original quantity βloc,0(C32):\n˜βloc,0=/summationdisplay\nR˜UR+/summationdisplay\nRVR\n= Tr{P1K¯g2K}+Tr{¯g1KP2K}\n+/summationdisplay\nRTr{¯g1γ12\nR¯g2γ21\nR}\n+/summationdisplay\nRTr/angbracketleftbig\nt1\nR(f12+¯g1γ12\nR¯g2)\n×t2\nR(f21+ ¯g2γ21\nR¯g1)/angbracketrightbig\n,(C45)\nwhere the tildes mark omission of terms not contributing\nto the summation over ( p,q) in Eq. (C1).\nLet us turn now to the contribution βloc,1(C33). It\ncan be reformulated by expressing the quantity ϑ12(and15\nϑ21) in terms of the quantities γ12andζ12(andγ21and\nζ21) from the sum rule (C17) and by using the identities\n(C20). The resultingformcan be written compactlywith\nhelp of an auxiliary operator ̺12(and̺21) defined as\n̺12= ˆχ12γ12+ζ12. (C46)\nThe result is\nβloc,1=βnl,vc+{ˆχ12γ12;γ21}\n−{ˆw12̺12;̺21}, (C47)\nwhere the first term has been defined in (C23). For the\nsecond term in (C47), we use the relation\nˆχ12γ12=/summationdisplay\nRΠR¯g1(γ12−γ12\nR)¯g2ΠR,(C48)\nwhich follows from the site-diagonal nature of the opera-\ntorγ12(C8) and from the definition of the superoperator\nˆχ12(C16). This yields:\n{ˆχ12γ12;γ21}= Tr{¯g1γ12¯g2γ21}\n−/summationdisplay\nRTr{¯g1γ12\nR¯g2γ21\nR}.(C49)\nFor the third term in (C47), only the site-diagonalblocks\nof the operator ̺12(and̺21), Eq. (C46), are needed be-\ncauseofthesite-diagonalnatureofthesuperoperator ˆ w12\n(C15). These site-diagonal blocks are given by\nΠR̺12ΠR= ΠR/bracketleftbig\n¯g1(γ12−γ12\nR)¯g2+ζ12/bracketrightbig\nΠR\n=−ΠR(f12+ ¯g1γ12\nR¯g2)ΠR,(C50)\nwhichfollowsfromthepreviousrelations(C48)and(C9).This yields:\n{ˆw12̺12;̺21}=/summationdisplay\nRTr/angbracketleftbig\nt1\nR(f12+ ¯g1γ12\nR¯g2)\n×t2\nR(f21+ ¯g2γ21\nR¯g1)/angbracketrightbig\n.(C51)\nThe term βloc,1(C47) is then equal to\nβloc,1=βnl,vc+Tr{¯g1γ12¯g2γ21}\n−/summationdisplay\nRTr{¯g1γ12\nR¯g2γ21\nR}\n−/summationdisplay\nRTr/angbracketleftbig\nt1\nR(f12+¯g1γ12\nR¯g2)\n×t2\nR(f21+ ¯g2γ21\nR¯g1)/angbracketrightbig\n.(C52)\nThe total quantity βloc(C31) is thus equivalent to the\nsum of (C45) and (C52):\n˜βloc=˜βloc,0+βloc,1\n= Tr{P1K¯g2K}+Tr{¯g1KP2K}\n+Tr{¯g1γ12¯g2γ21}+βnl,vc,(C53)\nwhere the tildes mark omission of terms irrelevant for the\nsummation over ( p,q) in Eq. (C1).\n5. 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Tosi (North-Holland, New York, 1985) p. 59."    },    {        "title": "1403.3199v1.The_best_decay_rate_of_the_damped_plate_equation_in_a_square.pdf",        "content": "arXiv:1403.3199v1  [math.OC]  13 Mar 2014The best decay rate of the damped plate\nequation in a square\nKa¨ ıs Ammari∗Abdelkader Sa¨ ıdi†\nSeptember 18, 2021\nAbstract. In this paper we study the best decay rate of the solutions of a dam ped\nplate equation in a square and with a homogeneousDirichlet boundary conditions. We show\nthat the fastest decay rate is given by the supremum of the real p art of the spectrum of the\ninfinitesimal generator of the underlying semigroup, if the damping c oefficient is in L∞(Ω).\nMoreover, we give some numerical illustrations by spectral comput ation of the spectrum\nassociated to the damped plate equation. The numerical results ob tained for various cases\nof damping are in a good agreement with theoretical ones. Computa tion of the spectrum\nand energy of discrete solution of damped plate show that the best decay rate is given by\nspectral abscissa of numerical solution.\nKeywords : optimal decay rate, damped plate, spectrum.\nAMS subject classifications : 35A05, 35B40, 35B37, 93B07.\n1 Introduction\nLet Ω = (0 ,1)×(0,1)⊂IR2,∂Ω = Γ. We consider the plate equation with interior\ndissipation, more precisely we have the following partial d ifferential equations :\n∂2u\n∂t2(x,t)+∆2u(x,t)+a(x)∂u\n∂t(x,t) = 0,(x,t)∈Ω×(0,+∞),(1.1)\nwith boundary conditions :\nu(x,t) = 0,∆u(x,t) = 0,(x,t)∈Γ×(0,+∞), (1.2)\nand initial conditions :\nu(x,0) =u0(x),∂u\n∂t(x,0) =u1(x), x∈Ω, (1.3)\nwherea(x)∈L∞(Ω) is a nonnegative damping coefficient.\n∗UR Analysis and Control of Pde, UR 13ES64, Department of Math ematics, Faculty of Sciences\nof Monastir, University of Monastir, 5019 Monastir, Tunisi a, email: kais.ammari@fsm.rnu.tn\n†Institut de Recherche Math´ ematique Avanc´ ee, University of Strasbourg, 7 rue Ren´ e Descartes,\nF-67084 Strasbourg, France, email: saidi@math.unistra.f r\n1Ifuis a solution of (1.1)-(1.3) we define the energy of uat instant tby :\nE(t) =1\n2/integraldisplay\nΩ/parenleftBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂u\n∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+|∆u|2/parenrightBig\ndx. (1.4)\nSimple calculations show that a sufficiently smooth solution of (1.1)-(1.3) satisfies\nE(t)−E(0) =−/integraldisplayt\n0/integraldisplay\nωa(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂u\n∂s/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndsdx. ∀t≥0. (1.5)\nIn particular (1.5) implies that\nE(t)≤E(0),∀t≥0. (1.6)\nEstimate above suggests that the natural wellposedness spa ce for (1.1)-(1.3) is X=\nV×L2(Ω),with inner product /an}bracketle{t[f,g],[u,v]/an}bracketri}ht=/integraltext\nΩ(∆f∆¯u+g¯v)dx,whereV=\nH2(Ω)∩H1\n0(Ω).\nWe have the following wellposedness result\nProposition 1.1. Suppose that (u0,u1)∈V×L2(Ω). Then the problem (1.1)-(1.3)\nadmits a unique solution\nu∈C(0,+∞;V)∩C1(0,+∞;L2(Ω))\nMoreover usatisfies the energy estimate (1.5).\nIf we denote by U= [u,∂tu], we can rewrite (1.1)-(1.3) in the form :\n/braceleftbigg∂tU−AaU= 0, Q= Ω×(0,+∞),\nU(x,0) =U0(x),Ω,(1.7)\nwhereU0= (u0,u1),Aa:D(Aa)⊂X→Xis the operator defined by :\nAa=/parenleftbigg0Id\n−∆2−a(x)/parenrightbigg\n, (1.8)\nD(Aa) =/braceleftbig\n(u,v)∈[H4(Ω)∩V]×V,∆u|Γ= 0/bracerightbig\n.\nIn order to state the result on the optimal location of the act uator, we define the\ndecay rate, depending on a, as\nω(a) = inf{ω|there exists C=C(ω)>0 such that E(t)≤C(ω)e2ωtE(0),\nfor every solution of (1.1)-(1.3) with initial data in V×L2(Ω)},(1.9)\nand the spectral abscissa of Aa,\nµ(a) = sup{Reλ:λ∈σ(Aa)}, (1.10)\nwhereE(t) is defined in (1.4) and σ(Aa) denotes the spectrum of Aa. It follows\neasily that\nµ(a)≤w(a). (1.11)\nAccording to (1.6) we have that ω(a)≤0 for all a∈L∞(Ω) is nonnegative.\nMoreover, if a∈L∞(Ω) is nonnegative and satisfying the following condition:\n∃c >0s.t., a(x)≥c;a.e.,in an open subset ω⊂Ω, meas(ω)/ne}ationslash= 0.(1.12)\nWe have, according to [12, 1, 16] (see Section 2) for more deta ils) that ω(a)<0.\nThe main result, on the optimal decay rate, is\n2Theorem 1.1. Ifa∈L∞(Ω)then\nµ(a) =w(a). (1.13)\nMoreover, if the assumption in damping coefficient (1.12)is holds, then all finite\nenergy solutions of (1.1)-(1.3)are exponentially stable which implies that the fastest\ndecay rate of the solutions of (1.1)-(1.3)satisfies (1.13).\nThe problem of finding the optimal decay rate for beams with di stributed interior\ndamping is difficult and has not a complete answer in the case of a variable (in\nspace) damping coefficient. We refer to [2], [3], [4], [6], [14 ], [10], [9], [11] and to\nreferences therein. Recently C. Castro and S. Cox in [8] made adecisive contribution\nby showing that one can get an arbitrarily large decay rate by means of appropriate\ndamping. By this way, they answer by the negative to an old con jecture according\nto which the best decay rate should be provided by the best con stant damping. The\nmain novelties brought in by this paper is that, in the simple r case of a distributed\ninterior damping, we can give the precise optimal decay rate and we illustrate this\nresult numerically. One of the main ingredients of this stud y is a result showing\nthat the eigenfunctions of the associated dissipative oper ator form a Riesz basis\nwith parentheses in the energy space.\nWe remark that the corresponding optimal decay rate problem make sense since,\nin this case, the system is exponentially stable (see for ins tance [12], [1]).\nThe paperis organized as follows. Section 2 contains some ba ckground on optimal\ndecay rate of dissipative systems. In sectiontheorique we g ive the proof of the main\nresult. Section 4 is devoted to some illustrations of the mai n result an optimal\ndecay rate by numerical spectral analysis strategy. We pres ent numerical results of\ncomputation of the spectrum for different cases of damping. Th e discrete energy of\nsolution is computed in each case and compared to the spectra l abscissa. The latest\nis showed to be the best decay rate in each example presented.\n2 Some background on optimal decay rate for dissipative oper ators\nLetHbe a Hilbert space equipped with the norm ||.||H, and let A:D(A)⊂H→H\nbeself-adjoint, positiveandwithcompactinvertibleoper ator. Then, Ahasadiscrete\nspectrum, let ( µk)k≥1be its eigenvalues, each taken with its multiplicity. Denot e\na complete orthonormal system of eigenvectors of the operat orAthat correspond\nto these eigenvalues by ( ϕk)k≥1. Moreover, we suppose that ( µk)k≥1satisfies the\nfollowing generalized uniform gap:\n∃p∈N∗, c >0; such that µk+p−µk≥c,∀k∈N∗. (2.14)\nWe introduce the scale of Hilbert spaces Hβ,β∈IR, as follows: for every β≥0,\nHβ=D(Aβ), with the norm /bardblz/bardblβ=/bardblAβz/bardblH. The space H−βis defined by duality\nwith respect to the pivot space Has follows: H−β=H∗\nβforβ >0. The operator A\ncan beextended (or restricted) to each Hβ, such that it becomes a boundedoperator\nA:Hβ→Hβ−1∀β∈IR. (2.15)\n3Let a bounded linear operator B:U→H, whereUis another Hilbert space which\nwill be identified with its dual.\nThe systems we consider are described by\n¨x(t)+Ax(t)+BB∗˙x(t) = 0, (2.16)\nx(0) =x0,˙x(0) =x1. (2.17)\nWe can rewrite the system (2.16)-(2.17) as a first order differe ntial equation, by\nputtingz(t) =/parenleftbiggx(t)\n˙x(t)/parenrightbigg\n:\n˙z(t)−Az(t)+BB∗z(t) = 0,z(0) =z0=/parenleftbiggx0\nx1/parenrightbigg\n, (2.18)\nwhere\nA=/parenleftbigg0I\n−A0/parenrightbigg\n:D(A) =H1×H1\n2⊂ H=H1\n2×H→ H,B=/parenleftbigg0\nB/parenrightbigg\n∈ L(U,H).\nBy the same way the system (2.16)-(2.17) can be also rewritte n by :\n˙z(t)+Adz(t) = 0,z(0) =z0, (2.19)\nwhere\nAd=A−BB∗:D(Ad) =H1×H1\n2⊂ H → H .\nIt is clear that the operator Ais skew-adjoint on Hand hence, it generates a\nstrongly continous group of unitary operators on H, denoted by ( S(t))t∈IR.\nSinceAdis dissipative and onto, it generates a contraction semigro up onH,\ndenoted by ( Sd(t))t∈IR+.\nThe system (2.16)-(2.17) is well-posed. More precisely, th e following classical\nresult, holds.\nProposition 2.1. Suppose that (x0,x1)∈H1\n2×H. Then the problem (2.16)-(2.17)\nadmits a unique solution\n(x,˙x)∈C([0,∞);H1\n2×H).\nMoreover wsatisfies, for all t≥0, the energy estimate\nE(0)−E(t) =/integraldisplayt\n0/bardblB∗˙x(s)/bardbl2\nU, ds, (2.20)\nwhereE(t) =1\n2/bardbl(x(t),˙x(t))/bardbl2\nH.\nFrom (2.20) it follows that the mapping t/ma√sto→ /bardbl(x(t),˙x(t))/bardbl2\nHis non increasing.\nIn many applications it is important to know if this mapping d ecays exponentially\nwhent→ ∞, i.e. if the system (2.16)-(2.17) is exponentially stable. One of the\nmethods currently used for proving such exponential stabil ity results is based on\n4an observability inequality for the undamped system associ ated to the initial value\nproblem\n¨φ(t)+Aφ(t) = 0, (2.21)\nφ(0) =x0,˙φ(0) =x1. (2.22)\nItiswellknownthat(1.3)-(2.22)iswell-posedin H1×H1\n2andinH. Theresultbelow,\nproved in [12, 1], shows that the exponential stability of (2 .16)-(2.17) is equivalent\nto an observability inequality for (2.21)-(2.22).\nTheorem 2.1. The system described by (2.16)-(2.17)is exponentially stable in Hif\nand only if there exists T,CT>0such that\nCT/integraldisplayT\n0||B∗S(t)z0||2\nUdt≥ ||z0||2\nH∀z0∈ H1. (2.23)\nIn order to state the result on the optimal decay rate, we defin e the decay rate,\ndepending on B, as\nω(B) = inf{ω|there exists C=C(ω)>0 such that E(t)≤C(ω)e2ωtE(0),\nfor every solution of (2.16)-(2.17) with initial data in H1\n2×H}(2.24)\nand the spectral abscissa as\nµ(B) = sup{Reλ:λ∈σ(Ad)}, (2.25)\nwhereσ(Ad) denotes the spectrum of Ad.\nIt follows easily that\nµ(B)≤ω(B). (2.26)\nWe recall that a Riesz basis in a Hilbert space, is by definitio n, isomorphic to an\northonormal basis.\nDefinition 2.1. A system (φk)k≥1of a space His called a basis with parentheses\nif the series f=/summationdisplay\nk≥1ckφkconverges in the norm of Hfor anyf∈Hafter some\narrangement of parentheses that does not depend on f. If a system remains a ba-\nsis after any permutation of the sets of its vectors correspo nding to the terms of\nthe series enclosed in parentheses, then such a system is cal led a Riesz basis with\nparentheses.\nThe operator Adis a perturbed self-adjoint operator, with the self-adjoin t part is\nwith discrete spectrumwhich satisfies thegap condition (2. 14) and the perturbation,\nBB∗∈ L(H) is bounded. Then, according to [18, Theorem 2], we have that the\neigenvectors of Adforms a Riesz basis with parentheses and according to [5] we\nobtain on estimation of optimal decay rate. We have the follo wing:\nTheorem 2.2. ([5, Ammari-Dimassi-Zerzeri]) If the observability inequa lity(2.23)\nis holds then,\nω(B) =µ(B)<0. (2.27)\nIn other words if all finite energy solutions of (2.16)-(2.17)are exponentially stable\nthen the fastest decay rate of the solutions of (2.16)-(2.17)satisfies (2.27).\n53 Proof of Theorem 1.1\nThe eigenvalue problem for the non self-adjoint, quadratic operator pencil generated\nby (1.1)-(1.3) is obtained by replacing uin (1.1) by\nu(x,t) =eλtφ(x).\nWe obtain from (1.1) the standard form\n(Aa−λId)Φ = 0;Φ = [ φ,λφ] =φ[1,λ].\nThe condition for the existence of non trivial solutions is t hatλ∈σ(Aa) (the\nspectrumof Aa). SinceD(Aa) is compactly embeddedin theenergy space V×L2(Ω)\nthen the spectrum σ(Aa) is discrete and the eigenvalues of Aahave a finite algebraic\nmultiplicity. On the other hand, since Aais a bounded monotone perturbation of\na skew-adjoint operator (undamped A0), it follows from the Hille-Yosida theorem\nthatAagenerates a C0-semigroup of contractions on the energy space V×L2(Ω).\nAccording to [16, Proposition A.1] the spectra of Aasatisfies the gap assumption\n(2.14). So, by Theorem 2.2 we end the proof.\n/square\n4 Discretization of the eigenvalue problem\nThe domain Ω is approximated by a net of equidistant discrete points Ω h=Mij\nwithi,j= 1,2,...,NandN= (1/h)+1. The Laplacian operator is approximated\nin a standard way by a second order centered difference scheme :\n∆hu= (ui+1,j+ui−1,j+ui,j+1+ui,j−1−4ui,j)/h2(4.28)\nwhereui,j=u(xi,yj). The discrete approximation of the operator Aadefined by\n(1.8) is then given by a nonsymmetric block matrix, Ah, of the form :\nAh=/bracketleftbigg0−IdN\n∆2\nhah/bracketrightbigg\n(4.29)\nwhereIdNis the identity matrix of order Nand ∆2\nhis the discrete version of the\nbilaplacian. ahis a diagonal matrix with akk=a(xi,yj). The value of kis deter-\nmined by the numbering of the discrete points Mij. We use the implicitly restarted\nArnodi-Lanszos method of Sorenson [19] to compute the eigen values of matrix Ah.\nThis method is a generalization of an inverse power method wi th subspace iteration\n[6],[19].\n5 The case of square\nWe consider a domain Ω = (0 ,1)×(0,1) with a set of normalized eigenfunctions\nΦn,m(x,y) =√\n2sin(nπx)√\n2sin(mπy), for all ( x,y)∈Ω the solution u(x,y,t) of the\nproblem (1.1)-(1.2) is given by :\nu(x,y,t) =/summationdisplay\nn,mαn,m(t)Φn,m(x,y) (5.30)\n6replacing (6.32) in equation (1.1) and multiplying by a test eigenfunction Φ k,l(x,y)\nthe solution for the low frequencies is solution of the equat ion :\nd2αn,m(t)\ndt2+((nπ)2+(mπ)2)2αn,m(t)+dαn,m(t)\ndt= 0 (5.31)\nIn the first example we consider a damped plate with constant c oefficient a(x) = 1\nand a damping region covering all the domain ω= Ω. We compute the spec-\ntrum of the damped plate (figure 1) and the energy of the first ei genmodes :\nn=m=3 and n=m=12 (figure 2 ). The energy of the solution (line 1 ) is com-\npared to E0eRe(λ0)t(line 2), where E0=E(0) is the energy of initial conditions\nandRe(λ0) =inf{Re(λ)/λ∈σ(A)}, (σ(A) : the spectrum of operator A).\nIn the other examples we consider a damping in different domain sω. The spectrum\nof the damped plate is computed and energy for different eigenm odes is compared\ntoE0eRe(λ0)t(line 2 : dashed line).\n−0.55 −0.54 −0.53 −0.52 −0.51 −0.5 −0.49 −0.48 −0.47 −0.46 −0.45−4000−3000−2000−100001000200030004000\nreal axisimaginary axis\n−0.22 −0.2 −0.18 −0.16 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02−4000−3000−2000−100001000200030004000\nreal axisimaginary axis\nfigure 1 : spectrum of the damped plate a(x) = 1 ,ω= Ω (left) ,\na(x) = 1 ,ω= (0,1\n2)×(0,1\n2) (right).\n0 2 4 6 8 10 12 14 16 18 2000.511.522.533.544.55\ntimeenergyline1\nline2\n0 5 10 15 20 25 30 35 40020004000600080001000012000\ntimeenergyline1\nline2\nfigure 2 : energy of the damped plate, a(x) = 1 ,ω= Ω,n=m= 3 (left), and\nn=m= 12 (right).\n70 5 10 15 20 25 30 35 40 45 5000.511.522.533.544.55\ntimeenergyline1\nline2\n0 5 10 15 20 25 30 35 40 45 50020040060080010001200\ntimeenergyline1\nline2\nfigure 3 : energy of the damped plate, a(x) = 1,ω= (0,1\n2)×(0,1\n2),n=m= 3\n(left), and n=m= 8 (right).\nIn figures 4. 5. and 6. we consider a damping in the domain ω= (0,2\n5)×(0,2\n5) and\nω= (0,3\n5)×(0,3\n5) :\n−0.16 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0−4000−3000−2000−100001000200030004000\nreal axisimaginary axis\n−0.26 −0.24 −0.22 −0.2 −0.18 −0.16 −0.14 −0.12 −0.1 −0.08−4000−3000−2000−100001000200030004000\nreal axisimaginary axis\nfigure 4 : spectrum of the damped plate, a(x) = 1 ,ω= (0,0.4)×(0,0.4) (left),\nω= (0,3\n5)×(0,3\n5) (right).\n0 5 10 15 20 25 30 35 40 45 5000.10.20.30.40.50.60.7\ntimeenergyline1\nline2\n0 5 10 15 20 25 30 35 40 45 50020040060080010001200\ntimeenergyline1\nline2\nfigure 5 : energy of the damped plate, a(x) = 1,ω= (0,0.4)×(0,0.4),n=m= 2\n(left), and n=m= 8 (right).\n80 5 10 15 20 25 30 35 40 45 5000.511.522.533.544.55\ntimeenergyline1\nline2\n0 5 10 15 20 25 30 35 40 45 50020040060080010001200\ntimeenergyline1\nline2\nfigure 6 : energy of the damped plate, a(x) = 1,ω= (0,0.6)×(0,0.6),n=m= 3\n(left), and n=m= 8 (right).\nIn figures 7., 8. and 9. we consider a damping in the domain ω= (0,1\n4)×(0,1\n4) and\nω= (0,1\n5)×(0,1\n5) :\n−0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0−4000−3000−2000−100001000200030004000\nreal axisimaginary axis\n−0.07 −0.06 −0.05 −0.04 −0.03 −0.02 −0.01 0−4000−3000−2000−100001000200030004000\nreal axisimaginary axis\nfigure 7 : spectrum of the damped plate, a(x) = 1,ω= (0,1\n4)×(0,1\n4) (left) and\nω= (0,1\n5)×(0,1\n5) (right).\n0 5 10 15 20 25 30 35 40 45 5000.10.20.30.40.50.60.7\ntimeenergyline1\nline2\n0 5 10 15 20 25 30 35 40 45 5001020304050607080\ntimeenergyline1\nline2\nfigure 8 : energy of the damped plate, a(x) = 1,ω= (0,0.25)×(0,0.25),n=m= 2\n(left), and n=m= 5 (right).\n90 10 20 30 40 50 60 70 80 90 10000.511.522.533.544.55\ntimeenergyline1\nline2\n0 100 200 300 400 500 600 700 800020040060080010001200\ntimeenergyline1\nline2\nfigure 9 : energy of the damped plate, a(x) = 1,ω= (0,0.2)×(0,0.2),n=m= 3\n(left), and n=m= 8 (right).\nIn figures 10., 11. and 12. we consider a damping in the domain ω= (0,1)×(0,1\n2)\nandω= (0,1)×(0,1\n4) :\n−0.3 −0.29 −0.28 −0.27 −0.26 −0.25 −0.24 −0.23 −0.22 −0.21 −0.2−4000−3000−2000−100001000200030004000\nreal axisimaginary axis\n−0.18 −0.16 −0.14 −0.12 −0.1 −0.08 −0.06−4000−3000−2000−100001000200030004000\nreal axisimaginary axis\nfigure 10 : spectrum of the damped plate, a(x) = 1 and ω= (0,1)×(0,1\n2) (left) ,\na(x) = 1 and ω= (0,1)×(0,1\n4) (right).\n0 5 10 15 20 25 30 35 40 45 5001020304050607080\ntimeenergyline1\nline2\n0 5 10 15 20 25 30 35 40 45 5005001000150020002500300035004000\ntimeenergyline1\nline2\nfigure 11 : energy of the damped plate, a(x) = 1,ω= (0,1)×(0,0.5),n=m= 5\n(left), and n=m= 10 (right).\n100 5 10 15 20 25 30 35 40 45 5000.511.522.533.544.55\ntimeenergyline1\nline2\n0 5 10 15 20 25 30 35 40 45 5005001000150020002500300035004000\ntimeenergyline1\nline2\nfigure 12 : energy of the damped plate, a(x) = 1,ω= (0,1)×(0,1\n4),n=m= 3\n(left), and n=m= 10 (right).\n−0.16 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02−4000−3000−2000−100001000200030004000\nreal axisimaginary axis\n−0.2 −0.15 −0.1 −0.05 0−4000−3000−2000−100001000200030004000\nreal axisimaginary axis\nfigure 13 : spectrum of the damped plate, a(x) = 1,ω= (0.4,0.6)×(0.4,0.6) (left),\na(x) = 1,ω= (0.35,0.65)×(0.35,0.65) (right).\n0 10 20 30 40 50 60 70 80 90 10000.10.20.30.40.50.60.7\ntimeenergyline1\nline2\n0 10 20 30 40 50 60 70 80 90 100050100150200250\ntimeenergyline1\nline2\nfigure 14 : energy of the damped plate, a(x) = 1,ω= (0.4,0.6)×(0.4,0.6),\nn=m= 2 (left), and n=m= 6 (right).\n110 10 20 30 40 50 60 70 80 90 10000.10.20.30.40.50.60.7\ntimeenergyline1\nline2\n0 10 20 30 40 50 60 70 80 90 100050100150200250\ntimeenergyline1\nline2\nfigure 15 : energy of the damped plate, a(x) = 1,ω= (0.35,0.65)×(0.35,0.65),\nn=m= 2 (left), and n=m= 6 (right).\n−0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05−4000−3000−2000−100001000200030004000\nreal axisimaginary axis\n−0.16 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0−4000−3000−2000−100001000200030004000\nreal axisimaginary axis\nfigure 16 : spectrum of the damped plate, a(x) = 1,ω= (0.25,0.75)×(0.25,0.75)\n(left),a(x) = 1,ω= (0.5,0.75)×(0.3,0.6) (right).\n0 10 20 30 40 50 60 70 80 90 10000.10.20.30.40.50.60.7\ntimeenergyline1\nline2\n0 10 20 30 40 50 60 70 80 90 100020040060080010001200\ntimeenergyline1\nline2\nfigure 17 : energy of the damped plate, a(x) = 1,ω= (0.25,0.75)×(0.25,0.75),\nn=m= 2 (left), and n=m= 8 (right).\nIn figures 13., 14. and 15., for the value a(x) = 1 we consider a damping in the\ndomainω= (0.4,0.6)×(0.4,0.6) andω= (0.35,0.65)×(0.35,0.65). In figures 16.,\n17. and 18. , we consider a damping in the domain ω= (0.25,0.75)×(0.25,0.75),\nandω= (0.5,0.75)×(0.3,0.6).\n120 10 20 30 40 50 60 70 80 90 10000.10.20.30.40.50.60.7\ntimeenergyline1\nline2\n0 10 20 30 40 50 60 70 80 90 100020040060080010001200\ntimeenergyline1\nline2\nfigure 18 : energy of the damped plate, a(x) = 1,ω= (0.5,0.75)×(0.3,0.6),\nn=m= 2 (left), and n=m= 8 (right).\nIn the following examples the damping is a function given by a(x1,x2) =x1x2. In\nthe case of a square the spectrum of the damped plate is comput ed for three position\nof damping domain ω= Ω,ω= (0,0.5)×(0,0.5) andω= (0.35,0.75)×(0.35,0.75)\n−0.125 −0.12 −0.115 −0.11 −0.105 −0.1 −0.095 −0.09 −0.085 −0.08−4000−3000−2000−100001000200030004000\nreal axisimaginary axis\nfigure 19 : spectrum of the damped plate, a(x1,x2) =x1x2,ω= Ω .\n0 1 2 3 4 5 6 7 8 9 1001020304050607080\ntimeenergyline1\nline2\n0 1 2 3 4 5 6 7 8 9 100.60.811.21.41.61.822.22.4x 105\ntimeenergyline1\nline2\nfigure 20 : energy of the damped plate, a(x1,x2) =x1x2, damping in all the\ndomainn=m= 5 (left), and n=m= 20 (right).\n13−14 −12 −10 −8 −6 −4 −2\nx 10−3−4000−3000−2000−100001000200030004000\nreal axisimaginary axis\n0 10 20 30 40 50 60 70 80 90 10020304050607080\ntimeenergyline1\nline2\nfigure 21 : a(x1,x2) =x1x2,ω= (0,0.5)×(0,0.5) spectrum of the damped plate,\n(left), energy of the damped plate, n=m= 5, (right) .\n−0.07 −0.06 −0.05 −0.04 −0.03 −0.02 −0.01 0−4000−3000−2000−100001000200030004000\nreal axisimaginary axis\n0 10 20 30 40 50 60 70 80 90 10001020304050607080\ntimeenergyline1\nline2\nfigure 22 : a(x1,x2) =x1x2,ω= (0.35,0.75)×(0.35,0.75) , spectrum of the\ndamped plate, (left) , energy of the damped plate, n=m= 5, (right).\nIn the next example we consider a damping defined by a(x1,x1) =sin(x1)cos(x2)\nfor three region ω= Ω,ω= (0,0.5)×(0,0.5) andω= (0.3,0.7)×(0.3,0,7). We plot\nthe spectrum in the three situations and the energy of the dam ped plate compared\ntoE0eRe(λ0)t(line 2 : dashed line). figures 23., 24., and figure 25.\n14−0.195 −0.19 −0.185 −0.18 −0.175 −0.17−4000−3000−2000−100001000200030004000\nreal axisimaginary axis\n−0.07 −0.06 −0.05 −0.04 −0.03 −0.02 −0.01−4000−3000−2000−100001000200030004000\nreal axisimaginary axis\nfigure 23 : spectrum of the damped plate, a(x1,x2) =sin(x1)cos(x2),ω= Ω (left) ,\nω= (0,0.5)×(0,0.5) (right).\n0 5 10 15 20 25 30 35 40 45 5000.511.522.533.544.55\ntimeenergyline1\nline2\n0 10 20 30 40 50 60 70 80 90 100020040060080010001200\ntimeenergyline1\nline2\nfigure 24 : energy of the damped plate, a(x1,x2) =sin(x1)cos(x2),ω= Ω,\nn=m= 3 (left), and n=m= 8 (right).\n0 5 10 15 20 25 30 35 40 45 5000.511.522.533.544.55\ntimeenergyline1\nline2\n0 5 10 15 20 25 30 35 40 45 50050100150200250\ntimeenergyline1\nline2\nfigure 25: energy of the damped plate,\na(x1,x2) =sin(x1)cos(x2),ω= (0,0.5)×(0,0.5),n=m= 3(left), and n=m= 6\n(right).\n156 The case of rectangle\nIn this case we consider rectangular a domain Ω = (0 ,a)×(0,b) with a set of\nnormalized eigenfunctions Φ n,m(x,y) =/radicalBig\n2\nasin(nπx\na)/radicalBig\n2\nbsin(mπy\nb), for all ( x,y)∈Ω\nthe solution u(x,y,t) of the problem (1.1)-(1.2) is given by :\nu(x,y,t) =/summationdisplay\nn,mαn,m(t)Φn,m(x,y) (6.32)\nwith :\nd2αn,m(t)\ndt2+((nπ\na)2+(mπ\nb)2)2αn,m(t)+dαn,m(t)\ndt= 0 (6.33)\nas in the case of square we compute the spectrum and energy of t he first eigenmodes\nfor different domains ω. We compare in each case the energy of the solution (line 1)\ntoE0eRe(λ0)t(line 2), with Re(λ0) =inf{Re(λ)/λ∈σ(A)}.\n−0.55 −0.54 −0.53 −0.52 −0.51 −0.5 −0.49 −0.48 −0.47 −0.46 −0.45−4000−3000−2000−100001000200030004000\nreal axisimaginary axis\n−0.18 −0.16 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04−4000−3000−2000−100001000200030004000\nreal axisimaginary axis\nfigure 26 : spectrum of the damped plate a(x) = 1,ω= Ω (left), and\nω= (0,1\n2)×(0,1) (right).\n0 2 4 6 8 10 12 14 16 18 2000.511.522.533.544.55\ntimeenergyline1\nline2\n0 10 20 30 40 50 60 70 80 90 100020040060080010001200140016001800\ntimeenergyline1\nline2\nfigure 27 : energy of the damped plate , a(x) = 1,n=m= 3 (left), and\nn=m= 12 (right).\n160 10 20 30 40 50 60 70 80 90 10000.050.10.150.20.25\ntimeenergyline1\nline2\n0 10 20 30 40 50 60 70 80 90 1000500100015002000250030003500400045005000\ntimeenergyline1\nline2\nfigure 28 : energy of the damped plate, a(x) = 1,ω= (0,1\n2)×(0,1),n=m= 2\n(left), and n=m= 10 (right).\n−0.2 −0.18 −0.16 −0.14 −0.12 −0.1 −0.08−4000−3000−2000−100001000200030004000\nreal axisimaginary axis\n−0.1 −0.09 −0.08 −0.07 −0.06 −0.05 −0.04 −0.03 −0.02 −0.01 0−4000−3000−2000−100001000200030004000\nreal axisimaginary axis\nfigure 29 : spectrum of the damped plate a(x) = 1,,ω= (0,0.6)×(0,1), (left) ,\nω= (0,1\n5)×(0,1) (right).\n0 10 20 30 40 50 60 70 80 90 10000.20.40.60.811.21.41.61.82\ntimeenergyline1\nline2\n0 10 20 30 40 50 60 70 80 90 100020040060080010001200140016001800\ntimeenergyline1\nline2\nfigure 30 : energy of the damped plate, a(x) = 1,ω= (0,1\n5)×(0,1),n=m= 3\n(left), and n=m= 10 (right).\n170 10 20 30 40 50 60 70 80 90 10000.050.10.150.20.25\ntimeenergyline1\nline2\n0 10 20 30 40 50 60 70 80 90 100020040060080010001200140016001800\ntimeenergyline1\nline2\nfigure 31 : energy of the damped plate, a(x) = 1,ω= (0,0.6)×(0,1),n=m= 2\n(left), and n=m= 10 (right).\n7 Optimization of the position of the damped region ω\nStarting from a fixed damping domain ωwe try to see the influence of the position\non decay energy. In the first example we plot the energy for five different damping\nposition : ω= (0.,0.4)×(0,0.4),ω= (0.3,0.7)×(0.3,0.7) (dashed line), ω=\n(0.5,0.9)×(0.4,0.8) ,ω= (0.1,0.5)×(0.5,0.9),ω= (0.3,0.7)×(0,0.4). The best\nposition in this case is the middle of the plate.\n0 5 10 15 20 25 30 35 40020040060080010001200\ntimeenergy\nfigure 32 : energy of the damped plate, a(x) = 1,n=m= 8.\nIn the second example we plot the energy for five damping regio nω= (0.,0.3)×\n(0,0.3) (green), ω= (0.35,0.65)×(0.35,0.65) (red), ω= (0.1,0.4)×(0.6,0.9) (blue)\n,ω= (0.7,1)×(0.5,0.8) (black), ω= (0.7,1)×(0.1,0.4), yellow. The best position\nin this case is the corner : ω= (0.,0.3)×(0,0.3).\n180 10 20 30 40 50 60 70 80020040060080010001200\ntimeenergy\nfigure 33 : energy of the damped plate, a(x) = 1,n=m= 8.\n0 10 20 30 40 50 60 70 8001020304050607080\ntimeenergy\nfigure 34 : energy of the damped plate, a(x) = 1,n=m= 5.\nIn the third example we plot the energy for five damping region ω= (0.,0.2)×\n(0,0.2) (black), ω= (0.1,0.3)×(0.1,0.3) (green), ω= (0.2,0.4)×(0.2,0.4) (blue)\n,ω= (0.3,0.5)×(0.3,0.5) (yellow), ω= (0.4,0.6)×(0.4,0.6), (red). for n=m=\n5,n=m= 6 and n=m= 7,\n0 50 100 150 200 250 30001020304050607080\ntimeenergy\n0 50 100 150 200 250 300050100150200250\ntimeenergy\nfigure 35 : energy of the damped plate, a(x) = 1,n=m= 5 and n=m= 7.\n190 10 20 30 40 50 60 70 80020040060080010001200\ntimeenergy\nfigure 36 : energy of the damped plate, a(x) = 1,n=m= 7.\nReferences\n[1]K. Ammari and M. Tucsnak, Stabilization of second order evolution equa-\ntions by a class of unbounded feedbacks, ESAIM Control Optim. Calc. Var,\nESAIM Control Optim. Calc. Var, 6(2001), 361-386.\n[2]K. Ammari, A. Henrot and M. Tucsnak, Optimal location of the actuator\nfor the pointwise stabilization of a string, C. R. Acad. Sci. Paris S´ er I.Math. ,\n330(2000), 275-280.\n[3]K. Ammari, A. Henrot and M. Tucsnak, Asymptotic behaviour of the\nsolutions and optimal location of the actuator for the point wise stabilization of\na string, Asymptotic Analysis., 28(2001), 215-240.\n[4]K. Ammari and A. Sa ¨ıdi,Optimal location of the actuator at high frequency\nfor the pointwise stabilization of a Bernoulli-Euler beam, Control Cybernetics,\n31(2002), 57-66.\n[5]K. Ammari, M. Dimassi and M. Zerzeri, Best decay rate of some dissipatifs\nsystems, preprint.\n[6]M. Asch and G. Lebeau, The spectrum of the damped wave operator for a\nbounded domain in IR2,Experimental Mathematics, 12(2003), 227-241.\n[7]C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the\nobservation, control and stabilization of waves from the bo undary, SIAM J.\nControl Optim., 30(1992), 1024-1065.\n[8]C. Castro and S. Cox, Achieving arbitrarily large decay in the dampedwave\nequation, SIAM J. Control. Optim. ,39(2001), 1748–1755.\n[9]S. Cox and E. Zuazua, The rate at which energy decays in a string damped\nat one end, Indiana Univ. Math.J. ,44(1995), 545-573.\n[10]S. Cox and E. Zuazua, The rate at which energy decays in a damped string,\nComm. Partial Differential Equations ,19(1994), 213-243.\n20[11]P. Freitas, Optimizing the rate of decay of solutions of the wave equatio n\nusinggeneticalgorithms: acounterexampletotheconstant dampingconjecture,\nSIAM J. Control Optim ,37(1999), 376-387.\n[12]A. Haraux, Une remarque sur la stabilisation de certains syst` emes du\ndeuxi` eme ordre en temps, Port. Math. ,46(1989), 245-258.\n[13]P. Freitas, Optimizing the Rate of Decay of Solutions of the Wave Equa-\ntion Using Genetic Algorithms: A Counterexample to the Cons tant Damping\nConjecture SIAM J. Control Optim., ,37, No 2, (1998), 376-387.\n[14]G. Lebeau, Equation des ondes amorties, Algebraic and geometric methods\nin mathematical physics. Proceedings of the 1st Ukrainian- French-Romanian\nsummer school, Kluwer Academic Publishers. Math. Phys. Stu d.19(1996), 73-\n109.\n[15]J. L. Lions et E. Magenes, Probl` emes aux limites non homog` enes et appli-\ncations, Dunod, Paris, 1968.\n[16]K. Ramadani, T. Takahashi and M. Tucsnak , Internal stabilization of the\nplate equation in a square: the continuous and the semi-disc etized problems, J.\nMath. Pures Appl., 85(2006), 17-37.\n[17]D. L. Russell, Decay rates for weakly damped systems in Hilbert space ob-\ntained with control theoretic methods, J. Diff. Eq. ,19(1975), 344-370.\n[18]A. A. Shkalikov, On the basis property of root vectors of a perturbed\nself-adjoint operator, Proceedings of the Steklov Institute of Mathematics ,269\n(2010), 290 - 303.\n[19]D. C. Sorenson, Implicitly Restarted Arnoldi/Lanczos Methods for Large\nScale Eigenvalue Problems , MATLAB documentation, 1995.\n21"    },    {        "title": "2311.18125v1.Bayesian_interpretation_of_Backus_Gilbert_methods.pdf",        "content": "Bayesian interpretation of Backus-Gilbert methods\nLuigi Del Debbio,𝑎Alessandro Lupo,𝑏,∗Marco Panero𝑐and Nazario Tantalo𝑑\n𝑎Higgs Centre for Theoretical Physics, School of Physics and Astronomy, The University of Edinburgh,\nPeter Guthrie Tait Road, Edinburgh EH9 3FD, UK\n𝑏Aix-Marseille Université, Université de Toulon, CNRS, CPT, Marseille, France\n𝑐Department of Physics, University of Turin & INFN, Turin\nVia Pietro Giuria 1, I-20125 Turin, Italy\n𝑑University and INFN of Roma Tor Vergata\nVia della Ricerca Scientifica 1, I-00133, Rome, Italy\nE-mail: alessandro.lupo@cpt.univ-mrs.fr\nThe extraction of spectral densities from Euclidean correlators evaluated on the lattice is an\nimportant problem, as these quantities encode physical information on scattering amplitudes,\nfinite-volume spectra, inclusive decay rates, and transport coefficients. In this contribution, we\nshow that the Bayesian approach to this “inverse” problem, based on Gaussian processes, can\nbe reformulated in a way that yields a solution equivalent, up to statistical uncertainties, to the\none obtained in a Backus-Gilbert approach. After discussing this equivalence, we point out its\nimplications for a reliable determination of spectral densities from lattice simulations.\nThe 40th International Symposium on Lattice Field Theory (Lattice 2023)\nJuly 31st - August 4th, 2023\nFermi National Accelerator Laboratory\n∗Speaker\n©Copyright owned by the author(s) under the terms of the Creative Commons\nAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/arXiv:2311.18125v1  [hep-lat]  29 Nov 2023Bayesian interpretation of Backus-Gilbert methods Alessandro Lupo\n1. Introduction\nThestudyofthenumericalinversionoftheLaplacetransformhasbecomepopularinthelattice\ncommunity,duetoitsimportanceindetermininghadronicobservablesfromsimulationsofquantum\nchromodynamics,andgaugetheoriesingeneral. Theproblem,however,isaparticularlychallenging\none, and requires a careful treatment to overcome the difficulties related to the inherently limited\ninformation that lattice data can provide, due to systematic and statistical uncertainties. Several\napproaches have been devised [1–5] as a means to provide a stable and reliable solution to the\nproblem, leading to an increase in applications [6–17].\nIn this work, we focus on two popular methods to tackle this inverse problem: the variation\nof the Backus-Gilbert (BG) procedure introduced in Ref. [1] and a Bayesian approach based on\nGaussian Processes (GP) [3, 18–20]. Even though these two approaches are based on drastically\ndifferent philosophies we shall prove, by building on the results of Ref. [18] and expanding the\ndiscussion present in Ref. [11], that a suitable choice of the inputs produces a Bayesian solution\ncentredaroundthemodifiedBGprediction. AfterabriefintroductioninSection2wedescribethe\nBayesian solution to the inversion problem in Section 3 which we then generalise, in Section 4, to\nmatch the results from Ref. [1].\n2. Formulation of the problem\nInlatticesimulations,Euclideancorrelators 𝐶𝐿𝑇computedinafinitehypervolume 𝐿3×𝑇are\nrelated to the spectral density 𝜌𝐿𝑇(𝐸)via a (generalised) Laplace transform,\n𝐶𝐿𝑇(𝑡)=∫∞\n0𝑑𝐸 𝑏𝑇(𝑡,𝐸)𝜌𝐿𝑇(𝐸), (1)\nthat has to be inverted to extract 𝜌𝐿𝑇(𝐸). We define 𝐶𝐿(𝑡)as the correlator in the limit 𝑇→∞,\nwhere𝑏𝑇(𝑡,𝐸)→𝑒−𝑡𝐸. Spectraldensitiesareespeciallyhardtomanageinafinitevolume,where\nthey are a sum of Dirac 𝛿distributions across the discrete spectrum of the Hamiltonian. For this\nreason, numerical methods typically target a smeared version of the spectral density, whereby the\nfinite-volume function 𝜌𝐿(𝐸)is convoluted with a Schwartz function S𝜎(𝜔,𝐸),\n𝜌𝐿(𝜎;𝜔)=∫∞\n0𝑑𝐸S𝜎(𝜔,𝐸)𝜌𝐿(𝐸), (2)\nlim\n𝜎→0S𝜎(𝜔,𝐸)=𝛿(𝜔−𝐸). (3)\nEq. (2) provides a way to define the infinite-volume limit for the spectral density by the following\nnon-commuting double limit\n𝜌(𝜔)=lim\n𝜎→0lim\n𝐿→∞𝜌𝐿(𝜎;𝜔). (4)\nAswelldocumentedintheliterature,theproblemofextracting 𝜌𝐿(𝜎,𝜔)from𝐶𝐿(𝑡)isill-defined.\nTo clarify this point, we begin by noting that if one had access to an infinite set of discrete data\nthat were exact, i.e., unaffected by uncertainties, then the solution could be written as a linear\ncombination of the data\n𝜌exact(𝜎;𝜔)=∞∑︁\n𝜏=1𝑔exact\n𝜏(𝜎;𝜔)𝐶exact(𝑎𝜏). (5)\n2Bayesian interpretation of Backus-Gilbert methods Alessandro Lupo\nIt is important to note that the previous infinite sum is an exactrepresentation of the continuous\nsmeared spectral density even if the data only constitute a discrete set. A first obstruction arises\nfrom the fact that, in reality, the correlator is available only at a finite number of data points\n0<𝜏≤𝜏max<𝑇/𝑎(where𝜏=𝑡/𝑎), which results into a systematic error\n𝜌exact(𝜎;𝜔)=𝜏max∑︁\n𝜏=1𝑔𝜏(𝜎;𝜔)𝐶exact(𝑎𝜏)+𝛿sys(𝜎,𝜏 max). (6)\nThe coefficients 𝑔𝜏(𝜎;𝜔)from Eq. (6) are typically very large and change sign swiftly, a property\nthatisnecessaryinordertoreproduceasmoothfunctionoutoftheexponentiallydecayingkernels\nof the correlators. This leads to a major difficulty in inverting Eq. (1). The correlators obtained\nfromlatticesimulationsare,infact,unavoidablyaffectedbystatisticalandsystematicuncertainties,\nmaking Eq. (6) numerically unstable. A way to tackle this problem consists in “regularising” the\nsum\n𝜌(𝜎;𝜔)=𝜏max∑︁\n𝜏=1𝑔𝜏(𝜎;𝜔)𝐶(𝑎𝜏) (7)\nby reducing the size of the 𝑔𝜏coefficients, so that 𝜌(𝜎;𝜔)does not depend too strongly on\nthe noise of the correlators. At the same time, the “regularised” coefficients should still yield\nmeaningful results with controlled uncertainties. Before describing two of these regularisations in\nthe following section, we remark that any linear combination of correlators necessarily reproduces\na smeared spectral density,\n𝜏max∑︁\n𝜏=1𝑔𝜏(𝜎;𝜔)𝐶(𝑎𝜏)=∫\n𝑑𝐸 𝜏max∑︁\n𝜏=1𝑔𝜏(𝜎;𝜔)𝑏𝑇(𝑎𝜏,𝐸)!\n𝜌𝐿(𝐸), (8)\nthe smearing kernel being\nS𝜎(𝐸,𝜔)=𝜏max∑︁\n𝜏=1𝑔𝜏(𝜎;𝜔)𝑏𝑇(𝑎𝜏,𝐸). (9)\n3. Bayesian Inference with Gaussian Processes\nLetRbe a stochastic field described by the Gaussian Process centred around 𝜌priorand with\ncovarianceKprior\nΠ[R]=1\nNexp\u0012\n−1\n2\f\fR−𝜌prior\f\f2\nKprior\u0013\n, (10)\nwhere we introduced the norm\n|R−𝜌prior\f\f2\nKprior=∫\n𝑑𝐸1∫\n𝑑𝐸2\u0002\nR(𝐸1)−𝜌prior(𝐸1)\u0003\nK−1\nprior(𝐸1,𝐸2)\u0002\nR(𝐸2)−𝜌prior(𝐸2)\u0003\n,\n(11)\nand the normalisation\nN=∫\nDRΠ[R]. (12)\nIn the last expressions, D𝑅is the functional integration measure over the field variable R, which\nwe shall use to describe a continuous function such as the spectral density.\n3Bayesian interpretation of Backus-Gilbert methods Alessandro Lupo\nLetC∈R𝜏maxbe a vector of stochastic variables, with components\nC(𝑡)=∫\n𝑑𝐸𝑏𝑇(𝑡,𝐸)R(𝐸)+𝜂(𝑡). (13)\nThe vector 𝜼∈R𝜏max, which describes the statistical fluctuations of the lattice correlators, is\nrepresented as a real-valued stochastic variable, for which we assume a multivariate Gaussian\ndistribution with zero mean and covariance Cov 𝑑,\nG[𝜼,Cov𝑑]=1√︁\ndet(2𝜋Cov𝑑)exp\u0012\n−1\n2𝜼Cov−1\n𝑑𝜼\u0013\n. (14)\nEq. 13 allows the evaluation of the covariance:\nCov[C(𝑡1),C(𝑡2)]=Σ𝑡1𝑡2+(Cov𝑑)𝑡1𝑡2, (15)\nwhere we defined\nΣ𝑡1𝑡2=∫\n𝑑𝐸1∫\n𝑑𝐸2𝑏𝑇(𝑡1,𝐸1)Kprior(𝐸1,𝐸2)𝑏𝑇(𝑡2,𝐸2). (16)\nWe are interested in the value of the spectral density at some energy 𝜔, its covariance and its\ncorrelation withC; for this purpose we extend the dimensionality of the covariance of Eq. (15) as\nfollows. Let us introduce the vector 𝑭∈R𝜏max,\nCov[C(𝑡),R(𝜔)]=𝐹𝑡(𝜔), (17)\nand the scalar function 𝐹∗,\nCov[R(𝜔),R(𝜔)]=𝐹∗(𝜔). (18)\nThe extended covariance is then\nΣtot= \n𝐹∗ 𝑭𝑇\n𝑭Σ+Cov𝑑!\n. (19)\nLet𝐶obs(𝑡)bethecentralvalueofthecorrelatormeasuredonthelatticebyaveragingoveragauge\nensemble, and let us denote 𝑪priorthe vector of components\n𝐶prior(𝑡)=∫\n𝑑𝐸𝑏𝑇(𝑡,𝐸)𝜌prior(𝐸). (20)\nThe joint probability density for R(𝜔)andC(𝑡)is the multidimensional Gaussian\nG\u0002\nR−𝜌prior,C−𝑪prior;Σtot\u0003\n, (21)\nwhich can be factorised into the product of two Gaussian distributions by performing a block\ndiagonalisationofthetotalcovarianceinEq.(19). Ofthetworesultingdistributions,onerepresents\nthe posterior probability density for R(𝜔)given its prior distribution and the set of measurements\nfor the correlator, while the other is the likelihood of the data:\nG\u0002\nR−𝜌prior,C−𝑪prior;Σtot\u0003\n=G\u0002\nR−𝜌post;Kpost\u0003\nG\u0002\nC−𝑪prior;Σ+Cov𝑑\u0003\n.(22)\n4Bayesian interpretation of Backus-Gilbert methods Alessandro Lupo\nThe posterior Gaussian distribution for the spectral density is centred around:\n𝜌post(𝜔)\f\f\nC=𝐶obs=𝜌prior(𝜔)+𝑭𝑇 1\nΣ+Cov𝑑\u0010\n𝑪obs−𝑪prior\u0011\n, (23)\nand has variance\nKpost(𝜔,𝜔)\f\f\nC=𝐶obs=Kprior(𝜔,𝜔)−𝑭𝑇 1\nΣ+Cov𝑑𝑭. (24)\nIn order to make contact with Eq. (6), we introduce the coefficients\n𝒈GP(𝜔)=𝑭𝑇 1\nΣ+Cov𝑑. (25)\nLet us discuss the previous equations. Eq. (19) shows how the inverse problem is regularised once\nexpressed in terms of probability distributions: the numerical instability, which leads to the very\nlargecoefficientsofEq.(5),isduetotheill-conditioningofthemodelcovariance ΣinEq.(16). The\ncovarianceCov 𝑑,addedtothematrix Σ,cutsoffitsnear-zeroeigenvalues. Theresultingsolutionis\nthen stable within statistical uncertainties. The very same regularisation is used in Backus-Gilbert\nmethods[1,11,21],despitethefactthattheycontainnoformulationintermsofstochasticvariables.\nAspointedoutaroundEqs.(8)and(9),thecentreoftheposteriordescribesasmearedspectral\ndensity,evenifthesmearingwasnotoneoftheinitialassumptions. Thesmearingfunction,which\ncanbeinferredfromEq.(9),isdeterminedbythechoiceofthepriorsandthenoiseonthedata,as\nseeninEq.(25). Consequently,thesmearingkernelobtainedinthissetupisnotknownapriori. In\nthisregard,thesolutiongiveninthissectionissimilartotheoriginalBackus-Gilbertproposal[21]\nratherthanthemethodofRef.[1],wherethesmearingkernelischosen. Thedirectconnectionwith\nRef. [1] is given in the next section.\n4. Backus-Gilbert methods in the Bayesian framework\nThe first step is to target the probability density for a spectral density smeared with an input\nkernelS𝜎(𝜔,𝐸). To generalise the results of the previous section, we introduce the stochastic\nvariable\nR𝜎(𝜔)=∫\n𝑑𝐸S𝜎(𝜔,𝐸)R(𝐸). (26)\nThe same steps described in Section 3 lead to the following extended covariance,\nΣtot= \n𝐹𝜎∗𝑭𝜎\n𝑭𝜎Σ+Cov𝑑!\n, (27)\nwhereΣis as in Eq. (16), while the other functions now include reference to the smearing kernel:\n𝐹𝜎\n∗(𝜔)=∫\n𝑑𝐸1∫\n𝑑𝐸2S𝜎(𝜔,𝐸 1)Kprior(𝐸1,𝐸2)S𝜎(𝐸2,𝜔), (28)\n𝐹𝜎\n𝑡(𝜔)=∫\n𝑑𝐸1∫\n𝑑𝐸2𝑏𝑇(𝑎𝜏,𝐸 1)Kprior(𝐸1,𝐸2)S𝜎(𝐸2,𝜔). (29)\nTo match Ref. [1] one can select [11] a diagonal model covariance,\nKprior(𝐸1,𝐸2)=𝑒𝛼𝐸\n𝜆𝛿(𝐸1−𝐸2), (30)\n5Bayesian interpretation of Backus-Gilbert methods Alessandro Lupo\nwhere𝛼 < 2and𝜆∈ (0,∞)are, in this context, hyperparameters, which can be chosen by\nmaximising the data likelihood [3]. They are introduced to match the parameters appearing in\nEq. (A3) in Ref. [1].\nLetthesmearingkernelbe,forinstance,aGaussian 𝐺𝜎(𝜔,𝐸)=exp[−(𝜔−𝐸)2/2𝜎2]/√\n2𝜋𝜎.\nThe posterior Gaussian distribution for a smeared spectral density, given the prior distribution and\nthe observed data, is centred around\n𝜌post\n𝜎(𝜔)=𝜌prior\n𝜎(𝜔)+𝜏max∑︁\n𝜏=1𝑔BG\n𝜏(𝜎,𝜔)𝐶(𝑎𝜏), (31)\nand has variance\nKpost(𝜔,𝜔)=\u0012∫\n𝑑𝐸𝐺2\n𝜎(𝜔,𝐸)𝑒𝛼𝜔\n𝜆\u0013\n−𝜏max∑︁\n𝜏=1𝑔BG\n𝜏(𝜎,𝜔)𝐹𝜎\n𝜏(𝜔). (32)\nDue to the careful choice of model covariance in Eq. (30), the coefficients 𝑔BG\n𝜏(𝜎,𝜔)are the same\nthat were derived in Ref. [1]. We recall that in the latter, the coefficients 𝑔BG\n𝜏are determined in a\ndrastically different way, i.e. by minimising the following functional:\n(1−𝜆′)∫∞\n0𝑑𝐸𝑒𝛼𝐸\f\f\f\f\f𝜏max∑︁\n𝜏=1𝑔𝜏(𝜎;𝜔)𝑏𝑇(𝑎𝜏,𝐸)−𝐺𝜎(𝜔,𝐸)\f\f\f\f\f2\n+𝜆′𝜏max∑︁\n𝜏,𝜏′=1𝑔𝜏(𝜎;𝜔)Cov𝜏𝜏′𝑔𝜏′(𝜎;𝜔). (33)\nThe Backus-Gilbert parameter 𝜆′∈(0,1)is related to the 𝜆in Eq. (30) by 𝜆=𝜆′/(1−𝜆′).\nWe identified a setup in which Bayesian and Ref. [1], two frameworks with utterly different\nphilosophies,providethesamecentralvalueforthesmearedspectraldensities. Thereare,however,\nimportant differences between the two methods, that we shall now discuss. A first aspect concerns\nthe error on the smeared spectral density. In non-Bayesian methods, including the one introduced\nin Ref. [1], statistical uncertainties are often propagated from the data by bootstrap. In the context\nofGPs,ontheotherhand,byworkingwithGaussiandistributionsweareabletoprovideananalytic\nexpression for the error, which is inherited by Eq. (32), the variance of the probability density\nthat describes the smeared spectral densities. Another important difference is the way algorithmic\nparameters are determined. For the Backus-Gilbert method, a procedure that has been proven\neffective [11, 12, 22] was introduced in Ref. [9]: it consists in finding a range of parameters, 𝜆\nand𝛼, such that any shift in the smeared spectral density is smaller than statistical fluctuations. In\nthis way, one ensures that the result does not depend on unphysical parameters of the algorithm.\nIn Bayesian inference, 𝜆and𝛼have a different interpretation as they are hyperparameters that\ndetermine the prior distribution. The latter, however, are again chosen ad hocas inputs of the\nprocedure, hence they should not affect the final result (up to statistical fluctuations). In Bayesian\ninference, it is common to determine the hyperparameters by minimising the negative logarithmic\nlikelihood (NLL),\n𝜏max\n2Log(2𝜋)+1\n2Logdet(Σ+Cov𝑑)+1\n2(𝑪obs−𝑪prior)1\nΣ+Cov𝑑(𝑪obs−𝑪prior).(34)\n6Bayesian interpretation of Backus-Gilbert methods Alessandro Lupo\nFigure 1: Toppanel: spectraldensitysmearedwithaGaussian,fordifferentvaluesofthe 𝜆-parameter. The\ncentral value is obtained according to Ref. [1], or equivalently from Eq. (31). The error is computed with a\nbootstrapintheformercase(BGinthelegend),andisgivenbyEq.(32)inthelatter(GPinthelegend). The\nfigurealsoshowsthestabilityregiondescribedinRef.[9],whichpredictsavalueidentifiedbythehorizontal\nband. This is consistent with the value obtained by choosing the 𝜆that minimises the NLL (red star). The\nresults are obtained using lattice data from Ref. [12].\nInlightoftheanalogydescribedinthiswork,itisnaturaltoaskhowvaluesof {𝜆,𝛼}specified\nby the minimum of the NLL relate to values determined according to the non-Bayesian procedure\nof Refs. [1, 9]. To this end we show, in the top panel of Fig. 1, a comparison between a smeared\nspectraldensityobtainedinthetwoapproaches,asafunctionof 𝜆. Thecentralvaluesarethesame,\nas inferred from Eq. (31). The statistical errors are determined with a bootstrap procedure (BG in\nthe legend) or from the square root of half Eq. (32) (GP in the legend). In this example, which\nusesMonteCarlolatticedata 1fromRef.[12],theuncertaintiesarefoundtobeofthesameorderof\nmagnitude,whichisanon-trivialresult,giventheprofounddifferenceinthewaytheyareobtained.\nAnother striking observation is that the value of 𝜆determined from the minimum of the NLL (red\nstar in the bottom panel), lies in a region in which the smeared spectral density does not change,\nwithin statistical noise, by changing the value of 𝜆. The value for the smeared spectral density that\nonewouldobtainfollowingRef.[9],shownasahorizontalbandinthetoppanelofFig.1,isinfact\n1The correlator used is a two-point correlation function of pseudoscalar mesons, in the two-index antisymmetric\nrepresentation of 𝑆𝑈(4)gauge theory, corresponding to the ensemble 𝐵3of Ref. [12].\n7Bayesian interpretation of Backus-Gilbert methods Alessandro Lupo\nconsistent with the value obtained minimising the NLL. This provides a non-trivial validation for\nboth methods and, eventually, for the predictions they yield.\n5. Conclusions\nThe method described in Ref. [1] and Gaussian processes are popular choices to compute\nsmeared spectral densities from lattice correlators. We have shown that the approach proposed\nin Ref. [1] can be reformulated in a Bayesian framework. This leads to a drastically different\ninterpretation of the variables at hand, which become stochastic variables characterised by their\nprobability distributions. The solution, and its error, are understood as the central value and the\nwidth of probability distributions. Nonetheless, for a specific choice of priors, the final prediction\nis identical, within statistical uncertainties, to the one from Ref. [1]. In our numerical tests, based\non lattice correlation functions of meson-like states, we have found that the Bayesian error on the\nspectral reconstruction, coming from the width of its probability density, is compatible with the\nstatistical error that one obtains by bootstrapping in a frequentist fashion. The analogy extends to\nthe determination of the input parameters, where the prescription of Ref. [9] for the 𝜆parameter is\nfound to be compatible with the minimisation of the NLL. Further details on this comparison will\nbe presented in a forthcoming publication.\nAcknowledgments\nAL is funded in part by l’Agence Nationale de la Recherche (ANR), under grant ANR-22-\nCE31-0011. AL and LDD received funding from the European Research Council (ERC) under\nthe European Union’s Horizon 2020 research and innovation program under Grant Agreement\nNo. 813942. LDD is also supported by the UK Science and Technology Facility Council (STFC)\ngrant ST/P000630/1. MP was partially supported by the Spoke 1 “FutureHPC & BigData” of\nthe Italian Research Center on High-Performance Computing, Big Data and Quantum Computing\n(ICSC)fundedbyMUR(M4C2-19)–NextGenerationEU(NGEU),bytheItalianPRIN“Progetti\ndiRicercadiRilevanteInteresseNazionale–Bando2022”,prot. 2022TJFCYB,andbythe“Simons\nCollaboration on Confinement and QCD Strings” funded by the Simons Foundation.\n8Bayesian interpretation of Backus-Gilbert methods Alessandro Lupo\nReferences\n[1] Martin Hansen, Alessandro Lupo, and Nazario Tantalo. Extraction of spectral densities from\nlattice correlators. Phys. Rev. D , 99(9):094508, 2019.\n[2] Lukas Kades, Jan M. Pawlowski, Alexander Rothkopf, Manuel Scherzer, Julian M. 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D , 108(7):074513, 2023.\n10"    },    {        "title": "1605.06578v1.Landau_Lifshitz_theory_of_the_magnon_drag_thermopower.pdf",        "content": "Landau-Lifshitz theory of the magnon-drag thermopower\nBenedetta Flebus,1, 2Rembert A. Duine,1, 3and Yaroslav Tserkovnyak2\n1Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena,\nUtrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands\n2Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\n3Department of Applied Physics, Eindhoven University of Technology,\nPO Box 513, 5600 MB, Eindhoven, The Netherlands\nMetallic ferromagnets subjected to a temperature gradient exhibit a magnonic drag of the electric\ncurrent. We address this problem by solving a stochastic Landau-Lifshitz equation to calculate the\nmagnon-drag thermopower. The long-wavelength magnetic dynamics result in two contributions to\nthe electromotive force acting on electrons: (1) An adiabatic Berry-phase force related to the solid\nangle subtended by the magnetic precession and (2) a dissipative correction thereof, which is rooted\nmicroscopically in the spin-dephasing scattering. The \frst contribution results in a net force pushing\nthe electrons towards the hot side, while the second contribution drags electrons towards the cold\nside, i.e., in the direction of the magnonic drift. The ratio between the two forces is proportional\nto the ratio between the Gilbert damping coe\u000ecient \u000band the coe\u000ecient \fparametrizing the\ndissipative contribution to the electromotive force.\nThe interest in thermoelectric phenomena in ferromag-\nnetic heterostructures has been recently revived by the\ndiscovery of the spin Seebeck e\u000bect [1]. This e\u000bect is now\nunderstood to stem from the interplay of the thermally-\ndriven magnonic spin current in the ferromagnet and the\n(inverse) spin Hall voltage generation in an adjacent nor-\nmal metal [2]. Lucassen et al. [3] subsequently proposed\nthat the thermally-induced magnon \row in a metallic\nferromagnet can also produce a detectable (longitudinal)\nvoltage in the bulk itself, due to the spin-transfer mecha-\nnism of magnon drag. Speci\fcally, smooth magnetization\ntexture dynamics induce an electromotive force [4], whose\nnet average over thermal \ructuations is proportional to\nthe temperature gradient. In this Letter, we develop\na Landau-Lifshitz theory for this magnon drag, which\ngeneralizes Ref. [3] to include a heretofore disregarded\nBerry-phase contribution. This additional magnon drag\ncan reverse the sign of the thermopower, which can have\npotential utility for designing scalable thermopiles based\non metallic ferromagnets.\nElectrons propagating through a smooth dynamic tex-\nture of the directional order parameter n(r;t) [such that\njn(r;t)j\u00111, with the self-consistent spin density given\nbys=sn] experience the geometric electromotive force\nof [4]\nFi=~\n2(n\u0001@tn\u0002@in\u0000\f@tn\u0001@in) (1)\nfor spins up along nand\u0000Fifor spins down. The resul-\ntant electric current density is given by\nji=\u001b\"\u0000\u001b#\nehFii=~P\u001b\n2ehn\u0001@tn\u0002@in\u0000\f@tn\u0001@ini;(2)\nwhere\u001b=\u001b\"+\u001b#is the total electrical conductivity, P=\n(\u001b\"\u0000\u001b#)=\u001bis the conducting spin polarization, and eis\nthe carrier charge (negative for electrons). The averaging\nh:::iin Eq. (2) is understood to be taken over the steady-\nstate stochastic \ructuations of the magnetic orientation.The latter obeys the stochastic Landau-Lifshitz-Gilbert\nequation [5]\ns(1 +\u000bn\u0002)@tn+n\u0002(Hz+h) +X\ni@iji= 0;(3)\nwhere\u000bis the dimensionless Gilbert parameter [6], H\nparametrizes a magnetic \feld (and/or axial anisotropy)\nalong thezaxis, and ji=\u0000An\u0002@inis the magnetic spin-\ncurrent density, which is proportional to the exchange\nsti\u000bnessA. ForH > 0, the equilibrium orientation is\nn!\u0000 z, which we will suppose in the following. The\nLangevin \feld stemming from the (local) Gilbert damp-\ning is described by the correlator [7]\nhhi(r;!)h\u0003\nj(r0;!0)i=2\u0019\u000bs~!\u000eij\u000e(r\u0000r0)\u000e(!\u0000!0)\ntanh~!\n2kBT(r);\n(4)\nupon Fourier transforming in time: h(!) =R\ndtei!th(t).\nAt temperatures much less than the Curie tempera-\nture,Tc, it su\u000eces to linearize the magnetic dynamics\nwith respect to small-angle \ructuations. To that end, we\nswitch to the complex variable, n\u0011nx\u0000iny, parametriz-\ning the transverse spin dynamics. Orienting a uniform\nthermal gradient along the xaxis,T(x) =T+x@xT,\nwe Fourier transform the Langevin \feld (4) also in real\nspace, with respect to the yandzaxes. Linearizing\nEq. (3) for small-angle dynamics results in the Helmholtz\nequation:\nA(@2\nx\u0000\u00142)n(x;q;!) =h(x;q;!); (5)\nwhere\u00142\u0011q2+[H\u0000(1+i\u000b)s!]=A,h\u0011hx\u0000ihy, and qis\nthe two-dimensional wave vector in the yzplane. Solving\nEq. (5) using the Green's function method, we substitute\nthe resultant ninto the expression for the charge current\ndensity (2), which can be appropriately rewritten in thearXiv:1605.06578v1  [cond-mat.mes-hall]  21 May 20162\nfollowing form (for the nonzero xcomponent):\njx=~P\u001b\n2eZd2qd!\n(2\u0019)3!\n\u0002Re(1 +i\f)hn(x;q;!)@xn\u0003(x;q0;!0)i\n(2\u0019)3\u000e(q\u0000q0)\u000e(!\u0000!0): (6)\nTedious but straightforward manipulations, using the\ncorrelator (4), \fnally give the following thermoelectric\ncurrent density:\njx=\u000bsP\u001b@xT\n4eA2kBT2Zd2qd!\n(2\u0019)3(~!)3\nsinh2~!\n2kBTRe [(1 +i\f)I];\n(7)\nwhereI(\u0014)\u0011\u0014=j\u0014j2(Re\u0014)2, having made the convention\nthat Re\u0014>0.\nTo recast expression (7) in terms of magnon modes, we\nincorporate the integration over qxby noticing that, in\nthe limit of low damping, \u000b!0,\nI=2\n\u0019Z\ndqx1 +iq2\nx=\u000b~!\n(~!\u0000q2x\u0000q2\u0000\u0018\u00002)2+ (\u000b~!)2:(8)\nHere, we have introduced the magnetic exchange length\n\u0018\u0011p\nA=H and de\fned ~ !\u0011s!=A . After approximating\nthe Lorentzian in Eq. (8) with the delta function when\n\u000b\u001c1, Eq. (7) can \fnally be expressed in terms of a\ndimensionless integral\nJ(a)\u0011Z1\na=p\n2dxx5p\n2x2\u0000a2\nsinh2x2; (9)\nas\nj=\u0012\n1\u0000\f\n3\u000b\u0013\nJ\u0012\u0015\n\u0018\u0013kBP\u001b\n\u00192e\u0012T\nTc\u00133=2\nrT: (10)\nHere,Tis the ambient temperature, kBTc\u0011A(~=s)1=3\nestimates the Curie temperature, and \u0015\u0011p\n~A=skBT\nis the thermal de Broglie wavelength in the absence of\nan applied \feld. We note that \u000b;\f\u001c1 while\u000b\u0018\f, in\ntypical transition-metal ferromagnets [8].\nFor temperatures much larger than the magnon gap\n(typically of the order of 1 K in metallic ferromagnets),\n\u0015\u001c\u0018and we can approximate J(\u0015=\u0018)\u0019J(0)\u0018\n1. This limit e\u000bectively corresponds to the gapless\nmagnon dispersion of \u000fq\u0011~!q\u0019~Aq2=s. Within\nthe Boltzmann phenomenology, the magnonic heat cur-\nrent induced by a uniform thermal gradient is given by\njQ=\u0000rTR\n[d3q=(2\u0019)3](@qx!q)2\u001c(!q)\u000fq@TnBE, where\n\u001c\u00001(!q) = 2\u000b!qis the Gilbert-damping decay rate of\nmagnons (to remain within the consistent LLG phe-\nnomenology) and nBE= [exp(\u000fq=kBT)\u00001]\u00001is the Bose-\nEinstein distribution function. By noticing that\n\u000fq@TnBE=kB\u0014~!q=2kBT\nsinh( ~!q=2kBT)\u00152\n; (11)\nrThydrodynamic\nr⌦<0geometricˆxˆzˆy\n⌦\ne\u0000e\u0000e\u0000e\u0000e\u0000e\u0000e\u0000e\u0000\nFIG. 1. Schematics for the two contributions to the electron-\nmagnon drag. In the absence of decay (i.e., \u000b!0), magnons\ndrifting from the hot (left) side to the cold (right) side drag\nthe charge carriers viscously in the same direction, inducing\na thermopower /\f. The (geometric) Berry-phase drag gov-\nerned by the magnon decay is proportional to \u000band acts in\nthe opposite direction. It is illustrated for a spin wave that is\nthermally emitted from the left. As the spin wave propagates\nto the right, the solid angle \n subtended by the spin preces-\nsion shrinks, inducing a force oriented to the left for spins\nparallel to n.\nit is easy to recast the second, /\fcontribution to\nEq. (10) in the form\nj(\f)=\f~P\u001b\n2eAjQ; (12)\nwhich reproduces the main result of Ref. [3].\nThe magnon-drag thermopower (Seebeck coe\u000ecient),\nS=\u0000@xV\n@xT\f\f\f\f\njx=0; (13)\ncorresponds to the voltage gradient @xVinduced under\nthe open-circuit condition. We thus get from Eq. (10):\nS=\u0012\f\n3\u000b\u00001\u0013\nJkBP\n\u00192e\u0012T\nTc\u00133=2\n= (\f\u00003\u000b)~P\u0014m\n2eA;\n(14)\nwhere\u0014m= (2=3\u00192)JkBA(T=Tc)3=2=\u000b~is the magnonic\ncontribution to the heat conductivity. Such magnon-drag\nthermopower has recently been observed in Fe and Co\n[9], with scaling/T3=2over a broad temperature range\nand opposite sign in the two metals. Note that the sign\ndepends on \f=\u000b and the e\u000bective carrier charge e.\nEquations (10) and (14) constitute the main results of\nthis paper. In the absence of Gilbert damping, \u000b!0,\nthe magnon-drag thermopower Sis proportional to the\nheat conductivity. This contribution was studied in\nRef. [3] and is understood as a viscous hydrodynamic\ndrag. In simple model calculations [8], \fP > 0 and this\nhydrodynamic thermopower thus has the sign of the ef-\nfective carrier charge e. WhenP > 0, so that the ma-\njority band is polarized along the spin order parameter3\nn, the/\u000bcontribution to the thermopower is opposite\nto the/\fcontribution. (Note that \u000bis always>0,\nin order to yield the positive dissipation.) The underly-\ning geometric meaning of this result is sketched in Fig. 1.\nNamely, the spin waves that are generated at the hot end\nand are propagating towards the cold end are associated\nwith a decreasing solid angle, @x\n<0. The \frst term\nin Eq. (1), which is rooted in the geometric Berry con-\nnection [10], is proportional to the gradient of this solid\nangle times the precession frequency, /!@i\n, resulting\nin a net force towards the hot side acting on the spins\ncollinear with n.\nNote that we have neglected the Onsager-reciprocal\nbackaction of the spin-polarized electron drift on the\nmagnetic dynamics. This is justi\fed as including the\ncorresponding spin-transfer torque in the LLG equation\nwould yield higher-order e\u000bects that are beyond our\ntreatment [11]. The di\u000busive contribution to the See-\nbeck e\u000bect,/T=EF, whereEFis a characteristic Fermi\nenergy, which has been omitted from our analysis, is ex-\npected to dominate only at very low temperatures [9].\nThe conventional phonon-drag e\u000bects have likewise been\ndisregarded. A systematic study of the relative impor-\ntance of the magnon and phonon drags is called upon in\nmagnetic metals and semiconductors.\nThis work is supported by the ARO under Contract\nNo. 911NF-14-1-0016, FAME (an SRC STARnet center\nsponsored by MARCO and DARPA), the Stichting voor\nFundamenteel Onderzoek der Materie (FOM), and the\nD-ITP consortium, a program of the Netherlands Orga-\nnization for Scienti\fc Research (NWO) that is funded by\nthe Dutch Ministry of Education, Culture, and Science\n(OCW).[1] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae,\nK. Ando, S. Maekawa, and E. Saitoh, Nature 455,\n778 (2008); K. Uchida, J. Xiao, H. Adachi, J. Ohe,\nS. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa,\nH. Kawai, G. E. W. Bauer, S. Maekawa, and E. Saitoh,\nNat. Mater. 9, 894 (2010).\n[2] J. Xiao, G. E. W. Bauer, K. Uchida, E. Saitoh, and\nS. Maekawa, Phys. Rev. B 81, 214418 (2010).\n[3] M. E. Lucassen, C. H. Wong, R. A. Duine, and\nY. Tserkovnyak, Appl. Phys. Lett. 99, 262506 (2011).\n[4] R. A. Duine, Phys. Rev. B 77, 014409 (2008);\nY. Tserkovnyak and M. Mecklenburg, ibid.77, 134407\n(2008).\n[5] S. Ho\u000bman, K. Sato, and Y. Tserkovnyak, Phys. Rev. B\n88, 064408 (2013).\n[6] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n[7] W. F. Brown, Phys. Rev. 130, 1677 (1963).\n[8] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, J.\nMagn. Magn. Mater. 320, 1282 (2008).\n[9] S. J. Watzman, R. A. Duine, Y. Tserkovnyak, H. Jin,\nA. Prakash, Y. Zheng, and J. P. Heremans, \\Magnon-\ndrag thermopower and Nernst coe\u000ecient in Fe and Co,\"\narXiv:1603.03736.\n[10] M. V. Berry, Proc. R. Soc. London A 392, 45 (1984);\nG. E. Volovik, J. Phys. C: Sol. State Phys. 20, L83\n(1987); S. E. Barnes and S. Maekawa, Phys. Rev. Lett.\n98, 246601 (2007); Y. Tserkovnyak and C. H. Wong,\nPhys. Rev. B 79, 014402 (2009).\n[11] The backaction by the spin-transfer torque would be ab-\nsent when the longitudinal spin current, ji=\u001b(PE i+\nFi=e)n, vanishes, where Eiis the electric \feld and Fi\nis the spin-motive force (1). Understanding Eq. (10) as\npertaining to the limit of the vanishing spin current ji\nrather than electric current ji=\u001b(Ei+PFi=e)nwould,\nhowever, result in higher-order (in T=T c) corrections to\nthe Seebeck coe\u000ecient (13). These are beyond the level\nof our approximations."    },    {        "title": "2003.12967v1.Stability_results_for_an_elastic_viscoelastic_waves_interaction_systems_with_localized_Kelvin_Voigt_damping_and_with_an_internal_or_boundary_time_delay.pdf",        "content": "arXiv:2003.12967v1  [math.AP]  29 Mar 2020STABILITY RESULTS FOR AN ELASTIC-VISCOELASTIC WAVES INTER ACTION\nSYSTEMS WITH LOCALIZED KELVIN-VOIGT DAMPING AND WITH AN INT ERNAL\nOR BOUNDARY TIME DELAY\nMOUHAMMAD GHADER1, RAYAN NASSER1,2, AND ALI WEHBE1\nAbstract. We investigate the stability of a one-dimensional wave equa tion with non smooth localized internal\nviscoelastic damping of Kelvin-Voigt type and with boundar y or localized internal delay feedback. The main\nnovelty in this paper is that the Kelvin-Voigt and the delay d amping are both localized via non smooth\ncoefficients. In the case that the Kelvin-Voigt damping is loc alized faraway from the tip and the wave is\nsubjected to a locally distributed internal or boundary del ay feedback, we prove that the energy of the system\ndecays polynomially of type t−4. However, an exponential decay of the energy of the system is established\nprovided that the Kelvin-Voigt damping is localized near a p art of the boundary and a time delay damping\nacts on the second boundary. While, when the Kelvin-Voigt an d the internal delay damping are both localized\nvia non smooth coefficients near the tip, the energy of the syst em decays polynomially of type t−4. Frequency\ndomain arguments combined with piecewise multiplier techn iques are employed.\nContents\n1. Introduction 1\n2. Wave equation with local Kelvin-Voigt damping and with bo undary delay feedback 7\n2.1. Wave equation with local Kelvin-Voigt damping far from the boundary and with boundary\ndelay feedback 8\n2.1.1. Well-posedness of the problem 9\n2.1.2. Strong Stability 12\n2.1.3. Polynomial Stability 16\n2.2. Wave equation with local Kelvin-Voigt damping near the boundary and boundary delay\nfeedback 21\n3. Wave equation with local internal Kelvin-Voigt damping a nd local internal delay feedback 24\n3.1. Well-posedness of the problem 25\n3.2. Polynomial Stability 28\nAppendix A. Notions of stability and theorems used 37\nReferences 38\n1Lebanese University, Faculty of sciences 1, Khawarizmi Lab oratory of Mathematics and Applications-KALMA,\nHadath-Beirut, Lebanon.\n2Université de Bretagne-Occidentale, France.\nE-mail addresses :mhammadghader@hotmail.com, rayan.nasser94@hotmail.co m, ali.wehbe@ul.edu.lb .\n1991Mathematics Subject Classification. 35L05; 35B35; 93D15; 93D20.\nKey words and phrases. Wave equation; Kelvin-Voigt damping; Time delay; Semigrou p; Stability.\niWAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\n1.Introduction\nViscoelastic materials feature intermediate characteris tics between purely elastic and purely viscous behav-\niors,i.e.they display both behaviors when undergoing deformation. I n wave equations, when the viscoelastic\ncontrolling parameter is null, the viscous property vanish es and the wave equation becomes a pure elastic wave\nequation. However, time delays arise in many applications a nd practical problems like physical, chemical, bi-\nological, thermal and economic phenomena, where an arbitra ry small delay may destroy the well-posedness of\nthe problem and destabilize it. Actually, it is well-known t hat simplest delay equations of parabolic type,\nut(x,t) = ∆u(x,t−τ),\nor hyperbolic type\nutt(x,t) = ∆u(x,t−τ),\nwith a delay parameter τ >0,are not well-posed. Their instability is due to the existenc e of a sequence of\ninitial data remaining bounded, while the corresponding so lutions go to infinity in an exponential manner at a\nfixed time (see [ 16,23]).\nThe stabilization of a wave equation with Kelvin-Voigt type damping and internal or boundary time delay\nhas attracted the attention of many authors in the last five ye ars. Indeed, in 2016 Messaoudi et al. studied\nthe stabilization of a wave equation with global Kelvin-Voi gt damping and internal time delay in the multidi-\nmensional case (see [ 30]), and they obtained an exponential stability result. In th e same year, Nicaise et al. in\n[33] considered the multidimensional wave equation with local ized Kelvin-Voigt damping and mixed boundary\ncondition with time delay. They obtained an exponential dec ay of the energy regarding that the damping is\nacting on a neighborhood of part of the boundary via a smooth c oefficient. Also, in 2018, Anikushyn et al. in\n[15] considered the stabilization of a wave equation with globa l viscoelastic material subjected to an internal\nstrong time delay where a global exponential decay rate was o btained. Thus, it seems to us that there are\nno previous results concerning the case of wave equations wi th internal localized Kelvin-Voigt type damping\nand boundary or internal time delay, especially in the absen ce of smoothness of the damping coefficient even\nin the one dimensional case. So, we are interested in studyin g the stability of elastic wave equation with local\nKelvin–Voigt damping and with boundary or internal time del ay (see Systems ( 1.1) and ( 1.2)).\nThis paper investigates the study of the stability of a strin g with Kelvin-Voigt type damping localized via\nnon-smooth coefficient and subjected to a localized internal or boundary time delay. Indeed, in the first part of\nthis paper, we study the stability of elastic wave equation w ith local Kelvin–Voigt damping, boundary feedback\nand time delay term at the boundary, i.e.we consider the following system\n(1.1)\n\nUtt(x,t)−/bracketleftbig\nκUx(x,t)+δ1χ(α,β)Uxt(x,t)/bracketrightbig\nx= 0,(x,t)∈(0,L)×(0,+∞),\nU(0,t) = 0, t∈(0,+∞),\nUx(L,t) =−δ3Ut(L,t)−δ2Ut(L,t−τ), t ∈(0,+∞),\n(U(x,0),Ut(x,0)) = (U0(x),U1(x)), x ∈(0,L),\nUt(L,t) =f0(L,t), t ∈(−τ,0),\nwhereL, τ, δ 1andδ3are strictly positive constant numbers, δ2is a non zero real number and the initial data\n(U0,U1,f0)belongs to a suitable space. Here 0≤α<β <L andU=uχ(0,α)+vχ(α,β)+wχ(β,L), withχ(a,b)is\nthe characteristic function of the interval (a,b).We assume that there exist strictly positive constant numbe rs\nκ1, κ2, κ3, such that κ=κ1χ(0,α)+κ2χ(α,β)+κ3χ(β,L).In fact, here we will consider two cases. In the first\ncase, we divide the bar into 3 pieces; the first piece is an elas tic part, the second piece is the viscoelastic part\nand in the third piece, the time delay feedback is effective at the ending point of the piece, i.e.we consider\nthe caseα>0(see Figure 1). While, in the second case, we divide the bar into 2 pieces; t he first piece is the\nviscoelastic part and in the second piece the time delay feed back is effective at the ending point of the piece,\ni.e.we consider the case α= 0(see Figure 2). Remark, here, in both cases, the Kelvin–Voigt damping is\neffective on a part of the piece and the time delay is effective a tL.\n1WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\n• •α βv(x) u(x) w(x)\n0 LElastic part Viscoelastic part Boundary delay feedback\nFigure 1. K-V damping is acting localized in the internal of the body an d time delay feedback\nis effective at L\n•\n0 βv(x) w(x)\nLViscoelastic part Boundary delay feedback\nFigure 2. K-V damping is acting localized near the boundary of the body and time delay\nfeedback is effective at L\nIn the second part of this paper, we study the stability of ela stic wave equation with local Kelvin–Voigt damping\nand local internal time delay. This system takes the followi ng form\n(1.2)\n\nUtt(x,t)−/bracketleftbig\nκUx(x,t)+χ(α,β)(δ1Uxt(x,t)+δ2Uxt(x,t−τ))/bracketrightbig\nx= 0,(x,t)∈(0,L)×(0,+∞),\nU(0,t) =U(L,t) = 0, t∈(0,+∞),\n(U(x,0),Ut(x,0)) = (U0(x),U1(x)), x ∈(0,L),\nUt(x,t) =f0(x,t), (x,t)∈(0,L)×(−τ,0),\nwhereL, τandδ1are strictly positive constant numbers, δ2is a non zero real number and the initial data\n(U0,U1,f0)belongs to a suitable space. Here 0<α<β <L andU=uχ(0,α)+vχ(α,β)+wχ(β,L), withχ(a,b)is\nthe characteristic function of the interval (a,b).We assume that there exist strictly positive constant numbe rs\nκ1, κ2, κ3, such that κ=κ1χ(0,α)+κ2χ(α,β)+κ3χ(β,L).In fact, here we will divide the bar into 3 pieces; the\nfirst piece is an elastic part, in the second piece the Kelvin– Voigt damping and the time delay are effective and\nthe third piece is an elastic part (see Figure 3). So, the Kelvin–Voigt damping and the time delay are effecti ve\non(α,β).\n• •α βv(x) u(x) w(x)\n0 LElastic part Viscoelastic part &delay feedback Elastic part\nFigure 3. Local internal K-V damping and Local internal delay feedbac k\nIn the literature, Datko et al. studied in 1985 the one dimensional wave equation which mode ls the vibrations\n2WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nof a string fixed at one end and free at the other one (see [ 14]). The system is given by the following:\n(1.3)\n\nutt(x,t)−uxx(x,t)+2aut(x,t)+a2u(x,t) = 0,(x,t)∈(0,1)×(0,+∞),\nu(0,t) = 0, t∈(0,+∞),\nux(1,t) =−κut(1,t−τ), t ∈(0,+∞),\nwhere the delay parameter τis strictly positive, a>0andκ>0. So, the above system models a string having\na boundary feedback with delay at the free end. They showed th at ifκ/parenleftbig\ne2a+1/parenrightbig\n<e2a−1,then System ( 1.3)\nis strongly stable for all small enough delays. However, if κ/parenleftbig\ne2a+1/parenrightbig\n>e2a−1,then there exists an open set D\ndense in (0,+∞), such that for all τinD, System ( 1.3) admits exponentially unstable solutions. Moreover, in\nthe absence of delay in System ( 1.3) (i.eτ= 0) anda≥0,κ≥0, its energy decays exponentially to zero under\nthe condition a2+κ2>0(see [12]). In 1990, Datko in [ 13] considered the boundary feedback stabilization of a\none-dimensional wave equation with time delay (see Example 3.5 in [ 13]). The system is given by the following:\n(1.4)\n\nutt(x,t)−uxx(x,t)−δuxxt(x,t) = 0,(x,t)∈(0,1)×(0,+∞),\nu(0,t) = 0, t ∈(0,+∞),\nux(1,t) =−κut(1,t−τ), t ∈(0,+∞),\nwhereτ >0, κ >0andδ >0. He proved that System ( 1.4) is unstable for an arbitrary small value of τ.\nIn 2006, Xu et al. in [44] investigated the following closed loop system with homoge neous Dirichlet boundary\ncondition at x= 0and delayed Neumann boundary feedback at x= 1:\n(1.5)\n\nutt(x,t)−uxx(x,t) = 0, (x,t)∈(0,1)×(0,+∞),\nu(0,t) = 0, t ∈(0,+∞),\nux(1,t) =−κut(1,t)−κ(1−µ)ut(1,t−τ), t∈(0,+∞),\n(u(x,0),ut(x,0)) = (u0(x),u1(x)), x ∈(0,1),\nut(1,t) =f0(1,t), t ∈(−τ,0).\nThe above system represents a wave equation that is fixed at on e end and subjected to a boundary control\ninput possessing a partial time delay of weight (1−µ)at the other end. They proved the following stability\nresults:\n1. Ifµ>2−1,then System ( 1.5) is uniformly stable.\n2. Ifµ= 2−1andτ∈Q∩(0,1), then System ( 1.5) is unstable.\n3. Ifµ= 2−1andτ∈(R\\Q)∩(0,1), then System ( 1.5) is asymptotically stable.\n4. Ifµ<2−1,then System ( 1.5) is always unstable.\nLater on, in 2008, Guo and Xu in [ 18] studied the stabilization of a wave equation in the 1-D case where it is\neffected by a boundary control and output observation sufferi ng from time delay. The system is given by the\nfollowing:\n\n\nutt(x,t)−uxx(x,t) = 0, (x,t)∈(0,1)×(0,+∞),\nu(0,t) = 0, t ∈(0,+∞),\nux(1,t) =w(t), t ∈(0,+∞),\n(u(x,0),ut(x,0)) = (u0(x),u1(x)), x∈(0,1),\ny(t) =ut(1,t−τ), t ∈(0,+∞),\nwherewis the control and yis the output observation. Using the separation principle, the authors proved\nthat the above delayed system is exponentially stable. In 20 10, Gugat in [ 17] studied the wave equation which\nmodels a string of length Lthat is rigidly fixed at one end and stabilized with a boundary feedback and constant\n3WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\ndelay at the other end. The problem is described by the follow ing system\n\n\nutt(x,t)−c2uxx(x,t) = 0, (x,t)∈(0,L)×(0,+∞),\nu(0,t) = 0, t ∈(0,+∞),\nux(L,t) = 0, t ∈(0,2Lc−1),\nux(L,t) =c−1λut/parenleftbig\nL,t−2Lc−1/parenrightbig\n, t ∈(2Lc−1,+∞),\n(u(x,0),ut(x,0),u(0,0)) = (u0(x),u1(x),0), x∈(0,L),\nwhereλis a real number and c >0. Gugat proved that the above system is exponentially stable . In 2011,\nJ. Wang et al. in [41] studied the stabilization of a wave equation under boundar y control and collocated\nobservation with time delay. The system is given by the follo wing:\n\n\nutt(x,t)−uxx(x,t) = 0, (x,t)∈(0,1)×(0,+∞),\nu(0,t) = 0, t ∈(0,+∞),\nux(1,t) =κut(1,t−τ), t ∈(0,+∞),\n(u(x,0),ut(x,0)) = (u0(x),u1(x)), x∈(0,1).\nThey showed that if the delay is equal to even multiples of the wave propagation time, then the above closed\nloop system is exponentially stable under sufficient and nece ssary conditions for κ. Else, if the delay is an odd\nmultiple of the wave propagation time, thus the closed loop s ystem is unstable. In 2013, H. Wang et al. in [43],\nstudied System ( 1.5) under the feedback control law ut(1,t) =w(t)provided that the weight of the feedback\nwith delay is a real βand that of the feedback without delay is a real α. They found a feedback control law\nthat stabilizes exponentially the system for any |α| /\\e}atio\\slash=|β|, by modifying the velocity feedback into the form\nu(t) =βwt(1,t)+αf(w(.,t),wt(.,t)), wherefis a linear functional. Finally, in 2017, Xu et al. in [42], studied\nthe stability problem of a one dimensional wave equation wit h internal control and boundary delay term\n\n\nutt(x,t)−uxx(x,t)+2αut(x,t) = 0,(x,t)∈(0,1)×(0,+∞),\nu(0,t) = 0, t ∈(0,+∞),\nux(1,t) =κut(1,t−τ), t ∈(0,+∞),\n(u(x,0),ut(x,0)) = (u0(x),u1(x)), x∈(0,1),\nut(1,t) =f0(1,t), t ∈(−τ,0),\nwhereτ >0,α>0andκis real. Based on the idea of Lyapunov functional, they prove d exponential stability\nof the above system under a certain relationship between αandκ.\nGoing to the multidimensional case, the stability of wave eq uation with time delay has been studied in\n[32,6,37,30,33,15,3,4]. In 2006, Nicaise and Pignotti in [ 32] studied the multidimensional wave equation\nconsidering two cases. The first case concerns a wave equatio n with boundary feedback and a delay term at\nthe boundary\n(1.6)\n\nutt(x,t)−∆u(x,t) = 0, (x,t)∈Ω×(0,+∞),\nu(x,t) = 0, (x,t)∈ΓD×(0,+∞),\n∂u\n∂ν(x,t) =−µ1ut(x,t)−µ2ut(x,t−τ),(x,t)∈ΓN×(0,+∞),\n(u(x,0),ut(x,0)) = (u0(x),u1(x)), x ∈Ω,\nut(x,t) =f0(x,t), (x,t)∈ΓN×(−τ,0).\n4WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nThe second case concerns a wave equation with an internal fee dback and a delayed velocity term ( i.ean internal\ndelay) and a mixed Dirichlet-Neumann boundary condition\n(1.7)\n\nutt(x,t)−∆u(x,t)+µ1ut(x,t)+µ2ut(x,t−τ) = 0,(x,t)∈Ω×(0,+∞),\nu(x,t) = 0, (x,t)∈ΓD×(0,+∞),\n∂u\n∂ν(x,t) = 0, (x,t)∈ΓN×(0,+∞),\n(u(x,0),ut(x,0)) = (u0(x),u1(x)), x ∈Ω,\nut(x,t) =f0(x,t), (x,t)∈Ω×(−τ,0).\nIn both systems, τ, µ1, µ2are strictly positive constants, ∂u/∂ν is the partial derivative, Ωis an open bounded\ndomain of RNwith a boundary Γof classC2andΓ = ΓD∪ΓN, such that ΓD∩ΓN=∅. Under the assumption\nthat the weight of the feedback with delay is smaller than tha t without delay (µ2< µ1), they obtained an\nexponential decay of the energy of both Systems ( 1.6) and ( 1.7). On the contrary, if the previous assumption\ndoes not hold (i.eµ2≥µ1), they found a sequence of delays for which the energy of some s olutions does not\ntend to zero (see [ 10] for the treatment of Problem ( 1.7) in more general abstract form). In 2009, Nicaise et\nal.in [34] studied System ( 1.6) in the one dimensional case where the delay time τis a function depending on\ntime and they established an exponential stability result u nder the condition that the derivative of the decay\nfunction is upper bounded by a constant d<1and assuming that µ2<√\n1−d µ1. In 2010, Ammari et al. in\n[6] studied the wave equation with interior delay damping and d issipative undelayed boundary condition in an\nopen domain ΩofRN, N≥2.The system is given by the following:\n(1.8)\n\nutt(x,t)−∆u(x,t)+aut(x,t−τ) = 0,(x,t)∈Ω×(0,+∞),\nu(x,t) = 0, (x,t)∈Γ0×(0,+∞),\n∂u\n∂ν(x,t) =−κut(x,t), (x,t)∈Γ1×(0,+∞),\n(u(x,0),ut(x,0)) = (u0(x),u1(x)), x ∈Ω,\nut(x,t) =f0(x,t), (x,t)∈Ω×(−τ,0),\nwhereτ >0,a>0andκ >0. Under the condition that Γ1satisfies the Γ-condition introduced in [ 25], they\nproved that System ( 1.8) is uniformly asymptotically stable whenever the delay coe fficient is sufficiently small.\nIn 2012, Pignotti in [ 37] considered the wave equation with internal distributed ti me delay and local damping\nin a bounded and smooth domain Ω⊂RN,N≥1. The system is given by the following:\n(1.9)\n\nutt(x,t)−∆u(x,t)+aχωut(x,t)+κut(x,t−τ) = 0,(x,t)∈Ω×(0,+∞),\nu(x,t) = 0, (x,t)∈Γ×(0,+∞),\n(u(x,0),ut(x,0)) = (u0(x),u1(x)), x ∈Ω,\nut(x,t) =f(x,t), (x,t)∈Ω×(−τ,0),\nwhereκreal,τ >0anda >0. System ( 1.9) shows that the damping is localized, indeed, it acts on a\nneighborhood of a part of the boundary of Ω. Under the assumption that |κ|<κ0<a,the author established\nan exponential decay rate. Later, in 2016, Messaoudi et al. in [30] considered the stabilization of the following\nwave equation with strong time delay\n\n\nutt(x,t)−∆u(x,t)−µ1∆ut(x,t)−µ2∆ut(x,t−τ) = 0,(x,t)∈Ω×(0,+∞),\nu(x,t) = 0, (x,t)∈Γ×(0,+∞),\n(u(x,0),ut(x,0)) = (u0(x),u1(x)), x ∈Ω,\nut(x,t) =f0(x,t), (x,t)∈Ω×(−τ,0),\nwhereµ1>0andµ2is a non zero real number. Under the assumption that |µ2|< µ1, they obtained an\nexponential stability result. In addition, in the same year , Nicaise et al. in [33] studied the multidimensional\n5WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nwave equation with localized Kelvin-Voigt damping and mixe d boundary condition with time delay\n(1.10)\n\nutt(x,t)−∆u(x,t)−div(a(x)∇ut) = 0,(x,t)∈Ω×(0,+∞),\nu(x,t) = 0, (x,t)∈Γ0×(0,+∞),\n∂u\n∂ν(x,t) =−a(x)∂ut\n∂ν(x,t)−κut(x,t−τ),(x,t)∈Γ1×(0,+∞),\n(u(x,0),ut(x,0)) = (u0(x),u1(x)), x ∈Ω,\nut(x,t) =f0(x,t), (x,t)∈Γ1×(−τ,0),\nwhereτ >0,κis real,a(x)∈L∞(Ω)anda(x)≥a0>0onωsuch thatω⊂Ωis an open neighborhood of Γ1.\nUnder an appropriate geometric condition on Γ1and assuming that a∈C1,1(Ω),∆a∈L∞(Ω), they proved\nan exponential decay of the energy of System ( 1.10). Finally, in 2018, Anikushyn et al. in [15] considered an\ninitial boundary value problem for a viscoelastic wave equa tion subjected to a strong time localized delay in a\nKelvin-Voigt type. The system is given by the following:\n\n\nutt(x,t)−c1∆u(x,t)−c2∆u(x,t−τ)−d1∆ut(x,t)−d2∆ut(x,t−τ) = 0,(x,t)∈Ω×(0,+∞),\nu(x,t) = 0, (x,t)∈Γ0×(0,+∞),\n∂u\n∂ν(x,t) = 0, (x,t)∈Γ1×(0,+∞),\n(u(x,0),ut(x,0)) = (u0(x),u1(x)), x∈Ω,\nu(x,t) =f0(x,t), (x,t)∈Ω×(−τ,0).\nUnder appropriate conditions on the coefficients, a global ex ponential decay rate is obtained. We can also\nmention that Ammari et al. in [7] considered the stabilization problem for an abstract equa tion with delay and\na Kelvin-Voigt damping in 2015. The system is given by the fol lowing:\n\n\nutt(t)+aBB∗ut(t)+BB∗u(t−τ), t∈(0,+∞),\n(u(0),ut(0)) = (u0,u1),\nB∗u(t) =f0(t), t ∈(−τ,0),\nfor an appropriate class of operator Banda >0.Using the frequency domain approach, they obtained an\nexponential stability result. Finally, the transmission p roblem of a wave equation with global or local Kelvin-\nVoigt damping and without any time delay was studied by many a uthors in the one dimensional case (see\n[26,2,21,1,20,19,35,39,28]) and in the multidimensional case (see [ 31,45,40,27]) and polynomial and\nexponential stability results were obtained. In addition, the stability of wave equations on tree with local\nKelvin-Voigt damping has been studied in [ 5].\nThus, as we confirmed in the beginning, the case of wave equati ons with localized Kelvin-Voigt type damping\nand boundary or internal time delay; as in our Systems ( 1.1) and ( 1.2), where the damping is acting in a non-\nsmooth region is still an open problem. The aim of the present paper consists in studying the stability of the\nSystems ( 1.1) and ( 1.2). For System ( 1.1), we consider two cases. Case one, if α >0(see Figure 1), then\nusing the semigroup theory of linear operators and a result o btained by Borichev and Tomilov, we show that\nthe energy of the System ( 1.1) has a polynomial decay rate of type t−4. Case two, if α= 0(see Figure 2),\nthen using the semigroup theory of linear operators and a res ult obtained by Huang and Prüss, we prove an\nexponential decay of the energy of System ( 1.1). For System ( 1.2), by using the semigroup theory of linear\noperators and a result obtained by Borichev and Tomilov, we s how that the energy of the System ( 1.2) has a\npolynomial decay rate of type t−4.\nThis paper is organized as follows: In Section 2, we study the stability of System ( 1.1). Indeed, in Subsection\n2.1, we consider the case α>0. First, we prove the well-posedness of System ( 1.1). Next, we prove the strong\nstability of the system in the lack of the compactness of the r esolvent of the generator. Then, we establish\na polynomial energy decay rate of type t−4(see Theorem 2.7). In addition, in Subsection 2.2, we consider\nthe caseα= 0and we prove the exponential stability of system ( 1.1) (see Theorem 2.14). In Section 3, we\nstudy the stability of System ( 1.2). First, we prove the well-posedness of System ( 1.2). Next, we establish a\npolynomial energy decay rate of type t−4(see Theorem 3.2).\n6WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\n2.Wave equation with local Kelvin-Voigt damping and with boun dary delay feedback\nThis section is devoted to our first aim, that is to study the st ability of a wave equation with localized Kelvin-\nVoigt damping and boundary delay feedback (see System ( 1.1)). For this aim, let us introduce the auxiliary\nunknown\nη(L,ρ,t) =Ut(L,t−ρτ), ρ∈(0,1), t>0.\nThus, Problem ( 1.1) is equivalent to\n(2.1)\n\nUtt(x,t)−/bracketleftbig\nκUx(x,t)+δ1χ(α,β)Uxt(x,t)/bracketrightbig\nx= 0,(x,t)∈(0,L)×(0,+∞),\nτηt(L,ρ,t)+ηρ(L,ρ,t) = 0, (ρ,t)∈(0,1)×(0,+∞),\nU(0,t) = 0, t∈(0,+∞),\nUx(L,t) =−δ3Ut(L,t)−δ2η(L,1,t), t ∈(0,+∞),\n(U(x,0),Ut(x,0)) = (U0(x),U1(x)), x ∈(0,L),\nη(L,ρ,0) =f0(L,−ρτ), ρ ∈(0,1).\nLetUbe a smooth solution of System ( 2.1), we associate its energy defined by\n(2.2) E(t) =1\n2/integraldisplayL\n0/parenleftbig\n|Ut|2+κ|Ux|2/parenrightbig\ndx+τ\n2/integraldisplay1\n0|η|2dρ.\nMultiplying the first equation of ( 2.1) byUt, integrating over (0,L)with respect to x, then using by parts\nintegration and the boundary conditions in ( 2.1) atx= 0and atx=L, we get\n(2.3)1\n2d\ndt/integraldisplayL\n0/parenleftbig\n|Ut|2+κ|Ux|2/parenrightbig\ndx=−δ1/integraldisplayβ\nα|Uxt|2dx−κ3δ3|Ut(L,t)|2−κ3δ2η(L,1,t)Ut(L,t).\nMultiplying the second equation of ( 2.1) byη, integrating over (0,1)with respect to ρ, then using the fact that\nη(L,0,t) =Ut(L,t), we get\n(2.4)τ\n2d\ndt/integraldisplay1\n0|η|2dρ=−1\n2|η(L,1,t)|2+1\n2|Ut(L,t)|2.\nAdding ( 2.3) and ( 2.4), we get\n(2.5)dE(t)\ndt=−δ1/integraldisplayβ\nα|Uxt|2dx−/parenleftbigg\nκ3δ3−1\n2/parenrightbigg\n|Ut(L,t)|2−κ3δ2η(L,1,t)Ut(L,t)−1\n2|η(L,1,t)|2.\nFor allp>0, we have\n(2.6) −κ3δ2η(L,1,t)Ut(L,t)≤κ3|δ2||η(L,1,t)|2\n2p+κ3|δ2|p|Ut(L,t)|2\n2.\nInserting ( 2.6) in (2.5), we get\n(2.7)dE(t)\ndt≤ −δ1/integraldisplayβ\nα|Uxt|2dx−/parenleftbigg1\n2−κ3|δ2|\n2p/parenrightbigg\n|η(L,1,t)|2−/parenleftbigg\nκ3δ3−1\n2−κ3|δ2|p\n2/parenrightbigg\n|Ut(L,t)|2.\nIn the sequel, the assumption on κ3, δ1, δ2andδ3will ensure that\n(H) κ3>0, δ1>0, δ3>0, δ2∈R∗, δ3>1\n2κ3,|δ2|<1\nκ3/radicalbig\n2κ3δ3−1.\nIn this case, we easily check that there exists a strictly pos itive number psatisfying\n(2.8) κ3|δ2|<p<2\nκ3|δ2|/parenleftbigg\nκ3δ3−1\n2/parenrightbigg\n,\nsuch that\n1\n2−κ3|δ2|\n2p>0andκ3δ3−1\n2−κ3|δ2|p\n2>0,\nso that the energies of the strong solutions satisfy E′(t)≤0.Hence, System ( 2.1) is dissipative in the sense\nthat its energy is non increasing with respect to the time t.\nFor studying the stability of System ( 2.1), we consider two cases. In Subsection 2.1, we consider the first case,\n7WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nwhen the Kelvin-Voigt damping is localized in the internal o f the body, i.e.α >0. While in Subsection 2.2,\nwe consider the second, when the Kelvin-Voigt damping is loc alized near the boundary of the body, i.e.α= 0.\n2.1.Wave equation with local Kelvin-Voigt damping far from the b oundary and with boundary\ndelay feedback. In this subsection, we assume that there exist αandβsuch that 0< α < β < L , in this\ncase, the Kelvin-Voigt damping is localized in the internal of the body (see Figure 1). For this aim, we denote\nthe longitudinal displacement by Uand this displacement is divided into three parts\nU(x,t) =\n\nu(x,t),(x,t)∈(0,α)×(0,+∞),\nv(x,t),(x,t)∈(α,β)×(0,+∞),\nw(x,t),(x,t)∈(β,L)×(0,+∞).\nIn this case, System ( 2.1) is equivalent to the following system\nutt−κ1uxx= 0,(x,t)∈(0,α)×(0,+∞), (2.9)\nvtt−(κ2vx+δ1vxt)x= 0,(x,t)∈(α,β)×(0,+∞), (2.10)\nwtt−κ3wxx= 0,(x,t)∈(β,L)×(0,+∞), (2.11)\nτηt(L,ρ,t)+ηρ(L,ρ,t) = 0,(ρ,t)∈(0,1)×(0,+∞), (2.12)\nwith the following boundary and transmission conditions\nu(0,t) = 0, t∈(0,+∞), (2.13)\nwx(L,t) =−δ3wt(L,t)−δ2η(L,1,t), t∈(0,+∞), (2.14)\nu(α,t) =v(α,t), t∈(0,+∞), (2.15)\nv(β,t) =w(β,t), t∈(0,+∞), (2.16)\nκ2vx(α,t)+δ1vxt(α,t) =κ1ux(α,t), t∈(0,+∞), (2.17)\nκ2vx(β,t)+δ1vxt(β,t) =κ3wx(β,t), t∈(0,+∞), (2.18)\nand with the following initial conditions\n(u(x,0),ut(x,0)) = (u0(x),u1(x)), x∈(0,α), (2.19)\n(v(x,0),vt(x,0)) = (v0(x),v1(x)), x∈(α,β), (2.20)\n(w(x,0),wt(x,0)) = (w0(x),w1(x)), x∈(β,L), (2.21)\nη(L,ρ,0) =f0(L,−ρτ), ρ∈(0,1), (2.22)\nwhere the initial data (u0,u1,v0,v1,w0,w1,f0)belongs to a suitable Hilbert space. So, using ( 2.2), the energy\nof System ( 2.9)-(2.22) is given by\nE(t) =1\n2/integraldisplayα\n0/parenleftbig\n|ut|2+κ1|ux|2/parenrightbig\ndx+1\n2/integraldisplayβ\nα/parenleftbig\n|vt|2+κ2|vx|2/parenrightbig\ndx+1\n2/integraldisplayL\nβ/parenleftbig\n|wt|2+κ3|wx|2/parenrightbig\ndx+τ\n2/integraldisplay1\n0|η|2dρ.\nSimilar to ( 2.5) and ( 2.7), we get\ndE(t)\ndt=−δ1/integraldisplayβ\nα|vxt|2dx−1\n2|η(L,1,t)|2−κ3δ2η(L,1,t)wt(L,t)−/parenleftbigg\nκ3δ3−1\n2/parenrightbigg\n|wt(L,t)|2,\n≤ −δ1/integraldisplayβ\nα|vxt|2dx−/parenleftbigg1\n2−κ3|δ2|\n2p/parenrightbigg\n|η(L,1,t)|2−/parenleftbigg\nκ3δ3−1\n2−κ3|δ2|p\n2/parenrightbigg\n|wt(L,t)|2,\nwherepis defined in ( 2.8). Thus, under hypothesis (H), the System ( 2.9)-(2.22) is dissipative in the sense that\nits energy is non increasing with respect to the time t.Now, we are in position to prove the existence and\nuniqueness of the solution of our system.\n8WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\n2.1.1. Well-posedness of the problem. We start this part by formulating System ( 2.9)-(2.22) as an abstract\nCauchy problem. For this aim, let us define\nHm=Hm(0,α)×Hm(α,β)×Hm(β,L), m= 1,2,\nL2=L2(0,α)×L2(α,β)×L2(β,L),\nH1\nL={(u,v,w)∈H1|u(0) = 0, u(α) =v(α), v(β) =w(β)}.\nRemark 2.1. The Hilbert space L2is equipped with the norm:\n/ba∇dbl(u,v,w)/ba∇dbl2\nL2=/integraldisplayα\n0|u|2dx+/integraldisplayβ\nα|v|2dx+/integraldisplayL\nβ|w|2dx.\nAlso, it is easy to check that the space H1\nLis Hilbert space over Cequipped with the norm:\n/ba∇dbl(u,v,w)/ba∇dbl2\nH1\nL=κ1/integraldisplayα\n0|ux|2dx+κ2/integraldisplayβ\nα|vx|2dx+κ3/integraldisplayL\nβ|wx|2dx.\nMoreover, by Poincaré inequality we can easily verify that the re existsC >0, such that\n/ba∇dbl(u,v,w)/ba∇dblL2≤C/ba∇dbl(u,v,w)/ba∇dblH1\nL.\n/square\nWe now define the Hilbert energy space Hby\nH=H1\nL×L2×L2(0,1)\nequipped with the following inner product\n/a\\}b∇acketle{tU,˜U/a\\}b∇acket∇i}htH=κ1/integraldisplayα\n0ux˜uxdx+κ2/integraldisplayβ\nαvx˜vxdx+κ3/integraldisplayL\nβwx˜wxdx+/integraldisplayα\n0y˜ydx+/integraldisplayβ\nαz˜zdx+/integraldisplayL\nβφ˜φdx+τ/integraldisplay1\n0η(L,ρ)˜η(L,ρ)dρ,\nwhereU= (u,v,w,y,z,φ,η (L,·))∈ Hand˜U= (˜u,˜v,˜w,˜y,˜z,˜φ,˜η(L,·))∈ H. We use /ba∇dblU/ba∇dblHto denote the\ncorresponding norm. We define the linear unbounded operator A:D(A)⊂ H −→ H by:\nD(A) =/braceleftbigg\n(u,v,w,y,z,φ,η (L,·))∈H1\nL×H1\nL×H1(0,1)|(u,κ2v+δ1z,w)∈H2,\nκ2vx(α)+δ1zx(α) =κ1ux(α), κ2vx(β)+δ1zx(β) =κ3wx(β),\nwx(L) =−δ3η(L,0)−δ2η(L,1), φ(L) =η(L,0)/bracerightbigg\nand for all U= (u,v,w,y,z,φ,η (L,·))∈D(A)\nAU=/parenleftbig\ny,z,φ,κ 1uxx,(κ2vx+δ1zx)x,κ3wxx,−τ−1ηρ(L,·)/parenrightbig\n.\nIfU= (u,v,w,u t,vt,wt,η(L,·))is a regular solution of System ( 2.9)-(2.22), then we transform this system into\nthe following initial value problem\n(2.23)/braceleftBigg\nUt=AU,\nU(0) =U0,\nwhereU0= (u0,v0,w0,u1,v1,w1,f0(L,−·τ))∈ H.We now use semigroup approach to establish well-posedness\nresult for the System ( 2.9)-(2.22). According to Lumer-Phillips theorem (see [ 36]), we need to prove that the\noperator Ais m-dissipative in H. Therefore, we prove the following proposition.\nProposition 2.2. Under hypothesis (H), the unbounded linear operator Ais m-dissipative in the energy space\nH.\nProof. For allU= (u,v,w,y,z,φ,η (L,·))∈D(A),we have\nRe/a\\}b∇acketle{tAU,U/a\\}b∇acket∇i}htH=κ1Re/integraldisplayα\n0(yxux+uxxy)dx+Re/integraldisplayβ\nα(κ2zxvx+(κ2vx+δ1zx)xz)dx\n+κ3Re/integraldisplayL\nβ/parenleftbig\nφxwx+wxxφ/parenrightbig\ndx−Re/integraldisplay1\n0ηρ(L,ρ)η(L,ρ)dρ.\n9WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nHere Re is used to denote the real part of a complex number. Usi ng by parts integration in the above equation,\nwe get\n(2.24)Re/a\\}b∇acketle{tAU,U/a\\}b∇acket∇i}htH=−δ1/integraldisplayβ\nα|zx|2dx−1\n2|η(L,1)|2+1\n2|η(L,0)|2+κ3Re/parenleftbig\nwx(L)φ(L)/parenrightbig\n−κ1Re(ux(0)y(0))+ Re(κ1ux(α)y(α)−κ2vx(α)z(α)−δ1zx(α)z(α))\n+Re/parenleftbig\nκ2vx(β)z(β)+δ1zx(β)z(β)−κ3wx(β)φ(β)/parenrightbig\n.\nOn the other hand, since U∈D(A), we have\n(2.25)\n\ny(0) = 0, y(α) =z(α), z(β) =φ(β),\nκ1ux(α)−κ2vx(α)−δ1zx(α) = 0, κ2vx(β)+δ1zx(β)−κ3wx(β) = 0,\nwx(L) =−δ3η(L,0)−δ2η(L,1), φ(L) =η(L,0).\nInserting ( 2.25) in (2.24), we get\n(2.26) Re /a\\}b∇acketle{tAU,U/a\\}b∇acket∇i}htH=−δ1/integraldisplayβ\nα|zx|2dx−1\n2|η(L,1)|2−/parenleftbigg\nκ3δ3−1\n2/parenrightbigg\n|η(L,0)|2−κ3δ2Re(η(L,0)η(L,1)).\nUnder hypothesis (H), we easily check that there exists p>0such that\nκ3|δ2|<p<2\nκ3|δ2|/parenleftbigg\nκ3δ3−1\n2/parenrightbigg\n.\nBy Young’s inequality, we get\n−κ3δ2Re(η(L,0)η(L,1))≤κ3|δ2||η(L,1)|2\n2p+κ3|δ2|p|η(L,0)|2\n2.\nInserting the above inequality in ( 2.26), we get\n(2.27) Re /a\\}b∇acketle{tAU,U/a\\}b∇acket∇i}htH≤ −δ1/integraldisplayβ\nα|zx|2dx−/parenleftbigg1\n2−κ3|δ2|\n2p/parenrightbigg\n|η(L,1)|2−/parenleftbigg\nκ3δ3−1\n2−κ3|δ2|p\n2/parenrightbigg\n|η(L,0)|2.\nFrom the construction of p, we have\n1\n2−κ3|δ2|\n2p>0andκ3δ3−1\n2−κ3|δ2|p\n2>0.\nTherefore, from ( 2.27), we get\nRe/a\\}b∇acketle{tAU,U/a\\}b∇acket∇i}htH≤0,\nwhich implies that Ais dissipative. Now, let us go on with maximality. Let F= (f1,f2,f3,f4,f5,f6,f7(L,·))∈\nHwe look for U= (u,v,w,y,z,φ,η (L,·))∈D(A)solution of the equation\n(2.28) −AU=F.\nEquivalently, we consider the following system\n−y=f1, (2.29)\n−z=f2, (2.30)\n−φ=f3, (2.31)\n−κ1uxx=f4, (2.32)\n−(κ2vx+δ1zx)x=f5, (2.33)\n−κ3wxx=f6, (2.34)\nηρ(L,ρ) =τf7(ρ). (2.35)\nIn addition, we consider the following boundary conditions\nu(0) = 0, u(α) =v(α), v(β) =w(β), (2.36)\nκ2vx(α)+δ1zx(α) =κ1ux(α), κ2vx(β)+δ1zx(β) =κ3wx(β), (2.37)\nwx(L) =−δ3η(L,0)−δ2η(L,1), (2.38)\nη(L,0) =φ(L). (2.39)\n10WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nFrom ( 2.29)-(2.31) and the fact that F∈ H, it is clear that (y,z,φ)∈H1\nL. Next, from ( 2.31), (2.39) and the\nfact thatf3∈H1(β,L), we get\nη(L,0) =φ(L) =−f3(L).\nFrom the above equation and Equation ( 2.35), we can determine\nη(L,ρ) =τ/integraldisplayρ\n0f7(ξ)dξ−f3(L).\nIt is clear that η(L,·)∈H1(0,1)andη(L,0) =φ(L) =−f3(L). Inserting the above equation in ( 2.38), then\nSystem ( 2.29)-(2.39) is equivalent to\ny=−f1, z=−f2, φ=−f3, η(L,ρ) =τ/integraldisplayρ\n0f7(ξ)dξ−f3(L), (2.40)\n−κ1uxx=f4, (2.41)\n−(κ2vx+δ1zx)x=f5, (2.42)\n−κ3wxx=f6, (2.43)\nu(0) = 0, u(α) =v(α), v(β) =w(β), (2.44)\nκ2vx(α)+δ1zx(α) =κ1ux(α), κ2vx(β)+δ1zx(β) =κ3wx(β), (2.45)\nwx(L) = (δ3+δ2)f3(L)−τδ2/integraldisplay1\n0f7(ξ)dξ. (2.46)\nLet(ϕ,ψ,θ)∈H1\nL. Multiplying Equations ( 2.41), (2.42), (2.43) byϕ,ψ,θ, integrating over (0,α),(α,β)and\n(β,L)respectively, taking the sum, then using by parts integrati on, we get\n(2.47)κ1/integraldisplayα\n0uxϕxdx+/integraldisplayβ\nα(κ2vx+δ1zx)ψxdx+κ3/integraldisplayL\nβwxθxdx+κ1ux(0)ϕ(0)\n−κ1ux(α)ϕ(α)+(κ2vx(α)+δ1zx(α))ψ(α)−(κ2vx(β)+δ1zx(β))ψ(β)+κ3wx(β)θ(β)\n=/integraldisplayα\n0f4ϕdx+/integraldisplayβ\nαf5ψdx+/integraldisplayL\nβf6θdx+κ3wx(L)θ(L).\nFrom the fact that (ϕ,ψ,θ)∈H1\nL,we have\nϕ(0) = 0, ϕ(α) =ψ(α), θ(β) =ψ(β).\nInserting the above equation in ( 2.47), then using ( 2.40) and ( 2.44)-(2.46), we get\n(2.48)κ1/integraldisplayα\n0uxϕxdx+κ2/integraldisplayβ\nαvxψxdx+κ3/integraldisplayL\nβwxθxdx=/integraldisplayα\n0f4ϕdx\n+/integraldisplayβ\nα/parenleftbig\nδ1(f2)xψx+f5ψ/parenrightbig\ndx+/integraldisplayL\nβf6θdx+κ3/parenleftbigg\n(δ3+δ2)f3(L)−τδ2/integraldisplay1\n0f7(ξ)dξ/parenrightbigg\nθ(L).\nWe can easily verify that the left hand side of ( 2.48) is a bilinear continuous coercive form on H1\nL×H1\nL,\nand the right hand side of ( 2.48) is a linear continuous form on H1\nL. Then, using Lax-Milgram theorem,\nwe deduce that there exists (u,v,w)∈H1\nLunique solution of the variational Problem ( 2.48). Using stan-\ndard arguments, we can show that (u,κ2v+δ1z,w)∈H2. Finally, by seting y=−f1, z=−f2, φ=−f3\nandη(L,ρ) =τ/integraldisplayρ\n0f7(ξ)dξ−f3(L)and by applying the classical elliptic regularity we deduce thatU=\n(u,v,w,y,z,φ,η (L,·))∈D(A)is solution of Equation ( 2.28). To conclude, we need to show the uniqueness of\nU. So, letU= (u,v,w,y,z,φ,η (L,·))∈D(A)be a solution of ( 2.28) withF= 0, then we directly deduce that\ny=z=φ=η(L,ρ) = 0 and that (u,v,w)∈H1\nLsatisfies Problem ( 2.48) with zero in the right hand side. This\nimplies that u=v=w= 0, in other words, ker A={0}and0belongs to the resolvent set ρ(A)ofA. Then,\nby contraction principale, we easily deduce that R(λI− A) =Hfor sufficiently small λ >0. This, together\nwith the dissipativeness of A, imply that D(A)is dense in Hand that Ais m-dissipative in H(see Theorems\n4.5, 4.6 in [ 36]). The proof is thus complete. /square\nThanks to Lumer-Philips theorem (see [ 36]), we deduce that Agenerates a C0−semigroup of contractions etA\ninHand therefore Problem ( 2.9)-(2.22) is well-posed. Then we have the following result:\n11WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nTheorem 2.3. Under hypothesis (H), for any U0∈ H,Problem (2.23)admits a unique weak solution,\nU(x,ρ,t) =etAU0(x,ρ), such that\nU∈C0(R+,H).\nMoreover, if U0∈D(A),then\nU∈C1(R+,H)∩C0(R+,D(A)).\n2.1.2. Strong Stability. Our main result in this part is the following theorem.\nTheorem 2.4. Under hypothesis (H), theC0−semigroup of contractions etAis strongly stable on the Hilbert\nspaceHin the sense that\nlim\nt→+∞||etAU0||H= 0, ∀U0∈ H.\nFor the proof of Theorem 2.4, according to Theorem A.2, we need to prove that the operator Ahas no pure\nimaginary eigenvalues and σ(A)∩iRcontains only a countable number of continuous spectrum of A. The\nargument for Theorem 2.4relies on the subsequent lemmas.\nLemma 2.5. Under hypothesis (H), forλ∈R,we haveiλI−Ais injective, i.e.\nker(iλI−A) ={0},∀λ∈R.\nProof. From Proposition 2.2, we have 0∈ρ(A).We still need to show the result for λ∈R∗.Suppose that there\nexists a real number λ/\\e}atio\\slash= 0andU= (u,v,w,y,z,φ,η (L,·))∈D(A)such that\n(2.49) AU=iλU.\nFirst, similar to Equation ( 2.27), we have\n0 =Re/a\\}b∇acketle{tAU,U/a\\}b∇acket∇i}htH≤ −δ1/integraldisplayβ\nα|zx|2dx−/parenleftbigg1\n2−κ3|δ2|\n2p/parenrightbigg\n|η(L,1)|2−/parenleftbigg\nκ3δ3−1\n2−κ3|δ2|p\n2/parenrightbigg\n|η(L,0)|2≤0.\nThus,\n(2.50) zx= 0 in(α,β)andη(L,1) =η(L,0) = 0.\nNext, writing ( 2.49) in a detailed form gives\ny=iλu, x∈(0,α), (2.51)\nz=iλv, x∈(α,β), (2.52)\nw=iλφ, x∈(β,L), (2.53)\nκ1uxx=iλy, x∈(0,α), (2.54)\n(κ2vx+δ1zx)x=iλz, x∈(α,β), (2.55)\nκ3wxx=iλφ, x∈(β,L), (2.56)\nηρ(L,ρ) =−iλτη(L,ρ), ρ∈(0,1). (2.57)\nFrom ( 2.57) and ( 2.50), we get\n(2.58) η(L,·) =η(L,0)e−iλτ·= 0 in(0,1).\nCombining ( 2.50) with ( 2.52), we get that\n(2.59) vx=zx= 0 in(α,β).\nThus,\nvxx=zxx= 0 in(α,β).\nInserting the above result in ( 2.55), then taking into consideration ( 2.52), we obtain\n(2.60) v=z= 0 in(α,β).\nFrom the definition of D(A)and using ( 2.58)-(2.60), we get\n\n\nu(α) =v(α) = 0, w(β) =v(β) = 0, y(α) =z(α) = 0, φ(β) =z(β) = 0,\nκ1ux(α) =κ2vx(α)+δ1zx(α) = 0, κ3wx(β) =k2vx(β)+δ1zx(β) = 0,\nw(L) =iλφ(L) =iλη(L,0) = 0, wx(L) =−δ3η(L,0)−δ2η(L,1) = 0.\n12WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nCombining ( 2.51) with ( 2.54) and ( 2.53) with ( 2.56) and using the above equation as boundary conditions, we\nget\n\nuxx+λ2\nκ1u= 0, x∈(0,α),\nu(0) =u(α) =ux(α) = 0,\n\nwxx+λ2\nκ3w= 0, x∈(β,L),\nw(β) =w(L) =wx(β) =wx(L) = 0.\nThus,\n(2.61) u(x) = 0∀x∈(0,α)andw(x) = 0∀x∈(β,L).\nCombining ( 2.61) with ( 2.51) and ( 2.53), we obtain\ny(x) = 0∀x∈(0,α)andφ(x) = 0∀x∈(β,L).\nFinally, from the above result, ( 2.58), (2.60) and ( 2.61), we get that U= 0.The proof is thus complete. /square\nLemma 2.6. Under hypothesis (H), forλ∈R,we haveiλI−Ais surjective, i.e.\nR(iλI−A) =H,∀λ∈R.\nProof. Since0∈ρ(A),we still need to show the result for λ∈R∗.For anyF= (f1,f2,f3,f4,f5,f6,f7(L,·))∈ H\nandλ∈R∗,we prove the existence of U= (u,v,w,y,z,φ,η (L,·))∈D(A)solution for the following equation\n(iλI−A)U=F.\nEquivalently, we consider the following problem\ny=iλu−f1inH1(0,α), (2.62)\nz=iλv−f2inH1(α,β), (2.63)\nφ=iλw−f3inH1(β,L), (2.64)\niλy−κ1uxx=f4inL2(0,α), (2.65)\niλz−(κ2vx+δ1zx)x=f5inL2(α,β), (2.66)\niλφ−κ3wxx=f6inL2(β,L), (2.67)\nηρ(L,·)+iτλη(L,·) =τf7(L,·)inL2(0,1), (2.68)\nwith the following boundary conditions\nu(0) = 0, u(α) =v(α), v(β) =w(β), (2.69)\nκ2vx(α)+δ1zx(α) =κ1ux(α), κ2vx(β)+δ1zx(β) =κ3wx(β), (2.70)\nwx(L) =−δ3η(L,0)−δ2η(L,1), (2.71)\nη(L,0) =φ(L). (2.72)\nIt follows from ( 2.68), (2.72) and ( 2.64) that\n(2.73) η(L,ρ) = (iλw(L)−f3(L))e−iτλρ+τ/integraldisplayρ\n0eiτλ(ξ−ρ)f7(L,ξ)dξ.\nInserting ( 2.62)-(2.64) and ( 2.73) in (2.65)-(2.72) and deriving ( 2.63) with respect to x, we get\n−λ2u−κ1uxx=iλf1+f4, (2.74)\n−λ2v−(κ2vx+δ1zx)x=iλf2+f5, (2.75)\n−λ2w−κ3wxx=iλf3+f6, (2.76)\nzx=iλvx−(f2)x, (2.77)\nu(0) = 0, u(α) =v(α), κ1ux(α) =κ2vx(α)+δ1zx(α), (2.78)\nw(β) =v(β), κ3wx(β) =κ2vx(β)+δ1zx(β), (2.79)\nwx(L) =−iλ/parenleftbig\nδ3+δ2e−iτλ/parenrightbig\nw(L)+/parenleftbig\nδ3+δ2e−iτλ/parenrightbig\nf3(L)−τδ2/integraldisplay1\n0eiτλ(ξ−1)f7(L,ξ)dξ. (2.80)\n13WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nLet(ϕ,ψ,θ)∈H1\nL. Multiplying Equations ( 2.74), (2.75), (2.76) byϕ,ψ,θ, integrating over (0,α),(α,β)and\n(β,L)respectively, taking the sum, then using by parts integrati on, we get\n(2.81)κ1/integraldisplayα\n0uxϕxdx+/integraldisplayβ\nα(κ2vx+δ1zx)ψxdx+κ3/integraldisplayL\nβwxθxdx+κ1ux(0)ϕ(0)\n−κ1ux(α)ϕ(α)+(κ2vx(α)+δ1zx(α))ψ(α)−(κ2vx(β)+δ1zx(β))ψ(β)+κ3wx(β)θ(β)\n−λ2/integraldisplayα\n0uϕdx−λ2/integraldisplayβ\nαvψdx−λ2/integraldisplayL\nβwθdx−κ3wx(L)θ(L)\n=/integraldisplayα\n0(iλf1+f4)ϕdx+/integraldisplayβ\nα(iλf2+f5)ψdx+/integraldisplayL\nβ(iλf3+f6)θdx.\nFrom the fact that (ϕ,ψ,θ)∈H1\nL,we have\nϕ(0) = 0, ϕ(α) =ψ(α), θ(β) =ψ(β).\nInserting the above equation in ( 2.81), then using ( 2.77)-(2.80), we get\n(2.82) a((u,v,w),(ϕ,ψ,θ)) = F(ϕ,ψ,θ),∀(ϕ,ψ,θ)∈H1\nL,\nwhere\nF(ϕ,ψ,θ) =/integraldisplayα\n0(iλf1+f4)ϕdx+/integraldisplayβ\nα(iλf2+f5)ψdx+δ1/integraldisplayβ\nα(f2)xψxdx\n+/integraldisplayL\nβ(iλf3+f6)θdx+κ3/parenleftbigg/parenleftbig\nδ3+δ2e−iτλ/parenrightbig\nf3(L)−τδ2/integraldisplay1\n0eiτλ(ξ−1)f7(L,ξ)dξ/parenrightbigg\nθ(L)\nand\na((u,v,w),(ϕ,ψ,θ)) =a1((u,v,w),(ϕ,ψ,θ))+a2((u,v,w),(ϕ,ψ,θ)),\nsuch that\n\na1((u,v,w),(ϕ,ψ,θ)) =κ1/integraldisplayα\n0uxϕxdx+(κ2+iδ1λ)/integraldisplayβ\nαvxψxdx+κ3/integraldisplayL\nβwxθxdx,\na2((u,v,w),(ϕ,ψ,θ)) =−λ2/integraldisplayα\n0uϕdx−λ2/integraldisplayβ\nαvψdx−λ2/integraldisplayL\nβwθdx+iκ3λ/parenleftbig\nδ3+δ2e−iτλ/parenrightbig\nw(L)θ(L).\nLet/parenleftbig\nH1\nL/parenrightbig′be the dual space of H1\nL. We define the operators A,A1andA2by\n/braceleftBigg\nA :H1\nL→/parenleftbig\nH1\nL/parenrightbig′\n(u,v,w)→A(u,v,w)/braceleftBigg\nA1:H1\nL→/parenleftbig\nH1\nL/parenrightbig′\n(u,v,w)→A1(u,v,w)/braceleftBigg\nA2:H1\nL→/parenleftbig\nH1\nL/parenrightbig′\n(u,v,w)→A2(u,v,w)\nsuch that\n(2.83)\n\n(A(u,v,w))(ϕ,ψ,θ) =a((u,v,w),(ϕ,ψ,θ)),∀(ϕ,ψ,θ)∈H1\nL,\n(A1(u,v,w))(ϕ,ψ,θ) =a1((u,v,w),(ϕ,ψ,θ)),∀(ϕ,ψ,θ)∈H1\nL,\n(A2(u,v,w))(ϕ,ψ,θ) =a2((u,v,w),(ϕ,ψ,θ)),∀(ϕ,ψ,θ)∈H1\nL.\nOur aim is to prove that the operator Ais an isomorphism. For this aim, we proceed the proof in three steps.\nStep 1. In this step we proof that the operator A1is an isomorphism. For this aim, according to ( 2.83), we\nhave\na1((u,v,w),(ϕ,ψ,θ)) =κ1/integraldisplayα\n0uxϕxdx+(κ2+iδ1λ)/integraldisplayβ\nαvxψxdx+κ3/integraldisplayL\nβwxθxdx\nWe can easily verify that a1is a bilinear continuous coercive form on H1\nL×H1\nL. Then, by Lax-Milgram lemma,\nthe operator A1is an isomorphism.\nStep 2. In this step we proof that the operator A2is compact. First, for1\n2<r<1, we introduce the Hilbert\nspaceHr\nLby\nHr\nL={(ϕ,ψ,θ)∈Hr(0,α)×Hr(α,β)×Hr(β,L)|ϕ(0) = 0, ϕ(α) =ψ(α), ψ(β) =θ(β)}.\n14WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nThus by trace theorem, there exists C >0, such that\n(2.84) |θ(L)| ≤C/ba∇dbl(ϕ,ψ,θ)/ba∇dblHr\nL.\nNow, according to ( 2.83), we have\na2((u,v,w),(ϕ,ψ,θ)) =−λ2/integraldisplayα\n0uϕdx−λ2/integraldisplayβ\nαvψdx−λ2/integraldisplayL\nβwθdx+iκ3λ/parenleftbig\nδ3+δ2e−iτλ/parenrightbig\nw(L)θ(L).\nThen, by using ( 2.84), we get\n|a2((u,v,w),(ϕ,ψ,θ))| ≤C1/ba∇dbl(u,v,w)/ba∇dblH1\nL/ba∇dbl(ϕ,ψ,θ)/ba∇dblL2+C1/ba∇dbl(u,v,w)/ba∇dblH1\nL/ba∇dbl(ϕ,ψ,θ)/ba∇dblHr\nL,\nwhereC1>0. Therefore, for all r∈(1\n2,1)there exists C2>0, such that\n|a2((u,v,w),(ϕ,ψ,θ))| ≤C2/ba∇dbl(u,v,w)/ba∇dblH1\nL/ba∇dbl(ϕ,ψ,θ)/ba∇dblHr\nL,\nwhich implies that\nA2∈ L/parenleftbig\nH1\nL,(Hr\nL)′/parenrightbig\n.\nFinally, using the compactness embedding from (Hr\nL)′into/parenleftbig\nH1\nL/parenrightbig′we deduce that A2is compact.\nFrom steps 1 and 2, we get that the operator A = A 1+A2is a Fredholm operator of index zero 0. Consequently,\nby Fredholm alternative, proving the operator Ais an isomorphism reduces to proving ker(A) = {0}.\nStep 3. In this step we proof that the ker(A) = {0}. For this aim, let (˜u,˜v,˜w)∈ker(A) ,i.e.\na((˜u,˜v,˜w),(ϕ,ψ,θ)) = 0,∀(ϕ,ψ,θ)∈H1\nL.\nEquivalently,\n/integraldisplayα\n0/parenleftbig\nκ1˜uxϕx−λ2˜uϕ/parenrightbig\ndx+/integraldisplayβ\nα/parenleftbig\n(κ2+iδ1λ)˜vxψx−λ2˜vψ/parenrightbig\ndx+/integraldisplayL\nβ/parenleftbig\nκ3˜wxθx−λ2˜wθ/parenrightbig\ndx\n+iκ3λ/parenleftbig\nδ3+δ2e−iτλ/parenrightbig\n˜w(L)θ(L) = 0,∀(ϕ,ψ,θ)∈H1\nL.\nThen, we find that\n\n−λ2˜u−κ1˜uxx= 0,\n−λ2˜v−(κ2+iδ1λ)˜vxx= 0,\n−λ2˜w−κ3˜wxx= 0,\n˜u(0) = 0,˜u(α) = ˜v(α), κ1˜ux(α) = (κ2+iδ1λ)˜vx(α),\n˜w(β) = ˜v(β), κ3˜wx(β) = (κ2+iδ1λ)˜vx(β),\n˜wx(L) =−iλ/parenleftbig\nδ3+δ2e−iτλ/parenrightbig\n˜w(L).\nTherefore, the vector ˜Vdefine by\n˜V=/parenleftbig\n˜u,˜v,˜w,iλ˜u,iλ˜v,iλ˜w,iλ˜w(L)e−iτλ·/parenrightbig\nbelongs toD(A)and we have\niλ˜V−A˜V= 0.\nThus,˜V∈ker(iλI−A), therefore by Lemma 2.5, we get ˜V= 0, this implies that ˜u= 0,˜v= 0and˜w= 0, so\nker(A) = {0}.\nTherefore, from step 3 and Fredholm alternative, we get that the operator Ais an isomorphism. It easy to\nsee that the operator Fis continuous form on H1\nL. Consequently, Equation ( 2.82) admits a unique solution\n(u,v,w)∈H1\nL. Thus, using ( 2.62)-(2.64), (2.73) and a classical regularity arguments, we conclude that (iλI−\nA)U=Fadmits a unique solution U∈D(A). The proof is thus complete. /square\nProof of Theorem 2.4.Form Lemma 2.5, we have that the operator Ahas no pure imaginary eigenvalues and\nby Lemma 2.6,R(iλI−A) =Hfor allλ∈R.Therefore, the closed graph theorem implies that σ(A)∩iR=∅.\nThus, we get the conclusion by applying Theorem A.2of Arendt and Batty. The proof is thus complete. /square\n15WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\n2.1.3. Polynomial Stability. In this part, we will prove the polynomial stability of Syste m (2.9)-(2.22). Our\nmain result in this part is the following theorem.\nTheorem 2.7. Under hypothesis (H), for all initial data U0∈D(A),there exists a constant C >0independent\nofU0such that the energy of System (2.9)-(2.22)satisfies the following estimation\n(2.85) E(t)≤C\nt4/ba∇dblU0/ba∇dbl2\nD(A),∀t>0.\nFrom Lemma 2.5and Lemma 2.6, we have seen that iR⊂ρ(A),then for the proof of Theorem 2.7, according\nto Theorem A.5(part (ii)), we need to prove that\n(2.86) sup\nλ∈R/vextenddouble/vextenddouble/vextenddouble(iλI−A)−1/vextenddouble/vextenddouble/vextenddouble\nL(H)=O/parenleftBig\n|λ|1\n2/parenrightBig\n.\nWe will argue by contradiction. Indeed, suppose there exist s\n{(λn,Un:= (un,vn,wn,yn,zn,φn,ηn(L,·)))}n≥1⊂R∗\n+×D(A),\nsuch that\n(2.87) λn→+∞,/ba∇dblUn/ba∇dblH= 1\nand there exists sequence Fn:= (f1,n,f2,n,f3,n,f4,n,f5,n,f6,n,f7,n(L,·))∈ H, such that\n(2.88) λℓ\nn(iλnI−A)Un=Fn→0inH.\nIn case that ℓ=1\n2, we will check condition ( 2.86) by finding a contradiction with /ba∇dblUn/ba∇dblH= 1such as/ba∇dblUn/ba∇dblH=\no(1).From now on, for simplicity, we drop the index n. By detailing Equation ( 2.88), we get the following\nsystem\niλu−y=λ−ℓf1inH1(0,α), (2.89)\niλv−z=λ−ℓf2inH1(α,β), (2.90)\niλw−φ=λ−ℓf3inH1(β,L), (2.91)\niλy−κ1uxx=λ−ℓf4inL2(0,α), (2.92)\niλz−(κ2vx+δ1zx)x=λ−ℓf5inL2(α,β), (2.93)\niλφ−κ3wxx=λ−ℓf6inL2(β,L), (2.94)\nηρ(L,·)+iτλη(L,·) =τλ−ℓf7(L,·)inL2(0,1). (2.95)\nRemark that, since U= (u,v,w,y,z,φ,η (L,·))∈D(A), we have the following boundary conditions\n(2.96)/braceleftBigg\n|ux(α)|=κ−1\n1|κ2vx(α)+δ1zx(α)|,|y(α)|=|z(α)|,\n|wx(β)|=κ−1\n3|κ2vx(β)+δ1zx(β)|,|z(β)|=|φ(β)|\nand\n(2.97) wx(L) =−δ3η(L,0)−δ2η(L,1), φ(L) =η(L,0).\nThe proof of Theorem 2.7is divided into several lemmas.\nLemma 2.8. Under hypothesis (H), for allℓ≥0, the solution (u,v,w,y,z,φ,η (L,·))∈D(A)of Equations\n(2.89)-(2.95)satisfies the following asymptotic behavior estimations\n/integraldisplayβ\nα|zx|2=o/parenleftbig\nλ−ℓ/parenrightbig\n, (2.98)\n|φ(L)|2=|η(L,0)|2=o/parenleftbig\nλ−ℓ/parenrightbig\n,|η(L,1)|2=o/parenleftbig\nλ−ℓ/parenrightbig\n, (2.99)\n/integraldisplayβ\nα|vx|2dx=o/parenleftbig\nλ−ℓ−2/parenrightbig\n, (2.100)\n|wx(L)|2=o/parenleftbig\nλ−ℓ/parenrightbig\n. (2.101)\n16WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nProof. Taking the inner product of ( 2.88) withUinH, then using the fact that Uis uniformly bounded in H,\nwe get\n−Re/a\\}b∇acketle{tAU,U/a\\}b∇acket∇i}htH=Re/a\\}b∇acketle{t(iλI−A)U,U/a\\}b∇acket∇i}htH=o/parenleftbig\nλ−ℓ/parenrightbig\n,\nNow, under hypothesis (H), similar to Equation ( 2.27), we get\n(2.102) 0≤δ1/integraldisplayβ\nα|zx|2dx+C1|η(L,1)|2+C2|η(L,0)|2≤ −Re/a\\}b∇acketle{tAU,U/a\\}b∇acket∇i}htH=o/parenleftbig\nλ−ℓ/parenrightbig\n,\nwhere\nC1=1\n2−κ3|δ2|\n2p>0andC2=κ3δ3−1\n2−κ3|δ2|p\n2>0.\nTherefore, from ( 2.102), we get ( 2.98) and ( 2.99). Next, from ( 2.90), (2.98) and the fact that (f2)x→0in\nL2(α,β), we get ( 2.100). Finally, from ( 2.97) and ( 2.99), we obtain ( 2.101). Thus, the proof of the lemma is\ncomplete. /square\nLemma 2.9. Under hypothesis (H), for allℓ≥0, the solution (u,v,w,y,z,φ,η (L,·))∈D(A)of Equations\n(2.89)-(2.95)satisfies the following asymptotic behavior estimation\n(2.103)/integraldisplay1\n0|η(L,ρ)|2dρ=o/parenleftbig\nλ−ℓ/parenrightbig\n.\nProof. It follows from ( 2.95) that\nη(L,ρ) =η(L,0)e−iτλρ+τλ−ℓ/integraldisplayρ\n0eiτλ(ξ−ρ)f7(L,ξ)dξ∀ρ∈(0,1).\nBy using Cauchy Schwarz inequality, we get\n|η(L,ρ)|2≤2|η(L,0)|2+2τ2λ−2ℓ/parenleftbigg/integraldisplay1\n0|f7(L,ξ)|dξ/parenrightbigg2\n≤2|η(L,0)|2+2τ2λ−2ℓ/integraldisplay1\n0|f7(L,ξ)|2dξ∀ρ∈(0,1).\nIntegrating over (0,1)with respect to ρ, then using ( 2.99) and the fact that f7(L,·)→0inL2(0,1), we get\n/integraldisplay1\n0|η(L,ρ)|2dρ≤2|η(L,0)|2+2τ2λ−2ℓ/integraldisplay1\n0|f7(L,ξ)|2dξ=o/parenleftbig\nλ−ℓ/parenrightbig\n,\nhence, we get ( 2.103). Thus, the proof of the lemma is complete. /square\nLemma 2.10. Under hypothesis (H), for allℓ≥0, the solution (u,v,w,y,z,φ,η (L,·))∈D(A)of Equations\n(2.89)-(2.95)satisfies the following asymptotic behavior estimations\n/integraldisplayL\nβ|φ|2dx=o/parenleftbig\nλ−ℓ/parenrightbig\n,/integraldisplayL\nβ|wx|2dx=o/parenleftbig\nλ−ℓ/parenrightbig\n, (2.104)\n|wx(β)|2=o/parenleftbig\nλ−ℓ/parenrightbig\n,|φ(β)|2=o/parenleftbig\nλ−ℓ/parenrightbig\n, (2.105)\n|κ2vx(β)+δ1zx(β)|2=o/parenleftbig\nλ−ℓ/parenrightbig\n,|z(β)|2=o/parenleftbig\nλ−ℓ/parenrightbig\n. (2.106)\nProof. Multiplying Equation ( 2.94) byxwxand integrating over (β,L),we get\n(2.107) iλ/integraldisplayL\nβxφwxdx−κ3/integraldisplayL\nβxwxxwxdx=λ−ℓ/integraldisplayL\nβxf6wxdx.\nFrom ( 2.91), we deduce that\niλwx=−φx−λ−ℓ(f3)x.\nInserting the above result in ( 2.107), then using the fact that φ, wxare uniformly bounded in L2(β,L)and\n(f3)x, f6converge to zero in L2(β,L)gives\n−/integraldisplayL\nβxφφxdx−κ3/integraldisplayL\nβxwxxwxdx=o/parenleftbig\nλ−ℓ/parenrightbig\n.\nTaking the real part in the above equation, then using by part s integration, we get\n1\n2/integraldisplayL\nβ|φ|2dx+κ3\n2/integraldisplayL\nβ|wx|2dx+β\n2/parenleftbig\nκ3|wx(β)|2+|φ(β)|2/parenrightbig\n=L\n2/parenleftbig\nκ3|wx(L)|2+|φ(L)|2/parenrightbig\n+o/parenleftbig\nλ−ℓ/parenrightbig\n.\n17WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nInserting ( 2.99) and ( 2.101) in the above equation, we get\n1\n2/integraldisplayL\nβ|φ|2dx+κ3\n2/integraldisplayL\nβ|wx|2dx+β\n2/parenleftbig\nκ3|wx(β)|2+|φ(β)|2/parenrightbig\n=o/parenleftbig\nλ−ℓ/parenrightbig\n,\nhence, we get ( 2.104) and ( 2.105). Finally, from ( 2.96) and ( 2.105), we obtain ( 2.106). The proof is thus\ncomplete. /square\nLemma 2.11. Under hypothesis (H), for allℓ≥0, the solution (u,v,w,y,z,φ,η (L,·))∈D(A)of Equations\n(2.89)-(2.95)satisfies the following asymptotic behavior estimations\n/integraldisplayβ\nα|z|2dx=o/parenleftBig\nλ−min(2ℓ+1\n2,ℓ+1)/parenrightBig\n, (2.108)\n|z(α)|2=o/parenleftBig\nλ−min(2ℓ,ℓ+1\n2)/parenrightBig\n,|z(β)|2=o/parenleftBig\nλ−min(2ℓ,ℓ+1\n2)/parenrightBig\n, (2.109)\n|κ2vx(α)+δ1zx(α)|2=o/parenleftBig\nλ−min(2ℓ−1,ℓ−1\n2)/parenrightBig\n. (2.110)\nProof. Letg∈C1([α,β])such that\ng(β) =−g(α) = 1,max\nx∈[α,β]|g(x)|=cgandmax\nx∈[α,β]|g′(x)|=cg′,\nwherecgandcg′are strictly positive constant numbers independent from λ. The proof is divided into three\nsteps.\nStep 1. In this step, we prove the following asymptotic behavior est imate\n(2.111) |z(β)|2+|z(α)|2≤/parenleftBigg\nλ1\n2\n2+2cg′/parenrightBigg/integraldisplayβ\nα|z|2dx+o/parenleftBig\nλ−min(2ℓ,ℓ+1\n2)/parenrightBig\n.\nFirst, from ( 2.90), we have\n(2.112) zx=iλvx−λ−ℓ(f2)xinL2(α,β).\nMultiplying ( 2.112) by2gzand integrating over (α,β),then taking the real part, we get\n/integraldisplayβ\nαg(x)(|z|2)xdx=Re/braceleftBigg\n2iλ/integraldisplayβ\nαg(x)vxzdx/bracerightBigg\n−Re/braceleftBigg\n2λ−ℓ/integraldisplayβ\nαg(x)(f2)xzdx/bracerightBigg\n,\nusing by parts integration in the left hand side of above equa tion, we get\n/bracketleftbig\ng(x)|z|2/bracketrightbigβ\nα=/integraldisplayβ\nαg′(x)|z|2dx+Re/braceleftBigg\n2iλ/integraldisplayβ\nαg(x)vxzdx/bracerightBigg\n−Re/braceleftBigg\n2λ−ℓ/integraldisplayβ\nαg(x)(f2)xzdx/bracerightBigg\n,\nconsequently,\n(2.113) |z(β)|2+|z(α)|2≤cg′/integraldisplayβ\nα|z|2dx+2λcg/integraldisplayβ\nα|vx||z|dx+2λ−ℓcg/integraldisplayβ\nα|(f2)x||z|dx.\nOn the other hand, we have\n2λcg|vx||z| ≤λ1\n2|z|2\n2+2λ3\n2c2\ng|vx|2and2λ−ℓcg|(f2)x||z| ≤cg′|z|2+c2\ngλ−2ℓ\ncg′|(f2)x|2.\nInserting the above equation in ( 2.113), then using ( 2.100) and the fact that (f2)x→0inL2(α,β), we get\n|z(β)|2+|z(α)|2≤/parenleftBigg\nλ1\n2\n2+2cg′/parenrightBigg/integraldisplayβ\nα|z|2dx+o/parenleftBig\nλ−min(2ℓ,ℓ+1\n2)/parenrightBig\n,\nhence, we get ( 2.111).\nStep 2. In this step, we prove the following asymptotic behavior est imate\n(2.114) |κ2vx(α)+δ1zx(α)|2+|κ2vx(β)+δ1zx(β)|2≤λ3\n2\n2/integraldisplayβ\nα|z|2dx+o/parenleftBig\nλ−ℓ+1\n2/parenrightBig\n.\n18WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nFirst, multiplying ( 2.93) by−2g(κ2vx+δ1zx)and integrating over (α,β),then taking the real part, we get\n/integraldisplayβ\nαg(x)/parenleftBig\n|κ2vx+δ1zx|2/parenrightBig\nxdx= 2Re/braceleftBigg\niλ/integraldisplayβ\nαg(x)z(κ2vx+δ1zx)dx/bracerightBigg\n−2λ−ℓRe/braceleftBigg/integraldisplayβ\nαg(x)f5(κ2vx+δ1zx)dx/bracerightBigg\n,\nusing by parts integration in the left hand side of above equa tion, we get\n/bracketleftBig\ng(x)|κ2vx+δ1zx|2/bracketrightBigβ\nα=/integraldisplayβ\nαg′(x)|κ2vx+δ1zx|2dx+2Re/braceleftBigg\niλ/integraldisplayβ\nαg(x)z(κ2vx+δ1zx)dx/bracerightBigg\n−2λ−ℓRe/braceleftBigg/integraldisplayβ\nαg(x)f5(κ2vx+δ1zx)dx/bracerightBigg\n,\nconsequently,\n|κ2vx(β)+δ1zx(β)|2+|κ2vx(α)+δ1zx(α)|2≤cg′/integraldisplayβ\nα|κ2vx+δ1zx|2dx+2λcg/integraldisplayβ\nα|z||κ2vx+δ1zx|dx\n+2λ−ℓcg/integraldisplayβ\nα|f5||κ2vx+δ1zx|dx.\nNow, using Cauchy Schwarz inequality, Equations ( 2.98), (2.100) and the fact that f5→0inL2(α,β)in the\nright hand side of above equation, we get\n(2.115) |κ2vx(β)+δ1zx(β)|2+|κ2vx(α)+δ1zx(α)|2≤2λcg/integraldisplayβ\nα|z||κ2vx+δ1zx|dx+o/parenleftbig\nλ−ℓ/parenrightbig\n.\nOn the other hand, we have\n2λcg|z||κ2vx+δ1zx| ≤λ3\n2\n2|z|2+2λ1\n2c2\ng|κ2vx+δ1zx|2.\nInserting the above equation in ( 2.115), then using Equations ( 2.98) and ( 2.100), we get\n|κ2vx(α)+δ1zx(α)|2+|κ2vx(β)+δ1zx(β)|2≤λ3\n2\n2/integraldisplayβ\nα|z|2dx+o/parenleftBig\nλ−ℓ+1\n2/parenrightBig\n,\nhence, we get ( 2.114).\nStep 3. In this step, we prove the asymptotic behavior estimations o f (2.108)-(2.110). First, multiplying ( 2.93)\nby−iλ−1zand integrating over (α,β),then taking the real part, we get\n/integraldisplayβ\nα|z|2dx=−Re/braceleftBigg\niλ−1/integraldisplayβ\nα(κ2vx+δ1zx)xzdx/bracerightBigg\n−Re/braceleftBigg\niλ−ℓ−1/integraldisplayβ\nαf5zdx/bracerightBigg\n,\nconsequently,\n(2.116)/integraldisplayβ\nα|z|2dx≤λ−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayβ\nα(κ2vx+δ1zx)xzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+λ−ℓ−1/integraldisplayβ\nα|f5||z|dx.\nFrom the fact that zis uniformly bounded in L2(α,β)andf5→0inL2(α,β), we get\n(2.117) λ−ℓ−1/integraldisplayβ\nα|f5||z|dx=o/parenleftbig\nλ−ℓ−1/parenrightbig\n.\n19WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nOn the other hand, using by parts integration and ( 2.98), (2.100), we get\n(2.118)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayβ\nα(κ2v+δ1z)xxzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle[(κ2vx+δ1zx)z]β\nα−/integraldisplayβ\nα(κ2vx+δ1zx)zxdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤ |κ2vx(β)+δ1zx(β)||z(β)|+|κ2vx(α)+δ1zx(α)||z(α)|+/integraldisplayβ\nα|κ2vx+δ1zx||zx|dx\n≤ |κ2vx(β)+δ1zx(β)||z(β)|+|κ2vx(α)+δ1z(α)||z(α)|+o/parenleftbig\nλ−ℓ/parenrightbig\n.\nInserting ( 2.117) and ( 2.118) in (2.116), we get\n(2.119)/integraldisplayβ\nα|z|2dx≤λ−1|κ2vx(β)+δ1zx(β)||z(β)|+λ−1|κ2vx(α)+δ1zx(α)||z(α)|+o/parenleftbig\nλ−ℓ−1/parenrightbig\n.\nNow, forζ=βorζ=α, we have\nλ−1|κ2vx(ζ)+δ1zx(ζ)||z(ζ)| ≤λ−1\n2\n2|z(ζ)|2+λ−3\n2\n2|κ2vx(ζ)+δ1zx(ζ)|2.\nInserting the above equation in ( 2.119), we get\n/integraldisplayβ\nα|z|2dx≤λ−1\n2\n2/parenleftbig\n|z(α)|2+|z(β)|2/parenrightbig\n+λ−3\n2\n2/parenleftBig\n|κ2vx(α)+δ1zx(α)|2+|κ2vx(β)+δ1zx(β)|2/parenrightBig\n+o/parenleftbig\nλ−ℓ−1/parenrightbig\n.\nNext, inserting Equations ( 2.111) and ( 2.114) in the above inequality, we obtain\n/integraldisplayβ\nα|z|2dx≤/parenleftbigg1\n2+cg′\nλ1\n2/parenrightbigg/integraldisplayβ\nα|z|2dx+o/parenleftBig\nλ−min(2ℓ+1\n2,ℓ+1)/parenrightBig\n,\nconsequently,/parenleftbigg1\n2−cg′\nλ1\n2/parenrightbigg/integraldisplayβ\nα|z|2dx≤o/parenleftBig\nλ−min(2ℓ+1\n2,ℓ+1)/parenrightBig\n.\nSinceλ→+∞, by choosing λ>4c2\ng′, we get\n0</parenleftbigg1\n2−cg′\nλ1\n2/parenrightbigg/integraldisplayβ\nα|z|2dx≤o/parenleftBig\nλ−min(2ℓ+1\n2,ℓ+1)/parenrightBig\n,\nhence, we get ( 2.108). Finally, inserting ( 2.108) in (2.111) and ( 2.114) and using the first asymptotic estimates\nof (2.106), we get ( 2.109) and ( 2.110). Thus, the proof of the lemma is complete. /square\nRemark 2.12. An example about g, we can take g(x) = cos/parenleftbigg(β−x)π\nβ−α/parenrightbigg\nto get\ng(β) =−g(α) = 1, g∈C1([α,β]),max\nx∈[α,β]|g(x)|= 1,max\nx∈[α,β]|g′(x)|=π\nβ−α.\nAlso, we can take\ng(x) =/parenleftbiggβ−x\nβ−α/parenrightbigg2\n−3/parenleftbiggβ−x\nβ−α/parenrightbigg\n+1.\n/square\nLemma 2.13. Under hypothesis (H), for allℓ≥0, the solution (u,v,w,y,z,φ,η (L,·))∈D(A)of Equations\n(2.89)-(2.95)satisfies the following asymptotic behavior estimations\n(2.120)/integraldisplayα\n0|y|2dx=o/parenleftBig\nλ−min(2ℓ−1,ℓ−1\n2)/parenrightBig\nand/integraldisplayα\n0|ux|2dx=o/parenleftBig\nλ−min(2ℓ−1,ℓ−1\n2)/parenrightBig\n.\nProof. Multiply Equation ( 2.92) byxuxand integrating over (0,α),we get\n(2.121) iλ/integraldisplayα\n0xyuxdx−κ1/integraldisplayα\n0xuxxuxdx=λ−ℓ/integraldisplayα\n0xf4uxdx.\nFrom ( 2.89), we deduce that\niλux=−yx−λ−ℓ(f1)x.\n20WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nInserting the above result in ( 2.121), then using the fact that ux, yare uniformly bounded in L2(0,α)and\n(f1)x, f4converge to zero in L2(0,α)gives\n−/integraldisplayα\n0xyyxdx−κ1/integraldisplayα\n0xuxxuxdx=o/parenleftbig\nλ−ℓ/parenrightbig\n.\nTaking the real part in the above equation, then using by part s integration, we get\n(2.122)1\n2/integraldisplayα\n0|y|2dx+κ1\n2/integraldisplayα\n0|ux|2dx−α\n2/parenleftbig\nκ1|ux(α)|2+|y(α)|2/parenrightbig\n=o/parenleftbig\nλ−ℓ/parenrightbig\n.\nInserting the boundary conditions ( 2.96) atx=αin (2.122) gives\n1\n2/integraldisplayα\n0|y|2dx+κ1\n2/integraldisplayα\n0|ux|2dx=α\n2/parenleftbig\nκ−1\n1|κ2vx(α)+δ1zx(α)|2+|z(α)|2/parenrightbig\n+o/parenleftbig\nλ−ℓ/parenrightbig\n.\nInserting ( 2.109)-(2.110) in the above equation, we obtain the first and the second asym ptotic estimates of\n(2.120). The proof is thus complete. /square\nProof of Theorem 2.7.From Lemma 2.8, Lemma 2.9, Lemma 2.10, Lemma 2.11and Lemma 2.13, we get\n/ba∇dblU/ba∇dbl2\nH=κ1/integraldisplayα\n0|ux|2dx+κ2/integraldisplayβ\nα|vx|2dx+κ3/integraldisplayL\nβ|wx|2dx+/integraldisplayα\n0|y|2dx\n+/integraldisplayβ\nα|z|2dx+/integraldisplayL\nβ|φ|2dx+τ/integraldisplay1\n0|η(L,ρ)|2dρ=o/parenleftBig\nλ−min(2ℓ−1,ℓ−1\n2)/parenrightBig\n.\nTo obtain /ba∇dblU/ba∇dblH=o(1), we need min/parenleftbigg\n2ℓ−1,ℓ−1\n2/parenrightbigg\n≥0, so we choose ℓ=1\n2as the optimal value. Hence,\nwe obtain that /ba∇dblU/ba∇dblH=o(1)which contradicts ( 2.87). Therefore, the energy of System ( 2.9)-(2.22) satisfies\nestimation ( 2.85) for all initial data U0∈D(A). /square\n2.2.Wave equation with local Kelvin-Voigt damping near the boun dary and boundary delay\nfeedback. In this subsection, we study the stability of System ( 2.1), but in the case that the Kelvin-Voigt\ndamping is near the boundary, i.e.α= 0and0<β <L (see Figure 2). For this aim, we denote the longitudinal\ndisplacement by Uand this displacement is divided into two parts\nU(x,t) =/braceleftBigg\nv(x,t),(x,t)∈(α,β)×(0,+∞),\nw(x,t),(x,t)∈(β,L)×(0,+∞).\nIn this case, System ( 2.1) is equivalent to the following system\n(2.123)\n\nvtt−(κ2vx+δ1vxt)x= 0, (x,t)∈(0,β)×(0,+∞),\nwtt−κ3wxx= 0, (x,t)∈(β,L)×(0,+∞),\nτηt(L,ρ,t)+ηρ(L,ρ,t) = 0, (ρ,t)∈(0,1)×(0,+∞),\nv(0,t) = 0, t ∈(0,+∞),\nwx(L,t) =−δ3wt(L,t)−δ2η(L,1,t), t∈(0,+∞),\nv(β,t) =w(β,t), t ∈(0,+∞),\nκ2vx(β,t)+δ1vxt(β,t) =κ3wx(β,t), t∈(0,+∞),\n(v(x,0),vt(x,0)) = (v0(x),v1(x)), x ∈(α,β),\n(w(x,0),wt(x,0)) = (w0(x),w1(x)), x∈(β,L),\nη(L,ρ,0) =f0(L,−ρτ), ρ ∈(0,1),\nwhere the initial data (v0,v1,w0,w1,f0)belongs to a suitable space. Similar to Section 2.1, we define\nXm=Hm(0,β)×Hm(β,L), m= 1,2,\nX0=L2(0,β)×L2(β,L),X1\nL={(v,w)∈X1|v(0) = 0, v(β) =w(β)},\n21WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nwhere the Hilbert space X0is equipped with the norm:\n/ba∇dbl(v,w)/ba∇dbl2\nX0=/integraldisplayβ\n0|v|2dx+/integraldisplayL\nβ|w|2dx.\nMoreover, it is easy to check that the space X1\nLis Hilbert space over Cequipped with the norm:\n/ba∇dbl(v,w)/ba∇dbl2\nX1\nL=κ2/integraldisplayβ\n0|vx|2dx+κ3/integraldisplayL\nβ|wx|2dx.\nIn addition, by Poincaré inequality we can easily verify tha t there exists C >0, such that\n/ba∇dbl(v,w)/ba∇dblX0≤C/ba∇dbl(v,w)/ba∇dblX1\nL.\nWe now define the Hilbert energy space by\nH1=X1\nL×X0×L2(0,1)\nequipped with the following inner product\n/a\\}b∇acketle{tU,˜U/a\\}b∇acket∇i}htH1=κ2/integraldisplayβ\n0vx˜vxdx+κ3/integraldisplayL\nβwx˜wxdx+/integraldisplayβ\n0z˜zdx+/integraldisplayL\nβφ˜φdx+τ/integraldisplay1\n0η(L,ρ)˜η(L,ρ)dρ,\nwhereU= (v,w,z,φ,η (L,·))∈ H1and˜U= (˜v,˜w,˜z,˜φ,˜η(L,·))∈ H1. We use /ba∇dblU/ba∇dblH1to denote the corresponding\nnorm. We define the linear unbounded operator A1:D(A1)⊂ H1−→ H 1by:\nD(A1) =/braceleftbigg\nU= (v,w,z,φ,η (L,·))∈X1\nL×X1\nL×H1(0,1)|(κ2v+δ1z,w)∈X2,\nκ2vx(β)+δ1zx(β) =κ3wx(β), wx(L) =−δ3η(L,0)−δ2η(L,1), φ(L) =η(L,0)/bracerightbigg\nand for all U= (v,w,z,φ,η (L,·))∈D(A1)\nA1U=/parenleftbig\nz,φ,(κ2vx+δ1zx)x,κ3wxx,−τ−1ηρ(L,·)/parenrightbig\n.\nIfU= (v,w,v t,wt,η(L,·))is a regular solution of System ( 2.123), then we transform this system into the\nfollowing initial value problem\n(2.124)/braceleftBigg\nUt=A1U,\nU(0) =U0,\nwhereU0= (v0,w0,v1,w1,f0(L,−·τ))∈ H1.Note thatD(A1)is dense in H1and that for all U∈D(A1), we\nhave\n(2.125) Re /a\\}b∇acketle{tA1U,U/a\\}b∇acket∇i}htH1≤ −δ1/integraldisplayβ\n0|zx|2dx−/parenleftbigg1\n2−κ3|δ2|\n2p/parenrightbigg\n|η(L,1)|2−/parenleftbigg\nκ3δ3−1\n2−κ3|δ2|p\n2/parenrightbigg\n|η(L,0)|2,\nwherepis defined in ( 2.8). Consequently, under hypothesis (H), the system becomes d issipative. We can easily\nadapt the proof in Subsection 2.1.1to prove the well-posedness of System ( 2.124).\nTheorem 2.14. Under hypothesis (H), for all initial data U0∈ H1,the System (2.123)is exponentially stable.\nAccording to Theorem A.5(part (i)), we have to check if the following conditions hold ,\n(2.126) iR⊆ρ(A1)\nand\n(2.127) sup\nλ∈R/vextenddouble/vextenddouble/vextenddouble(iλI−A1)−1/vextenddouble/vextenddouble/vextenddouble\nL(H1)=O(1).\nProof. First, we can easily adapt the proof in Subsection 2.1.2to prove the strong stability (condition ( 2.126))\nof System ( 2.123). Next, we will prove condition ( 2.127) by a contradiction argument. Indeed, suppose there\nexists\n{(λn,Un:= (vn,wn,zn,φn,ηn(L,·)))}n≥1⊂R∗\n+×D(A1),\nsuch that\n(2.128) λn→+∞,/ba∇dblUn/ba∇dblH1= 1\n22WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nand there exists sequence Gn:= (g1,n,g2,n,g3,n,g4,n,g5,n(L,·))∈ H1, such that\n(2.129) (iλnI−A1)Un=Gn→0inH1.\nWe will check condition ( 2.127) by finding a contradiction with /ba∇dblUn/ba∇dblH1= 1such as/ba∇dblUn/ba∇dblH1=o(1).From now\non, for simplicity, we drop the index n. By detailing Equation ( 2.129), we get the following system\niλv−z=g1inH1(0,β), (2.130)\niλw−φ=g2inH1(β,L), (2.131)\niλz−(κ2vx+δ1zx)x=g3inL2(0,β), (2.132)\niλφ−κ3wxx=g4inL2(β,L), (2.133)\nηρ(L,·)+iτλη(L,·) =τg5(L,·)inL2(0,1). (2.134)\nRemark that, since U= (v,w,z,φ,η (L,·))∈D(A1), we have the following boundary conditions\n|wx(β)|=κ−1\n3|κ2vx(β)+δ1zx(β)|,|z(β)|=|φ(β)|, (2.135)\nwx(L) =−δ3η(L,0)−δ2η(L,1), φ(L) =η(L,0). (2.136)\nTaking the inner product of ( 2.129) withUinH1, then using ( 2.125), hypothesis (H) and the fact that Uis\nuniformly bounded in H1, we obtain\n(2.137)/integraldisplayβ\n0|zx|2=o(1),|φ(L)|2=|η(L,0)|2=o(1),|η(L,1)|2=o(1).\nFrom ( 2.130), then using the first asymptotic estimate of ( 2.137) and the fact that (g1)x→0inL2(0,β), we\nget\n(2.138)/integraldisplayβ\n0|vx|2dx=o/parenleftbig\nλ−2/parenrightbig\n.\nFrom the first asymptotic estimate of ( 2.136), then using the second and the third asymptotic estimates o f\n(2.137), we obtain\n(2.139) |wx(L)|2=o(1).\nSimilar to Lemma 2.9, withℓ= 0, from ( 2.134), then using the second and the third asymptotic estimates o f\n(2.137), we obtain\n(2.140)/integraldisplay1\n0|η(L,ρ)|2dρ=o(1).\nSimilar to Lemma 2.10, withℓ= 0, multiplying Equation ( 2.133) byxwxand integrating over (β,L),after that\nusing the fact that iλwx=−φx−(g2)x, then using the fact that φ, wxare uniformly bounded in L2(β,L)and\n(g2)x, g4converge to zero in L2(β,L)gives\n−/integraldisplayL\nβxφφxdx−κ3/integraldisplayL\nβxwxxwxdx=o(1).\nTaking the real part in the above equation, then using by part s integration, Equation ( 2.139) and the second\nasymptotic estimate of ( 2.137), we obtain\n1\n2/integraldisplayL\nβ|φ|2dx+κ3\n2/integraldisplayL\nβ|wx|2dx+β\n2/parenleftbig\nκ3|wx(β)|2+|φ(β)|2/parenrightbig\n=o(1),\nhence, we get\n(2.141)/integraldisplayL\nβ|φ|2dx=o(1),/integraldisplayL\nβ|wx|2dx=o(1),|wx(β)|2=o(1),|φ(β)|2=o(1).\nInserting the third and the fourth asymptotic estimates of ( 2.141) in (2.135), we get\n(2.142) |κ2vx(β)+δ1zx(β)|=o(1),|z(β)|=o(1).\n23WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nSimilar to step 3 of Lemma 2.11, withα= 0andℓ= 0, multiplying ( 2.132) by−iλ−1zand integrating over\n(0,β),taking the real part, then using the fact that zis uniformly bounded in L2(0,β)andg3→0inL2(0,β),\nwe get\n(2.143)/integraldisplayβ\n0|z|2dx≤λ−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayβ\n0(κ2vx+δ1zx)xzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+o/parenleftbig\nλ−1/parenrightbig\n.\nOn the other hand, using by parts integration, the fact that z(0) = 0 , and Equations ( 2.137)-(2.138), (2.142),\nwe get\n(2.144)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayβ\n0(κ2vx+δ1zx)xzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle[(κ2vx+δ1zx)z]β\n0−/integraldisplayβ\n0(κ2vx+δ1zx)zxdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤ |κ2vx(β)+δ1zx(β)||z(β)|+/integraldisplayβ\n0|κ2vx+δ1zx||zx|dx=o(1).\nInserting ( 2.144) in (2.143), we get\n(2.145)/integraldisplayβ\n0|z|2dx=o/parenleftbig\nλ−1/parenrightbig\n.\nFinally, from ( 2.138), (2.140), (2.141) and ( 2.145), we get\n/ba∇dblU/ba∇dblH1=o(1),\nwhich contradicts ( 2.128). Therefore, ( 2.127) holds and the result follows from Theorem A.5(part (i)). /square\n3.Wave equation with local internal Kelvin-Voigt damping and local internal delay\nfeedback\nIn this section, we study the stability of System ( 1.2). We assume that there exists αandβsuch that\n0< α < β < L , in this case, the Kelvin-Voigt damping and the time delay fe edback are locally internal (see\nFigure 3). For this aim, we denote the longitudinal displacement by Uand this displacement is divided into\nthree parts\nU(x,t) =\n\nu(x,t),(x,t)∈(0,α)×(0,+∞),\nv(x,t),(x,t)∈(α,β)×(0,+∞),\nw(x,t),(x,t)∈(β,L)×(0,+∞).\nFurthermore, let us introduce the auxiliary unknown\nη(x,ρ,t) =vt(x,t−ρτ), x∈(α,β), ρ∈(0,1), t>0.\nIn this case, System ( 1.2) is equivalent to the following system\nutt−κ1uxx= 0,(x,t)∈(0,α)×(0,+∞), (3.1)\nvtt−(κ2vx+δ1vxt(x,t)+δ2ηx(x,1,t))x= 0,(x,t)∈(α,β)×(0,+∞), (3.2)\nwtt−κ3wxx= 0,(x,t)∈(β,L)×(0,+∞), (3.3)\nτηt(x,ρ,t)+ηρ(x,ρ,t) = 0,(x,ρ,t)∈(α,β)×(0,1)×(0,+∞), (3.4)\nwith the Dirichlet boundary conditions\n(3.5) u(0,t) =w(L,t) = 0, t∈(0,+∞),\nwith the following transmission conditions\n(3.6)\n\nu(α,t) =v(α,t), v(β,t) =w(β,t), t ∈(0,+∞),\nκ1ux(α,t) =κ2vx(α,t)+δ1vxt(α,t)+δ2ηx(α,1,t), t∈(0,+∞),\nκ3wx(β,t) =κ2vx(β,t)+δ1vxt(β,t)+δ2ηx(β,1,t), t∈(0,+∞),\n24WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nand with the following initial conditions\n(3.7)\n\n(u(x,0),ut(x,0)) = (u0(x),u1(x)), x∈(0,α),\n(v(x,0),vt(x,0)) = (v0(x),v1(x)), x ∈(α,β),\n(w(x,0),wt(x,0)) = (w0(x),w1(x)), x∈(β,L),\nη(x,ρ,0) =f0(x,−ρτ), (x,ρ)∈(α,β)×(0,1),\nwhere the initial data (u0,u1,v0,v1,w0,w1,f0)belongs to a suitable space. To a strong solution of System\n(3.1)-(3.7), we associate the energy defined by\nE(t) =1\n2/integraldisplayα\n0/parenleftbig\n|ut(x,t)|2+κ1|ux(x,t)|2/parenrightbig\ndx+1\n2/integraldisplayβ\nα/parenleftbig\n|vt(x,t)|2+κ2|vx(x,t)|2/parenrightbig\ndx\n+1\n2/integraldisplayL\nβ/parenleftbig\n|wt(x,t)|2+κ3|wx(x,t)|2/parenrightbig\ndx+τ|δ2|\n2/integraldisplay1\n0/integraldisplayβ\nα|ηx(x,ρ,t)|2dρdx.\nMultiplying ( 3.1), (3.2), (3.3) and ( 3.4)xbyut,yt,wtand|δ2|ηx, integrating over (0,α),(α,β),(β,L)and\n(α,β)×(0,1)respectively, taking the sum, then using by parts integrati on and the boundary conditions in\n(3.5)-(3.6), we get\nE′(t) =/parenleftbigg\n−δ1+|δ2|\n2/parenrightbigg/integraldisplayβ\nα|vxt(x,t)|2dx−|δ2|\n2/integraldisplayβ\nα|ηx(x,1,t)|2dx−δ2/integraldisplayβ\nαvxt(x,t)ηx(x,1,t)dx.\nUsing Young’s inequality for the third term in the right, we g et\nE′(t)≤(−δ1+|δ2|)/integraldisplayβ\nα|vxt(x,t)|2dx.\nIn the sequel, the assumption on δ1andδ2will ensure that\n(H1) δ1>0, δ2∈R∗,|δ2|<δ1.\nIn this case, the energies of the strong solutions satisfy E′(t)≤0.Hence, the System ( 3.1)-(3.7) is dissipative\nin the sense that its energy is non increasing with respect to the timet.\n3.1.Well-posedness of the problem. We start this part by formulating System ( 3.1)-(3.7) as an abstract\nCauchy problem. For this aim, let us define\nL2\n∗=L2(0,α)×L2\n∗(α,β)×L2(β,L),\nH1\n∗={(u,v,w)∈H1(0,α)×H1\n∗(α,β)×H1(β,L)|u(0) = 0, u(α) =v(α), v(β) =w(β), w(L) = 0},\nH2=H2(0,α)×H2(α,β)×H2(β,L).\nHere we consider\nL2\n∗(α,β) =/braceleftBigg\nz∈L2(α,β)|/integraldisplayβ\nαzdx= 0/bracerightBigg\nandH1\n∗(α,β) =H1(α,β)∩L2\n∗(α,β).\nThe spaces L2\n∗andH1\n∗are obviously a Hilbert spaces equipped respectively with t he norms\n/ba∇dbl(u,v,w)/ba∇dbl2\nL2∗=/integraldisplayα\n0|u|2dx+/integraldisplayβ\nα|v|2dx+/integraldisplayL\nβ|w|2dx\nand\n/ba∇dbl(u,v,w)/ba∇dbl2\nH1∗=κ1/integraldisplayα\n0|ux|2dx+κ2/integraldisplayβ\nα|vx|2dx+κ3/integraldisplayL\nβ|wx|2dx.\nIn addition by Poincaré inequality we can easily verify that there exists C >0, such that\n/ba∇dbl(u,v,w)/ba∇dblL2∗≤C/ba∇dbl(u,v,w)/ba∇dblH1∗.\nLet us define the energy Hilbert space H2by\nH2=H1\n∗×L2\n∗×L2/parenleftbig\n(0,1),H1\n∗(α,β)/parenrightbig\n25WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nequipped with the following inner product\n/a\\}b∇acketle{tU,˜U/a\\}b∇acket∇i}htH2=κ1/integraldisplayα\n0ux˜uxdx+κ2/integraldisplayβ\nαvx˜vxdx+κ3/integraldisplayL\nβwx˜wxdx\n+/integraldisplayα\n0y˜ydx+/integraldisplayβ\nαz˜zdx+/integraldisplayL\nβφ˜φdx+τ|δ2|/integraldisplay1\n0/integraldisplayβ\nαηx(x,ρ)˜ηx(x,ρ)dxdρ,\nwhereU= (u,v,w,y,z,φ,η (·,·))∈ H2and˜U= (˜u,˜v,˜w,˜y,˜z,˜φ,˜η(·,·))∈ H2. We use /ba∇dblU/ba∇dblH2to denote the\ncorresponding norm. We define the linear unbounded operator A2:D(A2)⊂ H2−→ H 2by:\nD(A2) =/braceleftbigg\n(u,v,w,y,z,φ,η (·,·))∈ H2|(y,z,φ)H1\n∗\n(u,κ2v+δ1z+δ2η(·,1),w)∈H2, κ2vx(α)+δ1zx(α)+δ2ηx(α,1) =κ1ux(α),\nκ2vx(β)+δ1zx(β)+δ2ηx(β,1) =κ3wx(β), η, ηρ∈L2/parenleftbig\n(0,1),H1\n∗(α,β)/parenrightbig\n, z(·) =η(·,0)/bracerightbigg\nand for all U= (u,v,w,y,z,φ,η (·,·))∈D(A2)\nA2U=/parenleftbig\ny,z,φ,κ 1uxx,(κ2vx+δ1zx+δ2ηx(·,1))x,κ3wxx,−τ−1ηρ(·,·)/parenrightbig\n.\nIfU= (u,v,w,u t,vt,wt,η(·,·))is a regular solution of System ( 3.1)-(3.7), then we transform this system into\nthe following initial value problem\n(3.8)/braceleftBigg\nUt=A2U,\nU(0) =U0,\nwhereU0= (u0,v0,w0,u1,v1,w1,f0(·,−·τ))∈ H2.We now use semigroup approach to establish well-posedness\nresult for the System ( 3.1)-(3.7). We prove the following proposition.\nProposition 3.1. Under hypothesis (H1), the unbounded linear operator A2is m-dissipative in the energy\nspaceH2.\nProof. For allU= (u,v,w,y,z,φ,η (·,·))∈D(A2),we have\nRe/a\\}b∇acketle{tA2U,U/a\\}b∇acket∇i}htH2=κ1Re/integraldisplayα\n0(yxux+uxxy)dx+Re/integraldisplayβ\nα(κ2zxvx+(κ2vx+δ1zx+δ2ηx(·,1))xz)dx\n+κ3Re/integraldisplayL\nβ/parenleftbig\nφxwx+wxxφ/parenrightbig\ndx−|δ2|Re/integraldisplayβ\nα/integraldisplay1\n0ηxρ(x,ρ)ηx(x,ρ)dxdρ.\nUsing by parts integration in the above equation, we get\n(3.9)Re/a\\}b∇acketle{tA2U,U2/a\\}b∇acket∇i}htH2=−δ1/integraldisplayβ\nα|zx|2dx−δ2Re/integraldisplayβ\nαηx(·,1)zxdx+|δ2|\n2/integraldisplayβ\nα|ηx(x,0)|2dx\n−|δ2|\n2/integraldisplayβ\nα|ηx(x,1)|2dx−κ1Re(ux(0)y(0))+κ3Re/parenleftbig\nwx(L)φ(L)/parenrightbig\n+Re(κ1ux(α)y(α)−κ2vx(α)z(α)−δ1zx(α)z(α)−δ2ηx(α,1)z(α))\n+Re/parenleftbig\nκ2vx(β)z(β)+δ1zx(β)z(β)+δ2ηx(β,1)z(β)−κ3wx(β)φ(β)/parenrightbig\n.\nSinceU∈D(A2), we have\n/braceleftBigg\ny(0) =φ(0) = 0, y(α) =z(α), z(β) =φ(β), z(x) =η(x,0),\nκ1ux(α)−κ2vx(α)−δ1zx(α)−δ2ηx(α,1) = 0, κ2vx(β)+δ1zx(β)+δ2ηx(β,1)−κ3wx(β) = 0.\nSubstituting the above boundary conditions in ( 3.9), then using Young’s inequality, we get\n(3.10)Re/a\\}b∇acketle{tA2U,U/a\\}b∇acket∇i}htH2=/parenleftbigg\n−δ1+|δ2|\n2/parenrightbigg/integraldisplayβ\nα|zx|2dx−|δ2|\n2/integraldisplayβ\nα|ηx(x,1)|2dx−δ2Re/integraldisplayβ\nαηx(·,1)zxdx\n≤(−δ1+|δ2|)/integraldisplayβ\nα|zx|2dx,\n26WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nhence under hypothesis (H1), we get\nRe/a\\}b∇acketle{tA2U,U/a\\}b∇acket∇i}htH2≤0,\nwhich implies that A2is dissipative. To prove that A2is m-dissipative, it is enough to prove that 0∈ρ(A2)\nsinceA2is a closed operator and D(A2) =H2. LetF= (f1,f2,f3,f4,f5,f6,f7(·,·))∈ H2.We should prove\nthat there exists a unique solution U= (u,v,w,y,z,φ,η (·,·))∈D(A2)of the equation\n−A2U=F.\nEquivalently, we consider the following system\n−y=f1, (3.11)\n−z=f2, (3.12)\n−φ=f3, (3.13)\n−κ1uxx=f4, (3.14)\n−(κ2vx+δ1zx+δ2ηx(·,1))x=f5, (3.15)\n−κ3wxx=f6, (3.16)\nηρ(x,ρ) =τf7(x,ρ). (3.17)\nIn addition, we consider the following boundary conditions\nu(0) = 0, u(α) =v(α), v(β) =w(β), w(L) = 0, (3.18)\nκ2vx(α)+δ1zx(α)+δ2ηx(α,1) =κ1ux(α), κ2vx(β)+δ1zx(β)+δ2ηx(β,1) =κ3wx(β), (3.19)\nη(·,0) =z(·). (3.20)\nFrom ( 3.11)-(3.13) and the fact that F∈ H, we obtain (y,z,φ)∈H1\n∗. Next, from ( 3.12), (3.20) and the fact\nthatf2∈H1\n∗(α,β), we get\nη(·,0) =z(·) =−f2(·)∈H1\n∗(α,β).\nFrom the above equation and Equation ( 3.17), we can determine\n(3.21) η(x,ρ) =τ/integraldisplayρ\n0f7(x,ξ)dξ−f2(x).\nSincef2∈H1\n∗(α,β)andf7∈L2/parenleftbig\n(0,1),H1\n∗(α,β)/parenrightbig\n, then it is clear that η, ηρ∈L2((0,1),H1\n∗(0,1)). Now, let\n(ϕ,ψ,θ)∈H1\n∗. Multiplying Equations ( 3.14), (3.15), (3.16) byϕ,ψ,θ, integrating over (0,α),(α,β)and(β,L)\nrespectively, taking the sum, then using by parts integrati on, we get\n(3.22)κ1/integraldisplayα\n0uxϕxdx+/integraldisplayβ\nα(κ2vx+δ1zx+δ2ηx(·,1))ψxdx+κ3/integraldisplayL\nβwxθxdx+κ1ux(0)ϕ(0)−κ3wx(L)θ(L)\n−κ1ux(α)ϕ(α)+(κ2vx(α)+δ1zx(α)+δ2ηx(α,1))ψ(α)−(κ2vx(β)+δ1zx(β)+δ2ηx(β,1))ψ(β)\n+κ3wx(β)θ(β) =/integraldisplayα\n0f4ϕdx+/integraldisplayβ\nαf5ψdx+/integraldisplayL\nβf6θdx.\nFrom the fact that (ϕ,ψ,θ)∈H1\n∗,we have\nϕ(0) = 0, ϕ(α) =ψ(α), θ(β) =ψ(β), θ(L) = 0.\nInserting the above equation in ( 3.22), then using ( 3.12), (3.19) and ( 3.21), we get\n(3.23)κ1/integraldisplayα\n0uxϕxdx+κ2/integraldisplayβ\nαvxψxdx+κ3/integraldisplayL\nβwxθxdx\n=/integraldisplayα\n0f4ϕdx+/integraldisplayβ\nαf5ψdx+/integraldisplayL\nβf6θdx+/integraldisplayβ\nα/parenleftbigg\n(δ1+δ2)(f2)x−δ2τ/integraldisplay1\n0(f7(·,ξ))xdξ/parenrightbigg\nψxdx.\nWe can easily verify that the left hand side of ( 3.23) is a bilinear continuous coercive form on H1\n∗×H1\n∗, and\nthe right hand side of ( 3.23) is a linear continuous form on H1\n∗. Then, using Lax-Milgram theorem, we deduce\nthat there exists (u,v,w)∈H1\n∗unique solution of the variational Problem ( 3.23). Using standard arguments,\nwe can show that (u,κ2v+δ1z+δ2η(·,1),w)∈H2. Thus, from ( 3.11)-(3.13), (3.21) and applying the classical\nelliptic regularity we deduce that U= (u,v,w,y,z,φ,η (·,·))∈D(A2). The proof is thus complete. /square\n27WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nThanks to Lumer-Philips theorem (see [ 36]), we deduce that A2generates a C0−semigroup of contractions etA2\ninH2and therefore Problem ( 3.1)-(3.7) is well-posed.\n3.2.Polynomial Stability. The main result in this subsection is the following theorem.\nTheorem 3.2. Under hypothesis (H1), for all initial data U0∈D(A2),there exists a constant C >0indepen-\ndent ofU0such that the energy of System (3.1)-(3.7)satisfies the following estimation\nE(t)≤C\nt4/ba∇dblU0/ba∇dbl2\nD(A2),∀t>0.\nAccording to Theorem A.5(part (ii)), we have to check if the following conditions hol d,\n(3.24) iR⊆ρ(A2)\nand\n(3.25) sup\nλ∈R/vextenddouble/vextenddouble/vextenddouble(iλI−A2)−1/vextenddouble/vextenddouble/vextenddouble\nL(H2)=O/parenleftBig\n|λ|1\n2/parenrightBig\n.\nThe next proposition is a technical result to be used in the pr oof of Theorem 3.2given below.\nProposition 3.3. Under hypothesis (H1), let(λ,U:= (u,v,w,y,z,φ,η (·,·)))∈R∗×D(A2),such that\n(3.26) (iλI−A2)U=F:= (f1,f2,f3,f4,f5,f6,g(·,·))∈ H2,\ni.e.\niλu−y=f1 inH1(0,α), (3.27)\niλv−z=f2 inH1\n∗(α,β), (3.28)\niλw−φ=f3 inH1(β,L), (3.29)\niλy−κ1uxx=f4 inL2(0,α), (3.30)\niλz−(κ2vx+δ1zx+δ2ηx(·,1))x=f5 inL2\n∗(α,β), (3.31)\niλφ−κ3wxx=f6 inL2(β,L), (3.32)\nηρ(·,·)+iτλη(·,·) =τg(·,·)inL2/parenleftbig\n(0,1),H1\n∗(α,β)/parenrightbig\n. (3.33)\nThen, we have the following inequality\n(3.34) /ba∇dblU/ba∇dbl2\nH2≤K1λ−4(|λ|+1)6/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n.\nIn addition, if |λ| ≥M >0,then we have\n(3.35) /ba∇dblU/ba∇dbl2\nH2≤K2/parenleftbigg√\nM+1√\nM/parenrightbigg2\n|λ|1\n2/parenleftBig\n1+|λ|−1\n2/parenrightBig8/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n.\nHere and below we denote by Kja positive constant number independent of λ.\nBefore stating the proof of Proposition 3.3, leth∈C1([α,β])such that\nh(α) =−h(β) = 1,max\nx∈[α,β]|h(x)|=Chandmax\nx∈[α,β]|h′(x)|=Ch′,\nwhereChandCh′are strictly positive constant numbers independent of λ. An example about h, we can take\nh(x) =−2(x−α)\nβ−α+1to get\nh(α) =−h(β) = 1, h∈C1([α,β]), Ch= 1, Ch′=2\nβ−α.\nFor the proof of Proposition 3.3, we need the following lemmas.\n28WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nLemma 3.4. Under hypothesis (H1), the solution (u,v,w,y,z,φ,η (·,·))∈D(A2)of Equation (3.26)satisfies\nthe following estimations\n/integraldisplayβ\nα|zx|2dx≤K3/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2, (3.36)\n/integraldisplayβ\nα|vx|2dx≤K4λ−2/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n, (3.37)\n/integraldisplayβ\nα/integraldisplay1\n0|ηx(x,ρ)|2dxdρ≤K5/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n, (3.38)\n/integraldisplayβ\nα|κ2vx+δ1zx+δ2ηx(·,1)|2dx≤K6/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n, (3.39)\nwhere /braceleftBigg\nK3= (δ1−|δ2|)−1, K4= 2max/parenleftbig\nK3,κ−1\n2/parenrightbig\n,\nK5= 2max/parenleftbig\nK3,τ|δ2|−1/parenrightbig\n, K6= 3max/parenleftbig\nκ2\n2K4,δ2\n1K3+δ2\n2K5/parenrightbig\n.\nProof. First, taking the inner product of ( 3.26) withUinH2, then using hypothesis (H1), arguing in the same\nway as ( 3.10), we obtain\n/integraldisplayβ\nα|zx|2dx≤ −1\nδ1−|δ2|Re/a\\}b∇acketle{tA2U,U/a\\}b∇acket∇i}htH2=1\nδ1−|δ2|Re/a\\}b∇acketle{tF,U/a\\}b∇acket∇i}htH2≤1\nδ1−|δ2|/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2,\nhence we get ( 3.36). Next, from ( 3.28), (3.36) and the fact that κ2/integraldisplayβ\nα|(f2)x|2dx≤ /ba∇dblF/ba∇dbl2\nH2, we obtain\n/integraldisplayβ\nα|vx|2dx≤2λ−2/integraldisplayβ\nα|zx|2dx+2λ−2/integraldisplayβ\nα|(f2)x|2dx\n≤2λ−2/parenleftBig\nK3/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+κ−1\n2/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n≤2λ−2max/parenleftbig\nK3,κ−1\n2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n.\ntherefore we get ( 3.37). Now, from ( 3.33) and using the fact that U∈D(A2) (i.e.η(·,0) =z(·)), we obtain\n(3.40) η(x,ρ) =z(x)e−iτλρ+τ/integraldisplayρ\n0eiτλ(ξ−ρ)g(x,ξ)dξ(x,ρ)∈(α,β)×(0,1),\nconsequently, we obtain\n/integraldisplayβ\nα/integraldisplay1\n0|ηx(x,ρ)|2dxdρ≤2/integraldisplayβ\nα|zx|2dx+2τ2/integraldisplayβ\nα/integraldisplay1\n0|gx(x,ξ)|2dξdx.\nInserting ( 3.36) in the above equation, then using the fact that τ|δ2|/integraldisplayβ\nα/integraldisplay1\n0|gx(x,ξ)|2dξdx≤ /ba∇dblF/ba∇dbl2\nH2, we obtain\n/integraldisplayβ\nα/integraldisplay1\n0|ηx(x,ρ)|2dxdρ≤2K3/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+2τ|δ2|−1/ba∇dblF/ba∇dbl2\nH2\n≤2max/parenleftbig\nK3,τ|δ2|−1/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nhence we get ( 3.38). On the other hand, from ( 3.40), we get\nηx(x,1) =zx(x)e−iτλ+τ/integraldisplay1\n0eiτλ(ξ−1)gx(x,ξ)dξ x∈(α,β).\nFrom the above equation and ( 3.36), we obtain\n/integraldisplayβ\nα|ηx(x,1)|2dx≤2/integraldisplayβ\nα|zx|2dx+2τ2/integraldisplayβ\nα/integraldisplay1\n0|gx(x,ξ)|2dξdx\n≤2K3/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+2τ|δ2|−1/ba∇dblF/ba∇dbl2\nH2\n≤2max/parenleftbig\nK3,τ|δ2|−1/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n.\n29WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nFinally, from ( 3.36), (3.37) and the above inequality, we get\n/integraldisplayβ\nα|κ2vx+δ1zx+δ2ηx(·,1)|2dx≤3κ2\n2/integraldisplayβ\nα|vx|2dx+3δ2\n1/integraldisplayβ\nα|zx|2dx+3δ2\n2/integraldisplayβ\nα|ηx(·,1)|2dx\n≤3/parenleftbig\nκ2\n2K4λ−2+δ2\n1K3+δ2\n2K5/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n≤3max/parenleftbig\nκ2\n2K4,δ2\n1K3+δ2\n2K5/parenrightbig/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nhence we get ( 3.39). /square\nLemma 3.5. Under hypothesis (H1), for alls1, s2∈Randr1, r2∈R∗\n+, the solution (u,v,w,y,z,φ,η (·,·))∈\nD(A2)of Equation (3.26)satisfies the following estimations\n(3.41) |z(β)|2+|z(α)|2≤/parenleftBigg\nCh′+|λ|1\n2−s1\nr1/parenrightBigg/integraldisplayβ\nα|z|2dx+K7r1C2\nh|λ|−1\n2+s1/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\nand\n(3.42)|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)|2+|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)|2\n≤|λ|3\n2−s2\nr2/integraldisplayβ\nα|z|2dx+K8/parenleftBig\nCh′+Ch+r2C2\nh|λ|1\n2+s2/parenrightBig/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nwhere\nK7= 2/parenleftbig\nK4+κ−1\n2/parenrightbig\n, K8=K6+1.\nProof. First, from Equation ( 3.28), we have\n−zx= (f2)x−iλvx.\nMultiplying the above equation by 2hz, integrating over (α,β)and taking the real parts, then using by parts\nintegration and the fact that h(α) =−h(β) = 1, we get\n(3.43)|z(β)|2+|z(α)|2=−/integraldisplayβ\nαh′|z|2dx−2Re/braceleftBigg/integraldisplayβ\nαh(iλvx−(f2)x)zdx/bracerightBigg\n≤Ch′/integraldisplayβ\nα|z|2dx+2Ch|λ|/integraldisplayβ\nα|vx||z|dx+2Ch/integraldisplayβ\nα|(f2)x||z|dx.\nOn the other hand, for all s1∈Randr1∈R∗\n+, we have\n2Ch|λ||vx||z| ≤|λ|1\n2−s1|z|2\n2r1+2r1C2\nh|λ|3\n2+s1|vx|2and2Ch|(f2)x||z| ≤|λ|1\n2−s1|z|2\n2r1+2r1C2\nh|λ|−1\n2+s1|(f2)x|2.\nInserting the above equation in ( 3.43), then using ( 3.37) and the fact that/integraldisplayβ\nα|(f2)x|2dx≤κ−1\n2/ba∇dblF/ba∇dbl2\nH2,we get\n|z(β)|2+|z(α)|2\n≤/parenleftBigg\nCh′+|λ|1\n2−s1\nr1/parenrightBigg/integraldisplayβ\nα|z|2dx+2r1C2\nh|λ|s1/parenleftBigg\n|λ|3\n2/integraldisplayβ\nα|vx|2dx+|λ|−1\n2/integraldisplayβ\nα|(f2)x|2dx/parenrightBigg\n≤/parenleftBigg\nCh′+|λ|1\n2−s1\nr1/parenrightBigg/integraldisplayβ\nα|z|2dx+2r1C2\nh|λ|s1−1\n2/parenleftBig\nK4/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n+κ−1\n2/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n≤/parenleftBigg\nCh′+|λ|1\n2−s1\nr1/parenrightBigg/integraldisplayβ\nα|z|2dx+2r1C2\nh|λ|s1−1\n2/parenleftbig\nK4+κ−1\n2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nhence we get ( 3.41). Next, multiplying Equation ( 3.31) by2h(κ2vx+δ1zx+δ2ηx(·,1)), integrating over (α,β)\nand taking the real parts, then using by parts integration, w e get\n|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)|2+|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)|2\n=−/integraldisplayβ\nαh′|κ2vx+δ1zx+δ2ηx(·,1)|2dx+2Re/integraldisplayβ\nαh(f5−iλz)(κ2vx+δ1zx+δ2ηx(·,1))dx,\n30WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nconsequently, we have\n(3.44)|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)|2+|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)|2\n≤Ch′/integraldisplayβ\nα|κ2vx+δ1zx+δ2ηx(·,1)|2dx+2Ch/integraldisplayβ\nα|f5||κ2vx+δ1zx+δ2ηx(·,1)|dx\n+2Ch|λ|/integraldisplayβ\nα|z||κ2vx+δ1zx+δ2ηx(·,1)|dx.\nOn the other hand, for all s2∈Randr2∈R∗\n+, we have\n\n\n2Ch|f5|||κ2vx+δ1zx+δ2ηx(·,1)| ≤Ch|f5|2+Ch|κ2vx+δ1zx+δ2ηx(·,1)|2,\n2Ch|λ||z|||κ2vx+δ1zx+δ2ηx(·,1)| ≤|λ|3\n2−s2\nr2|z|2+r2C2\nh|λ|1\n2+s2||κ2vx+δ1zx+δ2ηx(·,1)|2.\nInserting the above equation in ( 3.44), we get\n|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)|2+|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)|2≤|λ|3\n2−s2\nr2/integraldisplayβ\nα|z|2dx\n+/parenleftBig\nCh′+Ch+r2C2\nh|λ|1\n2+s2/parenrightBig/bracketleftBigg/integraldisplayβ\nα||κ2vx+δ1zx+δ2ηx(·,1)|2dx+/integraldisplayβ\nα|f5|2dx/bracketrightBigg\n.\nInserting ( 3.39) in the above equation, then using the fact that\n/integraldisplayβ\nα|f5|2dx≤ /ba∇dblF/ba∇dbl2\nH2≤/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nwe get\n|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)|2+|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)|2\n≤|λ|3\n2−s2\nr2/integraldisplayβ\nα|z|2dx+(K6+1)/parenleftBig\nCh′+Ch+r2C2\nh|λ|1\n2+s2/parenrightBig/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nhence we get ( 3.42). /square\nLemma 3.6. Under hypothesis (H1), for alls1, s2∈R, the solution (u,v,w,y,z,φ,η (·,·))∈D(A2)of Equation\n(3.26)satisfies the following estimation\n(3.45)/integraldisplayα\n0|y|2dx+κ1/integraldisplayα\n0|ux|2dx+/integraldisplayL\nβ|φ|2dx+κ3/integraldisplayL\nβ|wx|2dx\n≤K9/parenleftBig\n1+|λ|1\n2−s1+|λ|3\n2−s2/parenrightBig/integraldisplayβ\nα|z|2dx\n+K10/parenleftBig\n1+|λ|−1\n2+s1+|λ|1\n2+s2/parenrightBig/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nwhere\nK9= max/bracketleftbig\nmax(α,L−β),max/parenleftbig\nκ−1\n1α,κ−1\n3(L−β)/parenrightbig/bracketrightbig\nmax(Ch′,1)\nand\nK10= 2max/bracketleftBig\nK7C2\nhmax(α,L−β), K8max/parenleftbig\nκ−1\n1α,κ−1\n3(L−β)/parenrightbig\nmax/parenleftbig\nCh′+Ch,C2\nh/parenrightbig\n,4/parenleftBig\nακ−1\n2\n1+(L−β)κ−1\n2\n3/parenrightBig/bracketrightBig\n.\nProof. First, multiplying Equation ( 3.30) by2xux, integrating over (0,α)and taking the real parts, then using\nby parts integration, we get\n(3.46) 2Re/braceleftbigg\niλ/integraldisplayα\n0xyuxdx/bracerightbigg\n+κ1/integraldisplayα\n0|ux|2dx=κ1α|ux(α)|2+2Re/braceleftbigg/integraldisplayα\n0xf4uxdx/bracerightbigg\n.\nFrom ( 3.27), we deduce that\niλux=−yx−(f1)x.\n31WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nInserting the above result in ( 3.46), then using by parts integration, we get\n/integraldisplayα\n0|y|2dx+κ1/integraldisplayα\n0|ux|2dx=κ1α|ux(α)|2+α|y(α)|2+2Re/braceleftbigg/integraldisplayα\n0xf4uxdx/bracerightbigg\n+2Re/braceleftbigg/integraldisplayα\n0xy(f1)xdx/bracerightbigg\n,\nconsequently, we get\n(3.47)/integraldisplayα\n0|y|2dx+κ1/integraldisplayα\n0|ux|2dx≤κ1α|ux(α)|2+α|y(α)|2+2α/parenleftbigg/integraldisplayα\n0|ux||f4|dx+/integraldisplayα\n0|y||(f1)x|dx/parenrightbigg\n.\nUsing Cauchy Schwarz inequality, we get\n(3.48)/integraldisplayα\n0|ux||f4|dx+/integraldisplayα\n0|y||(f1)x|dx≤2κ−1\n2\n1/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2.\nOn the other hand, since U∈D(A2), we have\n(3.49)/braceleftBigg\nκ1|ux(α)|=|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)|,|y(α)|=|z(α)|,\nκ3|wx(β)|=|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)|,|φ(β)|=|z(β)|.\nSubstituting ( 3.48) and the boundary conditions ( 3.49) atx=αin (3.47), we obtain\n(3.50)/integraldisplayα\n0|y|2dx+κ1/integraldisplayα\n0|ux|2dx≤ακ−1\n1|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)|2+α|z(α)|2+4ακ−1\n2\n1/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2.\nNext, by the same way, we multiply equation ( 3.32) by2(x−L)wxand integrate over (β,L),then we use ( 3.29).\nArguing in the same way as ( 3.47), we get\n/integraldisplayL\nβ|φ|2dx+κ3/integraldisplayL\nβ|wx|2dx≤κ3(L−β)|wx(β)|2+(L−β)|φ(β)|2+4(L−β)κ−1\n2\n3/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2.\nSubstituting the boundary conditions ( 3.49) atx=βin the above equation, we obtain\n/integraldisplayL\nβ|φ|2dx+κ3/integraldisplayL\nβ|wx|2dx≤(L−β)/bracketleftBig\nκ−1\n3|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)|2+|z(β)|2+4κ−1\n2\n3/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2/bracketrightBig\n.\nNow, adding the above equation and ( 3.50), we get\n/integraldisplayα\n0|y|2dx+κ1/integraldisplayα\n0|ux|2dx+/integraldisplayL\nβ|φ|2dx+κ3/integraldisplayL\nβ|wx|2dx≤max(α,L−β)/parenleftbig\n|z(α)|2+|z(β)|2/parenrightbig\n+max/parenleftbig\nκ−1\n1α,κ−1\n3(L−β)/parenrightbig/parenleftbig\n|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)|2+|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)|2/parenrightbig\n+4/parenleftBig\nακ−1\n2\n1+(L−β)κ−1\n2\n3/parenrightBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2.\nInserting ( 3.41) and ( 3.42) withr1=r2= 1in the above estimation, we get\n/integraldisplayα\n0|y|2dx+κ1/integraldisplayα\n0|ux|2dx+/integraldisplayL\nβ|φ|2dx+κ3/integraldisplayL\nβ|wx|2dx\n≤max/bracketleftbig\nmax(α,L−β),max/parenleftbig\nκ−1\n1α,κ−1\n3(L−β)/parenrightbig/bracketrightbig/parenleftBig\nCh′+|λ|1\n2−s1+|λ|3\n2−s2/parenrightBig/integraldisplayβ\nα|z|2dx\n+max(α,L−β)K7C2\nh|λ|−1\n2+s1/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n+max/parenleftbig\nκ−1\n1α,κ−1\n3(L−β)/parenrightbig\nK8/parenleftBig\nCh′+Ch+C2\nh|λ|1\n2+s2/parenrightBig/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n+4/parenleftBig\nακ−1\n2\n1+(L−β)κ−1\n2\n3/parenrightBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2.\nIn the above equation, using the fact that\n\n\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2≤/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2≤/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\n32WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nwe get\n/integraldisplayα\n0|y|2dx+κ1/integraldisplayα\n0|ux|2dx+/integraldisplayL\nβ|φ|2dx+κ3/integraldisplayL\nβ|wx|2dx\n≤max/bracketleftbig\nmax(α,L−β),max/parenleftbig\nκ−1\n1α,κ−1\n3(L−β)/parenrightbig/bracketrightbig\nmax(Ch′,1)/parenleftBig\n1+|λ|1\n2−s1+|λ|3\n2−s2/parenrightBig/integraldisplayβ\nα|z|2dx\n+K7C2\nhmax(α,L−β)|λ|−1\n2+s1/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n+K8max/parenleftbig\nκ−1\n1α,κ−1\n3(L−β)/parenrightbig\nmax/parenleftbig\nCh′+Ch,C2\nh/parenrightbig/parenleftBig\n1+|λ|1\n2+s2/parenrightBig/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n+4/parenleftBig\nακ−1\n2\n1+(L−β)κ−1\n2\n3/parenrightBig/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nhence, we get ( 3.45). /square\nLemma 3.7. Under hypothesis (H1), for alls1, s2, s3∈Randr1, r2,r3∈R∗\n+, the solution (u,v,w,y,z,φ,η (·,·))∈\nD(A2)of Equation (3.26)satisfies the following estimations\n(3.51)/ba∇dblU/ba∇dbl2\nH2≤K11/parenleftBig\n1+|λ|1\n2−s1+|λ|3\n2−s2/parenrightBig/integraldisplayβ\nα|z|2dx\n+K12/parenleftBig\n1+|λ|1\n2+s2+|λ|−1\n2+s1/parenrightBig/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\nand\n(3.52) R1,λ/integraldisplayβ\nα|z|2dx≤K13R2,λ/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nsuch that\n\nR1,λ= 1−1\n2/parenleftbigg|λ|−s3−s2\nr2r3+r3|λ|−s1+s3\nr1+r3Ch′|λ|s3−1\n2/parenrightbigg\n,\nR2,λ=C2\nh|λ|−1(r1r3|λ|s1+s3+r2r−1\n3|λ|s2−s3)+r−1\n3|λ|−s3−3\n2(Ch′+Ch)+|λ|−1,\nwhere\nK11=K9+1, K12=K10+max(κ2K4,τ|δ2|K5), K13=max(K3+K6+2,max(K7,K8))\n2.\nProof. First, from ( 3.37), (3.38) and ( 3.45), we get\n/ba∇dblU/ba∇dbl2\nH2≤/parenleftBig\nK9+1+K9/parenleftBig\n|λ|1\n2−s1+|λ|3\n2−s2/parenrightBig/parenrightBig/integraldisplayβ\nα|z|2dx\n+K10/parenleftBig\n1+|λ|1\n2+s2+|λ|−1\n2+s1/parenrightBig/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n+max(κ2K4,τ|δ2|K5)/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nhence we get ( 3.51). Next, multiplying ( 3.31) by−iλ−1zand integrating over (α,β),then taking the real part,\nthen using by parts integration, we get\n/integraldisplayβ\nα|z|2dx=−Re/braceleftBigg\niλ−1/integraldisplayβ\nαf5zdx/bracerightBigg\n+Re/braceleftBigg\niλ−1/integraldisplayβ\nα(κ2v+δ1z+δ2η(·,1))xzxdx/bracerightBigg\n−Re/braceleftbig\niλ−1(κ2vx(β)+δ1zx(β)+δ2ηx(β,1))z(β)/bracerightbig\n+Re/braceleftbig\niλ−1(κ2vx(α)+δ1zx(α)+δ2ηx(α,1))z(α)/bracerightbig\n,\nconsequently,\n(3.53)/integraldisplayβ\nα|z|2dx≤ |λ|−1/integraldisplayβ\nα|f5||z|dx+|λ|−1/integraldisplayβ\nα|κ2vx+δ1zx+δ2ηx(·,1)||zx|dx\n+|λ|−1|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)||z(β)|+|λ|−1|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)||z(α)|.\nUsing Cauchy Schwarz inequality, we have\n(3.54) |λ|−1/integraldisplayβ\nα|f5||z|dx≤ |λ|−1/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2≤ |λ|−1/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n.\n33WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nFrom ( 3.36) and ( 3.39), we get\n/integraldisplayβ\nα|κ2vx+δ1zx+δ2ηx(·,1)||zx|dx\n≤1\n2/integraldisplayβ\nα|zx|2dx+1\n2/integraldisplayβ\nα|κ2vx+δ1zx+δ2ηx(·,1)|2dx\n≤K3\n2/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+K6\n2/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n≤K3+K6\n2/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n.\nInserting ( 3.54) and the above estimation in ( 3.53), we get\n(3.55)/integraldisplayβ\nα|z|2dx≤K3+K6+2\n2|λ|−1/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n+|λ|−1|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)||z(β)|+|λ|−1|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)||z(α)|.\nNow, for all s3∈R,r3∈R∗\n+and forζ=αorζ=β, we get\n|λ|−1|κ2vx(ζ)+δ1zx(ζ)+δ2ηx(ζ,1)||z(ζ)| ≤r3|λ|s3−1\n2\n2|z(ζ)|2+|λ|−s3−3\n2\n2r3|κ2vx(ζ)+δ1zx(ζ)+δ2ηx(ζ,1)|2.\nFrom the above inequality, we get\n|λ|−1|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)||z(β)|+|λ|−1|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)||z(α)|\n≤|λ|−s3−3\n2\n2r3/parenleftBig\n|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)|2+|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)|2/parenrightBig\n+r3|λ|s3−1\n2\n2/parenleftbig\n|z(α)|2+|z(β)|2/parenrightbig\n.\nInserting ( 3.41) and ( 3.42) in the above estimation, we obtain\n|λ|−1|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)||z(β)|+|λ|−1|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)||z(α)|\n≤1\n2/parenleftbigg|λ|−s3−s2\nr2r3+r3|λ|−s1+s3\nr1+r3Ch′|λ|s3−1\n2/parenrightbigg/integraldisplayβ\nα|z|2dx\n+max(K7,K8)\n2R3,λ/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nwhere\nR3,λ=C2\nh|λ|−1(r1r3|λ|s1+s3+r2r−1\n3|λ|s2−s3)+r−1\n3|λ|−s3−3\n2(Ch′+Ch).\nFinally, inserting the above equation in ( 3.55), we get\n/bracketleftbigg\n1−1\n2/parenleftbigg|λ|−s3−s2\nr2r3+r3|λ|−s1+s3\nr1+r3Ch′|λ|s3−1\n2/parenrightbigg/bracketrightbigg/integraldisplayβ\nα|z|2dx\n≤max(K3+K6+2,max(K7,K8))\n2/parenleftbig\nR3,λ+|λ|−1/parenrightbig/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nhence we get ( 3.52). /square\nProof of Proposition 3.3.We now divide the proof in two steps:\nStep 1. In this step, we prove the asymptotic behavior estimate of ( 3.34). Takings3=s1=−s2=1\n2,\nr1=1\nCh′, r2= 9Ch′andr3=1\n3Ch′in Lemma 3.7, we get\n\n\n1\n2/integraldisplayβ\nα|z|2dx≤K13λ−4/parenleftbiggC2\nh\n3C2\nh′λ2+|λ|+3Ch′/parenleftbig\n9C2\nhCh′+Ch+Ch′/parenrightbig/parenrightbigg/parenleftbig\nλ2+1/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\n/ba∇dblU/ba∇dbl2\nH2≤2K11/parenleftbig\nλ2+1/parenrightbig/integraldisplayβ\nα|z|2dx+3K12λ−2/parenleftbig\nλ2+1/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n.\n34WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nIn the above equation, using the fact that\nC2\nh\n3C2\nh′λ2+|λ|+3Ch′/parenleftbig\n9C2\nhCh′+Ch+Ch′/parenrightbig\n≤max/parenleftbiggC2\nh\n3C2\nh′,3Ch′/parenleftbig\n9C2\nhCh′+Ch+Ch′/parenrightbig\n,1/parenrightbigg/parenleftbig\nλ2+|λ|+1/parenrightbig\n≤max/parenleftbiggC2\nh\n3C2\nh′,3Ch′/parenleftbig\n9C2\nhCh′+Ch+Ch′/parenrightbig\n,1/parenrightbigg\n(|λ|+1)2\nand\nλ2+1≤(|λ|+1)2,\nwe get\n(3.56)/integraldisplayβ\nα|z|2dx≤K14λ−4(|λ|+1)4/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\nand\n(3.57) /ba∇dblU/ba∇dbl2\nH2≤2K11(|λ|+1)2/integraldisplayβ\nα|z|2dx+3K12λ−2(|λ|+1)2/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nwhere\nK14= 2K13max/parenleftbiggC2\nh\n3C2\nh′,3Ch′/parenleftbig\n9C2\nhCh′+Ch+Ch′/parenrightbig\n,1/parenrightbigg\n.\nInserting ( 3.56) in (3.57), we get\n/ba∇dblU/ba∇dbl2\nH2≤/parenleftBig\n2K11K14(|λ|+1)4+3K12λ2/parenrightBig\nλ−4(|λ|+1)2/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\n≤(2K11K14+3K12)λ−4(|λ|+1)6/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nhence we get ( 3.34).\nStep 2. In this step, we prove the asymptotic behavior estimate of ( 3.35). LetM∈R∗such that |λ| ≥M >0.\nIn this case, taking s1=s2=s3= 0,r1=3√\nM\n2Ch′,r2=3Ch′√\nMandr3=√\nM\n2Ch′in Lemma 3.7, we get\n(3.58)/ba∇dblU/ba∇dbl2\nH2≤K11|λ|3\n2/parenleftBig\n1+|λ|−1+|λ|−3\n2/parenrightBig/integraldisplayβ\nα|z|2dx\n+K12|λ|1\n2/parenleftBig\n1+|λ|−1\n2+|λ|−1/parenrightBig/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\nand\n(3.59)1\n2/parenleftBigg\n1−√\nM\n2|λ|1\n2/parenrightBigg/integraldisplayβ\nα|z|2dx\n≤K13|λ|−1/bracketleftBigg\n1+3C2\nhM\n4C2\nh′+6C2\nhC2\nh′\nM+2Ch′(Ch+Ch′)|λ|−1\n2√\nM/bracketrightBigg\n/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n.\nFrom the fact that |λ| ≥M, we get\n1\n2/parenleftBigg\n1−√\nM\n2|λ|1\n2/parenrightBigg\n≥1\n4>0.\nTherefore, from the above inequality and ( 3.59), we get\n(3.60)/integraldisplayβ\nα|z|2dx≤K15|λ|−1/parenleftBigg\n1+M+1\nM+|λ|−1\n2√\nM/parenrightBigg\n/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n.\nwhere\nK15= 4K13max/bracketleftbigg\n1,3C2\nh\n4C2\nh′,6C2\nhC2\nh′,2Ch′(Ch+Ch′)/bracketrightbigg\n.\n35WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nIn Estimation ( 3.59), using the fact that\n\n\n1+M+1\nM≤/parenleftbigg√\nM+1√\nM/parenrightbigg2\n,1√\nM≤/parenleftbigg√\nM+1√\nM/parenrightbigg2\n,\n1+λ−2≤/parenleftBig\n1+|λ|−1\n2/parenrightBig4\n.\nwe get\n(3.61)/integraldisplayβ\nα|z|2dx≤K15|λ|−1/parenleftbigg√\nM+1√\nM/parenrightbigg2/parenleftBig\n1+|λ|−1\n2/parenrightBig5/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n.\nInserting ( 3.61) in (3.58), then using the fact that\n1+|λ|−1+|λ|−3\n2≤/parenleftBig\n1+|λ|−1\n2/parenrightBig3\n,/parenleftBig\n1+|λ|−1\n2+|λ|−1/parenrightBig/parenleftbig\n1+λ−2/parenrightbig\n≤/parenleftBig\n1+|λ|−1\n2/parenrightBig6\n≤/parenleftBig\n1+|λ|−1\n2/parenrightBig8\n,\nwe get\n/ba∇dblU/ba∇dbl2\nH2≤max(K11K15,K12)|λ|1\n2/parenleftBig\n1+|λ|−1\n2/parenrightBig8/bracketleftBigg/parenleftbigg√\nM+1√\nM/parenrightbigg2\n+1/bracketrightBigg/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n≤2max(K11K15,K12)|λ|1\n2/parenleftBig\n1+|λ|−1\n2/parenrightBig8/parenleftbigg√\nM+1√\nM/parenrightbigg2/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nhence we get estimation of ( 3.35). The proof is thus complete. /square\nProof of Theorem 3.2.First, we will prove condition ( 3.24). Remark that it has been proved in Proposition\n3.1that0∈ρ(A2).Now, suppose ( 3.24) is not true, then there exists ω∈R∗such thatiω/\\e}atio\\slash∈ρ(A2). According\nto Lemma A.3and Remark A.4, there exists\n{(λn,Un:= (un,vn,wn,yn,zn,φn,ηn(·,·)))}n≥1⊂R∗×D(A2),\nwithλn→ωasn→ ∞,|λn|<|ω|and/ba∇dblUn/ba∇dblH2= 1, such that\n(iλnI−A2)Un=Fn:= (f1,n,f2,n,f3,n,f4,n,f5,6,f6,n,f7,n(·,·))→0inH2,asn→ ∞.\nWe will check condition ( 3.24) by finding a contradiction with /ba∇dblUn/ba∇dblH2= 1such as/ba∇dblUn/ba∇dblH2→0.According to\nEquation ( 3.34) in Proposition 3.3withU=Un, F=Fnandλ=λn, we obtain\n0≤ /ba∇dblUn/ba∇dbl2\nH2≤K1|λn|−4(|λn|+1)6/parenleftBig\n/ba∇dblFn/ba∇dblH2/ba∇dblUn/ba∇dblH2+/ba∇dblFn/ba∇dbl2\nH2/parenrightBig\n,\nasn→ ∞,we get/ba∇dblUn/ba∇dbl2\nH2→0,which contradicts /ba∇dblUn/ba∇dblH2= 1. Thus, condition ( 3.24) is holds true. Next, we\nwill prove condition ( 3.25) by a contradiction argument. Suppose there exists\n{(λn,Un:= (un,vn,wn,yn,zn,φn,ηn(·,·)))}n≥1⊂R∗×D(A2),\nwith|λn| ≥1without affecting the result, such that |λn| →+∞,and/ba∇dblUn/ba∇dblH2= 1and there exists a sequence\nGn:= (g1,n,g2,n,g3,n,g4,n,g5,6,g6,n,g7,n(·,·))∈ H2, such that\n(iλnI−A2)Un=λ−1\n2nGn→0inH2.\nWe will check condition ( 3.25) by finding a contradiction with /ba∇dblUn/ba∇dblH2= 1such as/ba∇dblUn/ba∇dblH2=o(1).According\nto Equation ( 3.35) in Proposition 3.3withU=Un, F=λ−1\n2Gn, λ=λnandM= 1, we get\n/ba∇dblUn/ba∇dbl2\nH2≤4K2/parenleftBig\n1+|λn|−1\n2/parenrightBig8/parenleftBig\n/ba∇dblGn/ba∇dblH2/ba∇dblUn/ba∇dblH2+|λn|−1\n2/ba∇dblGn/ba∇dbl2\nH2/parenrightBig\n,\nas|λn| → ∞,we get/ba∇dblUn/ba∇dbl2\nH2=o(1),which contradicts /ba∇dblUn/ba∇dblH2= 1. Thus, condition ( 3.25) is holds true. The\nresult follows from Theorem A.5(part (ii)). The proof is thus complete. /square\nRemark 3.8. In the case that α= 0andβ/\\e}atio\\slash=Lorβ=Landα/\\e}atio\\slash= 0, we can proceed similar to the proof of\nTheorem 3.2to check that the energy of System (3.1)-(3.7)decays polynomially of order t−4. /square\n36WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nAppendix A.Notions of stability and theorems used\nWe introduce here the notions of stability that we encounter in this work.\nDefinition A.1. Assume that Ais the generator of a C 0-semigroup of contractions/parenleftbig\netA/parenrightbig\nt≥0on a Hilbert space\nH. TheC0-semigroup/parenleftbig\netA/parenrightbig\nt≥0is said to be\n1.strongly stable if\nlim\nt→+∞/ba∇dbletAx0/ba∇dblH= 0,∀x0∈H;\n2.exponentially (or uniformly) stable if there exist two posit ive constants Mandǫsuch that\n/ba∇dbletAx0/ba∇dblH≤Me−ǫt/ba∇dblx0/ba∇dblH,∀t>0,∀x0∈H;\n3.polynomially stable if there exists two positive constants Candαsuch that\n/ba∇dbletAx0/ba∇dblH≤Ct−α/ba∇dblAx0/ba∇dblH,∀t>0,∀x0∈D(A).\n/square\nFor proving the strong stability of the C0-semigroup/parenleftbig\netA/parenrightbig\nt≥0, we will recall two methods, the first result\nobtained by Arendt and Batty in [ 8].\nTheorem A.2 (Arendt and Batty in [ 8]).Assume that Ais the generator of a C 0−semigroup of contractions/parenleftbig\netA/parenrightbig\nt≥0on a Hilbert space H. IfAhas no pure imaginary eigenvalues and σ(A)∩iRis countable, where\nσ(A)denotes the spectrum of A, then theC0-semigroup/parenleftbig\netA/parenrightbig\nt≥0is strongly stable. /square\nThe second one is a classical method based on Arendt and Batty theorem and the contradiction argument\n(see page 25 in [ 29]).\nLemma A.3. Assume that Ais the generator of a C 0−semigroup of contractions/parenleftbig\netA/parenrightbig\nt≥0on a Hilbert space\nH. Furthermore, Assume that 0∈ρ(A).If there exists ω∈R∗, such that iω/\\e}atio\\slash∈ρ(A), then\n(A.1)/braceleftBig\niλsuch thatλ∈R∗and|λ|</vextenddouble/vextenddoubleA−1/vextenddouble/vextenddouble−1≤ |ω|/bracerightBig\n⊂ρ(A)and sup\n|λ|</bardblA−1/bardbl−1≤|ω|/vextenddouble/vextenddouble/vextenddouble(iλ−A)−1/vextenddouble/vextenddouble/vextenddouble=∞.\nProof. Since0∈ρ(A), for any real number λwith|λ|</vextenddouble/vextenddoubleA−1/vextenddouble/vextenddouble−1, we deduce from the contraction mapping\ntheorem that operator iλI−A=−A−1(I−iλA)is invertible. Therefore, we get\n/braceleftBig\niλsuch thatλ∈R∗and|λ|</vextenddouble/vextenddoubleA−1/vextenddouble/vextenddouble−1/bracerightBig\n⊂ρ(A).\nIn addition, if there exists ω∈R∗, such that iω/\\e}atio\\slash∈ρ(A), then/vextenddouble/vextenddoubleA−1/vextenddouble/vextenddouble−1≤ |w|, hence we get the first estimation\nof (A.1). Next, since iω/\\e}atio\\slash∈ρ(A), then for all |λ|</vextenddouble/vextenddoubleA−1/vextenddouble/vextenddouble−1≤ |ω|the operator\niωI−A= (iλI−A)/parenleftBig\nI+i(ω−λ)(iλI−A)−1/parenrightBig\nis not invertible, hence from the contraction mapping theor em we deduce\n/vextenddouble/vextenddouble/vextenddouble(iλI−A)−1/vextenddouble/vextenddouble/vextenddouble≥1\n|ω−λ|,∀ |λ|</vextenddouble/vextenddoubleA−1/vextenddouble/vextenddouble−1≤ |ω|,\ntherefore\nsup\n|λ|</bardblA−1/bardbl−1≤|ω|/vextenddouble/vextenddouble/vextenddouble(iλI−A)−1/vextenddouble/vextenddouble/vextenddouble≥sup\n|λ|</bardblA−1/bardbl−1≤|ω|1\n|ω−λ|=∞,\nhence we get second estimation of ( A.1). /square\nRemark A.4. Condition (A.1)turns out that there exists {(λn,Un)}n≥1⊂R∗×D(A),withλn→ωas\nn→ ∞,|λn|<|ω|and/ba∇dblUn/ba∇dblH2= 1, such that\n(iλnI−A)Un=Fn→0inH, asn→ ∞.\nThen, we will check condition iR⊂ρ(A)by finding a contradiction with /ba∇dblUn/ba∇dblH= 1such as/ba∇dblUn/ba∇dblH=o(1)./square\nWe now recall the following standard result which is stated i n a comparable way (see [ 22,38] for part (i) and\n[9,11,9] for part (ii)).\n37WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nTheorem A.5. Assume that Ais the generator of a strongly continuous semigroup of contr actions/parenleftbig\netA/parenrightbig\nt≥0\nonH. Assume that iR⊂ρ(A).Then;\n(i)The semigroup etAis exponentially stable if and only if\nlim\nλ→∞/vextenddouble/vextenddouble/vextenddouble(iλI−A)−1/vextenddouble/vextenddouble/vextenddouble<∞.\n(ii)The semigroup etAis polynomially stable of order α>0if and only if\nlim\nλ→∞|λ|−1\nα/vextenddouble/vextenddouble/vextenddouble(iλ−A)−1/vextenddouble/vextenddouble/vextenddouble<∞.\n/square\nReferences\n[1] M. Alves, J. M. Rivera, M. Sepúlveda, and O. V. Villagrán. The Lack of Exponential Stability in Certain Transmission\nProblems with Localized Kelvin–Voigt Dissipation .SIAM Journal on Applied Mathematics , 74(2):345–365, Jan. 2014. 6\n[2] M. Alves, J. M. Rivera, M. Sepúlveda, O. V. Villagrán, and M. Z. Garay. The asymptotic behavior of the linear transmission\nproblem in viscoelasticity .Mathematische Nachrichten , 287(5-6):483–497, Oct. 2013. 6\n[3] K. Ammari and B. Chentouf. Asymptotic behavior of a delayed wave equation without disp lacement term .Zeitschrift für\nangewandte Mathematik und Physik , 68(5), Sept. 2017. 4\n[4] K. Ammari and B. Chentouf. On the exponential and polynomial convergence for a delayed wave equation without displace-\nment.Applied Mathematics Letters , 86:126–133, Dec. 2018. 4\n[5] K. Ammari, Z. Liu, and F. Shel. Stability of the wave equations on a tree with local Kelvin–V oigt damping .Semigroup Forum ,\nSept. 2019. 6\n[6] K. Ammari, S. Nicaise, and C. Pignotti. Feedback boundary stabilization of wave equations with int erior delay .Systems &\nControl Letters , 59(10):623–628, Oct. 2010. 4,5\n[7] K. Ammari, S. Nicaise, and C. Pignotti. Stability of an abstract-wave equation with delay and a Kelv in–Voigt damping .\nAsymptotic Analysis , 95(1-2):21–38, Oct. 2015. 6\n[8] W. Arendt and C. J. K. Batty. Tauberian theorems and stability of one-parameter semigro ups.Trans. Amer. Math. Soc. ,\n306(2):837–852, 1988. 37\n[9] C. J. K. 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Spectrum and Stability for Elastic Systems with Global or Lo cal Kelvin–Voigt Damping .SIAM\nJournal on Applied Mathematics , 59(2):651–668, jan 1998. 6\n38WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\n[27] K. Liu and B. Rao. Exponential stability for the wave equations with local Kel vin–Voigt damping .Zeitschrift für angewandte\nMathematik und Physik , 57(3):419–432, Jan. 2006. 6\n[28] Z. Liu and Q. Zhang. Stability of a String with Local Kelvin–Voigt Damping and No nsmooth Coefficient at Interface .SIAM\nJournal on Control and Optimization , 54(4):1859–1871, Jan. 2016. 6\n[29] Z. Liu and S. Zheng. Semigroups associated with dissipative systems , volume 398 of Chapman &Hall/CRC Research Notes\nin Mathematics . Chapman &Hall/CRC, Boca Raton, FL, 1999. 37\n[30] S. A. Messaoudi, A. Fareh, and N. Doudi. 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Stability of the heat and of the wave equations with boundary time-varying delays .\nDiscrete and Continuous Dynamical Systems - Series S , 2(3):559–581, June 2009. 5\n[35] H. P. Oquendo. Frictional versus Kelvin–Voigt damping in a transmission p roblem .Mathematical Methods in the Applied\nSciences , 40(18):7026–7032, July 2017. 6\n[36] A. Pazy. Semigroups of linear operators and applications to partial differential equations , volume 44 of Applied Mathematical\nSciences . Springer-Verlag, New York, 1983. 9,11,28\n[37] C. Pignotti. A note on stabilization of locally damped wave equations wit h time delay .Systems &Control Letters , 61(1):92–97,\nJan. 2012. 4,5\n[38] J. Prüss. On the spectrum of C0-semigroups .Trans. Amer. Math. Soc. , 284(2):847–857, 1984. 37\n[39] J. E. M. Rivera, O. V. Villagran, and M. Sepulveda. Stability to localized viscoelastic transmission problem .Communications\nin Partial Differential Equations , 43(5):821–838, May 2018. 6\n[40] L. Tebou. A constructive method for the stabilization of the wave equa tion with localized Kelvin–Voigt damping .Comptes\nRendus Mathematique , 350(11-12):603–608, June 2012. 6\n[41] J.-M. Wang, B.-Z. Guo, and M. Krstic. Wave Equation Stabilization by Delays Equal to Even Multipl es of the Wave Propa-\ngation Time .SIAM Journal on Control and Optimization , 49(2):517–554, Jan. 2011. 4\n[42] Y. Xie and G. Xu. 10(3):557–579, 2017. 4\n[43] Y. Xie and G. Xu. Exponential stability of 1-d wave equation with the boundar y time delay based on the interior control .\nDiscrete &Continuous Dynamical Systems - S , 10(3):557–579, 2017. 4\n[44] G. Q. Xu, S. P. Yung, and L. K. Li. Stabilization of wave systems with input delay in the bounda ry control .ESAIM: Control,\nOptimisation and Calculus of Variations , 12(4):770–785, Oct. 2006. 3\n[45] Q. Zhang. Polynomial decay of an elastic/viscoelastic waves interac tion system .Zeitschrift für angewandte Mathematik und\nPhysik , 69(4), June 2018. 6\n39"    },    {        "title": "1909.09085v1.Magnetization_dynamics_of_the_compensated_ferrimagnet__Mn__2_Ru__x_Ga_.pdf",        "content": "Magnetisation dynamics of the compensated ferrimagnet Mn 2RuxGa\nG. Bon\fglio\nRadboud University, Institute for Molecules and Materials, 6525 AJ Nijmegen, The Netherlands\nK. Rode, K. Siewerska, J. Besbas, G. Y. P. Atcheson, P. Stamenov, and J.M.D. Coey\nCRANN, AMBER and School of Physics, Trinity College Dublin, Ireland\nA.V. Kimel, Th. Rasing, and A. Kirilyuk\nRadboud University, Institute for Molecules and Materials, 6525 AJ Nijmegen, The Netherlands and\nFELIX Laboratory, Radboud University, Toernooiveld 7c, 6525 ED Nijmegen, The Netherlands\nHere we study both static and time-resolved dynamic magnetic properties of the compensated\nferrimagnet Mn 2RuxGa from room temperature down to 10 K, thus crossing the magnetic compen-\nsation temperature TM. The behaviour is analysed with a model of a simple collinear ferrimagnet\nwith uniaxial anisotropy and site-speci\fc gyromagnetic ratios. We \fnd a maximum zero-applied-\n\feld resonance frequency of \u0018160 GHz and a low intrinsic Gilbert damping \u000b\u00180:02, making it a\nvery attractive candidate for various spintronic applications.\nI. INTRODUCTION\nAntiferromagnets (AFM) and compensated ferrimag-\nnets (FiM) have attracted a lot of attention over the last\ndecade due to their potential use in spin electronics1,2.\nDue to their lack of a net magnetic moment, they are\ninsensitive to external \felds and create no demagnetis-\ning \felds of their own. In addition, their spin dynamics\nreach much higher frequencies than those of their ferro-\nmagnetic (FM) counterparts due to the contribution of\nthe exchange energy in the magnetic free energy3.\nDespite these clear advantages, AFMs are scarcely\nused apart from uni-directional exchange biasing rela-\ntively in spin electronic applications. This is because\nthe lack of net moment also implies that there is no\ndirect way to manipulate their magnetic state. Fur-\nthermore, detecting their magnetic state is also compli-\ncated and is usually possible only by neutron di\u000braction\nmeasurements4, or through interaction with an adjacent\nFM layer5.\nCompensated, metallic FiMs provide an interesting al-\nternative as they combine the high-speed advantages of\nAFMs with those of FMs, namely, the ease to manipu-\nlate their magnetic state. Furthermore, it has been shown\nthat such materials are good candidates for the emerging\n\feld of All-Optical Switching (AOS) in which the mag-\nnetic state is solely controlled by a fast laser pulse6{8.\nA compensated, half-metallic ferrimagnet was \frst en-\nvisaged by van Leuken and de Groot9. In their model\ntwo magnetic ions in crystallographically di\u000berent po-\nsitions couple antiferromagnetically and perfectly com-\npensate each-other, but only one of the two contributes\nto the states at the Fermi energy responsible for elec-\ntronic transport. The \frst experimental realisation of\nthis, Mn 2RuxGa (MRG), was provided by Kurt et al.10.\nMRG crystallises in the XAHeusler structure, space\ngroupF\u001643m, with Mn on the 4 aand 4csites11.\nSubstrate-induced bi-axial strain imposes a slight tetrag-\nonal distortion, which leads to perpendicular magneticanisotropy. Due to the di\u000berent local environment of\nthe two sublattices, the temperature dependence of their\nmagnetic moments di\u000ber, and perfect compensation is\ntherefore obtained at a speci\fc temperature TMthat\ndepends on the Ru concentration xand the degree of\nbiaxial strain. It was previously shown that MRG ex-\nhibits properties usually associated with FMs: a large\nanomalous Hall angle12, that depends only on the mag-\nnetisation of the 4 cmagnetic sublattice13; tunnel magne-\ntoresistance (TMR) of 40 %, a signature of its high spin\npolarisation14, was observed in magnetic tunnel junc-\ntions (MTJs) based on MRG15; and a clear magneto-\noptical Kerr e\u000bect and domain structure, even in the ab-\nsence of a net moment16,17. Strong exchange bias of a\nCoFeB layer by exchange coupling with MRG through\na Hf spacer layer18, as well as single-layer spin-orbit\ntorque19,20showed that MRG combined the qualities of\nFMs and AFMs in spin electronic devices.\nThe spin dynamics in materials where two distinct\nsublattices are subject to di\u000bering internal \felds (ex-\nchange, anisotropy, . . . ) is much richer than that of a\nsimple FM, as previously demonstrated by the obers-\nvation of single-pulse all-optical switching in amorphous\nGdFeCo21,22and very recently in MRG8. Given that the\nmagnetisation of MRG is small, escpecially close to the\ncompensation point, and the related frequency is high,\nnormal ferromagnetic resonance (FMR) spectroscopy is\nunsuited to study their properties. Therefore, we used\nthe all-optical pump-probe technique to characterize the\nresonance frequencies at di\u000berent temperatures in vicin-\nity of the magnetic compensation point. This, together\nwith the simulation of FMR, make it possible to deter-\nmine the e\u000bective g-factors, the anisotropy constants and\ntheir evolution across the compensation point. We found,\nin particular, that our ferrimagnetic half-metallic Heusler\nalloy has resonance frequency up to 160 GHz at zero-\feld\nand a relatively low Gilbert damping.arXiv:1909.09085v1  [cond-mat.mtrl-sci]  19 Sep 20192\nFIG. 1. Net moment measured by magnetometry and coercive\n\feld measured by static Faraday e\u000bect. The upturn of the net\nmoment below T\u001850 K is due to paramagnetic impurities\nin the MgO substrate. TMis indicated by the vertical dotted\nline. As expected the maximum available applied \feld \u00160H=\n7 T is insu\u000ecient to switch the magnetisation close to TM.\nII. EXPERIMENTAL DETAILS\nThin \flm samples of MRG were grown in a `Sham-\nrock' sputter deposition cluster with a base pressure of\n2\u000210\u00008Torr on MgO (001) substrates. Further infor-\nmation on sample deposition can be found elsewhere23.\nThe substrates were kept at 250\u000eC, and a protective\n\u00183 nm layer of aluminium oxide was added at room tem-\nperature. Here we focus on a 53 nm thick sample with\nx= 0:55, leading to TM\u001980 K as determined by SQUID\nmagnetometry using a Quantum Design 5 T MPMS sys-\ntem (see FIG. 1). We are able to study the magneto-\noptical properties both above and below TM.\nThe magnetisation dynamics was investigated using an\nall-optical two-colour pump-probe scheme in a Faraday\ngeometry inside a \u00160Hmax= 7 T superconducting coil-\ncryostat assembly. Both pump and probe were produced\nby a Ti:sapphire femtosecond pulsed laser with a cen-\ntral wavelength of 800 nm, a pulse width of 40 fs and\na repetition rate of 1 kHz. After splitting the beam in\ntwo, the high-intensity one was doubled in frequency by\na BBO crystal (giving \u0015= 400 nm) and then used as\nthe pump while the lower intensity 800 nm beam acted\nas the probe pulse. The time delay between the two was\nadjusted by a mechanical delay stage. The pump was\nthen modulated by a synchronised mechanical chopper\nat 500 Hz to improve the signal to noise ratio by lock-in\ndetection. Both pump and probe beams were linearly\npolarized, and with spot sizes on the sample of 150 µm\nand 70 µm, respectively. The pump pulse hit the sample\nat an incidence angle of \u001910\u000e. After interaction with\nthe sample, we split the probe beam in two orthogonally\npolarized parts using a Wollaston prism and detect the\nchanges in transmission and rotation by calculating the\nFIG. 2. Comparison of hysteresis loops obtained by Faraday,\nAHE, and magnetometry recorded at room temperature. The\ntwo former were recorded with the applied \feld perpendicular\nto the sample surface, while for the latter we show results for\nboth \feld applied parallel and perpendicular to the sample.\nsum and the di\u000berence in intensity of the two signals.\nThe external \feld was applied at 75\u000eto the easy axis of\nmagnetization thus tilting the magnetisation away from\nthe axis. Upon interaction with the pump beam the mag-\nnetisation is momentarily drastically changed24and we\nmonitor its return to the initial con\fguration via remag-\nnetisation and then precession through the time depen-\ndent Faraday e\u000bect on the probe pulse.\nThe static magneto-optical properties were examined\nin the same cryostat/magnet assembly.\nIII. RESULTS & DISCUSSION\nA. Static magnetic properties\nWe \frst focus on the static magnetic properties as\nobserved by the Faraday e\u000bect, and compare them to\nwhat is inferred from magnetometry and the anomalous\nHall e\u000bect. In FIG. 2 we present magnetic hysteresis\nloops as recorded using the three techniques. Due to the\nhalf metallic nature of the sample, the magnetotrans-\nport properties depend only on the 4 csublattice. As the\nmain contribution to the MRG dielectric tensor in the\nvisible and near infrared arises from the Drude tail16,\nboth AHE and Faraday e\u000bect probe essentially the same\nproperties (mainly the spin polarised conduction band of\nMRG), hence we observe overlapping loops for the two\ntechniques. Magnetometry, on the other hand, measures\nthe net moment, or to be precise the small di\u000berence\nbetween two large sublattice moments. The 4 asublat-\ntice, which is insigni\fcant for AHE and Faraday here\ncontributes on equal footing. FIG. 2 shows a clear di\u000ber-\nence in shape between the magnetometry loop and the3\nFIG. 3. Time resolved Faraday e\u000bect recorded at T= 290 K\nin applied \felds ranging from 1 T to 7 T. After the initial\ndemagnetisation seen as a sharp increase in the signal at t\u0018\n0 ps, magnetisation is recovered and followed by precession\naround the e\u000bective \feld until fully damped. The lines are\n\fts to the data. The inset shows the experimental geometry\nfurther detailed in the main text.\nAHE or Faraday loops. We highlight here that the ap-\nparent `soft' contribution that shows switching close to\nzero applied \feld, is not a secondary magnetic phase, but\na signature of the small di\u000berences in the \feld-behaviour\nof the two sublattices. We also note that this behaviour\nis a result of the non-collinear magnetic order of MRG.\nA complete analysis of the dynamic properties therefore\nrequires knowledge of the anisotropy constants on both\nsublattices as well as the (at least) three intra and in-\nter sublattice exchange constants. Such an analysis is\nbeyond the scope of this article, and we limit our anal-\nysis to the simplest model of a single, e\u000bective uniaxial\nanisotropy constant Kuin the exchange approximation\nof the ferrimagnet.\nB. Dynamic properties\nWe now turn to the time-resolved Faraday e\u000bect and\nspin dynamics. Time-resolved Faraday e\u000bect data were\nrecorded at \fve di\u000berent temperatures 10 K, 50 K, 100 K,\n200 K and 290 K, with applied \felds ranging from 1 T to\n7 T.\nFIG. 3 shows the \feld-dependence of the Faraday ef-\nfect as a function of the delay between the pump and\nthe probe pulses, recorded at T= 290 K. Negative de-\nlay indicates the probe is hitting the sample before the\npump. After the initial demagnetisation, the magneti-\nsation recovers and starts precessing around the e\u000bec-\ntive \feld which is determined by the anisotropy and the\napplied \feld. The solid lines in FIG. 3 are \fts to the\ndata to extract the period and the damping of the pre-cession in each case. The \ftting model was an expo-\nnentially damped sinusoid with a phase o\u000bset. We note\nthat the apparent evolution of the amplitude and phase\nwith changing applied magnetic \feld is due to the quasi-\nresonance of the spectrum of the precessional motion\nwith the low-frequency components of the convolution\nbetween the envelope of the probe pulse and the phys-\nical relaxation of the system. The latter include both\nelectron-electron and electron-lattice e\u000bects. A rudimen-\ntary model based on a classical oscillator successfully re-\nproduces the main features of the amplitude and phase\nobserved.\nIn two-sublattice FiMs, the gyromagnetic ratios of the\ntwo sublattices are not necessarily the same. This is par-\nticularly obvious in rare-earth/transition metal alloys,\nand is also the case for MRG despite the two sublat-\ntices being chemically similar; they are both Mn. Due\nto the di\u000berent local environment however, the degree\nof charge transfer for the two di\u000bers. This leads to two\ncharacteristic temperatures, a \frst TMwhere the mag-\nnetic moments compensate, and a second TAwhere the\nangular momenta compensate. It can be shown that for\nthe ferromagnetic mode, the e\u000bective gyromagnetic ratio\n\re\u000bcan then be written25\n\re\u000b=M4c(T)\u0000M4a(T)\nM4c(T)=\r4c\u0000M4a(T)=\r4a(1)\nsubscripti= 4a;4cdenotes sublattice i,Mi(T)\nthe temperature-dependent magnetisation, and \rithe\nsublattice-speci\fc gyromagnetic ratio. \re\u000bis related to\nthe e\u000bective g-factor\nge\u000b=\re\u000bh\n\u0016B(2)\nwherehis the Planck constant and \u0016Bthe Bohr magne-\nton.\nThe frequency of the precession is determined by the\ne\u000bective \feld, which can be inferred from the derivative\nof the magnetic free energy density with respect to M.\nFor an external \feld applied at a given \fxed angle with\nrespect to the easy axis this leads to the Smit-Beljers\nformula26\n!FMR =\re\u000bvuut1\nM2ssin2\u001e\"\n\u000e2E\n\u000e\u00122\u000e2E\n\u000e\u001e2\u0000\u0012\u000e2E\n\u000e\u0012\u000e\u001e\u00132#\n(3)\nwhere\u0012and\u001eare the polar and azimuthal angles of the\nmagnetisation vector, and Ethe magnetic free energy\ndensity\nE=\u0000\u00160H\u0001M+Kusin2\u0012+\u00160M2\nscos2\u0012=2 (4)\nwhere the terms correspond to the Zeeman, anisotropy\nand demagnetising energies, respectively, and Msis the\nnet saturation magnetisation. It should be mentioned\nthat the magnetic anisotropy constant Kuis related to\nM, which is being considered constant in magnitude, via\nKu=\f\u00160M2\ns=2,\fa dimensionless parameter.4\nFIG. 4. Observed precession frequency as a function of the\napplied \feld for various temperatures. The solid lines are \fts\nto the data as described in the main text.\nBased on Eqs. (1) through (4) we \ft our entire data set\nwith\re\u000bandKuas the only free parameters. The exper-\nimental data and the associated \fts are shown as points\nand solid lines in FIG. 4. At all temperatures our simple\nmodel with one e\u000bective gyromagnetic ratio \re\u000band a\nsingle uniaxial anisotropy parameter Kureproduces the\nexperimental data reasonably well. The model systemat-\nically underestimates the resonance frequency for inter-\nmediate \felds, with the point of maximum disagreement\nincreasing with decreasing temperature. We speculate\nthis is due to the use of a simple uniaxial anisotropy in\nthe free energy (see Eq. 4), while the real situation is\nmore likely to be better represented as a sperimagnet. In\nparticular, the non-collinear nature of MRG that leads\nto a deviation from 180\u000eof the angle between the two\nsublattice magnetisations, depending on the applied \feld\nand temperature.\nFrom the \fts in FIG. 4 we infer the values of ge\u000band\nthe anisotropy \feld \u00160Ha=2Ku=Ms. The result is shown\nin FIG. 5. The anisotropy \feld is monotonically increas-\ning with decreasing temperature as the magnetisation\nof the 4csublattice increases in the same temperature\nrange. We highlight here the advantage of determining\nthis \feld through time-resolved magneto-optics as op-\nposed to static magnetometry and optics. Indeed the\nanisotropy \feld as seen by static methods is sensitive to\nthe combination of anisotropy and the netmagnetic mo-\nment, as illustrated in FIG. 1, where the coercive \feld\ndiverges as T!TM. In statics one would expect a di-\nvergence of the anisotropy \feld at the same temperature.\nThe time-resolved methods however distinguish between\nthe net and the sublattice moments, hence better re\rect-\ning the evolution of the intrinsic material properties of\nthe ferrimagnet.\nThe temperature dependence of the anisotropy con-\nstants was a matter for discussion for many years27,28.\nFIG. 5. E\u000bective g-factor,ge\u000b, and the anisotropy \feld\nas determined by time-resolved Faraday e\u000bect. ge\u000b, orange\nsquares, increases from near the free electron value of 2 to 4\njust belowTM, while the anisotropy \feld, blue triangles, in-\ncreases near-linearly with decreasing temperature. A M3\ft,\nred dashes line, of the anisotropy behaviour shows the almost-\nmetallic origin of it, indicating the dominant character of the\n4c sublattice.\nWritten in spherical harmonics the 3 danisotropy can\nbe expressed as, k2Y0\n2(\u0012) +k4Y0\n4(\u0012)29wherek2/\nM(T)3andk4/M(T)10. The experimental measured\nanisotropy is then, K2(T) =ak2(T)+bk4(T), withaand\nbthe contributions of the respective spherical harmonics.\nFIG. 5 shows that a reasonable \ft of our data is ob-\ntained with M(T)3which means, \frst, that the contri-\nbution of the 4thorder harmonic can be neglected, and\nsecond, that the contribution of the TMand 2ndsublat-\ntice is negligible, indicating the dominant character of\nthe 4c sublattice.\nIn addition, we should note here that the high fre-\nquency exchange mode was never observed on our exper-\niments. While far from TMits frequency might be too\nhigh to be observable, in the vicinity of TM, in contrast,\nits frequency is expected to be in the detection range.\nMoreover, given the di\u000berent electronic structure of the\ntwo sublattices, it is expected that the laser pulse should\nselectively excite the sublattice 4c, and therefore lead to\nthe e\u000bective excitation of the exchange mode. We argue\nthat it is the non-collinearity of the sublattices (see sec-\ntion III A) that smears out the coherent precession at\nhigh frequencies.\nThe e\u000bective gyromagnetic ratio, ge\u000b, shows a non-\nmonotonic behaviour. It increases with decreasing Tto-\nwardsTM, reaching a maximum at about 50 K before\ndecreasing again at T= 10 K. We alluded above to\nthe di\u000berence between the magnetic and the angular mo-\nmenta compensation temperatures. We expect that ge\u000b\nreaches a maximum when T=TA30, here between the\nmeasurement at T= 50 K and the magnetic compensa-\ntion temperature TM\u001980 K.5\nFIG. 6. Intrinsic and anisotropic broadening in MRG across\ntheTM. The inset shows the evaluation process of the two\ndamping parameters. A linear \ft is used to evaluate intercept\n(anisotropic broadening) and slope (intrinsic damping) of the\nfrequencies versus the inverse of the decay time. The data\npoint are obtained from the \ft of time-resolved Faraday e\u000bect\nmeasurements (an example is shown in Fig.4).\nFrom XMCD data11, we could estimate spin and or-\nbital moment components of the magnetic moments of\nthe two sublattices, what allowed us to derive the ef-\nfective g-factors for the sublattices as g4a= 2:05 and\ng4c= 2:00. In this case we expect the angular momentum\ncompensation temperature TAto be below TM, opposite\nto what is observed for GdFeCo21. Given this small dif-\nference however, TAandTMare expected to be rather\nclose to each other, consistent with the limited increase\nofge\u000bacross the compensation points.\nWe turn \fnally to the damping of the precessional mo-\ntion of Maround the e\u000bective \feld \u00160He\u000b. Damping is\nusually described via the dimensionless parameter \u000bin\nthe Landau-Lifshiz-Gilbert equation, and it is a measure\nof the dissipation of magnetic energy in the system. In\nthis model, \u000bis a scalar constant and the observed broad-\nening in the time domain is therefore a linear function of\nthe frequency of precession31{33. We infer \u000b0, the total\ndamping, from our \fts of the time-resolved Faraday e\u000bect\nas\u000b0= (\u001cd)\u00001, where\u001cdis the decay time of the \fts. We\nthen, for each temperature, plot \u000b0as a function of the\nobserved frequency and regress the data using a straight\nline \ft. The intrinsic \u000bis the slope of this line, while the\nintercept represents the anisotropic broadening.\nFIG. 6 shows the intrinsic damping \u000band the\nanisotropic broadening as a function of temperature.\nAnisotropic broadening is usually attributed to a vari-\nation of the anisotropy \feld in the region probed by the\nprobe pulse34. For MRG this is due to slight lateral vari-\nations in the Ru content xin the thin \flm sample. Such a\nvariation leads to a variation in e\u000bective TMandTAand\ncan therefore have a large in\ruence on the broadening asa function of temperature. Despite this, the anisotropic\nbroadening is reasonably low in the entire temperature\nrange above TM, and a more likely explanation for its\nrapid increase below TMis that the applied magnetic\n\feld is insu\u000ecient to completely remagnetize the sam-\nple between two pump pulses. As observed in Fig.5, the\nanisotropy \feld reaches almost 4 T at low temperature,\ncomparable to our maximum applied \feld of 7 T. The\nintrinsic damping \u000bis less than 0.02 far from TM, but\nincreases sharply at T= 100 K. We tentatively attribute\nthis to an increasing portion of the available power be-\ning transferred into the high-energy exchange mode, al-\nthough we underline that we have not seen any direct\nevidence of such a mode in any of the experimental data.\nIV. CONCLUSION\nWe have shown that the time-resolved Faraday e\u000bect\nis a powerful tool to determine the spin dynamic proper-\nties in compensated, metallic ferrimagnets. The high spin\npolarisation of MRG enables meaningful Faraday data to\nbe recorded even near TMwhere the net magnetisation\nis vanishingly small, and the dependence of the dynamics\non the sublattice as opposed to the net magnetic prop-\nerties provides a more physical understanding of the ma-\nterial. Furthermore, we \fnd that the ferromagnetic-like\nmode of MRG reaches resonance frequencies as high as\n160 GHz in zero applied \feld, together with a small in-\ntrinsic damping. This value is remarkable if compared\nto well-known materials such as GdFeCo which, at zero\n\feld, resonates at tens of GHz21or [Co/Pt] nmultilay-\ners at 80 GHz35but with higher damping. We should\nhowever stress that, in the presence of strong anisotropy\n\felds, higher frequencies can be reached. Example of that\ncan be found for ferromagnetic Fe/Pt with \u0019280 GHz\n(Ha= 10T)36, and for Heusler-like ferrimagnet (Mn 3Ge\nand Mn 3Ga) with\u0019500 GHz (Ha= 20T)37,38. Never-\ntheless, the examples cited above show a considerably\nhigher intrinsic damping compared to MRG. In addi-\ntion, it was recently shown that MRG exhibits unusu-\nally strong intrinsic spin-orbit torque20. 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Yildirim,\net al. , Applied Physics Letters 109, 032403 (2016)."    },    {        "title": "1703.01879v5.Damping_dependence_of_spin_torque_effects_in_thermally_assisted_magnetization_reversal.pdf",        "content": "IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017  1 \nDamping dependence of spin-torque effects in thermally assisted \nmagnetization reversal   \nY.P. Kalmykov,1 D. Byrne,2 W.T. Coffey,3 W. J. Dowling,3 S.V.Titov,4 and J.E. Wegrowe5 \n1Univ. Perpignan Via Domitia, Laboratoire de Mathématiques et Physique, F -66860,  Perpignan, France  \n2School of Physics, University College Dublin, Belfield, Dublin 4, Ireland  \n3Department of Electronic and Electrical Engineering, Trinity College, Dublin 2, Ireland  \n4Kotel’nikov Institute of Radio Engineering and Electronics of the Russia n Academy of Sciences, Vvedenskii Square 1, \nFryazino, Moscow Region, 141120, Russia  \n5Laboratoire des Solides Irradiés, Ecole Polytechnique, 91128 Palaiseau Cedex, France  \nThermal fluctuations of nanomagnets driven by spin -polarized currents are treated via the Landau -Lifshitz -Gilbert equation as \ngeneralized to include both the random thermal noise field and Slonczewski spin -transfer torque (STT) term s. The magnetization \nreversal time of such a nanomagnet is then  evaluated for wide ranges of damping by using a method which  generalizes the solution of \nthe so -called Kramers turnover problem for mechanical Brownian particles  thereby  bridging the very low damping (VLD) and \nintermediate damping (ID) Kramers escape rates , to the analogous magnetic turnover problem. The reversal time is then evaluated for \na nanomagnet with the free energy density given in the standard form of superimposed easy -plane and in -plane easy -axis anisotropies \nwith the dc bias field along the easy axis.  \n \nIndex Terms — Escape rate, Nanomagnets, Reversal time of the magnetization, Spin -transfer t orque .  \n \nI. INTRODUCTION  \nue to the  spin-transfer torque (STT) effect [1 -6], the \nmagnetization of a nanoscale ferromagnet may be altered \nby spin -polarized currents . This phenomenon occurs because \nan electri c current with spin polarization in a ferromagnet has \nan associated flow of angular momentum [3,7] ther eby \nexerting a macroscopic spin torque.  The phenomenon  is the \norigin of the novel  subject of spintronics  [7,8], i.e., current -\ninduced control over magnet ic nanostructures . Common \napplications are very high-speed  current -induced \nmagnetization switching by  (a) reversing the orien tation of \nmagnetic bits [3,9 ] and (b) using spin polarized currents  to \ncontrol steady state microwave oscillations [9 ]. This is \naccomplished via the steady state magnetization precession \ndue to STT representing the conversion of DC input into an \nAC output voltage [3]. Unfortunately , thermal fluctuations \ncannot now be ignored due to the nanometric  size of STT \ndevices, e.g., leading to mainly noise -induced switching at \ncurrents far less than the critical switching current without \nnoise [10] as corroborated by experiments (e.g., [11]) \ndemonstrating that STT near room temperature significantly \nalters thermally activated switching processes . These now \nexhibit a pronounced dependence on both material and \ngeometrical parameters. Consequently, an accurate account of \nSTT switching effects at finite temperatures is necessary in \norder to achieve further improvements in the design and \ninterpretatio n of experiments, in view of the manifold practical applications in spintronics, random access memory \ntechnology, and so on.  \nDuring the last decade, various analytical and numerical \napproaches to the study of STT effects in the thermally \nassisted  magnetiza tion reversal (or switching) time in \nnanoscale ferromagnets  have been developed  [6,7,12 -26]. \nTheir objective being to generalize  methods originally \ndeveloped for zero STT  [12,27 -32] such as  stochastic \ndynamics simulations (e. g., Refs. [21 -25]) and  extensio ns to \nspin Hamiltonians  of the mean first passage time (MFPT) \nmethod (e.g., Refs. [16] and [17] ) in the Kramers escape rate \ntheory  [33,34]. However, unlike zero STT substantial progress \nin escape rate theory including STT effects has so far been \nachieved o nly in  the limit  of very low damping (VLD), \ncorresponding to vanishingly small values of the damping \nparameter \n  in the Landau -Lifshitz -Gilbert -Slonczewski \nequation  (see Eq. (5) below). Here  the pronounced time \nseparation between fast precessional and slow energy changes \nin lightly  damped  closed phase space trajectories (ca lled \nStoner -Wohlfarth orbits) has been exploited in Refs. \n[7,14, 16,17]  to formulate a one -dimensional Fokker -Planck \nequation for the energy distribution function  which may be \nsolved by quadratures. This equation is  essentially similar to \nthat derived by Kramers [ 33] in treating the VLD noise -\nactivated escape rate of a point Brownian particle from a \npotential well although the Hamiltonian of the magnetic \nproblem is no longer separable and additive and the barrier \nheight is now STT depend ent. The Stoner -Wohlfarth orbits \nand steady precession along such an orbit of constan t energy \noccur if the spin -torque is strong enough to cancel out the \ndissipative torque. The origin of the orbits arises from the \nbistable (or, indeed, in general multistable) structure of the \nanisotropy potential. This structure allows one to define a \nnonconservative “effective” potential with damping - and D \nManuscript received April 6, 2017; revised June 27, 2017; accepted July \n24, 2017. Date of publication July 27, 2017; date of current ver -sion \nSeptember 18, 2017. Correspondin g author: Y. P. Kalmykov (e -mail: \nkalmykov@univ -perp.fr).  \nColor versions of one or more of the figures in this paper are available \nonline at http://ieeexplore.ieee.org . \nDigital Object Identifier: 10.1109/TMAG.2017. 2732944  IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017  \n 2 \ncurrent -dependent potential barrier s between stationary self -\noscillatory states of the magnetization, thereby permitting one  \nto estimate the reversal (switching) time between these states . \nThe magnetizat ion reversal time in the VLD  limit  is then \nevaluated  [16,17,35 ] both for zero and nonzero STT. In \nparticular, for nonzero STT , the VLD reversal time has been \nevaluated analytically in Refs. [16,17 ]. Here it has been shown \nthat in the high barrier limit, an asymptotic equation for the \nVLD magnetization reversal time from a single  well in the \npresence of the STT is given by  \n \nVLD\nTST1\nCES . (1) \nIn Eq. (1), \n is the damping parameter  arising from the \nsurroundings , \nTST\nAE\nEfe  is the escape rate render ed by \ntransition state theory (TST) which ignores effects due to the \nloss of spins at the barrier [34], \nAEf  is the well precession \nfrequency, \nE  is the damping and spin -polarized -current \ndependent effective en ergy barrier, and \nCES  is the \ndimensionless action  at the saddle point C (the action is given \nby Eq. (13) below).  \nThe most essential fea ture of the results obtained in Refs. \n[16,17,35 ] and how they pertain to this  paper is that they apply \nat VLD only where  the inequality \n1\nCES  holds meaning \nthat the energy loss per cycle of the almost periodic motion at \nthe critical en ergy is much less than the thermal energy . \nUnfortunately  for typical values of the material parameters \nCES\n may be very high (\n310 ), meaning that this inequality  \ncan be fulfilled only for \n0.001 . In addition, both \nexperimental and theoretical estimates suggest  higher  values \nof  of the order of 0.001 -0.1 ( see, e.g., Refs. [6,36 -38]), \nimplying that the VLD asymptotic results are no longer valid \nas they will now differ substantially from the true  value of the \nreversal time . These considerations suggest that the \nasymptotic calculations for STT should be extended to include \nboth the VLD and intermediate damping (ID) regions. This is \nour primar y objective here . Now like point Brownian particles  \nwhich  are governed by a separable and additive Hamiltonian , \nin the escape rate problem as it pertains to magnetic moments \nof nanoparticles, three regimes of damping appear [ 12,33,34]. \nThese are  (i) very low damping \n( 1)\nCES , (ii) intermediate -\nto-high damping (IHD)  \n( 1)\nCES , and  (iii) a more or less \ncritically damped turnover regime \n( ~ 1)\nCES . Also , Kramers \n[33] obtained his now -famous VLD and IHD escape rate \nformulas for point Brownian particles by assuming in  both \ncases that the energy barrier is much greater than the thermal \nenergy so that the concept of an escape rate applies. He \nmentioned, however, that he could not find a general method \nof attack in order to obtain an escape rate formula valid for \nany damp ing regime. This problem, namely the Kramers \nturnover, was initially solved by Mel’nikov and Meshkov \n[39]. They obtained an escape rate that is valid for all values \nof the damping by a semi heuristic argument, thus constituting a solution of the Kramers tu rnover problem for point particles. \nLater, Grabert [40] and Pollak et al . [41] have presented  by \nusing a coupled oscillator model of the thermal bath , a \ncomplete solution of the Kramers turnover problem and have \nshown that the turnover escape rate formula can be obtained \nwithout  the ad hoc  interp olation between the VLD  and IHD \nregimes  as used by Mel’nikov and Meshkov . Finally, Coffey \net al. [42,43 ] have shown  for classical spins  that at zero STT , \nthe magnetization reversal  time for values of damping up to \nintermediate values, \n1,  can also be evaluated via the \nturnover formula for the escape rate bridging the  VLD  and ID  \nescape rates, namely,  \n \nTST1\n()\nCE AS , (2) \nwhere \n()Az  is the so-called depopulation factor, namely  [39-\n42] \n \n 2\n2\n0ln 1 exp[ ( 1/4)]1\n1/4()z\nd\nA z e\n   \n\n . (3) \nNow the ID reversal time (or the lower bound of the reversal \ntime) may always be evaluated via TST as [32,34]  \n \nID\nTST1 . (4) \nTherefore b ecause \n()\nCCEE A S S  is the energy loss per \ncycle at the critical energy  \n0\nCES  [39] (i.e. , in the VLD \nlimit) , Eq. (2) transparently reduces to the VLD Kramers \nresult, Eq.  (1). Moreover  in the ID range, where \n( ) 1\nCE AS , \nEq. (2) reduces to the TST Eq.  (4). Nevertheless in the high \nbarrier limit \n1,\nCES  \n given by Eq. (2) can substantially  \ndeviate in the damping range \n0.001 1  both from \nID , \nEq. (4), and \nVLD , Eq. (1). Now, the approach of Coffey et al. \n[42,43 ] generalizing the Kramers turnover results to classical \nspins (nanomagnets)  was developed for zero STT, \nnevertheless, it can also be used to account for STT effects. \nHere  we shall  extend th e zero STT results of Refs. \n[14,16,17,39 -42] treating the damping dependence of STT \neffects  in the magnetization reversal of  nano scaled \nferro magnets  via escape rate theory in the most important \nrange of damping comprising  the VLD and ID ranges , \n1.   \nII. MODEL  \nThe object of our study is the role played by STT effects in the  \nthermally assist ed magnetization reversal  using an adaptation \nof the theory of thermal fluctuations in nanomagnets \ndeveloped in the seminal work s of Néel [27] and Brown  \n[28,29]. The Néel -Brown theory i s effect ively  an adaptation of \nthe Kramers theory [ 33,34 ] originally given for point \nBrownian particles to magnetization  relaxa tion governed by a \ngyromagnetic -like equation  which is taken as the Langevin \nequation of the pro cess. Hence, the verification of that theory \nin the pure (i.e., without STT) nanomagnet context nicely \nillustrates the Kramers conception of a thermal relaxation \nprocess as escape over a potential barrier arising from the IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017  \n 3 \nshuttling action of the Brownian m otion. However, it should \nbe recalled throughout that unlike nanomagnets at zero STT \n(where the giant spin escape rate theory may be effectively \nregarded as fully developed),  devices  based on STT , due to the \ninjection of the spin -polarized current, invaria bly represent an \nopen system in an out-of-equilibrium steady state.  This is in \nmarked contrast to the conventional steady state of \nnanostructures characterized by the Boltzmann equilibrium \ndistribution that arises when STT is  omitted .  Hence both the \ngover ning Fokker -Planck and Langevin equations and the \nescape rate theory based on these must be modified .  \nTo facilitate our di scussion, we first describe a schematic \nmodel of the STT effect.  The archetypal model (Fig. 1  (a)) of \na STT device is a nanostructure  compr ising two magnetic \nstrata label ed the free and fixed layers and a nonmagnetic \nconducting spacer. The fixed layer is much more strongly \npinned along its orientation than the free one. If an electric \ncurrent  is passsed  through the fixed layer it become s spin -\npolarized . Thus , the current , as it encounters the free layer, \ninduces a STT . Hence, the magnetization \nM  of the free  layer  \nis altered . Both ferromagnetic layers are assumed to be \nuniformly  magnetized [3,6]. Although th is gia nt coherent spin  \napproximation cannot explain all observations of the \nmagnetization dynamics in spin -torque systems, nevertheless \nmany qualitative features needed to interpret experimental \ndata are satisfactorily reproduced. Indeed,  the current -induced \nmagnetization dynamics in the free layer may be described by \nthe Landau -Lifshitz -Gilbert -Slonczewski equation  including thermal fluctuations , i.e., the usual Landau -Lifshitz -Gilbert \nequation [ 44] incl uding STT, however  augmented by a \nrandom magnetic field  \n()tη  which is regarded as white noise. \nHence it now becomes a magnetic Langevin equation \n[3,6,7,12 ], viz.,  \n \nS             u u H η u u u u I\n . (5) \nHere  \n/SMuM  is the unit vector directed along \nM , \nSM  is \nthe saturation magnetization, and  is the gyromagnetic -type \nconstant . The effective  magnetic field \nH  comprising the \nanisotropy and external applied fields  is defined as  \n \n0SkT E\nvMHu . (6) \nHere  E is the  normalized free energy density of the free layer  \nconstituting a conservative potential, \nv  is the free layer  \nvolume , \n7 2 1\n04 10 JA m    in SI units,  and \nkT  is the \nthermal energy.  For purposes of illustration , we sh all take  \n,)(E\n in the standard form of superimposed easy -plane and \nin-plane easy -axis anisotropies  plus the Zeeman term due to \nthe applied  magnetic  field \n0H  [45] (in our notation):  \n \n22 2, ) sin cos sin cos )( ( 2 cos h E         . (7) \nIn Eq. (7)  and  are the polar and azimuthal angles  in the \nusual spherical polar coordinate system , \n0S/ (2 ) h H M D\n  \nand \n2\n0S / ( ) v M D kT\n  are the external field and anisotropy \nparameters, \n/1DD \n  is the biaxiality parameter \ncharacterized by  \nD\n  and \nD  thereby encompassing both \ndemagnetizing and magnet ocrystalline anisotropy effects \n(since \n  and \n  are determined by both  the volume an d the \nthickness of the free layer, th eir numerical values may vary \nthrough  a very large range, in particular, they can be very \nlarge , > 100  [45]). The form of Eq. (7) implies that  both the \napplied field \n0H  and the unit vector \nPe  identifying the \nmagnetization direction in the fixed layer are directed along \nthe easy X-axis (see Fig. 1(a)) . In general,  \n,()E  as \nrendered by Eq. (7) has two equivalent saddle points  C and \ntwo nonequivalent  wells  at \nA  and \nA  (see Fig.1(b) ). Finally , \nthe STT induced field \nSI  is given by   \n \n0S\nSkT\nvMIu , (8) \nwhere  \n is the normalized non conservative potential due to \nthe spin -polarized current, which in its simplest form i s \n \n ( , )PJ   eu . (9) \nIn Eq.  (9), \n()P J b I e kT\n  is the dimensionless STT \nparameter , I is the spin -polarized current regarded  as positive \nif electrons flow  from the free into the fixed layer, e is the \nelectronic charge, \n  is Planck’s reduced constant , and \nPb is a \nparameter determined  by the spin polarization factor \nP  [1]. \nAccompanying  the magnetic Langevin equation  (5) (i.e., the \nstochastic differential equation of the random magnetization \nprocess) , one has the Fokker -Planck equation  for th e evolution \nof the  associated probability density function \n( , , )Wt  of \norientations of \nM  on the unit sphere, viz., [ 6,12,16 ] \n \nX e   u Z \nY M  \n \neasy axis  H0  \nfixed layer  free layer   I eP \n(a)   \n \n \n       (b)  \nFig. 1. (a)  Geometry of the problem: A STT device consists of two \nferromagnetic strata labelled the free and fixed layers, respectively, and a \nnormal conducting spacer all sandwiched on a pillar between two ohmic \ncontact s [3,6]. Here I is the spin -polarized current, M is the magnetization of \nthe free layer, H0 is the dc bias magnetic field. The magnetization of the \nfixed layer is directed along  the unit vector  eP. (b) Free energy potential of \nthe free layer presented in the standard form of superimposed easy -plane and \nin-plane easy -axis anisotropies, Eq. (7), at  = 20 and h = 0.2 .  IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017  \n 4 \n \nFPLWWt , (10) \nwhere \nFPL  is the Fokker -Planck operator in phase space \n( , )\n defined via [6,12,26]  \n \n1\nN\n1\n11FP1 ( )L sin2 sin\n1 ( ) 1\nsin sin\n( ) ( )sinWEWW\nEW\nEEW   \n    \n\n\n\n\n\n         \n           \n \n\n \n    \n   (11) \nand \n0 N1\nS( ) / (2 ) v M kT     is the free diffus ion time \nof the magnetic moment. If \n0=  (zero STT), Eq. (10) \nbecomes the original Fokker -Planck equation derived by \nBrown  [33] for magnetic nanoparticles . \nIII. ESCAPE RATES AND REVE RSAL TIME IN THE DAM PING \nRANGE \n1  \nThe magnetization  reversal tim e can be calculated  exactly by \nevaluating  the smallest nonvanishing eigenvalue \n1  of the \nFokker -Planck operator L FP in Eq. (10) [32,34 ,42]. Thus \n1  is \nthe inverse of the longest relaxation time of the magnetization \n11/\n, which is usually associated with the reversal time . \nIn the manner of zero STT [42,43], the calculation of \n1  can \nbe approximately accomplished using the Mel’nikov -Meshkov \nformalism  [39]. This relies on the fact that  in the high barrier  \nand underda mped limit s, one may rewrite the Fokker -Planck \nequation, Eq. (10), as an energy -action diffusion equation. \nThis in turn is very similar to that for translating point \nBrownian particles moving along the x-axis in an external \npotential V(x) [7,17,42] . In the under damp ed case, which is the \nrange of interest, for the escape of spins from a single \npotential well with a minimum at a point A of the \nmagnetocristalline anisotropy over a single saddle point C, the \nenergy distribution function  \n()WE  for magnetic moments \nprecessing in the  potential well can  then be found via an \nintegral equation [42], which  can be  solved for \n()WE  by the \nWiener –Hopf method. Then, the flux -over-population method \n[33,34]  yields the decay ( escape ) rate as \n1/CAJN . Here \nconstCJ\n is the probability current density  over the sadd le \npoint and \n()C\nAE\nAEN W E dE  is the well population while the \nescape rate is rendered as the product of the depopulation \nfactor  \n( ),\nCE AS  Eq. (3), and the TST  escape rate \nTST\nAE\nEfe\n. In the preceding equation \nE  is the effective \nspin-polari zed current dependent energy barrier  given by  \n \n1\nAC\nCE\nE EAEVdE E E ES    , (12) \nwhere \nAE  is the energy at the bottom of the potential well, \nCE\n is the energy at the saddle  point, and the dimensionless \naction \nES  and the dimensionless work \nEV  done by the STT are defined as [7,17]  \n \nEEd SE  \nuuu\n , (13) \n \nEE d Vuuu\n , (14) \nrespectively. T he contour integrals in Eqs. (13) and (14) are \ntaken along the energy trajectory  \nconstE  and are to be \nevaluated in the vanishing damping sense.  \nFor the bistable potential, Eq. (7), having  two nonequivalent \nwells  \nA and \nA  with minima  \n( 1 2 ) Eh\n  at \n0A  \nand \nA , respectively,  and two equivalent saddle points C \nwith \n2\nCEh  at \ncosC h  (see Fig. 1(b)) we see that two \nwells and two escape routes over two saddle points  are \ninvolved in the relaxation process . Thus, a finite probability \nfor the magnetic dipole to return to the initi al well having \nalready visited the second one exists. This possibility cannot \nbe ignored in the underdamped regime  because then the \nmagnetic dipole having entered the second well loses its \nenergy so slowly that even after several precessions, thermal \nfluctuations may still reverse it back over the potential barrier. \nIn such a situation, on applying the Mel’nikov -Meshkov \nformalism  [39] to the free energy potential, Eq. (7), and the \nnonconservative potential, Eq. (9), the energy distribution \nfunction s \n()WE  and \n()WE  for magne tic moments \nprecessing in the  two potential well s can then be found by \nsolving two coupled integral equations for \n()WE  and \n()WE\n. These then yield the depopulation factor \n, ()\nCCEE A S S\n via the Mel’nik ov-Meshkov formula for two \nwells, viz., [39]  \n \n( ) ( )\n((), )CC\nCC\nCCEE\nEE\nEEA S A S\nA S SA S S\n\n\n .  \nHere \n()Az  is the depopulation factor for a single well \nintroduced in accordance with Eq. (3) above while \nCES  are the \ndimensionless action s at the energy saddle point s for two \nwells. These are to be calculated via Eq. (13) by integrating \nalong the energy trajectories \nC EE  between two saddle \npoints and are explicitly given by  \n     \n2\n3/2\n12\n21\n12(1 )\n(1(1 2 a4\n(1\nrct)\n)1an )(1 )\n)1 (1CCEEh\nhhhES\nhd\nh\nh \n\n\n\n\n\n      \n   \n \n\nuuu\n  (15) \n(at zero dc bias field, h = 0, these simplify to \nCCEESS  \n4\n). Furthermore, the overall TST escape rate \nTST  for \na bistable potential, Eq. (7), is estimated via the individual  \nescape rates \nTST\n  from each of the two wells as  \n \n TST TSTTST2.EEffee   \n      (16) \nIn Eq. (16), the factor 2 occurs because two magnetization \nescape r outes from each well over the two saddle points exist, \nwhile \nE  are the effective spin -polarized current dependent IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017  \n 5 \nbarrier heights  for two wells (explicit equations for \nE  are \nderived in Appendix A). In addit ion  \n \n01(1 )(1 )2f h h      (17) \nare the corresponding well precession frequencies, where \n1\n0S 2MD\n is a precession time  constant . Thus, the \ndecay rate \n1  becomes  \n2\n2(1 ) ( , )1\n0\n(1 ) ( , )(1 )(1 )\n(( ) ( )\n()\n, 1 )(1 )CC\nCCJh F hEE\nEE\nJh F hA S A S\nAh h e\nhhS\neS\n\n\n\n\n\n\n\n\n  \n\n \n\n \n\n(18) \nwhere both the functions \n( , )Fh  occurring  in each \nexponential are given by the analytical formula:  \n  \n22\n2\n1\n12\n2 1 1 2(1 ) (1( , ) 12 2 2 (1\n(1 21 arctan\n(1)(1 )\n)\n)1\n)(1 ) 1 (1 )hhFhhh h\nhh h\nh h h  \n\n\n\n\n       \n    \n\n\n\n  (19) \nand  0.38 is a numerical par ameter  (see Eq.  (A.6), etc. in \nAppendix A ). For zero STT, J = 0, Eq. (18) reduces to the \nknown results  of the Néel -Brown theory [32,43] for classical \nmagnetic moments with superimposed easy -plane and in -plane \neasy-axis anisotropies  plus the Zeeman term due to the applied  \nmagnetic  field. In contrast to zero STT, for  normalized spin \ncurrents J  0,  depends on \n  not only through the \ndepopulation factors \n()\nCE AS  but also through the spin-\npolarized current dependent  effective barrier heights \nE . \nThis i s so because part s of the arguments of the exponentials \nin Eq.  (18) , namely Eq. (19), are markedly dependent on the \nratio \n/J  and the dc bias field parameter.  The turnover Eq. \n(18) also yields a n asymptotic estimate for the inverse of the \nsmallest nonvanishing eigenvalue of the Fokker -Planck \noperator \nFPL  in Eq. (10). In additio n, one may  estimate two \nindividual  reversal times, namely, \n  from the deeper well \naround the energy minimum at \n0A  and \n  from the \nshallow well around the energy minimum at \nA  (see Fig. \n1(b))  as \n \n2(1 ) ( , )\n02\n( ) (1 )(1 )\nCJh F h\nEe\nA S h h\n\n\n  \n . (20) \nThe individual  times are in general unequal, i.e., \n . In \nderiving Eqs. (18) and (20), all terms of order \n22, , ,JJ  etc. \nare neglected. This hypothesis is true only for the \nunderdamped regime , α < 1, and weak  spin-polarized currents, \nJ<<1. ( Despite these restrictions as we will see below Eqs. \n(18) and (20) still yield accurate estimates for \n  for much \nhigher values of J). Now,  \n can also be calculated  \nnumerically  via the method of statistical moments developed  \nin Ref. [26] whereby t he solution of the Fokker -Planck \nequation (10) in configuration space  is reduced to the task of solving  an infinite hierarchy of differential -recurrence \nequations for the averaged spherical harmonics  \n( , ) ( )lmYt  \ngoverning the magnetization relaxation . (The \n( , )lmY  are the \nspherical harmonics [46 ], and the angular brackets denote the \nstatistical aver aging ). Thus one can evaluate  \n numerically \nvia \n1  of the Fokker -Planck operator L FP in Eq. (10) by using \nmatrix continued fr actions as described in Ref. [47 ]. We \nremark that the r anges of applicability of the escape rate \ntheory and the matrix continued -fraction method are in a sense  \ncomplementary because  escape rate theory cannot be used for  \nlow potential barrie rs, \n3E , while  the matrix continued -\nfraction method encounters substantial computation al \ndifficulties for  very high  potential barriers \n25E  in the \nVLD range, \n410 . Thus , in the foregoing se nse, numerical \nmethods and escape rate theory are very useful for the \ndetermination of τ for low and very high potential barriers, \nrespectively. Nevertheless , in certain  (wide) ranges of model \nparameters both methods yield accurate results for the reversal  \ntime ( here these ranges are \n5 30, 3,     and \n410 ). \nThen the numerically exact  benchmark solution provided by \nthe matrix continued fraction method  allows one  to test the \naccuracy of the analytical es cape rate equations given above.  \nIV. RESULTS AND DISCUSSIO N \nThroughout the calculations, the anisotropy and spin -\npolarization parameters will be taken as \n0.034 D\n , \n20 , \nand \n0.3P  (\n0.3 0.4P  are typical  of ferromagnetic \nmetals)  just as in Ref. 6. Thus for \n5 1 1mA s . 10 , 22  \n300T\nK\n, \n24~10v\n3m , and a current density of the order \nof \n7~ 10\n2A cm  in a 3 nm thick layer of cobalt with \n61\nS 1 1. Am 04 M\n, we have  the following estimates for the \nanisotropy (or inverse temperature ) parameter \n20.2 , \ncharacteristic time \n1\n0S2()MD\n0.48 ps, and spin -\npolarized current parameter \n( ) ~1P J b I e kT\n . In Figs. 2 \nand 3,  we compare  from the asymptotic escape rate Eq . (18) \nwith \n1\n1  of the Fokker –Planck operator as calculated \nnumerically via matrix continued fraction s [26]. Apparently,  \nas rendered by  the turnover equation (18) and \n1\n1  both lie \nvery close to each other in the high barrier limit, where the \nasymptotic Eq. (18) provides an  accurate approximation \nto\n1\n1.  In Fig. 2,  is plotte d as a function of \n  for various J. \nAs far as STT effects are concerned they are governed by the \nratio \n/J  so that by altering \n/J  the ensuing variation of  \nmay exceed  several or ders of magnitude (Fig. 2) . Invariably \nfor J << 1, which is a condition of applicability of the escape \nrate equations (1) and (18), STT effects on the magnetization \nrelaxation are pronounced only at very low damping,  << 1 . \nFor \n1 , i.e. high damping, STT influences  the reversal \nprocess very weak ly because the STT term in Eq. (5) is then \nsmall compared to the damping and random field terms . \nFurthermore,  may greatly exceed or, on the other hand, be  \nvery much less than the value for zero STT , i.e., J = 0 (see Fig. \n2). For example, as J decreases from positive values, \n  IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017  \n 6 \nexponentially increases  attaining a  maximum at a critical \nvalue of the spin -polarized current and then smoothly switches \nover to exponential decrease  as \nJ  is further increased \nthrough negative values of J [26]. Now, t he temperature , \nexternal d.c. bias field,  and dam ping dependence of  can \nreadily be understood in terms of  the effective potential \nbarriers  \nE  in Eq.  (18). For example, for  \n5,  the \ntemperature dependence  of  has the customary Arrhenius \nbehavior \n~,Ee  where \nE , Eq. (19), is markedly \ndependent on \n/J  (see Fig. 3 a). Furthermore, the slope of \n1()T\n significantly decreases as the dc bias field parameter h increases due to lowering  of the barrier height \nE  owing to \nthe action of  the external field  (see Fig. 3b) . Now, although \nthe range of applicability of  Eqs. (18) and (20) is ostensibly \nconfined to weak spin -polarized currents, J << 1, they can still \nyield accurate estimates for the reversal time for much higher \nvalues of J far exceeding this condition (see Fig. 3 a).  \nThus , the turnover formula for , Eqs. (18) and (20), \nbridgi ng the Kramers VLD  and ID escape rates as a function \nof the damping  parameter for point  particles [35,39 -41] as \nextended by Coffey et al.  [42,43] to the magnetization \nrelaxation in nanoscale ferromagnets allows us (via the further \nextension to include STT embodied in Eq. (18)) to accurately  \nevaluate STT effects in the magnetization reversal time of  a \nnanomagnet driven by spin -polarized current in the highly \nrelevant ID to VLD damping range. This (underdamped) range  \nis characterized by  \n1  and the asymptotic escape rates are \nin complete agreement with independent  numerical results  \n[17]. Two particular merits of the escape rate equations  for the \nreversal time are that (i)  they are relatively simple ( i.e., \nexpressed via elementary functions) and (ii) that they can be \nused in those parameter ranges, where numerical methods \n(such as matrix continued fractions [17]) may be no longer \napplicable , e.g., for very high barriers , \n25E . Hence , one \nmay conclude that the damping dependence of the \nmagnetization reversal time is very marked in the \nunderdamped regime \n1 , a fact which may be very \nsignificant  in int erpreting  many  STT experiments.  \nV. APPENDIX A: CALCULATION OF \n( , )Fh  IN EQ. (19) \nFor the bistable potential  given by  Eq. (7), and the \nnonconservative potential, Eq. (9), the spin -polarized current \ndependent effective barrier heights  \nE  for each of the two \nwells are given by (cf. Eq. (12)) \n \n21(1 ) ( , )h J F E h     \n , (A.1) \nwhere  \n \n( , )C\nAVFhSd\n\n \n\n\n , (A.2) \nwith \n/E , \n/ 1 2AAEh \n , \n2/CCEh . The \ndimensionless action  \nS and the dimensionless work  done by \nthe STT \nV for the deeper well can be calculated analytically \nvia elliptic integrals as described in detail in Ref. [17] yielding  \n   \n2 2\n0\n2\n22(1 )\n1\n2 ( )11 ( ) ( )\n)(2\n1\n(1 )( )\n() (142),(1 )( ( ( ) ) 1)p Ehd hpf\nEm hqq q m K m\nq h q mhpq q mS\nm\nKm\n\n\n\n\n   \n \n\n\n\n \n\n\n\n \n         \n   \n    \n\n  \n\n \n uuu\n  (A.3) \n\n54321: J = 0.2\n2: J = 0.1\n3: J = 0\n4: J = 0.1\n5: J = 0.2/ \nh =0.15\n =20\n = 201 \nFig. 2. Reversal time \n0/  vs the damping parameter \n  for various values \nof the  spin-polarized  current  parameter J. Solid lines : numerical calculations \nof the inverse of t he smallest nonvanishing eigenvalue \n1\n01()  of the \nFokker –Planck operator , Eq. (11). Asterisks: the turnover  formula, Eq. (18). \n \n54\n/ 3211: J = 1\n2: J = \n3: J = \n4: J = \n5: J = \nh = 0.1\n = 0.01\n  = 20\n(a)\n \n  (b)\n4\n/ 321 1: h = 0.0\n2: h = 0.1\n3: h = 0.2\n4: h = 0.3\n = 0.01\n  = 20\nJ  = \n\n \nFig. 3 . Reversal time \n0/  vs. the anisotropy  (inverse temperature) \nparameter  for various spin-polarized currents J (a) and dc bias field \nparameters h (b). Solid lines: numerical solution for the inverse of the \nsmallest nonvanishing eigenvalue \n1\n01() of the Fokker –Planck operator , \nEq. (11). Asterisks: Eq. (18). IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017  \n 7 \n    \n\n0\n2 23\n2\n2( 1)\n1 ( 1)\n( | )1\n2\n21 2 1()\n( ) (1\n(|)121 ( ) (,))phV\nm\nm\nqhdhf\nq hpK\nhp q E m m qqq q m K mm\n\n\n\n\n     \n\n\n\n \n\n    \n\n \n    \n\n \n    \n\nueu\n  (A.4) \nwhere  \n \n2\n2\n21( 1)ph\n\n , \n1\n1eqe\n ,  \n \n\n1 (1 )\n(1 ) 1eemee\n , \n2\n( 1)hepph\n\n  , \n()Km\n, \n()Em , and \n( | )am  are the complete  elliptic integrals \nof the fir st, seco nd, and third kinds, respectively [48], and  \nf \nis the precession frequency in the deeper well at a given \nenergy, namely,  \n \n0( 1)(1\n())(1 )\n8p e efKm\n\n     . (A.5) \nThe quantities \nS , \nV , and \nf  for the shallower well are \nobtained simply by replacing  the dc bias field parameter  \nh by \nh\n in all the equations for \nS , \nV , and \nf . We remark that  \nS\n and \nV in Eqs. (A.3) and (A.4) differ by a factor 2 from \nthose given in Ref. [17]. This is so because \nS  and \nV are \nnow calculated between the saddle points and not over the \nprecession period . When \n( , )C     , \nS  in Eqs. (A.3) \nreduces to \nCES , Eq. (15). \nIn the parameter ranges \n01h  and \n1 , the integral in \nEq. (A.2) can be accurately evaluated analytically using an \ninterpolation function for \n/VS  between t he two limiting \nvalues \n/\nAAVS  and \n/\nCCVS  at \n1A h\n  and \n2\nCh , \nnamely  \n \n11\nC AA\nA C AA\nCAV VV V\nS S S S\n \n   \n  \n          , (A.6) \nwhere  0.38 is an interpolation parameter yielding the best \nfit of \n/VS  in the interval \n.AC    These limiting \nvalues can be calculated from Eqs. (A.3) and (A.4) yielding \nafter tedious algebra:  \n \n1\n22A\nAV\nh S\n\n  (A.7) \nand  \n2\n2\n1\n12\n2 1 1 2)(1 ) 1 (112 (1\n(1 21 arct)\n)1an\n(1 (1 )(1 ) ) 1C\nCV h\nh Sh\nhh h\nh h h\n\n\n\n\n\n\n\n\n\n\n \n    \n. (A.8) \nHence with Eqs. (A.2) and (A.6), we have a simple a nalytic \nformula for  the current -dependent parts of the exponentials in \nEq. (18) \n( , )Fh , viz.   \n2( , ) (1 )C A\nACV V\nF h hSS \n    \n\n    , (A.9) \nwhich yields Eq. (19). For zero dc bias field, \n0h , Eq. (A.9) \nbecomes  \n \n1( ,0) ( ,0)2 4 ( 1)FF      . (A.10) \nThe maximum relative deviation bet ween the exact Eqs. (A.2) \nand approximate Eqs. (A.9) and (A.10) is less than 5% in the \nworst cases.  \nREFERENCES  \n[1] J. C. Slonczewski, “Current -driven excita tion of magnetic multilayers ”, \nJ. Magn. Magn. Mater ., vol.  159, p. L1, 1996 . \n[2] L. Berger, “Emission of spin waves by a magnetic multilayer traversed \nby a current ”, Phys. Rev. B , vol.  54, p. 9353 , 1996 . \n[3] M. D. Stiles and J. Miltat, “Spin-Transfer Torque and Dy namics ”, in: \nSpin Dynamics in Confined Magnetic Structures III , p. 225, Eds. B. \nHillebrandsn and A. Thiaville , Springer -Verlag, Berlin, 2006 . \n[4] D. C. Ralph and M. D. 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Mayergoyz, \n“Stochastic resonance in noise -induced transitions between self -\noscillations and eq uilibria in spin -valve nanomagnets ”, Phys. Rev. B , \nvol. 84, p. 214415 , 2011 . \n[26] Y. P. Kalmykov, W. T. Coffey, S. V. Titov, J. E. Wegrowe, and D. \nByrne , “Spin-torque effects in thermally assisted magnetization reversal: \nMethod of statistical moments ”, Phys. Re v. B, vol.  88, p. 144406 , 2013 ; \nD. Byrne , W. T. Coffey, Y. P. Kalmykov, S. V. Titov, and J. E. \nWegrowe, “Spin-transfer torque effects in the dynamic forced response \nof the magnetization of nanoscale ferromagnets in superimposed ac and \ndc bias fields in the  presence of thermal agitation ”, Phys. Rev. B , vol.  91, \np. 174406 , 2015.  \n[27] L. N éel, “Théorie du traînage magnétique des ferromagnétiques en \ngrains fins avec applications aux terres cuites  », Ann. Géophys ., vol.  5, p. \n99, 1949 . \n[28] W. F. Brown, Jr., “Thermal fl uctuations of a single -domain particle ”, \nPhys. 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Kramers, “Brownian motion in a field of force and the diffusion \nmodel of chemical reactions ”, Physica , vol.  7, p. 284, 1940. \n[34] P. Hänggi, P. Talkner, and M. Borkovec, “Reaction -Rate Theory: Fifty \nYears After Kramers ”, Rev. Mod. Phys ., vol.  62, p. 251, 1990 . \n[35] W.T. Coffey, Y.P. Kalmykov, and S.T. Titov, “Magnetization reversal \ntime of magnetic nanoparticles at very low damping ”, Phys. Rev. B , vol.  \n89, p. 054408 , 2014 ; D. Byrne, W.T. Coffey, W. J. Dowling, Y.P. \nKalmykov, and S.T. Titov, “On the Kramers very low damping escape \nrate for point particles and classical spins ”, Adv. Chem. Phys ., vol.  156, \np. 393, 2015 . \n[36] W. Wernsdorfer, E. Bonet Orozco, K. Hasselbach, A. Benoit, D. Mailly, \nO. Kubo, H. Nakano, and B. Barbara, “Macroscopic Quantum Tunneling \nof Magnetization of Single Ferrimagnetic Nanoparticles of Barium \nFerrite ”, Phys. Rev. Lett ., vol.  79, p. 4014 , 1997 ; W.T. Coffey, D.S.F . \nCrothers, J.L. Dormann, Yu.P. Kalmykov, E.C. Kennedy, and W. \nWernsdorfer, “Thermally Activated Relaxation Time of a Single \nDomain Ferromagnetic Particle Subjected to a Uniform Field at an \nOblique Angle to the Easy Axis: Comparison with Experimental \nObser vations ”, Phys. Rev. Lett ., vol.  80, p. 5655 , 1998 . \n[37] M. Oogane, T. Wakitani, S. Yakata, R. Yilgin, Y. Ando, A. Sakuma, and \nT. Miyazaki, “Magnetic Damping in Ferromagnetic Thin Films ”, Jpn. J. \nAppl. Phys ., vol.  45, p. 3889 , 2006 . \n[38] M. C. Hickey and J. S. Moode ra, “Origin of Intrinsic Gilbert Damping ”, \nPhys. Rev. Lett ., vol.  102, p. 137601 (2009).  \n[39] V. I. Mel’nikov and S. V. Meshkov, “Theory of activated rate processes: \nexact solution of the Kramers problem ”, J. Chem. Phys ., vol.  85, p. \n1018 , 1986.  \n[40] H. Grabert, “Escape from a metastable well: The Kramers turnover \nproblem ”, Phys. Rev. Lett. , vol. 61, p. 1683 , 1988 . \n[41] E. Pollak, H. Grabert, and P. Hänggi, “Theory of activated rate processes \nfor arbitrary frequency dependent friction: Solution of the turnover \nproblem ”, J. Chem. Phys ., vol.  91, p. 4073 , 1989 . \n[42] W. T. Coffey, D. A. Garanin, and D. J. McCarthy, “Crossover formulas \nin the Kramers theory of thermally activated escape rates – application \nto spin systems ”, Adv. Chem. Phys . vol. 117, p. 483, 2001 ; P.M. \nDéjardin, D .S.F. Crothers, W.T. Coffey, and D.J. McCarthy, \n“Interpolation formula between very low and intermediate -to-high \ndamping Kramers escape rates for single -domain ferromagnetic \nparticles ”, Phys. Rev. E , vol.  63, p. 021102 , 2001 . \n[43] Yu. P. Kalmykov, W.T. Coffey, B. Ouari, and S. V. Titov, “Damping \ndependence of the magnetization relaxation time of single -domain \nferromagnetic particles ”, J. Magn. Magn. Mater ., vol. 292, p. 372, 2005 ; \nYu. P. Kalmykov and B. 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IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017  \n 9 \n "    },    {        "title": "2010.00642v1.Modeling_coupled_spin_and_lattice_dynamics.pdf",        "content": "Modeling coupled spin and lattice dynamics\nMara Strungaru,1Matthew O A Ellis,2Sergiu Ruta,3Oksana Chubykalo-Fesenko,4Richard F L Evans,1and Roy W Chantrell1\n1Department of Physics, University of York, York, United Kingdom\n2Department of Computer Science, University of Sheffield, Sheffield, United Kingdom\n3Department of Physics,University of York, York, United Kingdom\n4Instituto de Ciencia de Materiales de Madrid, CSIC, Madrid, Spain\nA unified model of molecular and atomistic spin dynamics is presented enabling simulations both in micro-\ncanonical and canonical ensembles without the necessity of additional phenomenological spin damping. Trans-\nfer of energy and angular momentum between the lattice and the spin systems is achieved by a coupling term\nbased upon the spin-orbit interaction. The characteristic spectra of the spin and phonon systems are analyzed\nfor different coupling strength and temperatures. The spin spectral density shows magnon modes together with\nthe uncorrelated noise induced by the coupling to the lattice. The effective damping parameter is investigated\nshowing an increase with both coupling strength and temperature. The model paves the way to understanding\nmagnetic relaxation processes beyond the phenomenological approach of the Gilbert damping and the dynamics\nof the energy transfer between lattice and spins.\nI. INTRODUCTION\nWith the emergent field of ultrafast magnetisation\ndynamics1understanding the flow of energy and angular mo-\nmentum between electrons, spins and phonons is crucial for\nthe interpretation of the wide range of observed phenom-\nena2–5. For example, phonons strongly pumped in the THz\nregime by laser excitation can modulate the exchange field\nand manipulate the magnetisation as shown for the magnetic\ninsulator YIG6or in Gd7. The excitation of THz phonons\nleads to a magnetic response with the same frequency in\nGd7, proving the necessity of considering the dynamics of\nboth lattice and spins. Phonon excitations can modulate both\nanisotropy and exchange which can successfully manipulate\n8–10or potentially switch the magnetisation11,12, ultimately\nleading to the development of low-dissipative memories.\nMagnetisation relaxation is typically modeled using the\nphenomenological description of damping proposed by Lan-\ndau and Lifshitz13and later Gilbert14, where the precessional\nequation of motion is augmented by a friction-like term, re-\nsulting in the Landau-Lifshitz-Gilbert (LLG) equation. This\nrepresents the coupling of the magnetic modes (given pri-\nmarily by the localised atomic spin) with the non-magnetic\nmodes (lattice vibrations and electron orbits). The LLG equa-\ntion and its generalisations can be deduced from the quantum-\nmechanical approaches assuming an equilibrium phonon bath\nand the weak coupling of the spin to the bath degrees of\nfreedom15–17. Thus the standard approach works on the sup-\nposition that the time scales between the environmental de-\ngrees of freedom and the magnetic degrees of freedom are\nwell separated and reducing the coupling between the mag-\nnetization and its environment to a single phenomenological\ndamping parameter18,19. In reality, the lattice and magneti-\nsation dynamics have comparable time-scales, where the in-\nteraction between the two subsystems represents a source of\ndamping, hence the necessity of treating spin and lattice dy-\nnamics in a self-consistent way.\nTo investigate these phenomena, and aiming at predictive\npower for the design of competitive ultrafast magnetic nano-\ndevices, advanced frameworks beyond conventional micro-magnetics and atomistic spin dynamics20are needed21. A\ncomplete description of magnetic systems includes the inter-\naction between several degrees of freedom, such as lattice,\nspins and electrons, modeled in a self-consistent simulation\nframework. The characteristic relaxation timescales of elec-\ntrons are much smaller ( \u0019fs) in comparison to spin and lattice\n(100fs\u0000ps), hence magnetisation relaxation processes can be\ndescribed via coupled spin and lattice dynamics, termed Spin-\nLattice Dynamics (SLD)22–29. SLD models can be crucial in\ndisentangling the interplay between these sub-systems.\nSLD models have so far considered either micro-canonical\n(NVE - constant particle number, volume and energy)27,28or\ncanonical (NVT - constant particle number, volume and tem-\nperature) ensembles with two Langevin thermostats connected\nto both lattice and spin subsystems23,30. Damping due to spin-\nlattice interactions only within the canonical ensemble (NVT)\nhas not yet been addressed, but is of interest in future mod-\nelling of magnetic insulators at finite temperature. Here we\nintroduce a SLD model capable of describing both ensem-\nbles. Specifically, our model (i) takes into account the transfer\nof angular momentum from spin to lattice and vice-versa, (ii)\nworks both in a micro-canonical ensemble (constant energy)\nand in a canonical ensemble (constant temperature), (iii) al-\nlows a fixed Curie temperature of the system independent of\nthe spin-lattice coupling strength, (iv) disables uniform trans-\nlational motion of the system and additional constant energy\ndrift, which can be produced by certain spin-lattice coupling\nforms. Furthermore, in this work, the characteristics of the in-\nduced spin-lattice noise, the magnon-phonon induced damp-\ning and the equilibrium properties of the magnetic system has\nbeen systematically investigated.\nThe paper is organised as follows. We start by describ-\ning the computational model of Spin-Lattice Dynamics and\nthe magnetic and mechanical energy terms used in this frame-\nwork (Section II). We then explore the equilibrium properties\nof the system for both microcanonical and canonical simula-\ntions, proving that our model is able to efficiently transfer both\nenergy and angular momentum between the spin and lattice\ndegrees of freedom. In Section III we compute the equilibrium\nmagnetisation as function of temperature for both a dynamic\nand static lattice and we show that the order parameter is notarXiv:2010.00642v1  [cond-mat.mtrl-sci]  1 Oct 20202\ndependent on the details of the thermostat used. In Section IV\nwe analyse the auto-correlation functions and spectral char-\nacteristics of magnon, phonons and the coupling term prov-\ning that the pseudo-dipolar coupling efficiently mediates the\ntransfer of energy from spins to the lattice and vice-versa. We\nthen calculate the temperature and coupling dependence of the\ninduced magnon-phonon damping and we conclude that the\nvalues agree well with damping measured in magnetic insula-\ntors, where the electronic contributions to the damping can be\nneglected ( Section V).\nII. COMPUTATIONAL MODEL\nIn order to model the effects of both lattice and spin dy-\nnamics in magnetic materials an atomistic system is adopted\nwith localised atomic magnetic moments at the atomic coor-\ndinates. Within this framework there are now nine degrees of\nfreedom; atomic magnetic moment (or spin) S, atomic posi-\ntionrand velocity v. The lattice and the magnetic system can\ndirectly interact with each other via the position and spin de-\npendent Hamiltonians. The total Hamiltonian of the system\nconsists of a lattice Hlatand magnetic Hmagparts:\nHtot=Hlat+Hmag: (1)\nThe lattice Hamiltonian includes the classical kinetic and\npairwise inter-atomic potential energies:\nHlat=å\nimiv2\ni\n2+1\n2å\ni;jU(ri j): (2)\nOur model considers a harmonic potential (HP) defined as:\nU(ri j) =(\nV0(ri j\u0000r0\ni j)2=a2\n0ri j<rc\n0 ri j>rc:(3)\nwhere V0has been parametrised for BCC Fe in27anda0=1˚A\nis a dimension scale factor. To be more specific we consider\nthe equilibrium distances r0\ni jcorresponding to a symmetric\nBCC structure. The interaction cut-off is rc=7:8˚A. The pa-\nrameters of the potential are given in Table II. The harmonic\npotential has been used for simplicity, however it can lead to\nrather stiff lattice for a large interaction cutoff.\nAnother choice of the potential used in our model is an an-\nharmonic Morse potential (MP) parameterised in31for BCC\nFe and defined as:\nU(ri j) =(\nD[e\u00002a(ri j\u0000r0)\u00002e\u0000a(ri j\u0000r0)]ri j<rc\n0 ri j>rc(4)\nThe Morse potential approximates well the experimental\nphonon dispersion observed experimentally for BCC Fe32as\nshown in33. The phonon spectra for the choices of potential\nused in this work are given in Section IV. Other nonlinear\nchoices of potential can be calculated via the embedded atom\nmethod34,35.The spin Hamiltonian ( Hmag) used in our simulations con-\nsists of contributions from the exchange interaction, Zeeman\nenergy and a spin-lattice coupling Hamiltonian, given by the\npseudo-dipolar coupling term ( Hc), which we will describe\nlater:\nHmag=\u00001\n2å\ni;jJ(ri j)(Si\u0001Sj)\u0000å\nimiSi\u0001Happ+Hc;(5)\nwhere miis the magnetic moment of atom i,Siis a unit\nvector describing its spin direction and Happis an external ap-\nplied magnetic field. The exchange interactions used in our\nsimulations depend on atomic separation J(ri j). They were\ncalculated from first principle methods for BCC Fe by Ma et\nal23and follow the dependence:\nJ(ri j) =J0\u0012\n1\u0000ri j\nrc\u00133\nQ(rc\u0000ri j); (6)\nwhere rcis the cutoff and Q(rc\u0000ri j)is the Heaviside step\nfunction, which implies no exchange coupling between spins\nsituated at larger distance than rc.\nSeveral previous SLD models suffered from the fact that\nthey did not allow angular momentum transfer between lattice\nand spin systems28. This happened for magnetisation dynam-\nics in the absence of spin thermostat, governed by symmetric\nexchange only, due to total angular momentum conservation.\nTo enable transfer of angular momentum, Perera et al26have\nproposed local anisotropy terms to mimic the spin-orbit cou-\npling phenomenon due to symmetry breaking of the local en-\nvironment. Their approach was successful in thermalising the\nsubsystems, however, single site anisotropy spin terms with\na position dependent coefficients as employed in26can induce\nan artificial collective translational motion of the sample while\nthe system is magnetically saturated, due to the force \u0000¶Htot\n¶riproportional to spin orientation. To avoid large collective mo-\ntion of the atoms in the magnetic saturated state, we consider\na two-site coupling term, commonly known as the pseudo-\ndipolar coupling, described by\nHc=\u0000å\ni;jf(ri j)\u0014\n(Si\u0001ˆri j)(Sj\u0001ˆri j)\u00001\n3Si\u0001Sj\u0015\n: (7)\nThe origin of this term still lies in the spin-orbit interaction,\nappearing from the dynamic crystal field that affects the elec-\ntronic orbitals and spin states. It has been employed previ-\nously in SLD simulations22,27. It was initially proposed by\nAkhiezer36, having the same structure of a dipolar interac-\ntion, however with a distance dependence that falls off rapidly,\nhence the name pseudo-dipolar interaction. The exchange-\nlike term\u00001\n3Si\u0001Sjis necessary in order to preserve the Curie\ntemperature of the system under different coupling strengths\nand to ensure no net anisotropy when the atoms form a sym-\nmetric cubic lattice. For the mechanical forces, the exchange\nlike term eliminates the anisotropic force that leads to a non-\nphysical uniform translation of the system when the mag-\nnetic system is saturated. The magnitude of the interactions3\nis assumed to decay as f(ri j) =CJ0(a0=ri j)4as presented\nin27with Ctaken as a constant, for simplicity measured rel-\native to the exchange interactions and a0=1˚A is a dimen-\nsion scale factor. The constant Ccan be estimated from\nab-initio calculations26, approximated from magneto-elastic\ncoefficients27, or can be chosen to match the relaxation times\nand damping values, as in this work.\nSince the total Hamiltonian now depends on the coupled\nspin and lattice degrees of freedom ( vi,ri,Si), the following\nequations of motion (EOM) need to be solved concurrently to\nobtain the dynamics of our coupled system:\n¶ri\n¶t=vi; (8)\n¶vi\n¶t=\u0000hvi+Fi\nmi; (9)\n¶Si\n¶t=\u0000gSi\u0002Hi; (10)\nFi=\u0000¶Htot\n¶ri+Gi; (11)\nHi=\u00001\nmSm0¶Htot\n¶Si; (12)\nwhere FiandHirepresent the effective force and field, Girep-\nresents the fluctuation term (thermal force) and hrepresents\nthe friction term that controls the dissipation of energy from\nthe lattice into the external thermal reservoir. The strength of\nthe fluctuation term can be calculated by converting the dissi-\npation equations into a Fokker-Planck equation and then cal-\nculating the stationary solution. The thermal force has the\nform of a Gaussian noise:\nhGia(t)i=0; (13)\nhGia(t)Gjb(t0)i=2hkBT\nmidabdi jd(t\u0000t0): (14)\nTo prove that the complete interacting many-body spin-\nlattice framework presented in here is in agreement with the\nfluctuation-dissipation theorem, we have followed the ap-\nproach presented by Chubykalo et al37based on the Onsager\nrelations. Linearising the equation of motion for spins, we find\nthat the kinetic coefficients for the spin system are zero, due\nto the fact that the spin and internal field are thermodynamic\nconjugate variables. Hence, if the noise applied to the lattice\nobeys the fluctuation dissipation theory, the coupled system\nwill obey it as well, due to the precessional form of the equa-\ntion of motion for the spin.\nWe compare the SLD model presented here with other ex-\nisting model that do not take into account the lattice degrees\nof freedom (Atomistic Spin Dynamics - ASD). Particularly,\nin our case we assume a fixed lattice positions. The summary\nof the comparison is presented in Table I. Atomistic spin dy-\nnamics simulations (ASD)18,20,38,39have been widely used to\nstudy finite size effects, ultrafast magnetisation dynamics and\nnumerous other magnetic phenomena. Here the intrinsic spin\ndamping (the Gilbert damping - aG) is phenomenologically\nincluded. In our case since the lattice is fixed it is assumedModel Lattice Lattice Spin Intrinsic Spin\nthermostat thermostat damping\nSLD Dynamic On Off Phonon\ninduced\nASD Fixed Off On Electronic\nmainly\nTABLE I. Summary comparison of the SLD model developed here\nagainst other spin dynamics models.\nQuantity Symbol Value Units\nExchange23J0 0:904 eV\nrc 3:75 ˚A\nHarmonic potential27V0 0:15 eV\nrc 7:8 ˚A\nMorse potential31D 0:4174 eV\na 1:3885 ˚A\nr0 2:845 ˚A\nrc 7:8 ˚A\nMagnetic moment ms 2:22 mB\nCoupling constant C 0:5\nMass m 55:845 u\nLattice constant a 2:87 ˚A\nLattice damping h 0:6 s\u00001\nTABLE II. Parameters used in the spin-lattice model to simulate BCC\nFe.\nto come from electronic contributions. Consequently, only 3\nequations of motion per atom describing the spin dynamics\nare used:\n¶Si\n¶t=\u0000g\n(1+a2\nG)Si\u0002(Hi+aGSi\u0002Hi) (15)\nwith an additional field coming from the coupling to the\nfixed lattice positions. The temperature effects are introduced\nin spin variables by means of a Langevin thermostat. The spin\nthermostat is modeled by augmenting the effective fields by a\nthermal stochastic field ( Hi=xi\u0000¶H=¶Si) and its proper-\nties also follow the fluctuation-dissipation theorem:\nhxia(t)i=0; (16)\nhxia(t)xjb(t0)i=2aGkBT\ngmSda;bdi jd(t\u0000t0): (17)\nThe characteristics of the above presented models are sum-\nmarised in Table I.\nTo integrate the coupled spin and lattice equations of mo-\ntion we used a Suzuki-Trotter decomposition (STD) method40\nknown for its numerical accuracy and stability. The scheme\ncan integrate non-commuting operators, such as is the case of\nspin-lattice models and conserves the energy and space-phase\nvolume. The conservation of energy is necessary when deal-\ning with microcanonical simulations. Considering the gener-\nalized coordinate X=fr;v;Sgthe equations of motion can be4\nre-written using the Liouville operators:\n¶X\n¶t=ˆLX(t)= (ˆLr+ˆLv+ˆLS)X(t): (18)\nThe solution for the Liouville equation is X(t+Dt) =\neLDtX(t). Hence, following the form of this solution and ap-\nplying a Suzuki-Trotter decomposition as in Tsai’s work41,42,\nwe can write the solution as:\nX(t+Dt) =eˆLsDt\n2eˆLvDt\n2eˆLrDteˆLvDt\n2eˆLsDt\n2X(t)+O(Dt3);(19)\nwhere Ls;Lv;Lrare the Liouville operators for the spin, veloc-\nity and position. This update can be abbreviated as (s,v,r,v,s)\nupdate. The velocity and position are updated using a first\norder update, however the spin needs to be updated using a\nCayley transform43,44, due to the fact that the norm of each\nindividual spin needs to be conserved. Thus we have\neˆLvDtvi=vi+Dt\nmiFi; (20)\neˆLrDtri=ri+Dtvi; (21)\neˆLSDtSi=Si+DtHi\u0002Si+Dt2\n2\u0002\n(Hi\u0001Si)Hi\u00001\n2H2\niSi\u0003\n1+1\n4Dt2H2\ni:(22)\nThe spin equations of motions depend directly on the neigh-\nbouring spin orientations (through the effective field) hence\nindividual spins do not commute with each other. We need to\nfurther decompose the spin system ˆLs=åiˆLsi. The following\ndecomposition will be applied for the spin system:\neˆLs(Dt=2)=eˆLs1(Dt=4):::eˆLsN(Dt=2):::eˆLs1(Dt=4)+O(Dt3)(23)\nTests of the accuracy of the integration have been per-\nformed by checking the conservation of energy within the mi-\ncrocanonical ensemble. To ensure that the spin and lattice\nsub-systems have reached equilibrium, we calculate both the\nlattice temperature (from the Equipartition Theorem) and spin\ntemperature45. These are defined as:\nTL=2\n3NkBå\nip2\ni\n2m;TS=åi(Si\u0002Hi)2\n2kBåiSi\u0001Hi: (24)\nIII. SPIN-LATTICE THERMALISATION\nAs an initial test of our model we look at the thermalisation\nprocess within micro-canonical (NVE) and canonical (NVT)\nsimulations for a periodic BCC Fe system of 10\u000210\u000210 unit\ncells. No thermostat is applied directly to the spin system and\nits thermalisation occurs via transfer of energy and angular\nmomentum from the lattice, i.e. via the magnon-phonon inter-\naction. In the case of the NVE simulations, the energy is de-\nposited into the lattice by randomly displacing the atoms from\nan equilibrium BCC structure positions within a 0 :01˚A radius\nsphere and by initialising their velocities with a Boltzmann\nFIG. 1. NVE (top) and NVT (bottom) simulations for a 10 \u000210\u000210\nunit cell BCC Fe system. The spin system is randomly initialised with\na temperature of 1900 K, while the lattice velocities are initialised\nby a Boltzmann distribution at T=300 K. In both cases we obtain\nequilibration of the two subsystems on the ps timescale.\ndistribution at T=300 K. The spin system is initialised ran-\ndomly in the x\u0000yplane with a constant component of mag-\nnetisation of 0.5 in the out of plane ( z) direction. In the case\nof NVT simulations, the lattice is connected to a thermostat\nat a temperature of T=300 K. The parameters used in the\nsimulations are presented in Table II.\nFig. 1 shows the thermalisation process for the two types of\nsimulation. In both cases the spin system has an initial temper-\nature of T=1900 K due to the random initialisation. For the\nNVE simulations, the two subsystems are seen to equilibrate\nat a temperature of T=600 K, this temperature being depen-\ndent on the energy initially deposited into the system. In the\nNVT simulations, the lattice is thermalised at T=300 K fol-\nlowed by the relaxation of the spin towards the same temper-\nature. In both cases we observe that the relaxation of the spin\nsystem happens on a 100 ps timescale, corresponding to typi-\ncal values for spin-orbit relaxation. The corresponding change\nin the magnetisation is emphasized by the green lines in Fig.\n1 showing a transfer of angular momentum between the spin\nand lattice degrees of freedom.\nTo gain a better understanding of properties at thermal equi-\nlibrium within the Spin-Lattice Dynamics model, we have in-\nvestigated the temperature dependence of the magnetic order\nparameter in different frameworks that either enable or dis-\nable lattice dynamics, specifically: SLD or ASD. Tab. I il-\nlustrates the differences between the models. Since reaching\njoint thermal equilibrium depends strongly on the randomness\nalready present in the magnetic system this process is acceler-\nated by starting with a reduced magnetisation of M=MS=0:95\nFIG. 2. Magnetisation versus temperature curves for the SLD model\n(with different choices of lattice potential: MP-Morse Potential, HP-\nHarmonic Potential) and fixed lattice ASD model. The inset zooms\naround the ferromagnetic to paramagnetic phase transition tempera-\nture.\nforT>300 K.\nFig. 2 shows the comparison of the equilibrium magnetisa-\ntion using either the harmonic potential (HP), Morse potential\n(MP) or fixed lattice (ASD) simulations. The magnetisation\nis calculated by averaging for 200 ps after an initial equilibra-\ntion for 800 ps (for SLD type simulations) or 100 ps (for ASD)\nsimulations. We observe that even without a spin thermostat\n(in SLD model) the magnetisation reaches equilibrium via the\nthermal fluctuations of the lattice, proving that both energy\nand angular momentum can be successfully transferred be-\ntween the two sub-systems. Additionally, both the SLD and\nASD methods give the same equilibrium magnetisation over\nthe temperature range considered. This confirms that the equi-\nlibrium quantities are independent of the details of the thermo-\nstat used, in agreement with the fact that both SLD and ASD\nmodels obey the fluctuation-dissipation theorem.\nIn principle, since the strength of the exchange interaction\ndepends on the relative separation between the atoms, any\nthermal expansion of the lattice could potentially modify the\nCurie temperature. However, as highlighted in the inset of\nFig. 2, the same Curie temperature is observed in each model.\nWe attribute this to fact that the thermal lattice expansion is\nsmall in the temperature range considered due to two reasons:\ni) the Curie temperature of the system is well below the melt-\ning point of Fe (\u00191800K) and ii) we model a bulk, constant-\nvolume system with periodic boundary conditions that does\nnot present strong lattice displacements due to surfaces. We\nnote that Evans et al46found a reduction of TCin nanoparticles\ndue to an expansion of atomic separations at the surface that\nconsequently reduces the exchange interactions. For systems\nwith periodic boundary conditions we anticipate fluctuations\nin the exchange parameter due to changes in interatomic spac-\nings to be relatively small. Although the equilibrium proper-\nties are not dependent on the details of the thermostat or thepotential, the magnetisation dynamics could be strongly influ-\nenced by these choices.\nThe strength of the pseudo-dipolar coupling parameters C\ndetermines the timescale of the thermalisation process. Its\nvalue can be parametrised from magneto-elastic simulations\nvia calculations of the anisotropy energy as a function of\nstrain. The magneto-elastic Hamiltonian can be written for a\ncontinuous magnetisation Mand elastic strain tensor eas47,48:\nHm\u0000e=B1\nM2\nSå\niM2\nieii+B2\nM2så\niMiMjei j (25)\nwhere constants B1;B2can be measured experimentally49.\nThe pseudo-dipolar term acts as a local anisotropy, however,\nfor a lattice distorted randomly, this effective anisotropy is av-\neraged out to zero. At the same time under external strain\neffects, an effective anisotropy will arise due to the pseudo-\ndipolar coupling which is the origin of the magneto-elastic\neffects. To calculate the induced magnetic anisotropy energy\n(MAE), the BCC lattice is strained along the zdirection whilst\nfixed in the xyplane. The sample is then uniformly rotated and\nthe energy barrier is evaluated from the angular dependence of\nthe energy. Fig. 3 shows MAE for different strain values and\ncoupling strengths, with the magneto-elastic energy densities\nconstants B1obtained from the linear fit. The values of the ob-\ntained constants B1are larger than the typical values reported\nforBCC FeB1=\u00003:43 MJ m\u00003=\u00006:2415\u000210\u00006eV A\u0000349\nmeasured at T=300 K. Although the obtained magneto-\nelastic coupling constants for BCC Fe are larger than experi-\nmental values, it is important to stress that, as we will see later,\na large coupling is necessary in order to obtain damping pa-\nrameters comparable to the ones known for magnetic insula-\ntors where the main contribution comes from magnon-phonon\nscattering. In reality, in BCC Fe there is an important contribu-\ntion to the effective damping from electronic sources, which if\nconsidered, can lead to the smaller coupling strengths, consis-\ntent in magnitude with experimental magneto-elastic parame-\nters. Indeed, as we will show later, our finding suggests that\nphonon damping is a very small contribution in metallic sys-\ntems such as BCC Fe .\nIV . DYNAMIC PROPERTIES AT THERMAL\nEQUILIBRIUM\nSection III showed that the equilibrium magnetisation does\nnot depend on the details of the thermostat used and a success-\nful transfer of both energy and angular momentum is achieved\nbetween the spin and lattice sub-systems by the introduction\nof a pseudo-dipolar coupling term. In this section, we inves-\ntigate the properties of the magnons, phonons and the cou-\npling term that equilibrates the spin and phonon systems in\nthe absence of a phenomenological spin damping. Two types\nof simulations are presented here: i)magnon and phonon\nspectra calculated along the high symmetry path of a BCC\nlattice and ii)averaged temporal Fourier transform (FT) of\nindividual atoms datasets (spin, velocity, pseudo-dipolar cou-\npling field). The phonon - Fig. 4 and magnon - Fig. 5 spec-6\nFIG. 3. Magnetic anisotropy energy as function of strain for different\ncoupling strengths for T=0K.\ntra are calculated by initially equilibrating the system for 10\nps with a spin thermostat with aG=0:01 and a coupling of\nC=0:5, followed by 10 ps of equilibration in the absence\nof a spin thermostat. For the method i)the correlations are\ncomputed for a runtime of 20 ps after the above thermalisa-\ntion stage. For each point in k-space, the first three maxima\nof the auto-correlation function are plotted for better visual-\nisation. The auto-correlation function is projected onto the\nfrequency space and the average intensity is plotted for dif-\nferent frequencies. The phonon spectra are calculated from\nthe velocity auto-correlation function defined in Fourier space\nas33,50:\nAp(k;w) =Ztf\n0hvp\nk(t)vp\nk(t\u0000t0)ie\u0000iwtdt (26)\nwhere p=x;y;z,tfis the total time and vp\nk(t)is the spatial\nFourier Transform calculated numerically as a discrete Fourier\nTransform:\nvp\nk(t) =å\nivp\nie\u0000ik\u0001ri (27)\nThe same approach is applied for the magnon spectra, us-\ning the dynamical spin structure factor, which is given by\nthe space-time Fourier transform of the spin-spin correlation\nfunction defined as Cmn(r\u0000r0;t\u0000t0) =<Sm(r;t)Sn(r0;t0)>,\nwith m;ngiven by the x,y,z components51:\nSmn(k;w) =å\nr;r0eik\u0001(r\u0000r0)Ztf\n0Cmn(r\u0000r0;t\u0000t0)e\u0000iwtdt(28)\nThe second method ( ii)) to investigate the properties of the\nsystem involves calculating temporal Fourier transform of in-\ndividual atoms datasets, and averaging the Fourier response\nover 1000 atoms of the system. This response represents an\nintegrated response over the k-space. Hence, the projection of\nintensities on the frequency space presented by method i)has\nsimilar features as the spectra presented by method ii). For theresults presented in Fig. 6, a system of 10 \u000210\u000210BCC unit\ncells has been chosen. The system has been equilibrated for\na total time of 20 ps with the method presented in i)and the\nFast Fourier transform (FFT) is computed for the following\n100 ps.\nFig. 4 shows the phonon spectra for a SLD simulations at\nT=300K, C=0:5 for the Morse Potential - Fig. 4(a) and the\nHarmonic Potential - Fig. 4(b) calculated for the high sym-\nmetry path of a BCC system with respect to both energy and\nfrequency units. The interaction cutoff for both Morse and\nHarmonic potential is rc=7:8˚A. The Morse phonon spec-\ntrum agrees well with the spectrum observed experimentally32\nand with the results from33. The projection of the spectra onto\nthe frequency domain shows a peak close to 10.5 THz, due\nto the overlap of multiple phonon branches at that frequency\nand consequently a large spectral density with many k-points\nexcited at this frequency. Moving now to the harmonic poten-\ntial, parameterised as in Ref. 27, we first note that we observe\nthat some of the phonon branches overlap - Fig.4b). Secondly,\nthe projection of intensity onto the frequency domain shows a\nlarge peak at 8.6THz, due to a flat region in the phonon spec-\ntra producing even larger number of k-points in the spectrum\nwhich contribute to this frequency. Finally, the large cutoff\nmakes the Harmonic potential stiffer as all interactions are\ndefined by the same energy, V0, and their equilibrium posi-\ntions corresponding to a BCC structure. This is not the case\nfor the Morse potential which depends exponentially on the\ndifference between the inter-atomic distance and a constant\nequilibrium distance, r0. For a long interaction range, the har-\nmonic approximation will result in a more stiff lattice than the\nMorse parameterisation.\nIn principle, the harmonic potential with a decreased in-\nteraction cutoff and an increased strength could better repro-\nduce the full phonon spectra symmetry for BCC Fe. How-\never, in this work we preferred to use the parameterisation\nexisting in literature27and a large interaction cutoff for sta-\nbility purposes. Although the full symmetry of the BCC Fe\nphonon spectra is not reproduced by this harmonic potential,\nthe phonon energies/frequencies are comparable to the values\nobtained with the Morse potential. Nevertheless, we observed\nthe same equilibrium magnetisation and damping (discussed\nlater) for both potentials, hence the simple harmonic potential\nrepresents a suitable approximation, that has the advantage of\nbeing more computationally efficient.\nFig. 5 shows the magnon spectrum obtained within the SLD\nframework using the Morse potential together with its pro-\njection onto the frequency domain. The results agree very\nwell with previous calculations of magnon spectra28,52. For\nthe harmonic potential the magnon spectrum is found to be\nidentical to that for the Morse potential with only very small\nchanges regarding the projection of intensity onto the fre-\nquency domain. This is in line with our discussion in the pre-\nvious section where the choice of inter-atomic potential had\nlittle effect on the Curie temperature, which is closely linked\nto the magnonic properties. As the harmonic potential is more\ncomputationally efficient than the Morse, we next analyse the\nproperties of the system for a 10 \u000210\u000210 unit cells system\natT=300K with the harmonic potential.7\nFIG. 4. Phonon spectra calculated for a 32 \u000232\u000232 unit cell system at T=300K, C=0:5 for a) Morse potential, b)Harmonic potential. The\nspectra are calculated via method i).\nRight figure includes the projection of the intensity of the spectra onto the frequency domain. Solid lines are the experimental data of\nMinkiewicz et al32. For the Minkiewicz et al data there is only 1 datapoint for the N- Gpath for the second transverse mode which does not\nshow up on the line plots.\nFIG. 5. Magnon spectrum (x component) calculated for a 32 \u000232\u0002\n32 unit cell system at T=300K, C=0:5 for a Morse potential. The\nspectrum is calculated via method i).\nRight figure includes the projection of the intensity of the spectrum\nonto the frequency domain.\nThe power spectral density (auto-correlation in Fourier\nspace) of the magnon, phonons and coupling field at 300 K\nis shown in Fig. 6 computed using method iidetailed previ-\nously. The amplitude of the FFT spectra of velocities and\ncoupling field has been scaled by 0.12 and 0.05 respectively to\nallow for an easier comparison between these quantities. As\nshown in Fig. 6.a) the coupling term presents both magnon\nand phonon characteristics; demonstrating an efficient cou-\npling of the two sub-systems. The large peak observed at\na frequency of 8 :6 THz appears as a consequence of the flat\nphonon spectrum for a Harmonic potential, as observed in the\nspectrum and its projection onto the frequency domain in Fig.\n4.b). Additionally, Fig. 6.a) can give us an insight into the in-duced spin noise within the SLD framework. The background\nof the FFT of the coupling field is flat for the frequencies plot-\nted here, showing that the noise that acts on the spin is uncor-\nrelated. The inset shows a larger frequency domain where it\nis clear that there are no phonon modes for these frequencies,\nand only thermal noise decaying with frequency is visible. At\nthe same time an excitation of spin modes are visible for fre-\nquencies up to ca .100 THz.\nThe characteristics of the coupling field with respect to the\ncoupling strength for a dynamic (SLD) and fixed lattice simu-\nlations (ASD) are presented in Fig. 6(b). The only difference\nbetween the ASD and SLD simulations is given by the pres-\nence of phonons (lattice fluctuations) in the latter. Since the\nlarge peak at 8 :6 THz is due to the lattice vibrations, it is not\npresent in the ASD simulations. The smaller peaks are present\nin both models since they are proper magnonic modes. With\nincreasing coupling the width of the peaks increases suggest-\ning that the magnon-phonon damping has increased. Moving\ntowards the larger frequency regimes, Fig. 6.b) - (inset), we\nobserve that large coupling gives rise to a plateau in the spec-\ntra at around 150 THz, which is present as well for the fixed-\nlattice simulations (ASD). The plateau arises from a weak an-\ntiferromagnetic exchange that appears at large distances due to\nthe competition between the ferromagnetic exchange and the\nantiferromagnetic exchange-like term in the pseudo-dipolar\ncoupling.\nWe have also analysed the characteristics of the magnon\nand phonon spectra for different temperatures- Fig. 7. With8\nFIG. 6. The power spectral density of the auto-correlation function in the frequency domain for magnons, phonons and coupling field for a\nSLD simulations with a Harmonic lattice, calculated by method ii). Panel a) shows the power density of the auto-correlation function of the\nx component of the velocity vx, spin Sxand coupling field Hcx. Panel b) presents the power density of the auto-correlation function for the x\ncomponent of the coupling field for either static (ASD) or dynamic (SLD) lattice. The insets show the high-frequency spectra. For Panel a) the\nvelocity and the coupling field have been multiplied by a factor of 0.12 and 0.05 respectively for easier graphical comparison.\nFIG. 7. The power spectral density of the auto-correlation function in the frequency domain for magnons - Panel a) and phonons - Panel b) for\na SLD simulations with a Harmonic lattice, calculated by method ii), for three distinct temperatures and a coupling constant of C=0:5.\nincreasing temperature, the peaks corresponding to magnons\nshift to smaller frequencies. This is a typical situation known\nas a softening of low-frequency magnon modes due to the in-\nfluence of thermal population, see e.g.53- Panel a). The same\neffect can be observed by calculating the magnon spectra via\nmethod ifor various temperatures. In Panel b), the peak cor-\nresponding to phonons remains almost at the same frequency\nof about 8 :6 THz, as the phonon spectra is not largely affected\nby temperatures up to T=600K. The increase of the effec-\ntive damping (larger broadening) of each magnon mode with\ntemperature is clearly observed.\nV . MACROSOPIC MAGNETISATION DAMPING\nIn this section we evaluate the macroscopic damping pa-\nrameter experienced by magnetisation due to the magnon-\nphonon excitations for a periodic BCC system using our SLD\nmodel. This method for calculating the damping has been\npresented in54–56. The system is first thermalised at a non-\nzero temperature in an external field of Bext=50T applied in\nthezdirection, then the magnetisation is rotated coherently\nthrough an angle of 30\u000e. The system then relaxes back to\nequilibrium allowing the relaxation time to be extracted. The\naveraged zcomponent of magnetisation is then fitted to the\nfunction mz(t) =tanh(agBext(t+t0)=(1+a2))where arep-resents the macrosopic (LLG-like) damping, gthe gyromag-\nnetic ratio and t0a constant related to the initial conditions.\nThe model system consists of 10 \u000210\u000210 unit cells and the\ndamping value obtained from fitting of mz(t)is averaged over\n10 different simulations.\nFig. 8 shows the dependence of the average damping pa-\nrameter together with the values obtained from individual\nsimulations for different temperatures and coupling strengths\nfor two choices of mechanical potential. In our model, the\nspin system is thermalised by the phonon thermostat, hence\nno electronic damping is present. With increasing coupling,\nthe energy and angular momentum transfer is more efficient,\nhence the damping is enhanced. Since the observed value of\ninduced damping is small, calculating the damping at higher\ntemperature is challenging due to the strong thermal fluctua-\ntions that affect the accuracy of the results. Despite the low\ntemperatures simulated here, the obtained damping values (at\nT=50K, a=4:9\u000210\u00005) are of the same order as reported\nfor magnetic insulators such as YIG (1 \u000210\u00004to 1\u000210\u0000657,58\n) as well as in different SLD simulations (3 \u000210\u00005,27). Gener-\nally, the induced damping value depends on the phonon char-\nacteristics and the coupling term, that allows transfer of both\nenergy and angular momentum between the two subsystems.\nFig. 8(a) and (b) compare the calculated damping for the\nMorse and Harmonic potential for two values of the coupling\nstrength. We observe that the values are not greatly affected9\nFIG. 8. Damping parameter extracted from fitting the z component of the magnetisation for two different choices of potential: HP- Harmonic\nPotential (green open squares) and MP-Morse Potential (black open circles) as function of temperature Fig. a), b) and as function of the\ncoupling strength Fig. c), d); Fig a) and b) are calculated for a constant coupling strength of C=0:3,C=0:5 respectively. Fig c) and d)\nare calculated for temperatures of T=100K,T=300Krespectively. The black and green lines represents the average damping parameter\nobtained from the simulations using the Morse and the Harmonic Potentials, respectively.\nby the choice of potential. This arises due to the fact that only\nthe spin modes around Gpoint are excited and for this low k-\nvectors modes the inter-atomic distances between neighbour-\ning atoms do not vary significantly. The extracted damping\nparameter as a function of coupling strength for 100 K and\n300 K is presented in Fig. 8(c) and (d) respectively. The func-\ntional form of the variation is quadratic, in accordance with\nthe form of the coupling term. Measurements of damping in\nmagnetic insulators, such as YIG, show a linear increase in the\ndamping with temperature,58which agrees with the relaxation\nrates calculated by Kasuya and LeCraw59and the relaxation\nrates calculated in the NVE SLD simulations in Ref. 27. How-\never, Kasuya and LeCraw suggest that the relaxation rate can\nvary as Tn, where n=1\u00002 with n=2 corresponding to larger\ntemperature regimes. Nevertheless, the difference between\nthe quadratic temperature variation of the damping observed\nin our simulations and the linear one observed in experiments\nfor YIG can be attributed to the difference in complexity be-\ntween the BCC Fe model and YIG. The difference between\nthe trends may as well suggest that the spin-orbit coupling in\nYIG could be described better by a linear phenomenological\ncoupling term, such as the one used in Refs. 26 and 29, but\nwe note that such forms can lead to a uniform force in the di-\nrection of the magnetisation and so might need further adap-\ntation before being suitable. To test an alternate form of the\ncoupling we have changed the pseudo-dipolar coupling to an\non-site form, specifically Hc=\u0000åi;jf(ri j)((Si\u0001ˆri j)2\u00001\n3S2\ni)\ni.e a N ´eel-like anisotropy term. This leads to much smallerdamping as shown in Fig. 9 ( T=300 K, a=3:3\u000210\u00005,\naveraged over 5 realisations) making it difficult to accurately\ncalculate the temperature dependence of the damping, espe-\ncially for large temperatures. The magnon-phonon damping\ncan clearly have complex behavior depending on the proper-\nties of the system, especially the coupling term, hence no uni-\nversal behaviour of damping as function of temperature can\nbe deduced for spin-lattice models.\nNeglecting the lattice contribution, the temperature depen-\ndence of the macrosopic damping can be mapped onto the\nLandau-Lifshitz-Bloch formalism (LLB)54and theory17and\nASD simulations60have shown it to vary inversely with the\nequilibrium magnetisation. The LLB theory shows that the\nmacrosopic damping is enhanced for large temperatures due\nto thermal spin fluctuations. Using the equilibrium magneti-\nsation it is possible to approximate the variation of damping\nwith temperature produced due to thermal fluctuations within\nthe LLB model. From 100K to 400K the damping calculated\nvia the LLB model increases within the order of 10\u00005, which\nis considerably smaller than the results obtained via the SLD\nmodel. This shows that within the SLD model the temperature\nincrease of the damping parameter is predominantly due to\nmagnon-phonon interaction, and not due to thermal magnon\nscattering, as this process is predominant at larger tempera-\ntures.10\nFIG. 9. Temperature variation of the damping parameter for N ´eel-\nlike on-site coupling, Hc=\u0000åi;jf(ri j)((Si\u0001ˆri j)2\u00001\n3S2\ni). The val-\nues are extracted from mz(t)fittings for 10 realisations;\nVI. CONCLUSIONS AND OUTLOOK\nTo summarise, we have developed a SLD model that is able\nto transfer energy and angular momentum efficiently from\nthe spin to lattice sub-systems and vice-versa via a pseudo-\ndipolar coupling term. Our approached takes the best fea-\ntures from several previously suggested models and general-\nize them which allows modelling in both canonical and mi-\ncrocanonical ensembles. With only the lattice coupled to\na thermal reservoir and not the spin system, we reproduce\nthe temperature dependence of the equilibrium magnetisation,\nwhich agrees with the fact that the spin-lattice model obeys\nthe fluctuation-dissipation theorem. We are able to study the\ndynamic properties such as phonon and spin spectrum and\nmacrosopic damping, showing that the magnetic damping isnot greatly influenced by the choice of potential, however it\nis influenced by the form of the coupling term. This enables\nthe possibility of tailoring the form of the coupling term so it\ncan reproduce experimental dependencies of damping for dif-\nferent materials. In future, the addition of quantum statistics\nfor Spin Lattice Dynamics models61,62may also yield better\nagreement with experimental data.\nThe SLD model developed here opens the possibility of the\ninvestigation of ultrafast dynamics experiments and theoret-\nically studies of the mechanism through which angular mo-\nmentum can be transferred from spin to the lattice at ultrafast\ntimescales. As we have demonstrated that the model works\nwell in the absence of an phenomenological Gilbert damping,\nwhich consists mainly of electronic contributions, the SLD\nmodel can be employed to study magnetic insulators, such\nas YIG, where the principal contribution to damping is via\nmagnon-phonon interactions. Future application of this model\nincludes controlling the magnetisation via THz phonons7\nwhich can lead to non-dissipative switching of the magnetisa-\ntion11,12. With the increased volume of data stored, field-free,\nheat-free switching of magnetic bits could represent the future\nof energy efficient recording media applications. Another pos-\nsible application is more advanced modelling of the ultrafast\nEinstein-de-Haas effect2or phonon-spin transport63.\nVII. ACKNOWLEDGEMENTS\nWe are grateful to Dr. Pui-Wai Ma and Prof. Matt\nProbert for helpful discussions. Financial support of the Ad-\nvanced Storage Research Consortium is gratefully acknowl-\nedged. MOAE gratefully acknowledges support in part from\nEPSRC through grant EP/S009647/1. The spin-lattice simula-\ntions were undertaken on the VIKING cluster, which is a high\nperformance compute facility provided by the University of\nYork. 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Shaw1 \n \n1Quantum E lectromagnetics Division, National Institute of Standards and Technology, Boulder, CO \n80305 , USA  \n2Institute of Experimental and Applied Physics, University of Regensburg, 93053 Regensburg, \nGermany  \n3Department of Applied P hysics, Eindhoven University of Technology, 5600 MB Eindhoven, The \nNetherlands  \n \n \nDated: 01/05/2017  \n \n *Corresponding author: martin1.schoen@physik.uni -regensburg.de  \n \n \n \n \nAbstract  \nA systematic experimental study of Gilbert damping is performed via ferromagnet ic \nresonance for the disordered  crystalline  binary 3 d transition metal alloys Ni -Co, Ni -Fe and \nCo-Fe over the full range of alloy compositions. After accounting for inhomogeneous \nlinewidth  broadening , the damping shows clear evidence of both  interfacial da mping \nenhancement (by spin pumping) and radiative damping. We quantify these two extrinsic  \ncontributions  and thereby determine the intrinsic damping. The comparison of the intrinsic \ndamping to multiple theoretical calculation s yields good qualitative  and q uantitative  \nagreement  in most cases . Furthermore, the values of the damping obtained in this study are \nin good agreement with  a wide range of published experimental and theoretical values . \nAdditionally, we find a compositional dependence of the spin mixing  conductance.  \n \n \n \n \n 2  \n \n \n \n1  Introduction  \nThe magnetization dynamics in f erromagnetic films are phen omenologically well described \nby the Landau -Lifshitz -Gilbert formalism (LLG)  where  the damping  is described by a \nphenomenological damping parameter α.4,5 Over the past four decades, there have been \nconsiderable efforts to derive the phenomenological d amping parameter from first principles \ncalculations and to do so in a quantitative manner. One of the early promising  theories was that of \nKamberský, who introduced the so -called breathing Fermi surface model6–8. The name “breathing \nFermi  surface ” stems from the picture that the precessing magnetization, due to spin -orbit coupling, \ndistorts the Fermi surface. Re -populating the Fermi surface is delayed by the scattering time, \nresulting in a phase lag between the precession  and the Fermi sur face distortion. This  lag leads to  a \ndamping that is proportional to the scattering time. Although this approach describes the so-called \nconductivity -like behavior of the damping at low temperatures, it fails to describe  the high \ntemperature behavior of so me materials . The high temperature or resistivity -like behavior is \ndescribed by the so-called  “bubbling Fermi surface ” model. In the case of energetically shifted  \nbands , thermal  broaden ing can lead to a significant  overlap  of the spin-split bands  in 3d \nferromagnets.  A precessing magnetization can induce elec tronic transitions between such \noverlapping bands , leading to spin-flip process es. This process scales with the amount of band \noverlap. Since s uch overlap is further increased with the band broadening th at results from the finite \ntemperature of the sample, this contribution  is expected to increase as the temperature is increased. \nThis model for interband transition mediated damping describes the resistivity -like behavior of the \ndamping at higher temperatu res (shorter scattering times).  These two damping processes are  \ncombined in a torque correlation model by Gilmore , et al.9, as well as Thonig , et al.10, that  describes \nboth the low -temperature  (intra band transitio ns) and high -temperature  (inter band transitions) \nbehavior  of the damping . Another app roach via  scattering theory was successfully implemented by \nBrataas , et al.11 to describe damping in transition metals. Starikov , et al. ,2 applied the scattering \nmatrix approach to calculate the damping  of NixFe1-x alloys and  Liu, et al. ,12 expanded the formalism \nto include the influence of electron -phonon interaction s.  \nA numerical realization of the torque correlation model was performed  by Mankovsky , et \nal., for NixCo1-x, Ni xFe1-x, CoxFe1-x, and FexV1-x1. More recently , Turek , et al. ,3 calculated the \ndamping for NixFe1-x and CoxFe1-x alloys with the torque -correlation model, u tilizing non -local \ntorque correlators. It is important to stress that a ll of these  approaches  consider  only the intrinsic \ndamping. This complicates the quantitative comparison of calculated values for t he damping to \nexperimental data  since there are many extrinsic contributions to the damping that result from \nsample  structure , measurement geometr y, and/or sample properties . While some extrinsic \ncontributions to the damping  and linewidth  were discovered in the 1960 ’s and 1970 ’s, and are well \ndescribed  by the ory, e.g. eddy -current damping13,14, two -magnon scattering15–17, the slow rel axer \nmechanism18,19, or radiative damping20,21, interest in these mechanisms has been re -ignited \nrecently22,23. Further contributions, such as  spin-pumping, both extrinsic24,25 and intrinsic24,26, have 3 been discovered more recently and a re subject to extensive research27–31 for spintronics application.  \nTherefore , in order to allow a quantitative comparison to theoretical calculations  for intrins ic \ndamping, both the measurement and sample geometry must  be designed  to allow both the \ndetermination and possibl e minimization  of all additional contributions to the measured damping.  \nIn this study , we demonstrate methods to determine significant extrins ic contributions to the \ndamping , which  includ es a measurement of the effective spin mixing conductance for both the pure \nelements and select alloys. By precisely accounting for all of these extrinsic contributions, we \ndetermine the intrinsic damping  parame ters of the binary alloys Ni xCo1-x, Ni xFe1-x and Co xFe1-x and \ncompare them  to the calculation s by Mankovsky , et al. ,1, Turek , et al. , and Starikov , et al.2. \nFurthermore, we present the concentration -dependence of the  inhomogeneous linewidth \nbroadening, which for most alloys shows exceptionally small values, indicative  of the high \nhomogeneity of our samples.  \n2 Samples and method  \nWe deposited  NixCo1-x, Ni xFe1-x and Co xFe1-x alloys of varying composition (all compositio ns \ngiven in atomic percent)  with a thickness of 10 nm  on an oxidized (001)  Si substrate with  a Ta(3 \nnm)/Cu(3 nm) seed layer and  a Cu(3 nm) /Ta(3 nm) cap layer. In order to investigate interface \neffects , we also deposited multiple thickness series at 10 nm, 7 nm, 4 nm, 3 nm , and 2 nm of  both \nthe pure elements and select alloy s.  Structural characterization  was performed using X-ray \ndiffraction (XRD).  Field swept vector -network -analyzer  ferromagnetic resonance spectroscopy \n(VNA -FMR) was used in the out -of-plane geometry  to determine the total damping parameter αtot. \nFurther d etails of the deposition conditions, XRD, FMR measurement and fitting of the complex \nsusceptibility to the measured S21 parameter are reported in  Ref [66]. \nAn example of susceptibility fits t o the complex S21 data is shown in Fig.  1 (a) and (b). All \nfits were constrained  to a 3×  linewidth ΔH field window around the resonance field in order to \nminimize the influence of measurement drifts on the error in the susceptibility fits. The total \ndampin g parameter αtot and the inhomogeneous linewidth broadening Δ H0 are then determined  from \na fit to  the linewidth Δ H vs. frequency f plot22, as shown in Fig.  1 (c). \n ∆𝐻= 4𝜋𝛼𝑡𝑜𝑡𝑓\n𝛾𝜇0+ ∆𝐻0, (1) \nwhere γ=gμB/ħ is the gyro -magnetic ratio, μ0 is the vacuum permeability, μB is the Bohr -magneton , \nħ is the reduced Planck constant,  and g is the spectroscopic g-factor reported in Ref [ 66].  \n  4  \nFigure 1: (a) and (b) show respectively the real and imaginary part of the S21 transmission \nparameter (black squares)  measured at 20 GHz  with the complex susceptibility fit (red lines)  for \nthe Ni 90Fe10 sample . (c) The linewidths  from the suscept ibility fits  (symbols) and linear fits (solid \nlines) are plotted against frequency for different Ni -Fe compositions. Concentrations are denoted \non the right -hand  axis. The damping α and the inhomogeneous linewidth broadening Δ H0 for each \nalloy can be extracted from the fits via Eq.  (1). \n3 Results  \nThe first contribution t o the linewidth we discuss  is the  inhomogeneous linewidth broadening \nΔH0, which  is presumably  indicative of  sample inhomogeneity32,33. We plot Δ H0 for all the alloy \nsystem s against the respective concentrations in Fig.  2. For all alloys , ΔH0 is in the range of a few \nmT to 10  mT . There are only a limited  number of reports  for ΔH0 in the literature with which to \ncompare . For Permalloy (Ni 80Fe20) we measure Δ H0 = 0.35 mT, which is close to other reported \nvalues .34 For the  other  NixFe1-x alloys , ΔH0 exhibits a significant peak  near the fcc-to-bcc (face -\ncentered -cubic to body -centered -cubic) phase transition  at 30  % Ni , (see Fig.  2 (b)) which is easily \nseen in the raw data  in Fig. 1 (c). We speculate  that this  increase of  inhomogeneous broadening  in \nthe NixFe1-x is caused by the coexistence of the bcc and fcc  phases at the phase transition.  However,  \nthe CoxFe1-x alloys do not exhibit an increase in ΔH0 at the equivalent phase transition  at 70  % Co . \nThis suggests that the bcc and fcc phases of NixFe1-x tend to segregate near the phase transition, \nwhereas the same phases for CoxFe1-x remain intermixed throughout the transition.  \n5 One possible explanation for inhomogeneous broadening is magnetic anisotropy , as originally \nproposed  in Ref. [35]. However, this explanation does not account for our measured dependence of \nΔH0 on alloy concentration, since the perpendicular magnetic anisotropy , described in Ref [ 66] \neffectively  exhibits opposite behavior with alloy concentration.  For our alloys Δ H0 seems to roughly \ncorrelate to the inverse exchange constant36,37, which co uld be a starting point for future \ninvestigation of a quantitative theory of inhomogeneous broadening.  \n \n \n \n  \nFigure 2: The inhomogeneous linewidth -broadening Δ H0 is plotted vs. alloy composition for (a) \nNi-Co, (b) Ni -Fe and (c) Co -Fe. The alloy phases are denoted by color code described in Ref [ 66] \n \nWe plot the total measured damping αtot vs. composition for NixCo1-x, NixFe1-x and CoxFe1-x in \nFig. 3 (red crosses). The total damping of the NixCo1-x system increases monotonically  with \nincreased Ni content . Such smooth  behavior in the damping is  not surprising  owing to the absence \nof a phase transition for this alloy . In the NixFe1-x system , αtot changes very little from pure Fe to \napproximately 25 % Ni  where the bcc to fcc phase transition occurs . At the phase  transition,  αtot \nexhibits a step, increasing  sharply by approximately 30  %. For higher Ni concentrations , αtot \nincreases monotonically  with increasing Ni concentration . On the other hand, t he CoxFe1-x system \nshows a different behavior in the damping and d isplays a sharp minimum of (2.3 ± 0.1)×10-3 at 25 \n6 % Co as previously reported38. As the system changes to an fcc phase ( ≈ 70 % Co), αtot become \nalmost  constant.  \nWe compare our data to previously published values in Table I. However, direct c omparison \nof our data to previous report s is not trivial , owing to the variation in measurement conditions  and \nsample characteristic s for all the reported measurements . For example , the damping  can depend  on \nthe temperature .9,39 In addition, multiple intrinsic and intrinsic contributions to the total damping  \nare not always accounted for in the literature . This can be seen in the fact that the reported  damping \nin Ni80Fe20 (Permalloy) varies  from α=0.0 055 to α=0.04  at room tem perature  among studies . The \nlarge variation  for these reported data  is possibly the result of  different uncontrolled contributions  \nto the extrinsic damping  that add  to the total damping in the different experiments , e.g. spin-\npumping40–42, or roughness41. Therefore , the value for the intrinsic damping of Ni 20Fe80 is expected \nto be at the low end of this scatter . Our measured value of α=0.007 2 lies within the range of reported \nvalues. Similarly , many of our measured damping values for different alloy compositions lie within \nthe range  of reported values22,43 –48. Our measured  damping of the pure elements and the Ni 80Fe20 \nand Co 90Fe10 alloys is compared to room temperature values found in literature in Table 1, Col umns  \n2 and 3 . Column 5 contains theoretically calculated values .   \n \nTable 1: The total measured damping α tot (Col. 2)  and the intrinsic damping (C ol. 4) f or Ni 80Fe20, \nCo90Fe10, and the pure elements are compared to both experimental (Col. 3) and theoretical (Col. \n5) values from the literature . All values of the damping  are at room temperature if not noted \notherwise . \nMaterial  αtot (this study)  \n Liter ature values  αint (this study)  Calculated literature \nvalues  \nNi 0.029 (fcc)  0.06444 \n0.04549 0.024 (fcc)  0.0179 (fcc) at 0K  \n0.02212 (fcc) at 0K  \n0.0131 (fcc)  \nFe 0.0036 (bcc)  0.001944 \n0.002746 0.0025 (bcc)  0.00139 (bcc) at 0K  \n0.001012 (bcc)  at 0K  \n0.00121 (bcc) at 0K  \nCo 0.0047 (fcc)  0.01144  \n 0.0029 (fcc)  0.00119 (hcp)  at 0K  \n0.0007312 (hcp)  at 0K  \n0.0011 (hcp)  \nNi80Fe20 0.0073 (fcc)  0.00844 \n0.008 -0.0450 \n0.007848 \n0.00751 \n0.00652 \n0.00647 \n0.005553 0.0050 (fcc)  0.00462,54 (fcc) at 0K  \n0.0039 -0.00493 (fcc)  at 0K  \nCo90Fe10 0.0048 (fcc)  0.004344 \n0.004855 0.0030 (fcc)   7  \n \n \n \n  \nFigure 3: (color online) The measured damping αtot of all the alloys is plotted against the alloy \ncompositi on (red crosses) for (a) Ni -Co, (b) Ni -Fe and (c) Co -Fe (the data in (c) are taken from \nRef.[38]). The black squares  are the intrinsic damping  αint after correction for spin pumping and \nradiative contributions to the measured damping. The blue line is the intr insic damping calculated \nfrom the Ebert -Mankovsky theory ,1 where the blue circles are the values for the pure elements  at \n300K . The green  line is the calculated damping for the Ni -Fe alloys  by Starikov , et al.2 The inset \nin (b) depicts the damping in a smaller concentration window in order to better depict the small \nfeatures in the damping around the ph ase transition. The damping  for the Co -Fe alloys, calculated \nby Turek et al.3 is plotted as the orange  line. For the Ni -Co alloys the damp ing calculated by th e \nspin density of the respective alloy weighted bulk damping55 (purple dashed line).  \n8  \n \nThis scatter in the experimental data reported in the literature and its divergence from calculated \nvalues of the damping shows th e necessity to determine  the intrinsic damping  αint by quantification \nof all extrinsic contributions to the measured total damping α tot.  \nThe first extrinsic  contribution to the damping  that we consider  is the radiative damping α rad, \nwhich is caused by ind uctive coupling between sample and  waveguide , which results in energy \nflow from the sample back into the microwave circuit.23 αrad depends directly  on the measurement \nmethod and geometry. The effect is easily understood , since the strength of the inductive coupling \ndepends on the inductance of the FMR mode itself , which is in turn determined by the saturation \nmagnetization, sampl e thickness, sample  length, and waveguide width.  Assuming a homogeneous \nexcitation field , a uniform magnetization profile throughout the sample , and negligible spacing \nbetween the waveguide and sample , αrad is well approximated by23  \n 𝛼rad=𝛾𝑀𝑠𝜇02𝛿𝑙\n16 𝑍0𝑤𝑐𝑐, (2) \nwhere l (= 10 mm  in our case) is the sample length on the waveguide, wcc (= 100µm) is the width \nof the co -planar wave guide center conductor and Z0 (= 50 Ω), the impedance of the waveguide. \nThough inh erently small for most thin films , αrad can become  significant  for alloys with \nexceptionally small intrinsic damping and /or high saturation magnetization.  For example, it plays \na significant role (values of αrad ≈ 5x10-4) for the whole composition range of  the Co -Fe alloy system \nand the Fe -rich side of the Ni -Fe system. On the other hand, for pure Ni and Permalloy (Ni 80Fe20) \nαrad comprises only 3 % and 5  % of αtot, respectively.  \nThe second non -negligible contribution to the damping  that we consider  is the interfacial \ncontribution to the measured  damping , such as  spin-pumping into the adjacent Ta/Cu bilayers . Spin \npumping  is proportional to the reciprocal sample thickness as described in24 \n 𝛼sp= 2𝑔eff↑↓𝜇𝐵𝑔\n4𝜋𝑀s𝑡. (3) \nThe spectroscopic  g-factor and the saturation magnetization Ms of the alloys were reported  in \nRef [66] and the factor of 2 accounts for the presence of two nominally identical interfaces of the \nalloys in the cap and seed layers.  In Fig.  4 (a)-(c) we plot the damping dependence on reciprocal \nthickness  1/t for select alloy concentrations, which allows us to determine  the effective spin mixing \nconductance 𝑔eff↑↓  through fits to Eq. (3) . The effective spin m ixing conductance contains details of \nthe spin transport in the adjacent non -magnetic layers, such as the interfacial spin mixing \nconductance, both the conductivity and spin diffusion for all the non -magnetic layers with a non -\nnegligible spin accumulation,  as well as the details of the spatial profile for the net spin \naccumulation .56,57 The values of 𝑔eff↑↓, are plotted versus the alloy concentration in Fig. 4 (d), and are \nin the range of previously reported values  for samples prepare d under similar growth conditions55–\n59. Intermediate values of 𝑔eff↑↓ are determined by a guide to the eye interpolation [ grey lines, Fig.  4 \n(d)] and αsp is calculated for all alloy concentrations utilizing  those interpolated values.  \nThe data for 𝑔eff↑↓  in the NixFe1-x alloys shows approximately a factor two  increase of 𝑔eff↑↓ between \nNi concentrations of 30  % Ni and 50  % Ni, which we speculate to occur at the fcc to bcc phase \ntransition around 30  % Ni. According to this line of speculation , the previously mentioned step  \nincrease in the measured total damping at the NixFe1-x phase transition can be fully attributed to the \nincrease in spin pumping at the phase transition.  In CoxFe1-x, the presence of a step in  𝑔eff↑↓   at the \nphase transition is not confirmed, given the measurement precision, although  we do observe an \nincrease in the effective spin mixing conductance when transitioning from the bcc to fcc phase.  The 9 concentration dependence of 𝑔eff↑↓  requires further thorough investigation and we therefore restrict \nourselves to reporting the expe rimental findings.  \n \n \n           \n \nFigure 4: The damping for the thickness series at select alloy compositions vs. 1/ t for (a) Ni -Co, \n(b) Ni -Fe and (c) Co -Fe (data points, concentrations denoted in the plots), with linear fits to Eq. \n(3) (solid lines). (d) The extracted effective spin mixing conductance 𝑔eff↑↓  for the measured alloy \nsystems, where the gray lines show the  linear  interpolations for intermediate alloy concentrations. \nThe data for the Co -Fe system are taken from Ref.[38].  \n \n \n Eddy -current damping13,14 is estimated  by use of  the equations given in Ref.  [23]  for films  \n10 nm thick or less . Eddy currents are neglected because they are found to be less than 5  % of the \ntotal damping. Two -magnon scattering  is disregarded  because the mechanism is largely e xcluded \nin the out -of-plane measurement geometry15–17. The total measured damping  is therefore well \napproximated as the sum   \n 𝛼tot≅𝛼int+𝛼rad+𝛼sp, (4) \nWe determine  the intrinsic damping of the material by subtracting α sp and α rad from the measured \ntotal damping , as shown in Fig. 3 .  \n10 The intrinsic damping increases monotonically with Ni concentration  for the NixCo1-x alloys . \nIndicative of the importance of extrinsic sources of damping, αint is approximately 40 % smaller \nthan αtot for the Fe -rich alloy, though the difference decreases to only 15 % for pure Ni. This \nbehavior is expected, given that both αrad and αsp are proportional to  Ms. A comparison of αint to the \ncalculations by Mankovsky , et al. ,1 shows excellent quantitative agreement  to within 30 %.  \nFurthermore, w e compare αint of the NixCo1-x alloys to the spin density weighted average of the \nintrinsic damping of Ni and C o [purple dashed line in Fig. 3 (a)] , which gives good agreement with \nour data, as previously reported .55  \nαint for NixFe1-x (Fig. 3 (b)) also increases with Ni concentration after a small initial decrease \nfrom pure Fe to the first NixFe1-x alloys. The step increase found in αtot at the bcc to fcc phase \ntransition is fully attributed to αsp, as detailed in the previous section, and therefore does not occur \nin αint. Similar to the NixCo1-x system αint is significantly lower than αtot for Fe -rich alloys. With in \nerror bars, a  comparison to the calculations by Mankovsky , et al.1 (blue line) and Starikov , et al.2 \n(green line) exhibit excellent agreement in the fcc phase, with marginally larger deviations in the \nNi rich regime. Starikov , et al.2 calculated the damping over the ful l range of compositions, under \nthe assumption of continuous  fcc phase. This calcu lation deviates further from  our measured αint in \nthe bcc phase exhibiting qualitatively different behavior.  \nAs previously reported, t he dependence of αint on alloy compositio n in the CoxFe1-x alloys \nexhibits strongly non -monotonic behavior, differing from the two previously discussed alloys.38  \nαint displays a minimum at 25 % Co concentration with a, for conducting ferromagnets \nunprecedented, low value of int (5±1.8)  × 10-4. With increasing Co concentration , αint grows up \nto the phase transition, at which point  it increases by 10  % to 20 % unt il it reaches  the value for \npure Co.  It was shown  that αint scales with the density of states  (DOS)  at the Fermi energy n( EF) in \nthe bcc phase38, and the DOS also exhibits a sharp  minimum for Co 25Fe75. This scaling is \nexpected60,61 if the damping is dominated by the breathing Fermi surface process. With the \nbreathing surface model,  the intraband scattering  that leads to damping  directly scales with n( EF). \nThis scaling is particularly pronounc ed in the Co -Fe alloy system due to the small concentration  \ndependence of the spin -orbit coupling on alloy composition. The special properties of the CoxFe1-x \nalloy system are discussed  in greater detail in Ref.[38]. \nComparing αint to the calculations by Mankovsky  et al.1, we find good quantitative \nagreement with the value of the minimum. However, t he concentration of the minimum is \ncalculated to occur  at approximately 10 % to  20% Co, a slightly lower value than 25 % Co t hat we \nfind in this study. Furthermore , the strong concentration dependence around the minimum is not \nreflected in the calculations. More recent calculations by Turek et al.3, for the bcc CoxFe1-x alloys \n[orange line in Fig. 3 (c)] find the a minimum of the damping of 4x10-4 at 25 % Co concentration \nin good agreement with our experiment, but there is some deviation in concentration dependenc e \nof the damping around the minimum. Turek et al.3 also reported on the damping in the NixFe1-x \nalloy system, with similar qualitative and  quantitative results as the other two presented quantitative \ntheories1,2 and the results are therefore not plotted  in Fig.  3 (b) for the sake of comprehensibility of \nthe figure.  For both NixFe1-x and the CoxFe1-x alloys , the calculated spin density weighte d intrinsic \ndamping of the pure elements (not plotted) deviates significantly from the determined intrinsic \ndamping of the alloys, in contrary to the  good agreement  archived  for the  CoxNi1-x alloys. We \nspeculate that this difference between the alloy syste ms is caused by the non -monotonous \ndependence of the density of states at the Fermi Energy in the CoxFe1-x and NixFe1-x systems.  \nOther  calculated damping values for the pure elements and the Ni80Fe20 and Co 90Fe10 alloys \nare compared to the determined intr insic damping in Table 1. Generally , the calculations \nunderestimate the damping significantly, but our data are in good agreement with more recent \ncalculations for Permalloy ( Ni80Fe20).   11 It is important to point out that n one of the theories  considered he re include thermal \nfluctuations . Regardless, we find exceptional agreement with the calculations to αint at intermediate \nalloy concentrations . We speculate that the modeling of atomic disorder in the alloys in the \ncalculations, by the coherent potential approximation (CPA)  could be responsible for this \nexceptional agreement. The effect of disorder on the  electronic band structure possibly dominates \nany effect s due to nonzero temperature. Indeed, both effects cause a broadening of the bands due \nto enhanced  momentum scattering rates. This directly correlates to a change of the damping \nparameter  according to  the theory of Gilmore and Stiles9. Therefore , the inclusion of the inherent  \ndisorder  of solid -solution alloys  in the calculations  by Mankovsky et al1 mimic s the effects of \ntemperature on damping  to some extent . This argument is corroborated by the fact that the \ncalculations  by Mankovsky et al1 diverge for diluted alloys and pure elements  (as shown in Fig. 2 \n(c) for pure Fe) , where no  or to little disorder is introduced to account for temperature effects. \nMankovsky et al.1 performed temperature dependent calculations of the damping for pure bcc Fe, \nfcc Ni and hcp Co  and the values for 300 K are shown in Table 1 and Fig. 3. These calculations  for \nαint at a temperature of 300 K  are approximately a factor of two  less than our measured values , but \nthe agreement is  significantly improved relative to those  obtained by calculations that neglect \nthermal fluctuations . \n \n \n Figure 5: The intrinsic damping α int is plotted against ( g-2)2 for \nall alloys. We do not observe a proportionality between α int and \n(g-2)2. \n12 Finally, i t has been  reported45,64 that there is a  general proportionality between αint and (g-\n2)2 , as contained in the original microscopic BFS model proposed by  Kambersky .62 To examine \nthis relationship, w e plot αint versus (g-2)2 (determined in Ref [66]) for all samples measured here  \nin Figure 5 . While some samples with large values for ( g-2)2 also exhib it large αint, this is not a \ngeneral trend for all the measured samples . Given  that the damping is not purely a function of the \nspin-orbit strength, but also depends on the details of the band structure , the result  in Fig.  5 is \nexpected .  For example , the amount of  band overlap  will determine the amount of interband \ntransition leading to that damping channel.  Furthermore, the density of states at the Fermi energy \nwill affect the intraband contribution to the damping9,10.   Finally , the ratio of inter - to intra -band \nscattering that mediate s damping contributions at a fixed temperature (RT for our measurements) \nchanges  for different elements9,10 and therefore with alloy concentration. None of these f actors are \nnecessarily proportional to the spin -orbit coupling .  Therefore , we conclude that this simple \nrelation, which originally traces to an order of magnitude estimate  for the case of spin relaxation \nin semiconductors65, does not hold for  all magnetic systems  in general.   \n \n4  Summary  \nWe determined  the damping for the full compositi on range of the binary  3d transition  metal all oys \nNi-Co, Ni -Fe, and Co -Fe and showed that the measured damping can be explained by three \ncontributions to the damping: Intrinsic damping, radiative damping and damping due to spin \npumping. By quantifying all  extrinsic contributions to the measured damping, we determine  the \nintrinsic damping over the whole range of alloy compositions .  These values are compared  to \nmultiple theoretical calculations and yield excellent qualitative and good quantitative agreement for \nintermediate alloy concentrations. 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Ebert, H., Mankovsky, S., Ködderitzsch, D. & Kelly, P. J. Ab Initio Calculation of the Gilbert Damping Parameter \nvia the Linear Response Formalism. Phys. Rev. Lett.  107, 66603 –66607 (2011).  \n61. Lounis, S., Santos Dias, M. dos & Schweflinghaus, B. Transverse dynamical magnetic susceptibilities from \nregular static density functional theory: Evaluation of damping and g shifts of spin excitations. Phys. Rev . B 91, \n104420 (2015).  \n62. Kamberský, V. On the Landau -Lifshitz relaxation in ferromagnetic metals. Can. J. Phys.  48, 2906 (1970).  \n63. Korenman, V. & Prange, R. E. Anomalous Damping of Spin Waves in Magnetic Metals. Phys. Rev. B  6, 2769 –\n2777 (1972).  \n64. Devolder, T. et al.  Damping of Co xFe80-xB20 ultrathin films with perpendicular magnetic anisotropy. Applied \nPhysics Letters  102, (2013).  \n65. Elliott, R. J. Theory of the Effect of Spin -Orbit Coupling on Magnetic Resonance in Some Semiconductors. Phys. \nRev. 96, 266 –279 (1954).  \n66. Schoen, Martin A. W. , Lucassen, Juriaan , Nembach, Hans T. , Silva, T. J. , Koopmans, Bert , Back, Christian H. , \nShaw, Justin M.  Magnetic properties of ultra -thin 3d transition -metal binary alloys I: spin and orbital mo ments, \nanisotropy, and confirmation of Slater -Pauling behavior . arXiv:1701.02177  (2017 ).  \n \n \n "    },    {        "title": "1510.06793v1.Laser_induced_THz_magnetization_precession_for_a_tetragonal_Heusler_like_nearly_compensated_ferrimagnet.pdf",        "content": "arXiv:1510.06793v1  [cond-mat.mtrl-sci]  23 Oct 2015Laser-induced THz magnetization precession for a tetragon al Heusler-like nearly\ncompensated ferrimagnet\nS. Mizukami,1,a)A. Sugihara,1S. Iihama,2Y. Sasaki,2K. Z. Suzuki,1and T. Miyazaki1\n1)WPI Advanced Institute for Materials Research, Tohoku Univ ersity,\nSendai 980-8577, Japan\n2)Department of Applied Physics, Tohoku University, Sendai 9 80-8579,\nJapan\n(Dated: 23 July 2018)\nLaser-inducedmagnetizationprecessional dynamicswasinvestiga tedinepitaxialfilms\nof Mn 3Ge, which is a tetragonal Heusler-like nearly compensated ferrimag net. The\nferromagnetic resonance (FMR) mode was observed, the preces sion frequency for\nwhich exceeded 0.5 THz and originated from the large magnetic anisot ropy field of\napproximately 200 kOe for this ferrimagnet. The effective damping c onstant was\napproximately 0.03. The corresponding effective Landau-Lifshitz c onstant of approx-\nimately 60 Mrad/s and is comparable to those of the similar Mn-Ga mate rials. The\nphysical mechanisms for the Gilbert damping and for the laser-induc ed excitation of\nthe FMR mode were also discussed in terms of the spin-orbit-induced damping and\nthe laser-induced ultrafast modulation of the magnetic anisotropy , respectively.\na)Electronic mail: mizukami@wpi-aimr.tohoku.ac.jp\n1Among the various types of magnetization dynamics, coherent mag netization precession,\ni.e.,ferromagneticresonance(FMR),isthemostfundamentaltype, andplaysamajorrolein\nrf spintronics applications based on spin pumping1–5and the spin-transfer-torque (STT).6,7\nSpin pumping is a phenomenon through which magnetization precessio n generates dc and rf\nspin currents in conductors that are in contact with magnetic films. The spin current can be\nconverted into anelectric voltage throughthe inverse spin-Hall eff ect.8The magnitude of the\nspin current generatedvia spinpumping is proportionaltothe FMRf requency fFMR;4,5thus,\nthe output electric voltage is enhanced with increased fFMR. In the case of STT oscillators\nand diodes, the fFMRvalue for the free layer of a given magnetoresistive devices primarily\ndetermines the frequency range for those devices.9,10An STT oscillator and diode detector\nat a frequency of approximately 40 GHz have already been demonst rated;11–13therefore, one\nof the issues for consideration as regards practical applications is the possibility of increasing\nfFMRto hundreds of GHz or to the THz wave range (0.1-3 THz).11,14\nOnesimple methodthroughwhich fFMRcanbeincreased utilizes magneticmaterials with\nlarge perpendicular magnetic anisotropy fields Heff\nkand small Gilbert damping constants\nα.13,15,16This is because fFMRis proportional to Heff\nkand, also, because the FMR quality\nfactor and critical current of an STT-oscillator are inversely and d irectly proportional to α,\nrespectively. The Heff\nkvalue is determined by the relation Heff\nk= 2Ku/Ms−4πMsfor thin\nfilms, where KuandMsare the perpendicular magnetic anisotropy constant and saturat ion\nmagnetization, respectively. Thus, materials with a small Ms, largeKu, and low αare\nvery favorable; these characteristics are similar to those of mate rials used in the free layers\nof magnetic tunnel junctions integrated in gigabit STT memory applic ation.17We have\npreviously reported that the Mn-Ga metallic compound satisfies the above requirements,\nand that magnetization precession at fFMRof up to 0.28 THz was observed in this case.18\nA couple of research groups have studied magnetization precessio n dynamics in the THz\nwave range for the FePt films with a large Heff\nk, and reported an αvalue that is a factor of\nabout 10 larger than that of Mn-Ga.19–21Thus, it is important to examine whether there are\nmaterialsexhibiting properties similartothoseofMn-Gaexist, inorde r tobetter understand\nthe physics behind this behavior.\nIn this letter, we report on observed magnetization precession at fFMRof more than 0.5\nTHz for an epitaxial film of a Mn 3Ge metallic compound. Also, we discuss the relatively\nsmall observed Gilbert damping. Such THz-wave-range dynamics ca n be investigated by\n2means of a THz wave22or pulse laser. Here, we use the all-optical technique proposed\npreviously;23therefore, the mechanism of laser-induced magnetization preces sion is also dis-\ncussed, because this is not very clearly understood.\nMn3Ge has a tetragonal D0 22structure, and the lattice constants are a= 3.816 and\nc= 7.261˚A in bulk materials [Fig. 1(a)].24,25The Mn atoms occupy at two non-equivalent\nsites in the unit-cell. The magnetic moment of Mn I(∼3.0µB) is anti-parallel to that of\nMnII(∼1.9µB), because of anti-ferromagnetic exchange coupling, and the net magnetic\nmoment is ∼0.8µB/f.u. In other words, this material is a nearly compensated ferrima gnet\nwith a Curie temperature Tcover 800 K.26The tetragonal structure induces a uniaxial\nmagnetic anisotropy, where the c-axis is the easy axis.24The D0 22structure is identical to\nthat of tetragonally-distorted D0 3, which is a class similar to the L2 1Heusler structure;\nthus, D0 22Mn3Ge is also known as a tetragonal Heusler-like compound, as is Mn 3Ga.27\nThe growth of epitaxial films of D0 22Mn3Ge has been reported quite recently, with these\nfilms exhibiting a large Kuand small Ms, similar to Mn-Ga.28–30Note that Mn 3Ge films\nwith a single D0 22phase can be grown for near stoichiometric compositions.29,30Further, an\nextremely large tunnel magnetoresistance is expected in the magn etic tunnel junction with\nMn3Ge electrodes, owing to the fully spin-polarized energy band with ∆ 1symmetry and the\nBloch wave vector parallel to the c-axis at the Fermi level.29,31These properties constitute\nthe qualitative differences between the Mn 3Ge and Mn 3Ga compounds from the material\nperspective.\nAll-optical measurement for the time-resolved magneto-optical K err effect was employed\nusing a standard optical pump-probe setup with a Ti: sapphire laser and a regenerative\namplifier. The wavelength and duration of the laser pulse were appro ximately 800 nm and\n150 fs, respectively, while the pulse repetition rate was 1 kHz. The p ulse laser beam was\ndivided into an intense pump beam and a weaker probe beam; both bea ms weres-polarized.\nThe pump beam was almost perpendicularly incident to the film surface , whereas the angle\nof incidence of the probe beam was ∼6◦with respect to the film normal [Fig. 1(b)].\nBoth laser beams were focused on the film surface and the beam spo ts were overlapped\nspatially. The probe and pump beams had spot sizes with 0.6 and 1.3 mm, respectively.\nThe Kerr rotation angle of the probe beam reflected at the film surf ace was analyzed using\na Wollaston prism and balanced photodiodes. The pump beam intensity was modulated\nby a mechanical chopper at a frequency of 360 Hz. Then, the volta ge output from the\n3FIG. 1. (a) Illustration of D0 22crystal structure unit cell for Mn 3Ge. (b) Diagram showing\ncoordinate system used for optical measurement and ferroma gnetic resonance mode of magnetiza-\ntion precession. The net magnetization (= MII−MI) precesses about the equilibrium angle of\nmagnetization θ, whereMI(MII) is the magnetization vector for the Mn I(MnII) sub-lattice. (b)\nOut-of-plane normalized hysteresis loop of the Kerr rotati on angle φkmeasured for the sample.\nphotodiodes was detected using a lock-in amplifier, as a function of d elay time of the pump-\nprobe laser pulses. The pump pulse fluence was ∼0.6 mJ/cm2. Note that the weakest\npossible fluence was used in order to reduce the temperature incre ase while maintaining the\nsignal-to-noise ratio. A magnetic field Hof 1.95 T with variable direction θHwas applied\nusing an electromagnet [Fig. 1 (b)].\nThec-axis-oriented Mn 3Ge epitaxial films were grown on a single-crystalline (001) MgO\nsubstrate with a Cr seed layer, and were capped with thin MgO/Al lay ers at room tempera-\nture using a sputtering method with a base pressure below 1 ×10−7Pa. The characteristics\nof a 130-nm-thick film with slightly off-stoichiometric composition (74 a t% Mn) deposited\nat 500◦C are reported here, because this sample showed the smallest coer civity (less than\n1 T) and the largest saturation magnetization (117 emu/cm3) of a number of films grown\nwith various thicknesses, compositions, and temperatures. Thes e properties are important\nto obtaining the data of time-resolved Kerr rotation angle φkwith a higher signal-to-noise\nratio, because, as noted above, Mn 3Ge films have a large perpendicular magnetic anisotropy\n4field and a small Kerr rotation angle.30Figure 1(c) displays an out-of-plane hysteresis loop\nofφkobtained for a sample without pump-beam irradiation. The loop is norm alized by the\nsaturation value φk,sat 1.95 T. The light skin depth is considered to be about 30 nm for the\nemployed laser wavelength, so that the φkvalue measured using the setup described above\nwas almost proportional to the out-of-plane component of the ma gnetization Mzwithin the\nlight skin depth depth. The loop shape is consistent with that measur ed using a vibrating\nsample magnetometer, indicating that the film is magnetically homogen eous along the film\nthickness and that value of φk/φk,sapproximates to the Mz/Msvalue.\nFigure 2(a) shows the pump-pulse-induced change in the normalized Kerr rotation angle\n∆φk/φk,s(∆φk=φk−φk,s) as a function of the pump-probe delay time ∆ twith an applied\nmagnetic field Hperpendicular to the film plane. ∆ φk/φk,sdecreases quickly immediately\nafter the pump-laser pulse irradiation, but it rapidly recovers within ∼2.0 ps. This change\nis attributed to the ultrafast reduction and ps restoration of Mswithin the light skin depth\nregion, and is involved in the process of thermal equilibration among t he internal degrees of\nfreedom, i.e., the electron, spin, and lattice systems.32. After the electron system absorbs\nlight energy, the spin temperature increases in the sub-ps timesca le because of the heat\nflow from the electron system, which corresponds to a reduction in Ms. Subsequently, the\nelectron and spin systems are cooled by the dissipation of heat into t he lattices, which have\na high heat capacity. Then, all of the systems reach thermal equilib rium. This process is\nreflected in the ps restoration of Ms. Even after thermal equilibrium among these systems is\nreached, the heat energy remains within the light skin depth region a nd the temperature is\nslightly higher than the initial value. However, this region gradually co ols via the diffusion\nof this heat deeper into the film and substrate over a longer timesca le. Thus, the remaining\nheat causing the increased temperature corresponds to the sma ll reduction of ∆ φk/φk,safter\n∼2.0 ps.\nWith increasing θHfrom out-of-plane to in-plane, a damped oscillation becomes visible\nin the ∆φk/φk,sdata in the 2-12 ps range [Fig. 2(b)]. Additionally, a fast Fourier tran sform\nof this data clearly indicates a single spectrum at a frequency of 0.5- 0.6 THz [Fig. 2(c)].\nThese damped oscillations are attributed to the temporal oscillation ofMz, which reflects\nthe damped magnetization precession,23because the zcomponent of the magnetization\nprecession vector increases with increasing θH. Further, the single spectrum apparent in\nFig. 2(c) indicates that there are no excited standing spin-waves ( such as those observed in\n5thick Ni films), even though the film is thicker than the optical skin de pth.23\nFerrimagnets generally have two magnetization precession modes, i.e., the FMR and\nexchange modes, because of the presence of sub-lattices.33In the FMR mode, sub-lattice\nmagnetization vectors precess while maintaining an anti-parallel dire ction, as illustrated in\nFig. 1(b), such that their frequency is independent of the exchan ge coupling energy between\nthe sub-lattice magnetizations. On the other hand, the sub-lattic e magnetization vectors\nare canted in the exchange mode; therefore, the precession fre quency is proportional to the\nexchange coupling energy between them and is much higher than tha t of the FMR mode.\nAs observed in the case of amorphous ferrimagnets, the FMR mode is preferentially excited\nwhen the pump laser intensity is so weak that the increase in tempera ture is lower than the\nferrimagnet compensation temperature.34No compensation temperature is observed in the\nbulk Mn 3Ge.25,26Also, the temperature increase in this experiment is significantly sma ller\nthanTcbecause the reduction of Msis up to 4 %, as can be seen in Fig. 2(a). Therefore,\nthe observed magnetization precession is attributed to the FMR mo de. Further, as the\nmode excitation is limited to the light skin depth, the amplitude, freque ncy, and etc., for\nthe excited mode are dependent on the film thickness with respect t o the light skin depth.\nThis is because the locally excited magnetization precession propaga tes more deeply into\nthe film as a spin wave in cases where fFMRis in the GHz range.23Note that it is reasonably\nassumed that such a non-local effect is negligible in this study, becau se the timescale of the\ndamped precession discussed here ( ∼1-10 ps) is significantly shorter than that relevant to a\nspin wave with wavelength comparable to the light skin depth ( ∼100 ps).\nThe FMR mode in the THz-wave range is quantitatively examined below. When the ex-\nchangecouplingbetween thesub-latticemagnetizationsissufficient ly strongandthetemper-\nature is well below both Tcand the compensation temperature, the magnetization dynamics\nfor a ferrimagnet can be described using the effective Landau-Lifs hitz-Gilbert equation35\ndm\ndt=−γeffm×/bracketleftbig\nH+Heff\nk(m·z)z/bracketrightbig\n+αeffm×dm\ndt, (1)\nwheremis the unit vector of the net magnetization parallel (anti-parallel) to the magnetiza-\ntion vector MII(MI) for the Mn II(MnI) sub-lattice [Fig. 1(b)]. Here, the spatial change of\nmis negligible, as mentioned above. Heff\nkis the effective value of the perpendicular magnetic\nanisotropyfieldincluding thedemagnetizationfield, even thoughthe demagnetizationfieldis\nnegligibly small for thisferrimagnet (4 πMs= 1.5 kOe). Further, γeffandαeffaretheeffective\n6FIG. 2. Change in Kerr rotation angle ∆ φknormalized by the saturation value φk,sas a function\nof pump-probe delay time ∆ t: (a) for a short time-frame at θH= 0◦and (b) for a relatively long\ntime-frame and different values of θH. The solid curves in (a) and (b) are a visual guide and values\nfitted to the data, respectively. The data in (b) are plotted w ith offsets for clarity. (c) Power\nspectral density as a function of frequency fand magnetic field angle θH.\n7values of the gyromagnetic ratio and the damping constant, respe ctively, which are defined\nasγeff= (MII−MI)/(MII/γII−MI/γI) andαeff= (αIIMII/γII−αIMI/γI)/(MII/γII−MI/γI),\nrespectively, using the gyromagnetic ratio γI(II)and damping constant αI(II)for the sub-\nlattice magnetization of Mn I(II). In the case of Heff\nk≫H,fFMRand the relaxation time of\nthe FMR mode τFMRare derived from Eq. (1) as\nfFMR=γeff/2π/parenleftbig\nHeff\nk+Hz/parenrightbig\n, (2)\n1/τFMR= 2παefffFMR. (3)\nHere,Hzis the normal component of H. Figure 3(a) shows the Hzdependence of the\nprecession frequency fp. This is obtained using the experimental data on the oscillatory\npart of the change in ∆ φk/φk,svia least-square fitting to the damped sinusoidal func-\ntion, ∆φk,p/φk,sexp(−t/τp)sin(2πfp+φp), with an offset approximating the slow change\nof ∆φk/φk,s[solid curves, Fig. 2(b)]. Here, ∆ φk,p/φk,s,τp, andφpare the normalized am-\nplitude, relaxation time, and phase for the oscillatory part of ∆ φk/φk,s, respectively. The\nleast-square fitting of Eq. (2) to the fpvs.Hzdata yields γeff/2π= 2.83 GHz/kOe and\nHeff\nk= 183 kOe [solid line, Fig. 3(a)]. The γeffvalue is close to 2.80 GHz/kOe for the free\nelectron. The value of Heff\nkis equal to the value determined via static measurement (198\nkOe)30within the accepted range of experimental error. Thus, the analy sis confirms that\nthe THz-wave range FMR mode primarily results from the large magne tic anisotropy field in\nthe Mn 3Ge material. The αeffvalues, which are estimated using the relation αeff= 1/2πfpτp\nfollowing Eq. (3), are also plotted in Fig. 3(a). The experimental αeffvalues are indepen-\ndent ofHzwithin the accepted range of experimental error, being in accorda nce with Eq.\n(3); the mean value is 0.03. This value of αefffor D0 22Mn3Ge is slightly larger than the\npreviously reported values for for D0 22Mn2.12Ga (∼0.015) and L1 0Mn1.54Ga (∼0.008).18\nIn the case of metallic magnets, the Gilbert damping at ambient tempe rature is primarily\ncaused by phonon and atomic-disorder scattering for electrons a t the Fermi level in the\nBloch states that are perturbed by the spin-orbit interaction. Th is mechanism, the so-\ncalled Kambersky mechanism,36,37predicts α∝M−1\ns, so that it is more preferable to use\nthe Landau-Lifshitz constants λ(≡αγMs) for discussion of the experimental values of α\nfor different materials. Interestingly, λeff(≡αeffγeffMs) for Mn 3Ge was estimated to be 61\nMrad/s, which is almost identical to the values for D0 22Mn2.12Ga (∼81 Mrad/s) and L1 0\nMn1.54Ga(∼66Mrad/s). The λfortheKamberkymechanism isapproximatelyproportional\n8FIG. 3. (a) Normal component of magnetic field Hdependence on precession frequency fpand\neffective dampingconstant αeffforMn 3Gefilm. (b)Oscillation amplitudeoftheKerrrotation angle\n∆φk,p/φk,scorresponding to the magnetization precession as a functio n of the in-plane component\nofH. The solid line and curve are fit to the data. The dashed line de notes the mean value of αeff.\ntoλ2\nSOD(EF), whereλSOisthespin-orbitinteractionconstant and D(EF)isthetotaldensity\nof states at the Fermi level.37The theoretical values of D(EF) for the above materials are\nroughly identical, because of the similar crystal structures and co nstituent elements, even\nthough the band structures around at the Fermi level differ slight ly, as mentioned at the\nbeginning.18,29Furthermore, the spin-orbit interactions for Ga or Ge, depending on the\natomic number, may not differ significantly. Thus, the difference in αefffor these materials\ncan be understood qualitatively in terms of the Kambersky mechanis m. Further discussion\nbased on additional experiments is required in order to obtain more p recise values for αeff\nand to examine whether other relaxation mechanisms, such as extr insic mechanisms (related\nto the magnetic inhomogeneities), must also be considered.\nFinally, the excitation mechanism of magnetization precession in this s tudy is discussed\nbelow, in the context of a previously proposed scenario for laser-in duced magnetization\n9precession in Ni films.23The initial equilibrium direction of magnetization θis determined\nby thebalance between HandHeff\nk[Fig. 1(b)]. Duringtheperiodinwhich thethree internal\nsystems are not in thermal equilibrium for ∆ t <∼2.0 ps after the pump-laser irradiation\n[Fig. 2(a)], not only the value of Ms, but also the value of the uniaxial magnetic anisotropy,\ni.e.,Heff\nk, is altered. Thus, the equilibrium direction deviates slightly from θand is restored,\nwhich causes magnetization precession. This mechanism may be exam ined by considering\nthe angular dependence of the magnetization precession amplitude . Because the precession\namplitudemaybeproportionaltoanimpulsive torquegeneratedfro mthemodulationof Heff\nk\nin Eq. (1), the torque has the angular dependence |m0×(m0·z)z|, wherem0is the initial\ndirection of the magnetization. Consequently, the z-component of the precession amplitude,\ni.e., ∆φk,p/φk,s, is expressed as ∆ φk,p/φk,s=ζcosθsin2θ∼ζ/parenleftbig\nHx/Heff\nk/parenrightbig2, whereζis the\nproportionalityconstant and Hxisthe in-plane component of H. The experimental values of\n∆φk,p/φk,sare plotted as a function of Hxin Fig. 3(b). The measured data match the above\nrelation, which supports the above-described scenario. Although ζcould be determined via\nthe magnitude and the period of modulation of Heff\nk, it is necessary to consider the ultrafast\ndynamics of the electron, spin, and lattice in the non-equilibrium stat e in order to obtain a\nmore quantitative evaluation;38,39this is beyond the scope of this report.\nIn summary, magnetization precessional dynamics was studied in a D 022Mn3Ge epitaxial\nfilm using an all-optical pump-probe technique. The FMR mode at fFMRup to 0.56 THz\nwas observed, which was caused by the extremely large Heff\nk. A relatively small damping\nconstant of approximately 0.03 was also obtained, and the corresp onding Landau-Lifshitz\nconstant for Mn 3Ge were shown to be almost identical to that for Mn-Ga, being in quali-\ntatively accordance with the prediction of the Kambersky spin-orb it mechanism. 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When the damping is effective in the\nwhole domain ( −1,1) it was proven in [17] that the energy is decreasing over the time with a rate\nequal to t−1\n2. In this paper, using the frequency domain method we show the effect of the coupling\nand the non smoothness of the damping coefficient on the energy decay. Actually, as expected we\nshow the lack of exponential stability, that the semigroup l oses speed and it decays polynomially\nwith a slower rate then given in [17], down to zero at least as t−1\n12.\nContents\n1. Introduction 1\n2. Well-posedness 2\n3. Strong stability 4\n4. Lack of exponential stability 6\n5. Polynomial stabilization 13\nReferences 16\n1.Introduction\nWhen a vibrating source disturbs the first particular of a med ium, a wave is created. This\nphenomena begins to travel from particle to particle along t he medium, which is typically modelled\nby a wave equation. In order to suppress those vibrations, th e most common approach is adding\ndamping. It’s more likely to use one of two types:\n1) The linear viscous damping or ”external damping”, it does mostly model an external frictional\nforce, such that the auto-mobile shock absorber.\n2) The Kelvin-Voigt damping, it’s also called the ”internal damping” or the ”material damping”,\nwhich is originated from the extension or compression of the vibrating particles.\nIn therecent years, many researchers showed interest in pro blems involving this kind of damping. In\ncontrol theory for instance it was shown that when the Kelvin -Voigt dampingcoefficient is satisfying\nsomegeometrical controlconditionsthesemigroupcorresp ondingtothissystemisexponentialstable\n(see [14, 19]). Nonetheless, when the damping is arbitrary l ocalized with singular coefficient, it’s\nnot the case any more (see [2, 13]). Actually, in one-dimensi onal case we can consider the following\nproblem\n(1.1)\n\nutt−[ux(x,t)+b(x)uxt]x= 0−1< x <1, t≥0,\nu(t,−1) =u(t,1) = 0 t≥0,\nu(0,x) =u0(x),ut(0,x) =u1(x)−1≤x≤1,\nwithb∈L∞(−1,1) And\nb(x) =/braceleftbigg\n0 for x∈[0,1)\na(x) forx∈(−1,0).\nUnder the assumption that the damping coefficient has a singul arity at the interface of the damped\nand undamped regions, and behaves like xαnear the interface, it was proven by Liu abd Zhang\n[15] that the semigroup corresponding to the system is polyn omially or exponentially stable and\n2010Mathematics Subject Classification. 35B35, 35B40, 93D20.\nKey words and phrases. Coupled system, Kelvin-Voigt damping, frequency domain ap proach.\n12 FATHI HASSINE AND NADIA SOUAYEH\nthe decay rate depends on the parameter α∈(0,1]. When α= 0, Liu and Rao [13] showed that\nsystem (1.1) is polynomially stable with an order equal to 2 w here few years ago Liu and Liu [12]\nproved the lack of the exponential stability.\nWhen dealing with systems involving quantities described b y several components, pretending\nto control or observe all the state variables it turns out tha t certain systems possess an internal\nstructure that compensates the lack of control variables. S uch a phenomenon is referred to as\nindirect stabilization or indirect control. For instance A labo et al. did study in [1] the coupled\nwaves with partial frictional damping\n/braceleftbigg\nutt−∆u+αv= 0 x∈Ω, t≥0,\nvtt−∆v+αu+βvt= 0x∈Ω, t≥0,\nsubjected to Dirichlet boundary conditions. It was proven t hen the semigroup corresponding to this\nsystem is not exponentially stable, but it’s polynomially w ith the rate t−1\n2. In 2016, Oquendo and\nPacheco studied the wave equation with internal coupled ter ms where the Kelvin-Voigt damping is\nglobal in one equation and the second equation is conservati ve. Although the damping is stronger\nthan the frictional one, they had shown that the semigroup lo ses speed with a slower rate of\nt−1\n4. For this kind of coupled visco-elastic models we distingui sh what is called the transmission\nproblems which have been intensively studied by the first aut hor, Ammari and their collaborators\nin [2, 6, 7, 8, 9, 3] (see also [4]) where the systems studied in these papers are the wave or the\nplate equation or a coupled wave-plate equation. Assuming a non smooth and singular damping\ncoefficient it was shown in these works a uniform and a non-unif orm decay rates of the energy. In\nthiswork, weexaminethebehaviourofacoupledwaves system withapartialKelvin-Voigt damping,\nnamely we consider the following system where the first wave i s dissipative and the second one is\nconservative\n(1.2)\n\nutt(x,t)−[ux(x,t)+a(x)uxt(x,t)]x+vt(x,t) = 0 (x,t)∈(−1,1)×(0,+∞),\nvtt(x,t)−cvxx(x,t)−ut(x,t) = 0 ( x,t)∈(−1,1)×(0,+∞),\nu(0,t) =v(0,t) = 0,u(1,t) =v(1,t) = 0 ∀t >0,\nu(x,0) =u0(x),ut(x,0) =u1(x) ∀x∈(−1,1),\nv(x,0) =v0(x),vt(x,0) =v1(x) ∀x∈(−1,1),\nwherec >0 anda∈L∞(−1,1) is non-negative function. In this paper we assume that the damping\ncoefficient is piecewise function in particular we suppose th atahave the following form a=d1[0,1],\nwheredis a strictly positive constant. Since the damping is singul ar, this system can be seen as a\ncoupling of a conservative wave equation and a transmission wave equation.\nThe natural energy of (u,v) solution of (1.2) at an instant tis given by\nE(t) =1\n2/integraldisplay1\n−1/parenleftbig\n|ut(x,t)|2+|vt(x,t)|2+|ux(x,t)|2+c|vx(x,t)|2/parenrightbig\ndx,∀t >0.\nMultiplying the first equation of (1.2) by ¯ u, the second by ¯ v, integrating over (-1,1) and then taking\nthe real part leads to\nE′(t) =−/integraldisplay1\n−1a(x)|uxt(x,t)|2dx,∀t >0.\nTherefore, the energy is a non-increasing function of the ti me variable t. We show the lack of the\nexponential stability and prove that the semigroup corresp onding to this system is polynomially\nstable for regular initial data and with a slower rate, down t ot−1\n12.\nThis paper is organized as follows. In section 2, we prove tha t system (1.2) is well-posed. In\nsection 3, we show that the energy of the system is the strong s tability. In section 4, we prove the\nlack of exponential stability. In section 5, we prove a polyn omial stability decay of the energy.\n2.Well-posedness\nIn this section, we discuss the well-posesness of the proble m (1.2) using the semigroup theory.\nLetH= (H1\n0(−1.1))2×(L2(−1,1))2be the Hilbert space endowed with the inner product define,STABILITY FOR COUPLED WAVES WITH LOCALLY DISTURBED KELVIN- VOIGT DAMPING 3\nforU1= (u1,v1,w1,z1)∈ HandU2= (u2,v2,w2,z2)∈ H, by\n/an}bracketle{tU1,U2/an}bracketri}htH=/angbracketleftbig\nu1\nx,u2\nx/angbracketrightbig\nL2(−1,1)+/angbracketleftbig√cv1\nx,√cv2\nx/angbracketrightbig\nL2(−1,1)+/angbracketleftbig\nw1,w2/angbracketrightbig\nL2(−1,1)+/angbracketleftbig\nz1,z2/angbracketrightbig\nL2(−1,1).\nBy setting y(t) = (u(t),v(t),ut(t),vt(t)) andy0= (u0,v0,u1,v1) we can rewrite system (1.2) as a\nfirst order differential equation as follow\n(2.1) ˙ y(t) =Ay(t), y(0) =y0,\nwhere\nA(u1,v1,u2,v2) =/parenleftbig\nu2,v2,/parenleftbig\nu1\nx+au2\nx/parenrightbig\nx−v2,cv1\nxx+u2/parenrightbig\n,\nwith\n(u1,v1,u2,v2)∈ D(A) =/braceleftbig\n(u1,v1,u2,v2)∈ H,(u2,v2)∈(H1\n0(−1,1))2,\nv1∈H2(−1,1)∩H1\n0(−1,1),/parenleftbig\nu1\nx+au2\nx/parenrightbig\nx∈L2(−1,1)/bracerightbig\n.\nFor the well-posedness of system (2.1) we have the following proposition:\nProposition 2.1. For an initial datum y0= (u0,v0,u1,v1)∈ H, there exists a unique solution\ny= (u,v,ut,vt)∈C([0,+∞),H)to problem (2.1). Moreover, if y0∈ D(A), then\ny= (u,v,ut,vt)∈C([0,+∞),D(A))∩C1([0,+∞),H).\nProof.By Lumer-Phillips’ theorem (see [16]), it suffices to show tha tAis dissipative and maximal.\n(1) We first prove that Ais dissipative. Take Z= (u,v,w,z)∈ D(A). Then\n/an}bracketle{tAZ,Z/an}bracketri}htH=/an}bracketle{twx,ux/an}bracketri}htL2(−1,1)+c/an}bracketle{tzx,vx/an}bracketri}htL2(−1,1)+/an}bracketle{t(ux+awx)x,w/an}bracketri}htL2(−1,1)\n+/an}bracketle{tcvxx+w,z/an}bracketri}htL2(−1,1).\nBy integration by parts and using the boundary conditions, i t holds:\n(2.2) ( AZ,Z)H=−/an}bracketle{tawx,wx/an}bracketri}htL2(−1,1)=−/integraldisplay1\n−1a|wx|2dx≤0.\nThis shows that Ais the dissipative.\n(2) Let us now prove that Ais maximal, i.e., that λI−Ais surjective for some λ >0. So, for\nany given ( f,g,f1,g1)∈ H, we solve the equation A(u,v,w,z) = (f,g,f1,g1), which is recast on\nthe following way\n(2.3)\n\nw=f\nz=g\nuxx+(afx)x=f1+g\ncvxx=g1−f.\nIt is well known that by Lax-Milgram’s theorem the system (2. 3) admits a unique solution ( u,v)∈\nH1\n0(−1,1)×H1\n0(−1,1). Moreover by multiplying the second and the third lines of (2.3) by u,vre-\nspectively andintegrating over ( −1,1) andusingPoincar´ e inequality andCauchy-Schwarz inequ ality\nwe find that there exists a constant C >0 such that/integraldisplay1\n−1/parenleftbig\n|ux(x)|2+|vx(x)|2/parenrightbig\ndx≤C/integraldisplay1\n−1/parenleftbig\n|fx(x)|2+|gx(x)|2+|f1(x)|2+|g1(x)|2/parenrightbig\ndx.\nIt follows that ( u,v,w,z)∈ D(A) and we have\n/bardbl(u,v,w,z)/bardblH≤C/bardbl(f,g,f1,g1)/bardblH.\nThis imply that 0 ∈ρ(A) and by contraction principle, we easily get R(λI−A) =Hfor sufficient\nsmallλ >0. The density of the domain of Afollows from [16, Theorem 1.4.6]. Then thanks to\nLumer-Phillips Theorem (see [16, Theorem 1.4.3]), the oper atorAgenerates a C0-semigroup of\ncontractions on the Hilbert Hdenoted by ( etA)t≥0.\n/square4 FATHI HASSINE AND NADIA SOUAYEH\n3.Strong stability\nTheorem 3.1. The semigroup (etA)t≥0is strongly stable in the energy space Hi.e.,\nlim\nt→+∞/bardbletAy0/bardbl= 0,∀y0∈ D(A).\nProof.To show that the semigroup ( etA)t≥0is strongly stable we only have to prove that the\nintersection of σ(A) withiRis an empty set. Since the resolvent of the operator Ais not compact\n(see [14]) but 0 ∈ρ(A) we only need to prove that ( iµI−A) is a one-to-one correspondence in the\nenergy space Hfor allµ∈R∗. The proof will be done in two steps: In the first step we prove t he\ninjective property of ( iµI−A) and in the second step we prove the surjective property of th e same\noperator.\nStep 1. Let ( u,v,w,z)∈ D(A) such that\n(3.1) A(u,v,w,z) =iµ(u,v,w,z).\nor equivalently,\n(3.2)\n\nw=iµu in (−1,1),\nz=iµv in (−1,1),\n(ux+awx)x−z=iµw in (−1,1),\ncvxx+w=iµz in (−1,1),\nu(−1) =u(1) = 0, v(−1) =v(1) = 0.\nThen taking the real part of the scalar product of (3.1) with ( u,v,w,z) we get\nRe(iµ/bardbl(u,v,w,z)/bardbl2\nH) = Re/an}bracketle{tA(u,v,w,z),(u,v,w,z)/an}bracketri}htH=−d/integraldisplay1\n0|wx|2dx= 0.\nWhich implies that\nwx= 0 in (0 ,1).\nThis implies that from the first equation (3.2) that\nux= 0 in (0 ,1),\nwhich means that uis a constant in (0 ,1) and since u(1) = 0 we obtain that\nu=w= 0 in (0 ,1),\nHence, from the third and the second equation of (3.2) one get s\n(3.3) u=w=v=z= 0 in (0 ,1),\nUsing (3.3) then (3.2) is reduced to the following problem\n(3.4)\n\nw=iµu in (−1,0),\nz=iµv in (−1,0),\nµ2u+uxx−iµv= 0 in ( −1,0),\nµ2v+cvxx+iµu= 0 in ( −1,0),\nu(−1) =u(0) = 0, v(−1) =v(0) = 0.\nLety= (u,v,ux,vx) andyx= (ux,vx,uxx,vxx) then (3.4) is recast as follow\n(3.5)/braceleftbigg\nyx=Aµyin(−1,0)\nY(0) = 0\nwhere\nAµ=\n0 0 1 0\n0 0 0 1\n−µ2iµ0 0\n−iµ\nc−µ2\nc0 0\n.\nSinceAµis a bounded operator then the unique solution of (3.5) is y= 0 therefore u=v= 0 in\n(−1,0). Moreover, from the fist and the second equation of (3.4) we havew=z= 0 in (−1,1).\nCombining all this with(3.3), we deduce that u=v=w=z= 0 in (−1,1). This conclude the fist\npart of this proof.STABILITY FOR COUPLED WAVES WITH LOCALLY DISTURBED KELVIN- VOIGT DAMPING 5\nStep 2. Now given ( f,g)∈ H, we solve the equation\n(iµI−A)(u,v,w,z) = (f,g,f1,g1).\nOr equivalently,\n(3.6)\n\nw=iµu−f\nz=iµv−g\nµ2u+uxx+iµ(aux)x−iµv= (afx)x−iµf−f1−g=F\nµ2v+cvxx+iµu=−µg+f−g1=G.\nLet’s define the operator\nA: (H1\n0(−1,1))2−→(H−1(−1,1))2\n(u,v)/ma√sto−→(−uxx−iµ(aux)x+iµv,−cvxx−iµu).\nFirst we are going to show that Ais an isomorphism. For this purpose we consider the two opera tor\n˜AandCsuch that\n˜A: (H1\n0(−1,1))2−→(H−1(−1,1))2\n(u,v)/ma√sto−→(−uxx−iµ(aux)x,−cvxx),\nandCsuch that A=C+˜A. It’s easy to show that ˜Ais an isomorphism, then we could rewrite\nA=˜A(Id−˜A−1(−C)). To begin with, thanks to the compact embedding\nH1\n0(−1,1)2֒→L2(−1,1)2andL2(−1,1)2֒→H−1(−1,1)2,\nwe see that ˜A−1is a compact operator. Secondly, it’s clear that Cis a bounded operator, therefore,\nthanks to Fredholms alternative, we only need to prove that ( Id−˜A−1(−C)) is injective.\nLet (u,v)∈(H1\n0(−1,1))2such that ( Id−˜A−1(−C))(u,v) = 0, which implies that\n(˜A−(−C))(u,v) = 0.\nOr equivalently\n(3.7)\n\nuxx+iµ(aux)x−iµv= 0 in ( −1,1)\ncvxx+iµu= 0 in ( −1,1)\nu(−1) =u(1) = 0, v(−1) =v(1) = 0.\nMultiplying the first equation of (3.7) by ¯ uand the conjugate of the second by v, after integration\nover (−1,1), it follows\n−/integraldisplay1\n−1|ux|2dx+c/integraldisplay1\n−1|vx|2dx−iµ/integraldisplay1\n−1a|ux|2dx= 0\nNext, by taking the imaginary part, we can deduce that ux= 0 in (0 ,1) thenuis constant in (0 ,1)\nwhere with the boundary condition u(1) = 0 we have u= 0 in (0,1). Moreover, using the second\nequation of (3.7) we obtain v= 0 in (0 ,1), which implies that (3.7) that\n(3.8)\n\nuxx=iµv in(−1,1)\nvxx=−iµ\ncu in (−1,1)\nu(0) =u(−1) = 0, v(0) =v(−1) = 0\nLety= (u,v,ux,vx) andyx= (ux,vx,uxx,vxx), using the trace theorem we have:\n/braceleftbigg\nyx=Dµyin (−1,0)\ny(0) = 0,\nwhere\nDµ=\n0 0 1 0\n0 0 0 1\n0iµ0 0\n−iµ2\nc0 0 0\n.\nWith a same approach as in the first step, we can have the result that we are looking for (i.e. A is\nan isomorphism).6 FATHI HASSINE AND NADIA SOUAYEH\nNow, rewriting the third and the fourth lines of (3.6) one get s\n(u,v)−µ2A−1(u,v) =A−1(F,G).\nLet (u,v)∈ker(Id−µ2A−1), i.e.µ2(u,v)−A(u,v) = 0, so we can see that:\n(3.9)/braceleftbigg\nµ2u+uxx+iµ(aux)x−iµv= 0 in ( −1,1)\nµ2v+cvxx+iµu= 0 in ( −1,1).\nFurthermore, multiplying the first equation of (3.9) by ¯ uand the conjugate of the second by v, after\nintegration over ( −1,1) and taking the imaginary part, we deduce that\n/integraldisplay1\n−1a|ux|2dx=d/integraldisplay1\n0|ux|2dx= 0.\nSo, we get the same system as in the first step (see (3.2)). Thus , ker(I−µ2A−1) ={0(H−1(−1,1))2}.\nIn another hand, thanks to the compact embeddings H1\n0(−1,1)2֒→L2((−1,1))2andL2(−1,1)2֒→\nH−1(−1,1)2, we see that A−1is a compact operator. Now, thanks to the Fredholm’s alterna tive,\nthe operator ( Id−µ2A−1) is bijective in ( H1\n0(−1,1))2.Finally, the equation (3.6) have a unique\nsolution in H1\n0(−1,1)2. This completes the proof. /square\n4.Lack of exponential stability\nNow, we prove the lack of exponential stability given by the f ollowing theorem\nTheorem 4.1. The semigroup (etA)t≥0, is not exponentially stable in the energy space provided\nthatc >1and that\n(4.1) sin(2√cnπ)/ne}ationslash=O(n−1\n2),\nNoting that the assumption c >1 is made here just to make the calculation readable. The seco nd\nassumption (4.1) can be fulfilled for instance by taking csuch that 2√cis an integer number. To\nprove (4.1) we mainly use the following theorem\nTheorem 4.2. (see[10, 18]) LetetBbe a bounded C0-semigroup on a Hilbert space Hwith generator\nBsuch that iR⊂ρ(B). ThenetBis exponentially stable if and only if There exist a >0andM >0,\nsuch that\n/bardbletB/bardblL(H)≤Me−at,∀t≥0\nif and only if\nlimsup\nω∈R,|ω|→∞/bardbl(iωI−B)−1/bardblL(H)<∞.\nNow, based on the Theorem 4.2 we prove the Theorem 4.1.\nProof.Our main objective is to show that:\n(4.2) /bardbl(λI−A)−1/bardblL(H)is unbounded on the imaginary axis .\nForn∈Nlarge enough let λ=λn=iωn, where\n(4.3)ωn=/radicaligg\n8c(c+1)n2π2+2c+√\n∆\n4cwith ∆ = (8 c(c−1)π2n2)2+32(c+1)(cπn)2+4c2.\nIt’s clear that ωn−→+∞and in particular we have\n(4.4) ωn=√c/parenleftbigg\n2nπ+n−1\n4π(c−1)−cn−3\n32π3(c−1)3+o(n−4)/parenrightbigg\nand\n(4.5)1\nωn=1\n2nπ√c−1\n16√c(c−1)(πn)3+o(n−4).\nDefine (F1,G1,F2,G2)∈(H1\n0(0,1))2×(L2(0,1))2, such that\nF1=F1(x,n) = 0 ∀x∈(−1,1),STABILITY FOR COUPLED WAVES WITH LOCALLY DISTURBED KELVIN- VOIGT DAMPING 7\nG1=G1(x,n) =/braceleftigg0 in (0 ,1)\ng1=sin(2nπx)\n2nπin (−1,0),\nF2=F2(x,n) = 0 ∀x∈(−1,1),\nG2=G2(x,n) =\n\n0 in (0 ,1)\ng2=csin(2nπx)\ni/radicalbigg\n2c\nc+1+/radicalBig\n(c−1)2+4c\nω2nin (−1,0).\nA straight forward calculation leads to\n(4.6) /bardbl(F1,G1,F2,G2)/bardbl2\nH=1\n2+1\n2µ−−→1\n2/parenleftbigg\n1+1√c/parenrightbigg\nasnր+∞.\nOur goal is to prove that lim\n|λ|→∞/bardbl(λI−A)−1/bardblL(H)=∞. That’s why, we solve the resolvent equation\n(4.7) ( λI−A)(u1,v1,u2,v2) = (F1,G1,F2,G2).\nStep 1. For allx∈(0,1) , we have\n\nλu1−u2= 0\nλv1−v2= 0\nλu2−(1+λd)u1\nxx+v2= 0\nλv2−cv1\nxx−u2= 0\nv1(1) =u1(1) = 0(4.8)\nLet\nη+=−λ(1+λd−c)+ωn√reiφ\n2\n2(1+λd)and η−=−λ(1+λd−c)−ωn√reiφ\n2\n2(1+λd)\nwhere\nr=/radicalbig\na2+b2,cos(φ) =a\nrand sin( φ) =b\nr,\nwith\na=−(1−c)2+d2ω2−4c\nω2n\nb=−2d/parenleftbigg\n(1−c)ωn+2c\nωn/parenrightbigg\n.\nIt is important to note that\n√a=dω−(c−1)2\n2dω−1−(c−1)4+16cd2\n8d3ω−3+o(ω−3),\nb\na=2(c−1)\nd+2(c−1)−4cd\ndω−3+o(ω−4)\nand\ni√r\ndeiφ\n2=λ−c−1\nd−(c−1)3−d2(c−1)+2cd2\nd3λ−2\n+d2(c−1)2−(c−1)4−2cd3(c−1)−2cd2\nd4λ−3+o(ω−3).\nThen we obtain\nη+=−λ+c\nd−c\nd2λ−1+(c−1)3+d2(c+1)+2c\n2d3λ−2(4.9)\n+(c−1)4−(c−1)3−d2(c−1)(c−2)−2c\n2d4+o(ω−3)\nand\nη−=−(c−1)3+d2(c+1)\n2d3λ−2+(c−1)3(2−c)+d2(c−1)(c−2−2cd)\n2d4λ−3+o(ω−3) (4.10)8 FATHI HASSINE AND NADIA SOUAYEH\nA straightforward calculation leads to\n(u1+η+v1)xx= (β+)2(u1+η+v1) (4.11)\n(u1+η−v1)xx= (β−)2(u1+η−v1), (4.12)\nwhere\n(β±)2=cλ2−λη±(1+λd)\nc(1+λd).\nSo, fornlarge enough we get\nβ±=ωn/radicalbig\n2c(1+(dωn)2)√r±eiφ±\n2,\nwhere\nr±=/radicalig\na2\n±+b2\n±,cos(φ±) =a±\nr±and sin( φ±) =b±\nr±\nwith\na±=−(1+c)−(dωn)2±√r/parenleftbigg\n−dωncos/parenleftbiggφ\n2/parenrightbigg\n+sin/parenleftbiggφ\n2/parenrightbigg/parenrightbigg\nb±=cdωn±√r/parenleftbigg\n−cos/parenleftbiggφ\n2/parenrightbigg\n−dωnsin/parenleftbiggφ\n2/parenrightbigg/parenrightbigg\n.\nNoting that\n|a+|= 2(dω)2+c2−3c+6\n2+o(ω−1),\n/radicalbig\n|a+|=√\n2dω+c2−3c+6\n4√\n2dω−1+o(ω−1),\nb+=/parenleftbigg\nd(2cd+1−c)+(c−1)3\nd/parenrightbigg\nω−1+o(ω−1),\nand\nb+\na+=o(ω−2).\nThen we obtain\n(4.13) β+=λ√c−(c−1)(c−2)\n8√\n2d2+o(ω−1),\nand\n(4.14) β2\n+=λ2\nc+o(1).\nSimilarly we have\nb−= 2cdω+/parenleftbigg\nd(c−1−2cd)−(c−1)3\nd/parenrightbigg\nω−1+o(ω−1),\n/radicalbig\nb−=√\n2cdω/parenleftbigg\n1+/parenleftbiggc−1−2cd\n4c−(c−1)3\n4cd2/parenrightbigg\nω−2/parenrightbigg\n+o(ω−2)\na−=−2c+o(ω−1),\nand\na−\nb−=−ω−1\nd+o(ω−2),\nthen consequently we obtain\n(4.15) β−=/radicalbiggω\ndeiπ\n4−ω−1\n2\n2d3\n2e−iπ\n4+o(ω−1),STABILITY FOR COUPLED WAVES WITH LOCALLY DISTURBED KELVIN- VOIGT DAMPING 9\nand\nβ2\n−=λ\nd−1\nd2+(c−1)3+d(c−1)+2c\n2cd3λ−1(4.16)\n−(c−1)3(2−c)+d(c−1)(c−2−2cd)+2c\n2cd4λ−2+o(ω−2)\nNext, from (4.11), we get\n(u1+η+v1) =c1exβ++c2e−xβ+\nand\n(u1+η−v1) =c3exβ−+c4e−xβ−.\nRecalling that u1(1) =v1(1) = 0 we can rewrite the last two equations as follow\n(4.17) ( u1+η+v1) =c1(exβ+−e(2−x)β+),\n(4.18) ( u1+η−v1) =c3(exβ−−e(2−x)β−).\nHence by combining (4.17) and (4.18) we obtain\n(4.19) u1(x) =−c1η−\nη+−η−/parenleftig\neβ+x−eβ+(2−x)/parenrightig\n+c3η+\nη+−η−/parenleftig\neβ−x−eβ−(2−x)/parenrightig\n,\nand\n(4.20) v1(x) =c1\nη+−η−/parenleftig\neβ+x−eβ+(2−x)/parenrightig\n−c3\nη+−η−/parenleftig\neβ−x−eβ−(2−x)/parenrightig\n.\nStep 2. For allx∈(−1,0) we have\n\nλu1−u2= 0\nλv1−v2=g1\nλu2−u1\nxx+v2= 0\nλv2−cv1\nxx−u2=g2\nv1(−1) =u1(−1) = 0.(4.21)\nFollowing to the third and the fourth equation of (4.8) and of (4.21) we can deduce, thanks to the\nregularity of the stats, that\n(1+λd)u1\nx(0+) =u1\nx(0−), (4.22)\nv1\nx(0+) =v1\nx(0−). (4.23)\nand\n(1+λd)u1\nxx(0+) =u1\nxx(0−), (4.24)\nv1\nxx(0+) =v1\nxx(0−). (4.25)\nWe denote by\n(4.26) α+=λ\n2/parenleftigg\nc−1+/radicaligg\n(1−c)2+4c\nω2n/parenrightigg\n= (c−1)λ−c\nc−1λ−1−c2\n(c−1)3+o(ω−3),\nand\n(4.27) α−=λ\n2/parenleftigg\nc−1−/radicaligg\n(1−c)2+4c\nω2n/parenrightigg\n=c\nc−1λ−1+o(ω−1)\nand we define for nlarge enough µ±as follow\nµ±=√\n2c/radicalbigg\nc+1−/parenleftig\n±/radicalig\n(c−1)2+4c\nω2n/parenrightig,\nin particular with the chose of ωnin (4.3) one get\nµ2\n±=λ\nλ−α±\nc.10 FATHI HASSINE AND NADIA SOUAYEH\nBesides, we have\n(4.28) µ+=√c/parenleftbigg\n1−c\n2(c−1)λ−2+o(ω−2)/parenrightbigg\n,\n(4.29) µ−= 1+λ−2\n2(c−1)+o(ω−2),\nand\n(4.30)µ+\nµ−=√c/parenleftbigg\n1−c+1\n2(c−1)λ−2+o(ω−2)/parenrightbigg\n.\nWe set\nω+\n1(x) = (u2+α+v2+µ+(u1\nx+α+v1\nx), (4.31)\nω−\n1(x) = (u2+α+v2−µ+(u1\nx+α+v1\nx)), (4.32)\nω+\n2(x) = (u2+α−v2+µ−(u1\nx+α−v1\nx)), (4.33)\nω−\n2(x) = (u2+α−v2−µ−(u1\nx+α−v1\nx)). (4.34)\nNow, define Y= (ω+\n1,ω−\n1,ω+\n2,ω−\n2)tandZ= (g1x,g2)t. Then we have\n(4.35) Yx=AY+BZ\nwhere\nA=\nµ+(λ−α+\nc) 0 0 0\n0µ+(−λ+α+\nc) 0 0\n0 0 µ−(λ−α−\nc) 0\n0 0 0 µ−(−λ+α−\nc)\n\nand\nB=\n−α+−µ+α+\nc\n−α+µ+α+\nc\n−α−−µ−α−\nc\n−α−µ−α−\nc\n.\nThen, a straightforward calculation leads to:\nµ+(λ−α+\nc) = 2inπ.\nUsing the boundary condition at −1 we get\n(4.36) ω+\n1(−1) =−ω−\n1(−1) andω+\n2(−1) =−ω−\n2(−1),\nTaking into account of (4.36) then the solution of (4.35) is w ritten as follow\nω+\n1(x) =ω+\n1(−1)e2inπx−α+\n2/bracketleftbigg/parenleftbigg\n1−µ+\nµ−/parenrightbigg\n(x+1)e2inπx+1\n2nπ/parenleftbigg\n1+µ+\nµ−/parenrightbigg\nsin(2nπx)/bracketrightbigg\n, (4.37)\nω−\n1(x) =−ω+\n1(−1)e−2inπx−α+\n2/bracketleftbigg/parenleftbigg\n1−µ+\nµ−/parenrightbigg\n(x+1)e−2inπx+1\n2nπ/parenleftbigg\n1+µ+\nµ−/parenrightbigg\nsin(2nπx)/bracketrightbigg\n, (4.38)\nω+\n2(x) =ω+\n2(−1)eµ−(λ−α−\nc)(x+1)+α−\n2inπ+µ−/parenleftbig\nλ−α−\nc/parenrightbig/bracketleftig\ne−2inπx+eµ−(λ−α−\nc)(x+1)/bracketrightig\n, (4.39)\nω−\n2(x) =−ω+\n2(−1)e−µ−(λ−α−\nc)(x+1)−α−\n2inπ+µ−/parenleftbig\nλ−α−\nc/parenrightbig/bracketleftig\ne2inπx+e−µ−(λ−α−\nc)(x+1)/bracketrightig\n. (4.40)\nTaking the trace of ω+\n1andω−\n1in (4.37)-(4.38) and in (4.31)-(4.32) on the boundary 0 and u sing\nthe continuity of the states u2andv2we obtain\n(ω+\n1+ω−\n1)(0−) =α+/parenleftbiggµ+\nµ−−1/parenrightbigg\n= 2u2(0−)+2α+v2(0−)\n= 2λ(u1(0−)+α+v1(0−)) = 2λ(u1(0+)+α+v1(0+))\n=2λ\nη+−η−/parenleftig\nc1(1−e2β+)(α+−η−)+c3(1−e2β−)(η+−α+)/parenrightig\n,STABILITY FOR COUPLED WAVES WITH LOCALLY DISTURBED KELVIN- VOIGT DAMPING 11\nwhere we have used the the expressions of u1andv1in (4.19) and (4.20). This implies that\n(4.41) c3=1−e2β+\n1−e2β−Anc1+Bn\n1−e2β−\nwhere\nAn=η−−α+\nη+−α+=c−1\nc/parenleftbigg\n1+λ−1\nd−λ−2\nc−1+o(ω−2)/parenrightbigg\n=c−1\nc/parenleftbigg\n1+n−1\n2iπd√c−n−2\n4π2c(c−1)+o(ω−2)/parenrightbigg\n, (4.42)\nand\nBn=α+(η+−η−)/parenleftig\nµ+\nµ−−1/parenrightig\n2λ(η+−α+)\n=(c−1)(√c−1)\n2c/parenleftbigg\n1−c−1\ndλ−1−/parenleftbigg1\n(c−1)2+√c(c+1)\n2(√c−1)(c−1)/parenrightbigg\nλ−2+o(ω−2)/parenrightbigg\n=(c−1)(√c−1)\n2c/parenleftbigg\n1−c−1\n2iπd√cn−1−/parenleftbigg1\n(c−1)2+√c(c+1)\n2(√c−1)(c−1)/parenrightbigg\n×n−2\n4π2c+o(n−2)/parenrightbigg\n.(4.43)\nwhere we used here (4.9), (4.10), (4.26), (4.27), (4.30) and (4.3).\nUsing (4.37)-(4.38) and (4.24)-(4.25), one gets\n(ω+\n1−ω−\n1)′(0−) = 2inπα+/parenleftbiggµ+\nµ−−1/parenrightbigg\n= 2µ+(u1+α+v1)xx(0−) = 2µ+((1+λd)u1+α+v1)xx(0+)\n=2µ+/bracketleftbig\nc1β2\n+(1−e2β+)(α+−(1+λd)η−)+c3β2\n−(1−e2β−)((1+λd)η+−α+)/bracketrightbig\nη+−η−.\nThen we obtain\n(4.44) c1=1−e2β−\n1−e2β+A′\nnc3+B′\nn\n1−e2β+\nwhere\nA′\nn=β2\n−(α+−(1+λd)η+)\nβ2\n+(α+−(1+λd)η−)\n=c\nc−1/parenleftbigg\n1−λ−1\nd+/parenleftbigg(c−1)3\n2cd2+c−1\n2cd+3−c\n2d2+1\n2/parenrightbigg\nλ−2+o(ω−2)/parenrightbigg\n=c\nc−1/parenleftbigg\n1−n−1\n2iπd√c+/parenleftbigg(c−1)3\n2cd2+c−1\n2cd+3−c\n2d2+1\n2/parenrightbiggn−2\n4π2c+o(n−2)/parenrightbigg\n, (4.45)\nand\nB′\nn=inπα+(η+−η−)/parenleftig\nµ+\nµ−−1/parenrightig\nµ+β2\n+(α+−(1+λd)η−)\n=nπ(c−√c)\n2ω/parenleftbigg\n−1+c\ndλ−1+/parenleftbiggc+√c+3\n2(c−1)2−c+1+d2\n2d2/parenrightbigg\nλ−2/parenrightbigg\n+o(ω−2)\n=√c−1\n2/parenleftbigg\n−1+√c\n2iπdn−1+/parenleftbigg√c+4\nc−1−c+1+d\nd2/parenrightbiggn−2\n8cπ2+o(n−2)/parenrightbigg\n. (4.46)\nwhere we used here (4.9), (4.10), (4.14), (4.16), (4.26), (4 .27), (4.28), (4.30) and (4.3).\nCombining (4.41) and (4.44) then we find that\n(4.47) c1=1\n1−e2β+×A′\nnBn+B′\nn\n1−AnA′n=c′\n1\n1−e2β+12 FATHI HASSINE AND NADIA SOUAYEH\nand\n(4.48) c3=1\n1−e2β−×AnB′\nn+Bn\n1−AnA′n=c′\n3\n1−e2β−,\nwhere following to (4.42), (4.43), (4.45) and (4.46) we have\n(4.49) c′\n1=O(1) and c′\n3=O(1).\nIn another hand, by denoting θ=−iµ−/parenleftbig\nλ−α−\nc/parenrightbig\nand by using the same argument as previously,\none gets\n(ω+\n2+ω−\n2)(0−) = 2isin(θ)ω+\n2(−1)+2α−\n2nπ−θsin(θ) = 2λ(u1+α−v1)(0−) = 2λ(u1+α−v1)(0+)\n=2λ\nη+−η−/parenleftbig\nc′\n1(α−−η−)+c′\n3(η+−α−)/parenrightbig\n.\nIt’s clear that θ/ne}ationslash= 0[π] then we can write\nω+\n2(−1) =λ\nisin(θ)(η+−η−)/bracketleftbig\nc′\n1(α−−η−)+c′\n3(η+−α−)/bracketrightbig\n−α−\n2inπ+iθ. (4.50)\nNoting that from (4.3), (4.4), (4.27) and (4.29) we have\n(4.51) θ=ω/parenleftbigg\n1−3\n2(c−1)ω−2+o(ω−2)/parenrightbigg\n=√c/parenleftbigg\n2nπ+cπ−12\n4cπ2(c−1)n−1+o(n−1)/parenrightbigg\nThen from (4.4), (4.9), (4.10), (4.27), (4.29) and (4.51) we deduce that\n(4.52) ω+\n2(−1)∼2πn√cc′\n3\nsin(θ)\nUsing (4.22)-(4.23), (4.31)-(4.32) and (4.37)-(4.38) we g et\nω+\n1(−1) =(ω+\n1−ω−\n1)(0−)\n2=µ+(u1+α+v1)x(0−) =µ+((1+λd)u1+α+v1)x(0+)\n=µ+\nη+−η−/bracketleftig\nc1β+(1+e2β+)(α+−(1+λd)η−)+c3β−(1+e2β−)((1+λd)η+−α+)/bracketrightig\n. (4.53)\nThen from (4.4), (4.9), (4.10), (4.13), (4.15), (4.27) and ( 4.28) we deduce that\n(4.54) ω+\n1(−1)∼c′\n3/radicalbiggc\nde−iπ\n4(2π√cn)3\n2.\nNext, for all x∈(−1,0) we have\nv1\nx(x) =1\n2µ−µ+(α+−α−)/bracketleftbig\nα−(ω+\n1(x)−ω−\n1(x))−α+(ω+\n2(x)−ω−\n2(x))/bracketrightbig\n=1\n2µ−µ+(α+−α−)/bracketleftigg\nµ−/parenleftbigg\n2ω+\n1(−1)cos(2nπx)−iα+/parenleftbigg\n1−µ+\nµ−/parenrightbigg\n(x+1)sin(2 nπx)/parenrightbigg\n(4.55)\n−µ+/parenleftbigg\n2ω+\n2(−1)cos(θ(x+1))+2α−\n2inπ+iθ(cos(2nπx)+cos(θ(x+1)))/parenrightbigg/bracketrightigg\n,\nwhere we have used (4.31)-(4.34) and (4.37)-(4.40). Thus fu rther leads to\n/bardblv1\nx/bardbl2\nL2(−1,0)≥max/braceleftbigg|ω+\n1(−1)|2\n2µ2\n+|α+−α−|2,|ω+\n2(−1)|2\nµ2\n−|α+−α−|2/bracerightbigg\n−|α+|2(µ+−µ−)2\n4µ2\n−µ2\n+|α+−α−|2(4.56)\n−min/braceleftbigg|ω+\n1(−1)|2\n2µ2\n+|α+−α−|2,|ω+\n2(−1)|2\nµ2\n−|α+−α−|2/bracerightbigg\n−2|α−|2\nµ2\n−µ2\n+(2nπ+θ)2|α+−α−|2.\nSince,\nsin(θ)/ne}ationslash=O(n−1\n2),\nasngoes to the infinity (by (4.51) assumption (4.1)) then by usin g (4.3), (4.28), (4.29), (4.26),\n(4.27), (4.52) and (4.54) we can show that the second and the f ourth terms of the right hand sideSTABILITY FOR COUPLED WAVES WITH LOCALLY DISTURBED KELVIN- VOIGT DAMPING 13\nof (4.56) are bounded while the sum of the fist and the third ter ms tends to the infinity as ngoes\nto +∞, therefore we obtain\n(4.57) /bardblv1\nx/bardbl2\nL2(−1,0)asnր+∞.\nLast but not least, we have\n(4.58)\n/bardbl(iωnI−A)−1(F1,G1,F2,G2)/bardblH=/bardbl(u1,v1,u2,v2)/bardbl2\nH≥/integraldisplay0\n−1|v1\nx(x)|2dx−→+∞,asnր+∞.\nFinally we conclude, using (4.58) and (4.6) that\nlimsup\nω∈R,|ω|→∞/bardbl(iωI−A)−1/bardblL(H)= +∞.\nSo,etAis not exponentially stable in the energy space. This comple tes the proof. /square\n5.Polynomial stabilization\nThis subsection aims to prove the polynomial stability give n by the following theorem:\nTheorem 5.1. The semigroup of contraction (eTA)t≥0is polynomially stable of order1\n12.\nOur method is based on the Borichev and Tomilov result given b y the following:\nTheorem 5.2. [5, Theorem 2.4] LetBbe a generator of a C0-semigroup of contraction in a Hilbert\nspaceXwith domain D(B)such that iR⊂σ(B)thenetBis polynomially stable with order1\nγ,γ >0\ni.e. there exists C >0such that\n/bardbletBU0/bardblX≤C\n(1+t)1\nγ/bardblU0/bardblD(B),∀t≥0,∀U0∈ D(B),\nif and only if\nlimsup\nβ∈R,|β|→∞/bardblβ−γ(iβ−B)−1/bardblL(X)<+∞.\nBased on Theorem 5.2 we are able now to prove our main result gi ven in Theorem 5.1 of this\nsection. For this purpose, let’s consider the following:\nProposition 5.1. The operator Adefined in (2.1)satisfies:\n(5.1) limsup\nβ∈R,|β|→∞/bardblβ−12(iβ−A)−1/bardblL(H)<+∞.\nProof.To prove (5.1) we use an argument of contradiction. In fact, i f (5.1) is false, then, there exist\nβn∈R+andYn= (u1\nn,v1\nn,u2\nn,v2\nn)∈ D(A) such that\n(5.2) /bardblYn/bardblH= 1, βnր+∞andβγ(iβnI−A)Yn:= (f1\nn,g1\nn,f2\nn,g2\nn)−→0 inHasnր+∞.\nEquivalently, we have\n(5.3) βγ\nn/parenleftbig\niβnu1\nn−u2\nn/parenrightbig\n=f1\nn−→0 inH1\n0(−1,1),\n(5.4) βγ\nn/parenleftbig\niβnv1\nn−v2\nn/parenrightbig\n=g1\nn−→0 inH1\n0(−1,1),\n(5.5) βγ\nn/parenleftbig\niβnu2\nn−/parenleftbig\nu1\nnx+au2\nnx/parenrightbig\nx+v2\nn/parenrightbig\n=f2\nn−→0 inL2(−1,1),\n(5.6) βγ\nn/parenleftbig\niβnv2\nn−cv1\nnxx−u2\nn/parenrightbig\n=g2\nn−→0 inL2(−1,1).\nWe denote by\nTn=u1\nnx+au2\nnx.\nTaking the real part of /an}bracketle{tβγ(iβnI−A)Yn,Yn/an}bracketri}htHthen by the dissipation property of the semigroup of\nthe operator Awe get\nβγ\nn/integraldisplay1\n0d.|u2\nnx|2dx−→0,14 FATHI HASSINE AND NADIA SOUAYEH\nwhich leads to\n(5.7) βγ\n2n/bardblu2\nnx/bardblL2(0,1)−→0.\nNow thanks to (5.3) and (5.7), we obtain\n(5.8) βγ\n2+1\nn/bardblu1\nnx/bardblL2(0,1)−→0.\nFrom (5.7) and (5.8), it follows\n(5.9) βγ\n2n/bardblTn/bardblL2(0,1)−→0.\nTaking the inner product of (5.5) with u2\nninL2(0,1) we get\n(5.10) β3γ\n4n/parenleftig\niβn/bardblu2\nn/bardbl2\nL2(0,1)+/an}bracketle{tTn,u2\nnx/an}bracketri}htL2(0,1)+Tn(0+)u2n(0+)+/an}bracketle{tv2\nn,u2\nn/an}bracketri}htL2(0,1)/parenrightig\n=o(1).\nThanks to (5.2), (5.7) and (5.9), it’s clear that the second a nd the last terms converge to zero.\nFurthermore, we have\nβ3γ\n4nTn(0+)u2n(0+)≤Cβγ\n2n/parenleftbigg\n/bardblTn/bardbl1\n2\nL2(0,1)./bardblu2\nnx/bardbl1\n2\nL2(0,1)./bardblT′\nn/bardbl1\n2\nL2(0,1)./bardblu2\nn/bardbl1\n2\nL2(0,1)/parenrightbigg\n.\nFrom (5.5) we can see that /bardblβnu2\nn+v2\nn/bardblL2(0,1)∼ /bardblT′\nn/bardblL2(0,1)which implies that\nβ3γ\n4n|Tn(0+)|.|u2n(0+)| ≤Cβ3γ\n4n/bardblTn/bardbl1\n2\nL2(0,1)./bardblu2\nnx/bardbl1\n2\nL2(0,1)×\n/parenleftbigg\n/bardblβnu2\nn/bardbl1\n2\nL2(0,1)+/bardblv2\nn/bardbl1\n2\nL2(0,1)+o(1)/parenrightbigg\n./bardblu2\nn/bardbl1\n2\nL2(0,1)\n≤C/bardblβγ\n2nTn/bardbl1\n2\nL2(0,1)./bardblβγ\n2nu2\nnx/bardbl1\n2\nL2(0,1)×\n/parenleftbigg\n/bardblβnu2\nn/bardbl1\n2\nL2(0,1)+/bardblv2\nn/bardbl1\n2\nL2(0,1)/parenrightbigg\n/bardblβγ\n2nu2\nn/bardbl1\n2\nL2(0,1)+o(1)\n≤/parenleftbigg\n1+β1\n2+γ\n4n./bardblu2\nn/bardblL2(0,1)/parenrightbigg\no(1). (5.11)\nCombining (5.10) and (5.11), one follows\n(5.12) β1\n2+3γ\n8n/bardblu2\nn/bardblL2(0,1)−→0.\nMoreover, multiplying (5.5) by β−γ\n2n(1−x)Tnand integrating over the interval (0 ,1) then by taking\naccount of (5.9), an integration by parts leads to\nRe/an}bracketle{tiβ1\n2+3γ\n8nu2\nn,(1−x)β1\n2−3γ\n8+γ\n2nTn/an}bracketri}htL2(0,1)+βγ\n2n\n2/parenleftig\n|Tn(0+)|2−/bardblTn/bardbl2\nL2(0,1)/parenrightig\n(5.13)\n+βγ\n2nRe/an}bracketle{tv2\nn,(1−x)Tn/an}bracketri}htL2(0,1)=o(1).\nWe suppose that γ≥4\n3. It’s clear from (5.2), (5.9) and (5.12) that the first, the th ird and the last\nterms of (5.13) converge to zero then one gets\n(5.14) βγ\n4n.|Tn(0+)| −→0.\nTaking into account to (5.8) then the trace formula gives\n(5.15) βγ\n2+1\nn.|u1\nn(0+)| −→0.\nSubstituting(5.4) into(5.5) andtakingtheinnerproductw ithβ3−γ\nnv1\nninL2(0,1) thenbyintegrating\nby parts we have\n(5.16) iβ4\nn/angbracketleftbig\nu2\nn,v1\nn/angbracketrightbig\nL2(0,1)+β3\nn/angbracketleftbig\nTn,v1\nn/angbracketrightbig\nL2(0,1)+iβ4\nn/bardblv1\nn/bardbl2\nL2(0,1)+β3\nnTn(0+)v1n(0+) =o(1).\nTakingγ≥12 and using (5.2), (5.9), (5.12) and (5.14) we can see that th e first, the second and the\nfourth terms of (5.16) converge to zero, therefore\n(5.17) β2\nn./bardblv1\nn/bardblL2(0,1)−→0.STABILITY FOR COUPLED WAVES WITH LOCALLY DISTURBED KELVIN- VOIGT DAMPING 15\nFrom (5.4) and (5.17) it follows\n(5.18) βn/bardblv2\nn/bardblL2(0,1)−→0.\nMultiplying (5.6) with β−γ\nn(1−x)v1nxand integrating over (0 ,1) then by taking the real part we\nfind\nc\n2/parenleftig\n|v1\nnx(0+)|2−/bardblv1\nnx/bardbl2\nL2(0,1)/parenrightig\n= Re/angbracketleftbig\nu2\nn,(1−x)v1\nnx/angbracketrightbig\nL2(0,1)\n−Re/an}bracketle{tiβnv2\nn,(1−x)v1\nnx/an}bracketri}htL2(0,1)+o(1).\nUsing (5.2), (5.12) and (5.18) leads to\n(5.19) |v1\nnx(0+)|2−/bardblv1\nnx/bardbl2\nL2(0,1)−→0.\nWe take the inner product of (5.6) with β−γ\nnxv1\nninL2(0,1) then we have\nc/parenleftbigg/integraldisplay1\n0x|v1\nnx(x)|2dx+/angbracketleftbig\nv1\nnx,v1\nn/angbracketrightbig\nL2(0,1)/parenrightbigg\n=/angbracketleftbig\nu2\nn,xv1\nn/angbracketrightbig\nL2(0,1)−iβn/an}bracketle{tv2\nn,xv1\nn/an}bracketri}htL2(0,1)+o(1).\nUsing (5.2), (5.12) and (5.18) we deduce that\n/integraldisplay1\n0x|v1\nnx(x)|2dx−→0.\nThis implies in particular that for every εin (0,1) we have\n(5.20) /bardblv1\nnx/bardblL2(ε,1)−→0 asnր+∞.\nMultiplying (5.6) with β−γ\nn(1−x)v1nxand integrating over (0 ,ε) then by taking the real part we\nfind\nc\n2/parenleftig\n|v1\nnx(ε)|2−/bardblv1\nnx/bardbl2\nL2(ε,1)/parenrightig\n= Re/angbracketleftbig\nu2\nn,(1−x)v1\nnx/angbracketrightbig\nL2(ε,1)−Re/an}bracketle{tiβnv2\nn,(1−x)v1\nnx/an}bracketri}htL2(ε,1)+o(1).\nBesides, from (5.2), (5.12), (5.18) and (5.20) we follow\n|v1\nnx(ε)| −→0 asnր+∞.\nThen we deduce that\n(5.21) v1\nnx(x)−→0 a.e. in [0 ,1] asnր+∞.\nNow, (5.2) and (5.21) allows the use of the dominated converg ence theorem and lead to\n(5.22) /bardblv1\nnx/bardblL2(0,1)−→0.\nTherefore, we obtain\n(5.23) |v1\nn(0+)| −→0.\nBy combining (5.19) and (5.22) we find\n(5.24) |v1\nnx(0+)| −→0.\nFurthermore, taking the inner product of (5.4) with β1−γ\nn(1−x)v1\nnxand then considering the\nimaginary part one gets\nβ2\nnRe(v1\nnx,(1−x)v1\nn)−Imβn(v2\nn,(1−x)v1\nnx) =o(1)\n=1\n2(β2\nn|v1\nn(0+)|2−β2\nn/bardblv1\nn/bardbl2)−βnIm/an}bracketle{tv2\nn,(1−x)v1\nnx/an}bracketri}ht\nAdding to this (5.23), (5.17) and (5.18) we can deduce that :\n(5.25) βn|v1\nn(0+)| −→016 FATHI HASSINE AND NADIA SOUAYEH\nThanks to (5.14), (5.15), (5.23) and (5.24) one gets\nβγ\n2+1\nn.u1\nn(0−)−→0, (5.26)\nβγ\n4.u1\nnx(0−)−→0, (5.27)\nβnv1\nn(0−)−→0, (5.28)\nv1\nnx(0−)−→0. (5.29)\nNext, inserting (5.3) into (5.5) and inserting (5.4) into (5 .6) and consider both equations in the\ninterval (0 ,1), leads to\n(5.30) −β2\nnu1\nn−u1\nnxx+v2\nn=β−γ\nnf2\nn+iβ1−γ\nnf1\nn,\nand\n(5.31) −β2\nnv1\nn−cv1\nnxx−u2\nn=β−γ\nng2\nn+iβ1−γ\nng1\nn.\nA straightforward calculation shows that the real part of th e inner productof (5.30) with ( x+1).u1\nnx\nand that the real part of the inner of (5.31) with ( x+1).v1\nnxleads to\n1\n2/integraldisplay0\n−1/parenleftbig\n|βnu1\nn|2+|u1\nnx|2/parenrightbig\ndx=1\n2/parenleftbig\n|u1\nnx(0−)|2+β2\nn|u1\nn(0−)|2/parenrightbig\n(5.32)\n−Re/an}bracketle{tv2\nn,(x+1)u1\nnx/an}bracketri}htL2(−1,0)+o(1),\nand\n1\n2/integraldisplay0\n−1/parenleftbig\n|βnv1\nn|2+c|v1\nnx|2/parenrightbig\ndx=1\n2/parenleftbig\nc|v1\nnx(0−)|2+β2\nn|v1\nn(0−)|2/parenrightbig\n(5.33)\n+Re/an}bracketle{tu2\nn,(x+1)v1\nnx/an}bracketri}htL2(−1,0)+o(1).\nWhere we have used (5.2)-(5.6). In another hand, from (5.2), (5.12), (5.18) and (5.26)-(5.29) we get\n(5.34)/integraldisplay0\n−1/parenleftbig\n|βnu1\nn|2+|u1\nnx|2/parenrightbig\ndx−→0,\nand\n(5.35)/integraldisplay0\n−1/parenleftbig\n|βnv1\nn|2+c|v1\nnx|2/parenrightbig\ndx−→0.\nNow by summing (5.8) (5.12), (5.17), (5.18), (5.34) and (5.3 5) we can see that\n(5.36) /bardblYn/bardblH−→0.\nThis contradicts (5.2) and so (5.1) holds true with γ≥12. This completes the proof. /square\nReferences\n[1] F. Alabau, P. Cannarsa, V. Komornik, Indirect internal s tabilization of weakly coupled evolution equations, J.\nEvol. Equ.2 (2002) 127150.\n[2]K. Ammari, F. Hassine and L. Robbiano , Stabilization for the wave equation with singular Kelvin- Voigt\ndamping, arXiv:1805.10430.\n[3] K. Ammari, Z. Liu and F. Shel, Stabilization for the wave e quation with singular Kelvin-Voigt damping,\narXiv:1805.10430.\n[4]K. Ammari and S. Nicaise, Stabilization of elastic systems by collocated feedback, 2124, Springer, Cham, 2015.\n[5] A. Borichev and Y. Tomilov, Optimal polynomial decay of f unctions and operator semigroups, Math. Ann., 347\n(2010), 455–478.\n[6]F. Hassine, Stability of elastic transmission systems with a local Kelv inVoigt damping, European Journal of\nControl, 23(2015), 84–93.\n[7]F. Hassine, Asymptotic behavior of the transmission Euler-Bernoulli p late and wave equation with a localized\nKelvin-Voigt damping, Discrete and Continuous Dynamical Systems - Series B, 21(2016), 1757–1774.\n[8]F. Hassine, Energy decay estimates of elastic transmission wave/beam s ystems with a local Kelvin-Voigt damp-\ning,Internat. J. Control, 89(10) (2016), 1933-1950.\n[9]F. Hassine, Logarithmic stabilization of the Euler-Bernoulli plate eq uation with locally distributed Kelvin-Voigt\ndamping, Evolution Equations and Control Theory, 455(2) (2017), 1765–1782.\n[10] F. Huang, Characteristic conditions for exponential s tability of linear dynamical systems in Hilbert space, Ann.\nDifferential Equations ,1(1985), 43–56.STABILITY FOR COUPLED WAVES WITH LOCALLY DISTURBED KELVIN- VOIGT DAMPING 17\n[11] F. Huang, On the mathematical model for linear elastic s ystems with analytic damping, SIAM J. Control Optim.,\n26(3) (1988), 714–724.\n[12]K. Liu and Z. Liu, Exponential decay of energy of the Euler–Bernoulli beam wit h locally distributed Kelvin–\nVoigt damping, SIAM Journal on Control and Optimization, 36(1998), 1086–1098.\n[13]K. S. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation,\nZeitschrift f¨ ur Angewandte Mathematik und Physik (ZAMP), 56(2005), 630–644.\n[14]K. S. Liu and B. Rao, Exponential stability for wave equations with local Kelvin -Voigt damping, Zeitschrift\nf¨ ur Angewandte Mathematik und Physik (ZAMP), 57(2006), 419–432.\n[15] Z. Liu and Q. Zhang, Stability of a string with local Kelv in-Voigt damping and nonsmooth coefficient at interface,\nSIAM J. Control Optim., 54(2016), 1859–1871.\n[16] A. Pazy, Semigroups of linear operators and applications to partial differential equations , Springer, New York,\n1983.\n[17] H. Portillo Oquendo and P. Snez Pacheco, Optimal decay f or coupled waves with KelvinVoigt damping, Applied\nMathematics Letters 67(2017), 16-20.\n[18] J. Pr¨ uss, On the spectrum of C0-semigroups, Trans. Amer. Math. Soc. ,248(1984), 847–857.\n[19]L. Tebou, A constructive method for the stabilization of the wave equa tion with localized KelvinVoigt damping,\nC. R. Acad. Sci. Paris , Ser. I,350(2012), 603–608.\nUR Analysis and Control of PDEs, UR 13ES64, Department of Mat hematics, Faculty of Sciences\nof Monastir, University of Monastir, Tunisia\nE-mail address :fathi.hassine@fsm.rnu.tn\nUR Analysis and Control of PDEs, UR 13ES64, Department of Mat hematics, Faculty of Sciences\nof Monastir, University of Monastir, Tunisia\nE-mail address :nadia.souayeh@fst.utm.tn"    },    {        "title": "1204.5342v1.Nonlocal_feedback_in_ferromagnetic_resonance.pdf",        "content": "Nonlocal feedback in ferromagnetic resonance\nThomas Bose and Steffen Trimper\nInstitute of Physics, Martin-Luther-University, D-06099 Halle, Germany\u0003\n(Dated: April 27, 2022)\nAbstract\nFerromagnetic resonance in thin films is analyzed under the influence of spatiotemporal feedback\neffects. The equation of motion for the magnetization dynamics is nonlocal in both space and time\nandincludesisotropic, anisotropicanddipolarenergycontributionsaswellastheconservedGilbert-\nand the non-conserved Bloch-damping. We derive an analytical expression for the peak-to-peak\nlinewidth. It consists of four separate parts originated by Gilbert damping, Bloch-damping, a mixed\nGilbert-Bloch component and a contribution arising from retardation. In an intermediate frequency\nregimetheresultsarecomparablewiththecommonlyusedLandau-Lifshitz-Gilberttheorycombined\nwith two-magnon processes. Retardation effects together with Gilbert damping lead to a linewidth\nthe frequency dependence of which becomes strongly nonlinear. The relevance and the applicability\nof our approach to ferromagnetic resonance experiments is discussed.\nPACS numbers: 76.50.+g; 76.60.Es; 75.70.Ak; 75.40.Gb\n\u0003thomas.bose@physik.uni-halle.de; steffen.trimper@physik.uni-halle.de\n1arXiv:1204.5342v1  [cond-mat.mes-hall]  24 Apr 2012I. INTRODUCTION\nFerromagnetic resonance enables the investigation of spin wave damping in thin or ul-\ntrathin ferromagnetic films. The relevant information is contained in the linewidth of the\nresonance signal [1–3]. Whereas the intrinsic damping included in the Gilbert or Landau-\nLifshitz-Gilbert equation [4, 5], respectively, predicts a linear frequency dependence of the\nlinewidth [6], the extrinsic contributions associated with two-magnon scattering processes\nshow a nonlinear behavior. Theoretically two-magnon scattering was analyzed for the case\nthat the static external field lies in the film plane [7, 8]. The theory was quantitatively\nvalidated by experimental investigations with regard to the film thickness [9]. Later the\napproach was extended to the case of arbitrary angles between the external field and the\nfilm surface [10]. The angular dependence of the linewidth is often modeled by a sum of\ncontributions including angular spreads and internal field inhomogeneities [11]. Among oth-\ners, two-magnon mechanisms were used to explain the experimental observations [12–17]\nwhereas the influence of the size of the inhomogeneity was studied in [18]. As discussed in\n[3, 14] the two-magnon contribution to the linewidth disappears for tipping angles between\nmagnetization and film plane exceeding a critical one \bcrit\nM=\u0019=4. Recently, deviations from\nthis condition were observed comparing experimental data and numerical simulations [17].\nSpin pumping can also contribute to the linewidth as studied theoretically in [19]. How-\never, a superposition of both the Gilbert damping and the two-magnon contribution turned\nout to be in agreement very well with experimental data illustrating the dependence of the\nlinewidth on the frequency [16, 20–23]. Based on these findings it was put into question\nwhether the Landau-Lifshitz-Gilbert equation is an appropriate description for ferromag-\nnetic thin films. The pure Gilbert damping is not able to explain the nonlinear frequency\ndependence of the linewidth when two-magnon scattering processes are operative [3, 24].\nAssuming that damping mechanisms can also lead to a non-conserved spin length a way\nout might be the inclusion of the Bloch equations [25, 26] or the the Landau-Lifshitz-Bloch\nequation [27, 28] into the concept of ferromagnetic resonance.\nAnother aspect is the recent observation [29] that a periodic scattering potential can alter\nthe frequency dependence of the linewidth. The experimental results are not in agreement\nwith those based upon a combination of Gilbert damping and two-magnon scattering. It\nwas found that the linewidth as function of the frequency exhibits a non monotonous be-\n2havior. The authors [29] suggest to reconsider the approach with regard to spin relaxations.\nMoreover, it would be an advantage to derive an expression for the linewidth as a measure\nfor spin damping solely from the equation of motion for the magnetization.\nTaking all those arguments into account it is the aim of this paper to propose a gener-\nalized equation of motion for the magnetization dynamics including both Gilbert damping\nand Bloch terms. The dynamical model allows immediately to get the magnetic susceptibil-\nity as well as the ferromagnetic resonance linewidth which are appropriate for the analysis\nof experimental observations. A further generalization is the implementation of nonlocal\neffects in both space and time. This is achieved by introducing a retardation kernel which\ntakes into account temporal retardation within a characteristic time \u001cand a spatial one\nwith a characteristic scale \u0018. The last one simulates an additional mutual interaction of\nthe magnetic moments in different areas of the film within the retardation length \u0018. Re-\ncently such nonlocal effects were discussed in a complete different context [30]. Notice that\nretardation effects were already investigated for simpler models by means of the Landau-\nLifshitz-Gilbert equation. Here the existence of spin wave solutions were in the focus of the\nconsideration [31]. The expressions obtained for the frequency/damping parameters were\nconverted into linewidths according to the Gilbert contribution which is a linear function\nof the frequency [31, 32]. In the present approach we follow another line. The propagating\npart of the varying magnetization is supplemented by the two damping terms due to Gilbert\nand Bloch, compare Eq. (9). Based on this equation we derive analytical expressions for the\nmagnetic susceptibility, the resonance condition and the ferromagnetic resonance linewidth.\nDue to the superposition of damping and retardation effects the linewidth exhibits a non-\nlinear behavior as function of the frequency. The model is also extended by considering\nthe general case of arbitrary angles between the static external field and the film surface.\nMoreover the model includes several energy contributions as Zeeman and exchange energy\nas well as anisotropy and dipolar interaction. The consequences for ferromagnetic resonance\nexperiments are discussed.\nII. DERIVATION OF THE EQUATION OF MOTION\nIn order to define the geometry considered in the following we adopt the idea presented\nin [10], i.e. we employ two coordinate systems, the xyz-system referring to the film surface\n3ΘMey\nex,eX\nezMS\neZeY\nΘHH0\n/Bullet\n/Bullet\n/BulletξM(z1)\nM(z2)\nM(z3)hrf\nd\nlxlzFIG. 1. (Color online) The geometry referring to the film and the magnetization. Further descrip-\ntion in the text.\nand the XYZ-system which is canted by an angle \u0002Mwith respect to the film plane. The\nsituation for a film of thickness dis sketched in Fig. 1. The angle \u0002Mdescribing the direction\nof the saturation magnetization, aligned with the Z-axis, originates from the static external\nfieldH0which impinges upon the film surface under an angle \u0002H. Therefore, it is more\nconvenient to use the XYZ-system for the magnetization dynamics. As excitation source\nwe consider the radio-frequency (rf) magnetic field hrfpointing into the x= X-direction. It\nshould fulfill the condition hrf\u001cH0. To get the evolution equation of the magnetization\nM(r;t),r= (x;y;z )we have to define the energy of the system. This issue is well described\nin Ref. [10], so we just quote the most important results given there and refer to the cited\nliterature for details. Since we consider the thin film limit one can perform the average along\nthe direction perpendicular to the film, i.e.\nM(rk;t) =1\ndZd=2\n\u0000d=2dyM(r;t); (1)\nwhere rk= (x;0;z)lies in the film plane. In other words the spatial variation of the\nmagnetization across the film thickness dis neglected. The components of the magnetization\npoint into the directions of the XYZ-system and can be written as [33]\nM(rk;t) =MX(rk)eX+MY(rk)eY+\u0012\nMS\u0000M2\nX(rk) +M2\nY(rk)\n2MS\u0013\neZ:(2)\n4Typically the transverse components MX;Yare assumed to be much smaller than the satu-\nration magnetization MS. Remark that terms quadratic in MX;Yin the energy will lead to\nlinear terms in the equation of motion. The total energy of the system can now be expressed\nin terms of the averaged magnetization from Eq. (1) and reads\nH=Hz+Hex+Ha+Hd: (3)\nThe different contributions are the Zeeman energy\nHz=\u0000Z\nd3rH0sin (\u0002 H\u0000\u0002M)MY(rk)\n\u0000Z\nd3rH0cos (\u0002 H\u0000\u0002M)\u0012\nMS\u0000MX(rk)2+MY(rk)2\n2MS\u0013\n;(4)\nthe exchange energy\nHex=D\n2MSZ\nd3r\u0002\nrMX(rk)\u00032+\u0002\nrMY(rk)\u00032; (5)\nthe surface anisotropy energy\nHa=HSMSV\n2sin2(\u0002M) +HS\n2sin(2\u0002 M)Z\nd3rM Y(rk)\n+HS\n2MScos(2\u0002 M)Z\nd3rM Y(rk)2\u0000sin2(\u0002M)Z\nd3rM X(rk)2;(6)\nand the dipolar energy\nHd=2\u0019M2\nSVsin2(\u0002M) +\u0019Z\nd3r\u001a\n2MSsin(2\u0002 M)MY(rk)\n+\u0012dk2\nz\nkksin2(\u0002M)\u0000(dkk\u00002) cos2(\u0002M)\u00002 sin2(\u0002M)\u0013\nMY(rk)2\n+\u0012dk2\nx\nkk\u00002 sin2(\u0002M)\u0013\nMX(rk)2\u00002dkxkz\nkksin(\u0002 M)MX(rk)MY(rk)\u001b\n:(7)\nIn these expressions V=lxlzdis the volume of the film, Ddesignates the exchange stiffness\nandHS/d\u00001represents the uniaxial out-of-plane anisotropy field. If HS<0the easy axis\nis perpendicular to the film surface. The in-plane anisotropy contribution to the energy is\nneglected but it should be appropriate for polycrystalline samples [16]. Moreover kk=jkkj\nis introduced where kk=kxex+kzezis the wave vector of the spin waves parallel to the\nfilm surface. Eqs. (3)-(7) are valid in the thin film limit kkd\u001c1. In order to derive Hdin\nEq. (7) one defines a scalar magnetic potential and has to solve the corresponding boundary\n5value problem inside and outside of the film [34]. As result [10] one gets the expressions in\nEq. (7).\nIn general if the static magnetic field is applied under an arbitrary angle \u0002Hthe mag-\nnetization does not align in parallel, i.e. \u0002M6= \u0002 H. The angle \u0002Mcan be derived from\nthe equilibrium energy Heq=H(MX= 0;MY= 0). Defining the equilibrium free energy\ndensity asfeq(\u0002M) =Heq=Vaccording to Eqs. (3)-(7) one finds the well-known condition\nsin(\u0002 H\u0000\u0002M) =4\u0019M S+HS\n2H0sin(2 \u0002 M) (8)\nby minimizing feqwith respect to \u0002M. We further note that all terms linear in MYin\nEqs. (3)-(7) cancel mutually by applying Eq. (8) as already pointed out in Ref. [10].\nThe energy contributions in Eqs. (3) and the geometric aspects determine the dynamical\nequation for the magnetization. The following generalized form is proposed\n@\n@tM(rk;t) =ZZ\ndr0\nkdt0\u0000(rk\u0000r0\nk;t\u0000t0)(\n\r\u0002\nHeff(r0\nk;t0)\u0002M(r0\nk;t0)\u0003\n+\u000b\u0014\nM(r0\nk;t0)\u0002@\n@t0M(r0\nk;t0)\u0015\n\u00001\nT2M?(r0\nk;t0))\n;(9)\nwhere\r=g\u0016B=~is the absolute value of the gyromagnetic ratio, T2is the transverse\nrelaxation time of the components M?=MXeX+MYeYand\u000bdenotes the dimensionless\nGilbertdampingparameter. Thelatterisoftentransformedinto G=\u000b\rM Srepresentingthe\ncorresponding damping constant in unit s\u00001. The effective magnetic field Heffis related to\nthe energy in Eqs. (3)-(7) by means of variational principles [35], i.e. Heff=\u0000\u000eH=\u000eM+hrf.\nHere the external rf-field hrf(t)is added which drives the system out of equilibrium.\nRegarding the equation of motion presented in Eq. (9) we note that a similar type was\napplied in [12] for the evaluation of ferromagnetic resonance experiments. In this paper\nthe authors made use of a superposition of the Landau-Lifshitz equation and Bloch-like\nrelaxation. Here we have chosen the part which conserves the spin length in the Gilbert form\nand added the non-conserving Bloch term in the same manner. That the combination of\nthesetwodistinctdampingmechanismsissuitablefortheinvestigationofultrathinmagnetic\nfilms was also suggested in [24]. Since the projection of the magnetization onto the Z-axis is\nnot affected by T2this relaxation time characterizes the transfer of energy into the transverse\ncomponents of the magnetization. This damping type is supposed to account for spin-spin\nrelaxation processes such as magnon-magnon scattering [33, 36]. In our ansatz we introduce\n6another possible source of damping by means of the feedback kernel \u0000(rk\u0000r0\nk;t\u0000t0). The\nintroduction of this quantity reflects the assumption that the magnetization M(rk;t2)is\nnot independent of its previous value M(rk;t1)providedt2\u0000t1< \u001c. Here\u001cis a time\nscale where the temporal memory is relevant. In the same manner the spatial feedback\ncontrols the magnetization dynamics significantly on a characteristic length scale \u0018, called\nretardation length. Physically, it seems to be reasonable that the retardation length differs\nnoticeably from zero only in z-direction which is shown in Fig. 1. As illustrated in the figure\nM(x;z1;t)is affected by M(x;z2;t)while M(x;z3;t)is thought to have negligible influence\nonM(x;z1;t)sincejz3\u0000z1j>\u0018. Therefore we choose the following combination of a local\nand a nonlocal part as feedback kernel\n\u0000(rk\u0000r0\nk;t\u0000t0) =\u0000 0\u000e(rk\u0000r0\nk)\u000e(t\u0000t0)\n+\u00000\n4\u0018\u001c\u000e(x\u0000x0) exp\u0014\u0000jz\u0000z0j\n\u0018\u0015\nexp\u0014\u0000(t\u0000t0)\n\u001c\u0015\n; t>t0:(10)\nThe intensity of the spatiotemporal feedback is controlled by the dimensionless retardation\nstrength \u00000. The explicit form in Eq. (10) is chosen in such a manner that the Fourier-\ntransform \u0000(kk;!)!\u00000for\u0018!0and\u001c!0, and in case \u00000= 1the ordinary equation\nof motion for the magnetization is recovered. Further,R\ndrkdt\u0000(rk;t) = \u0000 0<1, i.e. the\nintegral remains finite.\nIII. SUSCEPTIBILITY AND FMR-LINEWIDTH\nIf the rf-driving field, likewise averaged over the film thickness, is applied in X-direction,\ni.e.hrf(rk;t) =hX(rk;t)eX, the Fourier transform of Eq. (9) is written as\n\u0014i!\n\r\u0000(kk;!)+1\n\rT2+H21(kk)\u0015\nMX(kk;!) =\u0000\u0014\nH1(kk) +i\u000b!\n\r\u0015\nMY(kk;!);\n\u0014i!\n\r\u0000(kk;!)+1\n\rT2+H12(kk)\u0015\nMY(kk;!) =\u0014\nH2(kk) +i\u000b!\n\r\u0015\nMX(kk;!)\u0000MShX(kk;!):\n(11)\n7The effective magnetic fields are expressed by\nH1(kk) =H0cos(\u0002 H\u0000\u0002M) + (4\u0019M S+HS) cos(2 \u0002 M)\n+ 2\u0019dkkMS \nk2\nz\nk2\nksin2(\u0002M)\u0000cos2(\u0002M)!\n+Dk2\nk\nH2(kk) =H0cos(\u0002 H\u0000\u0002M)\u0000(4\u0019M S+HS) sin2(\u0002M)\n+ 2\u0019dM Sk2\nx\nkk+Dk2\nk;(12)\nand\nH12(kk) = 2\u0019dM Skxkz\nkksin(\u0002 M) =\u0000H21(kk): (13)\nThe Fourier transform of the kernel yields\n\u0000(kk;!) =\u00000(1 + i!\u001c) + \u0000 1\n2 (1 + i!\u001c)(!2\u001c2\u001c1)'\u00000+ \u0000 1\n2\u0000i\n2\u00001!\u001c;\n\u00001=\u00000\n1 +\f2; \f =\u0018kz;(14)\nwhere the factor 1=2arises from the condition t > t0when performing the Fourier trans-\nformation from time into frequency domain. In Eq. (14) we discarded terms !2\u001c2\u001c1.\nThis condition is fulfilled in experimental realizations. So, it will be turned out later the\nretardation time \u001c\u001810 fs. Because the ferromagnetic resonance frequencies are of the order\n10:::100 GHz one finds!2\u001c2\u001810\u00008:::10\u00006. The retardation parameter \f=\u0018kz, introduced\nin Eq. (14), will be of importance in analyzing the linewidth of the resonance signal. With\nregard to the denominator in \u00001, compare Eq. (14), the parameter \fmay evolve ponderable\ninfluence on the spin wave damping if this quantity cannot be neglected compared to 1.\nAs known from two-magnon scattering the spin wave modes can be degenerated with the\nuniform resonance mode possessing wave vectors kk\u0018105cm\u00001. The retardation length \u0018\nmay be estimated by the size of inhomogeneities or the distance of defects on the film sur-\nface, respectively. Both length scales can be of the order \u001810:::1000 nm, see Refs. [18, 29].\nConsequently the retardation parameter \fcould reach or maybe even exceed the order of 1.\nLet us stress that in case \f= 0,\u001c= 0,\u00000= 1and neglecting the Gilbert damping,\ni.e.\u000b= 0, the spin wave dispersion relation is simply \rp\nH1(kk)H2(kk)\u0000H2\n12(kk). This\nexpression coincides with those ones given in Refs. [7] and [10].\nProceeding the analysis of Eq. (11) by defining the magnetic susceptibility \u001fas\nM\u000b(kk;!) =X\n\f\u001f\u000b\f(kk;!)h\f(kk;!);f\u000b;\fg=fX;Yg;(15)\n8whereh\fplays the role of a small perturbation and the susceptibility \u001f\u000b\fexhibits the\nresponse of the system. Eq. (15) reflects that there appears no dependence on the direction\nofkk.\nSince the rf-driving field is applied along the eX-direction it is sufficient to focus the\nfollowing discussion to the element \u001fXXof the susceptibility tensor. From Eq. (11) we\nconclude\n\u001fXX(kk;!) =MSh\nH1(kk;!) +i\u000b!\n\ri\nh\nH1(kk;!) +i\u000b!\n\rih\nH2(kk;!) +i\u000b!\n\ri\n+h\ni!\n\r\u0000(kk;!)+1\n\rT2i2:(16)\nBecause at ferromagnetic resonance a uniform mode is excited let us set kk= 0in Eqs. (12)-\n(13). Considering the resonance condition we can assume \f=\u0018kz= 0. For reasons men-\ntioned above we have to take \f=\u0018kz6= 0when the linewidth as a measure for spin damping\nis investigated. Physically we suppose that spin waves with non zero waves vectors are not\nexcited at the moment of the ferromagnetic resonance. However such excitations will evolve\nduring the relaxation process. In finding the resonance condition from Eq. (16) it seems to\nbe a reasonable approximation to disregard terms including the retardation time \u001c. Such\nterms give rise to higher order corrections. In the same manner all the contributions orig-\ninated from the damping, characterized by \u000bandT2, are negligible. Let us justify those\napproximation by quantitative estimations. The fields H1,H2and!=\rare supposed to\nrange in a comparable order of magnitude. On the other hand one finds \u000b\u001810\u00003:::10\u00002,\n!T2\u001810\u00002and!\u001c\u001810\u00004. Under these approximations the resonance condition reads\n\u0012!r\n\r\u00132\n= \u00002\n0H1(kk= 0)H2(kk= 0): (17)\nThisresultiswellknownforthecasewithoutretardationwith \u00000= 1. Althoughtheretarda-\ntion time\u001cand the retardation length \u0018are not incorporated in the resonance condition, the\nstrength of the feedback may be important as visible in Eq. (17). Now the consequences for\nthe experimental realization will be discussed. To address this issue the resonance condition\nEq. (17) is rewritten in terms of the resonance field Hr=H0(!=!r)leading to\nHr=1\n2 cos(\u0002 H\u0000\u0002M)8\n<\n:s\n(4\u0019M S+HS)2cos4(\u0002M) +\u00121\n\u000002!r\n\r\u00132\n\u0000(4\u0019M S+HS)(1\u00003 sin2(\u0002M))9\n=\n;:(18)\n9ΘM[deg]\nΘH[deg]Γ0= 0 .7\nΓ0= 1 .0\nΓ0= 1 .3FIG. 2. (Color online) Dependence of the magnetization angle \u0002Mon the angle \u0002Hunder which the\nstatic external field is applied for !r=(2\u0019) = 10 GHz . The parameters are taken from [16]: 4\u0019MS=\n16980 G,HS=\u00003400 G;\r= 0:019 GHz=G.\nThe result is arranged in the in the same manner as done in [16]. The difference is the\noccurrence of the parameter \u00000in the denominator. In [16] the gyromagnetic ratio \rand\nthe sum (4\u0019M S+HS)were obtained from \u0002H-dependent measurements and a fit of the\ndata according to Eq. (18) with \u00000= 1under the inclusion of Eq. (8). If the saturation\nmagnetization can be obtained from other experiments [16] the uniaxial anisotropy field HS\nresults. Thus, assuming \u000006= 1the angular dependence \u0002M(\u0002H)and the fitting parameters\nas well would change. In Fig. 2 we illustrate the angle \u0002M(\u0002H)for different values of \u00000and\na fixed resonance frequency. If \u00000<1the curve is shifted to larger \u0002Mand for \u00000>1to\nsmaller magnetization angles. To produce Fig. 2 we utilized quantitative results presented\nin [16]. They found for Co films grown on GaAs the parameters 4\u0019M S= 16980 G ,HS=\n\u00003400 Gand\r= 0:019 GHz=G. As next example we consider the influence of HSand denote\nH(0)\nS=\u00003400 Gthe anisotropy field for \u00000= 1andH(R)\nSthe anisotropy field for \u000006= 1. The\nabsolute value of their ratio jH(R)\nS=H(0)\nSj, derived from Hr(H(0)\nS;\u00000= 1) =Hr(H(R)\nS;\u000006= 1),\nisdepictedinFig.3forvariousfrequencies. Inthisgraphweassumedthatallotherquantities\nremain fixed. The effect of a varying retardation strength on the anisotropy field can clearly\nbeseen. Thechangeinthesignoftheslopeindicatesthattheanisotropyfield H(R)\nSmayeven\nchange its sign. From here we conclude that the directions of the easy axis and hard axis\nare interchanged. For the frequencies 4 GHzand10 GHzthis result is not observed in the\nrange chosen for \u00000. Moreover, the effects become more pronounced for higher frequencies.\n10/vextendsingle/vextendsingle/vextendsingleH(R)\nS/H(0)\nS/vextendsingle/vextendsingle/vextendsingle\nΓ04 GHz\n10 GHz\n35 GHz\n50 GHz\n70 GHzFIG. 3. (Color online) Effect of varying retardation strength on the uniaxial anisotropy field for\nvarious frequencies and \u0002M=\u0019=3.4\u0019MS= 16980 G ,HS=\u00003400 G;\r= 0:019 GHz=G, see [16].\nIn Fig. 3 we consider only a possible alteration of the anisotropy field. Other parameters like\nthe experimentally obtained gyromagnetic ration were unaffected. In general this parameter\nmay also experiences a quantitative change simultaneously with HS.\nLet us proceed by analyzing the susceptibility obtained in Eq. (16). Because the following\ndiscussion is referred to the energy absorption in the film, we investigate the imaginary part\nofthesusceptibility \u001f00\nXX. SinceexperimentallyoftenaLorentziancurvedescribessufficiently\nthe resonance signal we intend to arrange \u001f00\nXXin the form A0=(1 +u2), whereA0is the\nabsolute value of the amplitude and uis a small parameter around zero. The mapping to a\nLorentzian is possible under some assumptions. Because the discussion is concentrated on\nthe vicinity of the resonance we introduce \u000eH=H0\u0000Hr, whereHris the static external\nfield when resonance occurs. Consequently, the fields in Eq. (12) have to be replaced by\nH1;2!H(r)\n1;2+\u000eHcos(\u0002 H\u0000\u0002M). Additionally, we take into account only terms of the order\np\n\u000f\u0015in the final result for the linewidth where f\u000f;\u0015g/f!=\r[\u000b+!\u001c] + 1=(\rT2)g. After a\nlengthy but straightforward calculation we get for \u000eH=H(r)\n1;2\u001c1and using the resonance\ncondition in Eq. (17)\n\u001f00\nXX(!) =A0\n1 +h\nH0\u0000Hr\n\u0001Ti2; A0=MS\n(1 +\u0014) cos(\u0002 H\u0000\u0002M) \u0001T; \u0014=H(r)\n2\nH(r)\n1:(19)\nHere we have introduced the total half-width at half-maximum (HWHM) \u0001Twhich can be\n11brought in the form\n\u0001T=1\ncos(\u0002 H\u0000\u0002M)q\n\u00012\nG+ \u00012\nB+ \u00012\nGB+ \u00012\nR: (20)\nThe HWHM is a superposition of the Gilbert contribution \u0001G, the Bloch contribution \u0001B,\na joint contribution \u0001GBarising from the combination of the Gilbert and Bloch damping\nparts in the equation of motion and the contribution \u0001Rwhich has its origin purely in the\nfeedback mechanisms introduced into the system. The explicit expressions are\n\u0001G=!\n\rs\n\u000b\u0014\n\u000b\u000016p\u0014\n(1 +\u0014)\u00000\u00001!\u001c\n(\u00000+ \u0000 1)3\u0015\n; (21a)\n\u0001B=4 \u00000\n(\u00000+ \u0000 1)p\u0014\n(1 +\u0014)s\n1\n(\rT2)2\u00004 \u00001\n(\u00000+ \u0000 1)2!\n\r!\u001c\n\rT2; (21b)\n\u0001GB=s\n8\u00000\n(\u00000+ \u0000 1)p\u0014\n(1 +\u0014)\u000b!\n\r2T2; (21c)\n\u0001R=8p\u0014\n(1 +\u0014)!\n\r\u00000\u00001!\u001c\n(\u00000+ \u0000 1)3: (21d)\nThe parameter \u00001is defined in Eq. (14). If the expressions under the roots in Eqs. (21a)\nand (21b) are negative we assume that the corresponding process is deactivated and does\nnot contribute to the linewidth \u0001HT. Typically, experiments are evaluated in terms of the\npeak-to-peak linewidth of the derivative d\u001f00\nXX=dH0, denoted as \u0001H\u0011. One gets\n\u0001H\u0011=2p\n3\u0001\u0011; (22)\nwhere the index \u0011stands for G(Gilbert contribution), B(Bloch contribution), GB(joint\nGilbert-Bloch contribution), R(pure retardation contribution) or Tdesignating the total\nlinewidth according to Eq. (20) and Eqs. (21a)-(21d). Obviously these equations reveal a\nstrong nonlinear frequency dependence, which will be discussed in the subsequent section.\nIV. DISCUSSION\nAs indicated in Eqs. (20) - (22) the quantity \u0001H\u0011consists of well separated distinct\ncontributions. Thebehaviorof \u0001H\u0011isshowninFigs.4-6asfunctionofthethreeretardation\nparameters, the strength \u00000, the spatial range \fand the time scale \u001c. In all figures the\nfrequencyf=!=(2\u0019)is used. In Fig. 4 the dependence on the retardation strength \u00000is\n12∆HT[G]4 GHz\n10 GHz\n35 GHz\n50 GHz\n70 GHz∆Hη[G]\nΓ0∆HG\n∆HB\n∆HGB\n∆HR\n∆HTf= 70 GHzFIG. 4. (Color online) Influence of the retardation strength \u00000on the peak-to-peak linewidth \u0001HT\nfor various frequencies (top graph) and on the single contributions \u0001H\u0011forf= 70 GHz (bottom\ngraph). \u0001B= 0is this frequency region. The parameters are: \u0002H= \u0002 M= 0,\f= 0:5,\u000b= 0:01,\nT2= 5\u000210\u00008s;\u001c= 1:7\u000210\u000014s. The other parameters are 4\u0019MS= 16980 G ,HS=\u00003400 G;\r=\n0:019 GHz=G, compare [16].\nshown. As already observed in Figs. 2 and 3 a small change of \u00000may lead to remarkable\neffects. Hence we vary this parameter in a moderate range 0:5\u0014\u00000\u00142. The peak-to-peak\nlinewidth \u0001HTas function of \u00000remains nearly constant for f= 4 GHz andf= 10 GHz ,\nwhereas for f= 35 GHz a monotonous growth-up is observed. Increasing the frequency\nfurther tof= 50 GHz and70 GHzthe curves offers a pronounced kink. The subsequent\nenhancement is mainly due to the Gilbert damping. In the region of negative slope we\nset\u0001HG(\u00000) = 0, while in that one with a positive slope \u0001HG(\u00000)>0grows and tends\nto2\u000b!=(p\n3\r)for\u00000!1. The other significant contribution \u0001HR, arising from the\nretardation decay, offers likewise a monotonous increase for growing values of the retardation\nparameter \u00000. This behavior is depicted in Fig. 4 for f= 70 GHz . Now let us analyze the\ndependence on the dimensionless retardation length \f=\u0018kz. Because\fis only nonzero if\n13∆HT[G]4 GHz\n10 GHz\n35 GHz\n50 GHz\n70 GHz∆Hη[G]\nβ∆HG\n∆HB\n∆HGB\n∆HR\n∆HTf= 70 GHzFIG. 5. (Color online) Influence of the dimensionless retardation length \f=\u0018kzon the total\npeak-to-peak linewidth \u0001HTfor various frequencies (top graph) and on the single contributions\n\u0001H\u0011forf= 70 GHz (bottom graph); \u0001B= 0in this range. The parameters are: \u0002H= \u0002 M= 0,\n\u00000= 1:1,\u000b= 0:01,T2= 5\u000210\u00008s;\u001c= 1:7\u000210\u000014s. The other parameters: 4\u0019MS= 16980 G ,\nHS=\u00003400 Gand\r= 0:019 GHz=Gare taken from [16].\nkz6= 0this parameter \u0018accounts the influence of excitations with nonzero wave vector. We\nargue that both nonzero wave vector excitations, those arising from two-magnon scattering\nand those originated from feedback mechanisms, may coincide. Based on the estimation\nin the previous section we consider the relevant interval 10\u00002\u0014\f\u001410. The results are\nshown in Fig.5. Within the range of \fone recognizes that the total peak-to-peak linewidths\n\u0001HTforf= 4 GHz andf= 10 GHz offer no alteration when \fis changed. The plotted\nlinewidths are characterized by a minimum followed by an increase which occurs when \f\nexceeds approximately 1. This behavior is the more accentuated the larger the frequencies\nare. The shape of the curve can be explained by considering the single contributions as\nis visible in the lower part in Fig. 5. While both quantities \u0001HG(\f)and\u0001HR(\f)remain\nconstant for small \f,\u0001HG(\f)tends to a minimum and increases after that. The quantity\n14∆HT[G]4 GHz\n10 GHz\n35 GHz\n50 GHz\n70 GHz∆Hη[G]\nτ[fs]∆HG\n∆HB\n∆HGB\n∆HR\n∆HTf= 70 GHzFIG. 6. (Color online) Influence of the retardation time \u001con the total peak-to-peak linewidth\n\u0001HTfor various frequencies (top graph) and on the single contributions \u0001H\u0011forf= 70 GHz\n(bottom graph). \u0001B= 0in this region. The parameters are \u0002H= \u0002 M= 0,\f= 0:5,\u000b= 0:01,\nT2= 5\u000210\u00008s;\u00000= 1:1; the other parameters are taken from [16]: 4\u0019MS= 16980 G ,HS=\n\u00003400 G;\r= 0:019 GHz=G.\n\u0001HR(\f)develops a maximum around \f\u00191. Thus, both contributions show nearly opposite\nbehavior. The impact of the characteristic feedback time \u001con the linewidth is illustrated\nin Fig. 6. In this figure a linear time scale is appropriate since there are no significant\neffects in the range 1 fs\u0015\u001c\u00150. The total linewidth \u0001HT(\u001c)is again nearly constant\nforf= 4 GHz andf= 10 GHz . In contrast \u0001HT(\u001c)reveals for higher frequencies two\nregions with differing behavior. The total linewidth decreases until \u0001HG(\u001c)becomes zero.\nAfter that one observes a positive linear slope which is due to the retardation part \u0001HR(\u001c).\nThis linear dependency is recognizable in Eq. (21d), too. Below we will present arguments\nwhy the feedback time \u001cis supposed to be in the interval 0< \u001c < 100 fs. Before let us\nstudy the frequency dependence of the linewidth in more detail. The general shape of the\ntotal linewidth \u0001HT(!)is depicted in Fig. 7. Here both the single contribution to the\n15∆Hη[G]\nf[GHz]∆HG\n∆HB\n∆HGB\n∆HR\n∆HTFIG. 7. (Color online) Frequency dependence of all contributions to the peak-to-peak linewidth for\n\u0002H= \u0002 M= 0,\f= 0:5,\u000b= 0:01,T2= 5\u000210\u00008s,\u001c= 1:7\u000210\u000014sand\u00000= 1:2. Parameters taken\nfrom Ref. [16]: 4\u0019MS= 16980 G ,HS=\u00003400 Gand\r= 0:019 GHz=G. The Bloch contribution\n\u0001HBis shown in the inset.\nlinewidth and the total linewidth are shown. Notice that the total linewidth is not simply\nthe sum of the individual contributions but has to be calculated according to Eq. (20). One\nrealizes that the Bloch contribution \u0001HBis only nonzero for frequencies f\u00146 GHzin the\nexamples shown. Accordingly \u0001HB= 0in Figs. 4-6 (lower parts) since these plots refer to\nf= 70 GHz . The behavior of the Gilbert contribution deviates strongly from the typically\napplied linear frequency dependence. Moreover, the Gilbert contribution will develop a\nmaximum value and eventually it disappears at a certain frequency where the discriminant\nin Eq. (21a) becomes negative. Nevertheless, the total linewidth is a nearly monotonous\nincreasing function of the frequency albeit, as mentioned before, for some combinations of\nthe model parameters there might exist a very small frequency region where \u0001HGreaches\nzero and the slope of \u0001HTbecomes slightly negative. The loss due to the declining Gilbert\npart is nearly compensated or overcompensated by the additional line broadening originated\nbytheretardationpartandthecombinedGilbert-Blochterm. Thelatteroneis \u0001HGB/pf\nand\u0001HR/f2, see Eqs. (21c)-(21d). In the frequency region where \u0001HG= 0only \u0001HGB\nand\u0001HRcontribute to the total linewidth, the shape of the linewidth is mainly dominated\nby\u0001HR. Thispredictionisanewresult. Thebehavior \u0001HR/f2, obtainedinourmodelfor\nhigh frequencies, is in contrast to conventional ferromagnetic resonance including only the\nsum of a Gilbert part linear in frequency and a two-magnon contribution which is saturated\n16at high frequencies. So far, experimentally the frequency ranges from 1 GHzto225 GHz,\nsee [21]. Let us point out that the results presented in Fig. 7 can be adjusted in such a\nmanner that the Gilbert contribution will be inoperative at much higher frequencies by the\nappropriate choice of the model parameters. Due to this fact we suggest an experimental\nverification in more extended frequency ranges. Another aspect is the observation that\nexcitations with a nonzero wave vector might represent one possible retardation mechanism.\nRegarding Eqs. (21a)-(21d) retardation can also influence the linewidth in case kz= 0\n(i.e.\f= 0and\u00001= \u0000 0). Only if\u001c= 0the retardation effects disappear. Therefore let us\nconsider the time domain of retardation and its relation to the Gilbert damping. The Gilbert\ndamping and the attenuation due to retardation can be considered as competing processes.\nSo temporal feedback can cause that the Gilbert contribution disappears. In the same\nsense the Bloch contribution is a further competing damping effect. In this regard temporal\nfeedback has the ability to reverse the dephasing process of spin waves based on Gilbert and\nBloch damping. On the other hand the retardation part \u0001Rin Eq. (21d) is always positive\nfor\u001c > 0. Thus, the retardation itself leads to linewidth broadening in ferromagnetic\nresonance and consequently to spin damping. Whether the magnitude of retardation is able\nto exceed the Gilbert damping depends strongly on the frequency. With other words, the\nfrequency of the magnetic excitation ’decides’ to which damping mechanisms the excitation\nenergyistransferred. Ourcalculationsuggeststhatforsufficienthighfrequenciesretardation\neffects dominate the intrinsic damping behavior. Thus the orientation and the value of the\nmagnetization within the retardation time \u001cplays a major role for the total damping.\nGenerally, experimental data should be fit according to the frequency dependence of the\nlinewidth in terms of Eqs. (20)-(22). To underline this statement we present Fig. 8. In this\ngraph we reproduce some results presented in [7] for the case \u0002H= \u0002 M= 0. To be more\nspecific, we have used Eq. (94) in [7] which accounts for the two-magnon scattering and\nthe parameters given there. As result we find a copy of Fig. 4 in [7] except of the factor\n2=p\n3. Further, we have summed up the conventional Gilbert linewidth /fwith the Gilbert\ndamping parameter \u000b1= 0:003. This superposition yields to the dotted line in Fig. 8. The\nresult is compared with the total linewidth resulting from our retardation model plotted as\nsolid line. To obtain the depicted shape we set the Gilbert damping parameter according\nto the retardation model \u000b2= 0:0075, i.e. to get a similar behavior in the same order of\nmagnitude of \u0001HTwithin both approaches we have to assume that \u000b2is more than twice\n17∆HT[G]\nf[GHz]retardation model\nGilbert+2-magnonFIG. 8. (Color online) Comparison with the two-magnon model. Frequency dependence of the total\npeak-to-peak linewidth \u0001HTfor\u0002H= \u0002 M= 0,\f= 0:5,\u000b1= 0:003,\u000b2= 0:0075,T2= 5\u000210\u00008s,\n\u001c= 1:22\u000210\u000014sand\u00000= 1:2. Parameters taken from [7]: 4\u0019MS= 21000 G ,HS=\u000015000 Gand\nfrom [37]:\r= 0:018 GHz=G(derived from g= 2:09for bulk Fe). The dotted line is a superposition\nof Fig. 4 in [7] reflecting the two-magnon contribution and the Gilbert contribution (denoted as\n\u000b1in the text) linear in the frequency.\nas large compared to \u000b1.\nFinally we discuss briefly the \u0002H-dependence of the linewidth which is shown in Fig. 9.\nIn the upper part of the figure one observes that \u0001HT(\u0002H)exhibits a maximum which is\nshifted towards lower field angles as well as less pronounced for increasing frequencies. The\nlower part of Fig. 9, referring to f= 10 GHz , displays that the main contribution to the total\nlinewidth arises from the Gilbert part \u0001HG. This result for f= 10 GHz is in accordance\nwith the results discussed previously, compare Fig. 7. For higher frequencies the retardation\ncontribution \u0001HRmay exceed the Gilbert part.\nV. CONCLUSIONS\nA detailed study of spatiotemporal feedback effects and intrinsic damping terms offers\nthat both mechanisms become relevant in ferromagnetic resonance. Due to the superposi-\ntion of both effects it results a nonlinear dependence of the total linewidth on the frequency\nwhich is in accordance with experiments. In getting the results the conventional model in-\ncluding Landau-Lifshitz-Gilbert damping is extended by considering additional spatial and\n18linewidth ∆ HT[G]\n4 GHz\n10 GHz\n35 GHz\n50 GHz\n70 GHzlinewidth ∆ Hη[G]\nΘH[deg]∆HB\n∆HR\n∆HGB\n∆HG\n∆HTf= 10 GHzFIG. 9. (Color online) Angular dependence of the total peak-to-peak linewidth \u0001HTfor various\nfrequencies (top graph) and all contributions \u0001H\u0011forf= 10 GHz (bottom graph) with \f= 0:5,\n\u000b= 0:01,T2= 5\u000210\u00008s,\u001c= 1:7\u000210\u000014sand\u00000= 1:1. The parameters are taken from\n[16]: 4\u0019MS= 16980 G ,HS=\u00003400 Gand\r= 0:019 GHz=G.\ntemporal retardation and non-conserved Bloch damping terms. Our analytical approach\nenables us to derive explicit expressions for the resonance condition and the peak-to-peak\nlinewidth. We were able to link our results to such ones well-known from the literature.\nThe resonance condition is affected by the feedback strength \u00000. The spin wave damping is\nlikewise influenced by \u00000but moreover by the characteristic memory time \u001cand the retar-\ndation length \u0018. As expected the retardation gives rise to an additional damping process.\nFurthermore, the complete linewidth offers a nonlinear dependence on the frequency which\nis also triggered by the Gilbert damping. From here we conclude that for sufficient high\nfrequencies the linewidth is dominated by retardation effects. Generally, the contribution of\nthedifferentdampingmechanismstothelinewidthiscomprisedofwellseparatedrateswhich\nare presented in Eqs. (20)-(22). Since each contribution to the linewidth is characterized\nby adjustable parameters it would be very useful to verify our predictions experimentally.\n19Notice that the contributions to the linewidth in Eqs. (20)-(22) depend on the shape of\nthe retardation kernel which is therefore reasonable not only for the theoretical approach\nbut for the experimental verification, too. One cannot exclude that other mechanisms as\nmore-magnon scattering effects, nonlinear interactions, spin-lattice coupling etc. are likewise\nrelevant. Otherwise, we hope that our work stimulates further experimental investigations\nin ferromagnetic resonance.\nWe benefit from valuable discussions about the experimental background with Dr. Khali\nZakeri from the Max-Planck-Institute of Microstructure Physics. 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Lett. 49, 658\n(2000)\n22"    },    {        "title": "1701.03083v2.The_Cauchy_problem_for_the_Landau_Lifshitz_Gilbert_equation_in_BMO_and_self_similar_solutions.pdf",        "content": "The Cauchy problem for the Landau–Lifshitz–Gilbert equation\nin BMO and self-similar solutions\nSusana Gutiérrez1and André de Laire2\nAbstract\nWe prove a global well-posedness result for the Landau–Lifshitz equation with Gilbert\ndamping provided that the BMO semi-norm of the initial data is small. As a consequence,\nwe deduce the existence of self-similar solutions in any dimension. In the one-dimensional\ncase, we characterize the self-similar solutions associated with an initial data given by some\n(S2-valued) step function and establish their stability. We also show the existence of multiple\nsolutions if the damping is strong enough.\nOur arguments rely on the study of a dissipative quasilinear Schrödinger equation ob-\ntained via the stereographic projection and techniques introduced by Koch and Tataru.\nKeywords and phrases: Landau–Lifshitz–Gilbert equation, global well-posedness, discontin-\nuous initial data, stability, self-similar solutions, dissipative Schrödinger equation, complex\nGinzburg–Landau equation, ferromagnetic spin chain, heat-flow for harmonic maps.\n2010Mathematics Subject Classification: 35R05, 35Q60, 35A01, 35C06, 35B35, 35Q55,\n35Q56, 35A02, 53C44.\nContents\n1 Introduction and main results 2\n2 The Cauchy problem 6\n2.1 The Cauchy problem for a dissipative quasilinear Schrödinger equation . . . . . . 6\n2.2 The Cauchy problem for the LLG equation . . . . . . . . . . . . . . . . . . . . . 15\n3 Applications 22\n3.1 Existence of self-similar solutions in RN. . . . . . . . . . . . . . . . . . . . . . . 22\n3.2 The Cauchy problem for the one-dimensional LLG equation with a jump initial\ndata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23\n3.2.1 Existence, uniqueness and stability. Proof of Theorem 1.2 . . . . . . . . . 24\n3.2.2 Multiplicity of solutions. Proof of Theorem 1.3 . . . . . . . . . . . . . . . 28\n3.3 A singular solution for a nonlocal Schrödinger equation . . . . . . . . . . . . . . . 30\n4 Appendix 34\n1School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom.\nE-mail: s.gutierrez@bham.ac.uk\n2Univ. Lille, CNRS, Inria, UMR 8524, Laboratoire Paul Painlevé, F-59000 Lille, France.\nE-mail: andre.de-laire@univ-lille.fr\n1arXiv:1701.03083v2  [math.AP]  20 Mar 20191 Introduction and main results\nWe consider the Landau–Lifshitz–Gilbert (LLG) equation\n@tm=\fm\u0002\u0001m\u0000\u000bm\u0002(m\u0002\u0001m);onRN\u0002R+; (LLG\u000b)\nwhere m= (m1;m2;m3) :RN\u0002R+\u0000!S2is the spin vector, \f\u00150,\u000b\u00150;\u0002denotes the usual\ncross-product in R3, and S2is the unit sphere in R3. This model introduced by Landau and\nLifshitz describes the dynamics for the spin in ferromagnetic materials [26, 16] and constitutes a\nfundamental equation in the magnetic recording industry [36]. The parameters \f\u00150and\u000b\u00150\nare respectively the so-called exchange constant and Gilbert damping, and take into account the\nexchange of energy in the system and the effect of damping on the spin chain. Note that, by\nperforming a time-scaling, we assume w.l.o.g. that\n\u000b2[0;1]and\f=p\n1\u0000\u000b2:\nThe Landau–Lifshitz family of equations includes as special cases the well-known heat-flow for\nharmonic maps and the Schrödinger map equation onto the 2-sphere. In the limit case \f= 0\n(and so\u000b= 1) the LLG equation reduces to the heat-flow equation for harmonic maps\n@tm\u0000\u0001m=jrmj2m;onRN\u0002R+: (HFHM)\nThe case when \u000b= 0(i.e. no dissipation/damping) corresponds to the Schrödinger map equation\n@tm=m\u0002\u0001m;onRN\u0002R+: (SM)\nIn the one-dimensional case N= 1, we established in [17] the existence and asymptotics of the\nfamilyfmc;\u000bgc>0of self-similar solutions of (LLG \u000b) for any fixed \u000b2[0;1], extending the results\nin Gutiérrez, Rivas and Vega [18] in the setting of the Schrödinger map equation and related\nbinormal flow equation. The motivation for the results presented in this paper first originated\nfrom the desire to study further properties of the self-similar solutions found in [17], and in\nparticular their stability. In the case \u000b= 0, the stability of the self-similar solutions of the\nSchrödinger map has been considered in the series of papers by Banica and Vega [5, 6, 7], but\nno stability result is known for these solutions in the presence of damping, i.e. \u000b > 0. One of\nthe key ingredients in the analysis given by Banica and Vega is the reversibility in time of the\nequation in the absence of damping. However, since (LLG \u000b) is a dissipative equation for \u000b>0,\nthis property is no longer available and a new approach is needed.\nIn the one-dimensional case and for fixed \u000b2[0;1], the self-similar solutions of (LLG \u000b)\nconstitute a uniparametric family fmc;\u000bgc>0where mc;\u000bis defined by\nmc;\u000b(x;t) =f\u0012xp\nt\u0013\n;\nfor some profile f:R\u0000!S2, and is associated with an initial condition given by a step function\n(at least when cis small) of the form\nm0\nc;\u000b:=A+\nc;\u000b\u001fR++A\u0000\nc;\u000b\u001fR\u0000; (1.1)\nwhere A\u0006\nc;\u000bare certain unitary vectors and \u001fEdenotes the characteristic function of a set E. In\nparticular, when \u000b>0, the Dirichlet energy associated with the solutions mc;\u000bgiven by\nkrmc;\u000b(\u0001;t)k2\nL2=c2\u00102\u0019\n\u000bt\u00111=2\n; t> 0; (1.2)\n2diverges as t!0+3.\nA first natural question in the study of the stability properties of the family of solutions\nfmc;\u000bgc>0is whether or not it is possible to develop a well-posedness theory for the Cauchy\nproblem for (LLG \u000b) in a functional framework that allows us to handle initial conditions of the\ntype (1.1). In view of (1.1) and (1.2), such a framework should allow some “rough” functions\n(i.e. function spaces beyond the “classical” energy ones) and step functions.\nA few remarks about previously known results in this setting are in order. In the case \u000b>0,\nglobal well-posedness results for (LLG \u000b) have been established in N\u00152by Melcher [31] and by\nLin, Lai and Wang [30] for initial conditions with a smallness condition on the gradient in the\nLN(RN)and the Morrey M2;2(RN)norm4, respectively. Therefore these results do not apply\nto the initial condition m0\nc;\u000b. When\u000b= 1, global well-posedness results for the heat flow for\nharmonic maps (HFHM) have been obtained by Koch and Lamm [22] for an initial condition\nL1-close to a point and improved to an initial data with small BMO semi-norm by Wang [35].\nThe ideas used in [22] and [35] rely on techniques introduced by Koch and Tataru [23] for the\nNavier–Stokes equation. Since m0\nc;\u000bhas a small BMO semi-norm if cis small, the results in [35]\napply to the case \u000b= 1.\nTherearetwomainpurposesinthispaper. Thefirstoneistoadaptandextendthetechniques\ndeveloped in [22, 23, 35] to prove a global well-posedness result for (LLG \u000b) with\u000b2(0;1]for\ndatam0inL1(RN;S2)with small BMO semi-norm. The second one is to apply this result to\nestablish the stability of the family of self-similar solutions fmc;\u000bgc>0found in [17] and derive\nfurther properties for these solutions. In particular, a further understanding of the properties of\nthe functions mc;\u000bwill allow us to prove the existence of multiple smooth solutions of (LLG \u000b)\nassociated with the same initial condition, provided that \u000bis close to one.\nIn order to state the first of our results, we introduce the function space Xas follows:\nX=fv:Rn\u0002R+!R3:v;rv2L1\nloc(RN\u0002R+)andkvkX:=supt>0kv(t)kL1+ [v]X<1g\nwhere\n[v]X:=supt>0p\ntkrvkL1+ sup\nx2RN\nr>0 \n1\nrN\u0002\nBr(x)\u0002[0;r2]jrv(y;t)j2dtdy!1\n2\n;\nandBr(x)denotes the ball with center xand radius r >0inRN. Let us remark that the first\nterm in the definition of [v]Xallows to capture a blow-up rate of 1=p\ntforkrv(t)kL1, ast!0+.\nThis is exactly the blow-up rate for the self-similar solutions (see (3.1) and (3.12)). The integral\nterm in [v]Xis associated with the space BMO as explained in Subsection 2.1, and it is also well\nadapted to the self-similar solutions (see Proposition 3.4 and its proof).\nWe can now state the following (global) well-posedness result for the Cauchy problem for the\nLLG equation:\nTheorem 1.1. Let\u000b2(0;1]. There exist constants M1;M2;M3>0, depending only on \u000band\nNsuch that the following holds. For any m02L1(RN;S2),Q2S2,\u000e2(0;2]and\"0>0such\nthat\"0\u0014M1\u000e6,\ninf\nRNjm0\u0000Qj2\u00152\u000eand [m0]BMO\u0014\"0; (1.3)\nthere exists a unique solution m2X(RN\u0002R+;S2)of(LLG\u000b)with initial condition m0such\nthat\ninf\nx2RN\nt>0jm(x;t)\u0000Qj2\u00154\n1 +M2\n2(M3\u000e4+\u000e\u00001)2and [m]X\u00144M2(M3\u000e4+ 8\u000e\u00002\"0):(1.4)\n3We refer the reader to Theorem A.5 in the Appendix and to [17] for precise statements of these results.\n4See footnote in Section 3.3 for the definition of the Morrey space M2;2(RN).\n3In addition, mis a smooth function belonging to C1(RN\u0002R+;S2). Furthermore, assume that\nnis a solution to (LLG\u000b)fulfilling (1.4), with initial condition n0satisfying (1.3). Then\nkm\u0000nkX\u0014120M2\n\u000e2km0\u0000n0kL1: (1.5)\nAs we will see in Section 2, the proof of Theorem 1.1 relies on the use of the stereographic pro-\njection to reduce Theorem 1.1 to establish a well-posedness result for the associated dissipative\n(quasilinear) Schrödinger equation (see Theorem 2.1). In order to be able to apply Theorem 1.1\nto the study of both the initial value problem related to the LLG equation with a jump initial\ncondition, and the stability of the self-similar solutions found in [17], we will need a more quanti-\ntative version of this result. A more refined version of Theorem 1.1 will be stated in Theorem 2.9\nin Subsection 2.2.\nTheorem 1.1 (or more precisely Theorem 2.9) has two important consequences for the Cauchy\nproblem related to (LLG \u000b) in one dimension:\n8\n<\n:@tm=\fm\u0002@xxm\u0000\u000bm\u0002(m\u0002@xxm);onR\u0002R+;\nm0\nA\u0006:=A+\u001fR++A\u0000\u001fR\u0000;(1.6)\nwhere A\u0006are two given unitary vectors such that the angle between A+andA\u0000is sufficiently\nsmall:\n(a)From the uniqueness statement in Theorem 1.1, we can deduce that the solution to (1.6)\nprovided by Theorem 1.1 is a rotation of a self-similar solution mc;\u000bfor an appropriate\nvalue ofc(see Theorem 3.3 for a precise statement).\n(b)(Stability) From the dependence of the solution with respect to the initial data established\nin (1.5) and the analysis of the 1d-self-similar solutions mc;\u000bcarried out in [17], we obtain\nthe following stability result: For any given m02S2satisfying (1.3) and close enough to\nm0\nA\u0006, the solution mof (LLG\u000b) associated with m0given by Theorem 1.1 must remain\nclose to a rotation of a self-similar solution mc;\u000b, for somec>0. In particular, mremains\nclose to a self-similar solution.\nThe precise statement is provided in the following theorem.\nTheorem 1.2. Let\u000b2(0;1]. There exist constants L1;L2;L3>0,\u000e\u00032(\u00001;0),#\u0003>0such\nthat the following holds. Let A+,A\u00002S2with angle#between them. If\n0<#\u0014#\u0003;\nthen there is c>0such that for every m0satisfying\nkm0\u0000m0\nA\u0006kL1\u0014cp\u0019\n2p\u000b;\nthere existsR2SO(3), depending only on A+,A\u0000,\u000bandc, such that there is a unique global\nsmooth solution mof(LLG\u000b)with initial condition m0that satisfies\ninf\nx2R\nt>0(Rm)3(x;t)\u0015\u000e\u0003and [m]X\u0014L1+L2c: (1.7)\nMoreover,\nkm\u0000Rmc;\u000bkX\u0014L3km0\u0000m0\nA\u0006kL1:\n4In particular,\nk@xm\u0000@xRmc;\u000bkL1\u0014L3p\ntkm0\u0000m0\nA\u0006kL1;\nfor allt>0.\nNotice that Theorem 1.2 provides the existence of a unique solution in the set defined by the\nconditions (1.7), and hence it does not exclude the possibility of the existence of other solutions\nnot satisfying these conditions. In fact, as we will see in Theorem 1.3 below, one can prove the\nexistence of multiple solutions of the initial value problem (1.6), at least in the case when \u000bis\nclose to 1.\nWe point out that our results are valid only for \u000b>0. If we let\u000b!0, then the constants M1\nandM3in Theorem 1.1 go to 0 and M2blows up. Indeed, we use that the kernel associated with\nthe Ginzburg–Landau semigroup e(\u000b+i\f)t\u0001belongs toL1and its exponential decay. Therefore\nour techniques cannot be generalized (in a simple way) to cover the critical case \u000b= 0. In\nparticular, we cannot recover the stability results for the self-similar solutions in the case of\nSchrödinger maps proved by Banica and Vega in [5, 6, 7].\nAs mentioned before, in [30] and [31] some global well-posedness results for (LLG \u000b) with\n\u000b2(0;1]were proved for initial conditions with small gradient in LN(RN)andM2;2(RN),\nrespectively (see footnote in Subsection 3.3 for the definition of the space M2;2(RN)). In view of\nthe embeddings\nLN(RN)\u001aM2;2(RN)\u001aBMO\u00001(RN);\nforN\u00152, Theorem 1.1 can be seen as generalization of these results since it covers the case\nof less regular initial conditions. The arguments in [30, 31] are based on the method of moving\nframes that produces a covariant complex Ginzburg–Landau equation. In Subsection 3.3 we give\nmore details and discuss the corresponding equation in the one-dimensional case and provide\nsome properties related to the self-similar solutions.\nOur existence and uniqueness result given by Theorem 1.1 requires the initial condition to be\nsmall in the BMO semi-norm. Without this condition, the solution could develop a singularity\nin finite time. In fact, in dimensions N= 3;4, Ding and Wang [13] have proved that for some\nsmooth initial conditions with small (Dirichlet) energy, the associated solutions of (LLG \u000b) blow\nup in finite time.\nIn the context of the initial value problem (1.6), the smallness condition in the BMO semi-\nnorm is equivalent to the smallness of the angle between A+andA\u0000. As discussed in [17], in the\none dimensional case N= 1for fixed\u000b2(0;1]there is some numerical evidence that indicates\nthe existence of multiple (self-similar) solutions associated with the same initial condition of the\ntype in (1.6) (see Figures 2 and 3 in [17]). This suggests that the Cauchy problem for (LLG \u000b)\nwith initial condition (1.6) is ill-posed for general A+andA\u0000unitary vectors.\nThe following result states that in the case when \u000bis close to 1, one can actually prove the\nexistenceofmultiplesmoothsolutionsassociatedwiththesameinitialcondition m0\nA\u0006. Moreover,\ngiven any angle #2(0;\u0019)between two vectors A+andA\u00002S2, one can generate any number\nof distinct solutions by considering values of \u000bsufficiently close to 1.\nTheorem 1.3. Letk2N,A+,A\u00002S2and let#be the angle between A+andA\u0000. If\n#2(0;\u0019), then there exists \u000bk2(0;1)such that for every \u000b2[\u000bk;1]there are at least kdistinct\nsmooth self-similar solutions fmjgk\nj=1inX(R\u0002R+;S2)of(LLG\u000b)with initial condition m0\nA\u0006.\nThese solutions are characterized by a strictly increasing sequence of values fcjgk\nj=1, withck!1\nask!1, such that\nmj=Rjmcj;\u000b; (1.8)\n5whereRj2SO(3). In particular\np\ntk@xmj(\u0001;t)kL1=cj;for allt>0: (1.9)\nFurthermore, if \u000b= 1and#2[0;\u0019], then there is an infinite number of distinct smooth self-\nsimilar solutions fmjgj\u00151inX(R\u0002R+;S2)of(LLG\u000b)with initial condition m0\nA\u0006. These\nsolutions are also characterized by a sequence fcjg1\nj=1such that (1.8)and(1.9)are satisfied.\nThis sequence is explicitly given by\nc2`+1=`p\u0019\u0000#\n2p\u0019; c 2`=`p\u0019+#\n2p\u0019;for`\u00150: (1.10)\nIt is important to remark that in particular Theorem 1.3 asserts that when \u000b= 1, given\nA+;A\u00002S2such that A+=A\u0000, there exists an infinite number of distinct solutions fmjgj\u00151\ninX(R\u0002R+;S2)of (LLG\u000b) with initial condition m0\nA\u0006such that [m0\nA\u0006]BMO = 0. This\nparticular case shows that a condition on the size of X-norm of the solution as that given in\n(1.4) in Theorem 1.1 is necessary for the uniqueness of solution. We recall that for finite energy\nsolutions of (HFHM) there are several nonuniqueness results based on Coron’s technique [11] in\ndimensionN= 3. Alouges and Soyeur [2] successfully adapted this idea to prove the existence of\nmultiple solutions of the (LLG \u000b), with\u000b>0, for maps m: \n\u0000!S2, with \na bounded regular\ndomain of R3. In our case, since fcjgk\nj=1is strictly increasing, we have at least kgenuinely\ndifferent smoothsolutions. Notice also that the identity (1.9) implies that the X-norm of the\nsolution is large as j!1.\nStructure of the paper. This paper is organized as follows: in Section 2 we use the ste-\nreographic projection to reduce matters to the study the initial value problem for the resulting\ndissipative Schrödinger equation, prove its global well-posedness in well-adapted normed spaces,\nand use this result to establish Theorem 2.9 (a more quantitative version of Theorem 1.1). In\nSection 3 we focus on the self-similar solutions and we prove Theorems 1.2 and 1.3. In Section 3.3\nwe discuss some implications of the existence of explicit self-similar solutions for the Schrödinger\nequation obtained by means of the Hasimoto transformation. Finally, and for the convenience of\nthe reader, we have included some regularity results for the complex Ginzburg–Landau equation\nand some properties of the self-similar solutions mc;\u000bin the Appendix.\nNotations. We write R+= (0;1). Throughout this paper we will assume that \u000b2(0;1]and\nthe constants can depend on \u000b. In the proofs A.Bstands forA\u0014CBfor some constant\nC > 0depending only on \u000bandN. We denote in bold the vector-valued variables.\nSince we are interested in S2-valued functions, with a slightly abuse of notation, we denote\nbyL1(RN;S2)(resp.X(RN;S2)) the space of function in L1(RN;R3)(resp.X(RN;R3)) such\nthatjmj=1 a.e. on RN.\n2 The Cauchy problem\n2.1 The Cauchy problem for a dissipative quasilinear Schrödinger equation\nOur approach to study the Cauchy problem for (LLG \u000b) consists in analyzing the Cauchy prob-\nlem for the associated dissipative quasilinear Schrödinger equation through the stereographic\nprojection, and then “transferring” the results back to the original equation. To this end, we\nintroduce the stereographic projection from the South Pole P:S2nf(0;0;\u00001)g!Cdefined for\nby\nP(m) =m1+im2\n1 +m3:\n6Letmbe a smooth solution of (LLG \u000b) withm3>\u00001, then its stereographic projection u=\nP(m)satisfies the quasilinear dissipative Schrödinger equation (see e.g. [25] for details)\niut+ (\f\u0000i\u000b)\u0001u= 2(\f\u0000i\u000b)\u0016u(ru)2\n1 +juj2: (DNLS)\nAt least formally, the Duhamel formula gives the integral equation:\nu(x;t) =S\u000b(t)u0+\u0002t\n0S\u000b(t\u0000s)g(u)(s)ds; (IDNLS)\nwhereu0=u(\u0001;0)corresponds to the initial condition,\ng(u) =\u00002i(\f\u0000i\u000b)\u0016u(ru)2\n1 +juj2\nandS\u000b(t)is the dissipative Schrödinger semigroup (also called the complex Ginzburg–Landau\nsemigroup) given by S\u000b(t)\u001e=e(\u000b+i\f)t\u0001\u001e, i.e.\n(S\u000b(t)\u001e)(x) =\u0002\nRNG\u000b(x\u0000y;t)\u001e(y)dy;withG\u000b(x;t) =e\u0000jxj2\n4(\u000b+i\f)t\n(4\u0019(\u000b+i\f)t)N=2:(2.1)\nOne difficulty in studying (IDNLS) is to handle the term g(u). Taking into account that\nj\f\u0000i\u000bj= 1anda\n1 +a2\u00141\n2;for alla\u00150; (2.2)\nwe see that\njg(u)j\u0014jruj2; (2.3)\nso we need to control jruj2. Koch and Taratu dealt with a similar problem when studying the\nwell-posedness for the Navier–Stokes equation in [23]. Their approach was to introduce some new\nspaces related to BMO and BMO\u00001. Later, Koch and Lamm [22] and Wang [35] have adapted\nthese spaces to study some geometric flows. Following these ideas, we define the Banach spaces\nX(RN\u0002R+;F) =fv:RN\u0002R+!F:v;rv2L1\nloc(RN\u0002R+);kvkX<1gand\nY(RN\u0002R+;F) =fv:RN\u0002R+!F:v2L1\nloc(RN\u0002R+);kvkY<1g;\nwhere\nkvkX:= sup\nt>0kvkL1+ [v]X;with\n[v]X:= sup\nt>0p\ntkrvkL1+ sup\nx2RN\nr>0 \n1\nrN\u0002\nQr(x)jrv(y;t)j2dtdy!1\n2\n;and\nkvkY= sup\nt>0tkvkL1+ sup\nx2RN\nr>01\nrN\u0002\nQr(x)jv(y;t)jdtdy:\nHereQr(x)denotes the parabolic ball Qr(x) =Br(x)\u0002[0;r2]andFis either CorR3. The\nabsolute value stands for the complex absolute value if F=Cand for the euclidean norm if\nF=R3. We denote with the same symbol the absolute value in FandF3. Here and in the\nsequel we will omit the domain in the norms and semi-norms when they are taken in the whole\nspace, for example k\u0001kLpstands fork\u0001kLp(RN), forp2[1;1].\n7The spaces XandYare related to the spaces BMO (RN)and BMO\u00001(RN)and are well-\nadapted to study problems involving the heat semigroup S1(t) =et\u0001. In order to establish\nthe properties of the semigroup S\u000b(t)with\u000b2(0;1], we introduce the spaces BMO \u000b(RN)and\nBMO\u00001\n\u000b(RN)as the space of distributions f2S0(RN;F)such that the semi-norm and norm\ngiven respectively by\n[f]BMO\u000b:= sup\nx2RN\nr>0 \n1\nrN\u0002\nQr(x)jrS\u000b(t)fj2dtdy!1\n2\n;and\nkfkBMO\u00001\n\u000b:= sup\nx2RN\nr>0 \n1\nrN\u0002\nQr(x)jS\u000b(t)fj2dtdy!1\n2\n;\nare finite.\nOntheonehand, theCarlesonmeasurecharacterizationofBMOfunctions(see[34, Chapter4]\nand [27, Chapter 10]) yields that for fixed \u000b2(0;1],BMO\u000b(RN)coincides with the classical\nBMO (RN)space5, that is for all \u000b2(0;1]there exists a constant \u0003>0depending only on \u000b\nandNsuch that\n\u0003[f]BMO\u0014[f]BMO\u000b\u0014\u0003\u00001[f]BMO: (2.4)\nOn the other hand, Koch and Tataru proved in [23] that BMO\u00001(or equivalently BMO\u00001\n1,\nusing our notation) can be characterized as the space of derivatives of functions in BMO. A\nstraightforward generalization of their argument shows that the same result holds for BMO\u00001\n\u000b\n(see Theorem A.1). Hence, using the Carleson measure characterization theorem, we conclude\nthat BMO\u00001\n\u000bcoincides with the space BMO\u00001and that there exists a constant ~\u0003>0, depending\nonly on\u000bandN, such that\n~\u0003kfkBMO\u00001\u0014kfkBMO\u00001\n\u000b\u0014~\u0003\u00001kfkBMO\u00001: (2.5)\nThe above remarks allows us to use several of the estimates proved in [22, 23, 35] in the case\n\u000b= 1to study the integral equation (IDNLS) by using a fixed-point approach.\nOur first result concerns the global well-posedness of the Cauchy problem for (IDNLS) with\nsmall initial data in BMO (RN).\nTheorem 2.1. Let\u000b2(0;1]. There exist constants C;K\u00151such that for every L\u00150,\">0,\nand\u001a>0satisfying\n8C(\u001a+\")2\u0014\u001a; (2.6)\nifu02L1(RN;C), with\nku0kL1\u0014Land [u0]BMO\u0014\"; (2.7)\nthen there exists a unique solution u2X(RN\u0002R+;C)to(IDNLS) such that\n[u]X\u0014K(\u001a+\"): (2.8)\nMoreover,\n5\nBMO (RN) =ff:RN\u0002[0;1)!F:f2L1\nloc(RN);[f]BMO <1g;\nwith the semi-norm\n[f]BMO = sup\nx2RN\nr>0 \nBr(x)jf(y)\u0000fx;rjdy;\nwhere fx;ris the average\nfx;r= \nBr(x)f(y)dy=1\njBr(x)j\u0002\nBr(x)f(y)dy:\n8(i)supt>0kukL1\u0014K(\u001a+L).\n(ii)u2C1(RN\u0002R+)and(DNLS) holds pointwise.\n(iii) lim\nt!0+u(\u0001;t) =u0as tempered distributions. Moreover, for every '2S(RN), we have\nk(u(\u0001;t)\u0000u0)'kL1!0;ast!0+: (2.9)\n(iv) (Dependence on the initial data) Assume that uandvare respectively solutions to (IDNLS)\nfulfilling (2.8)with initial conditions u0andv0satisfying (2.7). Then\nku\u0000vkX\u00146Kku0\u0000v0kL1: (2.10)\nAlthough condition (2.6) appears naturally from the fixed-point used in the proof, it may be\nno so clear at first glance. To better understand it, let us define for C > 0\nS(C) =f(\u001a;\")2R+\u0002R+:C(\u001a+\")2\u0014\u001ag: (2.11)\nWe see that if (\u001a;\")2S(C), then\u001a;\"> 0and\n\"\u0014p\u001ap\nC\u0000\u001a: (2.12)\nTherefore the set S(C)is non-empty and bounded. The shape of this set is depicted in Figure 1.\nIn particular, we infer from (2.12) that if (\u001a;\")2S(C), then\n1\n4C\n1\n4C1\nCρε\nFigure 1: The shape of the set S(C).\n\u001a\u00141\nCand\"\u00141\n4C: (2.13)\nIn addition, if ~C\u0015C, then\nS(~C)\u0012S(C): (2.14)\nMoreover, taking for instance \u001a= 1=(32C), Theorem 2.1 asserts that for fixed \u000b2(0;1], we can\ntake for instance \"= 1=(32C)(that depends on \u000bandN, but not on the L1-norm of the initial\ndata) such that for any given initial condition u02L1(RN)with [u0]BMO\u0014\", there exists a\nglobal (smooth) solution u2X(RN\u0002R+;C)of (DNLS). Notice that u0is allowed to have a\nlargeL1-norm as long as [u0]BMOis sufficiently small; this is a weaker requirement that asking\nfor theL1-norm ofu0to be sufficiently small, since\n[f]BMO\u00142kfkL1;for allf2L1(RN): (2.15)\n9Remark 2.2. The smallness condition in (2.8) is necessary for the uniqueness of the solution.\nAs we will see in Subsection 3.2.2, at least in dimension one, it is possible to construct multiple\nsolutions of (IDNLS) in X(RN\u0002R+;C), if\u000bis close enough to 1.\nThe aim of this section is to prove Theorem 2.1 using a fixed-point technique. To this pursuit\nwe write (IDNLS) as\nu(t) =Tu0(u)(t); (2.16)\nwhere\nTu0(u)(t) =S\u000b(t)u0+T(g(u))(t)andT(f)(t) =\u0002t\n0S\u000b(t\u0000s)f(s)ds: (2.17)\nIn the next lemmas we study the semigroup S\u000band the operator Tto establish that the appli-\ncationTu0is a contraction on the ball\nB\u001a(u0) =fu2X(RN\u0002R+;C) :ku\u0000S\u000b(t)u0kX\u0014\u001ag;\nfor some\u001a>0depending on the size of the initial data.\nLemma 2.3. There exists C0>0such that for all f2BMO\u00001\n\u000b(RN),\nsup\nt>0p\ntkS\u000b(t)fkL1(RN)\u0014C0kfkBMO\u00001\n\u000b: (2.18)\nProof.The proof in the case \u000b= 1is done in [27, Lemma 16.1]. For \u000b2(0;1), decomposing\nS\u000b(t) =S\u000b(t\u0000s)S\u000b(s)and using the decay properties of the kernel G\u000bassociated with the\noperatorsS\u000b(t)(see (2.1)), we can check that the same proof still applies.\nLemma 2.4. There exists C1\u00151such that for all f2Y(RN\u0002R+;C),\nkT(f)kX\u0014C1kfkY: (2.19)\nProof.Estimate (2.19) can be proved using the arguments given in [23] or [35]. For the conve-\nnience of the reader, we sketch the proof following the lines in [35, Lemma 3.1]. By scaling and\ntranslation, it suffices to show that\njT(f)(0;1)j+jrT(f)(0;1)j+ \u0002\nQ1(0)jrT(f)j2!1=2\n.kfkY: (2.20)\nLetBr=Br(0). SettingW=T(f), we have\nW(0;1) =\u00021\n0\u0002\nRNG\u000b(\u0000y;1\u0000s)f(y;s)dyds\n= \u00021\n1=2\u0002\nRN+\u00021=2\n0\u0002\nB2+\u00021=2\n0\u0002\nRNnB2!\nG\u000b(\u0000y;1\u0000s)f(y;s)dyds\n:=I1+I2+I3:\nSincejG\u000b(y;1\u0000s)j=e\u0000\u000bjyj2\n4(1\u0000s)\n(4\u0019(1\u0000s))N=2;we obtain\njI1j\u0014\u00021\n1=2\u0002\nRNjG\u000b(\u0000y;1\u0000s)jjf(y;s)jdyds\n\u0014sup\n1\n2\u0014s\u00141kf(s)kL1 \u00021\n1\n2\u0002\nRnjG\u000b(\u0000y;1\u0000s)jdyds!\n.kfkY;\n10jI2j\u0014\u00021=2\n0\u0002\nB2jG\u000b(\u0000y;1\u0000s)jjf(y;s)jdyds\n.sup\n0\u0014s\u00141\n2kG\u000b(\u0001;1\u0000s)kL1(RN)\u0002\nB2\u0002[0;1\n2]jf(y; s)jdyds.kfkY\nand\njI3j\u0014\u00021=2\n0\u0002\nRNnB2jG\u000b(\u0000y;1\u0000s)jjf(y;s)jdyds\n\u0014C\u00021\n2\n0\u0002\nRNnB2e\u0000\u000bjyj2\n4jf(y; s)jdyds\n\u0014C 1X\nk=2kn\u00001e\u0000\u000bk2\n4! \nsup\ny2RN\u0002\nQ1(y)jf(y; s)jdyds!\n.kfkY:\nThe quantityjrT(f)(0;1)jcan be bounded in a similar way. The last term in the l.h.s. of (2.20)\ncan be controlled using an energy estimate. Indeed, Wsatisfies the equation\ni@tW+ (\f\u0000i\u000b)\u0001W=if (2.21)\nwith initial condition W(\u0001;0) = 0. Let\u00112C1\n0(B2)be a real-valued cut-off function such that\n0\u0014\u0011\u00141onRNand\u0011= 1onB1. By multiplying (2.21) by \u0000i\u00112W, integrating and taking\nreal part, we get\n1\n2@t\u0002\nRN\u00112jWj2+\u000b\u0002\nRN\u00112jrWj2+ 2 Re\u0012\n(\u000b+i\f)\u0002\nRN\u0011r\u0011WrW\u0013\n=\u0002\nRN\u00112Re(fW):\nUsing thatj\u000b+i\fj= 1and integrating in time between 0and1, it follows that\n1\n2\u0002\nRN\u00112jW(x;1)j2+\u000b\u0002\nRN\u0002[0;1]\u00112jrWj2\u0014\u0002\nRN\u0002[0;1](2\u0011jr\u0011jjWjjrWj+\u00112jfjjWj):\nFrom the inequality ab\u0014\"a2+b2=(4\");witha=\u0011jrWj,b= 2jr\u0011jjWjand\"=\u000b=2, we deduce\nthat\u000b\n2\u0002\nRN\u0002[0;1]\u00112jrWj2\u0014\u0002\nRN\u0002[0;1]\u00002\n\u000bjr\u0011j2jWj2+\u00112jfjjWj\u0001\n:\nBy the definition of \u0011, this implies that\nkrWk2\nL2(B1\u0002[0;1]).kWk2\nL1(B2\u0002[0;1])+kWkL1(B2\u0002[0;1])kfkL1(B2\u0002[0;1]):(2.22)\nFrom the first part of the proof, we have\nkWkL1(B2\u0002[0;1])\u0014CkfkY:\nUsing also that\nkfkL1(B2\u0002[0;1]).kfkY;\nwe conclude from (2.22) that\nkrWkL2(B1\u0002[0;1]).kfkY;\nwhich finishes the proof.\n11Lemma 2.5. Let\u000b2(0;1]and\u001a;\";L> 0. There exists C2\u00151, depending on \u000bandN, such\nthat for all u02L1(RN)\nkS\u000b(t)u0kX\u0014C2(ku0kL1+ [u0]BMO ): (2.23)\nIf in additionku0kL1\u0014Land[u0]BMO\u0014\", then for all u2B\u001a(u0)we have\nsup\nt>0kukL1\u0014C2(\u001a+L)and [u]X\u0014C2(\u001a+\"): (2.24)\nProof.We first controlkS\u000b(t)u0kX. On the one hand, using the definition of G\u000band the relation\n\u000b2+\f2= 1, we obtain\nkS\u000b(t)u0kL1=kG\u000b\u0003u0kL1\u0014kG\u000bkL1ku0kL1=\u000b\u0000N\n2ku0kL1;8t>0:\nThus\nsup\nt>0kS\u000b(t)u0kL1\u0014\u000b\u0000N\n2ku0kL1: (2.25)\nOn the other hand, using Lemma 2.3, Theorem A.1 and (2.4),\n[S\u000b(t)u0]X= sup\nt>0p\ntkrS\u000b(t)u0kL1+ sup\nx2RN\nr>0 \n1\nrN\u0002\nQr(x)jrS\u000b(t)u0j2dtdy!1\n2\n.kru0kBMO\u00001\n\u000b+ [u0]BMO\u000b\n.[u0]BMO\u000b\n.[u0]BMO:(2.26)\nThe estimate in (2.23) follows from (2.25) and (2.26), and we w.l.o.g. can choose C2\u00151.\nFinally, using (2.25), given u0such thatku0kL1\u0014Land[u0]BMO\u0014\", for allu2B\u001a(u0)we\nhave\nkukL1\u0014ku\u0000S\u000b(t)u0kL1+kS\u000b(t)u0kL1\u0014ku\u0000S\u000b(t)u0kX+kS\u000b(t)u0kL1\u0014C2(\u001a+L);\nand, using (2.26),\n[u]X\u0014[u\u0000S\u000b(t)u0]X+ [S\u000b(t)u0]X\u0014ku\u0000S\u000b(t)u0kX+ [S\u000b(t)u0]X\u0014C2(\u001a+\");\nwhich finishes the proof of (2.24).\nNow we proceed to bound the nonlinear term\ng(u) =\u00002i(\f\u0000i\u000b)\u0016u(ru)2\n1 +juj2:\nLemma 2.6. For allu2X(RN\u0002R+;C), we have\nkg(u)kY\u0014[u]2\nX:\nProof.Letu2X(RN\u0002R+;C). Using (2.3) and the definitions of the norms in YandX, it\nfollows that\nkg(u)kY\u0014\u0012\nsup\nt>0p\ntkrukL1\u00132\n+ sup\nx2RN\nr>01\nrN\u0002\nQr(x)jruj2dtdy\u0014[u]2\nX:\n12Now we have all the estimates to prove that Tu0is a contraction on B\u001a(u0).\nProposition 2.7. Let\u000b2(0;1]and\u001a;\"> 0. Given any u02L1(RN)with [u0]BMO\u0014\", the\noperatorTu0given in (2.17)defines a contraction on B\u001a(u0), whenever \u001aand\"satisfy\n8C1C2\n2(\u001a+\")2\u0014\u001a: (2.27)\nMoreover, for all u;v2X(RN\u0002R+;C),\nkT(g(u))\u0000T(g(v))kX\u0014C1(2[u]2\nX+ [u]X+ [v]X)ku\u0000vkX: (2.28)\nHere,C1\u00151andC2\u00151are the constants in Lemmas 2.4 and 2.5, respectively.\nRemark 2.8. Using the notation introduced in (2.11), the hypothesis (2.27) means that (\u001a;\")2\nS(8C1C2\n2). Therefore, by (2.13),\n\u001a\u00141\n8C1C2\n2;and\"\u00141\n32C1C2\n2; (2.29)\nso\u001aand\"are actually small. Since C1;C2\u00151, we have\nC2(\u001a+\")\u00145\n32: (2.30)\nProof.Letu02L1(RN)withku0kL1\u0014Land[u0]BMO\u0014\", andu2B\u001a(u0). Using Lemma 2.4,\nLemma 2.5 and Lemma 2.6, we have\nkTu0(u)\u0000S\u000b(t)u0kX=kT(g(u))kX\u0014C1kg(u)kY\u0014C1[u]2\nX\u0014C1C2\n2(\u001a+\")2:\nThereforeTu0mapsB\u001a(u0)into itself provided that\nC1C2\n2(\u001a+\")2\u0014\u001a: (2.31)\nNotice that by (2.14), the condition (2.27) implies that (2.31) is satisfied.\nTo prove (2.28), we use the decomposition\ng(u)\u0000g(v) =\u00002i(\f\u0000i\u000b)\u0014\u0012\u0016u\n1 +juj2\u0000\u0016v\n1 +jvj2\u0013\n(ru)2+\u0016v\n1 +jvj2((ru)2\u0000(rv)2))\u0015\n:\nSince \f\f\f\f\u0016u\n1 +juj2\u0000\u0016v\n1 +jvj2\f\f\f\f\u0014ju\u0000vj1 +jujjvj\n(1 +juj2)(1 +jvj2)\u0014ju\u0000vj;\nand using (2.2), we obtain\njg(u)\u0000g(v)j\u00142ju\u0000vjjruj2+jru\u0000rvj(jruj+jrvj):\nTherefore\nkg(u)\u0000g(v)kY\u00142kju\u0000vjjruj2kY+kjru\u0000rvj(jruj+jrvj)kY:=I1+I2:(2.32)\nForI1, it is immediate that\nI1\u00142 sup\nt>0ku\u0000vkL12\n64\u0010\nsup\nt>0p\ntkrukL1\u00112\n+ sup\nx2RN\nr>01\nrN\u0002\nQr(x)jruj2dtdy3\n75\u00142ku\u0000vkX[u]2\nX:\n(2.33)\n13Similarly, using the Cauchy–Schwarz inequality,\nI2\u0014\u0012\nsup\nt>0p\ntkru\u0000rvkL1\u0013\u0012\nsup\nt>0p\nt(krukL1+krvkL1)\u0013\n+ sup\nx2RN\nr>01\nrN\u0000\nkru\u0000rvkL2(Qr(x))\u0001\u0000\nkrukL2(Qr(x))+krvkL2(Qr(x))\u0001\n\u0014ku\u0000vkX([u]X+ [v]X):(2.34)\nUsing Lemma 2.4, (2.32), (2.33) and (2.34), we conclude that\nkT(g(u))\u0000T(g(v))kX\u0014C1(2[u]2\nX+ [u]X+ [v]X)ku\u0000vkX: (2.35)\nLetu;v2B\u001a(u0), by Lemma 2.5 and (2.30)\n[u]X\u0014C2(\u001a+\")\u00145\n32; (2.36)\nso that\n2[u]2\nX+ [u]X+ [v]X\u001437\n16C2(\u001a+\")<3C2(\u001a+\"): (2.37)\nThen (2.35) implies that\nkTu0(u)\u0000Tu0(v)kX\u00143C1C2(\u001a+\")ku\u0000vkX: (2.38)\nFrom (2.29), we conclude that\n3C1C2(\u001a+\")\u001415\n32\u00141\n2; (2.39)\nand then (2.38) yields that the operator Tu0defined in (2.17) is a contraction on B\u001a(u0). This\nconcludes the proof of the proposition.\nProof of Theorem 2.1. Let us setC=C1C2\n2andK=C2, whereC1andC2are the constants\nin Lemma 2.4 and Lemma 2.5 respectively. Since \u001asatisfies (2.6), Proposition 2.7 implies that\nthere exists a solution uof equation (2.16) in the ball B\u001a(u0), and in particular from Lemma 2.5\nsup\nt>0kukL1\u0014K(\u001a+L)and [u]X\u0014K(\u001a+\"):\nTo prove the uniqueness part of the theorem, let us assume that uandvare solutions of (IDNLS)\ninX(RN\u0002R+;C)such that\n[u]X;[v]X\u0014K(\u001a+\"); (2.40)\nwith the same initial condition u0. By the definitions of CandK, (2.6) and (2.40), the estimates\nin (2.29) and (2.30) hold. It follows that (2.36), (2.37) and (2.39) are satisfied. Then, using\n(2.28),\nku\u0000vkX=kT(g(u))\u0000T(g(v))kX\u0014C1(2[u]2\nX+ [u]X+ [v]X)ku\u0000vkX\n\u00141\n2ku\u0000vkX:\nFrom which it follows that u=v.\nTo prove the dependence of the solution with respect to the initial data (part (iv)), consider\nuandvsolutions of (IDNLS) satisfying (2.40) with initial conditions u0andv0. Then, by\ndefinition,u=Tu0(u),v=Tv0(v)and\nku\u0000vkX=kTu0(u)\u0000Tv0(v)kX\u0014kS\u000b(u0\u0000v0)kX+kT(g(u))\u0000T(g(v))kX:\n14Using (2.15), (2.23) and (2.28) and arguing as above, we have\nku\u0000vkX\u0014C2(ku0\u0000v0kL1+ [u0\u0000v0]BMO ) +C1(2[u]2\nX+ [u]X+ [v]X)ku\u0000vkX\n\u00143C2ku0\u0000v0kL1+1\n2ku\u0000vkX:\nThis yields (2.10), since K=C2.\nThe assertions in (ii)and(iii)follow from Theorem A.3.\n2.2 The Cauchy problem for the LLG equation\nBy using the inverse of the stereographic projection P\u00001:C!S2nf0;0;\u00001g, that is explicitly\ngiven by m= (m1;m2;m3) =P\u00001(u), with\nm1=2 Reu\n1 +juj2; m 2=2 Imu\n1 +juj2; m 3=1\u0000juj2\n1 +juj2; (2.41)\nwe will be able to establish the following global well-posedness result for (LLG \u000b).\nTheorem 2.9. Let\u000b2(0;1]. There exist constants C\u00151andK\u00154, such that for any\n\u000e2(0;2],\"0>0and\u001a>0such that\n8K4C\u000e\u00004(\u001a+ 8\u000e\u00002\"0)2\u0014\u001a; (2.42)\nifm0= (m0\n1;m0\n2;m0\n3)2L1(RN;S2)satisfies\ninf\nRNm0\n3\u0015\u00001 +\u000eand [m0]BMO\u0014\"0; (2.43)\nthen there exists a unique solution m= (m1;m2;m3)2X(RN\u0002R+;S2)of(LLG\u000b)such that\ninf\nx2RN\nt>0m3(x;t)\u0015\u00001 +2\n1 +K2(\u001a+\u000e\u00001)2and [m]X\u00144K(\u001a+ 8\u000e\u00002\"0):(2.44)\nMoreover, we have the following properties.\ni)m2C1(RN\u0002R+;S2).\nii)jm(\u0001;t)\u0000m0j\u0000! 0inS0(RN)ast\u0000!0+.\niii) Assume that mandnare respectively smooth solutions to (IDNLS) satisfying (2.44)with\ninitial conditions m0andn0satisfying (2.43). Then\nkm\u0000nkX\u0014120K\u000e\u00002km0\u0000n0kL1: (2.45)\nRemark 2.10. The restriction (2.42) on the parameters is similar to (2.27), but we need to\ninclude\u000e. To better understand the role of \u000e, we can proceed as before. Indeed, setting for\na;\u000e> 0,\nS\u000e(a) =f(\u001a;\"0)2R+\u0002R+:a\u000e\u00004(\u001a+ 8\u000e\u00002\"0)2\u0014\u001ag;\nwe see that its shape is similar to the one in Figure 1. It is simple to verify that for any\n(\u001a;\"0)2S\u000e(a), we have the bounds\n\u001a\u0014\u000e4\naand\"0\u0014\u000e6\n32a; (2.46)\nand the maximum value \"\u0003\n0=\u000e6\n32ais attained at \u001a\u0003=\u000e4\n4a. Also, the sets are well ordered, i.e. if\n~a\u0015a>0, thenS\u000e(~a)\u0012S\u000e(a).\n15We emphasize that the first condition in (2.43) is rather technical. Indeed, we need the\nessential range of m0to be far from the South Pole in order to use the stereographic projection.\nIn the case\u000b= 1, Wang [35] proved the global well-posedness using only the second restriction in\n(2.43). It is an open problem to determinate if this condition is necessary in the case \u000b2(0;1).\nThe choice of the South Pole is of course arbitrary. By using the invariance of (LLG \u000b) under\nrotations, we have the existence of solutions provided that the essential range of the initial\ncondition m0is far from an arbitrary point Q2S2. Precisely,\nCorollary 2.11. Let\u000b2(0;1],Q2S2,\u000e2(0;2], and\"0;\u001a> 0such that (2.42)holds. Given\nm0= (m0\n1;m0\n2;m0\n3)2L1(RN;S2)satisfying\ninf\nRNjm0\u0000Qj2\u00152\u000eand [m0]BMO\u0014\"0;\nthere exists a unique smooth solution m2X(RN\u0002R+;S2)of(LLG\u000b)with initial condition m0\nsuch that\ninf\nx2RN\nt>0jm(x;t)\u0000Qj2\u00154\n1 +K2(\u001a+\u000e\u00001)2and [m]X\u00144K(\u001a+ 8\u000e\u00002\"0):(2.47)\nFor the sake of clarity, before proving Theorem 2.9, we provide a precise meaning of what we\nrefer to as a weak and smooth global solution of the (LLG \u000b) equation. The definition below is\nmotivated by the following vector identities for a smooth function mwithjmj= 1:\nm\u0002\u0001m= div( m\u0002rm);\n\u0000m\u0002(m\u0002\u0001m) = \u0001m+jrmj2m:\nDefinition 2.12. LetT2(0;1]andm02L1(RN;S2). We say that\nm2L1\nloc((0;T);H1\nloc(RN;S2))\nis a weak solution of (LLG\u000b)in(0;T)with initial condition m0if\n\u0000hm;@t'i=\fhm\u0002rm;r'i\u0000\u000bhrm;r'i+\u000bhjrmj2m;'i;\nand\nk(m(t)\u0000m0)'kL1!0;ast!0+;for all'2C1\n0(RN\u0002(0;T)): (2.48)\nIfT=1, and in addition m2C1(RN\u0002R+), we say that mis a smooth global solution of\n(LLG\u000b)inRN\u0002R+with initial condition m0. Hereh\u0001;\u0001istands for\nhf1;f2i=\u00021\n0\u0002\nRNf1\u0001f2dxdt:\nWith this definition, we see the following: Assume that mis a smooth global solution of\n(LLG\u000b) with initial condition m0and consider its stereographic projection P(m). IfP(m)and\nP(m0)are well-defined, then P(m)2C1(RN\u0002R+;C)satisfies (DNLS) pointwise, and\nlim\nt!0+P(m) =P(m0)inS0(RN):\nTherefore,ifinaddition P(m)2X(RN\u0002R;C),thenP(m)isasmoothglobalsolutionof (DNLS)\nwith initial condition P(m0). Reciprocally, suppose that u2X(RN\u0002R+;C)\\C1(RN\u0002R+)\nis a solution of (IDNLS) with initial condition u02L1(RN)such that (2.9) holds. If P\u00001(u)\nandP\u00001(u0)are in appropriate spaces, then P\u00001(u)is a global smooth solution of (LLG \u000b) with\ninitial conditionP\u00001(u0). The above (formal) argument allows us to obtain Theorem 2.9 from\nTheorem 2.1 once we have established good estimates for the mappings PandP\u00001. In this\ncontext, we have the following\n16Lemma 2.13. Letu;v2C1(RN;C),m= (m1;m2;m3);n= (n1;n2;n3)2C1(RN;S2).\na) Assume that inf\nRNm3\u0015\u00001+\u000eandinf\nRNn3\u0015\u00001+\u000efor some constant \u000e2(0;2]. Ifu=P(m)\nandv=P(n), then\nju(x)\u0000v(x)j\u00144\n\u000e2jm(x)\u0000n(x)j; (2.49)\n[u]BMO\u00148\n\u000e2[m]BMO; (2.50)\njru(x)j\u00144\n\u000e2jrm(x)j; (2.51)\nfor allx2RN.\nb) Assume thatkukL1\u0014M,kvkL1\u0014M, for some constant M\u00150. Ifm=P\u00001(u)and\nn=P\u00001(v), then\ninf\nRNm3\u0015\u00001 +2\n1 +M2; (2.52)\njm(x)\u0000n(x)j\u00143ju(x)\u0000v(x)j; (2.53)\njrm(x)j\u00144jru(x)j; (2.54)\njrm(x)\u0000rn(x)j\u00144jru(x)\u0000rv(x)j+ 12ju(x)\u0000v(x)j(jru(x)j+jrv(x)j):(2.55)\nProof.In the proof we will use the notation \u0014m:=m1+im2. To establish (2.49), we write\nu(x)\u0000v(x) =\u0014m(x)\u0000\u0014n(x)\n1 +m3(x)+\u0014n(x)(n3(x)\u0000m3(x))\n(1 +m3(x))(1 +n3(x)):\nHence, sincej\u0014nj\u00141,m3(x) + 1\u0015\u000eandn3(x) + 1\u0015\u000e,8x2RN,\nju(x)\u0000v(x)j\u0014j\u0014m(x)\u0000\u0014n(x)j\n\u000e+jn3(x)\u0000m3(x)j\n\u000e2:\nUsing that\nj\u0014m\u0000\u0014nj\u0014jm\u0000nj (2.56)\nand that\nmax\u001a1\na;1\na2\u001b\n\u00142\na2;for alla2(0;2];\nwe obtain (2.49). The same argument also shows that\nju(y)\u0000u(z)j\u00144\n\u000e2jm(y)\u0000m(z)j;for ally;z2RN: (2.57)\nTo verify (2.50), we recall the following inequalities in BMO (see [10]):\n[f]BMO\u0014sup\nx2RN \nBr(x) \nBr(x)jf(y)\u0000f(z)jdydz\u00142[f]BMO: (2.58)\nEstimate (2.50) is an immediate consequence of this inequality and (2.57). To prove (2.51) it is\nenough to remark that\njruj\u00142\n\u000e2(jrm1j+jrm2j+jrm3j)\u00144\n\u000e2jrmj:\n17We turn into (b). Using the explicit formula for P\u00001in (2.41), we can write\nm3=\u00001 +2\n1 +juj2:\nSincekukL1\u0014M, we obtain (2.52).\nTo show (2.53), we compute\n\u0014m\u0000\u0014n=2u\n1 +juj2\u00002v\n1 +jvj2=2(u\u0000v) + 2uv(\u0016v\u0000\u0016u)\n(1 +juj2)(1 +jvj2); (2.59)\nm3\u0000n3=1\u0000juj2\n1 +juj2\u00001\u0000jvj2\n1 +jvj2=2(jvj2\u0000juj2)\n(1 +juj2)(1 +jvj2): (2.60)\nUsing the inequalities\na\n1 +a2\u00141\n2;1 +ab\n(1 +a2)(1 +b2)\u00141;anda+b\n(1 +a2)(1 +b2)\u00141;for alla;b\u00150;(2.61)\nfrom (2.59) and (2.60) we deduce that\nj\u0014m\u0000\u0014nj\u00142ju\u0000vjandjm3\u0000n3j\u00142ju\u0000vj: (2.62)\nHence\njm\u0000nj=p\nj\u0014m\u0000\u0014nj2+jm3\u0000n3j2\u0014p\n8ju\u0000vj\u00143ju\u0000vj:\nTo estimate the gradient, we compute\nr\u0014m=2ru\n1 +juj2\u00004uRe(\u0016uru)\n(1 +juj2)2; (2.63)\nfrom which it follows that\njr\u0014mj\u0014jruj\u00122\n1 +juj2+4juj2\n(1 +juj2)2\u0013\n\u00143jruj;\nsince4a\n(1+a)2\u00141;for alla\u00150. Form3, we have\nrm3=\u00002 Re(\u0016uru)\n1 +juj2\u00002 Re(\u0016uru)(1\u0000juj2)\n(1 +juj2)2=\u00004 Re(\u0016uru)\n(1 +juj2)2;\nand thereforejrm3j\u00142jruj, since\na\n(1 +a2)2\u00141\n2;for alla\u00150: (2.64)\nHence\njrmj=p\njrm1j2+jrm2j2+jrm3j2\u0014p\n13jruj\u00144jruj;\nwhich gives (2.54).\nIn order to prove (2.55), we start differentiating (2.59)\nr\u0014m\u0000r\u0014n=2r(u\u0000v) +r(uv)(\u0016v\u0000\u0016u) +uvr(\u0016v\u0000\u0016u)\n(1 +juj2)(1 +jvj2)\n\u00004((u\u0000v) +uv(\u0016v\u0000\u0016u))(Re(\u0016uru)(1 +jvj2) + Re(\u0016vrv)(1 +juj2))\n(1 +juj2)2(1 +jvj2)2;\n18Hence, setting R= maxfjru(x)j;jrv(x)jg,\njr\u0014m\u0000r\u0014nj\u00142jru\u0000rvj\u00121 +jujjvj\n(1 +juj2)(1 +jvj2)\u0013\n+ 2Rju\u0000vj\u0012juj+jvj\n(1 +juj2)(1 +jvj2)\u0013\n+ 4Rju\u0000vj\u0012juj(1 +jujjvj)\n(1 +juj2)2(1 +jvj2)+jvj(1 +jujjvj)\n(1 +juj2)(1 +jvj2)2\u0013\n:\nUsing again (2.61), we get\njuj(1 +jujjvj)\n(1 +juj2)2(1 +jvj2)\u0014juj\n(1 +juj2)\u00141\n2:\nBy symmetry, the same estimate holds interchanging ubyv. Therefore, invoking again (2.61),\nwe obtain\njr\u0014m\u0000r\u0014nj\u00142jru\u0000rvj+ 6Rju\u0000vj: (2.65)\nSimilarly, writing juj2\u0000jvj2= (u\u0000v)\u0016u+ (\u0016u\u0000\u0016v)v, from (2.60) we have\njrm3\u0000rn3j\u00142jru\u0000rvj+ 6Rju\u0000vj: (2.66)\nTherefore, sincep\na2+b2\u0014a+b;8a;b\u00150;\ninequalities (2.65) and (2.66) yield (2.55).\nNow we have all the elements to establish Theorem 2.9.\nProof of Theorem 2.9. We continue to use the constants CandKdefined in Theorem 2.1. We\nrecall that they are given by C=C1C2\n2andK=C2, whereC1\u00151andC2\u00151are the constants\nin Lemmas 2.4 and 2.5, respectively. In addition, w.l.o.g. we assume that\nK=C2\u00154; (2.67)\nin order to simplify our computations.\nFirst we notice that by Remark 2.10, any \u001aand\"0fulfilling the condition (2.42), also satisfy\n8C(\u001a+ 8\u000e\u00002\"0)2\u0014\u001a; (2.68)\nsince\u000e4=K4\u00141(notice that K\u00154and\u000e2(0;2]).\nLetm0as in the statement of the theorem and set u0=P(m0). Using (2.50) in Lemma 2.13,\nwe have\nku0kL1\u0014\r\r\r1\n1 +m0\n3\r\r\r\nL1\u00141\n\u000eand [u0]BMO\u00148\"0\n\u000e2:\nTherefore, bearing in mind (2.68), we can apply Theorem 2.1 with\nL:=1\n\u000eand\":= 8\u000e\u00002\"0;\nto obtain a smooth solution u2X(RN\u0002R+;C)to (IDNLS) with initial condition u0. In\nparticularusatisfies\nsup\nt>0kukL1\u0014K(\u001a+\u000e\u00001)and [u]X\u0014K(\u001a+ 8\u000e\u00002\"0): (2.69)\n19Defining m=P\u00001(u), we infer that mis a smooth solution to (LLG \u000b) and, using the fact that\nk(u(\u0001;t)\u0000u0)'kL1!0(see (2.9)) and (2.53),\njm(\u0001;t)\u0000m0j\u0000! 0inS0(RN);ast!0+:\nNotice also that applying Lemma 2.13 we obtain\ninf\nx2RN\nt>0m3(x;t)\u0015\u00001 +2\n1 +K2(\u001a+\u000e\u00001)2and [m]X\u00144[u]X\u00144K(\u001a+ 8\u000e\u00002\"0);\nwhich yields (2.44).\nLet us now prove the uniqueness. Let nbe a another smooth solution of (LLG \u000b) with initial\nconditionu0satisfying\ninf\nx2RN\nt>0n3(x;t)\u0015\u00001 +2\n1 +K2(\u001a+\u000e\u00001)2and [n]X\u00144K(\u001a+ 8\u000e\u00002\"0); (2.70)\nand letv=P(n)be its stereographic projection. Then by (2.51),\n[v]X\u0014\u0010\n1 +K2(\u001a+\u000e\u00001)2\u00112\n[n]X: (2.71)\nWe continue to control the upper bounds for [v]Xand[u]Xin terms of \u000eand the constants\nC1\u00151andC2\u00154. Notice that since \u001aand\"0satisfy (2.42), from (2.46) with a= 8K4C, it\nfollows that\n\u001a\u0014\u000e4\n8K4Cand\"0\u0014\u000e6\n28K4C;\nor equivalently (recall that K=C2andC=C1C2\n2)\n\u001a\u0014\u000e4\n8C1C6\n2and8\"0\n\u000e2\u0014\u000e4\n32C1C6\n2: (2.72)\nHence\nK(\u001a+ 8\u000e\u00002\"0)\u00145\u000e4\n32C1C5\n2: (2.73)\nAlso, using (2.72), we have\n1 +K2(\u001a+\u000e\u00001)2=1 +C2\n2\n\u000e2(\u001a\u000e+ 1)2=C2\n2\n\u000e2\u0012\u000e2\nC2\n2+ (\u001a\u000e+ 1)2\u0013\n\u0014C2\n2\n\u000e2 \n\u000e2\nC2\n2+\u0012\u000e5\n8C1C6\n2+ 1\u00132!\n\u00142C2\n2\n\u000e2;(2.74)\nsinceC1\u00151,C2\u00154and\u000e\u00142.\nFrom the bounds in (2.73) and (2.74), combined with (2.69), (2.70) and (2.71), we obtain\n[u]X\u0014K(\u001a+ 8\u000e\u00002\"0)\u00145\u000e4\n32C1C5\n2\u00145\n211C1\nand\n[v]X\u0014(1+K2(\u001a+\u000e\u00001)2)2[n]X\u0014(1+K2(\u001a+\u000e\u00001)2)24K(\u001a+8\u000e\u00002\"0)\u0014\u0012\n2C2\n2\n\u000e2\u0013220\u000e4\n32C1C5\n2\u00145\n8C1;\n20since\u000e\u00142andC2\u00154. Finally, since uandvare solutions to (IDNLS) with initial condition\nu0, (2.28) and the above inequalities for [u]Xand[v]Xyield\nku\u0000vkX\u0014C1(2[u]2\nX+ [u]X+ [v]X)ku\u0000vkX\n\u0014C1 \n2\u00125\n211C1\u00132\n+5\n211C1+5\n8C1!\nku\u0000vkX;\nwhich implies that u=v, bearing in mind that the constant on the r.h.s. of the above inequality\nis strictly less that one. This completes the proof of the uniqueness.\nIt remains to establish (2.45). Let mandntwo smooth solutions of (LLG \u000b) satisfying (2.44).\nAs a consequence of the uniqueness, we see that mandnare the inverse stereographic projection\nof some functions uandvthat are solutions of (IDNLS) with initial condition u0=P(m0)and\nv0=P(n0), respectively. In particular, uandvsatisfy the estimates in (2.69). Using also (2.53)\nand (2.55), we deduce that\nkm\u0000nkX\u00143 sup\nt>0ku\u0000vkL1+ 4[u\u0000v]X+ 12 sup\nt>0ku\u0000vkL1([u]X+ [v]X])\n\u00144ku\u0000vkX+ 24C2(\u001a+ 8\u000e\u00002\"0)ku\u0000vkX;\n\u00145ku\u0000vkX;\nwhere we have used (2.73) in obtaining the last inequality. Finally, using also (2.43) and (2.49),\nand applying (2.10) in Theorem 2.1,\nkm\u0000nkX\u001430Kku0\u0000v0kL1\n\u0014120K\u000e\u00002km0\u0000n0kL1;\nwhich yields (2.45).\nProof of Corollary 2.11. LetR2SO(3)such thatRQ= (0;0;\u00001), i.e.Ris the rotation that\nmapsQto the South Pole. Let us set m0\nR=Rm0. Then\njm0\u0000Qj2=jR(m0\u0000Q)j2=jm0\nR\u0000(0;0;\u00001)j2= 2(1 +m0\n3;R):\nHence,\ninf\nx2RNm0\n3;R\u0015\u00001 +\u000e\nand\n[m0\nR]BMO = [m0]BMO\u0014\"0:\nTherefore, Theorem2.9providestheexistenceofauniquesmoothsolution mR2X(RN\u0002R+;S2)\nof (LLG\u000b) satisfying (2.44). Using the invariance of (LLG \u000b) and setting m=R\u00001mRwe obtain\nthe existence of the desired solution. To establish the uniqueness, it suffices to observe that if n\nis another smooth solution of (LLG \u000b) satisfying (2.47), then nR:=Rnis a solution of (LLG \u000b)\nwith initial condition m0\nRand it fulfills (2.44). Therefore, from the uniqueness of solution in\nTheorem 2.9, it follows that mR=nRand then m=n.\nProof of Theorem 1.1. In Theorem 2.9 and Corollary 2.11, the constants are given by C=C1C2\n2\nandK=C2. As discussed in Remark 2.10, the value\n\u001a\u0003=\u000e4\n32C1C2\n2\n21maximizes the range for \"0in (2.27) and this inequality is satisfied for any \"0>0such that\n\"0\u0014\u000e6\n256C1C2\n2:\nTaking\nM1=1\n256C1C2\n2; M 2=C2andM3=1\n32C1C2\n2;\nso that\u001a\u0003=M3\u000e4, the conclusion follows from Theorem 2.9 and Corollary 2.11.\nRemark2.14. Wefinallyremarkthatispossibletostatelocal(intime)versionsofTheorems2.1\nand 2.9 as it was done in [23, 22, 35]. In our context, the local well-posedness would concern\nsolutions with initial condition m02VMO (RN), i.e. such that\nlim\nr!0+sup\nx2RN \nBr(x)jm0(y)\u0000m0\nx;rjdy= 0: (2.75)\nMoreover, some uniqueness results have been established for solutions with this kind of initial\ndata by Miura [32] for the Navier–Stokes equation, and adapted by Lin [29] to (HFHM). It is\nalso possible to do this for (LLG \u000b), for\u000b > 0. We do not pursuit here these types of results\nbecause they do not apply to the self-similar solutions mc;\u000b. This is due to the facts that the\nfunction m0\nA\u0006does not belong to VMO (R)and that\nlim\nT!0+sup\n0<t<Tp\ntk@xmc;\u000bkL16= 0:\n3 Applications\n3.1 Existence of self-similar solutions in RN\nThe LLG equation is invariant under the scaling (x;t)!(\u0015x;\u00152t), for\u0015 > 0, that is if m\nsatisfies (LLG \u000b), then so does the function\nm\u0015(x;t) =m(\u0015x;\u00152t); \u0015> 0:\nTherefore is natural to study the existence of self-similar solutions (of expander type), i.e. a\nsolution msatisfying\nm(x;t) =m(\u0015x;\u00152t);8\u0015>0; (3.1)\nor, equivalently,\nm(x;t) =f\u0012xp\nt\u0013\n;\nfor some f:RN\u0000!S2profile of m. In particular we have the relation f(y) =m(y;1), for all\ny2RN. From (3.1) we see that, at least formally, a necessary condition for the existence of a\nself-similar solution is that initial condition m0be homogeneous of degree 0, i.e.\nm0(\u0015x) =m0(x);8\u0015>0:\nSince the norm in X(RN\u0002R+;R3)is invariant under this scaling, i.e.\nkm\u0015kX=kmkX;8\u0015>0;\nwhere m\u0015is defined by (3.1), Theorem 2.9 yields the following result concerning the existence\nof self-similar solutions.\n22Corollary 3.1. With the same notations and hypotheses as in Theorem 2.9, assume also that\nm0is homogeneous of degree zero. Then the solution mof(LLG\u000b)provided by Theorem 2.9 is\nself-similar. In particular there exists a smooth profile f:RN!S2such that\nm(x;t) =f\u0012xp\nt\u0013\n;\nfor allx2RNandt>0, andfsatisfies the equation\n\u00001\n2y\u0001rf(y) =\ff(y)\u0002\u0001f(y)\u0000\u000bf(y)\u0002(f(y)\u0002\u0001f(y));\nfor ally2RN. Herey\u0001rf(y) = (y\u0001rf1(y);:::;y\u0001rfN(y)).\nRemark 3.2. Analogously, Theorem 2.1 leads to the existence of self-similar solutions for\n(DNLS), provided that u0is a homogeneous function of degree zero.\nFor instance, in dimensions N\u00152, Corollary 3.1 applies to the initial condition\nm0(x) =H\u0012x\njxj\u0013\n;\nwithHa Lipschitz map from SN\u00001toS2\\f(x1;x2;x3) :x3\u0015\u00001=2g, provided that the Lipschitz\nconstant is small enough. Indeed, using (2.58), we have\n[m0]BMO\u00144kHkLip;\nso that taking\n\u000e= 1=2; \u001a =\u000e4\n32K4C; \" 0=\u000e6\n256K4CandkHkLip\u0014\"0;\nthe condition (2.42) is satisfied and we can invoke Corollary 3.1.\nOther authors have considered self-similar solutions for the harmonic map flow (i.e. (LLG \u000b)\nwith\u000b= 1) in different settings. Actually, equation (HFHM) can be generalized for maps\nm:M\u0002R+!N, withMandNRiemannian manifolds. Biernat and Bizoń [8] established\nresults whenM=N=Sdand3\u0014d\u00146:Also, Germain and Rupflin [15] have investigated the\ncaseM=RdandN=Sd, ind\u00153. In both works the analysis is done only for equivariant\nsolutions and does not cover the case M=RNandN=S2.\n3.2 The Cauchy problem for the one-dimensional LLG equation with a jump\ninitial data\nThis section is devoted to prove Theorems 1.2 and 1.3 in the introduction. These two results con-\ncern the question of well-posedness/ill-posedness of the Cauchy problem for the one-dimensional\nLLG equation associated with a step function initial condition of the form\nm0\nA\u0006:=A+\u001fR++A\u0000\u001fR\u0000; (3.2)\nwhere A+andA\u0000are two given unitary vectors in S2.\n233.2.1 Existence, uniqueness and stability. Proof of Theorem 1.2\nAs mentioned in the introduction, in [17] we proved the existence of the uniparametric smooth\nfamily of self-similar solutions fmc;\u000bgc>0of (LLG\u000b) for all\u000b2[0;1]with initial condition of the\ntype (3.2) given by\nm0\nc;\u000b:=A+\nc;\u000b\u001fR++A\u0000\nc;\u000b\u001fR\u0000; (3.3)\nwhere A\u0006\nc;\u000b2S2are given by Theorem A.5. For the convenience of the reader, we collect some\nof the results proved in [17] in the Appendix. The results in this section rely on a further\nunderstanding of the properties of the self-similar solutions mc;\u000b.\nIn Proposition 3.4 we show that\nmc;\u000b= (m1;c;\u000b;m2;c;\u000b;m3;c;\u000b)2X(R\u0002R+;S2);\nthatm3;c;\u000bis far from the South Pole and that [mc;\u000b]Xis small, if cis small enough. This\nwill yield that mc;\u000bcorresponds (up to a rotation) to the solution given by Corollary 3.1. More\nprecisely, using the invariance under rotations of (LLG \u000b), we can prove that, if the angle between\nA+andA\u0000is small enough, then the solution given by Corollary 3.1 with initial condition m0\nA\u0006\ncoincides (modulo a rotation) with mc;\u000b, for somec. We have the following:\nTheorem 3.3. Let\u000b2(0;1]. There exist L1;L2>0,\u000e\u00032(\u00001;0)and#\u0003>0such that the\nfollowing holds. Let A+,A\u00002S2and let#be the angle between them. If\n0<#\u0014#\u0003; (3.4)\nthen there exists a solution mof(LLG\u000b)with initial condition m0\nA\u0006. Moreover, there exists\n0< c <p\u000b\n2p\u0019, such that mcoincides up to a rotation with the self-similar solution mc;\u000b, i.e.\nthere existsR2SO(3), depending only on A+,A\u0000,\u000bandc, such that\nm=Rmc;\u000b; (3.5)\nandmis the unique solution satisfying\ninf\nx2R\nt>0m3(x;t)\u0015\u000e\u0003and [m]X\u0014L1+L2c: (3.6)\nIn order to prove Theorem 3.3, we need some preliminary estimates for mc;\u000bin terms of c\nand\u000b. To obtain them, we use some properties of the profile profile fc;\u000b= (f1;c;\u000b;f2;c;\u000b;f3;c;\u000b)\nconstructed in [17] using the Serret–Frenet equations with initial conditions\nf1;c;\u000b(0) = 1; f 2;c;\u000b(0) =f3;c;\u000b(0) = 0:\nAlso,\njf0\nj;c;\u000b(s)j\u0014ce\u0000\u000bs2=4;for alls2R;\nforj2f1;2;3gand\nmc;\u000b(x;t) =fc;\u000b\u0012xp\nt\u0013\n;for all (x;t)2R\u0002R+: (3.7)\nHence, for any x2R,\njf3;c\u000b(x)j=jf3;c\u000b(x)\u0000f3;c\u000b(0)j\u0014\u0002jxj\n0ce\u0000\u000b\u001b2=4d\u001b\u0014cp\u0019p\u000b:\n24Since the same estimate holds for f2;c;\u000b, we conclude that\njm2;c;\u000b(x;t)j\u0014cp\u0019p\u000b;andjm3;c;\u000b(x;t)j\u0014cp\u0019p\u000bfor all (x;t)2R\u0002R+:(3.8)\nMoreover, since\nA\u0006\nc;\u000b= lim\nx!\u00061fc;\u000b(x);\nwe also get\njA\u0006\nj;c;\u000bj\u0014cp\u0019p\u000b;forj2f2;3g: (3.9)\nWe now provide some further properties of the self-similar solutions.\nProposition 3.4. For\u000b2(0;1]andc>0, we have\nkm0\n2;c;\u000bkL1\u0014cp\u0019p\u000b;km0\n3;c;\u000bkL1\u0014cp\u0019p\u000b;sup\nt>0km3;c;\u000bkL1\u0014cp\u0019p\u000b;(3.10)\n[m0\nc;\u000b]BMO\u00142cp\n2\u0019p\u000b; (3.11)\np\ntk@xmc;\u000bk1=c;for allt>0; (3.12)\nsup\nx2R\nr>01\nr\u0002\nQr(x)j@ymc;\u000b(y;t)j2dtdy\u00142p\n2\u0019c2\np\u000b: (3.13)\nIn particular, mc;\u000b2X(R\u0002R+;S2)and\n[mc;\u000b]X\u00144c\n\u000b1\n4: (3.14)\nProof of Proposition 3.4. The estimates in (3.10) follow from (3.8) and (3.9). To prove (3.11),\nwe use (2.58), (3.3), (3.10) and the fact that\nA\u0000\nc;\u000b= (A+\n1;c;\u000b;\u0000A+\n2;c;\u000b;\u0000A+\n3;c;\u000b); (3.15)\n(see Theorem A.5) to get\n[m0\nc;\u000b]BMO\u0014sup\nx2RN \nBr(x) \nBr(x)jm0\nc;\u000b(y)\u0000m0\nc;\u000b(z)jdydz\n\u00142q\n(A+\n2;c;\u000b)2+ (A+\n3;c;\u000b)2sup\nx2RN \nBr(x) \nBr(x)dydz\n\u00142cp\n2\u0019p\u000b:\nFrom (A.12) we obtain the equality in (3.12) and also\nIr;x:=1\nr\u0002\nQr(x)j@ymc;\u000b(y;t)j2dtdy =c2\nr\u0002x+r\nx\u0000r\u0002r2\n0e\u0000\u000by2\n2t\ntdtdy: (3.16)\nPerforming the change of variables z= (\u000by2)=(2t), we see that\n\u0002r2\n0e\u0000\u000by2\n2t\ntdt=E1\u0012\u000by2\n2r2\u0013\n; (3.17)\n25whereE1is the exponential integral function\nE1(y) =\u00021\nye\u0000z\nzdz:\nThis function satisfies that limy!0+E1(y) =1andlimy!1E1(y) = 0(see e.g. [1, Chapter 5]).\nMoreover, taking \u000f>0and integrating by parts,\n\u00021\n\u000fE1(y2)dy=yE1(y2)\f\f1\n\u000f+ 2\u00021\n\u000fe\u0000y2dy; (3.18)\nso L’Hôpital’s rule shows that the first term in the r.h.s. of (3.18) vanishes as \u000f!0+. Therefore,\nthe Lebesgue’s monotone convergence theorem allows to conclude that E1(y2)2L1(R+)and\n\u00021\n0E1(y2) =p\u0019: (3.19)\nBy using (3.16), (3.17), (3.19), and making the change of variables z=p\u000by=(rp\n2), we obtain\nIr;x=c2\nr\u0002x+r\nx\u0000rE1\u0012\u000by2\n2r2\u0013\ndy=p\n2c2\np\u000b\u0002p\u000bp\n2(x\nr+1)\np\u000bp\n2(x\nr\u00001)E1(z2)dz\u0014p\n2c2\np\u000b\u00012p\u0019; (3.20)\nwhich leads to (3.13). Finally, the bound in (3.14) easily follows from those in (3.12) and (3.13)\nand the elementary inequality\n\u0010\n1 +\u00102p\n2\u0019p\u000b\u00111=2\u0011\n\u00141\n\u000b1\n4\u0000\n1 + (2p\n2\u0019)1=2\u0001\n\u00144\n\u000b1\n4; \u000b2(0;1]:\nProof of Theorem 3.3. First, we consider the case when A+=A+\nc;\u000bandA\u0000=A\u0000\nc;\u000b(i.e. when\nm0\nA\u0006=m0\nc;\u000b) for some c >0. We will continue to show that the solution provided by Theo-\nrem 2.9 is exactly mc;\u000b, forcsmall. Indeed, bearing in mind the estimates in Proposition 3.4,\nwe consider\nc\u0014p\u000b\n2p\u0019;\nso that\ninf\nx2Rm0\n3;c;\u000b(x)\u0015\u00001\n2: (3.21)\nIn view of (3.11), (3.21) and Remark 2.10, we set\n\"0:= 4cp\u0019p\u000b; \u000e :=1\n2; \u001a :=\u000e4\n8K4C=1\n27K4C; (3.22)\nwhereC;K\u00151are the constants given by Theorem 2.9. In this manner, from (3.11), (3.21) and\n(3.22), we have\ninf\nRm0\n3\u0015\u00001 +\u000eand [m0]BMO\u0014\"0;\nand the condition (2.42) is fulfilled if\n\"0\u0014\u000e6\n256K4C;\nor equivalently, if c\u0014~c, with\n~c:=p\u000b\n216K4Cp\u0019:\n26Observe that in particular ~c<p\u000b\n2p\u0019.\nFor fixed 0<c< ~c, we can apply Theorem 2.9 to deduce the existence and uniqueness of a\nsolution mof (LLG\u000b) satisfying\ninf\nx2R\nt>0m3(x;t)\u0015\u00001 +2\n1 +K2(\u001a+ 2)2and [m]X\u00144K\u001a+29Kcp\u0019p\u000b:(3.23)\nNow by Proposition 3.4, for fixed 0<c\u0014~c, we have the following estimates for mc;\u000b\n[mc;\u000b]X\u00144c=\u000b1\n4and inf\nx2R\nt>0m3;c;\u000b(x;t)\u0015\u00001\n2;\nso in particular mc;\u000bsatisfies (3.23). Thus the uniqueness of solution implies that m=mc;\u000b,\nprovided that c\u0014~c. Defining the constants L1,L2and\u000e\u0003by\nL1= 4K\u001a; L 2=29Kp\u0019p\u000band\u000e\u0003=\u00001 +2\n1 +K2(\u001a+ 2)2;(3.24)\nthe theorem is proved in the case A\u0006=A\u0006\nc;\u000b.\nFor the general case, we would like to understand which angles can be reached by varying the\nparametercin the range (0;~c]. To this end, for fixed 0< c\u0014~c, let#c;\u000bbe the angle between\nA+\nc;\u000bandA\u0000\nc;\u000b. From Lemma A.6,\n#c;\u000b\u0015arccos\u0012\n1\u0000c2\u0019+ 32c3p\u0019\n\u000b2\u0013\n;for allc2\u0010\n0;\u000b2p\u0019\n32i\n:\nNow, it is easy to see that the function F(c) = arccos\u0010\n1\u0000c2\u0019+ 32c3p\u0019\n\u000b2\u0011\nis strictly increasing\non the interval [0;\u000b2p\u0019\n48]so that\nF(c)>F(0) = 0;for allc2\u0010\n0;\u000b2p\u0019\n48i\n: (3.25)\nLetc\u0003= min(~c;\u000b2p\u0019\n48)and consider the map T\u000b:c\u0000!#c;\u000bon[0;c\u0003]. By Lemma A.6, T\u000bis\ncontinuous on [0;c\u0003],T\u000b(0) = lim c!0+T\u000b(c) = 0and, bearing in mind (3.25), T(c\u0003) =#c\u0003;\u000b>0.\nThus, from the intermediate value theorem we infer that for any #2(0;#c\u0003;\u000b), there exists\nc2(0;c\u0003)such that\n#=T\u000b(c) =#c;\u000b:\nWe can now complete the proof for any A+,A\u00002S2. Let#be the angle between A+and\nA\u0000. From the previous lines, we know that there exists #\u0003:=#c\u0003;\u000bsuch that if #2(0;#\u0003),\nthere exists c2(0;c\u0003)such that#=#c;\u000b. For this value of c, consider the initial value problem\nassociated with m0\nc;\u000band the constants defined in (3.24). We have already seen the existence\nof a unique solution mc;\u000bof the LLG equation associated with this initial condition satisfying\n(3.6). LetR2SO(3)be the rotation on R3such that A+=RA+\nc;\u000bandA\u0000=RA\u0000\nc;\u000b. Then\nm:=Rmc;\u000bsolves (LLG \u000b) with initial condition m0\nA\u0006. Finally, recalling the above definition\nofL1,L2and\u000e\u0003, using the invariance of the norms under rotations and the fact that mc;\u000bis the\nunique solution satisfying (3.23), it follows that mis the unique solution satisfying the conditions\nin the statement of the theorem.\nWe are now in position to give the proof of Theorem 1.2, the second of our main results\nin this paper. In fact, we will see that Theorem 1.2 easily follows from Theorem 3.3 and the\nwell-posedness for the LLG equation stated in Theorem 2.9.\n27Proof of Theorem 1.2. Let#\u0003,\u000e\u0003,L1andL2betheconstantsdefinedintheproofofTheorem3.3.\nGiven A+andA\u0000such that 0<#<#\u0003, Theorem 3.3 asserts the existence of\n0<c<p\u000b\n2p\u0019(3.26)\nandR2SO(3)such thatRmc;\u000bis the unique solution of (LLG \u000b) with initial condition m0\nA\u0006\nsatisfying (3.6), and in particular m0\nA\u0006=Rm0\nc;\u000b. By hypothesis m0satisfies\nkm0\u0000m0\nA\u0006kL1\u0014cp\u0019\n2p\u000b: (3.27)\nHence, defining m0\nR=R\u00001m0, recalling that [f]BMO\u00142kfkL1and using the invariance of the\nnorms under rotations, we deduce from (3.27) that\nkm0\nRkL1\u0014km0\nc;\u000bkL1+cp\u0019\n2p\u000band [m0\nR]BMO\u0014[m0\nc;\u000b]BMO +cp\u0019p\u000b:\nThen, by Proposition 3.4,\nkm0\n3;RkL1\u00142cp\u0019p\u000band [m0\nR]BMO\u00144cp\u0019p\u000b: (3.28)\nFrom (3.26) and (3.28), it follows that\nm0\n3;R(x)\u0015\u00001=2;for allx2R:\nTherefore, as in the proof of Theorem 3.3, we can apply Theorem 2.9 with the values of \"0,\u000e\nand\u001agiven in (3.22) to deduce the existence of a unique (smooth) solution mRof (LLG\u000b) with\ninitial condition m0\nRsatisfying\ninf\nx2R\nt>0m3;R(x;t)\u0015\u00001 +2\n1 +K2(\u001a+ 2)2=\u000e\u0003and [mR]X\u00144K\u001a+29Kcp\u0019p\u000b=L1+L2c:\nSince we have taken the values for \"0,\u000eand\u001aas in the proof Theorem 3.3, Theorem 2.9 also\nimplies that\nkmR\u0000mc;\u000bkX\u0014480Kkm0\nR\u0000m0\nc;\u000bkL1:\nThe conclusion of the theorem follows defining m=RmRandL3= 480K, and using once\nagain the invariance of the norm under rotations.\n3.2.2 Multiplicity of solutions. Proof of Theorem 1.3\nAs proved in [17], when \u000b= 1, the self-similar solutions are explicitly given by\nmc;1(x;t) = (cos(cErf(x=p\nt));sin(cErf(x=p\nt));0);for all (x;t)2R\u0002R+;(3.29)\nfor everyc>0, where Erf(\u0001)is the non-normalized error function\nErf(s) =\u0002s\n0e\u0000\u001b2=4d\u001b:\nIn particular,\n~A\u0006\nc;1= (cos(cp\u0019);\u0006sin(cp\u0019);0)\n28#c;1\n\u0019\nc\nFigure 2: The angle #c;\u000bas a function of cfor\u000b= 1.\nand the angle between A+\nc;1andA\u0000\nc;1is given by\n#c;1= arccos(cos(2 cp\u0019)): (3.30)\nFormula (3.30) and Figure 2 show that there are infinite values of cthat allow to reach any\nangle in [0;\u0019]. Therefore, using the invariance of (LLG \u000b) under rotations, in the case when\n\u000b= 1, one can easily prove the existence of multiple solutions associated with a given initial\ndata of the form m0\nA\u0006for any given vectors A\u00062S2(see argument in the proof included below).\nIn the case that \u000bis close enough to 1, we can use a continuity argument to prove that we still\nhave multiple solutions. More precisely, Theorem 1.3 asserts that for any given initial data of\nthe form m0\nA\u0006with angle between A+andA\u0000in the interval (0;\u0019), if\u000bis sufficiently close\nto one, then there exist at leastk-distinct solutions of (LLG \u000b) associated with the same initial\ncondition, for any given k2N.\nThe rest of this section is devoted to the proof of Theorem 1.3.\nProof of Theorem 1.3. Letk2N,A\u00062S2and#2(0;\u0019)be the angle between A+andA\u0000.\nUsing the invariance of (LLG \u000b) under rotations, it suffices to prove the existence of \u000bk2(0;1)\nsuch that for every \u000b2[\u000bk;1]there exist 0< c1<\u0001\u0001\u0001< cksuch that the angle #cj;\u000bbetween\nA+\ncj;\u000bandA\u0000\ncj;\u000b, satisfies\n#cj;\u000b=#;for allj2f1;:::;kg: (3.31)\nIn what follows, and since we want to show the existence of at least k-distinct solutions, we will\nassume without loss of generality that kis large enough.\nFirst observe that, since A\u0000\nc;\u000b= (A+\n1;c;\u000b;\u0000A+\n2;c;\u000b;\u0000A+\n3;c;\u000b), we have the explicit formula\ncos(#c;\u000b) = 2(A+\n1;c;\u000b)2\u00001;\nand using Lemma A.8 in the Appendix, we get\njcos(#c;\u000b)\u0000cos(#c;1)j=j2((A+\n1;c;\u000b)2\u0000(A+\n1;c;1)2)j\u00144jA+\n1;c;\u000b\u0000A+\n1;c;1j\u00144h(c)p\n1\u0000\u000b;(3.32)\nfor all\u000b2[1=2;1], withh:R+\u0000!R+an increasing function satisfying lims!1h(s) =1.\nForj2N, we setaj= (2j+ 1)p\u0019=2andbj= (2j+ 2)p\u0019=2, so that (3.30) and (3.32) yield\ncos(#aj;\u000b)\u0014\u00001 + 4h(aj)p\n1\u0000\u000band cos(#bj;\u000b)\u00151\u00004h(bj)p\n1\u0000\u000b;8\u000b2[1=2;1]:\n(3.33)\n29Definel= cos(#)and\n\u000bk= max\u0012\n1\u0000\u00101\u0000l\n8h(bk)\u00112\n;1\u0000\u00101 +l\n8h(bk)\u00112\u0013\n:\nNotice that, since #2(0;\u0019), we have\u00001< l < 1and thus\u000bk<1. Also, since hdiverges to\n1, we can assume without loss of generality that \u000bk2[1=2;1), and from the definition of \u000bkwe\nhave\n0<p1\u0000\u000bk<min\u00121\u0000l\n8h(bk);1 +l\n8h(bk)\u0013\n:\nTherefore, from (3.33) and h(aj)< h(bj)\u0014h(bk)(sincehis a strictly increasing function), we\nget\ncos(#aj;\u000b)\u0014\u00001 +l\n2and1 +l\n2\u0014cos(#bj;\u000b);8j2f1;:::;kg;8\u000b2[\u000bk;1];\nand thus\ncos(#aj;\u000b)<l< cos(#bj;\u000b);8j2f1;:::;kg;8\u000b2[\u000bk;1]; (3.34)\nsincel2(\u00001;1).\nLet us fix\u000b2[\u000bk;1]andj2f1;:::;kg. By Lemma A.8, c!cos(#c;\u000b)is a continuous\nfunction on c. Therefore (3.34) and the intermediate value theorem yield the existence of cj2\n[aj;bj]such that\ncos(#cj;\u000b) =l= cos(#);\nor equivalently, such that #cj;\u000b=#.\nFinally, for each j2f1;:::;kg, letRj2SO(3)be such that m0\nA\u0006=Rjm0\ncj;\u000b, and define mj\nbymj=Rjmcj;\u000b. Then mjsolves (LLG \u000b) with initial data m0\nA\u0006and the control of @xmjin\n(1.9) follows from the definition of mjin terms of mcj;\u000band the analogous property established\nin Theorem A.5 for the self-similar solution mcj;\u000b(see (A.12)).\nIn the case when \u000b= 1and#2[0;\u0019], formula (3.30) shows that sequence fcjgj\u00151in (1.10)\nsatisfies#cj;1=#for allj2N\u0003. The result follows by considering the sequence of solutions\nfmjgj\u00151described above.\nRemark 3.5. Notice that the proof given above also shows how close \u000bkneeds to be to 1 in\nterms of the (fixed) angle #2(0;\u0019), and in particular \u000bk!1ask!1, and\u000bk!1as#!0\n(i.e. whenl!1).\nRemark 3.6. For\u000b= 1andc>0, the function\nu(x;t) =P(mc;1) = exp\u0000\nicErf(x=p\nt)\u0001\nis a solution of (DNLS) with initial condition\nu0=eicp\u0019\u001fR++e\u0000icp\u0019\u001fR\u0000:\nTherefore there is also a multiplicity phenomenon for the equation (DNLS).\n3.3 A singular solution for a nonlocal Schrödinger equation\nWe have used the stereographic projection to establish a well-posedness result for (LLG \u000b).\nMelcher [31] showed a global well-posedness result, provided that\nkrm0kLN\u0014\";m0\u0000Q2H1(RN)\\W1;N(RN); \u000b> 0; N\u00153;\n30for some Q2S2and\">0small. Later, Lin, Lan and Wang [30] improved this result and proved\nglobal well-posedness under the conditions\nkrm0kM2;2\u0014\";m0\u0000Q2L2(RN); \u000b> 0; N\u00152;\nfor some Q2S2and\">0small.6In the context of Theorem 1.1 and using the characterization\nofBMO\u00001in Theorem A.1, the second condition in (1.3) says that krm0kBMO\u00001is small. In\nview of the embeddings\nLN(RN)\u001aM2;2(RN)\u001aBMO\u00001(RN);\nforN\u00152, we deduce that Theorem 1.1 includes initial conditions with less regularity, as long\nas their essential range is not S2. The argument in [30, 31] is based on the method of moving\nframes that produces a covariant complex Ginzburg–Landau equation. One of the aims of this\nsubsection is to compare their approach in the context of the self-similar solutions mc;\u000b, and in\nparticular to draw attention to a possible difficulty in using it to study these solutions.\nIn the sequel we consider the one-dimensional case N= 1and\u000b2[0;1]. Then the moving\nframes technique can be recast as a Hasimoto transformation as follows. Assume that mis the\ntangent vector of a curve in R3, i.e.m=@xX, for some curve X(x;t)2R3parametrized by the\narc-length. It can be shown (see [12]) that if mevolves under (LLG \u000b), then the torsion \u001cand\nthe curvature cofXsatisfy\n@t\u001c=\f\u0012\nc@xc +@x\u0010@xxc\u0000c\u001c2\nc\u0011\u0013\n+\u000b\u0012\nc2\u001c+@x\u0010@x(c\u001c) +\u001c@xc\nc\u0011\u0013\n;\n@tc =\f(\u0000@x(c\u001c)\u0000\u001c@xc) +\u000b\u0000\n@xc\u0000c\u001c2\u0001\n:\nHence, defining the Hasimoto transformation [19] (also called filament function)\nv(x;t) = c(x;t)ei\u0001x\n0\u001c(\u001b;t)d\u001b; (3.35)\nwe verify that vsolves the following dissipative Schrödinger (or complex Ginzburg–Landau)\nequation\ni@tv+ (\f\u0000i\u000b)@xxv+v\n2\u0012\n\fjvj2+ 2\u000b\u0002x\n0Im(\u0016v@xv)\u0000A(t)\u0013\n= 0; (3.36)\nwhere\f=p\n1\u0000\u000b2and\nA(t) =\u0012\n\f\u0012\nc2+2(@xxc\u0000c\u001c2)\nc\u0013\n+ 2\u000b\u0012@x(c\u001c) +\u001c@xc\nc\u0013\u0013\n(0;t):\nThe curvature and torsion associated with the self-similar solutions mc;\u000bare (see [17]):\ncc;\u000b(x;t) =cp\nte\u0000\u000bx2\n4tand\u001cc;\u000b(x;t) =\fx\n2p\nt: (3.37)\nTherefore in this case\nA(t) =\fc2\nt(3.38)\n6We recall that v2M2;2(RN)ifv2L2\nloc(RN)and\nkvkM2;2:= sup\nx2RN\nr>01\nr(N\u00002)=2kvkL2(Br(x))<1:\n31and the Hasimoto transformation of mc;\u000bis\nvc;\u000b(x;t) =cp\nte(\u0000\u000b+i\f)x2\n4t:\nIn particular vc;\u000bis a solution of (3.36) with A(t)as in (3.38), for all \u000b2[0;1]andc > 0.\nMoreover, the Fourier transform of this function (w.r.t. the space variable) is\nbvc;\u000b(\u0018;t) = 2cp\n\u0019(\u000b+i\f)e\u0000(\u000b+i\f)\u00182t;\nso thatvc;\u000bis a solution of (3.36) with a Dirac delta as initial condition:\nvc;\u000b(\u0001;0) = 2cp\n\u0019(\u000b+i\f)\u000e:\nHere\u000edenotes the delta distribution at the point x= 0andpzdenotes the square root of a\ncomplex number zsuch that Im(pz)>0.\nIn the limit cases \u000b= 0and\u000b= 1, the first three terms in equation (3.36) lead to a cubic\nSchrödinger equation and to a linear heat equation, respectively. The Cauchy problem with\na Dirac delta for these kind of equations associated with a power type non-linearity has been\nstudied by several authors (see e.g. [4] and the reference therein). We recall two classical results.\nTheorem 3.7 ([9]).Letp\u00152andu2Lp\nloc(R\u0002R+)be a solution in the sense of distributions\nof\n@tu\u0000@xxu+jujpu= 0onR\u0002R+: (3.39)\nAssume that\nlim\nt!0+\u0002\nRu(x;t)'(x)dx= 0;for all'2C0(Rnf0g); (3.40)\nwhereC0(Rnf0g)denotes the space of continuous functions with compact support in Rnf0g.\nThenu2C2;1(R\u0002[0;1))andu(x;0) = 0for allx2R. In particular there is no solution of\n(3.39)such that\nlim\nt!0+\u0002\nRu(x;t)'(x)dx='(0);for all'2C0(RN):\nIn[9]itisalsoprovedthatif 1<p< 2, equation(3.39)hasaglobalsolutionwithaDiracdelta\nas initial condition, as in the case of the linear parabolic equation. Concerning the Schrödinger\nequation, we have the following ill-posedness result due to Kenig, Ponce and Vega [21].\nTheorem 3.8 ([21]).Letp\u00152. Either there is no solution in the sense of distributions of\ni@tu+@xxu+jujpu= 0onR\u0002R+; (3.41)\nwith\nlim\nt!0+u(\u0001;t) =\u000einS0(R);\nin the class u;jujpu2L1(R+;S0(R)), or there is more than one.\nAfter performing an appropriate change of variables, equation (3.36) leads to the following\nequation\ni@tu+ (\f\u0000i\u000b)@xxu+u\n2\u0012\n\fjuj2+ 2\u000b\u0002x\n0Im(\u0016u@yu)dy\u0013\n= 0: (3.42)\nSince\u000b2[0;1], the above equation can be seen as an intermediate model between (3.39) and\n(3.41). Therefore one could expect that when \u000b2(0;1], the solutions of (3.42) share similar\nproperties to those established in Theorem 3.7 for the equation (3.39). We will continue to\nobserve that this is not necessarily the case. To this end, we need the following:\n32Proposition 3.9. For all\u000b2[0;1]and for all c2Cnf0gthe function wc;\u000b:R\u0002R+!Cgiven\nby\nwc;\u000b(x;t) =cp\ntexp\u0012i\fjcj2\n2ln(t) + (i\f\u0000\u000b)x2\n4t\u0013\nis a solution of (3.42). In addition,\ni) If\u000b2[0;1), thenwc;\u000b(\u0001;t)does not converge in S0(R)ast!0+.\nii) If\u000b2(0;1], then\nlim\nt!0+\u0002\nRwc;\u000b(x;t)'(x)dx= 0;for all'2C0(Rnf0g): (3.43)\nProof.A straightforward computation shows that wc;\u000bsatisfies (3.42). The proof that wc;\u000b(\u0001;t)\ndoes not converge in S0(R)ast!0+ifc6= 0is the same as in [21]. Indeed, for '2S(R), by\nParseval’s theorem,\u0002\nR\u0016wc;\u000b(x;t)'(x)dx=1\n2\u0019\u0002\nRbwc;\u000b(\u0018;t)b'(\u0018)d\u0018\n=\u0016ce\u0000i\fjcj2ln(t)=2\np\u0019p\n\u000b+i\f\u0002\nRe\u0000(\u000b+i\f)\u00182tb'(\u0018)d\u0018:\nBy the dominated convergence theorem, the last integral converges:\nlim\nt!0+\u0002\nRe\u0000(\u000b+i\f)\u00182tb'(\u0018)d\u0018=\u0002\nRb'(\u0018)d\u0018= 2\u0019'(0):\nSince\f6= 0,e\u0000i\fjcj2ln(t)=2does not admit a limit at 0inS0(R). We conclude that wc;\u000b(\u0001;t)does\nnot converge in S0(R)ast!0+.\nIt remains to prove (3.43). Since now '2C0(Rnf0g), we cannot proceed as before. However,\nusing the change of variables x=p\nty, we have\nlim\nt!0+\u0002\nRwc;\u000b(x;t)'(x)dx=cei\fjcj2ln(t)=2\u0002\nRe(\u0000\u000b+i\f)y2=4'(p\nty)dy:\nTherefore, since \u000b>0and'(0) = 0, the dominated convergence theorem implies that\nlim\nt!0+\u0002\nRe(\u0000\u000b+i\f)y2=4'(p\nty)dy='(0)\u0002\nRe(\u0000\u000b+i\f)y2=4dy= 0:\nSincejei\fjcj2ln(t)=2j= 1, we obtain (3.43).\nThe results in Proposition 3.9 lead to the following remarks:\n1. Observe that if \u000b2(0;1),wc;\u000bprovides a solution to the dissipative equation (3.42).\nMoreover, form part (ii)in Proposition 3.9, wc;\u000bsatisfies the condition (3.40). However,\nnotice that wc;\u000bcannot be extended to C2;1(R\u0002[0;1))due to the presence of a logarithmic\noscillation. This is in contrast with the properties for solutions of the cubic heat equation\n(3.39) established in Theorem 3.7.\n2. In the case \u000b= 0, equation (3.42) corresponds to (3.41) with p= 2, i.e. to the equation\ncubic NLS equation that is invariant under the Galilean transformation. The proof of the\nill-posedness result given in Theorem 3.8 relies on this invariance and part (i)of Proposi-\ntion 3.9 with \u000b= 0. Although when \u000b > 0, equation (3.42) is no longer invariant under\nthe Galilean transformation, part (i)of Proposition 3.9 could be an indicator that that the\nCauchy problem (3.42) with a delta as initial condition is still ill-posed. This question rests\nopen for the moment and it seems that the use of (3.36) (or (3.42)) can be more difficult\nto formulate a Cauchy theory for (LLG \u000b) including self-similar solutions.\n334 Appendix\nThe characterization of BMO\u00001\n1(RN)as sum of derivatives of functions in BMO was proved by\nKoch and Tataru in [23]. A straightforward generalization of their proof leads to the following\ncharacterization of BMO\u00001\n\u000b(RN).\nTheorem A.1. Let\u000b2(0;1]andf2S0(RN). Thenf2BMO\u00001\n\u000b(RN)if and only if there exist\nf1;:::;fN2BMO\u000b(RN)such thatf=PN\nj=1@jfj. In addition, if such a decomposing holds,\nthen\nkfkBMO\u00001\n\u000b.NX\nj=1[fj]BMO\u000b:\nThe next results provide the equivalence between the weak solutions and the Duhamel for-\nmulation. We first need to introduce for T >0the spaceL1\nuloc(RN\u0002(0;T))defined as the space\nof measurable functions on RN\u0002(0;T)such that the norm\nkfkuloc;T:= sup\nx02RN\u0002\nB(x0;1)\u0002T\n0jf(y;t)jdtdy\nis finite. We refer the reader to Lemarié–Rieusset’s book [27] for more details about these kinds\nof spaces. In particular, we recall the following result corresponding to Lemma 11.3 in [27] in\nthe case\u000b= 1. It is straightforward to check that the same proof still applies if \u000b2(0;1).\nLemma A.2. Let\u000b2(0;1],T2(0;1)andw2L1\nuloc(RN\u0002(0;T)). Then the function\nW(x;t) :=\u0002t\n0S\u000b(t\u0000s)w(x;s)ds\nis well defined and belongs to L1\nuloc(RN\u0002(0;T)). Moreover,\ni@tW+ (\f\u0000i\u000b)\u0001W=winD0(RN\u0002R+);\nand the application\n[0;T]!R\nt7! kW(\u0001;t)kL1(B1(x0))\nis continuous for any x02R, withkW(\u0001;t)kL1(B1(x0))!0, ast!0+, uniformly in x0.\nFollowing the ideas in [27], we can establish now the equivalence between the notions of\nsolutions as well as the regularity.\nTheorem A.3. Let\u000b2(0;1]andu2X(RN\u0002R+;C). Then the following assertions are\nequivalent:\ni) The function usatisfies\niut+ (\f\u0000i\u000b)\u0001u= 2(\f\u0000i\u000b)\u0016u(ru)2\n1 +juj2inD0(RN\u0002R+): (A.1)\nii) There exists u02S0(RN)such thatusatisfies\nu(t) =S\u000b(t)u0\u00002(\f\u0000i\u000b)\u0002t\n0S\u000b(t\u0000s)\u0016u(ru)2\n1 +juj2ds:\n34Moreover, if (ii) holds, then u2C1(RN\u0002R+)and\nk(u(t)\u0000u0)'kL1(RN)!0;ast!0+; (A.2)\nfor any'2S(RN).\nProof.In view of Lemma A.2, we need to prove that the function\ng(u) =\u00002(\f\u0000i\u000b)\u0016u(ru)2\n1 +juj2\nbelongs toL1\nuloc(RN\u0002(0;T)), for allT >0. Indeed, by (2.3) we have\nkg(u)kuloc;T\u0014kjruj2kuloc;T: (A.3)\nIfT\u00141, then\nkjruj2kuloc;T\u0014sup\nx02RN\u0002\nQ1(x0)jru(y;t)j2dtdy\u0014kuk2\nX: (A.4)\nIfT\u00151, using that\njruj\u0014[u]Xp\nt;for anyt>0;\nwe get\nkjruj2kT;uloc\u0014sup\nx02RN\u0002\nQ1(x0)jru(y;t)j2dtdy + sup\nx02RN\u0002T\n1\u0002\nB1(x0)jruj2dydt\n\u0014kuk2\nX+ [u]2\nXjB1(0)j\u0002T\n11\ntdt\n\u0014kuk2\nX(1 +jB1(0)jln(T)):(A.5)\nIn conclusion, we deduce from (A.3), (A.4) and (A.5) that g(u)2L1\nuloc(RN\u0002(0;T))and then\nit follows from Lemma A.2 that (ii) implies (i). The other implication can be established as in\n[27, Theorem 11.2]. Moreover, we deduce that the function\nW(x;t) :=T(g(u))(x;t) =\u0002t\n0S\u000b(t\u0000s)g(u)ds\nsatisfieskW(\u0001;t)kL1(B1(x0))!0, ast!0+, uniformly in x02RN. Let us take '2S(RN)and\na constantC'>0such thatj'(x)j\u0014C'(2 +jxj)\u0000N\u00001. Then\n\u0002\nRNj'(y)W(y;t)jdy\u0014X\nk2ZN\u0002\nB1(k)C'\n(2 +jxj)N+1jW(y;t)jdy\n\u0014sup\nx02RNkW(\u0001;t)kL1(B1(x0))X\nk2ZNC'\n(1 +jkj)N+1;\nso thatk'W(\u0001;t)kL1(RN)!0ast!0+, i.e.\nk(u(t)\u0000S\u000b(t)u0)'kL1(RN)!0;ast!0+: (A.6)\nOn the other hand, since u02L1(RN),\nkS\u000b(t)u0\u0000u0kL1(Br(0))!0;ast!0+; (A.7)\n35for anyr>0(see e.g. [3, Corollary 2.4]). Given \u000f>0, we fixr\u000f>0such that\n2ku0k1k'kL1(Bcr\u000f(0))\u0014\u000f:\nUsing (A.7), we obtain\nlim\nt!0+k(S\u000b(t)u0\u0000u0)'kL1(Br\u000f(0))= 0:\nThen, passing to limit in the inequality\nk(S\u000b(t)u0\u0000u0)'kL1(RN)\u0014k(S\u000b(t)u0\u0000u0)'kL1(Br\u000f(0))+ 2ku0kL1(RN)k'kL1(Bcr\u000f(0));(A.8)\nwe obtain\nlim sup\nt!0+k(S\u000b(t)u0\u0000u0)'kL1(RN)\u0014\u000f: (A.9)\nTherefore\nlim\nt!0+k(S\u000b(t)u0\u0000u0)'kL1(RN)= 0:\nCombining with (A.6), we conclude the proof of (A.2).\nIt remains to prove that uis smooth for t >0. Sinceu2X(RN\u0002R+;C), we get that\nu;ru2L1\nloc(RN\u0002R+). Theng(u)2L2\nloc(RN\u0002R+)so theLp-regularity theory for parabolic\nequations implies that a function usatisfying (A.1) belongs to u2H2;1\nloc(RN\u0002R+)(see [28, 24]\nand [33, Remark 48.3] for notations and more details). Since the space Hk\\L1is stable under\nmultiplication (see e.g. [20, Chapter 6]), we can use a bootstrap argument to conclude that\nu2C1(RN\u0002R+).\nRemark A.4. Several authors have studied further properties of the solutions found by Koch\nand Tataru for the Navier–Stokes equations. For instance, analyticity, decay rates of the higher-\norder derivatives in space and time have been investigated by Miura and Sawada [32], Germain,\nPavlović and Staffilani [14], among others. A similar analysis for the solution uof (DNLS) is\nbeyond the scope of this paper, but it can probably be performed using the same arguments\ngiven in [32, 14].\nWe end this appendix with some properties of the self-similar found in [17].\nTheorem A.5 ([17]).LetN= 1. For every \u000b2[0;1]andc >0, there exists a profile fc;\u000b2\nC1(R;S2)such that\nmc;\u000b(x;t) =fc;\u000b\u0012xp\nt\u0013\n;for all (x;t)2R\u0002R+;\nis a smooth solution of (LLG\u000b)onR\u0002R+. Moreover,\n(i) There exist unitary vectors A\u0006\nc;\u000b= (A\u0006\nj;c;\u000b)3\nj=12S2such that the following pointwise con-\nvergence holds when tgoes to zero:\nlim\nt!0+mc;\u000b(x;t) =8\n<\n:A+\nc;\u000b;ifx>0;\nA\u0000\nc;\u000b;ifx<0;(A.10)\nandA\u0000\nc;\u000b= (A+\n1;c;\u000b;\u0000A+\n2;c;\u000b;\u0000A+\n3;c;\u000b).\n(ii) There exists a constant C(c;\u000b;p ), depending only on c,\u000bandpsuch that for all t>0\nkmc;\u000b(\u0001;t)\u0000A+\nc;\u000b\u001f(0;1)(\u0001)\u0000A\u0000\nc;\u000b\u001f(\u00001;0)(\u0001)kLp(R)\u0014C(c;\u000b;p )t1\n2p; (A.11)\nfor allp2(1;1). In addition, if \u000b>0,(A.11)also holds for p= 1.\n36(iii) Fort>0andx2R, the derivative in space satisfies\nj@xmc;\u000b(x;t)j=cp\nte\u0000\u000bx2\n4t: (A.12)\n(iv) Let\u000b2[0;1]. Then A+\nc;\u000b!(1;0;0)asc!0+.\nLemma A.6. Letc >0,\u000b2(0;1],A+\nc;\u000b;A\u0000\nc;\u000bbe the unit vectors given in Theorem A.5 and\n#c;\u000bthe angle between A+\nc;\u000bandA\u0000\nc;\u000b. Then, for fixed \u000b2(0;1],#c;\u000bis a continuous function\ninc. Also, for 0<c<\u000b2p\u0019=32,\n#c;\u000b\u0015arccos\u0012\n1\u0000c2\u0019+32c3p\u0019\n\u000b2\u0013\n: (A.13)\nRemark A.7. If\u000b2(0;1]andc2(0;\u000b2p\u0019=32), then 1\u0000c2\u0019+32c3p\u0019\n\u000b22(0;1), so that its\narccosis well-defined.\nProof.The continuity was proved in [17]. To show the estimate (A.13), we use Theorem 1.3 in\n[17], to get\n\f\f\f\f\fA+\n2;c;\u000b\u0000cp\n\u0019(1 +\u000b)p\n2\f\f\f\f\f\u0014c2\u0019\n4+c2\u0019\n\u000bp\n2 \n1 +c2\u0019\n8+cp\n\u0019(1 +\u000b)\n2p\n2!\n+\u0012c2\u0019\n2p\n2\u000b\u00132\n:\nSince\u000b2(0;1], we have for any c2(0;1],\n\u0019\n4+\u0019\n\u000bp\n2 \n1 +c2\u0019\n8+cp\n\u0019(1 +\u000b)\n2p\n2!\n+c2\u00192\n8\u000b2\u00141\n\u000b2\u0012\u0019\n4+\u0019p\n2\u0012\n1 +\u0019\n8+p\u0019\n2\u0013\n+\u00192\n8\u0013\n\u00148\n\u000b2:\nWe deduce that for all \u000b;c2(0;1],\nA+\n2;c;\u000b\u0015cp\n\u0019(1 +\u000b)p\n2\u00008c2\n\u000b2\u0015cp\u0019p\n2\u00008c2\n\u000b2:\nIn particular A+\n2;c;\u000b\u00150ifc\u0014\u000b2p\u0019=(8p\n2). Thus\n(A+\n2;c;\u000b)2\u0015c2\u0019\n2\u000016c3p\u0019p\n2\u000b2+64c4\n\u000b4\u0015c2\u0019\n2\u000016c3p\u0019\n\u000b2;\nso that\ncos(#c;\u000b) = 1\u00002((A+\n2;c;\u000b)2+ (A+\n3;c;\u000b)2)\u00141\u0000c2\u0019+32c3p\u0019\n\u000b2;\nwhich implies (A.13).\nThe following lemma is a slightly refinement of Theorem 1.4 in [17].\nLemma A.8 ([17]).Letc >0,\u000b2[0;1]andA+\nc;\u000bbe the unit vector given in Theorem A.5.\nThenA+\nc;\u000bis a continuous function of \u000bin[0;1]and\njA+\nc;\u000b\u0000A+\nc;1j\u0014h(c)p\n1\u0000\u000b;for all\u000b2[1=2;1]; (A.14)\nwhereh:R+!R+is a strictly increasing function satisfying\nlim\ns!1h(s) =1:\n37Proof.Inviewof[17, Theorem1.4], weonlyneedtoprovethattheconstant C(c)inthestatement\nof the Theorem 1.4 (notice that c0in [17] corresponds to cin our notation) is polynomial in c\nwith nonnegative coefficients. Looking at the proof of [17, Theorem 1.4], we see that the constant\nC(c)behaves like the constant in inequality (3.108) in [17]. In view of (3.17), the estimate (3.23)\nin [17] can be written as\njf(s)j\u0014p\n2andjf0(s)j\u0014c\n2e\u0000\u000bs2=4;\nand then (3.18) can be recast as\njgj\u0014 \nc\n4+c2p\n2\n8!\u0012s\n\fe\u0000\u000bs2=4+s2e\u0000\u000bs2=2\u0013\n:\nThen, it can be easily checked that the function his a polynomial with nonnegative coefficients.\nAcknowledgments. A.deLairewaspartiallysupportedbytheLabexCEMPI(ANR-11-LABX-\n0007-01) and the MathAmSud program. S. Gutierrez was partially supported by the EPSRC\ngrant EP/J01155X/1 and the ERCEA Advanced Grant 2014 669689 - HADE.\nReferences\n[1] M.AbramowitzandI.A.Stegun. Handbook of mathematical functions with formulas, graphs,\nand mathematical tables , volume 55 of National Bureau of Standards Applied Mathematics\nSeries. For sale by the Superintendent of Documents, U.S. Government Printing Office,\nWashington, D.C., 1964.\n[2] F.AlougesandA.Soyeur. OnglobalweaksolutionsforLandau-Lifshitzequations: existence\nand nonuniqueness. 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Springer Berlin Heidelberg, 2012.\n40"    },    {        "title": "1806.04782v3.Dynamical_and_current_induced_Dzyaloshinskii_Moriya_interaction__Role_for_damping__gyromagnetism__and_current_induced_torques_in_noncollinear_magnets.pdf",        "content": "arXiv:1806.04782v3  [cond-mat.other]  9 Dec 2020Dynamical and current-induced Dzyaloshinskii-Moriya int eraction: Role for damping,\ngyromagnetism, and current-induced torques in noncolline ar magnets\nFrank Freimuth1,2,∗Stefan Bl¨ ugel1, and Yuriy Mokrousov1,2\n1Peter Gr¨ unberg Institut and Institute for Advanced Simula tion,\nForschungszentrum J¨ ulich and JARA, 52425 J¨ ulich, German y and\n2Institute of Physics, Johannes Gutenberg University Mainz , 55099 Mainz, Germany\nBoth applied electric currents and magnetization dynamics modify the Dzyaloshinskii-Moriya in-\nteraction (DMI), which we call current-induced DMI (CIDMI) and dynamical DMI (DDMI), respec-\ntively. We report a theory of CIDMI and DDMI. The inverse of CI DMI consists in charge pumping\nbyatime-dependentgradient ofmagnetization ∂2M(r,t)/∂r∂t, while theinverseofDDMIdescribes\nthe torque generated by ∂2M(r,t)/∂r∂t. In noncollinear magnets CIDMI and DDMI depend on\nthe local magnetization direction. The resulting spatial g radients correspond to torques that need\nto be included into the theories of Gilbert damping, gyromag netism, and current-induced torques\n(CITs) in order to satisfy the Onsager reciprocity relation s. CIDMI is related to the modification of\norbital magnetism induced by magnetization dynamics, whic h we call dynamical orbital magnetism\n(DOM), and spatial gradients of DOM contribute to charge pum ping. We present applications of\nthis formalism to the CITs and to the torque-torque correlat ion in textured Rashba ferromagnets.\nI. INTRODUCTION\nSince the Dzyaloshinskii-Moriya interaction (DMI)\ncontrols the magnetic texture of domain walls and\nskyrmions, methods to tune this chiral interaction by\nexternal means have exciting prospects. Application of\ngatevoltage[1–3]orlaserpulses[4]arepromisingwaysto\nmodify DMI. Additionally, theory predicts that in mag-\nnetic trilayer structures the DMI in the top magnetic\nlayer can be controlled by the magnetization direction\nin the bottom magnetic layer [5]. Moreover, methods to\ngeneratespin currentsmaybe usedto induceDMI, which\nis predicted by the relations between the two [6, 7]. Re-\ncent experiments show that also electric currents modify\nDMI in metallic magnets, which leads to large changes in\nthe domain-wallvelocity [8, 9]. However,a rigoroustheo-\nretical formalism for the investigation of current-induced\nDMI (CIDMI) in metallic magnets has been lacking so\nfar, and the development of such a formalism is one goal\nof this paper.\nRecently, a Berry phase theory of DMI [6, 10, 11]\nhas been developed, which formally resembles the mod-\nern theory of orbital magnetization [12–14]. Orbital\nmagnetism is modified by the application of an electric\nfield, which is known as the orbital magnetoelectric re-\nsponse [15]. In the case of insulators it is straightfor-\nward to derive the expressions for the magnetoelectric\nresponse directly. However, in metals it is much easier\nto derive expressions instead for the inverse of the mag-\nnetoelectric response, i.e., for the generation of electric\ncurrents by time-dependent magnetic fields [16]. The in-\nverse current-induced DMI (ICIDMI) consists in charge\npumping by time-dependent gradients of magnetization.\nDue to the analogies between orbital magnetism and the\nBerryphasetheoryofDMIonemayexpectthatinmetals\nit is convenient to obtain expressions for ICIDMI, which\ncanthen be used todescribe the CIDMI byexploitingthereciprocity between CIDMI and ICIDMI. We will show\nin this paper that this is indeed the case.\nIn noncentrosymmetric ferromagnets spin-orbit inter-\naction (SOI) generates torques on the magnetization –\nthe so-called spin-orbit torques (SOTs) – when an elec-\ntric current is applied [17]. The Berry phase theory of\nDMI [6, 10, 11] establishes a relation to SOTs. The for-\nmal analogies between orbital magnetism and DMI have\nbeen shown to be a very useful guiding principle in the\ndevelopment of the theory of SOTs driven by heat cur-\nrents [18]. In particular, it is fruitful to consider the DMI\ncoefficients as a spiralization, which is formally analo-\ngous to magnetization. In the theory of thermoelectric\neffects in magnetic systems the curl of magnetization de-\nscribes a bound current, which cannot be measured in\ntransport experiments and needs to be subtracted from\nthe Kubo linear response in order to obtain the mea-\nsurable current [19–21]. Similarly, in the theory of the\nthermal spin-orbit torque spatial gradients of the DMI\nspiralization, which result from the temperature gradi-\nent together with the temperature dependence of DMI,\nneed to be subtracted in order to obtain the measurable\ntorqueandto satisfyaMott-likerelation[10, 18]. In non-\ncollinear magnets the question arises whether gradients\nof the spiralization that are due to the magnetic texture\ncorrespond to torques like those from thermal gradients.\nWe will show that indeed the spatial gradients of CIDMI\nneed to be included into the theory of current-induced\ntorques (CITs) in noncollinear magnets in order to sat-\nisfy the Onsager reciprocity relations [22].\nWhen the system is driven out of equilibrium by mag-\nnetization dynamics rather than electric current one may\nexpect DMI to be modified as well. The inverse effect of\nthis dynamical DMI (DDMI) consists in the generation\nof torques by time-dependent magnetization gradients.\nIn noncollinear magnets the DDMI spiralization varies\nin space. We will show that the resulting gradient cor-2\nresponds to a torque that needs to be considered in the\ntheory of Gilbert damping and gyromagnetism in non-\ncollinear magnets.\nThis paper is structured as follows. In section IIA\nwe give an overview of CIT in noncollinear magnets and\nintroduce the notation. In section IIB we describe the\nformalism used to calculate the response of electric cur-\nrent to time-dependent magnetization gradients. In sec-\ntion IIC we show that current-induced DMI (CIDMI)\nand electric current driven by time-dependent magneti-\nzation gradients are reciprocal effects. This allows us\nto obtain an expression for CIDMI based on the formal-\nism of section IIB. In section IID we discuss that time-\ndependent magnetization gradients generate additionally\ntorques on the magnetization and show that the inverse\neffect consists in the modification of DMI by magnetiza-\ntion dynamics, which we calldynamical DMI (DDMI). In\nsection IIE we demonstrate that magnetization dynam-\nics induces orbital magnetism, which we call dynamical\norbitalmagnetism (DOM) and showthat DOM is related\nto CIDMI. In section IIF we explain how the spatial gra-\ndients of CIDMI and DOM contribute to the direct and\nto the inverse CIT, respectively. In section IIG we dis-\ncuss how the spatial gradients of DDMI contribute to the\ntorque-torque correlation. In section IIH we complete\nthe formalism used to calculate the CIT in noncollinear\nmagnets by adding the chiral contribution of the torque-\nvelocity correlation. In section III we finalize the theory\nof the inverse CIT by adding the chiral contribution of\nthe velocity-torque correlation. In section IIJ we fin-\nish the computational formalism of gyromagnetism and\ndamping by adding the chiral contribution of the torque-\ntorque correlation and the response of the torque to the\ntime-dependent magnetization gradients. In section III\nwe discuss the symmetry properties of the response to\ntime-dependent magnetizationgradients. In section IVA\nwe present the results for the chiral contributions to the\ndirectand the inverseCITin the Rashbamodel andshow\nthat both the perturbation by the time-dependent mag-\nnetization gradient and the spatial gradients of CIDMI\nand DOM need to be included to ensure that they are\nreciprocal. In section IVB we present the results for the\nchiralcontributiontothe torque-torquecorrelationinthe\nRashba model and show that both the perturbation by\nthe time-dependent magnetization gradient and the spa-\ntialgradientsofDDMI need tobe included toensurethat\nit satisfies the Onsager symmetry relations. This paper\nends with a summary in section V.II. FORMALISM\nA. Direct and inverse current-induced torques in\nnoncollinear magnets\nEven in collinearmagnets the application of an electric\nfieldEgenerates a torque TCIT1on the magnetization\nwhen inversion symmetry is broken [17, 23]:\nTCIT1\ni=/summationdisplay\njtij(ˆM)Ej, (1)\nwheretij(ˆM) is the torkance tensor, which depends on\nthe magnetization direction ˆM. This torque is called\nspin-orbit torque (SOT), but we denote it here CIT1,\nbecause it is one contribution to the current-induced\ntorques (CITs) in noncollinear magnets. Inversely, mag-\nnetization dynamics pumps a charge current JICIT1ac-\ncording to [24]\nJICIT1\ni=/summationdisplay\njtji(−ˆM)ˆej·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\n,(2)\nwhereˆejis a unit vector that points into the j-th spa-\ntial direction. Generally, JICIT1can be explained by\nthe inverse spin-orbit torque [24] or the magnonic charge\npumping [25]. We denote it here by ICIT1, because it\nis one contribution to the inverse CIT in noncollinear\nmagnets. In the special case of magnetic bilayers one im-\nportantmechanism responsiblefor JICIT1arisesfrom the\ncombination of spin pumping and the inverse spin Hall\neffect [26, 27].\nIn noncollinear magnets there is a second contribution\nto the CIT, which is proportional to the spatial deriva-\ntives of magnetization [28]:\nTCIT2\ni=/summationdisplay\njklχCIT2\nijklEjˆek·/bracketleftBigg\nˆM×∂ˆM\n∂rl/bracketrightBigg\n.(3)\nThe description of noncollinearity by the derivatives\n∂ˆM/∂rlisonlyapplicablewhenthe magnetizationdirec-\ntion changes slowly in space like in magnetic skyrmions\nwith large radius and in wide magnetic domain walls. In\norder to treat noncollinear magnets such as Mn 3Sn [29],\nwhere the magnetization direction varies strongly on the\nscale of one unit cell, Eq. (3) needs to be modified, which\nis beyond the scope of the present paper. The adia-\nbatic and the non-adiabatic [30] spin transfer torques\nare two important contributions to χCIT2\nijkl, but the in-\nterplay between broken inversion symmetry, SOI, and\nnoncollinearity can lead to a large number of additional\nmechanisms [22, 31]. Similarly, the current pumped\nby magnetization dynamics contains a contribution that\nis proportional to the spatial derivatives of magnetiza-3\ntion [22, 32, 33]:\nJICIT2\ni=/summationdisplay\njklχICIT2\nijklˆej·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\nˆek·/bracketleftBigg\nˆM×∂ˆM\n∂rl/bracketrightBigg\n.\n(4)\nTCIT2\niandJICIT2\nican be considered as chiral contribu-\ntionsto the CIT and to the ICIT, respectively, because\nthey distinguish between left- and right-handed spin spi-\nrals. Due to the reciprocity between direct and inverse\nCIT [22, 24] the coefficients χICIT2\nijklandχCIT2\njiklare related\naccording to\nχICIT2\nijkl(ˆM) =χCIT2\njikl(−ˆM). (5)\nB. Response of electric current to time-dependent\nmagnetization gradients\nIn order to compute JICIT2based on the Kubo lin-\near response formalism it is necessary to split it into\ntwo contributions, JICIT2aandJICIT2b. While JICIT2a\nis obtained as linear response to the perturbation by\natime-dependent magnetization gradient in a collinear\nferromagnet, JICIT2bis obtained as linear response to\nthe perturbation by magnetization dynamics in a non-\ncollinear ferromagnet. Therefore, as will become clear\nbelow,JICIT2acan be expressed by a correlation func-\ntion of two operators, because it describes the response\nof the current to a time-dependent magnetization gradi-\nent: A time-dependent magnetization gradient is a single\nperturbation, which is described by a single perturbing\noperator. In contrast, JICIT2binvolves the correlation\nof three operators, because it describes the response of\nthe current to magnetization dynamics in the presence of\nperturbation by noncollinearity. These are twoperturba-\ntions: One perturbation by the magnetization dynamics,\nandasecondperturbationtodescribethenoncollinearity.\nIn the Kubo formalism the expressions for the response\nonethe onehand toatime-dependent magnetizationgra-\ndient, which is described by a single perturbing operator,\nand the response on the other hand to a time-dependent\nmagnetization in the presence of a magnetization gradi-\nent, which is described by two perturbing operators, are\ndifferent. Therefore, we split JICIT2into these two con-\ntributions, which we call JICIT2aandJICIT2b. In the\nremainder of this section we discuss the calculation of\nthe contribution JICIT2a. The contribution JICIT2bis\ndiscussed in section III below.\nJICIT2ais determined by the second derivative of mag-\nnetization with respect to time and space variables and\ncan be written as\nJICIT2a\ni=/summationdisplay\njkχICIT2a\nijk∂2ˆMj\n∂rk∂t. (6)\nAnonzerosecondderivative∂2ˆMj\n∂rk∂tis what we referto asa\ntime-dependent magnetization gradient . Wewillshowbe-low that in special cases∂2ˆMj\n∂rk∂tcan be expressed in terms\nof the products∂ˆMl\n∂rk∂ˆMl\n∂t, which will allow us to rewrite\nJICIT2a\niin the form of Eq. (4) in the cases relevant for\nthe chiral ICIT. However, as will become clear below,\nEq. (6) is the most general expression for the response\nto time-dependent magnetization gradients, and it can-\nnot generally be rewritten in the form of Eq. (4): This\nis only possible when it describes a contribution to the\nchiral ICIT.\nJICIT2aoccurs in two different situations, which need\nto be distinguished. In one case the magnetization gra-\ndient varies in time like sin( ωt) everywhere in space. An\nexample is\nˆM(r,t) =\nηsin(q·r)sin(ωt)\n0\n1\n, (7)\nwhereηis the amplitude and the derivatives at t= 0 and\nr= 0 are\n∂ˆM(r,t)\n∂ri/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=t=0=∂ˆM(r,t)\n∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=t=0= 0 (8)\nand\n∂2ˆM(r,t)\n∂ri∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=t=0=\nηqiω\n0\n0\n. (9)\nIn the other case the magnetic texture varies like a\npropagating wave, i.e., proportional to sin( q·r−ωt). An\nexample is given by\nˆM(r,t) =\nηsin(q·r−ωt)\n0\n1−η2\n2sin2(q·r−ωt)\n,(10)\nwhere the derivatives at t= 0 and r= 0 are\n∂ˆM(r,t)\n∂ri/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=t=0=\nηqi\n0\n0\n, (11)\n∂ˆM(r,t)\n∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=t=0=\n−ηω\n0\n0\n (12)\nand\n∂2ˆM(r,t)\n∂ri∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=t=0=\n0\n0\nη2qiω\n. (13)\nIn the latter example, Eq. (10), the second derivative,\nEq. (13), is along the magnetization ˆM(r= 0,t= 0),\nwhile in the former example, Eq. (7), the second deriva-\ntive, Eq. (9), is perpendicular to the magnetization when\nr= 0 andt= 0.4\nWe assume that the Hamiltonian is given by\nH(r,t) =−/planckover2pi12\n2me∆+V(r)+µBˆM(r,t)·σΩxc(r)+\n+1\n2ec2µBσ·[∇V(r)×v],\n(14)\nwhere the first term describes the kinetic energy, the sec-\nond term is a scalar potential, Ωxc(r) in the third term is\nthe exchange field, and the last term describes the spin-\norbit interaction. Around t= 0 and r= 0 we can de-\ncompose the Hamiltonian as H(r,t) =H0+δH(r,t),\nwhereH0is obtained from H(r,t) by replacing ˆM(r,t)\nbyˆM(r= 0,t= 0) and\nδH(r,t) =∂H0\n∂ˆMxηsin(q·r)sin(ωt)\n=µBΩxc(r)σxηsin(q·r)sin(ωt)(15)\nin the case of the first example, Eq. (7). In the case of\nthe second example, Eq. (10),\nδH(r,t)≃∂H\n∂ˆMxηsin(q·r−ωt)\n+∂H\n∂ˆMzη2sin(q·r)sin(ωt),(16)\nwhereforsmall randtonlythesecondterm ontheright-\nhand side contributes to∂2H(r,t)\n∂rk∂t. We consider here only\nthe time-dependence of the exchange field direction and\nignore the time-dependence of the exchange field mag-\nnitude Ωxc(r) that is induced by the time-dependence\nof the exchange field direction. While the variation of\nthe exchange field magnitude drives currents and torques\nas well, as shown in Ref. [34], the variation of the ex-\nchange field magnitude is a small response and therefore\nthese secondary responses are suppressed in magnitude\nwhen compared to the direct primary responses of the\ncurrent and torque to the variation in the exchange field\ndirection. We will use the perturbations Eq. (15) and\nEq. (16) in order to compute the response of current and\ntorque within the Kubo response formalism. An alterna-\ntive approach for the calculation of the response to time-\ndependent fields is variational linear-response, which has\nbeen applied to the spin susceptibility by Savrasov [35].\nThe perturbation by the time-dependent gradient can\nbe written as\nδH=∂H\n∂ˆM·∂2ˆM\n∂ri∂tsin(qiri)\nqisin(ωt)\nω,(17)\nwhich turns into Eq. (15) when Eq. (9) is inserted. When\nEq. (13) is inserted it turns into the second term in\nEq. (16).\nIn Appendix A we derive the linear response to pertur-\nbations of the type of Eq. (17) and show that the corre-\nsponding coefficient χICIT2a\nijkin Eq. (6) can be expressedas\nχICIT2a\nijk=ie\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nviRvkRROjR+viRRvkROjR+\n−viRROjRvkR−viRvkROjAA\n+viROjAvkAA+viROjAAvkA\n−viRvkRROjA−viRRvkROjA\n+viRROjAvkA+viAvkAOjAA\n−viAOjAvkAA−viAOjAAvkA/bracketrightBig\n,(18)\nwhereR=GR\nk(E) andA=GA\nk(E) are shorthands for the\nretarded and advanced Green’s functions, respectively,\nandOj=∂H/∂ˆMj.e >0 is the positive elementary\ncharge.\nIn the case of the perturbation of the type Eq. (7) the\nsecond derivative∂2ˆM\n∂ri∂tis perpendicular to M. In this\ncase it is convenient to rewrite Eq. (6) as\nJICIT2a\ni=/summationdisplay\njkχICIDMI\nijkˆej·/bracketleftBigg\nˆM×∂2ˆM\n∂rk∂t/bracketrightBigg\n,(19)\nwhere the coefficients χICIDMI\nijkare given by\nχICIDMI\nijk=ie\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nviRvkRRTjR+viRRvkRTjR+\n−viRRTjRvkR−viRvkRTjAA\n+viRTjAvkAA+viRTjAAvkA\n−viRvkRRTjA−viRRvkRTjA\n+viRRTjAvkA+viAvkATjAA\n−viATjAvkAA−viATjAAvkA/bracketrightBig\n,(20)\nand\nT=ˆM×∂H\n∂ˆM(21)\nis the torque operator. In Sec. IIC we will explain that\nχICIDMI\nijkdescribes the inverse of current-induced DMI\n(ICIDMI).\nIn the case of the perturbation of the type of Eq. (10)\nthe second derivative∂2ˆMj\n∂rk∂tmay be rewritten as product\nof the first derivatives∂ˆMl\n∂tand∂ˆMl\n∂rk. This may be seen5\nas follows:\n∂H\n∂ˆM·∂2ˆM\n∂ri∂t=∂2H\n∂t∂ri=\n=∂\n∂t/bracketleftBigg/parenleftbigg\nˆM×∂H\n∂ˆM/parenrightbigg\n·/parenleftBigg\nˆM×∂ˆM\n∂ri/parenrightBigg/bracketrightBigg\n=\n=/bracketleftBigg/parenleftBigg\n∂ˆM\n∂t×∂H\n∂ˆM/parenrightBigg\n·/parenleftBigg\nˆM×∂ˆM\n∂ri/parenrightBigg/bracketrightBigg\n=\n=/bracketleftBigg/parenleftBigg/parenleftBigg\nˆM×∂ˆM\n∂t/parenrightBigg\n׈M/parenrightBigg\n×∂H\n∂ˆM/bracketrightBigg\n·/bracketleftBigg\nˆM×∂ˆM\n∂ri/bracketrightBigg\n=\n=−/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\n·/bracketleftBigg\nˆM×∂ˆM\n∂ri/bracketrightBigg/bracketleftbigg\nˆM·∂H\n∂ˆM/bracketrightbigg\n=\n=−∂ˆM\n∂t·∂ˆM\n∂ri/bracketleftbigg\nˆM·∂H\n∂ˆM/bracketrightbigg\n.\n(22)\nThis expression is indeed satisfied by Eq. (11), Eq. (12)\nand Eq. (13):\n∂ˆM\n∂ri·∂ˆM\n∂t=−∂2ˆM\n∂ri∂t·ˆM (23)\natr= 0,t= 0. Consequently, Eq. (6) can be rewritten\nas\nJICIT2a\ni=/summationdisplay\njkχICIT2a\nijk∂2ˆMj\n∂rk∂t=\n=−/summationdisplay\njklχICIT2a\nijk∂ˆMl\n∂rk∂ˆMl\n∂t[1−δjl]\n=/summationdisplay\njklχICIT2a\nijklˆej·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\nˆek·/bracketleftBigg\nˆM×∂ˆM\n∂rl/bracketrightBigg\n,\n(24)\nwhere\nχICIT2a\nijkl=−/summationdisplay\nmχICIT2a\niml[1−δjm]δjk.(25)\nThus, Eq. (24) and Eq. (25) can be used to express\nJICIT2a\niin the form of Eq. (4).\nC. Direct and inverse CIDMI\nEq. (20) describes the response of the electric current\nto time-dependent magnetization gradients of the type\nEq. (15). The reciprocal process consists in the current-\ninduced modification of DMI. This can be shown by ex-\npressing the DMI coefficients as [10]\nDij=1\nV/summationdisplay\nnf(Ekn)/integraldisplay\nd3r(ψkn(r))∗Dijψkn(r)\n=1\nV/summationdisplay\nnf(Ekn)/integraldisplay\nd3r(ψkn(r))∗Ti(r)rjψkn(r),\n(26)where we defined the DMI-operator Dij=Tirj. Using\nthe Kubo formalism the current-induced modification of\nDMI may be written as\nDCIDMI\nij=/summationdisplay\nkχCIDMI\nkijEk (27)\nwith\nχCIDMI\nkij=1\nVlim\nω→0/bracketleftBige\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tDij;vk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightBig\n,(28)\nwhere\n∝an}bracketle{t∝an}bracketle{tDij;vk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) =−i∞/integraldisplay\n0dteiωt∝an}bracketle{t[Dij(t),vk(0)]−∝an}bracketri}ht(29)\nis the Fourier transform of a retarded function and Vis\nthe volume of the unit cell.\nSince the position operator rin the DMI operator\nDij=Tirjis not compatible with Bloch periodic bound-\nary conditions, we do not use Eq. (28) for numerical\ncalculations of CIDMI. However, it is convenient to use\nEq. (28) in order to demonstrate the reciprocity between\ndirect and inverse CIDMI.\nInverseCIDMI (ICIDMI) describes the electric current\nthat responds to the perturbation by a time-dependent\nmagnetization gradient according to\nJICIDMI\nk=/summationdisplay\nijχICIDMI\nkijˆei·/bracketleftBigg\nˆM×∂2ˆM\n∂t∂rj/bracketrightBigg\n.(30)\nThe perturbation by a time-dependent magnetization\ngradient may be written as\nδH=−/summationdisplay\njm·∂2ˆM\n∂t∂rjrjΩxc(r)sin(ωt)\nω=\n=/summationdisplay\njT·/bracketleftBigg\nˆM×∂2ˆM\n∂t∂rj/bracketrightBigg\nrjsin(ωt)\nω\n=/summationdisplay\nijDijˆei·/bracketleftBigg\nˆM×∂2ˆM\n∂t∂rj/bracketrightBigg\nsin(ωt)\nω.(31)\nConsequently, the coefficient χICIDMI\nkijis given by\nχICIDMI\nkij=1\nVlim\nω→0/bracketleftBige\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tvk;Dij∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightBig\n.(32)\nUsing\n∝an}bracketle{t∝an}bracketle{tDij;vk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω,ˆM) =−∝an}bracketle{t∝an}bracketle{tvk;Dij∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω,−ˆM) (33)\nwe find that CIDMI and ICIDMI are related through the\nequations\nχCIDMI\nkij(ˆM) =−χICIDMI\nkij(−ˆM). (34)\nIn order to calculate CIDMI we use Eq. (20) for ICIDMI\nand then use Eq. (34) to obtain CIDMI.6\nThe perturbation Eq. (16) describes a different kind\nof time-dependent magnetization gradient, for which the\nreciprocaleffect consists in the modification of the expec-\ntation value ∝an}bracketle{tσ·ˆMrj∝an}bracketri}ht. However, while the modification\nof∝an}bracketle{tTirj∝an}bracketri}htby an applied current can be measured [8, 9]\nfrom the change of the DMI constant Dij, the quantity\n∝an}bracketle{tσ·ˆMrj∝an}bracketri}hthas not been considered so far in ferromagnets.\nIn noncollinear magnets the quantity ∝an}bracketle{tσrj∝an}bracketri}htcan be used\ntodefinespintoroidization[36]. Therefore,whiletheper-\nturbation of the type of Eq. (15) is related to CIDMI and\nICIDMI, which are both accessible experimentally [8, 9],\nin the case of the perturbation of the type of Eq. (16)\nwe expect that only the effect of driving current by the\ntime-dependent magnetization gradient is easily accessi-\nble experimentally, while its inverse effect is difficult to\nmeasure.\nD. Direct and inverse dynamical DMI\nNot only applied electric currents modify DMI, but\nalso magnetization dynamics, which we call dynamical\nDMI (DDMI). DDMI can be expressed as\nDDDMI\nij=/summationdisplay\nkχDDMI\nkijˆek·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\n.(35)\nIn Sec. IIG we will show that the spatial gradient of\nDDMI contributes to damping and gyromagnetism in\nnoncollinear magnets. The perturbation used to describe\nmagnetization dynamics is given by [24]\nδH=sin(ωt)\nω/parenleftBigg\nˆM×∂ˆM\n∂t/parenrightBigg\n·T.(36)\nConsequently, the coefficients χDDMI\nkijmay be written as\nχDDMI\nkij=−1\nVlim\nω→0/bracketleftbigg1\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tDij;Tk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightbigg\n.(37)\nSince the position operator in Dijis not compatible\nwith Bloch periodic boundary conditions, we do not use\nEq. (37) for numerical calculations of DDMI, but instead\nwe obtain it from its inverse effect, which consists in the\ngeneration of torques on the magnetization due to time-\ndependent magnetization gradients. These torques can\nbe written as\nTIDDMI\nk=/summationdisplay\nijχIDDMI\nkijˆei·/bracketleftBigg\nˆM×∂2ˆM\n∂t∂rj/bracketrightBigg\n,(38)\nwhere the coefficients χIDDMI\nkijare\nχIDDMI\nkij=1\nVlim\nω→0/bracketleftbigg1\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tTk;Dij∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightbigg\n,(39)becausethe perturbationby the time-dependent gradient\ncan be expressed in terms of Dijaccording to Eq. (31)\nand because the torque on the magnetizationis described\nby−T[23]. Consequently,DDMIandIDDMIarerelated\nby\nχDDMI\nkij(ˆM) =−χIDDMI\nkij(−ˆM). (40)\nFor numerical calculations of IDDMI we use\nχIDDMI\nijk=i\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nTiRvkRRTjR+TiRRvkRTjR+\n−TiRRTjRvkR−TiRvkRTjAA\n+TiRTjAvkAA+TiRTjAAvkA\n−TiRvkRRTjA−TiRRvkRTjA\n+TiRRTjAvkA+TiAvkATjAA\n−TiATjAvkAA−TiATjAAvkA/bracketrightBig\n,(41)\nwhichisderivedinAppendix A. InordertoobtainDDMI\nwecalculateIDDMIfromEq.(41)andusethereciprocity\nrelation Eq. (40).\nEq.(38)is validfortime-dependent magnetizationgra-\ndients that lead to perturbations of the type of Eq. (15).\nPerturbations of the second type, Eq. (16), will induce\ntorques on the magnetization as well. However, the in-\nverse effect is difficult to measure in that case, because it\ncorresponds to the modification of the expectation value\n∝an}bracketle{tσ·ˆMrj∝an}bracketri}htby magnetization dynamics. Therefore, while\nin the case of Eq. (15) both direct and inverse response\nare expected to be measurable and correspond to ID-\nDMI and DDMI, respectively, we expect that in the case\nof Eq. (16) only the direct effect, i.e., the response of the\ntorque to the perturbation, is easy to observe.\nE. Dynamical orbital magnetism (DOM)\nMagnetization dynamics does not only induce DMI,\nbut also orbital magnetism, which we call dynamical or-\nbital magnetism (DOM). It can be written as\nMDOM\nij=/summationdisplay\nkχDOM\nkijˆek·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\n,(42)\nwhere we introduced the notation\nMDOM\nij=e\nV∝an}bracketle{tvirj∝an}bracketri}htDOM, (43)\nwhich defines a generalized orbital magnetization, such\nthat\nMDOM\ni=1\n2/summationdisplay\njkǫijkMDOM\njk (44)7\ncorresponds to the usual definition of orbital magnetiza-\ntion. The coefficients χDOM\nkijare given by\nχDOM\nkij=−1\nVlim\nω→0/bracketleftBige\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tvirj;Tk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightBig\n,(45)\nbecause the perturbation by magnetization dynamics is\ndescribed by Eq. (36). We will discuss in Sec. IIF that\nthe spatial gradient of DOM contributes to the inverse\nCIT. Additionally, we will show below that DOM and\nCIDMI are related to each other.\nIn order to obtain an expression for DOM it is conve-\nnient to consider the inverse effect, i.e., the generation of\natorquebythe applicationofa time-dependent magnetic\nfieldB(t) that actsonly onthe orbitaldegreesoffreedom\nof the electrons and not on their spins. This torque can\nbe written as\nTIDOM\nk=1\n2/summationdisplay\nijlχIDOM\nkijǫijl∂Bl\n∂t, (46)\nwhere\nχIDOM\nkij=−1\nVlim\nω→0/bracketleftBige\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tTk;virj∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightBig\n,(47)\nbecause the perturbation by the time-dependent mag-\nnetic field is given by\nδH=−e\n2/summationdisplay\nijkǫijkvirj∂Bk\n∂tsin(ωt)\nω.(48)\nTherefore, thecoefficientsofDOMandIDOM arerelated\nby\nχDOM\nkij(ˆM) =−χIDOM\nkij(−ˆM). (49)\nIn Appendix A we show that the coefficient χIDOM\nijkcan\nbe expressed as\nχIDOM\nijk=−ie\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nTiRvkRRvjR+TiRRvkRvjR+\n−TiRRvjRvkR−TiRvkRvjAA\n+TiRvjAvkAA+TiRvjAAvkA\n−TiRvkRRvjA−TiRRvkRvjA\n+TiRRvjAvkA+TiAvkAvjAA\n−TiAvjAvkAA−TiAvjAAvkA/bracketrightBig\n.(50)\nEq. (50) and Eq. (20) differ only in the positions of\nthe two velocity operators and the torque operator be-\ntween the Green functions. As a consequence, IDOM\nare ICIDMI are related. In Table I and Table II we list\nthe relations between IDOM and ICIDMI for the Rashba\nmodel Eq. (83). We will explain in Sec. III that IDOM\nandICIDMI arezeroin the Rashbamodel when themag-\nnetization is along the zdirection. Therefore, we discussin Table I the case where the magnetization lies in the xz\nplane, and in Table II we discuss the case where the mag-\nnetization lies in the yzplane. According to Table I and\nTable II the relation between IDOM and ICIDMI is of\nthe formχIDOM\nijk=±χICIDMI\njik. This is expected, because\nthe indexiinχIDOM\nijkis connected to the torque operator,\nwhile the index jinχICIDMI\nijkis connected to the torque\noperator.\nTABLEI:Relations betweentheinverseofthemagnetization -\ndynamics induced orbital magnetism (IDOM) and inverse\ncurrent-inducedDMI (ICIDMI)in the 2d Rashbamodel when\nˆMlies in the zxplane. The components of χIDOM\nijk(Eq. (50))\nandχICIDMI\nijk(Eq. (20)) are denoted by the three indices ( ijk).\nICIDMI IDOM\n(211) (121)\n(121) (211)\n-(221) (221)\n(112) (112)\n-(212) (122)\n-(122) (212)\n(222) (222)\n(231) (321)\n(132) (312)\n-(232) (322)\nTABLE II: Relations between IDOM and ICIDMI in the 2d\nRashba model when ˆMlies in the yzplane.\nICIDMI IDOM\n(111) (111)\n-(211) (121)\n-(121) (211)\n(221) (221)\n-(112) (112)\n(212) (122)\n(122) (212)\n-(131) (311)\n(231) (321)\n(132) (312)\nF. Contributions from CIDMI and DOM to direct\nand inverse CIT\nIn electronic transport theory the continuity equation\ndetermines the current only up to a curl field [37]. The\ncurl of magnetization corresponds to a bound current\nthat cannot be measured in electron transport experi-\nments such that\nJ=JKubo−∇×M (51)\nhastobeusedtoextractthetransportcurrent Jfromthe\ncurrentJKuboobtained from the Kubo linear response.8\nThe subtraction of ∇×Mhas been shown to be impor-\ntant when calculating the thermoelectric response [37]\nand the anomalous Nernst effect [20]. Similarly, in the\ntheory of the thermal spin-orbit torque [10, 18] the gra-\ndients of the DMI spiralization have to be subtracted in\norder to obtain the measurable torque:\nTi=TKubo\ni−/summationdisplay\nj∂\n∂rjDij, (52)\nwhere the spatial derivative of the spiralization arises\nfrom its temperature dependence and the temperature\ngradient.\nSince CIDMI and DOM depend on the magnetization\ndirection, they vary spatially in noncollinear magnets.\nSimilar to Eq. (52) the spatial derivatives of the current-\ninduced spiralization need to be included into the theory\nof CIT. Additionally, the gradients of DOM correspond\ntocurrentsthatneedtobeconsideredinthetheoryofthe\ninverse CIT, similar to Eq. (51). In section IV we explic-\nitly show that Onsager reciprocity is violated if spatial\ngradients of DOM and CIDMI are not subtracted from\nthe Kubo response expressions. By trial-and-error we\nfind that the following subtractions are necessary to ob-\ntain response currents and torques that satisfy this fun-\ndamental symmetry:\nJICIT\ni=JKubo\ni−1\n2/summationdisplay\nj∂ˆM\n∂rj·∂MDOM\nij\n∂ˆM(53)\nand\nTCIT\ni=TKubo\ni−1\n2/summationdisplay\nj∂ˆM\n∂rj·∂DCIDMI\nij\n∂ˆM,(54)\nwhereJICIT\niis the current driven by magnetization dy-\nnamics, and TCIT\niis the current-induced torque.\nInterestingly, we find that also the diagonal elements\nMDOM\niiare nonzero. This shows that the generalized def-\ninition Eq. (43) is necessary, because the diagonal ele-\nmentsMDOM\niido not contribute in the usual definition\nofMiaccording to Eq. (44). These differences in the\nsymmetry properties between equilibrium and nonequi-\nlibrium orbital magnetism can be traced back to sym-\nmetry breaking by the perturbations. Also in the case\nof the spiralization tensor Dijthe nonequilibrium cor-\nrectionδDijhas different symmetry properties than the\nequilibrium part (see Sec. III).\nThe contribution of DOM to χICIT2\nijklcan be written as\nχICIT2c\nijkl=−1\n2ˆek·/bracketleftBigg\nˆM×∂χDOM\njil\n∂ˆM/bracketrightBigg\n(55)\nand the contribution of CIDMI to χCIT2\nijklis given by\nχCIT2b\nijkl=−1\n2ˆek·/bracketleftBigg\nˆM×∂χCIDMI\njil\n∂ˆM/bracketrightBigg\n.(56)G. Contributions from DDMI to gyromagnetism\nand damping\nThe response to magnetization dynamics that is de-\nscribed by the torque-torque correlation function con-\nsists of torques that are related to damping and gyro-\nmagnetism [24]. The chiral contribution to these torques\ncan be written as\nTTT2\ni=/summationdisplay\njklχTT2\nijklˆej·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\nˆek·/bracketleftBigg\nˆM×∂ˆM\n∂rl/bracketrightBigg\n,\n(57)\nwhere the coefficients χTT2\nijklsatisfy the Onsager relations\nχTT2\nijkl(ˆM) =χTT2\njikl(−ˆM). (58)\nSinceDDMIdependsonthemagnetizationdirection,it\nvaries spatially in noncollinear magnets and the resulting\ngradients of DDMI contribute to the damping and to the\ngyromagnetic ratio:\nTTT\ni=TKubo\ni−1\n2/summationdisplay\nj∂ˆM\n∂rj·∂DDDMI\nij\n∂ˆM.(59)\nThe resulting contribution of the spatial derivatives of\nDDMI to the coefficient χTT2\nijklis\nχTT2c\nijkl=−1\n2ˆek·/bracketleftBigg\nˆM×∂χDDMI\njil(ˆM)\n∂ˆM/bracketrightBigg\n.(60)\nH. Current-induced torque (CIT) in noncollinear\nmagnets\nThe chiral contribution to CIT consists of the spatial\ngradient of CIDMI, χCIT2b\nijklin Eq. (56), and the Kubo\nlinear response of the torque to the applied electric field\nin a noncollinear magnet, χCIT2a\nijkl:\nχCIT2\nijkl=χCIT2a\nijkl+χCIT2b\nijkl. (61)\nIn orderto determine χCIT2a\nijkl, we assume that the magne-\ntization direction ˆM(r) oscillates spatially as described\nby\nˆM(r) =\nηsin(q·r)\n0\n1\n1/radicalBig\n1+η2sin2(q·r),(62)\nwherewewilltakethelimit q→0attheendofthecalcu-\nlation. Since the spatial derivative of the magnetization\ndirection is\n∂ˆM(r)\n∂ri=\nηqicos(q·r)\n0\n0\n+O(η3),(63)9\nthe chiralcontributiontothe CIToscillatesspatiallypro-\nportional to cos( q·r). In order to extract this spatially\noscillating contribution we multiply with cos( q·r) and\nintegrate over the unit cell. The resulting expression for\nχCIT2a\nijklis\nχCIT2a\nijkl=−2e\nVηlim\nq→0lim\nω→0/bracketleftBigg\n1\nql/integraldisplay\ncos(qlrl)Im∝an}bracketle{t∝an}bracketle{tTi(r);vj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n/planckover2pi1ωd3rd3r′/bracketrightBigg\n,\n(64)\nwhereVis the volume of the unit cell, and\nthe retarded torque-velocity correlation function\n∝an}bracketle{t∝an}bracketle{tTi(r);vj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) needs to be evaluated in the\npresence of the perturbation\nδH=Tkηsin(q·r) (65)\ndue to the noncollinearity (the index kin Eq. (65) needs\nto match the index kinχCIT2a\nijkl).\nIn Appendix B we show that χCIT2a\nijklcan be written as\nχCIT2a\nijkl=−2e\n/planckover2pi1Im/bracketleftBig\nW(surf)\nijkl+W(sea)\nijkl/bracketrightBig\n,(66)\nwhere\nW(surf)\nijkl=1\n4π/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)Tr/bracketleftBigg\nTiGR\nk(E)vlGR\nk(E)vjGA\nk(E)TkGA\nk(E)\n+TiGR\nk(E)vjGA\nk(E)vlGA\nk(E)TkGA\nk(E)\n−TiGR\nk(E)vjGA\nk(E)TkGA\nk(E)vlGA\nk(E)\n+/planckover2pi1\nmeδjlTiGR\nk(E)GA\nk(E)TkGA\nk(E)/bracketrightBigg(67)\nis a Fermi surface term ( f′(E) =df(E)/dE) and\nW(sea)\nijkl=1\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)/bracketleftBigg\n−Tr[TiRvlRRvjRTkR]−Tr[TiRvlRTkRRvjR]\n−Tr[TiRRvlRvjRTkR]−Tr[TiRRvjRvlRTkR]\n+Tr[TiRRvjRTkRvlR]+Tr[TiRRTkRvjRvlR]\n+Tr[TiRRTkRvlRvjR]−Tr[TiRRvlRTkRvjR]\n−Tr[TiRvlRRTkRvjR]+Tr[TiRTkRRvjRvlR]\n+Tr[TiRTkRRvlRvjR]+Tr[TiRTkRvlRRvjR]\n−/planckover2pi1\nmeδjlTr[TiRRRTkR]−/planckover2pi1\nmeδjlTr[TiAAATkA]\n−/planckover2pi1\nmeδjlTr[TiAATkAA]/bracketrightBigg(68)\nis a Fermi sea term.I. Inverse CIT in noncollinear magnets\nThe chiral contribution JICIT2(see Eq. (4)) to the\ncharge pumping is described by the coefficients\nχICIT2\nijkl=χICIT2a\nijkl+χICIT2b\nijkl+χICIT2c\nijkl,(69)\nwhereχICIT2a\nijkldescribes the response to the time-\ndependentmagnetizationgradient(seeEq.(18),Eq.(25),\nand Eq. (24)) and χICIT2c\nijklresults from the spatial gra-\ndient of DOM (see Eq. (55)). χICIT2b\nijkldescribes the re-\nsponseto the perturbation bymagnetizationdynamics in\na noncollinear magnet. In order to derive an expression\nforχICIT2b\nijklwe assume that the magnetization oscillates\nspatially as described by Eq. (62). Since the correspond-\ning response oscillates spatially proportional to cos( q·r),\nwe multiply by cos( q·r) and integrate over the unit cell\nin order to extract χICIT2b\nijklfrom the retarded velocity-\ntorque correlation function ∝an}bracketle{t∝an}bracketle{tvi(r);Tj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω), which\nis evaluated in the presence of the perturbation Eq. (65).\nWe obtain\nχICIT2b\nijkl=2e\nVηlim\nq→0lim\nω→0/bracketleftBigg\n1\nql/integraldisplay\ncos(qlrl)Im∝an}bracketle{t∝an}bracketle{tvi(r);Tj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n/planckover2pi1ωd3rd3r′/bracketrightBigg\n,\n(70)\nwhich can be written as (see Appendix B)\nχICIT2b\nijkl=2e\n/planckover2pi1Im/bracketleftBig\nV(surf)\nijkl+V(sea)\nijkl/bracketrightBig\n,(71)\nwhere\nV(surf)\nijkl=1\n4π/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)Tr/bracketleftBig\nviGR\nk(E)vlGR\nk(E)TjGA\nk(E)TkGA\nk(E)\n+viGR\nk(E)TjGA\nk(E)vlGA\nk(E)TkGA\nk(E)\n−viGR\nk(E)TjGA\nk(E)TkGA\nk(E)vlGA\nk(E)/bracketrightBig(72)\nis the Fermi surface term and\nV(sea)\nijkl=1\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\n−Tr[viRvlRRTjRTkR]−Tr[viRvlRTkRRTjR]\n−Tr[viRRvlRTjRTkR]−Tr[viRRTjRvlRTkR]\n+Tr[viRRTjRTkRvlR]+Tr[viRRTkRTjRvlR]\n+Tr[viRRTkRvlRTjR]−Tr[viRRvlRTkRTjR]\n−Tr[viRvlRRTkRTjR]+Tr[viRTkRRTjRvlR]\n+Tr[viRTkRRvlRTjR]+Tr[viRTkRvlRRTjR]/bracketrightBig(73)\nis the Fermi sea term.\nIn Eq. (70) we use the Kubo formula to describe the\nresponse to magnetization dynamics combined with per-\nturbation theory to include the effect of noncollinearity.10\nThereby, the time-dependent perturbation and the per-\nturbation by the magnetization gradient are separated\nand perturbations of the form of Eq. (15) or Eq. (16)\nare not automatically included. For example the flat cy-\ncloidal spin spiral\nˆM(x,t) =\nsin(qx−ωt)\n0\ncos(qx−ωt)\n (74)\nmoving inxdirection with speed ω/qand the helical spin\nspiral\nˆM(y,t) =\nsin(qy−ωt)\n0\ncos(qy−ωt)\n (75)\nmovinginydirectionwith speed ω/qbehavelikeEq.(10)\nwhentandraresmall. Thus, these movingdomainwalls\ncorrespond to the perturbation of the type of Eq. (10)\nand the resulting contribution JICIT2afrom the time-\ndependent magnetization gradient is not described by\nEq. (70) and needs to be added, which we do by adding\nχICIT2a\nijklin Eq. (69).\nJ. Damping and gyromagnetism in noncollinear\nmagnets\nThe chiral contribution Eq. (57) to the torque-torque\ncorrelation function is expressed in terms of the coeffi-\ncient\nχTT\nijkl=χTT2a\nijkl+χTT2b\nijkl+χTT2c\nijkl, (76)\nwhereχTT2c\nijklresults from the spatial gradient of DDMI\n(see Eq. (60)), χTT2a\nijkldescribes the response to a time-\ndependent magnetization gradient in a collinear magnet,\nandχTT2b\nijkldescribes the response to magnetization dy-\nnamics in a noncollinear magnet.\nIn order to derive an expression for χTT2b\nijklwe as-\nsume that the magnetization oscillates spatially accord-\ning to Eq. (62). We multiply the retarded torque-torque\ncorrelation function ∝an}bracketle{t∝an}bracketle{tTi(r);Tj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) with cos(qlrl)\nand integrate over the unit cell in order to extract the\npart of the response that varies spatially proportional to\ncos(qlrl). We obtain:\nχTT2b\nijkl=2\nVηlim\nql→0lim\nω→0/bracketleftBigg\n1\nql/integraldisplay\ncos(qlrl)Im∝an}bracketle{t∝an}bracketle{tTi(r);Tj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n/planckover2pi1ωd3rd3r′/bracketrightBigg\n.\n(77)\nIn Appendix B we discuss how to evaluate Eq. (77) in\nfirst order perturbation theory with respect to the per-\nturbation Eq.(65) and showthat χTT2b\nijklcan be expressedas\nχTT2b\nijkl=2\n/planckover2pi1Im/bracketleftBig\nX(surf)\nijkl+X(sea)\nijkl/bracketrightBig\n,(78)\nwhere\nX(surf)\nijkl=1\n4π/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)Tr/bracketleftBigg\nTiGR\nk(E)vlGR\nk(E)TjGA\nk(E)TkGA\nk(E)\n+TiGR\nk(E)TjGA\nk(E)vlGA\nk(E)TkGA\nk(E)\n−TiGR\nk(E)TjGA\nk(E)TkGA\nk(E)vlGA\nk(E)/bracketrightBigg(79)\nis a Fermi surface term and\nX(sea)\nijkl=1\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBigg\n−(TiRvlRRTjRTkR)−(TiRvlRTkRRTjR)\n−(TiRRvlRTjRTkR)−(TiRRTjRvlRTkR)\n+(TiRRTjRTkRvlR)+(TiRRTkRTjRvlR)\n+(TiRRTkRvlRTjR)−(TiRRvlRTkRTjR)\n−(TiRvlRRTkRTjR)+(TiRTkRRTjRvlR)\n+(TiRTkRRvlRTjR)+(TiRTkRvlRRTjR)/bracketrightBigg\n(80)\nis a Fermi sea term.\nThe contribution χTT2a\nijklfrom the time-dependent gra-\ndients is given by\nχTT2a\nijkl=−/summationdisplay\nmχTT2a\niml[1−δjm]δjk,(81)\nwhere\nχTT2a\niml=i\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nTiRvlRROmR+TiRRvlROmR+\n−TiRROmRvlR−TiRvlROmAA\n+TiROmAvlAA+TiROmAAvlA\n−TiRvlRROmA−TiRRvlROmA\n+TiRROmAvlA+TiAvlAOmAA\n−TiAOmAvlAA−TiAOmAAvlA/bracketrightBig\n,(82)\nwithOm=∂H/∂ˆMm(see Appendix A).\nIII. SYMMETRY PROPERTIES\nIn this section we discuss the symmetry properties of\nCIDMI, DDMI and DOM in the case of the magnetic\nRashba model\nHk(r) =/planckover2pi12\n2mek2+α(k׈ez)·σ+∆V\n2σ·ˆM(r).(83)11\nAdditionally, we discuss the symmetry properties of the\ncurrents and torques induced by time-dependent magne-\ntization gradients of the form of Eq. (10).\nWe consider mirror reflection Mxzat thexzplane,\nmirror reflection Myzat theyzplane, and c2 rotation\naround the zaxis. When ∆ V= 0 these operations leave\nEq. (83) invariant, but when ∆ V∝ne}ationslash= 0 they modify the\nmagnetization direction ˆMin Eq. (83), as shown in Ta-\nble III. At the same time, these operations affect the\ntorqueTandthecurrent Jdrivenbythe time-dependent\nmagnetization gradients (see Table III). In Table IV and\nTable V we show how ˆM×∂ˆM/∂rkis affected by the\nsymmetry operations.\nAflat cycloidalspin spiralwith spinsrotatingin the xz\nplane is mapped by a c2 rotation around the zaxis onto\nthe same spin spiral. Similarly, a flat helical spin spiral\nwith spins rotating in the yzplane is mapped by a c2 ro-\ntationaroundthe zaxisontothesamespinspiral. There-\nfore, when ˆMpoints inzdirection, a c2 rotation around\nthezaxis does not change ˆM×∂ˆM/∂ri, but it flips the\nin-plane current Jand the in-plane components of the\ntorque,TxandTy. Consequently, ˆM×∂2ˆM/∂ri∂tdoes\nnot induce currents or torques, i.e., ICIDMI, CIDMI, ID-\nDMI and DDMI are zero, when ˆMpoints inzdirection.\nHowever, they become nonzero when the magnetization\nhas an in-plane component (see Fig. 1).\nSimilarly, IDOM vanishes when the magnetization\npoints inzdirection: In that case Eq. (83) is invariant\nunder the c2 rotation. A time-dependent magnetic field\nalongzdirection is invariant under the c2 rotation as\nwell. However, TxandTychange sign under the c2 rota-\ntion. Consequently, symmetryforbidsIDOM inthiscase.\nHowever, when the magnetization has an in-plane com-\nponent, IDOM and DOM become nonzero (see Fig. 2).\nThat time-dependent magnetization gradients of the\ntype of Eq. (7) do not induce in-plane currents and\ntorqueswhen ˆMpoints inzdirectioncan alsobe seendi-\nrectly from Eq. (7): The c2 rotation transforms q→ −q\nandMx→ −Mx. Since sin( q·r) is odd in r, Eq. (7) is in-\nvariantunder c2rotation, whilethe in-planecurrentsand\ntorques induced by time-dependent magnetization gradi-\nents change sign under c2 rotation. In contrast, Eq. (10)\nis not invariant under c2 rotation, because sin( q·r−ωt)\nis not odd in rfort>0. Consequently, time-dependent\nmagnetization gradients of the type of Eq. (10) induce\ncurrents and torques also when ˆMpoints locally into\nthezdirection. These currents and torques, which are\ndescribed by Eq. (24) and Eq. (82), respectively, need to\nbe added to the chiral ICIT and the chiral torque-torque\ncorrelation. While CIDMI, DDMI, and DOM are zero\nwhen the magnetization points in zdirection, their gra-\ndients are not (see Fig. 1 and Fig. 2). Therefore, the gra-\ndients of CIDMI, DOM, and DDMI contribute to CIT, to\nICIT and to the torque-torque correlation, respectively,\neven when ˆMpoints locally into the zdirection.TABLE III: Effect of mirror reflection Mxzat thexzplane,\nmirror reflection Myzat theyzplane, and c2 rotation around\nthezaxis. The magnetization Mand the torque Ttransform\nlike axial vectors, while the current Jtransforms like a polar\nvector.\nMxMyMzJxJyTxTyTz\nMxz−MxMy−MzJx−Jy−TxTy−Tz\nMyzMx−My−Mz−JxJyTx−Ty−Tz\nc2-Mx-MyMz-Jx−Jy−Tx−TyTz\nTABLE IV: Effect of symmetry operations on the magneti-\nzation gradients. Magnetization gradients are described b y\nthree indices ( ijk). The first index denotes the magnetiza-\ntion direction at r= 0. The third index denotes the di-\nrection along which the magnetization changes. The second\nindex denotes the direction of ∂ˆM/∂rkδrk. The direction of\nˆM×∂ˆM/∂rkis specified by the number below the indices\n(ijk).\n(1,2,1) (1,3,1) (2,1,1) (2,3,1) (3,1,1) (3,2,1)\n3-2 -3 1 2-1\nMxz(-1,2,1)(-1,-3,1) (2,-1,1) (2,-3,1) (-3,-1,1) (-3,2,1)\n-3 -2 3 -1 2 1\nMyz(1,2,1) (1,3,1)(-2,-1,1) (-2,3,1) (-3,-1,1) (-3,2,1)\n3-2 -3 -1 2 1\nc2(-1,2,1)(-1,-3,1) (-2,1,1) (-2,-3,1) (3,1,1) (3,2,1)\n-3 -2 3 1 2-1\n.\nTABLE V: Continuation of Table IV\n(1,2,2) (1,3,2) (2,1,2) (2,3,2) (3,1,2) (3,2,2)\n3 -2 -3 1 2-1\nMxz(-1,-2,2) (-1,3,2) (2,1,2) (2,3,2)(-3,1,2)(-3,-2,2)\n3 2-3 1-2 -1\nMyz(1,-2,2) (1,-3,2) (-2,1,2)(-2,-3,2) (-3,1,2)(-3,-2,2)\n-3 2 3 1-2 -1\nc2(-1,2,2) (-1,-3,2) (-2,1,2)(-2,-3,2) (3,1,2) (3,2,2)\n-3 -2 3 1 2-1\nA. Symmetry properties of ICIDMI and IDDMI\nInthefollowingwediscusshowTableIII,TableIV,and\nTable V can be used to analyze the symmetry of ICIDMI\nandIDDMI.AccordingtoEq.(19)thecoefficient χICIDMI\nijk\ndescribes the response of the current JICIT2a\nito the time-\ndependent magnetization gradient ˆej·[ˆM×∂2ˆM\n∂rk∂t]. Since\nˆM×∂2ˆM\n∂rk∂t=∂\n∂t[ˆM×∂ˆM\n∂rk] fortime-dependent magnetiza-\ntion gradients of the type Eq. (7) the symmetry proper-\nties ofχICIDMI\nijkfollow from the transformation behaviour\nofˆM×∂ˆM\n∂rkandJunder symmetry operations.\nWe consider the case with magnetization in xdirec-\ntion. The component χICIDMI\n132describes the current in x\ndirection induced by the time-dependence of a cycloidal\nmagnetizationgradientin ydirection(withspinsrotating12\nFIG. 1: ICIDMI in a noncollinear magnet. (a) Arrows illus-\ntrate the magnetization direction. (b) Arrows illustrate t he\ncurrentJyinduced by a time-dependent magnetization gra-\ndient, which is described by χICIDMI\n221. When ˆMpoints in z\ndirection, χICIDMI\n221andJyare zero. The sign of χICIDMI\n221and\nofJychanges with the sign of Mx.\nFIG. 2: DOM in a noncollinear magnet. (a) Arrows illustrate\nthe magnetization direction. (b) Arrows illustrate the orb ital\nmagnetization induced by magnetization dynamics (DOM).\nWhenˆMpoints in zdirection, DOM is zero. The sign of\nDOM changes with the sign of Mx.\nin thexyplane).Myzflips both ˆM×∂ˆM\n∂yandJx, but\nit preserves ˆM.Mzxpreserves ˆM×∂ˆM\n∂yandJx, but it\nflipsˆM. A c2 rotation around the zaxis flips ˆM×∂ˆM\n∂y,\nˆMandJx. Consequently, χICIDMI\n132(ˆM) is allowed by\nsymmetry and it is even in ˆM. The component χICIDMI\n122\ndescribes the current in xdirection induced by the time-\ndependence of a helical magnetization gradient in ydi-\nrection (with spins rotating in the xzplane).Myzflips\nˆM×∂ˆM\n∂yandJx, but it preserves ˆM.MzxflipsˆM×∂ˆM\n∂y\nandˆM, but it preserves Jx. A c2 rotation around the z\naxis flipsJxandˆM, but it preserves ˆM×∂ˆM\n∂y. Conse-\nquently,χICIDMI\n122is allowed by symmetry and it is odd in\nˆM. The component χICIDMI\n221describes the current in y\ndirection induced by the time-dependence of a cycloidal\nmagnetization gradient in xdirection (with spins rotat-\ning in thexzplane).Mzxpreserves ˆM×∂ˆM\n∂x, but it flipsJyandˆM.Myzpreserves ˆM,Jy, andˆM×∂ˆM\n∂x. The\nc2 rotation around the zaxis preserves ˆM×∂ˆM\n∂x, but\nit flipsˆMandJy. Consequently, χICIDMI\n221is allowed by\nsymmetry and it is odd in ˆM. The component χICIDMI\n231\ndescribes the current in ydirection induced by the time-\ndependence of a cycloidal magnetization gradient in xdi-\nrection (with spins rotating in the xyplane).Mzxflips\nˆM×∂ˆM\n∂x,ˆM, andJy.Myzpreserves ˆM×∂ˆM\n∂x,ˆMand\nJy. The c2 rotation around the zaxis flips ˆM×∂ˆM\n∂x,Jy,\nandˆM. Consequently, χICIDMI\n231is allowed by symmetry\nand it is even in ˆM.\nThese properties are summarized in Table VI. Due to\nthe relations between CIDMI and DOM (see Table I and\nTable II), they can be used for DOM as well. When the\nmagnetization lies at a general angle in the xzplane or in\ntheyzplaneseveraladditionalcomponentsofCIDMIand\nDOMarenonzero(seeTableIandTableII,respectively).\nTABLE VI: Allowed components of χICIDMI\nijkwhenˆMpoints\ninxdirection. + components are even in ˆM, while - compo-\nnents are odd in ˆM.\n132 122 221 231\n+ - - +\nSimilarly, one can analyze the symmetry of DDMI. Ta-\nble VII lists the components of DDMI, χDDMI\nijk, which are\nallowed by symmetry when ˆMpoints inxdirection.\nTABLEVII:Allowedcomponentsof χDDMI\nijkwhenˆMpointsin\nxdirection. +componentsareevenin ˆM, while -components\nare odd in ˆM.\n222 232 322 332\n- + + -\nB. Response to time-dependent magnetization\ngradients of the second type (Eq. (10))\nAccording to Eq. (13) the time-dependent magneti-\nzation gradient is along the magnetization. Therefore,\nin contrast to the discussion in section IIIA we can-\nnot use ˆM×∂2ˆM\n∂rk∂tin the symmetry analysis. Eq. (24)\nand Eq. (25) show that χICIT2a\nijjldescribes the response of\nJICIT2a\nitoˆej·/bracketleftBig\nˆM×∂ˆM\n∂t/bracketrightBig\nˆej·/bracketleftBig\nˆM×∂ˆM\n∂rl/bracketrightBig\nwhileχICIT2a\nijkl=\n0 forj∝ne}ationslash=k. According to Eq. (23) the symmetry prop-\nerties of/bracketleftBig\nˆM×∂ˆM\n∂t/bracketrightBig\n·/bracketleftBig\nˆM×∂ˆM\n∂rl/bracketrightBig\nagree to the symmetry\nproperties of ˆM·∂2ˆM\n∂rl∂t. Therefore, in order to under-\nstand the symmetry properties of χICIT2a\nijjlwe consider\nthe transformation of JandˆM·∂2ˆM\n∂rl∂tunder symmetry\noperations.\nWe consider the case where ˆMpoints inzdirection.\nχICIT2a\n1jj1describes the current driven in xdirection, when13\nthe magnetization varies in xdirection. MxzflipsˆM,\nbut preserves JxandˆM·∂2ˆM/(∂x∂t).MyzflipsˆM,Jx,\nandˆM·∂2ˆM/(∂x∂t). c2 rotation flips ˆM·∂2ˆM/(∂x∂t)\nandJx, but preserves ˆM. Consequently, χICIT2a\n1jj1is al-\nlowed by symmetry and it is even in ˆM.\nχICIT2a\n2jj1describes the current flowing in ydirection,\nwhen magnetization varies in xdirection. MxzflipsˆM\nandJy, but preserves ˆM·∂2ˆM/(∂x∂t).MyzflipsˆM,\nandˆM·∂2ˆM/(∂x∂t), but preserves Jy. c2 rotation\nflipsˆM·∂2ˆM/(∂x∂t) andJy, but preserves ˆM. Conse-\nquently,χICIT2a\n2jj1is allowed by symmetry and it is odd in\nˆM.\nSimilarly, one can show that χICIT2a\n1jj2is odd in ˆMand\nthatχICIT2a\n2jj2is even in ˆM.\nAnalogously, one can investigate the symmetry prop-\nerties ofχTT2a\nijjl. We find that χTT2a\n1jj1andχTT2a\n2jj2are odd\ninˆM, whileχTT2a\n2jj1andχTT2a\n1jj2are even in ˆM.\nIV. RESULTS\nIn the following sections we discuss the results for the\ndirect and inverse chiral CIT and for the chiral torque-\ntorque correlation in the two-dimensional (2d) Rashba\nmodel Eq. (83), and in the one-dimensional (1d) Rashba\nmodel [38]\nHkx(x) =/planckover2pi12\n2mek2\nx−αkxσy+∆V\n2σ·ˆM(x).(84)\nAdditionally, we discuss the contributions of the time-\ndependent magnetization gradients, and of DDMI, DOM\nand CIDMI to these effects.\nWhile vertex corrections to the chiral CIT and to\nthe chiral torque-torque correlation are important in the\nRashba model [38], the purpose of this work is to show\nthe importance ofthe contributionsfrom time-dependent\nmagnetization gradients, DDMI, DOM and CIDMI. We\ntherefore consider only the intrinsic contributions here,\ni.e., we set\nGR\nk(E) =/planckover2pi1[E −Hk+iΓ]−1, (85)\nwhere Γ is a constant broadening, and we leave the study\nof vertex corrections for future work.\nThe results shown in the following sections are ob-\ntained for the model parameters ∆ V= 1eV,α=2eV˚A,\nand Γ = 0 .1Ry = 1.361eV, when the magnetization\npoints inzdirection, i.e., ˆM=ˆez. The unit of χCIT2\nijkl\nis charge times length in the 1d case and charge in the\n2d case. Therefore, in the 1d case we discuss the chiral\ntorkance in units of ea0, wherea0is Bohr’s radius. In the\n2d case we discuss the chiral torkance in units of e. The\nunit ofχTT2\nijklis angular momentum in the 1d case and\nangular momentum per length in the 2d case. Therefore,\nwe discussχTT2\nijklin units of /planckover2pi1in the 1d case, and in units\nof/planckover2pi1/a0in the 2d case.-2 -1 0 1 2\nFermi energy [eV]-0.02-0.0100.010.020.030.040.05χijklCIT2 [ea0]2121\n1121\n2121 (gauge-field)\n1121 (gauge-field)\nFIG. 3: Chiral CIT in the 1d Rashba model for cycloidal gra-\ndients vs. Fermi energy. General perturbation theory (soli d\nlines) agrees to the gauge-field approach (dashed lines).\nA. Direct and inverse chiral CIT\nIn Fig. 3 we show the chiral CIT as a function of the\nFermi energyfor cycloidalmagnetization gradients in the\n1d Rashba model. The components χCIT2\n2121andχCIT2\n1121are\nlabelled by 2121 and 1121, respectively. The component\n2121ofCITdescribesthe non-adiabatictorque, while the\ncomponent 1121 describes the adiabatic STT (modified\nby SOI). In the one-dimensional Rashba model, the con-\ntributionsχCIT2b\n2121andχCIT2b\n1121(Eq. (56)) from the CIDMI\nare zero when ˆM=ˆez(not shown in the figure). For cy-\ncloidal spin spirals, it is possible to solve the 1d Rashba\nmodel by a gauge-field approach [38], which allows us to\ntest the perturbation theory, Eq. (66). For comparison\nwe show in Fig. 3 the results obtained from the gauge-\nfield approach, which agree to the perturbation theory,\nEq. (66). This demonstrates the validity of Eq. (66).\nIn Fig. 4 we show the chiral ICIT in the 1d Rashba\nmodel. The components χICIT2\n1221andχICIT2\n1121are labelled\nby 1221and 1121, respectively. The contribution χICIT2a\n1221\nfrom the time-dependent gradient is of the same order of\nmagnitude as the total χICIT2\n1221. Comparison of Fig. 3 and\nFig. 4 shows that CIT and ICIT satisfy the reciprocity\nrelationsEq. (5), that χCIT2\n1121is odd in ˆM, and thatχCIT2\n2121\nis even in ˆM, i.e.,χCIT2\n2121=χICIT2\n1221andχCIT2\n1121=−χICIT2\n1121.\nThe contribution χICIT2a\n1221from the time-dependent gradi-\nents is crucial to satisfy the reciprocity relations between\nχCIT2\n2121andχICIT2\n1221.\nIn Fig. 5 and Fig. 6 we show the CIT and the ICIT, re-\nspectively, for helical gradients in the 1d Rashba model.\nThe components χCIT2\n2111andχCIT2\n1111are labelled 2111 and\n1111, respectively, in Fig. 5, while χICIT2\n1211andχICIT2\n1111\nare labelled 1211 and 1111, respectively, in Fig. 6. The\ncontributions χCIT2b\n2111andχCIT2b\n1111from CIDMI are of the14\n-2 -1 0 1 2\nFermi energy [eV]-0.0200.020.04χijklICIT2 [ea0]1221 \n1121\nχ1221ICIT2a\nFIG. 4: Chiral ICIT in the 1d Rashba model for cycloidal\ngradients vs. Fermi energy. Dashed line: Contribution from\nthe time-dependent gradient.\nsame order of magnitude as the total χCIT2\n2111andχCIT2\n1111.\nSimilarly, the contributions χICIT2c\n1211andχICIT2c\n1111from\nDOM are of the same order of magnitude as the to-\ntalχICIT2\n1211andχICIT2\n1111. Additionally, the contribution\nχICIT2a\n1111from the time-dependent gradient is substantial.\nComparisonofFig.5andFig.6showsthatCITandICIT\nsatisfy the reciprocity relation Eq. (5), that χCIT2\n2111is odd\ninˆM, and thatχCIT2\n1111is even in ˆM, i.e.,χCIT2\n1111=χICIT2\n1111\nandχCIT2\n2111=−χICIT2\n1211. These reciprocity relations be-\ntween CIT and ICIT are only satisfied when CIDMI,\nDOM, and the response to time-dependent magnetiza-\ntion gradients are included. Additionally, the compar-\nison between Fig. 5 and Fig. 6 shows that the contri-\nbutions of CIDMI to CIT ( χCIT2b\n1111andχCIT2b\n2111) are re-\nlated to the contributions of DOM to ICIT ( χICIT2c\n1111and\nχICIT2c\n1211). These relations between DOM and ICIT are\nexpected from Table I.\nIn Fig. 7 and Fig. 8 we show the CIT and the ICIT,\nrespectively, for cycloidal gradients in the 2d Rashba\nmodel. In this case there are contributions from CIDMI\nand DOM in contrast to the 1d case with cycloidal gra-\ndients (Fig. 3). Comparison between Fig. 7 and Fig. 8\nshows that χCIT2\n1121andχCIT2\n2221are odd in ˆM, thatχCIT2\n1221\nandχCIT2\n2121are even in ˆM, and that CIT and ICIT sat-\nisfy the reciprocity relation Eq. (5) when the gradients\nof CIDMI and DOM are included, i.e., χCIT2\n1121=−χICIT2\n1121,\nχCIT2\n2221=−χICIT2\n2221,χCIT2\n1221=χICIT2\n2121, andχCIT2\n2121=χICIT2\n1221.\nχCIT2\n1121describesthe adiabatic STT with SOI, while χCIT2\n2121\ndescribes the non-adiabatic STT. Experimentally, it has\nbeen found that CITs occur also when the electric field\nis applied parallel to domain-walls (i.e., perpendicular to\ntheq-vector of spin spirals) [39]. In our calculations, the\ncomponents χCIT2\n2221andχCIT2\n1221describe such a case, where\nthe applied electric field points in ydirection, while the-2 -1 0 1 2\nFermi energy [eV]-0.04-0.0200.020.040.06χijklCIT2 [ea0]1111\n2111\nχ1111CIT2b\nχ2111CIT2b\nFIG. 5: Chiral CIT for helical gradients in the 1d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nCIDMI.\n-2 -1 0 1 2\nFermi energy [eV]-0.0200.020.040.06χijklICIT2 [ea0]1111\n1211\nχ1111ICIT2a\nχ1111ICIT2c\nχ1211ICIT2c\nFIG. 6: Chiral ICIT for helical gradients in the 1d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nDOM. Dashed-dotted line: Contribution from the time-\ndependent magnetization gradient.\nmagnetization direction varies with the xcoordinate.\nIn Fig. 9 and Fig. 10 we show the chiral CIT and\nICIT, respectively, for helical gradients in the 2d Rashba\nmodel. The component χCIT2\n2111describes the adiabatic\nSTT with SOI and the component χCIT2\n1111describes the\nnon-adiabatic STT. The components χCIT2\n2211andχCIT2\n1211\ndescribe the case when the applied electric field points\ninydirection, i.e., perpendicular to the direction along\nwhich the magnetization direction varies. Comparison\nbetween Fig. 9 and Fig. 10 shows that χCIT2\n1111andχCIT2\n2211\nare even in ˆM, thatχCIT2\n1211andχCIT2\n2111are odd in ˆMand\nthat CIT andICIT satisfythe reciprocityrelationEq.(5)\nwhenthegradientsofCIDMIandDOMareincluded, i.e.,15\n-2 -1 0 1 2\nFermi energy [eV]-0.00200.0020.0040.006χijklCIT2 [e]1121\n2221\n1221\n2121\nχ2221CIT2b\nχ1221CIT2b\nχ2121CIT2b\nFIG. 7: Chiral CIT for cycloidal gradients in the 2d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nCIDMI.\n-2 -1 0 1 2\nFermi energy [eV]-0.00200.0020.0040.006χijklICIT2 [e]1121\n1221\n2121\n2221\nχ2221ICIT2a\nχ1221ICIT2a\nχ2121ICIT2c\nχ1221ICIT2c\nχ2221ICIT2c\nFIG. 8: Chiral ICIT for cycloidal gradients in the 2d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nDOM. Dashed-dotted lines: Contributions from the time-\ndependent gradients.\nχCIT2\n1111=χICIT2\n1111,χCIT2\n2211=χICIT2\n2211,χCIT2\n1211=−χICIT2\n2111, and\nχCIT2\n2111=−χICIT2\n1211.\nB. Chiral torque-torque correlation\nIn Fig. 11 we show the chiral contribution to the\ntorque-torque correlation in the 1d Rashba model for\ncycloidal gradients. We compare the perturbation the-\nory Eq. (78) plus Eq. (82) to the gauge-field approach\nfrom Ref. [38]. This comparison shows that perturba-\ntion theory provides the correct answer only when the\ncontribution χTT2a\nijkl(Eq. (82)) from the time-dependent-2 -1 0 1 2\nFermi energy [eV]-0.00200.0020.0040.006χijklCIT2 [e]2211\n1111\n1211\n2111\nχ2111CIT2b\nχ1211CIT2b\nχ2211CIT2b\nχ1111CIT2b\nFIG. 9: Chiral CIT for helical gradients in the 2d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nCIDMI.\n-2 -1 0 1 2\nFermi energy [eV]-0.004-0.00200.0020.0040.006χijklICIT2 [e]1111\n1211\n2111\n2211\nχ1111ICIT2a\nχ2221ICIT2a\nχ1111ICIT2c\nχ2111ICIT2c\nχ1211ICIT2c\nχ2211ICIT2c\nFIG. 10: Chiral ICIT for helical gradients in the 2d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nDOM. Dashed-dotted lines: Contributions from the time-\ndependent gradient.\ngradients is taken into account. The contributions χTT2a\n1221\nandχTT2a\n2221fromthe time-dependent gradientsarecompa-\nrable in magnitude to the total values. In the 1d Rashba\nmodel the DDMI-contribution in Eq. (60) is zero for cy-\ncloidal gradients (not shown in the figure). The compo-\nnentsχTT2\n2121andχTT2\n1221describe the chiral gyromagnetism\nwhile the components χTT2\n1121andχTT2\n2221describe the chi-\nral damping [38, 40, 41]. The components χTT2\n2121and\nχTT2\n1221are odd in ˆMand they satisfy the Onsagerrelation\nEq. (58), i.e., χTT2\n2121=−χTT2\n1221.\nIn Fig. 12 we show the chiral contributions to the\ntorque-torque correlation in the 1d Rashba model for\nhelical gradients. In contrast to the cycloidal gradients16\n-2 -1 0 1 2\nFermi energy [eV]-0.00500.0050.01χijklTT2 [h_]2121\n1221\n2221\n1121\nχ1221TT2a\nχ2221TT2a\n2121 (gf)\n1221 (gf)\n1121 (gf)\n2221 (gf)\nFIG. 11: Chiral contribution to the torque-torque correla-\ntion for cycloidal gradients in the 1d Rashba model vs. Fermi\nenergy. Perturbation theory (solid lines) agrees to the gau ge-\nfield (gf) approach (dotted lines). Dashed lines: Contribut ion\nfrom the time-dependent gradient.\n(Fig. 11) there are contributions from the spatial gra-\ndients of DDMI (Eq. (60)) in this case. The Onsager\nrelation Eq. (58) for the components χTT2\n2111andχTT2\n1211is\nsatisfied only when these contributions from DDMI are\ntaken into account, which are of the same order of mag-\nnitude as the total values. The components χTT2\n2111and\nχTT2\n1211are even in ˆMand describe chiral damping, while\nthe components χTT2\n1111andχTT2\n2211are odd in ˆMand de-\nscribe chiral gyromagnetism. As a consequence of the\nOnsager relation Eq. (58) we obtain χTT2\n1111=χTT2\n2211= 0\nfor the total components: Eq. (58) shows that diagonal\ncomponents of the torque-torque correlation function are\nzero unless they are even in ˆM. However, χTT2a\n1111,χTT2c\n1111,\nandχTT2b\n1111=−χTT2a\n1111−χTT2c\n1111are individually nonzero.\nInterestingly, the off-diagonal components of the torque-\ntorquecorrelationdescribechiraldampingforhelicalgra-\ndients, while for cycloidal gradients the off-diagonal ele-\nments describe chiral gyromagnetism and the diagonal\nelements describe chiral damping.\nIn Fig. 13 we show the chiral contributions to the\ntorque-torque correlation in the 2d Rashba model for cy-\ncloidal gradients. In contrast to the 1d Rashba model\nwith cycloidal gradients (Fig. 11) the contributions from\nDDMIχTT2c\nijkl(Eq.(60))arenonzerointhiscase. Without\nthesecontributionsfromDDMI theOnsagerrelation(58)\nχTT2\n2121=−χTT2\n1221is violated. The DDMI contribution is\nof the same order of magnitude as the total values. The\ncomponents χTT2\n2121andχTT2\n1221are odd in ˆMand describe\nchiral gyromagnetism, while the components χTT2\n1121and\nχTT2\n2221are even in ˆMand describe chiral damping.\nIn Fig. 14 we show the chiral contributions to the\ntorque-torque correlation in the 2d Rashba model for he-\nlical gradients. The components χTT2\n1211andχTT2\n2111are even-2 -1 0 1 2\nFermi energy [eV]-0.00500.0050.01χijklTT2 [h_]1111\n2111\n1211\n2211\nχ1111TT2c\nχ2111TT2c\nχ1211TT2c\nχ2211TT2c\nχ1111TT2a\nχ2111TT2a\nFIG. 12: Chiral contribution to the torque-torque correla-\ntion for helical gradients in the 1d Rashba model vs. Fermi\nenergy. Dashed lines: Contributions from DDMI. Dashed-\ndotted lines: Contributions from the time-dependent gradi -\nents.\n-2 -1 0 1 2\nFermi energy [eV]-0.000500.00050.001χijklTT2 [h_/a0]1121\n2121\n1221\n2221\nχ1221TT2a\nχ2221TT2a\nχ2121TT2c\nχ1221TT2c\nFIG. 13: Chiral contribution to the torque-torque correla-\ntion for cycloidal gradients in the 2d Rashba model vs. Fermi\nenergy. Dashed lines: Contributions from DDMI. Dashed-\ndotted lines: Contributions from the time-dependent gradi -\nents.\ninˆMand describe chiral damping, while the compo-\nnentsχTT2\n1111andχTT2\n2211are odd in ˆMand describe chiral\ngyromagnetism. The Onsager relation Eq. (58) requires\nχTT2\n1111=χTT2\n2211= 0 andχTT2\n2111=χTT2\n1211. Without the\ncontributions from DDMI these Onsager relations are vi-\nolated.17\n-2 -1 0 1 2\nFermi energy [eV]-0.000500.00050.001χijklTT2 [h_ /a0]1111\n2111\n1211\n2211\nχ1111TT2a\nχ2111TT2a\nχ1111TT2c\nχ1211TT2c\nχ2211TT2c\nFIG. 14: Chiral contribution to the torque-torque correla-\ntion for helical gradients in the 2d Rashba model vs. Fermi\nenergy. Dashed lines: Contributions from DDMI. Dashed-\ndotted lines: Contributions from the time-dependent gradi -\nents.\nV. SUMMARY\nFinding ways to tune the Dzyaloshinskii-Moriya inter-\naction (DMI) by external means, such as an applied elec-\ntriccurrent,holdsmuchpromiseforapplicationsinwhich\nDMI determines the magnetic texture of domain walls or\nskyrmions. In order to derive an expression for current-\ninduced Dzyaloshinskii-Moriya interaction (CIDMI) we\nfirst identify its inverse effect: When magnetic textures\nvary as a function of time, electric currents are driven by\nvarious mechanisms, which can be distinguished accord-\ningtotheirdifferentdependenceonthetime-derivativeof\nmagnetization, ∂ˆM(r,t)/∂t, and on the spatial deriva-\ntive∂ˆM(r,t)/∂r: One group of effects is proportional\nto∂ˆM(r,t)/∂t, a second group of effects is propor-\ntional to the product ∂ˆM(r,t)/∂t ∂ˆM(r,t)/∂r, and\na third group is proportional to the second derivative\n∂2ˆM(r,t)/∂r∂t. We show that the response of the elec-\ntric current to the time-dependent magnetization gradi-\nent∂2ˆM(r,t)/∂r∂tcontais the inverse of CIDMI. We\nestablish the reciprocity relation between inverse and di-\nrectCIDMI and therebyobtainan expressionforCIDMI.\nWe find that CIDMI is related to the modification of\norbital magnetism induced by magnetization dynamics,\nwhich we call dynamical orbital magnetism (DOM). We\nshow that torques are generated by time-dependent gra-\ndients of magnetization as well. The inverse effect con-\nsists in the modification of DMI by magnetization dy-\nnamics, which we call dynamical DMI (DDMI).\nAdditionally, we develop a formalism to calculate the\nchiral contributions to the direct and inverse current-\ninduced torques (CITs) and to the torque-torque correla-tion in noncollinear magnets. We show that the response\nto time-dependent magnetization gradients contributes\nsubstantially to these effects and that the Onsager reci-\nprocityrelationsareviolated when it is not takeninto ac-\ncount. InnoncollinearmagnetsCIDMI,DDMIandDOM\ndepend on the local magnetization direction. We show\nthat the resulting spatial gradients of CIDMI, DDMI\nand DOM have to be subtracted from the CIT, from\nthe torque-torque correlation, and from the inverse CIT,\nrespectively.\nWe apply our formalism to study CITs and the torque-\ntorque correlation in textured Rashba ferromagnets. We\nfind that the contribution of CIDMI to the chiral CIT is\noftheorderofmagnitudeofthe totaleffect. Similarly, we\nfind that the contribution of DDMI to the chiral torque-\ntorque correlation is of the order of magnitude of the\ntotal effect.\nAcknowledgments\nWeacknowledgefinancialsupportfromLeibnizCollab-\norative Excellence project OptiSPIN −Optical Control\nofNanoscaleSpin Textures. Weacknowledgefundingun-\nder SPP 2137 “Skyrmionics” of the DFG. We gratefully\nacknowledge financial support from the European Re-\nsearch Council (ERC) under the European Union’s Hori-\nzon 2020 research and innovation program (Grant No.\n856538, project ”3D MAGiC”). The work was also sup-\nported by the Deutsche Forschungsgemeinschaft (DFG,\nGerman Research Foundation) −TRR 173 −268565370\n(project A11). We gratefully acknowledge the J¨ ulich\nSupercomputing Centre and RWTH Aachen University\nfor providing computational resources under project No.\njiff40.\nAppendix A: Response to time-dependent gradients\nIn this appendix we derive Eq. (18), Eq. (20), Eq. (41),\nand Eq. (82), which describe the response to time-\ndependent magnetization gradients, and Eq. (50), which\ndescribesthe responsetotime-dependentmagneticfields.\nWe consider perturbations of the form\nδH(r,t) =Bb1\nqωsin(q·r)sin(ωt).(A1)\nWhenweset B=∂H\n∂ˆMkandb=∂2ˆMk\n∂ri∂t, Eq.(A1)turnsinto\nEq. (17), while when we set B=−eviandb=1\n2ǫijk∂Bk\n∂t\nwe obtain Eq. (48). We need to derive an expression for\nthe response δA(r,t) of an observable Ato this pertur-\nbation, which varies in time like cos( ωt) and in space like\ncos(q·r), because∂2ˆM(r,t)\n∂ri∂t∝cos(q·r)cos(ωt). There-\nfore, weusethe Kubolinearresponseformalismtoobtain18\nthe coefficient χin\nδA(r,t) =χcos(q·r)cos(ωt), (A2)\nwhich is given by\nχ=i\n/planckover2pi1qωV/bracketleftBig\n∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n−∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htR(−/planckover2pi1ω)/bracketrightBig\n,(A3)\nwhere∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) is the retarded\nfunction at frequency ωandVis the volume of the unit\ncell.\nThe operator Bsin(q·r) can be written as\nBsin(q·r) =1\n2i/summationdisplay\nknm/bracketleftBig\nB(1)\nknmc†\nk+nck−m−B(2)\nknmc†\nk−nck+m/bracketrightBig\n,\n(A4)\nwherek+=k+q/2,k−=k−q/2,c†\nk+nis the cre-\nation operator of an electron in state |uk+n∝an}bracketri}ht,ck−mis the\nannihilation operator of an electron in state |uk−m∝an}bracketri}ht,\nB(1)\nknm=1\n2∝an}bracketle{tuk+n|[Bk++Bk−]|uk−m∝an}bracketri}ht(A5)\nand\nB(2)\nknm=1\n2∝an}bracketle{tuk−n|[Bk++Bk−]|uk+m∝an}bracketri}ht.(A6)\nSimilarly,\nAcos(q·r) =1\n2/summationdisplay\nknm/bracketleftBig\nA(1)\nknmc†\nk+nck−m+A(2)\nknmc†\nk−nck+m/bracketrightBig\n,\n(A7)\nwhere\nA(1)\nknm=1\n2∝an}bracketle{tuk+n|/bracketleftbig\nAk++Ak−/bracketrightbig\n|uk−m∝an}bracketri}ht(A8)\nand\nA(2)\nknm=1\n2∝an}bracketle{tuk−n|/bracketleftbig\nAk++Ak−/bracketrightbig\n|uk+m∝an}bracketri}ht.(A9)\nIt is convenient to obtain the retarded response func-\ntion in Eq. (A3) from the correspondingMatsubarafunc-\ntion in imaginary time τ\n1\nV∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htM(τ) =\n=1\n4i/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/bracketleftBig\nA(1)\nknmB(2)\nkn′m′Z(1)\nknmn′m′(τ)\n−A(2)\nknmB(1)\nkn′m′Z(2)\nknmn′m′(τ)/bracketrightBig\n,\n(A10)\nwhered= 1,2 or 3 is the dimension,\nZ(1)\nknmn′m′(τ) =∝an}bracketle{tTτc†\nk+n(τ)ck−m(τ)c†\nk−n′(0)ck+m′(0)∝an}bracketri}ht\n=−GM\nm′n(k+,−τ)GM\nmn′(k−,τ),\n(A11)Z(2)\nknmn′m′(τ) =∝an}bracketle{tTτc†\nk−n(τ)ck+m(τ)c†\nk+n′(0)ck−m′(0)∝an}bracketri}ht\n=−GM\nm′n(k−,−τ)GM\nmn′(k+,τ),\n(A12)\nand\nGM\nmn′(k+,τ) =−∝an}bracketle{tTτck+m(τ)c†\nk+n′(0)∝an}bracketri}ht(A13)\nis the single-particle Matsubara function. The Fourier\ntransform of Eq. (A10) is given by\n1\nV∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htM(iEN) =\n=i\n4/planckover2pi1β/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\np/bracketleftBig\nA(1)\nknmB(2)\nkn′m′GM\nm′n(k+,iEp)GM\nmn′(k−,iEp+iEN)\n−A(2)\nknmB(1)\nkn′m′GM\nm′n(k−,iEp)GM\nmn′(k+,iEp+iEN)/bracketrightBig\n,\n(A14)\nwhereEN= 2πN/βandEp= (2p+ 1)π/βare bosonic\nandfermionicMatsubaraenergypoints, respectively, and\nβ= 1/(kBT) is the inverse temperature.\nIn order to carry out the Matsubara summation over\nEpwe make use of\n1\nβ/summationdisplay\npGM\nmn′(iEp+iEN)GM\nm′n(iEp) =\n=i\n2π/integraldisplay\ndE′f(E′)GM\nmn′(E′+iEN)GM\nm′n(E′+iδ)\n+i\n2π/integraldisplay\ndE′f(E′)GM\nmn′(E′+iδ)GM\nm′n(E′−iEN)\n−i\n2π/integraldisplay\ndE′f(E′)GM\nmn′(E′+iEN)GM\nm′n(E′−iδ)\n−i\n2π/integraldisplay\ndE′f(E′)GM\nmn′(E′−iδ)GM\nm′n(E′−iEN),(A15)\nwhereδis a positive infinitesimal. The retarded function\n∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htR(ω) is obtained from the Mat-\nsubara function ∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htM(iEN) by the\nanalytic continuation iEN→/planckover2pi1ωto real frequencies. The\nright-hand side of Eq. (A15) has the following analytic\ncontinuation to real frequencies:\ni\n2π/integraldisplay\ndE′f(E′)GR\nmn′(E′+/planckover2pi1ω)GR\nm′n(E′)\n+i\n2π/integraldisplay\ndE′f(E′)GR\nmn′(E′)GA\nm′n(E′−/planckover2pi1ω)\n−i\n2π/integraldisplay\ndE′f(E′)GR\nmn′(E′+/planckover2pi1ω)GA\nm′n(E′)\n−i\n2π/integraldisplay\ndE′f(E′)GA\nmn′(E′)GA\nm′n(E′−/planckover2pi1ω).(A16)\nTherefore, we obtain\nχ=−i\n8π/planckover2pi12qω/integraldisplayddk\n(2π)d[Zk(q,ω)−Zk(−q,ω)\n−Zk(q,−ω)+Zk(−q,−ω)],(A17)19\nwhere\nZk(q,ω) =\n=/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−(E′+/planckover2pi1ω)BkGR\nk+(E′)/bracketrightBig\n+/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−(E′)BkGA\nk+(E′−/planckover2pi1ω)/bracketrightBig\n−/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−(E′+/planckover2pi1ω)BkGA\nk+(E′)/bracketrightBig\n−/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGA\nk−(E′)BkGA\nk+(E′−/planckover2pi1ω)/bracketrightBig\n.(A18)\nWe consider the limit lim q→0limω→0χ. In this limit\nEq. (A17) may be rewritten as\nχ=−i\n2π/planckover2pi12/integraldisplayddk\n(2π)d∂2Zk(q,ω)\n∂q∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq=ω=0.(A19)\nThe frequency derivative of Zk(q,ω) is given by\n1\n/planckover2pi1∂Zk\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0=/integraldisplay\ndE′f(E′)Tr/bracketleftBigg\nAk∂GR\nk−(E′)\n∂E′BkGR\nk+(E′)/bracketrightBigg\n−/integraldisplay\ndE′f(E′)Tr/bracketleftBigg\nAkGR\nk−(E′)Bk∂GA\nk+(E′)\n∂E′/bracketrightBigg\n−/integraldisplay\ndE′f(E′)Tr/bracketleftBigg\nAk∂GR\nk−(E′)\n∂E′BkGA\nk+(E′)/bracketrightBigg\n+/integraldisplay\ndE′f(E′)Tr/bracketleftBigg\nAkGA\nk−(E′)Bk∂GA\nk+(E′)\n∂E′/bracketrightBigg\n.\n(A20)\nUsing∂GR(E)/∂E=−GR(E)GR(E)//planckover2pi1we obtain\n∂Zk\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0=−/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−GR\nk−BkGR\nk+/bracketrightBig\n+/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−BkGA\nk+GA\nk+/bracketrightBig\n+/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−GR\nk−BkGA\nk+/bracketrightBig\n−/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGA\nk−BkGA\nk+GA\nk+/bracketrightBig\n.\n(A21)\nMaking use of\nlim\nq→0∂GR\nk+\n∂q=1\n2GR\nkv·q\nqGR\nk (A22)we finally obtain\nχ=−i\n2π/planckover2pi12/integraldisplayddk\n(2π)dlim\nq→0lim\nω→0∂2Z(q,ω)\n∂q∂ω=\n=−i\n4π/planckover2pi12q\nq·/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nAkRvRRBkR+AkRRvRBkR\n−AkRRBkRvR−AkRvRBkAA\n+AkRBkAvAA+AkRBkAAvA\n−AkRvRRBkA−AkRRvRBkA\n+AkRRBkAvA\n+AkAvABkAA−AkABkAvAA\n−AkABkAAvA/bracketrightBig\n,(A23)\nwhere we use the abbreviations R=GR\nk(E) andA=\nGA\nk(E). When we substitute B=∂H\n∂ˆMj,A=−evi, and\nq=qkˆek, we obtain Eq. (18). When we substitute B=\nTj,A=−evi, andq=qkˆek, we obtain Eq. (20). When\nwe substitute A=−Ti,B=Tj, andq=qkˆek, we obtain\nEq. (41). When we substitute B=−evj,A=−Ti,\nandq=qkˆek, we obtain Eq. (50). When we substitute\nB=∂H\n∂ˆMj,A=−Ti, andq=qkˆek, we obtain Eq. (82).\nAppendix B: Perturbation theory for the chiral\ncontributions to CIT and to the torque-torque\ncorrelation\nIn this appendix we derive expressionsfor the retarded\nfunction\n∝an}bracketle{t∝an}bracketle{tAcos(q·r);C∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) (B1)\nwithin first-orderperturbation theory with respect to the\nperturbation\nδH=Bηsin(q·r), (B2)\nwhich may arise e.g. from the spatial oscillation of the\nmagnetization direction. As usual, it is convenient to ob-\ntain the retarded response function from the correspond-\ning Matsubara function\n∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(τ) =−∝an}bracketle{tTτcos(q·r)A(τ)C(0)∝an}bracketri}ht.\n(B3)\nThe starting point for the perturbative expansion is\nthe equation\n−∝an}bracketle{tTτcos(q·r)A(τ1)C(0)∝an}bracketri}ht=\n=−Tr/bracketleftbig\ne−βHTτcos(q·r)A(τ1)C(0)/bracketrightbig\nTr[e−βH]=\n=−Tr/braceleftbig\ne−βH0Tτ[Ucos(q·r)A(τ1)C(0)]/bracerightbig\nTr[e−βH0U],(B4)20\nwhereH0is the unperturbed Hamiltonian and we con-\nsider the first order in the perturbation δH:\nU(1)=−1\n/planckover2pi1/integraldisplay/planckover2pi1β\n0dτ1Tτ{eτ1H0//planckover2pi1δHe−τ1H0//planckover2pi1}.(B5)\nThe essentialdifference between Eq. (A3) and Eq. (B4) is\nthat in Eq. (A3) the operator Benters together with the\nfactor sin( q·r)sin(ωt) (see Eq. (A1)), while in Eq. (B4)\nonly the factor sin( q·r) is connected to Bin Eq. (B2),\nwhile the factor sin( ωt) is coupled to the additional op-\neratorC.\nWe use Eq. (A4) and Eq. (A7) in order to express\nAcos(q·r) andBsin(q·r) in terms of annihilation and\ncreation operators. In terms of the correlators\nZ(3)\nknmn′m′n′′m′′(τ,τ1) =\n∝an}bracketle{tTτc†\nk−n(τ)ck+m(τ)c†\nk+n′(τ1)ck−m′(τ1)c†\nk−n′′ck−m′′∝an}bracketri}ht\n(B6)\nand\nZ(4)\nknmn′m′n′′m′′(τ,τ1) =\n∝an}bracketle{tTτc†\nk−n(τ)ck+m(τ)c†\nk+n′(τ1)ck−m′(τ1)c†\nk+n′′ck+m′′∝an}bracketri}ht\n(B7)\nand\nZ(5)\nknmn′m′n′′m′′(τ,τ1) =\n∝an}bracketle{tTτc†\nk+n(τ)ck−m(τ)c†\nk−n′(τ1)ck+m′(τ1)c†\nk+n′′ck+m′′∝an}bracketri}ht\n(B8)\nand\nZ(6)\nknmn′m′n′′m′′(τ,τ1) =\n∝an}bracketle{tTτc†\nk+n(τ)ck−m(τ)c†\nk−n′(τ1)ck+m′(τ1)c†\nk−n′′ck−m′′∝an}bracketri}ht\n(B9)\nEq. (B4) can be written as\n∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(τ1) =\n=ηV\n4i/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay/planckover2pi1β\n0dτ/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′/bracketleftBigg\n−B(2)\nknmA(1)\nkn′m′Ck−n′′m′′Z(3)\nknmn′m′n′′m′′(τ,τ1)\n−B(2)\nknmA(1)\nkn′m′Ck+n′′m′′Z(4)\nknmn′m′n′′m′′(τ,τ1)\n+B(1)\nknmA(2)\nkn′m′Ck+n′′m′′Z(5)\nknmn′m′n′′m′′(τ,τ1)\n+B(1)\nknmA(2)\nkn′m′Ck−n′′m′′Z(6)\nknmn′m′n′′m′′(τ,τ1)/bracketrightBigg(B10)\nwithin first-order perturbation theory, where we de-\nfinedCk−n′′m′′=∝an}bracketle{tuk−n′′|C|uk−m′′∝an}bracketri}htandCk+n′′m′′=\n∝an}bracketle{tuk+n′′|C|uk+m′′∝an}bracketri}ht.\nNote that Z(5)can be obtained from Z(3)by replac-\ningk−byk+andk+byk−. Similarly, Z(6)can be\nobtained from Z(4)by replacing k−byk+andk+by\nk−. Therefore, we write down only the equations forZ(3)andZ(4)in the following. Using Wick’s theorem\nwe find\nZ(3)\nknmn′m′n′′m′′(τ,τ1) =\n=−GM\nm′n(k−,τ1−τ)GM\nmn′(k+,τ−τ1)GM\nm′′n′′(k−,0)\n+GM\nmn′(k+,τ−τ1)GM\nm′′n(k−,−τ)GM\nm′n′′(k−,τ1)\n(B11)\nand\nZ(4)\nknmn′m′n′′m′′(τ,τ1) =\n=−GM\nmn′(k+,τ−τ1)GM\nm′n(k−,τ1−τ)GM\nm′′n′′(k+,0)\n+GM\nmn′′(k+,τ)GM\nm′n(k−,τ1−τ)GM\nm′′n′(k+,−τ1).\n(B12)\nThe Fourier transform\n∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(iEN) =\n=/integraldisplay/planckover2pi1β\n0dτ1ei\n/planckover2pi1ENτ1∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(τ1)(B13)\nof Eq. (B10) can be written as\n∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(iEN) =\n=ηV\n4i/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′/bracketleftBigg\n−B(2)\nknmA(1)\nkn′m′Ck−n′′m′′Z(3a)\nknmn′m′n′′m′′(iEN)\n−B(2)\nknmA(1)\nkn′m′Ck+n′′m′′Z(4a)\nknmn′m′n′′m′′(iEN)\n+B(1)\nknmA(2)\nkn′m′Ck+n′′m′′Z(5a)\nknmn′m′n′′m′′(iEN)\n+B(1)\nknmA(2)\nkn′m′Ck−n′′m′′Z(6a)\nknmn′m′n′′m′′(iEN)/bracketrightBigg(B14)\nin terms of the integrals\nZ(3a)\nknmn′m′n′′m′′(iEN) =/integraldisplay/planckover2pi1β\n0dτ/integraldisplay/planckover2pi1β\n0dτ1ei\n/planckover2pi1ENτ1×\n×GM\nmn′(k+,τ−τ1)GM\nm′′n(k−,−τ)GM\nm′n′′(k−,τ1) =\n=1\n/planckover2pi1β/summationdisplay\npGM\nk+mn′(iEp)GM\nk−m′′n(iEp)GM\nk−m′n′′(iEp+iEN)\n(B15)\nand\nZ(4a)\nknmn′m′n′′m′′(iEN) =/integraldisplay/planckover2pi1β\n0dτ/integraldisplay/planckover2pi1β\n0dτ1ei\n/planckover2pi1ENτ1×\n×GM\nmn′′(k+,τ)GM\nm′n(k−,τ1−τ)GM\nm′′n′(k+,−τ1) =\n=1\n/planckover2pi1β/summationdisplay\npGM\nk+mn′′(iEp)GM\nk−m′n(iEp)GM\nk+m′′n′(iEp−iEN),\n(B16)\nwhereEN= 2πN/βis a bosonic Matsubara energy point\nand we used\nGM(τ) =1\n/planckover2pi1β∞/summationdisplay\np=−∞e−iEpτ//planckover2pi1GM(iEp),(B17)21\nwhereEp= (2p+1)π/βis a fermionic Matsubara point.\nAgain Z(5a)is obtained from Z(3a)by replacing k−by\nk+andk+byk−andZ(6a)is obtained from Z(4a)in\nthe same way.\nSummation overMatsubarapoints Epin Eq.(B15) and\nin Eq. (B16) and analytic continuation iEN→/planckover2pi1ωyields\n2πi/planckover2pi1Z(3a)\nknmn′m′n′′m′′(/planckover2pi1ω) =\n−/integraldisplay\ndEf(E)GR\nk+mn′(E)GR\nk−m′′n(E)GR\nk−m′n′′(E+/planckover2pi1ω)\n+/integraldisplay\ndEf(E)GA\nk+mn′(E)GA\nk−m′′n(E)GR\nk−m′n′′(E+/planckover2pi1ω)\n−/integraldisplay\ndEf(E)GA\nk+mn′(E−/planckover2pi1ω)GA\nk−m′′n(E−/planckover2pi1ω)GR\nk−m′n′′(E)\n+/integraldisplay\ndEf(E)GA\nk+mn′(E−/planckover2pi1ω)GA\nk−m′′n(E−/planckover2pi1ω)GA\nk−m′n′′(E)\n(B18)\nand\n2πi/planckover2pi1Z(4a)\nknmn′m′n′′m′′(/planckover2pi1ω) =\n−/integraldisplay\ndEf(E)GR\nk+mn′′(E)GR\nk−m′n(E)GA\nk+m′′n′(E−/planckover2pi1ω)\n+/integraldisplay\ndEf(E)GA\nk+mn′′(E)GA\nk−m′n(E)GA\nk+m′′n′(E−/planckover2pi1ω)\n−/integraldisplay\ndEf(E)GR\nk+mn′′(E+/planckover2pi1ω)GR\nk−m′n(E+/planckover2pi1ω)GR\nk+m′′n′(E)\n+/integraldisplay\ndEf(E)GR\nk+mn′′(E+/planckover2pi1ω)GR\nk−m′n(E+/planckover2pi1ω)GA\nk+m′′n′(E).\n(B19)\nIn the next step we take the limit ω→0 (see Eq. (64),\nEq. (70), and Eq. (77)):\n−1\nVlim\nω→0Im∝an}bracketle{t∝an}bracketle{tAcos(q·r);C∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n/planckover2pi1ω=\n=η\n4/planckover2pi1Im/bracketleftBig\nY(3)+Y(4)−Y(5)−Y(6)/bracketrightBig\n,(B20)where we defined\nY(3)=1\ni/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′B(2)\nknmA(1)\nkn′m′Ck−n′′m′′×\n×∂Z(3a)\nknmn′m′n′′m′′(/planckover2pi1ω)\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0,\nY(4)=1\ni/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′B(2)\nknmA(1)\nkn′m′Ck+n′′m′′×\n×∂Z(4a)\nknmn′m′n′′m′′(/planckover2pi1ω)\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0,\nY(5)=1\ni/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′B(1)\nknmA(2)\nkn′m′Ck+n′′m′′×\n×∂Z(5a)\nknmn′m′n′′m′′(/planckover2pi1ω)\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0,\nY(6)=1\ni/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′B(1)\nknmA(2)\nkn′m′Ck−n′′m′′×\n×∂Z(6a)\nknmn′m′n′′m′′(/planckover2pi1ω)\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0,\n(B21)\nwhich can be expressed as Y(3)=Y(3a)+Y(3b)and\nY(4)=Y(4a)+Y(4b), where\n2π/planckover2pi1Y(3a)=1\n/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)×\n×Tr/bracketleftBigg\nAkGR\nk−(E)Ck−GA\nk−(E)BkGA\nk+(E)GA\nk+(E)\n+AkGR\nk−(E)GR\nk−(E)Ck−GA\nk−(E)BkGA\nk+(E)\n+AkGR\nk−(E)Ck−GA\nk−(E)GA\nk−(E)BkGA\nk+(E)/bracketrightBigg\n=/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)×\n×Tr/bracketleftBig\nAkGR\nk−(E)Ck−GA\nk−(E)BkGA\nk+(E)/bracketrightBig\n(B22)\nand\n2π/planckover2pi1Y(3b)=−1\n/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)×\n×Tr/bracketleftBigg\nAkGA\nk−(E)Ck−GA\nk−(E)BkGA\nk+(E)GA\nk+(E)\n+AkGR\nk−(E)GR\nk−(E)Ck−GR\nk−(E)BkGR\nk+(E)\n+AkGA\nk−(E)Ck−GA\nk−(E)GA\nk−(E)BkGA\nk+(E)/bracketrightBigg\n.(B23)22\nSimilarly,\n2π/planckover2pi1Y(4a)=1\n/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)×\n×Tr/bracketleftBigg\nAkGR\nk−(E)BkGR\nk+(E)Ck+GA\nk+(E)GA\nk+(E)\n−AkGR\nk−(E)GR\nk−(E)BkGR\nk+(E)Ck+GA\nk+(E)\n−AkGR\nk−(E)BkGR\nk+(E)GR\nk+(E)Ck+GA\nk+(E)/bracketrightBigg\n=/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)×\n×Tr/bracketleftBig\nAkGR\nk−(E)BkGR\nk+(E)Ck+GA\nk+(E)/bracketrightBig\n(B24)\nand\n2π/planckover2pi1Y(4b)=−1\n/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)×\n×Tr/bracketleftBigg\nAkGA\nk−(E)BkGA\nk+(E)Ck+GA\nk+(E)GA\nk+(E)\n+AkGR\nk−(E)GR\nk−(E)BkGR\nk+(E)Ck+GR\nk+(E)\n+AkGR\nk−(E)BkGR\nk+(E)GR\nk+(E)Ck+GR\nk+(E)/bracketrightBigg\n.(B25)\nWe call Y(3a)andY(4a)Fermi surface terms and Y(3b)\nandY(4b)Fermi sea terms. Again Y(5)is obtained from\nY(3)by replacing k−byk+andk+byk−andY(6)is\nobtained from Y(4)in the same way.\nFinally, we take the limit q→0:\nΛ =−2\n/planckover2pi1VηIm lim\nq→0lim\nω→0∂\n∂ω∂\n∂qi∝an}bracketle{t∝an}bracketle{tAcos(q·r);C∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n=1\n2/planckover2pi1lim\nq→0∂\n∂qiIm/bracketleftBig\nY(3)+Y(4)−Y(5)−Y(6)/bracketrightBig\n=1\n2/planckover2pi1Im/bracketleftBig\nX(3)+X(4)−X(5)−X(6)/bracketrightBig\n,\n(B26)\nwhere we defined\nX(j)=∂\n∂qi/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq=0Y(j)(B27)\nforj= 3,4,5,6. Since Y(4)andY(6)are related by\nthe interchange of k−andk+it follows that X(6)=\n−X(4). Similarly, since Y(3)andY(5)arerelated by the\ninterchange of k−andk+it follows that X(5)=−X(3).\nConsequently, we need\nΛ =1\n/planckover2pi1Im/bracketleftBig\nX(3a)+X(3b)+X(4a)+X(4b)/bracketrightBig\n,(B28)\nwhere X(3a)andX(4a)are the Fermi surface terms and\nX(3b)andX(4b)are the Fermi sea terms. The Fermisurface terms are given by\nX(3a)=−1\n4π/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)Tr/bracketleftBigg\nAkGR\nk(E)vkGR\nk(E)CkGA\nk(E)BkGA\nk(E)\n+AkGR\nk(E)CkGA\nk(E)vkGA\nk(E)BkGA\nk(E)\n−AkGR\nk(E)CkGA\nk(E)BkGA\nk(E)vkGA\nk(E)\n+AkGR\nk(E)∂Ck\n∂kGA\nk(E)BkGA\nk(E)/bracketrightBigg(B29)\nand\nX(4a)=−/bracketleftBig\nX(3a)/bracketrightBig∗\n. (B30)\nThe Fermi sea terms are given by\nX(3b)=−1\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBigg\n−(ARvRRCRBR)+(AACAABAvA)\n−(ARRvRCRBR)−(ARRCRvRBR)\n+(ARRCRBRvR)−(AAvACABAA)\n−(AACAvABAA)+(AACABAvAA)\n+(AACABAAvA)−(AAvACAABA)\n−(AACAvAABA)−(AACAAvABA)\n−(ARR∂C\n∂kRBR)−(AA∂C\n∂kAABA)\n−(AA∂C\n∂kABAA)/bracketrightBigg(B31)\nand\nX(4b)=−/bracketleftBig\nX(3b)/bracketrightBig∗\n. (B32)\nIn Eq. (B31) we use the abbreviations R=GR\nk(E),A=\nGA\nk(E),A=Ak,B=Bk,C=Ck. It is important\nto note that Ck−andCk+depend on qthrough k−=\nk−q/2 andk+=k+q/2 . Theqderivative therefore\ngenerates the additional terms with ∂Ck/∂kin Eq. (B29)\nand Eq. (B31). In contrast, AkandBkdo not depend\nlinearly on q.\nEq. (B28) simplifies due to the relations Eq. (B30) and\nEq. (B32) as follows:\nΛ =2\n/planckover2pi1Im/bracketleftBig\nX(3a)+X(3b)/bracketrightBig\n. (B33)\nIn order to obtain the expression for the chiral con-\ntribution to the torque-torque correlation we choose the\noperators as follows:\nB→ Tk\nA→ −Ti\nC→ Tj\n∂C\n∂k= 0\nv→vl.(B34)23\nThis leads to Eq. (78), Eq. (79) and Eq. (80) of the main\ntext.\nIn order to obtain the expression for the chiral contri-\nbution to the CIT, we set\nB→ Tk\nA→ −Ti\nC→ −evj\n∂C\n∂k→ −e/planckover2pi1\nmδjl\nv→vl.(B35)\nThis leads to Eq. (66), Eq. (67) and Eq. (68).\nIn order to obtain the expression for the chiral contri-\nbution to the ICIT, we set\nB→ Tk\nA→ −evi\nC→ Tj\n∂C\n∂k→0\nv→vl.(B36)\nThis leads to Eq. (71), Eq. (72) and Eq. (73).\n∗Corresp. author: f.freimuth@fz-juelich.de\n[1] K. Nawaoka, S. 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B 93, 214429 (2016)."    },    {        "title": "2306.04617v2.Helicity_dependent_optical_control_of_the_magnetization_state_emerging_from_the_Landau_Lifshitz_Gilbert_equation.pdf",        "content": "1  Helicity-dependent optical control of the magnetization state emerging from the Landau-Lifshitz-Gilbert equation  Benjamin Assouline, Amir Capua*  Department of Applied Physics, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel   *e-mail: amir.capua@mail.huji.ac.il  Abstract:   It is well known that the Gilbert relaxation time of a magnetic moment scales inversely with the magnitude of the externally applied field, 𝑯, and the Gilbert damping, 𝜶. Therefore, in ultrashort optical pulses, where 𝑯 can temporarily be extremely large, the Gilbert relaxation time can momentarily be extremely short, reaching even picosecond timescales. Here we show that for typical ultrashort pulses, the optical control of the magnetization emerges by merely considering the optical magnetic field in the Landau-Lifshitz-Gilbert (LLG) equation. Surprisingly, when circularly polarized optical pulses are introduced to the LLG equation, an optically induced helicity-dependent torque results. We find that the strength of the interaction is determined by 𝜼=𝜶𝜸𝑯/𝒇𝒐𝒑𝒕, where 𝒇𝒐𝒑𝒕 and 𝜸 are the optical frequency and gyromagnetic ratio. Our results illustrate the generality of the LLG equation to the optical limit and the pivotal role of the Gilbert damping in the general interaction between optical magnetic fields and spins in solids.  2  The ability to control the magnetization order parameter using ultrashort circularly polarized (CP) optical pulses has attracted a great deal of attention since the early experiments of the all-optical helicity dependent switching (AO-HDS) [1-4]. This interaction was found intriguing since it appears to have all the necessary ingredients to be explained by a coherent transfer of angular momentum, yet it occurs at photon energies of 1\t−\t2\t𝑒𝑉, very far from the typical resonant transitions in metals. The technological applications and fundamental scientific aspects steered much debate and discussion [5,6], and the experiments that followed found dependencies on a variety of parameters including material composition [7-9], magnetic structure [10-12], and laser parameters [1,3,13], that were often experiment-specific [4]. Consequently, a multitude of mechanisms that entangle photons [14,15], spins [16,17], and phonons [18,19] have been discovered. References [4,20] provide a state of the art review of the theoretical and experimental works of the field. Ferromagnetic resonance (FMR) experiments are usually carried out at the 𝐺𝐻𝑧 range. In contrast, optical fields oscillate much faster, at ~\t400−800\t𝑇𝐻𝑧. Therefore, it seems unlikely that such fast-oscillating fields may interact with magnetic moments. However, the amplitude of the magnetic field in ultrashort optical pulses can, temporarily, be very large such that the magnetization may respond extremely fast. For example, in typical experiments having 40\t𝑓𝑠−1\t𝑝𝑠 pulses at 800\t𝑛𝑚, with energy of 0.5\t𝑚𝐽 that are focused to a spot size of ~0.5\t𝑚𝑚$, the peak magnetic flux density can be as high as ~\t5\t𝑇, for which the corresponding Gilbert relaxation time reduces to tens of picoseconds in typical ferromagnets. Here we show that ultrashort optical pulses may control the magnetization state by merely considering the optical magnetic field in the Landau-Lifshitz-Gilbert (LLG) equation. We find that the strength of the interaction is determined by 𝜂=𝛼𝛾𝐻/𝑓%&', where 𝑓%&' and 𝛼 are the angular optical frequency and the Gilbert damping, respectively, and 𝛾 is the gyromagnetic ratio. Moreover, we show that for circularly polarized (CP) pulses, the polarity of the optically induced torque is determined by the optical helicity. From a quantitative analysis, we find that a sizable effective out-of-plane field is generated which is comparable to that measured experimentally in ferromagnet/heavy-metal (FM/HM) material systems. 3  The LLG equation is typically not applied in the optical limit, and hence requires an alternative mathematical framework whose principles we adopt from the Bloch equations for semiconductor lasers [21,22]. We exploit the analogy between the magnetization state and the Bloch vector of a two-level system (TLS) [23,24] by transforming the LLG equation under a time-varying magnetic field excitation to the dynamical Maxwell-Bloch (MB) equations in the presence of an electrical carrier injection. In this transformation, the reversal of the magnetization is described in terms of population transfer between the states. The paper is organized as follows: We begin by transforming the LLG equation to the density matrix equations of a TLS. We then identify the mathematical form of a time-dependent magnetic field in the LLG equation, 𝐻AA⃗&()&↓↑, that is mapped to a time-independent carrier injection rate into the TLS. Such excitation induces a population transfer that varies linearly in time and accordingly to a magnetization switching profile that is also linear in time. The mathematical 𝐻AA⃗&()&↓↑ field emerges naturally as a temporal impulse-like excitation. We then show that when 𝛼 is sizable, 𝐻AA⃗&()&↓↑ acquires a CP component whose handedness is determined by the direction of the switching. By substituting 𝐻AA⃗&()&↓↑ for an experimentally realistic picosecond CP Gaussian optical magnetic pulse, we show that it can also exert a net torque on the magnetization. In this case as well, the helicity determines the polarity of the torque. Finally, we present a quantitative analysis that is based on experimental data. The LLG equation describing the dynamics of the magnetization, 𝑀AA⃗, where the losses are introduced in the Landau–Lifshitz form is given by [25]: 𝑑𝑀AA⃗𝑑𝑡=\t−𝛾1+𝛼$𝑀AA⃗×𝐻AA⃗−𝛾𝛼1+𝛼$1𝑀,𝑀AA⃗×𝑀AA⃗×𝐻AA⃗.(1) Here 𝑀, and 𝐻AA⃗ are the magnetization saturation and the time dependent externally applied magnetic field, respectively. We define 𝐻AA⃗-.. by: 𝐻AA⃗-..≜K𝐻AA⃗−\t𝛼𝑀,𝐻AA⃗×𝑀AA⃗\tL,(2) and in addition, 𝜅≜/012!O𝐻-..\t4−𝑗𝐻-..\t5Q/2 and 𝜅6\t≜/012!𝐻-..\t7, where 𝜅 and 𝜅6 can be regarded as effective AC and DC magnetic fields acting on 𝑀AA⃗, respectively. We 4  transform 𝑀AA⃗ to the density matrix elements of the Bloch state in the TLS picture and compare it to the Bloch equations describing a semiconductor laser that is electrically pumped [26]:  ⎩⎪⎨⎪⎧𝜌̇00=𝛬0−𝛾0𝜌00+𝑗2[(𝜌0$−𝜌$0)(𝑉0$+𝑉$0)−(𝜌0$+𝜌$0)(𝑉0$−𝑉$0)]𝜌̇$$=𝛬$−𝛾$𝜌$$−𝑗2[(𝜌0$−𝜌$0)(𝑉0$+𝑉$0)−(𝜌0$+𝜌$0)(𝑉0$−𝑉$0)]𝜌̇0$=\t−(𝑗𝜔89:+𝛾;<=)𝜌0$+𝑗(𝜌00−𝜌$$)𝑉0$\t.\t\t(3) In this reference model, 𝛬0 and 𝛬$ are injection rates of carriers to the ground and excited states of the TLS, respectively. They are assumed to be time independent and represent a constant injection of carriers from an undepleted reservoir [27]. 𝛾0 and 𝛾$ are the relaxation rates of the ground and excited states, and 𝛾;<= is the decoherence rate due to an inhomogeneous broadening. 𝑉0$ is the interaction term and 𝜔89: is the resonance frequency of the TLS. Figure 1(a) illustrates schematically the analogy between the magnetization dynamics and the electrically pumped TLS. We find the connection between the LLG equation expressed in the density matrix form and the model of the electrically pumped TLS: ]𝛬0−𝛾0𝜌00+[𝑀5ℜ{𝑉0$}+𝑀4ℑ{𝑉0$}]=\t−𝑗𝜅𝜌$0+𝑐.𝑐.𝛬$−𝛾$𝜌$$−[𝑀5ℜ{𝑉0$}+𝑀4ℑ{𝑉0$}]=\t𝑗𝜅𝜌$0+𝑐.𝑐.−(𝑗𝜔89:+𝛾;<=)𝜌0$+𝑗𝑀7𝑉0$=\t−𝑗𝜅6𝜌0$\t+𝑗𝜅𝑀7.(4) The pumping of the excited and ground states by the constant 𝛬0 and 𝛬$ rates implies that the reversal of the magnetization along the ∓\t𝑧̂ direction is linear in time. Using Eq. (4) we find 𝜅, and hence a field 𝐻AA⃗, that produces such 𝛬0 and 𝛬$. We define this field as 𝐻AA⃗&()&↓↑: 𝐻AA⃗&()&↓↑=\t±𝛬&𝑀,$−𝑀7$f𝑀5−\t𝑀40g.(5) 𝐻AA⃗&()&↓↑ depends on the temporal state of 𝑀AA⃗ while 𝛬&=𝛾𝛬0/(1+𝛼$) is the effective field strength parameter. 𝐻AA⃗&()&↓ and 𝐻AA⃗&()&↑ induce a linear transition of 𝑀AA⃗ towards the –𝑧̂ and +𝑧̂ direction, respectively. 5  Figure 1(b) presents the outcome of the application of 𝐻AA⃗&()&↓↑ by numerically integrating the LLG equation. The Figure illustrates 𝐻AA⃗(𝑡), 𝑀7(𝑡), and the 𝑧̂ torque, O−𝑀AA⃗×𝐻AA⃗Q7, for alternating 𝐻AA⃗&()&↓ and 𝐻AA⃗&()&↑ that switch 𝑀AA⃗ between ∓𝑀,𝑧̂. The magnitude of 𝛬& determines the switching time, 𝛥𝜏↓↑, chosen here to describe a femtosecond regime. Equation (4) yields 𝛥𝜏↓↑=(1+𝛼$)𝑀,/(\t𝛾𝛬&)≈𝑀,/𝛾𝛬& in which 𝑀7 is driven from 𝑀7=0 to 𝑀7≅±𝑀, (for derivation, see Supplemental Material Note 1). It is seen that O−𝑀AA⃗×𝐻AA⃗Q7 is constant when 𝐻AA⃗&()&↓ or 𝐻AA⃗&()&↑ are applied so that the switching profile of 𝑀7 is linear in time. It is also seen that 𝐻AA⃗&()&↓↑ requires that m𝐻AA⃗m diverge as 𝑀7 approaches ±𝑀,, which is not experimentally feasible. To account for a more realistic excitation, in Fig. 1(c) we simulated a pulse whose trailing edge was taken as a reflection in time of 𝐻AA⃗&()&↓↑, and that is shorter by an order of magnitude as compared to the leading edge. In this case 𝑀AA⃗ remains in its final state when 𝐻AA⃗ is eventually turned off. The polarization state of 𝐻AA⃗&()&↓↑ is determined from the polarization state of the transverse components of 𝑀AA⃗. Next, we show that for larger 𝛼, 𝑀5(𝑡) becomes appreciable such that 𝐻AA⃗&()&↓↑ acquires an additional CP component. This result emerges naturally from the Bloch picture: we recall that the transverse components of 𝑀AA⃗ are expressed by the off-diagonal density matrix element. According to Eq. (3), 𝜌0$ oscillates at 𝜔89: and decays at the rate 𝛾;<=, whereas the sign of 𝜔89: determines the handedness of the transverse components of 𝑀AA⃗. Namely, the ratio between 𝜔89: and 𝛾;<= determines the magnitude of the circular component in the n𝑀4(𝑡),𝑀5(𝑡)o trajectory. Under the application of 𝐻AA⃗&()&↓↑, Eq. (4) yields 𝜔89:=±𝛾𝛬&𝛼𝑀,/[(𝑀,$−𝑀7$)(1+𝛼$)] and 𝛾;<==∓𝛾𝛬&𝑀7/[(𝑀,$−𝑀7$)(1+𝛼$)] readily showing that |𝜔89:/𝛾;<=|=𝛼𝑀,/𝑀7 increases with 𝛼, so that 𝐻AA⃗&()&↓↑ acquires an additional CP component (see Supplemental Note 2 for full derivation). Figure 2 illustrates these results. Panel (a) presents the components of 𝑀AA⃗(𝑡) for the same simulation in Fig. 1(b). It is seen that 𝑀5(𝑡) is negligible and thus 𝐻AA⃗&()&↓↑ remains linearly polarized. When 𝛼 is increased, an elliptical trajectory of 𝑀AA⃗ in the 𝑥−𝑦 plane emerges, while the constant transition rate of 𝑀7 persists as illustrated in Fig. 2(b). In this case, 6  𝐻AA⃗&()&↓↑ acquires a right-CP (RCP) or left-CP (LCP) component depending on the choice of 𝐻AA⃗&()&↓ or 𝐻AA⃗&()&↑. The coupling between the handedness and reversal direction in a femtosecond excitation is reminiscent of the switching reported in AO-HDS experiments and emerges naturally in our model. These results call to examine the interaction of the CP magnetic field of a short optical pulse with 𝑀AA⃗. Figure 3(a) presents the calculation for experimental conditions [4]. The results are shown for an 800\t𝑛𝑚 optical magnetic field of an RCP Gaussian optical pulse 𝐻AA⃗%&'(𝑡). The pulse has a duration determined by 𝜏&, an angular frequency 𝜔%&', and a peak amplitude 𝐻&->? that is reached at 𝑡=𝑡&->?. In our simulations 𝜏&=3\t𝑝𝑠 and 𝑡&->?=10\t𝑝𝑠. The pulse energy was ~\t5\t𝑚𝐽 and assumed to be focused to a spot size of ~\t100\t𝜇𝑚$, for which 𝐻&->?=8⋅10@\t𝐴/𝑚. Here we take 𝛼=0.035 [28,29]. For such conditions, the Gilbert relaxation time corresponding to 𝐻&->? is 𝜏2=02/A\"#$%≈16\t𝑝𝑠 [30]. It is readily seen that for such 𝜏2 the magnetization responds within the duration of the optical pulse indicating that the interaction between the optical pulse and 𝑀AA⃗ becomes possible by the LLG equation. Following the interaction, 𝑀7=−5×10BC⋅𝑀:, namely a sizable net longitudinal torque results. In agreement with the prediction of the TLS model, pulses of the opposite helicity induce an opposite transition as shown in Fig. 3(b). The results are compared to the measured data discussed in Supplemental Material Note 3.  To this end we simulate the same conditions of the measurements including optical intensity and sample parameters. Accordingly, we find from our calculations an effective field which is of the same order of magnitude as measured.   For a given pulse duration, we define the interaction strength parameter 𝜂=2𝜋𝛼𝛾𝐻&->?/𝜔%&', which expresses the ratio between 𝜏2 and the optical cycle and is 2.5⋅10BC in Fig. 3(a). The principles of the interaction can be better understood at the limit where 𝜂→1 and for which the interaction can be described analytically. To this end, we set 𝜂=1. The higher optical magnetic fields required for this limit are achievable using conventional amplified femtosecond lasers, for example by focusing a ~\t5\t𝑚𝐽 pulse into a spot size of ~\t1\t𝜇𝑚$. Figure 3(c) illustrates the results for an RCP 𝐻AA⃗%&' pulse of a duration of 20\t𝑓𝑠 determined by the full width at half-maximum of the 7  intensity. The Figure reveals the different stages of the interaction. During the leading edge, for 𝑡<~\t40\t𝑓𝑠, the relative phase between 𝐻AA⃗%&' and 𝑀AA⃗ seems arbitrary. As 𝑡&->? is reached, the Gilbert relaxation time becomes as short as the optical cycle allowing 𝑀AA⃗ to follow 𝐻AA⃗%&' until it is entirely locked to 𝐻AA⃗%&'. In this case, 𝑀AA⃗ undergoes a right-circular trajectory about 𝑧̂. The switching of 𝑀AA⃗ takes place at the final stage of the interaction: During the trailing edge of the pulse, the amplitude of 𝐻AA⃗%&' reduces and 𝜏2 extends, thereby releasing the locking between 𝑀AA⃗ and 𝐻AA⃗%&'. In this case, the switching profile of 𝑀7 is monotonic linear-like in time, closely resembling the transition stemming from a constant carrier injection rate in the Bloch picture. The optically induced transition can be described analytically following the calculation presented in Supplemental Note 4, from which we find the transition rate:  𝛤/𝑀,=∓32√2𝑙𝑛K43L1𝜏&}~𝑙𝑛K𝐻&->?0.27𝐻'=L−~𝑙𝑛𝐻&->?𝐻'=/√2,(6) where 𝐻'==D&\"'$E/2 is the value of 𝐻&->? at 𝜂=1. The rate 𝛤/𝑀, is plotted as well in Fig. 3(c) and reproduces the numerical calculation. 𝛤 depends on the ratio between 𝐻&->? and 𝐻'= and is only weekly dependent on 𝐻&->?. Namely, when 𝐻&->?≫𝐻'=, the circular trajectory of 𝑀AA⃗ in the 𝑥−𝑦 plane persists longer after 𝑡&->?, but as the amplitude of the pulse decays below 𝐻'=/√2, 𝑀AA⃗ is driven out of the 𝑥−𝑦 plane and the reversal takes place (see Supplemental Material Note 5). This analysis also holds for LCP pulses, which result in an opposite reversal of 𝑀AA⃗, as shown in Fig. 3(d).  To summarize, in this work we demonstrated that the control of the magnetization by an optical field arises from first principles by introducing the magnetic part of the optical radiation to the LLG equation. This was seen from the comparison between the case where 𝜂≪1 and the case of 𝜂=1. Using the TLS model, we demonstrated the coupling between the optical helicity state and the polarity of the longitudinal torque. A quantitative analysis of the optically induced torque revealed that it can be comparable to that observed in experiments.   8  Figure 1 \n       Fig. 1. (a) Left panel: Illustration of 𝑴AAA⃗ on the Bloch sphere. Right panel: Illustration of the electrically pumped TLS. (b) Interaction with 𝑯AAA⃗𝒑𝒖𝒎𝒑↓↑ of Eq. (5). The Figure illustrates the temporal plots of 𝑴𝒛/𝑴𝒔, 𝑯AAA⃗𝒑𝒖𝒎𝒑↓↑,𝒚 and O−𝑴AAA⃗×𝑯AAA⃗Q𝒛 normalized to unity. (c) Interaction with 𝑯AAA⃗𝒑𝒖𝒎𝒑↓↑ and a more realistic trailing edge, for the same conditions in (b). Full lines correspond to 𝑯AAA⃗𝒑𝒖𝒎𝒑↓ and dashed lines correspond to 𝑯AAA⃗𝒑𝒖𝒎𝒑↑.    \n9  Figure 2 \n Fig. 2. Temporal evolution of the components of 𝑴AAA⃗ under the influence of alternating 𝑯AAA⃗𝒑𝒖𝒎𝒑↓ and 𝑯AAA⃗𝒑𝒖𝒎𝒑↑ for (a) small and (b) large damping. Black dashed lines indicate the alternation between 𝑯AAA⃗𝒑𝒖𝒎𝒑↓ and 𝑯AAA⃗𝒑𝒖𝒎𝒑↑.   \n10   \n11  Fig. 3. (a) Magnetization reversal induced by an RCP Gaussian pulse for\t𝜼=𝟐.𝟓⋅𝟏𝟎B𝟒. Top and middle panels depict the temporal evolution of the 𝒙 and 𝒚 components of 𝑴AAA⃗ and 𝑯AAA⃗𝒐𝒑𝒕\tin normalized units. Bottom panel depicts 𝑴𝒛/𝑴𝒔. (b) 𝑴𝒛/𝑴𝒔, for the application of an LCP pulse. (c) Magnetization reversal induced by an RCP Gaussian pulse for\t𝜼=𝟏. Top panel presents the temporal behavior of m𝑯AAA⃗𝒐𝒑𝒕m and 𝝉𝜶, where 𝑯𝒄𝒓𝒊𝒕=𝑯𝒕𝒉/√𝟐 and 𝑯𝟏/𝟐=𝟎.𝟐𝟕𝑯𝒕𝒉. (d) 𝑴𝒛/𝑴𝒔, for the application of an LCP pulse. In (c) and (d), black solid lines represent the analytical solution of 𝜞/𝑴𝒔.   References  [1] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and T. Rasing, \"All-Optical Magnetic Recording with Circularly Polarized Light\", Physical Review Letters 99, 047601 (2007). [2] J. Hohlfeld, C. D. Stanciu, and A. Rebei, \"Athermal all-optical femtosecond magnetization reversal in GdFeCo\", Applied Physics Letters 94, 152504 (2009). [3] D. Steil, S. Alebrand, A. Hassdenteufel, M. Cinchetti, and M. 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Bailey2\n1)SPINTEC, UMR(8191) CEA / CNRS / UJF / Grenoble INP ; INAC,\n17 rue des Martyrs, 38054 Grenoble Cedex, France\n2)Dept. of Applied Physics & Applied Mathematics, Columbia Un iversity,\nNew York NY 10027, USA\n(Dated: 7 September 2021)\nWe have measured the size effect in nonlocal Gilbert relaxation rate in FM(tFM) /\nCu (5nm) [/ Pt (2nm)] / Al(2nm) heterostructures, FM = {Ni81Fe19, Co60Fe20B20,\npure Co}. Common behavior is observed for three FM layers, where the addit ional\nrelaxation obeys botha strict inverse power law dependence ∆ G=Ktn,n=−1.04±\n0.06 and a similar magnitude K= 224±40 Mhz·nm. As the tested FM layers span\nan order of magnitude in spin diffusion length λSDL, the results are in support of\nspin diffusion, rather than nonlocal resistivity, as the origin of the e ffect.\n1Theprimarymaterialsparameter which describes thetemporal res ponseofmagnetization\nMto applied fields His the Gilbert damping parameter α, or relaxation rate G=|γ|Msα.\nUnderstanding of the Gilbert relaxation, particularly in structures of reduced dimension, is\nan essential question for optimizing the high speed / Ghz response o f nanoscale magnetic\ndevices.\nExperiments over the last decade have established that the Gilbert relaxation of ferro-\nmagnetic ultrathin films exhibits a size effect, some component of whic h is nonlocal. Both\nα(tFM) =α0+α′(tFM) andG(tFM) =G0+G′(tFM) increase severalfold with decreasing FM\nfilm thickness tFM, from near-bulk values α0,G0fortFM>∼20 nm. Moreover, the damp-\ning size effect can have a nonlocal contribution responsive to layers or scattering centers\nremoved, through a nonmagnetic (NM) layer, from the precessing FM. Contributed Gilbert\nrelaxation has been seen from other FM layers1as well as from heavy-element scattering\nlayers such as Pt.2\nThe nonlocal damping size effect is strongly reminiscent of the electr ical resistivity in\nferromagnetic ultrathin films. Electrical resistivity ρis size-dependent by a similar factor\noverasimilarrangeof tFM; theresistivity ρ(tFM)issimilarlynonlocal,dependentuponlayers\nnot in direct contact.3–5. It isprima facie plausible that the nonlocal damping and nonlocal\nelectrical resistivity share a common origin in momentum scattering ( with relaxation time\nτM) by overlayers. If the nonlocal damping arises from nonlocal scat teringτ−1\nM, however,\nthere should be a marked dependence upon FM layer type. Damping in materials with\nshort spin diffusion length λSDLis thought to be proportional to τ−1\nM(ref.6); the claim for\n”resistivity-like” damping hasbeenmadeexplicitly forNi 81Fe19byIngvarsson7et al. ForFM\nwith along λSDL, onthe other hand, relaxation Giseither nearly constant withtemperature\nor ”conductivity-like,” scaling as τM.\nInterpretation of the nonlocal damping size effect has centered in stead on a spin current\nmodel8advanced by Tserkovnyak et al9. An explicit prediction of this model is that the\nmagnitude of the nonlocal Gilbert relaxation rate ∆ Gis only weakly dependent upon the\nFM layer type. The effect has been calculated10as\n∆G=|γ|2¯h/4π/parenleftBig\ng↑↓\neff/S/parenrightBig\nt−1\nFM (1)\n, where the effective spin mixing conductance g↑↓\neff/Sis given in units of channels per area.\nAb-initio calculationspredictaveryweakmaterialsdependencefortheinter facialparameters\n2g↑↓/S, with±10% difference in systems as different as Fe/Au and Co/Cu, and neglig ible\ndependence on interfacial mixing.11\nIndividual measurements exist of the spin mixing conductance, thr ough the damping,\nin FM systems Ni 81Fe1912, Co13, and CoFeB14. However, these experiments do not share\na common methodology, which makes a numerical comparison of the r esults problematic,\nespecially given that Gilbert damping estimates are to some extent mo del-dependent15. In\nour experiments, we have taken care to isolate the nonlocal dampin g contribution due to Pt\noverlayers only, controlling for growth effects, interfacial interm ixing, and inhomogeneous\nlosses. The only variable in our comparison of nonlocal damping ∆ G(tFM), to the extent\npossible, has been the identity of the FM layer.\nGilbert damping αhas been measured through ferromagnetic resonance (FMR) fro m\nω/2π= 2-24 Ghz using a broadband coplanar waveguide (CPW) with broad c enter conduc-\ntor width w= 400µm, using field modulation and lock-in detection of the transmitted signa l\nto enhance sensitivity. The Gilbert damping has been separated fro minhomogeneous broad-\nening inthe filmsmeasured using the well-known relation∆ Hpp(ω) = ∆H0+/parenleftBig\n2/√\n3/parenrightBig\nαω/|γ|.\nWe have fit spectra to Lorenzian derivatives with Dysonian compone nts at each frequency,\nfor each film, to extract the linewidth ∆ Hppand resonance field Hres;αhas been extracted\nusing linear fits to ∆ H(ω).\nFor the films, six series of heterostructures were deposited of th e form Si/ SiO 2/\nX/ FM(tFM)/ Cu(3nm)[ /Pt(3nm)]/ Al(3nm), FM = {Ni81Fe19(”Py”), Co 60Fe20B20\n(”CoFeB”), pure Co }, andtFM= 2.5, 3.5, 6.0, 10.0, 17.5, 30.0 nm, for 36 heterostruc-\ntures included in the study. For each ferromagnetic layer type FM, one thickness series tFM\nwas deposited with the Pt overlayer and one thickness series tFMwas deposited without the\nPt overlayer. This makes it possible to record the additional damping ∆α(tFM) introduced\nby the Pt overlayer alone, independent of size effects present in th e FM/Cu layers deposited\nbelow. In the case of pure Co, a X=Ta(5nm)/Cu(5nm) underlayer w as necessary to sta-\nbilize low-linewidth films, otherwise, depositions were carried out direc tly upon the in-situ\nion-cleaned substrate.\nField-for-resonance data are presented in Figure 1. The main pane l showsω(H/bardbl\nB) data\nfor Ni 81Fe19(tFM). Note that there is a size effect in ω(H/bardbl\nB): the thinner films have a\nsubstantially lower resonance frequency. For tFM= 2.5 nm, the resonance frequency is\ndepressed by ∼5 Ghz from ∼20 Ghz resonance HB≃4 kOe. The behavior is fitted through\n3the Kittel relation (lines) ω(H/bardbl\nB) =|γ|/radicalbigg/parenleftBig\nH/bardbl\nB+HK/parenrightBig/parenleftBig\n4πMeff\ns+H/bardbl\nB+HK/parenrightBig\n, and the inset\nshows a summary of extracted 4 πMeff\ns(tFM) data for the three different FM layers. Samples\nwith (open symbols) and without (closed symbols) Pt overlayers sho w negligible differences.\nLinear fits according to 4 πMeff\ns(tFM) = 4πMs−(2Ks/Ms)t−1\nFMallow the extraction of bulk\nmagnetization 4 πMsand surface anisotropy Ks; we find 4 πMPy\ns= 10.7 kG, 4 πMCoFeB\ns=\n11.8 kG, 4 πMCo\ns= 18.3 kG, and KPy\ns= 0.69 erg/cm2,KCoFeB\ns= 0.69 erg/cm2,KCo\ns=\n1.04 erg/cm2. The value of gL/2 =|γ|/(e/mc),|γ|= 2π·(2.799 Mhz/Oe) ·(gL/2) is found\nfrom the Kittel fits subject to this choice, yielding gPy\nL= 2.09,gCoFeB\nL= 2.07,gCo\nL= 2.15.\nThe 4πMsandgLvalues, taken to be size-independent, are in good agreement with b ulk\nvalues.\nFMR linewidth as a function of frequency ∆ Hpp(ω) is plotted in Figure 2. The data\nfor Py show a near-proportionality, with negligble inhomogeneous co mponent ∆ H0≤4 Oe\neven for the the thinnest layers, facilitating the extraction of intr insic damping parameter\nα. The size effect in in α(tFM) accounts for an increase by a factor of ∼3, fromαPy\n0=\n0.0067 (GPy\n0= 105 Mhz) for the thickest films ( tFM= 30.0 nm) to α= 0.021 for the\nthinnest films ( tFM= 2.5 nm). The inset shows the line shapes for films with and without\nPt, illustrating the broadening without significant frequency shift o r significant change in\npeak asymmetry.\nA similar analysis has been carried through for CoFeB and Co (not pict ured). Larger\ninhomogeneous linewidths are observed for pure Co, but homogene ous linewidth still ex-\nceeds inhomogeneous linewidth by a factor of three over the frequ ency range studied, and\ninhomogeneous linewidths agree within experimental error for the t hinnest films with and\nwithout Pt overlayers. We extract for these films αCoFeB\n0= 0.0065 ( GCoFeB\n0= 111 Mhz)\nandαCo\n0= 0.0085( GCo\n0= 234 Mhz). The latter value is in very good agreement with the\naverage of easy- and hard-axis values for epitaxial FCC Co films mea sured up to 90 Ghz,\nGCo\n0= 225 Mhz.16\nWe isolate the effect of Pt overlayers on the damping size effect in Figu re 3. Values\nofαhave been fitted for each deposited heterostructure: each FM t ype, at each tFM,\nfor films with and without Pt overlayers. We take the difference ∆ α(tFM) for identical\nFM(tFM)/Cu(5nm)/Al(3nm) depositions with and without the insertion of Pt (3nm) after\nthe Cu deposition. Data, as shown on the logarithmic plot in the main pa nel, are found\n4to obey a power law ∆ α(tFM) =Ktn, withn= -1.04±0.06. This is excellent agreement\nwith an inverse thickness dependence ∆ α(tFM) =KFM/tFM, where the prefactor clearly\ndepends on the FM layer, highest for Py and lowest for Co. Note tha t efforts to extract\n∆α(tFM) =Ktnwithout the FM( tFM)/Cu baselines would meet with significant errors;\nnumerical fits to α(tFM) =KtFMnfor the FM( tFM)/Cu/Pt structures yield exponents\nn≃1.4.\nExpressing now the additional Gilbert relaxation as ∆ G(tFM) =|γ|Ms∆α(tFM) =\n|γFM|MFM\nsKFM/tFM, we plot ∆ G·tFMin Figure 4. We find ∆ G·tPy= 192±40 Mhz,\n∆G·tCoFeB= 265±40 Mhz, and ∆ G·tCo= 216±40 Mhz. The similarity of values for\n∆G·tFMis in good agreement with predictions of the spin pumping model in Equa tion 1,\ngiven that interfacial spin mixing parameters are nearly equal in diffe rent systems.\nThe similarity of the ∆ G·tFMvalues for the different FM layers is, however, at odds\nwith expectations from the ”resistivity-like” mechanism. In Figure 4 ,inset, we show the\ndependence of ∆ G·tFMupon the tabulated λSDLof these layers from Ref17. It can be seen\nthatλCo\nSDLis roughly an order of magnitude longer than it is for the other two FM layers,\nPy and CoFeB, but the contribution of Pt overlayers to damping is ve ry close to their\naverage. Since under the resistivity mechanism, only Py and CoFeB s hould be susceptible\nto a resistivity contribution in ∆ α(tFM), the results imply that the contribution of Pt to\nthe nonlocal damping size effect has a separate origin.\nFinally, we compare the magnitude of the nonlocal damping size effect with that pre-\ndicted by the spin pumping model in Ref.10. According to ∆ G·tFM=|γ|2¯h/4π=\n25.69 Mhz ·nm3(gL/2)2/parenleftBig\ng↑↓\neff/S/parenrightBig\n, our experimental ∆ G·tFMandgLdata yield effective\nspin mixing conductances g↑↓\neff/S[Py/Cu/Pt ] = 6.8 nm−2,g↑↓\neff/S[Co/Cu/Pt ] = 7.3 nm−2,\nandg↑↓\neff/S[CoFeB/Cu/Pt ] = 9.6 nm−2. The Sharvin-corrected form, in the realistic limit\nofλN\nSDL≫tN11is (g↑↓\neff/S)−1= (g↑↓\nF/N/S)−1−1\n2(g↑↓\nN,S/S)−1+ 2e2h−1ρ tN+ (˜g↑↓\nN1/N2/S)−1.\nUsing conductances 14.1nm−2(Co/Cu), 15.0nm−2(Cu), 211nm−2(bulkρCu,tN= 3nm), 35\nnm−2(Cu/Pt) would predict a theoretical g↑↓\neff,th./S[Co/Cu/Pt ] = 14.1 nm−2. Reconciling\ntheory and experiment would require an order of magnitude larger ρCu≃20µΩ·cm, likely\nnot physical.\nTo summarize, a common methodology, controlling for damping size eff ects and intermix-\ning in single films, has allowed us to compare the nonlocal damping size eff ect in different\nFM layers. We observe, for Cu/Pt overlayers, the same power law in thickness t−1.04±0.06,\n5the same materials independence, but roughly half the magnitude th at predicted by the spin\npumping theory of Tserkovnyak10. The rough independence on FM spin diffusion length,\nshown here for the first time, argues against a resistivity-based in terpretation for the effect.\nWe would like to acknowledge the US NSF-ECCS-0925829, the Bourse Accueil Pro n◦\n2715 of the Rhˆ one-Alpes Region, the French National Research A gency (ANR) Grant ANR-\n09-NANO-037, and the FP7-People-2009-IEF program no 252067 .\nREFERENCES\n1R. Urban, G. Woltersdorf, and B. Heinrich, “Gilbert damping in single a nd multilayer\nultrathin films: role of interfaces in nonlocal spin dynamics,” Physical Review Letters 87,\n217204–7 (2001).\n2S. Mizukami, Y. Ando, and T. Miyazaki, “Effect of spin diffusion onGilber t damping for a\nverythinpermalloylayer inCu/permalloy/Cu/Pt films,”Phys. Rev. B 66, 104413 (2002).\n3B. Dieny, J. Nozieres, V. Speriosu, B. Gurney, and D. Wilhoit, “Chan ge in conductance\nis the fundamental measure of spin-valve magnetoresistance,” Ap plied Physics Letters 61,\n2111–3 (1992).\n4W. H. Butler, X. G. Zhang, D. 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Pratt, W.P., “Spin-diffusion lengths in metals and alloys, and spin-flipping\nat metal/metal interfaces: an experimentalist’s critical review,” Jo urnal of Physics: Con-\ndensed Matter 19, 41 pp. –(2007); C. Ahn, K.-H. Shin, andW. Pratt, “Magnetotran sport\nproperties of CoFeB and Co/Ru interfaces in the current-perpen dicular-to-plane geome-\ntry,” Applied Physics Letters 92, 102509 – 1 (2008).\n7FIGURES\nω/ 2π (Ghz)\nH  (Oe)B     1 / t      (nm  ) FM -1 \nFIG. 1. Fields for resonance ω(HB) for in-plane FMR, FM=Ni 81Fe19, 2.5 nm ≤tFM≤30.0 nm;\nsolid lines are Kittel fits. Inset:4πMeff\nsfor all three FM/Cu, with and without Pt overlayers.\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48\n/s49/s52/s48/s48 /s49/s53/s48/s48 /s49/s54/s48/s48 /s49/s55/s48/s48/s45/s48/s46/s48/s48/s48/s49/s48/s46/s48/s48/s48/s48/s48/s46/s48/s48/s48/s49/s48/s46/s48/s48/s48/s50/s48/s46/s48/s48/s48/s51\n/s49/s50/s32/s71/s72/s122/s72\n/s114/s101/s115\n/s72\n/s112/s112/s32 /s32/s78/s105\n/s56/s49/s70/s101\n/s49/s57/s40/s54/s110/s109/s41\n/s32 /s32/s78/s105\n/s56/s49/s70/s101\n/s49/s57/s40/s54/s110/s109/s41/s45/s80/s116\n/s32 /s32/s67/s111\n/s54/s48/s70/s101\n/s50/s48/s66\n/s50/s48/s40/s54/s110/s109/s41\n/s32 /s32/s67/s111\n/s54/s48/s70/s101\n/s50/s48/s66\n/s50/s48/s40/s54/s110/s109/s41/s45/s80/s116/s39/s39/s47 /s72/s40/s97/s46/s117/s46/s41\n/s70/s105/s101/s108/s100/s32/s40/s79/s101/s41\n/s32/s50/s46/s53/s110/s109\n/s32/s51/s46/s53/s110/s109\n/s32/s54/s110/s109\n/s32/s49/s48/s110/s109\n/s32/s49/s55/s46/s53/s110/s109\n/s32/s51/s48/s110/s109/s112/s112/s40/s79/s101/s41\n/s84/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41\nFIG. 2. Frequency-dependent peak-to-peak FMR linewidth ∆ Hpp(ω) for FM=Ni 81Fe19,tFM\nas noted, films with Pt overlayers. Inset:lineshapes and fits for films with and without Pt,\nFM=Ni 81Fe19, CoFeB.\n8      t     (nm)FM       t     (nm) FM α\nFIG. 3. Inset:αno Pt(tFM) andαPtfor Py, after linear fits to data in Figure 2. Main panel:\n∆α(tFM) =αPt(tFM)−αno Pt(tFM) for Py, CoFeB, and Co. The slopes express the power law\nexponent n= -1.04±0.06.\nλPy \nCoFeB\nCo \n      t     (nm)FM ∆G·t      (Mhz·nm) FM \n∆G·t     FM \n(nm) SDL \nFIG. 4. The additional nonlocal relaxation due to Pt overlay ers, expressed as a Gilbert relaxation\nrate - thickness product ∆ G·tFMfor Py, CoFeB, and Co. Inset:dependence of ∆ G·tFMon spin\ndiffusion length λSDLas tabulated in17.\n9"    },    {        "title": "1911.00728v1.Tuning_Non_Gilbert_type_damping_in_FeGa_films_on_MgO_001__via_oblique_deposition.pdf",        "content": "Tuning Non -Gilbert -type damping in  FeGa film s on MgO(001)  via oblique \ndeposition  \nYang Li1,2, Yan Li1,2, Qian Liu3, Zhe Yuan3, Qing -Feng Zhan4, Wei He1, Hao-Liang \nLiu1, Ke Xia3, We i Yu1, Xiang-Qun Zhang1, Zhao -Hua Cheng1,2,5 a) \n1State Key Laboratory of Magnetism and Beijing National Laboratory for Condensed \nMatter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, \nChina  \n2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing \n100049, China  \n3The Center for Advanced Quantum Studies and Department of Physics, Beijing \nNormal University, 100875 China  \n4State Key Laboratory of Precision Spectroscopy, School of Physics and Materials \nScience, East Ch ina Normal University, Shanghai 200241, China  \n5Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China   \na) Corresponding author , e-mail: zhcheng@iphy.ac.cn   \nAbstract  \nThe ability to tailor the damping factor is essential for spintronic and spin- torque \napplication s. Here, we report an approach to manipulate the damping factor of \nFeGa/MgO(001) films by oblique deposition. Owing to  the defects at the surface or \ninterface in thin films , two -magnon scatterin g (TMS)  acts as a non -Gilbert damping \nmechanism in magnetization relaxation.  In this work, the contribution of TMS was \ncharacterized  by in-plane angul ar dependent  ferromagnetic resonance (FMR) . It is \ndemonstrated that the intrinsic Gilbert damping is isotropic and invariant , while the \nextrinsic mechanism  related to TMS is anisotropic  and can be tuned by oblique \ndeposition. Furthermore, the two and fourfold TMS related to  the uniaxial magnetic \nanisotropy  (UMA)  and m agnetocrystalline anisotropy were discussed. Our result s open \nan avenue to manipulate  magnetization relaxation in spintronic devices.  \n1 \n Keywords : Gilbert damping , two -magnon scattering, FMR, oblique deposition, \nmagnetic anisotropy  \n2 \n 1. Introduction  \nIn the past decades, controlling magnetization dynamics in magnetic \nnanostructures has been extensively studied due to its great importance for spintronic \nand spin- torque applications  [1,2] . The magnetic relaxation is described within the \nframework of the Landa u-Lifshitz  Gilbert (LLG)  phenomenology  using the Gilbert \ndamping factor  α [3]. The intrinsic  Gilbert damping depends primarily on the spin- orbit \ncoupling (SOC) [4,5] . It has been demonstrated that alloying or doping with non-\nmagnetic transition metals  provides an opportunity to tune the intrinsic damping  [6,7] . \nUnfortunately, in this way the soft magnetic properties will reduce . In addition to the \nintrinsic damping, the two-magnon scatterin g (TMS) process se rves as a n important  \nextrinsic  mechanism i n magnetization relaxation  in ultrathin films  due to the defects at \nsurface or interface [8,9] . This process  describes the scattering between the uniform \nmagnons and degener ate final -state spin wave modes  [10]. The existence of TMS has \nbeen demonstrated in many systems of ferrites  [11-13]. Since the anisotropic scattering \ncenters , the angular dependence of the extrinsic TMS process  exhibi ts a strong in -plane \nanisotropy  [14], which allows  us to adjust the overall magnetic relaxation , including \nboth the int ensity of relaxation rate and  the anisotropic behavior.  \nHere, we report an approach to engineer  the damping factor of Fe81Ga19 (FeGa ) \nfilms by oblique deposition. The FeGa alloy exhibits large  magnetostriction and narrow \nmicrowave resonance linewidth  [15] , which  could assure  it as a  promising material for \nspintronic devices. For the geometry of off -normal deposition, it has been demonstrated \nto provoke shadow effects and create a periodic stripe defect matrix. This can introduce \na strong uniaxial magnetization anisotropy (UMA) pe rpendicular to the projection of \nthe atom flux  [16-19]. Even though some reports have shown oblique deposition \nprovokes a  twofold TMS channel  [20-22], the oblique angle dependence of the intrinsic \n3 \n Gilbert damping and the TMS  still remain in doubt. For our case, on the basis of the \nfirst-principles calculation and the in -plane angular -dependent FMR measurements, we \nfound that the intrinsic Gilbert damping is isotropic and invariant with varying oblique \ndeposition angles, while the extrinsic mechanism related to the two -magnon -scattering \n(TMS ) is anisotropic and can be tuned by oblique deposition. In addition, importantly \nwe firstly observe a phenomenon that the cubic magnetocrystalline anisotropy \ndetermines the area including degenerate magnon modes, as well as the intensity of fourfold TMS. In general , the strong connection between the extrinsic TMS and the \nmagnetic anisotropy , as well their direct impact on the damping constants , are \nsystem ically investigated, which offer us a useful approach to tailor the damping factor.\n \n2. Experimental details  \nFeGa thin films with a thickness of 20 nm were grown on MgO(001) substrates in \na magnetron sputtering system with a base pressure below 3 × 10−7 Torr. Prior to \ndeposition, t he substrates were annealed at 700 °C for 1 h in a vacuum chamber to \nremove surface contaminations and then held at 250 °C during deposition. The incident \nFeGa beam was at different obl ique angles of ψ =0°, 15°, 30°, and 45°, with respect to \nthe surface normal , and named S1, S2, S3, and S4 in this paper , respectively.  The \nprojection of FeGa beam  on the plane of the substrates  was set perpendicular to the \nMgO[110] direction, which induces a UMA perpendicular to the projection of FeGa \nbeam , i.e., parallel to the MgO[110] direction, due to the we ll-known  self-shadowing \neffect.  Finally , all the samples  were covered with a 5 nm  Ta capping layer to avoid \nsurface oxidation [see figure  1(a)]. The epitaxial relation  of \nFeGa(001)[ 110]||MgO (001)[ 100] was characterized by using t he X -ray in -plane Φ-\nscans , as described elsewhere [23]. Magnetic hy steresis loops were measured  at various \nin-plane magnetic field orientations φ H with respect to the FeGa [100] axis using \n4 \n magneto -optical Kerr effect (MOKE) technique at room temperature . The d ynamic \nmagnetic properties were investigated by broadband FMR measurements  based on a \nbroadband vector network analyzer  (VNA) with a  transmission geometry coplanar \nwaveguide (VNA- FMR ) [24]. This setup allows both frequency and field- sweeps \nmeasurements with external field applied parallel to the sample plane.  During \nmeasurements,  the sampl es were placed face down on the coplanar wavegu ide and the \ntransmission coefficient S 21 was recorded.  \n3. Results and discussion \nFigure  1(b) displays the Kerr  hysteresis loops  of sample  S1 and S4 recorded along \nwith the main crystallographic directions of FeGa [100], [110],  and [010] . The sample \nS1 exhibit s rectangular hysteresis curves with sm all coercivities for the magnetic  field \nalong  [100] and [010]  easy axes. In contrast, the S4 displays a  hysteresis curve with \ntwo step s for the magnetic field along the [010]  axis, which indicates  a UMA along the \nFeGa[100 ] axis superimposed on the four fold magnetocrystalline anisotropy . As a \nresult,  with increasing the oblique angle, the angular dependence of normalized remnant \nmagnetization ( Mr/Ms) gradually reveals a four fold symmetry combined with a uniaxial \nsymmetry, as shown i n the inset of f igure 1(b).  \nSubsequently,  the magnetic anisotropic properties can be further precisely \ncharacterized  by the in -plane angular -dependent FMR measurements.  Figure 1(c) and \n1(d) show  typical  FMR spectra for the real and imaginary part s of coefficient S 21 for \nthe sample S2 . Recorded  FMR spectra contain a symmetric and an antisymmetric \nLorentzian peak , from which the resonant field H r with linewidth ∆𝐻𝐻 can be obtained  \n[24,25] . \nFigure 2(a) shows the in -plane angular dependence of H r measured at 13 .0 GHz  \nand can be fitted  by the following expression [26,27] : \n5 \n 𝑓𝑓=𝛾𝛾𝜇𝜇0\n2𝜋𝜋�𝐻𝐻𝑎𝑎𝐻𝐻𝑏𝑏                                                     (1 ) \nHere,𝐻𝐻𝑎𝑎=𝐻𝐻4(3+𝑐𝑐𝑐𝑐𝑐𝑐4𝜑𝜑M)/4+𝐻𝐻u𝑐𝑐𝑐𝑐𝑐𝑐2𝜑𝜑M+𝐻𝐻r𝑐𝑐𝑐𝑐𝑐𝑐(𝜑𝜑M−𝜑𝜑H)+𝑀𝑀eff and 𝐻𝐻𝑏𝑏=\n𝐻𝐻4𝑐𝑐𝑐𝑐𝑐𝑐4𝜑𝜑M+𝐻𝐻u𝑐𝑐𝑐𝑐𝑐𝑐2𝜑𝜑M+𝐻𝐻r𝑐𝑐𝑐𝑐𝑐𝑐(𝜑𝜑M−𝜑𝜑H), H4 and Hu represent the fourfold \nanisotropy field and the UMA field caused by the self -shadowing effect , respectively. \n𝜑𝜑H(𝜑𝜑M) is the  azimuthal angles of the applied field ( the tipped magnetization ) with \nrespect to the [100] direction , as depicted  in figure 1(a). 𝜇𝜇0𝑀𝑀eff=𝜇𝜇0𝑀𝑀𝑠𝑠−2𝐾𝐾out\n𝑀𝑀𝑠𝑠, Ms is \nthe saturation magnetization  and Kout is the out -of-plane uniaxial anisotropy constant . \n𝑓𝑓 is the resonance frequency , 𝛾𝛾 is the gyromagnetic ratio and here used as the accepted \nvalue for Fe films, 𝛾𝛾=185 rad GHz/T  [28] . \nThe angular dependent  Hr reveals  only a  fourfold symmetry for the none -\nobliquely deposited sample , which indicates  the cubic lattice texture of FeGa on MgO . \nWith increasing the oblique angle , a uniaxial symmetry is found  to be superimposed \non the four fold symmetry, clearly confirming a UMA is produced by the oblique \ngrowth , which agrees with the MOKE’ results.  The fitted  parameter  𝜇𝜇0𝑀𝑀eff=1.90±\n 0.05T is found to be independent on the oblique deposition and close to 𝜇𝜇0𝑀𝑀𝑠𝑠=\n1.89 ± 0.02 T  estimated using VSM, which is almost same as the value  of the  \nliterature  [29] . This indicates negligible out- of-plane ma gnetic anisotropy in the thick \nFeGa films . As shown in figure 2(b), it is observed  that the UMA (Ku=HuMs/2) exhibits \na general increasing  trend with oblique angle , which coincides with the fact  the \nshadowing effect is stronger  at larger  angles of incidence [16-19]. Interestingly the \noblique deposition also affects the cubic anisotropy  K4 (K4=H4Ms/2). Different from  \nthe K4 increases slightly with deposition angle  in Co/Cu system  [16], here the value of \nK4 is the lowest at a n oblique angle of 15°. It is well known th at film stress significantly \ninfluences  the crystallization tendency  [30,31] . FeGa alloy is highly stress ed sensitive \n6 \n due to its larger magnetostriction . Thus, t he change in  K4 of FeGa films  may be \nattributed to the anisotropy dispersion created due to the stress variations  during grain \ngrowth. It should be mentioned that the best way to determine magnetic parameters is \nto measure the out -of-plane  FMR . But the effective saturation magnetization 𝜇𝜇0𝑀𝑀eff=\n1.90T of FeGa alloy leads to the perpendicular applied field beyond our instrument \nlimit. Meanwhile, t he results obtained above are also in accord with those extracted \nby fitting field dependence of the resonance frequency with H//FeGa[100]  shown in \nfigure 2(c).  \nThe effective Gilbert damping 𝛼𝛼eff is extracted by linearly fitting the dependence  \nof linewidth  on frequency : 𝜇𝜇0∆H=𝜇𝜇0∆H0+2𝜋𝜋𝜋𝜋𝛼𝛼𝑒𝑒ff\n𝛾𝛾, where ∆𝐻𝐻0 is the inhomogeneous \nbroadening. For the sake of clarity , figure 3(a) only shows the frequency  dependence \nof linewidth for the  samples S1 and S2 along  [110]  and [100]  axes. It is evident that, \nfor the sample  S1, both linear slopes of two direction s are almost  same. While w ith \nregard to the sample S2 , the slope of the ∆H-f curve along  the easy axis is approximately \na factor of 2 greater than that of the hard axis. The obtained values of  𝛼𝛼eff are shown in \nfigure 3(b). Firstly, the results clearly indicate that the effective damping exhibits \nanisotropy , with  higher value along the easy axis . Secondly, f or the easy  axis, the \noblique angle dependence on the damping parameter indicates an extraordinary trend \nand has a peak at  deposition angle 15°. However, the damping shows an increasing \ntrend  with the oblique angle for the field along the hard axis. In the following part, we \nwill explore the effect of oblique deposition on the mechanism of the anisotropic \ndamping and the magnetic relaxation pr ocess.     \nSo far, convincing experimental evidence is still lacking to prove the existence of \nanisotropic damping in bulk magnets. Chen et al. have shown the emergence of \nanisotropic Gilbert damping in ultrathin Fe (1.3nm)/GaAs and its anisotropy disappears \n7 \n rapidly when the Fe thickness  increases  [32]. We perform  the first-principles \ncalculation of the Gilber t damping of Fe Ga alloy considering  the effect induced by the \nlattice distortion. W e artificially make a tetragonal lattice with varying the lattice \nconstant of the c -axis. The electronic structure of Fe -Ga alloy is calculated self -\nconsistently using the coherent potential approx imation implemented with the tight-\nbinding linear muffin- tin orbitals. Then the atomic potentials of Fe and Ga are randomly \ndistributed in a 5× 5 lateral supercell, which is connected to two semi -infinite Pd leads . \nA thermal lattice disorder is included via  displacing atoms randomly from the perfect \nlattice sites following a Gaussian type of distribution [ 33]. The root -mean -square \ndisplacement at room temperature is determined by the Debye model with the Debye \ntemperature 470 K. The length of the supercell is variable and the calculated total \ndamping is scaled linearly with this length. Thus, a linear least- squares fitting can be \nperformed to extract the bulk damping of the Fe -Ga alloy  [34].  The calculated Gilber t \ndamping is plotted in f igure 3(c) as a funct ion of the lattice distortion (𝑐𝑐−𝑎𝑎)𝑎𝑎⁄.  The \nGilbert damping is nearly independent of the lattice distortion and there is no evidence of anisotropy in t he intrinsic bulk damping of Fe Ga alloy.  \nSo the extrinsic contributions  are responsible for  the anisotropic behavior of \ndamping , which can be separated from the in -plane angular dependent linewidth. The \nrecorded FMR  linewidth  have the following different cont ributions  [11] :  \n     𝜇𝜇0∆𝐻𝐻=𝜇𝜇0∆𝐻𝐻inh+2𝜋𝜋𝛼𝛼𝐺𝐺𝑓𝑓\n𝛾𝛾𝛾𝛾+�𝜕𝜕𝐻𝐻r\n𝜕𝜕𝜑𝜑H∆𝜑𝜑H�+�Γ<𝑥𝑥𝑖𝑖>𝑓𝑓�𝜑𝜑H−𝜑𝜑<𝑥𝑥𝑖𝑖>�\n<𝑥𝑥𝑖𝑖>𝑎𝑎𝑎𝑎𝑐𝑐𝑐𝑐𝑎𝑎𝑎𝑎 \n                �(�𝜔𝜔2+(𝜔𝜔0\n2)2−𝜔𝜔0\n2)/(�𝜔𝜔2+(𝜔𝜔0\n2)2+𝜔𝜔0\n2)+Γtwofoldmaxcos4(φM- φtwofold)   (2) \n∆Hinh is both frequency and angle independent  term due to the  sample  \ninhomogeneity . The second term is the intrinsic Gilbert damping  (𝛼𝛼𝐺𝐺) contribution. 𝛾𝛾  \n8 \n is a correction factor owing to the field dragging effect caused by magnetic anisotropy  \n[12], 𝛾𝛾 =cos (φM-φH). The 𝜑𝜑M as a function of φH  for the sample S2 at fixed 13 GHz \nis calculated  and show n in figure 4(a).  Note that the draggi ng effect  vanishes  (𝜑𝜑M=\nφH) when the field is along the hard or  easy axes . The third term  describes the mosaicity \ncontribution originating from the angular dispersion of the crystallographic cubic axes \nand yield s a broader linewidth  [35]. The four th term is the TMS contribution. The \nΓ<𝑥𝑥𝑖𝑖> signifies  the intensity  of the TMS along the principal in -plane crystallographic \ndirection  <𝑥𝑥𝑖𝑖>. The 𝑓𝑓�𝜑𝜑H−𝜑𝜑<𝑥𝑥𝑖𝑖>� term indica tes the TMS contribution depending \non the in- plane direction of the field rel ative to <𝑥𝑥𝑖𝑖> and commonly expressed as \ncos2[2(φM-φ<xi>)] [14].  In addition, 𝜔𝜔 is the angular resonant frequency and  𝜔𝜔0=\n𝛾𝛾𝜇𝜇0𝑀𝑀eff. In our case, besides the fourfold TMS caused by expected lattice geometric \ndefects, the other twofold TMS channel is induced by the dipolar fields emerging from \nperiodic stripelike defects  [20,21] . This term is parameterized by its strength Γtwofoldmax \nand the axis of maximal scattering  rate φtwofold.  \nAs an example, t he angle- dependent linewidth measured  at 13 .0 GHz for  the \nsample S2 is  shown in f igure 4(b).  It clearly exhibits a strong in -plane anisotropy, and  \nthe linewidth along the [100] direction is significantly  larger than that along the [110] \ndirection . Taking only  isotropic Gilbert damping  into account , the dragging effect \nvanishes  with field applied  along the hard and easy axes . Meanwhile, the mosaicity \nterm gives an angular variation of the linewidth proportional to |𝜕𝜕𝐻𝐻𝑟𝑟𝜕𝜕𝜑𝜑𝐻𝐻|⁄ , which is \nalso zero along with the principal <100> and < 110>  directions. This gives direct \nevidence that the rel axation is not exclusively governed only considering the intrinsic \nGilbert mechanism and mosaicity term. Because the probability of defect formation \nalong with <100> directions is higher than that  along the <110> directions  [12], the \n9 \n TMS contribution is stronger along the easy axes , which is in accordance with t he fact \nthat the linewidth s along the [100] and [110]  direction s are non -equivalent. Moreover , \nthe linewidth of [010] direction is slightly  larger  than that along the [100] dire ction, \nsuggesting that  another twofold TMS channel is induce d by oblique deposition. As \nindicated by the red solid line  in figure 4(b), the linewidth can be well fitted. D ifferent \nparts making sense to the  linewidth can therefore be sepa rated  and summarized in Tab le \nI. As we know, the TMS predicts the curved non- linear frequency dependence of \nlinewidth, which not appear in a small frequency range for our case (as shown in f igure \n3(a)).  The linewidth as function of frequency was also well fitted including the TMS -\ndamping using the parameters in Table I  (not shown here) . \nThe larger strength of TMS along the easy axis can clearly explain the anisotropic \nbehavior of da mping , with higher value along the easy axis shown in f igure 3(b). The \nobtained Gilbert damping  factor  of ~ 7×10-3 is isotropic and invariant  with different \noblique angle s. The value of damping is slightly larger than the bulk value of 5.5×10-3 \n[29], which may be attributed to spin pumping of the Ta capping layer.   \n The obtained maxi ma of twofold TMS exhibits  an increasing  trend  with the \noblique angle  [shown in f igure 4(c)]. According  to previous works  on the shadowing \neffect [16-19], the larger  deposition angle  makes  the shadow ing effect  stronger , and the \ndipolar fields within stripe like defects  increase just like the UMA. This  can clearly \nexplain that the intensity of two fold TMS  follows exactly the same trend with the \ndeposition angle as the UMA . The axis of the maximal intensity of two fold TMS is \nparal lel to the projection of the  FeGa atom flux  from the fitting data.   As shown in Table  \nI, amazingly the modified growth conditions also influence  the fourfold TMS, \nespecially the strength of TMS along the <100>  axis. Figure 4(c) also presents the \nchanges of the fourfold TMS intensity as the  deposition angle  and shows a peak at 15° , \n10 \n which follows a similar trend as that of 𝛼𝛼eff along [100] axis as shown in f igure 3(b). \nThis indeed confirm s TMS -damping plays an important role in FeGa thin films.    \nFor the dispersion relation ω(k∥) in thin magnetic films , the propagation angle \n𝜑𝜑𝑘𝑘∥����⃗ defined as the angle between k∥���⃗ and the projection of the saturation magnetization \nMs into the sample plane is less than the critical value : 𝜑𝜑max =\n𝑐𝑐𝑎𝑎𝑎𝑎−1�𝜇𝜇0𝐻𝐻r(𝜇𝜇0𝐻𝐻r+𝜇𝜇0𝑀𝑀eff) ⁄  [9,36,37] . This implies  no degenerate modes are \navailable for the angle 𝜑𝜑𝑘𝑘∥����⃗ larger than φmax. Based on this theory, we propose a \nhypothesis that the crystallographic anisotropy  determine s the area including \ndegenerate magnon modes , as well as  the intensity of the fourfold TMS.  The resonance \nfield along <100>  axis change s due to the various crystallographic anisotropy , which \nhas a great effect on the φmax. The values of φmax of samples are shown in f igure 4(d). \nThe data follow the  same trend with the oblique angle as Γ<100>. During the grain \ngrowth,  the cubic anisotropy is  influenced possibly  since the anisotropy dispersion due \nto the stress. For the lower anisotropy of  sample S2 , a relatively larger  amount of stress \nand defects present in the sample and lead to a larger  four fold TMS.  \n4. Conclusions  \nIn conclusion, the effects of oblique deposition on the dynamic properties of FeGa \nthin films have been investigated  systematically . The pronounced TMS  as non-Gilbert \ndamping  results in an anisotropic magnetic relaxation . As the oblique angle increases, \nthe magnitude of the twofold TMS  increases due to the larger shadowing effect . \nFurthermore, the cubic anisotropy dominates  the area including degenerate magnon \nmodes, as well as the intensity of fourfold TMS. The reported results  confirm  that the \nmodified anisotropy can influence the extrinsic relaxation pr ocess  and open a n avenue \nto tailor magnetic  relaxation in spintronic devices. \n11 \n Acknowledgments  \nThis work is supported by the National Key Research Program of China (Grant Nos. \n2015CB921403, 2016YFA0300701, and 2017YFB0702702), the National Natural \nSciences Foundation of China (Grant Nos. 91622126, 51427801, and 51671212)  and \nthe Key Research Program of Frontier Sciences, CAS (Grant Nos. QYZDJ -SSW -\nJSC023, KJZD -SW-M01 and ZDYZ2012 -2). The work at Beijing Norma l University \nis partly supported by the National Natural Sciences Foundation of China (Grant Nos. 61774017, 61704018, and 11734004), the Recruitment Program of Global Youth \nExperts and the Fundamental Research Funds for the Central Universities (Grant No. \n2018EYT03).  \n12 \n References  \n[1] Slonczewski J C 1996 J. Magn. Magn. Mater. 159  L1 \n[2] Žutić  I, Fabian J and Das Sarma S 2004 Rev. Mod. Phys. 76 323 \n[3] Gilbert T L2004 IEEE Trans. Magn. 40 3443  \n[4] He P, Ma  X, Zhang  J W, Zhao  H B, Lüpke  G, Shi  Z and Zhou S M 2013 Phys. \nRev. Lett.  110 077203 \n[5] Heinrich B, Meredith D J and Cochran J F 1979 J. Appl. Phys. 50 7726 \n[6] Lee A J, Brangham  J T, Cheng  Y, White  S P, Ruane  W T, Esser  B D, \nMcComb  D W, Hammel  P C and Yang  F Y 2017 Nat. Commun. 8 234 \n[7] Scheck C, Cheng L, Barsukov I, Frait Z and Bailey W E 2007 Phys. Rev. Lett.  \n98 117601 \n[8] Azzawi  S, Hindmarch A and Atkinson D 2017 J. Phys. D: Appl. 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Phys. 98 033901   \n[29] Kuanr  B K, Camley  R E, Celinski  Z, McClure  A and Idzerda  Y 2014 J. Appl. \nPhys.  115 17C112 \n[30] Jhajhria  D, Pandya  D K and Chaudhary  S 2018 J. Alloy Compd. 763 728 \n[31] Jhajhria  D, Pandya  D K and  Chaudhary  S 2016 RSC Adv.  6 94717  \n[32] Chen L et al  2018 Nat. Phys . 14 490 \n[33] Liu Y, Starikov A A, Yuan Z and Kelly P J 2011 Phys. Rev. B  84 014412  \n14 \n [34] Starikov A A, Liu  Y, Yuan  Z and Kelly  P J 2018 Phys. Rev. B 97 214415  \n[35] McMichael R D, Twisselmann D J and Kunz A 2003 Phys. Rev. Lett. 90 227601  \n[36] Arias R and Mills D L  2000 J. Appl. Phys. 87 5455 \n[37] Lindner J, Barsukov I, Raeder C, Hassel C, Posth O, Meckenstock R, Landeros \nP and Mills D L 2009 Phys. Rev. B  80 224421  \n \n \n  \n   \n            \n15 \n Figure Captions  \nFigure  1 (color online) (a) Schematic illustration of the film deposition geometry and \ncoordinate system (b) I n-plane hysteresis loops of samples S1 and S4 with the field \nalong [100], [110], and [010]. The inset shows the polar plot of the normalized \nremanence (M r/Ms) as a functi on of the in- plane angle.  FMR spectrum for the sample \nS2 with H along [100] and [110] axes showing the real (c ) and imaginary (d ) part s of \nthe S 21.  \nFigure  2 (color online) (a) H r vs. φH for FeGa films. (b) The  anisotropy  constants  K4 \nand Ku vs. deposition angle. (c) f  vs. Hr plots measured at H //[100], Symbols are \nexperimental data and the solid lines are the fitted results.  \nFigure  3 (color online) (a) ∆H as a function of f  for samples S1 and S2 with  field along \neasy and hard axis. (b) The dependence of the damping parameter on  the oblique angle \nwith field along [100] and [110] directions. (c) The calculated damping of FeGa alloy \nas a function of lattice distortion.  Figure  4 (color online)  (a) φ\nM and (b) ∆H as a function of φH for the  sample S2  \nmeasured at 13.0 GHz. (c) Oblique angle dependences of Γ<100> and Γtwofoldmax. (d) The \nlargest angle including degenerate magnon modes as a function of the oblique angle \nwith the applied field along <100> direction. \nTable Caption  \nTable I. The magnetic relaxation parameters of the FeGa films prepared via oblique \ndeposition (with experimental errors in parentheses).  \n \n  \n16 \n Figure 1  \n \n \n  \n \n  \n \n  \n \n  \n \n17 \n Figure  2 \n \n \n \n \n \n  \n \n  \n \n  \n \n  \n \n18 \n Figur e 3 \n \n \n  \nFigure  4 \n \n \n \n \n \n \n \n19 \n TableⅠ \nSample  𝜇𝜇0ΔHinh\n(mT) 𝛼𝛼G Δ𝜑𝜑H \n(deg.) Γ<100> \n(107Hz) Γ<110> \n(107Hz) Γtwofoldmax\n(107Hz) 𝜑𝜑twofold  \n(deg. ) \nS1 0 0.007  0.62 17(3)  5.8(1.8)  0(2) 90 \nS2 0.7 0.007  1.2 81.4(3.7)  9.3(1.9)  7.4(3) 90 \nS3 0 0.007  1.0 59.2(4.5)  11.1(2) 13(3.7)  90 \nS4 0 0.007  1.1 33.3(6)  14.8(3.7)  26(4)  90 \n \n20 \n "    },    {        "title": "2211.07744v2.Magnetization_Dynamics_in_Synthetic_Antiferromagnets_with_Perpendicular_Magnetic_Anisotropy.pdf",        "content": "1 \n Magnetization Dynamics in Synthetic Antiferromagnets with Perpendicular \nMagnetic Anisotropy  \n \nDingbin Huang1,*, Delin Zhang2, Yun Kim1, Jian-Ping Wang2, and Xiaojia Wang1,* \n1Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455, USA  \n2Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN \n55455, USA  \n \n \nABSTRACT:   \nUnderstanding the rich physics of magnetization dynamics in  perpendicular  synthetic \nantiferromagnets  (p-SAFs)  is crucial  for developing next-generation  spintronic devices.  In this \nwork, we systematically investigate  the magnetization dynamics in p-SAFs  combining time-\nresolved magneto -optical Kerr effect (TR -MOKE)  measurements with theoretical modeling . \nThese model analyses, based on a Landau -Lifshitz -Gilbert approach incorporating exchange \ncoupling , provide detail s about the  magnetization dynamic characteristics  including  the amplitude s, \ndirections, and phases of the precession  of p-SAFs under varying magnetic fields . These model -\npredicted characteristics are in excellent quantitative agreement with TR-MOKE measurements \non an asymmetric p -SAF. We further reveal the damping mechanisms of two precession modes  \nco-existing in the p -SAF and successfully identify individual contributions from different sources , \ninclud ing Gilbert damping of each ferromagnetic layer , spin  pumping, and inhomogeneous \nbroadening . Such a comprehensive understanding of magnetization dynam ics in p -SAFs, obtained \n                                                 \n*Author s to whom correspondence should be addressed : huan1746@umn.edu  and wang4940@umn.edu  2 \n by integrating high -fidelity TR -MOKE measurements and theoretical modeling, can guide  the \ndesign of p-SAF-based architectures for spintronic  applications . \n \nKEYWORDS:  Synthetic antiferromagnets; Perpendicular magnetic anisotropy; Magnetization \nDynamics; Time -resolved magneto -optical Kerr effect; Spintronics   3 \n 1 INTRODUCTION  \nSynthetic antiferromagnet ic (SAF) structures have attracted considerable  interest for \napplications in spin mem ory and logic devices  because of their unique magnetic  configuration s [1-\n3]. The SAF structures are composed of two ferromagnetic (FM) layers anti -parallelly coupled \nthrough a non -magnetic (NM) spacer, offer ing great  flexibilit ies for the  manipulat ion of magnetic \nconfigurations  through  external stimuli  (e.g., electric -field and spin-orbit torque , SOT) . This \npermit s the design of new architecture s for  spintronic applications , such as magnetic tunnel \njunct ion (MTJ), SOT  devices, domain wall devices, sky rmion devices, among others  [4-7]. The \nSAF structures  possess many advantages for such applications , including  fast switching speeds \n(potentially in the THz regimes),  low off set fields, small switching current s (and thus low energy \nconsumption) , high thermal stability,  excellent resilience  to perturbations from external magnetic  \nfields, and large turnabilit y of magnetic properties  [3,8-16].  \nA comprehensive study  of the magnetization dynami cs of SAF structures  can facilitate the \nunderstanding of the switching behavior of spintronic  devices , and ultimately guide the design of \nnovel device architectures . Different from a single FM free layer, magnetization dynamics of the \nSAF structures involv es two modes of precession , namely high -frequency (HF) and low -frequency  \n(LF) modes,  that result from the hybridization of magnetizations  precession  in the two FM layers . \nThe relative phase and precession amplitude in two FM layers can significantly affect the spin-\npumping enhancement of magnetic damping  [17], and thus play an important role in determining \nthe magnetization dynamic behaviors in SAFs. Heretofore, the exchange -coupling strength and \nmagnetic damping constant of SAFs have been studied by ferromagnetic resonance (F MR)  [18-\n21] and optical metrolog y [22-25]. Most FMR -based experimental studies were limited to SAFs \nwith in -plane magnetic anisotropy (IM A). For device applications, perpendicular magnetic 4 \n anisotropy (PMA) gives  better scalability  [3,26] . Therefore , the characteristics of magnetization \ndynamics of perpendicular SAF (p-SAF) structures  are of much valu e to investigat e. In addition, \nprior  studies mainly focus ed on the mutual  spin pumping between two FM layers  [22,27,28] . A \nmore thorough understanding of the contribution s from various sources, including inhomogeneous \nbroadening  [29], remains elusive . \nIn this paper, we report  a comprehensive study of the magnetization dynamics of p -SAFs  by \nintegrating high -fidelity experiments and theoretical modeling to detail the characteristic \nparameters. These parameters describe the amplitude, phase, and direction of magnetization \nprecess ion of both the HF and LF modes for the two exchange -coupled FM layers in a p -SAF.  We \nconduct all -optical time -resolved magneto -optical Kerr effect (TR -MOKE) measurements  [30-33] \non an asymmetric p -SAF structure with two different FM layers. The field-dependent  amplitude \nand phase of TR -MOKE signal s can be well captured by our theoretical  model, which in turn \nprovid es comprehensive physical insights into the magnetization dynamics of p -SAF structures.  \nMost importantly, we show that inhomogeneous broadening  plays a  critical  role in determining \nthe effective damping of both HF and LF modes, especially at low fields.  We demonstrate the \nquantification of contributions from inhomogeneous broadening  and mutual spin pumping  (i.e., \nthe exchange of angular momentum between two FM layers via pumped spin currents ) [21] to the \neffect ive damping, enabl ing accurate determination of the Gilbert damping for individual FM \nlayers. Results of this work  are beneficial  for designing  p-SAF-based architectures  in spintronic  \napplication s. Additionally, this work also serves as a successful example demonstrating that  TR-\nMOKE, as an all -optical met rology, is a powerful tool to capture the magnetization dynamics and \nreveal the rich physics of complex structures that involve multilayer coupling . \n 5 \n 2 METHODOLOTY  \n2.1 Sample preparation and characterization  \nOne SAF  structure  was deposited onto thermally oxidize d silicon  wafers with a 300 -nm SiO 2 \nlayer by magnetron sputtering at room  temperature (RT) in a six -target ultra -high vacuum (UHV) \nShamrock sputtering system. The  base pressure is below  5×10−8 Torr. The stacking structure of \nthe SAF is:  [Si/SiO 2]sub/[Ta(5)/Pd(3)] seed/[Co(0.4)/Pd(0.7)/Co(0.4)] FM1/[Ru(0.6)/Ta(0.3)] NM/ \nCoFeB(1) FM2/[MgO(2)/Ta(3)] capping . The numbers in parentheses denote the layer thicknesses in \nnanometers. After deposition, the sample was annealed at 250 ℃ for 20 minutes by a rapid -\nthermal-annealing process.  The two FM layers are CoFeB and Co/Pd/Co layers, separated by a \nRu/Ta spacer, forming an asymmetric p -SAF structure ( i.e., two FM layers having different \nmagnetic properties). The M-Hext loops were characterized by a physical propert y measurement \nsystem (PPMS) with a vibrating -sample magnetometer (VSM) module. The resulting M-Hext loops \nare displayed in Fig. 1(a). Under low out -of-plane fields ( Hext < 500 Oe), the total magnetic \nmoments in two FM layers of the SAF stack perfectly cancel out each other: M1d1 = M2d2 with Mi \nand di being the magnetization and thickness of each FM layer ( i = 1 for the top CoFeB layer and \ni = 2 for the bottom Co/Pd/Co laye r). The spin-flipping field ( Hf  ≈ 500 Oe ) in the out -of-plane \nloop indicates the bilinear interlayer -exchange -coupling (IEC) J1 between the two FM layers : J1 = \n−HfMs,1d1 ≈ −0.062 erg cm-2 [34]. The values of Ms,1, Ms,2, d1, and d2 can be found in Table SI of \nthe Supplemental Material (SM)  [35]. \n \n2.2 Theoretical foundation  of magnetization dynamics for a p -SAF structure  \nThe magnetic free energy per unit area for a p -SAF structure with uniaxial PMA can be \nexpressed as [36]: 6 \n 𝐹=−𝐽1(𝐦1⋅𝐦2)−𝐽2(𝐦1⋅𝐦2)2\n+∑2\n𝑖=1𝑑𝑖𝑀s,𝑖[−1\n2𝐻k,eff,𝑖(𝐧⋅𝐦𝑖)2−𝐦𝑖⋅𝐇ext] (1) \nwhere J1 and J2 are the strength of the bilinear and biquadratic IEC. mi = Mi / Ms,i are the normalized \nmagnetization vectors for individual FM layers ( i = 1, 2). di, Ms,i, and Hk,eff, i denote, respectively, \nthe thickness, saturation magnetization, and the effective anisotropy field of the i-th layer. n is a \nunit vector indicating the sur face normal direction of the film. For the convenience of derivation \nand discussion, the direction of mi is represented in the spherical coordinates by the polar angle θi \nand the azimuthal angle φi, as shown in Fig. 1 (b). \nThe equilibrium direction of magne tization in each layer (𝜃0,𝑖,𝜑0,𝑖) under a given Hext is \nobtained by minimizing F in the (𝜃1,𝜑1,𝜃2,𝜑2) space. The magnetization precession is governed \nby the Landau -Lifshitz -Gilbert  (LLG)  equation considering the mutual spin pumping between two \nFM layers [27,37 -40]: \n𝑑𝐌𝑖\n𝑑𝑡=−𝛾𝑖𝐌𝑖×𝐇eff,𝑖+(𝛼0,𝑖+𝛼sp,𝑖𝑖)\n𝑀s,𝑖𝐌𝒊×𝑑𝐌𝒊\n𝑑𝑡−𝛼sp,𝑖𝑗\n𝑀s,𝑖𝐌𝒊×(𝐦𝐣×𝑑𝐦𝒋\n𝑑𝑡)×𝐌𝒊 (2) \nOn the right -hand side of Eq. (2), the first term describes the precession with the effective field \nHeff,i in each layer, given by the partial derivative of the total free energy in the M space via 𝐇eff,𝑖=\n−∇𝐌𝑖𝐹. The second term represents the relaxation induced by Gilbert damping ( α) of the i-th layer, \nwhich includes the intrinsic ( 𝛼0,𝑖) and spin -pumping -enhanced ( 𝛼sp,𝑖𝑖) damping. For TR -MOKE \nmeasurements, 𝛼0,𝑖 and 𝛼sp,𝑖𝑖 are indistinguishable. Hence, we def ine 𝛼𝑖=𝛼0,𝑖+𝛼sp,𝑖𝑖 to include \nboth terms. The last term in Eq. (2) considers the influence of pumped spin currents from the layer \nj on the magn etization dynamics of the layer  i.  7 \n The time evolution of Mi can be obtained by solving the linearized Eq. (2). Details are provided \nin Note 1  of the SM  [35]. The solutions to Eq. (2) in spherical coordinates are:  \n[𝜃1(𝑡)\n𝜑1(𝑡)\n𝜃2(𝑡)\n𝜑2(𝑡)]=[𝜃0,1\n𝜑0,1\n𝜃0,2\n𝜑0,2]+[Δ𝜃1(𝑡)\nΔ𝜑1(𝑡)\nΔ𝜃2(𝑡)\nΔ𝜑2(𝑡)]=[𝜃0,1\n𝜑0,1\n𝜃0,2\n𝜑0,2]+\n[    𝐶𝜃,1HF\n𝐶𝜑,1HF\n𝐶𝜃,2HF\n𝐶𝜑,2HF]    \nexp(𝑖𝜔HF𝑡)+\n[    𝐶𝜃,1LF\n𝐶𝜑,1LF\n𝐶𝜃,2LF\n𝐶𝜑,2LF]    \nexp(𝑖𝜔LF𝑡) (3) \nwith Δ𝜃𝑖 and Δ𝜑𝑖 representing the deviation angles of magnetization from its equilibrium direction \nalong the polar and azimuthal directions . The last two terms are the linear combination of two \neigen -solutions, denoted by  superscripts HF (high -frequency mode) and LF (low -frequency mode). \nω is the complex angular frequencies of two modes, with the real and imaginary parts representing \nthe precession angular frequency ( 𝑓/2𝜋) and relaxation rate (1/ τ), respectively. For each mode, \nthe complex prefactor vector [𝐶𝜃,1,𝐶𝜑,1,𝐶𝜃,2,𝐶𝜑,2]𝑇 contains detailed information about the \nmagnetization dynamics. As illustrated in Fig. 1 (c), the moduli, |𝐶𝜃,𝑖| and |𝐶𝜑,𝑖| correspond to the \nhalf cone angles of t he precession in layer i along the polar and azimuthal directions for a given \nmode immediately after laser heating, as shown by Δ𝜃 and Δ𝜑 in Figs. 1 (b-c). The phase \ndifference between Δ𝜃𝑖 and Δ𝜑𝑖, defined as Arg(Δ𝜃𝑖/Δ𝜑𝑖)=Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖) with Arg  \nrepresenting the argument of complex numbers, determines the direction of precession. If Δ𝜃𝑖 \nadvances Δ𝜑𝑖 by 90°, meaning Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖)=90°, the precession is counter -clockwise (CCW) \nin the θ-φ space  (from a view against Mi).Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖)=−90°, on the contrary, suggests \nclockwise (CW) precession [ Fig. 1 (d)]. Further, the argument of 𝐶𝜃,2/𝐶𝜃,1 provides the relative \nphase in two FM layers. Arg(𝐶𝜃,2/𝐶𝜃,1)=0° corresponds to the precession motions in two FM \nlayers that are in -phase (IP) in terms of θ for a given mode. While the out -of-phase (OOP) \nprecession in terms of θ is represented by Arg(𝐶𝜃,2/𝐶𝜃,1)=180°  [Fig. 1 (e)]. Given the precession 8 \n direction in each  layer and the phase difference between the two FM layers in terms of θ, the phase \ndifference in terms of φ can be automatically determined.   \n \n \nFIG. 1 (a) Magnetic hysteresis ( M-Hext) loops of the p -SAF stack. The magnetization is n ormalized \nto the saturation magnetization ( M/Ms). (b) Schematic illustration of the half cone angles (Δ θ and \nΔφ) and precession direction of magnetization. The precession direction is defined from a view \nagainst the equilibrium direction ( 0, φ0) of M. The representative precession direction in the \nschematic is counterclockwise (CCW). (c) The relation between precession half cone angles and \nthe prefactors. (d) The relation between precession direction and the prefactors. (e) The relative \nphase between two FM layers for different prefactor values.  \n \nAs for the effective damping 𝛼eff=1/2𝜋𝑓𝜏, in addition to the intrinsic damping ( α0,i) and the \nspin-pumping contribution ( αsp,ii and αsp,ji) considered in Eq. (2), inhomogeneities can also bring \nsubstantial damping enhancement [32,33,41,42] . Here, we m odel the total relaxation rate as \nfollows:  \n9 \n 1\n𝜏Φ=−Im(𝜔Φ)+1\n𝜏inhomoΦ (4) \nThe superscript Φ = HF or LF, representing either the high -frequency or low -frequency precession \nmodes. 𝜔Φ includes both the intrinsic and spin -pumping contributions. The inhomogeneous \nbroadening is calculated as:  \n1\n𝜏inhomoΦ=∑1\n𝜋|𝜕𝑓Φ\n𝜕𝐻k,eff,𝑖|\n𝑖Δ𝐻k,eff,𝑖+∑1\n𝜋|𝜕𝑓Φ\n𝜕𝐽𝑖|\n𝑖Δ𝐽𝑖 (5) \nwhere the first summation represents the contrib ution from the spatial variation of the effective \nanisotropy field of individual FM layers (Δ Hk,eff, i). The second summation denotes the contribution \nfrom the spatial fluctuations of the bilinear and biquadratic IEC (Δ J1 and Δ J2). According to \nSlonczewski’s “thickness fluctuations” theory, Δ J1 generates J2 [43,44] . Therefore,  the fact that  J2 \n= 0 for our sample suggests that  ΔJ1 is sufficiently small, allowing us to neglect  the inhomogeneous \nbroadening from th e fluctuations of both the bilinear and biquadratic IEC  in the following analyses . \n \n2.3 Detection of magnetization dynamics  \nThe magnetization dynamics of the p -SAF sample is detected by TR -MOKE, which is \nultrafast -laser -based metrology utilizing a pump -probe configuration. In TR -MOKE, pump laser \npulses interact with the sample, initiating magnetization dynamics in magnetic layers via inducing \nultrafast thermal demagnetization.  The laser -induced heating brings a rapid decrease to  the \nmagnetic anisotropy  fields  and IEC [45,46] , which changes 𝜃0,𝑖, 𝜑0,𝑖 and initiates the precession.  \nThe magnetizati on dynamics due to pump excitation is detected by a probe beam through the \nmagneto -optical Kerr effect. In our setup, the incident probe beam is normal to the sample surface \n(polar MOKE); therefore, the Kerr rotation angle ( 𝜃K) of the reflected probe beam  is proportional \nto the z component of the magnetization [47]. More details about the experimental setup can be 10 \n found in Refs. [30,32] . For p -SAF, TR -MOKE signals contain two oscillating frequencies that \ncorrespond to the HF and LF  modes  (𝑓HF>𝑓LF). The signals  are proportional to the change in \n𝜃K and can be analyzed as follows:  \nΔ𝜃K(𝑡)=𝐴+𝐵𝑒−𝑡/𝜏T+𝐶HFcos(2𝜋𝑓HF𝑡+𝛽HF)𝑒−𝑡/𝜏HF+𝐶LFcos(2𝜋𝑓LF𝑡+𝛽LF)𝑒−𝑡/𝜏LF (6) \nwhere the exponential term  𝐵𝑒−𝑡/𝜏T is related to the thermal background  with 𝜏T being the time \nscale of heat dissipation . The rest two terms on the right -hand side are the precession terms  with \nC, f, β, and τ denoting , respectively, the amplitude, frequency, phase, and relaxation time of the \nHF and LF  modes.  \nAfter excluding the thermal background from TR -MOKE signals, the precession is modeled \nwith the initial conditions of step -function de creases in 𝐻k,eff,𝑖 and 𝐽𝑖, following the ultrafast laser \nexcitation [48]. This is a reasonable  approximation since the precession period (~15 -100 ps for \nHext > 5 kOe) is much longer than the time scales of the laser excitation (~1.5 ps) and subsequent \nrelaxations among electrons, magnons, and lattice (~ 1 -2 ps) [49], but much shorter than the time \nscale of heat dissipation -governed recovery (~400 ps). With these  initial conditions , the prefactors \nin Eq. ( 3) can be determined  (see m ore details in Note 1  of the SM  [35]).  \nFor our SAF structure, 𝜃K detected by the probe beam contain s weighted contributions from \nboth the top and  bottom FM layers:  \n𝜃K(𝑡)\n𝜃K,s=𝑤cos𝜃1(𝑡)+(1−𝑤)cos𝜃2(𝑡) (7) \nwhere 𝜃K,s represents the Kerr rotation angle when the SAF s tack is saturated along the positive \nout-of-plane ( z) direction. w is the weighting factor, considering the different contributions to the \ntotal MOKE signals from two FM layers. w can be obtained from static MOKE measurements [50], \nwhich gives 𝑤= 0.457 (see more details in Note 2  of the SM  [35]). 11 \n  \n3 RESULTS AND DISCUSSION  \n3.1 Field -dependent p recession frequencies and equilibrium magnetization directions  \nTR-MOKE signals measured at varying  Hext are depicted in Fig. 2 (a). The external field is \ntilted 15 ° away from in-plane [θH = 75°, as defined by Fig. 2 (c)] to achieve larger amplitdues of  \nTR-MOKE signals [51]. The signals can be fitted to Eq.  (6) to extract the LF and HF precession \nmodes. The field -dependen t precession frequenc ies of both modes  are summarized in Fig. 2 (b). \nFor simplicity, when analyzing precession frequencies, magnetic damping  and mutual spin \npumping  are neglected due to its insignificant impacts on  precession frequencies.  By comparing \nthe experimental data and the prediction of ωHF/2π and ωLF/2π based on E q. (3), the effective \nanisotropy fields and the IEC strength  are fitted as Hk,eff, 1 = 1.23 ± 0.28  kOe, Hk,eff, 2 = 6.18 ± 0.13  \nkOe, J1 = −0.050 ± 0.020  erg cm−2, and J2 = 0. All parameters and their determination methods are \nsummarized in Table SI of the SM  [35]. The fitted J1 is close to that obtained from the M-Hext \nloops (~−0.062 erg cm-2). The inset of Fig 2 (b) shows the zoom ed-in view of field -dependent \nprecession frequencies around Hext = 8 kOe, where a n anti -crossing feature is observed: a  narrow \ngap (~2  GHz) open s in the frequency dispersion curves of the HF and LF modes owing to the weak \nIEC between two FM layers. Without a ny IEC, the precession frequencies of two FM layers would \ncross at Hext = 8 kOe, as indicated by the green dashed line and blue dashed line in the figure. We \nrefer to t hese two sets of crossing frequencies as the single -layer natural frequencies of two FM \nlayers (FM 1 and FM 2) in the following discussions .  12 \n  \nFIG. 2 (a) TR -MOKE signals under varying Hext when θH = 75° [as defined in panel (c)]. Circles \nare the experimental data and black lines are the fitting curves based on Eq. (6). (b) The precession \nfrequencies of the HF and LF modes as functions of Hext. Circles are experimental data and solid \nlines are fitting curves. The  inset highlights the zoomed -in view of the field -dependent frequencies \naround 8 kOe, where the green dashed line and blue dashed line are the single -layer (SL) \nprecession frequencies of FM 1 and FM 2 without interlayer exchange coupling. (c) Schematic \nillustration of the definition of the equilibrium polar angles ( θ0,1 and θ0,2), and the direction of the \nexternal magnetic field ( θH). The illustration is equivalent to Fig. 1(b) due to symmetry. (d) θ0,1 \nand θ0,2 as functions of Hext. The dash -dotted line plots the difference between the two equilibrium \npolar angles.  \n \n13 \n Based on the fitted stack properties ( Hk,eff,1, Hk,eff,2, J1, and J2), the equilibrium magnetization \ndirections in the two layers can be calculated. For SAFs with weak IEC compared with uniaxial \nPMA, the azimuthal angles of the magnetization in two FM layers are always the same as that of \nthe external field at equilibrium status. Therefore, two polar angles will be sufficient to describe \nthe equilibrium magnetization con figuration. Figure 2(c) illustrates the definition of the \nequilibrium polar angles of two FM layers ( θ0,1, θ0,2) and the external field ( θH). The values of θ0,1, \nθ0,2, and the difference between these two polar angles as functions of Hext are shown in Fig. 2(d). \nWhen Hext is low (<  1.6 kOe), magnetic anisotropy and antiferromagnetic coupling are dominant \nand |θ0,1 − θ0,2| is larger than 90 °. As Hext increases, both θ0,1 and θ0,2 approach θH. When Hext is \nhigh (> 15 kOe), the Zeeman energy becomes dominant and both M1 and M2 are almost aligned \nwith Hext. \n \n3.2 Cone angle, direction, and phase of magnetization  precession revealed by modeling  \nBesides the equilibrium configuration, using sample properties extracted from Fig. 2 (b) as \ninput parameters, the LLG -based modeling (described in section 2.2) also provide s information o n \nthe cone angle, direction, and phase of magnetization precession for each mode ( Fig. 1 ). The \ndiscussion in this section  is limited to the case  without damping an d mutual spin pumping . They \nwill be  considered  in Note 4 of the SM  [35], sections 3.3, and 3.4. The calculation results are \nshown in  Fig. 3 , which are categorized into three regions. At high external fields ( Hext > 1.6 kOe, \nregions 2 and 3), both FM layers precess CCW [ Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖)=90°], and the polar angles of \nmagnetization in two layers are  in-phase [Arg(𝐶𝜃,2/𝐶𝜃,1)= 0°] for the HF mode and  out-of-phase \n[Arg(𝐶𝜃,2/𝐶𝜃,1) = 180° ] for the LF mode. This is the reason for the HF mode (LF mode) also \nbeing called the acoustic mode ( optical mode) in the literature  [23]. The criterion to differentiate 14 \n region 2 from region 3 is the FM layer that dominat es a given precessional mode  (i.e., the layer \nwith larger precession cone angles) . In region 2  (1.6 kOe < Hext < 8 kOe) , the HF mode is \ndominated by FM 2 because FM 2 has larger cone angles than FM 1. This is reasonable since the \nhigher precession frequency is closer to the natural frequency of FM 2 [see Fig. 2(b)] in region 2. \nSimilarly, in region 3, the HF mode is dominated by FM 1 with larger precession cone angles.  \nWhen Hext is low (region 1), the angle between two magnetizations is larger than 90° [ Fig. 2 (d)] \nowing to the  more  dominan t AF-exchange -coupling  energy as compared with the Zeeman energy . \nIn this region, magnetization dynamics exhibits some unique features.  Firstly, CW [ Arg(𝐶𝜃,𝑖/\n𝐶𝜑,𝑖)=−90°] precession emerges: for each mode, the dominant layer precesses CCW (FM 2 for \nthe HF mode and FM 1 for the  LF mode) and the subservient layer precesses CW (FM 1 for the HF \nmode and FM 2 for the LF mode). This is because the effective field for the subservient layer [ e.g., \nHeff,1 for the HF mode, see Eq.  (2)] precesses CW owing to the CCW precession of the dominant \nlayer when |𝜃0,1−𝜃0,2|>90° [Fig. 2(d)] . In other words, a low Hext that makes |𝜃0,1−𝜃0,2|>\n90° is a necessary condition for the CW precession.  However, it is not a sufficient condition. In \ngeneral, certain degrees  of symmetry breaking ( Hk,eff,1 ≠ Hk,eff,2 or the field is tilted away from the \ndirection normal to the easy axis ) are also needed to generate CW precession.  For example, for \nsymmetric a ntiferromagnets ( Hk,eff,1 = Hk,eff,2) under fields perpendicular to the easy axis, CW \nprecession does not appear even at low fields (Fig. 2(a) in Ref. [52]). See Note 5 of the SM  [35] \nfor more details.  Secondly, as shown in  Fig. 3 , the precession motions in two FM layers are always \nin-phase for both HF and LF modes; thus, there is no longer a clear differentiation between \n“acoustic mode” and “optical mo de”. Instead, the two modes can be differentiated as “right -handed” \nand “left -handed” based on the chirality [53]. Here, we define the chirality with respect to a \nreference direction taken as the projection of Hext or M2 (magnetization direction of the layer with 15 \n a higher Hk,eff) on the easy axis [ -z direction in Fig. 3 (c)]. Lastly, the shape of the precession cone \nalso varies in different regions. Δ θi and Δφi are almost the same for both modes in region 3, \nindic ating the precession trajectories are nearly circular. While in regions 1 and 2, Δ θi and Δφi are \nnot always equal, suggesting the precession trajectories may have high ellipticities.  \n \n \nFIG. 3 The calculated  half cone angle, direction, and phase of magnetization precession for (a) the \nHF mode and (b) the LF mode. In the top row, four curves represent the polar and azimuthal  half \ncone angles of precession in two FM layers. All  half cone angles are normalized with r espect to \nΔθ1. The middle row shows the value of Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖) under different Hext. A value of 90°  (−90°) \nrepresents CCW (CW) precession. The bottom row is the phase difference of the polar angles in \ntwo layers. A value of 0° (180°) corresponds to the polar angles of the magnetization in two layers \nare IP (OOP ) during precession.  Dashed lines correspon d to the reference case where damping is \nzero in both layers. (c) Schematic illustrations of the cone angle, direction, and phase of \n16 \n magnetization precession for the HF and LF modes in different regions, and their corresponding \ncharacteristics regarding ch irality and phase difference.  \n \n3.3 Amplitude and phase of TR -MOKE signals  \nActual magnetization dynamics is resolvable  as a linear combination of the two eigenmodes  \n(the HF and  the LF modes ). By taking into account the initial conditions (i.e., laser excitation , see \nNote 1  of the SM  [35]), we can determine the amplitude and phase of the two modes in TR -MOKE \nsignals . Figure 4 (a) summarizes the amplitudes of both HF and LF modes [CHF and CLF in Eq.  (3)] \nunder different  Hext. Noted that  the y-axis represents Kerr angle ( θK) instead of the cone angle of \nprecession.  The LF mode  has a local minimum near 8 kOe, where the two FM layers have similar \nprecession cone angles but opposite phase s for the LF mode [ Fig. 3 (b)]. The amplitude s of both \nmodes decrease with Hext in the high -field region. This is similar to the single -layer case, where \nthe amplitudes of TR -MOKE signals decrease with Hext because the decrease  in Hk,eff induced by \nlaser heating is not able to significantly  alternate  the equilibrium magnetization direction when the \nZeeman energy dominates  [51]. The LF mode also has an amplitude peak at low fields ( Hext < 3 \nkOe), where the dominant layer of FM 1 changes its equilibrium direction dramatically with Hext \n(from ~75° to 170°) as shown in Fig. 2(d).  \nTo directly compare the amplitudes of TR -MOKE signals and the LLG -based calculations , the \nweighting factor w and the initial conditions are needed. The initial conditions are determined by \n𝐻k,eff,1′,𝐻k,eff,2′, and 𝐽1′, representing the instantan eous effective anisotropy fields and IEC strength \nupon laser heating. These instantaneous properties are different from their corresponding room -\ntemperature values ( Hk,eff,1, Hk,eff,2, and J1). The accurate determination of𝐻k,eff,1′,𝐻k,eff,2′, and 𝐽1′ \ndemands the modeling of the laser heating process as well as the temperature dependence of stack \nproperties, which are challenging. Here, we treat these three variables as adjustable parameters and 17 \n determine their values by fitting the field -dependent amp litudes of TR -MOKE signals , which \nyields 𝐻k,eff,1′𝐻k,eff,1⁄=0.90±0.01, 𝐻k,eff,2′𝐻k,eff,2⁄=0.95±0.01, and 𝐽1′𝐽1⁄=0.83±0.01. \nIt is apparent that the field dependence of TR -MOKE signal amplitude is in excellent agreement \nwith the theoretical modeling , as s hown in Fig. 4 (a). \nFigure 4 (b) shows the calculated  half polar cone angles for each mode in each FM layer. In \nTR-MOKE signals, the optical mode (the LF mode in regions 2 and 3) tends to be partially \ncanceled out  because the two layers precess out -of-phase.  Therefore, compared with Fig. 4 (a), the \ninformation in Fig. 4 (b) better reflects the actual intensity of both modes in FM 1 and FM 2. In Fig. \n4(b), the precession cone angles of both modes in FM 1 (Δ𝜃1HF,Δ𝜃1LF) have local maxima at the \nanti-crossing field (Hext ≈ 8 kOe). On the contrary, Δ𝜃2LF and Δ𝜃2HF of FM 2 have their maxima \neither above or below  the anti-crossing field. This is because FM 2 has larger precession amplitudes \n(cone angles) than FM 1 at the anti-crossing field if there is no IEC [the dotted  lines of FM 1 (SL) \nand FM 2 (SL) in Fig. 4 (b)]. With IEC, FM 2 with larger cone angles can drive the precession motion \nin FM 1 significantly near the anti-crossing field, where IEC is effective. Subsequently, the \nprecession amplitudes of FM 1 exhibit local maxima as its cone angle peaks at the anti-crossing  \nfield [solid lines in Fig. 4(b)]. Also, compared with the uncoupled case [FM 1 (SL) in Fig. 4(b)], \nFM 1 in the SAF structure has a much larger cone angle at the boundary between regions 1 and  2 \n(Hext ≈ 1.6 kOe).  This corresponds to the case where FM 1 fast switch ing is driven by Hext, as shown \nin Fig. 2( d). The energy valley of FM 1 created by IEC and uniaxial anisotropy is canceled out by \nHext. As a result, any perturbation in Hk,eff,1 or IEC can induce a large change in 𝜃1. \nBesides amplitude, the phase of TR -MOKE signals [ HF and LF in Eq. (6)] also provides \nimportant information about the magnetization dynamics in SAF  [Fig. 4 (c)]. In Fig. 4 (c), the phase \nof the HF mode stays constant around π. However, the LF mode goes through a π-phase shift at 18 \n the transition from region 2 to region 3. Th is phase shift can be explained by the change of the \ndominant layer from region 2 to region 3 for the LF mode [ Fig. 3(c)]. As illustrated in Fig. 4 (d), \nthe LF  mode (optical mode in regions 2 and 3) has opposite phases in FM 1 (~0°) and FM 2 (~180°). \nConsidering the two FM layers have comparable optical contributions to TR -MOKE signals ( w ≈ \n0.5), TR -MOKE signals will reflect the phase of the dominant layer for each mode. In region 3, \nFM 2 has larger p recession cone angles than FM 1 for the LF  mode ; therefore, LF TR-MOKE signals \nhave the same phase as FM 2 (~180°). However, in region 2, the dominant layer shifts from FM 2 to \nFM 1 for the LF mode. Hence, the phase of LF TR-MOKE signals also change s by ~180° t o be \nconsistent with the phase of FM 1 (~0°). As for the HF mode, since the two layers always have  \nalmost  the same phase ( ~180°), the change of the dominant layer does not cause a shift in the phase \nof TR -MOKE signals.   \nBy comparing Fig. 4 (d) and Fig. 3 (a-b), one can notice that the phase difference between two \nFM layers could deviate from 0° or 180° when damping  and mutual spin pumping  is considered \n[Fig. 4(d)]. The deviation of phase allows energy  to be transferred from one FM layer to the other \nduring precession via exchange coupling  [54]. In our sample system, FM 2 has a higher damping \nconstant ( 𝛼1= 0.020 and 𝛼2=0.060); therefore, the net transfer of energy is from FM 1 to FM 2. \nMore details can be found in Note 4 of the SM  [35], which shows the phase of TR -MOKE signals \nis affected by Gilbert damping in both layers and the mutual spin pumping . By fitting the phase \n[Fig. 4(c)] and the damping [ Fig. 5(a) ] of TR -MOKE signals simultaneously, we obtained 𝛼sp,12 \n= 0.010 ± 0.004, 𝛼sp,21=0.007−0.007+0.009, 1= 0.020 ± 0.002, and 2 = 0.060 ± 0.008. Nonreciprocal \nspin pumping damping ( 𝛼sp,12≠𝛼sp,21) has been reported in asymmetric FM 1/NM/FM 2 trilayers \nand attributed to the different spin -mixing conductance ( 𝑔𝑖↑↓) at the two FM/NM interfaces [27], \nfollowing 𝛼sp,𝑖𝑗=𝑔𝑖𝜇B𝑔𝑗↑↓/(8𝜋𝑀s,𝑖𝑑𝑖), with 𝑔𝑖 the 𝑔-factor of the i-th layer and 𝜇B the Bohr 19 \n magneton [55]. The above equation neglects the spin -flip scattering in NM and assumes that the \nspin accumulation in the NM spacer equally flows back to FM 1 and FM 2  [37]. However, the \nuncertainties of our 𝛼sp,𝑖𝑗 are too high to justify the nonreciprocity of 𝛼sp,𝑖𝑗 (see Note 3 of the SM  \n[35] for detailed uncertainty analyses). In fact, if the spin backflow to FM i is proportional  to 𝑔𝑖↑↓, \nthen 𝛼sp,𝑖𝑗=𝑔𝑖𝜇B𝑔𝑖↑↓𝑔𝑗↑↓/[4𝜋𝑀s,𝑖𝑑𝑖(𝑔𝑖↑↓+𝑔𝑗↑↓)] (Eq. 1.14 in Ref. [56]). In this case, the different \nspin-mixing conductance at two FM/NM interfaces ( 𝑔1↑↓≠𝑔2↑↓) will not lead to nonreciprocal \n𝛼sp,𝑖𝑗. Although differences in 𝑔𝑖 and magnetic moment per area ( 𝑀s,i𝑑𝑖) can potentially  lead to \nnonreciprocal 𝛼sp,𝑖𝑗, the values of 𝑔𝑖 and 𝑀s,i𝑑𝑖 for the two FM layers are expected to be similar \n(the net magnetization of SAF is zero without external fields). Therefore, nearly reciprocal 𝛼sp,𝑖𝑗 \nare plausible for our SAF stack. Assu ming 𝑔𝑖↑↓ values are similar at the two FM/NM interfaces \n(𝑔1↑↓≈𝑔2↑↓=𝑔↑↓), this yields 𝑔↑↓ =8𝜋𝑀s,𝑖𝑑𝑖𝛼sp,𝑖𝑗/(𝑔𝑖𝜇B) = 1.2 ~ 1.7 × 1015 cm−2. 𝑔↑↓ can also \nbe estimated from the free electron density per spin ( n) in the NM layer: 𝑔↑↓ ≈ 1.2𝑛2/3 [57]. With \nn = 5.2 × 1028 m−3 for Ru [58] (the value of n is similar for Ta [59]), 𝑔↑↓ is estimated to be 1.7  × 1015 \ncm−2, the same order as the 𝑔↑↓ value from TR -MOKE measurements, which justifies the 𝛼sp,𝑖𝑗 \nvalues derived from TR -MOKE are within a reasonable range. The values of 𝛼1 and 𝛼2 will be \ndiscussed in section 3.4.  \n 20 \n  \nFIG. 4 (a) Amplitudes of TR -MOKE signals a s functions of Hext. The circles and curves represent \nexperimental data and modeling fitting , respectively. (b) The calculated precession  half cone \nangles at different Hext. Red curves and black curves represent the cone angles of the HF  mode and \nthe LF mode in FM 1 (solid lines) and FM 2 (dash ed lines). Dotted lines are the precession cone \nangles of single -layer (SL) FM 1 and FM 2 without IEC. (c) Phases of TR -MOKE signals at  varying  \nHext. Circles and curves are experimental data and modeling fitting (𝛼sp,12=0.010, 𝛼sp,21=\n0.007, 𝛼1=0.020, 𝛼2=0.060). (d) Simulated precession phase of the HF mode (red curves) and \nthe LF mode (black curves) in FM 1 (solid lines) and FM 2 (dash ed lines).  \n \n3.4 Magnetic damping of the HF and LF precession modes  \nIn addition to the amplitude and phase of TR -MOKE signals for the p -SAF stack, the model \nanalyses also provide a better understanding of magnetic damping. Figure 5 (a) shows the effective \ndamping constant ( 𝛼eff=1/2𝜋𝑓𝜏) measured at different Hext (symbols), in comparison with \n21 \n model ing fitting (solid lines). The general Hext dependence of αeff can be well captured by the \nmodel. The fitted Gilbert damping, 1= 0.020 ± 0.002 and 2 = 0.060 ± 0.008 are close to the \nGilbert damping of Ta/CoFeB(1 nm)/MgO thin films (~0.017) [41,60]  and Co/Pd multilayers with \na similar tCo/tPd ratio (~0.085)  [61]. Other fitted parameters are  Δ𝐻k,eff,1=0.26±0.02 kOe, \nΔ𝐻k,eff,2= 1.42±0.18 kOe, 𝛼12sp=0.010±0.004  𝛼21sp=0.007−0.007+0.009. Δ𝐽1 and Δ𝐽2 are set to be \nzero, as explained in Sec. 2.2. More details regarding the values and determination methods of all \nparameters involved in our data reduction are provided in Note 3 of the SM  [35]. Dashed lines \nshow the calculated 𝛼eff without inhomogeneous broadening. At high Hext, the difference between \nthe solid lines  and dashed lines approaches zero because the inhomogeneous broadening is \nsuppressed. At low Hext, the solid lines are  significantly higher than the dashed lines , indicating \nsubstantial inhomogeneous broadening contributions .  \nThe effective  damping shows interesting features  near the anti-crossing field. As shown in \nFig. 5(b), due to the effective coupling  between two FM layers near the anti-crossing field, the \nhybridization of precession in two FM layers leads to a mix of damping with contributions from \nboth layers. The effective damping  of the FM 1-dominant mode  reaches a maximum within the \nanti-crossing region ( 7  Hext  10 kOe) and is higher than the single -layer (SL)  FM 1 case. \nSimilarly, the hybridized HF and LF  modes at 8.5 kOe exhibit  a lower 𝛼eff (~0.073) compared to \nthe SL FM 2 case. eff consists of contributions from Gilbert damping ( 𝛼𝑖), mutual spin pumping \n(𝛼sp,𝑖𝑗, 𝑖≠𝑗), and inhomogeneous broadening ( Δ𝐻k,eff,𝑖 and Δ𝐽𝑖). To better understand the mixing \ndamping behavior, Fig. 5 (c) shows eff after excluding the inhomogeneous contribution ( 𝛼effinhomo). \nCompared to the SL layer c ase (green and blue dashed lines), the HF and LF modes (red and black \ndashed lines) clearly suggest that IEC effectively mixes the damping in two layers around the anti -\ncrossing field. Without the IEC, precession in FM 2 with a higher damping relaxes faster  than that 22 \n in FM 1. However, the IEC provides a channel to transfer energy from FM 1 to FM 2, such that the \ntwo layers have the same precession relaxation rate for a given mode. Near the anti -crossing field, \ntwo layers have comparable precession cone angles; therefore, the damping values of the \nhybridized modes are roughly the average of two FM layers. In addition to the static IEC, dynamic \nspin pumping can also modify the damping of individual modes. The black and red solid lines \nrepresent the cases with mutu al spin pumping ( 𝛼sp,12 = 0.01 and 𝛼sp,21 = 0.007). Generally, in \nregions 2 & 3, mutual spin pumping reduces the damping of the HF mode and increases the \ndamping of the LF mode because the HF (LF) mode is near in -phase (out -of-phase). Overall, the \nstatic  IEC still plays the essential role for the damping mix near the anti -crossing field.  \n \n \nFIG. 5 (a) Effective damping constant under varying Hext. Circles are experimental data. Solid \nlines are fitting curves based on Eqs. (4 -5). Dashed lines denote eff after the removal of \ninhomogeneous -broadening contribution. (b) A zoomed -in figure of panel (a) between 5 kOe and \n15 kOe. Blue and green circles are measured effective damping of the mode dominated by FM 1 \nand FM 2, respectively. Blu e and green dashed lines are the 𝛼eff of FM 1 and FM 2 single layer \nwithout IEC. (c) Effective damping after excluding the inhomogeneous contribution as a function \nof Hext. The HF mode (red curves) and the LF mode (black curves) are represented by solid (o r \ndashed) curves when the mutual spin pumping terms ( 𝛼sp,12 and 𝛼sp,21) are considered (or \nexcluded). The d ashed green and blue lines are the SL cases for FM 1 and FM 2, respectively.  \n \n \n \n23 \n 4 CONCLUSION  \nWe systematically investigate d the magnetization dynamics excited by ultrafast laser pulses in \nan asymmetric p -SAF sample both theoretically and experimentally. We obtained d etailed \ninformation regarding magnetization dynamics, including the cone angles, directions, and phases \nof spin precession in each layer under different Hext. In particular, the dynamic features in the low -\nfield region  (region 1)  exhibiting  CW precession, were  revealed. The r esonance between the \nprecession of two FM layers occurs at the boundary between regions 2 an d 3, where an anti -\ncrossing feature is present in the frequency vs. Hext profile . The dominant FM layer for a given \nprecession mode also switches from region 2 to region 3. The amplitude and phase of TR -MOKE \nsignals are well captured by theoretical modeling . Importantly , we successfully quantified the \nindividual contributions from various sources to the effective damping , which enables the \ndetermination of Gilbert damping for both FM layers.  At low Hext, the contribution of \ninhomogeneous broadening to the effective damping is significant. Near the anti-crossing field, \nthe effective damping of  two coupled  modes contains substantial contributions from both FM \nlayers owing to the strong hybridization  via IEC . Although the analyses were made for an \nasymme tric SAF sample, this approach can be directly  applied to study magnetization dynamics \nand magnetic properties of  general complex material systems with  coupled multilayers , and thus  \nbenefits the design and optimization of spintronic materials via structural engineering.  \n \nAcknowledgements  \nThis work is  primarily  supported by the National Science Foundation ( NSF, CBET - 2226579). \nD.L.Z gratefully acknowledges the funding support from  the ERI program (FRANC) “Advanced \nMTJs for computation in and near ra ndom access memory” by DARPA, and ASCENT, one of six 24 \n centers in JUMP (a Semiconductor Research Corporation program, sponsored by MARCO and \nDARPA).  J.P.W  and X.J.W also appreciate the partial support from the UMN MRSEC Seed \nprogram (NSF, DMR -2011401 ). D.B.H . would like to thank the support from the UMN 2022 -2023 \nDoctoral Dissertation Fellowship. 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Magn. 48, 3288 (2012).  \n 1 \n Supplement al Material  for \nMagnetization Dynamics in Synthetic Antiferromagnets  with Perpendicular \nMagnetic Anisotropy   \n \nDingbin Huang1,*, Delin Zhang2, Yun Kim1, Jian -Ping Wang2, and Xiaojia Wang1,* \n1Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455, USA  \n2Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN \n55455, USA  \n \nSupplement al Note 1: Analyses of the magnetization precession in each  Ferromagnetic  (FM) \nlayer  \nFor the convenience of derivation, mi is represented in  the spherical coordinate s with the polar \nangle θi and the azimuthal angle φi, as shown in Fig. 1(b): \n𝐦𝑖=(sin𝜃𝑖cos𝜑𝑖,sin𝜃𝑖sin𝜑𝑖,cos𝜃𝑖) (S1) \nAccordingly, t he expressi on of  Eq. ( 2) in the spherical coordinate s is:  \n{        𝜃̇1=−𝛾1\n𝑑1𝑀s,1sin𝜃1∂𝐹\n∂𝜑1−𝛼1sin𝜃1𝜑̇1+𝛼sp,12sin𝜃2cos(𝜃2−𝜃1)𝜑̇2\n𝜑̇1=𝛾1\n𝑑1𝑀s,1sin𝜃1∂𝐹\n∂𝜃1+𝛼1\nsin𝜃1𝜃̇1−𝛼sp,12\nsin𝜃1𝜃̇2\n𝜃̇2=−𝛾2\n𝑑2𝑀s,2sin𝜃2∂𝐹\n∂𝜑2−𝛼2sin𝜃2𝜑̇2+𝛼sp,21sin𝜃1cos(𝜃1−𝜃2)𝜑̇1\n𝜑̇2=𝛾2\n𝑑2𝑀s,2sin𝜃2∂𝐹\n∂𝜃2+𝛼2\nsin𝜃2𝜃̇2−𝛼sp,21\nsin𝜃2𝜃̇1 (S2) \n \n*Author s to whom correspondence should be addressed : huan1746@umn.edu  and wang4940@umn.edu  2 \n where, a dot over variables  represents  a derivative with respect to time. When Mi  precesses  around \nits equilibrium direction:  \n{𝜃𝑖=𝜃0,𝑖+Δ𝜃𝑖\n𝜑𝑖=𝜑0,𝑖+Δ𝜑𝑖 (S3) \nwith \ni  and \ni  representing the deviation angles of  Mi from  its equilibrium direction  along the \npolar and azimuthal directions.  Assuming the deviation is small, under the first -order \napproximation, the first -order partial derivative of F in Eq. (S2) can be expanded as:  \n{    ∂𝐹\n∂𝜃𝑖≈∂2𝐹\n∂𝜃𝑖2Δ𝜃𝑖+∂2𝐹\n∂𝜑𝑖∂𝜃𝑖Δ𝜑𝑖+∂2𝐹\n∂𝜃𝑗∂𝜃𝑖Δ𝜃𝑗+∂2𝐹\n∂𝜑𝑗∂𝜃𝑖Δ𝜑𝑗\n∂𝐹\n∂𝜑𝑖≈∂2𝐹\n∂𝜃𝑖𝜕𝜑𝑖Δ𝜃𝑖+∂2𝐹\n∂𝜑𝑖2Δ𝜑𝑖+∂2𝐹\n∂𝜃𝑗∂𝜑𝑖Δ𝜃𝑗+∂2𝐹\n∂𝜑𝑗∂𝜑𝑖Δ𝜑𝑗 (S4) \nBy substituting Eq. ( S4), Equation ( S2) is linearized  as [1]: \n[   Δ𝜃̇1\nΔ𝜑̇1\nΔ𝜃̇2\nΔ𝜑̇2]   \n=𝐊[Δ𝜃1\nΔ𝜑1\nΔ𝜃2\nΔ𝜑2] (S5) \nwhere, K is a 4×4 matrix, con sisting of  the properties  of individual FM layers  and the second -\norder derivatives of F in terms of 𝜃1,𝜑1,𝜃2,and𝜑2. Equation (S5) has four eigen -solutions, in the \nform of 𝐶exp(𝑖𝜔𝑡), corresponding to four precession frequencies:  ±𝜔HF and ±𝜔LF. A pair of \neigen -solutions with the same absolute precession frequency are physically equivalent. Therefore, \nonly two eigen -solutions need to be considered:  \n{Δ𝜃𝑖=𝐶𝜃,𝑖HFexp(𝑖𝜔HF𝑡)\nΔ𝜑𝑖=𝐶𝜑,𝑖HFexp(𝑖𝜔HF𝑡) and {Δ𝜃𝑖=𝐶𝜃,𝑖LFexp(𝑖𝜔LF𝑡)\nΔ𝜑𝑖=𝐶𝜑,𝑖LFexp(𝑖𝜔LF𝑡) (S6) \nAfter r earrange ment , the full solutions in the spherical coordinates are expressed as below (also \nEq. (3) in the main paper).  3 \n [𝜃1(𝑡)\n𝜑1(𝑡)\n𝜃2(𝑡)\n𝜑2(𝑡)]=\n[   𝜃0,1\n𝜑0,1\n𝜃0,2\n𝜑0,2]   \n+[Δ𝜃1(𝑡)\nΔ𝜑1(𝑡)\nΔ𝜃2(𝑡)\nΔ𝜑2(𝑡)]=[𝜃0,1\n𝜑0,1\n𝜃0,2\n𝜑0,2]+\n[    𝐶𝜃,1HF\n𝐶𝜑,1HF\n𝐶𝜃,2HF\n𝐶𝜑,2HF]    \nexp(𝑖𝜔HF𝑡)+\n[    𝐶𝜃,1LF\n𝐶𝜑,1LF\n𝐶𝜃,2LF\n𝐶𝜑,2LF]    \nexp(𝑖𝜔LF𝑡)    (S7) \nThe prefactors of these  eigen -solutions  provide information about magnetization dynamics of both \nthe HF and LF modes. Directly from solving Eq. (S2), one can obtain the relative ratios of these \nprefactors , which are [𝐶𝜑1HF/𝐶𝜃1HF,𝐶𝜃2HF/𝐶𝜃1HF,𝐶𝜑2HF/𝐶𝜃1HF] and [𝐶𝜑1LF /𝐶𝜃1LF ,𝐶𝜃2LF /𝐶𝜃1LF ,𝐶𝜑2LF /𝐶𝜃1LF ]. \nThese ratios provide  precession information of each mode, as presented in Fig. 3.   \nObtaining the absolute values of [𝐶𝜃,1,𝐶𝜑,1,𝐶𝜃,2,𝐶𝜑,2]𝑇 for each mode requires the initial \nconditions of precession , which i s necessary for fitting the actual precession amplitudes in TR -\nMOKE signals . In TR -MOKE measurements, magnetization precession is initiated by laser \nheating, which reduces the magnetic anisotropy of each  FM layer and the interlayer exchange \ncoupling streng th between two FM layers  [2]. Considering the laser heating process is ultrafast \ncompared with magnetization precession while the following cooling due to heat dissipation is \nmuch slower than magnetization dynamics, we approximately model the temporal profiles of \neffective anisotropy fields and exchange coupling as step functions.  Owing to the sudden change \nin magnetic properties  induced by laser heating , magnetization in each layer will establish a new \nequilibrium direction (𝜃0,𝑖′,𝜑0,𝑖′). In other words, M i deviates from its new eq uilibrium direction \nby Δ𝜃𝑖=𝜃0,𝑖−𝜃0,𝑖′, Δ𝜑𝑖=𝜑0,𝑖−𝜑0,𝑖′. Substituting 𝑡=0 to Eq. ( S7), one can get the initial \nconditions for magnetization dynamics:  \nΔ𝜃𝑖(𝑡=0)=𝐶𝜃,𝑖HF+𝐶𝜃,𝑖LF=𝜃0,𝑖−𝜃0,𝑖′  \nΔ𝜑𝑖(𝑡=0)=𝐶𝜑,𝑖HF+𝐶𝜑,𝑖LF=𝜑0,𝑖−𝜑0,𝑖′=0 (S8) \nOnce the initial conditions are set, the absolute values of all prefactors can be obtained . \n 4 \n Supplementa l Note 2: Estimation of each layer’s contribution to total TR -MOKE signals  \nThe contribution from each FM layer is estimated by static MOKE measurement. According \nto Ref. [3], the resu lt from this method matches well with that from the optical calculation. The \nsample is perpendicularly saturated before the static MOKE measurement. Then the out -of-plane \nM-Hext loop ( Fig. S1) is measured by static MOKE. As shown in the figure, two different \nantiferromagnetic (AF) configurations have different normalized MOKE signals, indicating the \ndifferent contribution s to the total signals by two layers.  The weighting factor is calculated by:   \n−𝑤+(1−𝑤)=0.085 (S9) \nwhich gives 𝑤=0.457. Considering the relatively small layer thicknesses [FM 1: CoFeB(1), spacer: \nRu(0.6)/Ta(0.3), and FM 2: Co(0.4)/Pd(0.7)/Co(0.4)], it is reasonable that FM 1 and FM 2 make \ncomparable contributions to the total TR -MOKE signals ( i.e., w ≈ 0.5). \n \nFIG. S1 Static MOKE hysteresis loop. Magnetic fields are applied along the out -of-plane direction.  \n \n \n \n5 \n Supplemental Note 3: Summary of the parameters and uncertainties for data reduction  \nGiven that a number of variables are involved in the analysis, TABLE SI summarizes the major \nvariables discussed in the manuscript, along with their values and determinatio n methods.  \nTABLE SI. Summary of the values and determination methods of parameters used in the data \nreduction. The reported uncertainties are one -sigma uncertainties from the mathematical model \nfitting to the TR -MOKE measurement data.  \nParameters  Values  Determination Methods  \nHf ~500 Oe  VSM  \nMs,1 1240 emu cm−3 VSM  \nMs,2  827 emu  cm−3 VSM  \nd1 1 nm  Sample structure  \nd2 1.5 nm  Sample structure  \nHk,eff,1  1.23 ± 0.28 kOe  Fitted from f vs. Hext [Fig. 2(b)]  \nHk,eff,2 6.18 ± 0.13 kOe  Fitted from f vs. Hext [Fig. 2(b)]  \nγ1 17.79 ± 0.04  \nrad ns−1 kOe−1 Fitted from f vs. Hext [Fig. 2(b)]  \nγ2 17.85 ± 0.04  \nrad ns−1 kOe−1 Fitted from f vs. Hext [Fig. 2(b)]  \nJ1 −0.050 ± 0.020  \nerg cm−2 Fitted from f vs. Hext [Fig. 2(b)]  \nJ2 0 Fitted from f vs. Hext [Fig. 2(b)]  \nw 0.457  Static MOKE  \n𝐻k,eff,1′/𝐻k,eff,1 0.90 ± 0.01  Fitted from Amp vs. Hext [Fig. 4(a)]  \n𝐻k,eff,2′/𝐻k,eff,2 0.95 ± 0.01  Fitted from Amp vs. Hext [Fig. 4(a)]  \n𝐽1′/𝐽1 0.83 ± 0.01  Fitted from Amp vs. Hext [Fig. 4(a)]  \n𝛼1 0.020 ± 0.002  Fitted from eff vs. Hext [Fig. 5(a)] and  vs. Hext [Fig. 4(c)]  \n𝛼2 0.060 ± 0.008  Fitted from eff vs. Hext [Fig. 5(a)] and  vs. Hext [Fig. 4(c)]  \nΔ𝐻k,eff,1 0.26 ± 0.02 kOe  Fitted from eff vs. Hext [Fig. 5(a)]  \nΔ𝐻k,eff,2 1.42 ± 0.18 kOe  Fitted from eff vs. Hext [Fig. 5(a)]  \n𝛼sp,12 0.010 ± 0.004  Fitted from eff vs. Hext [Fig. 5(a)] and  vs. Hext [Fig. 4(c)]  \n𝛼sp,21 0.007−0.007+0.009 Fitted from eff vs. Hext [Fig. 5(a)] and  vs. Hext [Fig. 4(c)]  \n 6 \n Supplemental Note  4: Impacts of 𝜶𝟏, 𝜶𝟐, and mutual spin pumping on the phase  \nWithout damping, the phase difference in the precession polar angles of two FM layers \n[Arg(𝐶𝜃2/𝐶𝜃1)] is always 0° or 180°, as shown in Fig. 3 of the main article. However, this does not \nnecessarily hold if either the damping or mutual spin pumping is considered. The changes in the \nphase difference due to damping are depicted in Fig. S2. When 1 = 2, the phase difference \nbetween two layers stays at 0° or 180° [ Fig. S2(a)], identical to the lossless case ( 1 = 2 = 0) in \nFig. 3. As a result, the initial phase of TR -MOKE signals ( ) also stays at 0° or 180° [ Fig. S2(b)]. \nHowever, when 𝛼1≠𝛼2, Arg(𝐶𝜃2/𝐶𝜃1) deviates from 0° or 180° especially at high fields ( Hext > \n5 kOe) [ Fig. S2(c,e)]. The layer with a higher damping [FM 1 in (c) or FM 2 in (e)] tends to have a \nmore advanced phase at high fields (regions 2 and 3). For example, in Fig. S2(e), 0° < Arg( 𝐶𝜃2/𝐶𝜃1) \n< 180° for both HF and LF modes in regions 2 and 3. The deviation from the perfect in -phase (0°) \nor out -of-phase (180°) condition allows the IEC to transfer energy from the low -damping layer to \nthe high -damping layer, such that the precession in  both layers can damp at the same rate [4]. As \na result, the initial phase of the TR -MOKE signals also changes, which opens a negative or positive  \ngap at high fields (> 10 kOe) for both modes, as shown in Fig. S2(d,f). This enables us to determine \nthe difference between 1 and 2 by analyzing the in itial phase of TR -MOKE signals.  7 \n  \nFIG. S 2 Impact of 𝛼1 and 𝛼2 on the phase without mutual spin pumping. (a,c,e) The phase \ndifference between the polar angles in two layers for HF and LF modes. (b,d,f) The calculated \ninitial phase of TR -MOKE signals for each mode with 1 = 2 = 0.02 (a,b), 1 = 0.06 and 2 = \n0.02 (c,d), and 1 = 0.02 and 2 = 0.06 (e,f). The mutual spin pumping is set as 𝛼sp,12=𝛼sp,21 = \n0 for all three cases. The rest of the parameters used in this calculation can be found in TABLE SI.  \n \nThe impact of mutual spin pumping on the precession phase is illustrated in Fig. S3, where \nthree different cases of either the one -way (𝛼sp,12 or 𝛼sp,21) or two -way (both 𝛼sp,12 and 𝛼sp,21) \nspin pumping are considered. A reference case without the consideration of mutual spin pumping \n(1 = 0.02, 2 = 0.06, and 𝛼sp,12= 𝛼sp,21 = 0) is also plotted (dashed curves) for the ease of \ncomparison. In general, it can be seen that mutual spin pumping could also change the phase \ndifference in the precession polar angles of two layers, and thus the initial phase o f TR -MOKE \nsignals noticeably. This can be explained by the damping modification resulting from spin \npumping. In regions 2 and 3, Eq. (2) can be approximately rearranged as:  \n \n8 \n 𝑑𝐦𝑖\n𝑑𝑡≈−𝛾𝑖𝐦𝑖×𝐇eff,𝑖+𝛼𝑖𝐦𝑖×𝑑𝐦𝑖\n𝑑𝑡−𝐶𝑗\n𝐶𝑖𝛼sp,𝑖𝑗cos(𝜃0,1−𝜃0,2)𝐦𝑖×𝑑𝐦𝑖\n𝑑𝑡\n≈−𝛾𝑖𝐦𝑖×𝐇eff,𝑖+[𝛼𝑖−𝐶𝑗\n𝐶𝑖𝛼sp,𝑖𝑗cos(𝜃0,1−𝜃0,2)]𝐦𝑖×𝑑𝐦𝑖\n𝑑𝑡\n=−𝛾𝑖𝐦𝑖×𝐇eff,𝑖+𝛼̅𝑖𝐦𝑖×𝑑𝐦𝑖\n𝑑𝑡 (S10) \nwhere 𝐶𝑗/𝐶𝑖 represents the ratio of the cone angles in the j-th FM layer to the i-th FM layer. 𝐶𝑗/𝐶𝑖 \nis positive for the in -phase mode and negative for the out -of-phase mode.  θ0,1 and θ0,2 are the \nequilibrium polar angle s of M1 and M2, as defined in Fig. 2(c).  Therefore, the mutual spin -pumping \nterm either enhances or reduces the damping depending on the mode. 𝛼̅𝑖 = 𝛼𝑖−\n𝐶𝑗\n𝐶𝑖𝛼sp,𝑖𝑗cos(𝜃0,1−𝜃0,2) represents the effective Gilbert damping in the i-th FM layer after \nconsidering the mutual spin -pumping effect. This modification to damping is more significant \nwhen the i-th layer is subservient with a smaller cone angle ( e.g., FM 2 for the HF mode in region \n3), while the j-th layer is dominant with a mu ch larger precession cone angle ( e.g., FM 1 for the LF \nmode in region 3), leading to a large ratio of |𝐶𝑗/𝐶𝑖|. \nIn Fig. S 3(a), only the spin current injected from FM 1 to FM 2 is considered. According to the \nabove analysis, 𝛼sp,21 can only bring noticeable modifications to the damping of FM 2 when FM 1 \nis the dominant layer. Based on Fig. 3 in the main article, the LF mode in region 2 and HF mode \nin region 3 satisfy this condition (FM 1 dominant and FM 2 subservient). As shown in Fig. S 3(a), \nthe phase difference  noticeably deviates from the reference case without mutual spin pumping \n(dashed curves) in region 2 for the LF mode (black curves) and in region 3 for the HF mode (red \ncurves). For the LF mode in region 2, the precession motions in two layers are nearly o ut-of-phase \n(negative C1/C2); therefore, the spin pumping from FM 1 enhances the damping in FM 2. Since 1 \n(0.02) is less than 2 (0.06), the spin pumping from FM 1 to FM 2 further increases |  𝛼̅1 − 𝛼̅2| \nbetween the two layers. Consequently, the phase difference shifts further away from 180°. While 9 \n for the HF mode in region 3, 𝛼sp,21 reduces the damping of FM 2 because C1/C2 is positive resulting \nfrom the near in -phase feature of this mode. Hence, |  𝛼̅1 − 𝛼̅2| becomes smaller and the phase \ndifference gets closer to 0°. In Fig. S 3(c), only 𝛼sp,12 is considered, which requires FM 2 as the \ndominant layer (the HF mode in region 2 and LF mode in region 3) for noticeable changes in |  𝛼̅1 \n− 𝛼̅2|. For the HF mode  in region 2, spin pumping from FM 2 reduces 𝛼̅1 given that the precession \nmotions in two layers are nearly in phase (positive C2/C1). Therefore, |  𝛼̅1 − 𝛼̅2| increases and the \nphase difference in Fig. S 3(c) shifts further away from 0° in region 2. However,  for the LF mode \nin regions 3, the nearly out -of-phase precession in two FM layers (negative C1/C2) increases 𝛼̅1 \nand reduces |  𝛼̅1 − 𝛼̅2|. As a result, the phase difference in Fig. S 3(c) shifts toward 180°. When \nboth 𝛼sp,12 and 𝛼sp,21 are considered [ Fig. S 3(e)], a combined effect is expected for the phase \ndifference with noticeable changes for both the HF and LF modes in regions 2 and 3.  \nThe impacts of mutual spin pumping on the phase difference between the HF and LF modes \nare reflected by the initial phase of TR -MOKE signals [  in Fig. S 3(b,d,f)]. Compared with the \nreference case without mutual spin pumping (dashed curves), the introduction of mutual spin \npumping tends to change the gap in  between the two modes. As shown in Fig. S3(e,f), the values \nof two mutual -spin-pumping induced damping terms are chosen as 𝛼sp,12 = 0.013 and 𝛼sp,21 = \n0.004, such that the  gap of the initial phase of TR -MOKE signals is closed at high fields \n(region  3). Therefore, the initial phase of TR -MOKE signals provides certain measurement \nsensitivities to 𝛼sp,12 and 𝛼sp,21, which enables us to extract the values of 𝛼sp,𝑖𝑗 from \nmeasurement fitting. Here, we acknowledge that the measurement sensitivity to 𝛼sp,𝑖𝑗 from TR -\nMOKE is limited, which subsequently leads to relatively large error bars for 𝛼sp,𝑖𝑗 (see Table SI).  \n 10 \n  \nFIG. S3 Impact of mutual spin pumping on the phase with fixed damping values of 1 = 0.02 and \n2 = 0.06. (a,c,e) The phase difference between the polar angles in two layers for HF and LF modes. \n(b,d,f) The calculated initial phase of TR -MOKE signals ( ) for each mode with 𝛼sp,12= 0 and \n𝛼sp,21 = 0.01 (a,b), 𝛼sp,12 = 0.01 and 𝛼sp,21 = 0 (c,d), and  𝛼sp,12 = 0.013 and 𝛼sp,21 = 0.004 (e,f). \nFor the third case (e,f), the values of mutual spin pumping are chosen to close the  gap in panel \n(f) for Hext > 15 kOe. The rest of the parameters used in this calculation can be found in TABLE \nSI. Dashed lines  represent the reference case without mutual spin pumping ( 1 = 0.02, 2 = 0.06, \nand 𝛼sp,12= 𝛼sp,21 = 0).  \n \nSupplemental Note 5: Region diagram s for p -SAFs with different degrees of asymmetries  \nFigure S4 shows the region diagrams for p -SAFs with different degrees of asymmetries , \nrepresented by the difference of Hk,eff in two FM layers. Hk,eff,1 = Hk,eff,2 corresponds to the \nsymmetric case (lowest asymmetry), as shown by Fig. S4(c). While  the SAF  in Fig. S4(a) has the \nhighest asymmetry: Hk,eff,1 = 2 kOe, Hk,eff,2 = 6 kOe. Figure S4 clearly shows that |𝜃0,1−𝜃0,2|>\n90° is a necessary but not sufficient condition for region 1 (CW precession). Because regions 2 or \n3 also appear to the left of the red cu rve (where |𝜃0,1−𝜃0,2|>90°), especially when θH is close \nto 90° and Hk,eff,1 is close to Hk,eff,2. \n11 \n  \nFIG. S4 Region diagrams of p -SAFs with different  degrees of  asymmetries: Hk,eff,1 = 2 kOe, Hk,eff,2 \n= 6 kOe (a), Hk,eff,1 = 4 kOe, Hk,eff,2 = 6 kOe (b), Hk,eff,1 = 6 kOe, Hk,eff,2 = 6 kOe (c). The blue \nbackground represents region 1. The green background covers regions 2 and 3. The red curve \nshows the conditions where |𝜃0,1−𝜃0,2|=90°. |𝜃0,1−𝜃0,2|>90° to the left of the red curve. 𝛼1, \n𝛼2, 𝛼sp,12, and 𝛼sp,21 are set as zero. 𝛾1=𝛾2=17.8 rad ns−1 kOe−1. Values of the rest parameters are \nthe same as those in Table SI.  \n \nReferences  \n[1] Z. Zhang, L. Zhou, P. E. Wigen, and K. Ounadjela, Angular dependence of ferromagnetic \nresonance in exchange -coupled Co/Ru/Co trilayer structures, Phys. Rev. B 50, 6094 (1994).  \n[2] W. Wang, P. Li, C. Cao, F. Liu, R. Tang, G. Chai, and C. Jiang, Temperature dependence \nof interlayer exchange coupling and Gilbert damping in synthetic antiferromagnetic \ntrilayers investigated using broadband ferromagnetic resonance, Appl. Phys. Lett. 113, \n042401 (2018).  \n[3] G. Malinowski, F. Dalla Longa, J. H. H. Rietjens, P. V. Paluskar, R. Huijink, H. J. M. \nSwagten, and B. Koopmans, Control of speed and efficiency of ultrafast demagnetization \nby direct transfer of spin angular momentum, Nat. Phys. 4, 855 (2008).  \n[4] D. H. Zanette, Energy exchange between coupled mechanical oscil lators: linear regimes, J. \nPhys. Commun. 2, 095015 (2018).  \n \n"    },    {        "title": "2105.09879v2.On_the_the_critical_exponent_for_the_semilinear_Euler_Poisson_Darboux_Tricomi_equation_with_power_nonlinearity.pdf",        "content": "arXiv:2105.09879v2  [math.AP]  19 Mar 2024On the critical exponent for the semilinear\nEuler-Poisson-Darboux-Tricomi equation with power\nnonlinearity\nAlessandro Palmieri\nDepartment of Mathematics, University of Bari, Via E. Orabo na 4, 70125 Bari, Italy\nMarch 20, 2024\nAbstract\nIn this note, we derive a blow-up result for a semilinear gene ralized Tricomi equation with\ndamping and mass terms having time-dependent coefficients. W e consider these coefficients with\ncritical decay rates. Due to this threshold nature of the tim e-dependent coefficients (both for the\ndamping and for the mass), the multiplicative constants app earing in these lower order terms\nstrongly influence the value of the critical exponent, deter mining a competition between a Fujita-\ntype exponent and a Strauss-type exponent.\nKeywords Critical exponent, Fujita exponent, Strauss exponent, Blo w-up, Power nonlinearity\nAMS Classification (2020) 35B33, 35B44, 35L15, 35L71.\n1 Introduction\nIn the present note, we prove a blow-up result for local in tim e weak solutions to the following\nsemilinear Cauchy problem with power nonlinearity |u|p\n\n\n∂2\ntu−t2ℓ∆u+µt−1∂tu+ν2t−2u=|u|p, x ∈Rn, t> 1,\nu(1,x) =εu0(x), x ∈Rn,\n∂tu(1,x) =εu1(x), x ∈Rn,(1)\nwhereℓ>−1,µ,ν2are nonnegative real constants, p>1 andεis a positive constant describing the\nsize of Cauchy data. We consider the case with initial data ta ken at the time t= 1, nevertheless, the\nresults that we are going to prove are valid for data taken at a ny initial time t=t0>0.\nForℓ= 0 andν2= 0, the linearized equation associated with the equation in (1) is known as\nEuler-Poisson-Darboux equation (see the introduction of [6] for a detailed overview on the li terature\nregarding this model), while for µ=ν2= 0 the second-order operator ∂2\nt−t2ℓ∆ on the left-hand\nside of the equation in (1) is called generalized Tricomi operator . Motivated by these special cases\nand for the sake of brevity, we will call the equation in (1) semilinear Euler-Poisson-Darboux-Tricomi\nequation (semilinear EPDT equation).\nOver the last two decades, several papers have been devoted t o the study of semilinear models\nwith power nonlinearity, that are special cases of (1) for gi ven values of the parameters ℓ,µ,ν2.\nConcerning the Cauchy problem associated with the semiline ar generalized Tricomi equation\n\n\n∂2\ntu−t2ℓ∆u=f(u,∂ tu), x ∈Rn, t> 0,\nu(0,x) =εu0(x), x ∈Rn,\n∂tu(0,x) =εu1(x), x ∈Rn,(2)\nwe recall [10, 42, 43] for the first results in the case with pow er nonlinearity f(u,∂ tu) =|u|p. More\nspecifically, in [10] some blow-up results are proved for wea k solutions to the Cauchy problem associ-\nated with the generalized Tricomi operator (and, more gener ally, for Grushin-type operators) by mean\n1of the test function method. The so-called quasi-homogeneous dimension of the generalized Tricomi\noperator Q= (ℓ+ 1)n+ 1 made its appearance in the upper bound for pin these results (see [7, 25]\nfor the definition of the quasi-homogeneous dimension for a m ore general partial differential opera-\ntor). On the other hand, in [42, 43] the fundamental solution for the generalized Tricomi operator\n(sometimes called also Gellerstedt operator in the literat ure) is employed to derive an integral repre-\nsentation formula, that is used in turn to prove (under suita ble assumptions on p) the local existence\nof solutions and the global existence of small data solution s, respectively.\nAfterwards, in the series of papers [16, 17, 18, 41, 19, 35] it was established that the critical\nexponent for (2) with f(u,∂ tu) = |u|pandn/greaterorequalslant2 is the Strauss-type exponent (cf. [34] and the\nliterature citing this renowned paper), that we denote pStr(n,ℓ) in the present paper, given by the\nbiggest root of the quadratic equation\n/parenleftbiggn−1\n2+ℓ\n2(ℓ+ 1)/parenrightbigg\np2−/parenleftbiggn+ 1\n2−3ℓ\n2(ℓ+ 1)/parenrightbigg\np−1 = 0. (3)\nWe recall that for us the fact that pStr(n,ℓ) is the critical exponent for (2) with f(u,∂ tu) =|u|pmeans\nthe following: for any 1 <p<p Str(n,ℓ) local in time solutions blow up in finite time under suitable\nsign assumptions for the Cauchy data and regardless of their size, while for p > p Str(n,ℓ) (technical\nupper bounds for pmay appear, depending on the space for the solutions) a globa l in time existence\nresult for small data solutions holds.\nVery recently, even the cases with derivative type nonlinea rityf(u,∂ tu) =|∂tu|pand with mixed\nnonlinearity f(u,∂ tu) =|u|q+|∂tu|phave been studied from the point of view of blow-up dynamics i n\n[26, 3, 23, 13]. In particular, in [26] a blow-up result when f(u,∂ tu) =|∂tu|pis proved for 1 <p/lessorequalslantQ\nQ−2.\nRegarding the semilinear Euler-Poisson-Darboux equation and, more in general, the semilinear\nwave equation with scale-invariant damping and mass (i.e. ( 1) forℓ= 0), combining the contributions\nfrom many different authors (see [5, 8, 27, 20, 31, 33, 9, 24, 6, 4] and references therein) we may\nreasonably conjecture that for nonnegative µ,ν2such that the quantity\nδ.= (µ−1)2−4ν2(4)\nsatisfiesδ/greaterorequalslant0 the critical exponent is given by\nmax/braceleftBig\npStr(n+µ,0), pFuj/parenleftBig\nn+µ−1\n2−√\nδ\n2/parenrightBig/bracerightBig\n. (5)\nHerepFuj(n).= 1 +2\nndenotes the so-called Fujita exponent (cf. [11]) and its presence can be justified\nas a consequence of diffusion phenomena between the solution s to the corresponding linearized model\nand those to some suitable parabolic equation. We point out t hat the condition δ/greaterorequalslant0 implies somehow\nthat the damping term µt−1∂tuhas a dominant influence over the mass term ν2t−2u. Although the\nproof of the necessity part of this conjecture is fully demon strated (see [31, 33, 24]), for the sufficiency\npart only the one-dimensional case was recently completely clarified [6], while in the higher dimensional\ncase only a few special cases have been studied (often in the r adially symmetric case).\nWe emphasize that for ℓ=−2\n3andµ= 2,ν2= 0 the model in (1) is the semilinear wave equation\nin the Einstein-de Sitter spacetime with power nonlinearity (see [12]). Hence, for the semiline ar wave\nequation in the generalized Einstein-de Sitter spacetime, i.e., whenℓ∈(−1,0) andµ/greaterorequalslant0,ν2= 0 in\n(1), in the series of papers [29, 30, 37, 38] several blow-up r esults are proved provided that p >1 is\nbelow\nmax/braceleftBig\npStr/parenleftBig\nn+µ\nℓ+1,ℓ/parenrightBig\n, pFuj((ℓ+ 1)n)/bracerightBig\n. (6)\nMoreover, under the same conditions for the parameters as ab ove, also the case with derivative type\nnonlinearity |∂tu|pis studied in [14, 15, 40]. Furthermore, we point out that in [ 39, 40] even the case\nℓ/lessorequalslant−1 withν2= 0 is studied for different semilinear terms.\nThe purpose of the present paper is twofold: on the one hand, w e want to prove the necessity\npart for the one-dimensional case in (1) that together with t he sufficiency part from [6] (cf. Corollary\n5.1) will show the optimality of our result for n= 1; on the other hand, we want to generalize the\ncondition that determines the critical exponent for (1) in t he generaln-dimensional case, obtaining as\na candidate to be critical an exponent that is consistent wit h all previously mentioned special cases.\nFinally, we mention that in [2] the semilinear Cauchy proble m with derivative type nonlinearity and\nthe same linear partial differential operator as in (1) is con sidered, and a Glassey-type exponent is\nfound as the upper bound for the exponent in the blow-up range .\n2Notations Throughout the paper we denote by\nφℓ(t).=tℓ+1\nℓ+ 1(7)\nthe primitive of the speed of propagation tℓthat vanishes for t= 0. In particular, the amplitude of\nthe light-cone for the Cauchy problem with data prescribed a t the initial time t0= 1 is given by the\nfunctionφℓ(t)−φℓ(1). The ball in Rnwith radius Raround the origin is denoted BR. The notation\nf/lessorsimilargmeans that there exists a positive constant Csuch thatf/lessorequalslantCgand, analogously, f/greaterorsimilarg.\nFinally, as in the introduction, we will denote by pFuj(n) the Fujita exponent and by pStr(n,ℓ) the\nStrauss-type exponent.\n1.1 Main results\nLet us begin this section by introducing the notion of weak so lutions to (1) that we will employ\nthroughout the entire paper.\nDefinition 1.1. Letu0,u1∈L1\nloc(Rn) such that supp u0,suppu1⊂BRfor someR>0. We say that\nu∈C/parenleftbig\n[1,T),W1,1\nloc(Rn)/parenrightbig\n∩C1/parenleftbig\n[1,T),L1\nloc(Rn)/parenrightbig\n∩Lp\nloc/parenleftbig\n(1,T)×Rn/parenrightbig\nis aweak solution to (1) on [1 ,T) ifu(1,·) =εu0inL1\nloc(Rn),ufulfills the support condition\nsuppu(t,·)⊂BR+φℓ(t)−φℓ(1) for anyt∈(1,T), (8)\nand the integral identity\nˆ\nRn∂tu(t,x)φ(t,x) dx+ˆt\n1ˆ\nRn/parenleftbig\n−∂tu(s,x)φs(s,x) +s2ℓ∇u(s,x)· ∇φ(s,x)/parenrightbig\ndxds\n+ˆt\n1ˆ\nRn/parenleftbig\nµs−1∂tu(s,x)φ(s,x) +ν2s−2u(s,x)φ(s,x)/parenrightbig\ndxds\n=εˆ\nRnu1(x)φ(1,x) dx+ˆt\n1ˆ\nRn|u(s,x)|pφ(s,x) dxds (9)\nholds for any t∈(1,T) and any test function φ∈C∞\n0/parenleftbig\n[1,T)×Rn/parenrightbig\n.\nWe notice that, performing further steps of integration by p arts in (9), we get the integral relation\nˆ\nRn/parenleftbig\n∂tu(t,x)φ(t,x)−u(t,x)φt(t,x) +µt−1u(t,x)φ(t,x)/parenrightbig\ndx\n+ˆt\n1ˆ\nRnu(s,x)/parenleftbig\nφss(s,x)−s2ℓ∆φ(s,x)−µs−1φs(s,x) + (µ+ν2)s−2φ(s,x)/parenrightbig\ndxds\n=εˆ\nRn/parenleftbig\n(u1(x) +µu0(x))φ(1,x)−u0(x)φt(1,x)/parenrightbig\ndx+ˆt\n1ˆ\nRn|u(s,x)|pφ(s,x) dxds (10)\nfor anyt∈(1,T) and any test function φ∈C∞\n0/parenleftbig\n[1,T)×Rn/parenrightbig\n.\nLet us state our result in the sub-critical case.\nTheorem 1.2. Letℓ>−1andµ,ν2/greaterorequalslant0such thatδ/greaterorequalslant0. Let us assume that the exponent pof the\nnonlinear term satisfies\n1<p< max/braceleftBig\npStr/parenleftBig\nn+µ\nℓ+1,ℓ/parenrightBig\n,pFuj/parenleftBig\n(ℓ+ 1)n+µ−1\n2−√\nδ\n2/parenrightBig/bracerightBig\n.\nLetu0,u1∈L1\nloc(Rn)be nonnegative, nontrivial and compactly supported functi ons with supports\ncontained in BRfor someR>0such that\nu1+µ−1−√\nδ\n2u0/greaterorequalslant0. (11)\nLetu∈C/parenleftbig\n[1,T),W1,1\nloc(Rn)/parenrightbig\n∩C1/parenleftbig\n[1,T),L1\nloc(Rn)/parenrightbig\n∩Lp\nloc/parenleftbig\n(1,T)×Rn/parenrightbig\nbe a weak solution to (1)\naccording to Definition 1.1 with lifespan T=T(ε).\n3Then, there exists a positive constant ε0=ε0(u0,u1,n,p,ℓ,µ,ν2,R)such that for any ε∈(0,ε0]\nthe weak solution ublows up in finite time. Furthermore, the upper bound estimat es for the lifespan\nT(ε)/lessorequalslant\n\nCε−p(p−1)\nθ(n,ℓ,µ,p ) ifp<p Str/parenleftbig\nn+µ\nℓ+1,ℓ/parenrightbig\n,\nCε−/parenleftbig\n2\np−1−/parenleftbig\n(ℓ+1)n+µ−1\n2−√\nδ\n2/parenrightbig/parenrightbig−1\nifp<p Fuj/parenleftbig\n(ℓ+ 1)n+µ−1\n2−√\nδ\n2/parenrightbig\n,(12)\nholds, where the positive constant Cis independent of εand\nθ(n,ℓ,µ,p ).=ℓ+ 1 +/parenleftBig\nn+1\n2(ℓ+ 1) +µ−3ℓ\n2/parenrightBig\np−/parenleftBig\nn−1\n2(ℓ+ 1) +ℓ+µ\n2/parenrightBig\np2. (13)\nRemark 1.In the previous statement it might happen that the argument o f the Fujita exponent is\na nonpositive number (however, only for ℓ <0 andµ∈[0,1)). Whenever this happens, we do not\nrequire any upper bound for p>1, meaning formally that pFuj(k) =∞fork/lessorequalslant0.\nRemark 2.The exponent pStr/parenleftbig\nn+µ\nℓ+1,ℓ/parenrightbig\nis obtained from the Strauss-type exponent pStr(n,ℓ) defined\nthrough (3) by a shift of magnitudeµ\nℓ+1in the space dimension. Equivalently, pStr/parenleftbig\nn+µ\nℓ+1,ℓ/parenrightbig\nis the\npositive root to the quadratic equation\n/parenleftbiggn−1\n2(ℓ+ 1) +ℓ+µ\n2/parenrightbigg\np2−/parenleftbiggn+ 1\n2(ℓ+ 1) +µ−3ℓ\n2/parenrightbigg\np−(ℓ+ 1) = 0.\nFinally, we provide a blow-up result when we consider the cri tical Fujita-type exponent.\nTheorem 1.3. Letℓ>−1andµ,ν2/greaterorequalslant0such thatδ/greaterorequalslant0. Let us assume that the exponent pof the\nnonlinear term satisfies\np=pFuj/parenleftBig\n(ℓ+ 1)n+µ−1\n2−√\nδ\n2/parenrightBig\n.\nLetu0,u1∈L1\nloc(Rn)be nonnegative, nontrivial and compactly supported functi ons with supports\ncontained in BRfor someR>0. Letu∈C/parenleftbig\n[1,T),W1,1\nloc(Rn)/parenrightbig\n∩C1/parenleftbig\n[1,T),L1\nloc(Rn)/parenrightbig\n∩Lp\nloc/parenleftbig\n(1,T)×Rn/parenrightbig\nbe a weak solution to (1)according to Definition 1.1 with lifespan T=T(ε).\nThen, there exists a positive constant ε0=ε0(u0,u1,n,p,ℓ,µ,ν2,R)such that for any ε∈(0,ε0]\nthe weak solution ublows up in finite time. Moreover, the upper bound estimates f or the lifespan\nT(ε)/lessorequalslant/braceleftBigg\nexp/parenleftbig\nEε−(p−1)/parenrightbig\nifδ >0,\nexp/parenleftbig\nEε−(p−1)/p/parenrightbig\nifδ= 0,(14)\nholds, where the positive constant Eis independent of ε.\n2 Proof of Theorem 1.2\nIn this section, we prove Theorem 1.2 by deriving a sequence o f lower bound estimates for the space\naverage of a weak solution uto (1). More precisely, introducing the functional\nU(t).=ˆ\nRnu(t,x) dx fort∈[1,T),\nour aim is to determine estimates from below for U. Letting to ∞the index of this sequence of lower\nbounds, we establish that the space average of ucannot be globally in time defined and we determine\nas a byproduct an upper bound estimate for the lifespan. In pa rticular, the first two steps that we\nneed to carry out in order to apply the iteration argument are determining the iteration frame for\nU(namely, an integral inequality, where Uappears both on the left and right-hand side) and a first\nlower bound estimate for U. In order to establish such a first lower bound estimate for Uwe consider\na suitable positive solution to the adjoint equation to the l inear EPDT equation. Finally, we employ\nthis first lower bound estimate for Uto begin the iteration procedure, plugging it in the iterati on\nframe. Then, repeating iteratively the procedure we determ ine the desired sequence of lower bounds.\n42.1 Derivation of the iteration frame\nIn this subsection we derive the iteration frame for U. To this purpose we employ the double multiplier\ntechnique from [33] (see also [21, 32]). Given t∈(1,T), let us begin by choosing as test function in\n(9) a cut-off function that localizes the forward light-cone , namely, we take φ∈C∞\n0/parenleftbig\n[1,T)×Rn/parenrightbig\nsuch\nthatφ= 1 in {(s,x)∈[1,t]×Rn:|x|/lessorequalslantR+φℓ(s)−φℓ(1)}. Therefore, from (9) we get\nˆ\nRn∂tu(t,x) dx+ˆt\n1ˆ\nRn/parenleftbig\nµs−1∂tu(s,x) +ν2s−2u(s,x)/parenrightbig\ndxds\n=εˆ\nRnu1(x) dx+ˆt\n1ˆ\nRn|u(s,x)|pdxds.\nDifferentiating with respect to tthe previous equality, we obtain\nˆ\nRn|u(t,x)|pdx=ˆ\nRn∂2\ntu(t,x) dx+µt−1ˆ\nRn∂tu(t,x) dx+ν2t−2ˆ\nRnu(t,x) dx\n=U′′(t) +µt−1U′(t) +ν2t−2U(t). (15)\nIf we denote by r1,r2the roots of the quadratic equation\nr2−(µ−1)r+ν2= 0,\nthen, we can rewrite the right-hand side of the last relation as follows\nU′′(t) +µt−1U′(t) +ν2t−2U(t) =t−(r2+1)d\ndt/parenleftBig\ntr2+1−r1d\ndt/parenleftBig\ntr1U(t)/parenrightBig/parenrightBig\n. (16)\nWe emphasize that the role of r1andr2is fully interchangeable in the previous identity. Combini ng\n(15) and (16), after some straightforward steps we arrive at\nU(t) =Ulin(t) +ˆt\n1/parenleftBigs\nt/parenrightBigr1ˆs\n1/parenleftBigτ\ns/parenrightBigr2+1ˆ\nRn|u(τ,x)|pdxdτds, (17)\nwhere\nUlin(t).=/braceleftBigg\nr1t−r2−r2t−r1\nr1−r2U(1) +t−r2−t−r1\nr1−r2U′(1) ifδ>0,\nt−r1(1 +r1lnt)U(1) +t−r1lntU′(1) ifδ= 0.(18)\nLet us set\nr1.=µ−1−√\nδ\n2, r 2.=µ−1+√\nδ\n2,\nwhere the definition of δis given in (4). From here on, these will be the fixed values of r1,r2.\nConsequently, from (17) we have a twofold result. On the one h and, we get the lower bound estimate\nforU\nU(t)/greaterorequalslantIεt−r1fort∈[1,T), (19)\nwhere the multiplicative constant Idepends on the positive quantities´\nRnu0(x) dxand´\nRnu1(x) dx.\nOn the other hand, by using again the nonnegativity and nontr iviality ofu0,u1, we find\nU(t) =ˆt\n1/parenleftBigs\nt/parenrightBigr1ˆs\n1/parenleftBigτ\ns/parenrightBigr2+1ˆ\nRn|u(τ,x)|pdxdτds (20)\n/greaterorsimilarˆt\n1/parenleftBigs\nt/parenrightBigr1ˆs\n1/parenleftBigτ\ns/parenrightBigr2+1\n(R+φℓ(τ)−φℓ(1))−n(p−1)(U(τ))pdτds\n/greaterorsimilart−r1ˆt\n1sr1−r2−1ˆs\n1τr2+1(1 +τ)−n(ℓ+1)( p−1)(U(τ))pdτds.\nNote that in the last chain of inequalities we used the suppor t condition for uand Jensen’s inequality.\nHence, we obtained the following the iteration frame for U\nU(t)/greaterorequalslantCt−r1ˆt\n1sr1−r2−1ˆs\n1τr2+1τ−n(ℓ+1)( p−1)(U(τ))pdτds (21)\nfor a suitable positive constant Cthat depends on n,p,ℓ .\n52.2 Solution of the adjoint equation\nIn the previous subsection we established a first lower bound estimate for Uin (19). This estimate will\nbe the starting point for the proof of the blow-up result for p<p Fuj/parenleftbig\n(ℓ+1)n+µ−1\n2−√\nδ\n2/parenrightbig\n. However, to\nprove the blow-up result for p<p Str/parenleftbig\nn+µ\nℓ+1,ℓ/parenrightbig\nwe need to determine a further lower bound estimate\nforU. According to this purpose, we introduce a second auxiliary time-dependent functional, which\nis a certain weighted spatial average of u. In particular, we are going to choose the weight function\nto be a positive solution of the adjoint equation to the linea r EPDT equation, namely,\n∂2\nsψ−s2ℓ∆ψ−∂\n∂s(µs−1ψ) +ν2s−2ψ= 0. (22)\nIn order to determine a suitable solution to (22), we follow t he approach from [30, Subsection 2.1]. We\nlook for a solution to (22) with separate variables. As x-dependent function we consider the function\nϕ(x).=/braceleftBigg\nex+ e−xifn= 1,´\nSn−1ex·ωdσωifn/greaterorequalslant2,\nthat has been introduce for the study of blow-up phenomena fo r semilinear hyperbolic models in [44].\nThe positive function ϕ∈C∞(Rn) satisfies ∆ ϕ=ϕand\nϕ(x)∼cn|x|−n−1\n2e|x|as|x| → ∞, (23)\nwherecnis a positive constant depending on n.\nOn the other hand, as s-dependent function we look for a positive solution to the se cond-order\nlinear ODE\n̺′′(s)−s2ℓ̺(s)−µs−1̺′(s) + (µ+ν2)s−2̺(s) = 0. (24)\nLet us perform the change of variables σ=φℓ(s). Then,̺solves the previous equation if and only if\nσ2d2̺\ndσ2+ℓ−µ\nℓ+ 1σd̺\ndσ+/parenleftbiggµ+ν2\n(ℓ+ 1)2−σ2/parenrightbigg\n̺= 0. (25)\nNext, we consider the transformation ̺(σ) =σαη(σ) withα.=µ+1\n2(ℓ+1). Hence,̺solves (25) if and only\nifηis a solution to\nσ2d2η\ndσ2+σdη\ndσ−/parenleftBig\nσ2+δ\n4(ℓ+ 1)2/parenrightBig\nη= 0. (26)\nAs solution to (26) we choose the modified Bessel function of t he second kind K√\nδ\n2(ℓ+1)(σ). Consequently,\nwe set as positive solutions to (24)\n̺(s).=sµ+1\n2K√\nδ\n2(ℓ+1)(φℓ(s)). (27)\nNote that̺=̺(s;ℓ,µ,ν2), but for the sake of brevity we will skip the dependence on ℓ,µ,ν2in the\nnotations hereafter. Therefore, we may define now the follow ing function as positive solution to (22)\nψ(s,x).=̺(s)ϕ(x). (28)\nBy using the weight function ψ, we introduce the auxiliary functional\nU0(t).=ˆ\nRnu(t,x)ψ(t,x) dx.\nNotice that thanks to the support condition for the solution ufrom Definition 1.1, it is possible to\nemployψas test function in (10). Consequently, using (22), we get\nˆ\nRn/parenleftbig\n∂tu(t,x)ψ(t,x)−u(t,x)ψt(t,x) +µt−1u(t,x)ψ(t,x)/parenrightbig\ndx\n=εˆ\nRn/parenleftbig\n̺(1)(u1(x) +µu0(x))−̺′(1)u0(x)/parenrightbig\nϕ(x)dx+ˆt\n1ˆ\nRn|u(s,x)|pψ(s,x) dxds. (29)\n6Applying the recursive relation for the derivative of the mo dified Bessel function of the second kind\n∂zKγ(z) =−Kγ+1(z) +γ\nzKγ(z) (cf. [28, Equations (10.29.1)]), we have\n̺′(s) =−sµ+1\n2+ℓK√\nδ\n2(ℓ+1)+1(φℓ(s)) +µ+1+√\nδ\n2sµ−1\n2K√\nδ\n2(ℓ+1)(φℓ(s)).\nThus,\nIℓ,µ,ν2[u0,u1].=ˆ\nRn/parenleftbig\n̺(1)u1(x) + (̺(1)µ−̺′(1))u0(x)/parenrightbig\nϕ(x)dx\n=ˆ\nRn/parenleftBig\nK√\nδ\n2(ℓ+1)+1(φℓ(1))u0(x) + K√\nδ\n2(ℓ+1)(φℓ(1))/parenleftbig\nu1(x) +µ−1−√\nδ\n2u0(x)/parenrightbig/parenrightBig\nϕ(x)dx>0,\nwhere we employed the nonnegativity and the nontriviality o fu0and (11).\nHence, we can rewrite (29) as\nεIℓ,µ,ν2[u0,u1] +ˆt\n1ˆ\nRn|u(s,x)|pψ(s,x) dxds=U′\n0(t) +µt−1U0(t)−2̺′(t)\n̺(t)U0(t)\n=̺2(t)\ntµd\ndt/parenleftbiggtµ\n̺2(t)U0(t)/parenrightbigg\n.\nSince both ψand the nonlinear term are nonnegative, from the previous re lation we obtain\nU0(t)/greaterorequalslantεIℓ,µ,ν2[u0,u1]̺2(t)\ntµˆt\n1sµ\n̺2(s)ds>0 for any t∈[1,T).\nThanks to the assumptions on the Cauchy data we have shown tha t the functional U0is nonnegative.\nNext, we shall determine a lower bound estimate for U0for large times. For this reason we recall the\nasymptotic behavior of K γfor large arguments, namely, K γ(z) =/radicalbig\nπ/(2z)e−z(1 +O(z−1)) asz→ ∞\nforz >0 (cf. [28, Equation (10.25.3)]). So, there exists T0=T0(ℓ,µ,ν2)>1 such that for s/greaterorequalslantT0it\nholds\n1\n4π(ℓ+ 1) e−2φℓ(s)sµ−ℓ/lessorequalslant̺2(s)/lessorequalslantπ(ℓ+ 1) e−2φℓ(s)sµ−ℓ. (30)\nThen, fort/greaterorequalslantT0we have\nU0(t)/greaterorequalslantεIℓ,µ,ν2[u0,u1]̺2(t)\ntµˆt\nT0sµ\n̺2(s)ds\n/greaterorequalslantε\n4Iℓ,µ,ν2[u0,u1]t−ℓe−2φℓ(t)ˆt\nT0sℓe2φℓ(s)ds.\nFort/greaterorequalslant2T0, it results\nU0(t)/greaterorsimilarεIℓ,µ,ν2[u0,u1]t−ℓe−2φℓ(t)ˆt\nt/2sℓe2φℓ(s)ds\n/greaterorsimilarεIℓ,µ,ν2[u0,u1]t−ℓ/parenleftbig\n1−e2φℓ(t/2)−2φℓ(t)/parenrightbig\n=εIℓ,µ,ν2[u0,u1]t−ℓ/parenleftBig\n1−e−2\nℓ+1(2ℓ+1−1)tℓ+1/parenrightBig\n/greaterorsimilarεIℓ,µ,ν2[u0,u1]t−ℓ.\nWe emphasize that in the last step we use the condition ℓ >−1 to estimate from below the factor\ncontaining the exponential term with a positive constant.\nSummarizing, we proved\nU0(t)/greaterorsimilarεt−ℓfort/greaterorequalslant2T0, (31)\nwhere the unexpressed multiplicative constant depends on t he Cauchy data and on the parameters\nℓ,µ,ν2. Finally, we show how from (31) we derive a second lower bound estimate for U. By Hölder’s\ninequality, we find for t/greaterorequalslant2T0\nεt−ℓ/lessorsimilarU0(t)/lessorequalslant/parenleftbiggˆ\nRn|u(t,x)|pdx/parenrightbigg1/p/parenleftBiggˆ\nBR+φℓ(t)−φℓ(1)(ψ(t,x))p′dx/parenrightBigg1/p′\n,\n7wherep′denotes the conjugate exponent of p. Following [31, Section 3] and employing (30), we can\nestimate for t/greaterorequalslant2T0\nˆ\nRn|u(t,x)|pdx/greaterorsimilarεpt−ℓp/parenleftBiggˆ\nBR+φℓ(t)−φℓ(1)(ψ(t,x))p′dx/parenrightBigg−(p−1)\n/greaterorsimilarεpt−ℓp(̺(t))−pe−p(R+φℓ(t)−φℓ(1))(R+φℓ(t)−φℓ(1))−(n−1)(p−1)+n−1\n2p\n/greaterorsimilarεpt(−n−1\n2(ℓ+1)−ℓ+µ\n2)p+(n−1)(ℓ+1).\nPlugging this last estimate from below for the space integra l of the nonlinear term in (20), for any\nt/greaterorequalslantT1.= 2T0+ 1 we get\nU(t)/greaterorequalslantKεpt(−n−1\n2(ℓ+1)−ℓ+µ\n2)p+(n−1)(ℓ+1)+2, (32)\nwhere the multiplicative constant Kdepends on the Cauchy data and on n,p,ℓ,µ,ν2,R, which is the\ndesired lower bound estimate for U.\n2.3 Iteration argument\nIn this section we establish the following sequence of lower bound estimates for U\nU(t)/greaterorequalslantCjt−αj(t−T1)βjfort/greaterorequalslantT1, (33)\nwhere {αj}j∈N,{βj}j∈Nand{Cj}j∈Nare sequences of positive real numbers that we will determin e\niteratively during the proof.\nLet us begin by considering the case 1 <p<p Str/parenleftbig\nn+µ\nℓ+1,ℓ/parenrightbig\n. As we have previously mentioned, we\nconsider (32) as first lower bound estimate for Uforpbelow the Strauss-type exponent. Therefore, (33)\nforj= 0 is satisfied provided that we set C0.=Kεp,α0.= [n−1\n2(ℓ+1)+ℓ+µ\n2]pandβ0.= (n−1)(ℓ+1)+2.\nNext, we assume that (33) holds true for some j/greaterorequalslant0 and we want to prove it for j+ 1. If we plug\nthe lower bound estimate for Uin (33) in the iteration frame (21), we get\nU(t)/greaterorequalslantCt−r1ˆt\nT1sr1−r2−1ˆs\nT1τr2+1−n(ℓ+1)( p−1)(U(τ))pdτds\n/greaterorequalslantCCp\njt−r1ˆt\nT1sr1−r2−1ˆs\nT1(τ−T1)r2+1+ pβjτ−n(ℓ+1)( p−1)−pαjdτds\n/greaterorequalslantCCp\njt−r2−1−n(ℓ+1)( p−1)−pαjˆt\nT1ˆs\nT1(τ−T1)r2+1+ pβjdτds\n/greaterorequalslantCCp\nj\n(r2+ 3 +pβj)2t−r2−1−n(ℓ+1)( p−1)−pαj(t−T1)r2+3+ pβj,\nwhere we used the relations r1−r2−1 =−(√\nδ+ 1)<0 andr2+ 1>0. Hence, if we define\nCj+1.=CCp\nj(r2+ 3 +pβj)−2, (34)\nαj+1.=r2+ 1 +n(ℓ+ 1)(p−1) +pαj, β j+1.=r2+ 3 +pβj, (35)\nthen, we proved (33) for j+ 1 as well. Let us determine explicitly αjandβj. By using recursively\nthe definition in (35), we find\nαj=r2+ 1 +n(ℓ+ 1)(p−1) +pαj−1=...= (r2+ 1 +n(ℓ+ 1)(p−1))j−1/summationdisplay\nk=0pk+pjα0\n=/parenleftbiggr2+ 1\np−1+n(ℓ+ 1) +α0/parenrightbigg\npj−r2+ 1\np−1−n(ℓ+ 1), (36)\nand, analogously,\nβj=/parenleftbiggr2+ 3\np−1+β0/parenrightbigg\npj−r2+ 3\np−1. (37)\n8In particular, combining (34) and (37), we get\nCj=CCp\nj−1(r2+ 3 +pβj−1)−2=CCp\nj−1β−2\nj/greaterorequalslantC/parenleftbiggr2+ 3\np−1+β0/parenrightbigg−2\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright.=DCp\nj−1p−2j=DCp\nj−1p−2j\nfor anyj/greaterorequalslant1. Applying the logarithmic function to both sides of the ine qualityCj/greaterorequalslantDCp\nj−1p−2jand\nemploying iteratively the resulting relation, we have\nlnCj/greaterorequalslantplnCj−1−2jlnp+ lnD\n/greaterorequalslantp2lnCj−2−2(j+ (j−1)p) lnp+ (1 +p) lnD\n/greaterorequalslant.../greaterorequalslantpjlnC0−2 lnp/parenleftBiggj−1/summationdisplay\nk=0(j−k)pk/parenrightBigg\n+ lnDj−1/summationdisplay\nk=0pk\n=pj/parenleftbigg\nlnC0−2plnp\n(p−1)2+lnD\np−1/parenrightbigg\n+2 lnp\np−1j+2plnp\n(p−1)2−lnD\np−1,\nwhere we used the identity\nj−1/summationdisplay\nk=0(j−k)pk=1\np−1/parenleftbiggpj+1−p\np−1−j/parenrightbigg\n.\nLetj0=j0(n,ℓ,µ,ν2,p)∈Nbe the smallest integer such that\nj0/greaterorequalslantlnD\n2 lnp−p\np−1.\nThen, for any j/greaterorequalslantj0it results\nlnCj/greaterorequalslantpj/parenleftbigg\nlnC0−2plnp\n(p−1)2+lnD\np−1/parenrightbigg\n=pjln/parenleftbig/tildewideDεp/parenrightbig\n, (38)\nwhere/tildewideD.=KD1/(p−1)p−(2p)/(p−1)2.\nFinally, from (33) we can prove that Ublows up in finite time since the lower bound diverges\nasj→ ∞ and, besides, we can derive an upper bound estimate for the li fespan of the solution.\nCombining (36), (37) and (38), for j/greaterorequalslantj0andt/greaterorequalslantT1we obtain\nU(t)/greaterorequalslantexp/parenleftBig\npjln/parenleftbig/tildewideDεp/parenrightbig/parenrightBig\nt−αj(t−T1)βj\n/greaterorequalslantexp/parenleftBig\npj/parenleftBig\nln/parenleftbig/tildewideDεp/parenrightbig\n−/parenleftBig\nr2+1\np−1+n(ℓ+ 1) +α0/parenrightBig\nlnt+/parenleftBig\nr2+3\np−1+β0/parenrightBig\nln(t−T1)/parenrightBig/parenrightBig\ntr2+1\np−1+n(ℓ+1)(t−T1)−r3+1\np−1.\nFort>2T1it holds the relation ln( t−T1)/greaterorequalslantlnt−ln 2, consequently, for j/greaterorequalslantj0we get\nU(t)/greaterorequalslantexp/parenleftBig\npj/parenleftBig\nln/parenleftbig/hatwideDεp/parenrightbig\n+/parenleftBig\n2\np−1+β0−α0−n(ℓ+ 1)/parenrightBig\nlnt/parenrightBig/parenrightBig\ntr2+1\np−1+n(ℓ+1)(t−T1)−r3+1\np−1,\nwhere/hatwideD.= 2−r2+3\np−1−β0/tildewideD.\nLet us write explicitly the factor that multiplies ln tin the previous estimate\n2\np−1+β0−α0−n(ℓ+ 1) =2\np−1+ (n−1)(ℓ+ 1) + 2 −/bracketleftBig\nn−1\n2(ℓ+ 1) +ℓ+µ\n2/bracketrightBig\np−n(ℓ+ 1)\n=1\np−1/braceleftBig\n−/bracketleftBig\nn−1\n2(ℓ+ 1) +ℓ+µ\n2/bracketrightBig\np(p−1)−(ℓ+ 1)(p−1) + 2p/bracerightBig\n=1\np−1/braceleftBig\n−/bracketleftBig\nn−1\n2(ℓ+ 1) +ℓ+µ\n2/bracketrightBig\np2+/bracketleftBig\nn+1\n2(ℓ+ 1) +µ−3ℓ\n2/bracketrightBig\np+ℓ+ 1/bracerightBig\n=θ(n,ℓ,µ,p )\np−1,\nwhereθ(n,ℓ,µ,p ) is defined in (13) and is a positive quantity thanks to the con ditionp<p Str(n+µ\nℓ+1,ℓ)\non the exponent of the nonlinear term. Then, for j/greaterorequalslantj0andt/greaterorequalslant2T1we arrived at\nU(t)/greaterorequalslantexp/parenleftBig\npj/parenleftBig\nln/parenleftBig\n/hatwideDεptθ(n,ℓ,µ,p )\np−1/parenrightBig/parenrightBig/parenrightBig\ntr2+1\np−1+n(ℓ+1)(t−T1)−r3+1\np−1. (39)\n9Let us fixε0=ε0(u0,u1,n,p,ℓ,µ,ν2,R) such that 0 <ε 0<(2T1)−θ(n,ℓ,µ,p )/(p(p−1))/hatwideD−1/p. Then, for\nanyε∈(0,ε0] and anyt>/hatwideD−(p−1)/θ(n,ℓ,µ,p )ε−p(p−1)/θ(n,ℓ,µ,p )we have\nt>2T1and ln/parenleftBig\n/hatwideDεptθ(n,ℓ,µ,p )\np−1/parenrightBig\n>0,\nso, lettingj→ ∞ in (39), the lower bound for Ublows up. Thus, we proved that Uis not finite for\nt/greaterorsimilarε−p(p−1)/θ(n,ℓ,µ,p ), that is, we showed the lifespan estimate in (12) for p<p Str(n+µ\nℓ+1,ℓ).\nThe case 1<p<p Fuj/parenleftbig\n(ℓ+ 1)n+µ−1\n2−√\nδ\n2/parenrightbig\ncan be treated in a completely analogous way. Indeed,\nemploying (19) in place of (32), so that, C0=Iεandβ0−α0=1−µ\n2+√\nδ\n2, we determine the following\nlower bound estimate for Uinstead of (39)\nU(t)/greaterorequalslantexp/parenleftBig\npj/parenleftBig\nln/parenleftBig\n/tildewideCεt2\np−1−(ℓ+1)n+β0−α0/parenrightBig/parenrightBig/parenrightBig\ntr2+1\np−1+n(ℓ+1)(t−T1)−r3+1\np−1\nfor anyt/greaterorequalslant2T1and forjgreater than a suitable j1(n,ℓ,µ,ν2,p)∈N, where/tildewideCis a certain positive\nconstant. Thanks to the assumption on p, from this estimate we conclude the validity of the second\nupper bound estimate in (12) by repeating the same argument a s in the first case.\nRemark 3.In the case δ= 0 and for p < p Fuj((ℓ+ 1)n+r1), from (18) we see that we can actually\nimprove the lower bound estimate (19) by a logarithmic facto r. By using a slicing procedure as in the\nnext section, it is possible to prove the following upper bou nd estimate for the lifespan\nT(ε)2\np−1−((ℓ+1)n+r1)lnT(ε)/lessorsimilarε−1.\n3 Proof of Theorem 1.3\nIn order to prove Theorem 1.3 we derive a sequence of lower bou nd estimates for U(t) with additional\nlogarithmic factors. According to this purpose, we introdu ce{ℓj}j∈Nsuch thatℓj.= 2−2−(j+1). We\nare going to use this sequence to apply a slicing procedure in the iteration argument, following the\nideas introduced in the paper [1]. More precisely, we establ ish the following estimates\nU(t)/greaterorequalslantKjt−r1/parenleftbigg\nln/parenleftbiggt\nℓj/parenrightbigg/parenrightbiggγj\n(40)\nfor anyj∈Nand anyt/greaterorequalslantℓj, where {Kj}j∈N,{γj}j∈Nare sequences of nonnegative numbers to be\ndetermined iteratively. From (18) we obtain that\nU(t)/greaterorequalslant/braceleftBigg\nIεt−r1 ifδ>0,\nIεt−r1lntifδ= 0,\nwhich implies the validity of (40) for j= 0 provided that K0.=Iεand\nγ0.=/braceleftBigg\n0 ifδ>0,\n1 ifδ= 0.\nLet us proceed with the inductive step. We plug (40) for some j∈Nin (21) and we prove the validity\nof (40) forj+ 1, prescribing suitable recursive relations for the terms Kj+1andγj+1. Therefore, for\nt/greaterorequalslantℓjwe have\nU(t)/greaterorequalslantCKp\njt−r1ˆt\nℓjsr1−r2−1ˆs\nℓjτr2+1−n(ℓ+1)( p−1)−r1p/parenleftBig\nln/parenleftBig\nτ\nℓj/parenrightBig/parenrightBigpγj\ndτds\n/greaterorequalslantCKp\njt−r1ˆt\nℓjsr1−r2−1−[(ℓ+1)n+r1](p−1)ˆs\nℓjτr2−r1+1/parenleftBig\nln/parenleftBig\nτ\nℓj/parenrightBig/parenrightBigpγj\ndτds.\nFort/greaterorequalslantℓj+1, we can shrink the interval of integration in the τ-integral as follows\nU(t)/greaterorequalslantCKp\njt−r1ˆt\nℓj+1sr1−r2−1−[(ℓ+1)n+r1](p−1)ˆs\nℓjs\nℓj+1τr2−r1+1/parenleftBig\nln/parenleftBig\nτ\nℓj/parenrightBig/parenrightBigpγj\ndτds\n/greaterorequalslantCKp\njt−r1ˆt\nℓj+1sr1−r2−1−[(ℓ+1)n+r1](p−1)/parenleftBig\nln/parenleftBig\ns\nℓj+1/parenrightBig/parenrightBigpγjˆs\nℓjs\nℓj+1/parenleftBig\nτ−ℓjs\nℓj+1/parenrightBigr2−r1+1\ndτds\n=C(r2−r1+ 2)−1/parenleftBig\n1−ℓj\nℓj+1/parenrightBigr2−r1+2\nKp\njt−r1ˆt\nℓj+1s1−[(ℓ+1)n+r1](p−1)/parenleftBig\nln/parenleftBig\ns\nℓj+1/parenrightBig/parenrightBigpγj\nds.\n10Due top=pFuj((ℓ+ 1)n+r1), the power of sis actually −1 in the last integral, so, for t/greaterorequalslantℓj+1we\nobtain\nU(t)/greaterorequalslantC(r2−r1+ 2)−1/parenleftBig\n1−ℓj\nℓj+1/parenrightBigr2−r1+2\nKp\njt−r1ˆt\nℓj+1s−1/parenleftBig\nln/parenleftBig\ns\nℓj+1/parenrightBig/parenrightBigpγj\nds\n=C(r2−r1+ 2)−1/parenleftBig\n1−ℓj\nℓj+1/parenrightBigr2−r1+2\nKp\nj(pγj+ 1)−1t−r1/parenleftBig\nln/parenleftBig\nt\nℓj+1/parenrightBig/parenrightBigpγj+1\n,\nwhich is exactly (40) for j+ 1, provided that\nγj+1=pγj+ 1,\nKj+1=C(r2−r1+ 2)−1/parenleftBig\n1−ℓj\nℓj+1/parenrightBigr2−r1+2\n(pγj+ 1)−1Kp\nj.\nApplying iteratively the recursive relation γj=pγj−1+ 1, we obtain\nγj=pjγ0+j−1/summationdisplay\nk=0pk=/parenleftBig\nγ0+1\np−1/parenrightBig\npj−1\np−1(41)\n/lessorequalslant/parenleftBig\nγ0+1\np−1/parenrightBig\npj.\nMoreover, 1 −ℓj−1\nℓj>2−(j+2). Consequently, combining the previous considerations, fo r anyj/greaterorequalslant1 we\nhave\nKj/greaterorequalslantMQ−jKp\nj−1,\nwhereM.=C(r2−r1+ 2)−12−2(r2−r1+2)/parenleftBig\nγ0+1\np−1/parenrightBig−1\nandQ.= 2r2−r1+2p. Applying the logarithmic\nfunction to both sides of the previous inequality and employ ing recursively the obtained estimate, we\nget\nlnKj/greaterorequalslantplnKj−1−jlnQ+ lnM\n/greaterorequalslantp2lnKj−2−(j+ (j−1)p) lnQ+ (1 +p) lnM\n/greaterorequalslant.../greaterorequalslantpjlnK0−lnQ/parenleftBiggj−1/summationdisplay\nk=0(j−k)pk/parenrightBigg\n+ lnMj−1/summationdisplay\nk=0pk\n=pj/parenleftbigg\nlnK0−plnQ\n(p−1)2+lnM\np−1/parenrightbigg\n+lnQ\np−1j+plnQ\n(p−1)2−lnM\np−1.\nLetj2=j2(n,µ,ν2,ℓ)∈Nbe the smallest integer such that j1/greaterorequalslantlnM\nlnQ−p\np−1. Therefore, for any\nj∈N,j/greaterorequalslantj2it holds\nlnKj/greaterorequalslantpjln(/tildewiderMε), (42)\nwhere/tildewiderM.=IM1/(p−1)Q−p/(p−1)2.\nFinally, we combine (40), (41) and (42), so for any j/greaterorequalslantj2and anyt/greaterorequalslant2/greaterorequalslantℓjwe find\nU(t)/greaterorequalslantKjt−r1/parenleftbig\nln/parenleftbigt\n2/parenrightbig/parenrightbig(γ0+1\np−1)pj−1\np−1\n= exp/bracketleftBig\nlnKj+/parenleftBig\nγ0+1\np−1/parenrightBig\npjln/parenleftbig\nln/parenleftbigt\n2/parenrightbig/parenrightbig/bracketrightBig\nt−r1/parenleftbig\nln/parenleftbigt\n2/parenrightbig/parenrightbig−1\np−1\n/greaterorequalslantexp/bracketleftBig\npj/parenleftBig\nln(/tildewiderMε) +/parenleftBig\nγ0+1\np−1/parenrightBig\nln/parenleftbig\nln/parenleftbigt\n2/parenrightbig/parenrightbig/parenrightBig/bracketrightBig\nt−r1/parenleftbig\nln/parenleftbigt\n2/parenrightbig/parenrightbig−1\np−1\n= exp/bracketleftBig\npjln/parenleftBig\n/tildewiderMε/parenleftbig\nln/parenleftbigt\n2/parenrightbig/parenrightbigγ0+1/(p−1)/parenrightBig/bracketrightBig\nt−r1/parenleftbig\nln/parenleftbigt\n2/parenrightbig/parenrightbig−1\np−1.\nSince ln(t\n2)/greaterorequalslant1\n2lntfort/greaterorequalslant4, forj/greaterorequalslantj2andt/greaterorequalslant4 we arrive at\nU(t)/greaterorequalslantexp/bracketleftBig\npjln/parenleftBig\n/hatwiderMε(lnt)γ0+1/(p−1)/parenrightBig/bracketrightBig\nt−r1/parenleftbig\nln/parenleftbigt\n2/parenrightbig/parenrightbig−1\np−1, (43)\nwhere/hatwiderM.= 2−(γ0+1/(p−1))/tildewiderM.\n11Let us fixε0=ε(u0,u1,n,ℓ,µ,ν2,R)>0 such that ε0/lessorequalslant(2 ln 2)−(γ0+1/(p−1))/hatwiderM−1. Then, for any\nε∈(0,ε0] and anyt>exp/parenleftBig\n(/hatwiderMε)−(γ0+1/(p−1))−1/parenrightBig\nwe have\nt/greaterorequalslant4 and ln/parenleftBig\n/hatwiderMε(lnt)γ0+1/(p−1)/parenrightBig\n>0,\nthus, letting j→ ∞ in (43) we obtain that the lower bound for U(t) blows up. Hence, we proved that\nfor lnt/greaterorsimilarε−(γ0+1/(p−1))−1the average U(t) may not be finite. This completes the proof and shows\nthe upper bound estimate (14) for the lifespan.\n4 Concluding remarks\nIn this final section, we analyze the result obtained in Theor em 1.2. Setting\npc(n,ℓ,µ,ν2).= max/braceleftbigg\npStr/parenleftBig\nn+µ\nℓ+1,ℓ/parenrightBig\n,pFuj/parenleftbigg\n(ℓ+ 1)n+µ−1\n2−√\n(µ−1)2−4ν2\n2/parenrightbigg/bracerightbigg\n,\nfrom Theorem 1.2 we know that a blow-up result holds for 1 <p<p c(n,ℓ,µ,ν2) provided that δ/greaterorequalslant0\nand that the compactly supported Cauchy data fulfill suitabl e sign assumptions. As in the case of the\nsemilinear wave equation with scale-invariant damping and mass, the condition δ/greaterorequalslant0 implies that the\ndamping term is dominant over the mass one. From a technical v iewpoint, this assumption guarantees\nthe possibility to use the double multiplier technique (cf. [33, 21]) while deriving the iteration frame\n(21).\nCombining the blow-up result from this paper with the global existence result for small data\nsolutions in [6, Corollary 5.1], it follows that pc(n,ℓ,µ,ν2) is the critical exponent for (1) when n= 1.\nMoreover, it is reasonable to conjecture that the exponent pc(n,ℓ,µ,ν2) is critical even for higher\ndimensions. Indeed, for µ=ν2= 0 we have that pc(n,ℓ,0,0) =pStr(n,ℓ) forn/greaterorequalslant2 andpc(n,ℓ,0,0) =\npFuj(ℓ) forn= 1 (see [19, Remark 1.6] for the one-dimensional case) accor ding to the results for (2)\nwith power nonlinearity that we recalled in the introductio n. In particular, when ℓ= 0 too we find\nthatpc(n,0,0,0) is the solution to the quadratic equation ( n−1)p2−(n+ 1)p−2 = 0, namely, the\ncelebrated exponent named after the author of [34] which is t he critical exponent for the semilinear\nwave equation.\nFurthermore, we point out that for ℓ= 0 and for ν2= 0,ℓ∈(−1,0) the exponent pc(n,ℓ,µ,ν2)\ncoincides with (5) and (6), respectively.\nIn Theorem 1.3, the blow-up of local solutions is proved (und er suitable assumptions for the Cauchy\ndata) in the critical case with Fujita-type exponent. In the forthcoming paper [22], the blow-up in the\nother critical case of Strauss-type is considered, with the approach for the critical classical semilinear\nwave equation developed in [36].\nAcknowledgments\nA. Palmieri is member of the Gruppo Nazionale per L’Analisi Matematica, la Probabilità e le loro\nApplicazioni (GNAMPA) of the Instituto Nazionale di Alta Matematica (INdAM). A. Palmieri has\nbeen partially supported by INdAM - GNAMPA Project 2024 “Mod elli locali e non-locali con pertur-\nbazioni non-lineari” CUP E53C23001670001 and by ERC Seeds U niBa Project “NWEinNES” CUP\nH93C23000730001.\nReferences\n[1] Agemi R., Kurokawa Y., Takamura H., Critical curve for p−qsystems of nonlinear wave equations\nin three space dimensions. J. Differential Equations 167(1):87–133 (2000).\n[2] Ben Hassen M.F., Hamouda M., Hamza M.A., Teka H.K., Nonex istence result for the generalized\nTricomi equation with the scale-invariant damping, mass te rm and time derivative nonlinearity.\nAsymptotic Analysis 128(4): 495–515, (2022). DOI: 10.3233/ASY-211714\n[3] Chen W., Lucente S., Palmieri A., Nonexistence of global solutions for generalized Tricomi\nequations with combined nonlinearity. Nonlinear Anal. 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Appl. 514(1), 126266 (2022).\n[33] Palmieri A., Tu Z., Lifespan of semilinear wave equatio n with scale invariant dissipation and mass\nand sub-Strauss power nonlinearity. J. Math. Anal. Appl. 470(1): 447–469 (2019). https://doi:\n10.1016/j.jmaa.2018.10.015.\n[34] Strauss W.A., Nonlinear scattering theory at low energ y.J. Funct. Anal. 41(1): 110–133 (1981).\n[35] Sun Y., Sharp lifespan estimates for subcritical gener alized semilinear Tricomi equations. Math.\nMeth. Appl. Sci. 1–13 (2021). https://doi.org/10.1002/mma.7402\n[36] Takamura H., Wakasa K., The sharp upper bound of the life span of solutions to critical semilinear\nwave equations in high dimensions, J. Differential Equations 251(4/5): 1157–1171 (2011).\n[37] Tsutaya K., Wakasugi Y., Blow up of solutions of semilin ear wave equations in\nFriedmann-Lemaître-Robertson-Walker spacetime. J. Math. 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Probl. 2021 : 94 (2021)\nhttps://doi.org/10.1186/s13661-021-01571-0\n[41] Tu Z., Lin J., Lifespan of semilinear generalized Trico mi equation with Strauss type exponent.\nPreprint, arXiv:1903.11351v2 (2019).\n[42] Yagdjian K., A note on the fundamental solution for the T ricomi-type equation in the hyperbolic\ndomain, J. Differential Equations 206(1), 227–252 (2004).\n[43] Yagdjian K., Global Existence for the n-Dimensional Se milinear Tricomi-Type Equations. Comm.\nPartial Differential Equations 31(6), 907–944 (2006).\n14[44] Yordanov B.T., Zhang Q.S., Finite time blow up for criti cal wave equations in high dimensions.\nJ. Funct. Anal. 231(2): 361-374 (2006).\n15"    },    {        "title": "2105.03576v1.A_second_order_numerical_method_for_Landau_Lifshitz_Gilbert_equation_with_large_damping_parameters.pdf",        "content": "A SECOND-ORDER NUMERICAL METHOD FOR\nLANDAU-LIFSHITZ-GILBERT EQUATION WITH LARGE\nDAMPING PARAMETERS\nYONGYONG CAI, JINGRUN CHEN, CHENG WANG, AND CHANGJIAN XIE\nAbstract. A second order accurate numerical scheme is proposed and imple-\nmented for the Landau-Lifshitz-Gilbert equation, which models magnetization\ndynamics in ferromagnetic materials, with large damping parameters. The\nmain advantages of this method are associated with the following features:\n(1) It only solves linear systems of equations with constant coe\u000ecients where\nfast solvers are available, so that the numerical e\u000eciency has been greatly im-\nproved, in comparison with the existing Gauss-Seidel project method. (2) The\nsecond-order accuracy in time is achieved, and it is unconditionally stable for\nlarge damping parameters. Moreover, both the second-order accuracy and the\ngreat e\u000eciency improvement will be veri\fed by several numerical examples\nin the 1D and 3D simulations. In the presence of large damping parameters,\nit is observed that this method is unconditionally stable and \fnds physically\nreasonable structures while many existing methods have failed. For the do-\nmain wall dynamics, the linear dependence of wall velocity with respect to the\ndamping parameter and the external magnetic \feld will be obtained through\nthe reported simulations.\n1.Introduction\nFerromagnetic materials are widely used for data storage due to the bi-stable\nstates of the intrinsic magnetic order or magnetization. The dynamics of magneti-\nzation has been modeled by the Landau-Lifshitz-Gilbert (LLG) equation [9,13]. In\nparticular, two terms are involved in the dynamics of the LLG equation: the gyro-\nmagnetic term, which is energetically conservative, and the damping term, which\nis energetically dissipative.\nThe damping term is important since it strongly a\u000bects the energy required and\nthe speed at which a magnetic device operates. A recent experiment on a magnetic-\nsemiconductor heterostructure [25] has indicated that the Gilbert damping constant\ncan be adjusted. At the microscopic level, the electron scattering, the itinerant\nelectron relaxation [11], and the phonon-magnon coupling [16, 17] are responsible\nto the damping, which can be obtained from electronic structure calculations [19].\nFor the application purpose, tuning the damping parameter allows one to optimize\nthe magneto-dynamic properties in the material, such as lowering the switching\ncurrent and increasing the writing speed of magnetic memory devices [23].\nWhile most experiments have been devoted to small damping parameters [4,14,\n22], large damping e\u000bects are observed in [10,18]. The magnetization switching time\nDate : May 11, 2021.\n2010 Mathematics Subject Classi\fcation. 35K61, 65N06, 65N12.\nKey words and phrases. Micromagnetics simulations, Landau-Lifshitz-Gilbert equation,\nsecond-order method, large damping parameter.\n1arXiv:2105.03576v1  [physics.comp-ph]  8 May 20212 Y. CAI, J. CHEN, C. WANG, AND C. XIE\ntends to be shorter in the presence of the large damping constant [18]. Extremely\nlarge damping parameters ( \u00189) are presented in [10].\nThe LLG equation is a vectorial and nonlinear system with the \fxed length of\nmagnetization in a point-wise sense. Signi\fcant e\u000borts have been devoted to design\ne\u000ecient and stable numerical methods for micromagnetics simulations; see [6, 12]\nfor reviews and references therein. Among the existing numerical works, semi-\nimplicit schemes have been very popular since they avoid a complicated nonlinear\nsolver while preserving the numerical stability; see [2, 7, 24], etc. In particular,\nthe second-order accurate backward di\u000berentiation formula (BDF) scheme is con-\nstructed in [24], with a one-sided interpolation. In turn, a three-dimensional lin-\near system needs to be solved at each time step, with non-constant coe\u000ecients.\nMoreover, a theoretical analysis of the second order convergence estimate has been\nestablished in [5] for such a BDF2 method. As another approach, a linearly implicit\nmethod in [2] introduces the tangent space to deal with the length constraint of\nmagnetization, with the \frst-order temporal accuracy. As a further extension, high-\norder BDF schemes have been constructed and analyzed in a more recent work [1].\nAn unconditionally unique solvability of the semi-implicit schemes has been proved\nin [1,5], while the convergence analysis has required a condition that the temporal\nstep-size is proportional to the spatial grid-size. However, an obvious disadvantage\nhas been observed for these semi-implicit schemes: the vectorial structure of the\nLLG equation leads to a non-symmetric linear system at each time step, which\ncannot be implemented by an FFT-based fast solver. In fact, the GMRES is often\nused, while its e\u000eciency depends heavily on the temporal step-size and the spatial\ngrid-size, and extensive numerical experiments have indicated much more expensive\ncomputational costs than standard Poisson solvers [24].\nThe Gauss-Seidel projection method (GSPM) is another popular set of numerical\nalgorithms since only linear systems with constant coe\u000ecients need to be solved at\neach time step [8,15,21]. This method is based on a combination of a Gauss-Seidel\nupdate of an implicit solver for the gyromagnetic term, the heat \row of the harmonic\nmap, and a projection step to overcome the sti\u000bness and the nonlinearity associated\nto the LLG equation. In this numerical approach, the implicit discretization is only\napplied to the scalar heat equation implicitly several times; therefore, the FFT-\nbased fast solvers become available, due to the symmetric, positive de\fnite (SPD)\nstructures of the linear system. The original GSPM method [20] turns out to be\nunstable for small damping parameters, while this issue has been resolved in [8] with\nmore updates of the stray \feld. Its numerical e\u000eciency has been further improved\nby reducing the number of linear systems per time step [15]. One little de\fciency\nof GSPM is its \frst-order accuracy in time.\nMeanwhile, in spite of these improvements, the GSPM method is computation-\nally more expensive than the standard Poisson solver, because of the Gauss-Seidel\niteration involved in the algorithm. An additional de\fciency of the GSPM is its\n\frst-order accuracy in time. Moreover, most of the above-mentioned methods have\nbeen mainly focused on small damping parameters with the only exception in a\ntheoretical work [1]. In other words, there has been no numerical method designed\nspeci\fcally for real micromagnetics simulations with large damping parameters. In\nthis paper, we propose a second-order accurate numerical method to solve the LLG\nequation with large damping parameters, whose complexity is also comparable toA SECOND-ORDER METHOD FOR LLG EQUATION 3\nsolving the scalar heat equation. To achieve this goal, the LLG system is refor-\nmulated, in which the damping term is rewritten as a harmonic mapping \row. In\nturn, the constant-coe\u000ecient Laplacian part is treated by a standard BDF2 tem-\nporal discretization, and the associated dissipation will form the foundation of the\nnumerical stability. Meanwhile, all the nonlinear parts, including both the gyro-\nmagnetic term and the remaining nonlinear expansions in the damping term, are\ncomputed by a fully explicit approximation, which is accomplished by a second\norder extrapolation formula. Because of this fully explicit treatment for the nonlin-\near parts, the resulting numerical scheme only requires a standard Poisson solver at\neach time step. This fact will greatly facilitate the computational e\u000borts, since the\nFFT-based fast solver could be e\u000eciently applied, due to the SPD structure of the\nlinear system involved at each time step. In addition, the numerical stability has\nbeen demonstrated by extensive computational experiments, and these experiments\nhas veri\fed the idea that the dissipation property of the heat equation part would\nbe able to ensure the numerical stability of the nonlinear parts, with large damping\nparameters.\nThe rest of this paper is organized as follows. In section 2, the micromagnetics\nmodel is reviewed, and the numerical method is proposed, as well as its comparison\nwith the GSPM and the semi-implicit projection method (SIPM). Subsequently,\nthe numerical results are presented in section 3, including the temporal and spa-\ntial accuracy check in both the 1D and 3D computations, the numerical e\u000eciency\ninvestigation (in comparison with the GSPM and SIPM algorithms), the stability\nstudy with respect to the damping parameter, and the dependence of domain wall\nvelocity on the damping parameter and the external magnetic \feld. Finally, some\nconcluding remarks are made in section 4.\n2.The physical model and the numerical method\n2.1.Landau-Lifshitz-Gilbert equation. The LLG equation describes the dy-\nnamics of magnetization which consists of the gyromagnetic term and the damping\nterm [3,13]. In the nondimensionalized form, this equation reads as\nmt=\u0000m\u0002he\u000b\u0000\u000bm\u0002(m\u0002he\u000b) (2.1)\nwith the homogeneous Neumann boundary condition\n(2.2)@m\n@\u0017\f\f\f\n@\n= 0;\nwhere \n is a bounded domain occupied by the ferromagnetic material and \u0017is unit\noutward normal vector along @\n.\nIn more details, the magnetization m: \n\u001aRd!R3;d= 1;2;3 is a three-\ndimensional vector \feld with a pointwise constraint jmj= 1. The \frst term on the\nright-hand side in (2.1) is the gyromagnetic term and the second term stands for\nthe damping term, with \u000b>0 being the dimensionless damping coe\u000ecient.\nThe e\u000bective \feld he\u000bis obtained by taking the variation of the Gibbs free energy\nof the magnetic body with respect to m. The free energy includes the exchange\nenergy, the anisotropy energy, the magnetostatic energy, and the Zeeman energy:\n(2.3)F[m] =\u00160M2\ns\n2\u001aZ\n\n\u0000\n\u000fjrmj2+q\u0000\nm2\n2+m2\n3\u0001\n\u00002he\u0001m\u0000hs\u0001m\u0001\ndx\u001b\n:4 Y. CAI, J. CHEN, C. WANG, AND C. XIE\nTherefore, the e\u000bective \feld includes the exchange \feld, the anisotropy \feld, the\nstray \feldhs, and the external \feld he. For a uniaxial material, it is clear that\nhe\u000b=\u000f\u0001m\u0000q(m2e2+m3e3) +hs+he; (2.4)\nwhere the dimensionless parameters become \u000f=Cex=(\u00160M2\nsL2) andq=Ku=(\u00160M2\ns)\nwithLthe diameter of the ferromagnetic body and \u00160the permeability of vacuum.\nThe unit vectors are given by e2= (0;1;0),e3= (0;0;1), and \u0001 denotes the\nstandard Laplacian operator. For the Permalloy, an alloy of Nickel (80%) and\nIron (20%), typical values of the physical parameters are given by: the exchange\nconstantCex= 1:3\u000210\u000011J/m, the anisotropy constant Ku= 100 J/m3, the sat-\nuration magnetization constant Ms= 8:0\u0002105A/m. The stray \feld takes the\nform\nhs=1\n4\u0019rZ\n\nr\u00121\njx\u0000yj\u0013\n\u0001m(y)dy: (2.5)\nIf \n is a rectangular domain, the evaluation of (2.5) can be e\u000eciently done by the\nFast Fourier Transform (FFT) [20].\nFor brevity, the following source term is de\fned\nf=\u0000Q(m2e2+m3e3) +hs+he: (2.6)\nand the original PDE system (2.1) could be rewritten as\nmt=\u0000m\u0002(\u000f\u0001m+f)\u0000\u000bm\u0002m\u0002(\u000f\u0001m+f): (2.7)\nThanks to point-wise identity jmj= 1, we obtain an equivalent form:\n(2.8)mt=\u000b(\u000f\u0001m+f) +\u000b\u0000\n\u000fjrmj2\u0000m\u0001f\u0001\nm\u0000m\u0002(\u000f\u0001m+f):\nIn particular, it is noticed that the damping term is rewritten as a harmonic map-\nping \row, which contains a constant-coe\u000ecient Laplacian di\u000busion term. This fact\nwill greatly improve the numerical stability of the proposed scheme.\nFor the numerical description, we \frst introduce some notations for discretization\nand numerical approximation. Denote the temporal step-size by k, andtn=nk,\nn\u0014\u0004T\nk\u0005\nwithTthe \fnal time. The spatial mesh-size is given by hx=hy=hz=\nh= 1=N, andmn\ni;j;`stands for the magnetization at time step tn, evaluated at the\nspatial location ( xi\u00001\n2;yj\u00001\n2;z`\u00001\n2) withxi\u00001\n2=\u0000\ni\u00001\n2\u0001\nhx,yj\u00001\n2=\u0000\nj\u00001\n2\u0001\nhyand\nz`\u00001\n2=\u0000\n`\u00001\n2\u0001\nhz(0\u0014i;j;`\u0014N+ 1). In addition, a third order extrapolation\nformula is used to approximate the homogeneous Neumann boundary condition.\nFor example, such a formula near the boundary along the zdirection is given by\nmi;j;1=mi;j;0;mi;j;N +1=mi;j;N:\nThe boundary extrapolation along other boundary sections can be similarly made.\nThe standard second-order centered di\u000berence applied to \u0001 mresults in\n\u0001hmi;j;k=mi+1;j;k\u00002mi;j;k+mi\u00001;j;k\nh2x\n+mi;j+1;k\u00002mi;j;k+mi;j\u00001;k\nh2y\n+mi;j;k+1\u00002mi;j;k+mi;j;k\u00001\nh2z;A SECOND-ORDER METHOD FOR LLG EQUATION 5\nand the discrete gradient operator rhmwithm= (u;v;w )Treads as\nrhmi;j;k=2\n64ui+1;j;k\u0000ui\u00001;j;k\nhxvi+1;j;k\u0000vi\u00001;j;k\nhxwi+1;j;k\u0000wi\u00001;j;k\nhxui;j+1;k\u0000ui;j\u00001;k\nhyvi;j+1;k\u0000vi;j\u00001;k\nhywi;j+1;k\u0000wi;j\u00001;k\nhyui;j;k +1\u0000ui;j;k\u00001\nhzvi;j;k +1\u0000vi;j;k\u00001\nhzwi;j;k +1\u0000wi;j;k\u00001\nhz3\n75:\nSubsequently, the GSPM and the SIPM numerical methods need to be reviewed,\nwhich could be used for the later comparison.\n2.2.The Gauss-Seidel projection method. The GSPM is based on a combi-\nnation of a Gauss-Seidel update of an implicit solver for the gyromagnetic term,\nthe heat \row of the harmonic map, and a projection step. It only requires a series\nof heat equation solvers with constant coe\u000ecients; as a result, the FFT-based fast\nsolvers could be easily applied. This method is \frst-order in time and second-order\nin space. Below is the detailed outline of the GSPM method in [8].\nStep 1. Implicit Gauss-Seidel:\ngn\ni= (I\u0000\u000f\u0001t\u0001h)\u00001(mn\ni+ \u0001tfn\ni); i= 2;3;\ng\u0003\ni= (I\u0000\u000f\u0001t\u0001h)\u00001(m\u0003\ni+ \u0001tf\u0003\ni); i= 1;2; (2.9)\n(2.10)0\n@m\u0003\n1\nm\u0003\n2\nm\u0003\n31\nA=0\n@mn\n1+ (gn\n2mn\n3\u0000gn\n3mn\n2)\nmn\n2+ (gn\n3m\u0003\n1\u0000g\u0003\n1mn\n3)\nmn\n3+ (g\u0003\n1m\u0003\n2\u0000g\u0003\n2m\u0003\n1)1\nA:\nStep 2. Heat \row without constraints:\n(2.11) f\u0003=\u0000Q(m\u0003\n2e2+m\u0003\n3e3) +h\u0003\ns+he;\n(2.12)0\n@m\u0003\u0003\n1\nm\u0003\u0003\n2\nm\u0003\u0003\n31\nA=0\n@m\u0003\n1+\u000b\u0001t(\u000f\u0001hm\u0003\u0003\n1+f\u0003\n1)\nm\u0003\n2+\u000b\u0001t(\u000f\u0001hm\u0003\u0003\n2+f\u0003\n2)\nm\u0003\n3+\u000b\u0001t(\u000f\u0001hm\u0003\u0003\n3+f\u0003\n3)1\nA:\nStep 3. Projection onto S2:\n(2.13)0\n@mn+1\n1\nmn+1\n2\nmn+1\n31\nA=1\njm\u0003\u0003j0\n@m\u0003\u0003\n1\nm\u0003\u0003\n2\nm\u0003\u0003\n31\nA:\nHerem\u0003denotes the intermediate values of m, and stray \felds hn\nsandh\u0003\nsare\nevaluated at mnandm\u0003, respectively.\nRemark 2.1. Two improved versions of the GSPM have been studied in [15], which\nturn out to be more e\u000ecient than the original GSPM. Meanwhile, it is found that\nboth improved versions become unstable when \u000b > 1, while the original GSPM\n(outlined above) is stable even when \u000b\u001410. Therefore, we shall use the original\nGSPM in [8]for the numerical comparison in this work.6 Y. CAI, J. CHEN, C. WANG, AND C. XIE\n2.3.Semi-implicit projection method. The SIPM has been outlined in [5,24].\nThis method is based on the second-order BDF temporal discretization, combined\nwith an explicit extrapolation. It is found that SIPM is unconditionally stable and\nis second-order accurate in both space and time. The algorithmic details are given\nas follows.\n(2.14)8\n>>>>>><\n>>>>>>:3\n2~mn+2\nh\u00002mn+1\nh+1\n2mn\nh\nk=\u0000^mn+2\nh\u0002\u0000\n\u000f\u0001h~mn+2\nh+^fn+2\nh\u0001\n\u0000\u000b^mn+2\nh\u0002\u0010\n^mn+2\nh\u0002(\u000f\u0001h~mn+2\nh+^fn+2\nh)\u0011\n;\nmn+2\nh=~mn+2\nh\nj~mn+2\nhj;\nwhere ~mn+2\nhis an intermediate magnetization, and ^mn+2\nh,^fn+2\nhare given by the\nfollowing extrapolation formula:\n^mn+2\nh= 2mn+1\nh\u0000mn\nh;\n^fn+2\nh= 2fn+1\nh\u0000fn\nh;\nwithfn\nh=\u0000Q(mn\n2e2+mn\n3e3) +hn\ns+hn\ne. The presence of cross product in the\nSIPM yields a linear system of equations with non-symmetric structure and vari-\nable coe\u000ecients. In turn, the GMRES solver has to be applied to implement this\nnumerical system. The numerical evidence has revealed that, the convergence of\nGMRES solver becomes slower for larger temporal step-size kor smaller spatial\ngrid-sizeh, which makes the computation more challenging.\n2.4.The proposed numerical method. The SIPM in (2.14) treats both the\ngyromagentic and the damping terms in a semi-implicit way, i.e., \u0001 mis computed\nimplicitly, while the coe\u000ecient functions are updated by a second order accurate,\nexplicit extrapolation formula. The strength of the gyromagnetic term is controlled\nby \u0001m+fsince the length of mis always 1. Meanwhile, the strength of the\ndamping term is controlled by the product of \u0001 m+fand the damping parameter\n\u000b. For small \u000b, say\u000b\u00141, it is reasonable to treat both the gyromagentic and\nthe damping terms semi-implicitly. However, for large \u000b, an alternate approach\nwould be more reasonable, in which the whole gyromagentic term is computed by\nan explicit extrapolation, while the nonlinear parts in the damping term is also\nupdated by an explicit formula, and only the constant-coe\u000ecient \u0001 mpart in the\ndamping term is implicitly updated. This idea leads to the proposed numerical\nmethod. To further simplify the presentation, we start with (2.8), and the numerical\nalgorithm is proposed as follows.\n(2.15)8\n>>>>>>>>>><\n>>>>>>>>>>:3\n2~mn+2\nh\u00002mn+1\nh+1\n2mn\nh\nk=\u0000^mn+2\nh\u0002\u0010\n\u000f\u0001h^mn+2\nh+^fn+2\nh\u0011\n+\u000b\u0010\n\u000f\u0001h~mn+2\nh+^fn+2\nh\u0011\n+\u000b\u0010\n\u000fjrh^mn+2\nhj2\u0000^mn+2\nh\u0001^fn+2\nh\u0011\n^mn+2\nh;\nmn+2\nh=~mn+2\nh\nj~mn+2\nhj;A SECOND-ORDER METHOD FOR LLG EQUATION 7\nwhere\n^mn+2\nh= 2mn+1\nh\u0000mn\nh;\n^fn+2\nh= 2fn+1\nh\u0000fn\nh:\nTable 1 compares the proposed method, the GSPM and the SIPM in terms\nof number of unknowns, dimensional size, symmetry pattern, and availability of\nFFT-based fast solver of linear systems of equations, and the number of stray \feld\nupdates. At the formal level, the proposed method is clearly superior to both the\nGSPM and the SIPM algorithms. In more details, this scheme will greatly improve\nthe computational e\u000eciency, since only three Poisson solvers are needed at each\ntime step. Moreover, this numerical method preserves a second-order accuracy in\nboth space and time. The numerical results in section 3 will demonstrate that the\nproposed scheme provides a reliable and robust approach for micromagnetics simu-\nlations with high accuracy and e\u000eciency in the regime of large damping parameters.\nTable 1. Comparison of the proposed method, the Gauss-Seidel\nprojection method, and the semi-implicit projection method.\nProperty or number Proposed method GSPM SIPM\nLinear systems 3 7 1\nSize N3N33N3\nSymmetry Yes Yes No\nFast Solver Yes Yes No\nAccuracy O(k2+h2)O(k+h2)O(k2+h2)\nStray \feld updates 1 4 1\nRemark 2.2. To kick start the proposed method, one can apply a \frst-order al-\ngorithm, such as the \frst-order BDF method, in the \frst time step. An overall\nsecond-order accuracy is preserved in this approach.\n3.Numerical experiments\nIn this section, we present a few numerical experiments with a sequence of damp-\ning parameters for the proposed method, the GSPM [8] and the SIPM [24], with\nthe accuracy, e\u000eciency, and stability examined in details. Domain wall dynamics\nis studied and its velocity is recorded in terms of the damping parameter and the\nexternal magnetic \feld.\n3.1.Accuracy and e\u000eciency tests. We set\u000f= 1 andf= 0 in (2.8) for conve-\nnience. The 1D exact solution is given by\nme= (cos(X) sint;sin(X) sint;cost)T;\nand the corresponding exact solution in 3D becomes\nme= (cos(XYZ ) sint;sin(XYZ ) sint;cost)T;\nwhereX=x2(1\u0000x)2,Y=y2(1\u0000y)2,Z=z2(1\u0000z)2. In fact, the above exact\nsolutions satisfy (2.8) with the forcing term g=@tme\u0000\u000b\u0001me\u0000\u000bjrmej2+me\u0002\n\u0001me, as well as the homogeneous Neumann boundary condition.8 Y. CAI, J. CHEN, C. WANG, AND C. XIE\nFor the temporal accuracy test in the 1D case, we \fx the spatial resolution\nash= 5D\u00004, so that the spatial approximation error becomes negligible. The\ndamping parameter is taken as \u000b= 10, and the \fnal time is set as T= 1. In the 3D\ntest for the temporal accuracy, due to the limitation of spatial resolution, we take\na sequence of spatial and temporal mesh sizes: k=h2\nx=h2\ny=h2\nz=h2= 1=N0\nfor the \frst-order method and k=hx=hy=hz=h= 1=N0for the second-\norder method, with the variation of N0indicated below. Similarly, the damping\nparameter is given by \u000b= 10, while the \fnal time Tis indicated below. In turn,\nthe numerical errors are recorded in term of the temporal step-size kin Table 2. It\nis clear that the temporal accuracy orders of the proposed numerical method, the\nGSPM, and the SIPM are given by 2, 1, and 2, respectively, in both the 1D and\n3D computations.\nThe spatial accuracy order is tested by \fxing k= 1D\u00005,\u000b= 10,T= 1 in 1D\nandk= 1D\u00003,\u000b= 10,T= 1 in 3D. The numerical error is recorded in term of\nthe spatial grid-size hin Table 3. Similarly, the presented results have indicated\nthe second order spatial accuracy of all the numerical algorithms, including the\nproposed method, the GSPM, and the SIPM, respectively, in both the 1D and 3D\ncomputations.\nTo make a comparison in terms of the numerical e\u000eciency, we plot the CPU time\n(in seconds) vs. the error norm kmh\u0000mek1. In details, the CPU time is recorded\nas a function of the approximation error in Figure 1a in 1D and in Figure 1b in\n3D, with a variation of kand a \fxed value of h. Similar plots are also displayed in\nFigure 1c in 1D and Figure 1d in 3D, with a variation of hand a \fxed value of k. In\nthe case of a \fxed spatial resolution h, the proposed method is signi\fcantly more\ne\u000ecient than the GSPM and the SIPM in both the 1D and 3D computations. The\nSIPM is slightly more e\u000ecient than the GSPM, while such an advantage depends\non the performance of GMRES, which may vary for di\u000berent values of kandh. In\nthe case of a \fxed time step size k, the proposed method is slightly more e\u000ecient\nthan the GSPM, in both the 1D and 3D computations, and the GSPM is more\ne\u000ecient than the SIPM.\n3.2.Stability test with large damping parameters. To check the numerical\nstability of these three methods in the practical simulations of micromagnetics with\nlarge damping parameters, we consider a thin \flm of size 480 \u0002480\u000220 nm3with\ngrid points 100\u0002100\u00024. The temporal step-size is taken as k= 1 ps. A uniform\nstate along the xdirection is set to be the initial magnetization and the external\nmagnetic \feld is set to be 0. Three di\u000berent damping parameters, \u000b= 0:01;10;40,\nare tested with stable magnetization pro\fles shown in Figure 2. In particular, the\nfollowing observations are made.\n\u000fThe proposed method is the only one that is stable for very large damping\nparameters;\n\u000fAll three methods are stable for moderately large \u000b;\n\u000fThe proposed method is the only one that is unstable for small \u000b.\nIn fact, a preliminary theoretical analysis reveals that, an optimal rate convergence\nestimate of the proposed method could be theoretically justi\fed for \u000b>3. Mean-\nwhile, extensive numerical experiments have implied that \u000b > 1 is su\u000ecient to\nensure the numerical stability in the practical computations.A SECOND-ORDER METHOD FOR LLG EQUATION 9\nTable 2. The numerical errors for the proposed method, the\nGSPM and the SIPM with \u000b= 10 andT= 1. Left: 1D with\nh= 5D\u00004; Right: 3D with k=h2\nx=h2\ny=h2\nz=h2= 1=N0\nfor GSPM and k=hx=hy=hz=h= 1=N0for the proposed\nmethod and SIPM, with N0speci\fed in the table.\n1D 3D\nkk\u0001k1k\u0001k 2k\u0001kH1k=hk\u0001k1k\u0001k 2k\u0001kH1\n4.0D-2 4.459D-4 5.226D-4 5.588D-4 1/20 6.171D-4 4.240D-4 4.246D-4\n2.0D-2 1.147D-4 1.345D-4 1.436D-4 1/24 4.381D-4 3.010D-4 3.014D-4\n1.0D-2 2.899D-5 3.402D-5 3.631D-5 1/28 3.268D-4 2.245D-4 2.248D-4\n5.0D-3 7.192D-6 8.529D-6 9.119D-6 1/32 2.531D-4 1.739D-4 1.741D-4\n2.5D-3 1.699D-6 2.321D-6 2.518D-6 1/36 2.017D-4 1.386D-4 1.387D-4\norder 2.007 1.961 1.957 { 1.902 1.903 1.903\n(a)Proposed method\n1D 3D\nkk\u0001k1k\u0001k 2k\u0001kH1k=h2k\u0001k1k\u0001k 2k\u0001kH1\n2.5D-3 2.796D-4 2.264D-4 1.445D-3 1/36 4.194D-4 2.683D-4 2.815D-4\n1.25D-3 1.425D-4 1.174D-4 7.720D-4 1/64 2.388D-4 1.399D-4 1.500D-4\n6.25D-4 7.170D-5 5.940D-5 4.026D-4 1/144 1.069D-4 6.106D-5 6.736D-5\n3.125D-4 3.591D-5 2.971D-5 2.069D-4 1/256 6.021D-5 3.442D-5 3.860D-5\n1.5625D-4 1.799D-5 1.488D-5 1.054D-4 1/400 3.855D-5 2.208D-5 2.501D-5\norder 0.991 0.984 0.945 { 0.992 1.032 1.000\n(b)GSPM\n1D 3D\nkk\u0001k1k\u0001k 2k\u0001kH1k=hk\u0001k1k\u0001k 2k\u0001kH1\n4.0D-2 4.315D-4 5.111D-4 8.774D-4 1/20 6.170D-4 4.240D-4 4.249D-4\n2.0D-2 1.128D-4 1.334D-4 2.255D-4 1/24 4.380D-4 3.010D-4 3.016D-4\n1.0D-2 2.872D-5 3.399D-5 5.706D-5 1/28 3.268D-4 2.245D-4 2.251D-4\n5.0D-3 7.174D-6 8.552D-6 1.433D-5 1/32 2.531D-4 1.739D-4 1.743D-4\n2.5D-3 1.721D-6 2.333D-6 3.784D-6 1/36 2.017D-4 1.386D-4 1.389D-4\norder 1.991 1.951 1.969 { 1.902 1.903 1.902\n(c)SIPM\nUnder the same setup outlined above, we investigate the energy dissipation of\nthe proposed method, the GSPM, and the SIPM. The stable state is attainable at\nt= 2 ns, while the total energy is computed by (2.3). The energy evolution curves\nof di\u000berent numerical methods with di\u000berent damping parameters, \u000b= 2;5;8;10,\nare displayed in Figure 3. One common feature is that the energy dissipation rate\nturns out to be faster for larger \u000b, in all three schemes. Meanwhile, a theoretical\nderivation also reveals that the energy dissipation rate in the LLG equation (2.1)\ndepends on \u000b, and a larger \u000bleads to a faster energy dissipation rate. Therefore,\nthe numerical results generated by all these three numerical methods have made a\nnice agreement with the theoretical derivation.10 Y. CAI, J. CHEN, C. WANG, AND C. XIE\nTable 3. The numerical errors of the proposed method, the\nGSPM and the SIPM with \u000b= 10 andT= 1. Left: 1D with\nk= 1D\u00005; Right: 3D with k= 1D\u00003.\n1D 3D\nhk\u0001k1k\u0001k 2k\u0001kH1hk\u0001k1k\u0001k 2k\u0001kH1\n4.0D-2 7.388D-3 7.392D-3 8.243D-3 1/2 4.261D-3 2.472D-3 2.472D-3\n2.0D-2 1.848D-3 1.848D-3 2.061D-3 1/4 9.822D-4 5.595D-4 5.753D-4\n1.0D-2 4.621D-4 4.621D-4 5.153D-4 1/8 2.453D-4 1.390D-4 1.424D-4\n5.0D-3 1.155D-4 1.155D-4 1.288D-4 1/16 6.137D-5 3.471D-5 3.554D-5\norder 2.000 2.000 2.000 { 2.035 2.047 2.037\n(a)Proposed method\n1D 3D\nhk\u0001k1k\u0001k 2k\u0001kH1hk\u0001k1k\u0001k 2k\u0001kH1\n4.0D-2 7.388D-3 7.392D-3 8.244D-3 1/2 4.256D-3 2.470D-3 2.470D-3\n2.0D-2 1.848D-3 1.848D-3 2.061D-3 1/4 9.810D-4 5.589D-4 5.744D-4\n1.0D-2 4.619D-4 4.622D-4 5.158D-4 1/8 2.447D-4 1.388D-4 1.423D-4\n5.0D-3 1.153D-4 1.156D-4 1.302D-4 1/16 6.103D-5 3.468D-5 3.613D-5\norder 2.000 2.000 1.995 { 2.037 2.047 2.030\n(b)GSPM\n1D 3D\nhk\u0001k1k\u0001k 2k\u0001kH1hk\u0001k1k\u0001k 2k\u0001kH1\n4.0D-2 7.388D-3 7.392D-3 8.243D-3 1/2 4.261D-3 2.472D-3 2.472D-3\n2.0D-2 1.848D-3 1.848D-3 2.061D-3 1/4 9.822D-4 5.595D-4 5.753D-4\n1.0D-2 4.621D-4 4.621D-4 5.153D-4 1/8 2.453D-4 1.390D-4 1.424D-4\n5.0D-3 1.155D-4 1.155D-4 1.288D-4 1/16 6.137D-5 3.471D-5 3.554D-5\norder 2.000 2.000 2.000 { 2.035 2.047 2.037\n(c)SIPM\nMeanwhile, we choose the same sequence of values for \u000b, and display the energy\nevolution curves in terms of time up to T= 2 ns in Figure 4. It is found that the\nproposed method have almost the same energy dissipation pattern with the other\ntwo methods for moderately large damping parameters \u000b= 2;5;8. In the case of\n\u000b= 10, the SIPM has a slightly di\u000berent energy dissipation pattern from the other\ntwo numerical methods.\n3.3.Domain wall motion. A Ne\u0013 el wall is initialized in a nanostrip of size 800 \u0002\n100\u00024 nm3with grid points 128 \u000264\u00024. An external magnetic \feld of he= 5 mT\nis then applied along the positive xdirection and the domain wall dynamics is\nsimulated up to 2 ns with \u000b= 2;5;8. The corresponding magnetization pro\fles are\nvisualized in Figure 5. Qualitatively, the domain wall moves faster as the value of\n\u000bincreases. Quantitatively, the corresponding dependence is found to be linear;\nsee Figure 6. The slopes \ftted by the least-squares method in terms of \u000bandhe\nare recorded in Table 4.A SECOND-ORDER METHOD FOR LLG EQUATION 11\n10-610-510-410-3100101102\nProposed method\nGSPM\nSIPM\n(a)Varyingkin 1D up to\nT= 1\n1.8 2 2.2 2.4 2.6 2.8 3 3.2\n10-7101102103\nProposed method\nGSPM\nSIPM(b)Varyingkin 3D up to T=\n0:1\n10-510-410-310-2101102103\nProposed method\nGSPM\nSIPM\n(c)Varyinghin 1D up to\nT= 1\n10-410-310-210-1100101102103\nProposed method\nGSPM\nSIPM(d)Varyinghin 3D up to\nT= 1\nFigure 1. CPU time needed to achieve the desired numerical ac-\ncuracy, for the proposed method, the GSPM and the SIPM, in\nboth the 1D and 3D computations. The CPU time is recorded as\na function of the approximation error by varying korhindepen-\ndently. CPU time with varying k: proposed method <SIPM<\nGSPM; CPU time with varying h: proposed method /GSPM<\nSIPM.\n4.Conclusions\nIn this paper, we have proposed a second-order accurate numerical method to\nsolve the Landau-Lifshitz-Gilbert equation with large damping parameters. For the\nnumerical convenience, the LLG system is reformulated so that in which the damp-\ning term is rewritten as a harmonic mapping \row .This numerical scheme is based on\nthe second-order backward-di\u000berentiation formula approximation for the temporal\nderivative, combined with an implicit treatment of the constant-coe\u000ecient di\u000busion\nterm, and the fully explicit extrapolation approximation of the nonlinear terms, in-\ncluding the gyromagnetic term and the nonlinear part of the harmonic mapping\n\row. Thanks to the large damping parameter, the proposed method is veri\fed12 Y. CAI, J. CHEN, C. WANG, AND C. XIE\nFigure 2. Stable structures in the absence of magnetic \feld at\n2 ns when\u000b= 0:01;10;40. The color denotes the angle between the\n\frst two components of the magnetization vector. Top: Proposed\nmethod; Middle: GSPM; Bottom: SIPM. Left: \u000b= 40; Middle:\n\u000b= 10; Right: \u000b= 0:01.\n(a)Proposed\n (b)GSPM\n (c)SIPM\nFigure 3. Energy evolution curves of three numerical methods,\nwith di\u000berent damping constants, \u000b= 2;5;8;10, up tot= 2 ns in\nthe absence of external magnetic \feld. Left: Proposed numerical\nmethod; Middle: GSPM; Right: SIPM. One common feature is\nthat the energy dissipation rate is faster for larger \u000b, which is\nphysically reasonable.A SECOND-ORDER METHOD FOR LLG EQUATION 13\n(a)\u000b= 2\n (b)\u000b= 5\n(c)\u000b= 8\n (d)\u000b= 10\nFigure 4. Energy evolution curves in terms of time, for the nu-\nmerical results created by three numerical methods up to t= 2 ns\nin the absence of external magnetic \feld for (a) \u000b= 2, (b)\u000b= 5,\n(c)\u000b= 8, and (d) \u000b= 10. The energy dissipation pattern of the\nproposed method is consistent with the other two methods for (a),\n(b), and (c), and the SIPM has a slightly di\u000berent energy dissipa-\ntion pattern from the other two methods for (d).\nto be unconditionally stable. The proposed method is much more e\u000ecient than\nother semi-implicit schemes since only symmetric, positive de\fnite linear systems\nof equations with constant coe\u000ecients need to be solved. Meanwhile, the proposed\nmethod is more accurate than the standard Gauss-Seidel projection method, due\nto its second-order accuracy in time. Numerical results in 1D and 3D are pro-\nvided to demonstrate the accuracy and the e\u000eciency of the proposed numerical\nmethod. In addition, micromagnetics simulations using the proposed method have\nprovided physically reasonable structures and captured the linear dependence of\nthe domain wall velocity with respect to the damping parameter. Therefore, the\nproposed method could be e\u000eciently used for challenging practical simulations of\nmicromagnetics with large damping parameters.14 Y. CAI, J. CHEN, C. WANG, AND C. XIE\n(a)Magnetization for initial state\n (b)Magnetization with \u000b= 2 at\n2 ns\n(c)Magnetization with \u000b= 5 at\n2 ns\n(d)Magnetization with \u000b= 8 at\n2 ns\nFigure 5. Magnetization pro\fles of Ne\u0013 el wall motion in the pres-\nence of a magnetic \feld he= 5 mT, with \u000b= 2;5;8 at 2 ns for\nthe proposed numerical method. The in-plane arrow denotes the\n\frst two components of the magnetization vector. The wall moves\nfaster for larger values of \u000band its velocity depends linearly on \u000b.\nFigure 6. Linear dependence of the wall velocity with respect to\nthe damping parameter \u000b(left) and the external magnetic \feld he\n(right).\nAcknowledgments\nThis work is supported in part by the grants NSFC 11971021 (J. Chen), NSF\nDMS-2012669 (C. Wang), NSFC 11771036 (Y. Cai).A SECOND-ORDER METHOD FOR LLG EQUATION 15\nTable 4. Linear dependence of the domain wall velocity Vin\nterms of the external magnetic \feld heand the damping parameter\n\u000b.\n\u000bV(m/s) he(mT)5 6 7 8 9 10 Slope\n3 76 91 109 123 139 154 1.024\n4 105 118 139 157 179 196 0.928\n5 129 145 169 192 217 244 0.932\n6 153 169 200 227 256 286 0.927\n7 177 196 232 263 294 333 0.927\n8 200 222 263 303 333 385 0.954\n9 230 250 294 345 385 435 0.954\n10 253 270 323 370 417 476 0.943\nSlope 0.984 0.910 0.910 0.933 0.917 0.950 {\nReferences\n[1] G. Akrivis, M. Feischl, B. Kov\u0013 acs, and C. Lubich, Higher-order linearly implicit full dis-\ncretization of the Landau-Lifshitz-Gilbert equation , Math. Comp. 90(2021), 995{1038.\n[2] F. Alouges and P. Jaisson, Convergence of a \fnite element discretization for the Landau-\nLifshitz equations in micromagnetism , Math. Models Methods Appl. Sci. 16(2006), no. 02,\n299{316.\n[3] W.F. Brown, Micromagnetics , Interscience Tracts on Physics and Astronomy. Interscience\nPublishers (John Wiley and Sons), New York-London, 1963.\n[4] S. Budhathoki, A. Sapkota, K.M. Law, B. Nepal, S. Ranjit, S. Kc, T. Mewes, and A. Hauser,\nLow Gilbert damping and linewidth in magnetostrictive FeGa thin \flms , J. Magn. Magn.\nMater. 496(2020), 165906.\n[5] J. Chen, C. Wang, and C. Xie, Convergence analysis of a second-order semi-implicit pro-\njection method for Landau-Lifshiz equation , Appl. Numer. Math. (2021). Submitted and in\nreview.\n[6] I. Cimr\u0013 ak, A survey on the numerics and computations for the Landau-Lifshitz equation of\nmicromagnetism , Arch. Comput. Methods Eng. 15(2008), no. 3, 277{309.\n[7] H. Gao, Optimal error estimates of a linearized Backward Euler FEM for the Landau-Lifshitz\nequation , SIAM J. Numer. Anal. 52(2014), no. 5, 2574{2593.\n[8] C.J. Garc\u0010a-Cervera and W. E, Improved Gauss-Seidel projection method for micromagnetics\nsimulations , J. Comput. Phys. 171(2001), no. 1, 357{372.\n[9] T.L. Gilbert, Phys. Rev. 100(1955), 1243. [Abstract only; full report, Armor Research Foun-\ndation Project No. A059, Supplementary Report, May 1, 1956 (unpublished)].\n[10] T.L. Gilbert and J.M. Kelly, Anomalous rotational damping in ferromagnetic sheets , Armour\nResearch Foundation of Illinois Institute of Technology (1955). (unpublished).\n[11] B. Heinrich, D. Fra\u0013 \u0010tov\u0013 a, and V. Kambersk\u0013 y, The in\ruence of s-d exchange on relaxation of\nmagnons in metals , Phys. Stat. Solidi B-basic Solid Stat. Phys. 23(1967), 501{507.\n[12] M. Kruz\u0013 \u0010k and A. Prohl, Recent developments in the modeling, analysis, and numerics of\nferromagnetism , SIAM Rev. 48(2006), no. 3, 439{483.\n[13] L.D. Landau and E.M. Lifshits, On the theory of the dispersion of magnetic permeability in\nferromagnetic bodies , Phys. Z. Sowjet. 63(1935), no. 9, 153{169.\n[14] D.M. Lattery, D. Zhang, J. Zhu, X. Hang, J. Wang, and X. Wang, Low Gilbert damping\nconstant in perpendicularly magnetized W/CoFeB/MgO \flms with high thermal stability ,\nSci. Rep. 8(2018), 13395.\n[15] P. Li, C. Xie, R. Du, J. Chen, and X. Wang, Two improved Gauss-Seidel projection methods\nfor Landau-Lifshitz-Gilbert equation , J. Comput. Phys. 401(2020), 109046.\n[16] T. Nan, Y. Lee, S. Zhuang, Z. Hu, J. Clarkson, X. Wang, C. Ko, H. Choe, Z. Chen, D. Budil,\nJ. Wu, S. Salahuddin, J. Hu, R. Ramesh, and N. Sun, Electric-\feld control of spin dynamics\nduring magnetic phase transitions , Sci. Adv. 6(2020), no. 40, eabd2613.16 Y. CAI, J. CHEN, C. WANG, AND C. XIE\n[17] H. Suhl, Theory of the magnetic damping constant , IEEE Trans. Magn. 34(1998), 1834{\n1838.\n[18] T. Tanaka, S. Kashiwagi, Y. Otsuka, Y. Nozaki, Y. Hong, and K. Matsuyama, Microwave-\nassisted magnetization reversal of exchange-coupled composite nanopillar with large Gilbert\ndamping constant , IEEE Tran. Magn. 50(2014), 1{3.\n[19] H. Tang and K. Xia, Gilbert damping parameter in MgO-based magnetic tunnel junctions\nfrom \frst principles , Phys. Rev. Applied 7(2017), 034004.\n[20] C. Wang and J.-G. Liu, Convergence of gauge method for incompressible \row , Math. Comp.\n69(2000), 1385{1407.\n[21] X. Wang, C.J. Garc\u0013 \u0010a-Cervera, and W. E, A Gauss-Seidel projection method for micromag-\nnetics simulations , J. Comput. Phys. 171(2001), no. 1, 357{372.\n[22] R. Weber, D. Han, I. Boventer, S. Jaiswal, R. Lebrun, G. Jakob, and M. Kl aui, Gilbert\ndamping of CoFe-alloys , J. Phys. D 52(2019), 325001.\n[23] D. Wei, Micromagnetics and recording materials , Springer Briefs in Apllied Sciences and\nTechnology, Springer Berlin Heidelberg, 2012.\n[24] C. Xie, C.J. Garc\u0013 \u0010a-Cervera, C. Wang, Z. Zhou, and J. Chen, Second-order semi-implicit\nmethods for micromagnetics simulations , J. Comput. Phys. 404(2020), 109104.\n[25] D. Zhang, M. Li, L. Jin, C. Li, Y. Rao, X. Tang, and H. Zhang, Extremely large magnetiza-\ntion and Gilbert damping modulation in NiFe/GeBi bilayers , ACS Appl. Electron. Mater. 2\n(2020), no. 1, 254{259.\nSchool of Mathematical Sciences, Beijing Normal University, Beijing, China.\nEmail address :yongyong.cai@bnu.edu.cn\nSchool of Mathematical Sciences, Soochow University, Suzhou, China.\nEmail address :jingrunchen@suda.edu.cn\nMathematics Department, University of Massachusetts, North Dartmouth, MA 02747,\nUSA.\nEmail address :cwang1@umassd.edu\nSchool of Mathematical Sciences, Soochow University, Suzhou, China.\nEmail address :20184007005@stu.suda.edu.cn"    },    {        "title": "1003.1577v1.A_single_ion_nonlinear_mechanical_oscillator.pdf",        "content": "A single-ion nonlinear mechanical oscillator\nN. Akerman, S. Kotler, Y. Glickman, Y. Dallal, A. Keselman, and R. Ozeri\nPhysics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel\nWe study the steady state motion of a single trapped ion oscillator driven to the nonlinear regime.\nDamping is achieved via Doppler laser-cooling. The ion motion is found to be well described by\nthe Du\u000eng oscillator model with an additional nonlinear damping term. We demonstrate a unique\nability of tuning both the linear as well as the nonlinear damping coe\u000ecients by controlling the\ncooling laser parameters. Our observations open a way for the investigation of nonlinear dynamics\non the quantum-to-classical interface as well as mechanical noise squeezing in laser-cooling dynamics.\nPACS numbers: 37.10.Ty 37.10.Vz\nNonlinear dynamics prevails in many dynamical sys-\ntems in nature, introducing a rich behavior such as criti-\ncality, bifurcations and chaos. Nonlinear dynamics on the\nmicroscopic scale is especially interesting as it can shed\nlight on the quantum-to-classical transition as well as\nprovide a mean to suppress thermal and quantum noise.\nAll mechanical oscillators will show nonlinearity when\ndriven far enough from equilibrium. The simplest such\nnonlinear oscillator is the Du\u000eng oscillator which in-\ncludes a cubic term in the restoring force [1]. Recently,\nsuch Du\u000eng nonlinear dynamics has been extensively\nstudied with nano-electromechanical beam resonators.\nThe basins of attraction of a nano beam oscillator were\nmapped [2]. Noise squeezing and stochastic resonances\nwere observed close to the Du\u000eng instability [3, 4]. Noise\nsqueezing was predicted to enable mass and force detec-\ntion with precision below the standard thermal limit [5{\n7] and possibly below the standard quantum limit when\noperating close to the oscillator ground state [8].\nThe mechanical motion of trapped ions is highly con-\ntrollable and can be e\u000eciently laser-cooled to the quan-\ntum ground state [9]. High \fdelity production of Fock,\nsqueezed, and Schr odinger-cat states was demonstrated\nwith a single trapped-ion [10, 11]. At the temperature\nrange obtained with laser-cooling techniques, quadruple\nRF Paul traps are excellently approximated as harmonic.\nNonlinearity in ion motion was observed when several\nions are trapped due to their mutual Coulomb repulsion.\nHere nonlinearity couples between the ion-crystal normal\nmodes, even at the single quantum level [12, 13]. Trap\nnonlinearities are important in the context of resonance\nejection in high resolution mass spectrometry [14]. How-\never, in these experiments ions are typically not laser-\ncooled and furthermore the e\u000bect of Coulomb nonlinear-\nities in the large ion cloud is intertwined with that of\nthe trap. Recently, ampli\fcation saturation of a single-\nion \\phonon laser\", resulting from optical forces that are\nnonlinear in the ions velocity, was demonstrated [15].\nHere, we study the nonlinear mechanical response of\na single laser-cooled88Sr+ion, in a linear RF-Paul\ntrap. The nonlinearity originates from the higher than\nquadrupolar order terms in the trapping potential. We\nEMCCD \nLasers S1/2 422nm P1/2 \nD3/2 1092nm \nPMT 88 Sr+\nDrive force Trap RF FIG. 1: Schematic diagram of the experimental set-up and\nthe relevant energy levels of88Sr+ion. The positively biased\ntrap end-caps produce a static trapping potential in the axial\ndirection with a small anharmonicity. The ion oscillator is\ndriven to the nonlinear regime by a small oscillating voltage\non one of the trap end-caps. Violet and infra-red laser beams\nprovide laser cooling and optical pumping. Scattered violet\nphotons are collected by an imaging system and directed ei-\nther to an EMCCD camera or to a PMT.\n\fnd that the ion steady state response is well described by\nthe Du\u000eng model with an additional nonlinear damping\nterm [16]. Unlike other realizations of nonlinear mechani-\ncal oscillators, both the linear and the nonlinear damping\ncomponents can be precisely controlled.\nOur trap has the canonical linear four rods and two\nend-caps con\fguration shown in Fig. 1. The distance of\nthe ion to the end-caps and rod-electrodes is 0 :65 mm and\n0:27 mm respectively. Here we examine only the motion\nalong the axial direction of the trap. In this direction,\ntrapping is dominated by the static electric potential\ndue to a positive constant voltage on the trap end-caps.\nThis potential is well approximated to be harmonic with\n!0=2\u0019= 438 KHz. However, as the trap end-caps do not\nsatisfy the pure electric quadruple boundary condition, a\nsmall octupolar contribution to the electric \feld results a\npositive cubic term in the restoring force and an energy\nlevel di\u000berence of ~!0+~\u0001nlnwherenis the harmonic\noscillator quantum number and \u0001 nl=2\u0019= 0:8 mHz is thearXiv:1003.1577v1  [quant-ph]  8 Mar 20102\nnonlinear dispersion. This nonlinearity becomes increas-\ningly important with growing oscillation amplitude. The\nion is driven to the nonlinear regime by adding a small\noscillating voltage to one of the trap end-caps. The ion is\nDoppler-cooled by scattering photons from a single laser\nbeam, slightly red-detuned from the S1=2!P1=2tran-\nsition at 422 nm. To prevent population accumulation\nin theD3=2meta-stable level we repump the ion on the\nD3=2!P1=2transition at 1092 nm.\nWe measure the steady-state oscillation amplitude of\nthe ion as we slowly scan the drive frequency, !, across\nthe harmonic resonance, !0. The scan is from lower to\nhigher frequencies (positive sweep) or vice versa (negative\nsweep). Photons scattered during the cooling process\nare collected by an imaging system (N.A. = 0.31), and\nare either directed towards an Electron-Multiplying CCD\n(EMCCD) camera or a Photo Multiplier Tube (PMT).\nWe measure the amplitude of motion by taking time-\naveraged images of the ion as shown in Fig.2(a). The\nimage is then integrated along the direction perpendicu-\nlar to motion to produce a single curve. Every column in\nFig. 2(b) corresponds to a curve produced this way, for\na positive frequency sweep. As seen, the oscillation am-\nplitude increases as the drive frequency approaches !0,\ncontinues to increase passed !0, until at a given critical\ndrive frequency, !m, abruptly collapses to a signi\fcantly\nlower value. We extract the ion oscillation amplitude\nby \ftting the curve to the expected time-averaged po-\nsition distribution. The blue and red lines in Fig. 3\nare the measured amplitudes for positive and negative\nsweeps respectively. Di\u000berent curves correspond to dif-\nferent drive amplitudes. The asymmetry and hysteresis\nas well as the abrupt amplitude changes at speci\fc criti-\ncal drive frequencies are a clear deviation from the driven\nharmonic oscillator response. As expected from a posi-\ntive nonlinearity, the oscillator self-frequency is \\pulled\"\nto higher values at higher oscillation amplitudes.\nIn order to measure the phase di\u000berence between the\nion-oscillator and the driving force, we time-stamp each\nphoton measured by the PMT within a single drive pe-\nriod to allocate it with a corresponding drive phase.\nThe instantaneous photon scattering rate from the cool-\ning laser beam is determined by the ions' instantaneous\nvelocity through its associated Doppler shift [17]. A\nhistogram of the measured photon phases is shown in\nFig.2(c). A clear sinusoidal oscillation of the photon scat-\ntering rate yields the ion-oscillator phase. The columns\nin Fig.2(d) are photon phase histograms for a positive\ndrive frequency sweep. As seen, at the critical frequency,\n!m, a phase jump of 1 :2 radians in the oscillator motion\naccompanies the sudden change in oscillation amplitude.\nOur observations are well accounted for by the Du\u000eng\noscillator model. The Du\u000eng equation of motion is,\nx+ 2\u0016_x+!2\n0x+\u000bx3=kcos(!t): (1)\n         10001200140016001800\n0 π 2π\nDrive phase[rad]Counts(c)\nσ/2π[KHz]Drive phase[rad](d)\n  0  π2π\n−1−0.500.5                  \nσ/2π[KHz]X [µm](b)\n−1−0.500.5−20−1001020\n(a)\nFigure 1:\n1FIG. 2: Driven ion-oscillator amplitude and phase. (a) Time-\naveraged ion images taken at various drive frequencies. (b)\nColumns are time-averaged images, integrated along the di-\nrection perpendicular to ion-motion, during a positive fre-\nquency scan. (c) A histogram of the number of photons de-\ntected at di\u000berent driving force phases. (d) Columns are pho-\nton phase histograms taken during a positive frequency scan.\nThe solid line is the theoretical phase given by Eq.3 shifted\nby a constant to match the peak in the histograms.\nHerexis the displacement of the ion from its equilib-\nrium position, \u000bis the an-harmonic coe\u000ecient, \u0016is the\nlinear damping coe\u000ecient and kis the drive amplitude.\nThe recoil noise inherent to the spontaneous photon scat-\ntering process, which would appear as a Langevin force\nterm, is neglected. An approximate solution to Eq.1 can\nbe obtained by the multiple scale method [1]. Here the\nsolution has the from x(t) =a(t) cos(!t\u0000\u001e) , wherea(t)\nis a slowly-varying oscillation amplitude and \u001eis the os-\ncillator phase. The steady-state solution for asolves,\n\u001b=3\u000b\n8!0a2\u0006s\nk2\n4!2\n0a2\u0000\u00162; (2)\nwhere\u001b=!\u0000!0is the drive detuning. The steady-state\nsolution for a, at a \fxed k, vs. drive frequency is shown\nby the black line in the inset of Fig.3. Above a criti-\ncal amplitude ac,atrifurcates into three solutions. One\nsolution with small and one with large amplitude, are sta-\nble, while the third, intermediate amplitude solution, is\nunstable and is positioned on the state-space separatrix.\nThis bistablity persists until the high amplitude solution\nreaches a maximal value, am, at which the drive force is\noverwhelmed by damping and the oscillator is forced into\na single stable solution. Positive and negative frequency\nscans carry the oscillator into the bistability region along\ndi\u000berent stable attractors leading to the observed hys-\nteresis as illustrated by the arrows in the inset. To com-\npare with our data we independently measure all the pa-3\nrameters in Eq.1. The driving force amplitude, k, is mea-\nsured by observing ion displacement vs. end-cap voltage,\n!0is measured via ion response in the linear regime. A\nvalue of\u000b=4\u00192= 1:24\u00060:03\u00011018Hz2=m2is measured\nusing the observed dependence of amon\u001bm=!m\u0000!0,\nthe instability detuning, am=p\n8!0\u001bm=3\u000b. A value\nof\u0016=2\u0019= 39:2\u00060:3 Hz, which result in a quality factor\nQ= 5590, is evaluated using the variation of amwith the\ndrive amplitude, am=k=(2\u0016!0). The blue and red cir-\ncles in the inset are the measured amplitudes, for positive\nand negative scans respectively, showing good agreement\nwith the theoretical curve.\n−1−0.500.51012345\n02040\nk[Hz2m]\nσ/2π[KHz]a[µm] −1−0.5 00.511.5010203040\nσ/2π[KHz]a[µm]\nx104σm\nam\nFIG. 3: Measured oscillator amplitude vs. drive frequency\nfor various driving force amplitudes and both positive (blue)\nand negative (red) scans. The inset shows the Du\u000eng model\ncalculation (black solid line) and our measured amplitudes\n(blue circles - positive scan; red circles - negative scan)\nThe Du\u000eng oscillator steady state phase is given by,\ntan(\u001e) =8\u0016!0\n3\u000ba2\u0000!0\u001b: (3)\nThe white solid line in Fig.2(d) shows the theoretical\nphase curve vs. drive frequency for our experimental pa-\nrameters, showing good agreement with our data.\nLinear damping is a very good approximation for most\nmechanical oscillators, as typically dissipation originates\nfrom coupling of the oscillator to an ohmic bath. Re-\ncently, the contribution of non-linear damping to the mo-\ntion of a nano beam resonator was studied [19]. In our\nexperiment damping results from the change in radia-\ntion pressure vs. ion velocity. When the laser frequency\nis tuned below the cooling transition, the leading con-\ntribution is indeed linear in the ions' velocity. However,\nas the oscillation amplitude increases or the cooling-laser\ndetuning reduced, the e\u000bect of damping force terms that\nare nonlinear in the ions' velocity increases [17].\nTo account for nonlinear damping, Eq.1 is modi\fed to\ninclude a term which is cubic in the oscillator velocity,\nx+ 2\u0016_x+\r_x3+!2\n0x+\u000bx3=kcos(!t): (4)Here\ris the cubic damping coe\u000ecient. The steady-state\namplitude is now a solution of [18, 19],\n9\n16(\u000b2+\r2!6\n0)a6+3!0(\u0016\r!3\n0\u0000\u001b\u000b)a4\n+4!2\n0(\u001b2+\u00162)a2\u0000k2= 0:(5)\nWhen\r > 0, nonlinear damping acts to e\u000bectively in-\ncrease dissipation for larger oscillation amplitudes. Un-\nlike the linear damping case, amdoes not increase linearly\nwithkbut is rather limited by the growing dissipation.\nWe \fnd\u0016and\rby a maximum likelihood \ft of the mea-\nsuredamvs.kcurve to the solution of Eq. 5. It is\ninstructive to look at the responsivity, \u001f= 2\u0016!0a=k, in\norder to distinguish linear from nonlinear damping [18].\nIn Fig.4 we plot the measured \u001ffor positive scans and\nvarious drive amplitudes, k, for two di\u000berent cooling-laser\ndetuning values, \u000e. In Fig.4(a) \u000e=2\u0019=\u0000420 MHz,\r= 0\nand the maximal responsivity is seen to be independent of\nk. In Fig.4(b) \u000e=2\u0019=\u0000160 MHz,!2\n0\r=2\u0019= 0:09\u00060:002\n\u0016m\u00002Hz and the maximal responsivity decreases as kin-\ncreases. The linear dissipation term, \u0016, is similar in both\ncases. The solid lines are the solutions of Eq.5 showing\ngood agreement with the data.\n−0.500.511.500.20.40.60.81\nσ/2π[KHz]χ(a)\n−0.5 00.5 100.20.40.60.81\nσ/2π[KHz]χ(b)\nFIG. 4: Calculated and measured responsivity, \u001f= 2\u0016!0a=k,\nfor positive drive scans. Di\u000berent curves correspond to di\u000ber-\nent drive amplitudes. Due to small drifts in !0(<100Hz)\neach curve was separately shifted on the frequency axis to \ft\nthe theoretical curve. (a) Linear damping, the cooling laser\ndetuning\u000e=2\u0019=\u0000420 MHz and \r= 0. The maximal re-\nsponsivity is independent of drive amplitude k. (b) Nonlinear\ndamping,\u000e=2\u0019=\u0000160 MHz and !2\n0\r=2\u0019= 0:09\u0016m\u00002Hz.\nThe maximal responsivity decreases as kincreases.\nWe next repeat the measurement of \u0016and\rfor various\ncooling-laser detunings at a \fxed repump-laser frequency\nand lasers intensities. The measured \u0016and\rvs.\u000eare\nshown in \fgures 5(a) and 5(b) respectively. To com-\npare with the theoretically predicted values we write the\ncooling-laser scattering force,\nFs( _x) =~kc\u0000\u001ap(\u000ec+kc_x;\u000er+kr_x); (6)\nwherekc=rand\u000ec=rare the wave-vectors and detunings\nof the cooling and repump lasers respectively, \u0000 = 2 \u0019\u0002\n21 MHz is the spectral linewidth of the P1=2level and4\n\u001apis theP1=2population. The damping coe\u000ecients are\ntherefore given by the appropriate derivatives,\n\u0016=1\n2mdFs\nd_x;\r=1\n6md3Fs\nd_x3: (7)\nHeremis the ion mass. We calculate \u001aPby numerically\nsolving the eight coupled Bloch equations, corresponding\nto the population in all states in the S1=2,P1=2andD3=2\nlevels coupled by the cooling and repump lasers. The\ncubic damping coe\u000ecient is highly sensitive to di\u000berent\nlaser parameters due to the presence of dark resonances.\nThe solid lines in \fgures 5(a) and 5(b) are the calculated\n\u0016and\rshowing good agreement with our measured val-\nues. The two lasers intensities and the repump-laser de-\ntuning were used as \ft parameters, yielding values that\nagree within 20% with their measured value.\n−200−150−100020406080100120\nδc/2π[MHz]µ/2π [Hz](a)\n−200−150−10000.050.10.150.20.25\nδc/2π[MHz]ω02γ/2π [µm−2Hz](b)\nFIG. 5: (a) Linear and (b) cubic damping coe\u000ecients for\nvarious cooling-beam detunings. Filled circles are measured\nvalues and solid lines are calculated using equations 6 and 7.\nAn additional nonlinear damping term, proportional\ntox2_x, results from the laser beam \fnite size (100 \u0016m\nFWHM) and has an identical e\u000bect to that of \ron the\nsteady-state motion [16, 18]. This term was calculated\nto be small relative to \r[20] and was taken into account\nin Fig.5.\nIn conclusion, we have driven a single-ion oscillator to\nthe nonlinear regime. The ion steady-state motion, show-\ning a bifurcation into two stable attractors and hysteresis,\nis well described by the Du\u000eng oscillator model with an\nadditional nonlinear damping term. Unlike previously\nstudied nonlinear mechanical oscillators, here both the\nlinear and nonlinear parts of dissipation can be tuned\nwith the cooling laser parameters.\nThe study of the nonlinear motion of trapped laser-\ncooled ions opens several exciting research avenues. Since\ntrapped atomic-ions can be cooled to the quantum\nground state, they are an excellent platform to study\nnonlinear behavior in the quantum regime. As shown\nin [21], unlike the simple harmonic oscillator, a Du\u000eng\noscillator will demonstrate a clear quantum-to-classical\ntransition even when classically driven. Moreover, as theion-spin can be entangled with its motion, it will be pos-\nsible to form a coherent superposition of the two attrac-\ntors states of motion. Laser-cooling of a nonlinear driven\nion-oscillator has several interesting aspects that can be\nfurther explored. Since the Doppler shifts associated\nwith the oscillation amplitudes in the nonlinear regime\nare signi\fcant compared with the cooling transition line-\nwidth, the laser-cooling force is largely nonlinear in the\noscillator velocity. Furthermore, the thermal state gen-\nerated by laser-cooling is the result of balance between\nthe damping force and the inherent heating due to the\nrecoil noise from spontaneous photon scattering. Close\nto the Du\u000eng instability, the ion-oscillator response to\nnoise is quadrature dependent. One noise quadrature is\nlargely enhanced whereas the other quadrature is sup-\npressed [3]. Laser-cooling in this case is likely to produce\nsqueezed states of motion.\nThis work was partially supported by the ISF Morasha\nprogram and the Minerva foundation.\n[1] A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations,\nser.Wiley Classics Library. New York: Wiley, 1995.\n[2] I. Kozinsky et. al. , Phys. Rev. Lett. 99, 207201 (2007).\n[3] R. Almog, S. Zaitsev, O. Shtempluck, and E. Buks, Phys.\nRev. Lett. 98, 078103 (2007).\n[4] R. L. Badzey and P. Mohanty, Nature 437, 995 (2005).\n[5] B. Yurke, D. S. Greywall, A. N. Pargellis, and\nP. A. Busch, Phys. Rev. A 51, 4211 (1995).\n[6] E. Buks and B. Yurke, Phys. Rev. E 74, 046619 (2006).\n[7] J. S. Aldridge and A. N. Cleland, Phys. Rev. Lett. 94,\n156403 (2005).\n[8] E. Babourina-Brooks, A. Doherty, and G. J. Milburn,\nNew. J. Phys. 10, 105020 (2008).\n[9] D. Leibfried, R. Blatt, C. Monroe, and D. J. Wineland,\nRev. Mod. Phys. 75, 281 (2003).\n[10] D. M. Meekhof et. al. , Phys. Rev. Lett. 76, 1796 (1996).\n[11] C. Monroe, D. M. Meekhof, B. E. King, D. J. Wineland,\nScience 272, 1131 (1996).\n[12] C. Marquet, F. Schmidt-Kaler, D. F. V. James, Appl.\nPhys. B 76, 199 (2003).\n[13] C. F. Roos et. al. , Phys. Rev. A 77, 040302(R) (2008).\n[14] A. A. Makarov, Anal. Chem. 68, 4257 (1996)\n[15] K. Vahala et. al. , Nature Physics 5, 682 (2009).\n[16] B. Ravindra and A. K. Mallik Phys. Rev. E 49, 4950\n(1994)\n[17] As the life-time of the P 1=2level (8 ns) is much shorter\nthan the oscillation period (228 ns), the photon scattering\nrate instantaneously adjusts as the ions' velocity varies.\n[18] R. Lifshitz and M. C. Cross, Review of Nonlinear Dy-\nnamics and Complexity 1, 1 (2008)\n[19] S. Zaitsev, O. Shtempluck, E. Buks, and O. Gottlieb\narXiv:cond-mat/053130v1 (2005).\n[20] The ratio between the two terms depends on the cooling\nlaser detuning and is always below 0 :2.\n[21] I. Katz, A. Retzker, R. Straub, and R. Lifshitz, Phys.\nRev. Lett. 99, 040404 (2007)."    },    {        "title": "1010.0478v2.Thermal_fluctuation_field_for_current_induced_domain_wall_motion.pdf",        "content": "Thermal fluctuation field for current-induced domain wall motion\nKyoung-Whan Kim and Hyun-Woo Lee\nPCTP and Department of Physics, Pohang University of Science and Technology, Pohang 790-784, Korea\n/H20849Received 18 May 2010; revised manuscript received 23 August 2010; published 20 October 2010 /H20850\nCurrent-induced domain wall motion in magnetic nanowires is affected by thermal fluctuation. In order to\naccount for this effect, the Landau-Lifshitz-Gilbert equation includes a thermal fluctuation field and literatureoften utilizes the fluctuation-dissipation theorem to characterize statistical properties of the thermal fluctuationfield. However, the theorem is not applicable to the system under finite current since it is not in equilibrium. Toexamine the effect of finite current on the thermal fluctuation, we adopt the influence functional formalismdeveloped by Feynman and Vernon, which is known to be a useful tool to analyze effects of dissipation andthermal fluctuation. For this purpose, we construct a quantum-mechanical effective Hamiltonian describingcurrent-induced domain wall motion by generalizing the Caldeira-Leggett description of quantum dissipation.We find that even for the current-induced domain wall motion, the statistical properties of the thermal noise isstill described by the fluctuation-dissipation theorem if the current density is sufficiently lower than theintrinsic critical current density and thus the domain wall tilting angle is sufficiently lower than\n/H9266/4. The\nrelation between our result and a recent result /H20851R. A. Duine, A. S. Núñez, J. Sinova, and A. H. MacDonald,\nPhys. Rev. B 75, 214420 /H208492007/H20850/H20852, which also addresses the thermal fluctuation, is discussed. We also find\ninteresting physical meanings of the Gilbert damping /H9251and the nonadiabaticy parameter /H9252; while /H9251charac-\nterizes the coupling strength between the magnetization dynamics /H20849the domain wall motion in this paper /H20850and\nthe thermal reservoir /H20849or environment /H20850,/H9252characterizes the coupling strength between the spin current and the\nthermal reservoir.\nDOI: 10.1103/PhysRevB.82.134431 PACS number /H20849s/H20850: 75.78.Fg, 75.60.Ch, 05.40.Ca\nI. INTRODUCTION\nCurrent-induced domain wall /H20849DW/H20850motion in a ferro-\nmagnetic nanowire is one of representative examples tostudy the effect of spin-transfer torque /H20849STT/H20850. The motion of\nDW is generated by the angular momentum transfer betweenspace-time-dependent magnetization m\n/H6023/H20849x,t/H20850and conduction\nelectrons, of which spins interact with m/H6023by the exchange\ncoupling. This system is usually described by the Landau-Lifshitz-Gilbert /H20849LLG/H20850equation,\n1–3\n/H11509m/H6023\n/H11509t=/H92530H/H6023eff/H11003m/H6023+/H9251\nmsm/H6023/H11003/H11509m/H6023\n/H11509t+jp/H9262B\nems/H20875/H11509m/H6023\n/H11509x−/H9252\nmsm/H6023/H11003/H11509m/H6023\n/H11509x/H20876,\n/H208491/H20850\nwhere /H92530is the gyromagnetic ratio, jpis the spin-current\ndensity, ms=/H20841m/H6023/H20841is the saturation magnetization, and /H9262Bis the\nBohr magneton. /H9251is the Gilbert damping coefficient, and /H9252\nis the nonadiabatic coefficient representing the magnitude ofthe nonadiabatic STT.\n4In Eq. /H208491/H20850, the effective magnetic\nfield Heffis given by\nH/H6023eff=A/H116122m/H6023+H/H6023ani+H/H6023th, /H208492/H20850\nwhere Ais stiffness constant, H/H6023anidescribes the effect of the\nmagnetic anisotropy, and H/H6023this the thermal fluctuation field\ndescribing the thermal noise. In equilibrium situations, the\nmagnitude and spatiotemporal correlation of H/H6023thare gov-\nerned by the fluctuation-dissipation theorem,5–7\n/H20855Hth,i/H20849x/H6023,t/H20850Hth,j/H20849x/H6023/H11032,t/H11032/H20850/H20856=4/H9251kBT\n/H6036/H9267/H9254/H20849x/H6023−x/H6023/H11032/H20850/H9254/H20849t−t/H11032/H20850/H9254ij,/H208493/H20850\nwhere /H20855¯/H20856represents the statistical average, i,jdenote x,y,\norzcomponent, kBis the Boltzmann constant, Tis the tem-perature, and /H9267=ms//H9262Bis the spin density. Equation /H208493/H20850\nplays an important role for the study of the magnetizationdynamics at finite temperature,\n8\nEquation /H208493/H20850has been also used in literature9–13to exam-\nine effects of thermal fluctuations on the current-inducedDW motion. In nonequilibrium situations, however, the\nfluctuation-dissipation theorem does not hold generally.Since the system is not in equilibrium any more when thecurrent is applied, it is not clear whether Eq. /H208493/H20850may be still\nused. Recalling that H\n/H6023this estimated to affect the magnetiza-\ntion dynamics considerably in many experimentalsituations\n14–17of the current-driven DW motion, it is highly\ndesired to properly characterize H/H6023thin situations with non-\nzero jp. Recently, Duine18attempted this characterization and\nshowed that Eq. /H208493/H20850is not altered by the spin current up to\nfirst order in the spin-current magnitude. This analysis how-ever is limited to situations where the spin-flip scattering isthe main mechanism responsible for\n/H9252. In this paper, we\ngeneralize this analysis by using a completely different ap-proach which does not assume any specific physical origin of\n/H9252.\nHtharises from extra degrees of freedom /H20849other than mag-\nnetization /H20850, which are not included in the LLG equation. The\nextra degrees of freedom /H20849phonons for instance /H20850usually have\nmuch larger number of degrees of freedom than magnetiza-tion and thus form a heat reservoir. Thus properties of H\nthare\ndetermined by the heat reservoir. The heat reservoir playsanother role. In the absence of the extra degrees of freedom,the Gilbert damping coefficient\n/H9251should be zero since the\ntotal energy should be conserved when all degrees of free-dom are taken into account. Thus the heat reservoir is re-sponsible also for finite\n/H9251. These dual roles of the heat res-\nervoir are the main idea behind the Einstein’s theory of thePHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n1098-0121/2010/82 /H2084913/H20850/134431 /H2084916/H20850 ©2010 The American Physical Society 134431-1Brownian motion.19There are also claims that /H9251is correlated\nwith/H9252/H20849Refs. 18and20–22/H20850in the sense that mechanisms,\nwhich generate /H9252, also contribute to /H9251. Thus the issue of H/H6023th\nand the issue of /H9251and/H9252are mutually connected. Recalling\nthat the main mechanism responsible for /H9251varies from ma-\nterial to material, it is reasonable to expect that the main\nmechanism for H/H6023thand/H9252may also vary from material to\nmaterial. Recently, various mechanisms of /H9252were examined\nsuch as momentum transfer,23–25spin mistracking,26,27spin-\nflip scattering,18,21,22,25,28and the influence of a transport\ncurrent.29This diversity of mechanisms will probably apply\ntoH/H6023thas well.\nInstead of examining each mechanism of H/H6023thone by one,\nwe take an alternative approach to address this issue. In1963, Feynman and Vernon\n30proposed the so-called influ-\nence functional formalism, which allows one to take accountof damping effects without detailed accounts of dampingmechanisms. This formalism was later generalized by Smith\nand Caldeira.\n31This formalism has been demonstrated to be\na useful tool to address dissipation effects /H20849without specific\naccounts of detailed damping mechanisms /H20850on, for instance,\nquantum tunneling,32nonequilibrium dynamic Coulomb\nblockade,33and quantum noise.34To take account of damp-\ning effects which are energy nonconserving processes in gen-eral, the basic idea of the influence functional formalism is tointroduce infinite number of degrees of freedom /H20849called en-\nvironment /H20850behaves like harmonic oscillators which couple\nwith the damped system. /H20851See Eq. /H2084913/H20850./H20852Caldeira and\nLeggett\n32suggested the structure of the spectrum of environ-\nment Eq. /H2084913/H20850and integrated out the degrees of freedom of\nenvironment to find the effective Hamiltonian describing theclassical damping Eq. /H2084912/H20850. For readers who are not familiar\nwith the Caldeira-Leggett’s theory of quantum dissipation,we present the summary of details of the theory in Sec. II B.\nIn order to address the issue of H\n/H6023th, we follow the idea of\nthe influence functional formalism and construct an effectiveHamiltonian describing the magnetization dynamics. The ef-fective Hamiltonian describes not only energy-conservingprocesses but also energy-nonconserving processes such asdamping and STT. From this approach, we find that Eq. /H208493/H20850\nholds even in nonequilibrium situations with finite j\np, pro-\nvided that jpis sufficiently smaller than the so-called intrin-\nsic critical current density23so that the DW tilting angle /H9278\n/H20849to be defined below /H20850is sufficiently smaller than /H9266/4. We\nremark that in the special case where the spin-flip scattering\nmechanism of /H9252is the main mechanism of H/H6023th, our finding is\nconsistent with Ref. 18, which reports that the spin flip scat-\ntering mechanism does not alter Eq. /H208493/H20850at least up to the first\norder in jp. But our calculation indicates that Eq. /H208493/H20850holds\nnot only in situations where the spin flip scattering is the\ndominant mechanism of H/H6023thand/H9252but also in more diverse\nsituations as long as the heat reservoir can be described bybosonic excitations /H20849such as electron-hole pair excitations or\nphonon /H20850, i.e., the excitations effectively behave like har-\nmonic oscillators to be described by Caldeira-Leggett’stheory. We also remark that in addition to the derivation ofEq./H208493/H20850in nonequilibrium situations, our calculation also re-\nveals an interesting physical meaning of\n/H9252, which will be\ndetailed in Sec. III.This paper is organized as follows. In Sec. II, we first\nintroduce the Caldeira-Leggett’s version of the influencefunctional formalism and later generalize this formalism sothat it is applicable to our problem. This way, we construct aHamiltonian describing the DW motion. In Sec. III, some\nimplications of this model is discussed. First, a distinct in-sight on\n/H9252is emphasized. Second, as an application, statisti-\ncal properties of the thermal fluctuation field are calculatedin the presence of nonzero j\np, which verifies the validity of\nEq./H208493/H20850when jpis sufficiently smaller than the intrinsic criti-\ncal density. It is believed that many experiments16,17are in-\ndeed in this regime. Finally, in Sec. IV, we present some\nconcluding remarks. Technical details about the quantumtheory of the DW motion and methods to obtain solutions areincluded in Appendices.\nII. GENERALIZED CALDEIRA-LEGGETT DESCRIPTION\nA. Background\nInstead of full magnetization profile m/H6023/H20849x,t/H20850, the DW dy-\nnamics is often described2,23,35–37by two collective coordi-\nnates, DW position x/H20849t/H20850and DW tilting angle /H9278/H20849t/H20850. When\nexpressed in terms of these collective coordinates, the LLGEq./H208491/H20850reduces to the so-called Thiele equations,\ndx\ndt=jp/H9262B\nems+/H9251/H9261d/H9278\ndt+/H92530K/H9261\nmssin 2/H9278+/H9257x/H20849t/H20850,/H208494a/H20850\n/H9261d/H9278\ndt=−/H9251dx\ndt+/H9252jp/H9262B\nems+/H9257p/H20849t/H20850. /H208494b/H20850\nHere Kis the hard-axis anisotropy, /H9261is the DW thickness.\n/H9257x/H20849t/H20850and/H9257p/H20849t/H20850are functions describing thermal noise field\nHth,i/H20849x,t/H20850. By definition, the statistical average of the thermal\nnoise field Hth,i/H20849x,t/H20850is zero and similarly the statistical aver-\nages of /H9257x/H20849t/H20850and/H9257p/H20849t/H20850should also vanish regardless of\nwhether the system is in equilibrium. The question of theircorrelation function is not trivial however. If the thermalnoise field H\nth,i/H20849x,t/H20850satisfies the correlation in Eq. /H208493/H20850,i tc a n\nbe derived from Eq. /H208493/H20850that/H9257x/H20849t/H20850and/H9257p/H20849t/H20850satisfy the cor-\nrelation relation12\n/H20855/H9257i/H20849t/H20850/H9257j/H20849t/H11032/H20850/H20856/H11008/H9251kBT/H9254ij/H9254/H20849t−t/H11032/H20850, /H208495/H20850\nfor/H20853i,j/H20854=/H20853x,p/H20854. But as mentioned in Sec. I, Eq./H208493/H20850is not\nguaranteed generally in the presence of the nonzero current.Then Eq. /H208495/H20850is not guaranteed either. The question of what\nshould be the correlation function /H20855\n/H9257i/H20849t/H20850/H9257j/H20849t/H11032/H20850/H20856in such a situ-\nation will be discussed in Sec. III.\nWhen the spin-current density jpis sufficiently smaller\nthan the so-called intrinsic critical density /H20841e/H92530K/H9261//H9262B/H20841,23/H9278\nstays sufficiently smaller than /H9266/4. In many experimental\nsituations,38–40this is indeed the case,41so we will confine\nourselves to the small /H9278regime in this paper. Then, one can\napproximate sin 2 /H9278/H110152/H9278to convert the equations into the\nfollowing form:42\ndx\ndt=vs+/H9251S\n2KMdp\ndt+p\nM+/H92571/H20849t/H20850, /H208496a/H20850KYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-2dp\ndt=−2/H9251KM\nSdx\ndt+2/H9252KM\nSvs+/H92572/H20849t/H20850, /H208496b/H20850\nwhere p=2KM/H9261/H9278/S,Sis the spin angular momentum at\neach individual magnetic site, and vs=jp/H9262B/emsis the adia-\nbatic velocity,43which is a constant of velocity dimension\nand proportional to jp. The yet undetermined constant Mis\nthe effective DW mass42–44which will be fixed so that the\nnew variable pbecomes the canonical conjugate to x./H92571/H20849t/H20850\nand/H92572/H20849t/H20850are the same as /H9257x/H20849t/H20850and/H9257p/H20849t/H20850except for propor-\ntionality constants.\nWhen the thermal noises /H92571/H20849t/H20850and/H92572/H20849t/H20850are ignored, one\nobtains from Eq. /H208496/H20850the time dependence of the DW posi-\ntion,\nx/H20849t/H20850=x/H208490/H20850+/H9252\n/H9251vst+S\n2KM/H92512/H208491−e−2K/H9251t/S/H208491+/H92512/H20850/H20850\n/H11003/H20851/H9251p/H208490/H20850−Mvs/H20849/H9251−/H9252/H20850/H20852. /H208497/H20850\nNote that after a short transient time, the DW speed ap-\nproaches the terminal velocity /H9252vs//H9251. Thus the ratio /H9252//H9251is\nan important parameter for the DW motion. When the ther-mal noises are considered, they generate a correction to Eq./H208497/H20850. However, from Eq. /H208496/H20850, it is evident that the statistical\naverage of x/H20849t/H20850should still follow Eq. /H208497/H20850. Thus as far as the\ntemporal evolution of the statistical average is concerned, wemay ignore the thermal noises. In the rest of Sec. II,w ea i m\nto derive a quantum mechanical Hamiltonian, which repro-duces the same temporal evolution as Eq. /H208497/H20850in the statistical\naverage level. In Sec. III, we use the Hamiltonian to derive\nthe correlation function /H20855\n/H9257i/H20849t/H20850/H9257j/H20849t/H11032/H20850/H20856in the presence of the\nnonzero current.\nNow, we begin our attempt to construct an effective\nHamiltonian that reproduces the DW dynamics Eq. /H208496/H20850/H20851or\nequivalently Eq. /H208497/H20850/H20852. We first begin with the microscopic\nquantum-mechanical Hamiltonian Hs-d,\nHs-d=−J/H20858\niS/H6023i·S/H6023i+1−A/H20858\ni/H20849S/H6023i·zˆ/H208502+K/H20858\ni/H20849S/H6023i·yˆ/H208502+HcS,\n/H208498/H20850\nwhich has been used in previous studies20of the DW dynam-\nics. Here Jrepresents the ferromagnetic exchange interac-\ntion, AandKrepresent longitudinal /H20849easy-axis /H20850and trans-\nverse/H20849hard-axis /H20850anisotropy, respectively. The last term HcS\nrepresents the coupling of the spin system with the spin-\npolarized current,\nHcS=−/H20858\ni,/H9251=↑,↓/H20851t/H20849ci/H9251†ci+1/H9251+ci+1/H9251†ci/H9251/H20850−/H9262ci/H9251†ci/H9251/H20852−JH/H20858\niS/H6023ci·S/H6023i,\n/H208499/H20850\nwhere JHis the exchange interaction between conduction\nelectron and the localized spins, ci/H9251is the annihilation opera-\ntor of the conduction electron at the site i,S/H6023ciis the electron-\nspin operator, tis the hopping integral, and /H9262is the chemical\npotential of the system.Recently Kim et al.43analyzed Hs-din detail in the small\ntilting angle regime and found that Hs-dcontains gapless\nlow-lying excitations and also high-energy excitations with afinite energy gap. The gapless excitations of H\ns-dare de-\nscribed by a simple Hamiltonian H0,\nH0=vsP+P2\n2M/H2084910/H20850\nwhile the high-energy excitations have a finite energy gap\n2S/H20881A/H20849A+K/H20850.I nE q . /H2084910/H20850,Pis the canonical momentum of\nthe DW position operator Q, and M=/H60362\nK/H208812A\nJa4is the effective\nDW mass called Döring mass.44Here, ais the lattice spacing\nbetween two neighboring spins. /H20849See, for details, Appendix\nA./H20850Below we will neglect the high energy excitations and\nfocus on the low-lying excitations described by Eq. /H2084910/H20850. For\nthe analysis of the high-energy excitation effects on the DW,See Ref. 42.\nFrom Eq. /H2084910/H20850, one obtains the following Heisenberg’s\nequation of motion:\ndQ\ndt=vs+P\nM, /H2084911a/H20850\ndP\ndt=0 . /H2084911b/H20850\nNote that the current /H20849proportional to vs/H20850appears in the equa-\ntion fordQ\ndt. Thus the current affects the DW dynamics by\nintroducing a difference between the canonical momentum P\nand the kinematic momentum P+Mvs. In this sense, the ef-\nfect of the current is similar to a vector potential /H20851canonical\nmomentum P/H6023and kinematic momentum P/H6023+/H20849e/c/H20850A/H6023/H20852. The\nvector potential /H20849difference between the canonical momen-\ntum and the kinetic momentum /H20850allows the system in the\ninitially zero momentum state to move without breaking thetranslational symmetry of the system. In other words, thecurrent-induced DW motion is generated without any forceterm in Eq. /H2084911b/H20850violating the translational symmetry of the\nsystem. This should be contrasted with the effect of the mag-netic field or magnetic defects, which generates a force termin Eq. /H2084911b/H20850.\nThe solution of Eq. /H2084911/H20850is trivial, /H20855Q/H20849t/H20850/H20856=/H20855Q/H208490/H20850/H20856\n+/H20849/H20855P/H208490/H20850/H20856/M+\nvs/H20850t. Here, the statistical average /H20855¯/H20856is de-\nfined as /H20855¯/H20856=Tr/H20849/H9267¯/H20850/Tr/H20849/H9267/H20850, where /H9267denotes the density\nmatrix at t=0. Associating /H20855Q/H20849t/H20850/H20856=x/H20849t/H20850,/H20855P/H20849t/H20850/H20856=p/H20849t/H20850, one\nfinds that Eq. /H2084911/H20850is identical to Eq. /H208496/H20850if/H9251=/H9252=0. This\nimplies that the effective Hamiltonian H0/H20851Eq./H2084910/H20850/H20852fails to\ncapture effects of nonzero /H9251and/H9252. In the next three sections,\nwe attempt to resolve this problem.\nB. Caldeira-Leggett description of damping\nTo solve the problem, one should first find a way to de-\nscribe damping. A convenient way to describe finite dampingwithin the effective Hamiltonian approach is to adopt theCaldeira-Leggett description\n32of the damping. Its main idea\nis to introduce a collection of additional degrees of freedom/H20849called environment /H20850and couple them to the original dy-\nnamic variables so that energy of the dynamic variables canTHERMAL FLUCTUATION FIELD FOR CURRENT-INDUCED … PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-3be transferred to the environment. For instance, for a one-\ndimensional /H208491D/H20850particle subject to damped dynamics,\ndx\ndt=p\nM, /H2084912a/H20850\ndp\ndt=−dV/H20849x/H20850\ndx−/H9253dx\ndt. /H2084912b/H20850\nCaldeira and Leggett32demonstrated that its quantum-\nmechanical Hamiltonian can be constructed by adding damp-ing Hamiltonian H\n1to the undamped Hamiltonian H0\n=P2/2M+V/H20849Q/H20850. The damping Hamiltonian H1contains a\ncollection of environmental degrees of freedom /H20853xi,pi/H20854be-\nhaving like harmonic oscillators /H20851see Eq. /H2084914/H20850/H20852, which couple\nto the particle through the linear coupling term /H20858iCixiQbe-\ntween Qand the environmental variables xi. Here, Ciis the\ncoupling constant between xiandQ. The implication of the\ncoupling is twofold: /H20849i/H20850the coupling to the environment gen-\nerates damping, whose precise form depends on Ci,mi, and\n/H9275i. It is demonstrated in Ref. 32that the coupling generates\nthe simple damping of the form in Eq. /H2084912b/H20850ifCi,mi, and/H9275i\nsatisfy the following relation of the spectral function J/H20849/H9275/H20850:\nJ/H20849/H9275/H20850/H11013/H9266\n2/H20858\niCi2\nmi/H9275i/H9254/H20849/H9275−/H9275i/H20850=/H9253/H9275. /H2084913/H20850\n/H20849ii/H20850The coupling also modified the potential Vby generating\nan additional contribution − /H20858iCi2Q2/2mi/H9275i2. This implies that\nV/H20849x/H20850in Eq. /H2084912b/H20850should not be identified with V/H20849Q/H20850inH0\n/H20849even though the same symbol Vis used /H20850but should be iden-\ntified instead with the total potential that includes the contri-bution from the environmental coupling. If we express thetotal Hamiltonian Hin terms of the effective V/H20849x/H20850that ap-\npears in Eq. /H2084912b/H20850, it reads\nH=H\n0+H1, /H2084914a/H20850\nH0=P2\n2M+V/H20849Q/H20850, /H2084914b/H20850\nH1=/H20858\ni/H20875pi2\n2mi+1\n2mi/H9275i2/H20873xi+Ci\nmi/H9275i2Q/H208742/H20876./H2084914c/H20850\nBy identifying x/H20849t/H20850=/H20855Q/H20849t/H20850/H20856,p/H20849t/H20850=/H20855P/H20849t/H20850/H20856, the equations of\nmotion obtained from Eqs. /H2084913/H20850and/H2084914/H20850reproduce Eq. /H2084912/H20850.\nC. Generalization to the DW motion: /H9251term\nHere we aim to apply the Caldeira-Leggett approach to\nconstruct an effective Hamiltonian of the DW dynamics sub-ject to finite damping /H20849\n/H9251/HS110050/H20850. To simplify the problem, we\nfirst focus on a situation, where only /H9251is relevant and /H9252is\nirrelevant. This situation occurs if there is no current /H20849vs\n=0/H20850. Then Eq. /H208496/H20850reduces to\ndx\ndt=/H9251S\n2KMdp\ndt+p\nM, /H2084915a/H20850dp\ndt=−2/H9251KM\nSdx\ndt. /H2084915b/H20850\nNote that /H9252does not appear. Note also that these equations\nare slightly different from Eq. /H2084912/H20850, where a damping term is\ncontained only in the equation ofdp\ndt. However, in the equa-\ntions of the DW /H20851Eq./H2084915/H20850/H20852, damping terms appear not only in\nthe equation ofdp\ndt/H20851Eq./H2084915b/H20850/H20852but also in the equation ofdx\ndt\n/H20851Eq./H2084915a/H20850/H20852.\nThus the Caldeira-Leggett description in the preceding\nsection is not directly applicable and should be generalized.To get a hint, it is useful to recall the conjugate relation\nbetween QandP. The equations ofdQ\ndtanddP\ndtare obtained\nby differentiating Hwith respect to Pand − Q, respectively.\nOf course, it holds for /H20849xi,pi/H20850, also. Thus, one can obtain\nanother set of Heisenberg’s equation of motion by exchang-ing/H20849Q,x\ni/H20850↔/H20849−P,−pi/H20850. By this canonical transformation, the\nposition coupling /H20858iCixiQchanges to a momentum coupling\nterm, and the damping term in the equation ofdP\ndtis now in\nthat ofdQ\ndt. This mathematical relation that the momentum\ncoupling generates a damping term in the equation ofdQ\ndt\nmakes it reasonable to expect that the momentum coupling\n/H20858iDipiPis needed45to generate the damping in the equation\nfordQ\ndt. Here Diis the coupling constant between Pandpi.\nThe reason why, in the standard Caldeira-Leggett approach,the damping term appears only in Eq. /H2084912b/H20850is that Eq. /H2084914/H20850\ncontains only position coupling terms /H20858\niCixiQ. It can be eas-\nily verified that the implications of the momentum couplingare again twofold: /H20849i/H20850the coupling indeed introduces the\ndamping term in the equation ofdQ\ndt./H20849ii/H20850it modifies the DW\nmass. The mass renormalization arises from the fact that inthe presence of the momentum coupling /H20858\niDipiP, the kine-\nmatic momentum midxi\ndtof an environmental degree of free-\ndom xiis now given by /H20849pi+DimiP/H20850instead of pi. Then the\nterm/H20858i/H20851pi2\n2mi+DipiP/H20852can be decomposed into two pieces,\n/H20858i/H20849pi+DimiP/H208502\n2mi, which is the kinetic energy associated with xi,\nand/H20851−/H20858iDi2mi\n2/H20852P2. Note that the second piece has the same\nform as the DW kinetic termP2\n2M. Thus this second piece\ngenerates the renormalization of the DW mass. Due to thismass renormalization effect, Min Eq. /H2084915/H20850should be inter-\npreted as the renormalized mass that contains the contribu-tion from the environmental coupling. If MinH\n0in Eq. /H2084910/H20850\nis interpreted as the renormalized mass, the environmentHamiltonian H\n2for the DW dynamics becomes\nH2=/H20858\ni/H208751\n2mi/H20849pi+DimiP/H208502+1\n2mi/H9275i2/H20873xi+Ci\nmi/H9275i2Q/H208742/H20876.\n/H2084916/H20850\nHere,/H20858i/H20849pi+DimiP/H208502/2micoupling is equivalent to the origi-\nnal form /H20858i/H20849pi2/2mi+DipiP/H20850under the mass renormalization\n1/M→1/M−/H20858iDi2mi/2. Note that in H2, the collective co-\nordinates QandPof the DW couple to the environmental\ndegrees of freedom /H20853xi,pi/H20854through two types of coupling,\n/H20858iCixiQand/H20858iDipiP.\nFinally, one obtains the total Hamiltonian describing the\nDW motion in the absence of the current,KYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-4H=H0/H20841vs=0+H2=P2\n2M+/H20858\ni/H208751\n2mi/H20849pi+DimiP/H208502\n+1\n2mi/H9275i2/H20873xi+Ci\nmi/H9275i2Q/H208742/H20876. /H2084917/H20850\nNow, the renormalized mass Min the above equation is iden-\ntical to the mass in Eq. /H2084915/H20850. To make the physical meaning\nofxiclearer, we perform the canonical transformation,\nxi→−Ci\nmi/H9275i2xi,pi→−mi/H9275i2\nCipi. /H2084918/H20850\nDefining /H9253i=CiDi\n/H9275i2, and redefining a new miasmi/H20849new/H20850=Ci2\nmi/H9275i4,\nthe Hamiltonian becomes simpler as\nH=P2\n2M+/H20858\ni/H208751\n2mi/H20849pi−/H9253iP/H208502+1\n2mi/H9275i2/H20849xi−Q/H208502/H20876.\n/H2084919/H20850\nNow, the translational symmetry of the system and the physi-\ncal meaning of xibecome obvious.\nThe next step is to impose proper constraints on /H9253iandmi,\nso that the damping terms arising from Eq. /H2084919/H20850agree exactly\nwith those in Eq. /H2084915/H20850. For this purpose, it is convenient to\nintroduce Laplace transformed variables Q˜/H20849/H9261/H20850,P˜/H20849/H9261/H20850,x˜i/H20849/H9261/H20850,\np˜i/H20849/H9261/H20850, where Q˜/H20849/H9261/H20850=/H208480/H11009e−/H9261t/H20855Q/H20849t/H20850/H20856dt, and other transformed\nvariables are defined in a similar way. Then the variables x˜i\nandp˜ican be integrated out easily /H20849see Appendix B /H20850. After\nsome tedious but straightforward algebra, it is verified thatwhen the following three constraints on\n/H9253i,/H9275i,miare satis-\nfied for any positive /H9261,\n/H20858\ni/H9253i/H9275i2\n/H92612+/H9275i2=0 , /H2084920a/H20850\n/H20858\ni/H9253i2/H9261\nmi/H20849/H92612+/H9275i2/H20850=/H9251S\n2KM, /H2084920b/H20850\n/H20858\nimi/H9275i2/H9261\n/H92612+/H9275i2=2/H9251KM\nS, /H2084920c/H20850\nthe DW dynamics satisfies the following equation:\n/H20898/H9261 −1\nM−/H9251S/H9261\n2KM\n2/H9251KM\nS/H9261 /H9261/H20899/H20873Q˜\nP˜/H20874\n=/H20873/H20855Q/H208490/H20850/H20856\n/H20855P/H208490/H20850/H20856/H20874+/H20898−/H9251S\n2KM/H20855P/H208490/H20850/H20856\n2/H9251KM\nS/H20855Q/H208490/H20850/H20856/H20899, /H2084921/H20850\nwhich is nothing but the Laplace transformation of the DW\nequation /H20851Eq./H2084915/H20850/H20852if/H20855Q/H20856and/H20855P/H20856are identified with xandp.\nThus we verify that the Hamiltonian Hin Eq. /H2084919/H20850indeed\nprovides a generalized Caldeira-Leggett-type quantumHamiltonian for the DW motion. As a passing remark, we\nmention that in the derivation of Eq. /H2084921/H20850, the environmental\ndegrees of freedom at the initial moment /H20849t=0/H20850are assumed\nto be in their thermal equilibrium so that\n/H20855xi/H208490/H20850/H20856=/H20855Q/H208490/H20850/H20856, /H2084922a/H20850\n/H20855pi/H208490/H20850/H20856=/H9253i/H20855P/H208490/H20850/H20856. /H2084922b/H20850\nEquation /H2084922/H20850can be understood as follows. First, one ob-\ntains Eq. /H2084922/H20850by following Appendix D which describes the\nstatistical properties of Eq. /H2084919/H20850at high temperature. In Ap-\npendix D, /H20855xi/H208490/H20850−Q/H208490/H20850/H20856=/H20855pi/H208490/H20850−/H9253iP/H208490/H20850/H20856is reduced to an in-\ntegration of an odd function so it is shown to vanish. Thesecond way is probably easier to understand and does notrequire the classical limit or high-temperature limit. TheHamiltonian /H20851Eq./H2084919/H20850/H20852is symmetric under the canonical\ntransformation Q/H208490/H20850→−Q/H208490/H20850,P/H208490/H20850→−P/H208490/H20850,x\ni/H208490/H20850→−xi/H208490/H20850,\nand pi/H208490/H20850→−pi/H208490/H20850. Due to this symmetry, one obtains\n/H20855xi/H208490/H20850−Q/H208490/H20850/H20856=/H20855Q/H208490/H20850−xi/H208490/H20850/H20856and/H20855pi/H208490/H20850−/H9253iP/H208490/H20850/H20856=/H20855/H9253iP/H208490/H20850\n−pi/H208490/H20850/H20856, which lead to /H20855xi/H208490/H20850/H20856=/H20855Q/H208490/H20850/H20856and/H20855pi/H208490/H20850/H20856=/H9253i/H20855P/H208490/H20850/H20856,\nrespectively.\nHere physical origin of the momentum coupling /H20849/H9253/H20850be-\ntween the DW and environment deserves some discussion.Equation /H2084919/H20850is reduced to the original Caldeira-Leggett\nHamiltonian if\n/H9253i=0. However, Eq. /H2084920b/H20850implies that the\nmomentum coupling as well as the position coupling is in-dispensable to describe the Gilbert damping. To understandthe origin of the momentum coupling\n/H9253i, it is useful to recall\nthat since P/H11008/H9278/H11008/H20849tilting/H20850, one can interpret Pand Qas\ntransverse and longitudinal spin fluctuation of the DW state,respectively. /H20849See, for explicit mathematical relation, Appen-\ndix A. /H20850Thus, if there is rotational symmetry on spin interac-\ntion with the heat bath /H20849or environment /H20850, the existence of the\nposition coupling requires the existence of the momentumcoupling. Thus the appearance of the damping terms both inEqs./H2084915a/H20850and/H2084915b/H20850is natural in view of the rotational sym-\nmetry of the spin exchange interaction and also in view ofthe physical meaning of PandQas transverse and longitu-\ndinal spin fluctuations.\nD. Coupling with the spin current: /H9252term\nIn this section, we aim to construct a Caldeira-Leggett-\ntype effective quantum Hamiltonian that takes account of notonly\n/H9251but also /H9252. Since /H9252becomes relevant only when there\nexists finite spin current, we have to deal with situations withfinite current /H20849\nvs/HS110050/H20850. Then the system is notin thermal equi-\nlibrium.\nAs demonstrated in Eq. /H2084910/H20850, the spin current couples with\nthe DW linear momentum, i.e., vsP. Here, adiabatic velocity\nvsacts as the coupling constant proportional to spin current.\nThe spin current may also couple directly to the environmen-tal degrees of freedom. Calling this coupling constant\nv, one\nintroduces the corresponding coupling term /H20858ivpi. Later we\nfind that this coupling is crucial to account for nonzero /H9252.A t\nthis point we will not specify the value of v. Now, the total\neffective Hamiltonian in the presence of the spin current ob-tained by adding the coupling term /H20858\nivpito Eq. /H2084919/H20850. Then,THERMAL FLUCTUATION FIELD FOR CURRENT-INDUCED … PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-5Htot=H+Hcurrent =P2\n2M+vsP+/H20858\nivpi\n+/H20858\ni/H208751\n2mi/H20849pi−/H9253iP/H208502+1\n2mi/H9275i2/H20849xi−Q/H208502/H20876./H2084923/H20850\nIn order to illustrate the relation between Eqs. /H208496/H20850and\n/H2084923/H20850, we consider a situation, where the current is zero until\nt=0 and turned on at t=0 to a finite value. This situation is\ndescribed by the following time-dependent Hamiltonian:\nHtot=P2\n2M+vs/H20849t/H20850P+/H20858\niv/H20849t/H20850pi\n+/H20858\ni/H208751\n2mi/H20849pi−/H9253iP/H208502+1\n2mi/H9275i2/H20849xi−Q/H208502/H20876,/H2084924/H20850\nwhere vs/H20849t/H20850=vs/H9008/H20849t/H20850andv/H20849t/H20850=v/H9008/H20849t/H20850. And /H9008/H20849t/H20850is\n/H9008/H20849t/H20850=/H208771fort/H110220,\n0fort/H110210./H20878 /H2084925/H20850\nTo make a quantitative comparison between Eqs. /H208496/H20850and\n/H2084924/H20850, one needs to integrate out environmental degrees of\nfreedom /H20853xi,pi/H20854, which requires one to specify their initial\nconditions. Since the system is in thermal equilibrium untilt=0, we may still impose the constraint in Eq. /H2084922/H20850to exam-\nine the DW dynamics for t/H110220. By following a similar pro-\ncedure as in Sec. II C and by using the constraints in Eq.\n/H2084920/H20850,\n46one finds that the effective Hamiltonian H/H20851Eq./H2084924/H20850/H20852\npredicts/H20855Q/H20849t/H20850/H20856=/H20855Q/H208490/H20850/H20856+vt+S\n2KM/H92512/H208491−e−2K/H9251t/S/H208491+/H92512/H20850/H20850\n/H11003/H20851/H9251/H20855P/H208490/H20850/H20856−M/H9251/H20849vs−v/H20850/H20852. /H2084926/H20850\nThis is exactly the same as Eq. /H208497/H20850if\n/H9252\n/H9251=v\nvs. /H2084927/H20850\nSo by identifying vwith vs/H9252//H9251, we obtain a Caldeira-\nLeggett-type effective quantum Hamiltonian of the DW dy-namics.\nOne needs to consider an external force on Eq. /H208496b/H20850/H20851or\nEq./H208494b/H20850/H20852when the translational symmetry of the system is\nbroken by some factors such as external magnetic field andmagnetic defects. To describe this force, one can add a posi-tion dependent potential V/H20849Q/H20850/H20849Ref. 47/H20850to Eq. /H2084924/H20850. Consid-\nering the Heisenberg’s equation, the potential V/H20849Q/H20850generates\nthe term − V\n/H11032/H20849Q/H20850in Eq. /H208496b/H20850.\nIII. IMPLICATIONS\nA. Insights on the physical meaning of /H9252\nEquation /H2084927/H20850provides insights on the physical meaning\nof/H9252./H9252depends largely on the coupling between the envi-\nronment and current, not on the damping form. Recallingthat\nvsdescribes the coupling between the current and the\nDW, we find that /H9252//H9251, which describes the asymptotic be-\nhavior of the DW motion, is the ratio between the current-magnetization /H20849DW in the present case /H20850coupling and\ncurrent-environment coupling. That is,\n/H9252\n/H9251=/H20849Coupling between the current and the environment /H20850\n/H20849Coupling between the current and the DW /H20850. /H2084928/H20850\nTo make the physical meaning of Eq. /H2084928/H20850more transpar-\nent, it is useful to examine consequences of the nonzero cou-pling\nvbetween the current and the environment. One of the\nimmediate consequences of the nonzero vappears in the ve-\nlocities of the environmental degrees of freedom. It can beverified easily that the initial velocities of environmental co-\nordinates are given by exactly\nv,/H20855x˙i/H208490/H20850/H20856=v. Recalling that\nthe terminal velocity of the DW, /H20855Q˙/H20849t/H20850/H20856approaches vs/H9252//H9251,\none finds from Eq. /H2084927/H20850that the terminal velocity of the DW\nis nothing but the environment velocity. This result is verynatural since the total Hamiltonian H\ntot/H20851Eq./H2084923/H20850/H20852is Galilean\ninvariant and the total mass of the environment /H20849or reservoir /H20850\nis much larger than the DW mass.48A very similar conclu-\nsion is obtained by Garate et al.29By analyzing the Kamber-\nsky mechanism,49which is reported50to be the dominant\ndamping mechanism in transition metals such as Fe, Co, Ni,they found that the ratio\n/H9252//H9251is approximately given by the\nratio between the drift velocity of the Kohn-Sham quasipar-\nticles and vs. Since the collection of Kohn-Sham quasiparti-cles play the role of the environment in case of the Kamber-\nsky mechanism, the result in Ref. 29is consistent with ours.\nIt is interesting to note that our calculation, which is largelyindependent of details of damping mechanism, reproducesthe result for the specific case.\n29This implies that the result\nin Ref. 29can be generalized if the drift velocity of the\nKohn-Sham quasiparticles is replaced by the general cou-pling constant\nvbetween the current and the environment.\nOur claim that the origin of /H9252is the direct coupling be-\ntween the current and environment has an interesting con-ceptual consistency with the work by Zhang and Li.\n4Zhang\nand Li derived the nonadiabatic term by introducing a spin-relaxation term in the equation of motion of the conduction\nelectrons. A clear consistency arises from generalizing thespin relaxation in Ref. 4to the coupling with environment in\nour work. In Ref. 4, Gilbert damping /H20849\n/H9251/H20850and the nonadia-\nbatic STT /H20849/H9252/H20850are identified as the spin relaxation of magne-\ntization and conduction electrons, respectively. Generalizingthe spin relaxation to environmental coupling,\n/H9251and/H9252areKYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-6now identified as the coupling of the environment with the\nmagnetization /H20849i.e., the DW in our model /H20850and the coupling\nof the environment with current, respectively. It is exactlyhow we identified\n/H9251and/H9252, and this gives the conceptual\nconsistency between our work and Ref. 4. As an additional\ncomment, while some magnitudes and origins of /H9252claimed\nin different references, such as Refs. 4and29, seem to be\nbased on completely independent phenomena, our work andinterpretation on\n/H9252provide a connection between them\nthrough the environmental degrees of freedom.\nB. Effect of environment on stochastic forces\nUntil now, our considerations has been limited to the evo-\nlution of the expectation values /H20855Q/H20849t/H20850/H20856and/H20855P/H20849t/H20850/H20856and thus\nthermal fluctuation effects have been ignored. In this section,we address the issue of thermal fluctuations. For this pur-pose, we need to go beyond the expectation values and so wederive the following operator equations from the Hamil-\ntonian Eq. /H2084923/H20850:\nQ˙=\nvs+/H9251S\n2KMP˙+P\nM+/H92571/H20849t/H20850, /H2084929a/H20850\nP˙=−2/H9251KM\nSQ˙+2/H9251KM\nSv+/H92572/H20849t/H20850, /H2084929b/H20850\nwhere\n/H92571/H20849t/H20850=/H20858\ni/H9253i/H9275i/H20873/H9004xisin/H9275it−/H9004pi\nmi/H9275icos/H9275it/H20874,/H2084930a/H20850\n/H92572/H20849t/H20850=/H20858\ni/H20849mi/H9275i2/H9004xicos/H9275it+/H9275i/H9004pisin/H9275it/H20850./H2084930b/H20850\nHere/H9004xi/H11013xi/H208490/H20850−Q/H208490/H20850and/H9004pi/H11013pi/H208490/H20850−/H9253iP/H208490/H20850. The deriva-\ntion of Eqs. /H2084929/H20850and/H2084930/H20850utilizes constraints Eqs. /H2084920/H20850and\n/H2084927/H20850. We remark that the result in Sec. II D can be recovered\nfrom Eqs. /H2084929/H20850and/H2084930/H20850by taking the expectation values of\nthe operators. When Eq. /H2084929/H20850is compared to Eq. /H208496/H20850,i ti s\nevident that /H92571/H20849t/H20850and/H92572/H20849t/H20850defined in Eq. /H2084930/H20850carry the\ninformation about the thermal noise. It is easy to verify thatthe expectation values of\n/H92571/H20849t/H20850and/H92572/H20849t/H20850vanish, thus repro-\nducing the results in the earlier section. Here it should benoticed that Eq. /H2084930/H20850relates\n/H92571/H20849t/H20850and/H92572/H20849t/H20850in the nonequi-\nlibrium situations /H20849after the current is turned on or t/H110220/H20850to\nthe operators /H9004xiand/H9004pi, which are defined in the equilib-\nrium situation /H20849right before the current is turned on or t=0/H20850.\nThus by combining Eq. /H2084930/H20850with the equilibrium noise char-\nacteristics of /H9004xiand/H9004pi, we can determine the thermal\nnoise characteristic in the nonequilibrium situation /H20849t/H110220/H20850.\nTo extract information about the noise, one needs to\nevaluate the correlation functions /H20855/H20853/H9257i/H20849t/H20850,/H9257j/H20849t/H20850/H20854/H20856 /H20849i,j=1,2/H20850,\nwhere /H20853,/H20854denotes the anticommutator. Due to the relations in\nEq./H2084930/H20850, the evaluation of the correlation function reduces to\nthe expectation value evaluation of the operator products/H20853x\ni/H208490/H20850,pj/H208490/H20850/H20854,xi/H208490/H20850xj/H208490/H20850, and pi/H208490/H20850pj/H208490/H20850in the equilibrium\nsituation governed by the equilibrium Hamiltonian /H20851Eq.\n/H2084919/H20850/H20852.\nIn the classical limit /H20849/H6036→0, see the next paragraph to find\nout when the classical limit is applicable /H20850, Eq./H2084919/H20850is just acollection of independent harmonic oscillators of /H20853/H9004xi,/H9004pi/H20854.\nHence, the equipartition theorem determines their correla-tions,\n/H20855/H9004x\ni/H20856=/H20855/H9004pi/H20856=/H20855/H9004xi/H9004pi/H20856=0 , /H2084931a/H20850\n/H20855/H9004xi/H9004xj/H20856=kBT\nmi/H9275i2/H9254ij, /H2084931b/H20850\n/H20855/H9004pi/H9004pj/H20856=mikBT/H9254ij. /H2084931c/H20850\nEquation /H2084920/H20850and/H2084931/H20850give the correlations of /H92571/H20849t/H20850and\n/H92572/H20849t/H20850. After some algebra, one straightforwardly gets\n/H20855/H9257i/H20849t/H20850/H20856=0 , /H2084932a/H20850\n/H20855/H92571/H20849t/H20850/H92572/H20849t/H11032/H20850/H20856=0 , /H2084932b/H20850\n/H20855/H92571/H20849t/H20850/H92571/H20849t/H11032/H20850/H20856=/H9251S\n2KMkBT/H9254/H20849t−t/H11032/H20850, /H2084932c/H20850\n/H20855/H92572/H20849t/H20850/H92572/H20849t/H11032/H20850/H20856=2/H9251KM\nSkBT/H9254/H20849t−t/H11032/H20850. /H2084932d/H20850\nThese relations are consistent with Eq. /H208495/H20850when/H92571/H20849t/H20850and\n/H92572/H20849t/H20850in Eq. /H2084930/H20850are identified with those in Eq. /H208496/H20850. Thus\nthey confirm that the relations /H20851Eq./H2084932/H20850/H20852assumed in many\npapers9–13indeed hold rather generally in the regime where\nthe tilting angle remains sufficiently smaller than /H9266/4.\nNext we consider the regime where the condition of the\nclassical limit is valid. Since statistical properties of the sys-\ntem at finite temperature is determined bykBT\n/H6036, the classical\nlimit/H20849/H6036→0/H20850is equivalent to the high-temperature limit /H20849T\n→/H11009/H20850. Thus, in actual experimental situations, the above cor-\nrelation relations, Eq. /H2084932/H20850, will be satisfied at high tempera-\nture. In this respect, we find that most experimental situa-tions belong to the high-temperature regime. See AppendixD for the estimation of the “threshold” temperature, abovewhich Eq. /H2084932/H20850is applicable. In Appendix D, the correlations\nin the high temperatures are derived more rigorously.\nFinally we comment briefly on the low-temperature quan-\ntum regime. In this regime, one cannot use the equipartitiontheorem since the system is not composed of independentharmonic oscillators, that is, /H20851/H9004x\ni,/H9004pj/H20852=i/H6036/H20849/H9254ij+/H9253j/H20850. Note\nthat the commutator contains an additional term i/H6036/H9253j. Here,\nthe additional term i/H6036/H9253jcomes from the commutator /H20851−Q,\n−/H9253jP/H20852. Then, Eq. /H2084932/H20850, which is assumed in other papers,9–13\nis not guaranteed any more.\nIV . CONCLUSION\nIn this paper, we examine the effect of finite current on\nthermal fluctuation of current-induced DW motion by con-structing generalized Caldeira-Leggett-type Hamiltonian ofthe DW dynamics, which describes not only energy-conserving dynamics processes but also the Gilbert dampingand STT. Unlike the classical damping worked out by Cal-deira and Leggett,\n32the momentum coupling is indispensable\nto describe the Gilbert damping. This is also related to theTHERMAL FLUCTUATION FIELD FOR CURRENT-INDUCED … PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-7rotational symmetry of spin-interaction nature. It is demon-\nstrated that the derived Caldeira-Leggett-type quantum-mechanical Hamiltonian reproduces the well-known DWequations of motion.\nOur Hamiltonian also illustrates that the nonadiabatic\nSTT is closely related with the coupling of the spin current tothe environment. Thus, the environmental degrees of free-dom are responsible for both the Gilbert damping /H20849\n/H9251/H20850and the\nnonadiabatic STT /H20849/H9252/H20850. By this process, the ratio of /H9252and/H9251\nwas derived to be the ratio of current-DW coupling and\ncurrent-environment coupling. The nonadiabatic term isnothing but the result of the direct coupling between thecurrent and environment in our theory.\nBy using the Calderia-Leggett-type Hamiltonian, which\ndescribes the time evolution of the system, we obtained theexpression of stochastic forces caused by thermal noise inthe presence of the finite current. By calculating the equilib-rium thermal fluctuation at high temperature, we verify thatwhen j\npis sufficiently smaller than the intrinsic critical den-\nsity, jpdoes not modify the correlation relations of thermal\nnoise unless the temperature is extremely low. The upperbound of the critical temperature, below which the aboveconclusion does not apply, is obtained by reexamining thesystem with Feynman path integral. The bound is muchlower than the temperature in most experimental situations.\nLastly we remark that the Joule heating\n51is an important\nfactor that affects the thermal fluctuation field since it raisesthe temperature of the nanowire. The degree of the tempera-ture rise depends on the thermal conductivities and heat ca-pacities of not only the nanowire but also its surroundingmaterials such as substrate layer materials of the nanowire.Such factors are not taken into account in this paper. Simul-taneous account of the Joule heating dynamics and the ther-mal fluctuation field /H20849in the presence of current /H20850goes beyond\nthe scope of the paper and may be a subject of future re-search.\nACKNOWLEDGMENTS\nWe acknowledge critical comment by M. Stiles, who\npointed out the importance of the momentum coupling andinformed us of Ref. 29. This work was financially supported\nby the NRF /H20849Grants No. 2007-0055184, No. 2009-0084542,\nand No. 2010-0014109 /H20850and BK21. K.W.K. acknowledges\nthe financial support by the TJ Park.\nAPPENDIX A: EFFECTIVE HAMILTONIAN OF THE DW\nMOTION FROM 1D s-dMODEL (Ref. 52)\nThe starting point is 1D s-dmodel,\nHs-d=−J/H20858\niS/H6023i·S/H6023i+1−A/H20858\ni/H20849S/H6023i·zˆ/H208502+K/H20858\ni/H20849S/H6023i·yˆ/H208502+HcS,\n/H20849A1/H20850\nas mentioned in Sec. II A.\nIn order to consider the DW dynamics, one first introduce\nthe classical DW profile initially given by\n/H20855S/H6023i·xˆ/H20856=Ssin/H9258/H20849zi/H20850, /H20849A2a/H20850/H20855S/H6023i·yˆ/H20856=0 , /H20849A2b/H20850\n/H20855S/H6023i·zˆ/H20856=Scos/H9258/H20849zi/H20850, /H20849A2c/H20850\nwhere ziis the position of the ith localized spin, and /H9258/H20849z/H20850\n=2 cot−1e−/H208812A/Ja2/H20849z−q/H20850. Here qis the classical position of the\nDW. Small quantum fluctuations of spins on top of the clas-sical DW profile can be described by the Holstein-Primakoffboson operator b\ni, to describe magnon excitations. Kim et\nal.43found eigenmodes of these quantum fluctuations in the\npresence of the classical DW background, which amount toquantum mechanical version of the classical vibration eigen-modes in the presence of the DW background reported longtime ago by Winter.\n53The corresponding eigenstates of this\nHamiltonian are composed of spin-wave states with the finite\neigenenergy Ek=/H20881/H20849JSa2k2+2AS/H20850/H20849JSa2k2+2AS+2KS/H20850\n/H20849/H113502S/H20881A/H20849A+K/H20850/H20850and so-called bound magnon states with\nzero energy Ew=0. Here, kis the momentum of spin wave\nstates and ais the lattice spacing between two neighboring\nspins. Let akandbwdenote proper linear combinations of bi\nandbi†, which represent the boson annihilation operators of\nfinite-energy spin-wave states and zero-energy bound mag-non states, respectively. In terms of these operators, Eq. /H208498/H20850\nreduces to\nH\ns-d=P2\n2M+/H20858\nkEkak†ak+HcS, /H20849A3/H20850\nwhere higher-order processes describing magnon-magnon in-\nteractions are ignored. Here Mis the so-called Döring\nmass,44defined as M=/H60362\nK/H208812A\nJa4, and Pis defined as\n−i/H6036/H208492AS2\nJa4/H208501/4/H20849bw†−bw/H20850. According to Ref. 43,Pis a translation\ngenerator of the DW position, that is, exp /H20849iPq 0//H6036/H20850shifts the\nDW position by q0. Thus Pcan be interpreted as a canonical\nmomentum of the DW translational motion. The first term inEq./H20849A3/H20850, which amounts to the kinetic energy of the DW\ntranslational motion, implies that Mis the DW mass. We\nidentify this Mwith the undetermined constant Min Eq. /H208496/H20850.\nAccording to Ref. 43,Pis also proportional to the degree of\nthe DW tilting, that is, /H20849b\nw†−bw/H20850/H11008Siy.\nIn the adiabatic limit, that is, when the DW width /H9261is\nsufficiently large in view of the electron dynamics, the re-maining term H\ncScan be represented in a simple way in\nterms of the bound magnon operators and the adiabatic ve-locity of the DW,\n20,43\nHcS=vsP. /H20849A4/H20850\nThen the effective s-dHamiltonian of the DW motion be-\ncomes\nHs-d=P2\n2M+vsP+/H20858\nkEkak†ak. /H20849A5/H20850\nNote that the bound magnon part and the spin-wave part are\ncompletely decoupled in Eq. /H20849A5/H20850since Pcontains only the\nbound magnon operators, which commute with the spin-wave operators.\nThe DW position operator should satisfy the following\ntwo properties: geometrical relation /H20855Q/H20856−q=a\n2S/H20858i/H20855S/H6023i·zˆ/H20856andKYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-8canonical relation /H20851Q,P/H20852=i/H6036. Then, one can show that Q\n=q−/H20849Ja4\n32AS2/H208501/4/H20849bw†+bw/H20850satisfies these two properties. Note\nthatQis expressed in terms of the bound magnon operators.\nThen as far as the Heisenberg equations of motion for Qand\nPare concerned, the last term in Eq. /H20849A5/H20850does not play any\nrole. This term will be ignored from now on. Thus, the ef-fective Hamiltonian for the DW motion is reduced to\nH\n0=P2\n2M+vsP /H20849A6/H20850\nso we got Eq. /H2084910/H20850, the effective Hamiltonian of the DW\nmotion.\nAPPENDIX B: SOLUTION FOR A GENERAL QUADRATIC\nDAMPING\nThis section provides the solution of the equation of mo-\ntion for a general quadratic damping. This is applicable notonly for the generalized Caldeira-Leggett description in thispaper but also for any damping type which quadraticallyinteracts with the DW.\nIn general, let us consider a general quadratic damping\nHamiltonian,\nH=P\n2\n2M+vsP+/H20858\ni/H9273iTAi/H9273i, /H20849B1/H20850\nwhere /H9273i=/H20849QPx ipi/H20850T, and Aii sa4/H110034 Hermitian matrix.\nNow, one straightforwardly gets the corresponding coupledequations,\ndQ\ndt=P\nM+vs+/H20858\ni/H20849B21iQ+B22iP+B23ixi+B24ipi/H20850,\n/H20849B2a/H20850\ndP\ndt=−/H20858\ni/H20849B11iQ+B12iP+B13ixi+B14ipi/H20850,/H20849B2b/H20850\ndxi\ndt=B41iQ+B42iP+B43ixi+B44ipi, /H20849B2c/H20850\ndpi\ndt=−/H20849B31iQ+B32iP+B33ixi+B34ipi/H20850./H20849B2d/H20850\nHere, Bii sa4 /H110034 real symmetric matrix defined as Bi\n=2 Re /H20851Ai/H20852, and Bjkiis the element of Biinjth row and kth\ncolumn.\nWith the Laplace transform of the expectation values of\neach operator, for example,\nQ˜/H20849/H9261/H20850/H11013L/H20851Q/H20849t/H20850/H20852/H20849/H9261/H20850=/H20885\n0/H11009\n/H20855Q/H20849t/H20850/H20856e−/H9261tdt, /H20849B3/H20850\nthe set of coupled equations transforms as\n/H9261Q˜−/H20855Q/H208490/H20850/H20856=P˜\nM+vs\n/H9261+/H20858\ni/H20849B21iQ˜+B22iP˜+B23ix˜i+B24ip˜i/H20850,\n/H20849B4a/H20850/H9261P˜−/H20855P/H208490/H20850/H20856=−/H20858\ni/H20849B11iQ˜+B12iP˜+B13ix˜i+B14ip˜i/H20850,\n/H20849B4b/H20850\n/H9261x˜i−/H20855xi/H208490/H20850/H20856=B41iQ˜+B42iP˜+B43ix˜i+B44ip˜i,/H20849B4c/H20850\n/H9261p˜i−/H20855pi/H208490/H20850/H20856=−/H20849B31iQ˜+B32iP˜+B33ix˜i+B34ip˜i/H20850.\n/H20849B4d/H20850\nRewriting these in matrix forms, the equations become sim-\npler as\n/H9261/H20873Q˜\nP˜/H20874−/H20898/H20855Q/H208490/H20850/H20856+vs\n/H9261\n/H20855P/H208490/H20850/H20856/H20899=/H20902/H2089801\nM\n00/H20899+/H20858\ni/H20873B21iB22i\n−B11i−B12i/H20874/H20903\n/H11003/H20873Q˜\nP˜/H20874+/H20858\ni/H20873B23iB24i\n−B13i−B14i/H20874\n/H11003/H20873x˜i\np˜i/H20874, /H20849B5a/H20850\n/H9261/H20873x˜i\np˜i/H20874−/H20873/H20855xi/H208490/H20850/H20856\n/H20855pi/H208490/H20850/H20856/H20874=/H20873B41iB42i\n−B31i−B32i/H20874/H20873Q˜\nP˜/H20874\n+/H20873B43iB44i\n−B33i−B34i/H20874/H20873x˜i\np˜i/H20874.\n/H20849B5b/H20850\nFrom Eq. /H20849B5b/H20850, one can calculate /H20849x˜ip˜i/H20850Tin terms of Q˜and\nP˜,\n/H20873x˜i\np˜i/H20874=/H20873/H9261−B43i−B44i\nB33i/H9261+B34i/H20874−1/H20873B41iB42i\n−B31i−B32i/H20874/H20873Q˜\nP˜/H20874\n+/H20873/H9261−B43i−B44i\nB33i/H9261+B34i/H20874−1/H20873/H20855xi/H208490/H20850/H20856\n/H20855pi/H208490/H20850/H20856/H20874. /H20849B6/H20850\nFrom Eqs. /H20849B5a/H20850and/H20849B6/H20850, one finally gets the equation of\n/H20849Q˜P˜/H20850T,THERMAL FLUCTUATION FIELD FOR CURRENT-INDUCED … PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-9/H20900/H20898/H9261−1\nM\n0/H9261/H20899−/H20858\ni/H20877/H20873B21iB22i\n−B11i−B12i/H20874+/H20873B23iB24i\n−B13i−B14i/H20874/H20873/H9261−B43i−B44i\nB33i/H9261+B34i/H20874−1/H20873B41iB42i\n−B31i−B32i/H20874/H20878/H20901/H20873Q˜\nP˜/H20874\n=/H20898/H20855Q/H208490/H20850/H20856+vs\n/H9261\n/H20855P/H208490/H20850/H20856/H20899+/H20858\ni/H20873B23iB24i\n−B13i−B14i/H20874/H20873/H9261−B43i−B44i\nB33i/H9261+B34i/H20874−1/H20873/H20855xi/H208490/H20850/H20856\n/H20855pi/H208490/H20850/H20856/H20874. /H20849B7/H20850\nInverting the matrix in front of /H20849Q˜P˜/H20850T, one can get the solu-\ntion of /H20849Q˜P˜/H20850T. Then, finally, the solution /H20849/H20855Q/H20856/H20855P/H20856/H20850Tis ob-\ntained by the inverse Laplace transform of /H20849Q˜P˜/H20850T,\n/H20873/H20855Q/H20849t/H20850/H20856\n/H20855P/H20849t/H20850/H20856/H20874=L−1/H20875/H20873Q˜/H20849/H9261/H20850\nP˜/H20849/H9261/H20850/H20874/H20876. /H20849B8/H20850\nAPPENDIX C: SOLUTION OF EQ. ( 24)\nIn the special case that current is applied at t=0,/H9008/H20849t/H20850in\nEq./H2084925/H20850becomes Heaviside step function. This is the case\nwe are interested in. In a real DW system, the DW velocityjumps from 0 to a finite value at the moment that the spincurrent starts to be applied. This jumping comes from thediscontinuity in Eq. /H2084925/H20850which makes the Hamiltonian dis-\ncontinuous. Right before the current is applied, the DW re-mains on the stable /H20849or equilibrium /H20850state described by Eq.\n/H2084922/H20850.\nSuppose that Eq. /H2084920a/H20850also holds for /H9261=0. Then, Eq. /H2084924/H20850\ntransforms as /H20849up to constant /H20850\nH\ntot=P2\n2M+vs/H20849t/H20850P+/H20858\ni/H208751\n2mi/H20849pi−/H9253iP+miv/H20849t/H20850/H208502/H20876\n+/H20858\ni1\n2mi/H9275i2/H20849xi−Q/H208502. /H20849C1/H20850\nPerforming the canonical transform pi→pi−miv/H20849t/H20850, one can\ntransform this Hamiltonian in the form of Eq. /H20849B1/H20850,\nHtot=P2\n2M+vs/H20849t/H20850P+/H20858\ni/H208751\n2mi/H20849pi−/H9253iP/H208502+1\n2mi/H9275i2/H20849xi−Q/H208502/H20876.\nHere, one of the constraints Eq. /H2084920a/H20850is generalized to hold\neven for /H9261=0, so that /H20858i/H9253i=0. Note that the discontinuity due\ntov/H20849t/H20850is absorbed in the new pi. Thus, Eq. /H2084922b/H20850should be\nwritten as\n/H20855pi/H208490+/H20850/H20856=/H20855pi/H208490−/H20850/H20856+miv=/H9253i/H20855P/H208490/H20850/H20856+miv./H20849C2/H20850\nThe initial condition of xiis the same as Eq. /H2084922a/H20850. Now,\nusing these initial conditions and Eqs. /H20849B7/H20850and/H20849B8/H20850under\nthe constraints in Eq. /H2084920/H20850, one gets the solution of this sys-\ntem as Eq. /H2084926/H20850.APPENDIX D: CORRELATIONS OF STOCHASTIC\nFORCES AT HIGH TEMPERATURE\nThis section provides the quantum derivation of correla-\ntion relations of stochastic forces at high temperature. Theclassical correlation relations in Eq. /H2084932/H20850are valid quantum\nmechanically at high temperature. Since Eq. /H2084931/H20850implies Eq.\n/H2084932/H20850, it suffices to show Eq. /H2084931/H20850in this section. The basic\nstrategy is studying statistical properties of the HamiltonianEq./H2084919/H20850/H20849under quadratic potential bQ\n2/H2085054by the Feynman\npath integral along the imaginary-time axis. The Feynmanpath integral of a system described by a quadratic Lagrang-ian is proportional to the exponential of the action valueevaluated at the classical solution. Hence, the key point ofthe procedure is to get the classical solution with imaginarytime.\n1. General relations\na. Classical action under high-temperature limit\nDefine a column vector /H9273=/H20849Qx1x2¯/H20850T. Let the Euclidean\nLagrangian of the system be LE=1\n2/H9273˙TA/H9273˙+1\n2/H9273B/H9273, where A\nandBare symmetric matrices. /H20849The symbols “ A” and “ B” are\nnot the same as those in Appendix B. /H20850Explicitly, L\n=1\n2/H20858nmx˙nAnmx˙m+1\n2/H20858nmxnBnmxm. Here x0/H11013Q./H11509LE\n/H11509x˙n=/H20858mAnmx˙m\n=A/H9273˙and/H11509LE\n/H11509xn=/H20858mBnmxm=B/H9273lead to the classical equation of\nmotion,\nA/H9273¨=B/H9273. /H20849D1/H20850\nThe classical action value Sc/H20849evaluated at the classical path /H20850\nis then, Sc=/H208480/H9270LEdt=1\n2/H208480/H9270/H20849/H9273˙TA/H9273˙+/H9273B/H9273/H20850dt=1\n2/H9273TA/H9273˙/H208410/H9270\n+/H208480/H9270/H20849−/H9273TA/H9273¨+/H9273B/H9273/H20850dt=1\n2/H9273TA/H9273˙/H208410/H9270. Here, /H9270=/H6036/kBT. Now, the\nonly thing one needs is to find /H9273˙at boundary points.\nIn the case of Eq. /H2084919/H20850,Ais invertible. Hence, the equa-\ntion becomes /H9273¨=A−1B/H9273. Suppose that A−1Bis diagonaliz-\nable, that is A−1B=C−1DC. Here Dnm=/H9261n/H9254nmis diagonal ma-\ntrix and /H9261nisnth eigenvalue of A−1B. Define a new vector\n/H9264=C/H9273. Finally, we get the equation,\n/H9264¨=/H20898/H926100¯\n0/H92611¯\n]]/GS/H20899/H9264. /H20849D2/H20850\nImposing the boundary condition /H9273/H208490/H20850=/H9273i,/H9273/H20849/H9270/H20850=/H9273fand de-\nfining the corresponding /H9264i=C/H9273i,/H9264f=C/H9264f, then one gets the\nsolution of /H9264and its derivative straightforwardly,KYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-10/H9264n=/H9264fn+/H9264in\n2cosh/H20881/H9261n/H20873t−/H9270\n2/H20874\ncosh/H20881/H9261n/H9270\n2+/H9264fn−/H9264in\n2sinh/H20881/H9261n/H20873t−/H9270\n2/H20874\nsinh/H20881/H9261n/H9270\n2,\n/H20849D3/H20850\n/H9264˙n=/H20881/H9261n/H20900/H9264fn+/H9264in\n2sinh/H20881/H9261n/H20873t−/H9270\n2/H20874\ncosh/H20881/H9261n/H9270\n2\n+/H9264fn−/H9264in\n2cosh/H20881/H9261n/H20873t−/H9270\n2/H20874\nsinh/H20881/H9261n/H9270\n2/H20901. /H20849D4/H20850\nNow, /H9264˙at boundary points are obtained as\n/H9264˙n/H208490/H20850=/H20881/H9261n/H20873−/H9264fn+/H9264in\n2tanh/H20881/H9261n/H9270\n2+/H9264fn−/H9264in\n2coth/H20881/H9261n/H9270\n2/H20874,\n/H20849D5/H20850\n/H9264˙n/H20849/H9270/H20850=/H20881/H9261n/H20873/H9264fn+/H9264in\n2tanh/H20881/H9261n/H9270\n2+/H9264fn−/H9264in\n2coth/H20881/H9261n/H9270\n2/H20874.\n/H20849D6/H20850\nIf/H20881/H20841/H9261n/H20841/H9270\n2=/H20881/H20841/H9261n/H20841/H6036\n2kBT/H112701, tanh/H20881/H9261n/H9270\n2/H11015/H20881/H9261n/H9270\n2. Then,\n/H9264˙n/H208490/H20850/H11015−/H9264fn+/H9264in\n2/H9261n/H9270\n2+/H9264fn−/H9264in\n/H9270, /H20849D7/H20850\n/H9264˙n/H20849/H9270/H20850/H11015/H9264fn+/H9264in\n2/H9261n/H9270\n2+/H9264fn−/H9264in\n/H9270. /H20849D8/H20850\nIn matrix form,\n/H9264˙/H208490/H20850/H11015−D/H9264f+/H9264i\n2/H9270\n2+/H9264f−/H9264i\n/H9270=−DC/H9273f+/H9273i\n2/H9270\n2+C/H9273f−/H9273i\n/H9270,\n/H20849D9/H20850\n/H9264˙/H20849/H9270/H20850/H11015D/H9264f+/H9264i\n2/H9270\n2+/H9264f−/H9264i\n/H9270=DC/H9273f+/H9273i\n2/H9270\n2+C/H9273f−/H9273i\n/H9270.\n/H20849D10/H20850\nUsing A−1B=C−1DC, it leads to\n/H9273˙/H208490/H20850/H11015−A−1B/H9273f+/H9273i\n2/H9270\n2+/H9273f−/H9273i\n/H9270, /H20849D11/H20850\n/H9273˙/H20849/H9270/H20850/H11015A−1B/H9273f+/H9273i\n2/H9270\n2+/H9273f−/H9273i\n/H9270. /H20849D12/H20850\nFinally one can obtain the classical action,Sc=1\n2/H9273TA/H9273˙/H208410/H9270=/H20873/H9273f+/H9273i\n2/H20874T\nB/H20873/H9273f+/H9273i\n2/H20874/H9270\n2\n+/H20873/H9273f−/H9273i\n2/H20874T\nA/H20873/H9273f−/H9273i\n2/H208742\n/H9270. /H20849D13/H20850\nThis is valid even if some eigenvalues are zero. /H20849By taking\nlimit of /H9261i→0, cosh and sinh becomes constant and linear,\nrespectively. /H20850\nb. Propagator and its derivatives\nThe propagator is given by the Feynman path integral,\nK/H20849/H9273f,/H9273i;/H9270/H20850=/H20855/H9273f/H20841e−H/kBT/H20841/H9273i/H20856=/H20848D/H9273e−/H20848LEdt//H6036, where D/H9273=/H20863iDxi.\nFor quadratic Lagrangian, it is well known that/H20848D\n/H9273e−/H20848LEdt/H6036=F/H20849/H9270/H20850e−Sc//H6036. Here F/H20849/H9270/H20850is a smooth function de-\npendent on /H9270only.\nNow we aim to calculate K/H20849/H9273i+/H9254/H9273,/H9273i;/H9270/H20850. It is easy to\nobtain the corresponding classical action by replacing /H9273f\n=/H9273i+/H9254/H9273in Eq. /H20849D13/H20850,\nSc/H20849/H9273i+/H9254/H9273,/H9273i;/H9270/H20850=/H9270\n2/H9273iTB/H9273i+/H9270\n2/H9273iTB/H9254/H9273+/H9270\n8/H9254/H9273TB/H9254/H9273\n+1\n2/H9270/H9254/H9273TA/H9254/H9273. /H20849D14/H20850\nThen, K/H20849/H9273i+/H9254/H9273,/H9273i;/H9270/H20850is/H20849up to second order of /H9254/H9273/H20850,\nK/H20849/H9273i+/H9254/H9273,/H9273i;/H9270/H20850=F/H20849/H9270/H20850e−Sc//H6036=F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i\n/H11003/H208751−1\n/H6036/H20873/H9270\n2/H9273iTB/H9254/H9273+/H9270\n8/H9254/H9273TB/H9254/H9273\n+1\n2/H9270/H9254/H9273TA/H9254/H9273/H20874+1\n2/H60362/H20873/H9270\n2/H9273iTB/H9254/H9273/H208742/H20876.\nZeroth order: F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i.\nFirst order:−/H9270\n2/H6036F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H9273iTB/H9254/H9273\n=−/H9270\n2/H6036F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H20858nmxinBnm/H9254xm.\nSecond order: F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H20877−1\n/H6036/H20873/H9270\n8/H9254/H9273TB/H9254/H9273\n+1\n2/H9270/H9254/H9273TA/H9254/H9273/H20874+1\n2/H60362/H20873/H9270\n2/H9273iTB/H9254/H9273/H208742/H20878\n=F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H20877−1\n2/H6036/H20858\nnm/H9254xn/H20873/H9270\n4Bnm+1\n/H9270Anm/H20874/H9254xm\n+/H92702\n8/H60362/H20873/H20858\nklmnxikBkn/H9254xnxilBlm/H9254xm/H20874/H20878. /H20849D15/H20850\nBy the relation, K/H20849/H9273i+/H9254/H9273,/H9273i;/H9270/H20850=K/H20849/H9273i,/H9273i;/H9270/H20850+/H20858m/H11509K\n/H11509xfm/H9254xm\n+/H20858nm1\n2/H115092K\n/H11509xfn/H11509xfm/H9254xn/H9254xm+O/H20849/H9254/H92733/H20850,\nK/H20849/H9273i,/H9273i;/H9270/H20850=F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i, /H20849D16/H20850THERMAL FLUCTUATION FIELD FOR CURRENT-INDUCED … PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-11/H20879/H11509K\n/H11509xfm/H20879\n/H9273i=/H9273f=−/H9270\n2/H6036F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H20858\nnBnmxin,/H20849D17/H20850\n/H20879/H115092K\n/H11509xfn/H11509xfm/H20879\n/H9273i=/H9273f=F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H20877−1\n/H6036/H20873/H9270\n4Bnm+1\n/H9270Anm/H20874\n+/H92702\n4/H60362/H20873/H20858\nklBknxikBlmxil/H20874/H20878\n=F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H20877−1\n/H6036/H20873/H9270\n4Bnm+1\n/H9270Anm/H20874\n+/H92702\n4/H60362/H20873/H20858\nkBknxik/H20874/H20873/H20858\nkBkmxik/H20874/H20878./H20849D18/H20850\nc. Correlations\nStatistical average of an operator Ais given byTr/H20849Ae−H/kBT/H20850\nTr/H20849e−H/kBT/H20850.\nWhat we want to find are the averages of /H9004xn/H9004xm,/H9004pn/H9004pm,\nand/H20853/H9004xn,/H9004pm/H20854for/H9004xn/H11013xn−Qand/H9004pn/H11013pn−/H9253nP,\nTr/H20849/H9004xn/H9004xme−H/kBT/H20850=/H20885d/H9273i/H20855/H9273i/H20841/H9004xn/H9004xme−H/kBT/H20841/H9273i/H20856\n=/H20885d/H9273i/H20849xin−Qi/H20850/H20849xim−Qi/H20850/H20855/H9273i/H20841e−H/kBT/H20841/H9273i/H20856\n=/H20885d/H9273i/H20849xin−Qi/H20850/H20849xim−Qi/H20850K/H20849/H9273i,/H9273i;/H9270/H20850,\n/H20849D19/H20850\nTr/H20849/H9004pn/H9004pme−H/kBT/H20850=/H20885d/H9273i/H20855/H9273i/H20841/H9004pn/H9004pme−H/kBT/H20841/H9273i/H20856\n=−/H60362/H20885/H20879d/H9273i/H20873/H11509\n/H11509xfn−/H9253n/H11509\n/H11509Qf/H20874\n/H11003/H20873/H11509\n/H11509xfm−/H9253m/H11509\n/H11509Qf/H20874K/H20849/H9273f,/H9273i;/H9270/H20850/H20879\n/H9273i=/H9273f,\n/H20849D20/H20850\nTr/H20849/H9004xn/H9004pme−H/kBT/H20850=/H20885d/H9273i/H20855/H9273i/H20841/H9004xn/H9004pme−H/kBT/H20841/H9273i/H20856\n=−i/H6036/H20885/H20879d/H9273i/H20849xin−Qi/H20850/H20873/H11509\n/H11509xfm\n−/H9253m/H11509\n/H11509Qf/H20874K/H20849/H9273f,/H9273i;/H9270/H20850/H20879\n/H9273i=/H9273f,\n/H20849D21/H20850\nTr/H20849e−H/kBT/H20850=/H20885d/H9273i/H20855/H9273i/H20841e−H/kBT/H20841/H9273i/H20856=/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850,\n/H20849D22/H20850\nwhere d/H9273i=/H20863ndxin.2. Correlations under quadratic potential\nUnder potential bQ2, the matrices AandBcorresponding\nthe Hamiltonian Eq. /H2084919/H20850are\nA=/H20898MM /H92531 M/H92532 ¯\nM/H92531M/H925312+m1M/H92531/H92532¯\nM/H92532M/H92532/H92531M/H925322+m2¯\n]] ] /GS /H20899,/H20849D23/H20850\nB=/H20898b+/H20858\nnmn/H9275n2\n−m1/H927512−m2/H927522¯\n−m1/H927512m1/H927512 0 ¯\n−m2/H9275220 m2/H927522¯\n]] ] /GS/H20899./H20849D24/H20850\nThen, e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273iis written as e−/H20849/H9270/2/H6036/H20850/H20851/H20858nmn/H9275n2/H20849Qi−xin/H208502+bQi2/H20852.\na. x-x correlations\nSince K/H20849/H9273,/H9273;/H9270/H20850is an even function of /H20849xn−Qi/H20850,i ti s\ntrivial that Tr /H20849/H9004xn/H9004xme−H/kBT/H20850=0 unless n=m.\nFor n=m,T r /H20849/H9004xn2e−H/kBT/H20850=/H20848d/H9273i/H20849xin−Qi/H208502K/H20849/H9273i,/H9273i;/H9270/H20850.\nThus,\nTr/H20849/H9004xn2e−H/kBT/H20850\nTr/H20849e−H/kBT/H20850=/H20885dxin/H20849xin−Qi/H208502e−/H20849/H9270/2/H6036/H20850mnwn2/H20849Qi−xin/H208502\n/H20885dxine−/H20849/H9270/2/H6036/H20850mnwn2/H20849Qi−xin/H208502\n=/H6036\n/H9270mnwn2=kBT\nmnwn2. /H20849D25/H20850\nSo, finally one gets /H20855/H9004xn/H9004xm/H20856=kBT\nmnwn2/H9254nm.\nb. x-p correlations\nExplicitly rewriting the derivative of K,\n/H20879/H11509K\n/H11509xfm/H20879\n/H9273i=/H9273f=−/H9270\n2/H6036K/H20849/H9273i,/H9273i;/H9270/H20850mm/H9275m2/H20849xim−Qi/H20850for/H20849m\n/HS110050/H20850, /H20849D26/H20850\n/H20879/H11509K\n/H11509Qf/H20879\n/H9273i=/H9273f=−/H9270\n2/H6036K/H20849/H9273i,/H9273i;/H9270/H20850/H20877bQi2+/H20858\nnmn/H9275n2/H20849Qi−xin/H20850/H20878.\n/H20849D27/H20850\nUsing the above relations,KYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-12Tr/H20849/H9004xn/H9004pme−H/kBT/H20850=−i/H6036/H20885/H20879d/H9273i/H20849xin−Qi/H20850/H20873/H11509\n/H11509xfm−/H9253m/H11509\n/H11509Qf/H20874K/H20849/H9273f,/H9273i;/H9270/H20850/H20879\n/H9273i=/H9273f\n=−i/H9270\n2/H20885d/H9273i/H20849xin−Qi/H20850/H20877/H9253mbQi2+/H9253m/H20858\nlml/H9275l2/H20849Qi−xil/H20850+mm/H9275m2/H20849Qi−xim/H20850/H20878K/H20849/H9273i,/H9273i;/H9270/H20850\n=−i/H9270\n2/H20885d/H9273i/H20849xin−Qi/H20850/H20877/H9253m/H20858\nlml/H9275l2/H20849Qi−xil/H20850+mm/H9275m2/H20849Qi−xim/H20850/H20878K/H20849/H9273i,/H9273i;/H9270/H20850\n=i/H9270\n2/H9253m/H20858\nlml/H9275l2Tr/H20849/H20849/H9004xin/H9004xile−H/kBT/H20850/H20850+mm/H9275m2Tr/H20849/H9004xin/H9004xime−H/kBT/H20850. /H20849D28/H20850\nIn the third line, it is used that /H20848dxin/H20849xin−Qi/H20850\n/H11003/H20851even function of /H20849xin−Qi/H20850/H20852=0.\nOne can now write the x-pcorrelations in terms of x-x\ncorrelations.\n/H20855/H9004xn/H9004pm/H20856=i/H9270\n2/H20873/H9253m/H20858\nlml/H9275l2/H20855/H9004xin/H9004xil/H20856+mm/H9275m2/H20855/H9004xin/H9004xim/H20856/H20874\n=i/H9270kBT\n2/H20873/H9253m/H20858\nl/H9254nl+/H9254nm/H20874=i/H6036\n2/H20849/H9253m+/H9254nm/H20850,/H20849D29/H20850\nwhich is purely imaginary. Thus, /H20855/H20853/H9004xn,/H9004pm/H20854/H20856=/H20855/H9004xn/H9004pm/H20856\n+/H20855/H9004xn/H9004pm/H20856/H11569=0.\nc. p-p correlations\nIt is convenient to calculate /H20848d/H9273i/H115092K\n/H11509xfn/H11509xfm/H20841/H9273i=/H9273f. The trickiest\npart is /H20848d/H9273i/H20858kBknxik/H20858kBkmxikK/H20849/H9273i,/H9273i;/H9270/H20850,\nn/HS110050,m/HS110050:/H20858\nkBknxik/H20858\nkBkmxik=mn/H9275n2/H20849xin−Qi/H20850mm/H9275m2/H20849xim\n−Qi/H20850,\nn=0 , m/HS110050:/H20858\nkBknxik/H20858\nkBkmxik=/H20873/H20858\nkmk/H9275k2/H20849Qi−xik/H20850\n+bQi/H20874mm/H9275m2/H20849xim−Qi/H20850,\nn=0 , m=0 :/H20858\nkBknxik/H20858\nkBkmxik=/H20873/H20858\nkmk/H9275k2/H20849Qi−xik/H20850+bQi/H20874\n/H11003/H20873/H20858\nkmk/H9275k2/H20849Qi−xik/H20850+bQi/H20874.\nAfter integrating over xik, odd terms with respect to /H20849xik\n−Q/H20850vanish. Taking only even terms, one obtains\nn/HS110050,m/HS110050→mn2/H9275n4/H20849xin−Qi/H208502/H9254nm=mm/H9275m2/H20849xin−Qi/H208502Bnm,\nn=0 , m/HS110050→−mm2/H9275m4/H20849Qi−xim/H208502=mm/H9275m2/H20849xim−Qi/H208502Bnm,n=0 , m=0→/H20858\nkmk2/H9275k4/H20849Qi−xik/H208502+b2Qi2.\nIntegrating out and using the identity /H20848duu2e−u2/2/H9251\n=/H9251/H20848due−u2/2/H9251for/H9251/H110220, one finds\nn/HS110050,m/HS110050:/H20885d/H9273imm/H9275m2/H20849xin−Qi/H208502BnmK/H20849/H9273i,/H9273i;/H9270/H20850\n=/H6036\n/H9270Bnm/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850,\nn=0 , m/HS110050:/H20885d/H9273imm/H9275m2/H20849xim−Qi/H208502BnmK/H20849/H9273i,/H9273i;/H9270/H20850\n=/H6036\n/H9270Bnm/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850,\nn=0 , m=0 :/H20885d/H9273i/H20875/H20858\nkmk2/H9275k4/H20849Qi−xik/H208502+b2Qi2/H20876K/H20849/H9273i,/H9273i;/H9270/H20850\n=/H6036\n/H9270/H20873/H20858\nkmk/H9275k2+b/H20874/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850\n=/H6036\n/H9270Bnm/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850.\nThe result is/H6036\n/H9270Bnm/H20848d/H9273iKindependent of the cases. Finally,\none can obtain\n/H20885/H20879d/H9273i/H115092K\n/H11509xfn/H11509xfm/H20879\n/H9273i=/H9273f=/H20877−1\n/H6036/H20873/H9270\n4Bnm+1\n/H9270Anm/H20874\n+/H9270\n4/H6036Bnm/H20878/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850\n=−Anm\n/H9270/H6036/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850,/H20849D30/H20850\nor equivalently,THERMAL FLUCTUATION FIELD FOR CURRENT-INDUCED … PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-13/H20883/H115092\n/H11509xn/H11509xm/H20884=−kBT\n/H60362Anm=−kBT\n/H60362/H20849M/H9253n/H9253m+mn/H9254nm/H20850,\n/H20849D31/H20850\nwhere /H92530=1,m0=0. Finally, p-pcorrelation is obtained\n/H20855/H9004pn/H9004pm/H20856=−/H60362/H20883/H20873/H11509\n/H11509xn−/H9253n/H11509\n/H11509Q/H20874/H20873/H11509\n/H11509xm−/H9253m/H11509\n/H11509Q/H20874/H20884\n=kBT/H20849Anm−/H9253mAn0−/H9253nAm0+/H9253n/H9253mA00/H20850\n=kBT/H20849M/H9253n/H9253m+mn/H9254mn−M/H9253m/H9253n−M/H9253n/H9253m\n+M/H9253n/H9253m/H20850=mnkBT/H9254mn. /H20849D32/H20850\nThe above three results of x-x,x-p, and p-pcorrelations\nare the same as Eq. /H2084931/H20850.3. Sufficient condition for “high” temperature\nWe assumed the high-temperature approximationkBT\n/H6036\n/H11271/H20881/H20841/H9261n/H20841\n2. Indeed, the temperature should satisfykBT\n/H6036/H11271/H20881/H9261M\n2,\nwhere /H9261Mis the absolute value of maximum eigenvalue of\nA−1B. It is known that, for eigenvalue /H9261of a matrix A,/H20841/H9261/H20841is\nnot greater than maximum column /H20849or row /H20850sum,55\n/H20841/H9261/H20841/H11349max\nj/H20858\ni/H20841aij/H20841/H11013/H20648A/H20648. /H20849D33/H20850\nAccording to the above definition of /H20648·/H20648, It is not hard to see\nthat/H20648AB/H20648/H11349/H20648A/H20648/H20648B/H20648.\nThe above argument says\n/H9261M/H11349/H20648A−1B/H20648/H11349/H20648A−1/H20648/H20648B/H20648. /H20849D34/H20850\nIt is not hard to obtain A−1with the following LDU factorization.\n/H20898MM /H92531 M/H92532 ¯\nM/H92531M/H925312+m1M/H92531/H92532¯\nM/H92532M/H92532/H92531M/H925322+m2¯\n]] ] /GS /H20899=/H2089810 0 ¯\n/H9253110 ¯\n/H9253201 ¯\n]] ]/GS/H20899/H20898M 0 0¯\n0m10¯\n00 m2¯\n]]]/GS/H20899/H208981/H92531/H92532¯\n01 0 ¯\n00 1 ¯\n]] ]/GS/H20899. /H20849D35/H20850\nInverting the factorized matrices,\nA−1=/H208981/H92531/H92532¯\n01 0 ¯\n00 1 ¯\n]] ]/GS/H20899−1\n/H20898M 0 0¯\n0m10¯\n00 m2¯\n]]]/GS/H20899−1\n/H2089810 0 ¯\n/H9253110 ¯\n/H9253201 ¯\n]] ]/GS/H20899−1\n=/H208981−/H92531−/H92532¯\n01 0 ¯\n00 1 ¯\n]] ]/GS/H20899/H208981\nM0 0¯\n01\nm10¯\n001\nm2¯\n]]] /GS/H20899\n/H11003/H2089810 0 ¯\n−/H9253110 ¯\n−/H9253201 ¯\n]] ] /GS/H20899=/H208981\nM+/H20858\nn/H9253n2\nmn−/H92531\nm1−/H92532\nm2¯\n−/H92531\nm11\nm10¯\n−/H92532\nm201\nm2¯\n]] ] /GS/H20899. /H20849D36/H20850KYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-14Thus, the maximum column sum of A−1is\n/H20648A−1/H20648= max\nn/H208731\nM+/H20858\ni/H9253i2\nmi+/H20858\ni/H20841/H9253i/H20841\nmi,1+/H20841/H9253n/H20841\nmn/H20874./H20849D37/H20850\nIf/H9253iare on the order of 1 or larger,1\nM+/H20858i/H9253i2\nmi+/H20858i/H20841/H9253i/H20841\nmiis the\nmaximum value. And, in this limit, it is smaller than1\nM\n+2/H20858i/H9253i2\nmi. So one can get\n/H20648A/H20648/H113491\nM+2/H20858\ni/H9253i2\nmi. /H20849D38/H20850\nSince Bis given by\nB=/H20898b+/H20858\nnmn/H9275n2\n−m1/H927512−m2/H927522¯\n−m1/H927512m1/H927512 0 ¯\n−m2/H9275220 m2/H927522¯\n]] ] /GS/H20899,/H20849D39/H20850\nthe maximum column sum of Bis\n/H20648B/H206481= max\nn/H20873b+2/H20858\nimiwi2,2mnwn2/H20874/H11349/H20841b/H20841+2/H20858\nimiwi2.\n/H20849D40/H20850\nFinally, one obtains the upper bound of /H9261M,\n/H9261M/H11349/H20648A−1/H20648/H20648B/H20648/H11349/H208731\nM+2/H20858\ni/H9253i2\nmi/H20874/H20873/H20841b/H20841+2/H20858\nimi/H9275i2/H20874.\n/H20849D41/H20850\nIn order to evaluate the expression on the right-hand side of\nthe inequality Eq. /H20849D41/H20850, we use the constraints Eq. /H2084920/H20850.T o\nconvert the summations to known quantities, we generalizethe constraint to the Caldeira-Legget-type continuous formwith the following definitions of spectral functions,\nJ\np/H20849/H9275/H20850/H11013/H9266\n2/H20858\ni/H9253i2/H9275i\nmi/H9254/H20849/H9275i−/H9275/H20850=/H9251S\n2KM/H9275,/H20849D42/H20850\nJx/H20849/H9275/H20850/H11013/H9266\n2/H20858\nimi/H9275i3/H9254/H20849/H9275i−/H9275/H20850=2/H9251KM\nS/H9275./H20849D43/H20850\nChecking the constraints,/H20858\ni/H9253i2/H9261\nmi/H20849/H92612+/H9275i2/H20850=2/H9261\n/H9266/H20885d/H9275Jp/H20849/H9275/H20850\n/H9275/H20849/H92612+/H92752/H20850\n=2/H9261\n/H9266/H9251S\n2KM/H20885d/H92751\n/H92612+/H92752=/H9251S\n2KM,\n/H20849D44/H20850\n/H20858\nimi/H9275i2/H9261\n/H92612+/H9275i2=2/H9261\n/H9266/H20885d/H9275Jx/H20849/H9275/H20850\n/H9275/H20849/H92612+/H92752/H20850\n=2/H9261\n/H92662/H9251KM\nS/H20885d/H92751\n/H92612+/H92752=2/H9251KM\nS.\n/H20849D45/H20850\nFinally,\n/H20858\ni/H9253i2\nmi=2\n/H9266/H20885d/H9275/H9251S\n2KM=/H9251S\n/H9266KM/H9275c, /H20849D46/H20850\n/H20858\nimi/H9275i2=2\n/H9266/H20885d/H92752/H9251KM\nS=4/H9251KM\nS/H9266/H9275c,/H20849D47/H20850\nwhere /H9275cis the critical frequency of the environmental exci-\ntations.\nTherefore, /H9261M/H11349/H208491\nM+2/H9251S\n/H9266KM/H9275c/H20850/H20849/H20841b/H20841+8/H9251KM\nS/H9266/H9275c/H20850. Hence, one fi-\nnally finds that the sufficient condition of the high tempera-\nture is T/H11271Tc, where the critical temperature Tcis defined as\nTc/H11013/H6036\n2kB/H20881/H208731\nM+2/H9251S\n/H9266KM/H9275c/H20874/H20873/H20841b/H20841+8/H9251KM\nS/H9266/H9275c/H20874.\n/H20849D48/H20850\nNow, we check if the above condition is satisfied in ex-\nperimental situations. Ignoring /H20841b/H20841, the critical temperature\nbecomes /H20881/H208491+2/H9251S\n/H9266K/H9275c/H208502/H9251K\nS/H9266/H9275c. Since the environmental excita-\ntion is caused by magnetization dynamics, one can note thatthere is no need to consider the environmental excitationwith frequencies far exceeding the frequency scale of mag-netization dynamics. This concludes that\n/H9275cis on the order\nof the frequency of magnetization dynamics, which is knownas about 10 GHz or less.\n56,57With conventional scale /H9251\n/H110110.01, K/H1101110−4eV, and 2 S/H11011/H6036, the critical temperature is\nestimated as Tc/H1101130 mK. Therefore, our calculation is con-\ncluded to be well satisfied in most experimental situation.\n1A. Thiaville, Y . Nakatani, J. Miltat, and Y . Suzuki, Europhys.\nLett. 69, 990/H208492005/H20850.\n2G. Tatara, T. Takayama, H. Kohno, J. Shibata, Y . Nakatani, and\nH. Fukuyama, J. Phys. Soc. Jpn. 75, 064708 /H208492006/H20850.\n3M. D. Stiles, W. M. Saslow, M. J. Donahue, and A. Zangwill,\nPhys. Rev. B 75, 214423 /H208492007/H20850.\n4S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 /H208492004/H20850.\n5W. F. Brown, Jr., Phys. Rev. 130, 1677 /H208491963/H20850.\n6R. Kubo and N. Hashitsume, Suppl. Prog. Theor. Phys. 46, 210/H208491970/H20850.\n7J. Foros, A. Brataas, Y . Tserkovnyak, and G. E. W. Bauer, Phys.\nRev. B 78, 140402 /H20849R/H20850/H208492008/H20850.\n8One of the well-known situations where thermal fluctuations\nplays a crucial role is the DW creep. See, for instance, S. Le-merle, J. Ferre, C. Chappert, V . Mathet, T. Giamarchi, and P. LeDoussal, Phys. Rev. Lett. 80, 849/H208491998/H20850; P. Chauve, T. Giama-\nrchi, and P. Le Doussal, Phys. Rev. B 62, 6241 /H208492000/H20850; K. Kim,\nJ.-C. Lee, S.-M. Ahn, K.-S. Lee, C.-W. Lee, Y . J. Cho, S. Seo,THERMAL FLUCTUATION FIELD FOR CURRENT-INDUCED … PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-15K.-H. Shin, S.-B. Choe, and H.-W. Lee, Nature /H20849London /H20850458,\n740/H208492009/H20850.\n9R. A. Duine, A. S. Núñez, and A. H. MacDonald, Phys. Rev.\nLett. 98, 056605 /H208492007/H20850.\n10R. A. Duine and C. M. Smith, Phys. Rev. B 77, 094434 /H208492008/H20850.\n11J.-V . Kim and C. Burrowes, Phys. Rev. B 80, 214424 /H208492009/H20850.\n12E. Martinez, L. Lopez-Diaz, L. Torres, C. Tristan, and O. Alejos,\nPhys. Rev. 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Nazarov, in Single Charge Tunneling\nCoulomb Blockade Phenomena in Nanostructures , edited by H.\nGrabert and M. Devoret /H20849Plenum, New York, 1992 /H20850.\n34H. Lee and L. S. Levitov, Phys. Rev. B 53, 7383 /H208491996/H20850.\n35N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406 /H208491974/H20850.\n36A. P. Malozemoff and J. C. Slonczewski, Magnetic Domains\nWalls in Bubble Materials /H20849Academic, New York, 1979 /H20850.\n37S.-W. Jung, W. Kim, T.-D. Lee, K.-J. Lee, and H.-W. Lee, Appl.Phys. Lett. 92, 202508 /H208492008/H20850; J. Ryu and H.-W. Lee, J. Appl.\nPhys. 105, 093929 /H208492009/H20850.\n38M. Kläui, P.-O. Jubert, R. Allenspach, A. Bischof, J. A. C. Bland,\nG. Faini, U. Rüdiger, C. A. F. Vaz, L. Vila, and C. V ouille, Phys.\nRev. Lett. 95, 026601 /H208492005/H20850.\n39G. S. D. Beach, C. Knutson, C. Nistor, M. Tsoi, and J. L. Ersk-\nine,Phys. Rev. Lett. 97, 057203 /H208492006/H20850.\n40M. Hayashi, L. Thomas, C. Rettner, R. Moriya, Y . B. Bazaliy,\nand S. S. P. Parkin, Phys. Rev. Lett. 98, 037204 /H208492007/H20850.\n41For permalloy, /H20841e/H92530K/H9261//H9262B/H20841/H11011109A/cm2, which is about an or-\nder larger than the current density of /H11011108A/cm2used in many\nexperiments /H20849Refs. 38–40/H20850.\n42Y . Le Maho, J.-V . Kim, and G. Tatara, Phys. Rev. B 79, 174404\n/H208492009/H20850.\n43T. Kim, J. Ieda, and S. Maekawa, arXiv:0901.3066 /H20849unpub-\nlished/H20850.\n44V . W. Döring, Z. Naturforsch. A 3A, 373/H208491948/H20850.\n45We thank M. Stiles for pointing out this point.\n46To solve this system, one of the constaints Eq. /H2084920a/H20850is general-\nized to hold even for /H9261=0. That is, /H20858i/H9253i=0. See, for a detail,\nAppendix C.\n47To consider a force on Eq. /H208496a/H20850, the potential should be general-\nized to depend on the momentum.\n48Forv=0, the terminal velocity of the DW vanishes indepen-\ndently of its the initial velocity since the environmental mass ismuch larger than the DW mass. With\nv/H110220, one can perform the\nGalilean transformation to make /H20855x˙i/H208490/H20850/H20856=0 instead of /H20855x˙i/H208490/H20850/H20856\n=v. Since the system is Galilean invariant, one expect that the\nDW also stops in this frame, just as v=0. It implies that the\nterminal velocity of the DW in the lab frame is also v.\n49V . Kamberský, Czech. J. Phys., Sect. B 26, 1366 /H208491976/H20850; Can. J.\nPhys. 48, 2906 /H208491970/H20850;Czech. J. Phys., Sect. B 34, 1111\n/H208491984/H20850.\n50K. Gilmore, Y . U. Idzerda, and M. D. Stiles, Phys. Rev. Lett. 99,\n027204 /H208492007/H20850.\n51C.-Y . You, I. M. Sung, and B.-K. Joe, Appl. Phys. Lett. 89,\n222513 /H208492006/H20850; C.-Y . You and S.-S. Ha, ibid. 91, 022507\n/H208492007/H20850.\n52This section summarizes the work by Kim et al./H20849Ref. 43/H20850.\n53J. M. Winter, Phys. Rev. 124, 452/H208491961/H20850.\n54By the same argument, Eq. /H2084932/H20850is obtained under an arbitrary\npotential V/H20849Q/H20850. Since the system was in equilibrium before ap-\nplying current, we assume V/H11032/H20849Q/H20850=0. At high temperature limit,\n/H9273moves in very short /H20849imaginary /H20850time interval. Therefore, we\ncan take quadratic approximation and V/H20849Q/H20850to be the form of\nbQ2.\n55See, for example, G. Strang, Linear Algebra and its Applications\n/H20849Thomson, USA, 1988 /H20850, Chap. 7.\n56A. Mourachkine, O. V . Yazyev, C. Ducati, and J.-Ph. Ansermet,\nNano Lett. 8, 3683 /H208492008/H20850.\n57C. Boone, J. A. Katine, J. R. Childress, J. Zhu, X. Cheng, and I.\nN. Krivorotov, Phys. Rev. B 79, 140404 /H20849R/H20850/H208492009/H20850.KYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-16"    },    {        "title": "1810.06471v1.Localized_spin_waves_in_isolated__kπ__skyrmions.pdf",        "content": "Localized spin waves in isolated k\u0019skyrmions\nLevente Rózsa,1,\u0003Julian Hagemeister,1Elena Y. Vedmedenko,1and Roland Wiesendanger1\n1Department of Physics, University of Hamburg, D-20355 Hamburg, Germany\n(Dated: October 16, 2018)\nThelocalizedmagnonmodesofisolated k\u0019skyrmionsonafield-polarizedbackgroundareanalyzed\nbasedontheLandau–Lifshitz–Gilbertequationwithinthetermsofanatomisticclassicalspinmodel,\nwith system parameters based on the Pd/Fe biatomic layer on Ir(111). For increasing skyrmion\norderka higher number of excitation modes are found, including modes with nodes in the radial\neigenfunctions. Itisshownthatatlowfields 2\u0019and 3\u0019skyrmionsaredestroyedviaaburstinstability\nconnected to a breathing mode, while 1\u0019skyrmions undergo an elliptic instability. At high fields all\nk\u0019skyrmions collapse due to the instability of a breathing mode. The effective damping parameters\nof the spin waves are calculated in the low Gilbert damping limit, and they are found to diverge\nin the case of the lowest-lying modes at the burst and collapse instabilities, but not at the elliptic\ninstability. It is shown that the breathing modes of k\u0019skyrmions may become overdamped at higher\nGilbert damping values.\nI. INTRODUCTION\nMagneticskyrmionsarelocalizedparticle-likespincon-\nfigurations [1], which have become the focus of intense\nresearch activities over the last years due to their promis-\ning applications in spintronic devices [2–5]. While their\nparticle-like properties make them suitable to be used\nas bits of information, the collective excitations of the\nspins constituting the magnetic skyrmion, known as spin\nwaves or magnons, open possible applications in the field\nof magnonics [6].\nThese spin wave modes were first investigated theo-\nretically [7–10] and experimentally [11–13] in skyrmion\nlattice phases, where the interactions between the\nskyrmions lead to the formation of magnon bands. If\na skyrmion is confined in a finite-sized nanoelement, it\nwill possess discrete excitation frequencies [14–17]. Al-\nthough such geometries have also been successfully ap-\nplied to the time-resolved imaging of the dynamical mo-\ntion of magnetic bubble domains [18, 19], in such a case\nit is not possible to distinguish between the excitations\nof the particle-like object itself and spin waves forming\nat the edges of the sample [14]. In order to rule out\nboundary effects, the excitations of isolated skyrmions\nhave to be investigated, as was performed theoretically\nin Refs. [20–23]. It was suggested recently [24] that\nthe experimentally determined excitation frequencies in\nthe Ir/Fe/Co/Pt multilayer system may be identified as\nspin wave modes of isolated skyrmions, rather than as\nmagnons stemming from an ordered skyrmion lattice.\nIn most investigations skyrmions correspond to sim-\nple domains with the magnetization in their core point-\ning opposite to the collinear background. However, it\nwas shown already in Ref. [25] that the Dzyaloshinsky–\nMoriya interaction [26, 27] responsible for their stabiliza-\ntion may also lead to the formation of structures where\nthe direction of the magnetization rotates multiple times\n\u0003rozsa.levente@physnet.uni-hamburg.debetween the center of the structure and the collinear re-\ngion. Such target states or k\u0019skyrmions, where kis the\nnumber of sign changes of the out-of-plane magnetization\nwhen moving along the radial direction, have also been\ninvestigated in constricted geometries [28–32]. The ex-\nperimental observation of localized spin structures with\nmultiple rotations has been mainly restricted to systems\nwith negligible Dzyaloshinsky–Moriya interaction so far\n[19, 33, 34], where the formation of domain structures is\nattributed to the magnetostatic dipolar interaction.\nThe collapse of isolated k\u0019skyrmions and their cre-\nationinnanodotsbyswitchingtheexternalfielddirection\nwas recently investigated in Ref. [35]. It was found that\nduring the creation process the skyrmions display signif-\nicant size oscillations resembling breathing eigenmodes.\nIn Ref. [25], the stability of k\u0019skyrmions was studied\nin a system with a ferromagnetic ground state, and it\nwas found that applying the external field opposite to\nthe background magnetization leads to a divergence of\nthe skyrmion radius at a critical field value, a so-called\nburst instability. This instability can be attributed to a\nsign change of one of the eigenvalues of the energy func-\ntional expanded around the k\u0019skyrmion configuration,\nintrinsically related to the dynamics of the system. How-\never, the spin wave frequencies of isolated k\u0019skyrmions\nremain unexplored.\nBesides the excitation frequencies themselves, the life-\ntime of spin waves is also of crucial importance in\nmagnonics applications. This is primarily influenced by\nthe Gilbert damping parameter \u000b[36], the value of which\ncan be determined experimentally based on resonance\nlineshapes measured in the collinear state [11, 19, 24].\nIt was demonstrated recently [23] that the noncollinear\nspin structure drastically influences the effective damp-\ning parameter acting on the spin waves, leading to mode-\ndependent and enhanced values compared to the Gilbert\ndamping parameter. This effect was discussed through\nthe example of the 1\u0019skyrmion in Ref. [23], but it is also\nexpected to be observable for k\u0019skyrmions with higher\norderk.\nHere the localized spin wave frequencies of isolated k\u0019arXiv:1810.06471v1  [cond-mat.mes-hall]  15 Oct 20182\nskyrmions are investigated in a classical atomistic spin\nmodel. The parameters in the Hamiltonian represent the\nPd/Fe/Ir(111) model-type system, where the properties\nof skyrmions have been studied in detail both from the\nexperimental [37, 38] and from the theoretical [35, 39–\n41] side. The paper is organized as follows. The classical\natomistic spin Hamiltonian and the method of calculat-\ning the eigenmodes is introduced in Sec. IIA, while the\nangular momentum and nodal quantum numbers charac-\nterizing the excitations are defined in Sec. IIB within the\nframework of the corresponding micromagnetic model.\nEigenfrequencies equal to or approaching zero are dis-\ncussed in Sec. IIC, and the effective damping param-\neters are introduced in Sec. IID. The eigenmodes of k\u0019\nskyrmions with k= 1;2;3are compared in Sec. IIIA, the\ninstabilities occurring at low and high field values are dis-\ncussed in connection to magnons with vanishing frequen-\ncies in Sec. IIIB, and the effective damping parameters\nof the different modes are calculated for vanishing and\nhigher values of the Gilbert damping in Secs. IIIC and\nIIID, respectively. A summary is given in Sec. IV.\nII. METHODS\nA. Atomistic model\nThe system is described by the classical atomistic\nmodel Hamiltonian\nH=\u00001\n2X\nhi;jiJSiSj\u00001\n2X\nhi;jiDij(Si\u0002Sj)\n\u0000X\niK(Sz\ni)2\u0000X\ni\u0016sBSi; (1)\nwith theSiunit vectors representing the spins in\na single-layer triangular lattice; J,Dij, andKde-\nnoting nearest-neighbor Heisenberg and Dzyaloshinsky–\nMoriya exchange interactions and on-site magnetocrys-\ntalline anisotropy, respectively; while \u0016sandBstand\nfor the spin magnetic moment and the external mag-\nnetic field. The numerical values of the parameters are\ntaken from Ref. [35], being J= 5:72meV;D=jDijj=\n1:52meV;K= 0:4meV, and\u0016s= 3\u0016B, describing the\nPd/Fe/Ir(111) system. The energy parameters were de-\ntermined based on measuring the field-dependence of 1\u0019\nskyrmion profiles in the system by spin-polarized scan-\nning tunneling microscopy in Ref. [38].\nDuring the calculations the external field Bis ori-\nented along the out-of-plane zdirection. The equilib-\nriumk\u0019skyrmion structures are determined from a rea-\nsonable initial configuration by iteratively rotating the\nspinsSitowards the direction of the effective magnetic\nfieldBeff\ni=\u00001\n\u0016s@H\n@Si. The iteration is performed un-\ntil the torque acting on the spins, Ti=\u0000Si\u0002Beff\ni,\nbecomes smaller at every lattice site than a predefinedvalue, generally chosen to be 10\u00008meV=\u0016B. The calcula-\ntions are performed on a lattice with periodic boundary\nconditions, with system sizes up to 256\u0002256for the\nlargestk\u0019skyrmions in order to avoid edge effects and\nenable the accurate modeling of isolated skyrmions.\nOnce the equilibrium configuration S(0)\niis determined,\nthe spins are rotated to a local coordinate system ~Si=\nRiSiusing the rotational matrices Ri. In the local coor-\ndinatesystemtheequilibriumspindirectionsarepointing\nalong the local zaxis, ~S(0)\ni= (0;0;1). The Hamiltonian\nin Eq. (1) is expanded up to second-order terms in the\nsmall variables ~Sx\ni;~Sy\nias (cf. Ref. [23])\nH\u0019H0+1\n2\u0010\n~S?\u0011T\nHSW~S?\n=H0+1\n2\u0002~Sx~Sy\u0003\u0014A1A2\nAy\n2A3\u0015\u0014~Sx\n~Sy\u0015\n:(2)\nThematrix products areunderstoodtorunoverlattice\nsite indices i, with the matrix components reading\nA1;ij=\u0000~Jxx\nij+\u000eij X\nk~Jzz\nik\u00002~Kxx\ni+ 2~Kzz\ni+\u0016s~Bz\ni!\n;(3)\nA2;ij=\u0000~Jxy\nij\u0000\u000eij2~Kxy\ni; (4)\nA3;ij=\u0000~Jyy\nij+\u000eij X\nk~Jzz\nik\u00002~Kyy\ni+ 2~Kzz\ni+\u0016s~Bz\ni!\n:(5)\nThe energy terms in the Hamiltonian are ro-\ntated to the local coordinate system via ~Jij=\nRi[JI\u0000Dij\u0002]RT\nj;~Ki=RiKRT\nj;and ~Bi=RiB,\nwhereIis the 3\u00023identity matrix, Dij\u0002is the ma-\ntrix describing the vector product with Dij, andKis\nthe anisotropy matrix with the only nonzero element be-\ningKzz=K.\nThe spin wave frequencies are obtained from the lin-\nearized Landau–Lifshitz–Gilbert equation [36, 42]\n@t~S?=\r0\n\u0016s(\u0000i\u001by\u0000\u000b)HSW~S?=DSW~S?;(6)\nwith\u001by=\u0014\n0\u0000iIs\niIs0\u0015\nthe Pauli matrix in Cartesian\ncomponents and acting as the identity matrix Isin the\nlattice site summations. The symbol \r0denotes the gyro-\nmagnetic ratio \r=ge\n2mdivided by a factor of 1+\u000b2, with\ngthe electron gfactor,ethe elementary charge, mthe\nelectron’s mass, and \u000bthe Gilbert damping parameter.\nEquation (6) is rewritten as an eigenvalue equation by\nassuming the time dependence ~S?(t) =e\u0000i!qt~S?\nqand\nperforming the replacement @t!\u0000i!q.\nSince thek\u0019skyrmions represent local energy minima,\nHSWin Eq. (2) is a positive semidefinite matrix. For\n\u000b= 0the!qfrequenciesof DSWarerealandtheyalways\noccurin\u0006!qpairsonthesubspacewhere HSWisstrictly\npositive, for details see, e.g., Ref. [23]. In the following,\nwe will only treat the solutions with Re !q>0, but their3\nRe!q<0pairs are also necessary for constructing real-\nvalued eigenvectors of Eq. (6). The zero eigenvalues are\ndiscussed in Sec. IIC.\nAs is known from previous calculations for 1\u0019\nskyrmions [21–23], the localized excitation modes of k\u0019\nskyrmions are found below the ferromagnetic resonance\nfrequency!FMR =\r\n\u0016s(2K+\u0016sB). During the numerical\nsolution of Eq. (6) these lowest-lying eigenmodes of the\nsparse matrix DSWare determined, as implemented in\nthemontecrystal atomistic spin simulation program\n[43].\nB. Micromagnetic model\nThe atomistic model described in the previous Sec-\ntion enables the treatment of noncollinear spin structures\nwhere the direction of the spins significantly differs be-\ntween neighboring lattice sites. This is especially impor-\ntant when discussing the collapse of k\u0019skyrmions on the\nlattice as was performed in Ref. [35]. Here we will dis-\ncuss the micromagnetic model which on the one hand is\napplicable only if the characteristic length scale of non-\ncollinear structures is significantly larger than the lattice\nconstant, but on the other hand enables a simple classi-\nfication of the spin wave modes.\nThe free energy functional of the micromagnetic model\nis defined as\nH=Z\nAX\n\u000b=x;y;z(rS\u000b)2+K(Sz)2\u0000MBSz\n+D(Sz@xSx\u0000Sx@xSz+Sz@ySy\u0000Sy@ySz)dr;\n(7)\nwhere for the Pd/Fe/Ir(111) system the following pa-\nrameter values were used: A= 2:0pJ/m is the ex-\nchange stiffness,D=\u00003:9mJ/m2is the Dzyaloshinsky–\nMoriya interaction describing right-handed rotation [39],\nK=\u00002:5MJ/m3is the easy-axis anisotropy, and M=\n1:1MA/m is the saturation magnetization.\nThe equilibrium spin structure S(0)=\n(sin \u0002 0cos \b 0;sin \u0002 0sin \b 0;cos \u0002 0)ofk\u0019skyrmions will\nbe cylindrically symmetric, given by \b0(r;') ='+\u0019\ndue to the right-handed rotational sense and\n\u00020(r;') = \u0002 0(r), which is the solution of the\nEuler–Lagrange equation\nA\u0012\n@2\nr\u00020+1\nr@r\u00020\u00001\nr2sin \u0002 0cos \u0002 0\u0013\n+jDj1\nrsin2\u00020\n+Ksin \u0002 0cos \u0002 0\u00001\n2MBsin \u0002 0= 0: (8)\nThe skyrmion order kis encapsulated in the bound-\nary conditions \u00020(0) =k\u0019;\u00020(1) = 0. Equation (8) is\nsolved numerically in a finite interval r2[0;R]signifi-\ncantly larger than the equilibrium k\u0019skyrmion size. A\nfirst approximation to the spin structure is constructed\nbased on the corresponding initial value problem usingthe shooting method [25], then iteratively optimizing the\nstructure using a finite-difference discretization.\nThe spin wave Hamiltonian may be determined anal-\nogously to Eq. (2), by using the local coordinate system\n\u0002 = \u0002 0+~Sx;\b = \b 0+1\nsin \u0002 0~Sy. ThematricesinEqs.(3)-\n(5) are replaced by the operators\nA1=\u00002A\u0012\nr2\u00001\nr2cos 2\u0002 0\u0013\n\u00002jDj1\nrsin 2\u0002 0\n\u00002Kcos 2\u0002 0+MBcos \u0002 0; (9)\nA2= 4A1\nr2cos \u0002 0@'\u00002jDj1\nrsin \u0002 0@'; (10)\nA3=\u00002A\u001a\nr2+\u0014\n(@r\u00020)2\u00001\nr2cos2\u00020\u0015\u001b\n\u00002jDj\u0012\n@r\u00020+1\nrsin \u0002 0cos \u0002 0\u0013\n\u00002Kcos2\u00020+MBcos \u0002 0: (11)\nDue to the cylindrical symmetry of the structure, the\nsolutions of Eq. (6) are sought in the form ~S?(r;';t ) =\ne\u0000i!n;mteim'~S?\nn;m(r), performing the replacements @t!\n\u0000i!n;mand@'!im. For each angular momentum\nquantum number m, an infinite number of solutions in-\ndexed bynmay be found, but only a few of these are\nlocated below !FMR =\r\nM(\u00002K+MB), hence repre-\nsenting localized spin wave modes of the k\u0019skyrmions.\nThe different nquantum numbers typically denote solu-\ntions with different numbers of nodes, analogously to the\nquantum-mechanical eigenstates of a particle in a box.\nBecause of the property HSW(m) =H\u0003\nSW(\u0000m)and\nHSWbeing self-adjoint, the eigenvalues of HSW(m)\nandHSW(\u0000m)coincide, leading to a double degeneracy\napart from the m= 0modes. The\u0006!qeigenvalue pairs\nofDSWdiscussed in Sec. IIA for the atomistic model at\n\u000b= 0in this case can be written as !n;m=\u0000!n;\u0000m.\nHowever, considering only the modes with Re !n;m>0,\none has!n;m6=!n;\u0000mindicating nonreciprocity or an\nenergy difference between clockwise ( m < 0) and coun-\nterclockwise ( m> 0) rotating modes [17, 23].\nFor finding the eigenvectors and eigenvalues of the\nmicromagnetic model, Eq. (6) is solved using a finite-\ndifference method on the r2[0;R]interval. For treat-\ning the Laplacian r2in Eqs. (9) and (11) the improved\ndiscretization scheme suggested in Ref. [44] was applied,\nwhich enables a more accurate treatment of modes with\neigenvalues converging to zero in the infinite and contin-\nuous micromagnetic limit.\nThe spin wave modes of the atomistic model discussed\nin Sec. IIA were assigned the (n;m)quantum numbers,\nwhich are strictly speaking only applicable in the mi-\ncromagnetic limit with perfect cylindrical symmetry, by\nvisualizingthe real-spacestructureofthe numericallyob-\ntained eigenvectors.4\nC. Goldstone modes and instabilities\nSince the translation of the k\u0019skyrmions on the\ncollinear background in the plane costs no energy, the\nspin wave Hamiltonian HSWpossesses two eigenvectors\nbelonging to zero eigenvalue, representing the Goldstone\nmodes of the system. Within the micromagnetic descrip-\ntion of Sec. IIB, these may be expressed analytically as\n[21–23]\n\u0010\n~Sx;~Sy\u0011\n=e\u0000i'\u0012\n\u0000@r\u00020;i1\nrsin \u0002 0\u0013\n;(12)\n\u0010\n~Sx;~Sy\u0011\n=ei'\u0012\n\u0000@r\u00020;\u0000i1\nrsin \u0002 0\u0013\n:(13)\nEquations (12) and (13) represent eigenvectors of the\ndynamical matrix DSWas well. From Eqs. (2) and (6) it\nfollows that the eigenvectors of HSWandDSWbelong-\ning to zero eigenvalue must coincide, HSW~S?=0,\nDSW~S?=0, because (\u0000i\u001by\u0000\u000b)in Eq. (6) is an in-\nvertible matrix. Because from the solutions of the equa-\ntion of motion (6) we will only keep the ones satisfying\nRe!n;m>0, the eigenvectors from Eqs. (12) and (13)\nwill be denoted as the single spin wave mode !0;\u00001= 0.\nSince the eigenvectors and eigenvalues are determined\nnumerically in a finite system by using a discretization\nprocedure, the Goldstone modes will possess a small fi-\nnite frequency. However, these will not be presented\nin Sec. IIIA together with the other frequencies since\nthey represent a numerical artifact. For the 1\u0019and\n3\u0019skyrmions the !0;1eigenmode has a positive fre-\nquency and an eigenvector clearly distinguishable from\nthat of the !0;\u00001translational mode. However, for the\n2\u0019skyrmion both the !0;\u00001and the!0;1eigenfrequen-\ncies ofDSWare very close to zero, and the correspond-\ning eigenvectors converge to Eqs. (12) and (13) as the\ndiscretization is refined and the system size is increased.\nThis can occur because DSWis not self-adjoint and its\neigenvectors are generally not orthogonal. In contrast,\nthe eigenvectors of HSWremain orthogonal, with only a\nsingle pair of them taking the form of Eqs. (12) and (13).\nIn contrast to the Goldstone modes with always zero\nenergy, the sign change of another eigenvalue of HSW\nindicates that the isolated k\u0019skyrmion is transformed\nfrom a stable local energy minimum into an unstable\nsaddle point, leading to its disappearance from the sys-\ntem. Such instabilities were determined by calculating\nthe lowest-lying eigenvalues of HSWin Eq. (2). Due\nto the connection between the HSWandDSWmatrices\nexpressed in Eq. (6), at least one of the precession fre-\nquencies!qwill also approach zero at such an instability\npoint.\nD. Effective damping parameters\nFor finite values of the Gilbert damping \u000b, the spin\nwaves in the system will decay over time as the systemrelaxes to the equilibrium state during the time evolu-\ntion described by the Landau–Lifshitz–Gilbert equation.\nThe speed of the relaxation can be characterized by the\neffective damping parameter, which for a given mode q\nis defined as\n\u000bq;eff=\f\f\f\fIm!q\nRe!q\f\f\f\f: (14)\nAs discussed in detail in Ref. [23], \u000bq;effis mode-\ndependent and can be significantly higher than the\nGilbert damping parameter \u000bdue to the elliptic polar-\nization of spin waves, which can primarily be attributed\nto the noncollinear spin structure of the k\u0019skyrmions.\nFor\u000b\u001c1,\u000bq;effmay be expressed as\n\u000bq;eff\n\u000b=X\ni\f\f\f~S(0);x\nq;i\f\f\f2\n+\f\f\f~S(0);y\nq;i\f\f\f2\nX\ni2Imh\u0010\n~S(0);x\nq;i\u0011\u0003~S(0);y\nq;ii;(15)\nwheretheeigenvectorsinEq.(15)arecalculatedat \u000b= 0\nfrom Eq. (6). Equation (15) may also be expressed by\nthe axes of the polarization ellipse of the spins in mode\nq, see Ref. [23] for details.\nForhighervaluesof \u000b, thecomplexfrequencies !qhave\nto be determined from Eq. (6), while the effective damp-\ning parameters can be calculated from Eq. (14). Also for\nfinite values of \u000bfor each frequency with Re !q>0there\nexists a pair with Re !q0<0such that!q0=\u0000!\u0003\nq[23].\nThe spin waves will be circularly polarized if A1=A3\nandAy\n2=\u0000A2in Eq. (2), in which case the dependence\nof!qon\u000bmay simply be expressed by the undamped\nfrequency!(0)\nqas\nRe!q(\u000b) =1\n1 +\u000b2!(0)\nq; (16)\njIm!q(\u000b)j=\u000b\n1 +\u000b2!(0)\nq: (17)\nThese relations are known for uniaxial ferromagnets;\nsee, e.g., Ref. [45]. In the elliptically polarized modes of\nnoncollinear structures, such as k\u0019skyrmions, a devia-\ntion from Eqs. (16)-(17) is expected.\nIII. RESULTS\nA. Eigenmodes\nThe frequencies of the localized spin wave modes of the\n1\u0019,2\u0019, and 3\u0019skyrmion, calculated from the atomistic\nmodel for\u000b= 0as described in Sec. IIA, are shown in\nFig. 1. For the 1\u0019skyrmion six localized modes can be\nobserved below the FMR frequency of the field-polarized\nbackground in Fig. 1(a), four of which are clockwise ro-\ntating modes ( m < 0), one is a gyration mode rotating\ncounterclockwise ( m= 1), while the final one is a breath-\ning mode (m= 0). The excitation frequencies show good5\n0.7 0.8 0.9 1.0 1.1 1.20255075100125150175\n(a)\n0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.150255075100125150175\n(b)\n0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.150255075100125150175\n(c)\nFIG. 1. Frequencies of localized spin wave modes at \u000b= 0for\n(a) the 1\u0019, (b) the 2\u0019, and (c) the 3\u0019skyrmion. Selected spin\nwave modes are visualized in contour plots of the out-of-plane\nspin component and denoted by open symbols connected by\nlines in the figure, the remaining modes are denoted by con-\nnected dots.quantitative agreement with the ones calculated from the\nmicromagnetic model for the same system in Ref. [23].\nCompared to Ref. [21], the additional appearance of the\neigenmodes with m= 1;\u00004;\u00005can be attributed to the\nfinite value of the anisotropy parameter Kin the present\ncase. Increasing the anisotropy value makes it possible\nto stabilize the skyrmions at lower field values, down to\nzerofieldatthecriticalvalueinthemicromagneticmodel\njKcj=\u00192D2\n16A, where the transition from the spin spiral\nto the ferromagnetic ground state occurs at zero exter-\nnal field [46]. Since the excitation frequencies decrease\nat lower field values as shown in Fig. 1(a), this favors\nthe appearance of further modes. Simultaneously, the\nFMR frequency increases with K, meaning that modes\nwith higher frequencies become observable for larger uni-\naxial anisotropy. For each angular momentum quantum\nnumberm, only a single mode ( n= 0) appears.\nIn the case of the 2\u0019skyrmion an increased number of\neigenmodes may be seen in Fig. 1(b). This can mainly be\nattributed to the appearance of spin waves with higher\nangular momentum quantum numbers both for clockwise\n(up tom=\u000017) and counterclockwise (up to m= 12)\nrotational directions. Furthermore, in this case modes\nwithn= 1node in the eigenfunction can be observed\nas well. The same trend continues in the case of 3\u0019\nskyrmionsinFig.1(c), thelargenumberofinternaleigen-\nmodes can be attributed to angular momentum quantum\nnumbers ranging from m=\u000022tom= 16, as well as to\nspin wave eigenvectors with up to n= 2nodes. The dif-\nferent rotational directions and numbers of nodes are il-\nlustrated in Supplemental Videos 1-4 [47] via the square-\nshaped modes ( n= 0;1,m=\u00064) of the 3\u0019skyrmion at\nB= 0:825T.\nThe increase of possible angular momentum quantum\nnumbers for higher skyrmion order kas well as for de-\ncreasing magnetic field Bmay be qualitatively explained\nby an increase in the skyrmion size. Modes with a given\nvalue ofmindicate a total of jmjmodulation periods\nalong the perimeter of the skyrmion; for larger skyrmion\nsizes this corresponds to a modulation on a longer length\nscale, which has a smaller cost in exchange energy.\nThe breathing modes of the 3\u0019skyrmion with dif-\nferent numbers of nodes are visualized in Fig. 2 at\nB= 1T. The results shown in Fig. 2 are obtained\nfrom the micromagnetic model in Sec. IIB, which is in\ngood quantitative agreement with the atomistic calcu-\nlations at the given field. All the eigenmodes display\nthree peaks of various heights, while they decay expo-\nnentially outside the 3\u0019skyrmion. As can be seen in\nFig. 2, the peaks are localized roughly around the re-\ngions where the spins are lying in-plane, indicated by\nthe domain walls (DW) between pairs of dashed lines.\nThe widths of the domain walls were determined by ap-\nproximating the 3\u0019skyrmion profile with linear func-\ntions close to the inflection points rj;\u00020;j;j= 1;2;3\nwhere the spins are lying in-plane, and calculating where\nthese linear functions intersect integer multiples of \u0019in6\n0 10 20 30 40 50-2-023\n-0.100.000.10\nFIG. 2. Comparison between the 3\u0019skyrmion profile (left\nvertical axis) and the eigenvectors of the breathing modes\n(m= 0) with different numbers of nodes n= 0;1;2(right\nvertical axis). The calculations were performed using the mi-\ncromagnetic model described in Sec. IIB at B= 1T, the\nlattice constant is a= 0:271nm. Double arrows between ver-\ntical dashed lines indicate the extensions of the domain walls\nin the structure.\n\u00020. Thus, the domain walls are located between the in-\nnerRin;j=rj+[@r\u00020(rj)]\u00001[(4\u0000j)\u0019\u0000\u00020;j]and outer\nRout;j=rj+ [@r\u00020(rj)]\u00001[(3\u0000j)\u0019\u0000\u00020;j]radii. Such\na description was used to calculate the skyrmion radius\nin, e.g., Ref. [46], and it was also applied for calculating\nthe widths of planar domain walls [48].\nThe nodes of the eigenmodes are located roughly be-\ntween these domain walls, meaning that typically excita-\ntion modes with n= 0;:::;k\u00001nodes may be observed\nink\u0019skyrmions, in agreement with the results in Fig. 1.\nA higher number of nodes would require splitting a single\npeak into multiple peaks, the energy cost of which gen-\nerally exceeds the FMR frequency, thereby making these\nmodes unobservable. The sign changes in the ~Sx\nn;meigen-\nvectors mean that the different modes can be imagined\nas the domain walls breathing in the same phase or in\nopposite phase, as can be seen in Supplemental Videos\n5-7 [47]. Note that eigenmodes with higher nquantum\nnumbers may also be observed for skyrmions confined in\nnanodots [14–16] where the peaks of the eigenmodes may\nalso be localized at the edge of the sample, in contrast\nto the present case where isolated k\u0019skyrmions are dis-\ncussed on an infinite collinear background.\nItisalsoworthnotingthatthelowest-lyingnonzerogy-\nration mode is n= 0;m= 1for the 1\u0019and3\u0019skyrmions,\nwhile it isn= 1;m= 1for the 2\u0019skyrmion, see Fig. 1.\nAs already mentioned in Sec. IIC, numerical calculations\nfor the 2\u0019skyrmion indicate both in the atomistic and\nthemicromagneticcasethatbyincreasingthesystemsize\nor refining the discretization the eigenvectors of both the\nn= 0;m=\u00001and then= 0;m= 1modes ofDSW\nin Eq. (6) converge to the same eigenvectors in Eqs. (12)\nand(13)and 0eigenvalue, whichcorrespondtothetrans-\nlational Goldstone mode in the infinite system. This dif-ference can probably be attributed to the deviation in\nthe value of the topological charge, being finite for 1\u0019\nand3\u0019skyrmions but zero for the 2\u0019skyrmion [35].\nB. Instabilities\nSkyrmions with different order kdeviate in their low-\nfield behavior. Since the considered Pd/Fe/Ir(111) sys-\ntem has a spin spiral ground state [38], decreasing the\nmagnetic field value will make the formation of domain\nwallsenergeticallypreferableinthesystem. Inthecaseof\nthe1\u0019skyrmion this means that the lowest-lying eigen-\nmode ofHSWin Eq. (2), which is an elliptic mode with\nm=\u00062, changes sign from positive to negative, occur-\nring between B= 0:650T andB= 0:625T in the present\nsystem. This is indicated in Fig. 1(a) by the fact that the\nfrequency of the n= 0;m=\u00002eigenmode of DSWin\nEq. (6) converges to zero. This leads to an elongation of\nthe skyrmion into a spin spiral segment which gradually\nfills the ferromagnetic background, a so-called strip-out\nor elliptic instability already discussed in previous publi-\ncations [21, 46]. In contrast, for the 2\u0019and3\u0019skyrmions\nthe lowest-lying eigenmode of HSWis a breathing mode\nwithm= 0, which tends to zero between B= 0:800T\nandB= 0:775Tforbothskyrmions. Thisisindicatedby\nthe lowest-lying n= 0;m= 0mode ofDSWin Fig. 1(b)\nfor the 2\u0019skyrmion, which is the second lowest after the\nn= 0;m= 1mode for the 3\u0019skyrmion in Fig. 1(c). This\nmeans that the radius of the outer two rings of 2\u0019and\n3\u0019skyrmions diverges at a finite field value, leading to a\nburst instability. Such a type of instability was already\nshown to occur in Ref. [25] in the case of a ferromagnetic\nground state at negative field values, in which case it also\naffects 1\u0019skyrmions.\nAt the burst instability, modes with n= 0and all\nangular momentum quantum numbers mappear to ap-\nproach zero because of the drastic increase in skyrmion\nradius decreasing the frequency of these modes as dis-\ncussed in Sec. IIIA. A similar effect was observed for the\n1\u0019skyrmion in Ref. [22] when the critical value of the\nDzyaloshinsky–Moriya interaction, jDcj=4\n\u0019p\nAjKj, was\napproached at zero external field from the direction of\nthe ferromagnetic ground state. In contrast, the ellip-\ntic instability only seems to affect the n= 0;m=\u00002\nmode, while other mvalues and the nonreciprocity are\napparently weakly influenced.\nIn the atomistic model, skyrmions collapse when their\ncharacteristic size becomes comparable to the lattice\nconstant. For the 1\u0019,2\u0019, and 3\u0019skyrmions the col-\nlapse of the innermost ring occurs at Bc;1\u0019\u00194:495T,\nBc;2\u0019\u00191:175T, andBc;3\u0019\u00191:155T, respectively [35].\nAs can be seen in Figs. 1(b), 1(c), and 3, this instabil-\nity is again signaled by the n= 0;m= 0eigenfrequency\ngoing to zero, but in contrast to the burst instability,\nthe other excitation frequencies keep increasing with the\nfield in this regime. Figure 3 demonstrates that close to\nthe collapse field the excitation frequency may be well7\n4.45 4.46 4.47 4.48 4.49 4.50020406080100\nFIG. 3. Frequency of the breathing mode n= 0;m = 0of\nthe 1\u0019skyrmion close to the collapse field. Calculation data\nare shown by open symbols, red line denotes the power-law\nfitf0;0=Af(Bc;1\u0019\u0000B)\ff.\napproximated by the power law f0;0=Af(Bc;1\u0019\u0000B)\ff,\nwithAf= 175:6GHz\nT\ff,Bc;1\u0019= 4:4957T, and\ff= 0:23.\nC. Effective damping parameters in the limit of\nlow\u000b\nThe effective damping parameters \u000bn;m;effwere first\ncalculated from the eigenvectors obtained at \u000b= 0fol-\nlowing Eq. (15). The results for the 1\u0019,2\u0019, and 3\u0019\nskyrmions are summarized in Fig. 4. As discussed in\nRef. [23], the \u000bn;m;effvalues are always larger than the\nGilbert damping \u000b, and they tend to decrease with in-\ncreasing angular momentum quantum number jmjand\nmagnetic field B. The spin wave possessing the high-\nest effective damping is the n= 0;m= 0breathing\nmode both for the 1\u0019and 2\u0019skyrmion, but it is the\nn= 0;m= 1gyration mode for the 3\u0019skyrmion for a\nlarge part of the external field range where the struc-\nture is stable. Excitation pairs with quantum num-\nbersn;\u0006mtend to decay with similar \u000bn;m;effvalues to\neach other, with \u000bn;jmj;eff< \u000bn;\u0000jmj;eff, where clockwise\nmodes (m < 0) have lower frequencies and higher effec-\ntive damping due to the nonreciprocity.\nThe effective damping parameters drastically increase\nand for the lowest-lying modes apparently diverge close\nto the burst instability, while no such sign of nonan-\nalytical behavior can be observed in the case of the\n1\u0019skyrmion with the elliptic instability. For the same\nn;mmode, the effective damping parameter tends to in-\ncrease with skyrmion order kaway from the critical field\nregimes; for example, for the n= 0;m= 0mode at\nB= 1:00T one finds \u000b0;0;eff;1\u0019= 2:04,\u000b0;0;eff;2\u0019= 5:87,\nand\u000b0;0;eff;3\u0019= 10:09.\nClose to the collapse field, the effective damping pa-\nrameter of the n= 0;m= 0breathing mode tends to\n0.7 0.8 0.9 1.0 1.1 1.21.01.52.02.5\n0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15125102050100\n0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15125102050100FIG.4. Effectivedamping parameters calculatedaccordingto\nEq. (15) for the eigenmodes of the (a) 1\u0019, (b) 2\u0019, and (c) 3\u0019\nskyrmions, plotted on a logarithmic scale. The corresponding\nexcitation frequencies are shown in Fig. 1.\ndiverge as shown in Figs. 4(b), 4(c), and 5 for the 2\u0019,\n3\u0019, and 1\u0019skyrmions, respectively. Similarly to the\neigenfrequency converging to zero in Fig. 3, the criti-\ncal behavior of the effective damping may be approxi-\nmated by a power-law fit \u000b0;0;eff=A\u000b(Bc;1\u0019\u0000B)\u0000\f\u000b\nas shown in Fig. 5, this time with a negative exponent8\n4.45 4.46 4.47 4.48 4.49 4.50024681012\nFIG. 5. Effective damping parameter \u000b0;0;effof the breathing\nmoden= 0;m= 0of the 1\u0019skyrmion close to the collapse\nfield. The corresponding excitation frequencies are shown in\nFig. 3. Calculation data are shown by open symbols, red line\ndenotes the power-law fit \u000b0;0;eff=A\u000b(Bc;1\u0019\u0000B)\u0000\f\u000b.\ndue to the divergence. The fitting yields the parameters\nA\u000b= 0:96T\f\u000b,Bc;1\u0019= 4:4957T, and\f\u000b= 0:23. Natu-\nrally, the critical field values agree between the two fits,\nbut interestingly one also finds \ff=\f\u000bup to two digits\nprecision. Rearranging Eq. (14) yields\n\u000b0;0;eff\n\u000bRe!0;0=1\n\u000bjIm!0;0j; (18)\nwhere the left-hand side is proportional to\n(Bc;1\u0019\u0000B)\ff\u0000\f\u000bwhich is approximately constant\ndue to the exponents canceling. This indicates that\nwhile Re!0;0diverges close to the collapse field,\njIm!0;0j=\u000bremains almost constant at low \u000bvalues.\nD. Damping for higher \u000bvalues\nDue tothe divergences oftheeffective damping param-\neters found at the burst instability and collapse fields, it\nis worthwhile to investigate the consequences of using a\nfinite\u000bvalue in Eq. (6), in contrast to relying on Eq. (15)\nwhich is determined from the eigenvectors at \u000b= 0. The\n\u000bdependence of the real and imaginary parts of the !0;0\nbreathingmodefrequencyofthe 1\u0019skyrmionisdisplayed\nin Fig. 6, at a field value of B= 1T far from the el-\nliptic and collapse instabilities. As shown in Fig. 6(a),\nunlike circularly polarized modes described by Eq. (16)\nwhere Re!qdecreases smoothly and equals half of the\nundamped value at \u000b= 1, the Re!0;0value for the ellip-\ntically polarized eigenmode displays a much faster decay\nand reaches exactly zero at around \u000b\u00190:58. According\ntoEq.(14), thisindicatesthatthecorrespondingeffective\ndamping parameter \u000b0;0;effdiverges at this point.\nSince the real part of the frequency disappears, the\n!q0=\u0000!\u0003\nqrelation connecting Re !q>0and Re!q0<0\n0.0 0.2 0.4 0.6 0.8 1.00102030405060\n0.0 0.2 0.4 0.6 0.8 1.00.00.10.20.30.40.50.6FIG. 6. (a) Frequency f0;0=Re!0;0=2\u0019and (b) inverse\nlifetimejIm!0;0jof then= 0;m = 0breathing mode of\nthe 1\u0019skyrmion at B= 1T as a function of the Gilbert\ndamping parameter \u000b. The solutions of Eq. (6) for the ellip-\ntically polarized eigenmode of the 1\u0019skyrmion are compared\nto Eqs. (16)-(17) which are only valid for circularly polarized\nmodes.\nsolutions of Eq. (6) discussed in Sec. IID no longer holds,\nand two different purely imaginary eigenfrequencies are\nfound in this regime as shown in Fig. 6(b). This is analo-\ngous to overdamping in a classical linear harmonic oscil-\nlator, meaningthatthepurelyprecessionalfirst-orderdif-\nferential equation describing circularly polarized modes\nis transformed into two coupled first-order differential\nequations [23] with an effective mass term for the breath-\ning mode of k\u0019skyrmions. This implies that when per-\nforming spin dynamics simulations based on the Landau–\nLifshitz–Gilbert equation, the value of the Gilbert damp-\ning parameter has to be chosen carefully if the fastest\nrelaxation to the equilibrium spin structure is required.\nThe high effective damping of the breathing mode in the\n\u000b\u001c1limit (cf. Fig. 4(a)) ensures that the inverse\nlifetime of the elliptically polarized excitations remains\nlarger for a wide range of \u000bvalues in Fig. 6(b) than what\nwould be expected for circularly polarized modes based9\n0.85 0.90 0.95 1.00 1.05 1.10 1.1505101520\n0.85 0.90 0.95 1.00 1.05 1.10 1.150.000.020.040.060.080.10\nFIG. 7. (a) Frequency f0;0=Re!0;0=2\u0019and (b) inverse\nlifetimejIm!0;0jof then= 0;m= 0breathing mode of the\n2\u0019skyrmion at \u000b= 0:1as a function of the external magnetic\nfieldB. The solutions of Eq. (6) for the elliptically polarized\neigenmode of the 2\u0019skyrmion are compared to Eqs. (16)-(17)\nwhich are only valid for circularly polarized modes.\non Eq. (17). Note that contrary to Sec. IIIB, Re !0;0be-\ncomingzeroinFig.6(a)doesnotindicateaninstabilityof\nthe system, since stability is determined by the eigenval-\nues of the matrix HSWin Eq. (2) which are independent\nof\u000b.\nSince the disappearance of Re !0;0and the bifurcation\nof Im!0;0occurs as the excitation frequency becomes\nsmaller, it is expected that such an effect may also be ob-\nservedatafixed \u000bvalueastheexternalfieldisdecreased.\nThis is illustrated for the n= 0;m= 0breathing mode\nof the 2\u0019skyrmion in Fig. 7 at \u000b= 0:1. For this interme-\ndiate value of the damping, the breathing mode becomes\noverdamped around B= 0:875T, which is significantly\nhigher than the burst instability between B= 0:775T\nandB= 0:800T (cf. Fig. 1(b) and the circularly polar-\nized approximation in Fig. 7(a)). This means that the\nlowest-lying breathing mode of the 2\u0019skyrmion cannot\nbe excited below this external field value. In Fig. 7(b) it\ncan be observed that contrary to the circularly polarizedapproximation Eq. (17) following the field dependence of\nthe frequency, for the actual elliptically polarized eigen-\nmodejIm!0;0jisalmostconstantforallfieldvaluesabove\nthebifurcationpoint. Althoughasimilarobservationwas\nmade at the end of Sec. IIIC as the system approached\nthe collapse field at \u000b= 0, it is to be emphasized again\nthat no instability occurs where Re !0;0disappears in\nFig. 7(a).\nIV. CONCLUSION\nIn summary, the localized spin wave modes of k\u0019\nskyrmions were investigated in an atomistic spin model,\nwith parameters based on the Pd/Fe/Ir(111) system. It\nwas found that the number of observable modes increases\nwith skyrmion order k, firstly because of excitations with\nhigher angular momentum quantum numbers mforming\nalong the larger perimeter of the skyrmion, secondly be-\ncause of nodes appearing between the multiple domain\nwalls. It was found that the 2\u0019and3\u0019skyrmions un-\ndergo a burst instability at low fields, in contrast to the\nelliptic instability of the 1\u0019skyrmion. At high field val-\nues the innermost ring of the structure collapses in all\ncases, connected to an instability of a breathing mode.\nThe effective damping parameters of the excitation\nmodes were determined, and it was found that for the\nsamen;mmode they tend to increase with skyrmion\norderk. The effective damping parameter of the n=\n0;m= 0breathing mode diverges at the burst and\ncollapse instabilities, but no such effect was observed\nin case of the elliptic instability. For higher values of\nthe Gilbert damping parameter \u000ba deviation from the\nbehavior of circularly polarized modes has been found,\nwith the breathing modes becoming overdamped. It was\ndemonstrated that such an overdamping may be observ-\nable in 2\u0019and3\u0019skyrmions for intermediate values of\nthe damping significantly above the burst instability field\nwhere the structures themselves disappear from the sys-\ntem.\nThe results presented here may motivate further ex-\nperimental and theoretical studies on k\u0019skyrmions, of-\nfering a wider selection of localized excitations compared\nto the 1\u0019skyrmion, thereby opening further possibilities\nin magnonics applications.\nACKNOWLEDGMENTS\nThe authors would like to thank A. Siemens for fruit-\nful discussions. Financial support for this work from the\nAlexander von Humboldt Foundation, from the Deutsche\nForschungsgemeinschaft via SFB 668, from the European\nUnion via the Horizon 2020 research and innovation pro-\ngram under Grant Agreement No. 665095 (MAGicSky),\nand from the National Research, Development and Inno-\nvation Office of Hungary under Project No. K115575 is\ngratefully acknowledged.10\n[1] A. N. Bogdanov and D. A. Yablonski ˘i, Sov. Phys. JETP\n68, 101 (1989).\n[2] A. Fert, V. Cros, and J. Sampaio, Nat. Nanotechnol. 8,\n152 (2013).\n[3] W. Jiang, X. Zhang, G. Yu, W. Zhang, X. Wang, M. B.\nJungfleisch, J. E. Pearson, X. Cheng, O. Heinonen, K. L.\nWang, Y. Zhou, A. Hoffmann, and S. G. E. te Velthuis,\nNat. Phys. 13, 162 (2017).\n[4] P.-J. Hsu, A. Kubetzka, A. Finco, N. Romming, K. von\nBergmann, and R. Wiesendanger, Nat. Nanotechnol. 12,\n123 (2017).\n[5] F. Büttner, I. Lemesh, M. Schneider, B. Pfau, C. M.\nGünther, P. Hessing, J. Geilhufe, L. Caretta, D. Engel,\nB. Krüger, J. 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Schäfer, Magnetic Domains (Springer,\nBerlin, 1998)."    },    {        "title": "2401.00486v1.Molecular_Hybridization_Induced_Antidamping_and_Sizable_Enhanced_Spin_to_Charge_Conversion_in_Co20Fe60B20__β__W_C60_Heterostructures.pdf",        "content": "Molecular Hybridization Induced Antidamping and Sizable Enhanced Spin-to-Charge\nConversion in Co 20Fe60B20/β-W/C 60Heterostructures\nAntarjami Sahoo 1, Aritra Mukhopadhyaya 2, Swayang Priya Mahanta 1, Md. Ehesan Ali 2, Subhankar Bedanta 1,3\n1 Laboratory for Nanomagnetism and Magnetic Materials (LNMM),\nSchool of Physical Sciences, National Institute of Science Education and Research (NISER),\nAn OCC of Homi Bhabha National Institute (HBNI), Jatni 752050, Odisha, India\n2 Institute of Nano Science and Technology, Knowledge City, Sector-81, Mohali, Punjab 140306, India and\n3 Center for Interdisciplinary Sciences (CIS), National Institute of Science Education and Research (NISER),\nAn OCC of Homi Bhabha National Institute (HBNI), Jatni, Odisha 752050, India\nMd. Ehesan Ali∗and Subhankar Bedanta†\nDevelopment of power efficient spintronics devices has been the compelling need in the post-CMOS\ntechnology era. The effective tunability of spin-orbit-coupling (SOC) in bulk and at the interfaces of\nhybrid materials stacking is a prerequisite for scaling down the dimension and power consumption of\nthese devices. In this work, we demonstrate the strong chemisorption of C 60molecules when grown\non the high SOC β-W layer. The parent CFB/ β-W bilayer exhibits large spin-to-charge intercon-\nversion efficiency, which can be ascribed to the interfacial SOC observed at the Ferromagnet/Heavy\nmetal interface. Further, the adsorption of C 60molecules on β-W reduces the effective Gilbert\ndamping by ∼15% in the CFB/ β-W/C 60heterostructures. The anti-damping is accompanied by a\ngigantic ∼115% enhancement in the spin-pumping induced output voltage owing to the molecular\nhybridization. The non-collinear Density Functional Theory calculations confirm the long-range en-\nhancement of SOC of β-W upon the chemisorption of C 60molecules, which in turn can also enhance\nthe SOC at the CFB/ β-W interface in CFB/ β-W/C 60heterostructures. The combined amplifica-\ntion of bulk as well interfacial SOC upon molecular hybridization stabilizes the anti-damping and\nenhanced spin-to-charge conversion, which can pave the way for the fabrication of power efficient\nspintronics devices.\nI. INTRODUCTION\nSpintronic logic and memory devices have proven to\nbe one of the most suitable research domains to meet\nthe ultra-low power consumption demand in the post-\nComplementary Metal Oxide Semiconductor (CMOS)\ntechnology era. Especially, with the advent of artificial\nintelligence and the Internet of Things (IoT), the further\nscaling down of CMOS technology can reach its physi-\ncal limits in size, speed, and static energy consumption.\nThe conceptualized spin orbit torque magnetic random\naccess memory (SOT-MRAM) devices which take the ad-\nvantage of spin Hall effect (SHE) can bring down the\nenergy consumption to femto Joule from the pico Joule\nscale [1, 2]. The SHE based magnetization switching\nmechanism in SOT-MRAMs also offers much improved\nendurance owing to the separation in data writing and\nreading paths. Though these potentials of SOT-MRAMs\nhave attracted major foundries, several challenges need\nto be addressed before the commercialization of SOT-\nMRAMs [1, 2]. The increase of writing efficiency to re-\nduce power consumption is one of those aspects which\nrequires significant consideration. In this context, the\nspin Hall angle, θSH(JS⁄JC) of the nonmagnetic layer\npresent in the SOT-MRAMs, where J Cand J Sare the\ncharge and spin current densities, respectively, plays a\n∗ehesan.ali@inst.ac.in\n†sbedanta@niser.ac.incritical role in determining the writing efficiency [3]. The\nefficient charge to spin interconversion can lead to the\nfaster switching of magnetization of the adjacent mag-\nnetic layer via SHE. Hence, various types of heavy metals\n(HMs), like Pt, Ta, W, Ir etc. have been investigated in\nthe past two decades to reduce the power consumption\nof future spintronic devices [4–6]. On a similar note, the\nRashba-Edelstein effect (REE) occurring at the interfaces\nwith spatial inversion symmetry breaking and high spin\norbit coupling (SOC) has also the potential for the man-\nifestation of efficient charge to spin interconversion[7–9].\nHence, the combination of SHE and REE can be the most\nsuitable alternative for the development of power efficient\nspintronics application.\nAmong all the heavy metals, highly resistive ( ρβ−W∼\n100−300µΩ cm) metastable β-W possesses the largest\nθSH∼-0.3 to -0.4 [10–14], which makes it a strong can-\ndidate for SOT-MRAM devices. Usually, additional re-\nactive gases, like O 2, N2, and F are employed to stabilize\nthe A15 crystal structure of β-W [11] and consequently, a\nlarger θSHis realized. For example, Demasius et al., have\nbeen able to achieve θSH∼-0.5 by incorporating the oxy-\ngen into the tungsten thin films [12]. Interface engineer-\ning also acts as a powerful tool for enhancing the writing\nefficiency in β-W based SOT-MRAM devices [15–17]. For\ninstance, the presence of an interfacial atomically thin\nα-W layer in CoFeB/ α-W/β-W trilayer suppresses the\nspin backflow current, resulting in a 45% increase in the\nspin mixing conductance [15]. Further, the REE evolved\nat the W/Pt interface owing to the charge accumulationarXiv:2401.00486v1  [cond-mat.mtrl-sci]  31 Dec 20232\ngenerates an additional spin orbit field on the adjacent\nferromagnet (FM) NiFe (Py) layer [18]. The coexistence\nof SHE and REE has also been reported in CoFeB/ β-Ta\nand NiFe/Pt bilayers, where the interfacial SOC arising\nat the FM/HM interface plays a vital role in the spin-\nto-charge interconversion phenomena [19, 20]. More in-\nterestingly, a recent theoretical work has predicted the\ninterfacial SOC mediated spin Hall angle of Pt can be 25\ntimes larger than the bulk value in NiFe/Pt heterostruc-\nture [21]. The interfacial SOC mediated spin accumula-\ntion has also been reported to occur at the Rashba-like\nβ-Ta/Py interface without flowing the DC current [22].\nThe spin pumping induced by the ferromagnetic reso-\nnance results in non-equilibrium spin accumulation at the\ninterface which consequently reduces the effective Gilbert\ndamping of the β-Ta/Py bilayer. The reduction in ef-\nfective damping, also termed as antidamping, is similar\nto the interfacial Rashba like SOT, observed in various\nHM/FM heterostructures [22]. The anti-damping phe-\nnomena without the requirement of DC current depends\non several factors, like SOC of HM, strength of built in\nelectric field at the interface, interface quality etc. Hence,\nthe interface engineering via tuning the interfacial SOC\ninβ-W based HM/FM heterostructures can be the path\nforward for developing power efficient SOT-MRAM de-\nvices.\nTill the date, most of the interface engineering re-\nsearch have been focused on employing an additional\nmetallic or oxide layer in the HM/FM system for the\nenhancement of spin-to-charge interconversion efficiency.\nWhereas, the organic semiconductors (OSCs) can also\nbe incorporated in the HM/FM system to fabricate hy-\nbrid power efficient spintronic devices owing to their\nstrong interfacial hybridization and charge transfer na-\nture at metal/OSC interface [23]. Recently, the SOC\nof Pt has been found to be enhanced due to the on-\nsurface physical adsorption of C 60(fullerene) molecules\nin YIG/Pt/C 60trilayer [24]. However, the θSHof Pt is\nusually found to be smaller compared to β-W and it is\nimportant to investigate the magnetization dynamics and\nspin to charge conversion phenomena in FM/ β-W/C 60\nheterostructures. Hence, in this article, we report the\neffect of molecular hybridization at β-W/C 60interface\non magnetization dynamics and spin-to-charge conver-\nsion phenomena in Co 20Fe60B20(CFB)/ β-W/C 60het-\nerostructures. The molecular hybridization reduces the\neffective Gilbert damping and also enhances the spin-to-\ncharge conversion efficiency owing to the enhanced SOC\nofβ-W and consequent strengthening of possible Rashba-\nlike interaction at the CFB/ β-W interface. The strong\nchemisorption at the β-W/C 60interface and evolution of\nenhanced SOC of β-W upon the molecular hybridization\nhave also been confirmed by the first principle density\nfunctional theory (DFT) based calculations.II. EXPERIMENTAL AND COMPUTATIONAL\nMETHODS\nFour different types of heterostructures with CFB (7\nnm)/ β-W (2.5, 5 nm) (Figure 1 (a)) and CFB (7 nm)/ β-\nW (2.5, 5 nm)/C 60(13 nm) (Figure 1 (b)) stackings were\nfabricated on Si/SiO 2(300 nm) substrates for the inves-\ntigation of magnetization dynamics and spin pumping\nphenomena. In addition, the CFB (7 nm)/ β-W (10, 13\nnm) heterostructures were also fabricated to reaffirm the\nstabilization of β-W. The heterostructure stackings and\ntheir nomenclatures are mentioned in Table I. The CFB\nandβ-W layers were grown by DC magnetron sputter-\ning, while the Effusion cell equipped in a separate cham-\nber (Manufactured by EXCEL Instruments, India) was\nused for the growth of the C 60over layers in the CFWC\nseries. While preparing the CFWC1 and CFWC2, the\nsamples were transferred in-situ into the chamber with\nEffusion cell in a vacuum of ∼10−8mbar for the de-\nposition of C 60. Before the fabrication of heterostruc-\ntures, thin films of CFB, β-W, and C 60were prepared for\nthickness calibration and study of magnetic and electrical\nproperties. The base pressure of the sputtering chamber\nand chamber with Effusion cells were usually maintained\nat∼4×10−8mbar and ∼6×10−9mbar, respectively.\nThe structural characterizations of individual thin films\nand heterostructures were performed by x-ray diffrac-\ntion (XRD), x-ray reflectivity (XRR), and Raman spec-\ntrometer. The magneto-optic Kerr effect (MOKE) based\nmicroscope and superconducting quantum interference\ndevice based vibrating sample magnetometer (SQUID-\nVSM) were employed for the static magnetization char-\nacterization and magnetic domain imaging. The mag-\nnetization dynamics was investigated by a lock-in based\nferromagnetic resonance (FMR) spectrometer manufac-\ntured by NanOsc, Sweden. The heterostructures were\nkept in a flip-chip manner on the co-planner waveguide\n(CPW). The FMR spectra were recorded in the 4-17 GHz\nrange for all the samples. The FMR spectrometer set-up\nis also equipped with an additional nano voltmeter using\nwhich spin-to-charge conversion phenomena of all the de-\nvices were measured via inverse spin Hall effect (ISHE)\nwith 5-22 dBm RF power. The contacts were given at\nthe two opposite ends of 3 mm ×2 mm devices using\nsilver paste to measure the ISHE induced voltage drop\nacross the samples. The details of the ISHE measure-\nment set-up are mentioned elsewhere [25, 26].\nDensity functional theory (DFT)-based electronic\nstructure calculations were performed in the Vienna Ab-\ninitio simulation package (VASP) [27, 28] to understand\nthe interface’s chemical bonding and surface reconstruc-\ntions. The plane wave basis sets expand the valance\nelectronic states, and the core electrons are treated\nwith the pseudopotentials. The core-valance interac-\ntions are considered with the Projected Augmented Wave\nmethod. The exchange-correlation potentials are treated\nwith Perdew, Bruke and Ernzerof (PBE) [29] functional\nwhich inherits the Generalized Gradient Approximation3\nTABLE I. Details of the heterostructures and their nomenclatures\nSl. No. Stacking Nomenclature\n1 Si/SiO 2(300 nm)/CFB (7 nm)/ β-W (2.5 nm) CFW1\n2 Si/SiO 2(300 nm)/CFB (7 nm)/ β-W (5 nm) CFW2\n3 Si/SiO 2(300 nm)/CFB (7 nm)/ β-W (2.5 nm)/C 60(13 nm) CFWC1\n4 Si/SiO 2(300 nm)/CFB (7 nm)/ β-W (5 nm)/C 60(13 nm) CFWC2\n(GGA). This functional produces a reliable understand-\ning of similar kinds of interfaces. The convergences in the\nself-consistent field iterations were ensured with a plane-\nwave cutoff energy of 500 eV and a tolerance of 10−6\neV/cycle. A D3 dispersion correction term, devised by\nGrimme, accounts for the long-range interaction terms\nwas employed in the calculations. The optimized unit\ncell parameter obtained from the aforementioned meth-\nods for the cubic A15 crystal of the β-W is 5.014 ˚A,\nwhich resembles the experimental parameter of 5.036 ˚A.\nA 5×2×1 repetition is used to construct the (210) surface\nunit cell of the β-W to model the surface supercell. The\nlower two atomic layers were fixed at the bulk, and the\nremaining three layers were allowed to relax during the\ngeometry optimization. The surface layer of β-W con-\ntains the C 60molecules. To understand the effect of the\nspin-orbit coupling interactions, we have performed the\nnon-collinear DFT calculations as implemented in VASP.\nThe E SOC calculated from these calculations quantifies\nthe strength of the SOC term in the Hamiltonian.\nIII. RESULTS AND DISCUSSION\nThe grazing incidence x-ray diffraction (GIXRD) was\nperformed for all the heterostructures. The XRD pat-\nterns of CFB/ β-W heterostructures with different thick-\nnesses of β-W are shown in Figure 1 (c). The presence of\n(200), (210) and (211) Bragg’s peaks of W at 35.5◦, 39.8◦\nand 43.5◦indicate the stabilization of metastable βphase\nof W (A-15 crystal structure) [5, 30]. In addition, we\nhave also observed the (320) and (321) Bragg’s peaks of\nW, which further suggests the growth of polycrystalline\nβ-W. The relative intensity of (320) and (321) Bragg’s\npeaks of W is lower compared to (200), (210) and (211)\nBragg’s peaks, consistent with previous reports [30]. The\nBragg’s peaks are more prominent for heterostructures\nwith thicker W layers as diffraction intensity increases\nwith the increase in W thickness. The XRD patterns for\nCFWC1 and CFWC2 are similar to that for CFW1 and\nCFW2, respectively, as the thickness of β-W are same.\nHere, we have not used the reactive gases like, O 2and\nN2for the growth of β-W unlike some previous report\n[11]. The resistivity of W films with thicknesses 2.5, 5,\n10 nm were measured by standard four probe methods.\nThe resistivity decreases with increase in thickness of W\nand were found in between ∼300-100 µΩ-cm, further\nconfirming the growth of βphase of W [5, 30]. We donot also observe the (110), (200), (210) Bragg’s peaks for\nthe bcc α-W in the XRD patterns and the α-W would\nhave also exhibited one order less resistivity compared to\nwhat we have observed [30]. The stabilization of pure β\nphase of W is quite important for future SOT device fab-\nrication and hence, we can expect a high spin-to-charge\nconversion efficiency in our CFB/ β-W heterostructures\nowing to high SOC of β-W [10, 30].\nThe XRR measurements were performed for all the\nsamples in both the CFW and CFWC series to confirm\nthe desired thickness of individual layers and to investi-\ngate the interface quality. Figure S1 (Supporting Infor-\nmation) shows the XRR patterns of all the heterostruc-\ntures considered for the present study. The experimental\ndata were fitted using GenX software and the simulated\npatterns are also shown in Figure S1 (red curves). The\npresence of Kiessig oscillations for all the films infer the\nabsence of a high degree of interfacial disorder and dis-\nlocations. The relative peak positions and intensity of\nthe simulated patterns agree quite well with the experi-\nmentally observed low angel XRR data. The fit provides\nthe anticipated thickness of individual layer in each sam-\nple as mentioned in Table I. The interface roughness for\nall the heterostructures were found in between 0.2-0.5\nnm, further inferring the high quality growth of both the\nseries of samples. Figure S2 (Supporting Information)\ndisplays the Raman spectra of 13 nm C 60film grown on\nSi/SiO 2(300 nm) substrate with the same growth con-\ndition as in the heterostructures. The presence of A g(2)\nand H g(8) Raman modes of C 60around ∼1460 cm−1\nand 1566 cm−1, respectively confirms the growth of C 60\nfilm [31, 32]. In addition, the Raman mode around ∼\n495 cm−1corresponding to A g(1) mode of C 60is also ob-\nserved in the Raman spectrum. The anticipated thick-\nness of C 60in the C 60thin film, CFWC1 and CFWC2\nhas also been confirmed from the XRR measurements.\nThe Raman spectrum of our C 60film grown by effusion\ncells are quite similar to those prepared by different so-\nlution methods in HCl or N 2atmosphere [31, 32]. The\nsaturation magnetization and the magnetic domain im-\nages of all the heterostructures are found to be similar\n(see Supporting Information) as the bottom CFB layer\nis same for all the heterostructures.\nThe magnetization relaxation and propagation of spin\nangular momentum in the CFB thin film and the het-\nerostructures in both the CFW and CFWC series were\nstudied to explore the effect of high resistive β-W and\nβ-W/C 60bilayer by in-plane FMR technique. The het-4\nFIG. 1. Schematics of (a) Si/SiO 2/CFB/ β-W and (b) Si/SiO 2/CFB/ β-W/C 60heterostructures, (c) GIXRD patterns of\nSi/SiO 2/CFB/ β-W heterostructures with various thicknesses of β-W.\nerostructures are placed in a flip-chip manner on CPW\nas shown in the schematics in Figure S4 (a) (Support-\ning Information). Figure S4 (c) shows the typical FMR\nspectra of CFW1 and CFWC1 heterostructures measuredin the 4-17 GHz range. All the FMR spectra were fit-\nted to the derivative of symmetric and antisymmetric\nLorentzian function to evaluate the resonance field ( Hres)\nand linewidth (∆ H) [33]:\nFMRSignal =K14(∆H)(H−Hres)\n[(∆H)2+ 4(H−Hres)2]2−K2(∆H)2−4(H−Hres)2\n[(∆H)2+ 4(H−Hres)2]2+Offset, (1)\nwhere K 1and K 2are the antisymmetric and symmetric\nabsorption coefficients, respectively. The extracted Hres\nand ∆ Hvalues at different resonance frequencies ( f) of\nall the heterostructures are shown in Figure 2 (a-b). The\nfvsHresof different samples in the CFW and CFWC\nseries are plotted in Figure 2 (a). The fvsHresplots\nare fitted by using equation 2 [33]:\nf=γ\n2πq\n(HK+Hres)(HK+Hres+ 4πMeff),(2)\nwhere\n4πMeff= 4πMS+2KS\nMStFM\nand H K, KS, and t FMare the anisotropy field, perpen-\ndicular surface anisotropy constant, and the thicknessof FM, respectively. Here, γis the gyromagnetic ra-\ntio and 4 πMeffrepresents the effective magnetization.\nThe 4 πMeffextracted from the fitting gives similar val-\nues as compared with the saturation magnetization value\n(4πMS) calculated from the SQUID-VSM. Further, the\neffective Gilbert damping constant ( αeff) and hence, the\nmagnetization relaxation mechanism are studied from\nthe resonance frequency dependent FMR linewidth be-\nhavior. The ∆ Hvsfplots are shown in Figure 2 (b).\nThe linear dependency of ∆ Honfindicates the mag-\nnetic damping is mainly governed by intrinsic mechanism\nvia electron-magnon scattering rather than the extrinsic\ntwo magnon scattering. The ∆ Hvsfplots are fitted by5\nFIG. 2. (a) Frequency ( f) versus resonance field ( Hres) and (b) linewidth (∆ H) versus frequency ( f) behaviour for various\nheterostructures. The solid lines are the best fits to equation 2 and 3.\nthe following linear equation [33] to evaluate the αeff.\n∆H= ∆H0+4παeff\nγf, (3)\nwhere the ∆ H0is the inhomogeneous linewidth broad-\nening. The αeffvalues for all the heterostructures and\nCFB thin film obtained from the fitting are shown in Ta-\nble II. The αeffvalue for CFW series ( ∼0.0075 ±0.0001\nfor CFW1 and ∼0.0080 ±0.0001 for CFW2) are found\nto be larger compared to that of the CFB thin film\n(∼0.0059 ±0.0001). The enhancement of αeffindicates\nthe possible evolution of spin pumping mechanism in the\nCFB/ β-W bilayers. Interestingly, the αeffdecreases to\n∼0.0065 ±0.0001 upon the deposition of C 60molecules\non CFB/ β-W bilayers in CFWC series. The signifi-\ncant change in αefffor the CFB/ β-W/C 60heterostruc-\ntures compared to CFB/ β-W bilayers infers the modi-\nfication of physical properties of β-W layer in CFB/ β-\nW/C 60. The deposition of C 60molecules can lead to the\nmetal/molecule hybridization at the β-W/C 60interface,\nwhich in turn can alter the properties of β-W.\nThe DFT based first principle calculations were per-\nformed to elucidate further the molecular hybridization\nat the β-W/C 60interface and its consequences on the\nmagnetization dynamics of CFB/ β-W/C 60heterostruc-\ntures. The extended simulation supercell for the C 60on\nβ-W(210) are shown in Figure 3 (a). The C 60molecule\nis observed as strongly chemisorbed onto the β-W (210)\nsurface with an adsorption energy of -253.5 kcal/mol.\nThe adsorption energy is quite high as compared to the\nother substrates. For example, the adsorption energy for\nCo/C 60was found to be -90 kcal/mol [34] while for the\nPt/C 60interface it is reported to be -115 kcal/mol [35].\nThe chemisorption in case of β-W/C 60is quite strong\nand induces distortion to the spherical shape of the ad-\nsorbed C 60. The distance between two carbon atoms\nfrom two opposite hexagons of adsorbed C 60is shorter\nalong one direction compared to the other measured in\nthe plane (left panel of Figure 3 (a)). The diameter of C 60molecules decreases by 0.3 ˚Awhen it is measured perpen-\ndicular to the β-W (210) surface (right panel of Figure\n3 (a)). This distortion can be attributed to the W-C\nbond formation due to the strong chemisorption at the\nβ-W/C 60interface. This chemisorption strongly alters\nthe electronic structure of the β-W and C 60molecule\n(Figure 3 (b)). The pzorbital, which accommodates the\nπ-electrons of the C 60, hybridizes with the d-orbitals of\ntheβ-W atom and forms the hybridised interfacial states.\nThe out-of-plane d-orbitals ( dxz,dyzanddz2orbitals) are\nstrongly hybridized with the pzorbital of the carbon\natom over a large energy window near the Fermi energy\nlevel (Figure 3 and Figure S5 (Supporting Information)).\nThe sharp peaks observed in the DOS of free C 60layer\ngets significantly broadened, flattened, and shifted for β-\nW/C 60stacking. The strong metallo-organic hybridiza-\ntion also modifies the PDOS of various d-orbitals of β-\nW. The various d-orbitals become flattened and spread\nover larger energy spectrum around the Fermi level upon\nmolecular hybridization. The formation of the W-C bond\nalso costs a transfer of 3.25e−from the interfacial layer of\ntheβ-W to C 60molecule (Figure 3 (c)). This is relatively\nhigher compared to the previously reported the 0.25e−\ntransfer from Pt (111) and 3e−transfer from Cu (111) to\nthe adjacent C 60molecule, inferring the metallo-organic\nhybridization is quite stronger in case of β-W/C 60in-\nterface [35].Hence, the molecular hybridization of β-W\nis expected to alter its physical properties with greater\neffect and can be considered as an important tool to op-\ntimize the spintronics device performances.\nThe modified electronic structure was found to carry\na long-range effect on the strength of the spin-orbit cou-\npling. The E SOC of bare 2.5 nm β-W and 2.5 nm β-\nW covered with C 60molecules, and the variation of the\nESOC(∆E SOC) due to β-W/C 60hybridization are shown\nin Figure 4. The interfacial W atoms involved in the hy-\nbridization with C 60show a decrease in the E SOC. The\nrest of the W atoms from the surface layer exhibit an\nincrease in the E SOC. The lower atomic layers of W6\nFIG. 3. (a) The extended simulation supercell for the C 60onβ-W(210) substrate. The left panel shows the top view of the\nsurface supercell (along the z-axis), and the right panel shows the side view of the same. The pink balls of larger size and cyan\nballs of smaller size represent the tungsten and carbon atoms, respectively. The yellow bonds highlight the part of the C 60\nwhich takes part in the interface formation. The double-headed dotted arrows quantify the diameter of the C 60spheres in two\ndirections. (b-c) The modification in the electronic structure due to chemisorption of the C 60molecule on the β-W. (b) The\natom projected orbital resolved partial density of states of β-W(210), C 60, and β-W(210)/C 60, and (c) The electron density\nredistribution due to chemisorption. The red and green iso-surfaces depict electron density depletion and accumulation of the\nelectron density at the interface, respectively. The bi-coloured arrow depicts the direction of the electron transfer process.\nalso show an increment in the E SOC. The W layer, far-\nthest from the β-W/C 60interface (nearer to the CFB/ β-\nW interface), exhibits the most increased E SOC. Hence,\nthe hybridization at the β-W/C 60interface increases the\noverall spin-orbit coupling strength of the β-W layer.\nMore importantly, the SOC at the CFB/ β-W interface\nis enhanced for CFB/ β-W/C 60stacking compared to the\nCFB/ β-W bilayer. The enhanced bulk SOC of β-W and\nthe interfacial SOC at CFB/ β-W interface can facilitate\nan efficient spin to charge conversion in CFB/ β-W/C 60\nheterostructures.\nThe decrease in damping, usually know as anti-\ndamping, has been observed previously in FM/HM bilay-ers [22, 26, 30]. In those systems, the effective damping\nvalues become lower than the αeffof the FM layer and\nthis phenomenon has been attributed to the formation\nof Rashba like interfacial states [22, 30]. Similar type of\nevolution of Rashba like states at the CFB/ β-W inter-\nface can be expected due to structural inversion asym-\nmetry and large SOC of β-W. The spin accumulation\nat the CFB/ β-W interface can lead to evolution of the\nnon-equilibrium spin states. The non-equilibrium spin\nstates along with the enhanced SOC at CFB/ β-W inter-\nface due to molecular hybridization as confirmed from the\nDFT calculations can generate an additional charge cur-\nrent due to IREE and can also induce the antidamping7\nFIG. 4. The effect of the chemisorption of the C 60molecule at the β-W(210) surface on the E SOCof various atomic sites. The\npercentage change in the E SOC (∆E SOC) is calculated in terms of the change in the E SOC of the bare β-W(210) substrate.\nLayer 5 is the interfacial layer that interacts with the C 60, and layer 1 is the opposite to the β-W/C 60interface layer.\ntorque on the magnetization of FM layer. The antidamp-\ning torque can make the magnetization precession rela-\ntively slower and thus decreasing the αeffof the CFB/ β-\nW/C 60heterostructures compared to the CFB/ β-W bi-\nlayer. The control of Gilbert damping of FMs by inter-\nfacing with adjacent non-magnetic metal/organic bilay-\ners can also provide an alternative to the search for low\ndamping magnetic materials. Especially, the low cost and\nabundant availability of carbon based organic molecules\ncan be commercially beneficial in optimizing the mag-\nnetic damping for spintronic applications. Further, the\nGilbert damping modulation can also control the effec-\ntive spin mixing conductance ( g(↑↓)\neff) of the heterostruc-\ntures which also plays a vital role for efficient spin current\ntransport across the interface. Hence, the g(↑↓)\neffof all the\nheterostructures was calculated from the damping con-\nstant measurement by equation 4 [33]:\ng(↑↓)\neff=4πMstCFB\ngµB(αCFB/NM −αCFB), (4)\nwhere g, µBandtCFB are the Land´ e g factor (2.1),\nBohr’s magnetron, and thickness of CFB layer, respec-\ntively. αCFB/NM is the damping constant of bilayer ortri-layers and αCFB is the damping constant of the ref-\nerence CFB thin film. The g(↑↓)\nefffor CFW1 and CFW2\n(Table II) are relatively higher compared to the previ-\nous reports on FM/ β-W bilayers. Especially, the g(↑↓)\neffof\nCFW2 is one order higher than that reported for Py/ β-W\nbilayer (1.63 ×1018m−2) [30], and 2 order higher com-\npared to that of the YIG/ β-W (5.98 ×1017m−2) [14].\nThis indicates the absence of any significant amount of\nspin back flow from β-W layer and high SOC strength\nof parent β-W layer in our system. However, the g(↑↓)\neff\nvalues decrease for the CFWC1 and CFWC2 tri-layers\nowing to anti-damping phenomena.\nThe ISHE measurements were performed for all the\nheterostructures in CFW and CFWC series to gain more\ninsights about the effect of molecular hybridization in\nCFB/ β-W/C 60on the magnetization dynamics and spin\nto charge conversion efficiency. Figure 5 shows the typi-\ncal field dependent DC voltage ( Vdc) measured across the\nCFB (7 nm)/ β-W (5 nm)/C 60(13 nm) heterostructure\nunder FMR conditions. In order to separate the symmet-\nric (VSY M) and asymmetric ( VASY M ) components, the\nVdcvsHplots were fitted with the following Lorentzian\nfunction:\nVdc=VSY M(∆H)2\n(∆H)2+ (H−Hres)2+VASY M(∆H)(H−Hres)\n(∆H)2+ (H−Hres)2(5)\nThe extracted field dependent VSY M andVASY M are also plotted in Figure 5. Similar type of field depen-8\nFIG. 5. VMEAS ,VSY M andVASY M versus Hfor CFB (7)/ β-W(5)/C 60(13) [CFWC2] heterostructure with ϕ∼(a) 180◦and\n(b) 0◦measured at 15 dBm RF power. The red curve is Lorentzian fit with equation 5 to VdcvsHplot.\nTABLE II. Effective Gilbert damping, spin mixing conduc-\ntance, and symmetric component of measured DC voltage for\ndifferent heterostructures.\nHeterostructures αeff(±0.0001) g(↑↓)\neff(1019m−2)VSY M(µV)\nCFB 0.0059 - -\nCFW1 0.0075 0.87 1.08\nCFW2 0.0080 1.13 1.25\nCFWC1 0.0064 0.27 2.32\nCFWC2 0.0065 0.32 1.78\ndent VMEAS ,VSY M, and VASY M are also observed for\nother samples in both CFW and CFWC series. The\nVSY M is mainly contributed by the spin pumping voltage\n(VISHE ) and the spin rectification effects arising from\nthe anisotropic magnetoresistance (AMR) [ VAMR] [33].\nWhereas, the asymmetric component of the measured\nvoltage arises solely due to anomalous Hall effect and\nAMR [33]. The sign of VSY M is reversed when ϕ(angel\nbetween the perpendicular direction to the applied mag-\nnetic field ( H) and direction of voltage measurement) is\nchanged from 0◦to 180◦(Figure 5), confirming the pres-\nence of ISHE in our heterostructures. The field depen-\ndent VSY M for all the four heterostructures are plotted\nin Figure 6 (a-b). Interestingly, the VSY M value at the\nresonance field for CFB/ β-W/C 60trilayers is found to be\nincreased compared to that for CFB/ β-W bilayers. Theincrement is ∼115% for β-W thickness 2.5 nm, while it\nbecomes ∼20% for β-W thickness 5 nm. The gigantic\nenhancement of VSY M for CFB (7)/ β-W(2.5)/C 60(13)\ninfers the modification of SOC of β-W when capped with\norganic C 60molecules and the presence of an additional\nspin to charge conversion effect in the heterostructures.\nThe power dependent spin-to-charge conversion measure-\nments were also performed to further confirm the en-\nhancement of VSY M. The spin pumping induced voltage\nincreases linearly with the RF power as shown in Fig-\nure 6 (c) for both CFW1 and CFWC1. The VSY M at\ndifferent RF power is found to be increased for CFWC1\ncompared to CFW1, which further confirms the molec-\nular hybridization induced enhanced spin-to-charge con-\nversion. As the thickness, magnetic properties of bot-\ntom CFB layer is same for all the heterostructures, the\ncontribution of VAMR is expected to be same for CFB\n(7 nm)/ β-W(2.5 nm)/C 60(13 nm) and CFB (7 nm)/ β-\nW(2.5 nm). Hence, the sizable increase in the measured\nvoltage can be attributed to the enhanced SOC of β-W\ndue to molecular hybridization and additional charge cur-\nrent flowing at the CFB/ β-W interface due to IREE as\nshown in the Figure 6 (d). In order to understand the en-\nhanced spin-to-charge conversion phenomena further, we\nalso calculated the θSHof the heterostructures by using\nequations 6 and 7 [14, 33]:\nJs=g(↑↓)\neffγ2h2\nrfℏ[γ4πMs+p\n(γ4πMs)2+ 4ω2]\n8πα2\neff[(γ4πMs)2+ 4ω2]×(2e\nℏ), (6)9\nFIG. 6. VSY M versus applied magnetic field with ϕ∼180◦for (a) CFB (7)/ β-W(2.5) [CFW1] and CFB (7)/ β-W(2.5)/C 60\n(13) [CFWC1] and (b) CFB (7)/ β-W(5) [CFW2] and CFB (7)/ β-W(5)/C 60(13) [CFWC2] heterostructures measured at 15\ndBm RF power, (c) Power dependent VSY M for CFW1 and CFWC1 (The solid line is the linear fit), (d) Schematic showing\nthe spin-to-charge conversion phenomena in CFB/ β-W/C 60heterostructures.\nVISHE =wyLρNM\ntNMθSHλNMtanh(tNM\n2λNM)Js, (7)\nwhere the ρNMis the resistivity of the β-W measured by\nfour-probe technique and Lis the length of sample. The\nRF field ( hrf) and the width of the CPW transmission\nline ( wy) in our measurements are 0.5 Oe (at 15 dBm RF\npower) and 200 µm, respectively. The λNMfor the β-W\nhas been taken as ∼3 nm from the literature [36]. Angel\ndependent ISHE measurements were performed to sepa-\nrate the AMR contribution from the VSY M. The contri-\nbution of VAMR was found to be one order smaller com-\npared to V ISHE . For example, the VAMR and V ISHE for\nCFW2 heterostructure are found to be ∼0.15µV and ∼\n1.25µV, respectively (See Supporting Information). The\nρNMfor 5 nm β-W is found to be 250 µΩ cm. Hence, the\nθSHfor CFB (7 nm)/ β-W (5 nm) bilayer estimated using\nequations 6 and 7 is found to be ∼-0.6±0.01. A similar\ntype of calculation for CFB (7 nm)/ β-W (2.5 nm) bilayer\nestimates the θSHto be∼-0.67±0.01. The observed θSHvalue is larger compared to that reported in the literature\n[10–12]. The high SOC of our β-W and higher spin mix-\ning conductance could be responsible for this enhanced\nθSH. Further, the interfacial SOC at CFB/ β-W interface\ncan also induce an additive spin-to-charge conversion ef-\nfect, contributing to the enhancement of θSH. Such type\nof interfacial SOC mediated enhanced spin-to-charge con-\nversion has been reported previously for NiFe/Pt and\nCFB/ β-Ta [19, 20]. Here, it is important to note that\nit is difficult to disentangle the IREE and ISHE effect in\nthese type of FM/HM systems. On the other hand, the\ng(↑↓)\nefffor CFWC1 and CFWC2 decreases by 70 % due to\nthe anti-damping phenomena and hence, the reduction\ninJsaccording to equation 6. However, the VISHE for\nthe CFWC1 and CFWC2 are found to be larger than\nCFW1 and CFW2, respectively (Figure 6). This leads to\ntheθSHvalue >1, calculated using the equation 6 and10\n7 for CFB/ β-W/C 60heterostructures. This type of gi-\ngantic enhancement of θSHcannot be explained by mere\nbulk ISHE in β-W. The enhanced θSHcan be partly at-\ntributed to the enhanced bulk SOC of β-W upon molec-\nular hybridization as predicted by the DFT calculations.\nFurther, our DFT calculations also predict the enhance-\nment of SOC of β-W layer closer to the CFB/ β-W inter-\nface due to the molecular hybridization in the CFB/ β-\nW/C 60heterostructures. The larger interfacial SOC and\ninversion symmetry breaking at the CFB/ β-W interface\nmakes the scenario favorable for realizing an enhanced\ninterfacial charge current due to the IREE as depicted in\nFigure 6 (d). Hence, the combination of bulk and interfa-\ncial SOC enhancement owing to the strong chemisorption\nof C 60onβ-W can attribute to the sizable increase in the\nθSHin CFB/ β-W/C 60heterostructures.\nThe enhanced output DC voltage due to the spin\npumping upon the C 60deposition on β-W is also con-\nsistent with the reduced effective damping value as dis-\ncussed earlier. The enhanced SOC of β-W and the struc-\ntural inversion asymmetry at the CFB/ β-W interface\ncan stabilize the Rashba like states at FM/HM inter-\nface [19, 20]. The IREE mediated spin to charge con-\nversion has received considerable interest after it was\ndiscovered at the Ag/Bi interface [7]. Till the date,\nmost of the IREE effects have been experimentally re-\nalized at the all inorganic metal/metal, metal/oxide or\noxide/oxide interfaces [9]. Our experiments and theoret-\nical calculations show that the molecular hybridization\nat the HM/OSC interface can also help in strengthen-\ning the Rashba spin-orbit coupling at the FM/HM in-\nterface. The Rashba interaction leads to the spin split-\nting of bands, whose magnitude is dependent on the SOC\nstrength at the interface. Upon the molecular hybridiza-\ntion, the SOC strength of β-W is further enhanced. This\ncould have lead for a larger Rashba coefficient αRand\nhence, a relatively larger IREE at the FM/HM inter-\nface. The simultaneous observation of ISHE and IREE\nby engineering the HM interface with OSC can help in\nreducing the power consumption of future SOT-MRAM\ndevices. As the CFB/ β-W stacking is employed for fab-\nrication of spin Hall nano oscillators (SHNOs) [37], theincorporation organic molecules can also significantly en-\nhance their efficiency. Hence, the HM/C 60interface can\nreduce the power consumption for data storage as well as\nfacilitate in performing efficient spin logic operations.\nIV. CONCLUSION\nIn conclusion, we present that a strong interfacial SOC\ncan lead to the larger spin Hall angle in CFB/ β-W bi-\nlayer. The thermally evaporated organic C 60molecules\non CFB/ β-W bilayer leads to a strong chemisorption at\ntheβ-W/C 60interface. The experimental and theoreti-\ncal calculations confirm that the molecular hybridization\nenhances the bulk as well as interfacial SOC in CFB/ β-\nW/C 60heterostructures. The strengthening of techno-\nlogically important SOC manifests an anti-damping phe-\nnomena and gigantic ∼115% increase in spin-pumping\ninduced output voltage for CFB/ β-W/C 60stacking. The\ncontrol of magnetization dynamics and output efficiency\nin spintronics devices by the molecular hybridization can\nbe a viable alternative to the other interface engineering\nand surface alloying techniques. The stabilization of the\nanti-damping and enhanced spin-to-charge conversion by\ntuning the bulk as well interfacial SOC via employing\nthe cost effective, abundant organic molecule can pave\nthe way for the fabrication of next generation power effi-\ncient spintronics devices.\nV. ACKNOWLEDGEMENT\nWe acknowledge the Department of Atomic En-\nergy (DAE), the Department of Science and Technol-\nogy (DST) of the Government of India, and SERB\nproject CRG/2021/001245. A.S. acknowledges the DST-\nNational Postdoctoral Fellowship in Nano Science and\nTechnology. 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Kurebayas hi et al., Nature \nNanotechnology 9, 211 (2014). ] that have been widely studied for applications in magnetic \nmemory. We focus , in this article,  not on the spin orbit effect producing the above spin \ntorques, but on its magnifying the damping constant of all field like spin torques. As first \norder precession leads to second order damping, the Rashba constant is naturally co -opted, \nproducing a magnified field -like dam ping effect. The Landau -Liftshitz -Gilbert equations are \nwritten separately for the local magnet ization and the itinerant spin, allowing the \nprogression of magnetization to be self -consistently locked to  the spin.   \n \n  \n \n \n \nPACS: 03.65.Vf, 73.63. -b, 73.43. -f \n† Correspondence author:  \nSeng Ghee Tan  \nEmail: Tan_Seng_Ghee@dsi.a -star.edu.sg  \n Magnified Damping under Rashba Spin Orbit Coupling  \nNovember 1, 2015  \n \n \n2 \n 1. Introduction  \n      In spintronic and magnetic physics, magnetization switching and spin torque [1] have \nbeen well -studied. The advent of the Rashba spin-orbit coupling ( RSOC) [2,3]  due to \ninversion asymmetry at the i nterface of the ferromagnetic/heavy atom (FM/HA) \nheterostructure introduces new spin torque to the FM magnetization. The field -like [4-6] \nand the damping -like [7] SOC spin torque had been  theoretically derived  based on the gauge \nphysics and the Pancharatna m-Berry’s phase , as well as  experimentally verified  and resolved . \nThe numerous  observation s of spin -orbit generation of spin torque [8-10], are all related to \nthe experimental resolutions [6,7] of their field -like and damping -like nature, thus ushering \nin the possibility of spin -orbit based magnetic memory. While the damping -like spin torque  \ndue to Kurebayashi et al. [7] is dissipative in nature, the field -like due to Tan et al. [4,11], is \nnon-dissipative , and precession causing . Recent studies have even mo re clearly \ndemonstrated the physics and application promises of both the field -like and the damping -\nlike SOC spin torque [12-14]. Besides , similar SOC spin torque have also been studied \ntheoretically  in FM/3D -Rashba [15] and FM -topological -insulator materi al [16,17] , and \nexperimentally shown [18, 19 ] in topological insulator materials.  \n      The dissipative physics of all field -like magnetic torque terms have been  derived in \nsecond -order manifestation in a manner introduced by Gilbert  in the 1950 ’s. Conven tional \nstud y of magnetization dynamics is based on a Gilbert damping constant  which is  \nincorporated manually into the Landau -Lifshit z-Gilbert  (LLG)  equation. In this paper, we will \nfocus our attention not so much on the spin-orbit effect producing the SOC spin torque, as \non the spin -orbit effect magnifying the damping constant of all field -like spin torques. As \nfield -like spin torques,  regardless of origin s, generate first-order precession , the Rashba \nconstant will be co -opted in to the second -order damping effect, producing a mag nified \ndamping  constant . On the other hand, c onventional incorporation of the dissipative \ndamping physics into the LLG would fail t o account for the spin-orbit magnification of  the \ndamping strength . It would therefore be necessary to  deriv e the LLG equations from a \nHamiltonian  which  describes electron  due to  the local  FM magnetization (𝒎), and those \nitinerant (𝒔) and injected from external parts . We present a set of modified LLG  equation s \nfor the 𝒎 and the 𝒔. This will be necessary  for a more precise modeling of the 𝒎 trajectory  \nthat simultaneously tracks the  𝒔 trajectory. In summary, the two central themes of this Magnified Damping under Rashba Spin Orbit Coupling  \nNovember 1, 2015  \n \n \n3 \n paper is our presentation of  a self-consistent set of LLG equations under the Rashba SOC \nand the  derivation of the Ra shba -magnified d amping constant in the  second -order damping -\nlike spin torque .  \n \n2. Theory of Magnified Damping  \n    The system under consideration is a FM/HA hetero -structure  with inversion asymmetry \nprovided by the interface. F ree electron denoted by  𝒔, is injected in an in -plane manner into \nthe device . The FM equilibrium  electron  is denoted by  𝒎. One considers the external \nsource -drain bias  to inject  electron  of free-electron  nature 𝒔 into the FM with kinetic, \nscattering , magnetic, and spin -orbit  energ ies.  The Hamiltonian is \n𝐻𝑓=𝑝2\n2𝑚+𝑉𝑖𝑚𝑝𝑠+𝐽𝑠𝑑𝑺.𝑴 +𝜇0𝑴.𝑯𝒂𝒏𝒊+(2(𝜆+𝜆′)\nℏ)(𝒔+𝒎).(𝒑×𝑬𝒕)\n−𝑖(𝜆+𝜆′)(𝒔+𝒎).(∇×𝑬𝒕) \n(1) \nwhere  𝒔,𝒎  have  the unit s of angular momentum i.e.  𝑛ℏ\n2 ,  while 𝑴=(𝑔𝑠𝜇𝐵\nℏ)𝒎 has the unit \nof magnetic moment , and 𝜇𝐵=𝑒ℏ\n2𝑚 is the Bohr magneton . Note that  (2𝜆\nℏ) is the vacuum SOC \nconstant, while  (2𝜆′\nℏ=2𝜂𝑅\nℏ2𝐸𝑖𝑛𝑣) is the Rashba  SOC constant. The SOC part of the Hamiltonian \nillustrates th e simultaneous presence of vacuum and Rashba SOC. The proportion of the \nnumber of electron subject to each coupling would depend on the degree of hybridization.  \nBut s ince 𝜆′≫𝜆, the above can be written with just the Rashba SOC effect. Care is taken t o \nensure  𝜆,𝜆′ share the same dimension of 𝑇𝑒𝑠𝑙 𝑎−1, and  𝑬𝒕 is the total electric field ,  𝐽𝑠𝑑 is \nthe s -d coupling constant, 𝑉𝑖𝑚𝑝𝑠 denotes the spin flip scattering potential, 𝑯𝒂𝒏𝒊 denotes the \naniso tropy field of the FM  material. On the other hand, one needs to be aware that the \nabove is an e xpanded SOC expression that c omprises a momentum part as well as a \ncurvature part [20]. One can then consider  the physics of the electric  curvature as related to  \nthe time dynamic of the spin moment , which bears  a similar origin to the Faraday effect.  In \nthe modern context of  Rashba physics  [21], one considers electron spin to lock to the orbital \nangular momentum  𝑳 due to intrinsic spin orbit coupling at the  atomic level. Due to  broken  Magnified Damping under Rashba Spin Orbit Coupling  \nNovember 1, 2015  \n \n \n4 \n inversion symmetry , electric field (𝑬𝒊𝒏𝒗) points perpendicular to the plane of the FM/HA \nhost . Because of hybridization,  the 𝒔,𝑳,𝒑 of an electron is coupled in a complic ated way by \nthe electric field. In a simple way, one first considers 𝑳 to be  coupled as 𝐻=(2𝜆\nℏ)𝑳.(𝒑×\n𝑬𝒊𝒏𝒗). As spin 𝒔 is coupled via atomic spin orbit locking to 𝑳, an effective coupling of 𝒔 to \n𝑬𝒊𝒏𝒗 can be expected  to occur with strength as determined by the atomic electric field.   We \nwill now take things a step further to make an assumption that 𝒔 is also coupled  via 𝑳  to \nother sources of electric f ields  e.g. those  arising from spin dynamic  (𝒅 𝑴\n𝒅𝒕,𝒅 𝑺\n𝒅𝒕), in the same \nway that it is coupled to 𝑬𝒊𝒏𝒗 . The actual extent of coupling will , however,  be an \nexperimental parameter that measures  the efficiency of Rashba coupling to 𝑬𝒊𝒏𝒗 as opposed \nto electric fields (𝑬𝒎 ,𝑬𝒔) arising due to spin dynamic . The total electric field  in the system \nis now  𝑬𝒕=𝑬𝒊𝒏𝒗 +𝑬𝒎+𝑬𝒔 , where 𝑬𝒎 ,𝑬𝒔 arise due to 𝒅 𝑴\n𝒅𝒕,𝒅 𝑺\n𝒅𝒕, respectively. On the \nmomentum part of the Hamiltonian  2 𝜆′𝒔.(𝒌×𝑬𝒕), we only need to  consider  that \n𝑬𝒕=𝑬𝒊𝒏𝒗 as one can, for simplicity, consider 𝑬𝒎 and  𝑬𝒔 to simply vanish on average. Thus \nin this renewed treatment, the momentum part is : \n2𝜆′\nℏ𝒔.(𝒑×𝑬𝒊𝒏𝒗)=𝜂𝑅𝝈.(𝒌×𝒆𝒊𝒏𝒗) \n(2) \nwhere 𝜂𝑅=𝜆′ℏ𝐸𝑖𝑛𝑣 is the Rashba constant that has been vastly measured  in many material \nsystems  with  experimental values ranging from 0.1 to 2 𝑒𝑉𝐴̇. On the curvature part, one \nconsiders   𝑬𝒕=𝑬𝒎+𝑬𝒔 without the 𝑬𝒊𝒏𝒗 as 𝑬𝒊𝒏𝒗  is spatially uniform and thus would have \nzero curvature.  In summary, the theory of this paper has it that the time -dynamic of the \nspin in a Rashba system produces a curvature part o f  𝑖𝜆′(𝒔+𝒎).(∇×𝑬𝒕). Without the \nRashba effect, this energy term  would just  take on the vacuum constant of (2𝜆\nℏ) instead of \nthe magnified (2𝜆′\nℏ).  The key physics is that in a Rashba FM/HA system , curvature \n𝑖𝜆′(𝒔+𝒎).(∇×𝑬𝒕) is satisfied by the first-order precession due to 𝒅𝑴\n𝒅𝒕,𝒅𝑺\n𝒅𝒕  which provide \nthe electric field curvature  in the form of −𝜇0(1+𝜒𝑚)𝑑 𝑴\n𝑑𝑡=∇×𝑬𝒎 , and  −𝜇0(1+\n𝜒𝑠)𝑑𝑺\n𝑑𝑡=∇×𝑬𝒔 , where we remind reader again that 𝑴,𝑺 have the unit of magnetic \nmoment.  This results in  spin becoming couple d to its own time dynamic, producing a spin -Magnified Damping under Rashba Spin Orbit Coupling  \nNovember 1, 2015  \n \n \n5 \n orbit second -order damping -like spin torque.  The electric field effect is illustrated in Fig. 1 \nbelow:  \n         \n \n \n \n \n \n \n \n \nFig.1 . Magnetic precession under the effect of electric fields due to inv ersion asymmetry, self -dynamic of  𝑑𝑴\n𝑑𝑡 \nand the spin dynamic of  𝑑𝑺\n𝑑𝑡 . Projecting 𝑑𝑀 to the heterostructure surface, one could visualize the emergence \nof an induced electric field in the form of 𝛻𝑋𝐸 in such orientation as to satisfy the law of electromagnetism.  \n \n \n    One notes that the LLG equation is normally derived by letting 𝑺 satisfy the physical \nrequirements of spin transport . One example  of these requirements is assumed and \ndiscussed in REF 1 , with definitions  contained  therein :   \n𝑺(𝒓,𝑡)=𝑆0𝒏+𝜹𝑺 \n𝑱(𝒓,𝑡)=−𝜇𝐵𝑃\n𝑒 𝑱𝒆⊗𝒏−𝐷0∇𝜹𝑺 \n(3) \nwhere 𝒏  is the unit vector of 𝑴, and 𝐷0 is the spin diffusion constant.  Thus 𝑺=𝑺𝟎+𝜹𝑺 \nwould be the total spin density that contains , respectively,  the equilibrium, the non-\nequilibrium adiabatic,  non-adiabatic , and Rashba field -like terms , i.e. 𝜹𝑺=𝜹𝑺𝒂+𝜹𝑺𝒏𝒂+\n𝜹𝑺𝑹.  One notes that 𝑺𝟎 is the equilibrium part of 𝒔 that is aligned to 𝒎, meaning 𝒔𝟎 could  \nexist  in the absence of external field and current in the system. The conditions to satisfy are \nrepresented explicitly by the equations  of:   𝑑𝑀 \n𝐸 𝑓𝑖𝑒𝑙𝑑  𝑑𝑢𝑒 𝑡𝑜 \n𝑖𝑛𝑣𝑒𝑟𝑠𝑖𝑜𝑛  𝑎𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑦  \n𝛻𝑋𝐸 \n𝑑𝑀 Magnified Damping under Rashba Spin Orbit Coupling  \nNovember 1, 2015  \n \n \n6 \n 𝜕𝜹𝑺 \n𝜕𝑡=0, 𝐷0∇2𝜹𝑺=0,−𝜇𝐵𝑃\n𝑒 𝛁.𝑱𝒆𝑴\n𝑀𝑠=0, 𝑠0𝑴(𝒓,𝑡)\n𝑡𝑓𝑀𝑠=0 \n (4) \nIn the steady state treatment where  𝜕  𝜹𝑺\n𝜕𝑡=0, one recover s the adiabatic component of \n𝜹𝑺𝒂=𝒏×𝒋𝒆.𝛁𝒏 , and the non -adiabatic component of 𝜹𝑺𝒏𝒂=𝒋𝒆.𝛁𝒏. We also take the \nopportunity  here  to reconcile this with  the gauge physics of spin torque, in which case , the \nspin potential 𝐴𝜇𝑠𝑚=𝑒 [𝛼 𝑈𝐸𝑖𝜎𝑗𝜀𝑖𝑗𝜇𝑈†+𝑖ℏ\n𝑒𝑈𝜕𝜇𝑈†] would  correspond , respectively, to  \n𝜹𝑺𝑹+ 𝜹𝑺𝒂. In fact, t he emergent spin p otential [22, 23] can be considered to encapsulate  \nthe physics of electron interaction with the local magnetization under the effect of SOC [4, 5 , \n24-26]. Here  we caution that 𝜹𝒔𝑹 is restricted to the field -like spin -orbit effect  only .  \n    However, in this paper ,  𝑺 is defined to satisfy the transport equations in Eq.(4) except for \n𝜕𝜹𝑺 \n𝜕𝑡=0. Keeping the dynamic property of 𝑺 here allows  a self -consistent equation set \n𝑑𝑺\n𝑑𝑡,𝑑𝑴\n𝑑𝑡 to be introduced . The energy as experienced by the 𝑺,𝑴 electron are, respectively,  \n𝐻𝑓𝑠=𝑺.𝛿𝐻𝑓\n𝛿𝑺 ,       𝐻𝑓𝑚=𝑴.𝛿𝐻𝑓\n𝛿𝑴 \n(5) \nwith caution that  𝐻𝑓𝑠≠𝐻𝑓𝑚 . Upon rearrangement, the 𝒔,𝒎 centric energ ies are, \nrespectively,   \n𝐻𝑓𝑠=(𝑝2\n2𝑚+𝑉𝑖𝑚𝑝𝑠+𝐽𝑠𝑑𝑺.𝑴+𝑺.𝑩𝑹−𝒊𝜆′𝒔.(∇×𝑬𝒕)) \n𝐻𝑓𝑚=(𝐽𝑠𝑑𝑴.𝑺+𝜇0𝑴.𝑯𝒂−𝒊𝜆′𝒎.(∇×𝑬𝒕) ) \n(6) \nwhere 2𝜆′\nℏ𝒔.(𝒑×𝑬𝒕)=𝑺.𝑩𝑹, while  2𝜆′\nℏ𝒎.(𝒑×𝑬𝒕)  vanishes . We particularly note that \nthere have been recent discussions on the field -like [4,6,11 ] spin orbit torque as well as the \ndamping [7] version. With 𝑑𝒔\n𝑑𝑡=−𝟏\n𝒊ℏ[𝒔,𝐻𝑓𝑠] ,𝑑𝒎\n𝑑𝑡=−𝟏\n𝒊ℏ[𝒎,𝐻𝑓𝑚], one would now  have four \ndissipative torque t erms experienced by electron 𝒔,𝒎 as shown below : Magnified Damping under Rashba Spin Orbit Coupling  \nNovember 1, 2015  \n \n \n7 \n ( 𝝉𝑺𝑺 𝝉𝑺𝑴\n𝝉𝑴𝑺 𝝉𝑴𝑴)=𝑖𝜆′𝜇0(𝒔×(1+𝜒𝑠−1)𝑑𝑺\n𝑑𝑡𝒔×(1+𝜒𝑚−1)𝑑𝑴\n𝑑𝑡\n𝒎×(1+𝜒𝑠−1)𝑑𝑺\n𝑑𝑡𝒎×(1+𝜒𝑚−1)𝑑𝑴\n𝑑𝑡) \n(7) \nTo be consistent with conventional necessity to preserve magnetization norm in the physics \nof the LLG equation, we will drop the  off-diagonal terms which are  norm -breaking (non -\nconservation) . This is in order to keep the LLG equation in its conventional norm -conserving  \nform, simplifying physics and calculation therefrom. Nonetheless, the non -conserving parts \nrepresent new dynamic physics that can be analysed in the future with techniques other \nthan the familiar LLG  equations.  The self -consistent pair of spin torque equations in their \nopen forms are: \n𝜕𝑺\n𝜕𝑡=−(𝑺× 𝑩𝑹+𝑺\n𝑡𝑓)−1\n𝑒𝛻𝑎(𝑗𝑎𝒔 𝑺)−(𝑺×𝑴\n𝑚𝑡𝑒𝑥)−𝝉𝑺𝑺 \n𝜕𝑴\n𝜕𝑡=−𝛾𝑴×𝜇0𝑯𝒂−𝑴×𝑺\n𝑚 𝑡𝑒𝑥−𝝉𝑴𝑴  \n(8) \nwhere 𝐽𝑠𝑑=1\n𝑚𝑡𝑒𝑥 has been applied, 𝛾 is the gyromagnetic ratio, 𝜒𝑚 is the susceptibility.  For \nthe stud y of Rashba -magnified damping i n this paper, we only need to keep  the most \nrelevant term which is 𝝉𝑴𝑴=𝑖 𝜂𝑅\nℏ𝐸𝑖𝑛𝑣𝜇0(1+𝜒𝑚−1) 𝒎×𝑑 𝑴\n𝑑𝑡. In the phenomenological physics \nof Gilbert,  the first-order precession leads inevitably to the second -order dissipative terms \nvia  𝒔.𝒅𝑺\n𝒅𝒕 ,𝒎.𝒅𝑴\n𝒅𝒕. But in this paper, the general SOC physics had been expanded  as shown in \nearlier sections,  so that the dissipative terms  are to naturally arise fr om such expansion. The \nadvantage of the non -phenomenological approach is  that, as said earlier, the Rashba \nconstant will be co -opted into the second -order damping effect, resulting in the \nmagnification of the damping constant associated with all field -like spin torque.   \n \n Magnified Damping under Rashba Spin Orbit Coupling  \nNovember 1, 2015  \n \n \n8 \n 3. Conclusion  \n     The im portant result in this paper is that the damping constants have been magnified by \nthe Rashba effect. This would not be possible if the damping constant was incorporated \nmanually by standard means of Gilbert. As the Rashba constant is larger than the vacuum \nSOC constant  as can be deduced from Table 1 and shown below  \n𝛼𝑅=𝛼𝜆′\n𝜆 ,  \n(9) \nmagnetization dynamics in FM/HA hetero -structure with inversion asymmetry (interface, or \nbulk) might have to be modelled with the new equations. It is important to remind  that all \npreviously measured 𝜂𝑅 has had 𝐸𝑖𝑛𝑣 captured in the measured value.  But w hat is needed in \nour study is the coupling of 𝑺 to a dynamic electric field, and that requires the value of just \nthe coupling strength (𝜆′). As most measurement is carr ied out for 𝜂𝑅, the exact knowledge \nof 𝐸𝑖𝑛𝑣 corresponding to a specific 𝜂𝑅 will have a direct impact on the actual value of 𝜆′.  We \nwill, nonetheless, provide a quick, possibly exaggerated estimate. Noting that 𝜆=𝑒ℏ\n4𝑚2𝑐2  \nand 𝜆′=𝜂𝑅\nℏ𝐸𝑖𝑛𝑣, and taking one measured value  of 𝜂𝑅=1×10−10𝑒𝑉𝑚 , corresponding to a \n𝐸𝑖𝑛𝑣=1010𝑉/𝑚, the magnification of 𝛼 works out to 104 times in magnitude , which may \nseem unrealistically strong . The caveat lies  in the exact correspondence of 𝜂𝑅 to 𝐸𝑖𝑛𝑣, which \nremains to be determined experimentally.  For example, if an experimentally determined   \n𝜂𝑅 actual ly corresponds to a much larger 𝐸𝑖𝑛𝑣, that would mean that 𝜆′=𝜂𝑅\nℏ𝐸𝑖𝑛𝑣 which \nmagnifies the damping constant through  𝛼𝑅=𝛼𝜆′\n𝜆 might actually be much lower than \npres ent estimate.  Therefore, it is worth remembering, for simplicity sake  that 𝛼𝑅 actually \ndepends on the ratio of 𝜂𝑅\n𝐸𝑖𝑛𝑣 but not 𝜂𝑅.  It has  also been assumed that  𝑳 couples to  𝑬𝒔,𝑬𝒎 \nwith the same efficiency that it couples to 𝑬𝒊𝒏𝒗.  This is still uncertain as th e Rashba \nconstant with respect to 𝑬𝒔,𝑬𝒎 might actually be lower than  those  𝜂𝑅 values that have \nbeen experimentally measured mostly with respect to 𝑬𝒊𝒏𝒗. Last, we note that as damping \nconstant has been magnified here, and as increasingly high -precision, live monitoring of \nsimult aneous 𝒔,𝒎 evolution is no longer redundant in smaller devices, care has been taken Magnified Damping under Rashba Spin Orbit Coupling  \nNovember 1, 2015  \n \n \n9 \n to present the LLG equations  in the form of a self-consistent  pair of dynamic equations \ninvolving 𝑴 and 𝑺. This will be necessary for the accurate modeling of the simultaneous \ntrajectory of both 𝑴 and 𝑺.   \n \nTable 1. Summary of damping torque and damping con stant with and without Rashba effects.   \n Hamiltonian  Torque  Damping constant  \n1. 𝐻=(2𝜆\nℏ)𝒔.(𝒑×𝑬𝒕) \n𝜆=𝑒ℏ\n4𝑚2𝑐2  \n𝜕𝒎\n𝑑𝑡=𝑖𝜆𝜇0𝒎×(1+𝜒𝑚−1)𝜕𝑴\n𝑑𝑡  \n𝛼=𝑖𝜆\n2𝜇0𝑀𝑠(1+𝜒𝑚−1) \n2. 𝐻𝑅=(2𝜆′\nℏ)𝒔.(𝒑×𝑬𝒕) \n𝜆′=𝜂𝑅\nℏ𝐸𝑖𝑛𝑣  \n𝜕𝒎\n𝑑𝑡=𝑖𝜆′𝜇0𝒎×(1+𝜒𝑚−1)𝑑𝑴\n𝑑𝑡  \n𝛼𝑅=𝑖𝜆′\n2𝜇0𝑀𝑠(1+𝜒𝑚−1) \n    \n \n \n \n \nREFERENCES  \n \n \n[1] S. Zhang & Z. Li, “Roles of Non -equilibrium conduction electrons on the  magnetization dynamics \nof ferromagnets”, Phys. Rev. Letts 93, 127204 (2004).  \n[2] F.T. Vasko, “Spin splitting in the spectrum of two -dimensional electrons due to the surface \npotential”, Pis’ma Zh.  Eksp. Teor. Fiz.  30, 574 (1979) [ JETP Lett. , 30, 541].  \n[3] Y.A. Bychkov  & E.I. Rashba, “Properties of a 2D electron gas with lifted spectral degeneracy”, \nPis’ma Zh. Eksp. Teor. Fiz. , 39, 66 (1984) [ JETP Lett. , 39, 78].  \n[4] S. G. Tan, M. B. A. 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A. Jalil & Seng Ghee Tan, “Robustness of topological Hall effect of nontrivial spin \ntextures”, Scientific Reports 4, 5123 (2014).  \n[24] Y.B. Bazaliy, B.A. Jones, S. -C. Zhang, “Modific ation of the Landau -Lifshitz equation in the \npresence of a spin -polarized current in colossal - and giant -magnetoresistive materials”, Phys. Rev. B \n57 (1998) 3213(R).  \n[25] Takashi Fujita, MBA Jalil, SG Tan, Shuichi Murakami, “Gauge Field in Spintronics”, J.  Appl. Phys. \n[Appl. Phys. Rev.] 110, 121301 (2011).  \n[26] Seng Ghee Tan & Mansoor B.A. Jalil  “Introduction to the Physics of Nanoelectronic”, Woodhead \nPublishing Limited, Cambridge, U.K . Chapter -5 (2012).  \n \n  "    },    {        "title": "1805.11468v1.Gilbert_damping_in_non_collinear_magnetic_system.pdf",        "content": "arXiv:1805.11468v1  [cond-mat.mtrl-sci]  29 May 2018APS/123-QED\nGilbert damping in non-collinear magnetic systems\nS. Mankovsky, S. Wimmer, H. Ebert\nDepartment of Chemistry/Phys. Chemistry, LMU Munich,\nButenandtstrasse 11, D-81377 Munich, Germany\n(Dated: May 30, 2018)\nThe modification of the magnetization dissipation or Gilber t damping caused by an inhomoge-\nneous magnetic structure and expressed in terms of a wave vec tor dependent tensor α(/vector q) is in-\nvestigated by means of linear response theory. A correspond ing expression for α(/vector q) in terms of\nthe electronic Green function has been developed giving in p articular the leading contributions to\nthe Gilbert damping linear and quadratic in q. Numerical results for realistic systems are pre-\nsented that have been obtained by implementing the scheme wi thin the framework of the fully\nrelativistic KKR (Korringa-Kohn-Rostoker) band structur e method. Using the multilayered system\n(Cu/Fe 1−xCox/Pt)nas an example for systems without inversion symmetry we demo nstrate the\noccurrence of non-vanishing linear contributions. For the alloy system bcc Fe 1−xCoxhaving inver-\nsion symmetry, on the other hand, only the quadratic contrib ution is non-zero. As it is shown, this\nquadratic contribution does not vanish even if the spin-orb it coupling is suppressed, i.e. it is a direct\nconsequence of the non-collinear spin configuration.\nPACS numbers: 71.15.-m,71.55.Ak, 75.30.Ds\nI. INTRODUCTION\nThe magnetization dissipation in magnetic materi-\nals is conventionally characterized by means of the\nGilbert damping (GD) tensor αthat enters the Landau-\nLifshitz-Gilbert (LLG) equation [1]. This positive-\ndefinite second-rank tensor depends in general on the\nmagnetization direction. It is well established that in\nthe case of spatially uniformly magnetized ferromagnetic\n(FM) metals two regimes of slow magnetization dynam-\nics can be distinguished, which are governed by differ-\nent mechanisms of dissipation [2–4]: a conductivity-like\nbehaviour occuring in the limiting case of ordered com-\npounds that may be connected to the Fermi breathing\nmechanism and a resistivity-likebehaviourshown by ma-\nterials with appreciable structural, chemical or tempera-\nture induced disorder and connected to a spin-flip scat-\nteringmechanism. Animportantissueisthatbothmech-\nanisms are determined by the spin-orbit coupling in the\nsystem (see e.g. [2, 4, 5]). During the last years, it was\ndemonstrated by variousauthors that first-principles cal-\nculationsforthe GD parameterforcollinearferromagntic\nmaterials allow to cover both regimes without use of any\nphenomenological parameters. In fact, in spite of the dif-\nferences concerning the formulation for the damping pa-\nrameter and the corresponding implementaion [6–8], the\nnumerical results are in generalin rathergood agreement\nwith each other as well as with experiment.\nIn the case of a pronounced non-collinear magnetic\ntexture, e.g. in the case of domain walls or topologi-\ncally nontrivial magnetic configurations like skyrmions,\nthe description of the magnetization dissipation assum-\ning a spatial-invariant tensor αis incomplete, and a non-\nlocal character of GD tensor in such systems has to be\ntaken into account [9–11]. This implies that the dissipa-\ntive torque on the magnetization should be representedby the expression of the following general form [12]:\nτGD= ˆm(/vector r,t)×/integraldisplay\nd3r′α(/vector r−/vector r′)∂\n∂tˆm(/vector r′,t).(1)\nIn the case of a magnetic texture varying slowly in space,\nhowever, an expansion of the damping parameter in\nterms of the magnetization density and its gradients [11]\nis nevertheless appropriate:\nαij=αij+αkl\nijmkml+αklp\nijmk∂\n∂rlmp(2)\n+αklpq\nij∂\n∂rkml∂\n∂rpmq+... ,\nwhere the first term αijstands for the conventional\nisotropic GD and the second term αkl\nijmkmlis associated\nwith the magneto-crystalline anisotropy (MCA). The\nthird so-called chiral term αklp\nijmk∂\n∂rlmpis non-vanishing\nin non-centrosymmetric systems. The important role of\nthis contribution to the damping was demonstrated ex-\nperimentally when investigating the field-driven domain\nwall(DW)motioninasymmetricPt/Co/Pttrilayers[13].\nAs an alternative to the expansion in Eq. (2) one can\ndiscuss the Fourier transform α(/vector q) of the damping pa-\nrametercharacterizinginhomogeneousmagneticsystems,\nwhich enter the spin dynamics equation\n∂\n∂t/vector m(/vector q) =−γ/vector m(/vector q)×/vectorH−/vector m(/vector q)×α(/vector q)∂\n∂t/vector m(/vector q).(3)\nIn this formulation the term linear in qis the first chiral\nterm appearing in the expansion of α(/vector q) in powers of q.\nFurthermore, it is important to note that it is directly\nconnected to the αklp\nijmk∂\n∂rlmpterm in Eq. (2).\nBy applying a gauge field theory, the origin of the\nnon-collinear corrections to the GD can be ascribed to\nthe emergent electromagnetic field created in the time-\ndependent magnetic texture [14, 15]. Such an emergent2\nelectromagneticfieldgivesrisetoaspincurrentwhosedi-\nvergence characterizes the change of the angular momen-\ntum in the system. This allows to discuss the impact of\nnon-collinearity on the GD via a spin-pumping formula-\ntion[9,14,16]. Somedetailsofthephysicsbehind thisef-\nfect depend on the specific propertiesofthe materialcon-\nsidered. Accordingly, different models for magnetisation\ndissipation were discussed in the literature [9, 12, 14, 17–\n19]. Non-centrosymmetric two-dimensional systems for\nwhich the Rashba-like spin-orbit coupling plays an im-\nportant role havereceived special interest in this context.\nThey have been discussed in particular by Akosa et al.\n[19], in order to explain the origin of chiral GD in the\npresence of a chiral magnetic structure.\nThe fourth term on the r.h.s. of Eq. (2) corresponds\nto a quadratic term of an expansion of α(/vector q) with re-\nspect to q. It was investigated for bulk systems with\nnon-magnetic [20] and magnetic [9] impurity atoms, for\nwhich the authors have shown on the basis of model con-\nsideration that it can give a significant correction to the\nhomogeneous GD in the case of weak metallic ferromag-\nnets. In striking contrast to the uniform part of the GD\nthis contribution does not require a non-vanishing spin-\norbit interaction.\nTo our knowledge, only very few ab-initio investiga-\ntions on the Gilbert damping in non-collinear magnetic\nsystems along the lines sketched above have been re-\nported so far in the literature. Yuan et al. [21] calcu-\nlated the in-plane and out-of-plane damping parameters\nin terms of the scattering matrix for permalloy in the\npresence of N´ eel and Bloch domain walls. Freimuth et\nal. [22], discuss the properties of a q-dependent Gilbert\ndamping α(/vector q) calculated for the one-dimensional Rashba\nmodelinthepresenceofthe N´ eel-typenon-collinearmag-\nnetic exchange field, demonstrating different GD for left-\nhanded and right-handed DWs. Here we extend the for-\nmalism developed before to deal with the GD in ferro-\nmagnets [6], to get access to non-collinear system. The\nformalism based on linear response theory allows to ex-\npand the GD parameters with respect to a modulation\nof the magnetization expressed in terms of a wave vector\n/vector q. Correspondingnumerical results will be presented and\ndiscussed.\nII. GILBERT DAMPING FOR\nNON-COLLINEAR MAGNETIZATION\nIn the following we focus on the intrinsic contribution\nto the Gilbert damping, excluding spin current induced\nmagnetizationdissipationwhich occursin the presenceof\nan external electric field. For the considerations on the\nmagnetization dissipation an adiabatic variation of the\nmagnetization in the time and space domain is assumed.\nMoreover, it is assumed that the magnitude of the local\nmagnetic moments is unchanged during a change of the\nmagnetization, i.e. the exchange field should be strong\nenough to separate transverse and longitudinal parts ofthe magnetic susceptibility. With these restrictions, the\nnon-local Gilbert damping can be determined in terms of\nthe spin susceptibility tensor\nχαβ(/vector q,ω) =i1\nV∞/integraldisplay\n0dt∝angbracketleftˆSα(/vector q,t)ˆSα(−/vector q,0)∝angbracketright0ei(ω−δ)t,(4)\nwhereˆSα(/vector q,t) is the /vector q- andt-dependent spin operator\nand reduced units havebeen used ( /planckover2pi1= 1). With this, the\nFourier transformationofthe real-spaceGilbert damping\ncan be represented by the expression [23, 24]\nααβ(/vector q) =γ\nM0Vlim\nω→0∂ℑ[χ−1]αβ(/vector q,ω)\n∂ω.(5)\nHereγ=gµBis the gyromagneticratio, M0=µtotµB/V\nis the equilibrium magnetization and Vis the volume of\nthe system. In order to avoid the calculation of the dy-\nnamical magnetic susceptibility tensor χ(/vector q,ω), which is\nthe Fourier transformed of the real space susceptibility\nχ(/vector r−/vector r′,ω), it is convenient to represent χ(/vector q,ω) in Eq.\n(5), in terms of a correlation function of time deriva-\ntives ofˆS. As˙ˆScorresponds to the torque /vectorT, that may\ninclude non-dissipative and dissipative parts, one may\nconsider instead the torque-torque correlation function\nπ(/vector q,ω) [24–27].\nAssuming the magnetization direction parallelto ˆ zone\nobtains the expression for the Gilbert damping α(/vector q)\nα(/vector q) =γ\nM0Vlim\nω→0∂ℑ[ǫ·π(/vector q,ω)·ǫ]\n∂ω. (6)\nwhereǫ=/bracketleftbigg\n0 1\n−1 0/bracketrightbigg\nis the transverse Levi-Civita tensor.\nThisimpliesthefollowingrelationshipofthe αtensorele-\nments with the elements of the torque-torque correlation\ntensorπ:αxx∼ −πyyandαyy∼ −πxx[24].\nUsing Kubo’s linear response theory in the Matsubara\nrepresentation and taking into account the translational\nsymmetry of a solid the torque-torque correlation func-\ntionπαβ(/vector q,ω) can be expressed by (see, e.g. [28]):\nπαβ(/vector q,iωn) =1\nβ/summationdisplay\npm∝angbracketleftTαG(/vectork+/vector q,iωn+ipm)\nTβG(/vectork,ipm)∝angbracketrightc,(7)\nwhereG(/vectork,ip) is the Matsubara Green function and ∝angbracketleft...∝angbracketrightc\nindicates a configurational average required in the pres-\nence of any disorder (chemical, structural or magnetic)\nin the system. Using a Lehman representation for the\nGreen function [28]\nG(/vectork,ipm) =/integraldisplay+∞\n−∞dE\nπℑG+(/vectork,E)\nipm−E(8)\nwithG+(/vectork,E) the retarded Green function and using the\nrelation\n1\nβ/summationdisplay\npm1\nipm+iωn−E11\nipm−E2=f(E2)−f(E1)\niωn+E2−E13\nfor the sum over the Matsubara poles in Eq. (7), the torque-torq ue correlation function is obtained as:\nπαβ(/vector q,iωn) =1\nΩBZ/integraldisplay\nd3k+∞/integraldisplay\n−∞dE1\nπ+∞/integraldisplay\n−∞dE2\nπTr/angbracketleftbigg\nTαℑG(/vectork,E1)TβℑG(/vectork,E2)f(E2)−f(E1)\niωn+E2−E1/angbracketrightbigg\nc. (9)\nPerfoming finally the analytical continuation iωn→ω+iδone arrives at the expression\nΓαβ(/vector q,ω) =−π\nΩBZ/integraldisplay\nd3k+∞/integraldisplay\n−∞dE1\nπ+∞/integraldisplay\n−∞dE2\nπTr/angbracketleftbigg\nTαℑG(/vectork+/vector q,E1)TβℑG(/vectork,E2)/angbracketrightbigg\nc(f(E2)−f(E1))δ(ω+E2−E1)\n=−π\nΩBZ/integraldisplay\nd3k+∞/integraldisplay\n−∞dE\nπTr/angbracketleftbigg\nTαℑG(/vectork+/vector q,E)TβℑG(/vectork,E+ω)/angbracketrightbigg\nc(f(E)−f(E+ω)) (10)\nfor the imaginary part of the correlation function with Γ αβ(/vector q,ω) =−πℑπαβ(/vector q,ω). Accordingly one gets for the\ndiagonal elements of Gilbert damping tensor the expression\nααα(/vector q) =γ\nM0Vlim\nω→0∂[ǫ·Γ(/vector q,ω)·ǫ]\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nαα\n=γπ\nM0Vlim\nω→0∂\n∂ω1\nΩBZ/integraldisplay\nd3k+∞/integraldisplay\n−∞dE\nπ2(f(E+ω)−f(E))Tr/angbracketleftbigg\nTβℑG(/vectork+/vector q,E)TβℑG(/vectork,E+ω)/angbracketrightbigg\nc\n=γ\nM0V1\nΩBZ/integraldisplay\nd3k+∞/integraldisplay\n−∞dE\nπδ(E−EF)Tr/angbracketleftbigg\nTβℑG(/vectork+/vector q,E)TβℑG(/vectork,E)/angbracketrightbigg\nc\n=1\n4[ααα(/vector q,G+,G+)+ααα(/vector q,G−,G−)−ααα(/vector q,G+,G−)−ααα(/vector q,G−,G+)], (11)\nwhere the index βof the torque operator Tβis related to the index αaccording to Eq. 6, and the auxiliary functions\nααα(/vector q,G±,G±) =γ\nM0Vπ1\nΩBZ/integraldisplay\nd3kTr/angbracketleftbigg\nTβG±(/vectork+/vector q,EF)TβG±(/vectork,EF)/angbracketrightbigg\nc(12)\nexpressed in terms of the retarded and advanced Green function s,G+andG−, respectively.\nTo account properly for the impact of spin-orbit coupling when dealin g with Eqs. (11) and (12) a description of\nthe electronic structure based on the fully relativistic Dirac formalis m is used. Working within the framework of local\nspin density formalism (LSDA) this implies for the Hamiltonian the form [2 9]:\nˆHD=c/vectorα·/vector p+βmc2+V(/vector r)+β/vectorσ·ˆ/vector mBxc(/vector r). (13)\nHereαiandβare the standard Dirac matrices, /vectorσdenotes the vector of relativistic Pauli matrices, /vector pis the relativistic\nmomentum operator [30] and the functions V(/vector r) and/vectorBxc=/vectorσ·ˆ/vector mBxc(/vector r) are the spin-averaged and spin-dependent\nparts, respectively, of the LSDA potential [31] with ˆ/vector mgiving the orientation of the magnetisation.\nWith the Dirac Hamiltonian given by Eq. (13), the torque operator ma y be written as /vectorT=β[/vector σ׈/vector m]Bxc(/vector r).\nFurthermore, the Green functions entering Eqs. (11) and (12) a re determined using the spin-polarized relativistic\nversion of multiple scattering theory [29, 32] with the real space re presentation of the retarded Green function given\nby:\nG+(/vector r,/vector r′,E) =/summationdisplay\nΛΛ′Zn\nΛ(/vector r,E)τnm\nΛΛ′(E)Zm×\nΛ′(/vector r′,E)\n−δnm/summationdisplay\nΛ/bracketleftbig\nZn\nΛ(/vector r,E)Jn×\nΛ′(/vector r′,E)Θ(r′\nn−rn)\n+Jn\nΛ(/vector r,E)Zn×\nΛ′(/vector r′,E)Θ(rn−r′\nn)/bracketrightbig\n. (14)4\nHere/vector r,/vector r′refertoatomiccellscenteredatsites nandm, respectively,where Zn\nΛ(/vector r,E) =ZΛ(/vector rn,E) =ZΛ(/vector r−/vectorRn,E) isa\nfunction centered at the corresponding lattice vector /vectorRn. The four-component wave functions Zn\nΛ(/vector r,E) (Jn\nΛ(/vector r,E)) are\nregular (irregular) solutions to the single-site Dirac equation labeled by the combined quantum numbers Λ = ( κ,µ),\nwithκandµbeing the spin-orbit and magnetic quantum numbers [30]. Finally, τnm\nΛΛ′(E) is the so-called scattering\npath operator that transfers an electronic wave coming in at site minto a wave going out from site nwith all possible\nintermediate scattering events accounted for.\nUsing matrix notation with respect to Λ, this leads to the following exp ression for the auxilary damping parameters\nin Eq. (12):\nααα(/vector q,G±,G±) =γ\nM0Vπ1\nΩBZ/integraldisplay\nd3kTr/angbracketleftbigg\nTβτ(/vectork+/vector q,E±\nF)Tβτ(/vectork,E±\nF)/angbracketrightbigg\nc. (15)\nIn the case of a uniform magnetization, i.e. for q= 0 one obviously gets an expression for the Gilbert damping tensor\nas it was worked out before [7]. Assuming small wave vectors, the te rmτ(/vectork+/vector q,E±\nF) can be expanded w.r.t. /vector qleading\nto the series\nτ(/vectork+/vector q,EF) =τ(/vectork,E)+/summationdisplay\nµ∂τ(/vectork,E)\n∂kµqα+1\n2/summationdisplay\nµν∂τ(/vectork,E)\n∂kµ∂kνqµqν+... (16)\nthat results in a corresponding expansion for the Gilbert damping:\nα(/vector q) =α+/summationdisplay\nµαµqµ+1\n2/summationdisplay\nµναµνqµqν+... (17)\nwith the following expansion coefficients:\nα0±±\nαα=g\nπµtot1\nΩBZTrace/integraldisplay\nd3k/angbracketleftbigg\nTβτ(/vectork,E±\nF)Tβτ(/vectork,E±\nF)/angbracketrightbigg\nc(18)\nαµ±±\nαα=g\nπµtot1\nΩBZTrace/integraldisplay\nd3k/angbracketleftbigg\nTβ∂τ(/vectork,E±\nF)\n∂kµTβτ(/vectork,E±\nF)/angbracketrightbigg\nc(19)\nαµν±±\nαα=g\nπµtot1\n2ΩBZTrace/integraldisplay\nd3k/angbracketleftbigg\nTβ∂2τ(/vectork,E±\nF)\n∂kµ∂kνTβτ(/vectork,E±\nF)/angbracketrightbigg\nc, (20)\nand with the g-factor 2(1+ µorb/µspin) in terms of the spin and orbital moments, µspinandµorb, respectively, and the\ntotal magnetic moment µtot=µspin+µorb. The numerically cumbersome term in Eq. (20), that involves the sec ond\norder derivative of the matrix of /vectork-dependent scattering path operator τ(/vectork,E), can be reformulated by means of an\nintegration by parts:\n1\nΩBZ/integraldisplay\nd3kTβτ(/vectork,EF)Tβ∂2τ(/vectork,EF)\n∂kµ∂kν=/bracketleftBigg/integraldisplay /integraldisplay\ndkβdkγTi\nβτ(/vectork,E)Tj\nβ∂τ(/vectork,E)\n∂kβ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleKα\n2\n−Kα\n2/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n=0\n−/integraldisplay /integraldisplay /integraldisplay\ndkαdkβdkγTβ∂τ(/vectork,EF)\n∂kµTβ∂τ(/vectork,EF)\n∂kν/bracketrightBigg\n=−1\nΩBZ/integraldisplay\nd3kTβ∂τ(/vectork,EF)\n∂kµTβ∂τ(/vectork,EF)\n∂kν\nleading to the much more convenient expression:\nαµν±±\nαα=−g\n2πµtot/integraldisplay\nd3kTr/angbracketleftbigg\nTβ∂τ(/vectork,E±\nF)\n∂kµTβ∂τ(/vectork,E±\nF)\n∂kν/angbracketrightbigg\nc. (21)\nIII. RESULTS AND DISCUSSIONS\nThe scheme presented above to deal with the Gilbert\ndamping in non-collinear systems has been implementedwithin the SPR-KKR program package [33]. To exam-5\nine the importance of the chiral correction to the Gilbert\ndamping a first application of Eq. (19) has been made\nfor the multilayer system (Cu/Fe 1−xCox/Pt)nseen as a\nnon-centrosymmetricmodelsystem. Thecalculatedzero-\norder (uniform) GD parameter αxxand the correspond-\ning first-order (chiral) αx\nxxcorrection term for /vector q∝bardblˆxare\nplotted in Fig. 1 top and bottom, respectively, as a func-\ntion of the Fe concentration x. Both terms, αxxandαx\nxx,\n0 0.2 0.4 0.6 0.8 100.20.4αxx\n0 0.2 0.4 0.6 0.8 1xCo0123αxxx (a.u.)\nFIG. 1: The Gilbert damping parameters αxx(top) and\nαx\nxx(bottom) calculated for the model multilayer system\n(Cu/Fe 1−xCox/Pt)nusing Eqs. (18) and (19), respectively.\nincrease approaching the pure limits w.r.t. the Fe 1−xCox\nalloy subsystem. In the case of the uniform parame-\nterαxx, this increase is associated with the dominating\nbreathing Fermi-surface damping mechanism. This im-\nplies that the modification of the Fermi surface (FS) in-\nduced by the spin-orbit coupling (SOC) follows the mag-\nnetization direction that slowly varies with time. An ad-\nditional contribution to the GD, having a similar origin,\noccurs for the non-centrosymmertic systems with heli-\nmagnetic structure. In this case, the features of the elec-\ntronicstructure governedby the lackofinversionsymme-\ntry result in a FS modification dependent on the helicity\nof the magnetic structure. This implies a chiral contri-\nbution to the GD which can be associated with the term\nproportional to the gradient of the magnetization. Ob-\nviously, this additional modification of the FS and the\nassociated mechanism for the GD does not show up for\na uniform ferromagnet. As αis caused by the SOC one\ncan expect that it vanishes for vanishing SOC. This was\nindeed demonstrated before [5]. The same holds also for\nαxthat is cased by SOC as well.\nAnother system considered is the ferromagnetic alloy\nsystem bcc Fe 1−xCox. As this system has inversion sym-\nmetry the first-order term αµshould vanish. This expec-\ntation could also be confirmed by calculations that ac-count for the SOC. The next non-vanishing term of the\nexpansion of the GD is the term ∝q2. The correspond-\ning second-order term αxx\nxxis plotted in Fig. 2 (bottom)\ntogether with the zero-order term αxx(top). The bot-\n0 0.1 0.2 0.3 0.4 0.500.511.52αxx× 103\n0 0.1 0.2 0.3 0.4 0.5xCo012αxxxx ((a.u.)2)Fe1-xCox\nFIG. 2: The Gilbert damping terms αxx(top) and αxx\nxx(bot-\ntom) calculated for bcc Fe 1−xCox.\ntom panel shows in addition results for αxx\nxxthat have\nbeen obtained by calculations with the SOC suppressed.\nAs one notes the results for the full SOC and for SOC\nsuppressed are very close to each other. The small dif-\nference between the curves for that reason have to be as-\ncribed to the hybridization of the spin-up and spin-down\nsubsystems due to SOC. As discussed in the literature\n[9, 17, 20] a non-collinear magnetic texture has a corre-\nsponding consequence but a much stronger impact here.\nIn contrastto the GDin uniform FM systemswhereSOC\nisrequiredto breakthe totalspin conservationin the sys-\ntem,αxx\nxxis associated with the spin-pumping effect that\ncan be ascribed to an emergent electric field created in\nthe non-uniform magnetic system. In this case magnetic\ndissipation occurs due to the misalignment of the elec-\ntron spin following the dynamic magnetic profile and the\nmagnetization orientation at each atomic site, leading to\nthe dephasing of electron spins [16]\nIV. SUMMARY\nTo summarize, expressions for corrections to the GD\nofhomogeneoussystems werederived which areexpected\nto contribute in the case of non-collinear magnetic sys-\ntems. The expression for the GD parameter α(/vector q) seen\nas a function of the wave vector /vector qis expanded in powers\nofq. In the limit of weakly varying magnetic textures,\nthis leads to the standard uniform term, α, and the first-\nand second-order corrections, αµandαµν, respectively.\nModel calculations confirmed that a non-vanishing value6\nforαµcan be expected for systems without inversion\nsymmetry. In addition, SOC has been identified as the\nmajor source for this term. The second-order term, on\nthe other hand, may also show up for systems with inver-\nsion symmetry. In this case it was demonstrated by nu-\nmerical work, that SOC plays only a minor role for αµν,\nwhile the non-collinearity of the magnetization plays the\ncentral role.V. ACKNOWLEDGEMENT\nFinancial support by the DFG via SFB 1277 (Emer-\ngente relativistische Effekte in der Kondensierten Ma-\nterie) is gratefully acknowledged.\n[1] T. L. Gilbert, IEEE Transactions on Magnetics 40, 3443\n(2004).\n[2] V. 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